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. . .<V c:g "9 d. #:I. ~ 1~• -1""°""""""rilNJlflftfl!fl!lHl'l11llTl~!'HM'l'I 1.1.1 mn u1Au{)Ullfllfl11" 1.1.1
' ' ' .q t:i 1 #:I. #:I. "9 ""'!
!'lll!'ll1fll1ftfllJlrl'lfJ1J'l'l l l'll'l'll'l'I 1.1 !1M'l'I 1.1.1
i'.1~1nl11111 "m1n~1<1f'1lluh •Duflrn'1l-lli~'llil~Nu'i11111~~ liiltJfl~~~'~rl1urni'ltJh~1iJ1"1ll• •
m1&ii1lluhYium61" nT1~0J~n~1&ii1lluh~u ~u'i11111'~iiiil~cliu~i'.lm1l-lii'.loilil:J;i61~~Yi11Vii'.lm1l-l~uh
'Wil~'il~nii'1<if'1lluh m11iJn~1<1f'1lluhLni1'il1nni11m"i<lf'1lluh~i1i1b'l'111~1iJ~~eJll-ibbuh1u<iiil:J;J61 UiltJ
fl~'~llm"l1'11rnrnflni,y1 (case study) lu'l'll1d*~'~llm-,rl1111u'1n-llii.$lni,y1~iin$lni,y101:;ilei~'ln<iiil:J;Jmrn
111~eJ1lrn.iii1urnruiiul11iJmn Ll-i•wm'l'1eJ iin$lni,y1~noi:;ibrnruoilrn;161Ll-ifl"ilJm•flu'11c111GJ1'11LVfm1
'l•<i11~il'lJ1111iiu1iJ1"1~rnl'i1~m1 •-ilu.~mnu1um<1101'i~~1'11~1um1iJmm1n~oi~iln1"111 111l:J;J61~i'.l
~umut11u'1l-lu1rn '''1111i1n~'1l-1m-,1'111feJm1oii11ill-161i:l'il'luu~i1bYi1~11111.lulm'l,fl11~'11 bbil:;o/ii1lluh• • •
1u-,~<lfu111rt~ ~~N61~m'il'l~bn'1<111l-lmfeJm">L<il~uili~m1:>Jil1•1iu~'~111l~1111m1:>Ji111'ieJ.iJei:i;:61•~l-lb~l-I
b~ilm"><if'1iiuh~•bl-imh '1l-lJ"lffil-l1n~u111'iil<iiil:J;J61b~ilmwi1'1il!Jfll'><ifi1iiuh11]niiiil~b111m~'1l-IVl~il1iJ
111'ieJm'iloi~ullllrjm">.i'u1111.i'uffl'1*~11il:J;J61~oi~,l'JuiJ-,~LtJ'llurl1111~un1'>"11,uun'ilm1.ieJ1u1.l
1um1"i'1m1m1"11bttu~1unoirn">l-lil~hl<111:.J ii:n$lni,y101:;1i'eJ~1'11m1l-I•1111'ilnu'deJb~il,,Xu1
111~1lil'l}1111iiu11VftieJ,bbYfrieJui:iuiiieJ m1GiLOJ111t!'lVibb<11n b.iJ1h L'ii'Y11'i111'im1l1hi:J!1.J111111h1111'i1i'il~fll"i1Vi1'11
mJ1,1-,111'iilil'lJ1111~1Jm..n•lium.h'11"W-i'eJ:>J,ii~iiim•1l1h1ueill'1TI<i'1~'1 ~..~"Wu111'im~ilul'JJ.i1~11Vi~

-iiDl-ln~7m-ii1miiuriDui'lu<iimmmbbt1:;1Vimm11-ifol-1nLvrudJwnm.Yh1•"l~ -iiDl-lnLviml'lum1l-I
• • •
~"1,ilu bb<1:;vi1ni.i11t11-iinf1~bbt1rnbt1d1-iiil~n1'1-m~1mm'll~1m:S~tl"lmm vi"lD-iiD~n1viu~i'],~'ll:;Vi
1ub:S~mi1J11wii1~ fll'lbbt1rnbt1:;i1-iiDl-l<1m'li'lu"l'lllfll~ilm'lm'lliilbU'U~1uiiu1Vi'A'll"lilll 3Vf1-iiDV1'9'fl$iil' .1) Jl1ui:l•~m-llboii1 (input)
2) #11urn:;mum-. (process)
3) Jl1m.JnN!i<11 (output)
m1Vlm·:1t111nH.r1'llif11.011u11.Y1 (Input)
-iiil~n#i1ui:l'J~tJU1boii1vi"lD Input blJu-iiD~m~mnuil•~t1'1i1~'1 ~u1m1-ii'lum-.lii1bu'!4-l1u m-.
'A'll"lm1-iiD~<i~blJ'U Input iiu m-.,:;'A'J1"lm11Vi~vi1fl-iiD~n Input ~u1m1-tlaJIJfl<if1l~'Vl"lDii
fl1ll-1'1<11ilb'1~1l'U n'J:;'Yi11Virn:;u1um"lvi1~1rn'i'1Ltlbb<1:;/V1"lilN<1Nli<11bnilmll-1WilW<ll'111ill-1ltiLJI ,~11
•:;iirn:;mum-.~~'vi1'l'1n<111l-I ~~iiu ~~<ifml'.im-.fliurnmoifo~n'li1~'1 1vr~ ,r)D1Vil11l-11-.11u1~~~Jfl
<ifD1bvi1iiu~'J:;boii1m:;mum"l'Yi1~1uii"1Ltlum•1fliiu'11™•1-.m1m1mh~t!'llEN-iiil~n~•:;u11ti1-ifJl1t111
blJu-iiil~n~iim1i<'51•lJ'Ubvi1iiu ~~':;'Yi11Vim-.~'1m"lbn'1tl-,:;ifV1nNnbb<i:;tl;:;ifVInmw!~ m'lu1bm-ii€1~n
'51u1'Ul-11fl'l Vi)aJiim1l-lit1~tJl-11fli<1'-ii1rn:;u1um·N:;,tium•La.i~l-lnul111~m•:;~•:;tl7:;mn 1'fl'll:;Vi
-iiD~n'ii1~'1 m<11iiu1'1t11aJ'51blJ'U vi11VlL-ifV17wmrn1tim.h'1a.i~l-l~l
..:::.. 91 G 91
fll':i'l'l1ll':iil.!l'llfljll I'lllll'llil1~1J1'11fll':i (Process)
1fo~n#i1ufl;:;mum-,vi"liJ Process 1Jiwi -iiD~m~mnurn:;mum'lbb<il;~'U<11il'Umdtiijif~~m
'llmm;ii1bU'IWl'Um'liiml"im1i<tJ1111mD,l'.irn~1'Um'ltiijif~~1'Uil~1~h iin1'9''"n:;l'.im1l-llill-lldllb'Yi1l•
blJ'UV1~fl'llmm.U1-iiill-lf!Ji'1udm'lii'lVfbn'1tl;:;bti'l1U'lum•11~bbN'U m•~~~m m•G'i'1<11ll-I bbfl:;mu'1l-I
. '
m•l'i1buu~1u Jr~iiu 1uil,l"im•V1"lilV11'i1rn1um••:;ll-iiil~'<111 viu1m1ulln1'9',m•Nli111•vi1h l1
b'1~m~fl'lVl"lm'1~il~llD ""':;n1'9','1u~•rntln,V11il'lil,'lu Input 1<iimfluilmbfiL'Vlu -iiD~<ibV1<i1d•:;u1
1t11-iftl,:;fo'l1UlJi'D~l~mfll'Ufll'l1l"bN'Urll~'fll'lNfl111 fll~ilfll'lflillfllW bb<l:;fll';j~ilvilrilVl'Uilfll'l'Yil,l'U
(scheduling)'llm,1u'luviu1m1uV11flml"im•laJmm•11l1*1,n1'9',m•Nli1111Vl,f1t1mD'liDmmm.l1'U~boii1m
'il:;'Yi11Vibn'1m•<i1-if1 d'w1um-.Ni1111""':;1aJmm'll.ll1*1,Nm1u1111l-l~"lJiwl-ID1JV1mt11#1 'lurrnflt9'unu
vi1m1*1,n1'9'~m-.Nli<11mflbnu"lt1iim,tf11•:;mm·mt1ijifG'i~m"l<iiD~1~,f1t1~wil w;in'il:;bn'1m•11~1u (idle
time) 'llD1ri1~~m-.Nli111riilNmlilt1~1bb~fll"l'<11'YJ'U~LaJ~l-lfi1 m1:;'111'lJr11'9'1L'i!Yfum1u
-=>. v v ..::..
fll':i'l'l'il1':iil.!l'llfljlllll'lll'li:ll'li:lll (Output)
iiD~nJl1uN!'IJN!i111vi1il Output LJi'bbri -iiD~m~mnuN'<1Nli111vi1ilN<i'llmm•~1buu,1u "laJ11'il:;
'Yi1m'l,l'Uill;l'lV!lflllfll'l111'l1'ill11l1Jif1l~<I~blJ'UNflVl1il Output bb~111llJ'U11i1l-l~<if0~m'lVl1ilLaJ n'il:;bnil
m™'(l.llmbbn:;tl7ut11~rn:;u1um.'lrn1uV11Dm:;mufl1"lm"l'Yi1~1u'ii1''1 '~il~'il:;vi1'lvrbnilNnt9'w~~~

~1-1'limiiei1'liei1 nu11.1 1u.m.itli]u~'JJmm"lmu<iJl<<iJrn111Y<n<J~1'iim1i,iunY<~m'li1111umfim.,y11iii1~1-1
V1'J'.il-ifil-'.im1'J'.l'l1-1'1J1N~n.'11 (customer complaint) mdJ1-1-ifm;1fl~<J~tlwrn~mumw1111-1'li111 '1Jmm.'im-,
V11nwu1m11-1V1'J'.eJN'tlnu~Y1nwu11'JY1n.ius:lm1<i111<Jflil1J~1~<i11-1m1l11<J1n-ifill<fl"111-1 Output d Nfl~l"1'I 'l<J q q 'II
n<J~dJ1-1m1~m.h1ri111-1V1u1m11-1V1'J'.iJn1J1J.i.imX1-1 m11'l<Jl'i"llil-ifm.Jfl"111-1Nf1V17il Output d'lvri1.;f1"111-1• • •
~""~1i.i~ 1v1il'lVfi11Nfl~~i:i1-1m<J<J~u11t1'l-ifnuV1u1mm51-1~-if111~!N dlum"l'llml'JNfl m1i,i~1t1'l1-1
V1u1m1wl1-11 V11nil~hi1~l-i~n<J~1li1-1uY11~l'l1-1'lVl"1i1oii1h1u1-1eJ1.'im1i,iiml'J11-1il1.'im-,'liil11.1
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!'1-1!!'1-11fll'l"l'lfl'lllll'111U'l'l 1 ~0'111'1 l.l !10~'1'1 1.1.2

•..All'l-11'1 1.2
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,:;ei,Y 1.2.1 m1:i.i'Y1mmm:;11<1Jmm·;"11mnl'i1l.i'll:Nil:ii'li1mrn
,4ei,~ 1.2.2 m'ill1:;~n<11"l'6m1'i<.i'll:Ni<:ii,mmrn
<1mfi 1.2.3 <if1Lbuu'llei,m1'i<m1:;V1,:ii,mmrn
1. m'l'il'1'll:;VI':jl,tfimrnLllum111-TN<if1LbUlJm1:;m1vn~1<11eimnn<if1bblJlJ'~ei1.t1~1<11eiu
~11K'1u1JH1lum'l<if"1i1uh'Yll~Ij'lfi"J
z. 1•1nm11Lm1:Ni•:il~mmrnlieiri1«u"1~u•~ei11:i11'.Jft~<11'll:i.1111n.i~~~ftm L"1m~:i.i"J1nm1
~un.irn<11m11<11'1'b'11111Lb11u'1vil'1-rnii~11K~"1fiu<if1••uum11111•m1:i.i,:ii,•lfu:i.111-6
t 'j 2'A !UA 2' <v I ,.,,! J; fDI ~ "-'
'Yll~m'l'Ylvn'l <nnum"1:i.Jr;j<11"1'1U<il'lbblJU<111''1 L~:i.J'llU:i.JlluU!11"1lJ
3. m1li1:;EJn<11"l'6'i'b'1m1'ib.i'll:;Vib:ii~mmrn 1"1m;.~1:;1u~1uY1ncy"lfi"J•llum1l'i1•uum1
L~€1LiJi't;'fm1:i.i.J1111ei'11'lftUl'Yli1~"J:;1.t11111ii,Um'l<ifilftUl<J,~mnum'll'ill'UU!i"ln"J l~€1• •
vi11VIIl"n"111ii~~"1ey~~"1'llei~li.,:;il'Yllin1~1;';
4. <if1Lbuum1'lb<fl'll:;VIb:ii~li1mrnYi~;';i.t1Lftuei1U'J/"1'i'b'1d1;';mi <if1bblJUm11li1Lbrn:i.i':ii~,ffu
<if1bblJUnl'l'llufi, 'YI11i.;5m1<if"1i1uh m1<if<11ilu1<m1l'.J1~viml'.lnl:Jmru,o/ibbl1:;<if1bblJU
ci1ilu~ub:ii,1l!fl'll:;VI m'l'ibm1:;VIb"11ei.n1m1uLbft:;<if1Lbuum7"1ei'VJ <if1bblJU.im.im.1
v ,.I J
1llfi!u':i:::i:l'.:iri
b~ei0ni.;1<11eiu~ 1.2 "JUlli'i'l un0ni.;1mm'lll
1. eilirnl'.Jm1:i.i'Y1ml'.JLbci:;'l1'<1Jmm'l'llmrn'l'ib<fl'll:;VI,:ii,mmru.11K
2. eilirnl'.Jb~eiu1'll'llei~m'l1.t1m'l1•m1:;V1,:ii,111mrn1111i'lum'lbvi:i.ie.Ji1<11rn~1;';
3. eilirnmLu1~il'llei~<if'mlJlJ'111''1 lu'i·inm'l'il<fl'll:;VIb:ii,111:i.i1ru,11K

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vnnmh1li~i'lf1~111i'i1EJnl'i-'lbm1tVil'R~ll'hnru.LL.11 'J:;i'.l;m'lEJrn'iueJtj 3 ;€1 $iei m1'lbm1::Vi
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b'n~lliju&im1 'Vl'ieJ Operations Research iif~d~ueJtjnuLLU1'Y11~'Vl'l€ll1:J.J:J.J€1~ (approach) ~1i'lum1Yl,n1ru1
UeJn'J1ndl'J'~il'l'lf1m~fl.i'1EJfl~~nuLun1fl'l'lf1Lfl™::1ff1ilil4 L'i!Jffll 'l'lf1Lff11'!ll'l&i 'Vl'ieJ Econometrics
~~VimEJ!i~L'Ylflufl<il1~'1 ~1~um1'lbfldl::Vi'Yl1~bff"i1'1l'i11i1i14
fl113J'IUJ1£J911illl'IJ1'11Hl'lflfll"illfl"il:::Yl1:ti-:itJ13.11tu (Quantitative Analysis)
lULbU1'Y11~1i11:J-Jl'lf1m1'lLfldltVibon~mmru.d 1.-i!Jvf1LllLL.11bJlUbbU1'Y11~~L-ifnulW11'1l'lf11J"i'V11"l
Jfln'J riiN'J1nmi~nw:i1m-i~'Jru"lV11-iii-in'J~U 'J::L-ifiif~i'Ylm'f11ilil4mi:;i'li1ll:; (Science and Art) 1u
mTu1V11'1nm-i'Y11~i'YImmilil4m1i'lum-iu"lV11-iTI1n'J~u 'I::Ll'ium1&i.-ibL'1t<i'i.-ii1uh 1'1!J~U"11U'll€1~' ~
<i'i1b'1'll'Vl'ieJ.ifil:1;Ji1~l'.liltj m1'lbm1::Vi,:n,mmruV1mEJii' mm1ifJ1.ifm;Ji1~dJu<i'i1•'1'llmll-i::m!JWi!bllu.
il,f1m1:J.J-i' (knowledge) 'Vl1umwurnfl (information) xr!fl1-if<ilri1uhrrn1t1n'J EJn<i'11mh:1b'liu b";nl'.l
.if€1:1;Ji1in&iLueJ~lileJtj<;i1u1u'Vl'l~ mn&ililb€11:J.J1'Vl1'11b%~!J (arithmetic mean) V11'11fhub1'imblJU:J.Jllil"l.!jlU
(standard deviation) bbi1::dJum1:J.Jib~mnull-i:;'illfld'llil~m11l1')ru.'wu&imh~h byjeJ~'J::u1Ltll-if
<i'iii11uhb'R'Jl"n'J<ileJ11.J V1'ieJV11fflin'Jm1l1m1:J.Jii~mffuJnfi1'lleJ1n'Jm";d1 Jnfi1m1-ifu"lm11uilii111~
:J.JlnUeJ!J'lJUllilL'VIU bb'1:0n'Jm"Ji'.lm1:J.Ji1m1nlum11Viu"lm-i1Vi11.-iL11mn'11Ullil1'Vlu n'Jm1nmm1n~
'JtLm.ifei:J.J!1LV1fi1iXU:J.J1ll"i::m'1lVilli'im1:J.J-i'11imum1ill (situation) m11Vimm1'llei~n'JmdL!'iuilu11h• •
jnfl1'J:;~~v·mhLL'11'Vlu V1'ieJjnfl1'Jttl"i::6luiJ'lJV11mnihilmf1EJ~llil ,yjei'li1EJ1um1<i'iii11uh11lueiu1fllil
m"i~'J::1'1l1m1LYi:J.J~lilm1:J.Jmm1nh1m11Viu"lmdL~:J.J~u'Vl'lei1:J.J LL!1tV11rn~:J.Jmw"lm"ilb'11jnfl1'Jtl1
ml:J.J~~vmhmn~u'Vl'leiLl-lL<Wm1.-i
<i'i~iXu bl1eiv:l'J1-;ru.11uL'lf1'llil1m"iibfldl::ViL'lf1mmru. m:J.J1"innfi1111i'in111'1 11Li'.leii'.J<i'IJL!1'llLb'11
'JtLm<i'i1L!1'llmll1:;m.,tV1,n.-ieJ1'1m1:J.Ji•yjeJ1til~um.,il.-i1uh'Y111ii1n'J11i'ieJu11h luLeJnm1tl"itmi1
i1"i::'l'll1m"i'lbfl11::Vi,:n1mmru.d'J::nfi11ii,11lm"i<i11'1 ~'Jrn1::m'1.ifeJ:1;J'1lu~mJu<i11'1 LvleJ1Vllli'idJu
ml:J.J-i' ~1'Jtl-11iihum"ii-i'11<i'i1rnJu (model) 1uwLLuu"111'1 ,yjeJfln1't1LLi1:0'lbfldl::Vim:nun1"iru"111'1 ~. ' .L~EJ1num1~.-im"i 1u~d'lleJ1Viibnflnmb.if1h11<i'i1bL1JlJ (model) "11~'1 ~u1LiueJnululi111'11udiif1
VI'11EJbl'iun1"i"i1u-;1:J.Jimti1:;iun11ru'lleJ~un'l'lf1m"iLL"::ibn~.-im1"11~'l ~1'1tl1:;iunuiJtJ'VI1"":;fu
'V1u11lm"i'Vl1'111iieJlJ'lleJ~iJ'lJV11iXu'l ~e.hum f!J"i:;lii1~'1 LU'VIU~~eJli111rrnli'i1ud ~~bl'iuLY1EJ1<i'11eJu1~

'llm1ilm11h~;J,J'lt1.ifei;J,J11"il1m1mum1rn1uei~<ii~r-l1um11 b~eil'.l.ifei;J,J11~ueith,V1ii, 14fl'.lnwu1m
• •
1h~m111u~ni,rru~V1ii, bbl1~14fNl1bU'.fl,fi'm1;J,JVimm1mm1t11-ii,1w1ifl1t114f bY<'ll~>t~tXuwu,fifl4f1u
n111b'fl11~Vib'ii'ili;J,J1ru~'l-l1b<.1ueinuT<iitJvf11ti"il~blJu<if1fl~1'm'nh~m<1<if1bb1JlJ<ii1''1 ~'1u~dll14f'l-l1bm
<if1bbuutl1~mru 10 <if1bblJlJm1Vi0ni,r1 bY<11~>1~tXu"i'<iitltl1~mfi''llfl,m'l0ni,r11u1'1i1m'l1b'fl'll~Vib'ii'
'
ilimrud'.Jum'lfini,r1<il1fl~l''llfl'm~u1um'l;,i<il'llfl'iin1'1i1m·rn1mXn11'1mm1w1flufl~<il l<iiff'l<il*
vimtJ~bbVi"il~'1um'lfini,r1 <iffl,m'l1Viiin0ni,r1~'V1mtJ14f1'111u11 b~fl14fri1m1i'1ni,r11'1i1d1bii'1l'.lm1;J,J
b-ii1hm1;J,J;,i<iibb<1~1ilm1'llfl,iin1'1i1m1ml11~u ~'vi1ni1iinfini,r11~Y<uvi1flbN'iitlJnu<.1mum1rnb~mnu
IImm1t1'1-l1<if1uurnvil11tXumtl'l~tJn'iit'614f vi1flvi1ntl1~<.1ununmum'lrn~"11'Lu<J1nYi14fit1bi.1'.fl111u
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'Jl<il1'1i1d iinfinmm10nB1"il~mm'ln'l-l1mtu1um1;,iil'llfl'iin1'1i1m'lbV1Gi1tXu~r-i1umbbii'1 mbllu
<if1fl~1'1um1n1'1,m~mum1;,i<il'llm<iiubfl'b'V1flm'filtlJV11 vi1fl'l-l1m1-iim;111'llfl,<i1Ubm~l'.11unnl'.n11rn
tXumtl1~mmlium1;J,Jib'iilfl1-iit11~nflum1<ii'<iifluh1um1~1buu,1u<iifl1ti <if'~um10ni,r11u1'1i14f1ud
~'1l-J14fi'.l'llfl1Jb'll<ilb'V1rnbb.i<ii''lbblJlJ~Ulbi.1'.8Hb'Yil~'. 'llfl1Vifi1u'li'11<iifl1u"ilt'l-l11JJ1-iiri1eitl'l 1u~d'llfl
buu111Virhn11fini,r1m~mum1;,i<ilmnn11~"il~0nB1bV1rn<ii'1bb1JlJ'llflnilm1"11''1 ~l'.l~;,i<ii.'u~umbbii''l
•
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m1:1JID"JY1llu (prnhabilityl •W'j1niii.ld;j<Ji1~mfrir;u'11 '1ufii'.im1:1Jmbi'111n'111.9o •~Vi'd:l.1.275%
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~~l11h bb11~~1b~~um4'1eiW o/11bbuubbm~~t1 "i1~~''lnrn•flm:11lii~bbu~1'1lt1m"lL.ifm.,.~111~~lmoo1m- - -" - -·-- -- ---~- -"- -----.~-------- --··- ~------"-"'"-- ,.
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W11!1l1Jfll'ihhm1rn111i.:i1.Tu (Linear Programming)
<i11:1-1~1ef1mi11mbbii'1L11li•~1'~'ll1l''l'lf1m•"ibfll'1r,Vib~'li~:1-11rn11m-illidbbrn:1-1b~'bl11t'111LlJ11
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f11li.Jii'1bblJlJ'lli)lnl'l1lJ1LLn"J:rn'il1•iitt ~€1 "alloc.ation of limited resources for optimum use or result"
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resources 111m!'JP1 1111vm1rn~l'.l<i1fi.i LYl"lW'1%~1-I !l1fltJ.,:;n!l1J'llillf1l"l1tJ·mrni.JL'illbi'i1-1~1tJ"l:;nil1J111tJ,-._,.____ ~~- ---- ~ " ' - ~ ~
2 ri11-1 ri11-1LL"ln~m'11r'.li'wl'<ii1JU"l:::fi1fl (objective function) ~€1~1rj1l<J"l11-ll 1117€1 optimum result
1117€1Nl9'l1€11m.,~.i"""~,.,1tJ"l1"l11m •'liu ,.,111€11m•~.i"""1111'y;l'J1rn•~il111111111h<J1'l.i w1n<li'11
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Distribution vi'iil MODI 1Unl'l'Vll'1llllilU 1;;J1Ji'l-iiw'l!mltlllJlilcfl1bbUUnl'.'111'lbbOWb~ilbftUbLUUbb"iD~
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fl1111J1Jfll'HJ3'U'IUJ1£J~TU (Assignment Model)
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~'<i11bblJlJ'llil''Y1'11'1iim:wdmm"lt1'li1tJ'lVi4im;1'inum1'111 'lu'1mum"lm~iim"lbb'li,1iu 2 Hl!.id uforntJ
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11.J"lbbm:wb'll~bifu'lum"l'Vll'11"1fllJ
m1fl'Aiftti'ililll'Jii11'1rn11'Jfl~!flWorl (Multicriteria Programming)
m"l<if"11uhmtJ'lo/J'Vlmtmi)mrn.vi'J:;bbU~flflmlJu 31.J-;:;u1'Y1 ~fl 1.J-:;:;uwm-;n Preemptive Goal
Programming ~fl m"lu1<i11bblJ1Jm"l11.J"lbbm:m'll~bifUJ.J1~ni,;1<il1.ifff~1ni;i ni111~fl 'VllnNflur:iu<ifil
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•
<J:;'lVii11Jlum1:w'11fi'IJ (priority) 'llil,'i1u1u1'VIUJ.Jlnn11nu 'luii'ni,;rn:odb1tm11 d'lu Preemptive Goal
Programming
1.J-:;:;m~'i'il' ~il Multiple Objectives Goal Programming tln~m-;11.J'lbbn"ll-lb'll,bifu 'J:;Sl
y;j,fi'iiu"l<11!Jtl'l:;'i',fir'i'ub~mnuY!~n'iiu4iil~1ni;i5n"J1"1'Vl'if, bb'ii'lumnumm~l'11<111Jtl'l:;mfi~o/Jmm-;
lJ"l".il1'Vll1ltJ1<11t1tl"l:;mfi<J:;1.jj Multiple Objective Goal Programming bb'i:;o/lfl,rll'VIU1?1'11<if1Jm1:w• •
rl1firu'llil~bb'lil1:;1'<11t11.J"l:;'i''fi1~ <ii'1bblJ1Jm"l11.J"lbbn"l:WbUl'VlmtJ1.J"l:;m'Y1 Multiple Objectives Goal Pro-. .grammingf<J:;blJu<i11bblJlJ~<J:;'l'i1'lVim<iii;ii1uh~i;imwmmn~S1'i1n01b~ilN'ili''V1i~~~~i;i,'liub~tJ1nu
'ii/ 51~...
~'Yi~ Preemptive Goal Programming bb'i:; Multiple Objectives Goal Programming 'J:O b'll1llnl"l
bbniJ'IJ'Vll'llil'nl"l11.J"lbbn"l:Wb']l,bftU'Vil'11"1illJ
tl"l:;m'Yl~fil:J.J ~e:i Analytical Heirachical Process H1t1 AHP d'/u1lim'lfiu11u11flum11Vf
'1rnuuil<J~tJ<ii1''1 <111:wi11Jlum1:wrl1ffqJ <if1iltll,b'liu 'lum-:;<ifi;i1uhb~ilmifu'Y11,'lt11"1ti'i'l"l'Vl'ii~Lifu
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• •
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m''tum1si1Li1mj,j'iiJm1L~.h1'11' o/11iJ81,L'liu mn<llei"m'lL~'i-11'11,'illn'<jOlj,jii,11..it'.i'iin'<jOlj,jii, m'ill'lm"
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iJtl1"h~":;LillLi'i'l-l1'11'~~.h~Zj'11il~'l-ii f'iil Shortest Route Problem
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'11l~ml:i.JW~Wil L"J b111j1lfll b'1il1Jl~'1'Ylij''1
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{single queue single server) 'lrnlJVll111J'liil~ - Vll1ltJ~U<ililU (multiple queue - multiple server)
' .'l::1JlJVll111J'liil~ - VIU~ifU<ililU (single queue - multiple server)

fll'Hlltlt!~iitnttfll'HU (Simulation)
m1~1'1!l,'1mum·niidJun1111fi,i/11mJumuulli1bbm:1Jl'1il:1J1'i1b<11ei1 i/11ei1h"b-ihl m1ili'1bbUU
bbm01eitJ:w1b'iitJmtlu1111bbn1:w01ei:w1'i1b<il!l1 bbfl1~1'1ei"nw1-i1,1u'llei~bb11l'1eitJb.;;jei~1<iei'.rn~nT;;
bli~tJ'Ubbli<i,'llil,bbl111'1!ltJ b.;;j!l0ni;1~mif~TI'llmufornrnh"'l ~ri1111ul1m i/11mhb'li'U U'il'jii'Un'ilm1Sl
ri1u1Vium11 1111u1tJ bb111'1!ltJ 1 mn 1111m~:wri1u1Vium1-;btlu 2111u1tJ Slbbl111'1!ltJ~iuu1m1 1 bbm
'il::mm'l11<il1bm11'1eitJ bb<i::nm-;ei01eitJ<i'1#lb'l'i111111'iei'ii::%'1.lVIS1ci1u1Vium11 2111u1tJilmnl'1eitJ$uum1-;
2 bbm1'1eitJ 'iltbn01bbml'1eimb<itb1m1ei01eitJeitJ1,h 'iltb~u1'111n1'0~1<iei,<1mum1rn blluvrnbliein€in
1'11,Vlfi,l'Um">1111.i1<11eJU'illnili'1bbUUl1i1"'1
m">1111.i1<11eiu'llei,i/i'1bbuul1i1"'1 ~n<i11:w1mm1t11i1::1Jn~l'1ru<11m11<1111fuj"m1<Wi1.i1<11!lu 111'iei
1-iim1~1<iei,<1mum1rn (simulation) n'l::1#1.i1<11eirn~mnu bblliLurn,mrn ili'1bbuuilu'l vi'iei
11mum1rn'l1,ilu'l m"J"J::Slm1:w-iiu-iieiu~"mrnnun11'::L-ii'Vlqi;51'1ru<11m11<11~m1111.i1<11eiu1#1 i/11eitJ1,
b'li'U rnrn<if1bbUUbb1101!ltJluifni;ru~i1ci1u1Viu1m1111mtJ'l VIU1tJC,lmft!l,n'U (multi stage server) l'U
ii'ni;ru::Vi~mwu1m1 'l::4fo,iuu1m1'1nci1u1ViD1m1ri1u~111fi,,1nilu1liiuJJ1m-;"J1nri1u1Vi
u1m1ci1u~ 2 bb<i::ri1u1Viu1m1ci1u~ 3 11imftei,nu111<111:wli1<ifJJ 1umnh'liudm-;li1::~n<11"l-ii1'lqi;5
111'~m:wi/11bb1J1Jbb111'1eitJm"J"J::~' mnbb<i::-iiu.ifeiu bb<i::1urn,mrn'Vlll1"5m"<1n~<if,n<i11m"J"J::1a.J
mm·mvi1.i1<11eiJJ1Vibbt11l'1!ltJifni;ru::d'l#l bbl1it'i11-iim-;'<i1<im11mum1rn'1"1Ju11hbm">:wl'1ei:w.Y:hb<il!l1 n
"J::0ni;111mum1rnilubm::1111.i1<11ei1J1.1m-;~1<im11mum1rn~'bllub1'll'1i:il'1~'::bbni:Ji!J11111#ieitJ1,n1'1,'ll11'
.if,1uui!J1111~mm·m1.ifbm::1a.Jmm""'1t'Viqi;5m,ml<11m11<111~uI'bbnUi!J11111#1 m-;~1<iei,11mum1rn
~"bllm'Vll'1iil'1~fi1'1i!JL'Yll'1iim1fl,1um~"J::0ni;111mum1rnl1i1''1 Lu'.i'lnm-;1,1'1"'1::Vlb'ii,mmru1#1
ueirn111uei"J1nili'1bb1Jul1i1''1 bb<i::m~1<im11mum-;rn<111:w~1#ln<i11mbbfl1 t'.foilili'1bb1J1JLun~:w
'llmm-;11.hbbmm'ii,biiullnili'1bb1J1JV1ft"~fi1'1i!J 1#imi m-;11hbbm:w'<i1u1ub<li:w (Integer Programming)
iin0ni;1mm1t1~"J::0ni;1L#l"J1n111ii,i1ei111'iei.J111m"#11um11bl'1"'1tVl,'Ji,117:w1ru111'i!l'1111#11u1%tJb'lr'
115ii~m1vf1111 1'1tJ~i111bb1JJJm11111b1m:w~1mub<li:wbliuslibb1JU111ft,'ll!l"m11lidbm1:wb'ii,,ifubb11iSlci1u~
b~:w1~:w~'UJ.Jl"Jlni/11bb1J1Jm111.ldbmwb:ir,biiu flei 51-ifoii"ii'u111'iei'11'eiri1111u'111.i1<11!lu~1#1,::~m1#1bU'U
~1mu1<ii:w ~'1um11111bbm:wb'lr"'fiuilu 'l::ri1111'U'1'Jll'1:1Jii~~fi1'1i!J'llm'1:wm111i1''1 ~•::11~1,1tlui/11
• I ~ • I .~ <v J ~ v < v • I , .~ <v
bb1J1J11ei"1J1::m101ei u 1::mdbLdn "'"n'll'Ubb<lt11:1Jm-;•1mb'1l1"m1:w":w~ull'llei"mbb1J1<il1"'1 'Yjnn"n'll'U 'OJ::
blium1:wfl:wYiurrb'lr'6iiu111'ieim1:wfl:wYiuri'll!l'ili'1bbli1~tJnri1if'111ft"''l'i1ilu bm::li1::m~<1ei" f!!l w,n.:in.i,
11i1''1 1u11111bnd:1Jb'ii,bft'U"Jt~!l,bU'UW"rloii'Ubb1J1J11imflm (continuous function) i:iuf!ei ili'1bbli1€im::
'Yjnilf1mm1t1S1~11.i'!l~1,llimftei''YJn~lmtJ1urn!lu'llei,.i1<11!lu fl!l bU'U1#J.if,'<im1ub<li:wbb<i::'<i1u1u
iil1'11'Uvi'iei1'1l'fiitJ:IJ bb11i1urnrn'll!l,nl"l1111bbnd:IJ'<il'U1'Ub<ii:IJbll'Unl"l<l::bl'.1l1nlJbnill.fi'lJ!l"n111li"lbbnd:IJb:ir,
,fiu1uifni;ruv~11,::ii,f'i'JJ1Vi.i1<11eiJJVi't#iilii.S1~111Ju,YJm~1mub<ii:wfim111ileiu"J::li'.'i'JJ'i11'1,nifuLuili'1bbU1J
btluw,nifmbJJJJlaJllimft!l" (discrete function) bb'VIU~"J::blluw,nifu11uJJl1imftei,

m-;vrnhmiulVli'fum-;11hLm1:1-1"511,1,1uL<li:1-1 'il~LLtJrnlJu 2.ifu111l.b .ifu111l'ULL':in 'ii~1-iiL'Yl'1U'1'll1l~
m1t1hLm1:1-1L'il'Lffat1itJ1:iJ.-i1u'~~4i1l<i1i'f1i'llmm-;Lllu-oim1uL<li:1-1 (integer) 'il1ni'.iu,~1Nil'ilU11.-i1111llJ~•
1<ii,'.('ULlJ.bff1ift1UVl'i1l'-il'U'l'UL<li!-I fllVlln11.-i1111llJ'JIel'm-;t1hLLml-IL'ij,Lftui'.i.I,1<iidJwil'U'l.bL<Ii!-ILLfi'l1-11"1
.-il111llJ'lJ1l,nl':ittJ1LLf1';!-1-.il'U'l'UL<lil-1LLfi1l.bYJ.b.y] LLllnl'Vlln11.-i111fllJ'lJ1l'n1-;ttJ-,LLml-1L'il'Lft'ULfJ'ULl'l'lJ
ff1il'11uil--i1LDu<ifm'Yi1.ifu~n1l' tiiu1u.ifu~<!B,Jt 'il~U1Lm.-i1111llJ'll1l,m'ltti-,LLmrn'il,Lfiui'.iu:1-11.-i1mru
"11l<ii1um1Lv1:1-1c~Bu1'llL1i1m Lvl1l1Vil<ii.-i1111lu~LDw~1u1uL<li:1-1 'lllm-;V11'11"11lu1Vi'<ii1LLuum'l1t11Lmw
"51u1uL<li:1-11<iimi L'Yl'1U'1'llmm1w1i LL1lu#i rn1#i (Branch & Bound) Vl'i1lfi"1&i,L'l"ll1.b (Cutting Plane)
.1lUl'1111i11:1-1 <iiB,'Jlmuuiin.-i~'m"n1l,n:1-1m-;1um1Lti'lLLm:1-1"51u1m<li:1-1i:iu 1:iJ1<ii'Vlmum1:1-111rr1'wii1
LLuu~um1Vl:i.1LLl'I~~11LDu~m1uL<li:1-1LL"1<iiu mnn1'1'LlJu~1u1.bL~:1-1LL"11i1Y1,fi'iiul11''1 1u<if1LLUU'il~
nmml'JuV1~n'ifuLLuu1:iJllmd1l, *'1:iJmm'ln1-iiL'Yl'1U'1'lJB~m.,1ti1LLmrn'il'L~iM1' 1i,i:iuV1~nm-;fi1l m1
1t11i,#iu.-i111<JUiin'iiV1d~1Vi'LDu~1u1uL<li:1-1 LL"1Lu~'i!h.J1<ii'Vlm1l~'m"1tlu11Lfi1"'YlfiUtJ!-1 LDum-;1t1'Yi11Vi'
f.11um~mum-;iin.-i~'Vl'd~1Vll<ii,l'Ju-oi1u1uL<li:1-1 1um-;u11titi'l~!Jn<lli'funmum'lrn'il'l'i:iu m1t1hLLm:1-1
"51muL<li:1-1m'il'il~ilm1mlliitl!ri~1tin'llnl':i1t11LLml-IL'il'L~.I,1111.JL%'1"11~1JU1'ri,1.bm':i&i"1il'UhL'il,lj'lfi'il L'l"l'll~
m'l<ifiiiluhm~1pn'ilLu'Vlm1l'1 mm'il::<ii1l,<if1iiluhLDU~1.b1.bL~l-I <ii11lU1,L'liu <1:1-1:1-1&11t.,,.nuN111
Hbm~Ln15 mml111<llJ'lJ<l,m'lNi11ii1~~LLl'l~Ln151Vll<iiri1htPl'.!11 fi<JNi11i11~:; 4.6 <ii1LL1l:Ni11i1Lfll5 3.4
<i'1 1um1rnliu'il'l,i'.iuNi11ii1:iJ14l mn<ifm~'Ni1"1LV1'1<ii'il"i''1 <if<J,NiiIii1Vi'LDu"51u1uL<li:1-1 m'l1ti'lLLm:1-1
~l.1,1.bL~l-I~'Lllu&i1LLlJlJ~ill#it)mn1.bnl'l~11Vl~''illnm~1<ii.-i1lil1llJ'illnm'ltt11LLml-IL'il~Lft.bLLfi1 1ll'il
<i!<J,u11tii"1LLtim"11lLVi'LDu1t1·mrn:1-1~1u1.bL~:1-1Ly;1l'li1u1um'l1i"1iluhm~~-;fi'ii LLJ'.f11o/f1LLUUm'l
1t11LLm:1-1"51u1uL~l-l'il~1:iJ14lu1L<luei111U'lj111•1nlt LL"1iin~ni;-1rr1m1n~'il~~ni;-114l'i11n<ii11wrn<ii1um'l
1Lm1~L'il,tJ'lmruV1'i<J<i11'l1'Yl1'<ii1um'l1~tJL'il~tl{ili&im1rf11t1

•..WI'el'I-!'l'I 1.3
'
v "'
l'i1!'Hl'I
d d.
b'>eHYI 1.3.2 LLU1fi'1LL!1:;111i'nm'>'JJil,m'l'i''1m'>m'>~lbUU,1u
,4m~ 1.3.3 'JJil1J'li1mdeJV11ll!1:;~,~'11'>VUm':l<if'1m':in1'>~1bUU'm
lim-Yi 1.3.4 bbU1LU:W'JJil,m'><if'1m'>m':l~1bUU'1u
1. m'><if'1m':mTl~1bum1u 111mtJli' m':if!''1,.,m1J~lbUU,1Ulb!1:;m':lei1u1 tim'>'>l::1J1J
m':i~16umm'lVl1J'>':i!j1'1l)U':i:;""'JJil'il'.1m':i "11tim'l1'i1'lVibn'1:l;J!1!>i1'y;j:WiJ'iJ'1~':::
'i11LVfTI'ln,l'lri1hmn~f!'1
" "
2. V1i'nm·;;l1ffqi'JJil,m'l<il'1m'lm':l~1buu-nu i.J-,mil1J"11ti m':if!''1,'l:::1J1J~l6UUm':iLL!1:;
m'lei1mtJmd"l:::1J1J~lLUU,1u
3. m':i1''1":;1J1Jm'l~lLUU.:11u 1.J':l:::nmJ'111tJ n!1tJ1'1ITm'l~1LUU,1Ubl!1:::m'lililmL1J1J
= ... fl ""' ""' ~ = fl
N!1'1lllli'l'l 'l:;1Jum'lN!1'1lb!1:::u':lm'l m'lililmL1J1J'l:::uu'11Lum1u nwl'm1rnrn lb!1:::
nl'l11'6lNUfi1Gl'nl'lNii'1
4. m':lei1mtJm'l'l:01J1Jn1'>~llUU,1u u'lm1m'111tJ m'l<if'1m'lvi1,t'li'li.Jmu m'l<if'1m'l
1'1'l m'l<if'1ii1ii'rn1ub6!1:::m'lmu'):Wm'l~1buu,1u m'leiiJnbl1Jrn1Ull!1m1'll'1NlN1U
..... ~ ..... ""' ..., ""m'l..,'1n1'l'Jllill1'Y'I mw11"n1'flll!1:;nnN!1'1lb1J1J1'1unm'Y'lil'1
5. lbU1LU:w'JJmm'l<if'1m'lm'l~1buu'm":::ilajmtJ'l<iirnrnm'1'1n 4 '111ui'iil rn:;m1b4il,
')rnlll'Y'I n':l:olbf!m1:w<iimm'l~V1mn111!11tJ n':l:obLf!'JJil,m'l6U~tJubiJJ!1'61'1'1t UL!1ti lL!1:::
rn:::LLf!m'lLL'li,ifuL!1n

v -' '11ilt.JU':i!::l:l''lft
~<ifln1'fl<ilil.h~ 1.3 'J1Jlbii'1 ilnflnmi11m'11
1. ei1rnEJm1J.J'm.J1tm11ii1buu,1u rn:;mumdlbti.i,'1ml'J bb.i:;J;J.i.!1b~J.J1#1
2. 'llm1dJ;J.i'11l~J.J~ln<il'J1nm:;mum'bltlmi1ml'J1#1
3. ei1rnEJ~U<il<iw11mm:;mum1%'llm1lb'1:;~m'll1l'm'i1fi,Nli<iiml'J1#1
4. 1lTI1JlEJm:;mum,%<i1'1~1,,:;1J1Jm1lii1buu,m1#1
5. ilWlmmrnhEJ'll1l'm'el1u1EJm-;1:;1J1Jm':Jlii1bU.h,l.h1#1
6. 1li1rnmbmlui.Jm,~<ilm'm'lii1buu~1un1EJ1<ifm:;lli1"11~11#1

nm'1m·m1'l"'11L'UU~1U (Operation Management) VlmtJn~ m~'1lVl'Yl'i'Ym1mm'Yi1,1wi1l-JnU
byjilfl-i'1~iu1:;u~md'l":;uum-,>11bUU~1ui.Vfud'l"!j'i'<ii!Jli"l:;ft'.'i'l11Nil'.'im-, m•flnm#hum'l"~'1m<Jm<J
'11bi'.iu~1u d'!um<Jflnl!t1b~mnum•ii''11'iuh'lum•~m:;uum<J'11buu,1u (operation) m•ll,!llio/i~1u
'llmil~.'im-,bv:lil1Vibfi'1e111Ni1<il ~~ill<J<J:;bl'lul'iu.111111ilNi1<iiii'ru./ 1111ilu~m•~<J:;li~l-lilU1Vinu~n.111111il
N'm'l1JU~m•m•'11buu,1u1u~d bl'luli1u'11f1111i1u111<l,'llil~m.'im~1111'hJ11"J:;bl'lufi<Jmw<Jfi"l"ll'lim<J• •• •
111"fom.'im-,~1i-lbbrl1~1111ri11• ~'":;1irnl'.lm-,ilfi~e111eii'i<i1mi'1li,l-lilU'fiil~5ubv1€lu11111-ii'lViifi'1li<J:;fo'l!u
'fiil111 mh1L'1tJrf1Llibbii'1 mw~1'l"fi"lm•11n'liii'1"l:;<ifil~i'.1111ii1~~u31u (basic function) 3 iltl1~i'iil
nl"l<il!11'1 (marketing) 1Jibbrl mTIU'Yl"llUm1l-l<ifei,m<J'llil~~n.1'11111il~~1Ji'luei11li'l":;LtJ'lfu i,<fiilmll-1
1i'il~m-,ii'~n'1111118'li1u5u'llil~il,'1m'l"byjilflfi~N!1Ni'i<il u11:;~'1lVii'.lm'l"'llltJVl1illi~l-lilUN!1Nl1<iitXw1 1Vi
nu~n.11111~il~Yi~JiwN!'lli<J:;ltJ'liU nl'l"'11bUU,lU 1J1mi mm1bill-ifill;j!'lmll-lib~mnum1l-J<ifil,nl"l'llil~
~n.11111~€l~Yi~1iwe1!1li<J:;Lmrii mll•:;mmb!1:;~'11Vii'.lm•ilfi~N!1Ni'i<ilml-!m1l-l<ifil,m.,iiw1 bm:;m•C'1u
(finance) ~''Yi1111u1~~'1VllbbVl'1,(,lU1'JUbyjilmlmUm'i">11bUU~lU o/i'1<iill-I uufin bb!'l:;mlJ'll-lnl'l"lii:iu
1'JU'llei,fi"lnl'l1Vfil'luLli<iill-JUltJlJltJ nl'l"l'ilbuu,1u~~liml'luli1u~1u~'11'1!1-J~'l'1ft1U111rt~'llil,m'1m'l"
bY'l"ll:;il'luci1u,1u~nil1Vibfi'1N!'lNii<ii ~~bl'lu~u~nm''llil''lrn'11'llil~ei~.1nl'i 1i-111"l:;bl'luil,f;m-,
l.11:;ul'Y1l'1 1111n1i-Jmm<Jnil-i'1~N!'lNi1<il~<il'l"~num1l-l<ifNnl'l"iltJ1~i'.11.h:;i1'YITIJ11Y'Ibbii'1 il~.'im.,iiun"J:;
bl'luei~.'im~1-i''lru'11 ii'l-ibVl11 u11:;'1l-lilmtJ1111u~ij'1
mnn'1111ui'ir~<J:;uu (system approach) m1>i1bUU,lU<J:;V1:J.mJ1i~ m'l"Ulbill'Yl™mm (in-
put) mi;J1um:;mum<J (process) 'l111'11iil'iuN11Ni1<il (output) ~illm'l"mi,i-1ilu'fi€l~n.11111~il~'luN!1
li'l":;LtJ'llumh''i1~Y'lilh m:;mum'l"di~tJn11 mrn1um'l"bblil1'ilmY'I (transformation process) mi
o/11iJtJ1~b'llu fi"lm%11'i<i11'iu.11n<J:;i'.lmm1bm'Yl'i'Y<mm ?lei 'i'<ii!J$u '"~~1u Y'lii'~~1u '111-if<i1tJ Lfl~tJ m
N1Umrn1Unl'lbblil'l~illl1Y'I 'Vft1(dJufiu.J'1V1~ilmnl'l~illm'lnu1Lli~1VIU1tJl#f m:;mUnl'lbbl.1!1,illllY'I
~'lrnili-iuo/i~b'1i.;?iil iriili'.lm'lbblil'l~ilmY'l"l1n'i'<ii!J$u1um'lN1'i<il1VidJuilu;'·h1#1bbii'1 N!1Ni'i<i1V1~ill'iu.1'1
tiu,:;ili-111'11~~n-i1'YI™mm~u1m1-ifTil-lnumh1?i€lm:;mum•bbli!1,ilmY'lnfiilm'l'Yi11Vibll'1l-l11'111fii-i. . .( )
;; " • ~ ~ ~ ~ v ' v ' v ' ~. ,;;
value added 'llUm 'li'il1m'ln'Y11m1i-il'll1b"J l'1'<1n<il1iltJl~ilil'<il1ilm~'1~<ilil b11U
'v . . .
1!116!.IH'll 1.2
h"1uN1'i<ilL'YITYf'1uilm'lim'i'<ii!J$uim:;~uli1u1um'lNi1<il~'Vll-1'1'l1l-luil1i'.ll;J!1'11li'l:;mru2.000
'Ul'YI mbv:lil'11iuum1Ni1<iil'.lm'll-ifbl'l"1uLlil;J!1'11 1,000 'Ul'YI il'111il~tl1umoNl1<iiVl~il overhead 'l1l-I
1,500 rn'YI 'l1l-lilluiJ<J~t1m'lNl1<ii ~llib-ii1111 4,500 'Ul'YI iriBu1il~'1li'lmilu~n~11m~~Vll-!'1tXumw1u

mru11,ml'mtll1'1i1m'WL~d'lutmYf'i1uihc1'J~ 1 b'i14il, ~'1l'l"ll'i11'llltJ'J1nh'1'11u 6,0oo m'VI m111i1bU.I
m1lfoni1111~bnli1rnru1um·mtl;;i'1l1m'W'J1niJ'J~m111>11llll~l'l:i.mfl1'l1:i.i 4,500 m'VI d'iuN;;iNi1lll~l'l:>mfi1n'1
• •
a,ooo m'VJ ,,,1mni1J.Jl1fi1vvi:w~u 1,500 m'VI m111i1cuu1'i'ln'J~"1Jum1bbm'1mm1iitll1'1i1m'W'llil'1'lll11@iu. ' '
bb"''m m11:i.ii'uf.i1urn:;u'lum1bbUl1'1i1ml"l1VIblJuN11llliitwri~i1:w;;ifl1bYi:wbb'111i1n"l'l-ii:i.iflfl1'~;i,Xu. ' .1iJ~1mlJufi1t-ii~11.1'11'1'1 Lu~1u~u b'llu fi1t.if~11.1~mm1llmi1 fi11.if~11.1u'Jm1 blJu~u 'J.lmrn'i'lu
w1'Jbb'11~'1'J:;i'VJlimlJwilh l'i'1,X.I 1fl-ii'OJ'J:;i1'i111:wb.ifmi~'1 'VJ1B'J:;l1m1:i.imm1111um"l'¥11ri1h 'VJ1B'OJ:;
1~wm1;;u1.1:wmnifo1.1bri1ti1 n'J:;~uBajnum1~i1tVll'lm111i1buu,1u'VJ1Bn11i1fi,m:;u1ubbtlmi1ml"ld
1Vlmm111i1-i'i'1l;Jl1fl1b'Vi:w1~mn UiltJiltl1'1h cllml1'1qJ
v ' 4
1'1101'JN'11 l.3
i1Vlmbbl'l'VJd'1VIB'1bmiu11'1'llil'1m~'1 'J1n,Xu1tl~ili1~niim1'VJtl)m'Jinh'1'11.lL.l"ll'i11flil.ll1:; 12
m'VI bmrni1B'1!liln bb'11bmi1~11'1uuiu1uVIB'1 Mili111'i11'l111.1L~bllunBu;;i:; 15 m'VI bri1dnbllu
mru1um1bbtll1'1i1mmbuu'VJd, ~'1bnli1l;Jl1'11'Yi:w 3 UTYI 'VJi'''1'J1n,Xum'J'J:;&ii1c'i14B'1tlwmm'i1Lufiu
~1tVl~~ill'lm1;;i1:;mni1m1.1mn~u ~'1mm'lm'Vi;;11'i11'll1ml'lu 20 uw1 inli1l;Jl1fl1b'Vimj'1~.lbllu 5 1J1'VI
'J1n,Xu1tlbmil11.1:;liln'Jm'l~Wft'VITil1i1bU.1'11.1 (franchise) ~l1:;lmlimm~i1'VJU1-i'1u ~1lV1'1~''i'um1mi1
,:;}ilflB;;1n~u ~'1btl~tJ.l'll1tJ~'l1'i11 25 1J1'VI iiu?iil l;Jl1fllb~J.J~'1J'1~u:w11im1Ju 5 1J1'VJ 'Jdiu1~11 m1
i11'1'1m:;mum'lbbtlmi1ml"l,Xu mm'llli1'Vl1'1bli!lnl~'VJmtJ'l '1'11'1 bb'1;;i:;m'1hliBnm'J'J:;riB1Vlbnli1l;Jl1fll
'Yi;;~"llu'l1n:11u'llmm1;;m:w11111um'li11'1'1ri1h 'VJ1Bi1-i'1'1'Jmfi11Vlnun'Jm1 iln~li1m'lbbl1:;iin
u'J'VJ11m111i1iU.1'11.1~'1'J:;~B'1l'lm1:wim1;;b.iflt'J1.l'VJi''nm'l m~i1m'lm'llli1bUu'11.lbDum.h'1&1 ~'":;
mm'ln<ii"li15uhi11'1'1bbl1:;u'J'VJ11 ri1i'urn:;mum1bbtll1'1i1ml"l'llil'1n'OJm'l'llil'1ll.I 1VIbni1m'li11'1'1l;Jl1fll
' '~ ' ~
bl"i:WiltJ1'1i'VJ;Jl:;1;J'Vliji1
,-r;11:;w;,Xum.,~11l1n'OJ?iB m1bbl11'1mm:;u1um'lbbtll1'1i1ml"l~nil'lVIbnli1l'll;Jl1'11,~;; '"':;
i11m111~1ri1h"l1m;1;;ifi1,Yi;;~•ni1~.I m'l~i1m'lm'llli1'um1u~''~m'1iiJ'1num1<iii1ft.lL'J1ub4!l'1'lliJ'1
m'l~i1m'lm:;mum1bltlmi1mv; •'lfu m1bli!lmbl1:;iJiJmbuu1iuf11'VJ1iJu'Jm~N11ll m1cliiJni1mu~~'
tl1:;n!lum1 m1•liiJm'Vl'i1tut;;i~ b'i14iJ,~rn Btlnm m1iJiJmcuui1mutl1miJum1 m1~m1'1ri1'1''1'i1.I
'
""m1w'J'VJ1.,m.lm., ,l'Ju~u iJtJ1,hnll1:w m'lriiJ1VI•ni1i,i;;ifi1,~:i.i1J.J1~~1ni1iJajcv.im'VJu'ltJ"ll'ln",r11,Xu
bbilbb'1€1'1fim11:i.ibb"1'1'VJ1ri1hiJtl1'1'VJU1m1u11'lfm1 m1<'i1m1m1<1i1bU.1'11.l~~!lm~1l1'id'lli1m"JbbUl1'1
i1ml"lb'lfunu ~fo,:(u1:i.i11m.lm1lli1'l n'J:;~iJ'11"imm:i.ibbl11'1mmru1um1bbtlfMi1ml"l~i1:;Yf!l.ltVil'l
i,i;;ifi1c'Vi:i.ij''1ijli1 ~,1um'1'i1~'1m'J'J:;1:i.i1'lfi,i;;ifi1,Yi:i.ilJ'''lli1Lub~'ri1h bbl'ibllUl;J;;ifi1,.yi;;~rh1Vlm.lm1
vi1iJviu1rn1u d'JuiJ'1.lm1~ii')mfi111i1tJiim'lLoiiiJ'J..J'tJu1b'111vi1iJ Input UiJtJ~iji1 bvliJ1Vicni1Nl1N111llvi1!l
Output ~riiJ'lVi'ini1tl1:;fo'Jfull1J.J'llll'ltl1:;l1'1fi'lliJ'1iJ'1fim'lmn~iji1

'
t
Input
Operations
Output
191tJ$)1J
bb"l.:J.:JlU rnnnum"lLLti'".:i'1m'l'l
'l'l~.:J.:JlU flU~l/U'inl"l
1'1i1tJm"lN1i<1l
'
4.500 I
lJ'"fllb~l-1 = 1,500 1Jl'VI
6,000 1Jl'VI
"1.:J,Xu rnh11Ji'11V1ii1tJNii<1l'Vl~ll'VIU1tJtiJ.jlio/im.,dJu'Viii1t1~rl1;llJmnm.,1:::1iitJtino/io;inl'.Ju
V1u1m1u~'l1i'Vl~'V'Imm'Vl~ll'Vl~'V'l6i1umnbvillit1m'1~1.:Jm~mum"'''tl<1.:Jflrn'V'i1um.,li11'uu.:J1u ~.:Ji11u
L'VltJJ'l~'l1i'Vl~'V'lmrnmnn111>14.:J'Vlit.:J'llm'VI™mn~!l.:Jff.:Jn'lm., 'lurn.:J1pno;io;i~'lVfm1J-Jrl1;llJLiitJl'.lV1i1tJ
.:JlUbbtJnLfluLeini'VI'1lf1'VIW'VIU1miullJ.jlio/in11L%'V'il~ ni,;rn:;'IJ!l.:Jm.,'iliim.,J(1un11Nii<1lbbl'l~u'im.,,Xu
'iiiLfluei81.:JD.:J~'l~L.if1hm~mum., 'lrrm1 V1nm.,li11iuu.:J1u'lu.,ruuNiil'lbb<1~m1u'inwll!l.:J'wu1tJ.:i1u
ri!lu~.:JLii~.:J~u ~u1'1'11.,J11um1Niil'lLbl'I~m1u'im1o;i:;J;mL.if1hm~1num.,,j1.:J'l 'lVici!l.:JbbYl ,yjei~o;i~
mm.,ri'l1i'YI™mmiV1fi1,Xu1J(!ltJ1.:Jl'.ltl1~i1Y1rrrn'V'i bb<1~~,;fi1,j!lm1<1.:i11ueitJ1.:Jj.:J')ii1J(

• •.o:!:lq A QJ Q.I o A
!1fl~TI 1.3.2 HU1flVl!!f;l~'tHlflfll1'Ufl~fl11'ilVIfl11fll1VIlil!'U~lU
1urn"i~'ilrn"irn"i~11'lu,1ui:iu 'il~i1111Anrn"lL'liuL~mnu111Anrn"i~'ilrn"lvf1111 li"l~nt1uili1mm'
~uc1mu~'11qii11JWlt1 rn"lil11'"i~lJlJ (organizing) rn"Jei1u1tml"i (directing) nii11Wit1 lurn"i~'il~il
m~murn"liblifl,ilm'W~'il~ill;Jflfi1•~:i.ii.J'~'l'ilLu~m~:i.iiliun~!l <i!t1,ilfi,m~murn"l1Llifl,ilml'1mn!lu
~'1lt1,ilrn"im'YJ'Ul.bbf1~!l,~m ilmu~ !llfll"i ~iiu L'iun-)1 facility ilrn"i!l!lrnLurn1u ilrn"l'1"i"iV11
1Jf1mmmli.,~'ii1'lum~murn.,'11''1 tiei11LlJw11.,il11''lVILn"1m~murn"lLLlifl,ilml'1 111'l!lLn"1#nw111'1
1unl';jiblifl'ilml'1i:J'il%EJii1dh'lVlblJuNii'lliirn.ri111'leiu'ln1.,'il1:1-11lmrn"i m~u1urn"iLLlimilml'1~il-l'1,~u
d,~11~"1m1:1-1mm.,lli.J''l"1~"i~llu111rt, L'itm11 rhfi''rn.,Niilll (capacity) rn"iilTl'm~murn"iLLlim
ilm'WLVlil~umd nmm.,ll~'il~1'il~'ilm1:1-1m:i.i1"ill'llil,iin~'ilrn.,Lili"i~llu111rt,11unm~'ilrn"iLV1ii11'.11
1.bnl"lil11'm~lJ1.bnl"iiilimilm'1~i-1~"1ml:l.Jftl:l.Jl"illVl'iiln1A'f)l';jNiill~';j~llu111rt,i(u ,~11<M1oil
'Yl"f'Wmm111L'Yhh ~~il-'1,n1A'm.,Niilll~.,~llrn~mfiu LL.iilm.,1-ii1'111'1mm111iiBEJn-i1 8!l:i.i'il~1ili
Lmtimffoi1~"1m1:1-1mm"illLurn"lLL'li,oifui.J'n-;i1~~11!l''l-ii1'111'1mmmnn11 ~'liei1ili111liuG'iflli~'lurn"i
%"1m"lili1u111rt, rn"l~1•'lum"lilfi,d,~~ufiuil'il%ti111mtieitJ1, ~"~tJn<ihei81,,muuL'i-1uull,d
mti n. LLfl~ 'll. il~u 10 li'1urn'Ylwhnu LL'1fl~f1u111#,h,,1uNiillllltj 1"1tJ~.bltJ n. 111#,
h,,l'UNiillllltj 10 li'1urn'YI i1i1tJri1fi''rn.,Niilll 10,000 neiu/fa U11J 'll. L.ifL1u 10 li'1urn1'1#,h,,1u
Niillllltj~iJri1fi',Niilll 12,000 nei.b/1u lll.,,d~ei1'ilm1:1-1illm"il1Vl'iil1'illi.,~R1'1ITN'1 (effectiveness) l.b
rn"i%"1il'i'1,.,rnun~ei1"1111-ii~u 10 li'1urnmvhnu ,~1iliri1ii°,m"lNiilllLYi1h ~''~Li'lu11mtJ 'll. i1
li"l~ft'YIITNfli.J'n11 iiu~eiLiil"1''lVILi'luli,m1:1-1mm"i!l'llmrn"i%"1rn"i'lu~urn"i%"1il'i'1,.,~uu
'lu~u~ilm~m1"Jei1u1tJm"l'lVlm~murn.,~il-l'1,~ui1u '1'11,m'l..1iliNfl,1u111'ieiNf!Nl'illlLYi1fiu
111'l!l'lnli'L~rnfiun1A,Nii<i1111'leilll1:1-1#nu.nT'l~ilfi,lJ ll,i:iu 1'.li1ei'llm~u'i1111"lLuLL~'llil,m"lei1u1tirn"llu
~.blllil.bdn~ei 'il~ftl:l.Jl"illlJ'iVll"i n1fiurn"i~lLU.b,l.b'll!l,1pn'il'llmlllulVi1iliNflNiilllLnli'L~rn111'i!llll"i'llll:l.J
n1ii°,rn"lNiilll~#,1-Jmniieim.Wu,l"1 f1l"il'ilNff1l"iBl.b'll'Jnl"i'il~l"1l.bL'li'li"i~ft1'1Tii111'1 (efficiency) <ii'1eitJ1,
b'llu 'lumruT.,,,1u n. i1ri1'1,Nl'illl 10.000 nilull'u ilmw'l1111·"1u~ill.lJl'l"f.1N~'ilef~l<ii'1'u11~ 8,000 ntJu
'il~i'!'ilbU.bli"l~ft1'1Tii11'1lUm"iNiilll'i'iltJ'1~ 80 il1uh"1u 'll. ~,;/ri1fi',m"JNiilllli' 12,000 flil.b LL'1Niilll
Lili'il'i'Lwrn 7,200 nilu 111'leii'.1li"l~ft'YIITml'1'lurn"iNiilllLoWrn-l'iltJf!~ 60 ~'111mtJli,11 1"l"1U 'll. i'.1m1.lJ
mm·m'lurnw'i1111"i~1n'i1 ~!l l'.1li"i~ft1'1ITml'11urn"lNl'illl~~1n'i1
'l.bnl~'il~l"11lrn"J%"1m"irn"i~liUU,l.b1"11'J"i1:1.J°i111ili~LLfllVl.b'il~11ei,LillNf!'llm~' 2~.bll!J.b:l.Jl
Yi'il1"irn11wnu nii11~ilh,,1u n. Hfi'.Ju111 10 li'1urn1'1 i1Ti,ri1A'rn"Niilll1ili 10.000 nilu mm"ill
~liUU,l.bNl'illliltj'lVli'.111"J~ft'YITIJlll'1ij't1'1eitJf~ 80 ~,1iliN'1Niilll1.bfl~ 8,000 flilU il1uh"1u 'll 'loii11u
111 10 li'1urnm'llufiu ilfi,ri1A'rn"Niilll1ilill'ti' 12.000 nilu ib'1~1Li'lu,1uNl'illlilti'lVli1li"l~ft'YIITml'1• •

LV1m-'ll1.l1~ 60 ;:J,1.'1Nl1Nii<11LV1rn1ul1~ 7,200 nllu nl'llli1Ntml'lii1LUU~ULW11'1"i'l'l:J.J 'il~llilLULll,Nii<1lrn'l"i
(productivity) ~''Vl~W~,,;r~'ll'llll~m'lLfllilN1N'.i<11 (output) l'illli'imrn'Yl~'l"imn'lri1-ifl11 (input)
"11:J.J~<1l'l
Nfi<1lll1'1"1 (Productivity) =
Nl1N'.i<11 (Output)
'Yl~'l"imrn~1'1fl11 (Input)
J),,:iu1urnru'lleNt,,,,1u'1~,ii,'1m mi111.'1111.,,,1u n. ilNii<1lrn'l"i1u.'lmL~u"~'YJU L'Yi1nu 8,ooo
nllUl'ill 10 tll'UlJl'YI 'Vl7ll 800 neiul'ieiti'1urn'YI chuh"1u 'll. ilNii<1lrn'l"iLU~l'Ub~'Ul1~'YJ'U L'YilnlJ 7,200
neiul'iei 10 ti'1urn'VI V11ei 720 neiul'ieiti'mrn'YI ~''111iL~11h~,11u n. ilm'l,Joim'lm'lii1Luu,1u1.'I~
n11h,,1u 'll. iiui!iei m'l101~01m1:J.J'11m'l<iV11ll1oiN1'llll,nwJ'111m'lm'lii1Lum1u~~lum1:J.JLDu'il''mL'11
Ll'ltJvf'lL1ibbtl'lLi.J1~'illJLV1mueirnL'1i11h,,1u n. Nii<1lrn'l"i~n11h"1u 'lL nl'il~lJlln1Vf~nmn,~U<1JlJ'j,
'l18'nl~li1'1-l'1,,,~lJlJLL1:;m'lellU'l1.lnl"i ml"OJ~lJlln11fi1V1~h"1u n. ilil'lJ'V11"17'L4ll,li'l:;il'YITIN1LU
m'lli111.,:;uu~1 LLl'iLttei,'il1nilm•u"lV11"i~~i:l~ilti'l:;il'Yl'ilrn'l"I1um•Nii<1ln,-'ei1.11~ 80 V11rn~fmmTI'i°'IJJm
1Vll.'1N1N'.i<11LY1:J.J~U1115n m"i%111"i:;lJlJV11llli'l~fl'YITIN1rin111~1lU<1lll'U~UR'il~LDwiiei~1nl'lLVinuL7',11-1
1u'll~L~mnuh,~m 'll. Lttei~"OJ1nlil'l•~uu11'~ilti'l:;fl'YlllNl1M 1u'llrn:;dn~LLif11ilm•u'iV11"iri.'leitJ•
tl1:;fl'Vlllrn'l"ln11 LL'iin1V11ntl~ut11~m'lu'iV11•~1-11'ii:;m:J.J1"imv1:J.J N1Nii"11~mn~u11111.1i'.i'1i.J~1LDu
":;4lei~11i'llm1.1n'ilm•
1u;i1u~'llll~~~n1'l1'l'lf1m~l'lm"im"iii1Luu'1ud ~~nml~1bD1-14ill~L.if1hLubd~11N1'llll~m'l
%111m1m':iii1bu1-1~1u m'il1nm•'1-i'1~•:;uubL1rn1'lelm1tm1"i"i:;uui:l~4iei~LL1.ln1Vieiein11 Lrlein'ilm1ii
iJbJ'Vllb~1.l'lnuN1Nii<1l'llll~"1'1-1,:1u iJbJ'Vll,:i1-l:J.Jl'il1nm1'1-i'l~"i~lJlJ'Vl1llm1ellU'l1.lnl"i"i:;lJlJLD1-IUb!VilL:a~
tl•~fl'YITilll'l"l'Vl1llbll~tl"i~ft'YITIN1
L'l"i"il~%:;,:Cu 1Ub~ll~'llll~'l'lf1m'l'<J111m•m'lii1Lu1-1~1u~~LD1-1'l•1nri'1"i"iV11'lllm11um"i<iil'lfluh1u
Lll~'l'Ylmm'1<1l~'ll<Nm1%1i1'1-i'i~•:;uuLmrn1'lu'iV111rn~u11-1m':iLL1iil~'1rn'l"IL~1l1Vf1i1111m•1-ii'Yl~'l"lmn'
Ull1.1rirjl11Lb1:;L4i'l;Jl1fllLY1:J.J (value added) mn~rjlil Ji~LL'11il~L1'1W11'i"i~ 1.2

Organizing
Input
(Effectiveness)
- Capacity
YJ'l"i'1mm
flU
Clu
Directing
(Efficiency)
Output
[
Nii<ilm~
(Productivity)
=
N~Niilil (Output) J'Yl'l"i'1mmfi1~'b.J (Input)
. ' 'v <'!! ~ ~ ""'1 v "i'~I ~I""'""""'""'
'l'IM'iJlflftfl'lfl!'llti11ll!1~::1~M'l'l 1.3.2 mn rn~AufiUAfl'ilfln:tt 1.3.2
1
..s 1 .d. .d. ,A .d.
'11!!'111fll~ftflll1'11'1111J'l'l 1 lltl'll'l'l 1.3 !W~'l'l 1.3,2

. '
"" ""'~10'i'YI 1.3.3
W V o A
fll':i'lliltr':iH':i:i:1J1Jfl1':iil1!'ll'IJ~1'IJ
"lilb~:1-111u'118,rn:'.l.J1umi"bbtl'1'flrnl'1'V1~m.,fl-i'l'"S:;l.Jl.J fi8 "J:;flfi,rn:'.l.J1um-,bbtl'1'flrnl'181l1'h•
1u%u.~:1-111ulfi8 118,vi,1-,nn:hm118'n17LVirn:;mun1-,bbtl'1'611111'1,XUNi1ilftu;ii1.,,''foum1-,8:;1-, ~'
,:;118,•~:1-1,1nm1q:'n1'11bb'1:;V1,nru1n·:innmmbbl.Jubb'1:;v:i@u1 Ni'im'fru.vi ~'d'Ju"lil.~:1-111u'118'
rn:;u1um-,bbll'1,flrnl'1~"l:;fl'i'1,~u:1-11 11rufl:1-1ii~'ll8,Nl1iliiru.vi~118,fiTu'ii'fi8 "l:5118'm" 11run1'1
(quality) 11'1-l'YJ'l-I (cost) m:1-11ru~"J:;Ni1il (volume) bb'1:;m1:1-11'iil'V1aj'l-l'Um-,Ni1il (flexibility) 'u-,:;$fu'ii
~u1m-,118,mm111i'filft'l-Ih b~mfium-,rh'Vi'l-1il'jilLI'-1'118,ft'!-l;ii1bb'1:;u1m-, (product positioning) 'Jim
n"lm-, 1rn/l11ll&:;1~u~' 4 iXu1#1 mh1fi8 ftu;ii1'V178u1m~n,m-,11mm-,Ni1il ,:;LViil11run1V1-,:;$iu
L'Vi'l-15:111UYJ'!-1":;$iuL'Vi'l-1TlfllbrllLil mmru1ri1'lil 11rnm-,'lViilm1:1-11'iil'V1aj'l-l'lum-,Ni1il 1-liuil~bbl.Jl.J (model)
~~bbl.JU ilm1:1-1'V1mn'V1mtJ (variety) mnifombfiL'Vi'l-1 ~u1'V11-,118,-,:;'.n1$111m&il'jil1'iu'll8~ft'!-l;ii15:l
m1:1-1rl1fi11J"18n"lm-,d'Ju81l1'ri, b'1-,1:;blJU'jilb~:1-111u'118'U1vrnvllu'l ~118,il1:1-1m bb'1:;$18,.,:;'.n1#111
'lulj-,i1"l'l wd, n"lm-,"l:;fl1m-,11b'.8n'jil1'iu~~bU'l-lm-,Nfl:i-1Nfll'l-l'll8'flllil'l:1-1ii~'ll8'Ni1iliiru.vi~'~
tl-,:;,~u$1,mhi1#imnmti'V1mmbl.ll.J
1!181l1' 1l"n'~Ni1ilmYlmm"l"l:;blJun"lm~Ni1ilmYlm-,ifl1bb'1'LiltiNl'iil1l-lmmbl.J1l iinzj:1-1
jn;ii1bbil1l bb"1b'd8~•1milu1iu.'h-,1il1j~ 11run1V1J'~"iluftu;ii1~l'i1n1h'lVifi1ln•m-,1#imn Liltis:l
mmrum-,Nl'iilLl-l:1-11nii:n l'i1'lVin"lm7tl-,:;fl1Jil11:1-irl1.~.1#1#11tl''lil1'iuil: 'V1~'V11mU'l-lrnrn'118'1l"n'Nl'iil
mY!m11run1'1&:;$iutlmnm' -,1mtlmnm' lb"1Ni1il'ii1u1umn~u ii~bbl.ll.lmn~u lmm-,!1l'i1'lVi
n'lm-,tJ-,:;51Jffll:i-irllb~'ll#i '¥178'urnrn'118~m-,Ni1il'l-11Yfm'Jfilnl'1~1'111tJ'uTlmtJmb.iNl'iilmmrnmn
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1. mmm41l"fl0011YI (Quality Conscious)
2. mnbffm1:1-1<ilil'm~111mn111mtJ (Variety Requirement)
3. mmff'llmfll"lbD~tJUbbDiNL'Yll'l1ufotl (Technology Change)
4. mtbbfffll"lbb'li,1fu1l1n (Global Competition)
d
1. fl':i~lll:T!'Hl~~tl!.fll'l'l
n"ltbbff1m41l,'llil~flWllYI ~'LU'UNl1llrn~il,m'il1nYi@jmm"l'llil'''YI;;itu1l1tlbbl1tbf1'>1'Jlfi'il fi
-111Vfi.J1t·1nm1.inl'Jm1:1-1fiu&iilaj&imn~u ~,nm1:1-1<ilil~m11um11J'ltn;;i~l'1m1:1-1i.J1::dh1mn~um1:1-1
111l-11tJ'lli.l,m1:1-1<ili.l'm~i.J1trn'ilmn~ul'J 2 '1nm1.1t ?1i.l ;;i11:1-1<il1l~nT'l1f~1nl'lfiu.11~n'l0011Y1j'~uffu
m1:1-1<ifil,m1~l'lm1:1-1bU'Uft1'U<if1 (personalize) L'Uft1'U'llil,mll-J<ilmm"lb4il,l'lWllYl~ff,~ud <if1ei1.l1'. .
L'liu 1111nbUUN~i'JWnYIbft"l~fi'il1l-i~'ilt1J71nl'lm1111-:;LV1mLL'1m1tf,'liY11Ji"lff'J5liilbbl1tl'lWllYl!ll'lll"l'ilt1l-i. "' .~1~()! bYi1L111iin LL.ILriei;;iul'lbft"l1'Jlfi'ii&l~ul'Jm1l-IbU'UeJaj&i~u !<iiil'm"l~'ilt'i'ui.J-:;:;'YJ1um1111"l~!Jnh
"lff'J51~11nh w1'liLV1mmw71nmi11111-:;LYleii.J-:;:;~,m1:1-1~1 Lllu<ilu i),,:(mrimm~fililt111tJ"l1:i-i'll1l'1l1n
. "',,,.,% n I c,>,,;. <"I ,,;.){ :; a, A !!.< ,,J, Jf. ,:!
11'll'U1J"lt'J51mnl-l'Yll'bl11lll11'llU LY1-:;1t:WUnY1mtJ1l-lbl11lnn1-:;'i1ei1Jff'Uil~m1:1-1'ilil'm"'Yll''llwlf'm:;bbff
,4ei,fl0011Y1dn'1111Ji11blJuYI'1~Nii'niiufiuiiu""ln'llil~m"lfllf1,1:;uum"l~1bilu'1u
2. fl':i~l!l:Tfl11111'lti~fll':i~'IHllfl'tltlltl
1um1:1-1<ilmm-:;Lumw'11nl'l~l'Jm1:1-1l.l-:;trn'ilmn~u,Xu UilnL'llUil°OJ1nL41l~l'l0011Y1 mtLLfl'llei'•
fOt ..J v Ci.al Ji 2' ' A ,,,J 2' ='f _ -=!. &>.Ji a, &:! !!.<
m1:1-1buU-r;;1'U'il1 (personalize) rn-1mn'Jlu1111tJ nm11'1il bl-11lN1J"lU11'1:1-11llut11'11'Ufl'ilt:l-lm1:1-1'ilei,m-:;
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fiu<ll1~'il"l,num1:1-1<ifei,m"l'llil,'il'Ubil,'il'l''l mn~u LL'1l1tl'l'U'ilt'V'lmm:1-1l'Jm1:1-1bllucl1u<if1 mtLLfflu
,4il,d~,111N<i'niiu1Vfm"lNij'ilfiu<lf111111lnn'J111m-:;m-:;~1Luu~1w~1blJU<ilil~l'lm1:1-1111mn111mtJmn~u l'1
'JllbbulJ (model) '11~'1 mn~u <if1il1.ll~b'li'U 1j"lfi'Jl.J"lmillJ"l1tJ'Uiii 'lllmUubrieifliitJ 20-30 llrfou m-:;
N!i'il"lmmliiLL'1l1~"'tl11l-1Am LL'1l1t-iu<Jtal,v;mi1~u1l1u•V1m 4 ll LL'll1uil•"UUTI-:;fi<J"l1tJulii~1,1Ju~ ~ y~ q q «II q q
<il1l~llll1Viblleinmn~u m-:;Nij'ilwtJu!ii<ilml'Je:i1.l1~iie:itJ s ll bbl1t'ju'li1•0mlb&imffu~~~u (color key) Lu
-i1ue:i~b~mnuITTfi<Jthu~111ffTiffiitJb~:1-1m<J'ilta11ViLlleinuiltJbbuuLL.iil-..'iluu<ile:i~ntvrLllein111mn111mtJmn~u
1urn~l'I~~LL'1l1~bb1J1Jill<J'ilt<ilei~!Jill-1lVf1jn.'11'11ill.J'i'ubl.J~tJ'U1Ji1h~ b'V'i~t:t,X'Ubb-:;~11111i'lu'llei~m"l~lbu'U
~1ul?1ei o/rei~m"lflWll'V'ij~bbl1tllm1l-l111mn111m!Jmn ~~m1N!i'ilb~il1Vfllm1:1-1111mn111mm.nn,Xu<Jt

d'JuiltlJvi1ei81.:i'lvib)'VI'11.:i'llrnn'm"ibW'n::;l"1u011t1bb~1m.,i;11,uu.:imn'ilm':l:iin':;iiirn'11u.:iii~m.,1iii
fl1l:i.JU':J:;V1tf"1'ii€1'1l.lli1 (economy of scale) nli11$iei n1Nli"11.lflll'111l"11:i.Jbb'iibWmeith~b~tnbb'iiNli<ii
.-i~~l1:;mn--1 'i!:;l-11'lViil<ilu'YJULum·mli"1~1bbl1:;i'ltl1:;1r1ilmwj~ 'luii:iimim.1,m1vi1h~~11-1vi'ieim1vi1
n'ilm1l"1u011t1 n'il:;~<;1V1'9nm1L~m1:i.itl1:;vilf"1"1B'll.11"1 bb<iii:l'ii"JUl-ln'ilm':l'il:;vi1b-tlui:iu1:J1~ bw11:;
m1:i.i~iiiei~il.-i11:1.1V1l11nvimumn ~.:i"11:i.itln~bb~1vi1nilm1:i.ivimnvici1u:i.i1nn'ilm1n'il:;1:J1~fi'Vlt9n'llei~
m11~m1:i.itl':J:;viti"1<iiei'll.11i1 bw11:;,:;vi1'lViil<ifmm'1~~u bb'iim"ibb'l1~'1iu~ilei8ei81~.,ubb"i~'luil'il'iluu d'Juq ~ ~ ~ •
bb"j~f1"1#i'U'lvr~1u~mm1i;11bU.1~1uiiiei~mm1ll~'il:;Nli"1i1Uf11~ilm1:i.ivimnvimumn1~ ~1tJ<ifU'YJU~~l
bbl1:;'l!Wl1Wj~ #r~i:iu iu11d'Jub4ei.:iYl1mtJm1'1<11m1m1i;11buu~1u'1lm~u'im1'luti.-iil'il"JUUbfJueiEJ1~~~
3. 11-:;:::111:r'tle.:im-:;1tl~mrntln.:imflhihrl
m:;bb'1'1Jei~m1btl~tJ.lbbtl"~br1.-ituLl1ubfJub4ei.:i~YITvnum1{"1m"lm1i;i1bu1-1~11-1'l1-ltJ.-ii:I'"luulin
b4rnVlii~b11'i!:;ft~bf1"1b~U1~11m1bU~tJUbbUl1~V1'ieinw1'fGll.11'1lrnbr1.-i1.Ill1U'il:;i11.:i'J1 (cycle) '1J€l.Jnl"l
tl~rntl~tJ.lbY1'1Lull1u~u""4eiu'l li1eiEJ1~~b~u1~-if"1 $iei m"lbti~uuiu'llei~bYlflL1-1Ll1uflei:i.iYl1b"1€1~ #r~i:iu
'l1-1m"l~'i!:;l1.:iY1u'lfibYI'1lull1u'l"1 ru 1ud1eim'1~,:;~1'1:iiueiEJ1~"l1"1b~1i:iullm1mlJu1u1~'1.:imn m1' .{<;1m"lm"li;11bU.1~1.1~.J'J:;<ilei.:i'lVlf11l:i.Jftl4i'tjLub4eNd $iei n1'i!:;l1~'YJU'l.lbYlfll.lll1Uli1bYlfll.lll1~viii~<ifei~~
11bYl'1lUll1~i:i.lm'il'i1:;~1'1:iftJl.'1€18l~'J1i1b~1 iJ%:::i:iu<ifrniJ1u.li1bbl1:;iJ1unTl{i1mmubYlfllUll1ui:iub~eJ
~'J:;li<;1GUhbl1€lmYl'1t.lll1mei81.JbV1:i.J1:;('1j.J l4>1:;i:iu..~1V11n1tilli"111Uhbi'.eimY1'1lUll1~~Ni1~Um !1m'J
,:;iJNl1'iieim1i.Jmi.J1"ln'lum"lbb'li~'1iu1~1-11:;u:;m1
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4. ll'.i:::l!'i:rfll'.i!!'ll.:i'll'll!ftll
bD.I~Yl'JllJn.1~11Nl1'1J€~nwi'ilillu1bYI'1lU111u~Nl.ll-Jl'il.ln~i:l'J"JU.11~EJuEJei1:;vi11~tl"i:;bYlrf
vi'ieiti"l:;'1J1'1:i.i<ii1~'1 'l1-11"n'lVimm1mb11mtl~uunu1.'lei81~'ln~~i1'Jum:;0.:iii~'l"1~11111n1~w"lmb<11u
(boarderless world vi'iei one world) nl111$iei tl"i:;'1J1'1i.J'li1.:i'l 'll€~lfln'11m"itl~'J:;~<;1<i]mbflmti~tJ.I
m1'11.1bYIrr i'luff1 bbfl:;u'im1nu1.'leiEJ1.:i"w-i'vimtJ1:Jl'.l'li"11iu"ii1n"1l11.1ff1vi'ieiu'im~Nli"1'l1.1tl1:;'1J1fl:i.i
vii'.l.:im'J'1l:i.Jl'Jtlri~1ti"ii1VIU1tJ'l1.1tl1:;'1J1f)l-J5U'l 1.'lei81~.ieiwii1.:i~1tJ<i1ltJ 'lUlill1Uf11'Jmb'111.1d~Ni;i"1G.lfl1
vi'ieiu'im"l1:JbWm'J:;<if€~bboJi~'1f1.Il1.lnl'Jvill!"lfi'Jnu~51-1'l1.IU'J:;'1J1fll-Jb~tJ1nU blJ€~b~tJ1nu 'Vl'im.h:;bYII'
b~tJ1nu bb'Iitfo'J:;<if€1'bb'li''1fUn1Jl!"lfi'J'J1f1U"i:;'1J1fl:i.JVl'i€1'J1f1U'J:;bYI1'!5w1 .'11tJ€Ell.JV!i'.m~ml:JL~ b'1'<11:;
1!"n'""1ntl1nY11'15um'ilmm1llri.:ii'lm'ilm"ii1viu1uV1'ieim'lViu'1m1'lu"1mi1'1l€.Jb.,11'1 bbfl:;'l.l'llru:;
b~mnumnm'ilmm"lllri.:ii'lufl111N1viu1uvi'iei1t1'lViJJ°1m1'lu"1m"1'lu~5u'l 1~~1u b~mfJub-tludm.,
bb'li.:i-liu'lub~~ITTn'il~lblJ1.1'J:;<ilei.:iY11m1i.Jmbb"l.JV1'ieiY11m1i.JV1mnvimumn~ub4eitJ'l"11:i.im:;bb'1l'1m.fi1"1u
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~iJJ1'1-1'J:;iJi.J1n~ub4iltJ'l n'Jnl"i~,-;i1bUU'J:;<ifei.:i'1fi.:im1i.Jw~ei:i.i bb<l:;f~1.:i'lii1m1l-J'1ll-ll'JllL1.lf11'Jbb'li.:i'1iu
'lVij.:i~u'l1ti"11w111m1"lbb'li.:i'1iu~~'":;bllubb"l.Jn"1#i'ulin!l81.:iV1i'.l.:i~unm.,.;J<11m"lm1i;11bum11.1'J:;<i1rntl.,:;'1u
ni'i11L"1tJii':ltinfiei in{<;1mw'il'iluu'J:;<lfo"m'1J1.1:;m1:i.in"1lii'u~.:i 4 d'lvtl~ ~":;tl1:;'1JJm1i.J
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'11b~'i!1:;#iuj.:i1.1

ueim111ilei1u<J1nm:::bbl1n?lii'ui1i, 4 m:::bb11~nil11mbbfi11f1,o/lu 'lutjfiil"il'ij'Liudiidlli'l:::b~u
lAmn'IJm'l<i'1m'lm'l~1buu,1u'luipn"il'luJ11u5u'l ~,j1blluo/lei,'ii1u,fi~lln 2u'l:::b~utV111J'l $iei u'l:::b~u
bAmfl'Ul.Jf1mmbbi1:::iJ7:::b~t.1bAmn'IJnT1N'.1<11bb'IJ'IJil<ll1uii~111'ieibb'IJ'IJ1'i'1111aju
U"i~!W'lll~l'l1il'U!l'ltllfl1
m7<i'1m'lb~mn'Ul.Jf1mmlum'li.Jijlf~'1u<111i,Jbbt.11i'i'1bb'IJ'IJbnwei'i1 i,J'i-!1'fU,!mimni1,1u~
~'1,bliufi,11 (consistence) i,J'lttlU"il:::iim1i,Jb~iltJfi'1 bb'<1:::iinl1m71'i1'm1iJ<111i,JiJl'li,Jrn~'blJum'l:::'llrn
n1'l<i'1m'lbW11:::i,J'lt1'fUiii:ltiJV11mn bW'll:::%:::iiun<Jn1'l~~wmmi,JUlbillbf14ei~<Ynw1bb'Yl~'lt1'1UlVfmn
~M1b'Yil~"il:::mn1Ji '-'?!~ii''lif1umJRbW'll:::"il?11'i1,1udt!,l1m1i,J<.i1blJu~rn'l.iff1u<111i,JbbU1i'i111b~i,J~'i,Jil'q ll. 'IJ q
i1'1ubllubl1l1eiu111diiu (liability)
il1111-i'unT;<i111m11.Jf1mm'lum'lliijlf~m'l<111i,JbbU1i'i'1bb'IJ'IJ'l111JJmwh fiUbllUb!li'.ieiuiiu'Yl-l'wu
(asset) 1111tJi,JU1'fuiim1i,Jbb<11n.J1,"il1mf14ei,<sm $iei l1m1i,J~111crfi,l1'l'l"1 (creative) l1m1i,J1'i111111tiu
' '
(flexible) 'i'udn1ilti1' wi,d1'11llnilti1'1Ji bbi1:::i,J'lttlU<11ei'IJl1uei,.Jeim1i,JYi1mtJ1111tJ1JJii'Ji'1,j1fi111 o/11eiti1~
~bVlu'ii111 b'llu 11li~1u1iJ'.i,JlJnt!~l1m71'i1mtJl1li~ (break record) eJ~bili,JeJ ii',iiuil"il'ijUU b~eJi,JeJ,l1'1U
bl'Juiiu'Yl-J'wubbfi1 m'leieinbb'IJrnmlilrnwmmi,Jbm1u'Y17wum'loil'lvrbllui.J'l:::1mrunil11$iei f1u1'11,1u1~.if
m'll1fi,m1.'1'11ei,'1Ubil,bllu,1u~mm'lrJ~"l:::1'i111111aju1~l-Jn1,1u~1'111n,j1b"ilbbl'l:::iin'lleirn1u~Yi1mtJ ii~
iium1eieimb'IJrn1u Uob design) fl~rn~"ill'lrn1'lub4ei~1111i:i1dJl1tJ
""' ..... "' ~ .... ~ 'fll "iNtll'l!!UU!ll'l !'ll>ll'l!!ti~ fl11Ntll'l!!UU!J>l'l'li',!'ll
nil11nwi11'1tJ1iTii,J'lll~'llrnm7l'i1'1u'lu11iitJneiun1o/lei,m'lnl'lN'.1<11bb'IJ'IJil'1Luii~ (automation)
R"J:::~1'i111wiu (flexible) iiu$iei n1eimn1Jiu'l:::ii'Ylrrmw (efficiency) Miei,1l-J~?IVltiU n1ilmn~111111tiu
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Tio/I8~[JeJi,J11tJU'l:::ft'YIIT.fllW'lUnl'lN"1'111lu14ei~~[JeJj,j-J''IJnumu1Ubbfi11?I[J1l-Jlilei~1u1n11n~14ei,m'lLJl
l'l1ll,JU'l:::VltJ?1'1ei'llUl?I (economy of scale) 111'leim'lLJlm1i,JU'l:::V1ii'1"illnm'lN'.1'1ftUfllVlmtJ'llU?I
(economy of scope) n1o/lei,m1 flexibility fl~ei,iim'lN'.1<111ufi'1111mtJ'llU?l~'n11Vi'tJJ1Jim1i,J
u'l:::VIt1?l.ieionm?I 111'leim'lLJlm1i,JiJ'l:::V1ii?l"Jlnm'lN'.1<111ufi'1111mtJ'llu1111llub4ei,~lilrno/i111iiuh 1liein
b'V1milti1''l111eiti1,wd,m11'1tJ'1'<11l?llu14ei-:ionrnm'l<i'1m'lm'l'ii11uu'1u bb.iiJ<J'ijUUb~mmb'Vlfl1u1111'il11'l
l1Ub'Yl'1 (information technology) m'lmum'l'iilbUU,lU"Jlm:::'IJ'IJ~b'lmi1 CAD/CAM (Computer
v }i !'1
Aided Design and Computer Aided Manufacturing) W'lJUl'nUmbuU CIM (Computer Integrated
. '
,,;; :f A ""'"' ' ""1
Manufacturing) onrn:::um:::bbllbi1ni,Jlm'ij?l'Ylb'mn11 Flexible Manufacturing System Vl"ieJ Program-
mable Automation ~~VimtJn,1111iitJdmmi,Jl'l!1~"J:::11fi,'l:::lJ'IJm'lN"1'1bblJ'IJfi<111uii~~1'i111111aju~~11Jl
u11mf11li,J~11<11ei1m1.if bW'll:::%:::iiu bbu11ui,JiJUlf1'1"J:::jnrlei,dl11l'Y11lw11eitl1,mn "JlnL~i,J~bfitJ~?I
11m'lN'.1<11bb'IJ'IJil<111uii~n'IJm'lN'.1'1~1'i'1111~u1i.Jnu~111 111'leim'lLJlm1i,J1h:::111ii111.Jeionm?I 1mml''i111
m1i,Ju'l:::111tJ?l"J1nm'lN~'1iiufi'1111mtJ'llu1111i.Jnu1JJ1Jl onrn:::dti.Jnu1Jibbfi1 1111mn111bll1-1V1l1nm'l'ii1buu,1u
~b'itJffll Mass Customization ~'VlmtJn~ m'l'iilbUm1uV1'ieim'lN"1'1l'Urni,J1ru,mn'l (mass produc-
tion) bb.li11m'lriiim1i,J111mnV1mtJ~<il8'1Jl1'Urnm1~o/lei,mn1tJ'IJf1i1l11 (customization) IW'll~t:::iXu
'

~u'i'l'111'11urn1oi1•uu~1u~~mntJ'il~~miJ'i'rnihdhnu~~n1Vi1'1 vw11~;:i~iiumrn111Nii<ii1ueJU1'1<ii'il~
~il~ii~~1'i'11'1aju1'"~'l'1u1~i1'Ylilm-w'lu-w-l'ili.Jnu ~~~ftlf'i(!~'J'1n$'ilnl'J~•nil:tu'l'1 $'ilnl'J~'iinh~~nr;I
L-ifm'11ufatJ'1l"JilUb'Yl'1•-lf1lULum8nl1Nfi<ilnril'[,'! .J'Jumli.JYll'YllUYfforn~"i11'9n5nfi'uwd~
. ' 'v .<:!! ~ ~ ""' ii ~~I ~I""...,,,,.,"""
ttil~'illflllfllll!'IHlttltn'Ji::l'JM'l'l 1.3.4 !!il1 !1J1'1ui)Ullfl'ilfl'l1" 1.3.4
1t111t11m1ilnllmtl1uri 1 mmri 1.3 !~turi 1.3.4

Chase, Richard B., Jacobs F. Robert and Aquilano, Nicolas J. Operations Management for Competi-
tive Advantage. 11th ed., New York: McGraw-Hill Book Co., 2006.
Dilworth, James B. Production and Operations Management: Manufacturing and Services. 5th ed.,
Singapore: McGraw-Hill Book Co., 1993.
Gaither, Norman and Frazier, Greg. Production and Operations Management. 8th ed., International
Thomson Publishing, 1999.
Schroeder, Roger G.Operations Management: Decision Making in an Operations Function. 4th ed.,
Singapore: McGraw-Hill Book Co., 1993.
Taha, Hamdy A Operations Research: An Introduction to Management Science. 6th ed., New
Jersey: Prentice-Hall, Inc., 1999.
Taylor Ill, Bernard W. Introduction to Management Science. 5th ed., New Jersey: Prentice-Hall, Inc.,
1999.
Winstion, Wayne L. Operations Research: Applications and Algorithms. 3'd ed., Califormia: Wadsworth,
Inc. 1994.

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2.2.3 irr1'11fi1c;i8uLill'J'-iic;i1111nn'lmrl'
2.2.4 iii1bblJlJ1l1'l:J.18lJ1'1mm11-1
2.2.5 m11'11<i1c;it1uiif1bbuum1'll1-1rl1bbi1tiif1bbuum1:1-18U1'1mml'l-ILill'J1-if'll!lW<ii•n4
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~iu1ufiu,1u 111"lei~1u1U"lll~<J~1>1i1"1 1umrnilth,d,~cifei,11frn.,-1JJ.,-i,rn.,-;;J-4iu1mo/J.J (integer
programming) L-if1'Ji1tJ1um"lbbnU!;lJVll

"fillll}m'1i1''l ll'l tJ'il:::8'1:i.Jn ~11!,'ltm.,1111'11"1€JlJV11€1nl'rnnllllJVll
1. 1lll}Vllnl'lVl1ci'lU1.h:::f!l-lNii'lliill'l1 (product mix)
2. llll}Vllnl'lVllill'lii1unl'rnf!l-l€11Vll'l (feed-mix)
3. ll!]!Vllm'l<.il'l'ill'll"&iu'lri (bus-scheduling)
4. 1l!]!Vl1m'l<.il'lm'lJl1um'lb'.lu (inventory control)
5. ll!]!Vllnl'l'JJUii'llu.'h (transportation)
1'110UH~ 2.1 lJ!]!Vllnl'lVllii'lUl.h:::f!J.le-Ji'ilfi'ill.yj
' !I.> ' SI.I
lJ~'i:t'YINNli'ilbfl1€1,f!lJU1'lll'JVIU' 'Yi1m'lNli'ilbfl1€l'f!lJU1€1!ln~1111u1tJ 2 bblJlJ fi€1 bblJlJ A
• • •
m1::: B 'il:::'Yi1rhh1VinlJlJ~*v1111u1tJf!::: 5,000 lJl'YlftlVl"i'lJbblJlJ A bbf!::: 4,000 lJl'YlftlVl"i'JJbblJlJ B LU
m'le-ili'ilbfl~€J'f!lJ~1,)f,f!!l"blJlJd 'il:::o/l!J,~.hum:::mum'lNii'il'll€J"bNUn~ l bbNUn~ 2 bb.1'1o/1€1"'1'
•
hiiJ,bbNUn'Yll'lf!!llJ bDUbbNUnl'jl'lYll l'J
'll tJf!:::bstJm~mnlJfi''ll'llm'l1.ifnf!1'll€1'6fl4€1,%m1rn•e-iun~ 1 bbf!::: 2 bbf!:::b'lm1ubbNun
'Yll'lf!€Jmbf!l'l'~'m'll'
tl9l'ilni'i1'1l'mn (f1hn)
Tirvm1m !!lJlJ A !llJIJ B llrn1uf11m~il!!d
I V'iJ ~
i;11nr m11
bbNUn 1 12 18 160
bbNUn 2 20 10 170
bbNUrn'Wlf!8U 14 12 130
ri1h<iim'1i-bl'J (rn'YI) 5,000 4,000
.&>Vi->A ""2' l
lJ'l1'1'Yi ~ m1'il'lll'JU'il8,nl'l'Yl'lllJ'll
' JV ' '
m 'l'J:::~fo,Nii'iibfl18'f!lJU1€1!ln <hVIUl l'J bblJlJf!:::nb1'11€1''1i8
•
ilt1"11,_, ~":::1J1n1h"'""• •

Yl1!!tl1'UIHYl1!!UU rilVl'IWlLVf
xA = .J1u1ub'14€1,~uJ1LLuu A ~"il~wli'il ilwu1mlim'14€1Mil'i.J<i11,,'
x8 = .J1mm'14€1,~uJ1..uu B ~"il~Nli'il il111u1mlim'14€1,/il'i.J<ii1i
rf~fl'lfirJ'Atjtl1~tHfi "il1nUl!Vllb'Jlcii€1,fll'JVl1fi1h~'lj'il ,),i('U "il~b~LJ'Ur;f,n4fu1'iltJU'l"~i1,,
Maximize z =5,000 XA + 4,000 XB
rr~ri'lfwuti;l1fi'A <iiei.J1n'il'li€l,Ul!Vlldil€1~ 3 U'J~m• iii€! nm~ifo~'li€1,bbN'Un~ 1 bbW'Unfi
2 LL<iWN'Un'Yl'ili11llJ~'mm•m~vmtiur;t,fi<11uoii€1.J1fi'<i11WILtiu.1,d
'
12 XA + 18 X
8
S 160 (b1'11'1i1l'bbN'Unfi 1)
20 XA + 10 X
8 S 170 (b1'11'1i€1,bbN'Unfi 2)
14 XA + 12 XB s 130 (nm'li1l"bN'Un'Yl'ili11llJ)
o),i(u "11LL1JlJ'li1l'm'11.J"ILLnd:W':ii,•ffu'11111'i'ui11111111dmm•'1'~vu1#1ii',d
Maximize Z = 5,000 XA + 4,000 X
8
mv'liiioii€1.J1fi'<i1
12 XA + 18 XB s 160
20 XA + 10 X8 s 170
14 XA + 12 ~ s 130
XA, X8 ~ 0
Yl16UH~ 2.2 i11111111m"iV11il'<iifh'Um"iNi1:WmVl1"i (feed-mix)
'rl11m"il111!4im~m1ri 20,000 /11 1<ilv'loiinm s il'i.J<i11,,' rieum"i~1111u1LJ~fi1l,'ilm'il fi,
LLif11m11111fi1riw1iii~/11nu<>J~13-J··vhnu bb<i~tJ1mrnm1:w<iiei'm"i'lieJ,mvn"inbb'iln<ii1'num:wmv w1i
'
t'ilm>t~mbff11riw1iii~lli'1"il~fl'U€11Vll"iil'tl<i11,,'l1~ 0.5 ntiin'i':w 'i1'1€1'il"irn~mv 8 il'i.J<i11,,'
'
L~1l'lViln"1111L&iuL<iiLLii~il~1111iinmn'WeJ1111:w~<iieJ'm' m1111"i~Nl1:w'lVftn,,~iil1l,ilfi<iifi1ui11"l
m1111"lli"'l ~b111:w1~i1:w n"•if11m•m11111lli"'l ~1n<ii1l,m"iilmnmf.l111mv'l!i'l'il1u1~uLLii~1'rl•u€11
'l'i1umL~mnunu-l11ti~u~m:w1"imbmwi1:wbtiueJ11111"lil'<i11~,511w;111mLJ'l!i:t<ii ..lli'lu~d,,~~'l1"iill1L~v'
3 'lii'l'il i!ieJ LUi'11ln111ev <ii11L'W'il LL<i~mntf1L111i11N tJ1mrn'li1Ni11"iil1V11•~ilm.flu1'<iii:J~rn•lli<i~'l!i'l11
11'1€111'J'U'i"l'11'1iil,l'iltJ~ULblli'1~'Jii:11i11l-ll'J'1bbi111'lWl4(,'ill'i"l'llieJ1tJd

1J'imw <il1an'i'wii1an'i'l-f'U1M-ll'l1Ji>lu)
~
!ll'l<l!'ll~l-1 hh~'ll hhuai 'lll'll (llll'llfl.fl.)
bU~mrnBt1 0.38 - - 0.05
-ii111'V'lil 0.001 0.09 0.02 3.50
mnCi'1m~B-1 0.002 0.50 0.08 6.75
m1111'iail1~Nfl:1-1bbii'1<it11ei-:il'.iu1:1-11rnm'imvn'iiil-:io/iei1ud i!iei
'
1. 11ml'.ibbfl<U;tJ:l-lil~11Uill'J'Ylijil 08% bb'1t11ei-:ihibnU 1.2%
2. <J'fo,J:i1u,&iu1:iJ~1n11 22%
3. 11ei-:il'.ilvhuei.fl:iJbiiu 5%
''i111e:Mm1flli11o/11bbuum1Tu1•m1:1-1•'fi-:ib#fu b~ei1111u1mrn'llei11'1i1n~ubblii11t'lfi'.11i1'11111''uNfl:i-1
'
d'ium11111a1>rl'l1Xlh1m~1~l'lil 1'1tJ°r1111Xu1mrnfl11m11111~1n11B1m1i1ei~1"11uil1:1-1r11111-1il' .A1!!tl'i'U01A1!!UU ll1111Uil1iX
x1
= <J1u1uii111n1m1e:rnuli1ln1111ltJ~lt
x2 <J1u1uii111n'':1-1'111J1oii11h11i1~lt
X3
= <J1u1uii111n'i':1-1'lJ1l1rnn~1rnli1l1~1t
'llf1nil'u1'fl1Jtl1~tl1.i <i1nl:l11J1111m111l-:JmTvmh11u11u~1ijlil iii-:i,u U'IJ1111'lJ1l1b11iii1l m'l'V1
.-i1wi1ijil ~-:ii.t1m,~tJu•llu1'1-:in'liu1ii11u1tfl1fl1Jliii-:id
Minimize Z = 0.50X
1
+ 3.50X
2
+ 6.75X
3
b~1l 0.50X
1
fi1l <J1u1ub'.lu~111l1~1m~1l;mllli1lnmm X
1
n111n''i-1
3.50X
2
fi1l <J1u1ub'.lu~111l1-.l1m~1l;1Joii11lv'l"1 X
2
n111n'':1-1
6.75X fi1l <J1u1m'.lu~lif1l1-.l1m~1l;1loii111'V'l"1 X
3
n111n'':1-1
3 '
'llf1niiWU!l;)1OA ojJ1l<l1nii'lJil-:JUI]!vnU'ltmiuJI'1t1oiiil~1n'"111uu1mrnm1m11111Yii1lilllii'il1m'i
""~~1u1uei111111i11i1-l~lifei-:im11ti!f1111:1-1"1
fi1u'llei1m1111'ii11i11~11ei1m1 ,rtei1<i1nmi11n 20,ooo o/11 bblii11to/11<i~·ifo1iium11111i1u"11,,t;10t
0.5 n111n'':1-1 iii-:i,u l'ilbblii'1ti11l"111'lb'i111ei1i1m111T'li11i11l:iJ~1n11 10,000 n111n'':1-1 1'i'1n'liu'lJil1oiiil
~1n"1d~,mm1n.~tJ'il1J'•llu
x
1
+ x
2 + x
3
~ 10,000 n110n'':1-1
1uit1u'l1ei1m1m11111~lii'ei-:im1,u <i1nil1'i11~ bbl'l"11oii111ii'u<in~u1J'11m'ibbfl'1,;t1:1-1,u
1111nltblllie:m1111lt1 X
1 n111n'':1-1 oii111'V'llil X
2
n110n'':1-1 bbiO~mn~1b111liei1 X
3 nfon'':1-1 u1mrn'llil1
bbfliOb;tJ:1-1~1Jl<J1n1'lillJ~U~1mun:1-1nu<JtbU'il (O 3BX
1
+ o.001x
2
+ o 002X
3
) n111ni:i:1-1 bbi0t<i1n
oiiei~1n"1'111l11l1mrnbbfl10';tJ:1-1 b'il mm'im~tJmlJu1'1-:in-n'uoii1l~1nl'l'11111''ubbl'l'1,;tJ:1-11J'iii-:id

(0.38X
1
+ 0.001X
2
+ 0.002X) 2: 0.008 (X
1
+ X
2
+ X
3
)
bbfi~ (0.38X
1
+ 0.001X
2 + 0.002X
3) ,.:; 0.012 (X
1
+ X
2 + X
3
)
ftl-lm1oii1~1Ubb'YIU-iiei-oi1n<;imll-l1i:J~fll"i~l<j'1 fl'lUftl-lf11"i<i1~bb'YIU-iieJ-oiln'1mll-l1ei~fll"ij'~
~
<j'1'll1J~ bb'1flb'lftJl-l
ftl-lm1ilf~crei~cr1m111Yi11~~1tJfl'111blJU
0.372X
1
- 0.007X
2
- 0.006X
3
;:: 0
bbfi~ 0.368X
1
- 0.011X
2 - 0.010X
3
,.:; 0
LuYi1uei~b~mnu -iiei<>i1n'1'llei,mmmm1l-l1ei,m1liJ1~ubbfl~11'1b1mfo1m1m;tJublJuYi,r'iiiu
11.l"i~U 0.9X
2
+ 0.50X
3
;:: 0.22 (X
1
+ X
2
+ X
3
)
~
0.220X
1
+ 0.130X
2
- 0.280X
3
·wrn ,.:; 0
11'1blJ1l7 0.2X
2
+ 0.08X
3
,.:; 0.05 (X
1
+ X
2
+ X
3
)
'Vl'iil 0.050X
1 + 0.030X
2
- 0.030X
3
2: 0
ii'~iiu b11~,mm111b;tJu$11bblJlJm"i11.l"ibbnWb:ir,bftWJJ0,i:J!JJ'VllLu$11081,dl.#1bl.Ju
Minimize Z = 0.50X
1
+ 3.50X
2
+ 6.75X
3
m tJ111-ii0-oi1n'1
X1+X2+X3
0.372X
1
- 0.007X
2
- 0.006X
3
0.368X
1
- 0.011X
2
- 0.010X
3
0.220X
1
+ 0 130X
2
- 0.280X
3
,.:; 0
0.050X
1
+ 0.030X
2
- 0.030X
3
2: 0
x 2: 0 ; j = 1, 2, 3
' J
ni0dHn 2.3 i:J(j,J,,,1m1?i'1'1Jl"i1,b~u-,11
b'YI flrnflb!i1l'1milbbvi~,_,d~ri1ii''Yi1m1$lni±1fj,m 1l-lb11u1i.J1/i'lunwi.i11~lJlJ'lluft'l-l11il'15ub-ii1
m1·iib~mi'1U!JJmm"i"l"il"l7~~'1<if'1 1um-,flni±1d-oi1blJU1ieN'Yl"illJfi~<>i1u1wm~-oi1blJu10~1-ii'lum1
'llu<i~N'I'1tJftl"J "l1nm111JJ11l-loii1ll-l1ilb~0,iiiu~JJ-J1-oi1inu-,111'1tJft11~-oi1blJuiii0'il b~1lJJ1m1N'I'1tJftl"i
• • •
LUbbllifl~oJi1,nm bbft'1,01-iLM1~~ 2.1
•

12
12 -
10
8 I- 78
4 4 4
0 I I I I I
0: 00 4 :00 8: 00
d ~
12 : 00 16 : 00 20 : 00 24 : 00 b'l!M'1U
b'>l"J~<1-i'1~#111uum-,1ti·rnm:i<b:t~iflm~il'llll'iim1u<Jn~'lVfu7m-,'lmi.111~'Ji1~n111 L11 tJ'lVfl'.l
'1m1u'Jn~'lVfu7m'J'J1:>JnuLmb'ii11~1'ul1.!1~1iji1 nlVIUillVl"mbb.111~.'lu'lViu7m'J1.i'bYltJ~ 8 ~11:11~
'iiil1u ?i!l'lwl~1'11u11~ 1 n~ 1'1u'lvr~~n~11~ s ~1Tm bb!1~'>mb.111~.'iu<J~b1hn~1.i'b'1:w1~~.i'u'lJ!l~'Ji1~
•
nm?i!l b'i'.l'i'n~nm O.D1, 4.01, s.01, 12.01, 16 01 bb!1~ 20.01 m~mb1'i1uu
l'i'111tl':i'U!Hl'i'1111J1J nlVIUilLVl
x1
xz
x3
x4
x5
~ . d
= .j1u1wJ1il~'>nl'I b•inn~b:i<il'1m
d • d
= .j1u1u'lJil~'J1:1'lb'lJ1n~b:>Jilnm
d • d
= -'il1U1U'lJil~WYlb'lJlf1~b:>J!lb1l11
d . d
= .j1u1wJ1m'J1:1'Ylb'/Jln~b:i<ilnm
' '~ . ~
= .jm1u'lJ!N'J1:1'Ib'lJln~ b:l<ilnm
0:01 Ul~m
4:01 m~m
8:01 m~m
12:01 m~m
16:01 m~m
X6 = '11u1wJ1il~'>1:1~boii1n~b~ilnm 20:01 m~m
-~ iv CV ti d v ~"1 "-' "j ,Jo x c:o" 121 I o ,d ii .:::! d..~n'lflllllQ ':i:i:ll'~l'l "Jlnl'11UbJ'>'llbb'1~"11bbu"i'YlnlVIUi1'lJWilnVIU '111 "J1U1U"i1:1'llil!l~:l<b'Viil
u7m'>r0'1tJ<!l"i<J~4iil~l1'11mu1'u<i~ X1 + x2 + x3 + X4 + X5 + X6 .'iu bl1m"il4iil~nT;-1111.j1u1u
-:;i;i~mm-:;n'lViu7m-:;'lu'Ji1~nm'ii1~'1 1.1 liltJ'lVilH1u1u-,1:11'lu1b.111~1'uil.!1~1~ij'1 if~,:(.,,. v-i~i'i1iu
1'11Qtl1~<1~.1mm1m~tJu1.i'wr~d ?i!l
Minimize z = X1 + X2 + X3 + X4 + X5 + X6
w~~ii'u'1i'0'11flll 'illnoiiiJ.jlff'1'lJil~'Jn~4ill~nl'J'lmb'ii<i~'Ji1~b1l'11Wl~bb<ii~'lum'Vi~ "2.1 "J~b'VIU
1tii11'lu'Ji1~nm1~11111~ 0.01 - 4.00 m~m 4iil~il"lnil~1~~1~ij'1 4 ~u b~llu7m1r0'1u'111 f11m
mVl<J11ru1'lu'Ji1~nmWl~rnil1 1n~<J~'lVfu7m1'lu'Ji1~nmd<J~l'.J.j1u1u x1 + X6 ?i!l .j1u1u X1
~u

~~boif1n~b~m1m 0.01 bb'1~~1U1.I X6
rl.I ~~Loif1n~L~flnm 20.01 mYlm
ii1'Y1'iu~1mwm1wih~nm 0.01 - 4.0o mYlmif1m•m~tJu1~'1Ju
•
~,iu y;j~fioii'uoiffl~lnlil
x1
+ x6
;,,, 4
Lu'ii1ufl~L&imnu ii1'Y1w'1i1~nm">~w11, 4.01 - 8.oo mYlm ~1u1u">ti~i'.i~~Lu'1i1~nmd
iiifl x
1
flu ('J1nm">L-ii1n~•~flnm 0.01 mY!m) ~~i:iu y;j~fioii'uoiff1~1nlilii1'Y1'iu~1u1u•tiLu'1i1,nm
4.01 - 8.00 mYlm iiifl
' X1 +X2 28
'iirnfl',&tJ1nun1J'lf1~rnnliw-1 n1ma.il'rnmw~noii'uoiffl~lnlil'llfl~~1u1u">tiLu'1i1~ rnn.11~'1
1.'ltlilfJ'iTI~flilrntJmLbii1 ~~i:iu •'>1'11a.Jl'rn•~tJuif11HhLbuum•11hLbma.i•'ii~Lftu1~~~d
Minimize Z = X
1
+ X2
+ X
3
+ X4 + X5 + X6
mtJ111-iifl~1nlil
x1 + x6 2 4 (0.01 u. - 4.oo u)
x1 + xz 2 8 (4.01 u. - 8.oo u.)
xz + x3 2 10 (8 01 U. - 12.00 U.)
x3 + x4 2 7 (12 01 U. - 16.00 u.)
x4 + x5 2 12 (16 01 u. - 20.00 u.)
x5 + x6 2 4 (20.01 u. - 24.00 U.)
x 2 o· j = 1, 2, 3,...6
J
v ' "'
fl'J!l!!Hl'I 2.4 i1qJmm7%1ilm">111um7''.lu
'A I ""' OJ "" OJ <v ff I ,d ,::!. ""' Ji.I._, Ji.I
'1a.la.1Wl11J'>m'IL~U1'JU'Y16'1nm'Wmb'Y1~'Y1U,a.lb~U 8,000,000 1Jl'Y1 f;j'Jlilnl'>Glfl~m-,mL~.1'11.11.1
d1ufl'1'J"'1u'Y1ffrn1'i-w ~-~fl1VlNfllilflrnL'Y1u'J1nm•fl~1'ju~~ijlil lliltJ1Vfii•~~um1m~tJ'"1nm7'n1'ju
Lu'>~~u~u~i:fvitJfla.iw1~ lflmifLum'>fl~1'J"' Nfllilflrnrnwnnrn•fl~1'J"' bbfl~fi'lil'>lfl11a.JL~tJ~~~""lil'
Lu1i11"1'
i:lmlNllflllU!!l'l'U ' ..;
.. ll''J'U!U!MIU'U:IJlfl1~1U
l'll~!llllflfl11ll~'IJ'U
(1tlf!i1.Vuii1J 'llfl~NllflfJU!!l'l'U (O')
Ji.I ,., '''""
'1UQ1J'lli'YI 8.50 0.25
...'1UQ'l]lJ1CI 6.50 0.00
ljumiltjj'llil~mmrn A 10.90 6.50
il'urniltjj'llm1J1.11m1 B 11.00 3.50
il'u~~a.l~'Y11i'lJiJ~1J~~U'IJU 10.00 4.00

N'~'1m'l1<iiif'1i:luL"J11'1libL:I~tJri1~~lVIUn'll1J~l:J,~tJfl11:1-1L~tJ,V1'jE1ci1m~tJ'Luu:1-11<11'l'll'U. ~
'll1l,Ni1<11E1rnmu (cr) •~<iieN1:i-JLfiu 3.0 'llfl~L1ui1''Y1u~~V1:1-1'1 LL'1~<ii1l'm'"''Y1u1uVfun1l~1,{ieitJ 20%q q q w
LLlliLu~1u1ud,~<iJ1J,LfJuVfun'llm1"1mi11:i-Jiimm11 40% LL'1~'1,'Yl'U1uVfumi'.iru'IJE1'1i'Ulm'l A 1#1q w ~ q q .,
L:i-JLfi'U 2,000,000 lJl'Yl
ll1mh'llMVl·;u!'Uu rllVl'U'1LVl
X1
= ~rn1m'.lur:H11u1wt1'uQ''ll1l,lJ'iB'Y1
X2
= ~1u1uwu"''Ylu1ui1uQ''11El,1:Jmi1
X3
= ~l'U1'UL'.l'Ui1''YJ'Ul'U~'Uf!li'.fb)'JJEl,1i'Ulfll'l A
X4
= ~1u1uL'.lt1i1''YJ'U1u~umifb!'llmll'Ulfll'l B
x5
= ~1u1m'.lu"''YJ'U1u~ul.J'i:1-1i:l'Y1ii'll1l,u'ii:im'.lu'YJU
W~rl<U'irJftQlh~tl'~i{ "JlnUb)VllL'll<if1J,m'lVllfllNi1<11EllJLL'Y1W'Jlnm'li1''YJ'Ul'Ui1mh~Ln'Ylllil''l
1Vimn~')'1 ~'L~tJ'ULU'Uw~r'ifu1<11qtJ'l~ff'fl"l#i'4),d
Maximize Z = 8.5 X
1
+ 6.5 X
2
+ 10.9 X
3
+ 11~0 X
4
+ 10.0 X
5
W~rl<U'wiieHhn.-i .iJE1~lfl11'llfl'm'lsM'YJ'U'll1l'lJ'iBriiJ<iJ1tJn'UYJ~Vld-1'1 5 <ifm L#fLLfi
rl1~~lVUn'lJEi'i:I'"~tJfl1ld-IL~mVl'j(Jll1'UL~tl' LlJ'Ui.l1<11'l:Jl'U'11El'Ni1<111JlJLL'Yl'U ((j) Fi<11'l1shUnl'li1~'YJ'U
1u~uij' dE1ul'll'll1l~rn'li1''YJ'U1ui1uij''lleJ'1:Jmi1 deJuh'llfl~m•i1~YJUlu~umi'.ib)'118'1imm'l A LLi1~
L'.lu~mh~mrusi1111fom'll'M'YJ'U~il~~V1:11'1 4),~u ~~rr1:1-11'lm~tJuw'Miwifo~1n'1~'111:1-1'1L<ii41,d
.25 x1+ o x2+ 6.5 x3+ 3.5 x4
+ 4 x5s 3 (8,000,000) (m1m~mL'1rn>1~tJ)
x1 + x2 :?: 1,600,000 (m'li1''YJUlU~UQ 20%)
x2 :?: 0.40
x1 0.60 (il''1il1um'li1~'YJU1u
~
0.60 x2 :?: 0.40 x1 Viun1"1mi11:i-Jii1ltin11Vl'lfl
' .~
~
-0.40 x1+ 0.60 x2 :?: 0 40%)V'j"(]
x3 ::; 2,000,000 (nl'li1''YJ'ULU~Uffli'.fb!
'llfl~1i'Ulm• A)
x1 + xz + x3 + x4 + xs s 8,000,000 ('lJtJJ":;mrnm'lih1'YJ'U
•
Yl~Vld-111)

if1111J1Jm'll1Jmmm:ii,bftU'llil,lif1il~1,d ~'blluiif,d
Maximize z = 8.5 X1 + 6.5 X2 + 10.9 X3 + 11.0 X4 + 10.0 x5
mtJ14J.jj[)-;51n01
.25 x1
+ o x2
+ 6.5 x3
+ 3.5 x4
+ 4 x5
::> 3 (8,ooo.oooJ
x1 + xz
-0.40 x1
+ 0.60 x2
x2
<:: 1,600,000
<::O
:5 2,000,000
x1
+ x2
+ x3
+ x4
+ x5
::> 8,000.000
x <:: 0 ; j = 1, 2, 3,...6
' l
A''lsaNfi 2.s ilb!J1111m.,mu'fli.Jl1u.11rn11i',
fr'lnl'l'llltJilDmcl1'i10!1Ubbvi,111rt, 'J~#,;ilft'Ufllflil bfl~il'<if01111rif1'ililU4fUb~ilU'llil'bb'ii~~b~ilU
bb~~Li'-:fuftUfll"lln~'llltJ'ililU~Ub~ilU n>Jnl'l'J~'Jl1tJft'Ufl1~l'.Jiltj'l~1111l'b~ilU (ftUfl14fU'1'il + ft'Ufll
~#,;[)1ub~ilunilu) bb~~n"lm'lmi.J1'l11'll1tJi1u.11.ilf,111i.J01~l'.liltj
'llm;m1~~'llfll'lllmiJ~1.1mb1J~'il~b'1i.Jil m'lbD~1.1mblJ~''llil,'llml11111-:i1J'lW~b1m 6 b~ilu
bb'101'iif''ill'll,oif1'1,i1' uiln•1ndn,m'l-oi1i'il'li1u.1111""i''1ili'1,'101 150 111il1rn'Yi1iiu N''1101m'l'llil,n"lm'l
• • •" " '
bbv1,.n,41il,m'lri1111u01-oim1U1111.l1tJ'JIil'm'l:/lilbb~rnl'l'lJl tJ bb'ii~~b~il'U bYlil"l~1i'-:i1Jri1L'lJ"l01 n>Jm'l
hil'.li1u.1'11lm~ilUi.Jn'llfli.J bb~~1i-il'.li1u.1111'b111~il~.lb~!lul'lb)U11.1.1
.. ~ 1 r 9) ~
'.ill'll'IHWl6'1'1ln0 ( 'll!Ylll'll)
1 I I <lS
'.ilt'll'IJ10t'l!l'l'l'IJ10 ('.i~'l'l1N!Ylll'll)
!Yl!lll
(rl'llUl'l'I) (rl'llUl'l'I)
i.irn111i.i 50 -
. ' 50 60f1W111'1U1l
~
40 40a.Jlllf1a.J
bl.MJW 60 50
1'1f]~Jllfla.J 50 60
5'.jbJ'UlfJ'J,.I, - 70
A'lmh'IJs~A'l!!llll ri1111u011Vf
X1, X2, X3, X4, x5, x6
= -oi1u1ui1u.11#,;il01!lu4i'm~ilu i.Jm111i.i 11i.imY1u~
A .<!I. "" o ...,
>JUmb01!l.li.Jb)Ul1.IU '!11a.im011J
Y1, Y2, Y3, Y4, Y5, Y6
= -oi1mui1u.11~'lJ11.1'!lilu1lub~!lu i.Jn'llfli.J f1i.JmY1u~
"lUli"~!lUl'lb)Ul1.IU 'illi.J!i1iif1J

b<l1l'1'ill nn'il fll";hi:i1 ih1,~1lliub~<J'j,j,l.Jfl";1'1l-1~'11i.i:i1m-;'ln1.1liu~11m~<J'j,j,l.Jml '1 l.J
Y1 = o ""~n•m-,1i.i:i1m-,A'1;eiiiu~11ub~1lu:i1iim1Ju ~'1tXu x 1 = o
_'!( <v v _I <
TI~fl'lf1flV1Qll'J~iHfl
Maximize Z = 60Y
2
+ 40Y
3
+ 50Y
4
+ GOY
5
+ 70Y
6
- (50X1
+ 50X2
+ 40X3
+ 60X4
+ 50X5)
n~ <fcv SI o v
..~fl'lf'U'U0'11fl'1 b<lewil1nn'iln1-;:i1oii<J~lrl01JIl'U~'j,j,~l'Ufll";bnu1n1"lft'Ufll
•nu1rn,r1C1'10011~ 150 viu11J ~'1lliei'11'i1m-;'11u1mvi1fiu~1'1'1bV1fl1ltimm101'll1l'1bb'li<An~<iu ~'1'11-
• '
wim1~~i[
iiufi'1tlmm101
Lu~dri1111u01LVf I
""- ii/~ % I
= f!Ufl1111'U'110I + 'if1l - 'lJ11J";~Vl11'1'110I
= <ii1muiiufi'1'1'1bV1fl1ltimm101
II (ft'U~1'1'1bvifi1lb~rJUl.Jm1'1l.J) = x1
""' ~ ~ ,.!! .,,. ,;
!2 (f!'U'11'1'1bVl<AilbOl1l'Ufll.J11l'W'U1J) = 11 + X2 - y2
!
3
(ftU~lf1'1bVlflilb~1JU:iJ'Ul'1l.J) = 12 + x3 - Y3
!4 (ft'Ufl1'1'1bVlfl1l.~1JU.l.J1"1tJ'U) = 13 + x4 - Y4
15 (ft'Ufll '1'1bVIfl1lb~1l'U'Wl1"111'1l.J) = 14 + x5 - Y5
1
6
(fiufi'1'1'1bV1lim~rJu:i1iim1.1u) = I5 + X6 - Y6' .d v v
bl.J1l X
6
= 0 01--l'UU 16 = 15 - Y6
Y1'1n'1iuoil1l<ii1n.,,~..0.,,fi,m-,:ilfiufi'1'1'1 bvifiBi.Jm[J .~<iu'J''1oi1i.i•nu
II s 150
12 s 150
13 s 150
!4 s 150
15 s 150
16 s 150
0-;tl<if1bblJlJ'lJ1l'1nl"llti-;bbfl";l.J b'li'1bfi'U1~<ii'1d
'
Maximize Z = 60Y2 + 40Y3
+ 50Y4
+ 60Y5
+ 70Y6
- (50X1
+ 50X2
+ 40X3
+ 60X4 + 50X5)
m1.111foiiei<ii1n01
x1 s 150
11 + x2 - Y2 s 150
12 + x3 - Y3 s 150
13 + x4 - Y4 s 150
14 + x5 - Y5 s 150
15 + x6 - Y6 s 150

bl'.lilbb'YIUfll I Ji'1tJ x mi~ y ,t1~'<if1mJU ~il
x1
x1 + x2 - Y2
x1 + x2 + x3 - Y2 - Y3
x1 + x2 + x3 + x4 - Y2 - Y3 - Y4
~+~+~+~+~-~-~-~-~
$ 150
$ 150
$ 150
$ 150
$ 150
x1 + x2 + x3 + x4 + xa - Y2 - Y3 - Y4 - Y5 - Ya s 150
x1
?'. o, Y ?'. o ;i = 1, 2,...,6
' J
A'1'el~wn 2.6 lltJm<il1J'1<1m,Nfi<il
~<il1lm-iN1tJNl'i<iilii'il<1m'ln1V1u1l<iil'll.Jm'lNl'i<iii:1urhihwl'u 5 •~Buoif1<1wu1 '1noiim;Jil'lu
il~<il N'<il1lm'l'Yl'llU11n1l1<1mJ'Nl'i<ii~m~ilub'Yhnu 2,000 viu1u bblltfl1m'lnNl'i<i11Jl~n 600 V1u1u
•
'tw1h<1'11<1nm (overtime basis) liiu'Ylum·mfi<il~ilVIU1m'Yi1nu 10 'IJl'YllUt1<1nmiJn~ bbilt 15
'
Ul'YI 'lu'll1<1'11<1nmiftJtJl i1'tJtJln1'l;ili:1ufi1Lb~<i~b~ilul'.l'1<1d
.. .; ~ Ji '
!PlllU'll lflJlJlOW'IHl (111-110)
1 1,200
2 2,100
3 2,400
4 3,000
5 4,000
liiu'VjW1JiJ<1i:1ufi1fl.JLVli'lil~ilV1U1mY11nu 2 Ul'Yl~m~eiu ,;r.Jd~<iiiim'l1:iJliiei.Jm'l'lVia:lfiuf1
fl.JLvil'leiV1l1f"1rn~ilu~ 5 w1u~u1u •'lllilmn'Yl'l1u11,~nTVIU11bbNUmJ'Nl'i<iifiu.'11'lmb~ll~b~eiuLliu
-.i1u1m'Yi1liiL~il'lVio/iu'VjUnlJ'N11'11••<i ~1iu'VjUi:1u.'ilfl.JLVI~eiJ'1l-lnmLli'15'.l'11~l'Jii
A1!!U~'llMA1!11JU rllVIUii'Vi
il1bblJJ''llm~11Luu'JJei<1lltJV1ld -~•liu-.irn1um'lNl'i<iil1u.'i'1'llil<1'll1<1nmiJn~~eib~eJu -.irn1u
m-iNl1lli:1u.'i'1'l1ei<1'll1<1'11<1nm~m~eiu •m~-.i1u1ui:1ufi1fl<1ml'lei'luL~eiu~ 1 ii.J 4 fll-ll-l~bb'YIU~1u~1
bblJ,(?'f,d
X = -.i1mumJ'Nfi<il'lunmiJn~~m~eiu L11 tJ~ i = 1, 2, 3, 4, 5I
Y = ~1u1um·mfi<il'lu'll1<1l'i1<1nm~m~eiu LiltJ~ i = 1, 2, 3, 4, 5I
w = -.i1mui:1u.'i'1f1.JLV1l'lil<i1ElU~ub~eiu LiltJ~ i = 1, 2, 3, 4I

if~fiiru1~mh~1:r~fi N'~ii m• ~1uN!1<11o/I!l.:Jm•Y111uN<111:J.Jo/luY1um1N11<11"'"~o/luY1ulluff1q 'II q ~
¥ML1'1~!l~l'lil o/lu'Yjum1Ni1<11lli!l1'1u1uLunml.Jn~hYi1fi'u 10 lJ11'1 o/lu'YjunTm!1<1111i!l1'1u1uLw1h.:i~1.:i
nm•Yi1fi'u 15 rn1'1 LL<1~o/lu'Yjum1~LLr;illuff1¥1.:JL1'llim'Yi1fi'u 2 rnY111i!l1'1u1u >1~~m~uuw.:inifwl'<111J
iJ•~fl.:J'11.o/IdJu
Minimize Z = 10 (X
1
+ X
2
+ X
3
+ X
4
+ Xs) + 15 (Y
1
+ Y
2
+ Y
3
+ Y
4
+ Ys)
+ 2 (W
1
+ w2
+ w3
+ W
4
)
if~1Wu-«u'iilfl~ .iJ!l'illnil'ljii LL1m1mil'l!1'lldb~u1.iJ!l.:inu'il1u1unnN11<11lUL1mtln~LL"1~~1.:J
nmLmLlli<1~L~!lu oii!l'il1nilLu-il1.:inmtln~'il~L~uu1o/fLliu
xi ::; 2,000 i = 1, 2, 3, 4, 5
-ifD'il1nilLu-il1.:i~1.:inm'il~L~uu1o/l,1Ju
y ::; 600 i = 1, 2, 3, 4, 5
l ' ' '
oii!l'il1nil'ljilfifl!l.:J Lnu1oii!l.:inrnLNum·m!1<11lluff11uLLlli"1~L~!lu '11,,,:rm~!lufi 1 'iliu1u
m1N!1<11Lunmtln~"'"~li1.:inmn>inu Vino/11u'il1u1ulluff1.i-iL,,,(i!lL~!l~UL~!lu ,~o/I!l.:JL'Yi1num1:J.J
o/l!).:Jnl1~ill'lufi'11uL~!JU~ 1 ~.:il'l.-i1LUU 1,200 lJ11'1
x1
+ Y
1
- w1
= 1,200
LUL~!lu~ 2 '11uiunTm!1<11lurnni.Jn~LL"1~~1.:inmmno/11ul1uff1.i.:iL1'l~!l'nmL~!lu~ 1 Vin
o/11u11u4h¥1.:iL1'lli!lLuL~ilu~ z 'ii~o/lmLYi1fi'um1m<fil.:im11luff1LuL~ilu~ 2
x2
+ Y2
+ w1
- w2
= 2,100
'11,,,:ruoif!l'il1nilLm~ilu~ 3 ""~ 4 'il~L~uu1o/1Lliu
x3
+ Y
3
+ w2
- w3
= 2,400
X
4
+ Y
4
+ w3
- w4
= 3,ooo
-ifil'illnilLUL~ilU~ 5 'ii~1i.il'llluff1¥1.:Jb1'l~m~il~Ub~!JU %~~U'il~b~l'.JU.ji!J>j1fi'illo/fLliu
XS+ Ys + W4 = 4,000
<if1LLlJlJm1Tt11LLm>i1:il.:i1~u~.:i1~uu141LUU
Minimize Z = 10 (X
1
+ X
2
+ X
3
+ X
4
+ X
5
) + 15 (Y
1
+ Y
2
+ Y
3
+ Y
4
+ Y
5
)
+ 2 (W1
+ w2
+ w3
+ W4
J
s 2,000
s 600
= 1,200
X
2
+ Y
2
+ W
1
- w2
= 2,100
X
3
+ Y
3
+ w2
- w3
= 2,400
i = 1, 2, 3, 4, 5
i = 1, 2, 3, 4, 5

x5 + Ys + w4
xi, Yi, wi
= 4,000
;:::: 0
iil111aH~ 2.1 DlJJ'Vl1fi11-11.ht<i:wm1Lvntlljn
L~1'llml:fr'.ivitt.:i<ifil.:im•'r1•1'lJ'i1<Jt<if1Mll<1nYi'l!<ii1.:i'liulln1-1 3 'liull'lu~~u 3 Lbll<1.:imJ1.:ih~.:i• •
"Jt'Yi1'lVf1.<ifw~n1:W'llil.:Jn1h5.:J5ll L~l'llil.:J1'i11udii~@i>1'1l-1~.:J~j, 2,000 1'i LL'1tL~LLU.:J~~'l-lililnd'Ju. '
3 LLUm •nrnXu1~11~1.:i'li11h 3 '1>1m'Yi1m.,.vntlljn'lu~@iud L~l'liil.:JhLL>1ni.i'l~iJjnYi'li 3 'liUll
G'iil i11t'l'lll rf1~5.:J LL'1 trf1b'Vll1il.:i 'll'l-Illl'llil.:J~@imL<iii1nLU'1.:J)nnl'Vl'l-lll~Will:Wn1<1.:im•W~'ll'llil.:J
'li11L'iLL<iii1~flj, im;1m~mn'lJ'll'l-llll'lliNLLUi1.:J~@i1-1 ifl'~lnll'llil.:Jn11Ujn'~'libL<ii'1~'liUll mi~nlh
<iiilhii<K.:id
~~'111!1hi~~ 'IJ'lll>l (1-i)
1 500
2 800
3 700
. '.. .. d iJ (I' 111i1fiaH (111'1'1)fl jl 'IJ'lll>ll'l'll'llll'lW )0 1)
ii11Tw~ 900 600
f1115.:i 700 450
" ~
1,000 300trn111im~
Yi'lltt<ii<1~'llullm<J<J~lnuln'l1-1•Lll<1.:iLlln1<if Lb<lib~l'llil.:i1'i1~~iil~1nll5w1 .~:w.~l-1<11.:id "~
<ifm'Yi1nT>bw1~1ljnYi'libL<ii'1~'liulliltJ1,ifou 60% ,.yjil'l~Lbuh11'li111'ibb<ii'1tl'l>11~'Yi1m.,.w1~1ljn'll1l-I
n1li'.:im1:W'11:w11!1'llil.:J bb<li'1rn'l-lbbi1~'ll'l-llll~'l-I~L'1ltU<jn b~l'llil.:J1-i<ifil.:Jm1ii'1!) fi11-1'l1il.:Jn11b'l'll~U';]n
Yi'll'l1-1Lb<i1<1tbbll<1.:iii~iwhnu~.:ivil-1llml-1 bbll<1.:i 1'lli:iu•~<l.:i.,;'llil.:J,~1'llil.:J1-iMiil <ifil.:im1'1111'lJ11<J~<ifm
UjnYi'llbb<lii1~'liUll'l1-ILL<iii1~LLUi1.:J dl1-1~11-11mvhlll ~-l"J~1~nlL'l'1:Wj.:Jljll
' ' ' .1 ><i t d. """ <:!t ..::!.
'111!'111fl1•flfl'lfl11'1111'l'll 2 Vl!l'll'll 2.111!1~'11 2.1.2

m"lbLnll'l)mm~Jl"lfi'ii1l1tm1"lh.i'mmm:il~Li1ui:iu G1m"lmLu~~u111eiuV1i'w1 111 2 ~u111eiu
fiei nl"l'1~1~Gl1LLUU (model formulation) m1~m'l'Vllfll"18U'illnGi'1LLUU ~~m'l'Vllfl1"18U'illnGl1
bbUU~G~l~~Um'il141 'lilm1'rl (graphical method) n'lll'll8~Gl1bLUU~llGi1LLlhGlili'lUh 2 Gl1LL1h
'Vl7ei'lil:ill-IL'W~ni (Simplex method) V11rnlJumillll'l)'Vll~l'.l~~LL'1i 2 Gl1bblh~u11.I '11V1'i'u1u~ill'lll
;'.(,~Liium'l'Vll'11'11elU111tJ'lli'Bl-1 L'W~ni 1ilmieiu~'~mh1li~m~u1um'l'll8~'lil'Bl-IL'W~ni ii:n!'mn
fl'l'l'Yl'llUli~nl'lbLli'1~Wf1LLUUm'l11.i'lLLn'll-1L:if~Li1u1Vielaj1uiliGi1LLUUm m;i1u'llel~:ifl-1L'W~nirieiu
_f 6191~1 v .e....c:S.<::!t. a .t
1. m·rnuti'I 1'11!utt~1!11J1JttWl"l~1tt'U~'l1Ii'llmvrnll'll
m'lLLli.,~V'l~noiiwiiei~hnil~~'Vll-1ilLVfi.l aj1uili'll<M'1l-lm"lii'1LLuum'll•;im1l1tJ'lil'lll-1Lw~nill
v
'VA "-' ,d.
<llli'1l-IU"1il~U
1) '1l-Im'l-iiil~lnilYJnGl-lm'l"J~41i.l~l'.1fi1Y11~'l111l'.1il'llmGil-Im"lLlJufi1u1nL'1l-Iel
2) lii1LL1.i'l'Ylnlii14fi.l~l'.lfi1mnn11V17m'Yi1nufu~
' .3) V'i~noiiu1''11b)li"~"~f;m'ilLlJunl'iV11'11J~'lil'Vl"lilfi1~1')(i111111
1mhu'1iel11.ld "J~mh1li~m"lll'i'uGi1LLUUnl'l11i'lLLn"ll-IL~~Li1UL~el1Vielaj1uiliLLUUmm;i1u
il1V11'u'lil'Bmw~ni 1ilmmm1tJ'1~Lll tJil'llll-11'11-ifil fiel
-~ "<u v " <I.I
1.1 n~ll'llU'IJ<l'illfWI
1.t.1 V1~noiiuoiiei<l1nil~~elaj1uili'llil~el'1l-lm'l fel llLf1~el~1'1mti
LVlilaj1uili'llil~Gl-lm'lL"11iltJnl'l'1U'Vl"lelu1n1l1tilif1LL1l'llii1wd~ L'llu
x1 + zxz :<> 6
t'i'1'1i1lii1Lbll'l SI ~~ SI 2 0 mmrn.iJ1.iJ1~-if1tJ 'il~ri11Viil'1l-lm'lb1.l~tJULlJUGl-lm'l#i'~d
X1 + 2X2 + s1
= 6 '
V11nV1~noifu.ifiJ~lflilUULlJuYl~noifwifel<llnil'llil~'Yl'''W mmYiflilaj~lrlil Gl1 LLli'l S
1
"J~
d'iu<l1u1uY1-:i1•m1m~L'Vl~elelaj"J1nm'lL4i'N~'ll x1
Lm~ x 2
"J~L"ltinGi'1LLli'l'l!uild11Gi'1bLll'l'1Lb'1'1 (Slack
variable)
•
Li1FJ~'<!1n~1~~1 t.111 tl.JB~d-Jn1-a~crl1;w1n n11i1'1..;ier111 J)..;i,Xit.

X
1
+3X
2
-S2=4
<ihm.h S
2
l'l!Jffll <ihbbU'l"l'lfEl1'1<i'i1 (Surplus variable)
1.1.2 mmfi1'1,~'Y11,'lJ11l'.i!l'lm,rn:1-m1.,-iiil'ii1i'i<1l51fi1•1iul1u mm.,mh1V1,1Juu1n1~
L<?l!Jnl'l"lm -1 ~lliill1EJ<1l l'liu
X
1
+ 2X
2
- 3X
3
= -2
i'1m'l"nvi11Vl-ii1,'lJ'll'llEl'l'fl-im'l"lU'UlJ1nL<1ll'Jm'l"~lliill1il<1ll'fl-lm'l"~1l'Jfi1 -1 ,:;1~
-X-2X+3X=2
1 ' 2 ' 3
1.1.3 nl'l"lU!il'Jmfi'lil,Vlml'J 2 Ll.ilU'U ~ 111'lei'J1n ~ Ll.ilUu 2 mm'l"nvi11#i't<1ll'JU1
•m -1 ~lliill1il<1l~'w'r'l<iiu b'liu
X1 +2X2 ~5
•l.i~l'JullJu
- X
1
- 2X
2
2 -5
1u'Y11uei'•~milul11wl'ueii'l-lm'l" 2 mm-;;ml.i~!J'UlU'Uiliill-lm-;; ~ 1#i't<1lmm -1 ~rn
ill1€1<1l~'w'r'11iu •'liu
1~
X
1
-2X
2
22
•tl~!Jubuu
-X
1
+ 2X
2
2 -2
1.1.4 mm~w,r'11iwiiei'ii1i'i<1l51b'14€1,Vlmm'Yhi'iu (=) l'li'U
x1
+ x2
= 100
L'Uill'l"l'l~i<.;fw111:1,;l-l l1'1~nii1iu"iiJ:;ri1111u<1l<if'l bbl.i-;;#<1ll'luh1Vf5181bUu~u~~'Vll-l<?l
<ii,i1Cu mn x1
bbl1:; x2
<Uu~u~1'17ill-li'iubb.l'1 ,:;1'i11Vil'fl-lm-;;-ii1,.;Ju1iJ,1Ju'J'l' 1umm<'liud,:;.;J1:1,
ri1111u<1l<ii''lbbl.i-;;lYil'Jl-l (artificial variable) ~um b~1:11'i11Vlmm-;;11ri1V1u<1l X1
bbl1:; X2
51fi1bU'U~u~
,.:;i1iul11m'l"m~l'J'UW,fl<iiu-ii1:1'ii1i'i<1lL~111iJdJu
X1
+ x? + a = 100
<if'l..i.1.,,Yil'Jl-lYi1i171,~u<uu<Ylm<if'lbb1h~1oii<umf141:1~511:11um-;;ri1111u<1l1VI<if1bbl.I-;;#<11 l'lu
hl1fi1<1iu~u~um-;;•~l-l.;Ju'Jli;l-l•1'1~ni <if'lb•l.i-;;Jr,n~111iJ511:1~''l'1ui:111J1111 Jr,i1Cu 1um'l"bbni:lqJ1111
"J:;,;J!l,Vll'Yll''ll~<?l<if'l..11.,d1:1iln<J1n'1111€1lJ ~u~1:1 'Y111Vl<if1bbl.i-;;51fi1bUu~u~u'11il1:1u~~~lj<1l
1um-;;'ll~<1l<if1bbl.i-;;1Yil'Jl-lei1:1n'J1n'11111:1u~•111m:;iili<~lj<1li1iu 1'i11#i't<1ll'Jii1~1bbl.i-;;bYil'Jl-li~!Ju
.,,i<1uw,fl<iiu1111Jl.1":;!1,¥ L<1ll'J1Vii1l'l-ll.i'l"l'.l'l'Yl~'llil,<if1LLl.i'l"bYJl'Jl-l51fi1mn1 .~l'J'U~'l!Jiln1'Yi M m-;;-;;1l-l
#1LLU'l"bYi!Jl-llui:i~r'l<iiu1iiril.l'l":;ii~¥1Vi'Y111u'Yll~~'J:;'lJ~<1lii1bbl.I-,,Yi!Jl-l1:11:1n<J1n'11iiillJ~'111m:;i'l-l~1i1<1l
' '
i1iu {fouui:lqJ1111m-;;1111fill~lJ<1ln1Vl-;;1l-l<if1..i.1.,,fi!Jl-l l<1ll'Jm-;;mrnein"J1nw~r'l<iiu111lJl.i":;iil~¥ ''liu
Maximize Z = ~C?i - ~M.ah

i!TvrfoilrumnT>VI1'11~1'191 nl'>'ll%91</i'lLL'Ll1LfitJ:Wililn•nmi1'ileJlJL111mt~ft91 'Yi11#i'l:91tJ
m1u1<if'lLLD"iLfitJ:W~,'1~eJ•inlJft:W'l.!"iti1~i M 1J'lm.if1n1J'l1,noifu1'11ti'Ll"iti1,fi L'liu ". .. "
Minimize Z = LCX + LM.ah
J J
tvrfi,m'll11fi:w'Ll1~i11'li'lleJ,4!1LL'Ll1bfi1.1:w1'1n<if1i1'11L'Yi1nu fieJ M i!f,iit•wnil-11 M :i1
"
'11L111ajmn1111il:i1'11ml:uo/i Ji,,:(,,. M 'YJn'11:i1'11LY11nu
1.2 fl'1mh 1u<ii1LL1JlJ;Jl'il"i!J1U'lJ!J,1TI'il:WL'i'<~niri1V1U91l1 <if1LLD'i'YJnlii''l'iJt<ifo,j'.jq,i1mn'il~LUU
!>11<ilJ1a-J1'1 J(,,iu t'11tum1:wLUU"J~'iin11rnt'll!l,iJqJ1111l'.i<ii1LL'Ll"iLUufi1i11J~,,j1LUu<ilil''Yi1tVfi1!>11LUU
lJ'lmi1tJneJU 191tJi1.'1,<if1LL'Ll"i~Um1111a-J 2 <ii1 LL~1U1;Jli11Jfi'U lii1LL'Ll1ilf,i1ll,<if'l'ilt<ilmi1'11Lllumn
1'liu i1:w:J.J&i<if1LL'Ll"i Y l'.ii>i1L91'l mm'irn~uum1:w#:wi:fu1huu
-= s; Y 2". = tumrndmm1m~1.1u<if1mh Y LLY1u.111.1<1f11L'Ll1 x1 LL<it x2 tieJ
Y=X
1
-X
2
X
1,X
2 2".0
'iltL~u1'111 "J1n'111:w#:w'iiuii.if1,<ifumm.,i;m'Ll<i,'11
LL<it x2 l'ilu~<lf1LL'Ll"iilf,i1eJ"t1JeJ,l'.i.111uuJJ1m'Yi1,Xu
1.3 Vf~~.Vu11'ltj'lh~tH~ 1um'iLLnilqJ1111m1L'Ll'iLLn'i:J.J1'il'LRUL91t1'l'il'ilm'>'<~ni,Xu m:wTrn
1-ii'l.1nlJY1,noiiu1'lll)'Ll'iti1,.-;~LUUilf,m"iVl1fll~''J'iiLLi1~'11~1'l91 1urn''1~,n11LLnilt1J1'1l'il~'Yi11'1~1ti
~ut'11'Yi1m1LL'Ll<i,~'Ll'il1n~n11rn~111rt, L'liu m1L1'Lla:im"lVl1'11~''l911'LlLUUm"iV11!>11~1iJ'il 1111eJm1LL'Ll<i'
m11111'11~1i1911'LlLUum11111'11'1,i1'il
" . " '
1uY11,'1rn'llflli1'111 m"lVl1'11~''l'il'lleN'i'l,noiiuL111'1 l'.im1:J.J111mm111:i1eJunJJm11111'11'1i1')91'lleJ'
'11i11J'lleJ'Y1'noiiu,Xw1 L'liu
Minimize Z = 2X
1
+ 3X2
Subject to
x1
+ x2
= 6
-zx1
+ 3X
2
s; -4
3X
1
-3X
2
~8
xl :i1'11l'il'l n1'1
x2 2". o
'il1n<lf1LLlJlJm"i1'Ll111m:wL'il'LRuJi,mh1
L~<ill~~'U~DU~.Jd
1. ~ru,'l1,nifa.ifil'i1n'il~ 2 .11tJ -1 LL'1~U1<if1LLD"i S1 mJililn"Jln1'1W~1tJ'lJil''l1,noifu
2. lJ1n1'11,-ifltJ'lleJ'Y1'Mfu.ifeJ'ilff91~ 3 '11t1<if1LL'Ll"i 82
3. LLY1u.11<ii1LL'Ll1 x1 '111.1 x1 = Y1 - Y2 <i~tuY1,noiiu1''llbJ'Ll'~"'fiLL<i~i1:wm1.ifeJ,j1n111iii,
m:w J(,,i,,.,~1'1411LL1JlJ:W1'111!J1u'11111-i'u'i'il'ilm'r'<~ni Ji,d

Minimize z = 2Y1
+ 2Y2 + 3X2 + o.s1
+O.S2
Subject to
Y1 - Yz + x2 = 6
2Y
1
- 2Y
2
- 3X
2
- S
1
= 4
3Y
1
- 3Y
2
- 3X2
+ S2
= 8
YI, y2' x2' SI, s2 2: 0
11111'um·mni:lblJ1111m-:;ltJ·mm:1-1b'li,,i"1.-~'1tb"W1"iei'lu'lu~1eith'~ 2.s '1td'ium'l1111'11<i1eiu
1<i1t1'lll Basic Feasible Solution 11111'u~1mJ1,'IJ!l,m'lbbniJ'!JV1lm'llU'lbbn'lm'li'bftU~~m'l"U'1'
'11nm1:1-1m'lblluff:1-1m'll<ilt1'li'~1bbU'lbYit1:1-1 'lJibbff<ilAlb~mq:1-1'lmrn1m'l~ni,;1b~!l'~ 2.1.3
"-"' j-1 ...,.. ~ gJ t::::...e:l <:::., Q r1
2. flTmfluty ti1!ll1 u 1!1111:t.l!'ll'lItri.I~11'.111l'll:t.I!'l'HIfl'll
v.'1 1·' - v '[ -~- o < v !ff. fl ~ '[
m·rnnublJVll lJ'ibbn'l:i-1 b'll"ff'Ub<iltl11l'lfm'Vrnn'lf IJn'l'1'iJJU1'lJUbuUl'l'l"b'lm<ill'J Goerge B.
Dantzig 'luu fl.'1. 1947 1<ilt1!llfft1111flnm'l'lJmv:i'lJf!ru<ilb:l-IV17n'il (matrix algebra) 'lum'lV!1'11<il!lU
1<iltJmtmum'lilm1€1.ff!'IJ!l'bi.JVl'in'il bbi'.i-l1'lll:Jr:1-1b'l'l~n'il'iltiJbbfl1~m1:1-1~'mn w1i~ui11!lu'IJ!l''llld
~m1:1-1b1t1u~1t1 l<il1m1fft1rntu1um'l1111,flru<11'11ff<11.f'lJeNnwYi1~1'1 '1um~Yi','lJ''11<11!lub111m~ff:1-1
Yiff~
ll·;rndN~ 2.s m'l1111'11iJ":f<ilmrui:lblJV11n1'lV11R1UU'ltff:i-1Nl'i<11iirn~ (product mix)
u'lmii:f1'lJYimwdfi"ei.f 'ii1n<i1 Nl'i<11b'l'lei-ffi•'1dli1iirn1u 2 'lJU<il ~!l 1<il~fl!l:1-1Yi1•<11!l.fbbff~
1<ll~'li1'm ;tr,Nl'i<ii'11n'li'.ie1<i1bllu111fln l<iltJoiJ!ll;Jff~b~m.ifei,num'l~<ilfiuh Ni1<11fl~,dmm'li:mffil'111
•<V ~ I "'°
i1''1mm'u
~ .ll'Ufll
1~ - <<iltfleJ:l-l'Vrn<11eJ'l
1<llt'Yi1~1u
'il1mur1mmmif~~u
iilew
(~u.l'J.11'11)
4
6
48
11~rw1mhlfmNiiw
!!'l~~l'U!IN'U fl !l'l~~l'U!IN'Ufl ii1119i01'11 (tlll'l)
iJ • v
j~fl!llJ ('lflJ.fl!lfl1)
• • v
wm1m ('lflJ.fl!lfl1)
2 2 340
2 1 400
18 16

ri1V1ui1vr x1
= ~1u1u1~~flei:wf11•1i1eii'~"il~1'11m•Nl1il
x2
= ~m1u1~~1'11~1u~'il~°li1m•Nl1il
lii1bbuum•1i.J,.m•:w•ll~•'1'u'llei~u"li;f'Y1'1L'11YfmvJeii'1'lb'ild '51ni mm•ri•'lltm1Jflii,d
Max z = 340 x1
+ 400 x2
mt111r-iiei:.51ni
'
4 x1
+ 6 x2
s 48 ('Yli'l'lmn-,1~iliJ
2 X1
+ 2 X2
:S 18 ('Yli'l'lmn·mNuni.J'irneiu)
X
1
, X
2
~ 0
~''11:1<1'.lrib'llt1rnlluif1bbUumil'.l!jlU<l1V1iu'l1lm11l:wb'l'I~n.flJ'lii,d
Max z = 340 x1
+ 400 x2
+ os1
+ os2
+ os3
mtJ111-iiei~lni
4 x1
+ 6 x2
+ s1
= 48
2x1
+ x 2
+ s 3
= 16
X
1
, X2
, S1
, S2
, S3
~ O
<J1n<i'hbbUUm'.lLiJ1bbm:wbll~b.i'u-ii1,1fu u11i.Jb:zi t1um11~b~:w1fu'llei~i/r:wb'l'l~n'1f Liti'J~u1bi:1'11~
' ' q
iii1rn'11ifl-11.J1~ft'YIil'lleN'1:wm1YI'n:i!u'iIiitiiJ1~'1'~lbi1~iii1'i1'11if:wi.J•~li'YIil'llei~'1:wm1YI~n:i!uoihi'51ni •ii
'
"~1uil1"i1'6ii1i:iu Lit1<J~u1iii1m.h.!1~1 1ci11uuVi1il111~•~ei1Vf'1~i1n'llilm'.llln'1:wm1•~eim.-i1ileiu
~•Vlm~~lj'i il111,~:w4ru'1lei~lll-Il'l'l~n'1illfii~1Jllii~ill"il'~ 2.1
C, 340 400 0 0 0
J
1'11mh flll'l!lll
lllirl'l x1 x2 SI s2 s3 (RHS)
0 s, 4 6 1 0 0 48
0 s2 2 2 0 1 0 18
0 83 2 1 0 0 1 16
z 0 0 0 0 0 0
I
C-Z. 340 400 0 0 0
I I

~m11~b~lJ'li'u'lJti~!ElJb'WilniiJ~1iJ11'i'l1m1Nliii bn1l~'U ~~ri1111uoi1Vf"11bb1hi1bb10rn'li1mbUu'li'1
bbthb'U!lG'I (basic variable) ~.:i111m11ii~bU'U'li'1bblh~i1m1lJV1lJ11JV111l'li''lbb1h~"J:;bU'U.-i1ilfl'IJ (RH8)
' 2' ' 2' '6j -"'"" ,:!! A A ._, "-' .,i._, .":! I
b'Ul'l'U G'leJ 8
1
= 48, 8
2
= 18, 8
3
= 16 'lf.:i111m11rnY11';w1mY1~1111JoiY1mG>1.:ib111!1t11l!,J
fll'lJfl" Z. 'lJ<l"bb'1il1:;<!1llJll'i.'U11111" .-i1u1ru.11'11i"d
I
Z
1
= (0 x 4) + (O x 2) + (0 x 2) = 0
z
2
= (o x 6) + (o x 2) + (o x 1) = o
z
3
= (o x 1) + (o x o) + (o x o) = o
z
4
= (o x o) + (o x 1) + (o x o) = o
z
5
= (o x o) + (o x o) + (o x 1) = o
z
6
= (o x 48) + (o x 1s) + (o x 16)= o
~".11'lJei' z6 1u~diiiei '11'lJei,rilh<1,<!1l~"J:;11'11u L111JLu~dbri1nu o b-dei,"J1m'.fo1iJi1m1
"' = )I 'II ~
Nl1ilbn1l'll'U
'11 c - z 'lleJ,<!11lJll'i.11'1 iiiti Nl1.i1,.,:;11111, c bbl1:; z. ~iltJ1m11llJnb~tnnu t1111'11
..,; J 6J J 6'v 1 !:; A • "1 ,,J j £.-L J .:.'. 'll' v .d $. '
c. - z. lJb%Yn:; m<!1llJllil'lbbU'lbl'll'U'U bb<l11,rnm b'll'l"Jn111'1J"J1nm1b'WlJG'llil'lbbU'll'I j 'll'U 1 111m11
I I , , .Ji , .Ji
•'liu ci - zi 'lJm x 1 = 340 111m11m11J11t'l1b~1J.11 x 1 •~lJ'll'U 1 111'.1tJ"J:;'V11LVfri1h11m~lJ'lJ'U
340 'Ul'i1
~'Uillil'Ufi'111u'lleJ,'ili!ElJb'Wilnoii i'iei m1-W<rnmbvlei1111 rlw1011~11m1~~i{il (optimum solution)
~.:imm111~1•uum111'1 ~,d
1) 11111"J<1eiu-11'11111eiu~11'i'"J1n111111,:nmy.iljnoiiiXubtiuf11111eJu~•VIm :;~ij11bbii'11111eJ1iJ
2) 6liein"116bU"lU'Ub'IJ!lG>1 (non basic variable) 6-ii'1bU'U#1bbU'lb'IJ1lG'ibb10:;m"lbliein'li'16bU11lfln
"Jln'411'li'1bbti"lb'IJ!.i
3) tl1'ut11'111111~!Em"Wiln-ii•Ylil1111'11111eiu1111a.J'liei1u
n1rnm1#111uh Nli111'lJil~u1~<1hYitJw~du,"d -.J1n11 "':;11'1'V11m"l-W<rnmbyjilV11'11111ilu
•' ' ."'" ,,,. ._, ""
m111m:;Y1'l111l~u
1) lll"l'l"J<!tiu-11'11111eiu~11'i'"l1nm"l1,IE1J•y.iljnoiiiXmuufi1111eiu~•111m:;~<1116bii'11111ei1a.J L111J1VI•
~"J1"lrn1~•bm c. - z t11'11#1b10'JJ'li1''1 L'Ubbmdi1.J1ii:eitJn111111m'Yi1nu.ru~~~1111Joi bb<l11,11'11
J } 1 ' 'II
111eiu"J1n1111"l1,1EmY>1iinoii1111"l1~~~n'111•tiuf11111eiufJb111m:;Yi<1116bii'1 bb'lit11iJ~i1 c - z '11L1lmnn11
' I I q J I
;;iu~ bb<l11,1H:;4'ei'-W®m•v:leiV11'11111il'Ufl bVlm:;<11Jfl"!11Iiiei1ti
"J1n1111"ll~~ 2.1 "J:;rtiu-11.11 c - z ifol1'11mnn11.iu~iltl 2 '11 i'iti L'U<!1llJn X
1
,,.,:; x
2J l 'll 'll
~'i1'11•'Yi1nu 340 ''":; 400 11111Ji:i1~u bb<l11,11 '11111eiu'lJm'li'1mh tiil x1 = o, x 2 = o bbl1:; z = o
iJ,1a.J6Uufi1111t1u~•VIm:;~<!11 iJ~'Ii'ei~'V11ni-it11utl'l' ,yjei1111'11111eiu~b111m:;~<11l'liei1ti. . .
2) bliein#1bbU"lUum!•n-if1bU'Uli!'lbbU1b'IJ!lG'I (entering variable) bbl1m1"lblitin#1bbU"lililn
"J111'41llil'lbbU"lb'IJ!lG'I (leaving variable)

m7W'J17rn1Ll1iln<ihLu.bumuiifl'oihmdJu<ihLLU7blJiifl 1Viw<J17rn111<ii1LLtloii:u•uii.i11'1~eiti
•
m~nmh c - z ~i1fi1mn~i1'1 <il1bbtl1i:iu?iei<ii1bLti~<J:;nmiwif1dJu<il1bbtl1•uii.i'liei1ti bl'<'i"l:O<J:o
I ) J ~ I ' 'lJ '
Lwi.ifi1Vl~noifw'i'"1ntl1:;ct~.i"l~mnYii1'1 <J1n<i1111~Yi 2.1 <J:;1~11<ii1LLtl1 x <J:;,l'Ju<ii1b•tl1Yinml1einq q 2 'lJ
b.jfol'Ju<il1bbU1LlJiifl'liil1ti
nldbl1iln<ii1LLtl1eiiln<J1n'lf'1<ii1LLtidbuiifl ;;hum.,•11iln<ii1LLtioblJiifl b&'ii.ieiein<J1n'lf'1<ii1LLU'i"LlJiifl
I q ' ' q
~~ •t • •61 r1" v 'Ii ,; v _f ""' """' .:!>:! :v ~I v- _I ..,,.
'YIl-lilfJ '1tm1m1Lil1fl1'u trllitfll'll1llllJ'l'll1m11miutrlliifl'llll ~mum TI!'l'I~'1mt'I"m'1111uum llu.,tlJtrfl
t11tJi1'Vl'1nLnrn'l'l1umdbl1iln<il1bbtl1ililn<J1n'ljli1<ii1LLtidbuii.i iiiei <ii1LLtloli1~il~'ilo~nufi1'llil~r-;i1'11'<n
~i1'11u1nifotJ~iJi1 <il1bbtl1i:iui'iil <ii1LLtio~"ilrnnii1ilein<J1n'lfi1<iilbLtioLlJiifl. . '
C.
J
0
0
0
.::t ,,.. osd d,utl'"'~tl A "'~ti
lll'lHTI 2.2 lll'lH'IJIJ!'l'l!lfl'IH!trll~fll'l!!lllflll'JI! Wlll!ull11111 1!U!ltll!'1~fl11!!1llfl1l'JI! 'l
!l!lfl'ill fl'lj'1ii''JIIti'l!Uiffl
340 400 0 0 0
iiJ"Jutl; rllllllU
~
x1 x2 s, s2 s3 (RHS)!Utrfl
s1 4 1]]4- 1 0 0 48 48 + 6 = 8
I2 2 0 1 0 18 18 + 2 =982
83 2 1 0 0 1 16 16 + 1 =16
<iiomJ
zj
c-z.
I I
0 0 0 0
340
~ 0 0
I
G11~~11-:r~-ll1 fle:i x2
.. .m•i11vimu~~mm'[,,,
diu1 'lu~m~n~1u
flB S
1
0 0
0
m11~~ 2.2 LLi1i1~mdbl1ein<ii1LLtl.,i('UblJi1moif1LlJu<ii1LLtidluil.i "":;m·nl1ein<ii1LLUdililn<J1n
'lj1i1<il1LLti'>•uil'1 ~~1un1ru'llei~u~*"1'1t'llYimwdiiL<Jei1 ~ini1i:iu ililu1tJ1~111um-,ii1 x 2 •oif1Luu
<ii1LLti"HlJilmLi1i1~ll n1'i"W"ilT'llli1~"il:;Nii(;l x2 1 <ii1<J:;<ilei~1~iifili1 4 1'lJ.W. Jl~i:iu1il'fili1~l1«~~'U
48 fllJ.'J. <J:;i11m111Nii(;l x2
1~ 48 + 6 = 8 <ii1 uein<J1ni:iu m<JNli(;l X2
ii~<ilei~1oiinm1uLLN'Un
ti<Jrneiu 2 'lfl-1. 'liei<ii1 nm1uLLNuntid:;neiu:l:iei~«~~u 18 'lfl-1. <J:;Nli(;l X2
1~ 18 + 2 = 9 <ii1 LL'1:;1oif
nml'ULLN'Un(;lrnb'li~ 1 'lfi.i. 'liei<ii1 ~~nl.'l1'llrnLLN'Un(;lrnL'li~l1«~~u 16 'lll-1. <J:o1oiiNii(;l X2 14i 16 + 1
= 16 <ii1 4i1m'Vl<iJ~ x2 LlJu<ii1LLtl~•fimi1h1Vlu~*"11~mn~'11i1 Jl~i:ium1Nii(;l x2 mmvi111nri~
14in1hmn~mrhi:iu LL'li<J1n.i11i.i>i1ni1'llil~'Yl~l'<mm'li1~'1 ~.~moifei~ 1'1m%"t<1:01iifil'l ~~'Vi11Vi<J:o

Nll<ii X flil 11i:;hrm1Ji,1')1N 8 o/11,'f'i1iiu <!i,iiu o/f1mhb1Jil'il S1
~,'OJ:;4iiJ,,UUo/f1bbU"l~ililn"Jln
2 ' ' '
'Jl'1o/11":Ll"u1l'il "'°r x2
,:;,-ii11:L1,l:iuo/11":Ll"u1l'ilbb'YIU i!i11111u.11o/11rn'llYiB~LurnBufl,111l1tJlJ fiB.11
LUl'VllJltJ (pivot element) ~b'll'OJ:;qf!l,l'i1L~il'11LUU 1 LU<iil'll,t'i'11:Li
3) :Ll'lu:Ll1,<i1111,"llm1'1ilnoii,~B1111fi1<iiil1Jl9iil111 m1:L11u:L11,.-i111'1u~udqril,illfftJ 2 'ill"ll'
flil m11"'~m'10~<i1111'~':;1.1~1,~u1m.J 1'1tJ<iil"ll'1111:i.i~:L11u~1"'"11<J:;qfil,i'.ii'n1'!ru:;iii,d flei
'11Lu<i1111'~"lu'OJ'11l'i'11:;11111'<1'1lJ!fo<irnm~dluo/h":Ll"u1lmllmnu 111'l'il pivot element 'OJ:;qfil,
i'.i.11,tlu 1 ci1u.i~liw1 1u'1'1lJn'JJB,o/11":Ll'~"uui:iu ,u11'il,:;qrB,i'.i'11,1iu..ruu
c.J
l>'i'1tnh
miifl
400 x2
0 s2
0 83
zJ
c-z.J J
340 400 0 0
x1 x2 s, s2
4 2 1
Ii= 1 I.-5= - 0
3 6
2
2
~
2 0 1
1 0 0
~iiu~ 1uM1>Jnis
Wi'mvi1'l.Xdju~tl
~1-nLUU~i1~111~-il,,'JG'fila.Jfibbf!~
Lim~ilibmh x2
Ofm>11'lVibllu l
0
ftlfl61J
83 (RHS)
0
48
6 =8
0 18
1 16
ny;'i11~'11~u'l 1u'1'1:J.1n x2
il'11,l:iu..ruul'i11~'1tJm1u1'11,~lJ1u<ii1"111u,i:iu1:Li~runu
'11.J1''1 1uum'llil,ll'f1":Ll1,u1l'i1L111:i.i"ff1LillN<ii'i'1'1TIYi~o'i't:L110u!liln'OJ1n'111u"m~qril,nwvh1~fi11u
'1'1lJno/11":Ll11111:i.iLtlmJUU<iillJ111i'i'nm1'llei' Row Operation ~1Jlrn111m"ff11m4!l,~ 1.2.1 41,,:(,.
'i1111,"lllJ'1'1iln'1i111i'i',m1:L11u:Ll·pblif1'11m111~'0J:;""'1'iii'1U'ill'll'~ 2.4

C. 340 400 0 0 0
J
l>i'1m.h 1'111'161.J
mllri x1 x2 SI s2 s3 (RHS)
400 x2
2 1
- 1 - 0 0 8
3 6
0 s2
2
0
·1
1 0 2-
3 3
0 s3 4 0
·1 0 1 8-
3 6
z 800
400
200
0 0 3,200
3
-
J 3
220 -200
c-z. 3 0 3 0 0
J J
'11rnnn1~ 2.4 81'i1Bu~11il$iB :1m1Ni1'il x2
s lif1 11'1tJ11i!wri1hff-:iffl'lb'Yi1nu 3,200 m'YI.'
wi181<i1BlJfi't1i!dd'Jm~tJ.:J8l<i1illJ~bUUlllllii' (feasible solution) ii-:il:J.1'l'li81<i1illJ~b1'1m~ff:J.1~')1'1 b'l"l'il~
ifo:1'11 c - z m-:i'11mnn11ff1.1u bbftl'l.:J1181<i1BlJ~llii'ii-:i1iJ1i81<i1illJ~b1'1:m~ff:J.1~ffl'I ii-:i'1-:iifiB-:i:i1
J J lb <!,! I Ill 1 ill q I
m'Jll-JlJU"l.:J'ill'JWllillU 11'1tJ'Y11'1f1.l<i1il1.Yl 2 bbff~'if1.l<i1il1.Yl 3 'lll<iiillU lii.:J'ill'Jl.:JYl 2.5 bbff~'ill'Jl.:Jfi 2.6

Q "" " t! 4. ..... _1 v ~· "" -' ""'l'lljN'll 2.5 1'lljN•1m!VHlfl'll!!lll'l~flljrn!Jfll'l1!!uj!'1Jl!uUl'l1!!U'Jl'Ulll'l!!ll~
nij1ii1Jfl1l1uu..1J!Jfl'i1111'!vi1l'1uu1mllflfl~~;' 2
c. 340 400 0 0 0
I
1l1uuj flll'l!J'U
~
x1 xz SI sz s3 (RHS)!'Ullfl
400 x2
2
3
1
1
6
0 0 8
2
8 +- = 12
3
0 32
2
0
-1
1 0 2
2
3
- 2 +- = 3
3 3
0 83
4 1
-1
0 8
4 <ihmJ- 1 8 +- =5
3 6 3
t1£J s2
z 800
400
200
0
3 3
0 3,200
J
~
-200
C-Z. 0 3 0 0
J J
"I
c_ 340 400 0 0 0
J
11111uj flll'l!J'U
mllfl x1 xz SI s2 s3 (RHS)
400 x2 0 1
1 -1
2
0 6
340 x1 1 0 -1 3 0 3
2 2
0 33 0 0 1 -2 1 4
2
z 340 400 30 110 0 3,420
J
C-Z. 0 0 -30 -110 0
J J

'111imm c - z 'll!l"liilill.1111il'l bliilil-:in-:i ri1hiil'YITI~'il~b~l.l~u t'i1b~l.lfi1'llm~1bb1h1u
~ ., J 1 ~ ' ){
iilill-lniiu~u5n 1 viu1a '111uiilill.ltl'll!N~'lbl.h X biiloM11m11Y1l.l'llWll!l"I x 'il~1:i.Jl1N1111i!lri1h
' ){ 1 1
flll'Uiilill.ltl'llil"l~'lbblh X2 bbftiM1lfl1'Hf1l.l'll'U'llil"l~'lLbl.h X2
n1:i.Jl'.iNii11iilfillW1i'Ufi''U flll'U'fill.ln
'llil"l~'lbbl.h s1
Lbiil91111 m•~l1'Yl'l'wmn•'lu'i1-:inoifwiimhnil1b'ln1:i.J1<1i'l.1i-:i1m~l.l~u 1 viil1a 'il~'t11'lVi
ri1ln1'Ylii11il11-:i 30 rn'YI .111uflill-ltl'll!l"l~1bbl.h s2
bbiilil"l11m•~l1'Yl'l'wrnn11uYl;inilwifo'<l1nil~
2 1:i.J1<1i'l.1i-:i1ub~l-l~'U 1 viil1a 'il~1'111Viri1hfJ'YITI'1iliil"I 110 lJl'YI biil~flll'Uiil'1l.ltl'llil"l~'lbLU'l s3
bbiil91"111
m~l'l'Yl-lwrnm1u'i1;inilwiiei'<l1n.i~ 3 1:i.J1<1i'l1i;i1ub~l.l~u 1 viu1a 'il~1:i.Jl1r..ml1ieiri1hbbl1il.J•~m11il
li(;i~u 'illn'Yl-lw mn1~l1e:i tj '11<11e:iu ~bvi m~~'ijWlle:i-:i1l!1Jvi1m1vi1il1ui.J•~fll.INfl<11fi'rno/i
(product mix) 'llmu'l'il'.'Yl'ij1'l1Yiml'lei7ilb'<im1 '<l1nil Ll'i'bbrl m1Nf1<11T<l1~'1e:Jl.IV1'lb<11!l7 (X1
) ~l'U'l'U
3 ~1 1m~Nfi<111Gl'~'t11;i1u (X2
) '<l1mu 6 ~1 ri1hiil"lflil~Ll'i'-lu fi!l 3,420 1Jl'YI
• •
iil'JD~'U<11!lul1i1;i1 'll!l"llTI~mwlin.fl~$J"ld
1) OJ1n~1bb1J1Jm11i.J1b<ml.Ib:j);ibffu1Vib~ tJ'UbD'U~'lbb1J1J9inh<11'11ii1<111ml'l'ljl'Uff1V17u11l~l-lbWlini
'v ""' ii "" <;:; t:
2) fl'll"l91l'll"lb'll-l91'U'llfWlil-lbWiiln'll
3) l'l'l'l'ilftilu11'11l'l!l1J~L~'illMll'll"l~mwfnoBblJu'11l'le:iu~bvim~~'lii bbli'1vi'lmJ';i 1iltm1•
ilrh'lum11 c - z.
'I/ J J I I I
4) n1'11'1e:iuY1~blJu'111'1e:iuYbVlm~Yl'ijil 1Vivitjilm1m'1Jl1JV1l vi1nti-:i1:i.J1-il'il~~m't11m1Llie:in
~1LLU'lboif1Lbiil~ililn'il1n'ljil~1bLD'lL1Jfl'111ie:i1i.J
5) 1'11m•i.J-lui.J1;im11;i~mwflni
6) iie:iuniu1i.J't11~u.,e:iu~ 3) 'ilun11'il~1~'111'1e:J1J~LVlm~~'lil
' .3. ifw111rieirn1f~::Uui'i'um1tl1~a11vi'"l.Um11ti11m·nJ1:il~nruu •
fll'lbbn1Jru'V11'Yl-lwmm~l'.ieitl'<l1nill'i''lam'l11.J'lbbml-lb:j)"Ibfiul'i'1a'lll:j)l.Jbwiinii:iu. .
UtJV11lui1n~rn~l1ie:i1i.Jd'l~
3.1 ilq/mri1iHlf1ll'llltJ (Infeasibility) i';ibfl91L~'illf1911'll"lfJ91Yl1tJ'lle:J"l:j)mwflni t'i1'11'1e:J1J~
&i~iiiil~Ll'i'ii;il'.1'11l'l!l1J'll!l"l~'lbLU'lbYIal.Ie:i tJ~1tJ vi'fa5nilt!vitt-:it'f1ii-:il'.1~1bLD'lbfitJl-l~'llil~l'lrhmnn11
• •
r,JUJ uii1il-:i-l1iltj!vi1m'lllhbLn'l:!.ib:n"ILifulii'"lmi11UUVqJm~hii'.J<ill'lil1J
3.2 ifq/mrif1rn0tJ1iJi'.l'IJ0U!'IJl'l (Unboundness) i1"1bn91Ll'i''il1nm'l~l'Ul'll'll"l:fimwfln~il'l b'l1
wu11.i1ii<11'llil'l'U'l~'Vl11"1fl1'1"1~'Yl1"1'll'l1i1!l (RHS) nu~1biil'lll'Uiil'1l-ltl'll!l"I~'lbU'l~"l~boif1blJU~'lLD'l
blJft91 (entering variable) l1'11~iliil1J'Yl'le:i Infinity
3.3 ifq/mriilf1rn0tJri~riqm-nnt1All'l!llJ (Multiple Solution) i1"1bf11'1L~'illflb'll'il~Wu'<l1mu
'11.rutl'luLbm c - z l'Ul'll'll"l'ijilYll!J'lle:J"l:fil-!1wflnil1mnn11<l1u1u~1bbU'lb1Jft'1
I J!l.o J ' ! I !i.-
'U<ln'illn'Ulu;~vi11;im"li.J'luli1~m•11'15l.I1~flnibv:le:i'ii'IJJU1vi1'11'1e:iufi~fi'lilUu L'llill'il~lJ
u!.].!V11~l1<i'f1bbD'l~iil1m•m;im~e:imoif1blJu~1•bl.J'll1Jfl'1Ll'i'mnn11 1 ~'l vimai'~ b'ilwu11l1~1L1i.J•

mnn11 1 o/11~mm7mijilm.;J1,1Juo/11 bb1hb1Jilfl'lu<11171~~m'Y'l~n4n<111t1 'V1'foliniitJ'V1rt~l1o/11bbtl7
mnn11 1 o/11~l1fi1 c. - z. mn~iii<11'V1"liluiltJ~ft<11b'Yi1nu ,rimn<11i.Jru'V11d~u'i.,iiJno/11bbtl'i'l<11b.;r1J .ti l ~ q iJ "'
biJuo/11bbU'iblJRfl111~ '11niiun'Yhm7tl'i'llt11~<11171~~i.i ''Y'I~n4iiu'iiil1t1
'V1"iilllt]Jm~l1o/11bbtl7~ft1mwt,Jml'iilnililn'il1n'J!<11o/11bbtl'ib1Jlifl1~mnn11 1 o/11 ~''luYi1uil'
b~tJ1 nw11rn~~Yi1mw'i'utll:J'11171,~i.Jb'Y'I~n4 bv:lm:i~Ul'V11'1l<ilil1J~~~'l'"b711Jl'il'Y'l1Jlli1o/11LLU7mn
n11 1 o/11~iii1i.i11rru11J11n'11n'J!<11~1bbtl7b1Jlifl1~ fl1'V1-i'umrndmm7rmni.JtJJm1~,'lim~mnumru
~l'.1#1bblhti.Ibl'iilrnuu#1bbti'lb1Jilf11#i'mnn11 1 #1 $iil bl'i11n#1bb1h'l<11il11n<>a1n'lj<11#1bbtl7b1Jilfln1~
bb'iiLumrudm<>abn<11utJ!mii'nl:tm~'Vlfl,~wmi.im1~ iiu~il bn<11utJ!mm7'V1'JUb'itJunii'rn1il,'11<11ilu
n<i11~il <11171,~i.ib'Y'l~n4~1~<>a~1unii'u1tinii'um1<11tJhimm'lllm'11<11il1J~~~'l<11'iiei1ti1~ 'lumrn
do/11bbtl1rn'~1'wlf<11~1bbtl7buli"b'Yi1nuilm'i m,:;tJnii'nl:trn~~'11'llei,o/11mhLuilfillfi1LUui1u611Ln<11' . .Degeneracy 'iiim1LbnUtJ!'V1ld~il 'li.Yi1nl'ibtl~tJuo/11LLU'i~'il~t,Jnillil!lflJlfl'lj<11o/f1LLU'iLlJRfl'w1J LL<'i1
Yi1m1u'i'uu1,<11111,~i.i L'Y'I~n4'iiil1u
4. m11m11:!'tfo·niii1
m'i'iLm1~1'1m1i.i11 (sensitivity analysis) 'llil,m'i1u'lLmwL~'LffuLi'ium'i~fll:tlD,m'i
Lu~tJmLuiN1u'llei,'11<11eiu~~~'l'" Lrimn<11m'lLU~mmum'llil,fi1"'~ o/11LLu., LL11~V>J,n~u -ii'il~1n<11
'llei'~1LLlJlJm'itu'iLLmi.iL~'Llfu m'i'iLfl71~1'1mli.J11.f!L"ltJn!in~il'Vlfl,11 m1'iL'171~iXm!.mii''1~'11
<ilillJ~~~l'j<ll (post optimality analysis) 'lU'Yrnm'i~"1m'il'iil11LlJum1Yi1m1'iL'1'il~1'1 Wliat ... if
analysis
U'i~fo'llttfll~tJ!'llil,m'i'ib'1'll~1'1'111i.i11~il m1'l{flLm1~1'1li~m1i.ibU~tJULbU'1~lU'llil,l'il<11illJ
~~~'J<ll 1riill1m7LU~tJULLU11,fi1'ii1''1 'il1no/11Lb1Jrn@ii.i1<11tJ1:i.i~1LlJuqfil~L~i.i.lu'V11.i1•mu'lm-i ~1tJ
iJ ,,,: 2' = a f! v ' .d ~ ..l.:t..l '
nl'iiii'i1'J<ill'il'b'ii.J<ilU'llil,'li'i.Jb'Y'liifl'li' bb'1~'Y'l~Ullll'il'"11''1 b'Y'lil'Vllfll<ilillJ'YI"1'Yll'j<ll LL<ilftlm'ill'Vll
.1111illJ~~~'l<11'l'V1:i.it<11t1'Yh'iimnn<ill'il'l'j<11Yf1m@ii.i~'Yh111~L'1tJ ff1m1LU~tJUbLU"'l1N11m~Y11J'liil
fil91B'l.J~@1~~il~"'1L1tb~')
ft1'V1-i'1Jm1'ib'1'il~1'1m1i.i11'llil,m'itu'ibhfl'ii.Jb:il'Lffu~'il~n<i11li''lu'J!<11'i'll1d 'il~n<i11li~ii'nl:tm~
'llil,fll'lbU~t1UbbU'1''u 2 mru ;;iil fll'lbU~t1UbLU11,.J1'1i.iu'l~il'YITI'llil,o/i1LbU'l'uYJ,n~u1<illlU'l~ft,i. '
'Vl"iilm'ibU~t1uLLu11~'llil,ri1l'l'iiil'V1ii1tJ'V1"i11o/lm1u'ii11'V1u1tJ'llil'iiu~h LL11~m'ibu~tJm1tl11,.;ril~1n<11'llil'
Yl'i''Y'It11 m 'Vl"i!l!iniitmfl,~!lnl'ibU~tJUbLU'1''ll!l'fi1'"'~'Ym'll11i'.J!l'll!l,Yldi~uoif!l~ln<11 1'1tJ'il~n<i11
!i,#f1t1'iTim'l~lUN'1ii''Y'ITI'illflnl7'l.nu7LLfl'ii.Jflili.J~1L<ilil1'iiei1u'lm~!l~~ 2.1 A
. . .v .;!( ~ ~ A V ')_j -1""'- V<O>.Q.
ttll~'il1!lftfl'lfl!'ll0'111!!11~!1Mfl 2.1.3 !!'11 m1YlufilJA!l'il!l1'UI 2.1,3
' ' ' 'GI <':i ' ""' .<:!. <:!!. ""'
!'llll'll'Jlll'>ftfl'lfltt'll'J!lfl 2 Atl'llfl 2.1 !'>!Hf! 2.1.3

mor1-ii"1ila.JV-i1L'llil~'li1ti1um.,LLnU!)JVllm.,l1Jmma.J·~~Li'imi:lu'l1frn.,~1-iitum~l.J~li@i L-dt1~
<JlnU!)JVll<J'i~bVUU!)JVll~ll'llu11'1Lvib!)mn tUU'J~UU11i'.lmorYi@mll.JdLLml-Jcl1b7,~jJ.~tiLLnU!)JVI1
m.,ll.Jdbm.,a.JL~~Li'iuLi:lu<i1u1umn m m·rnLLnU!)JV11~ll'llu11'11vib!)ll#1 mhLLi'1dit1<i1nl'1<i1u1u
mn141 l1'1ti#1mJ1,ll.JdLLn7a.Jcl1L%'J1tltmhd 141mi 11.J.,LLma.JAlL'l'il'j;U LINDO (Linear Interactive
Discrete Optimizer) ll.JdLLn7i.!AlL'l'iJ'j;l.J AB: OM vi'foll.J.,LLma.ifi1L'l'ii~ OM for Windows vi'iil
<ll'J1'111:1.JdLLnW Spreadsheet L'liu Microsoft Excel 141
cl1V111JlU'ljl'll'lfld'il~1'111:1.JdLLma.JcllL'l'ii~U LINDO LLl'1~m1l-ii Microsoft Excel lUn11Vll
fil'llillJ
1. fll~'tl1fl1~6'.Uvl1t!hhmrnJIT11~'ll4V LINDO
m1vi1.-il'llillJ4l1till.J1LLma.icl1t%'Jdl.J LINDO ll~U'll<JU1f~d ;Ji)
•
1.1 "171,#1bLlJlJll.JdLLn7i.!L~~L1fU'llil~U!)JV11
1.2 llt1u#1bblJlJ1Vi'ti.JoLLma.J LINDO
1.3 ~AVi LINDO m.-i1'11'1lJ
v • "'
ffJ<JUN'l'l 2.9
'i11n#1mh'~ 2.8 m1V11.-i1~'~"1mruu!)JV1lm1V11fi1ul.J1~"1a.JN~'llfi'twn (product mix) 'll'1~
u'i~l1l'll11m'l'lEi-lfo'iiil-l <i1nl'1 Lri<ill<iu#1bLuu1Vfnull.Jdlm1a.Jcl1L'l'ii~l.J LINDO 11,d
Max 340 X1 + 400 X2
subject to
4X1+6X2<= 48
2 Xl + 2 X2 <= 18
2X1 + X2 <= 16
END
LP OPTIMUM FOUND AT STEP 2
OBJECTIVE FUNCTION VALUE

1) 3420.000
VARIABLE
X1
X2
VALUE
3.000000
6.000000
REDUCED COST
0.000000
0 000000
ROW SLACK OR SURPLUS DUAL PRICES
2)
3)
4)
0.000000
0.000000
4.000000
NO. ITERATIONS = 2
30.000000
110.000000
0.000000
RANGES JN WHICH THE BASIS JS UNCHANGED:
VARIABLE
X1
X2
ROW
2
3
4
OBJ COEFFICIENT RANGES
CURRENT ALLOWABLE
COEF INCREASE
340.000000 60.000000
400.000000 110.000000
RIGHTHAND SIDE RANGES
CURRENT ALLOWABLE
RHS INCREASE
48.000000 6.000000
18.000000 2.000000
16.000000 INFINITY
ALLOWABLE
DECREASE
73.333336
60.000000
ALLOWABLE
DECREASE
8.000000
2.000000
4.000000
'<illf1N'1il'l'1i'lrn~iJ'l!Vllnl'l'l1Jrnn•i.;1'll~1fiu'llil~~1ilth~~ 2.9 mm•ri~'<il~1bfldl~Yifi1111iJ'IJL~
LU 2 fl1UVlilf1"] )l~d
. . .I .::t "" <lo .::t.::t.:t
tl''Jll'll 1 ill'l'J!l'l'll~111'llil01J'llil'll'lil
fi1<111l'IJ~@l~'j<il'<il~lltj'lufi1u~vi-d~'llll~N<iil'l'1i ~~mm•ri11f!'l'1~V11~)l~d
1. ~1u1u•ilu'lum•vi1fi1111iJ'IJ1111 tJ1n'lli.; l''1~n~i:im111u 2 1111•1~111~.-i1'1liJ'IJ~lV1m~oii.;~11<i1
'
2. 'lumill'llll~~1m.h~~ 2.11 rhh'1~iJ<iiYl1~w.Yi1nu 3,420 rnYJ
3. fi1"1il'IJ~lVlm~l1i.J~'j<il $ill N~"1lqJ~flllUi'l1b"1ll-j'. (X1) ~1'U1'U 3 o/11 bl'1~N~"1lq(~'Y]1~1'U
(X2) ~1'U1'U 6 o/11

I ' ,£. o "-' '°'"-' ~ I
4. rn SLACK OR SURPLUS 'JJil' ROW 111''1 V1:1.rn.1m 'llU1U'Yl'l'Y11.llnTI1mfl,bVl'1ilill/
b'Y1'llt1'6Wvi:i.ilil (SLACK) 111"leiol1mu'Y1-l'1111.11nTiil'lbnWnn-iiilfllVIUlil (SURPLUS) 1m.11u~d'Yl-l''Y1tnm
1umoNl1'11 !iei ROW 2) 1iiillil ROW 3) iifl nm'lJil-lbbNUnth~nmJtXUIJn1.jfVl:i.Jlil11.1~,l'lfi1bllu
flu6~'fi Lu'JJru~~ ROW 4) !iei nm'llil,bbNUn'llmb.i,i:iutfornl'l•vi~eiml 4 ~11:w,'l,I 'ill ' 'I.I
T <Ol ""' .J lll.
ll'1'11'11 2 mn1R11::;11m1:i.1 n
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i=l i=l l] lj
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n = 'iil'U1Ubb'lllci-:il.Jm8'Yll-:J
= 1, 2, ...., m bbl'!~ j = 1, 2, ...., n
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n = ~1U1UbbVl<fo1imtJm"
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bl<il bi > 0, 1 = 1, 2, ..., n
u'ii:milrnrn'.t ~in111 l'l1"""1u 3 h""1u~"ei~'lu%"1'11'111'1i1"1 3 6bvi" .~eiNft<iiilu'11fk11'llti"
rn'1"1lu'11 3 6bvi"$iei 111'1"fiu'11 A l1l'1"fiu'11 B 66'1tl1l'1"fiu'11 c h,,1u~'C11:i.mvi"l'lri1'1,m1Nft'il1~
bi'iu 100 <iiu 75 <iiu 66'1~ 125 <iiu <iili<<i1.'i'u 66'1~11l'1,fiu'11~' 3 6bvi' i'.l~u~'11'V~u%1i1b!'luilu'111~
6iili<i'i6'Yi1nu 50 <iiu 170 <iiu 66'1~ so iiiu <i11i<<i1iiiu Jlu'YJuLumTIJufhilu#l1'ii1nh,"1u~' 3 6bvi"1'll
tfol1l'1,ilu#i'1~'mi<dJuiii.id

mnh~nu 1unK~li111'1'1
ii 1 v <>J
l'll!'IJl!fll1'1l'l!!H (Vl'l!lllfl/1'1'11)
1 A 3
1 B 2
1 c 4
2 A 1
2 B 5
2 c 2
3 A 4
3 B 6
3 c 3
•~<111~'l'l~n~u1'<il1J1h~<1~.11111d1~n~wiiei~1fi<il'llei~ilQJV11m1'llmt~d
l1:1J:1J~LVfli1mrnm'l'llU<i~•1nh~~m1t1'1ii'~luf11"11~'1 LlJuJi~d
1lif1ii'~l1ufl1 tl1mrn (ifu)
A XIA
B XIB
c XIC
2 A XZA
B
"2B
c x2C
3 A X3A
B
"3B
c X3C
mm'l~L~1.milQ!Vllnl'l'll'Uft~4i'1~1fuLll'Ul1:!Jm'l~liLi1J1JilQJVllnl'llU'lLLn'l:!JL:a~L~'U1#1Ji~d
'l'l~n~u1<il!JU'l~l1~'1 Min. Cost ~ 3x1A + 2x!B + 4x1c + x2A + 5x
2
B + 2x
2
c
+ 4x3A + 6x3B + 3x3C
'l'l~n~u4i'il~1n11
#JmijUYll'U X1A + XIB + X1C ~ 100
x2A + XZB + x2C ~ 75
x3A + x3B + x3c ~ 125

4!1ueiu"";'I x1A+ x2A+ x3A 2 50
xlB + XZB + x3B 2 170
x1c + x2c + x3c 2 so
x 201)
2. m'rnnifqJll1IYia'hl'lll11Hm1'll1-1n,;i
i = 1, 2
j =A, B, C
f1l'nbflUq)Vll'Vl"Jflm"l'V11'11111fl1JlV';rllJUq)Vl1f1l"l'l1Ufi" !1l:J-ll"l'1~-~'YilLcli'l:i11JLif1111"Jl,f1l'l'll'l-lfi"
(transportation table) ~"d'iu'i1lm"l'11U1bUuvrn;~'Yi1141~11.1bbi'1~'1'1 b~1ffllf1l'l'lif'irl'llfl"m"
Tu·mm1Jblff,bffu ~,,Xu '11n<il1fltl1"~ 2.11 b~flii1mll-l'1,blJu1111"J1,m'l'lJUfi"'11m"lri'Yi11#~"m"1"
fl 2.7
~
0 v ~
flltHflljl'illl'I
11il~ih1ft'1 A
v ~ •
flil-:Jil'Wrl'l c 'll!l~h~HJ.lj11Jllll~llJ!ll B
(1i''U)fl
h''l'i-11 L:_ ~ ~ 100
XlA xlB x!C
h,.Jl'l-12 ll_ ~ ~ 75
x2A "ZB "2C
h.:mi.13 l±_ ~ ~ 125
x3A "is ":ic
• 50 170 80
~
'1111Jl11<Hf1l?llil'
lj1f11'1':;11J
(ml 0
111l"ll"f1l"j']J'.l,fi,~bbMM~l,lifu bUU111l'll,f11'l'llUlf,ill'Vl"i'1Jm'l'llUci,'lliloi~tJfl(i)'j11J'llfl,flU'YllU
'
•'Yi1nutJ fli1'l11J'llfl'flU'1"fi Toio~ x ~'lct1flwilei"'11''1 -~blJuu1mrum'l'llUfi"~141'l'um'l~i1'1'l'l' 1)
'luf1'jll~bUuuq)Vllf)l'j'lJUfl"'lJU(i)~ tJfl (i)"l1IJ']Jfl"eiu'Yll'i-1hib'Yi1rl1J1Jfl(i)'j1IJ']J fl' eiu""fl111lIJ
~Ubb1J1JUqJ1'11f1l'l'!JUci'mm:.l'1u fl1u~b.;tj1J~udb~on11 ~IJi'.J (dummy) '11V1":iuliiu11um'l'llUfi"~1Ji'.J
i'.J<i1•lJu'1u~ (o) biifl"'1n1.Ji'.Jmnmbbiil1~u'1' ~1Ji'.J'I~'l'IXfl tJ'lubbmV1"ifl'1i11JnV1'1"rulL#i'~,d• •

1. mm~eiJJ11,r:ueitJn11eitlY11u <J~o/leJ,bvi:1-Jii1:1-Jl'.iY11,11(ii:i.JJ1 (column dummy)
' '
~10cl1-lil 2.13
u'l'.lffibb1'i,111tt,l'.iT"''1u 3 bb1'i'eJ~~bl'.ieJ' n. bl'.iei, 'll. bbiil~•l'.iei' fl. h"1'1-bbb<iiiil~•b1'i,$lri1K'
fll'l"N~.i 60 Ku 70 Ku bbiil~ 80 <Jiu.i1mhiilu ilieJ,fll'l"R,fiu.l1~N~.i1.'!1.J.Jtfofll1'llu.l1 3 bb1'i'~'
111:1-J1'l"mnullu.l11ili 50 <Jiu 65 <Jiu bbiil~ 85 <Jiu.i1mi1iilu l'.iii'm1'llui'l'iil'~
'illnh~~lll "h.lti~~i:lm'1
9.1 I .,,, <V
l'lll'(!llnl~'Ull«~ ('l'lllUl'l'l/l'lll)
n. 1 4
n. 2 3
fl. 3 5
'11. 1 1
'11. 2 2
'll. 3 4
'1. 1 3
fl. 2 2
'1. 3 6
<J1niJqJ1111i11,niii11u1m'ii<J1'l"m1tJei<11'l"1:1-JY1i'1huQtlY11'1-bbbiil~Qtl11,r:i11,~
l'JeJ<il'l"1:1-Jili1uQtlY11u = 60 + 70 + 80 Ku
=210
mJm.ibei1.1n11eJtlY11ub'Yi1nu 10 <Jiu i11,iXu ilieJ"vi:1-J<Ji:1-Jl'.iY11,'1<11:1-JJ1 10 <Jiu 'lf"11m.,n11-:f1,
' '
lill'l"1,f11'l"'l1Uft'1iliiii,lill'l"1'~ 2.8

' '
"' "" ~ "' cl
~nN'll 2.8 !!IT'1~1ll~!WIJ'11Ul'l1Nl!'11Jl1
~
• v ~
lllll~fl1'Nll'1
v ~ v v ~ v v ~ v v ~
-v<Mh~~rnn1Jfl!HIT'Ufll 1 111M1f'Ul'll 2 ft!Ml!'UI'll 3 '11JIJ
(ili''U)ll
h~~1UA ~ li ~ L<!_ 60
h"'11U B
~ ~ ~ ~ 70
t1~~1uC
~ l2- ~ ~ 80
• 50 65 85 10
~
'111i.Jllil~f11~il~
llm"111i.J
(<ii'u) 0
2. mrn~qtJ11,fimnn11qtlmu ,~ll';;i,bwi.Jlii'i.JJ'lm,L1m (row ctummvl
v • ~
ffJ!l!JH'll 2.14
'1n~hmh~fi 2.13 i1i.Ji.J~'lVim1i.Jmm•mnuliu~1'lli'l''1~~liu~1if, 3 bivi"liu 60 iiu
75 Wlu ibsi~ 85 Wiu<111i.J'<i1ii'u
l'Ji'l"1'l'1i.J~lUijUYl1U = 60 + 70 + 80 Wfu
qtl11~fimnn11qtlmu
m•'llu'1,1~1l1~<11l'l'1~~ 2.9
= 210
60 + 75 + 85 <ii'u

~
01il~m1Nlil'l
w ~ • w ~ •
fnl-.1i.fufl'1 3 'U!Hh~~ltl111Jl'lll~ll'l!l'll 1 l'llMll'l!l'll 2
(1'1'11)n
lm1un. G_ ~ l2_ 60
1..:i.JlU'lJ. l ~ l±_ 70
1·m1uf1. l2- ~ ~ 80
w A
l_g_ l_g_ l_g_ 10<il!Jl.J
• 60 75 85
~
'11l:>J<ilfil'lnl':i'll1l.J
<ilmm1:>1
(ci'u) 0
u1ln•1nd1um1'1-i'1.J<i1111.Jm1'llurl-Jm•vm11 UtlJ'l-4lfll'l'llUrl-Ju1.:iiltlJV11i'.liilu'YJUfll'l'llUrl.:i
'WrnU'YJf1'li1l.J'Yll-1b~1lfl btl1l-J•1ni'.l-if<i~1n'ilbUfll'l'lJUrl~lJl.JbCTU'Yll-1 1-ii1Vlfll?!JUrl-11UbCTU'Yll.Jitui'.liiiu'YJU~-1
mfih'lfudb~1l'1-i'1.J<il111-1fll'l'l1Url.Jm'l'l~l.JiifU'YJUfl1'l'llUrl-1lU'lf<l-1'Yll.Jb~1lflbCTU'Yll.Ji(U,~brlfi1 M b~1l
LL'R'ii.Jll Ltiwi!1l'l~.11u'YJUm 'l'llurl-1',J' LLsi~m'l'VI~n .~l'.Mm 1L~1ln'li1l'Id LUu.i1<ii1llJ
v ' ~
l'ldll!JH'll 2.15
u1~LL'Vi.JV1i:i,nt.,,,1u1l~ 3 LL'Vi.J LL"1<i~LL'Vi-1i'.ln1'1.Jm'll'1~<il 1,000 'VIU'ltJ 1,500 V1'!.i1ti ••<i~
2,000 'VIU'l ti <i11i.Jl'i1ci'u iif1l.Jfl1'l'llurl.:i w~<ilii'rn-r1~w~<ilb'l1,1,..i11111Lnu1u.-i'1.Jlluil1 2 LL'Vi.J~i'.lm1:;i
1i'1l.Jfl1'lbb'Vi.Jsi~ 2,000 'VIU'lrnL<i~ 2,500 'VIU'l1J l<iitii'.liifu'YJUfl1'l'llUrl-1"11l'VIU11Jcii'-Jd
1~~11~HI!
1 2
1 30 50
2 40 60
3 - 35

~f)
~ ~ .All~ll'l!Al 1
~ ~ .AllW'l!Al 2 iillmu'l111
h-:i-:11u 1 ~ ~ 1,000
1':.-:i,nu 2
~ ~ 1,500
1'l-:i.nu 3 ~ ~ 2,000
eium!'ni.i 2,000 2,500
~0
1~8<11'wm11-:imnu'1-:111~1 'il~i'i'.lu1'1-li1;r)1~1'181J'll8-:iill!Jj.jlnl1'llu<i-:i~ilfo-:inl''i~'i1l m1
fi1j.ju<11 1ffu'Yll-:JLL"~i.J~m rn~'il~'llu<i-:i1m1ii"~1fl'u'YI1-:J 1<11 til11'1ii1h~r1-:i#i~'il~.j1iffu'Yll-:J 11l1~i.J~mrn'll8-:J
nl'l'JJU1'-:1~'!11lVl~U'))Unl''i'l1Urf-:il1l.J~l'l11
l l I I
'
<:s I .::. .::!! .:!:!. .::!I
'IH!'lnfll'll'ffl~l'll'll11l't'l 2 mn.m 2.2 mHTI 2.2.2

m·mnilb)Jmm1'lluri~L<iltJltb'YlflUfl'll<i~ii'1Lbuum"i'llUiil,l'J~u<11ilU"1ti'n 2 ~u<11ilufiil
•• d •
1. nl"im'11l1fl1JblJiN<il'U
•0 . . "
1. !ll':ill11'11l'l61J!'.U6~I'!'Ii
'Jlimfil<ilflUbnil,1f'Ul'Unl'JbbnUb)Jmnl'J'll'Uiil,l'JJi'1tJn'UV'11tJ'JTI 'J~b!iilnlmlilflmJi' bbl'i'JTI~Ul
m1oifiXu<ifo,'W'l'i11ViLn<ilm1Ltl~tJ'Ubbtli1'1ueitlmu'll<i~'j<il<iium'bbl1;;f1tlft,'fl'llil'"lfllli11tJ'Yll' Lb'1'1;;Lm1
bbi1~bbl'ii1~ftfl:l,di 'ili"11fi1<11eirnnil,<ifud m,'Jjjill'J111~mmwn~ltJ'l bb1;;b.if1h1Ji'~1tJn'll'ili5w1 'ill
m1~';;bft'Uil1u~ddJu115~~1tJl'ieJm"ib'ihh bbi1~blJu'ili~.iem.if1''~1Vifi1<11ilu~1nl1Lfimfi1<11flu~&i~
'J<il ~,blJ'U'i'ii~utJ:w1-ifnuil~l,bbY<1"1mtJ 1Ji'm1 'iliueimL1ftiliflil~b'Uil~ (Northwest Comer Method)
bbi1;;11hh~:w1ru'11'llil'L'lbni1 (Vogel's Approximation Method 1117il VAM) ~,115~,@,d!l~'U<ilfl'U
Q,tj t! t! d tf"",ol f{ f{ ft If'° A.<11 ~ ..,,: ~ I .d
1.1 'Jli'll!l':i'l'll1'tf'l'lfl!l'lTIJ!l':i 'l1l'U!l"i1'1b'lft'Ylflil'Jb'Ufl'J biJ'U'lllVIlflll1<llJb'J:>J<il'UbbUrnltl'Yl'J<ilbbi1~
1oifm1fi1u1ruUiltJ~ftli1 i'l~u<11ilum11111fi1<11eiu 11'1tJb~:>Jm1~'J11ru1'J1rnlei'1u:;i;i,iuuoif1tJl:'iil'lJil,111"il'0 0
m1'lluri' ('lifl~ 1,1) -i'l'lft"J"itnmrum"i'll'Uri' x. 1Vimn~ft<ilbl'il~'J;;mn1Ji L<iltJ<iim1;i,Ji:Jwiiff-i1n<il~l] l q i ! I
i'liltj mh1fiei .ifil~1n<ilrrnili'1ueit1Y11ubbl1;;fltlm.'l~~"ft<il~iltjYltimmm1YI 1 bbi1~tli11tJft<i1;J,J/iY 1 'illn
iXu~~-i'<i1ft'l"itl'lmrum1'lluiil~1t1!'.i,bbmn<il1t1,,,7m1<il:wnfi'<i11t1 1iilmL4mJ'l <111;i,ibm'l'YILLmJ1;i,i'lliJ~<111"i1'
'JU~U'J<iiii,:11:;i'll'll'llil~11111' t<ii tJ~'ltift~'fl1117iltl'lmrum1;i,iiiiil~m'l bbi1~'lll'YIiu1117!lri1~'m.,w!il1~
i'J!ltjlJi'·fonl"i-1<iift'J'lfl'JUl1l;J,J.if!lfi111'U<ii

v . . .
m!laH11 2.16
•nn<111'l1~m'l'lluil~'llil~i11ilth~~
•
2.12 ~JU
~
.. ~
ftl!Mflnl'lllll
v ~ •
fli~iiufl1 B tio~ilut'1 c 'lleMh~~1unuAll~i.nlfl1 A
(i!lJ)fl
h""lU 1 ~ ~ l±_ 100
h""lUZ li l2_ lL 75
h""lU 3 ~ ~ ~ 125
• 50 170 80
~
m1:1<<11il"m'l'llil"
<11"10lTI:l<
(iiu) 0
0 ,;: !lJ !lJ ""'""' ti ti ft ti " """ .... ,%
nwvnm'ileiub1Jil~<11um m1mTmil'l1'1b1flY1flil'lbuei; 'l~m buum'l"1"u
1) 'it:l<J11rn1-l:i<'lfl1'lli'1mrnm"l'JJUS~'1~'lwfo~ (1,1) brieiv:l<J1"lrn1ri1fi'~m1Ni1<11•irn~h~~1u 1
•vi1i'u 100 o/iu bbllim1:1<4lei~m'l'llil~<11fi'~lu#i'1 A 'iu1.i,.Wm 50 o/iu >1~,Xutfhnrn~1.iwrn1-l:i<'li11d
'lwiiei~ (1,1) ~~•ri1nu 50 o/iu
2) btt1N<J1nm1:1<4lei"m'l'JJil~flfi'~lu#i'1 A m1m'11 ~,41ei,'l'l,1'lrn1~<'1ii'l'lli'1mrnm"l'JJ'US'
'lVi<11fi~lu#i'1 B lliil111 ~~mv:l<J1'lm1-l:i<'lfl"l'lli'1mrnm1'1Juil''1'Lui1<'1:i<n~ 2 ~'llei' (1, 2) ~'flfi'
lu#i'1 B iim1i.i4iei~m'leitj 170 o/iu 'JJrn~~h,,iu~ 1 f'.i',<11,b1>1~eiluf11~"l~-l:i"1i11d'lVloi5n 50 o/iu
$J,,X,. ~'lleN (1, 2) ,~1.i:l'Dm'l-l:i<'lfl'l"> 50 ilu ~'b1lh.iri1fi,Ni1<11'1Jei'h''1u~ 1 'l'>lil~ bblli<11fi',lu.11 B
..... ""' fV 1.<'1 ..., .,q '" 2-<
1-M'JJl<'lfl'Ufllil~iln 120 <llU'J~"l~b<ll:l<<lll:IH111:!<<111l'nl'l
3) ~,,Xu, ~,41il,~"l1'lrn1m'l-l:i"1ii"l"lllu.1'1"l1nt;,,11,1,~ 2 :1<1tfoflfi',fiu#i'1 B 'llei' (2, 2) ~'
t.,,,11-1~ 2 fllm"ln-l:i<'lfld1ftuf11'lVloi,.Wm 75 o/iu nb~mhfi,m'lNi1<11bbt11 $1~,X,. ~'lleN (2, 2) d~~ll
m:i<1rnluf11~"l~'Yhm"l'JJUS'1.i 75 o/iu <11fi,fiufl1 B fJ''lll<'lftU.115n 45 ilu ~'fl1'lloi'nm1'l~<'lfi'l'l
m"l1nh~,1u~ 3 ~'111l' (3, 2) ~'iill'1mrnm"l'JJUS''1nh,,1u~ 3 mf'.i'"flfi',lu.11 B bil<'l~u 45 ilu
bb'1~h,,1u~ 3 m~eifiufl1~"l~-l:i<'lfl'l'lL.i1ln so o/iu ~'bllutl'1mrn~-l:i<'li.'l"l"lLVf<111:1<m1:1<4f1l,m"l'll1l'
flfi,fiufl1 c 'lu'll1l' (3, 3) W1l~ li~:i<1rnm"l'JJuil'~bn11~'U'l1nt.,,,,u~' 3 bb'Vi' Ltltfoflfi,fiufl1~' 3
bb'Vi' fllm"lmbfl<'l,1.i~,'111">1'~ 2.11

~fl
h-:MlU 1
h""lU2
1-;,,lU 3
•m11J<ileMnw1m"
flifollu.'l1
mil
h"'lU 1
lm1u1
h'"1U2
1-;,,l'J.1.3
h~1u3
u Q •
flll~llllrll A
~
®---------
U_
~
50
m'follu.'l1 A
flfi,lluf;1 B
flfi"llu.'i1 B
flfi"lluf;1 B
flfi"llu.'l1 c
l'Jil?1~11J
u Q •
flM!lllfll B
--------+@)
~
'
~'
~
l l2_
®---------
170
50
50
75
45
80
300
u Q •
flll~llllfll c
-------~
80
' '(flll'llll11J)
3
2
5
6
3
~
~
~
0 u Q
fllll~fll1Nllfl
'U!l~h~~rnnii
(All)
100
75
125
~0
. 'fillflll'U Ill!~11l-I
('W'llm'l'I)
150
100
375
270
240
1,135

1.2 1limmwti1'1JeMh1mt (tt~e1li VAM) •lJu11'~ilil-lnumniltJteimn 11't1-,~mrnfi1'llei-l
bmti bllu1iin1'l"bbniJ(jJ1111~'loii111!1mrnn1'YImm'li1tJ t11mnn'l<11l)tl'i~fi-lfi~1frnm'i'il~V11Lifu'1'11-l66'1 ~
tl'hnrnm'i'llUfT-l~ri1'lV11iu'YIUn1''l'llUfT-l~1ff11 111!9nm'i'll8-l1TI VAM i'J~U<llilU'1'18ff'it11"141-ld
' ' '
1) u1o/f1rn'll1fUYJUnl'i'llUfT-l~~li'J11 2 ~1U1U~uajlUbbfl1 b~mfiu111'luff11J<llb~tJ1nu
2) bi.ieimmlli1-J~1.rlu~u 1) ~i1'11iJ-li'J1188nmV1'i11'irn111bUU'lleJ-16bm111'leiff1iw1"l11
Ni.'illi1-l'J-li'J1idbl'lufi1~mi1m1i<biltJh (regret) mn~i'J1i'il1nm'i~1;J'loii'liei-l~i'.i1fu11u~1'111'lum'i~11
ff·nm'i'llufi-J V1'ilTlrn1~11ff'i1tl7mrn'llUfi-1~mn~i'J11<111a<.]ei~1ff11~i1eiaj'lUb6m VI'leiff11a<iliiUi.'1-l'lmfo-J
~1i'u'Ylum1'l1ufi,~1ff11 'lumrn~il wi.1"11-JMff11 •'Vi1fiu 2 ~iu1u 'lVibi;'ieinf1'ilnrn1•&Ii<o/11bi.'i'llLu~ q 'II ~
bbll1V1'leiff11Ji<n~i11i'u'Ylum1'llufi-J~1n11 V11ntl7mrnm1'llufi-J~~1iff'l'ii.'l-J1tliiub~a<mi<ii11~1n1i#i'iu
'
eitl'Yl1uf'lVfcif11bbmiiueiein1t1 V1'famm~i<<111i<~11~1n11#i'iueitlff-Jfin'lVfo/l11ff1ii<niiu1rnn1t1 bbllii'i'10 '
eitimUbbi.'I~iltlff-1fl~timtJ'llu-l'liei-J41-lmh1bvhiiu 'lVi<ff11bbm111'.leiff<1la<Ii~!111u'YIUm7'l1Ufi-l'lubbm VI'lei0 ' 0
'luff1ii<limnn11eiein1t1
3) '11buum1~u~ 1) bbi.'I~ 2) Jl'ilun11'il~ri1m1~1iff'i'itl7mrnm1'lluii-l1"1.i1u<111i<
.jf8~1Tl11<li1ue:it1'Yl1Ubbi.'I~iJtlff-1fl
'il1niJ(jJ1111Luo/11mh-l~ 2.16 b~u1111!i1<11eirn~ei-11i'ut11JtJ1TI'll8-l VAM 'il~ri11"1"1-l<11111-J~ 2.13
' '
lll'iNTI 2,13 lll!>l~lll'il~fll'i'llitii~'llfl~~1arl1~n 2.16
~
0 w ~
flltNfll'iNllll
w ~ v
Af~«ufl1 B w ~ v
'll!Mh~~lUtllNl!Utll A tltHl!U!ll C
n
h-1-ll'.I 1 ~ ~ ~ 100
1'l-l~l'.I 2 Ll_ ~ ~ 75
h-l-ll'.13 l±- ~ li__ 125
• 50 170 80
~
m1:1-J<11D-lm1'llm
11'1-lf!u#l1
0

d
bb11111 2
~
bb11111 3
3
2
4
3
5
3
'
fi'191-:r11i11Gjil =
~ 'N'1lW'i!l
2 = 1
1 = 1
3 = 1
1 = 2
2 = @]
2 = 1
' ' .(2) 'il~b~Ullilil:l-IJlfi 2 l1Nl1ii''WTIYidil'l'illnilm11i11Gjil 2 'illU'lUl1Unul1fi11J''lil <ii''iiu
'il~bi:i!lnil11:1-1.il~ 2 Ylln'W'ilYHu1iiu'YJumnuri'LUill'IJ.Jn~ 2 'iln~u11'1l!l' (1, 2) lliiu'YJunn
'lJUfl,~l'Jl'I ~,bi:i!lmriu'll!l,~'il~'il'lil':f':f'.iJ'Smmm-:r'lJUfl'1VI bbli~bri!JW'ill':fllili)UYllU'lJ!J'h''lU 1 bUU
100 <ii'u 'l'.IJrr"1'1JeNl'1ii',l{uff1 B Lriu 170 <ii'u <11,,l;u,~'il'lrr-:r-:r1<1frwtN 100 <ii'u 1<l1:1-1m:1-1oii!l'il1n1Y1'1J!l'
mnfi 1
. '.,. ..... ""' 2' ..... ,,.
il'UU'il~'lJYlbilUl'li?lbb11YI 1 !l!ln
Y1!i'"1n<ii'1Y1Lbm~ 1 min1'.IJ..l'l'1 '11•uum-:r1911J.J%ul'l!lUb~J.J'li!lL'.IJ Nl1ii'Y<i~1ii'ii~•riu
2
4
4
6
3
'
ilwnli11GjYI
1
3
1
5
2
=
=
=
=
=
=
~ 'Nl1l1'W!i
1
1
[]]
1
1
'iln~u11Nliii'Y<TIJ''ll'I•nl'l'illnNl1'1i1'1urrl'l:1-1n~ 1 1ri!lw'il1-:rmliiu'YJUm-:r'llUfl'6bl'1'11i'.1u11'1lm
(2, 1) lliiu'YJUnl':f'lJUfl'~~l'liYILUili?IJ.Jn~ 1 <if,,);1,1,,~'i1Y1i11-:r'.IJ'SmmnwJJufl'1u'll!l' (2, 1) ~'~'.l.Jmu
'lJ!l,l;-~,1u 2 l'l!l~bwm 75 <ii'u Lbl1~i)'.i.Ji1,.l'IJ!l'l'1ii',i{uf11 A l'l!l~ 50 <ii'u ~'6qJ.J<ii'1Ll1'1JL~ 50 <ii'u bbli~
<ii'l'lill'l:1-1li~ 1 !l!lnL'.IJ m-:r?i11uum-:r1911:1-1~mh1mmi'11wifo (1) bb'1~ (2) bbill'l,1~<ii',1911-:r1'~ 2.14

~fl
l:n-:nu 1
h.:i.:i1u2
h.:i.:i1u3
•m1i.Jllil.:Jflld''llil.:J
11ii.:iilu~1
f1a'.:Jifuf1'1 A
®
50
2
3
~
U_
~
v Q •
l'l!lil''111'11 B
@
170
3
1
~
~
~
v Q •
l'l!lil!tll'll c
80
1
1
~
~
~
0 v Q
flllMfll1Nill'I
'IJ01h11m
100
25
125
~0
irt&:r,nnmn~ 1 ••1.1~11i1:wn~ 1 1<1f,l:woiie:r~1ni1bboi'1 ~.:i1.J1bm~"l11ru15n •nnil:u1Vfu1iifu11u'll8.:J
mn•rn~11i1:wn~L'l'1li81l~:w1~hiilum11'i8111if.:id
' ' <
fi'lll;1"1118.:i1.1.:im fi'lll'11"11<ji1 = N1.1&''11 TI
mnYi 2 5 2 = 3
"'mm1 3 6 3 = 3
'"' 5 1fioi:wrm 2 6 =
fti1:wn~ 3 3 2 = 1
"l~bilu11N1.1'1''V1flubbm~ 2 ""~mn~ 3 •'Yi1nu bb.-iLLm~ 2 i'.iiifu11um1'1Juol.:i~1n11~8 1u
LLm~ 2 t8.:i~iifu11um1'1Juol.:i~1<jli1~1'.i.:i1.J1<1i''i'um;~i11ld''lb'Yi1Nu 2 lJTYl/<ifu LLm~ 3 iifu11um1'11uol.:i
~11li1bliu 3 lJlW<ifu if.:iil:u'l~b~8flb~:W<if1b'1'lllumn~ 2 n8u 11i1rn~:wlUt8'1 (2, 3) ~.:Jl'.i81l'YllU'118.:J
' '
h.:i.:i1u 2 b1>1~88~ 25 <iiu i]11ll'1'1'11m!1'1'.:i11u~1 c i'.i so <iiu ~.:J~i1ft1'f11d''1Juol.:il1u~1'1.:ilu'li8.:iitl<ll
25 <ifu 1'i11Vfm1:wiif1l.:imd'1J8.:J!1'1.:il1u~1 c ii.:ifl.:Jb'l'1~88tl 55 <ifu ci1uh.:i.:i1u 2 •l:woiiil~1ni1<1f1u81I. '
mubboi'1~.:i<ifli1bbm~ 2 88f1111 1'i1lVlb'1'1li8mn~ 3 LY1mLLrrniiim V1"l11rn1~i1ft1'1l'hml!m1'1Juci.:i~
mli81.1.:ilut8.:i (3, 2) 1<11 70 <iiu bbi.1~til.:i (3, 3) 1<11 55 <ifu ~.:J"l~1'i1LVil<li'N1.1n:w<li'iuiJ1l1l.:i'1bb1.1~
i]11'Yllub<il:w:fli1~1ni1'1'miii ~.:J!'ll:wl'lblbLfti1.:Jf1111>1lfl1ll81Jbfi8.:iiilu1i1EJlTI VAM 1<1i'lil.:illl111.:J~ 2.15
1
1
1

~fl
h""lU 1
h.mu2
h"·ll', 3
"f111l-1'11B"nl'l'1Jil"
flml1ui11
fla'~if'hlA'l A
@)
00
2
3
~
ll_
~
~ ~ .i'lll~ll'l!tll B
@
@
170
3
1
1
[_:_
~
~
tiii'~ii'l!tl'l c
@
@
80
1
1
1
ti_
~
~
.~ ~
fllll~fllW<ll'l
'lleHh~~l'I!
100
25
125
~0
'J1rnJ'hmum;-nufi"~'<llil<i'l'llubbl9iii~'llei"~"m"1"~ 2.15 mm'l~u1mfi1mru~u'i'Junwllufi"
'J1nfi1'lleirnflei~~u1#1~"'111'll"~ 2.16
ii1 ~lfl'I! • • '~lfl l'l'l!'J'I! l'l'l!'J'l!'ll'l!ll~nu
(1'i!lr1u•ul (.w'I!111'1'1)
h""lU 1 fl~"l1ui11 B 100 2 200
h""1U2 fl~"l1ui11 A 00 1 00
h""1U3 fl~"l1ui11 c 25 2 50
1'm1u4 fl~"l1ui11 B 70 6 420
lm1u5 fli'.i"l1ui11 c 54 3 156
Llellil'l1l-I 299 876
1
1 1
1 1

m'mniJbJmnT>'ll'l.lfl'IL>ltJ'lTI vAM di.um'l1ll]ii~mm1nfi1u1rui.'.11'1111-:Jb&'im bb.ii.u~d
.;ril,m1illlmmliuiw1 i.Vib'i1wlivi•'u ~'lilllmrn•urn:rnmliu 2 <11111'l-if1,.;ru ,riilu1fi11'1illJ'f1141,1n
'l'B VAM 11h1l"lurnfforn'flJ.i11'11llJ~l41,1n'lnuil4m1l'fYif11l4bueJfo.11'l'B vAM iim~i.Vifi1V1illJ~
a1.;ru11um1'llufl'1~1n111'5udYJnl'!Yif1fflb'.ld~'~'uui1bJVllb~milu ~,dm11~'l'B vAM l1111iinm'l
m'1nm1111l'im~mm1,5tJh~'~hi141Yi1m1%"1'111.l1mrum1'll'.lfl'li1'li.wfo'I~a:l.;ruYJ'.1~1'"1 'lnd
~'11'1mm}l~'.I1111'11•iluteJml'fJ'l'!"1~-:i•n"1~u,1nm•il1.;ru11u'eJ'l'lj"1mi1lJfou~eJu111l~m1~~1111til'I
~a:llifu11um1'llurl'l~1i.um'V111}1'!"1"1il11.l '1nti1<11eJrnneJ-:ililu#i1u'l'B vAM bbf11 ,riill1m11u1111
ti11'!1llJ~~~'!"11"1tJ'lTil'fb'il1JU'I'll'l'.I (Stepping-stone Method) Vl11l'lTil}ll>l (MODI cieJm,1n
Modified Distribution Method) ~''~141nii11n'l"1il1ui:iu m'lf1~'1n'V1lJ11fi1<11ilrnneJ-:ilifu~141,1n
'ln VAM nb!Juri1<11ilrn~mnlJfil<il1llJ~~~'!"1~'Vl1141
,riil141fi1<11eJrnneJ,lifm,.11 mm•nYJ"1'illJti1<11eJlJil'lnii11141 ,~eJ1111m-:i~wm11.l~fi1<11eJlJ~
~~'!"1"1eJlU m1YJ"1l'feJlJ'11<11eJlJiulifumm1nYi1l~"1tm11YJ"1l'fillJ~1u1uY11'1bll1ln1117il~1u1util'I~
l41l1fll'>b~}l<Jhbi1'11~'1~1U1'.IY11'1bl'iilfl~l]ml'iilflL'.lfl11bbniJbJVl1fll'l'll'.lfl'lvf'llU'~bYhnlJ ~1U1'.lbbtn
+ '51u1uff>l}ln - 1 1111il (m+n-1) bl'f}iil v:i,11ru1,1n1i'1ilci1'1~ 2.16 ~1u1m•ml1 3 bbm ~1u1u
l'f>l}ln 3 l'f>l}ln il,i:iu'51u1uiil'lfi1<11illJ'~'liu 3 + 3 - 1 •rhillJ 5 til-:i '1'1}11tJm1}111,1n~1u1um'
bl'iiln~dJu1u141~'1Vl}l>l g imm"~l1m'l'i"1ff11<i'i1bi1'llmmrumTiurl'li1'1i.um11'1'llu<l'lbv:itJ-:i 5 teJ'I
iil'l~l4ll'.lm1'i"1l'!'l11i'1bi1'llLfli1'1lUdb1tJn11 basic cell '11util'l~ll-il1m1{"1'111l1bi1'llLfti1'1lU•1tJn
11 non-basic cell ~''~u111lri~u<11ilu'llmm1~wm1111fi1<11illJ~~~ff"1"1il11.l. "
t:::ot.<::! Q d..ctd.
2. 11ittlfl1V101J'l'll>l'l'lt!:Vl
briilVl1fi1<ililrnne:M41'.ll41bb.1'1 ~U.JiJlU°'~'i1-11llil'Vl1fi1<ilillJ~~~'!"1 ~'1l41bb!l fil<ilillJ~a:l
lifurmm1'llu<l-:i11}1~1'"1 m1~wmfi1<11ilu1u<lfi1<11eJu~~~ff"1~UtJ}Ji.-iinu 141•ni 'l'Bt}l1"1""~'l'Bq q ... q
m<111.JU'lffl<11u ~'lffil'l'lnl1111iinm1~'1'11tJnu ~il <i~YJ>ll'!iluim1111il cell ~1l-il1m1'lluft'l~'l•"ltJn11
non basic cell l"1tJv:i,11ru111 tl1bU~tJ'.lbbUi1'1fl11'llUft'llUtfo cell i:iw1 '~Yi1i.Vl/fu11um'l'llUfl'I
11}1l,Jf1MVl1il1l-i,1n cell ~l1m1%>l'11m1'llurl'll1lbbfi''l ~'lb"ltJn11 basic cell
;;11111-llJi.u.41N~ 2.2 .2 d,~UlbffUil bb"1fll'lVl1'11<ilillJ~&1~ff>lt>ltJ'lTil}ll>l '11u'l'Bffb<11UU'll'll<i1'.I
"
ii:n$1nm'1m1n$1nm141<J1mb'.l'lm1$1m,;1,41l'I~ 2.2.2 l"1tJii:n0mn'~.;;il,i.'6111iinm1i.um1{>lm1
mmru'llufl'l'l~'l'l11'l non basic cell nlJ basic cell ii11tJ'ln closed loop 'llil'l'lnm<111.lu-:il'fl<11umi.-6
ilu'lnt}l1"1411tJ
111-;mtim0u.r,:m'lilii.i1oi (MODI)
fll'lVl1fil<ilillJ~~~'!>liil1tJ'lTil}ll>l
'lnm<111.liJ'lffl<11u m11111fi1<111llJl"1tJ'lnt}l1"1

ri1'Yl1-w11vr u. <ii11<11<if1,,,tl~d'Ju o LUwi1<11L~J-11!ul 0
2) f{J-1J-11ii<i11bb'l.h~Ul-115n 1 <ii1 q;j;;:i k. fi1w~t1 non basic cell 'Yl1;;:ii;;:i~Yi'li.i1<ii:i1m1%<11
Q '
'1•·nJimrnnwJJmi-:i'Ylnim l<11tJri1'Ylu<11tVI k. = c.. - (u + v) 'Yl1!l c.. - u. - v. l<11tJYi k. 'YlmtJn~
' i Ji 1] lJ 1 J 1] x 1, J l] 2.>
<iiu11u1umTIJusi-:iYiboWl-l'll'U'Yl1mw1iN<Jlnm111<11<1ei~%<11'1""LVf:i1mTIJusi-:ibii<11'll'UYi non basic cell U'U'l
ti111n non basic cell :i1'11 k. 2 O bb'1<il~11l<iiri1.-i;;:iu~~~'1<ilbbii'1 bL'1li1 k. < o bb'1<il~11ri1.-i;;:irn~l-Iq lj q 1] '
LJ~li.iL'liri1'il!ltl~~~'l<il bbfi<il-:JllLJ~i'flm111~'J~<1<111lu11um1'llusi~<1-:il1l5n LVJ'Yll cell Yi:i1'11 k. bU'Ul]
'~ ' ~
'"1!l1JHfl 2.17
'J1nm11~ril<il!ltlb~!l-:i1lu<i11J-1iuu;;:i111nfi-Yil'l!l1bu;;:i1 ~'Yl1l<ii'l1n<ii1mh~~ 2.16 1.bmri1V1U<il
<ii1m.h u. bbl'!~ v. 'J~l<ii'11'll!l~ u. bbl'!~ v. <il-:Jbbl'l<il-:JLU<iildl-:J~ 2.17 n-:i.-i111-:i~ 2.20
u = 01
l l 1 J
' .11m1~fi 2.17 !!l!!ll~fll!llBll!i'l1m'u111:i.illlm1in11l!iil'l0inrni
V.
J
•m1:1-1'ilrnm1
'll!l-:irn'foliufi1
50
v =2
B
170 80
0 ~ ~
flll:M!ll1Nll!IJ
·1m~h~~m
100
75
125

mo<i1u1ru.-i1 u. mi~ v. 'lJil1bblii;o~ basic cell <11mo11<i1u1ru14iiii~d f'IJ.J:i.J&iLVf u. = o1 J 1
u1 +vA =3
O+vA =3
VA
u1 +VB
VB
U2 +VB
uz
U3 +VB
u3
u3 + ve
=
=
=
=
=
=
=
=
3
2
2
5
5
6
6
3
- 0 = 3
- 0 = 2
- 2 = 3
- 2 = 4
Ve = 3 - 4 = -1
liiil'JlrnXu-;;~dJumo<i1mruV11.-i1
14'iii~liiil111d
-;;1n k = cii - (ui + vi)1)
kle = 4 - (0 + (- 1)) = 5
~ = 1 - (3 + 3) = -5 *
kze = 2 - (3 + (-1)) = 0
1'iA = 4 - (4 + 3) = -3
-;;1n.-i1 k. ~<irn1ru141 V'llJ11~ti~fil1JuY1uma'lJUci~~ti~<11m·mllo~Vlil'11415n Vlln'Yilmo1) '
'1i'1<ioall~mrumo'lJUci~-;;1nh~~l'l.1.~ 2 111il~f1i'.i'~llu.i'1 A iil~iiu -;;~1lil~Yl'1'1il~'i'1<1Tlll~mru
mo'lJuci~LVfnu non basic (2, 1) -;;1ni:iuV11 closed loop 'lJil~mo%'1<i"ianl"i'lJUci1dmiJ'JTI'lJil1
Stepping-stone 1'11.l'J~1lrnf'lil'1'1tiJ1il1!Jb~ilul'lJi11uijll<i1fibbl1~ijllmu'lJil1bbmbbl1~n'1iJni:iu -;;~141

~
• A • ~A •
B
~A •
c 0 ~ A
fllNllllFl1 A fllH<lllfll Flll~llllfll fllll~fll'lNlll'l
'lrn~h~~1ufl
hmu1 li ~ l1_ 100
-
..§t~
h~~lU 2 + ~ ~ ~ 75
@i" . @-
l:n~1u3 ~ ~ ~ 125
45 80
• 50 170 80
~
m1:wlilmm'l
'llil~f1i-:Jfui!i'1 0
'• A
IMlill'il~YI 2.19
..i, " I I , '
1'111Hl'l 2.19 !llllil~fll'lfl1J1ll!fll u. !Ill~ v. 'llM!!l'lll~ basic cell 'Im
u.
•
u = 01
u = 32
•m1:wlilmm'l
'llil-mi~ilu.1'1
l J
50 170
f1tl.;iUufl1 c fi1tl~nl'lN~9l
'll1Hh~~lll
100
75
l_-3_
~ 125
80

k1
A = 3 - (0 + (- 2)) =5
k
1
C = 4 - (0 + (- 1)) = 5
k2
c = 2 - (3 + (- 1)) =o
~ = 4 - (4 + (- 2)) = 2
~1nf1 k. '1111 ~1;; "WJJ111~il&h-1'Ylunl''i'JJUli1L'1~'~tl'i~'V1t1'11;;lirnlfl'1 l111iiu "11'il'~1] < I q I
2.18 ~' LUu"11'l1,fi1"1EJJJYi~Yif!'1 11i1t1JJ~,;f'Ylf1umft <:11n'1 '~~'1'1'l'l1Vlilm'l'JJUli,1'1u.11,1nT.,,,mY
1 11.1t1,'1ii',1'1u'11 B 100 iiu •nnT.,,,1u~ 2 111!'.f,Flii',fiu.11 A 50 iiu fiii',fiu.11 B 25 iiu lll1~'1n
1.,"1u~ 3 l1ifiii''f1u'11 B 45 iiu lll1~'1ii'~f1u'11 c 80 iiu 1'1t1ilo/lu'YJUnl'i'JJUci'n:1J~1f!~rnaj~
'
885,000 lJl'YI i;l,"1111,Yl 2.20
1tl '1rn1u v '
11'u'lu (wum11)mn l'l'U'J'Uf'11'Udti
h""lU 1 '1A"l1u#l1 B 100 2 200
h""lU 2 '1A,l1ufi1 A 50 1 50
l,,,1U 2 '1A"l1u#l1 B 25 5 125
h""lU 3 1'1l,l1ufi1 B 45 6 270
t-,,,1U3 l'lii',l1u.11 c 80 3 240
tJEJi1TI~ 885
. ' .""" <'S'. <:!:i d!I .d. v 'i~I Al""' QJ«:>.""
'IHHllflftfllll!lltl'tllll'l1~!'iM1'1 2.2.3 uin !U'il'lu{JUl'lflllfl'i'i:IJ 2.2.3
' ' ' '
1
.:!! I d. c:I t$, <"I
'1!1!'1!11111ftfllJ1'11'1!1!111 2 Atll!11 2.2 !1tl~1'1 2.2.3

• •
"I "" <V
!".fijJfl 2.2.4 ?11H'U'Ufll".fi!ij'l.J'lU.Jltlll'l-8
if1Lb111Jnl'i'JJUri.J d:iu<if1LL1J1Jm111Jmm-:1-1 bt.Jbftu~1-ii'tunw~·111ff11iiufl1Vl'iil1ffq•nnbbVlli.J
~U'Yll.J (source) 1VlnrnLV1li11.JimJ'Yll.J (destination) 'ill:1-Jm1:1-1mm'ill'llil.J"ll1l~U'Yll.Jbbl'l:;m1:1-1~ei.Jm'i
'JJil.J~111tJm tJ'Yll.J t'foi'1if1bb1111m1ttJ1bLn'i:1-1 bt .Jb5u5ntJ1~m'YlVIrt,~1i'li1V1':irn~·n11ru~UtJV11~~ll'1m-i~'
fi'u1i'1bb1J1Jnl'i'JJUlf.J 1"1m1 <if1bb1JlJnl'i:l-lil1JVl:l-lltl.JlU (assignment model) bf1tJ.JLbi;l~il.JblJUm'i%111
ff'i'i~.J'JJil.J '1U b~4il.J%m- V15ei.;1u•nrnLVIli.J~u'Yll.JV1it111.Ju.;bbVIli.;tJmtJ'Yll.JVIit,bri1iXu 1111rn1Juffn11ru~
' 'OJ &.! I d:j
m 'i'J111ft'J'ibblJ1JVIU.J'ililVIU.J
if1 bb1111m1:1-1il11V1mmiuff1m1n'iii 1tJtJ1~~n"11i'm'lullJV11m.Jm111~V1111~uViml'Jn1m
<if1il u1wifu m1%111.J1um1Ni1'il1Vfnrn~4il.J%n1Lbi;ll'l~b~4il, m1%m11Xrn1u1VfYi11111l:1~i;l1.J'1 m1
%111.J1u'lVfL~4il,~il:1-1V11b'ilil{tJ-,~m'1N'1 Lliu1lu ,)f,.fil111 tJil1'1i1!)l.J'i~Nfl'JJmm1%1111111 l"iei b~il'lvrll
~u11u~1lj111 Vl1il'li'nm11:1-1uiltJ~lj111 Vl'im~il'lVfllri1hj.J~lj111
iii'.J~1"1n<i11m LLll'Ti1 if1bL1111mwil1JVlmm1uLliu<if1Lb1J1Jm11tJ1Lm1:1-1 bt.JL1iuL'ilunu 1111u
mm111~'~b~m11Vlei ~'lwitJtf11tJ1"1iii.J-d
ri1V1u1111Vf c ~il fl11i'~1tJ1um1%1111111
m n
Minimize z = i:i:cxi=1j=1 lJ lJ
Subject to
m
i:x = 1
1=1 1]
i = 1, 2, ..., m
n
i:x = oi=l lJ
i = 1, 2, ..., n
~ 0 .,, .,,
x.. = o Vl'iil 1 fflVl'i1J i bbl'l~ i 'Ylnml] '
lu~d th X .. i'Jfilbrllfl1J 1 Vl:l-lltJml:l-lll ilm7%111ft'i'i'Yl':f'YW1n'i'JlnbLVl<i.J~U'Y11.J i LVlfllJ
lJ
LLVl<i.Jl.JmtJ'Yll.J i LLl'l~V11n x.. = o V1muli.J'i11:i.J1"1%111ff11'Yl':i'l'lmm-'J1mLV1li.J~u'Y11.J i 1VlnrnLVl<i.Jl]
iJmum.J i
eiu1.Jhnm:i.i m1LLnullJV11V11ilm1V11.l1'iJei11'lVfn11if1 Lb1111m1:1-1ei11V1m tJ.J1u ,~'li''i1lm1~
L'itJn'll Hungarian Method ~.Ji'l~U'ililU1111tJffW iii.JU
'
~U'ililU~ 1 ~il m111-l'1.J'il111.Jfi1LiitJlilmft (opportunity table) 'lu~d'VlmtJn.Jm'i~'J~
V1'J11rui~11V11nb111:i.J1"1Yi1m1:i.iei11V1mtJVl'iei%111ft'i'i'Yl-i''l'l mm-'lVIbV1m~11:1-11~V111.J LLmi.;1lum.JLbl'l:;
bbV1li.;1.JmtJ'Y11.JLLll'1 ,:;i'lf11LiitJ1eimmn111~ub'Yi1h m1V111i11111'11LiitJ1eim11Yi11111l'J

1) vh row reduction nii11$iei Ul~16i1'll'11'ltl1i11J~~l'1lllD'11J"illn~16i1'll'11'lt11i1!.l'i'm
~1'llEl'LLm'l-lil'l-H$i1J1n'l-ll'i-1lll'll'4f'I-I11'1-1 l'1-1~iL1i'.iEl'i-11111i,-J11111nb111:i-J14J1'i1m'll-lil1JVIm 1J'Yi-'Wm~'l
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2) 1'i1 column reduction rnhi$iEI U1~1bi1'll'11'lt'i111J~~lljllLDi11J"illn~1bi1'llflllt'ii1tJ
.., Ii ...t <v 6j ~ d <"'I ,d 1 1 ,, I i 1j 2-<v v
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u111si,4fu'Yll,~iliiiu11u~1ljll4i11J "il:::i'.i.!1LfltJfommnll;mrhlll
~'1-1llil'l-I~ z "illnll111,.i1Lfll'JTeim<f'l:::u1mw"il11m111Lumhlleiu~$i~ljllLL~111111ll:i-J'111um1
~-l1'11m1Dvhm·rn1l1JVlmmiu14ifl11J1111i.i~1u1'1-l'lleMLL111<1,iiiu'Yll'LL'1l::LL111'4,iJm1J'Yll'LL~111111l th m1
:i<eiu111:1<1u,1wil:::1'i1Llll'JW"il11m1-J1llLl1'll o iJ11n{)~LLmuueiu11111l'1ll:i<n1im'i1'LL~1mN'Ylll'11l,1'i1m1
i.i1lu111m1.1'111um1ri1111ull.!1 o iXu1VinrnLV1si,4fu'Yl1'"":::"111si,iJm1.1'Yl1,ili1''1 Yil1:::~ V11nri1111ull14l
fl'i"UVl:i<lln LL'1ll,-J1141.-i1ll1l1J~~~ljll (optimum solution) u~1 1111eim"il"il:::1'i11Ji1lll'l'lli'Yllli1il'
mnLli'l-lll'l'Nim•mueiuLL'1:::'1ll:i<li L1>11J1t~1u1uLfiw11·Hueimfiu~ljlllViNl'l-IL'1'll~u!'i111:1<1>111n~1
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$i~Mi ~-:rn1"ilW"il11m11~,1n-J1V11n~1u1uLfiull1'~mnNl'l-lbi1'll.-ruiilull111'.!1Lfl1.1LEIrnftl'lumm-i1' .~iu1uLLmuE1u1117ff~1u1u<fll:i<ll , :::o/11l,<ii1•uum'l'li1l111 ¥i1l 'lVii11~16'1'll~iheiu~'lll~1:i-JllLfl'i-1llo'
lll'1lnLfl'l-INlU ll.ii11J1l1ln<J1n~1 Li1'll11n~1~hill Lfi'l-lll'l,mn N1ULL'1:::ii111.lu-inniu•<ihnu~1b'l'll
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lrJ!l!JH'l'I 2.18
1um1ri1111ullm<J11ii~<feiu 4 fl'U'lVi-'ue1ll'61l1J'11lU'i'61'1i1''1 4 'i'll1 <>i"L<fll''lum'l1'Lll1J~
1ll'Jl'lii~'11lui-inflULUU~~ Lfl1Jft1lu'lu'i'61,1i,~dmu~1 .;"111111:1LL'll'l-l,'i'lll'll1l,1ll"ill'lfl,if, 4 1~1-11
fll::LL'U'Ut·Ji1nl'i"U'l:OL:i1'1-11ll'Jl'l!'jLL'lii1:::fl'l-I~Lfll'Jl~-'1.J<Jlllnl'lftil'l-l'i'lll'lil''l LVsll,x'U (fl:::bLU'l-IL<i1:i< 100
fl:::LLUu) <ii"'"1>1'lulll11'miJ1:::n1lum1w<J11m1J.Jeiu111m1J'l'l-I
l'lll
A B c D
1 80 75 90 85
2 95 90 90 97
3 85 95 88 91
4 93 91 80 88

,i(!l,,1noifaim11u011'l1-ld:luf!~bbuum'i"1h~d'.lu~m'1'l~w1ir;i~f!u1~-'u ~-lflnmu~l'.ifi1mn•
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1~1'1iunu ~-ii:iutu~uwru,~Wf!l-lLbU'1-l-ifm;ir;i1u01ld1-itvr,iJumrn'll!l-l Minimization 1l'!tiri!lu ~1tJ
m'lbm -1 1i.J~rn111•'1'll1'JWl 1111umdl-l
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1 -80 -75 -90 -85
2 -95 -90 -90 -97
3 -85 -95 -88 -91
4 -93 -91 -80 -88
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'l'lil
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1 10 15 0 5
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n'LIWJ1 bb'IJ'IJfll'j'lJ'l,lci,1111tWl1bb'IJ'IJfll"jlJtJ'IJVlm tJ"l'l,I w;,1uW11mh"~ 2.20
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5!l.t Target Cell;
Equal To: r Max r. Mill r !£alueof: lo
;fiy Changing Cells:·
l$B$14:$D$16
Sy_bject to the Constraints:
$B$14:$D$16 =integer
$8$14:$0$16 >= 0
$B$9:$D$9 =$8$17:$0$17
$E$14:$E$16 <= $E$6:$E$8
8.dd
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;~~6~~7~B~- > >> co- - -

S!ll; Target Cell:
Equal To: r !;!ax r+ Miu r l(ai!e of: lo
:!!Y Chani;ii'1g Cells:
: 1$8$6:$1:$9 :ii ~s
LSillject ta the Coostra1nts: ·· · -··············--
$8$11:$E$11 = $8$10:$E$10
$8$6:$1:$9 >= 0
$(;$6:$6$9 = $F$6:$F$9
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Render, Barry, Stair Jr., Ralph and Hanna, Michael E. Quantitative Analysis for Management. 3th ed.
Upper Saddle River, New Jersey: Prentice-Hall, 2003.
Taylor III, Bernard W. Management Science. 6thed. Upper Saddle River, New Jersey: Prentice-Hall,
1999.
Winston, Wayne L. Operations Research: AppJjcations and Algorithms. 3'd ed. Belmont, Wadsworth
Publishing Company, 1994.

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<ii'1ft'Uh<iii1ft'Uhff~'t.,,,l'U'l111:i.i ,if,fib1"111:;ll'1iil~ri~-ili1L"J'Ulln'Jm7l1m7r11fft'J.!t'J.!lcl'l,VIU1L'Um7Nii'il
ilufi1'Ylilfl':1-1eiu'lVinu~nfi1'luu'imru~i,'p~u'YJn'l lJ 'lu'llrn:;~ri1i',m7Nii'il'llil'h"1uL~:1-1~l1ilaj1iJ
'Y1tMl"lil'iiilm'lNii'il 11,i:ium1:1-1~1Lllu'lum1!'f.J'1,h,,1u'l111i,J~,l1m1:1-1~1Lll'U <J1n.jfil:l-l!'l1i',mh1<J:;•
Vi<J1'l"rn11#i'11~<iii1iluL'J'Yl'l"lU'1iil~t1~'iii1L<J'U .'),,:(um'i<iii1ft'Uhff~l~h"1u'l111iJ~,Lllum'l"Vi<J1'lm11~
11m'l<iii1ilu1w111'J'l1!.i11:J.JLLU'Uil'U

3.2 rn"l<ii<ili1uhmtJlo/lm1:;,il~m (decision making under risk) blJUrn7iii'<ili1uh~N'v
'Yl1rn"l<ii<ili1uh 1:;,immmvmubbii'ii<il11w1"Jrn"lbll"l<ill UilUllfl'il,l'.bfl<il~u1Wle:ithibbiiue:iuw;imm"ltllfll<ilm"lrn
fornl1'V17eim1:;,iii1,~•llu~'~•n<ill'Vl<1Jrn"lrnl'Vlii1i:iu1#1 ~~m'J'Jl'.L.if.ifeil;Jiilue:i~'il 1.J"ll'.'1Urn"lrn 'Vl7€l
'l'Jl"lblqjlt.lllJil~~'llrn"liif<ilfiU1-0J
3.3 rn"l<ii<ili1uhmtJlo/lm1:;,ili.ibbUUilU (decision making under uncertainty) dJurn"l
o/i<ili1U~~1'11rn"l<ii<ili1Uh 1i.imm"l1'Yl"ll1JblU'if<ilill'Vl"Jnl"ltu"l<il1U€lUllfl'il'J~ln<il~Uvl#lil~l~bbUU€JU bbli~1i.i
mm"lnm<ilrn"lru'1e:irnl1'V17€llfl1wiil'Jl'.Lllu~<Jl'.ln<il•'Vl'ilrn"lrnl'Vlii1i:iu1#1 ,)i~dm,,fiil~m'J1n1i.iiioii€J:;,iii• v
LU€l~'il 'V17€lm'J'Jl'.lflui:l11J'Vllf1l"l<ii<ili1uhlu•~ml'Vli.i'l ~~o/l<ili1uh1i.ii1-ifol;Jm~:;,ie:itj .-;~i:iu~<ii<ili1uh<J~
li1€ln1.ifmru.fiVi<J1"lru1'1i1~'1 m'li1tJ1urn"l<ii<ili1uh 1Vli1i.J"l~i1'Yllim'V1 l<iltJ'V1mm:;,iii<ilm1:;,iLi.ibbiiU€lU'!liM
l'Vl"Jrn"lrn'11~'1 ii~ ,1'1€l1Vf•n<ilm1:;,ibbilu€lul~:;,i~u
4. mw'l'i~nml'm'li~.Uu m"l<ii<ili1uhbb1b€l€ln1#1,llu 21.J"l~Ul'YI 1Wlbbri
4.1 m"l<ii<ili1uhmtJ1o/1'1m1m"lrnri1i.il'JfiwiJ~-ii'u •llum"l<ii<ili1uh~N'<ii<ili1uhVi<J1"lru1v v
.if€ll;jl1D"ll'.nil1Jbv1€11.j(umw11lU~lUb%'V1ll'.i1ltJlU'!J€l~'i1Ul€l~l<iltJ1i.i<iim11M1u~ll~~<ii<ili1Uh"lltJ~U 'Vl7€l
~bb'li~-ii'U"lltJ~WJ ~~uu~e:i€ln1#1,llu 2 Ani,r~~01'111J $18
4.1.1 mru.rim"l<ii<ili1uh~ ~<ii<iliiluh 1i.i'l.if<i111:;,iii1'Jl'.bllum1.if1um"lVi'J11ru1
(nonprobabilistic decision criterion) bbli~
4.1.2 mru.rini-i<ii<ili1uh~~<ii<ili1uh 1.iffi1m1:;,ii(1,l'.Lllum.j(um™'J11ru1 (proba-
bilistic decision criterion)
4.2 rn1<ii<iliiluhmtJ1o/1'1m1m1rn~l'l~"'li~ dium1<ii<ili1uh1umrn~l'lm"lll'li~-iiuJi~"<;i 2
1'1u'Vl7e:i 2 NltJ~u1i.Jl'lwiiiJ1l'.LtJ'llib~ii<illlrJ~nu ~<ii<iliiluh'Jl'.l'Jm"J11~bbNulv1€lVi'J1'lru11um'lnl'llU<il
nii!J'YITI'!l€l~'ilulv1m.i11iJ1-ii'lum1bb'li~iiun1J5nN1!J'll-d~ 1umrud~<ii<ili1ul<JiNlm'lnu1'Y11w5•n:;,i (Game
Theory) m.j(um™'Jl"ltull'llm"l<ii<iliiluh
01'Vl'l'ui.J1~miim'll'.'1l<il'i'll1LU'llU1tJd mru.rim1Vi<J1"lrulm"l<ii<iliiluh'Jl'.t.ifmru.fi,)f~mru.ri#lm
f!1111m"lru"lun1"l<ii<iliiluh lllil'.lnrn.ri#11um1ll'li~iiuVi'J1"lru1':i1:;,inu1<iltJ~
1. m'l<ii<iliiluhmtJto/1'1m1m'lrn~1i.il'l~ll'li~iiu 'Jl'.Vi<J1"lru11WI 2mill$1e:i
1.1 mru.rim'liii'<iliiluh~1:;,i1.ifm1:;,iiil'Jl'.LlluVi<J11ru1 ~~'Jl'.€l~mtJlWll1m1m"lrn<ii<iliiluh
~bbiiue:iu lllil'.'1m1m1rn<ii<ili1uhri1i.illUUilU"lltJlil'.lBtJ<il<Jmi1111i'lub~€l~~ 3.1.2
1.2 mru.rim1<ii<ili1uh~1.ifm1:;,iii1'JnlluVi'J1"lru1 ~~'Jl'.€l~mtJ1o/1'1m1m1ru<ii<iliiluh~llv
' ' 'd d ' 1~ d d
l'111:;,i•l1m 11tJlil'.l€ltJ<il'Jl'.nm1 'l l'Ul'l€l~'YI 3.1.3
2 m1<ii<iliiluhmtJ1o/111m1m1ru~ll~ll'li~iiu 1umrud~<ii<iliiluhmm1n1.if'Ylqi,r5m:;,im'Ji1tJ
Vi<Jl'ltul1um"l<ii<ili1uh1#1 "lltJlil'.WtJ<il'J~mh11i'lu'll€lu~ 3.2

ifl'j·Hl'{H'IJfl~9J1!!'1J1Jfll<j9JliJihJi'il
1m~11ti~'llm<if1LLlJlJn10<ii"11luh (structure of decision model) L"1f.l,:i11t1<J:::i11n.1h:::mm
lii~<iiil1t1d
1. ~l'lviih11'il (Decision Maker) vimf.ln' l.J<fWlllrio/liJ~'i'lJ~"1'lfillJ<iiilNf.1m'l"<if91iiuh *'ill"J'<l:::
d'lmV1muflflf.1L~mvi1ilvimf.l'flun1"1 L'liu m'l"<if"1iluh'llil,Nli'l":::nilum.,'lum'l"Nli<i11lwfi1'lfi4"1LV1:i.lililn• •
<llilm"1 vi1ilm'l"<if"1l1uh<J1n'flrn:::i:'.iu"iV11'l"'JJil'm.m.,'luni-ifJ'11'l"rn1<if"1l1uh ,iiilntm,m'l"'lum'l"m"'u
. . "
2. flNIU0fl (Alternative Course of Actions) Vll-llf.ln' 'Yl1,Liiilnrid'lu1t11"1iif,Vll-1"1riN<if"1ilU
•
h<ii1,YJ<J1'l"rn111'lum'l"LLn1lqiV11vii't''l i1'Yl1,,iiilnL"1u1''lum'l"LLn1lqi1111 L"1f.ltin&i<J:::o/liJ,i1mnn11111ii~
'Vil'Liiiln~u1t1 l"1f.lm'l"v:i<J1'l"rn1Liiiln'Yll'LiiilnL"1'Y11'Liiilnvii't~iiu<J:::o/Iil~YJ<J1'l"rn1Nf.1IiiillJLL'YIUri~ri0"1•
,f A ttd ""' Ji' """ 61 .., 6J ,d '
3. tl'ill1fll'Hl! (State of Nature) 111mt.1rn 0m1m'l"lli'YILn"1'11U"J'l"~'l bUnl'l"ililGUL';J'lf'LLrnililn
3.1 m'l"<if"11luhmf.l'lo/111m1m'l"ruriLLUUilU (decision making under certainty) d'lum'l"
<if"1GUhri~<if"1GUhm1lJLLUUiJU11<J:::Li'i"1L'Vl"Jnl'l"l.l,1"1~Un1Lililn'YrnLiiilnriililaj<J:;1"1Nf.1ii'YITiil~1~h
3.2 m'l"<if'11luhmf.l'lo/111m1m'Jrurii1m1l-!L~f.l' (decision making under risk) d'lum'J
lii"1l1uhrii:'.i<ii"11luh1:i.im1rnLii-i1"1 11<J:::Ln"1LV1"1m'Jrn"l"1 LL<li'11m">t:1Loif-iiill-lmLf.1:::n1111u"1m1J-1u1<J:::d'lu• • •
(probability) ~<J:::Ln"1LV1<1Jm'l"ruL111~1iXu1"1
3.3 m'l"<if"1GUlWi1t.I'lo/111m1m'l"rurii11:i.iLLUUilU (decision making under uncertainty)
LlJUm'J<if"11Uhri~<if"1GUh 1:iJ111m'Jl:1'Yl'l"llJLLUUilU11LVl"Jnl'l"l.l,1"1"1:::Ln"1~U Lbf.1l;1:i.im<Jri1'VIU91 vi1ilfl1"1
fl:::Lum1J-1Ul"J:::LlJUlUn1'l"Ln"1L'Jlll1inl'Jruiiu1#1Lf.1fJ
•
m'l"<if"1iiuhmf.l'lo/111m1m.,ruriwmwh~nu <J:::'loifviilrnnrn'l'l"lun1™'1"rn1riLL<iln"11,nu
:01 ..J"" ~ .::!I. 61 <'!, A & V<v ~
4. l'ltlll01Jllfl1-I (Payoff) L1JUNf.1ilillJLL'YIU'YILn"1"J1nm'l"Lf.1iln'Yl1'Lf.1iln L"1'Yl1'Lf.1ilnVIU~'lf' ~ilil~U
h<J:::'l.i1Nf.1ii'Yl~fi1"1<J1nm'l"<if'1iluhit11titi'l":::Li.lu'f11Lliu<if1Lf.1'11L1m11 Nf.1"1illJLL'YIU*~"J:::fJ<Jl'JllilnU'lUL~
'llmn11'l"'Vl1il'l"lfJ1"1 vi1ilm<JYJ<J1'l"rn1'luLL~'llil~o/lu'l'luvi1il"11'loif.J1t.1n1"1
•
'lli1Illfli1'IJfl~fll'j9]i;iifui 'il
nl'l"~ilflUh'llm~u'11111'l""J:::il'iil~l'.i~u111iJU~UjlU~frl~qj ti'J:::nillJ1t1"11tJ 6 ~U"1iJU@l,d
1. m'l"'l":::i.in~uqi1111vi1m4mri<J:::1'i1mTw<J1"Jrn1
2. m'l"ri1viu"1'Yll~Liiiln"11~'1 rid'lu1u1"1
A 6' < .J. ""' Ji' 6]
3. nl'J"'l"l;lj!N'11111nl'l"lli"11~'1 'Yl"Jl;Ln"1'1JULUilUlfl"1
' ~
4. n1TI:::UNf.1<ilillJLL'YIU'lli1'LL"1f.1:::'Yl1~Lf.1iln
•
5. nl'l"LililnWhLLlJlJ~';):;'l-im1m'l"<if"1iluh
6. m'l"<if"1iluhLililn'Yll~Lililn

'11n~u<iiflu~~ 6 <il~ni111'J~'llflt1niii1mh~~ 3.1 u"l'lffi'<l1'llrft1 -ol1n<ii Yi~u"lV111'llfl~A,m1'~
1'11m1Y1'Jl'ltllliii"1fluL'Ji~t11n1J1ll'l'llmmi1-ii'~m'lNi<iifluri1'llfl~n'J1ll'l
.~ . "'V11011N'l'l 3.1
u"l'lffi'11'llrft1 -ol1ii<ii N'u"l11111'llfl~1J7'1ffi'J~1-11rn'iY10J11ru1iii<iifluh l~t11nurn1'llmt1ri1-ii'~rn·mi<ii
• •
'llmii<Jm'l L<iit1i'.1Y11~lfiflnfl~ 3 'Yll~ififln~fl
Y11~ififln~ 11'111111nll'11'1~11~~1u1mJ
'Yll~iflflfl~ 2'llEntJh~~l'Ui&il.I
Y11~ifi1m-Yi 3 ~1~u7iffimtJuflnNi<ii
L<iiu~~u"l11111111fl1<iifl~mfl11!.io/ifl~1111'11mln'11i~t11nufluri1'l111~A'111111
1. n1G1n'11ilm1l.lo/imrn'lfluri1'1~ m1n11'11'1~11~~1u1111:i.Ju"l'lffi'J~11iri1huG1~ 8 l11U1Jl'l'I
• •
m'l'llmuh~~mi&i:i.iu7'1ffi'J~11iri11W'1~ 6 l11urn111 il'1~i11~1~u7'lffimuu11nNii<iiu"l'lffi'il~11iri1hum; 3
2. n1'1nri1ilm1l.lo/i11"1111flurl11hunm~ m1flfi"h~"1u1111:i.J,~ilri11w"~ 5l11urn'YI 1111•
'llmu1'l""1ul&il.lu"l'lffi'J~11iri11'lUG'l~ 4 l11urn'YI ii'1~i11~1"u'hl'.111muu11nNii<iiu"i'lffi'il~11i·l'uri11WG1~ 2
3. n1'1nri1ilm1l.lo/i11~m'lflurl1u1lu rn1flfi~h"~1u1111:i.Ju7iffi'il~'ll1il'YJulJ"~ 4 l1mrn111• •
m1'llmtJL'l~~1ul&il.lu7'lffi'll1<ilrJUUG1~ 2 l11uu1111 n1~1~m'lffimuu1lnNiililu"l'lffi'il~11iri11wG1~ 1 l11u
rn111 N'u"i1111'l'JJfl~u"li:wiflhriu -ol1n"1 'il~iii<iifluh1urnntdmh"11• •
'il1niii1mh"~ 3.1 d 'il~'ll1l1lTI1JltJ1<iltJ1.if~ui11lU'll1l"rn'liii<ilfluh 11i<ii"d
~l.lll!ll.I~ 1 mn~1,1tl~ifty11111~m~M~'il~n11111i'fo1iTl.l1'il ~"m'il'JnDum1iii<iifluhi~moiim
rl1JNG1i11l1Jii'YIUbU'lU'll1l"'l1tJ11'1111;fori1h 111"lil'l'i'111ru11md'llmfi11.if~1u111'111o/iu111ui111 <nniii1il81~~ 3.1• •
L1ium'l'l'i'111ru1m'liii<iifluh 1um'l'llmmi1fl~m'lNiili111m'lilliln'Yll"ifiilnLil~LVING1<ilfl1Jil'YIUri~riffi1
~l.lll!ll.I~ 2 1111irnl.li'l'l'll~1~01wh~1 ~1ul.liU'l.1! 'Yll"lllilnLum'lllni:l11J1111111"lm4mri'il~~1m1
V1'111ru1iii<iiluhi'.11111"lflflnL<iiDl~ 1urn'iV1'il11rn1m1iii<iiluh1iluvi11u1111~lflilm~o/i1l~ilmnn11wd"
'Yll"lfi11n 'il1niii1eJ81"ri 3.1 N'u"l11111'l111"u'hl'.111f!L'llrftJ -ol1n<il i11111~lfifln1um'l'llml'Jn1-ii'~m1Ni<>1iJ831111~• • •
1111~lfi11nri 1 1'11rn1n1l'11'i~11~~1u1111:i.J
'Yll~ifl1Jf)ri 2'lll'JltJl'l"~lUl&il.I
A A Ill ""V ""
'Yll"l'11lfl'YI 3'11"1J'ili'YlilltJUflf1NG1'>1
:,. ,d ~ <!1 ,. .c,~, <to .d .,._j'51
'Ul.lll!ll.l'l'l 3 fll1'>~'1,1{1~1,'fJ111fll'JtllllN1 'l'l'il~l!Wl'lll.I l.l!ll.lli'lll flill11ll'llli'>IW1 'Yl'il~lfl<il'llUl'U
€lmflil'llmll'Ii"~'YIl~lfifin ~iii<iifluh 'il~o/i1l"'Yi'l'lllil1l8l"'lil1Jflil1Jn1lUri'J~lfiilnmm~,.y:jil1'1lm'liiii1lu
h1Vifleiilfll1mnu'1m1m1n'i<il"ni111 '11niii1mh"ri 3.1 N'u"l11111'l111"u"ijj111ff1'llrfl'J '51n<il 11'11~ut1"• • •
6'..J~l';j~l11!1A I 6'd! 'Y ;::} Ill l!I ~ l!l~I
'1m1rn'lnt'Yll1JUl1J <ill.lilF,J 3 lWIJm'lmflil m1:i.ii111~m1...urn!J~ mll.li11l"m1...urnu1unm~ ii'1~
'111l.Iillmm1flur!TU1ll'J

t ~ I <:!>I if, 1
'IJ'J.!Vlll'Ufl 4 mn::jN!IVl!llJ!lfl'U'IJM!!Vlll::flrnllllll 'illmVl"lnl'lll.1'1111'1 N'1'ilil1JLLVIULL'i161:;vi11
L~iln'llmLVl'ilm'lrn.i11'1 LUW~l'l-l'lULYilL11,:(U w<if111iluh'i1:;4lm'Yi1m"lflll11f!:;LUN'1'11€11JLLV1Uri'i1:;141w Lv1€1
" v
1J-;:;Ll'.lufi1ililnm LLi11Y1'ill'lnll<ifl11ftUh L~ilnVl11L~ilnriLVlm:;!Ji.J~i)11
'il1n<ihmh1~ 3.1 ~u'ivi1'l'llmu'ilffii)L'JIYitJ '>i1n111 1.i'm111f!:;LuN11'ililULLV1Uri'il:;1#11u'il1nm'l
'llmtJri1~1m-;Nli1111#i'Ji1e
il1<1nf11l1m1i.14i!l1m'lflufi1m m-;ff.;'11h111u1vi:iJ u'ilffi'il:;1#i'ri1hli11:; s i11uu1vi m'l'llmtJv v
h111uL~i.iu'ilffi'il:;LJiwri11'lll11:; s i11urnvi LL<1:;i'f1~11u'ilffimtJu1lnNli'il u'ilffi'il:;1.i'wri1h1J<1:; 3'911-1
UlVI
il1~nfi1l1m1i.14imm'lflufi1u1unm1 m'lffT11L'l111u1vi:iJ u'ilffi'il:;14lri11w<1:; 5i11urnvi m'l
'llmtJh111uL~i.iu'ilffi'il:;141ri11w<1:; 4i11urnvi LL<1:;f11~11u'ilffimtJu1lnNli.i u'ilffi'il:;1Jlri11w<1:; 2i11u
UlVI
il1~nfi1l1m1i.14i1l1m'lflufi1ii1ltJm'l!1";-11h111u1vi:iJu'ilffi'il:;'ll1.i11u1J11:; 4 '91urnvi m'l'llmtJ
h111uL~i.iu'ilffi'il:;'l1111111ull<1:; 2 i11urnvi Lm:;l11,;J11u'ilffimtJU1lnNli.iu'ilffi'il:;1Jlri1hll<1:; 1 i11urnvi
,9uvi0u~ 5 11111~011;}1uulJ~'1::1,r1111111;}vi!fu1;i ~<if111iluh'il:;.i1-ilm,1mf1mnum1L~ilnNi'I
"11lULLVIU'llB1LL.i<1:;vi11L~1lrn1mvi"lm-;rn.i11'1 i.11~111il~Lu~lJLLUU'llm'il1'll~m-;Jl.iiluh (decision table)
1111tJ'ill'l11ill'ilLL!11111lU~U'llil1N'1'111lULLVIU (payoff table) fi1L-if~1tJ (cost table) Vl':l!l.-i1LfttJl<im!l
(opportunity loss table) Ji'1LL!1111111'tu.i1'l11ri 3.1 vi':lilL-ifLL'llU1m'l<if111fluh (decision tree)
Ji'1LLff111111'tirnw1~ 3.1 n141
~
1 2 3 ... n
11
1 c,, c,2 c13 ... c1n
2 c21 c22 cz, ... c2n
3 c3, c3, c33 ... c3n
m c cm2 cmJ ... cml mn

d ,,!! ,:II. ' ,,J,.,,, ' ~ % , .... 0
'il1nm-,un 3.1 bbll1'.m.mn111~rnn1~bfliln<111~'1 'Yli.Jilf,J 'ill'.1'.bbll1'.il'.'ll'.il,jnlJ'ill'.1'.'Yll~
""" de:! c ..J r: dl r: ' d ... J1 ~ -ti!.
b<iiln'Yli.Jil!J ( i = 1, 2, 3•••m) '11'.bbll1ff<11~mmn111~1m1m1rn"lu1<1rn'l mn111'll'. 'ill'.11'.bbtnff<ili.Jll'll'.
ei~fiu~1u~unm1rn·n;l~,11111~u ( i = 1, 2, 3.••nl 11118<ii1NS1'11il1Jbb'Yl'.'llil~'Yl1~•l1iln~ i .~mn111b·w1irn-,cli
'ilmn111~<ii1 c ( ~~m'il'Vlm!!n~rh1-,V1~il<ifu'Ylu<iiilV1U18~L<if)
ii ' ~
'illn~hmh~fi 3.1 -iim.innn<ii'ilil'.h'llil~ ~lJ"l'Vll"llJ"li;'Yli;j1'llYf!! ~1n111 bbff<11~'111"ll~Nf!<ii1
'11il1Jbb'Ylu1Ji<ii~'111"l1~~ 3.2
(VIU11J : !11urn'YI)
8 5 -4
6 4 -2
3 2 1
<d
bVl"lfll"Jffi'YI 1 c,,
Gd
bVlljfll"Jffi'YI 2 c,2
~'. ·~ c,,~'i>'0
bVlljfll'iffi'YI 3
"'"~
<d
bVl"lnl"lrn'Yl 1 c21
Yl1.:n~e:JnY 2 ·~bVl"lfll'lffi'YI 2 c22
<d
c,,IJ7~tff
bVl"lfll"Jffi'YI 3
<de:!!)q' bVlljfll"Jffi'YI 1 c,,3
<d
bVlljfll"iffi'YI 2 c?~
<d
bVl"lfll"lrn'YI 3 c..,3
'illnrn'11~ 3.1 bb'llU~rn-,<if'11l'iuh bbff<11~-iiil:!Jfl1ufj°n1'lnt~bbN'.nl'Vm'llmrn"l<if'11iluh'iinh~nillJ
4i''l1.l"l<11'llmrn;<ii'11il'.l'ii (decision points or nodes) bbfl~'il'llil~l'lrn1rn"lrn<ii1~'1 ~bfl'il~'. (state of
nature points or nodes) <ii~un111~•Duifr1Jfj°n1'lrni¥i~~

A < '
LL'YIU"1j]'lJil~nl'JLn 11 LVlliJnl"inllll''1
LL'YIULL'llU' (branchs) 'Yll,Lliiln'llil,m"l<if1li'luh111'l'mLft11~
ft ' ..: """ $.
ftJ111nl"inllll''l 'YJLn1l'lJU
'Yll,Lliiln'llil'"j1l'llil,m'J<if1li'lUh<ifil,l1mnn11 1 1'11,Lliiln LLfl:O~UfntJLL'llU' 'Yll,LliilmJn'Yll'
Lliilffil:O<iiiJ,J'.lm111m"imLfi1l~UiltlHUill'J 1 LVlliJnl"im 1~wl1<if1Lfl'll ~flllil1JLL'YIU'llil'LL<iifl:::'Yll'Lliilmriil
Lfi1liV111Jm'>m'ii1''1 (cJ n1fi'uilaj~ummL'llU,'llil'LV1114mm.J1''1
,l]
•nn<ii1iltlWM 3.1 -i!!ll;jflnl"l<ii'1i'luh'llil,~1J'i1111"11J,~ij1'll'ifl'J ~ln'1 LLft'1,LL'llU,m'J<ii'1i'luh1#1
P
1
(0 2) 8
v
1"1U1Jl'Yl
p2 (0 5) 5
v
mulJlYJ
~'{.;, P
3
(0.3) -4
v
~ ~~'.,.
'111,lJl'Yl
,, ~" v
4">' P
1
(0.2) 6 '11U1Jl'Yl
""m'l'llml'Jh"1u P
2
(0.5) 4 •1"1U1Jl'Yl
P
3
(0.3) -2
v
l/1~v
"1U1Jl'Yl
v
)~~- P
1
(0 2) 3 1"1U1Jl'Yl
-~'<!))<;
p2 (0 5) 2
v
4&1)'t.'
murn'Yl
v<'l!§J p3 (0.3) 1 "11,lJl'Yl
nm~ 3.2 11'11u~m1w01lfu1o'll<i~'ll~il''llqhril11 <01001
~'t.IW<l'U~ 6 fll'JAO!U'Ulo1n<ifl'lll~lll6fl ~<ii'1flUh'ilt:Vi'ill"inl1-if!ll;jfl'illnlll"il,'11ll!llJLb'YIULLfl:::
Vi'il1"lrn1ftm1m"lm mi'1'5~Vi'il1"lrni m rn.ri LY1!l'111m"l<if'1i'luh Lliiln'Yll'Lli!ln~1Vi.11~flll!llJLL'Ylu~~!l11
1um"l11i'il1"lrn1<if1li'luhi:iry1111L~mnu"lll'J1#1111'l'!ln1h 'il:::Vi'il1"lrn1.i1ll!l1JLL'YIUf'i)'1 (maximum)
rl'lmnrn'i'l1Um"l<ii'1flUh LLfltLU'AlU!l'L~mnu rr1dJum"l<if'1i'lUhL~mnu<ii'U'YjUVl'l'!lfi1L'll'i11 l'J 'ill".
Vi'il1'Jn11.11ll!l1J LL'YIU~lij'1 (minimum) Li'lumrn'i'l1um'J<ii'1i'luh Lliiln'Yll' Lli!ln~LVlmtftl-1

'lum1~'il11rn1<ii'lflfiuh •1ieimrn•1iein~mm::;n:w~lll'l'llm~<il91l'luh~ei~'l<iinm1m1ru~biuueiu
•m::;mtJ'l<iinm1m1ru~1:iJbbuueiu-Au ~<il91l'luh'i!::;'l-i{mrn'l1m1<il9ll'luh~1:iJ<iiei"'l-i{'111:wu1'il:::d'lu
(nonprobabilistic decision criterion) Yi1m™'i111rn1<il9lftUh
mrn'11m1<il91l'luh~1:iJ'l-i{m1:wu1'il:::blJU'ii:::mhm1~'il11rn1eieirnliu 2 mill ~ei
1. rnillm1<il9ll'luhmtJ'l<iii11111m1ru~bbuueiu "ii:::
2. rnillm1<il9ll'luhmtJ'l<iinm1m1illfi):iJbbuueiu
•.::t q,/ .... Ci ~ 91 rf.::t '
1. ll':W.!fll1Iii'1l:l''I!Mlfll £.I!lll:l'fl11fll1W'l'll!'lfl-Ifl'I!
m1<il9ll'lu'lwntJ'l4inm1m1ru~bbuueiuvimd" m1<ii9ll'luh~N'Yi1m1<il9ll'luhi'.1-iiei:wn~n:wrnruv v v
b~mnub~il"~'il:::Yi1m1<i19lft'j,h bbfi::;ftlm11.~'il:::'Yl"illJn"Nfl~d'i91<t'U'illf1i11ll1f111ll~'1191rJl'il:::bn91~'J,L<ii
•llueith"&' vi"lmlium1<il9liiluh~Ln91~u'iieibitmbllmJo:::<i1ii"wnYi1'l~m1<il9liiluh~1m•n:::119lb~1 ,,im'il1n
i'.J·ifo:l;Jii~,v:imwmm-'u'iieim1<ii'911iluh b'liu~u"ivi11'11ei"n'ilm1<il9lfiuhn'l'1"h"~1u'lvi:iJ ~"dbi'.tmm<nn
n"lm1iltl"imrnm1'111tJ~b'fi:w~u'i11nm1Yi1if11!!1Jl~t1'llltJ11fi1"·wu1nlJjn~1 Ji~,:(u~<ii'1i!luh 'ii::;
~"Jl"lrnl'Yll~b!iein~bV1:w1:::n:w~'l'1 vi"iei'l~flliJil1Jbb'YIU~"lll'l (maximax) •liumrn'Yl'1um1<ii'l'liiluh 'lu
n1m~•llum1~'il11rn1'lu#l1u'llt1~11tJ1<iivi"iein1h rhu'l.um1~'i111rn1•1iein4iu'Yluvi"lt1'11'l-i{-o:J1tJ~1n9I• •
(minimin) dJumrn'Yl'1um1<il9lftUh
'lum1~'il11n1.1<ii'l'll'luhmtJ'l4inm1m1ru~"uut1u 4iei:wn'llei"un1mtl1:::1J1'Yld'ii:::i'.l'Yll-:n1ieinmJv ·~ v
vimtJ'Yll" bb'iin:::'Yll"•1ieirnvi<i1d'il:::•n9l~u'lunm1m1ill~..uueiu•V1tJ"eith"'&'tJ1'il1n4'ei:wn'l.uiJ1'1"~ 3.2 l'f1v
u"imi'Yl"illJbbuueiu11m1:w4iei"m'O'llil'Jrn'll"J::;i'.]bb'U11mi~" ~lJ"iVll"ilJ~ll1'11YitJ <iln91 "J:::n'l'1"h"~l'J,
'lvi:iJbw"il:::1<ii-l'uri1h~"'l9lf1ei s li'mrn'YI bb'iil'l1~u"ivi11u"imi'l1'11YitJ <i1nl'l ml'lm1ru11m1:w4Jei,m1
'llil"~n~1ueitJu.,mi'111~1"u~mtJueinwliiJ ~~'ii:::Lff-i'uwnn111 1 '11u1J1'Y1 bb'iil'l11tin·~'nh"'1u'lvi:iJ
vi"iei'llmtJh"~lUb~:W'il:::'lll91'YJU 4 '11urnm•n::: 2'11urn'Y1iJ1mi1Jl1J Jl"bb1.91"11Jr"mw~ 3.3

!!) u~.t.J uft'"<! yi.iurr-:i IL'!1'ff1 ::i!ltu.Ju ~e:~n..iiin iDiJ.i:I nibsh-l 1S~ii'
lJ~iQri:)l'!t:a@[Q.!;-"' JbE!ill <J.10. t.,E·j
Arial • 10 ... B .l ll -~ ~ m@ % J :og
M22 T fx 88 = Max (85 : 87)
-·-----"""'-------·A 8 c- -·-
1 I I
·-·
2....
3 :V11v1tlon ¥111J.JffiOJn1'i
2.1
2.2
4
5 :fl1'B1~1..J'i'iv:..11t.l'lv11...i
6 fl11'UU1Ui'i..JJ1Ul~l..J
T .n1'i.;i:1,,,u~l!Wirt1uuonwRm
8 f.ff~IJHJUUV1UN..J'1VI
9
10 --
bnrn~ Mnxjmax er Minin1in
'Lnill'li Maximin or Minimax
I
I
2.3 'lil Lap1.ace or Equrlly Likely
lJ1n
"-"·~
2.4 'lil Hurwicz or Criterion of ReaEsm
tJ1umnJ
8
6
3
8
I
2.5 'lil Minimax Regret m Criterion Regret
2.1 mw'l'i Maximax or Minimin
Uou
5
4
2
5
D
-4
-2
1
1
-

'illn<ihel tl1..:iu'i*vlij1'l1YiEl 'illn11 blJiml'llif"1liUh LAmnrn'h17Vl't"11u 41..:iiXu~..:iblJUm'li'i'ill'llli1
1if<?1!iuhLi'ieinm..:idieirnmmmu.fi Maximax 41..:ibb'1"1..:i11'lullll'll..:J~ 3.3
""' "" ""' "" ,.. .. 'lll'lHll 3.3 111nHl1'llllll'lflf'I-! U!m'lfll'IH!Mfll'IUJ!ftlll'll Maximax
(VlU1ti : ~1U1J11'1)
l'lrnli0111111~Piliu1;i m11Jili'Mm'l'11'1~!flA1 rirn0rnmul'~ljfl
u~ 1.hun~1~ U!JV
8 5 -4 8*
6 4 -2 6
3 2 1 3
'illnlll'll..:J~ 3.3 m"~einm'l'11°1..:ih"1uLVliJ 1'11ll<iUbb1'1U'llrnLL<iifl~fim1m'lruLlJu41il m1i;
#lrnm'l'llrnflnJilfi..:i m1i;o/11nm'l'llrnrinrh1hunm..:i bbfl~m1i;o/iei,m'l'llll,flm'hileiEJ ill'11Lri1nu s 5 Lm~'IJ 'II 'IS 'II
•!V • ..... ' "" """"' !V
-4 l11U1Jl1'l llll1l11ll1J '11llll1Jbb1'lUj..:Jij"1'llll'1'll"flllnU<ill 8 l11U1Jl1'l
m..:iLl'illnm'l'llmEJh,..:i1ub~l1 1'11lleiuu'Y1u'llrnLL<iifl~fim1m'lruLlJu41,d m1i;<ifei,m'l'lleNflnril
•
j..:J m1i;o/ieJ..:im'l'll€J..:Jjnfl1iJ1unm' bbfl~m1i;o/f€J,nl'l'llmjm'l1i1€JEJ l1'11Lri1nu 6 4 ufl~ -2 ~1urn'Y1
•• ..... t "" """11 !V
llll1l11"11J '11ll€J1Jbb1'1UJ'ij11'll€J..:J'Y11'bl1llnU<ill 6 l11U1Jl'Y1
'Vll.J b~Bnn11-ctr1..:iu1'lffin1uueJnN~91 f11918tJ bbYli..J/1.1€1-J bLlli'fl~~ri11n1":'in11.1Ju~..:id fl11~'91D..:in11
'llrnjnfl1)' m1i;o/irnm'l'llmjnfi11J1unm' Lm~m1i;o/1€J..:im'l'll1l'llnJ11i1€JEJ l1'11Lri1nu 3 2 Ufl~ 1 ~1u
•• ..... ' "" """" i!J
1Jl1'l llll1l'IW11J '11llll1Jbb1'1UJ"')11'll€J-J'Y1l"flllnU<ill 3 l11U1Jl'Y1
•nni11llll1JU1'1Uj..:Jij"1'llll-JU<iim~m''l'i1ln~ll 8 6 LLfl~ 3 ~1urn'Y1 1Vf1'i'ill'lllilb!i1lnl'11ll€J1Jbb'Y1U~
61..:Jfi11 ~€J s ~1uu11'1 41,,;(utfu™1'l'lleJ,u'ilffiil1'llYiEJ 'i1n"1 m'llif"1liuhLli1lnm..:iLli1ln~'il~fifnh..:i..:i1u'lJ q 'IJ q
1111.J L'ii€J..:i'il1n1"11u1'11lllll1JU'Y1Uj-J~ijll

'!:! oi. ~ ,~ iit;, ~ • "' • ,-; • ~i H !llll .If!
Arlal 10.,. B ,Z !! t~:i{:~m ~~~ 1 l:2 i~i~
2
,3
4
5
i
:,Vl1·,!l.fion
n11i1~1-J'i~lV1U1viW
:·m'itlU1!.J"i5..J..J1Ul~
n151:l1..JU~l!Wifl1UU£lfl~fim
'·Maximax
'
1'1Nl~!lOtll'liiJfllfll1'il
" 1 "fYl"'.iff~·1'1 ~~'11~·~'ll'Ulf1ll.'t:Jn
rrr'Sn-n~1~.,:i-:i1i~'l:!t;,1f1lr;~,1.J
" 1 ' 'nldf7'~1-J ':i.J-:Jllt>"llUlWI ~liqJ
won
8
6
3
' '
..
...-i11.w&iovn1'i NrWHJUlVIU
t.l1Ut1i'l'1V Uou RJR!7!
,? ___·4. 8
4 ·2 6
2 1 3
8
fo
.~
lfin1 0l'llll!lfl1 1'1ll0!1W
fl-l~ "'llflll~ l~'llll
2 2.5 3
2.5 3 3.5
-·---
3 3.5 4

...:. .............. ,.. .. ''11'.!Hfl 3.5 flWl'Hll'.11111'1'1'l'll IJ!a!lflflH!ll!lfl'1llJ!fllll'l'l Minimin
(VIU1f.J : fllU!Jl'Yl)
m~1illlnm'.!ii'vill'll111 •v ,;;
ti1'1llIJ!!'lllll'lll!'1'lil11fll'.!IU!lll'.!1'1llft!IJV
.; ...ll'1ll~ m'll 'l~'IJ'll
•
m•11'h1h-l'11U'lJU1rn~n 2 2.5 3 2*
m'l11-i'1-lh-l~1U'llU1'1mrn 2.5 3 3.5 2.5
nl"l11fi-lh-l-l1U'lJU111l'VllJ 3 3.5 4 3
;nrn;n-;1,:i~ 3.5 m"'i11lnm'li1-i'1,:it-;,:i,:i1u'lJUli1L~n <ilu11u1um.,-i1-i'1,:ih,:i,:i1u'lJ'Uli1L~n1uw1i11~
i1m1m1m•lJuJi,:itl i1m1m1ruel'<m1i11lrnf:l'J11i111,:i rn~ ""~ll'~u l11'i1•1'i1ilu 2 2.5 ""~ 3 i11urn11 1111:1-1
' .• .... ' " ..... """"' 2.J
!11ilU i111111lUbL'VIU<illji1'llm'Yll'ILl11l!lUi11l 2 l11UUl'Vl
111,:i1l11lnm-;i1ml-;,:i,:i1u'llm111nm,:i<ilu11u1um-;i1fi,:il-;,:i,:i1u'lJu1i1nm,:iLuw1i11~i1m1m1ruLlJuJi,:itl
r:.... JI ,,J x .<>( ' ' <v v • ..., '
flJ111n11lli1li1111111l!lbUl'Jfli1!1'1 il'l'YI Lbfi~j'l'lJU :1-iillL'Vll!lU 2.5 3 Lbfl~ 3.5 l1l'UUl'Ylill1:1-il11i1U '11
~ <"I. 1IA. v
illilUbL'YlU<illj11'lJm'Vll"'fl1l!lUi1EI 2.5 l1l'UUl'Vl
'Vll'ILi1E1nm'li1-i'1,:ih"1u'lJm1111V1aj<ilu11u1um-;Clfi'Ih"1u'lJmi1l'Vlf;JJ1uw1i11~i1m1m1m•lJuJi,:itl
t:<U Jf. d. x "" ' ' .... 2.J 0 <v '
flJ111nl'llli1l<ml11il!lbUl'Jfl11fl'I il'l'Vl LLfl~j'l'lJU :!JillL'YllllU 3 3.5 bLfi~ 4 l11UUl'Vlilll:!Jl11i1U illillilUbL'Vl'U
~ ..... 11..... 2.J
illljil'llEl'l'Yll'lbfl1l!lUi1EI 3 fll'UUl'Vl
<nnl'i1<11E1UbL'VIU~lfli1'llEl'IWlifl~l'lli1ElnflEI 2 2.5 Lbfl~ 3 i11U!Jl'Vl 1VIYi'i111llilLi1Elnl'i1111E1rnb'YIU
'' . .'111iji1 fiil 2 i11urn'Vl Ji,:iuu~u~V1T'ln'ilm1LLvi'lit m-;<ili1fluh•i1E1nm,:i1l1E1nm1i1fi,:ih"1u'llu1i1L~n
' '-='!. <=I l " .:'I IV
L'U1l'l'ill!l:i-1'11111EIUbL'VIU<illji1 ilil 2 l11UUl'Yl
'2.2 !flll!'l'l Maximin or Minimax
mrn.fi Maximin ~<ililfluh'ii~Li1ilnNflillEIULL'VIU~~1~fli1'lJEl'ILL'111~m'lli1E1nil1ln:i.i1 'i11nilu~,:i. '
1l1E1nN11<11E1ULL'VIU~l11'i1mn~ji1 1llm.d'il~u1m1-ii!'i1V1~ui:lqJV11m1<ili1fluh1u~'lJ1l'11ll'J1Jr rhh
Lumm mrn.ri Minimax ~<il11fluh'il~Li11ln<ilu11u~IJ'~'l11Lu11'111~m'lli1'-mil1lnm 'il1nilu~,:i
1l11ln<ilu'Vlu~l11'i1~1~i111 1llm.d,~u1m1-ii!'i1V1~ui:lrumm-;<il11fluh 1u1u'lJE1'l<i11-ii<i11'J <ilu'Vluq q ... 'ii q
<J1n<ii'11l!J1,:i'llmu~i:i'Vli1t'lJYil'J -oi1n11 LlJum-;<il11fluhL~mf'uri1h Ji,:iilu~'lblJum1Yi<J11rn1
'

ivi1v1flnn ..
!
!n11N~1Jl1.J..:nulviW
lfl11iJU1tl'i:'iVJ1Ul~lJ
3
4
5.
13
7
8
in1'i.;j1vu~t:Wi111uunnN&m
'.lvlaxirnin
9 l
10 i,,,,
11 'rn" ••• ••••••••••••••••••
12 l
8
6
3
IJ1n
5
4
2
,lfl11J.Ji)}t}Jfl1'i
LhUnfl1.J Uou
8 6
6 4
3 2
I
'
-4 -4
-2 -2
1 1'
!
N"lVILIUUVl'U
fi'JNVI
.4 -4
-2 -2
1 1
'
1

(VIU1tJ •!1uu1'I)
'~
. 'Tirnam1m~~flllu1'il
q
irill1fll~ 11rnm1fl1Jmuv All'llJIJ!!flUIJ~l)'fl
.; ;;;
~•rn~ l'l~'fl !~'Ill-I
mo61-1'1"h-J"lU'llUl1L~n 2 2.5 3 3*
mo61-1'1"h-J"lU'llUlilrnrn 2.5 3 3.5 3.5
mo61-i'1-Jh-J-J1u'llmillV1aj 3 3.5 4 4
'illn'ill'll-1~ 3.7 rrnL~e:Jnrno6111-Jh-J-JlU'JJU111L~n <iiuYJUlum16111-Jh-J-JlU'JJU111L~n'luw1i11~
61m1m1rndJu&l-Jd '1rmm1rni11111w1ilmiltJ111111-J <fl-J~ •rn~lJ"~u i1'11Lri1nu 2 2.5 u11~ 3 '11urn'VI
.1111;wii1Wlu '11111e:Jrnb'VlulJ-1')11'1lil-J'll-JL~ilnii~il 3 '11urn'VI
1'11-JL~e:Jnnl':il11-i-Jh-J-JlU'JJUiilmrn <iiuYJu'lum1611-i-J11-J-J1u'1Jm11nm-J'luw1i11~m1m1rndJu&l-id
v ' •
61m1m1rnfi11111111lmi:itJ111111-J <fl-Jfi m1~<1-J~u i1'11•Yi1nu 2.5 3 u.i~ 3.5 '11urn'11111:wii1&lu '11
••
'il!lrnb'Vlu61-Jft11'1lil-J'll-JL~e:Jnii~e:J 3.5 '11urn'I
;1-J:~e:Jnm1<11-i-Jh-J-J1u'1Ju111'lviajliiuYJu'lumo<l11-Jh-J-JlU'JJU111LVll!J'luw1i.i~<lm1m1rnLDu&l-Jd
. ' .<lrmm1rnilm111e:Jmi:itJ.i1111-J <fl-Jfi ''"~""~u i1'11Lri1nu 3 3.5 ''"~ 4 '11urn'11111;wii1&lu '11111iluul'lu•
'illn.11111e:JrnL'IUlJ-1')11'1Jil-JWli'1:;vJl-Jl~iln~ieJ 3 3.5 ''"~ 4 ~lUlJl'VI 'lVi'Yi'ill'i"Ulll~iln.11<11illJbl'VIU
lii1ij11 ~il 3 '11urn'VI &i-Ji'.iu~u1V11-;n'ilm1uvi-Jd m-;lii11!1uhL~e:Jn'Vl1-JL~imm1<111-Jh-J-JlU'JJU111L~n
' 'di "" ' • ~ 2.>
LUm'il1n:wrn111e:Jrnb'Vlu'il1')'il fiil 3 murn'I
2.3 1li Laplace or Equally Likely
'lll Laplace ~n1m1ii11!1uhYi'il11rn1'11-JL~iln~~~6111 t11tJYi'ilT>U11'il1n'11'1111V1l-Jft-Jft11 'lu'I.I q 'I.I q
ff'lrn~dJui:lqJVll nl';~11!1uh .~mnlJ'lltJ1<ii ri111 lll1~Yi'ill';UI1'1l<ill11Vll-1~1')11 'lumrn~LDuilqJVllnl'i"
ii<11!1uhL~mnu.11'l-ii~1tJ <iiuYJu 'l'ilm.,di1i'ini,;rn~<fl.i'1tJnum1ii11nuh'lu<lm1m1rn~i1m1;wL~tJ-J~'il~
<iieJ-Ji1m1ri1V1u11m1mb'il~•Du'1lmw1il1~•111111m1rn'1:1' t11tJi1-iiil<l;w:wGi~11'lVf'Vlnm111m1rni1m1:wu1'~'Du
' ' '
~'il~Lnil~uL'Yil'l nu (equally likely) L'liu Lumrn~i1<lm1m1rnilaj 3 <lm1m1rn <inla.JUl'il~LlJu'JJm
u'li11~61m1m1rn 'ilnri1nu 113 V1~m'i1i161m1m1rnilaj 2 '1m1m-;rn m1miH~•Du'1Jil-Jw1il1~
' ' .<lm1m1rn'il~L'11nu 112
,1nii1e:Jtl1-J'lleJ-Ju1l:friiJt'1lYitJ <j1fi'11 <11m-;nYi'il1-;rn1'11-JL~iln'lum1'111-J<fle:Jut11fautJ;w t11tJ'l-ii
mrno/im~m<1t11tJi1m1;wu1'il~LDu'luu'lil1~i'lm1m1rn •ri1nu 113 (Vl'Sil = o33) ,'lm,1ni1•w1im1rn<j1mu
<
3 <1rmm1rn

3-21
~ ~ I ''" ""<::!.
finH'll 3.8 fll>!Milfllfi0U!!mrnm1n Laplace
(WU1tJ : fi1'l-llJTYI)
.o!I ""'""~ w w OJ
nHmanfll>fi'1l;l''IH;i m 1:1J1101m>•1rn1In111 N~ '10uunu111V1111~
Arial
2
3
4..
5
M22 T
A
... ~
'V11JlfH'Jn< ~""'"""
}
!m>~hJ'l>0v1u1Mi.i
Jf11'l"llU1Ui:'l.JV1UtG'ilJ
10 .
/&
.
.
6..
7
:
'01'i~1-JU~i:Wlfl1UUUONflVI
8 :Laplace
:
9
":j'(:i";-·
B
ES =
lJ1fl
111 tlnm~H ~ea
5 ·4
6 4 ·2
3 2 1
I !!
="="
+.o .-00
J .oo +.-o
Max (E.~ : E7)
8 c D E...
lm1wo'i<Nm'i . ··'·
thumn0 Unu N~GlUUllVJU
0.33 0.33 0.33 ~0<1m
8 5 .4 3
6 4 -2 2.66
3 2 1 2
I I 3
b'li~ii E5 : = SUMPRODUCT (B4 : D4. 85 : D5) (ef'i"1fl1lflJ<il~hJif.:i•·~m E6 : E7)
3
2.6G
2
..F
,; Gjv .=< v v!;v,,,. ,,,...., "~, •"-'
e::nvn1·1.:i Excel b'lff1!1 E7 b'Vlvi'1<il81Jbb1'1U•J"1'1 '1€1 9 CllHU11'1 1>:.JUUr;JlJ':i'Vll'J'IJ'l1'Wl'jl'li'VltJ .,nnl':
A 1LJ 4'° 6j , _,J { V<v 91'il~bflilnm'J&i.:.:i C'J.:i.:i·nH'Vll-1 Lum;nn w11um 'Jj·l'JI':
2.4 'lll Hurwicz or Criterion of Realism
11 Hurwicz bthn1im'J'Vllflll'i+~tici1.:itlwun (wei<Jhted average) LllumoN'1d<Nl1lmm1
,,, y,_, ~ 6l .J. _6'61 _,;v_ ,d-=1 "-' suv ""6l
'111l.lf,il'llillW,ii11'1U b'J1'1l-lil.JbVl'11fll'Jrn bULnn.IN'•t11U1'1'1 (optimistic decisoon criterion) f:l.INvH1111H"
" ' "
~l.lil.Jrn11m-.r11u1.nrn.Y:;llm~1i.J&l (pessimistic decision criterion) 17i1l1#:'1l'.lnu li111.Jn1V1uillvfr'.i
' "

'Y11m7~'1liuhw1i'1~'1U117~1i'.iu"1u1m ll'1~ri1'VIU'1"J~Jirnm~m71~~N'11lifl'll1l""1udJufi1 a l'1tJ~fi1 a
l'.ifi17~-J1" Oti" 1 t'i'1ri1'Vlu'1'11 a 1lajln'1 o 11'1'1"11~~'1fl'l-lhi'.i11mfi'1~iim5tJ"1111um" Maximin
11'1~lu'Vl1""1'"nu-ii1l1n1ri1'Vlu1?1 a 1.if11n'11 ""'""11~~'1liuhi'.l1m1fi1?1~1iim5tJ"1111u'Vl1" Maximax
1~e:J~'Y11m74f1?1fl'l-ll<J14lri1'Vlu'1'11 a bb'11 <J1niXu<J~'Y11m7'11u1rn'Vl1fi1"1eJU11'VIU1%~tJri1"{i1,jii:n
~8-JLL<ii~~'Yll'1~~8n1ili'~'1d
' .fi1N'1<1le:Jrn1'Vlu1;;i,litJri1"u1'Vlii:n = a (fil<111lUll'VIWJ"'l'"'llil"'Vl1'1lie:Jn) +
(1 - a ) (N'1"1e:JU11'Vlu~1'l'1'llil''Vl1"'lie:Jn)
<J1n4i1mh,u'lli'Yl'll'llYitJ ~1n'1 t'i'i~u~m-,141 ri1'Vlu'1'11 a = 0.6 rn7'11mrnj,j1fi1"1e:Jrn1'Vlu
1%~mb,{i1,jii:rn1<W1~141Ji""1171'~ 3.9
(m.i1tJ : <'1UU1'VI)
ft1UlftM11WIJ!l~Jf1ftl
..;
· l'l~l'l!llJ!!l'IU!l!!U'.l
• .
I!~ um11~1~
' 0 v
1J!l11 m~lll'l11lf1
8 5 -4 3.2
6 4 -2 2.8
3 2 1 2.2
'l1f1"1l"ll~~ 3.9 fi1"1e:Jrnmrn;;i,~mh'{i1,jii:f1'j'~'l'1l'1tJlTI Hurwicz ~llflii'11'1.uGll'll"~ 3.9
fi<J 3.2 '1mrn'VI (,1mrnm1Ji<Jin (O 6) (8) + (1 oo-o 6) (-4) = 3.2 '11urn'VI)
J;,,:Cu~u~'Vl1'l'lle:J"u1i:i'V16ll'llYJl'I ~1nii m14liiflUh1lie:Jn~<J~fl~1'h'"1ul'Vll-J 1rte:J"<J1n1Vi'ri1
' . '
<118U11'Vlu1;;i,l1tJri1"ii.1'Vlii:ni;pi'i61ii
lijilii.fj,iij@M.lfifi!,i,;
~ Arial .,. 10 ... ! B I !! := .s. -e m:~ % , ~08 .;~ t~ i~ D .. & .,, ~ • ...
M22 fx HS = Max (HS · H7)
A B c I D E F G I H I
1 ---- ··-- I - -- " - ----~
1
2
3 V11Vt~.Lln ~111J&iDJn1'i' NNllliJULlVlU"1VriVI Wfl.VIDUUVlUffi1~Gl NNG'.IUUt VIUt;).~U
-
4 won tl1.U0~1J Uou 0.6 0.4
5 fl1'iffi1J'l'iJJIU1Mi.i 8 5 -4 8 -4 3.2
6 fl1'i'l!U1Ui'iVV1ULGiW 6 4 -2 6 -2 2.8
7 n11~"'hJU'5ci1.ilrr1uuunw~m 3 2 1 3 1 2.2
8 Hwwicz 3.2
9 :
.
- -

b'llii~ F5 : = Max(B5 : D5) (!'i~m!lnl,j<il•1iJii~b'll1~ F6 : F7)
b'll<l~ G5 : = Min(B5 : D5) (!'i"1mlflj'<il"l1iJif~b'll1~ G6 : G7)
b'lll'1~ H5 : = SUMPRODUCT (F4 : G4), (F5 : G5)
'l1f1'ill'l1~ Excel b'lli'l~ H7 lVJNi'l'il!llJbl'YIUj'1lJ"1 ~!l 11 li'mrn'YI 4!1i:iu~u~V11w~~lJt'llYitJ
'11n'" "Jtbiieinm"><1511h11m1V1a..i ,rt!l1'l1n1'1wri11-,ll1ll'"
2.5 'Jfi Minimax Regret or Criterion Regret
111 Minimax Regret or Criterion Regret ~111"1iiuh'lt~'l1'>1U1'l1mb'U1~"1l'Ub~!l1'll!l1'11
bfld!lm<! t'"l'J~'ll'Orn1li1"1iiuh biiein'11bfll'Jt!lm<!~~l~lJ'" 'l1n.11bRl'J1!lm<!~]'1~lJ"1l Ubbl'il'lt'Yl1"ii'm
m-,fi1u1rn.11bfll'J1flm<! !ll'il'l:;fi1u1rn'1rnn'l11Ni'l'il!llJbb'Yluri1L'>'Vl'fo'il1n'il1'l111iu'Ylun'l.1
•
2.5.1 f11'>fi1U11U'l1n'il1'011Ni'l'il!l1Jbb''1Un1h
.i1bfll'Jl!lm<1 = ri11~m">'ltLJw,riill11'"1iiuhJn1fm - ri1h~1Ji'fu,~1
• v
2.5.2 f11'01'11U11U'l1n'il1'l11'il'UYJU
.i1bftl'Jl!lm<1 = '11L1i'ii1tJ~bn"1~u"J~1 - '1111i'ii1tJ~m'>'ltdJubriflli1"1iiuhJn1i'fl1
'l1nl111m.h1~ 3.1 '111'l11Nl'l'ilflm•'Ylu'llmu~~ll1'llYitJ '11n"1
(m.bl'J : i:'1UU1'YI)
m~1ila11m~~~liu111 fl1llJv'M111~·1rn~~flftl
P tlm11m~ U'e.t1
nl"lfl11'h~~1UlVia.I 8 5 -4
m'l"nmtJh,~1ub~lJ 6 4 -2
v
nl"l"ll~UMflll'JU!lnNii<il 3 2 1
'il1n'il1'>1'~ 3.2 mm'l1"llll1fl1bftl'Jl!lm<!~~1lJ"1lUbbl'il'l:;ftm1m"lrn l'Un"lrnm1l-l1ffl~nl'>'llm
l'ln411''"1 m"lli11?1iiuhLiifln''111biiflnm"><l111h111u1mJ ,:;1Vi'11<ilfllJbb'YIUiW'"1?1!l s ii'1urn'YI 111~i:iut'f1tf'll ) 'll o~ 'I.I
u~'Vll"lU~lJl'llYitJ '<lln<'l bii!ln'Y111biiflnd,:;b'Jum"lli1"1iiuh~,V1m:;<1l-J~fl ,:;1a.il:'11bfttJlflmmn"1~u
bbl'it'f1li1"1iiul 'ilbiifln''111biiflnm"l'llm tJh111ub~l-IV17!lm1•iiflnmT•i'11u~~mtJU!lnNii'il 'ilt1Ji'fu.11
'llfllJbb'YIUbYi1nu 6 '11urn'Ylbbl'l:; 3 '11uu1'YI ~1'il:;l:.i1bftl'J1flmmn"1~u•Yi1nu 2 '11urn'YI (S-6 '11urn'YIJ
•rn:; 5 fl1urn'YI (8-3 '11urn'YI) 1uYi1uf11b~mnuturnrnm1l-l1i'il~nw11fl11Jnf11iJ1unm1m"lli1"1iiu
hbiiilnm1biiilnm"li1-i'Nh~11u1V1a.i 'ltLVf'11<il!llJbb'YIU<!1<1"1~!l 5 '11urn'YI 4!1i:iut'f1tfu~1w~~'l'llYitJ'I.I q w q
'11n'" biiiln'Yl11bi'l!ind'il:;blJum"lli1"1iiuh~•'Vll-llt'l-l?l!l<J:;1a.il:.-i1bfttJL!lmmn<'l~u bbl'it'i1li1"1iiuh •ii!In
1'111bi'l!inm"l'llmtJh111m~l-IV17!l'Y111bi'l!inm"l%1~u~~mtJu!lnNli<ii 'il:;1.lwNl'l'il!lrnb'YIUbYi1nu 4 '11u

V ,j A•,St ~x •~ V ( V y
lJTYlbm~ 2 murn'i'1 'lf''iil~:l.lrnL"!J eimffLn?i'llUL'i'11fl1J 1 muJJ1'i'1 5 - 4 murn'i'1) LLl1~ 3 mulJ1'1'1 (5 - 2
i11urn'i'1) Lm~1umrnm1:1.1o/lmm1'll1l'l;Jn.'11ueiLi m1<if?11uhLlieinm'Llieinm~1,JJ"i*'1rnLlueinN11'11
'iil~lVlNl1'1lil1JLL'i'1'Uj'il'"i"iei 1 i11urn'i'1 'il~LD'Um'l<iiwi1uh~LV1m~ff:1.1i"iei 'iil~1iJi'l'ii1Li'1!lTeimff LL~{h<if?Jiu
"L'iilLft1lfl'i'11'Lftilflfl1'l'ff-i'1'r.,,,1'U1ViiJ Vl~il''11,Lftilflfll'l''llm d.,,~1'UL~:IJ 'il~i'1'11Lft!JTeimmn?J~'ULYllnlJ
5 ii1'U1J1'1'1 (1- (-4) i11urn'i'1) LLl1~ 3 G1'1'U1J1'1'1 (1- (-2) ii1'U1J1'1'1) <ii~ff1111~<ii,'111TI~~ 3.10
(VIU18 : fi'1U1Jl'i'1)
m~ii'i!lflfll'j91'1lf'U,'i) fl11lJ9l'!MflU'lltMlflfll ;h!lia1!lfll'f
II'~ tl1u11~1~ '!Y!la
8 - 8 = 0 5 - 5 = 0 1-(-4)=5 5
8 - 6 = 2 5 - 4 = 1 1 - (-2) = 3 3*
8 - 3 = 5 5 - 2 = 3 1 - 1 = 0 5
'i'1l~Llieinm1fffi~h~~1ulV1iJ'ii1Li'1!lTeimffj'il'"i"iei 5 i11urn'i'1 'i'1l~Llieinm1'llm!lt.,,~1uL~:1.1'11Lft!lTeimff
ff~'1?Jflil 3 i11urn'i'1 'i'1l~Llieinm1~1~JJ"i*'1rnt.JueinN!1<ii'ii1Li'1!Jlilmffff~ffwifiei 5 i11urn'i'1 <J1n'ii1Lifoq_, q 'ti ~
fomff'1~'1?J'llil~m"wi1m~•liein~~V1:1.1wi1Vf~<J11m1m~1li•m~i'1'11Li'l!lTeim11~1ff?J 1umrn~i"iei 3 i11urn'i'1
.' '
<ii~iXu~JJ"iV111'llmJJ"ii:i'l'1ijt'lln!l 'il1n?J m'l'<iiwiiuh'llm!lh~~1u•~m~"1~'11Li'l!lTe:im"ll~il'"'lleJ~m~
.' '~ ~ 0 ' ~ &:!.
L11ilflU<illfl'l1'1'11~ Ll1ilfl1l'U"]

Arial • 10.BI!J:""=
M22 • /;< E15 = Min (E12: E14)
A B C
""'1'''T"
2
vi1i.nRnn
:n1'i"1'i11.1"i'l·,h.l'1U"lVlW
'n1'lt.ILI1Ui'lVV1ULG'llJ
"n1'l.:;1..illi'l!Wlfl1UUOnN00i
3
4
5
G
7
8
9
10
11
·12
13
14
15
16'
:Vl1>lL~£l0
-
:O'l'i'f1~1J"i~h11U"h-tW
:f1WtJLl1U'ioJJ1UlOilJ
:n11~1vu5t:Wlfl1UUOnNfiVI
1<1111''u'!omi'I
Mnximox
'
lJ1n
-
lJ1f1
8-8-0
8-5-2
8-3-5
D1NG1'~ilu1<:d]u~~t~-'.IT?JtJ1.~~'Iv v
.., = ~ "~""" .~,.,""
9Hllft'U~"1 b"J'Lfl8f)''JTI Maxirr:in
lfl1"'11Jffif:h1f'l1S'
thumrn
8
6
3
"1111''u'iomi'I
'(:1'l1wffinvn1'l
thunR1J
5-5-0
5-4-1
5-2-3
5
4
2
~ ~ • ~.f H tllJ ,Iii) 100%
% , :DS.t:&.t~~~ EE,..& ..
.. D E F
iiuu '
-4 I
-2 ~
1 '
i i
-
Hf-lG'lOUUV!U
'Uuu ~.J"1V!
1-i-41~5 5
1-f-21~3 3
1-1-0 5
3
'
5V ~/V (; ,;,, !!.rd.,,., = '"°' ' A .d ..'., t ({.J. "" !i ,,,;"f I ..,
lllN•l'1ftt;l<J d'.hm~1ll url11lvmfilun'11~ l'1!J:w?111&: b'l'eJ:wwi··•'M'i:mc::i'l'1ln'1'l!Hl: •ilnli'l'Yll'l m:v v '
~~~10'J~U1'<J~~on::::cr Laplace
5V 5V<v_~ ' r.;, ~-,,,l,,,. "51 A Y ~~j rt ~,"" O! ..0. ''~''°'
fil IN<ili,<iUl.'lbUUN'YIV<'ilnm•nnl"ji''11b'1 tJ :nm<llu'Ulflffi'·]!]"l~"1(i)'1U V•ch'1flfl11l Minimax Regret'IJ <u 'IJ ~ti ~

1urn-:imrn~<ii'lli1uhe:mrw<n;rui 1.ifmrno/inwii'1i1uh~-:i 5 mrno/i 'ii"l1';lrni 1iie:invn-:i1iiein~
mrno/im~"ll';lrn1<if<ili1uh1vii'leiunuvi~ei~1nuvimtJ'l mrno/i n"l:::1iieinV11-:i1iieirnXu1um,,<if<ili1uh li!-:i
<ii1mh-:i mrnum'll'nYJ[J ~lfl11 nl';i'ii"ll';it:mnrno/im';l<ii'lli1uh~-:i 5 mrno/i f11iJ1~1il-:id
<
1. lflffi'YI Maximax
<
2. mrum Maximin
<
3. mru'YI Laplace
4. ~nni.o/i Hurwicz
<
5. mru'YI Minimax Regret
'OJ1mnrno/im,,<if'lli1uh~-:i 5 mrno/i m1<ii'lli1uhf111-:ih-:i-:i1u1vii-i•:::i'.lmrno/im,,<if'lli1uh~ml'leiu
nuvi~ei~1nuei~ 3 mrno/i fi1um,,<if'1i1uhonmtJh-:i-:i1u1~lcJ m1:::m·Fii'1-:iu'l~mtJuein~li'll•dkwtJ-:ibb.i
mrn'l11~m lil-:i~u~u'lm,,u%'Yl'll<nYftJ ~1n'il '11';l<ii'lli1uhi111-:ih-:i-:i1u1vii-i
' • • l
, e! ' <::l <::l 4 ""
Ul!U1fll'lrl!l11111'1fltlfl 3 ll!li.lfl 3.1 !'lMfl 3.1.2

m-i<ii'1i1uh~1oiffl11J-Ju1'l~ll'lu Ll'ium-i<ii'1i1uhmtJLtill'm1m-mi~iim1:m~rn bl'lum'l<ii'1i1u
h~~<ii'1i1uh1~mm"l1111111lJ••u:S'111l'm1m"lrn"l'1'l~•n'1~u ••llimm111~'l~fl1'1fl~bu111'i€lri1111u'1m1;;;
u1'il~bl'lu (probability) lubblirin111"lm1rn'1WI l'1tJm4ftJ.jfil~fllu€l&i.i U"i~l'lJm-icl 111'iil'll1n"l'l11rn!j!1rn
'llm~1'hm-i41'1i1uh m~
m1;;;u1'l~bl'iu~'l~ln'1l'm1m1clli1~'1 bbti~min1Wi'Ll'iu 2 ti-i:;b.rwi $il
1. 1'111llU1'1~!il<uri't~m0ilNi'lttii'm11wofi (Objective Probability) 1Wim'l1n111'1rim111mtJ'l fl~~
111'iil'l1nm1liu~n.jfil;;;"l u<J~lil <ii18tJ1~•'liu lJ'l,;f1111il~ ..:i1n<il 1'1llm"l'l'11iu~n.jfil;;;rim"i'll1mfl4il~• •
flilllW1blilil.fluil~lil~N1u;;;1 $!~{(
tnmt11fll'i'IJ11Jvim;il0u ii1u1u1;il0u~'IJ11J1.li' 'il<:>lo '11'1111Jlllll~I 1ll'lll~lll111ll1J 1J
.;
(lfl'i!l~) tnmruviHi
30 3 3 + 50 =0.06
40 15 15 + 50 =0.30
50 20 20 + 50 =0.40
60 10 10 + 50 =0.20
70 2 2 + 50 =0.04
·nll 50 1.00
' "' < ~~mlll'Ul'IJ:;bu'U'llil~bWi)nl"lrn (Probability of an Event) 1111 bl'l'lln
P(E) = n(E)
n(S)
E bbYl'U b'Vl</lnl"lrn"'1'1
P(E) ll'Yl'U mlllU1'l:;bl'i'U'llil~b'Vl</lm"im1<11'1
0 "-' f{ I ({
n(E) bbYl'U 'll'U1'UNflfl'l'ill'llil~bblilfl:!b'Vl</lnl"ltl,i,
" "-' f{~ f{
n(S) bbYl'U 'll'U1'UN"fl'l'i1JYl~'Vlll<il'llml'Vl"lnl"irn
l<iltJ~ P(E) i'J.-i11~'Vl11~ 0 fi~ 1

2. A1rnu1'11:::1ilu~1~m;:nom111m1:::ttuu1!u:i.1 (Subjective Probability) fi<J m1:i.1'1-i1<J::
dJu~ii'l..'lri1viul'l~um<J1.i1~l'.lvii'nmru.ri bb"1ri1viul'l~umTl'!tJ1-ii1h::rium7rnm1:w'61mllJV17<J'l"l17ru-
llJ1ll'JJ<J"~'l'hm7<ii'l'li'iuh ,Ji~dbd<J~:w1"'l1n'1.a-Jl'.l1fol;lfl1uei~<11~~1umvi7mlium7<ii'l'li'iuh1ub'i<J~1vil-1'1 ~
ii~'ll-1hfw'Yi1m7<ii'l'li'iuhmri<Ju <ii1ei~1~ millu1*nl'jT'JJYitJ ~hn<11 'i.aJl'.l-iieil;J!11uei~<m~mnum7'1JmtJ
ri1!'~m.,~ii<11 ~~'i."1L7tJmh::'JJ:w;iru:;~u1VJ17'1Jei"u~m-J1:wYi"'l1">rui1'1tJml"itJ-iiei$1'!biu"11"'1 -iim.m~' . .
bbGi11::;iuu1mbr1uei1u~1h::'lj:w bbii'1-J1:wnuri1viul'!m1:wu1"'l:;dJu'JJmm1:i.1<ifo"m"''JJmjn.1111"'l:;l'.lm1:w
ii1"'l:;blJu1i~d m1:w<ifo~m-,i'lu.11'1Jei~jn.1111~ 1hunm~ bb11:;ueitJ ii1eimr1bn<11~u.'11tJ.11m1:wu1<J::blJu
Lvi1nu 0.2 o.5 bb<1:: o.3 <111:w<111iu 1i'~bbrl<11~'1.1'lu<1117l"~ 3.11
Ul'llilllN
8 5 -4
6 4 -2
3 2 1
0.2 0.5 0.3
m"><ii'<11i'luhrntJ1<irrim1m"Jrn~iim1:w•~tJ~ uein"'l1n"'l::1-ii'11m1:wii1"'l::Lllm•ii'1 ~<iil'li'luh,::
1-if'11rnl'IV11~ (expected value) 'Yi1m~"'ll'illl<iil'li'iuh
'11rnl'IV11'"'1Jm<ii1mJ"lri:w"'l:;bllu.11m~tJ'JJ<J~<ii1mhii:i.1~riuhiitrno11 fi1u1ru'1..1,1nrfil"J' • v
E(x)
xl
E(x) = Lx P(x)
l l
fiei '11rnl'!vi'l~'JJei~<ii'1bb'.lhrj:i.1 x
fiei <ii1mhrj:w<11iui1 i = 1, 2, 3,..,m
P(x) fiei '11m1:wii1"'l::bUU'JJ<J~<ii1m.hrj:w x} F l r
"'l1n<ii1mh~ u~L<Jfi 'i1nl'I ~1rn<ivi1'~"51mu•m<J~'1<J:i.1Yi1•<11<J~"'l::'JJ1tJ1#!1ubb"111:;,@i<Ju
E(x) = L x. P(x)
l l
= (30)(0.06)+ (40)(0.30)+ (50)(0.40)+ (60)(0.20)+ (70)(0.04)
= 48.6

ii:ufiei<ii1.i1"1vi1'-J'll<Mm1'll1UL1>14ei~1>1eii.J~1•111ei!J~,~'l11u1~mL<iifl~L~ilm'Yi1nu 48.6 L1>14ei,<iiei
L~eiuvi"lei:l'.1<ii11h~mru 49 L1'14ei,<iim~eiu
m14i'"1ft'UhmuL<if'1m1m1lli~:l'.1m1:i.1L~m~u~<Jil'lftUl"1;1.jfLLU1~"1L4e.N'llil,'11m"1V11' m1.jf
LUULlmvrn1um7n1viu"1mruo/im1<ii"11'iuh1<iiei~ 2 mruo/i~'11'1)J fiei
1. mruo/i.111>11"1V11'-l~lUWii1u1m'.lu (Expected Monetary Value; EMV)
2. mruo/i<ii1li1deim'1 (Expected Opportunity Loss; EOL)
1. mw'llri1A1Att'i'~~1if'wiirn1m~u
'111>11"1vi1',~lumhu1uL'.lu fiei Nfl"11lULl'YIU~1'11"111'~1<ii~ut'"m,;i~u1u-,~u~m1,1flm14i'"11'iu
1,Li1ilfl'Yll'Li1eifl~Lvim~'1i.J~'l'" ll'Iu~~<Ji"1l'lu1,,1fl mruo/i'11'11"1vi1'~LUmhu1m".lu~,1-,ru1Nfl
"1ilULLVIU'llil'LL<iifl~'Yll'Li9ilflfi1u1ru11il,lflNflU1fl'llmNfl;;Jlli•~Vl11-11'11ll-JU1,~LDU1Ubb<ii'1~'1rn1m-,lli
nuN<ll'iV'lii'lULL<iifl~'Yll,Li1ilfl lllLDumrn~'Yilm74f"1ftUh l~U'lflUdll'J14f rllh m-,~,111.blLi1ilfl'Yll'Li1ilfl
~&i~iJ'"'~~-1.,ru1,1fl'11m"11111'~~'l'" LL<iin11umrn~'Yi1m1~'1"ru14i'"11'luhL~mnu'111.jf~1tJ <ii'ii'YJii
r > ' I !
"" ""' ,,,.,,,,,,,. "" ' <V ,,,. ~ ,,,.
n11V'l,17llil'Yll,b<lilfl'Yl"1'Yl'J"1'~V'l-17llil'lfll'111'11"1Vl1-l'Yl"11'Ylij"1
EMV(i) =L (CJ P i =1, 2, 3 ..m,, l
j = 1, 2, 3...n
= .111'11"1V11'~~LDu4f1L'.JUEMV
c'l
""' ""- = .:
= Nfl"1ilUlL'YIU'llil'Vll'Lfliln i L:i.1mn"1'1m1m1ru j
=m1:i.iib,~•uu~'~Li'i"1'1rn1m1lli i ll'lu~ 2: P =1
l
p
l
m
n = ~1U1U'1i111n17rn
'lflollill;Jfl'llei'u11ffiijhYil'J ~lnl'I oilil:l;JflNfl"1ilULLVIU'llil''Yll~ Li1ein<iil~'l lL'1~'11m1ml1-~LDU
'll8-J'1m1m-,lli<ii1''1 i.t1mfi1u1ru<ii11>11"11111'~LUW·ilu1uL'.lu1Ji'1"L'1"1-J11'1.1.1111171'~ 3.12
(VIU11J : fllU!Jl'YI)
i;l11~1.l (EMV)
8 5 -4 2.9
6 4 -2 2.6
3 2 1 1.9
0.2 0.5 0.3

= (02) (ill + (05) (5) + (0 3) (-4)
= (02) (6) + (05) (4) + (0 3) (-2)
= 2.6 i.11u11 l'YI
EMV (m<J~1~u"imn'io%OnNli111) = (o 2) (3) + (o 5) (2) + (o 3) (1)
= 1.9 i.11U1Jl'YI
rr-r:Ff'ii>1f4114l;J~~bD2fl'Yll·J~Ran~1<f~1($181.J~~Yl~-~~j.J~qi41fl n1JN~1,J1'i.J.J·n.~~iv3-J ~..:iilf11 !fl1<?lVl1'<1~
•lluii11'.Ju (EMV) ij'16j·111"'8 2.9 •'frµ;:.m1
.~!!11.H.HllJIUllA!.fi' 11;I
@ uf!JJ ufft'!I lJ.IJJJO< LL!'.[•n ~yuuu lfl~u<Ou '.lu<Ji> ]iu1oi1< 25~'1 Ado!l.e PDF
IC) • .
Arial • 10 • B I JI ::: ::: =iill m•iill % , +.o .oo
.oo + .o
M22 .,. fx E9 = Max (E5 :-~·-7,_)--,-
A B C D E
ivni11nn 1'1111Jli!LlJn1'i;
I w1n 1.humnJ UflLI EMV
!n1'irl~1Jl'iJJ1U'lviiJ 8 5 -4 2.9
i n1'itJLI1Lil'iJJ1Ul~IJ 6 4 -2 2.6
:m'i~1Ju~!:Wimuuum1&m 3 2 1 1.9
.~V111Wti1~~lliu
;
0.30.2 0.5
iFi1A1r11vi1'>lYIlilum1Lluu"iif~ 2.9;
= SUMPRODUCT (B 5 : D5 . $B $8 : D$8)
v ~.Iv <
(m:ii8f1iJl'l'llut::1•2£l11 E6: E7)
E9 = Max (E5 : E7)
f

''41 4 ~
2. 11111.!'llfll!lW 0flll!'l'lfll"1'111~
fllbftd1Jmi'l .liJ fllbftJVlltJ~bii<i1'1lnfHS~rii<ilfluhN<il l<iltJ1~1~bi'11ln'Yll~bi'1iln~~~i'l<ilftlVIW
bw11m&ru~bii<il~U'1'l~ 111"iilfl1•fltJTeim "i>ill'i1•fltJVlltJ~bii<il'11nm'~tirii<ilfluL'1bftiln'Yll.J•iiilnN<il.bb11u~'1~• •
bi'1ilnl11.Jbftiln~~~'l<il mrn<li.-i1bfttJfom<!~!ill<ilVl1.Jd~rii<ilfluh'1~Yl'11&rn1.-i1bfltJLilm'1~!ill<il')j1~~n.-11
~l~i'l<il ~.J.-11LfltJle:Jmi'!~!'l1<ilV11'.Jfi1u1rn1~'11rn.mbbo/1n<i11.J'S~11111~N1ri11-sV1"iilNt'i<il8Ubb11u~~~fi<ilnu• •
N1'ilil!Jbmu11~~'.iu'1"i.JLUbb"111~'1m1m.,tii1<iltJnnwl'l'lri1'il1"1.JfllbfttJLilmi'!~mn (opportunityloss table)
bbfl1tlri1m.,rif"11uh'1~fim1rnfi1LntJL1lmi'!~!'l1<il1111.Jm~1riil'li1uh bfte:Jnm~biiiln~n.11 EOL ~~1~<!"1• •
EOL (i) =.2: (CJ P i =1, 2, 3 ..m
c,,
p
J
m
• <
n = 'J1U1'Ui'!1)11nl'S1b
,, J
j = 1, 2, 3...n
'11nim-,i~~ 3.10 b~e:Ji1m'l'61.um'Sfim1rn1111 '11bl!ltJfom<!'lJEJ.JU"ii:irl<jl'IJYJtJ "11n0
l'll~H~ 3.13 Alllf!Jt0nll!'ll0~U~iJ!1lj't'llTI!J <filOPI
('11U11'1 : i:i'1UUTYI)
0 0 5
2 1 3
5 3 0
0.2 0.5 0.3
'lln'ill'Sl~~ 3.13 mm·rnfi1u1rnfi1 EOL 1~Ji~d
EOL (m'Si'!fJ.Jh.J.Jl'UlVI~) = (0.2) (0) + (0 5) (0) + (0 3) (5)
~
= 1.5 !11'U1Jl'YI
= (0 2) (2) + (0 5) (1) + (0.3) (3)
= 1.8 A1um11
= (0 2) (5) + (0 5) (3) + (0.3) (0)
~
= 2.5 mUUl'l'l
1.5*
1.8
2.5

!'.! U!jlJ u.ii1'1l !,[IJIJiN Ul!l'ifl ~!l11uu lfl'llD•i'.lu ii.DJJ'l l![ihi;h• 11i~'it AdoQ.e PDF
D ~ lilll r8l ~ ~ [9. ~ I .lb ~ • T <f;t' I<') T I A •. " ' L T ~+ i+ ~
Arial T 10 T B I !! I - == ~ ~ % +.o .oo +:= •=-= EE °""' <> > .oo +.o . =;-- ,,
1
2
3
4
5
6
7
8
9
10
M22 fi, E9 = Min (E5 : E7)
A B c
1vi101ii<Jn lm1wGiovff11
I w1n 1humn0
;n1'i"<l'~1J'i'iJ.J1U'l~JJ 0
ln1'i'tJU1U'i1JJ1UL~W 2
•m1'i10u~!:WJ m uuunwi'ivi 5
-i~11wti1~~1Uu 0.2
1Fi1m0l>1'i•'llu•Fi;1i!ru1Dm ilVlohi'f"'
= SUMPRODUCT (B 5 1 D5, $B $8 : $D $8)
(ffr<~Bf:~'>l':il'iJ~-:J~/iii.'1~ E6 : E7}
= Min (E5 1 E7)
0
1
3
0.5
D E
I I
.
U<JU EOL
5 1.5
3 1.8
0 2.5
0.3
1,5

d f{ , • ..... '""' 61 ,,,,' -=I
'!JiJ1fT>f'iUY:f1'Ylf!lJj'lfi: (Expec:ed Value of Perfect bformat10n; EVPI) m1v1V"11'ilb'<:;i.J~J-J'1l'l-
.d I{ V I l <> 61 iU V 61 .,;:Jl o ' { V
f'ilb11GW!f;;J1J'JIH1':1fl!1l'Wc::N,tl11 bVlnl'Jlillilll'ib'J>'i'llU nl'll'll'i1'11m11 EVPI Vil Lv1,1n<1:1-Jfri'l
•
. ~ '1d"' ,,,; 6'
EVPI = rnmlili~1~·r.111m 'l',J.Jil:W'11%1i,V11Yl<liJj"irl+ (Expected Profit with Perfect
Infnmation; EPP! ) - 1)1 EMV ~1')1il
"'"" .,J • v o 1 J ..l ..J r> ",J,.,,,,J E
1•1m:fff':llilV11~'ni11rn b"ik14il:l-J<!l'li1'mr:f11'1'1:Nj'lill = (vm'~iJUL6'Yl'il'YliiWifi''~EJ1i'!m1rn1rn
,,,: &. > <:0• !i'd .s ,J#.,l f{ V I
'YIVIH1)('1Ti;}j'il 'J tbUH'lli'J~f{J111nT'.illil1~1'J,.J) +'''(Nflil€11JLL1'1U'YI 'ii'Yl:')i1'l1El11rn~ nnrnq11:1'11ti) (l'llllJ'ill
fl f{ v
'Jt bum:i'l1<111i-:nTrnl,ij'i:'YI1ti)
~1n'.jn"fl.J~~ 3.12
. .' ~ •1AI# d f{
rnrnv:m1'llEJ1m "ib:WEJ:W<!l'lH'ilb'Ylf1~•&1}LiJ'lm = ( 8) ( 0,2) + ( 5) ( 0 5) + ( 1) ( 0,3 )
= 4A li'l'ilUl'YI
= 4.4 - 2.9
: Arial
M22
10 T s I !I == m:~
fx E12 = E10 - Max (E5: E7)
1
21
3
4 I
A
V11Jl~Qf)
5
6
Im111i1o'i1001u1rnj
Imrnmu'i1001mGiw
7
8
9
10
11
12
13 I
If11'i~1JU~t:WJfl1tJUOfll-lfiOl
:vr11wU:1~~LLiu
I
i.NRVlDUU VIUloolifN1~ro.l!Vl71'
imiu1ru"'lum1efimiiu1~
1EVPI
1J1fl
8 C D
m1wd00m;:
U1Ufli'l1J Unu
8 5
6 4
3 2
0.2 0.5
8 5
.
. "' L • ~i H ID ·~
% 1 :o& .;~8 t~ t~ [
E F
EMV
-4 2.9
-2 2.6
1 1.9
0.3
1 4.4
1.5

L'lfl'lt; E5 = SUMPRODUCT (B5 • D5, $B $8 • $D$8)
(~lill'lilnj<il"ilUti"L'lf<lt; E6 • E7)
B10 = Max (B5 • B7) (~lil'1ilnj<m1uii"L'lfl'lt; ClO • DlO)
E10 = SUMPRODUCT (BlO • DlO, BS • DB)
(~lil"1illlj<il"iL'llii"L'lf<lt; E6 • E7)
E12 = ElO - Max (E5 • E7)
-onnmJJ" Excel L'li<A.1 E12 'lVi'11ff1lilWl,'lliN'111'1uL'Yl<i1~'1i.Jj'lrn ~il 1.5 li'1ulJ1'Yl
'1filif,ll1"1 "lllln1Ti1"ll'lffilfi1 EVPI 'OJtl'.ifi1LrilnlJ '11 EOL
' . . .q <:9: ' ..,, ,,,,, ~ .t::l
lll!!U1flUftfl111'11U11'J'l'l 3 110U'l'l 3.1 !10~'1'1 3.1.3
bb'11U,rn'l<ii'lil11uh (decision tree) d'Ju<ii'1LLlJlJiltll'1'1d,'llil"rn1<ii'(i)!1uh~'loiinlJrn1<ii'(i)!1uh
mi'!'l<11'1m1rn1rn~l1"11i.J1~m 11<111~il mi.J1111ri11'1u(i)m1i.Ju1'iin1Ju~"t•n(i)11'1~rn1rn<ii1''11iii L(i)tJ~
LL'llu,rn1<ii'(i)!1uh"l::ll'1(i),'1fm,J<A'ii1''1 1'llu 'Yll-J1fiiln 11'1~rn1rn N<A"1ilrn1'Ylu'luiu'llmllN'Ull11'1~-:n-ihh~1EJ
i.Jil,m1'1'llil'il'!Jm1iii'iililb'ilu ~-Jw11u,m1<ii'lil!1uh uiln"l1n"lt'loiinlJiJqJm~1fil,rn1m1<ii'1i1!1u'l'il<i1~'Liim
1Lli'1 i:i-Jmm111u11iJ'l·foniJqJm~11mm1m1<ii'lillluh1'1mi'l'l <i1~-J"11i.Jli1<1i'lJ1iiillniil1i'l
,YolJ11~'611'Ju~0m1lf~N!!'IJU~m1wlilifut'il 1.htnellJiif1EJ
1. 'Yl1,Lfiiln'llmm1<ii'lillluh
2. l1'1~m1rn<ii1''l ~d'Ju1u1iii~"lt Lnlil~ULUil'Ul'1"1 ~'"tl'.imli.Jifi.Jv'iunnlJ'Yll'.11ilmlllil'l:O'Yll"fiilll
3. Nl'lfl1'1TI~'ltliif'l111m111ililm~m1rn1Lllil'ltl1'1~m1rn
4. '111i.Ju1'ltLlJmm,m1Lnlil11'1~m1rn"lli"t•1'1~m1rn
~,'1fm;Jl'l<ii''~ii11'111,1fu iin"l::1iiim'l1nrn1<i1W1<iltm'llm~lJ'lm~'l::LlJu~<ii'lilAuh
v v '"'1"' v v ~ •
lft)!llfl11W'l'l 'IHUfll'llf'lN!!'IJU~fll'll'llillfU l'il
'lum1'1.J'i,Ll'llu-Jm1<11lilAuh'llil-JilqJm<ii1''l "::'l<ifif'!!flni.:;rn~1lJUi.J1.i1l1u11'1i1ilunu<ii-Jd

D
2. ii'11Jil'mn:U: O ••YJu 'J<i1'll1l-Jm-;;•n<11•11111Jm"ci<ii1''1
3. C!11J<in1'tcl b'i°tJffll bb'llU" (branch) bbYJU 1'11"bli!lrn!l"m"<if<i1nuh
(i'imflu111'i'm•'ll~~1l1lmn•nn 0) 111'i'm11111Jm-;;cl!ii1"'1 ~bn<i1~u (i'imftu111'i'mb'llU-1~!leJn:m
"J1nO)
m~£Yf1~11'U'IHfll':ifl'Yiiiu1ll~ilii1fl'1J-frmrnu
1. "l~b~:1-1ci~1""J1n4!1wihti1u<il1u'l1111<11m~:1-1~ D ~""lrne!<i1"1'1l"'~eJniii"111:1-11i1~d'Ju1111411<i1tJ 1
' '
fi" 1111€1 1 bb'llU" 111mtili" 11'1l"blieJn li'1i'.l n 'Ymb~eJn (A , A2
, ... , A ) "l~i'.l n 1'11-lb~eJn Wi-iml'I~ 3.41 n
A
n
2. 'j<i1U'1ltJ'llNbb'ii01~1-Jb~eJn "OJ~i'.l'j<i1'lleJ"b11111Jn1':lcl ~,11fii'11Ji'n1'fci 0 bW•]nl'lcl~d'Ju1111<ii'
"lrnYJu1<i1rnffu - 1<11m•'ii<i~Lffu111'i'mb'llU-1bbYJU 1 bWijm'lci b'llu m'l<if<i1!1uhi'.l 21'1l"b~eJn<L<i~w1i01~m"
,d. "" ft ..... ""
L'1€Jn :l-1 2 L11111Jnl'lnl (E1
, E) <i1"lll'i'l1'1 3.5
E,
E,

3. bb'iil1:;<J111Ul11LI'llmbb<iil1:;bWilnl"ll11 <):;1'1ri1m1:wu1<J:;bUU (P) 'llmbb'iil1:;b'/1111nlcm1 bbl1:;Nl1
~ ' q J ~
111eJ1Jbb'YlU (CJ l]'<if,m'l"lfi 3.6,,
(P
1)C
11
(P
2)C
12
(P
1
) C
21
mi<r-i'Hu'llmm>ii'oiiru1'll 1111Livf11umm">rn•u'1-'leieimuu 2 1]-,:;kf1'Y1 $iei
1. 11'Ill!~fln iioiiru1'llihttfoifi;ymm1ii'oiirul'll-0"ull01.11A!J1
'VimLI~' bb'llU,m"'i<if111nuh~bbl1i11'~'1l1JJ'Vllm"><if1111'uh ~i'.lm-,<if1111'uh ,y;m'1~'b&im <if1m.h~
«iiu DIJJ'Vllilm"~ein 3 m~•~ein $iei A1
, A2
, A3
cbl1:;ilrvi<1Jm">m~buu1u1"1'llei,bb'iil1:;''ll"~ein1"1 2
6&> <>! ' 6PJ. "'"
b'Vl'Jnl"'iW.'1eJ E 1' E 2bbl1:;:WNl1111eJlJbb'Y1U'llmWill1:;b'Vl<IJnl"l'W.'1eJ ell, c,,. c,,. c,,. c31' c32 nl"l'b'llLIU
bb'llU~m-,<i11111'uh bb1111,1-'l<ii~m'l"I~ 3.7
c,,
c,2
c,,
c22
c,,

2. !!'IJ'U~flll WYlif'll1lli:httfoflll i1JYIifu1ll'l•!l,ll!J,y'IJ'16'U
'Yli-m.1fi, bb'llUJm'l<if<111uh~LLi1<11'n'ilbJ!m~l:Jm7<if<111uh'YlmEJ<i~' Liu m'l<if<11lluh<i~'""n
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L~l-ll~:l,.llfi<il~'U 'Yli1"1nm'lLfi11L'Ylllf1l'lul E l:JvrnLflim5n 2 'YrnLfliln~il A LLIO~ A 111EJ~'YrnLli1Jn A
-=1 ({ .J"" J{ <"I ~ ,,} , S'a 3 4 3
,~m'YlliJf117rn mn1i'llu~1l E3
llli~ E4
LL10~;;.ie-.110111lU.L'Yl'ULLlll1~L'YlliJm"rn<i1l c12
, c21, c22
, C33
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c
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czz
m;'lm;1:::llH'IJ'll~ml"1>ilru1'il l:J~u<1mulif,f{
1. m71'i1mrw1h'iiw1 l'ULL'll'U,m7<if1illuh 'il~<ifil,~:J.J'illn'll'lllj11LU'YIWITTEJfj11 ~11L~:J.J<ifu'llmmY<
'Yl~~1lf1l'l<if<111l'Uh
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n1h "~.11mru'11<i11i'Y1:J.J11J'l11l,n1h
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- '11Llld1lmi1'il~l'i1mru'Yl1fi1Li1EJ11lmi1~'111l'Yl:i.m11<if
3. Ul<il'111l'Yl:J.Jl1J'llil,LL'iil1~'Yll'Lfiiln:J.JlLmEJlJLfl1JlJn'UL~il'il~<if1lft'Ul'iJLfliln'Yll'Lfiiln~~~i1<11111ti~
'
t'f1Ll'.iu'11.-i111Vl1''lJmn1h '~LflDnvrnLliDn~'lVi'11m1l'Vl:J.JlEJ'llD'n111~l:Jfi1i1'i111
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- t'f1Ll'.iu'11.-i1<11V11,'lJei,<ifu'Ylu'Yl~eifi1'loii'~1EJ 'il~Lfiilnm'Lfiein~'lVi'11m1l'Yl:J.Jlrn1m<ifu'Ylu
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V1'11l'11'loii'~1EJ~l:J'11~llj11
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'il1n<ihmJ1,1&i:w'lJei,1J'llffiijhYil'I 'ii1n'il ~1J'lV11"'11ei,1J'llffimm·:m1-ii~h111Jm1'lJu,nwif'ilnuh
Vi"Jl'lTUTYm11einiif, 31'11-:Jli11ln141"1,m'I"<~ 3.9
P
1
(0.2) •8 G11U1Jl1'1
p2 (0.5)
•5 G11U1Jl1'1
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p3 (0.3)
•
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•
m"i'lJmtJhmu P
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•
'ii1n'il <>J1m1'lJu~nwii'ilnuhil1u141"1,d
1. Yrn11ein1um">il-11'L.,,,1u1vii.J
= 8(0.2)+5(0 5)+(-4)(0 3)
•= 2.9 mUlJlYI
2. Yll'11einlUnl'l"'lJI'lll'I1-,,,lUL&ii.J
= 6(0.2)+4(0 5)+(-2)(0 3)
= 2.6 ii'1u1J11'1
3. Yrn1iiem1um-,%1,1J'llffiml'lu!lnN1i'il
= 3(02)+2(0 5)+1(0 3)
•= 1.9 mUlJ11'1
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m'l"4ilill1uh 'luf!rn1m'Jci~Slm'lll'li,oifu ~'~ll'li,oiful'.li;miJ'ldtJ'llu~-ii'lilllt'f,nu d'Ju1iniin~
111d,'llmm'l"4i'lill1uh'lurnru:i1'.~u1Vi1'lmm'lt1UlVll1"Dlrn-1 (Game Theory) m'li1tJ'lum1~1i1l1uh1.i'lm.1~
m1~lill1uh 'luiin1"rn~dtru1V1111l-lmm111~•~rn~Yi11'11l1.11'lllulim-~ ll'ii'1~~u1l~num'l4ilill'luh-nm• •
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m~n'ln• lvi1l'lVfN1tJ<i1Ul1l'1'1iJ'ldtJ'ilul-11n~fflil lllu<ifu• •
V1l1"Dlnl-llml~l-l~u•flu<ii~'ll'l"n'lu 'ii.rt 194411i1rn:hn1•1nm'l~;ei '11lViu yjeiu U'llll-IU (John
Von Neumann) llftrnilffm1 l-11l1llnum~Hu (Oscar Morganstern) 1liltJll1Jl11V1l1"Dlnl-l'1~1'11m1
V1•1'lrn1m'J4i'1i1'.'luhrntJ'l<ilffrn1m'lci~Slm'lll'li,oifu ~Sli;m1.i'NmiirnlluNft'ilillJllVl'U 1liltJ~~4ilil'.'iuh 2
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"" "' "' v "' " .1 T < A .'I "'NltJVl'U~ 111mtJrn nl'Jll'V'IVl'l1lnl'Jl'1tJNftu'JntJ'llWJJ1l,1lnN1tJVl'U'
'lum'Jll'li~oii'u<ii~~1111i~'l m••~Sle:ill'ii~oifw~1mw.J1nlil ~ei 'ii1u-miiu1<11 (finite) 11111lm'il•~Sltr• •
ll'li,oifumn-;,iuiiu1lJ{l1ul11<11 (infinite) {11Sl~ll'li~oif'Ul-11nl11'11'lVfi:lqJ1111oifu.if1lumn~uu1ln'11nd~ll'li~oifu
w1iij:;N1tJ!J,SJnii~VIIT (strategy) ~'l4r'lum'Jll'li~oifu'ii1u1u111d,~~m'ilulJ'iiT1-nu1<111111!luu1l-l1'1 llft~
'ii1u1un!jtJVITI<nil~ll'iiii~N1u1l-l'ii1lllu<ifil,•1'i1nu 'lum'l"Ll'li~oifue:ill'li,oifuwii!j:;N1tJ'1:;lliiln'l.ifnfttJVITillil• • •
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.,,i,,J Ci 2?1v~F- ..,. ~ :I fl'V .d l ,:;,,1, e;; v.d. ""' l Ci dv
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game)
i'rJl:IJ'tt:IJ1ti'lHM!fl:IJ
ml-I liiil ftmum'lcim~m'l"ll'li~oifu~Slrill'li,oifuJi~ll'ii 2 NltJ~u1iJ l1i1tJ~ll'iift~v11ul-i,1111~~-~1<11• •
~uoiftJ'll~l1'1U1l1inv11tJ ~~'luftmum1ci-nmm'lll'li~oifuml-l1'Jt1UlVlbl1"5ml-lm'l1i'lilu4i1lllJlJ'li1tJ'lum'J
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'Yi1m'J4i'lill'luhlli1lnrn1~V1IT~•:;u1m'l4r'lum'l"ll'li~oiiulliltJl'.l"llil*VlmtJlvi1l1<1i'~u'iiu'll~l1'1ii1l1inv11tJ
v
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ll'li~-ii'ul1'1m 2 NltJ'1:;l'5tJffll m:1-1'l:;Vl'Jl~ 2 NltJ (two-person game) ll'ii{i1m:1-1'l1i1~Slrill'li~-ii''Ul-11nn11 2•
• O .Ji ~.I A , , .'I ) v l'I ,JA ' .'I .Ji ~.I
v;ltJ'n'illu'1:;rn'Jml Lnl-l'l~Vl'll~VlmtJwltJ (n-person game blllu'illnl-IV1l-lmnn11 2 "ltJ'n'illu m1
" "° ' v v V JI
ri1mrn-;,i:;l-Jm1:1-1~~mnffijlJ'lJlJ'il!lumrnru

2. rn·m'li~'1iu1mrn.;,,,;1~'1 ~LL'li~'1iuw1iii:;N1l'.J"J:;il'Yll~L~iln1'i7il11irn'l""11~'1 flu1urn'l"l'ilLi'.lu
fl1'l"LL'li~'1iu1'i7il~L71'.Jnfiu-11 nmJ'VITI (strategy)
3. ~LL'li~'1iuLL"1ii:;N1l'.J"J:O';';m m:i.;.-i1II .-i:;mii1~,,,,xlii~nii~'VITI~!JnNll'.J'il:;i.i1m1-iiL~il~'ii:;i.i1
'1im.Jiiil~nii11mo/i'lliiuhL~ilnniiu'Vln~llULm~11'11Ll'Juniil'.J'YITI~L1'im:olifl-J~·M1 L~!lL~.'i'~1J'ii'l'.J'llu:;'i11n'I.I q ~ q
fll'l"bL'li~'1f'U
4. ml'.J1<11l'fm1rn'l"m~ilrn'l"LL'li,'1iu ,:;iJrnm1Lmo/11LL1J1J'V11,.-irullflllifll~m·1bu11i'lu~LL1J1J'IJm
lll'l"l'Niillil1JLL'Yl'U (payoff table) L~il1-iid'lu'1im;1ii1urn'l"o/i'lliiuhl'Ufl1'l"L~ilnnii~'YITI 1lll'.JY'itJll'.Jl:J-J~'il:;
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l iJ!!1J1J'IJfl'1!11:t-J
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• • iu ' ....
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1.2 m:i.;~ilzjLL'li~'1iu1'iml'.JNll'.J (n-person game) ~fl mnurn'iru'IJfl,fll'l"LLm'1iu~ilzjLL'li'
'1iumnn11 2 Nll'.J~'U11.1~L'1i1mvhm'l"LL'li''1iu
• ..... r: ' .....
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2.1 Ln:J-1'Yl:J-1Nfl'i'J:J-1'1Jmm'iLL'IJ~'ll'UL'Ylln1J~'Ul'.J (zero-sum game) 1'imrn:1~ lifl:ll'l.bfll'l"ill
'llil,fll'l"LL'li''1iu~Niifi'~lll)IIYf1l'.J'llil'fll'l"LL'li''1iu 'il:oilfi'rn.;m:;~ihlit1J~ilNiiL.'i''IJm Nll'.J'lJU:;'i1l-Jfiu'ii:;il'11
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2.2 m:i.;~ilNii'i1l-l'IJil'fll'l"LL'li,'1iul:J-JL'Vi1fi1J~'ULl (non-zero sum game) 1'il-lll'.Jn' lifmurn'l"rib
'IJ!l,rn'iLL'li''1iu~Niifi'v;llij'1Ylll'.J'IJei~rn"L'li''1iu ,:;i:Jfi'n'l'fm....,yjl11~t1Jflfl Niil.'i''llmN11'.J'llu:;n:i.;fiu"J:oil.J1
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t11'ii1LLUnlll:J-1nii~'Ylll~11i'lurn'iLL'li,olfu mm'imLi.J"n:i.;eifln1~d]u 2 ~tiLL1J1J1,,,t!!'l flei
3.1 m:i.;niil'.J'YlTILLYl (pure strategy game) 1'imuii' flmurn'l"rii'llmrn'l"LL'li''1iu~fiLL'li'-' .v , A ~ ~ <') < A 'f ~ ' v ~ A A ' -1 ' F< ~ ~
'IJ'Ulill~Liifln L'llflii~'YITI LllnmJ'YITI1'i'U' L111'.Jlliiilll L'Ufll'iLL'IJ,'ll'UL'ULn:J-11'i'U''l 'll'LL'ill'1:;..,1ulllWM]';fl L'iJ L'U
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3.2 m:wnl1fJ'YITINl1:W (mixed strategy game) 111mtiii" l1murn"irn'll€l"n1'ilb'li~iiu~tj
'~ ' ~ ~~ ' .:'1 '~ ~ d d - Ji~ '
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..,.. ' .., ' .., & ...! 6l 2.J & I</ .... It
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Nl1l-ILurn"ibb'li~iiuwh,Xu
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IV """"' I I
. <J1n<ill'l1~N'1ilillJLil'IU'llil~ A bb'1:;il1'l1~N'1ilillJLb'YIU'llm B il1b'1'llil1'l1~'Yl:l.1'111J'lnbbftil.:J11
b~1'llil~lil1"1.:idJuf11l'J1"11h::ll'J'llu $11b'1'lllil1"1.:il1fi111rnbftlil~11 b~1'llmlil1"1.:iblluf11mill'Jti~Li'J'llii
fll'J~liltrtt11l!~!J1fl'IJ'lltfl;~t111!!'1l~.Utt
m'l$11iii'lul<J b~mnlJ'Yl~1'llm1bb'li1oiiu lill'Jvf111.1bbfl1<J::Yi1m1YJ<J11ru1vi1'111i1mN1nllt:JJV11nn
bb'li1oiiu 14i'bbn
' ' v .I ~ v -~ ~ <'I v ,.J ' ~ .J ' ~ v A • I ., ' &
1. ~ bb'll~'ll'U>Jll'Jbb1n<J:; b'llnmj'Yl'D blil bb'1::<J:; b'll b'Uilil'l1.,1urn1 b'l'Yl<J:onei bVI Lnilu'l:O bl'J'll'UVl'lil
<v 6"..J,,.,,J • <v
N'1'1'V'l'D'YIil'Ylljii'11'Y1'l1.Jb'll1
2. ~'''li1oiiu5nN11'J<J:o't-iinm.J'YIITllil LL'1:0<J::'t-ififlil':l1S1ub'Yi1h~<J::nei'tVibfililtl"l::ll'J'llu'Y11ei N'1fi'V'ltl~
'""'-=I • <v
il'Ylljlilft1Vl"l1Jb'lll
3. .11'llmm:l.lm'lbb'li1oiiublJUbrllh~1'11'llmbn:!.ii"im-m14fvi1ilN'1bftl'Jlill'Jb>i~l'J<J1nm':m'li~oiiuviml'J'l
ttilflmruorl.ffii'ltt1111flliltrtt111
viiin mmo/i~'loii'lum'l~ili'luh<J::YJ"J11ru1"J1nil1"1~N'1'ilillJ'''Y!Um"lbb'li1oiiu
1. fiLL'li1oiiu~blJUL~1'llil1il1"1~N'1ilillJbb'YIU"J:O'l-ifvifirnnruo/i.11'1~'1lil<J1n.11~111il (Maximin) iliu'I.I '!.I q q
mru'Yl'tum1$11iil'luh
2. ~bb'li1oiiu5nf111'J "J::'t-iivifirnnruo/i.11~1ljlil<J1n.11j~'llil (Minimax) bllumru'Yl1um1$11iil'luh
- Maxim.in ;iiei n1'lbYil'JlJfl1~1ljlilLUbb.!'1::bbn1bliiln'11~1ljlil'llil~YJrnm114fbbfl1 ~~YJ"J11ru1bliein
.-11j1ljil<J1n.11~1ljil'llil~,,<;].,:;,,m bllu'.1finmru'Yl1um1$11iii'luh~*bUUbliein.11j~ljlil"J1n.11~1ljlil m"l
~flbb'li~oiiu~'t-ifmruo/i Maximin YJ"J11ru1~ili'luh "J:OYJ<J11ru1 N'1ilil1JLL'YIU~Uill'J~ftlil'tm,<;i11:;'Yl1'61iein~. '
'lViN'1ilillJbb'Yl'Uft~~ftil1UlJTIil1N"iiillJbb'YIUil tJ1~~1'llmbb'1111:O'Yl11bi'iln
. '
- Minimax ;iiei m1bYil'JlJfl1j1ljlilLUbb.i'1::ftlil:l.lnbrimi'iein.11j1ljil'llil1YJnftlil:!-m"l4f,bf!1 ~'
v:i<J11ru1b!iein.11~1ljlil<J1n.11j1ljil'llil1 ,,<;]11:;'1il:l.ln illuviiirnnru'11nT:;~Iiii'luh~aj~buuiiie:in.;1~1ljil<J1n
fiv•.:iftlil m"l~flbb'li1oiiu5nf11rni'iein'l-if,nruo/i Minimax YJ<J1"lm1~1iil'luhi:iu ,iim,1n.ibi'li1oiiui:iu'i1:;! q 'II '!.I
'V'lm m:l.l'tVibnlil Nmi'll'J~l~ljlil <J1nlJTIil1Nmi'!l'Jj1ljlil'llmbb.i11:;'Yl1.:ibliein
11im1'tl1ri1 Maximin
' ~ ' .... 6"
1) vi1'11illljlilneiu'llil~YJn bbtl1Vl'lilftlil:l.IJ1<J1n<111':l11N'1Iii1llJLb'YIu
2) briilVl1fl1~1"lil'llil-:J'Ylnmnvi1ei11il:l.IJ1"l4i'bbfl1 n<J:;'Y11fl1ft~ftil"illnfl1~1ftlil5nfiviii1~~ q 'll ~ ~
~1eitJ1.:i'tum"lbb'li1oiiw:;::vi11.:iN11'J A bb'1:;N11'J B bbftlil1N'1ilil1Jbb'YIU<J1nm1bb'li.:ioiiumw1-:i14141.:id

w1EJ B w1tJ B
-6 7 8 6 -7 -8
3 -2 5 w1tJ A -3 2 -5
-4 6 -3 4 -6 3
11m-,vnfi1 Maximin
. ' 'v A ' o <
'llU111 1 m-,v11'11.i1ql'lmiu 'il1m1nmnu!lu
. '
'1iuYi 1 l ; "
m111111'11.i1!'!l'l'il1n111nl'!l'll-111'llil.:J
' '
'i11n.i1-,i.:i~m.imJbb1llU '11"1'1l'l<l'1fl1JbLVIU
mnu!luri 1 -6 7 8 '11'il1!jl'I -6
,.,,,: 6',,,: .....,:
l'!l'll-111111 1 l'!l'll-111111 2 l'!l'll-111111 3
mnu!luri 2 3 -2 5 fi1'il1!jl'I -2 6 -7 -8
'
mnu!luri 1 -4 6 -3 fi1'il1!jl'I -4 ~ 2 ~
4 -6 3
' . ,,..""
1'11'11!jl'l'llil.:Jl'!l'll-1111'1fl -3 -7 -8
. ' '
'1iuYi 2 m11111'11rr.:il'!l'l'il1nfi1'il1l'!l'I
. ' '
. ' 'CV a I ! o
'llU111 2 n1'l"11111'11j.:J!jl'l'il1nl'11'11!jl'I
'i11nfi1'il1l'!l'l~.:in-<i11fei -6 -2 -4
'
I o V I ~
'il1nm.i1!jl'll'l.:Jnm1mi -3 -7 -8
b!1einfi1ey.:i!jl'l'il1nfi1~1ql'I 'il~1#1.J1fei -2 bfieinfi1ey.:iql'I -~1#1.J1fiil -3
• •
~.:iiiufi1 Maximin = -2 $1.:iiiufi1 Maximin = -3
1iltn'lttll'il Minimax
' ' .<II. 6'
1) 1111mill'qi1n eiw11fl.:JY!n bbm1111ill'll'll-ll1'il1n.i1-,1.:il'l<l'1il1Jbb'YIU
2) briil1111fl1ey.:iql'l'llil.:!1'!n bbm1111ill'Il'IJ-111'1#Ibbtl1n'l~1111fi1~1ql'l'illnfi1ij.:!ll'IiinYi111~.:i'l1n~heith.:i b~l-1
w1EJ 'll. w1EJ 'll.
-6 7 8 6 -7 -8
w1tJ n. 3 -2 5 ww n. -3 2 -5
-4 6 -3 4 -6 3

'inm"rn1'11 Minimax
~ ..,l ' ' F/
'll'IWI 1 m"rn1m'1,'1li1ml't-h'il1n111n'11i1:;JJl
'
-~'1ii:;JflVI 1
-6
3
-4
. " "
<d
'11i1l-IJlVI 3
8
5
-3
fll~'!lli1 3 7 8
. ' '
oiluY 2 m'l1111'11"11l'W1'il1n'11i1,'11i1" . "
•l'l1'11M Ill:lJ
~d '
'l1lnl'lVlli11 Minimax
. '
• d '
'llUVI 1 nl':iVl1i1l~'!lli1'illn'l'Jrnm'lU€JU'llil'
'i1111,mW1€JlJLLVIU
Lbrl'lUilUYl 1 6 -7 -8 fl1i1,MI 6
."
Lbrl'lUilUYl 2 -3 2 -5 fl1'1,'11i1 2
."
LLl:l'lUilUYl 3 4 -6 3 '11"11'1i1 4
"
. ' '
oiluY 2 m"lVl1'11"11ijli1'il1n'11~'!lli1 >J1n'11~'!jli1
' .,.,. ' .:!I. ""' t ~ ,:!( .., ....
i1,nl11'li1€J 6 2 4 L'11Jn'1llill!jli1i18 2 i1~UU
fll Minimax ;:'El 2
'11'llmm:;,i (value of the game) fiei Nl'lnl'llilillJUVIU~il~LolUNll'JVl'l,p.J1tJLli1'J~l~'l'u11i1tJn'l
,,.~tJ'1ieim'l1'ilu111if,'1~' '11'lleJ'ml-ldd'Ju'11fo'"~tJ'llei'm-;1<u111mtJ'1~~'1iei'i'lil~nu111'i€Jd'Ju'11mli1
Vi:;Jll'J~d'Julul~'JJ[NNll'JLli1Nll'JVl'l,..~l~mil'J~Lnt.1111"1,'Jln~LolULnl-li:iUVl'1ll'J'1~,.,Jm'iiil,nu
' .<iJAlJtltJ'Utltl1~
'Jli11i1'<Wfm'irh' (saddle point) ;:'El flllil!llJLLVlu~mli111'J~l~'J1nm'l~WlolULL"1l1~Nll'J1-if~ ~ 'ii '!J
nl1~Vl~li11'J1iJilm'llU~l'JULLUl1,m1~V1Till'll'J '"~111'nl1~Vlilli1R'J~111'nl1~VliiXulill18i1lu ~'Lum'lLL'li,oiJuLu
m:;.i111;i,'1 m'"'"~il~1i1"Jl1<rJm'irb,111'iil1iJl11~ Lli11'J~m:;,iL1i1~l1~1i1"Jl1<rJu~ri1''"~•nli1~u1umlli~'11 Maxi-
min = Minimax LLl1~l1'11L'Vi1faJ'11'llil"rn-J~'ll'J Lumlli~m:;,iLli1 '11 Maximin oft Minimax m:;,iiXu"J~
Li.Jl1"Jli11i1l1'1U~ri'l'
" " .u •
1'110£JH
Nl!J B Maximin
7 3 3*
-2 -5 -5
Minimax 7 3*

11im1 Maximin ~~bi'JuniltJVITI'IJ<i~e11ti A "l1nfi1~1lj"1fi~.'i' ~1l 3 bbil:: -5 u1'11mmfi1j'~lj"1
5n~111i'.l~ 1.'i'mi 3 ~~b"itiffllbUu.11 Maximin
"lii Minimax ~~bUUnW'VITI'IJ1l~ B "l1nf111~1"1~L.'i'~1l 7 mi:: 3u1mmfi1~1'1"15n~111ii~1.'fmi"ii 'IJ ~ ~
'i 1'1 2.f I I A I 1f.&:!. ({ I d '
"ll:bVl'Ub"1111'11 Maximin = Minimax 1'11l 3 bb1"1~11mi-imi'l"1'J<l~U1Jm~bb!1::i-1m'IJ1l~m:w = 3
..,::;, .:'1.612' .r".d ..i6]v ir..l1' jv.., :"f t:...i:
"1~UUi>ll1J A "ll:b!11ln b'lfff'1tJV11lVI 1 bb<ll: B "ll:b!11ln b'lfm1tJV11lVI 2 "11JNltJ A "l::b"11lJtlo::btJ'lfU 1'11l 3 bb!l::
e11tJ B '1::bi'lue11mfitJt11::fo'lfu~ei 3
. . ...., d: A: A: <::! v ,_, -'""'~""'
'ttll~lllflftfllfl!U0111'Tl1~!10~'fl 3.2.1 1!111 rn11'lu{JUl'lflllf171:1J 3.2.1
. . ' .
1 di ' <::I <::! <!I ""'-
l!l!l!1flUfffllfl'l1'1-fJtlfl 3 ll0l!fl 3.2 !10~fl 3.2.1
n!ltJVITibbYJ (pure strategy) ~<i mi~~bb'li~.zru,if~ 2 NltJ bliein1i°niltJVlfl"1nm~VliiVli'.l~bWm
m1tJVITIb~m11!1B"1nm1iJbtl~!JU1tiLoiun<itJVlii5u 1"1tJ~n!1tJVITILLYJ"l::i'.l'iln1'ii-1u~~l114!!!!~<i
- '11 Maximin = '11 Minimax = 'l"1'J!1~U~bl1~ (Saddle Point) = fll'IJB~m:wmobb'li~-iiu
(Value of the Game)
- m11111fii q;i111Bu11w;i!1::Nl!J"l::•ii1ln1in!1tJVIfl"1~~~ll"1bb!l::m11111fil'IJ1l~m:wmm1tlV11q;i1
'111llJ1.1-Yfufi
.if10~Nn 3.3
'1lnN!1<111llJbb'YIU'11nm·m'li~-iiu'IJm A bb!l:: B 1iX1111fi1'1J1l~bm.Jbb!1::n!1tJVl~~~!J"1'll1l~ A bb!l:: B
&:!. "" 6j!IJ
'Yl"l::b!11lrn'lf

NltJ B
2 8 7
NltJ A
-5 4 -6
1 -3 5
NltJ B
2 8 7 Maximin
e11tJ A
-5 4 -6
1 -3 5
Minimax 2* 8 7
bi'.hwnn.11 Maximin = Minimax iie:i 2
"" ~ ' ,,J d ir.d
:J.i<illbbVli~1:1~1'1 bbbni1:11-11'1 1 i1'1:J.IJ11'1 1
'1~1'111 mutld'JunfltJ1'1TibbYJ 1n'1~'1'Jfl~ii~thJ .11'!Jmmm'Yi1fiu 2
nfl!J1'1TI~~~i)'1 '!Jm A iiiil 11e:in1oiinfltj1'1TI~ 1
' ' '6'...i"'"" d. ""'51v t:A
nfl!J1'11l1'1'11'1i)'1 'lleJ~ B '11:1 me:im'linfl!J1'11l1'1 1
L"1tJNltJ A dJiiN1u1'11h~LtJ'liu NltJ B d'JiiN1rni'ltJ1h~Lu'liu
'v ' ~
m0111~n 3.4
ill"ll,Nfl1'11:11Jlb1'1Will"ll~wii,olfii'llil' A bbfl~ B 1Vlvi1'11'llmmm1fl~nfltJ1'1m~~'1<11'llm A bbfl~ B
' '

Nl!J B
3 9 -4
Nl!J A
6 -2 3
8 5 7
lill11,Nl11ii!l1JLlYIU'll!l' A
Nl!J B Maximin
3 9 -4 -4
Nl!J A
6 -2 3 -2
8 5 7 5*
Minimax 8 9 7*
bUB,'illnl'h Maximin "' Minimax
bb!iii,11bfl:l-l.f(tlJSl"Jiiiil1.-!Utlrl1' m:1.1.f(tl-l'l'lfn11tJ'YIITbbVl iJ,vi1'11'll!l'ml-I1l41~Yiufi bUm"l1mnl-Iq q 'ii q
d~,r;iu A ""~ B "~l~miEJ'YIITNfll-1 $i!ll~n!1EJ'YIIT~' 1 2 bbl1~ 3 Nfl:l-lnu 1i,,i1~b~Bnl~•Ylmnl1EJ'YIITb~m
41,,:(u~1bUU!l~1'ri,ri"~~mvi1i1iii11u'll1Nn1''lb~Eln1~n11~'Viiibblli11m11~'YIITbUUb'Yi1'lii bbl1~'11'll!l'•nl-1i'lfii
b'Yi11111tJl1~blltJii"lmr:i11li'lub4!l,ri 3.2.3
' l ' •
, 4 I d. <ti. 4 ~
1.1111.11m~rimn111.111rn 3 mmn 3.2 mi~n s.2.2

nritJ'l'1TIN<l:iJ (Mixed Strategy) ~il m1~fiLL~'oiiu.Ji, 2 NltJ l'l1J!LliilnLiin'11J'YIITL'1nritJ'YITIV1ti,q '>l ~ q
w1iLV1mnritJ'l'1TIL&im w1i'1:::Llir:in1>iinr11J'l'1TIV1m1.11liN<l:iJnu l'11J~nri1J'l'1TIN'1:iJ'1:::l1flrn'1:iJu&l~'114lru~r:iq q q q ""
- '11 Maximin ;e '11 Minimax
- 1:iil1~'1i)rlf)U~'11' (Saddle Point)
- f111Vllfll<ilr:llJ'11111.'.J'lJ~'1ftUh11LLlii61:::N11J'1:::Lliilnt>ifnm.J'YIITL'1 LL61:::m1V11fil'llil'bn;;J '1:::
!!..I 0 0 = ,,,.
<ilr:l,'1'11f1l'lfl1U1llim:iJL<il:iJ
- fi1'llmLn:lJ'1:::mh:::11111' Maximin s fll'llil"f1:iJ s Minimax
•
f111Vllfll'ilr:llJLYl:iJL&im'1111.'.J'lJm'l~OlftUh '1:::t>if111'1n~11~'1~&i~'f'1'1:::Ln'1~u.nr:lNrl'ilillJLb'YIU~
rn'11111',rl1111.'.J'lJ~•<iuN1m1ti,1JI•'Yi1nJJm1:iJj'llJ•litJ~rn'11111,'llil~~•<iu5nN11J l'11J1iJ'11u,11N11J'1'l'
oii1:lJ'1:::Lliiln1>iinii~'l'1ITL'1
•
oiftt'iJ!JUf1l'lfl1U1lli
1. '11u1rn1111'11<il!llJb~ilm'l~'1ftUh'llil'NlrnL1n1um1Llir:in1>ifnritj'l'1flm1LL~'oiiu
2. '11u1ru1111'11<il!llJb~!lm1lii'1i:iuh'llil'1lnN11J1um1•lir:in1>iinrifJ'YITif1l'lLL~'oiiu
3. '11u1ru1111fi1'll!l,Ln:iJm·m~'oiiu
'Jlifll'jftl'lfJtu!~0 'lll ftl ll01Jif1'll~1Jfll'jiillU'l!1-01uf)'j iii~!ii'I!flmrn~Ntl':IJ
m11111'11<ilr:ilJ'11m1n1>ii'illrnrn1fi1fl1'1'Yl1''111'iil1>ifm1.iu1'1:::dJutum-,1111'11111illJ1JI
nli11Lrir:iN'L~um:iJ.Ji, 2N1rnlir:in1>iin'11J'l'1TI<1i1''1N'1.inu111'1uN1u-,:::tJ:::nmwd"i'Ju11imJ1,<11mtir:i,111'iil. '
t>J1u111,i'Ju1:::1.1:::m1m'1 fi1rn'11111',111'iilm1.iu1'1:::LDtt'llr:i,fi1'llr:i"n.im1LL~'oiiu '11nm1Liiiln1>iin'11J'l'1TI
'
'llm~L~um.i.Ji, 2 Nll'J '1:::l1'111nil'tjfi1fl,~fi1111ti, ~"'"'"'1Jl<i11:iJ~1il~l'~ 3.5
ill10uN~ 3.5 JJ'i~'YI A LL'1:::JJ'ii:i'l'1 B vi1m1"~'oiiunu l'11JlJ'i~ A •Du•~1'llm<i1111,e-;r1
lJ~A
'ill'll,Nrl'ilr:llJbb'YIU'lleNlJ'i~ A
lJ'i~ B
5 -3
-2 10

1Vi~'ill'Hu1m'l<11'11'luh LLi11'11ilUfi1mi.ii1li(
1 u5i;m A ~in'1 m'l1i1Bn1-irnm~Y1tl'1u1' •m~1-irLLwi'1m'11.JYITii1i'mi111ul1'1fi1u•Yi1h
2 mi;m B ~ln'1 '"1'lii1ilnl-irfll11.J1'1tl'1U1' mi~loifbbii!1~fll11.JYITii1i'rni11lu#'1ci1ubri1h
3. .11'llemni.im'lwli,1iud'iuLYi1h
'iilno/i1il~l'~ 3.5 fll'lbb'li,1iuil'l1Ul'1 Matrix bblJlJ 2x2 m-rn1fil'llillJfllm'lnV11fi1'11il1Jl.'i't'1tJ
1rrm'lVl1'11'11'1V1'foV17ei1rrm.,1<iim1i.iu1'i1:;d'.Ju
1. fl'jrumfilV!l)lJfaaillfll'j'111ri1Al'1ttr~
~d •
1li1'11
rhfll'l'll'l1<Jfteiu11mi.iddium'l1-irm11.JVITI"Yrvi~ilf1'11.JVITINft;J,J
A1
A2
Minimax
- .11 Maximin "' Minimax
ii',Ju~"llum'l1-irn'11.JVITINftl-J
B1 B2
5 -3
-2 10
5* 10
1.1 fiimrnmfi1'11il1JL~ilm'l1i'11'lul'ii'llil~lJ~ A~ln'll
1Vi Q = €i'll'llff1U~1J'Turi A ~ln'1l<iifll11jYITI~ 1
1 - Q = D'll'llff1U~1J5i;m A ~1n'll1-irn!11JVITI~ 2
Maximin
-3
-2*
t11u~ B~in'11-irnl11JVITI~ 1 "11u1uri1l'lm'1V11','llei,u1i;m A~1fi''1 'il~•v1i.i~u
= 5 Q + (-2) (1 - Q)
t11u~ B-ii1n'll1-ilnmJVI~ 2 -ii1u1uri1hm'll1111','llmu1i;m A-ii1n'1<J~•v1i.i~u
= (- 3 )Q + 10 (1 - Q)
bbfl<f;J,jfll'lVll.il Q
5 Q + (-2 )(1 - Q)
7 Q - 2
20 Q
= (-3) Q + 10 (1 - Q)
=-130+10
= 12

Q = 12
20
=3
5
=1-3=2
5 5
li'i~~u u'llffi A"51n"1 m-;-1-iiniitJYITI~ 11uil<11"l1'1u LL'1:;'<i11'lLiiBn1-iin11tJY1TI~ 21u
,.,q 5 ' ,
il<11'lli11u .1.. V1mtJm1:1-J11ti1ilm"lLL~~4iu 5 .-i'fo u'llffi A ~ln"1 m"lLiiBn11in11tJY1TIYi 1 3 .-i1~ n11tJY1TI' ,., 5 q q
Y122.-i1~
1.2 qii1u1ruV11qii1<11BmY1Bm"l<if"1liuh'lJmlJ'llffi B "51n"1
tvrP = il<11"l1ri1u~u~ B ~1n"1 1-iinmJYITI~ 1
1 - p = il<1101ri1u~1J'lijYI B ~1n"1L-iin'11JYI~ 2
n11J'llffi A ~ln"1L<iin'11JYITI~ 1"51u1un1l"i'/Jil~1J~ B 'l:0-'1"1'1~
= 5P + (- 3) (1 - P)
n11J'llffi B ~1n"1L<iinm~YI~ 2 °ii1mun1h'lJil~lJ~ B '1:;''1"111~
= (-2) P + 10 (1 - P)
5P +(- 3 )(1 - P)
SP - 3
20P
p
= (-2) P + 10 (1 - P)
= -12P + 10
= 13
= 13
20
=1 - 13 =7- -
20 20
Ji~~ulJ'llffi B °il1n"1 m"ll-iinii"YI~ 11uil"1"l1ri1u 13 LL'1:;'fn'lLiiiln1-iin11tJY1~ 2 1u
~ 20 '

P= 13 1 - p = 7
20 20
0= 3 5 -3
5
1 - Q = 2 -2 10
5
NiWl1llJLL'YIU'llil1Lnl.J e'iil'llihu e'f<1mrl1u '11'llil1Lni.J
~ '
'llme;JLf!UA
~ '
'llil1e;JLf!U B
5 x 3 x 13 = 1. 95
5 20
-3 x 3 x 7 = -0.63
5 20
-2 x 2 x 13 = -0.52
5 20
10 x 2 x 7 = 1.4
5 20
'11'llmLni.J = 2.2
Vi'JiJ.il'll€11Lni.J.iiii,1"l,'il111LLm UilUVi'JiJfl?1i.Jn bb'11l?1bh11V1tt111#i'<J~lJ.J1Lvi1nu
d
mnuilU'YI 1 '11'llil1Lnl.J =
(5>[~~]+ (-3 l:o]
= 44
20
= 22
'
(-2l~~]+ (1ol;o]
m11UilUYi 2 '11'llmmi.J =
= 44
20
= 2.2

=
= 11
5
=
= 11
5
= 2.2
~,iXu'11'lJ!l~Ln:l-IL'Yhnu 2.2 111mtJm1:l-l'i1 '01nm1LL'li~oifu111mtJ'1~' l111rn>1~rnL'11 u'ii:m A
'il1n111 ":;1<1iri1hLYll-l~u 2.2'11um'Yl1'u'lJru:;~u~i:m B 'il1n111 '":;~f'lJLlitJri1hLYhnu 2.2 ~lum'YI
2. O'lWmfl1~011faui,'m1:JJu111~11Jii
m1fi1mru<li1tJirrd L11'":;iii!l~'1:!-1lJ&11~Lnum:l-liii~'1!l~fl1tJ'O:;i'.jm1Lli!lnHin"~'YltluLLlii":;'1~~
~nm1LL'li~oifudJimuutj:l-I (random) LL'1:;'il1u1un'1~'YITI'lJmiii,'11l,N1tJOJ:;<i1eJ~L'Yhnu ~1numlJ1m:1mrn
U1lUri1111u111'lViLlJU A LL1;1:;1111mLrni1'lllJ1iLlJU B NltJ A nn'1tJ'YITI,1i,111:l-1'1 m NltJ B nn'1tJ'YITiiii~111lJl'I n
' '
n'1tJ'Ylil ll'ltJfi
'
' '
'11111-l'um1L:iitJu Matrix fiLL'1'1~N'11111lrnL'YIU'lJ1l~LnlJ'O:;L111neJunufinn11mLL'11 w1knri1111u01'11
x LL1;1:; y lO!tJnl".iL:iitJUri1nul'i~Vf1'1J1l~W1i'1rnrnu1l.lbL1;1:;LL111;1:;LLrn'1oilJ/i~,dl ) ,... ,...
'11111-l'um1fi1mruloitJ'l-iiml:l-IUl"l:;LlJUL.lnl".i1111fi1'111JUU'U"l:;LLU~mlrnlJu 2 mrn~~u
2.1 Lil!li!iOJ11ru1~Lnu1111:l-ILLrnueJu (f11tJ A) ,::n~wmm~~d
2.1 .1 1111'11'11011111~ (expected value) 'lJ!l,N'1'1l!l1JLL'YIUNltJ A L~!lNltJ B Lli1ln'loii
, , m • ,
m1~'YITifi j loitJfi EV(j) = L a.. x, j=1, 2, 3. ' n ~~ii:u'11'11"1111l~'lJmN'1'111l1JLL'YIU'lJ!l~N1tJ A .n!JNltJ
i:::::1 lJ J

. Alf!Wl'M1~'ll!MNlll'lllU!!'n1l'll0~Nl!J A
.,;
1i'J0f1111 B ll'lJfll1'n
4 l ll <f4
!llllfl 'llmtl'j'nfi'n
1 a11 x1 + az1 xz + a31 x3 + · .. +a x
ml m
1
2 ai2 x1 + azz xz + a32 x3 + · .. +a x
m2 m
2
3 a13 x1 + a23 xz + a33 x3 + · .. +a xm3 m
3
n aln x1 + azn xz + 8 3n x3 + · .. +a x nmn m
2.1.2 V11.-i1'1lil~ x . x2
• • x L<ilmil'Vl'U<illVlfli.Jm"J (1)=(2), (1)=(3), (1)=(4) 1111 n
'4ei1'J'l <Jum~vf~l.'fi.Jm":i (1) bri1n1mi.im"J (n) bbfl~ x1
+x
2
+ ..+xm = 1 bb~'lbbrll.'fl-lm"JJi~mh1b~il'!11.-i1 x
1
,
x2
,. ., xm 'liei111
2.1.3 .-i1'11ei~m:w~~~l)<i1V11L#i1<i11'Jm"JLL'Yl'Ufll x, x,., x m1ul.'f:wm"JL<ilfl:wm":i'Vl;i~, I 2 m
(l.'f:wrn"Jfi 1, 2, 3 ... n) ~~R~il'11m<il'Vl1~'11mNfl<iiillJbb'Yl'UN11'! A uum~
2.2 ~ei~,1.,rn1~bl'lu<ii1:wbbm,Y~ <e-111'! BJ -~n~u<iieJUJi~d
2.z.1 V11'11m<ilV11'~'1lil~Nfl<i11l1Jbb'Yl'U'llil~N11'J B bi'.ieie-118 A bliein1'1infl~'YIJJ'.~ i L<i11'J~
m • ,
"" v v ' v ·" d ·" ~ 'l" 'Ev0J = .:. a.• X j=l, 2, 3 . ., n <il~'U'Umm<il'Vl'l~'llmNfl<ililrnb'Yl'UN18 B bl-l1lN18 A bfliln 'lfnfl8'Ylll
i=1 l J ~
~ ; b~8u1.i'Jl~<i11"Jl~~ 3.14
.; '!lJllN111 A Alflll'l'Ml~'ll!l~Nlll'lllUU'n'IJ'll!l~Nl!J B
.,;
ll'lJfll1'n
4 ~!II .,..,t
Ill!lfl 'IIflll!J'nfi'n
1 at1 Y1 + 8 12 Yz + ai3 Y-:; + · · · +a1n yn 1
2 8 21 Y1 + 8 22 Yz + 8 23 Y3 + · · · +azn yn 2
3 8 31Y1+ 8 3zY2+ 8 32 Y3+ · · · +a3n yn 3
m amt Y1 + 8mz Yz + 8m3 Y3 + · · · +amn Yn m

2.2.2 m'11'lJr:i~ y , y , ..., y L<ill'.ni1V1u<i1lVl!1~m1 (1)=(2), (1)=(3), (1)=(4) 111
, , 1 2 n ,
•~r:itJ"] 'lum:;Yf~11i-im1 (1) Lvi1nu11~m1 (m) "":; y +y2
+...+y = 1 bLi1'1Lbn!1~m1ii~nli11•Ylr:iV11.!1 y,
y' 'y <ilr:i1112 n
,,(
(!1~nl':i'YI 1, 2, 3
u'lli'.'YI A
x1
x2
1 n 1
u'lli'.'YI B
yl Yz
5 -3
-2 10
rhv1w111Vf x1
, x2
dJum1~'.i1,:;dJu~u1~ A Liir:in1-iln!1~'YITI~ 1 LL!1mmJ'YITI~ 2
y , y bf'lum1uli1'l:;d'Ju~u~~ B •iir:in1-ilnmJ'Yln~ 1 Lb!1:::n!1tJ'YITI~ 21 2 q q
m':il-11'11m<ilvi1'~'llr:J~f.l!1<il<l'IJLb'YIU'llmlJ'lli'.'YI A •~e:iu'lli'.'YI B i1m11-ilnmJ'YI~ 1 bL!1:;n!11jVI~
= 5x
1
- 2x
2
................... (1)
EV
2
= -3x
1
- 10x
2
.................. (2)
1oiti~ x
1
+ x
2
= 1
ril'vmoiLVf (1) = (2)
5x
1
- 2x
2
= -3x
1
+ 10x
2
8x
1
= 12x
2
Lb'Ylu.11 x
1
1u (3)
x, = 12 x 2
8
12 x2
+ x2
= 1
8
20x
2
= 8
(3)

x = 82 -
20
=2
5
x = 1- 2
1
= 3
5
5
nl"l'lllftl'll!l~bnl< V111#it~tml"lbb'11Uftl x
1
, x2
'1~LU'l<mTil (1) 'll~!l'l<mTil (2) !fl~
biYIU'11Lu'l<mTI1 (1)
'll~mmuLU'l<m"l~ (2)
'11'llmml1
= 15 4
5 5
= 11
5
= 2.2
= - 9 + 20
5 5
=11
5
= 2.2
iiufifllJ1~ A 1-li'nmmn~ 1 Luei'lil11fi1u 2 m1~1-li'ni1tJYJTI~ 21u51i111fi1u _g_ tiltJ
~ 5 q 5
ii'11'llfl~ml<cri1nlJ 2.2 V1mtJm1l<'l1'il1nm"lu'IJ~1iuV1mtJfl~~tilm%~m•li'1 1J1~ A 'il~1#1n1hc~l<~u
2.2 li'1ulJ1'Y1 1u'llru~u~ B 'ii1nil 'il~J'lJ•RtJn1h•ri1nlJ 2.2 li'1uu1YJ
nl"l'lllftlfl1il'111~'llme·mlil!l1Jbb'llU'll!l~1J~ B bSl!llJ~ Aiim"lL'6'nl.1~YJ~ 1 bbl.1~nl.1~YJ~
= 5yl - 3y2 ..... (1)
= -2yl + 10y2 ··············· ...... (2)
············· ······· (3)

ri1VIU'1LVf (1) = (2)
5y1 - 3y2
7 Y1
Yz
bbYIUfl1 y
1
LU (3)
JlYz + Yz
7
= -2y! + 10y2
= 13 y
2
= 13 y
- 2
7
= 1
=7
=7
20
= 1- 7
20
= 13
20
!11'l1111fl1'1Jmbm.J Vll1;'(t~tJ!111UYIUfl1 y
1
, y
2
rl~LU1"l.Jf11.,yj (1) Vl~81"l.Jf111~ (2) n1~
uY1ufi1Lu1":wmlfi (1)
Vl~8bbYIUlU1":Wf111~ (2)
f11'lJ0~ma.J
= 65 - 21
- -
20 20
= 44
20
= 2.2
= -2 [(~~j + 10 [;~J
= -26 + 70
- -
20 20
= 44
20
= 2.2

fi1'1le:mna-JrvhnJJ 2.2 vimEJm1a-J11 "J1nmm'li1'1iuvimEJ<1~1l91mu~mbii'1u~ B "J~!'lrubiiEJri1lrnti1nJJ
2.2 ii'1ulJ11'1191EJ~lJ~Jffi A "J~lJ'ri1hbfla-J~U 2.2 ii'1UlJ11'l " •
W-101.h~~ s.6 1un1·m'li~'1iuma-J'11U91vil'.t1mEJ n. uii~mEJ '11. i'Jnii~1'lil~1mum"Jbb'li1'1iu
<1uii~ 3 nii~1'1TI ~.mm·m'li~'1iUbb!'l'1~1'1!1vlJibb!i~l'lmfttJLU~ Matrix if1d
Ul1J '11.
1 2 3
1 -1 1 1
Ul1J n. 2 2 -2 2
3 3 3 -3
Lt1'~"Jl"Jihlfll"J@f91(1Uh bbii''l'fltllJ.-ilm;Jif~d
1. u1EJ n. lm~mEJ '11. m"Jbl1t1n1'1inii~1'ltl91u1~Lubna-Jfll"Jbb'li~{iu
2. fi1'1lmbna-Jfll"JLL'li1'1iubl'lubrhh
"J1n@f1t1~1~d ma-Jn1"JbL'li1'1iui'J'11Ul91 Matrix bblJlJ 3x3 fila-Jl"JtiVl1.-i1111t11J14i't91tJ"li'lm"JL'1i

UltJ ~-
1 2 3 Maximin
1
UltJ n. 2
3
Minimix
'11 Maximin ¢ Minimix
i;)'~~u~~btlum1t.j{n~l'J'YITIN<l:J.J
-1
2
3
3
1 1 -1 *
-2 2 -2
3 -3 -3
3 2*
rhviu'1tvr x
1
, x
2
, x
3
dJum1mb<J~btlu~mtJ n. "J~bi!ie:mtoii'n~l'J'YITI~ 1, 2 mi~ 3
f°I ' .f"! .=t ""' °IV ffp.
Y,. Yz' Y3 buUffll:l-JUl"J~buU'YIUll'J 'JI. "J~b~ilm'Jln~l'j'YITI'YI 1, 2 ..~~ 3
b11b~tJU x
1
, x
2
, x
3
, y
1
, y
2
, y
3
ri1nulJ~n~l'j'YITibb.i~~W11'J1;';'1~d
Y,
-1 1 1
mti n. 2 -2 2
3 3 -3
L
1. vi1'11rn'1Vll~'llil~N~'1ill.Jbb'Ylu'llil~ml'.I n. bi'.lilml'.I 'll. i'.lm'ltoii'n~tJ'YITI~ 1, 2, 31;';'1~d
EV(1) = -x
1
+ 2x
2
+ 3x
3
... (1)
EV(2) = x
1
- 2x
2
+ 3x
3
........... (2)
EV(3) = x + 2x
2
- 3x, ............. . . (3)1 c
... (4)

-x1
+ 2x
2
+ 3x
3
x3
=
=
=
=
x
1
- 2x
2
+ 3x
3
x,
2
3
<J1ndeJul11~'11 x
1
+ x
2
+ x
3
= 1 mmf1 x
2
mi::: x
3
ti~lu<r:i.im'i" (4)
x, + x, + x, = 1
2 3
x = 6
1
11
x2 = x, =
2
x, =
-5 =
3
= -x
1
+ 2x2
+ 3x
3
= -1~+ 2[:~)]+ 3[~2:]
•m:1-1'11<iiu cm:::fi1'lJmm:i.if~fi'J<11Lri1nu
lh:::Tt.iw
= 6
11
EV(1) = -y
1
+ Y
2
+ Y3
EV(2) = 2y
1
- 2y
2
+ 2y
3
EV(3) = 3y
1
+ 3y
2
- 3y
3
3
11
2
-
11
(5)
(6)
.. (7)
Y1
+ Y2
+ Y3
= 1 ............. (8)
nl'v1U<lllVl<f:i-in1'i" (5) =(6); -y1
+ Y2
+ Y
3
=
... (9)

m1:::l'li<m'l (5) = (7) ; -y1
+ Y
2
+ Y
3
Y3
= 3y + 3y - 3y
1 2 3
Y2
+ Y1
2
(10)
Y, 5y
2
bb'llU'1''lu<1i<m'l (9) b~eim yJ
8
= 3y, - 3 (5y,)
8
(9y,J
8
'illnb~ilUl'JJ~ll y
1
+ y
2
+ y
3
= 1 bb'YIUfll y
1
bb'1::: y2
"'Lu<!i<m'l (8)
5y
2
+ y
2
+ 9y
2
= 1
- -
8 8
= 8
22
= 5y = 5 1m::: y = 9y = 9
_ 2 _ 3 _ 2 _
8 22 8 22
fll'JJmbni<fll'il'il::wll'illflfll'11il111'~''1Jil,N'1ilillJbb'YIU'JJmU11J 'JJ. lil1Jnl'lbb'Y1Ufll y + y + y
' •2' 1 2 3
mLul'li<m'l (5) ~~ei (6) ~~il (7) 111#1 ~''il:::1#1.-i11vhnu 'lufiubb'Ylum'lul'li<m'l (5)
=-y,+y,+y3
= -5 + 8 + 9
22 22 22
= 12
22
= 6
11
i:iu;!ieimEJ 'JJ. 'il:::1~ilfl'l«flmJ'YITI~ 1, 2, 3 #11LJm1i-iu1'iidJmvhnJJ _[)_ ,_i:i_ ,_!!_ illi-Jii1JiJJ
I I V 6 .J. "°1"-'I l<v ... .., I <v 222222
u'1:::'11'1Jil"flrn'Yllfl1J ii 'lf''il:::1ilillb'YllflUb~i-JilUfl1Jm'lm'il1mnil1ilm~'JJmc-1'1ilillJbb'YIU'JJil' Ul1J fl.
lil1J~Ul1J fl. dJuN11J1#11.h~LEJ'lfu bb'1:::m1J •11 dJuN1ml'mh:::lEJ'lfU

d '
mru'I'llviu
mru'ii'bi;iu (dominance criteria) i'lr:i 111iimnru'ii'~ti'!um1M1'11'U1<1i'11iNUriJV1llVlb~nm L'1'171~
1mn~q;i~~~bb'li~l1m1~'Yltlum1bb'li~'iiu111mtJ'iTI w1im~m1~'YITii:ll'il'il~1iJH)mbmd:ie:in1<ii L'V'l':il~Nii
ii ilrnL'YIU'JJil~n"romiiiXu1iJ~1tin11n'1tJ'Ylufiu"' wr,i:iu L11~'mm111if<11n'1tJ'Ylu~m<111111,11 .!"'ll~-liu'il~• • I • 'II
1iJb'<iiln1-iinmmiii:iueitJ1,bLU'Ui:l'Uililn1ti •m~v11m1~'i111rn1b:IY>Jlm'1tJ'YI~LVl'<ir:ir:itJLYi1i:iu ''V1r:il1'1m1n• • •
~.'1 1~· ~Ji
6bf1JJrjVl1 <1i~ll'Jb6"~6'l1'11'U
f1110uHn 3.7
bf1:1-inl'lbb'li~1iu1~11111~Nbb'li~oiiu X lbi1~ Yill'll~Ni1ilil1Jbb'Yl'U'il1nm1bb'll'1iu d'Ju'JJJ.11<11 3x2•
'11tJ Y
6 5
'11ti x 4 22
2 2
1um:1-1dNL~u X 'ilthJl.jfniitJ'YI~ 3 ~~d,i".lei~'il1nnii"'YITI X, x lVlNi1'ili:llJLL'Yl'U~~n11'lJ ~ 'i 1 2
~~i:iu'JJ'U1<1i'JJiNbf1:1-ilV1:i-i'il~Lti~1J'U'illn 3x2 dJu 2x2
'11tJ Y
6 5
'11ti x 4 22
m'l'11mru1111'11'1lil~m:1-1 mrnm:1-1 2x2 [.j{1Jlm11111'11'11<11'1~•w1lil1ti 111"lr:i'iilm"[.jfm1:1-1u1'il~
LlJ'U'iiil1tJ
'
fll1<iti1~n 3.s
m:1-1m'llb'li~1iu'lt11111~ A ""~ B lbl1'<1i~ill'll~N'1'ili:llJbb'Yl'U'illnm'llb'li~1iu Lllu'JJ'U1<1i 3x2

<i1l'HJNiN11llJbb'VIU NltJ A
NltJ B
5 4
NltJ A 20 23
12 10
1uml-ld~biiUNltJ A "J:::Li.lLoiin;;i~'l'ITI A1
m1:::n'1~'l'ITI A 3
<i1mm'l'lu~~n11 i#i~.&u'llm.i'llei~m:J-1"J:::ci.l~tJuL'llbiJrnm.1'lluw1 1x2
..Yl~Ub'i'1ild1 nmrnn A2 LVie-i;;i
Nll'J B
Nll'J A 20 23
l'iei Nll'J A 'l:::Loiin'1~'l'ITI x2
emu B 1oiinm.J'l'I~ 1 .-i1'llmcmm'11nu 20 l'iei ill1tJ A 1#1.lfue-i"
i.h:::1rniii
v '
mo1J1~
m:J-1m1LL'li~-ifu1:::'l'l11~NLL'li~-ifu x bb'1::: Y
•
<ill'H:Je-iiW11llJbb'VIU X
-5 5 8 -3
1'11tJ x
-2 7 4 -20
L~Ln:J-1d~Lb'li~1iuf11tJ Y "J:::Li.ibliein1oiin;;i~'l'ITI Y2
m1::: Y3
iii~dc'i'111:::~bb'li~-ifuN1tJ Y "J:::1#1.lfue-i;;i
a .... , v a ' ft
<i1illJbb'l'IU'l'l<ilf111 bllb'1ilf1b'1Uf1'1~'l'lll Y1
bb'1::: Y4
i#i'~i:iu'llUl<il'llil~Lnl-lL'l'li.i"J:::c'll~tJWnn'llm<il 2x4 d'Ju'JJUWI 2x2

fl18 Y
-5 -3
fl18 x
-2 -20
w ' ...
fflfll'JH'l'l 3.9
.,,.,. J 2J ,,.,. ........
L!l:J.lf11'lbi"ll~"llW.~Vl'll~r;JLL'lWJJU A bii1~ B hLl11~Ni1>l!llJLL'YIU11~U
-1 2 -2
3 -1 0
0 -2 -1
'1nm"1~1>1!l1JLL'YIU"ll!l~ A •nnm'lLL-il~<ifunu B l'.i'llm11"11!l~mmliu 3x3 ii'1m'W'Yi1m'li111"llU111
"11!l~i:Jruvnm'lLL-il~<ifu L'll'~i111l~'Yi1mTI11~11>1!lrnih'Yl'iurn'l1i'111fiuh"llm e'.iL<iitLL'1ii1~N11.1'11l1u. .,,.i,1n1>11"1~Ni1'il!l1JLL'YIU A LUULn:J.J"llUl11 3x3 ill:J.Jl'lt1i111'l1Ul11"11!l~L!l:J.Jm1Ji~~d
Nl8 B
-1 2 -2
Nl8 A
3 -1 0
1um:wd 'W·fa111.1 B '~Lfi!ln1-ii'n11~'Ylfl11111>11:J.J Nl1J A ,~1i-iLfi!ln1-ii'ni1~'YIIT A3 'Yi1l~'l1Ul11"11!l~
U'lJ'YllLlinmLlJU"llUll>l"ll!l~Ln:J.J 2x3
u'1iii'1L'll'ifo~m'li111"llm111'll!l~m1LL~~<ifu'11l1ufin mli11m111'Yi11Ji'1i'~d

eh.1 B
2 -2
e11tJ A -1 0
<Jln<m'.n,N'1"181JLLVIU A elltJ B<J~Li.J1<iinmJVITI Bl 1iJ11 A<J~LoiinmJV1TI'l.in"1li.J
Ji,i:iu'JJm"1'JJ!l'U'!JV11<J1rnni.J zx3 <J~LUUbni.J'JJU1"1 2x2 Ji,i:ium"1111'11.ieJ1.1'111111'1m11<ii'1i1uh
L11<J~1mlm11111.11m'1'1nu111~!l'ilim11oiim1i.Ju1<Jnl'iulii!lLl.J
. ' '<V .cg .<!:!. <!!! .=t !ii 'l_I _,.,,.......,,,,,,,,,.
'llMlllfll'lfl'lfl!'llOl'lll:rl'l~!'HH'll 3.2.3 mn !U'lVllliJUV!llllll'j'jlJ 3.2.3
' ' . 'Ill ~ I <::! <::! ~ <::!
!'lll!'ll11ll'lftlllfl'll'll1!l'll 3 ~IO'll't'I 3.2 !'lM'll 3.2.3

Anderson, David R Sweeney, Dennis J. and Williams, Thomas A Quantitative Methods for Busi-
ness. 8th ed. Ohio: South - Western College Publishing. 2001.
Levin, Richard I and Rubin, David S. Statistics for Management 7th ed . New Jersey: Prentice Hall
International, Inc., 1998.
Render, Barry and Heizer, Jay. Prindples of Operation Management with Tutorials. 2nd ed. New
Jersey: Prentice Hall International, Inc., 1997.
Render, Barry and Stair, Ralph M. Jr. Quantitative Analysis for Management 6th ed . New Jersey:
Prentice Hall International, Inc., 1997.
Stevenson, William J. Production Operation Management 6th ed. Boston: Irwin McGraw - Hill,
1997.
Taylor ill, Barnard W. Introduction To Management Science. s'h ed. New Jersey: Prentice Hall

·4' "''tltt1Ul'I
~0 ~ori1E.Jfilflq]Tl'<>Jl'}tiL1it111'l5t1 ~~rliLL~~
l'" 1!UJ., M.Eng. (System Engineer)
!Jl1uttil-s rJ'l11t1filfl~Tl~11~tl~~1fl~1f1'1nna.if11fl(114 :!JVli1'Vlt1l~t1~Vii11~
'1u1v~iJ>uiJ1, 111.b~~ 4

•
,, 1 "'!fl1 fl'l.:J!'IHItt1
lleJUYi 4.1 m'l1U'lbbff'l:J.JLiJ1'Vlm1J (Goal Programming)
4.1.1 Lm1!i01.~u1i'um'l1u'lLLn'lmiJ1V1mu
4.1.2 <ihmJlJ1ll'lbbff'i:J.JbiJ1vimu'l!u01 Non-preemptive
4.1.3 <ii1LLUU1u·:mn'lm1l1vimu'l!u01 Preemptive
'
'ileiuil 4.2 L1'1f!Ufi AHP
4.2.1 Lm1!i01Ln1J1n1Jb'Ylf!Ufl AHP
4.2.2 <ii1bbUU<i1<iiu~U'llt11m'l<if01lluh
4.2.3 m'lU'l~La:lu101 tmnLu1tJrniluuil;o~tj
4.2.4 nl'l'il'l1"lffeJuml:J.J)n<iitl1'll1l~ NiOm'lll'l~La:lu
'
4.2.5 nl'lU'l~ Li'.lUVl1'Yl11Lilt1n-iLvim~ff:J.J
1. nn1ll'lLLn'lmiJ1V1mu'l-if<ii1LLUU1u·rnn'lm:il~Lffu'lum'lvi1.i1llt1u~rnm~ff:J.J~ff01
'
'lum'lLLrlUt)vi1~l11'11riu'l~ff1f:~<iiei~m'l'VIm tJ4irn101tJ'lViNi0~L4i~n1'ill:J.JLil1vimt1
vi1ei'ln<i'Li'imi'rnil1vi:J.J1tJmn~ij01 bbU1bUU LU'lbbn'l:J.JbUl'Vlmt1'l!u01 Non-preemptive
bbiO~ Preemptive LU'lbbn'l:J.JbU1'Vl:J.J11J'l!U01 Non-preemptive l1Lll1vimtJ~o/fei1m'l
~1'11<iium1:J.Jrl1fft11Lri1LYi1J:J.Jnu fi1u1u'lLLrn:J.JLU1'Vlmt1'l!u01 Preemptive l1Lil1'Vlmt1
~1LLu1<i1<iiuf!11:J.Jrl1ffti!~1iei1m'lU'l'lij<il111 nu 'Him'lLtniJ'IJV11'llei11u'lLLn'l:J.J
bUl'VlmtJ"J~bU~1JUtu'lbLn'l:J.JbU1Vlmrnuu<ii1bbUlJLU'lbbn'l:J.Jb:il1Lffu (Linear Program-
ming) bbfl1lo/JTim'lV11.il'il1J1Jb'Jim~t11fi'UUllJV11ti.J'lbLn'l:J.J b:if1Lffuff11u
2. L'Ylf!Ufl AHP 1-if'lum•<ii01lluh'Vlmtmi:JLnm<Vi ~1~<ii01llm1~4ft111hnilum1:J.J
L'Vlm~<l:J.J'll1l1'Yl11Lilein101t1l'.lmm<Vim'll.l'l~Lil1.1.vimt14!1u 'illm'l'll1l1L'Ylf1Ufl AHP <J~
" J '
ff-i'1~<ii1LLUu<i1<iiu-iiu (Heirachy) 'l!1LLff011ei1f:i.J7rneiu'l1ei1mm.vi101YLLu1eit1nLUU
ii1<iiu~u<i111 LL'11'loif'illm'lll'l~LiluJ1viiinm1:J.Jrl1fltl'll1l1Lnru.vi101 bbiO~u•~Lilu
flnm1.1.m1:J.J LVlm~<l:J.J'll ei1'Y111Lilein'lumru.vi1'01 101tJ'l~'illm'lbl.l1tJrnilurnnru.vi101

v _, '
1111Qu'.i:::l:l''ii'I
Lb'1 ::;Y11~ L~ilflifi<i~~'YillVifll'll.i'l~L!lUi'.11'111:!.iJnefiil~bb11 ~'Yi11"11W'11 fl 11Wl'lllloil
1Tifll'l'Yll~fl rul'1I'll11'1 ~lU fll'l'YI 1~1'YIUflfl11i-J ill q1)ljJ'll il~bflbib.yjbb11 ~fl~blUUfl11i-J
b'Ylm~fli.i'llil~'Yll~ b~ilfl1um'l<l'1'11ilum1i.ib'Ylm~11i.i'llil~'Yl1~ b~ilfl'lil~'l 1.1
b~eiiilflmY1il11.1~ 4 "ilULL~1 iifliilmnmm'lll
1. <J'ilrn1.1m'l11iwmmil1Y1m1.11.1
2. ei'ilu1rn11flUfl AHP 1'1

.
""l'ltl'l-l'YI 4.1
11111i11u111JJ1i11'ttint1 (Goal Programming)
l1eNVi 4.1.1 llu11'1111l~mnum.,tll·rnmi.ili:l1j.jm1.J
l~m~ 4.1.2 <if1ll1JD1ll'Jlln':li.JlU1j.jmtJ'liU?1 Non-preemptive
l1mVi 4.1.3 <if1ll1J1J1ll'Jllmmi:l1.jm1.J'liU'1 Preemptive
1. nTo1ll'i'llmmi:l1.jmtJ llJu<if1ll1J1J~L-ii"lum':l<if'1iluh li!himi1'181J~lj.jmtcii.i~l'j'1
1un':lrn~l1 Li:J1j.jmtJj.jmtJeith~~4lei~m.,-w-'m.1nu1'1tJtJEJi.i NiluflmtJ1Vim"u""l1
•
lU1j.jmtJ rn~•i11.jm1.1m,,t1i.J1i~nu~~l'J'1 llliil'1m1i.illil'1Ni1i'i'-wil~14f,tl1m1i.i
lritJ~l1JU"linlU11-ji.ii1.1~41il~n11'i'1i.ill'11ililtJ~s1i <if1ll1J1Jdllium':lll1ttJrn'i.1'11<if1ll1J1J. .
m':l1ll'i'llmi.il:ii~lfiu l'1m~i.1~1i1"11n1i (constraint) 1u411um':l1J'i''i''JlU1j.jmtJlii1~1
li111.l lli1tn11'1u'1'W~niiu1'<11i;:ill'Jts~f;llJuNm1mm~m1i.ilriml1JU"J1nlU1j.jm1.14f1u
'11~1 ll!i'1j.jifi1<11il1J~Yii1Vf'W~niiu1'<11!;)1.l':lts~<rl1~iuiltJ~l'J'1 ~~"JtllJufi1<11il1J~LVi
Ni'lnl'i'lri tJ~luw•nmiJ1.jm tJUil tJ~S'1iiulei~nnlll'Jbln'i'i.i lDlj.jm 1.1 llU~dlu•
2 U'JtlJl'YILj.j'l) 14fmi nl'Jlll'i'llmmiJ1j.jm1.Jll1J1J Non-preemptive lrntll1J1J
Preemptive
2. m'i'llhlm':lmiJ1j.jmml1J1J Non-preemptive llJU<if1ll1J1J1ll'i'lm':lmiJ1j.jmtJ~
li:l1.jm1.IVJnmJ1~l1ii1<ifum1i.irl14it),!~4i'il~nl"llVl1J'J'Jljlril'l nu loifi1Jm'JS-'1~<if1ll1J1J
ll1J1Jnl'i'lll'i'lm'i'i.ili~lfiu ~~l1~~niiu1'<11t)U'i'te!~<rlllJum1i.ilri1Nl1JU"J1nlDlj.jmtJ~
4i'il~nYrni1tj.j1fi1<11eiu~~1'i11Vfm1i.i,riml1JU"J1nlU1j.jmtJueitJ~l'j11 l'1tJl1~'1"11n1i
l~mGii.iLUl~il~lDlj.jmtJ'111uil1~1 ~4lei~nT'l dtei~"l1mi:l1.jmtJ1ii1~1 l1ii1.Jum1i.i
fl14it),! lril'l nu ~~UlmLifLu<if1ll1J1Jlli1t'l'11filli1il1J'W-'ili.inulUfl~~l~tJ1

3. ni-;Ltl·mmm{:h.,,mtmuu Preemptive bU'.il1bblJlJ1t1·mmmil11'imtJ~bil11'imtJ
'11~'1 il'11o/i'um1:1.1ll1'1l!~'~~ei~m-:;'lViunij•iJ1.,,:1.11tJriDu1'ifi~1iJ•'Yi1,;:1tJ:1.1nu m">
'1~1~o/11bbuu'"'~m">'Vll~1<iiillJ<J~l•ti~til11'i:1.11tJDeinlU'.-,~o/i'u'111:1.1ll14i'l!'11~'1 .~:1.1
fl~1~<ii'1lllJlJll1l~.,,1~1<i1eiu<J1nlil11'imtJ~~ei~m-:;'lVfrn·rnrieiu blii'1~~1t1'1~1~<if1tllJlJ•
lll1~'Vll~l<ilillJ'lmiJ1V1mtJ~ilm1:1.1~ei~m1'lVllJ'l"l"i'J'lugu~">8~11~1t1m:1.1'11#iu #11lllJlJ
bbl1~lTI'Vll~l<ili.l1JL'.ll'111~guilfini;rn~l'li'.l~tJ'l nunl"l"tt1-:mn1:J.1lUl1'i:l.lltJlllJlJ Non-
preemptive t~ei1'1~1<ili.llJ'lJu~g'.'Vl'ii~lt'11 <J:;.l1~1<ilulJ~1iii"lt101:1.1'lu4l1lblJ1J'llu~
ti111'imtJ'lu%u-,ei~11~1t1m:1.1'11o/i'u Lowmoltl~tJ'.~l<ileiui:iu 1t1tll'.~?1~1n?lL'.%u
'1D1ti <J'.1'1~1<iiu1J~'1WlJ'1lj:J.1YJmil11'imtJ'lu11Jn.,~o/i'um1:1.1ll14l]J~~ei~m-:;
v .I '
1l!Qu~~l:Mfl
l~i.lfni;1<ilu'.~ 4.1 <Jrnt.11 unfni;1mmoll
1. eillrnmm1~?l'iJu~m-,Ttl-,llmmi11'VlmtJ1'1
2. '1~1~/11lllJlJLti"Slbfl':i:J.ilill1'i:l.lltJ'n'U?I Non-preemptive ll1lrnllu1tJi'.<ilu'.'lum-:;
fl~1~<ii'1 ltlJ11ll1l~'V11~1<iiuu1'1
3. '111~411lllJlJltlolln"l":J.llil11'imtJ'n'U?I Preemptive bli.'l:;i.llrntJi'.<ilel'.L'.nl"l"'111~
<if1tlurnl11:;m~l<ileiu1'1

m'l"bbtlU'lJm1um'l"<ililft'Uh~hrnli1bbll1Jf11'l"1u·mmm'61.ffu (Linear Programming Model)
i:iu l'.i.iffl~1n'1'llfl1m'l"1-ii<ii1Lb1J1J111,..mm'61bfiwli11.11um'l"<ililfiuhV1m1.11i,:;m1 ~1tJ.iffl"ii1nil'llfl1
<i'l1bb1J1J11i'l"bLm;i.ib'61•ffmuub~fl1'llfl1 bthVlmtJ~1IB1m11J"'l 'il:ori1V1umihVlm1.11~bY11.11fltJ11b&°i1.11
w1i1Unl'l"<ii'1fl'Uh rm~lUnl'l"<ti'1nl"i'llfl1ii'iJnl'l"~Ufll'ili'.im1;i.i~1bUU~'il:;o/lfl17n1'll ·illVIlJ11.11'1i'lltJ'l
Ji'1u11 1Jl1fl~1billVIm tJ~o/ffl1nl'l"ill'ilJ'.i 011llJ.ifilbb{i1fi'mltj~1m'l~ b'lfu n'iJnl'l"ill'ili'.iLiJlVl:J.11tJ~o/ffl1
m'l"~·mri1l'l"bbl'!:;1U'lllli:;L&°imnu m'iil'.im1:J.11lfl1m'l"~'il:;1Vf<i1ubb1l1m'l"ili'llili'.ii'!ilrhu~mn~11.1 ti1
'il:Ol-if<i'f1Lb1J1J LP 'lf1tJlUrn'l"4filftUh W4JilflU1'J'iJ:;o/feNbliiJmiJ1V1mtJ~~lfl11Jb.v1tJ1fltJl~L&°im ci1u. .
bil1V1m1.15u'l ~i'.im1:J.1~1fitjJ'l"il1i11lJlill'i111fl1ri1V1Uillu~1l'llfl~l1lJm'l"~il~1nil (constraint) LLllm'l"
l11i'11<ii1Lb1J1Jb'liud,:;l'i11Vfl'11l:J.1'1iifl~<ii11um'>117rnil1V1m1.1~bUU~il"ii1nili:iuV1:J.1ill1.J
<ii1bblJlJ11i'l"bbmmil1V1mtJ bUu<ii1bb1J1J~U'l:OtJn111"loii111'1nmnfl1<ii1bb1J1Jl1hbbmm'E~Lfiu 1u
nl'l"bLtlU'lJVll~1.hmfl1J~11.11'ilQU'l":Ol1~.1~~il1nl'l"VImti ~1u'V'llffllJ'l nu Ufln'il1ndti~mm'l"m:;l.J~'1
"iilnil111~'1 lU<i'f1bb1J1Jl~b'liUb&°imn1J<i'f1bb1J1J1u'l"bbn'>l.lb'61bffUrf1'1 llJ
1Htl1.h:::n<1u'IJ<1~iii'111uu1u-:;11n·rnr1lnn.na <ii1bblJlJlli'>bbmlJbU1V1mtJi'.iiJ1.lli'lmil1J'lliN
<i'l1bL1J1J'lu~m'li1~'1 <ii~dfifl
1. rlhmna bUlV1:J.11tJ~li'l:onil1J~muu<ii1bb1J1J bUUbil1111mtJ~r11111-1il blJu<ii1bl'l'lli;.rnm1:J.1
<fieNnl'>1u~1u'li11'1 ill"Jl'.il~111mmil1111mt11u<il1bb1J1Jb&°itJ1b'limil1111mtJLub4il1Nl1ri1h ~11:J.11m'>
l'i1~1uii11 nmbbl1:;.,:;if1J1'l11ij011b111lifl LlJuo/lu
bUlVl:J.11l'J~rilVIU'1"J:OU'>:Onfl1J~11'Jlln1'llli:O~i11fi'tjJ 3 fltll~ 1~bbrl
1.1 ft11mrla.:nuu~1mil1111Jw Lil1111mtJ~1lfl1m'lltu4fl~m'lL1.J'lbbmmil1V1mtJi:iu blJu
~,m~&°i~'lil~o/ffl1nl''i Lb'liill"Jli,Jl1llJl'HllJ'l'l"'Jl~~1VIlJ11~1o/ffl1 j'.jnl'l'l:;l.Jn1011llJbri1.MblJ'UVl"iilmllJ
Nil'V'lmil'il1nbil1111m1.1~1lfl1m.,-)11'111:J.1 "ri tJ1L1JU'il1nbilT111mtJ~tJfllJtVibnil~ubriil b1.J"l1.1u "Vi1.1ui11J
bU1V1:J.11tJ~1lB1m'>bL.i'1i'.im1mL11in'li1~nufltJ1~h~~bb1l~1~LlJu 3 iln1'llli:;fiil
1) Nl'l~l~tJillJLVlUfltJffJlbUlVl:J.lltJl~
2) Nl1~l~tJil:J.1lVfmnn11Lil1111mtJ1~
3) N11~1~tJfllJ1ViLbiln'li11'il1mil1V1mtJ1~ ~11um1~;i.i1nn11111"lflufl1.1n11
" ~
3 illUflfl

Li1111mtJ 1: 4i1Nrn'l'wmlih vii1~il'i:w b~fl'U!l:; 2 f11uu1'YI
LU111:!-ill'J 2 ~1l:J-1~1'i1~1mh~nm #lv~1iiiilu L~fl'U!l:; 500 ~1ll-I'
LiJ111mtJ 3: 'l':;lii'u'J<i~.i'L11~fl rvhi'fu 200 11u1tJ
m1miJtJ,blJ'l-l'illmi1111mtJmi.11'l'mlJu1<11 2 LblJlJ ~!l mnn11LiJ111mtJ117flu!ll'J
=
mll-ILllrnLU'U'illmiJ111mmi11ii'u~ i (deviation variable)
~l'il~''ll!l,LiJ111ml'J - ~1Li1111mE.1~4ffl,rn1
Lriflfi~1~111bblJUfl!lnm1ul1n1'frn:;'llfl'111LLUU LP
11ibbUU'll!Nrn1mfi1"1€JU~1'i11Vfwm1l-!'ll!l~~1 d. U!l EJ~fiil,X'l-lrn'. ' '
111bblinfl,111 LLuv
'i'(!l,,.1n.i11l-! ,iJEJ'LlJ'l-l<llm'l'nLDu1111<11«,1u'Yll~~U!lE.Jn11111!lmnn11111!lLYi1n11
Lil111l-!1EJ Gl1!l1.i1wiiu Li1111mEJ~ 1 4f!l,rn1Li1111ml'J'lJiNN<lrllh!lt11~u!lm~!l'U!l:; 2 fi1urn'YI t'i1
Nilrllb'l'~bfi'1~'1-byJlnlJ 2.5 fll'UUl'YI
1VI ct1
= m1mlJEJ"uu'"1mi1111mEJ 1
•0.5 illUUl'YI
•1.5 ill'UUl'YI
111bblJlJ
!U 1 .<!! , cv ff ( <it A !Y e>IA<':t ol <
'ilt"1!l~mnm1w:immnrn,j'UE.1 x. ?: 0 11'l!l Non-negativity Constraint) 'il'"1fl~mim1'l'bbumm d.
' . '
1Vf,1Ju11iml-I Non-negativity Constraint ft ((J'llEJ!l:;b5tJil'illnm.iwu1n)
1u~ud m'Jm111<1111 b~!l1V1~1 d Lllu11iml-! Constraint d 'il:;n111u'11VI
' ''
ct+ Lri!l~1~4f!l,rn1mnn-l1Lil1~
m1m11tJ"uu =
' l ' '
~
ct~ L:i'lfl~1fi1<1i'umm11Lili'111l-IblJl'J"lJ'j, = l
d = ct++ ct-
l 1 1
l'1 EJ~ ct.. ct:'. ct.- ?: 0
l ' '
'W'n'iiu1'"1t)Uo:;i1'fi (Objective Function)
'l'1l-l'll!l,mll-ILllmLuuluLi1111l-!1t.1"11''l «,11l-lil
ct = ct+ mi:; ct- = oi i i
ct. = -ct- bbil:; ct+= 0
l ' '
'llfl'1111bblJlJ 'il:;bD'l-lnl'l Minimize N~
1.2 11i11v1''1'llV~!ill11JJW Li'lfl-:i'illn Objective Function 'll1M111LblJU'ilnDurn1
Minimize w;;i11l-l'llfl~m1l-!,iJl'WUU'il1mi1111mt.111mi1111ml'J t'i1 Li1111m tJ:i'l11u1 l'J":i'?1~'111-:inu b'Jiu

ltlT1-imtJJi'1uNmi1hi1111il1tJ-'fil•llui'i'1urn'VI •tl1111mtJ'llil-lm1<il1'1ll1-i'11ui'.l'f1u1tJ-'lil•llwil1u1mm m1
~"Jt11;.;l'i1'llil-lbi'.ltJ-lbuw1h-l'l l'1i1lllu Objective Function l&itJ1nuq;il-ll4i''liim1ri1'1J1'f1iin l'liu
r11111uilJ1'f1iinl'i1m1mlfomtJ (penalty) 'llil-lm1mritmuulul4ei-lliil-l'l l~il1Vfmm1m1;.;m1;;
.ritJ-lbuu1uJl1ulii1-i'l •'1i1Ji'1tJnu1JJ
1.3 1~.Yumwtl1fil)j'll!Ntfl11nn~ (Priority Level) ltl1'f1mmllii1tilUl-l!ll"il1imm1lVf
u11111JJJ11tJm1;.;ihf'i'ruhi•Yhnu rn-leitl1-im"Ji1m1;;-oi1•llu~"lt1iei-lv111~;' lu'llm~~•tl1'f1mtJrn-l' .
eitl1-li1m1;.;<l1f'i'(j!1ei-lin1t1~-lt'i1v111Ji'lh1t~ bb!iit'i11i.i1Ji''f11ei11ii'i.l-i~iinilif-lbllu~tJei;;''u •'liu ii"lm1
m"J"Jt1iei-lm11~JIN1r11heitJ1,ii:eitJtrhnulU1111mtJ•llui11Jlull1n fi1um1'i'mn1t#lum1<il1'1'11um"J
lllUlUlVIJ.JltJ~'l"eJ'1!1'11tidJui11Jlu~18'1
mrn~•tl1'f1mtmlii11rntJ1,l'Jm1;.;fi1'1!!•'Yi1•n1.1;.;nu 1iei,m1~"JtlVfu11'l•i11111mtrrri'ei;;'l nu
llluiihbbUU'll!l-ln111111LtmJ.JlUlV1mtJ~l1tJffll Non-preemptive Goal Programming
fi1umrn~lUl'VlmtJi'.lm1%ili11Jlum1J.Jfllfl(J~"ilt1iil-lnl'l"1VlU'l"'l"ljlUl'VlmWl!JUVlii'-lliil-l'l nu
d:lulii1bbuum11ti1bbmmi11111mu~l11'm11 Preemptive Goal Programming ~-l!ll"Jbbtl-lbUlVl:wltJ
d'lmt#lulii1-l'l bb'ii11~1~Jlum"Jti1tneiu/hmil1111mtJ1u.4ei-ilii1-i'l 1JJ111mm4EJ-i
2. fl'1m.hfl'vil!Ji1111 (Decision Variable) d:lulii1bb1i1~q;ei-im1m1ul'i1 i'.lii'm,;mn'liul~!J1nu
lii1bbtl11iiililuh1ulii1blUUnn1111bbmm:ii-lblfu
3. Constraints lulii1lbuum1lti1llmmil1111;;1mltl-Jei!lrnlJu 2 lllJU 1Jlmi
3.1 Constraints ~1:iJmm1l1LIilV1~u1111!ltJilJ.J1Vfi1m1lrimluu"J1n~il-oi1nil1JJ ii'ni,;mt
t'liul~1.11nu Constraint lulii1lluu LP ~-l"l~ri1111uill'il~il-oi1nilJJ1m.i4ei-i111mtJ >, <, = ~'
mm1m1;;11'lulii1lluu1Jl•'lim~tJ1nulii1•luu LP m;;tln~
3.2 Constraints ~m"il1rni11'f1ml'JluJl1ulii1-i'l (Goal Constraint) ~-lbt'1il-Jn-lli11111m1.1
~q;ei-im11JJ'l'u bb!ii!ll"JtJilJ.J1Vfilil"il1rni11111mtJ1JJ1h, lii1E1tl1-ll'liu Nl'i<11iimo/i 3 'llUl'I riEJlVflnilN1
r11hiiiil111u1m'Yi1nu 3, 5 ll1t 7 Ul'VI <111;.;i11<iiu t'i11Vf x
1
, x
2
, x
3
dJu-oi1u1uNl'i<iliimo/i 3 'llul'!ft
N1rllh11;; = 3x
1
+ 5x
2
+ 7x
3
tllrll'f1Uilllq;8-lnl'l"N1rll1wtl1'1UiltJ 100,000 Ul'VI lulii1lbUlJ LP 1in~"Jtrll'f1Uil
Constraint 1Ji'lUU
3x + 5x + 7x ~ 100,000
1 2 3 '
1ulii1bbuum11111Llmmi11'f1mu lU!l'1"J1nmm111tJilJ.J1Vfl'lm1NiltU1'f1mu1JJ
t'i1ri1'f1UillU1111mtJ'll1l'1Nmi1hL'Yi1nu 100,000 rn'VI ll1~lVfm1mritJ,tuu•'Yi1nu l'i1 d
1

(3x
1
+ 5x
2
+ 7x
3
) - 200,000 = di
~
(3x
1
+ 5x
2
+ 7x
3
) - d1
200,000vrrn =
"1"!11111 di ct+- d-=
I I
<J:::111 Constraint 'll1l~iil1111:J.J1mliu
(3x
1
+ 5x
2
+ 7x
3
) - (d7 - d~) = 200,000
1uo/11bilJUlu·mnrnbUlVl:J.JlLl~~lll:J.Jl"im:::u Constraints <J1mil1111m81~'U"iUbilJU• •
(mmrn'll1l~bil1vim1.1 i ~111) - (ct; - ct~) = •il1111m~ (i) ~<ii1l~m.,
.'!f J v v • I J v 'f. I h ~ v • I <~" '
4. n~fl'll'ifJflqu1~!Mfl l11bbUUbu"ibbn"i:J.JbulVl:J.Jl8:1n<ill)u"i:::'1~1'1'1'1l11l~m"i Minimize I'll
wfil"i'l:J.J'll1l~m1mn1.1~•uu~~111:J.J1111ViueJ1.1~'l111 Wl~~uii'i1bb1J.,fi1m1mnm•uu <ct; ibfil::: ct~) ,::;11.,1n{I
mJ1uYl~n-nu1<iin1J"i:::'1~fi
• •
li'1~<J17rn1b>h1"ll:::Lil1111:J.J181lLll~b~ m '1n1'lrn:::'lleJ"i11vimLJ<J:::i:imJefhm'fu 3 bbUU 111mi•
•i11vi:J.J11.1~ri1viu111Jl11.1m1:J.Jbn1.1~buu~:J.J1nn-l1 ii1l1.1n-)1 vi"lml'i1nrni11vim1.1 <il11ltJ1~b'liu
Lil1vim1.1 1 <ii'1l~m"iw1i1ri11.,1ltJ1~u1l1.1 100,000 Ul1'1
bU1111m1.1 2 L1'U1ii~'YJ'Ul>lbnu 500,000 Ul'l'I
Li11111m1.1 3 <J1mul'lu~1u•'Yi1nu 200 l'l'U
Li11111m1.1~ 1 ~u m1ri1h :::: 100,000 m'l'I li'1 ct1
•Dum1mnmbuu<J1mi11111m1.1
100,000 Ul'l'I
bbfilv{i'l d = d+ - d-v 1 1 1
1mi11vim1.1~ 1 ii fll'll1l~ ct- = o bil:J.Jmw>1:::wrnilh"J:::l.Jo/leJ~m.,1Vl~1n-l1Li11
" I
&i~iiufi1'll!l~ ct- 1.J<l1Lliu<ileJ~JJ-:nnn1u1'1~n-nu1'1ln1J"i:::'1~fillnllieJ11J
I "' •
l1l:J.Jbi11vim1.1~ 2 L,lum'YJ'U :,; soo,ooo Ul'l'I
li'11VI d2 = m1:J.JinmbUWillmi11 500,000 Ul'l'I
= ct+- ct-
' ...., 2 2 '
1u.l11vim1.1fi 2 nfi1 ct; •lirn1u"~'Ylufi•nu•ll1 500,000 m'YI ~~':::1.J1.1eJ:J.J1Vi•n'll~u
•
&i~iiu ct+ = o2 •
fi1 ct+ ~~1.Ju11n{l1u1'1~n-nu1<iin1J.,:::mfi .i~l'.l•b'1ib%1"11:::.i1 ct- i'Yi1uu
2 ' & ~ 2 ' ....,
'111115rnil1111:J.Jwfi 3 'll~<iieJ~m., <J1u1u.iu~1u = 200 .iu m<Ji:im1mutmuu111rr~

4'~iXu~1n1mwn1t11"1111u'111n1fu1'11t)li">~n14i'llEJ~o/11mJu"l~:rl'11 ct~ "''"~ ct~ 'llEJ~•i:l1vimtJ
~ i 1iltJ~unufini,;ru~'llEJ~.ifEJn1111ui11'llil,,i:l1111mtJ
t'11bi:l1111mtJr11111ui11"11rnl'l~il'V1mtJ ;;:: 1i.i:rl'11 ct~ 1u z bbi1~:rl'11 ct~ 1u z
n1•i:l1vimrni1111uiloi'1rnl'l'l'ei'vimtJ :S: 1i.i:rl'11 ct+ 1u z bbi1~:rl'11 ct-1u zI ~ J i
n1•i:l1111mtJri1111ui11"11rnl'l'l'ei'vimtJ = :r1Yi1'11 ct+ "''"~ ct~ 1u zl > 1 'V 1 I
1w11ruVi:rlvimrni:l1vimtJVi<ii'mm1"l~1·if'i1im11Vfu111114'nm1:1<il1<Fl!JJ'llEl~m1:J.J•ilmrnu"l1n
•i:l1111mtJw;i'1~mh' vi'l'Eiri1111ui11blluo/111'lruboihnu'11 ct ~1"l~m:w1,nu1'11 ct. ~~V1:J.Jilm'i"1:1<nmilu
. ' '
111,n1fu1'1lt.ili'~"~l'l1~
v d~ v " I v .~ <v ~ " I < ' .J i'I " I v
'll1bblJlJ'YI bil'J~u'i"nlillJ111 tJY1~f1'lfU1'11t)u'J~<i,mrn~ Constraint '1111'1 'lf1buU~u bblJlJ'llil11111
bblJU LP iiubil' ~1flll-il'HlVl1~1'1lilU1oi'b'liUb~tJ1flU1i'1bblJU LP rf11 Lli1i'1tJ'lTI Simplex vi'l'eim'Jloif
11i1'bf1'i":!-il'lEl:J.J'W1lIiiil~i111117u bbflD!JJVI1o/11bblJU11i1'bf1'i"m'61bl1'UbbtlD!JJVlllol
, . ' '
1 <:!ii ' "" """ <!{ ""
'Ull'lfllll1rlfl'lfl'll'U11J'l'l 4 l'Hl'U'l'l 4.1 mH'l'l 4.l.l

• •
~18.:!TI 4.1.2 vl1H1J1J1tl~Hfl1iJd:httiilttH'U'U Non-preemptive
11Jrnmmih111mtJ'llU<?i Non-preemptive d'iuii1bb'U'U~1<if1Nm'lLVl'U'l'lijLi11111mtJVlmtJ'l
eiEl1111.ly;!fei:w'iuT<?ItJ1i.il1iil<ifmn1:1.Jihl"iqJriilu111l11'llei1Li:l1111mtJ~<ifei1mo'U'l'lij m1ii!!f11$11bb'U'Ufll<J
n'1111Jiiiu~u'l <if~d
1. 'l:::l.JLU1111mtJ'luJl1w1h1'1 l<?1tJi1m1:1.J111mtJ'llil"i:l1m.11tJ~<ifei1m1'lu~liLL'U'U~bbii!<?11Ji'1tJ
,: ~
l'1'lel1Vll.Jll'J 2, :,; Vl'lel =
2. 'lVi ct. = lif1bbD'lfl11l.JLril'J1b'UU<JlnLU1Vll-lll'J~<ifil1nl'l
'
= '11'llei1•i:l1111mtJ~1Ji'l'U - 1:::<ifrniJ1111mtJ~ri1111u<?111'
bbf.'l:::tVi ct =
' i
l<?il'JYl ct: lbf.'I::: ct- 2 0
. ' '
<if1iu '11 ct, = ct; ~il ct, Liiumn
3. tvr
ct, = -ctl Li1il ct, bi'JU~'U
<if1Llli'Jii<?iliuhY1<ifei1mTYl'l1'Ufl1X. =
J
Constraint 'llmlif1LL'U'Ulio:::m1'Uiii'1tJ Constraint 'llil11ii'1Ll'U'U LP (n1i1) bbf.'I::; Constraint
~m<J1miJ1111mtJ<ii11'1 Constraint 'lliNlU1Vll.Jll'J~ i il~Lu~libb'U'U
fl1'lleJ1lU1Vll.Jll'J~ i (Liiu Linear Function 'llil1lii'1bbli'l<if<?1liuh x. - (ct - ctj
l J l l
= 1:::<ifrni:l1111mtJfln1111u<?1
'
Lfl'lm111mtJ = bii!l.Jil
4. ri1111u<?1'11i'.i1111inm1:1.Jil11"iqJ'll<Nm'JbrltJ1b'UU'llil1Li11111m1.11ii1~'1 .~ei'lVimm1m1:1.Jwf.'IG1y;il
'IJil1m1mrltJ1L'Uulii11'1 1Ji'LiiuYJ1niiu1'<?1fJ1.lo:::ii!1.1 l<?1tJm1ri11i'.i1111ii'rnil"i:l1111mtJJi'm<ii11'1 'lun1rn
~Li11111mtJVlmmi11111mmL<iiC1:::biJ1111:1.J1tJi1Vl'l.i1tJ1'<?11ii11'1 nu
5-, Yl~niiu1'<11rJ1h:::M.1 ,iiuwC111:1.J'IJei~fi1 ct, rimi11111mtJ~.l1mri1~~1Vl'IXnt<?1mi11111m1.1~1
1:::rniiuLfl'lil~111mtJ 2, :,; 111'lil = ":::n1111u<?1fi1 ct +LL!.'I::: ct -11luYJ~niiu1<11lllio:::ii!~.1 (Z) <111:1.J<111'J1~q I l q

' '
f11<i181JY$1Yi')'1'llil'1i'1bb1J1JhiJum'J Minimize '11 Z
6. b~81i'1bilh1i''1liuh mi:::1!1mhm1m~1Nb1JU i'.i')rnffl-JU~<illl-1 Non-negativity Con-
'V P1v J ~ i!i.lil=""
straint (x 2 0, ct+ 2 0, ct.- 2 0) <il1bi1J1J'J:;buU'11bblJlJ LP 'li'flll-11'"1Vllfll<i181J b'1'111'J11l Simplex
l l 1 '
7. fil'll81J'illfllTI Simplex 'J:;bbff'1,fi11!1bbth1i'1?1liuh (x) bbl'l:;fi11!1bblhm1;ibutmuu (ct+
• l l
bbl'I::: ct~) fi'1fi1 ct+ 11118 ct- L'Yi1nrn;rutlbbl11?1,11Lil1111mt1uufi1m'>n1J'>'>lJ1#1m1~limm1 bb~i'i'1 ct: 11118I l , , 1' , l
ct;- i'.i.11 fi1 ct; 11118 cti U'tlbbM1'ri'Nl'IYiLlf''1Jlubil1111mt1ii:u11~1''illmil1Yili8,m11i.Jd"im1rn1m'Yi1
rllJfll ct+ 11118 ct~
) i-' 1 SI.' IV
oiiu'll€lm1111'i1u'J:::$ni,i-11#i''il1n1!1mh'~€lLl.lu
9!1!>BN~ 4.1 Ll.l"ibbm:i.ail1111mt1'liU'1 Non-preemptive
u%i'Ylri1il"-¥1'ill"lrn111"bbNUNii'llllwn1111i.i 3 'liitl?l Ll'lt1'il:;1i1n-¥1'ill"llli1ri1111ui1B'll'Jlm1Nfi'II
'118"Nfi<llll'rno/i 3 'liit1?1d'lVIL111m:;ff;i Lil1111m1.1~1imm1i'.i 3 m.h"#i'1rn'i'u 1#1uri Nl'lri1Ln:;1.1:;m1
v v -
"l:::ill.Jnl'l'ill""l'tl bbl'l:::b"Uil"'YJU
iil1t1~1?1m11#lri1111u1?1bU1111m1.1'lub4€l" 3 #11udli'iifod
•
1. 1f€l,f11"lNl'ln1h'lu'l:;t1:;m1mh,ifo1.1 125 ii'1u1Jl'Yl'illf1Nfi<iill'rno/i 3 m.h,11,
2. ''ni,i-11:;ilum'l<i'1""rn1l~~rn1uv.urn1u 4,000 flU
3. b1Ul'l''YJULi.ibnU 55 ii'1urn'YI
!il1t.J~lilm'l1iil"m"~'il:;u11ljbll1111mt.J 3 mJ1,d'lrn1mb$1t.J1nu ~"ri1111u1?1Nil~'l:::1#1''u'J1n
Lil1111mtJ~l"'l 'lulU'll8"m-;111'~1V1Uf1'11€l"m1;ibff1.11111rn~ilv.m'1bth Li?lt1n1111uil~1111'.inm1;i11tJ
1111tJ~bni?l~Ub~ilV.flli?lbU1111mt1~1''1 il,d1i€l
1. Nl'ln111~uilt1n11Lil1111mt1 1 ii'1urn'YI m1mlll'l1111m'Yi1n1J 5 111u1t.J
2. m-;%1"'1mnun11Lil1111m1.1 100 flu Ln1?1m1;ibilt11111rn1'i1n1J 2 111u11.1
3. m1%1,,muilt.Jn11L1:i1111mt.J 100 flu bn1?1m1~1llt11111m'Yi1n1J 4 111u1t1
4. m"'""'l'Jubnun11Lil1111m1.1 1 ii'1urn'YI Ln1?1m1~bllt.J1111m'Yi1n1J 3 111u11.1
Nii'llilrno/i 3 8tJ1"d'J:;riil'lVi'Ln1?1Nl'l~mil1111m1.1~1''1 'lu#l1uNGin1hiiuNfi'llilrno/1'1iit1?1~ 1
'il:::n8'V,n1i1Nl'lr11h1rn:;m1 12 ii'mrn'Yl~il111u1l'Jm1Nfi'llNfi'llilrno/1'1iu1?1~ 2 'J:;nil'lVi'Ln1?1Nl'ln1h
'l:::tJrn11 9 fl1UlJ1'Yl~8VIU1l'Jnl"lNfi'll Nfi'llJlUlo/i'liuilrJ 3 'J:;nilkVlbn'1Nl'lrl1h'J:::tJnl11 15 fl1UlJ1'YI
~8VIU1l'Jnl"lNfi<il
'
'lu#l1um1'11'"m Nfi'llifrno/1'1iit1?1Yi 1 'il:::i'.im1'11""1u 500 flU Nfi'llifrno/i'liulilrl 2 'l:;<i'1"
= .o:I. ..... !I.I ' ' ""'
'lU 300 flU bbl'l:;'liUl?l'YI 3 'l:::~nl'J'l1,,1U 400 flU<il<l'YJflVIU'ltJnl"lNl'l<il
'lu#l1um'Jl'l''YJU Nfi'llifrno/i'liitlil~ 1 'l:::1f<J,'.if11ul'l''YJU 5 '1mrn'Yl~<J111u1l'Jm'JN1i'll Nfi'll-
ifrno/i'liitl'l~ 2 'l.ifb1Ul'l"'YIU 7 fl1UlJ1'Yl~ilVIU1l'Jnl"lNfi'llml:;Nii'llifrno/1'1iul?l~ 3 'l.ifb1UM'YIU 8 ii'1u, ,

0 ' "" = v i: "" &:I
x = 'ill.b1U'VIU11'Jnl'j"NiWINfl111.mwri'liU111'YI 3
3 •
bi:l1111ml'Jtu 3 4f1u'il~b~l'Ju14f~"u
1. Nfltllh
•2. nl"l'il1""1U
3. nl"lfl"'l'JU
tvr z =
125
5x
1
+ 3x
2
+ 4x
3
= 40
5x + 7x + Sx
3
:<:; 55
0 11"'12.J,,,,j'...J
'il1U1um1),! biEJ'V11 l'J'YI bMl'liUbl-lil'V<fll 111 bU1'VIm l'J
z = 5 (~1U1Ut·Jfltl1h~~1n-J1 125 ii'1UlJ1'YI: 'VIU11'J 1 ii'1U1J1'YI)
'
+ 2 (~1U1Um"1'111""1uYibnU 4,000 flU: 'VIU1EJ 100 flU)
+ 4 (~1u1um7'1J1""1uYioi1n11 4,000 111u: 111u11'J 100 111u)
+ 3 (~1'U'lUnl'lfl"'l'JUflbnU 55 ii1u: 'VIU1EJ 1 ii1U1J1'YI)
n1111u111.11<if1 mh~bbi111"m1:1.1N111v.f11111 ifll'J"b1JU'11niU1111m l'J
tvr ct, = Nfln1hirn4!1mi:l1111ml'J (111u11'J: iiiurn'YI)
= 12x1
+ 9x2
+ 15x3
- 125
ct2 = m"1%1""mfl1J4f1 l'J bi:l1111m l'J ('VIU11'J: 100 flU)
= 5x +3x +4x -401 2 3
ct3 = m"lfl"'l'JUfl1J4i1mi11111ml'J
' .('Vl'U1EJ: fl1U1J1'YI)
'
" 'mEJ"'il1nfl1
= ct+- ct-
1 1
= ct+- ct-
2 2
= ct+- ct-
3 3
ct+ - ct-
i' i
:?'. 0
ct. = ct+
' i

1.
bbYIU'11 ct
1
12x
1
+ 9x
2
+ 15x
3
- 125
(ct+ - ct-)
1 1
12x
1
+ 9x
2
+ 15x
3
- (ct~ - ct~)
2. 'l1n'11 ct
2
'I::1Ji
125
5x
1
+ 3x
2
+ 4x
3
- (ct;- ct;) = 40
3. '11n'11 ct
3
'l:O1Jr
5x + 7x + Bx - (ct+ - ct-) = 55
~ 1 2 £/' 3 3 3
<ii~iiu <if1bb1J1J'll<i~llt]Jmu1u~tlt1.11bbmm~~bi:lu ~ei
Minimize Z = 5ct; + 2ct; + 4ct~ + 3ct; Subject to
125
5x1
+ 3x
2
+ 4x
3
- (ct;- ct;) = 40
5x1
+ 7x
2
+ 8x
3
- (ct;- ct3) 55
x ;::: 0
J
ct+ ;::: 0
'
ct- ;::: 0
j
1umru~lltJJV11l'.1~111'51n1111m~B,llw1 Yi1iJb~mnrni11V1ml'l i11m1n1'1'lu<ii1bb1J1J1Jrb'lim~m
nu<if'lbb1J1J11.lobbmm~'bfti>rf'l'l 11.1 ~''l:ObUU~111~1nl1~l'.lb>1Yn:;<if1bbti1 X
IV ! J '
lltjjV1ltli1bbn'l~b~'bftUU bilil1oiitu1bbn'l~ LINDO bbfllltjjVll'l:OlJiq;i1ileJ1J.il~m'l"lfi 4.1 l'iB
111~ 1um11oiitu1bmrn LINDO 1Jrn1V1ul1'11<if·mtl1111''1 <if~~
D1PLUS
ct- D1MINUS
1
ct+ = D2PLUS
2
ct- = D2MINUS
2
ct+ = D3PLUS
3
ct- = D3MINUS
3

l'i1m1~•ll1J11111J~u1l1J~'l"1 z = 16.666 (16 ~)
x, = 8333(~). 3
x2 = 0
x3 = 1.666 ( ~ )
ct+
1
= 0
ct- = 0
1
8.333 (
2
3
5
)ct+ =
2
ct- = 0
2
ct+
3
= 0
•
ct-
3
= 0
lii1iiu ct, = 0
ct2 = 8.333
ct3 = 0
d
'lf1111~11Jm1~-l1
- biJ1111m1J~ 1
""~ 3
Lil1111m1J~ 3 mm-,111'111<il<ii1~ru1111~11J
- Lil1111m1J~'11l1 (m-,{111111.1) ~1n11111.1"111~ 4,000 '11.1 1i.Jmm"it11'111'1 w1i<J~1fmll
m"i{i1111ubnu11krhfiu 8.333 x 100 = 833 '11.1
bliemmiNli<ii lu~l'liU>1 1 8.33 111u1m1"1Nlilil

GOAL PROGRAMMING NON PREEMPTIVE 1
MIN 5 DIMINUS + 2D2PLUS + 4D2MINUS + 3 D3PLUS
ST
12X1 + 9X2 + 15X3 - D1PLUS + DIMINUS = 125
5X1 + 3X2 + 4X3 - D2PLUS + D2.MINUS = 40
5X1 + 7X2 + 8X3 - D3PLUS + D3MINUS = 55
END
1) 16.66667
VARIABLE VALUE REDUCED COST
DIMINUS 0.000000 3.095238
D2PLUS 8.333333 0.000000
D2MINUS 0.000000 6.000000
D3PLUS 0.000000 0.428571
X1 8.333333 0.000000
X2 0.000000 6.857143
X3 1.666667 0.000000
DIPLUS 0.000000 1.904762
D3MINUS 0.000000 2.571429
2)
3)
4)
0.000000
0.000000
0.000000
NO.ITERATIONS= 1
-1.904762
2.000000
2.571429
RANGES IN WHICH THE BASIS IS UNCHANGED:

VARIABIE CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
D1MINUS 5.000000 INFINITY 3.095238
D2PLUS 2.000000 0.333333 2.000000
D3PLUS 3.000000 INFINITY 0.428571
X1 0.000000 0.600000 3.6000000
X2 0.000000 INFINITY 6.857142
X3 0.000000 3.692307 0.750000
DIPLUS 0.000000 INFINITY 1.904762
RJGHTHAND SIDE RANGES
ROW CURRENT ALLOWABIE ALLOWABLE
RHS INCREASE DECREASE
2 125.000000 7.000000 8.750000
3 40.000000 8.333333 INFINITY
4 55.000000 6.481482 2.916667
mw~ 4.1 l!ll'lil1Nlllllflhhtm~IJ LINDO 'llM~11!UU Non-preemptive
u cS: Ji Ji .,; SI T-1 _.,,,. <V<:>t.""'
'l11t~'il111H11111mB'l1lll'l'l::;t'leH'l1 4.1.2 mn u'llilu{)Ul'lfl'ilfl'l'l:IJ 4.1.2
' ' ' 'q cS: t """ .<::{ <9. """
mmnfll'lflfllrnnnt1'11 4 l'l!l't.!'11 4.1 !'lB~'ll 4.1.2

• •
ijei~n 4.1.3 111n1111l1ln~fl'j:.J!Ul'ViiJlaH1J'U Preemptive
l1mn;u~'lJ!l'UlJ'V1111.J·:mmmth'VlmtJ'llUl'l Preemptive ~u bU1'VlmtJ~'i1-ol1u1u'Vlmmi:h
V1mtJ •~i1'11~mmml1#1qJ1iJb'l'i1nu~,qrei,i'.Jm':lr1viui'l'l~~um1:>Jil1'1jJ'llil''i11111mtJ (priority level)
•il1vi:>J1tJ~i'.l':l~~um1:>Jil1fiqJ'l~~lJbb'ln'l'll<i'll11•1iu•i11vimtJ~qril,m':l1VllJ'l'l'Jneiu•i11vimtJ'l~~mei'
"'11.J 'l~~lJm1:>Jil1'1qJdm:>JT'H1ri1viu'lll'l'11.J•liu'l~~lJ~uL~'Vlmm~~lJ •'liu •i11vimtJ'l~~lJ~lei'
qfei,m'l1VllJ'l'l'J•i11vimtJneiu.i:l1vimm~~lJ~l1:>J •liuqfu
lUbb'lll'l~..~~lJm1:>Jillflqj'lJ€1,bUlvimtJ~u m•i1bU1vimtJ1u..~~lJm1:>J;!l·1#ljj,@]m nuvimtJ
bU1VI:>J1uf1~ bblj~mm':lllr1viu'll~1viunm1:>J,f[tJvi1tJ'lJil'm'N'll'il1vimtJbb<illj~bU1l~u•~~lJ,@] tJ1
nu1~
'lllm':ibbnUi)JV11Ll.J':ibbmmi:l1vi:>J1tJ'llUl'l Preemptive i:iu 1u~d.~mi11fi,'lllLLniJjJV11~b1tJn
' . .' .s .o:I • .., .., .., ""'
11 Sequential Procedure 'll':>J<n'lllJ'lJWil!lU>l'U
1. uli'iftymm·m::; t~rnmvii1A'urn1:11i1fi'tymn~'!lilft!li! (Priority Level 1) bri!l"W'il1':llli1
b%·tn~bU1vimtJ~!ltjtu'j~~lJm1:>JlihfijJb~tJ1nu iJjj'Vl1d'il~b1irnvii1!lUnlJUjjvil'llUil Non-preemp-
tive .-il<ll!llJ (solution) Yi'l~'il~i'.l 2 '1n1'1lli~
1.1 t'h.-i1'1l€llJ (solution) ~1~i1.-i1<11!lrn~m (unique solution) 1iJ-ol1b1iuqf!l,'W'l1'llli1
n"i:l1vi:>J1tJ~U'l ~lltjlu'j~~lJ'l!l'lj'mlln
1.2 ii'1.-i1'1leJlJ~Ll'lblimblJlJ'VlmtJ.;1'1l!llJ (multiple solution) vi~!l.;1'1llllJi'.11~iJ-oj1fi''ll
~'<ii1,f1Vf.-hbrhnu mJ,bliU 2 mrn fiei
fl. ii'1fi1 Z = 0 'il~qfei,l.J'1,$11bblJlJLU Priority 'l~~lJ'leJ'ij'11.J l'lltJ1Vft<ifi1$11
bbl.J'lbbll'l,m1mllmblJU'J1rnil1vimtJlU'l~~lJ Priority d 1ufi1 Z brnnmfi1$11bbl.J':lbbl'll,m1mfltJ,-
blJU'i11rnil1'VlmtJ1m~~lJ~vitl,eiein•1nfi1 z LVl'Vl:>Ji'l
l:>Jm'lbb1l'll,:;i'll-ol1n'll'il1m~~lJ~vitl,iJ,'"'11'lu$11bblJlJ LP mJ
•
'lJ. ii'1fi1 Z = Z* ~~ Z* bliufi1$11blj'IJ ll'ltJ~ Z * > O$11bblJlJLU~U~1i!l,d'l~'ll1
bmbU1vimtJ~u~lei,dboif111.J'l1:>Jnrnil1vimtJ~u~viii, iiufi!l blii:>Jfi1$11bbl.J'lbb5"''m1:>JblltJ~buu'llll~
bi:l1'Vlmm~~Ul!l,d •oii11iJ1u$11bblJlJfi1 z
,yj!lLiJ1Vfi'.1$11LL1.J'l'll11mlltJ"lJu•1n'l~~rn••nl.J':l1fl{]!l tj1ufi1 z mm·rn$1"'
o/'1bbiJ'ldei!lnL~ l'llm~:>J1i:>Jm'l:;i'll-ol1n~1boii11i.Jlln 1 l:>Jm'l iiuiiiei .~:>Jl:>Jm•:;i'll-ol1nl'l
objective function 'lleJ'':l~~lJbb'>fl = Z*

'1:1-1m,,;i;i>l1ni;id,~1'111Vi'ft1m"lnlii'i;ilif1LLthm1:1-1•fltJ-JL1Ju"i11m~lifrn•'>nmm1u141
ti;itJ~tf-Jfl-Jdeiul'lll'il-l'l 1ulii'1 bL1J1J'>~lifrnL&flB~LV1lleim~:1-1
lii'1LL1J1J~"ii~l4l'ii~dJui1tjjVll LP ii'ulVllJ ~-lftl:l-ll'lbll~'lii Simplex bbflU!JJVlll4i
' . 'V ""' _if "" <=t.::I <> "" l.IL I""'- 'JJ <V IL A !IA=>.<=t , <V ""'- 1.11 V.d.
2. tlllJ!Ul'l1:1-ilffl~~U'l'l:l-li'l111.iilli'l'Jl"l!M!M !U!lfl illl.ll"l<l1Hli.!mlU !U!'lf11ifll':i 'll'V'll'l'l 1 !~011
<>Jun11<>J~1J.Jll Lil1V1mm~lif1J'>B-JM1u1Vi'V11.-11llil1J5n .-11llei1J~l4i'ilntJu.-i1lleiJJ~lii~iJll
ii'10rlH~ 4.2 "il1nlii'1mh-J~ 4.1 1J'Jiffl~"il17ill1LLG1'11Vlu11m7~1-J-JlUL~:1-1mnn11Ltl1V1mtJ
LLft~fll'>l'MYJU~mnn11 Lil1V1m tJ LUU~-J~u'Jiffllii'ei-Jm7Vli1nL~tJ-Jmn~ffll #i-Ji:imtl1V1mtJ'llil-J1J'JM111
' '
"il~LLU-l7~#i'1J<i11l:i-iill~!JJBilflbUU 2 7~#i'1J fiei
m1:1-1l11~tiJ!11#i'rnL'lfl
1. m7~1-J-Jm uiltJn11•il1V1mtJ 4,ooo flu
2. -l1Jft-l'YJU lii'il-JlJJ•nu 55 '11u
'
m1:1-11'11~t)Ji11#iu'iiffei-J
1. Nftrill'l"i~tJ~m1 1J.J~1n11 125 .11u
2. m"l~1-J-J1u Lnun11Lil1V1mtJ 4,ooo fl'U
ti;itJ~~lVIUflml:i-1LfitJVlltJ'llil-lfll"ibfl tJ-1 b1JU'illfl LlJ1>i1-J'l fl-l fl-lbVI:J'.i ilUb~:1-1
m'lLu'liJtiJVll'il~'1"11-Jlif1LL1J1J'"Yn ~LUlVIm tJ!110.um1:1-1l11fltiJ'Gi10.1J LL'>flrieiuiiufB'ldllif1-
LIU"ib>1'!'l1~
"1u1W111
ct2 = m7~1-l-ll'U~Uill'Jfl11Lil1Vl:l-lltJ (VIU11'J = 100 flU)
ct3 m'lft-l'YJ'U~1J.Jd1mil1V1mtJ (V1u1tJ:
,,
= ftl'U1Jl'YI)
,, 1 ,,
(ct+ - ct-) bbft~ ct = (ct+ - ct-)m VI ct2 = 2 2 3 3 3
"J~lJilii'1111J1J LP ,1Ju
Minimize z = 2ct+ + 3ct+
2 3
Subject to
5x1
+ 3x2 + 4x3 - (ct; - ct;J = 40
5x1
+ 7x2 + 8x3
- (ct; - ct:;J = 55
x <". 0
J
c' <". 0
ct; <". 0
1%iii Simplex l111'JtU'lLLm:1-1 LINDO LLni1t)J1111d "J~l4f Solution lll:l-lfn'l'lfi 4.2 ~-l
z = 0
ct+
2
= 0
ct+ = 03

5x
1
+ 3x
2
+ 4x
3
:5 40
mi~ 5x + 7x + Bx :5 55
2' 1 2 3,.., i/
Ji'-:iuu ~-:i<iiei-:i·i,b{h111:im.1 2 111'.l1-:iL1-,numdlu constraint 1wliuiilB'tu *-:irnTil11m1iJ1111:1.;1tJ
2 •ifoftmd'Ju Constraint "1~11L~fl-:ifi1 ct; = o Ill~ ct; = o Jr-:iiiu fi1 ct; Ill~ ct; "1~Ll-Ju·rm{IB~1u
i'l:imTniJ1111mtJ~ n
step 1 1st priority input
MIN 2 D2PLUS +3 D3PLUS
ST
5x1 + 3x2 + 4x3 - D2PLUS + D2MINUS = 40
5x1 + 7x2 + 8x3 - D2PLUS + D3MINUS = 55
END
1) O.OOOOOOOE+OO
D2PLUS 0.000000 2.000000
D3PLUS 0.000000 3.000000
Xl 0.000000 0.000000
X2 0.000000 0.000000
X3 0.000000 0.000000
D2MINUS 40.000000 0.000000
D3MINUS 55.000000 0.000000
2)
3)
0.000000
0.000000
0.000000
0.000000

NO. ITERATIONS= 0
RANGES IN WHJCH THE BASIS IS UNCHANGED:
VARIABLE
D2PLUS
D3PLUS
Xl
X2
X3
D2MINUS
D3MINUS
ROW
2
3
CURRENT AILOWABLE
COEF INCREASE
2.000000 INFINITY
3.000000 INFINITY
0.000000 INFINITY
0.000000 INFINITY
0.000000 INFINITY
0.000000 0.000000
0.000000 0.000000
RIGHTHAND SIDE RANGES
CURRENT ALLOWABLE
RHS INCREASE
40.000000 INFINITY
55.000000 INFINITY
ALLOWABLE
DECREASE
2.000000
3.000000
0.000000
0.000000
0.000000
2.000000
2.000000
AILOWABLE
DECREASE
40.000000
55.000000
ill'l'I~ 4.2 lll!Vl~IHllllfli1h111m.J LINDO 'JM,)''IJ~ 1
LblJlJ
Minimize Z = 5ct- + 4ct-
1 2
Subject to
12x1
+ 9x2
+ 15x3
- (ct;- ct~) = 125
5x1
+ 3x2
+ 4x3
+ ct; = 40
5x1
+ 7x2
+ 8x3
+ ct; = 55
x 2 0
'
'+ 2 0
cJ; 2 0

z = 43
3
4
x1 = 5
x2 = 0
15
x, = 4
ct+
1 = 0
ct-
35
=1 4
ct- = 0
2
ct- = 0
3
MIN 5 DIMINUS + 4 D2MINUS
ST
12 X1 + 9 X2 + 15 X3 - D1PLUS + DlMINUS 125
5 X1 + 3 X2 + 4 X3 + D2MINUS = 40
5X1 + 7 X2 + 8 X3 + D3MJNUS = 50
END
1) 43.75000
DlMJNUS 8.750000 0.000000
D2MJNUS 0.000000 9.250000
X1 5.000000 0.000000
X2 0.000000 18.000000
X3 3.750000 0.000000
D1PLUS 0.000000 5.000000
D3MJNUS 0.000000 6.750000

ROW
2)
3)
4)
SLACK OF SURPLUS
0.000000
0.000000
0.000000
DUAL PRICES
-5.000000
5.250000
6.750000
NO. ITERATIONS= 3
RANGES IN WHICH THE BASIS IS UNCHANGED:
VARIABLE
DlM!NUS
D2M!NUS
Xl
X2
X3
DlPLUS
D3M!NUS
ROW
2
3
4
CURRENT ALLOWABLE ALLOWABLE
COEF INCREASE DECREASE
5.000000 INFINl1Y 5.000000
4.000000 INFINITY 9.250000
0.000000 23.125000 33.750000
0.000000 INFINITY 18.000000
0.000000 18.000000 37.000000
0.000000 INFINITY 5.000000
0.000000 INFINITY 6.750000
R!GillHAND SIDE RANGES
CURRENT ALLOWABLE ALLOWABLE
RHS INCREASE DECREASE
125.000000 INFINITY 8.750000
40.000000 8.333334 12.500000
55.000000 6.481482 15.000000
. . ... ,_, ~ ~ ...fllVfll 4.3 !!lf'1~NIJ'llB~m1tm11J LINDO 911tnJ1J Preemptive 'll'll'lllfB~
x, 5
x2 = 0
15
x3 =
4

m1~'lrni:hvimtJ!i1.11Jm1:1-J1!1'1lJJbbon mmonlJoo')Llii'~~ 2 •ll1vimtJ (ct;, ct; = O) 'lmil1-
Vl:J-JltJli1.11J~fi!l~ mmo111Joo')bil1vimtJmo<t'1-J-Jl'U~L~u!ltJn'l1•il1vimtJ (ct~ = O) chu5mil1vimtJ
vitl,~!l N11ri1'loo~tJ~tn1iiu 'il~U!ltJn-l1~<1f!l,mo!l~ _;§__ i11u (ct- = 875)
• 4 1

'
""Vl0'1fi'I 4.2
'
v "'
'tt1!'HFl
' '
L1il-:iYi 4.2.1 bb't-l'lfiilbfltJ'lnrn'YlflUfl AHP
b4eJ.:J~ 4.2.2 lif1 bblJ1Jli1il1J~'U'IJil.:Jnl'i<ii"1ii'Uh
b~il.:J~ 4.2.3 m'i1h~L~uLil tJ'iiJLl.J1tJrnYi tJuYi;;i~.i•
~ei-:i~ 4.2.4 nl'ill'i1'ill1ilUm1:m;ino/lei-:i'IJei-:ii:-rnm'il.J'i~b~u
b~il.:J~ 4.2.5 m'il.J'i~b~UVl1'Yl1.:JLliein~mm~l1i.i
1. b'Ylf!Ufl AHP 'l-iiil1wfom"l<ii"1iiuhmtJ'lo/Ivim tJnlJ mnw'11"1 tJ'loii'lilm'i'lVi~1viiin
m1i.iil1fi'!J'llei-:imi:wriLbi'l~i.J1~L~'l-lfl~bbuu'llil-:irrnLliiln bb~1u1mll•~L~u•1i.inu'~ei
-iile'luliium1i.ibV1m~l1i.i'llil.:J'Yl1-:ibliein Lilm'Ylf!Ufl AHP 'loii'lilmilunT'lLVi~1viiin
m1i.i'11ff!! bbi'l~fl~bbuum-:iLlifln~mi.iT'll'l'111Jl~1tJbbi'l~!Jno/lfl-:in11'lilllu
2. lif1bblJUli1ilu~u (heirachy) 'llfl.:Jm'ilif"1ii'l-lL"Jbllulil1bblJUbbNWlillbl1"1.:Jfl.:J.il.J'i~nfllJ•
'llfl.:J mrn<Yi'li1'il1'irn1m-:iLliein~mm'ilbblfoei-:i.iu•~nflU'llei-:i mrn<YiililnLlluli1liiu~u
t.ifl tJ'l ;;i-:i1umi.ili1ilueit.i1-:i-ii'"1b<J't-1'1'11'lVil11m'ilU'i~b~u~1V1iinm1i.iil1ff(J'llil.:J Lbliii'l~
mrn<Yit.iil tJ ~.:J'il~u1m•1i.iLllu~1V1iin'llil.:Jm rn<Yi~eit.i'luli1liiul1.:J~u1Jl
• •
3. nl'iU'i~b~'t-1L"1tJ"lTibU1tJrnYitJuYi;;i~~ (pairwise comparison) 'loii'lumw'in~'t-IVll
~lVIUn'llil.:Jmrn.ri1"1 bbi'l~nl'iU'i~b~'t-lflrnuum1i.ivmh'llil.:J'Yll.:Jbi'lilnlii1-:i'l Lil tJu1
mrn<Yi1"1V11ei'Yl1-:iLliilnm'li1'il1'irn1Lll1tJubYitJunuYi;;i~~LiltJLll1tJu'YitJu~1V1iinm1i.i
cl1ff11Jvi1flm1u1•mh m1mm.n~l1i.i bbi'.11~-:i'lVlfl~bb't-IULllulil1L;;i'll 'il1ni1iuu1f!rnuu
.Id ~ ' ' ,; ~ <.I~ ~
bJJ'itJlJb'YI tJlJ'IJil.:Jbblli'l~~ l-ll'nl-1bbl1il.:JblJ'Ubbl-l'Yl'in'JfbLJ'itJlJb'YI tJlJ (pairwise compari-
son matrix) '1'11'lVi'l.oii"lllm'i'Yll.:Jfllll1'1ll1"11'l't-lm'ifi1u1rnV11~lVIUnflrnuu'i1l-!'11 fl.:J
mrn<Yi1"11ii1-:i'l bbi'lrn~Lmum1i.i~-:ivmh'llei-:i1111-:iLiieinlii1-:i'l 1Jl

4. Ni1fll"llh:;1i'.iuJ1V11Xnri:;llUUllll-l1&lWl"l1'illlil1Jrl11l-lt;Jn<if€l, (consistency) •irn,m&
1J&~tl'.iu1'111i11J'ii1u1rn'Vl1<1i1iif'l!um1:wt;jn<ilil' (consistency index) it11u11J'l1Ju-
tri1Junu<ri1iif'l!um1l-!Nli11"'11llil (random index) ~,d'Ju<ri1iif'l!umm~iu~ttlllil'm1l-I
Nli11"''1llil'JJil'm'iJ":;1l'.iu t'i1<ri1iif'l!um1l-lt;Jn<ifB,1i.Jd1u 10% 'JJ<J,<ri1 Random
Index lbMM11N1"m&lJ&:;1l'.iu{iul'.lm1l-ltJn<if1Mlu&:;iilu~tm:w'l'u11'1
5. m&lJ"l:;tl'.iu'Vll'Yll"liiln~1V1m:;ll:w'JJ€l"'Vl'1Ufl AHP 'il:O1-ifr<10'1:;1muJ1'Vlilnm1l-I
i114iljj'JJil'mrno/ilJ"i:;ti'.iuli'iui;'i1''1 ~rnnuf:;IUU'JJil''Yll'tf1€lnlUtnllonli'1uiXu'l
tt.i'1it1N1"~rnm&1l-lnud'Jucy]1111'JJ~"lllil'm1l-!tV1m:;lll-l'JJil~'Vll~1li€Jn{(u m'1li€Jn~
l'.lcyf1110'JJN11&1l-I~m n~llilttium'tliiln~t'Vlm:;ll:w~llil
t~B0nmililU~ 4.2 'ilrn1K1 iln0mnmm"ll1
1. Blirnmm1filil'JJil~l'Vlf!Ufl AHP 1li'
2. illiu11JB"'iu&:;nilU'JJil~cyf1lluuii1iifu~u'JJil~nT;icyfIiiiluh tt10:;t:iitJulii1ttuuii1iifu~u
'JJ1MUllJ'Vl1m&cyflilifuh11'1
3. illu11JlTitlJ'i1JrnVitJUVi11:;~l#f
4. ll'.11~1t:w'Vl'ln41u'itJrnfitJutt11:;V11'11ri:;tmuJiwun'JJ€l"muontt10:;m~tliiln1#1
5. il&1'illlilUm1l-l11n~il''JJil'N1"m11J1:;1i'.iulli'1tJm&V11iif'l!um1l-!11n<if€l'11'1• •
6. Bliu1tJ'llllJ&:;1l'.iuV11'V11~tliiln~t'Vlm:;lll-!'i11nfl:;lluuJ 1V1un'JJ<J~mm.vitt11:;f!:;ttuWllil~

1,,1,m'lo/I"1fiuhm1~'ln"il iin<Jdhih111:mtJ~<if€l1m•111111tJ<i11u~'l'1l:J-Jnu 5niif11,i:l1111:mtJ1um1
<if1u€l1"ild'Ju1,i111'1:J.11l'J1,'1l1'Jmn1~~1:i.il'.lml1tJ1'"1 d'Juo/11L'1'll~<lf"1 L<Ju m'i~:rlbU11'1:mtJ~<ifMm'l1'1'11l'J'l
<if1u Lriilw'i'eMm'l~<Jtibfl'l1t'llm-;o/fl?lftuh 1m'1l1li'hnrn ~1<if81:rl':itim1111'l€lo/11mJlJ~U1:mtoii'lum1
o/li?ittuh o/11iltl11'l1€l1m1o/11i1l1uh1,1,1J1Jd €ll"i1~1J1<iflui:IQJmm'lbl'iilnl'i1L'1~~1 ~1:l'.l€l1.'ili'ltnil1Jtu
m'l'Yi"ill'lrul~''€l:J-J'l nu1'1l111'Ji!flU1Jl1€ltll1lld'Jmi:l1mntJ1'1"iil€l1~li'lrn€l1Jb'il1mmru 1,'liu '11~~u
fi1'lluli1 1Jl1€ltl111,l'Ju1,i111'1:mm'll1'Jmfll~ 1,'llu '11'1ufl&i'll€l1'l!:J-J'llu~:l'.llii1ln"ilm'l •l'iu<iiu 1,1,u1~1?1tu
m'l1Lil';lt'llm'lo/li1ttuh1ul'in1'!rntd"ilt1oii'1'lm1ri11'1ui1mruo/lo/11i1l1uh1u<if1ulii11'1 ~1"ilrni11?11~1
1,i:l11'1:mtJ~<i'iil1m1 1,1,'1tui mruo/lo/fl?lftul"ii~11,l'Juiif~mruo/Ib'll1u'l:mru1,1,'1tmru.ri1,'ll1'Jrum~m1bf111t'll
11:J-Jnutum11J'ltdlurrn1,l'i€ln
~1!l1J1Jlll11til'l~!!UU~WIU!l (Scoring Model)
o/111,1,1J1J~uli1u~ill'J:J-J1-ii'lum1o/11i1l1uh~:rl111111tJn{Jbnrno/I 1,1,'1t:rliif1mru'111,'1l1ilrum~1,1,'1tmruo/I~ '
,'1l1u'l:mru-;1:1-Jnu 1,l'Juif1bL1J1J~1,'ltJn-)1o/11bb1J1Jm11VIilt•mui'.ii111un o/11bb1J1Jd:rl€l1.'iu-;tn1l1J~fi1.'fQJ
If ,¢1 'U i<> .:i= ~ I
1. mru'l'IVl'l€l1'11'll€lYl~"il1'lru1muiill1'1
2. m11,1€ln
1,m1~l?l'll€l1o/f1bb1J1Jm11VIflrnuuJ11111Xnd "iltri1111u1?1bnruo/111i1if11ulii11'1 iif11,'1l1f)rum~bL'1t
u'lmru N'o/11i1l1uh"iltbl'iuN'o/11?1fiu-l1mruo/11'1i1<if1ulii11'1 :rli'.f1111unm1:J-Jfi1.'iru (weight) :J-J1nii1lm'i1tJ1• • •
li?i l1i11'J"iit:l'.lm1i1111'1i1'llmi'.i11111'.hnm1:J-Jfi1.'il)J~ri1111ui1~um
J11'1unm1:J-Jfi1#ibl!~"iit1,l'Juo/11L'1'lll1i1'11bU11'1:J-J1l'J1'1"iilbnruo/I<ifrn,Xu'1 :rl'111:J-Jfi1;'(Q!Iii€lm1
o/11i1l1uhmnifom'i1tJ1llil ri1111u1?111?1mritJrnfmfi'l.m111,1,li11tiii'1Jm1:J-Jfi1.'iQJ un<lifo.1:J-JbbU11,l'Ju 5 ~u
iii'1bb'1i1~i1l:J-Ji1l'll1
.0 •
1'1111.Jlfll'lt)! 11Mifn
:J-J1n 5
•fiaw1111mn 4
l11umrn 3
' .. 2f!EJ'U'll11.lem
•'Uiltl 1

,~<if1Jm1:1-1i11ff!J!V17!l~1V1un·mrnnrn'IV:i111d m'ilri1V1u111mm11'111blJubbJJJJ5u'1 n1oi' l'i1u1'111
bli!ln~h''l iiu ~<ii111llu'il~~'il17rnl<mmnrn.YP:i111w1i;;i~Jl1u bbii'1ri1V1u111fllfl~bb'.u~rm,11flnii1-l'l 1Jr
~11 rmbiifln~~Vl7fl~~<i1111iuhllm1:1-1.W,1'1!lhmn "~Loi'flrnuu~~P fi1f!~bb'.'.~m'ilri1V1u111bllu
5 7tilm'lim~tJ1n11~1V1iinwnm~1ffb))V17m1Ju:1-11<1171"J1115u'1 nLJI b'liu fl~bb'.Ubcyjm'Yi1n1J 100
'l~bmu blluliiu
m7<if111llu'l'.b~fl-lJ l'VIUnm1ml1ffbJ!'llB-lmrnoii'Vl7flfl1.1hrnfl11bbIii tflt bb'.u'llfl1'Yrnb11fln'il~<ii111
lluhl111tJ'loiim1:1-1il'in'llfl,~<i1111lluh
t'i1'lVi wi = J1V1iim111:1-1i114ib))'llfl1mrnoiiV17flfl1.-11h~nfl11~ i
""' d = JU i:
r.. = fl~bb'.'.'llfl,1'11-lllilfln i b:l-lfl1'1'ill7rn11111'.bnrn'i'I i,,
Si fl~bb'.'.71:1-i'llfl1'Y111blifln
'iltV11fi1 fltbb'.UTl:l-l'lleHYll-lllifln i 1.l'i11n
fi1'll1l1fl~bbuu71:1-1 (S
1
) ~1.l'iitbllufi1~bblil1111m1m'VIm~lil:i-i'llfl1'Y111blifln 'Yll1bliflnfillfllfl~-
"" I ,:II e:I"" 2/ I
bbU'.71:1-imn 'i1~:1-1m1:1-1bV1m~lil:1-1mnm1V11,blilfln'Yl:1-1fl~bb'.U'l1:1-1ufltJm1
"11iiu m'l<i1111lluhm7'liBn'Yl11bli1ln~i1fi1flrn'.u'l1:1-1mn~!j111
l'l'll'IUl'I AHP (Analylitic Hierarchy Process)
'lum'l'loii<i11bb1J1Jm'l'lVlfl~bb'.'.J l'Vliin b~B<i1111lluh biifln'Yl11biiflmii1'1 iiu lloiffl'11l tJfl~
:LJ7~m'l'Vli,,;(ufifl ~<ii111lluL'iJ,,~ififl,'l-ii'l'il1'lrn!J!1rn'lum'lri1V1u111J1V1unm1:1-1li1ffb))'llfl1bnrnoii1111V17fl
B"'1hrn1l11 bblilrn1'lri1V1u111fllflrnuU'lJB,'Yll-lbiifln'111'1 m'lril'Vl'.111fi1d111,fl~'ll'1tJliln5tJ111l'i1u
U'ltn1l111u.11u'11''1 'VllilltJfl~11 ~1~<i1111lluh'ii~ififl1iilm~'ill'lrnl bbii'1.,1:1-1nub'V11lril'Vl'.l1lflflnm
blJufi1<i11blil'll bv:itJ,fllb~ tJ1
b'Ylf!Ufl AHP bU'.b'Ylflilfl~ii1111~u~umb~fl'li1tJ'lvrm.,'lV1J1V1unm1:1-1!:11ff!J! bblilrn1.,'lVifl~
"'.U'I'll'blifln lillm'l1rill.1~1tJbblilt!Jnifi'1Mmn~u l111tJl'.J'YI']1'Y~'Yll1flrn<ilf!llil<i17flii111tJf11:1-1!Jno/ffl-l
'llfl1~u<11flu1.1"u1ii111~!:11fftlJfl ~'lu~u'llmm'l1'i'il1'lrnlri1V1u111J l'Vliinm1:1-1i!i1fftlJ'Vl7BfltbbuUb'YlflUfl
AHP 'ilt1'*1llm71'i'ill'lrnlbiJ'fornfitJ1J'Yll1blifln'Vl~mnrnoii1111'11''1 filil~~ (pairwise comparison)
d ' fijv ..t .=i Gj!V ~ ..., ooj!V v 2f. .., ~ 6lv
'li"1t'll1tJ b'Vlnl'lbD'ltJ1Jb'YItJlJbblilt b'llfl~bb'.milumblil'll'Yll bl1l!Jn<ilfl1bblil~lil~1111nmrnm '111i11'illnuU'il~ b'll
"illm'l'Yll,flrn<11'li1<117'lum'l~'il~'l1:1-1fl~bb'.'.<i11b1ii'll'11''1 Bflnmbllufi1J1"11iinm1:1-1i!i1ff!J!bblil~fl~bbuu
'111:1-1b'llm~lil:i-i'llfl1'Yll1blifln b~BiilLiJ'loii'lum'l~111ei'u<if11'lu~u'ilfl'.!j111Yi1tJU"ilim'l'iltll;;u;i-1'.bbu1ii111
b'Jim~tJ1nJJ<i11bb1J1Jnn'lVlfltbbuui±ubfl1

.v ~
'll'Ul'lll'U'Ull~!'flfl'Ui'l AHP
h'Yl'1U'1 AHP LLU1LU'U%'U'ililUL1'1aj1"1'11d
1. lll'l1l'i'N~h!!1l1JtllA1J~'U (Hierarchy) 'llil1Lnn1o/i$i'ilftUh;fJ1"J:OLUU$11LLUU~bLIW11--J1i:JllJ1'11
m'l$i'il1'1uhu'l::niluJl1mnm.vi<if'il1'1uh (criteria) 1m4eJ1iii11'1 il th1h mm.vi1'ilJi1uiii11'1 rnli1
iXui1m1l-lif:1-1iiiuiinuiltJ11h u~n<J1ndii1LL<f'il1n1'Y111Lfiiln (alternative) iii11'1 ~i11l~
-· >">; Q <I.I ., u <1 ""'
2. lll'lu'l::!l-l'Ui'l::!lll'U'Ul't!'Ufli'l1Ul1l'lflty'lllJ~lflW'l'I (Criteria) Lb'1:0'1:0LL'U'U'111l-lb1'1:i.Jll;Sfl-JVl'lil
m1l-1~1V<ilh'llil1'Yl11Lfiiln (Alternative) iii11'1 Tm.11oiiiilm·nmE.Jml1E.Jul1ii::~ 1u%udmi;iil,i;i AHP
if1i1'lilfl'l1<J<filUm1l-1IJnJIil1'llil1ni-·nu"lE.Jubl1tJu11i1mll-llJnlii'il11'1"ieJ1i.JJf1!J
3. m'lml'im::ut1t1'>1t1'111Hul>i<1:;'1'lN1~1J11 'v1eJu11u'1i'ilii1i'i1.1m1:1-1,,,,m::'1l-l111"leJm1l-!~1weJ
h ;fJ1<J::L'if1'1fi'nm·n'6m&lmilu$11LLuunTiLVl.-irnuu
'l1E.JiinlitJ'il'IJeJ1%ufleJu<>i111 <J::nli111m4il1'1iil'l 1u
' ' . .1 4 ' "" "" ~ .,,,,
'UU'l.111ll'lflfl'lfl't!'U11'J'l'l 4 l'llJ'U'l'l 4.2 !'llJ~'ll 4.2.1
• • •
ij6'.ln 4.2.2 v111iu1rii1~u'1ru'lJ6'.lf11'lvl'~irui~
%uflil'ULL'lf1'llil1L'YI.-iil,i;i AHP LU'Unl'l'1'111<if1LLUU'llil1U'lJ1'11nl'l<if'il1'1uh 1'ilm~:1-1<J1rni111'1m1J
'l1l-l'llil1m'l<if'ill'iuh LLii'1LLU1ililrnlJuli'ni,rrn::Jf1uiii11'1 ~ii1m1oitlum'l$i'il1'1'Uh LfiilmmLfiiln"1111
,yjil1Vl1JiLU11'1m1Jflll-l~lii'eJ1m'l mrno/ili'ni,rm::Jl1u<>i111 d mm'lti~<J::Lbu1ililnLllLlluli'ni,rrn::tlil!J
m1u5n1Ji <if1i1iu mm.viii'ni,rm::~W"Jl'lm11um'l<if'ill'iuh~1m:1-11'lm11i11lilrnlJuii1Jlu%u (Hierachy)
bDU'll1J'1::L5!J'il<i11iJ 'illl-1~~<if'ilftUhlifil1nl'l
<if1LLuuii1'1uiwJJil1m'l<if'il1'1uhd il.tJrn~!JULD'ULLVlUIJi1 1'ilE.Jrf11l.l<J::i1li'ni,rm::J11mv;~ 4.4
iiiil1ud

bU1Vl:J.J1tJTIIJ
'"bflllb'l'll1f11'fru:'.
Yi'Yi"il1-;rn1
fim;·rn~lliltJrf111hm-J4f1bblJU!i1Jfu~u"il~Ll1ii1'1bthVlmtJ'n:w11ilf1uuu1!i1bb'1~"iil'1'1'1m"ilnb1!i1'1
'
ffm,;rnnrnwn"luii1JfuLL"in ~""~1oiflum·n'l"1-;rn14f1i1l1uhL~il1Vi1illbU1Vlm tJ"i1:J.J ~'1ill<rii~LbD'11ilf
dJuV1rn1bf1rn.;Jilf1tJnu b'liu ffm,;rn~~ 1 '1m1rn~~ 2 Llluiilu bf1rn'l'l"luii1Jlu~udm"l"l~l'i"1"rn1
lliemlliltJ1-ifflni,;rn~~iltJm1iJ5n ~'1ill"lbb1ii1~11luii1Jfu~ufli1'1~1iJilf1uti1~ b'liu ffni,;rn~~ 1 ill"l"l~
l'i"ll"iilll"J1rnnrn.;J 3 i'n1'trn~ 1'1mi ffnlffii~~ u 1.2 bbl1~ L3
'Yll~bi'1iln"11~'1 ~i1iltj"J~bb1li1'1114huli1~'ji1'llil'1bbNU1Ji'l m1Loifb'Ylf1Ufl AHP lUf11"i4ili1l1uh
"l~iliirntJi1f1 tJlif1il ~1-J.iei1i.Jd
l'l1llth~~ 4.3 1um"ilifli1l1uh bliiln~il"in ~u1m"l'llil-Ju1ifflbbvi~wd-J1'11'l"l1"irn1i'l-Jm'1biiiln
"11~'1 "il1mn~i1'll1tJLUYJil'1fll11i1bbi'1 '1'jiJ1ilf11l1·rnil~ 3 bblJlJ~b.if1'!J1tJLum"il'i"ill"lilll 1ilfmi "i!.liu
A, B bbl1~ c li1tJ~bf1ill'l'l"lum"il'i"il1"irn1bl'1iln"ild"i~niluilf1rnnrn.;Jilf1u.!1~1 4 '11u.11tJnu 1ilfbLri
1. 'llfll
2. 8'1i1'llfll"iloiib~ilb'>1l1'1
3. m1:J.J1!~mn1irntJ
4. ~iJ'Yl"i'1JlltJUilf1
<ifil:iJm~mnuwff-1 3 bllJU LL11i1~fll:J.Ji1l'll'1fi 4.1 "1il1i.Jd

..m~rnm1
n1A 111 B 111 c
"1'11 1,200,000 1,050,000 92,000
illl"1111'll-ii•~mVI~" (n:l-l.illl'l) 8.5 9.0 11.0
d~
1'1ifomu1"c/11.1 m1~e'.11 • ~ d
b1J1~V11.I" f.Jl i.imm~iJ
lfl4a"•1lrn 'iVll'Jll'1~lfl'iNLii1.1 DVD 'iVl~ll'1~lfl'imbi1.1 VCD 'iVll'Jll'1~lfl'i1l"lii1.1 CD
~fl1"llit:db1 '111.11llnm" 41h~'J '111.11llL~n 5 U'l~'J 'll1.111ll~n 4U'j;;llJ
Lum11i'l'~b!ium"'i11lnTm11.n'mrwri111"'1 0" 4 #11u~u mrn-Yi#l1u11mLL'1~il.i'l1m1l.n'b~€l
b'1"4ft" i'.l-iim.Jm~"tl'l:J.J1rn~'1~'li1EJ1Vi1hn!iu1#1 bblleJ81"hn.i11J mrn-Yim"eJ81"m'1<1i€l"1-iim1:J.J-r~n• •
1um1<iillfiu11'Yl1"bl'eJnL"1'1~i'.li!1i.ium.i1nn11nu Liu -iim;1"•~tnnu'l'1'11 w A '1~Lbw"n11111 B
€l~L'!'i1n1J 1,200,000 - 1,050,000 = 150,000 'Ul'YI ~<illlRUh'1~<1i€l"ti'l'~b!iU11m1:1.Jlb<1Jf1ll1"d'1~i'.l
l'l~lbUU'IJ€l"n1"l<ifllftU1'JLi'€ln A i.i'ieJ B blJUL'Yill11
rl1i.i1'umrn-Yi#11um1iJ<l~111n'11J1!J LLii~#(1uit1vi1"m!JU€lniiUbtlmnrn-Yi<iiII iiluh ~<Ii1n
oW'Jnrn11Vi.irnuurl11111'u'Yl1"bl'eJmLllii~#f1u L11!Jnl'JHm1:1.J~~m'Yi1~u iJqJ1111dmm'l'11b'llEJmlJu
<ii1bl1J1J li1;'iu~u (AHP) 1#1.i1:1.JJ11W~ 4.5 lleJ1tid
mrn.ri)1111uh "1'11 11ll'l'lm'll-iiL~mVil" !1111J'1~111f1'11Jl~ sU'Yl'l"ll1~1.lllf1
•
li1'1uofulb'lf1
~
'Yl"l'11lf1
'ltl A 'ltl A 'ltl A 'ltl A
'ltl B 'ltl B 'ltl B 'ltl B
'ltl c 'ltl c ·me 'ltl c

moiii(illh1h •~tnnumru'll".!"t~m m•<limY1•11ru1mru'lltlil m,jmtJ'l ~m1hrnilunu
•°liu •nru'li'~1u'i"1miXum<Ji'.lil~fl1htnilu1u~1u'ii1~'1 1~"ri 'i"lrrn<ll:J.J b-lu(1111ii fi'l'l'>l(ililrnlltJ '>rn~
nmm'i"t.JiJu'Jl11t il~iXu iii1LLuu AHP d il1<Jb~l'JuLL'1"1~mru'lliii(ili{uh1uii1"1u~flil~il(il1uihv1fo
mrn'lll>11u,1m11>1dJu ml'<~ 4.6 'iiil1ud
" .
li1ii'uoilui'i01m
'V11~"1iiln
'l!1 A
'l!1 B
'lll c
'l!1 A
'111 B
'1!1 c
'1Tv1'i'mnrno/irn,iii(ili{uhii14i'u~uLL'>n~m~ilBn 3 iltll~ 1~..n fi'"1Tim'i"L{i~mw~~ m1:J.1
fl~(il'Jnflu1mrn~iu'Vl'i"~mtJuilniXu m<Jb~tJumrno/iiii(ili{uh1uli14i'u~u~flil~L~:J.1L~:J.111>15rn'llunu bb'ii
hi1~LLfl(il~11'lumw~ 4.6 ,v:jil1i.i1Vlmw~~~mnmrnnu1u
'i"ltJ'1tLBtJ(il~U"1ilum·J"i•m1~ii''llil~•'Vli1Ui1 AHP 1u111u11'Jd<Jrnci11L%Wltm•1Li1'i"l~ii'~ll
li14i'u~U'llil~mrn'li'iii(iliuht1'il'J~li1eflu-i'utb'lmrhilu (W11:J.1mw~ 4 5)
~ "" J! .. .; " ~-· -·~~~'l1tHIJ1flflillfl!'IHl'l11ll'l':iii;IW~'fl 4,2,2 mn !ll':iYIU{)'lJl'lflllflnll 4.2.2
1 <:!l 1.d d Jd
'IJll'IJ1fll':iflfl'lfl'l1'1J1tl'fl 4 ll!J'IJ'fl 4,2 !':i!H'fl 4,2,2

L~il1'1-ilil'l'11<if1LL1Ju91<ifu%u'llel,m'l<ifilfiuh ~'1h::nelu<i11tJ'Yll'Ll'ieln.11''1 mi::mruo/i
<ifilfiuh<i11u"11''1 ~'1.h::rn1unudJu'9w'f11%umi'11um'l1Lm1::ViL~elVl'Yl1'Ll'ieln~$1~i'i'iliiu t'.f<ifilfiuh' .'ilt<ilel''1'11mnl'l::Li'.iufil'fl::LLuuJ:lVliinm1i-i;;11;'1ilJ (priority) 'llel'mru"1<ifilfiuh Lrn::m'Ll'i1ln"11''1
~''il::i'l'lm'lmh::Li'.iu1#H1m'Ll'ielnlildJum,Ll'iiln~$1~')il ~U'1elUm'l'°iL'fl'lltVl"lu~ud~,mm'lrimJ~
1<ifLlJu 2 ~u1Vlqj <iJ,d
1. m'l1h::Li'.iU'flnLuuJ:1V1iinm1ml1;'i(jJ'llil'mru"1 (criteria)
2. m'lthni'.iu'fl::Lmuil1V1Un'f111i<'l"JelhV7elm1i<LV1mtl1i<'ll!N'Yll,Ll'iiln (alternative)
'lltJG'inBtJil~U;lilUnl'lU'ltbfoL'fll:bbU'j,'llil'mru"1LLG'il:'Yll'Ll'iiln1i'ilim'l~bVl'lilunm7tJWJl'°iTI
m'lLmtJmfltJllfl<it~ (pairwise comparison) ~''ii::illirntJLiltJ<i::LBtJil1u~um'lti'l::Li'.iuJ:i111iinm1i-i
ii1ilt)J'llil'mru"11il ~U'1elud'il::1<if<if1Ll'i'll,li111iinm1i-i!l1;'itJJ'll!Nmru"11il ~''il::ui1t11oii'lum'l<if'1fiuh
fi1umnl'ltLi'.i'U'fl:OLLuum1i-iL111m::i'l'i<'lJil,'Yll'Liielniiu 'ill:ilTilJltJTiltinu-'1''1LUel~'illn1i'ilim'l
bburn$1mnu
_I'::! """ J ' .::!. f d'U
lll':l!u':l!J'U!'l'll'l'U!flfWllfiH1 flll~~ (Pairwise Comparison) 'll<l~!fllll'l'l1'1
1u~wd~<ifilfiuh'il::<ifel,,,i1mru"1.!1''1 ~LllwiielW"ll'ltlll<if'1fiuh mLti7tJlJlfl tiunu TiltJtin
mflm;~LL<i::w'ill'ltll1Lti1tiJJLfl tJJJ1Vi'fl::LLuuJ:1111iinm1i-i!l1;')'!!
m'lLmtJJJ'fltiumru"1f1"::~d ~<if'1fiu'il::Lti7tJJJLfltJJJfl~~"::i'l'el'mruo/i1ilLvhiiu1'1tJhiui
mru"1llumW"ll'ltll1<if1tJ mru"1<ifilftuh~u1mLti7tJuLfl tJJJill"lLlJmnrn"1L~'mmru11171lL~'flrnrn1'1
n1Ji !l1111'l'rnnrn"1L~'tl::lmrn L'liu 'llm !l1111'l'u~<ifilnuhLL"111rnu~1u1u'l1m~11JmL1'1'"11~nu m"l
1Viil1111iinm1i-i!l1;'iqJLlJL'Yi1nu L'liu 'l1m~.!1~nu 100,000 JJl'YI 1117el 5% 'llel~m,Ll'ieln 2 'I'll~ ~
<ifilnuh fl'U1111i~m"l LViu'i1:l1m1i-i!l1.lilJmn lu'llrn::~5n'fl'U11111~elH"l::Lvt'U111iJ"11~numnn1<if
<if~,xun~bb~-lHtbU'Umru."1b~~tl::lmrnn;lli-1
1uiu'1el'i-1dt'.f<if'1nul"l"l::l'lfoml'!~"l:::l1fi1utum'>ri1111uilfltL<uuJ:i111iinmii-i!l1;'iruL'liuL$1m• v
~ '~
nrnnrn'YIL'li~fjrn.nrn
nl'lLti'ltJrnfltJJJ.tf "lt<ifil,W"ll'lllilLU'U 2 ~WllilU ~el
1. W"ll'>rn1-l1mrn"11illil (tu 2 mrn"1dJ l'lfl11i.i!l1;'iqJmnn-l1nu ~~~<ililnuhfiil-l1
c OJ 1 ...i A J
mfltJJm1elnm'Li11ln111u'
2. t'.i<if'1nul"lW"ll'ltlllri1111uilJ:i111iinfl::LLuumii-i!l1;'itJJ'llil'm'lLti7tJJJLYitJJJmrn<rlf),Xu11:l1• •
il1111iinm1i.i!l1;'iilJLL'1n"11~numnuilmf1tJ'lil TiltJittJi-1ri1111uil1Vi:l1'l::<ifJJm1mL<11n.J1,111iin'l 5 'lt<ifJJ
<if1tJnu '1li.J'11'l1'~ 4.2 "11l1t1d

1~"111F111iJti1Aiy ft~A!'llU
d
mn'Y16jl1l 9
i.nn 7
•!leJU'lll~l.Jln 5
i.hurnrn 3
1nii'L¥irnnu 1
if1vifoil~Lmu 2, 4, 6, LL<!~ s 'liitr1wrun;m~~<il<?Jiluh1h~d'.ium1i.1'11ffqJilaj'l~vi11~
'l:;li!u~n1viuilli' L'liu 1h~Li1u11m1i.iif1ffqJ:i'.lm1mLWJnlli1~nuli1~ LLllim1mLWJnlli1~iXu1i.Jii1nu'l~
lii'uu1un<11~.Yh~m il~Lmu~'lVl'il~b'Yhnu 2 d'Jwifu <ii~iXu.,~ifuil~bmu~i'ttJi.i'liinu'l~i1'l1i.1~1vii.1il
~ . ~
9 'l~ilUil1tJnu
fl'J!lth~~ 4.4 <?ll:J.Jlll'W~ 4.5 'lJil~lihmJ11~ 4.3 bliU<if1bbUU AHP 'llmU'!JVllfll'llii'il!luh
LlieJm11~1i1'Yl11blieJn 3 'Yll~ 141wi 'l11 A, B, c 11iltJn1viuilmrn.;J~,~VJ'J1'lrn1 4 4!1u 14imi
1. 'llill
2. el<?l'llJ11'lb-ifL~ilbY<!~
3. il1l:J.J61~il1J161lJltJ
4. lU'Yl'l~mtJueJn
m'ldfoubYitJrnnrnon1il~~
il11i.iif1fl'!J'llil~bnrn.;1ilLU'U~'l <ii~d
'llill nu ii'WJ'llJ11"lb-il'b~mY<li1
'llill nu il1l:J.J61~il1J161lJltJ
71ill nu lU'Yl"l~llltJueJn
iflil"lllll"lb-ifL~ElbY<!i~ nu il1l:J.J61~il1J161lJltJ
il'WJ'l1m.,'loifb~m""li~ nu ~u'Yl'l~mtJuiln
m1i.161~mn61mtJ nu ~u'Yl"l~mtJuiln
'lum'lLU'ltJlJLilitJ1Jmrn.;1il~bl'ln ~lii'il!luh~lil11 "l1mi1m1i.iif1ff'!Jl111iJW111m'lL-il'b~mY<1i1
'l~<ilum1i.iif1ffqJd:i'.lm1m1WJnlli11nu'lu1~<iluu1unm1 <i11~uil~1mum1:J.Jl11ffqJ1'Yi1nu 3
1~mu'ltJrnYitJuil'llJ'YJn~"i11 mm1116111lN1"m'ltu'ltJrnYi!'JUmrn.;1il141m:J.Jill'l11~ 4.3
wie:i1ud

n111il111mn111J
<!.:;:!, c u I
!Olll'l'l'll!l'li'l!Jflll i'l~!l'U'U
•1ri1 - €1m1rn•l~L%mwi-:i '1i11 3
.im - m1i<!!~i1111!!u11.1 .im 2
.i111 -•tlm-:im1.1ua11•
'1111 2
€1<1mm•HL%mwi-:i - m1i<!!~'1111!!1J11.1 i111l-Jl1~'11l1!!1J11.I 4
€1<1mni.HL%m'V'ii-:i - ~1hmm1.1ua11 ~rmm1.11,811 4
m1i<l!~il1l1'11J11J -~'Y1'-:il111JU8l1 ~m-:im1.1uan 2
m1!f~N!!ll'l'l'lll'li1m11mn1m (Comparison Matrix)
bbi<'Yl'ln•fonl'.lmfil'Ju bUUbbi<'Yl'ln~~bbfilil~i1~Lmum1i<il1'1ll! '1~iXu 1.i•-:ifi-i'l~'lleNbbi<'Yl'ln~
'~bbfii1~Lnmo/i1'i1ifi'1u'li1-:i'l ,;(~1mLtnbbfl~fli1i<i ,)~,u Lnm.ri1'1~1u1u n Lnm.fi,~fi-i'1~bbi<'Yl'ln~
Liffornfiuu1ifii1'llu1i1 n x n mi<rn"I~ 4,7 <iiei1t1d bbl!i1~bbi<'Yl'ln~~i1bl1m.ri1'1 4 m.h~
•d
blllli'll'Yl 1 2 3 4
mm..~ 1 a,, a12 an a,,
2 a21 a22 a" a"
3 a,, 3
32 a33 a34
4 a,, a42 a43 a44
f1~"fli1~1mLi<'Yl'ln~Lliuf1Jl'llUnritbbuu~Lni1'1nm<Jibbnrn'll1uLL'lifltmn1tlLti'll'Ju'Yiuu
lllJ bb'lifltbl1lli'll1Uffi1l-Ji ti1tJ~bl1lli'i'l1mm1iXu'l fi1m1i<fi1!lqJmnn-J1
mrn~.i11i<il11l'llJ'llil~bm1i!u'l i1m1i<il1ffqJUe:JtJll'llbl1lli'i'l1uff'1i<i ri1J1'11Ul1,tbliurl1u
niiilJ'll!l~rllJl'llUn~bbl!il~1nU<ill'Jl~tl'Jtb:ilu
nn!!-i'1~bbi<'Yl'ln~ilmiirl1ritbbuuJl'llUn~tl•~d1u11m1rl1mLi<'Yl'ln~'1I'.!51%U<il!lu'1~d
1. f1Lmbu1mm~'1ni<i<uulfi1ufi1LL•n (a ) ,,_.ri~ (a ) i'.ifilbr1nu'llii~ Llt!l~'1ndJum'l~ ~ 11 nn
bti'Jl'Jrnfil'JlJblllli.fiUU'"] lllJ~1ifUbiN
2. '11ritbb'Uu~tl•tb:ilu11LLii'1il1'11~1Jm'JLti'lurnYil'J1Jbb'lililt~u1 mwor'i~ilfl11titi1fl'tyrrfoih1
~ ' ,.) ~ ' .. 1'1 ' • '"1"1,.1~ - '
!!tl1'UM!!l.l'l'l':iil'll '11UEJmnm'i'l'llU~'tbuU!!i1l-J11 Uli1tbb'UU'Yl i1 b~ b'Ubbi-l'Yl'Jl1'1i
3. rl1ri~Lmu~b'll~!l,X,_.,~Lliuiiinl'1mtm·ntl'lurnYiuu~niiiunu b'Jiu m<Jri1'llu"1-l1Lnmo/iifi1u
.i.i1i11lllJn-l1Lnm.riifi1u1t1"*1~rnl'Juein1'1l'JilfltbmuJl'llumr1nu 2 '/Imum1i<-J1 Lnm.ri1t1"*1-:i
rnl'.lu!ln,ti1J1viumr1nu l. onti~bl1m.ri~1u<J1m '1~iXur11fltLmuturl1u~L'll~ti'tmm-:;llLrldJu
2

l111.mi1m1eN8rnuu~i.h:;dlu11'l1.1m;1~ 11'111t1tl1~~1t.1$it1111:1-Jl7bl'i.ilf!Wuu~i1t1tjni111•.i~l11m•i11
•~:1-J1wfo~%~11:1-Jm<mi'rnm11'1"t.J~~ilrll8:;"uuvvhi'11V1d~
.~t1u1.-i1>J1nGJ1•1~~ 4.3 :1-J111~h~d'Jm•:1-J1'1'1mfoJ'1mwntJ11v;-fa:1-J~~1i1.-i111.1 Cell 1'1"m:l1:1-J
>J:;LJi":1-J1'1'1n.fiJl~m~~ 4.7 '1lt11i..Hi (•~t1m1:1-J11:;011m:;1oii111•'1'll 1, 2, 3, 4 LL1'11.lmrn"1~~ 4 t1tl1~
L'im<ill:l-J~lJi'll)
1
2
3
4
1
2. fi'GJ11
m11•1~m~fi~
3
1
4
4
1. lmm11'l"tM.-i1LmL:1-J1'1'1n.fiLvhi'11 1
3. '111:1-J
i:;mmirnt.1
2
1
2
4. ~m~
mwan
2
1
2. m1Lll'1t.1rnfit.11111mtl14i'blJn11ilGJ11m11oif,~m~fi~ Ji~.&1.11oii11md'lmu.1111r;i:;ifGJ11
m1Loii1~rn~fi~•llu11;i:1-J/i •hlmL:1-Jmni•vi1ff11 3 (a = 3)12 •
3. m11i.J'1t.1rnfit.11111mtl1fl'lJn11m1iJ11:;0111111111tJ Jl~iu1oii11m•llu••m "":;m1:1-J11:;;i1n
11rnml11.111;i:1-J/i .-11l'U":1-J1'1'1ni•vi1i'11 2 (a = 2)
0 ' & i.< 13 ! ft 2.- A o .... i,I F (0
,51iib~:1-J.-J1fiLVIllil'lJ~Ll'J'UUl'VIU118W1.1umrnflLnl'Vl'lJil~LbmU1.1'l :1-J811:1-J1118jj'Uilt.J1111 b11I'VI
l1.111<il:l-Jfi >J:;1ilLL:l-J1'1'1n.fi~ft:l-Jj1tU<i11:1-J111~'1iil1i.Jd
1
2
3
4
1
1
1
3
1
2
1
2
2
3
1
4
4
3
2
1
4
1
2
4
2
1
4
1
2
1

~~ .:t I .<!:!.
fll'J!u'UJ1J'l11l~~ (Pairwise Comparison) 'UlH'l1Nmo11
m.,-1hni:lu'Vl1,,;:!iiln'll!l,~<ii'1l'iu1•nXu L'1tJYi11u~<ii'1l'iul;i,~i11!l,ri1V1u'1.i~'mu1hwun1vr
nu'Vl1~b;:!i!lnl'h''1 t'1tJ~"l1.,-rn111'Vl1";:!i!ln~u'1 il.iJ!l~ -iimiitJmJ1~hih~ b'Vl'1U'1 AHP ,~1.jj'lllm.,
ri1V1'U'1'1~bb'UU'll!l''Vl1'b;:!i!ln<il1''1 db'l!Ub~tJ1num.,-ri1V1u'1~1V1un.i11:mh#i'!JJ'1l!l,mrn'l11''1<il1''1 tiltJ
"l~Wlil,U'J~dlu'Vil'b;:!iiln'"Yn~mrn'l'l1'1mrn.ri'VI~' liltJ1.jj~u111ilUb'lim$itJ1num1iJ1~bllu~1V1un'!J!l'
<v
bnlli'l'l1il
m.,-,iJ7tJrn-YitJu'VJ1";:!i!ln-Yil'l~~d "~1.jjrn•1Vi.inmu'111:J.J'l"lilh Lilmu7tJrntitJu'Vl1";:!i!ln
2 'V11~11~<ii'1l'iuhl1m1:J.J'l"lilh'Vl1";:!i!ln1'1mnn--h l'1tJ~'OJ1';-ru,11umrn'l1i111u'11''1 Yifl~ill1u m-;
bu7tJrntitJud'lrn'Li"flu 2 ~u111ilub'li'Ub~tJ'lnu ?'l!l
1) ~<J1'lrn111m"i?mn1'1 ~<ii'1l'iuhi1.i11:J.J'liil1J'Vl~!l~''l"lilhmnn11nmrlei~<Jnrn1
1umrn'11ill1u~u'1
2) ri1V1uili'.i1V1unm1:J.J~''l"l!lh btlui'.i1V1un.i~bb'UU'll!l,mJbu7tJrntitJu•~'Vl11~ 2 'Vil'
b;:!i!lniXu liltJ1.jj,~<iiu'1~bb'U'Ub'liub~mnu•~<ii'u'1~bbuu~1-ii'lurn.,-i.J,~bllmnru.'111ii
~
ill1 tJ'lllm7bbuu'~ mnumm'11 il'rn1u•~ bi:lumrn.ri1'1 '"~m:w1.,-n'111'1~bmuu1V1un.i11:J.J
'l"lilh'1lil,'Yrnb;:!i!lrnrl!l~'il1'lrn11umrn'l1ill1u<il1~'1 1'11
1'11ouN~ 4.5 <J1n<ii'1bb1JU AHP 'llil'U!JJ'Vllrn'Jb;:!iein-;mmm11w~ 4.5 i1'Vl1~b;:!i!ln 1illm1
w A, B bbl'l~ c l<i1tJl'.lmrn'l1m1<ii'1!1uh!laj 4 ill1u 1illm1 .,-1m e'i'•mm.,-1.jj,~m'l"I~' m1:J.Ji1~mn
11u1tJ bbfl~~u'Vl"mtJuein
<11,i:1u,~i11ei~l1m.,-bu7tJrn-YitJu1n 3 bbuud1ubd'll!l"nrn'l1 4 illiuill1m'iu
1u411u-;1m ~<ii<illluh1ili'W'il17lli1'0J1noiitli;Jl17lmbbli'1wu11rn.,-,iJ7tJrnYitJU';-1mbfJU1iJlll:J.J
m-,1,~ 4.4 <ilei1ud
m11U111mt111u !!11'111~~'1'101'il rlZUt!U
'mmn1
·mAnuB B 3
'lf;l Anu C c 4
7f;1 Bnu C c 2

A B c
A 1
1 1
3 4
B 3 1
1
2
c 4 2 1
lll'W~ 4.9 Comparison Matrix 'lrn~"ill'll
•
i'!lJltJbb6'~oti'Yl'l~m!JU8n <11m1rn:ii1.Ju Comparison Matrix 1Wllil~u
•
A B c
A 1
1 1
B(l)'llm•141.~m'W1~
4 6
B 4 1
1
3
c 6 3 1
A B c
A 1 2 8 m1M1~mrn1rn1.J
B
1
2
1 6
c 1 1
1
8 6
A B c
A 1
1
3
4 :itJm~m8uan
B 3 1 7
c 1 1
1
4 7
d
ill'W'll 4.10 Comparison Matrix

m~T1hum:m1f111JJW<Wl:"1f/Uf/ AHP 4-39
•• • v
fll'HilflUl:::!!'l-l'l-l'l-ll'tt'IHl (Priority)
m11'bb:J-J'Yl7mfoJ1tJU•YitJUdjum11~""'il~'l~iiiu~l'YIUn~J.J•~•:iluL'iltJm'lnJ1tJU•YitJUmru-Yi
'Yl1!l'Yll~dianYi61~fi 1~!l4fa.:m1'lil111.J'l'1f-;j1dju~'lt<ii'8~'Yl111fi1~1'Yliin'l7'~~'l1:J-JnlJW'll'lilli 'lu'Yln41m. '
~1a:1Jniuoil1<>'11tJnui1u1ilmvhL'il
l'i1~1'Yliin (priority) '1Jm11n!l~'1111~rnmn:1Jnui:lfi11'Yi1nu'Yli, ~~d.~amw1~!i,'1'"1ri1u~1'Yliin
'.. .., "" JU
ffll:IJf!lflllJ'Ylt1n'il8'. .
'lm'YlflUfl AHP mm1n~j'lU'Yll~flrn'ilmf!'ilfi.4111 fi1~blf!'il~iiliiium1:1Jcl11'1llJ'IJei,mru.fi
liil''l 'Yl1aii1iiluf111:1J~8h'1Ja~'Y11~1f1anlii1''l 'l~1$'l1nfi1 Eigen Vector (~'i:1Nm1:1J1ilu'Yld, 'Yl1a
<=! I J 12' n '
l'ltJn11 Normalize Eigen Vector) 'Ii~'~ llil'nflnl'lffl'U'llli'Yllm Eigen Value 'IJ!l~ Comparison
Matrix i1u m-m1fi1 Eigen Vector 6'61~ Eigen Value 'l~1:iJmh1li,'lltJ61~1lltJ1iil'U~d 'llii'l~lmrr
nT>m.11 Eigen Vector L'iltJV1tmru. ~"ttiXfi1~~!l'ltu11J.J1'6'lum•ii'il~uh1$1 111ii1ilu'lrrm'l~
1;;Jaj,mn fi1~141d'lt1tlul'i1'1J!l' Priority 'Yl"i!l~l'YIUnfltii'UU1%~tJ'llm!l,'11htnmJii1''l (priority
vector)
. ',,,..,. ~ "" &:I
11m1'l'YImlVl'Unfl~"um%61tJ
1. 'l1:1Jflll'Ubbl'i61t Column 'IJm Comparison Matrix
.1•1~ ' .1i .fl J ( "f •
2. lJ"ilJ 'YINl'111:1Jbb'il61t Column umnmlJ'U'Yl'U~ normalize) 1ilm!llN61"i1:1J'IJ!l"''il!I~
Column d1J.Jm1if1m'IJ'lu Column i1u'l
3. 'lu'lliim:: Row 'l1:1Jfi1'lu Row i1u'l 11i1$1tJnubb.i'1mfi11%~tJ fi11%~tJ'IJ!l"'l'i61t Row
bbf!lil~li,fi1~1'Yliinfl~bb'U'ULlilm%~tJ {priority) 'll!l' Row i1u'l
Vi'10a1~ m•'Yl1'11~1111iimu~ tJ'll!l"nru-Yi11ii,1nm~~ 4.11 ~"•f!'il"':J-J'Yl7n•foJ1tJrnYi tJ1J
fo
Lnill'l'l1il 4 $11u'luillj!V1114!l,m'llf1!lmn
1 2 3 4
1 1 3 2 2
1 1 1
2 1
4 43
3
1
4 1
1
2 2
4
1
2
4 2 1
2.33 12 5.25 3.75
.d. ,,. J Q
mTI11 4.t1 mm•n'lf1ml'luml'Ju

1. 'l'l:J.Jfi1luw1i'1t Column l~<iil:J.Jm''1~ 4.10 l'Ubbmii1'filil
2. U1N'1'l'l:J.l')J8,bblli'1t Column "6UVl1'lflllU Column ~U'l '1tl~bb:J.Jl'l'in.tlvii.Jil,d
1
2
3
4
1
0.429
0.143
0.214
0.214
2
0.25
0.083
0.333
0.333
3
0.381
0.048
0.190
0.381
4
0.533
0.067
0.133
0.267
0.398
0.085
0.218
0.299
3. tl1bm'11lmbmb~mnm1:J.Jnu b'liu w;n~ 1 (mrn.r11i1~1irrn•n) (0.429 + 0.25 + 0.381
+ 0 533) I 4 = 0.398
filb'il,~tJ~l~<Jtbbfflil,'11il1viiinm1:J.J<i1ff11J'JJB'bmu.fi11ii 4 mh' ~'m<Jff"jUl~il,d
fo
.0 ..., c OJ ,;...,
!fllW111'1 'lll ll'lllli'l1 l:UlYlAIJJ'U!l~!1111!'111'1
1. 'l1'11 0.398
2. eJ>i'llrn'l1-ihl1iiu 0.085
3. m1m'itlil1m1u1tJ 0.218
4. ~m~mtiuiln 0.299
<i1vi1'um'lVl1'1~bbU'Ufnl:J.l~'''1<l1'J'JJ8'bbU11'11''1~loif'.iih'Jiub~mnu lif,~'U lu4hmh,dbrie:i
v:i<J1'lrn1<J1mb:J.JmmblJ7tJrnVitJu'lJe:i,.,11 A, B, c 1mnrn~~mlli1~'1 4 ~iu brie:i1-ii11im'lb~mnu
'1tfflm'll1Vlll~11 '1tbbuum1:J.J~~''1eiL<J~:lllliB'll1 3 iu 1mrie:iv:i<J1'lrn1<J1mnrn.ri~1u111~'1 blJu1iJ
<i11:J.Jm'l1'~ 4.6 11ie:i1ud

'v v
mtu'l111'1!lm1
~1111
1• .11 ~flU 'lf!'lfll!'l'HM 1'1111H!~l'l1flll'!Jl!f li.lm~mmrnn
-:rll A 0.123 0.087 0.593 0.265
-:rll B 0.320 0274 0.341 0.656
-:rll c 0.557 0.639 0.065 0.080
' l ' '
1
4 I <:!. <:!. <5. q
tt!!tt11ll'll'lll'lfl'l'lll11'1'11 4 1'10'11'11 4.2 !>!1~'11 4.2.3
'lum-:r'lVffin1uu~1'V11Xm~m.bm1hnllu1?ItJl'Ylf1Ul'1 AHP ~~d'Jum-;;11.J'fou1ritJuei~f;1h:;
rn1uo/11~'1 ri'1::~iXu 'lumru~i.hi~f;1hrneiu~o/iei~{u~numn ~th::LllueiTOJi1'111:1-11i.Jm.imm'lum•
th::LllU~~rillVlNillll'llh::LllWif?ILLtJ~n'Ubi;' ihmhWlf'U n1Li.J'fornfitJrnmuo/i1?I A nu B A i1
m1:1-lft1#f11Jmnn11 B l?lt1i'.ifi1Li1nu 3 LL'1::L~mi.J'JtJrnfitJU B nu C B i1f111:1-lft1#fl!Jmnn11 C
l?ltJs:ifi1Lli1nu 2 <il~iXun1~?1'11:1-J!'i?lsh'U~llno/iei~LLli'1 A u1<J:;i1m1:1-1;;{1#frumnn-J1 C l?lt1i1fi1Lli1
nu 6 LL.J~<il?lfiuhm"l"l:;ll'l:;Lll'Ufi:;bLUUl'U~'lrud.J1~<Jln 6 L'lf'U LlJ'U 4 'Vl'i~ 5 111"1 'Vl'iillU1J1~f1~~
n1d'Jum'l{U~U'l::Lll'UblJu-oirn1u1.nn eju'lnll'Uill"lU'lnllulVf C i1m1:1-1R1#f11Jmnn-J1 B ~~<J:;
11ium'l'iiii LLU~nul?ILl~m~~
'lm'Ylfi'Ufi AHP i1'iiim'll'Ulll'l'1'l1"lililUm1:1-llJno/iei~ (consistency) 'JJil~lll'lU'l:;1lluLLUU
{ufid <J::tJB:1-1'lVii1m1:1-1i1'1Y1"1'1Ln'1~'Ul"1ii1~1'11.1m'l'V11fl1 Consistency Index (CI) •1rn~m-:rL1l'l1.1u
•
LYiuu11iu~ ~~<J::Lriu<il'l!uLLil'1~fi1ffi1:1-1Jno/iei~'JJei~m-:ru-:r:;1lluL1l'ltJu1YitJuiXu
i1'11 Consistency Index buLu'itJrnritJunufi1m~ri~1u~1'lm11 Random Index (RI) '11
RI dd'Ju<il'llu~'1-i'1~~mnnLL:l-l'Yl'7ni1u'ltJu1YitJu~~i1l'1rnuu~ll•::Lllu11iu'1miru:;'lrn~1'1'll<1:1-1 L'1uil•
LL:l-l'Yl'7niLu'itJu1ritJu~'1-11~~'U<J1n1'1'JJ~:1-1dJu-oi1u1umnY1Bf11'mli'1u1mmfi1 Random Index
{'1'11 Consistency Index 1i.J1iiu 10% 'JJil~fll Random Index LL'1'1~11m-:r11l'lt1rnritJu'1i'u
~,:(uilml:l-llno/iei~ei~i,wn:<ilu~'l-li'l"1

~ J ~.
bVl CR = Consistency Ratio 'lf1'i1:;1111 bil'illniJ11ra
R
. (CR) Consistency Index (CI)
Consistency at10 =
Random Index (RI)
n1 CR ~ 0.10 bbil<i11llm•~lJ~bti'ltJrnfitJ1J,:(UiJm1~1Jn<l'feM
I }!, I IV o ._, V (OJ .,,: o <:>. 2' <> "-' V f/
<ill'IJm RI 'IJU!l1,JnlJ'illU1UVl'l'IJ!l'IJ!l"mwmil1'1Ulm'W'ill•rn1 m n = 'illWlUVl'l'IJ!l'IJ!l"nrn'l'I
1"1 fll RI 'i):;bbilil1LWl<iil~"11111~ 4.7
" '11ll~H114.7 I'll Random Index
n 3 4 5 5 7 8
RI 0.58 0.90 1.12 1.24 1.32 1.41
.., ' "" 0
m~ rnnu<ill'IJ!11bb~1'11n'li
m•~rn'il:;ii1•m'11<ii:;bbUU~1111iinli1<ifu~ i 1t1~rnnu'11<if1b'1'1J'lu Column
':;14f"~mn.tl111a.J
'
fl i 61JfJ.jtb~'Yl;n~
. '
2. Ulbillfll<il:ObbUUUlVIUn<il'll~ftlilljJ'IJ!l1mrno/11"1lil<if1Jfi i 1t11111•'11 Weighted Sum
li1<ifufi i
3. 1111'11b>1~tJ'IJ!l1'11~1111~"1'lu~u~ 2 'lVib'Yhnu 'Amax
4. '11 Consistency Index (Cl) 111114i''il1rnJ<il•
')., -n
L5'.je:i n blJu-aj1't-b'J'Ubnru?l~cV:i~1~ru1
5. '11 Consistency Ratio 111114f'il1n
CR =
max
n - 1
CI
RI
l"1tJ~fi1 RI 'illn<il1111 n1 CR ~ 0.1 O bbil<111'i1m'btl'fornfitJuiiu:i1m1~i;in1fB1Lu•:;Wfu~
tJB~~lJ141

iil1eu1~n 4.6 'il1nfomJ1,~ 4.4 m'Y'lfi 4.8 lbl1ll'"i.J'Yl'inoiibu'h.1rnfitJu'll8"bmu-Yi'llll 4 ~1u
~"u1111'Y'I~ 4.8 mbbl11ll"5n'1~",;[)1ud
1 2 3 4
1 1 3 2 2
2
1
1
1 1
3 4 4
3
1
4 1 1
2 2
4 4 2 1
2
' • <:;
4 41'1u1~~._,dbbi1~l1li.JT:mm'11b'1iitJ'll8,UT1111Xnm1i.Jfi1~IJJ (priority) 'll[)"bml'Yl'YI"
'rn1w11 Priority
1. 'l'lm 0.398
2. iilll·:nm11-iib~m'Y'l1i" 0.085
3. m1:1-1l1~011nffiJ1tJ 0.218
4. 'li.J'Yl'l-lilltJU€ln 0.299
•
<ifo"m"ill"i1'ill1<iu11 m"ibU'l°tJubfi tJutl11'1ttnm1:1-1fi1~IJJ'll8"mrn.ri1lldilm1:1-1tJm'18,,.,''[)1i.i
• •0 OJ <V ""
'il~'Yllllll-l'lJUll[)Ull'U
i. u1'11tl1,,,un (priority)
ff<v .J
'lJ[)-lbnlli'Yl1ll'YI i ~rnnu'11<ii'1bi1'll1u Column ~ . - <
1 'll8'1bbJ.J'Yl"in'll
1 (0398) 3 (0.085) 2 (0218) 2 (0.299)
_1_(0398)
3
1 (0 085) _1_(0218)
4
_l(0.299)
4
~ (0398) 4 (0.218) 1 (0.218) ~ (0299)
~ (0.398) 4 (0.218) 2 (0218) 1 (0.299)

1
2
3
4
1
0.398
0.133
0.199
0.199
2
0.255
0.085
0.340
0.340
3
0.436
0.054
0.218
0.436
4
0.598
0.075
0.149
0.299
• ' ' ""' ~Ji ..... ' d
u1rnbb<ii<1rnm'lleJ~bbi.JY1'm'l!umnnu b'llU m11111 .398 + 0.255 + 0.436 + 0.598 = 1.687
~~1Jl.11 Weighted Sum i'l.11
2.
3.
1 1.687
2 0.347
3 0.907
4 1.274
u1'11 Priority 'llmmtwn~ i ltlvmfi1 Weight Sum
1.687/0.398 = 4.236
0.347/0.085 = 4.077
0.907/0.218 = 4.163
1.274/0.298 = 4.264
u1'11Lu~u~ 2 mfiw>>~tJ A,max
'Am =
4.236 + 4.077 + 4.163 + 4.264
4
4. 1111.11 Consistency Index 1<iltJ~ n = 4
'A - n
= 4.185
CI =
max
=
4.185 - 4
4 - 1
= 0.0616
n - 1

5. m'11 Consistency Ratio
CR = CI 'JlfWll'lTI n = 4 81 RI = 0.90
RI
CR =
0.0616
= 0.068
0.90
' . ' '<il ~ ' .:'1 "" <!!. "'1
nm1n111'.iflfl'tfl'l'lt11U'l'l 4 ~lltl'l'l 4.2 !'.ill~'l'l 4.2.4
''lf!Ufl AHP >J:;1VIN<i<11iJ1t1"1iu~1111unm1i.i~1~l]J'll!l~bmll.o/i':ffl~l~'l tb<i:;N<i<11iJ'llei~
f!:;Lbt1t1m7l.J7:;d1t1'll~b~ein~1~1 ~~bUu~n'l:rrn:;'llei~moWIflllul>JWf1bblJlJnl'>Wlflfl'Uhvi<i1 tm1:Jrnru.o/i
iit1bei~ iit1fieii;l11VI
w
' '
r.. = f!:;bmum1i.ivmh (rating) 'll!l~Vll~b~!ln j b5'.i!l'Y'1'J1oru.1"11rnnru.o/i i
'l
s = flnmt1'>1i.i'll!l~'Vll~b~!ln i
J
>J:;1111!11 S Vl~€1fl:;LL'Ut'J"1j.j1il>J1n
J
s = LWJ..
J l lj
wWl>lilt1h>J:;b~€ln'Vl1~1~ein~f!rnuwni.imn~<!fl. '
Wl1€11.i1~ 1rl€11-iimflilfl AHP V11'11~1V1unm1i.i~1~1]J'll€l~rnru.o/i'l>l
.w~..,.€1h~i'.1~€1'>!1 A, B, c 1t1"11u~1~1 4 "11u1"1m11<11iJ<mi.i11no1~~ 4.8

·mA
1!1 B
1!1 c
·~!nt1'1'11ll
Ji
ltlf11~il111Ullil1lfll ii'll111t'lllltTili~ m1mr~mmnJ111
W=0.398 W=0.085 W=0.218 W=0.299
0.123 0.087 0.593 0.265
0.320 0.274 0.341 0.656
0.557 0.639 0.065 0.080
iltLL'.l'.l'Tl:i.J'JJIN A = 0.398 (0 123)+0.085 (0.087)+0.218(0.593)+0.299(0.265)
= 0.265
6] 21=""""' ..... ' 1 i-'
Lmt L'll11lbiltnn1.•'iltVl1il1iltLL'.IU'llil~ B bb61t C Lil
ilnL'U'U11:i.J'llil~ B = 0.421
iltLL'.l'U'J1:i.J'llil~ C = 0.314
'1~tXu~~11:i.J1'lWililL'ltJ~Gi1'1u'llil~Y11~•lliln 3 m~1.l ii'~tXwm B 'iitd'lu'lll~L~iln~~~(j'il
. ' '..,., .:i a t:!!. .::t v 'l_I -l""' Q..k':>..,.
111Mllilllil'tll!'l!ll'ttlt'fl':i~!':ill~'l'I 4.2.5 mn !U':illUfllJllil'ilil':i':i:IJ 4.2.5
1
.::!j I d J di, d
'll!t'll1ill':iflil't1111'1l1tl'l'I 4 All'll'l'I 4.2 !':ill~'l'I 4.2.5

:1-1111'l'YIm iitJlilhYitJTI11m1l·nir i.h:;J.J1fii'f1'J:J'll(j/"i'lf1nn"iiflTI::ifif!JJ.fiJ.11mwi:;nwi'(j/nnm-,.i1iuuJ1u uu'YllJ'i
' ' '
:1-1111'i'Y1miimj1'JJYftJl!.,.,mTI'Jl'lf 2544
Anderson, David R., Sweeny, Dennis J.. Williams, Thomas A. and Martin, Kipp. An Introduction to
Management Science, Quantitative Approaches to Decision Making. 12th ed. Mason Ohio
South Western Thompson Company 2008.
Hillier, Federick S. and Lieberman, Gerald J. Introduction to Operations Research. 6th ed. New York
McGraw Hill, Inc. 1995.

.fllflNU1fl
l'lli'IU!lfl1W{Hlf11!111J LP ~~ll1mhi'Jfi1ti'l'IH11l1Ji'
4bmJu LP~-~1mil Simplex1um•blni:ln•'Vl11<11{iu,~1i'1l"€18lu'TIJ11uum'l•'llui±u$1€1~ti '!.! 'I.I "'5
Subject to
am1x1 + am2x2 + ··· + amnxn 5 bm
"; ;:o: 0
Lumrn~~'l~ln"1i11'1~€i"'VIJ.JlrnUu ='Vl'fo ;:o: n~1tuu1fei"Lolfiilrmb~J.Jliiirn~mi1lViif1t1J1J LP {iumJLu
~b1J1Jl-J1'1'>j1U111mnm~J.Jiii1mh1fi8J.J (artificial variable) boii11u 1'liu'iii Mmethod 1uwiiu •
rnJ.J'TIJL1uuJ.J1'1•'llu{iufi1iii1uu•iii"1i1uh (xJ {iu,~n1'Vlu"14i18~"1~1n"1-ifilft"1Yf18{iu$iil x <ilrn:ilfi11uu +'IJ «fl l • l
Vl'jililLll"umJ1i'1l"'Yi1nu.iuti (x ;:o: 0) 'Vl'jm18ffll Non-negativity Constraint
'iJ ,1 g.,
lui:lt1J'Vl1m•iii'1i11.ih1vh'l11iuu iii1L1u•iii"'iiuhdJufi1lum'l'1'l" ~"uniiillfi'1fi1~d}..i..,u•~'Wi1m1a<'Vlm8
#f"~u-ii1l~1n"1 x ;:o: o~"'uuoii€1~1n"'~"1l"1'1fi'mnu<!m1~1um•iii"'i1uh1"'8rf1'11u
.,,1 !U t..., ~2'.Jf !YO.< 6]<u
cl1'V1•1JUtl)'Vl1"11U Goal Programming uu il1bl1J1J'Ylft•1"'llum'Vl1n'il~m1'1i;rntuu 1u6'ni;ru~'ll1l"nn
lu,.1mrn~'1dim1fi'1 iii1uU"i1Jl'1fo'il~ftll-rn!l:flfi11uu6'1JlJi' ii11€1Lll.J'liu n1n1'VIU"1lVl x = u~mrui'u<i11~'il~Nli'II
•m 300
x, = 0
•~:ilm1J.J'Vll-J11Jll 'il~1imNli'l1~J.J'il1rnu1'Vlm8 300 'VIU18
ilm1mm18'i1Nli'l'l1J.JbU1'Vla<181"1€1&1
x = -200 i'lm1J.J'Vla<18-l1•~N(;'luimn'i1tll1111m8200111u18
' I •
mrnfi'11iii1llu•iil'li1ul"~i'lfi1dluim14'uu mfotuu 2mrn4i18nu$iei
1. iii1llu•4f"1iiuhi'l'lleirn'll'lfllu€18~!J'1bllufi1Giu
2 4i'1btu•41'111u°l';i'l.J:il'llilrn'll'llfl1Uil8~!J"'
1. mrn~4!1blu1iii"1i1u°l'li'l'llil1J'll'lfi1uiltJ~!l'11tlufi1iiumm1rn'i8U<!J.Jm•iiil~1n"1lJ1uu
x ;:o: L
l l
~il L 1uu4!11mi'lfi11uuiiu 1'liu
l
x2 ;:o: -10 L = -10
' 2

oii'mi1n<il'ikii-1TI(lbbllll"blJUoif11~1n<il~<if1bbl.l'Jl'.i.-i1bUU'1U!'iV1'l1lbUU1J1n~11ml.lbblJ1Ji-ll<il'l'ilU'lJ1l" LP1~. . ~
1'1ltJ
x+ ;:::
I
x+ =
j
0
x -L
j j
x++L
I jxi
ti1bb1'1Ufil x LUfobblJlJ LP. ~1tJfil (x++ L)
j J' l '
<i11bbll'J<if<ili1uh-.t1htn11u~1tJ.-i1 x+ :Jf-l<JtbuumnV1"lmvhnu111.11'ibili-J11 fhu<il1bbll"i x :J"m<J•uum11~j 'll j
<Jt'VlltJ111-.1n<il1mJu
straint
"
m<Jbbil<il-l~1tl.fl1~<ii,fi
0
-10
x+ = 8
2
L = -10
x,
x+
2
x,
x,
x,
~
=
=
-10 (L = -10)
2
x
2
- (-10) =x
2
+10
x+-10
2
x+ = 25
2
-2 x; = -2 + 10 = 8
15 x+ =
2
15 + 10 = 25
i:l1bb1'1U.-i1 x, = x;-101u4i'1bb1J1J LP <Jt1~fil
x+ :<:; 0 bili-111
2
2. n-rni~4i'1bbll'>4i'<ilftUl<J~i'.i'll1l1Jb'll<ilfilu1ltJ~il<il iiu~fo1;.Ji'.i:zi<il~ln<ilb~t11n11.-i1 x fi1 x d-.tril'VIU<ilLVf~ j i
x = X:'" -x.-
I I I
x+ ~ 0
I

<Jlnf)lli1"~1i~~"" Basic Feasible, Solution ~1Ji,1n'ill Simplex b11mmwi'lj•u"lJi"l1~ Basic
Feasible Solution '11~B" x.+ ''"~ x~ -~'liJmi-mmhmwuu Basic Variable LJi'l'l~fo~nu'lu solution •~mnu ~"
J J "
111mmrn~11 tll x.+ i'.J.11<ii1•11~ x.- -~·ri1nu11u!'iV1'iB<ti'1 X. i'.1'11<ii1m~ x -~·ri1nu11u!'i muff" x bbf1~ x,. -~•UU~ J 1 ' 'II ) j 'II )
~u!'iff"~~U'YJn Basic Feasible Solution ~"m":iitJulili'11
x+ =
j xi
0
x.- lx,1 ti'1 x2 0
J J '
= o 'lurnrnliu
if"iXu '11 x.+ ~"bUUfil~B" x. 'lu1fbu~lUUfi1mn (positive component) b!1~'11 x.- •~•uu.11~1M x. 'lui l l l l
rl1ufillJu.1111u (negative component)
'liimri'l.uu'l~u11Jl-tlumo<t-ll"<ii1bluumo1111llrn~•u1V1mtJJi1tJm1ri1V1u11'lVi.11m1~•i'lmluu~"m'
•l'iu1ili'iii".11mrn•11~11u ri1V11.111'lVi1hmBuili'1tJ<fB"ii1u (d) ¥1B ct.+ mi~ ct.- ~"'11"il.11~1nn11V1'imri1nu11u~I l l 'II

d
YB
1.U
fi1utttl~
":i8~~1~fil"fl"ilTifl"Jitu'Y1-S "!JntviYrut1~n~
1'1'L1J., 1J1i.:IJ. (nl":i~lilfll":i)
1a..:iier1r!(f)"il""J1'.iil'Ll"J":::-.51r11'llr1'll'11'Vl~1n1":i~~n1,.
:1Jvt11Yltn~ttf!1'11~t1rn:JJ1ITT!'l!'
,.,;,.~,Jfo•h• ""'"ii5

•
,, ' "'!l'll I l'l'I.:!!'IHJ't'il
""' i: "" '
5.1.2 fl1'l1bfl'l1:;Vlbfl'l1l'll1rn1iubuu Shortest Route
"" 6' "" '
5.1.3 fl1'l1bfl'Ol:;Vlbfl'l1l'll1rn1UbblJU Minimal Spanning Tree
A 6' "'1. I
5.1.4 fl1'l1bfl'l1:;Vlbfl'l1l'll1rn11-lbblJU Maximum Flow
'
<11eiuYi 5.2 m'l'i bfl'l1:;Vibfl1ei'li1t.1-J1u1um'l<Jilm'l1fl'l~m•
5.2.1 bm1flilbM t.11 num•<JilnT>1fl•~m•
5.2.2 fl1'l'1.hii1bblJU PERT/CPM m 1oi!'lum'l<Jilm'l1m~m'l
5.2.3 m•d~1fl'l~m•
"" ... <it '
5.2.4 fl1'l1 bfl'll:;VI bfl'l1l'lJ1 tJ~l'UbblJU PERT
<111luYi 5.3 <i'i1bbuum~fl1lW
5.3.1 bb'U1flilbMt.11nu<i'i1bbUU:m~fl1lW
5.3.2 m'l1J'l:;!Jn<11"l.jj<ii1bbuu:i.n1fl1lW
5.3.3 <i'i1bb1JUm1fleiWmtJl<iib~1l'Ul'll'Ji1 Wll'<'I
1. <ii1bLuubfl1il'li1rn11-lLDum•'ibm1:;Vib:n,i.r.mm~l'.Jn1•"ii1!11l~bVl'Jm•rn1111il"mum•rn
1vfotj1').~1.JbblJlJ'JJil-Jbfl11l"li1rn1u~'Yilfll'lb;1l:J.J11'.Mbill-'1'11~1"111lbb'lJ'U~'l:;1"11l-J11"11-lil
,,,11l!'mu~1w1h''l ~l'.Jfl11:J.Jii':J.J'l'iuilnu •'liu bfl11l"li1 t.1-J1um•11~mtJl'l'l'lff'l'IYi
.,:;,,,11-Jll":J.J!'1t.1"11''1 bfl1E1"li1f.l~l'Ub.lum-Jw1wwib%mJ1rn!'mu<ii1~1 bfl11l"li1t.1
-Jrnm111~Yim.h:::u1•:;,,,11~!'mu'li1''l d'Ju<iiu 1ilt.1l11'<11tJ'Li•:::fl-J.11um•~nB1
'ibm1:;Vi L~EJm'l:;tJ:;m~ 1111EJnm 1"11EJ.i11-if'ii1t.1~~u,,,1E1uilt.1~C)il 1um'lb~1l:J.J1m
b.1'um~'il1n1111ufl11111l!'mub~:J.J<iiu11.J,um:;Yi~11"1uil1"11il!'mu1Jmt.1m~

~ -• "1P!Qu':i::m~ri
2. mTil'1TI~~l'17'i'ii1m1u1um1{.im11m1m1 ~lJ~V111mm1nil1<1i1bllJlJ PERT/CPM
it1J.J1Loitlum1{.im11'11~m11.il'Il~J.J~1ll"1m1111llN'Um11'i1•iiun'Jm1J.J.i11'1 m1
{.im1J.Jf!J.Jwuil'lleJ1n'Jm1J.J.!1~'1 ll'1~m1mlJ!JJ.Jn>Jm1:J.J.i11'1 'lleJ1tm1m1 .~eJ
1iXllii'1lft~'J<i11:J.Jri1111U'1nm'lleJ11m1m1 1.il'J~<ii1bblJlJ PERT/CPM 141-l'um1w'ill'.11
mLUnmLnii'lfimfi'u i'Jmd1J1ttJniflm1l<if11'Ublii~i'Jli'ni;rut'll€l1'Ji1tN1U~'1i''11'1'1~1fi'U
bl.ibl<11n"111nu1u•4eJ1'llmnm~toitlum1'Vi1n'Jm1:J.J.i11'1 1.il'I~ CPM mm1n'YITilJ
nm1um1Y11n>Jm1m•"1'1:on>Jm1J.J141ml11lluueJu Lu'llru~rl PERT nm1um1
'V11n>Jm1:J.JLl-imm1nri1111u.inm~bluueJu141 ii1eJ~m'1l'lm11h~mrun!11
3. <ii1lllJlJJ.J1~'1eJ'i'Jd'Jum11•m1~~'W'l~m1:J.J'lleJ1/11mh~ft'Uh1um1~ni;1 l~eJ~'J~
Y11u1l'IVl11l'Wmmcl'W'1~m1:J.JL'U!l'U1'1<11'lleJ1<ii1mJ,,!u 1.il'l<J~141Niili''Wiltu~l.J'lleJ1
m1:J.Jul'J~•lJu ~1mm1m.bmi.J1~tJn<11"l<ilnuihlJ11111um1<ii.iflul'J1911:J.JVIU1~'Yll~
TI1n'J141 ~1md'Vi1ml'l1111ei'Wmmcl<ii1bli.J1'11mh1,!u ft1m1nm~Y11141,.;r,1m~l'l:o~u
'
,rieJ~ni,;1111u11'1~
1. €JTilJl!Jbli1~1 lf111~~lf11ei'li11'1~1'U141
2. €lTilJ1!Jbbi1~U1~1'jn<ii"l<il PERT/CPM 1unw'5'.im1tm1m11111
3. eJtrnmrn~i.J1~i;in<i1<il<ii1bllJrn.i1~'1eJ'i'J141

'..~0'1-!'l'l 5.1
"' .! "'! '
fll"J1!rl"Jl~l'l!rl"JO'Ultl~l'tt
' '
•7mri 5.1.2 m'llbfrll~Vibf17il'li1rn1Ubb!J!J Shortest Route
' '
b7il-:iri 5.1.3 nl'llbml~Vibf17il'lilrn1Ulb!J!J Minimal Spanning Tree
' '
b7il-:iri 5.1.4 nl'llbfrll~Vibf17il'lilrn1Ulb!J!J Maximum Flow
1. nl''i'lb'1'l1~Vi•'17il'li1l'mu dlum•'l•'1'Jl~V1'1~wl'lm'l~li'1il-:Jb'Vl"lm'lruV17!li'1mum•ru
1u(;tibb!J!J'JJ!l-:JbbN'U.fll'Vj'111!1m1YIn bbi'1<il-:J•Kum-:i'Vl1!lbb'JJU-:J~1'11m'lb'li!l:1-1Lm1~V111-:i
L'VIU<ii'Vl7!li'1mu-:i1ulii1-:i'l L<iil'Ji'.11<1mli•~i'1-:i<l•Y1!lV111~1'J~m-:i~Au~i'1<il 'Vl1!lt-iinmu!ll'J
' '
~i'1<il V11m~l'J'111-ii-.311'J~1~i'1<il1um'lb'1~!lui11 l'J~-:J'JJ!l-:J'l1nmnu•~:1-1/iu 'Vl7!li'1mu
' '
'Vld-:i111i'foiini'1muV1d-:i~!ll'J111 'lUm~rf-:iil-:i'l<il'Vlml'Jlimmrn
2. m1'lb'1"ll~Vib'17!l'lill'J-:J1Ubb!J!J Shortest Route Lliu'lilm'lbbntlqJV11~l'11'11tili•~m.-1
'VI~n ,.;'1il'111Lifum-:i'Vl1!l"'JJU-:J1um'lb&ium-:i'Vl7!lm'lb'1~!luif1l'J~-:J'JJ!l-:J'l1nmnu'Vld-:i
111ti-:imnullu'l L<iil'J1Vfl'.J7nJ~'Yl1-:i V11m1m V11il'11t-ii-.311'JAu~iJ<ilV11iluill'J~ij<il
3. m'l'lbm1~Vib'11il'lill'J-:J1Ubb!J!J Minimal Spanning Tree blJu'lilm·mntlqJ'lll~i'.l
1'11nli•~i'1-:i.-l'Vl~n ,Y1ili'1-i'1-:ibi!fu'Y11-:Jb'liil:1-1Lmi'1mu-:i1u'Vl1ilL'VIU<il'Ylnt'VIU<il1m'11il'li1rn1u
' '
l<iil'J1Vfi'.l•w~m-:i 'Vl1m1m V11il'111-ii-;J11'J~Au~i'1<il'Vl1!luill'J~i'1<il
' '
4. m•'l•m1~Vim1il'li1m1ubbuu Maximum Flow dlu"lilm·mntlqJV11~l'11'11tili•~i'1-:i.-l
V1~mY1!lV111i!fu'Yl1-:iV11ilbb'JJU-:J'lum'll'11'1'JJ!l-:J~-:J~i'1uhfini,;1 'li-:iill'lblJuli1mrn1n~
~-:iil~UubifU'YIWIJil-:Jnuu 'Vl7il111mrn~1 ~i:i1u bbrli'1~l'lli'1lli<m:>1'Yiil<i-:i~i'.lm1i'1-i'1-:i
b<il1il'lill'J11 ,;(,dJ'11'll!Jli"l~i'1-:Jfl*bUUlY1il1vrbi!fU'Yll-:J'Vl1mb'JJU-:JAU"] i'.1111mrnnTi-l'111"
'
j-:Jriij<il

" . I J
191Qu'l::~~lfl ,
<i'.1€1$rn,;1lliJU-ti 5.1 'ilrn<li'1 iin$nl:f1<11m·n1
1. ilrrrnmm1i'l•n~mnrnl'l"fo'li11J~1u141
2. 1<m1:;i);bl'l~EJ'li1rn1mrn11 Shortest Route 141
A ({ 11'!1 ! '1 V
3. 1bl'l'n:;Vlbl'l':lEl'lll1.mU"1.Jll Minimal Spanning Tree bll
4. lbl'l'n:;1.i'bl'l~il'li1rn1mbllll Maximum F1ow 141

'l'Um"llii1Li'i'U~1wml-l'Vlu1~1pn,"11~'1 ~u"l'V11"l"J::i'.Jm.,lii15'Uh'Vl'ia1LniJ(jJ'Vl1.i1''1 1Ji'Liri
m"lBan Lhllll"ltllllm"l'll'l-161~ m"lmm LLllll Lif1 ::'ltllllfll'l"fltu'VI fl m"l'Ji1 iii#i'm1'U'lla'lfl'l",:inn
wu"lm"lff1m"llU1li1LillllLfl'iil'li1tJ,1'ULLR::m1'lifl'l1::ViLfl'iil'li1tJ~1'UL.Ji1l-J1'l.Jil'Um"l'lLm1::Viilruvn. .
$1~mh1 ti1tJ~lii'lLLt1tJLfl'iil'li1tJ'1.bd<>J::'5''11'Vfil~'lW);ULLt1t1'llB~iLN'Um'V'l'Vl'iam1Yln li1tJm"lL~Bl-lttJ,
"lt'Vl'l1~'li1 'Vl'iil!'!!llU 'Vl'iBtWUi1 (Node) .11''1 ~'J::'l.Jil'Um"l'llUft~L.Jil'Jlnt'Vl'Ui1'Vlct~'lti!'.i~llnl'Vl.bi1
'Vlct~ 'Vl'iei<>J1nhmi1L~l-l/i'UNrn1u1'Vl'Ui1.i1~'1 <>J'Um::rf~~~'li1'Vll-l1tJtimtJ'Yl1~'Vl'iil'li1t'Vl'Ui1~'U'li1
4i'1LLlltJLfl'iil'li1tJ~l'U (network model) ilJ'U'liim"l~UtJl-l'l.n'n'Umnl'Uf11M1~'l'Ylmm1'5'i1m"l
(Management Science) ~,ddiei,m'JlnLWIJWf1~il1fi'(jJ~il 1) if1iillllL'1'iei'li1miumm111tl1::~nlll
'linuiJ(jJV11'Yll~lj'l"n'J1Ji'ilU1~n'l1,'ll11' bbf1:;~1 tJ'liilm1'lbfl'llrn1i1 tJi'.lnl'Ji~ tJ'Ulii'1bbllll1'1tJ'liLLW'UJ11'V'I
'Vl'ieim1'Wmifli1~"lltJf1tiBtJi111 bii1t 2) d'J'U'lTim'l"Linil(jJ'V11€1 U1~iU.b~Uilil'Uiif1tiU'U"ltllll
m1ti1lii1LLt1rnfl'iei'li1m1u <>Jtt;Jnt'.11m'linuil'!J'Vll'Yl1,m1'5'i1m1'lu'Vlu1~Y11~1l"n"1Ji' 3 1:t11Luu
~illfl(J ~il
L m'l'lLm1::Vi'li1m1mLllll Shortest Route
- ,, . . .
2. m'l"'lbff'llt'Vl'llltJ~lUibllll Mm1ma1 Spannmg Tree
3. m"l'lLm1::Vi'li1tJ~lULillll Maximum Flow
'liim1'lLfl"lltViL.i'iei'li1tMl.bLillll Shortest Route t;Jn'Uim'l-D'n1J'liim"lLinll'!J'Vll1i1 tJi'.l
LU1'VlmtJ~il11i'QJ~ei /iil~m'l"'Vl11::tJ::Y11~~~'U~'li1'l'Um'lL~Bl-lfo~<>J1n1111ui1111ct,1u!J,5n1111w;i'Vlct,~ei~
uu1fl'iil'li1tJ~1.b (~~1::tJt'Yll'~~u~'1i1m<>Jei ~'lwi;ti'll!l~1::tJt'Yll~ nm 111'iil'11'l-ii<J1tJ)
1ilm'l'lili1'l1::Vi1.i'iil'li1tJ~litiilJll Minimal Spanning Tree t;Jnti1m'l-iin1J'lilm1iinll(J'V11
1'1tJi'.lLi11111mtJ~il1fi''!J~il /imm7vn"lttJt'Yl1~1i1tJ71l-l~''Vll-li1'1Jil~m'l"i~ill-!btJ~1'Vl'Ui1.i1~'1 11nb'Vl'Ui1
...i1 «1.1 V V v
'YI ei ~ll'Ubfl"l1l'lll tJ~1m'llli1'ltJnu
'lilm'l"'lLm1::ViLfl'iei'li1 tJ~l'ULLllll Maximum Flow 1Jnti1m'linu'lilm1LLnilqJ1111~il11i'QJ
~ei Ltlum71111~rn1u'Vl'iiltl"lmrum'l"l'Vlf1~~'Vll-li1'llil'~'~fluhi!ln1'11 L'li'U tl'imrn~1iiu iinfl 111'i!l
~1~l'Vlf11i11l-l'Yiilft' ~1.b1wmtJ'Ulll~~~ei~t1'Um~li1'U onilm1l-l~ii'l'U"ltt1t11i'UL'Ylil1LUlil Ltlu/iu
li'f1 LLt1rnli1'iil'li1 tJ'l'U~~ 3 1:t1 LLt1rnli',mh1 'Jtt;Jn'li1m'liti1tJi'.l1u1111m tJ~LLliln"11~nu
'Uei n"l1nd!J, LLliln.11,num'l"'lili1"l1::Vi PERT/CPM ~Lu'Um11J"l111nl1i1.,,m.,~tli1.,,m.,.i1''1 ":::
ti'l"::nillJ1t1Ji'1tJn<>Jm'l"l-l'Vl'iill 'Vl'Ui1.i1~'1 ~<>J::/iil,'l-11'lVltm,m'l"Lfl~w11tJl 'U'l"w::nmnl-l~'~'U'llil'
11i11~m1 ti1tJ~u'i'Vl111m,m'l"'J::/iei,i'.Jm111,LLW'U ri1111'Ui1!i111ium1l-lill-lwuma~n<>JnTil-l LLf1t1i11t1f)l-lm1
iili'iuf'Jm'l"l-l'li1''1 lVlib-.11 Lfl~"JmtJlitfi1'VIUi1nm~/iiJ,f1l'l" 11tJf1tLBtJi1'JTim'l"'JLfl'JltViLfl'iil'liltJ

'1uiif, 3 11.JmJlJ 'il:;n~11l'i'lw:nmm~tJ"1L4il,~ 5.1.2-5.1.4 rl1um,-1L'1-n:;Vi PERT/CPM 'ilrn~11
11lu'l"1811:;LBtJ"1"1ilU~ 5.2
,_, .....
lNPlll'l":::flfl'IJ'UeM!Pl'l"fl'Ul!Nll'l-I
L'17'i'li1rn1u'il:::1JrnL'1"1'111.u~'llil'LL~Wl1'Y'IVl71ln-n'l'ln~JJ,-m!l1J1JJ#11mMfll.J"i:::rnm 2 rl1u fiei
1. l111U"1 (Node) bb'YIU#l1tJ 0
2. LL'llU,111:'maum' (Branch or Arcs) LL'Ym#l1tJ __
T111u"1LL'YIU'l"1L~ei:1-1LtJ,111:'eiT111u"1wi1''1 dh#l1w'lu rl1mL'llU,LL'YIULauY11,~L~!l:1-1Trn.,::11111,T111u"1
~1!JU1,L'JIU L'1~fo'li1m1uLaU'Yll'"it11ww1 BTS 'ii::JJ,-:;neiu1JJ#l1tJftmibn1WW1<ii1''1 ~'LiJ'JtJlJ1#1
nuLVIU"1 rl1mflU'Y11,"i!11WW1"i:::11111,ftmU"it11wV11LiJ'JtJlJ1#1nrnL'llU'111~ilLi1U'Yl1'
ii!'JlJUH ~ 5.1 n1"iL~tJULfl~il'li1m1Ubbft"1,'l":;tJ:;'Yl1,~1oifn1':H~U'Yl1,'Yl1,"it18U<il "i::11111'm'b'Yl'V'i-
'
:1-11111Ufl"i nlJ<iJ,WJ"1<ii1''1 ii''LL'1"1'11[Wl1'Y'i~ 5.1
15 25
2D
10 35
3
'il1nm'Y'i~ 5.1 L'1um,"iiitJu<i"i::11111,n1'•Y1'Y'l:1-11111u'1'l"nu<iJ,1111"1wi1''1 l.J"i:;neiu1JJ#l1tJ 1111u"1
(<if,1111"1) wi1''1 <51mu 4 T111u"1 "":;iiLL'llu' (Laum') ~1u1u 5 LflUY11' L"1tJri1111u"111~111u"1 1 LL'Ylu
n1'L'Yl'Y'i:i-11111U'1"i d'JU'l"1L4:1-1<1lu'llmL'1~il'liltJ'1U ff1UL111U"1~L111ii!l'il:;LllU'l"1111:1-11tJDmtJ'Yl1' ,if,d~u
il~nlJ11'il:Ori1111U"1'l"1111:1-11tJDmtJ'Yl1'11~<ii',1111"1l"1 'illnm'Y'i~ 5.1 LLwi11::T111U"1'il:::i1111mm'1'llil~'1uT111U"1
'ii::1oifL7tmbb'YIU~il<if,1111"1 L'liu LflU'Yl1,"it1tJU<il'il1nm"'Yl'Y'i:i-11111Um1iJ<ii--11111'1UU'YllJ7 'il:::L7tJnLL'lJU,11171l
' '
LflU'Y11' 1 - 2 Lllu<1lu
L"1tJl.Jn~LLfi1 '11ii'1L11'll~ei~lJULL'llU'111~ilLflU'Yll''ll!l--1L'171l'li1m1U'il::LLY1u#l1m::tJ::m' nm
111'Jei.111oif<J1tJ ~1oifL~il:l-ITm.,:;11111'T111u"1wi1''1 ~1mh"'Jlu "!:;tJ:::'Yl1,'il1nn1"'Yl'Y'i:l-l1111U'1"i11.Ji'.fo<if,1111"1
UU'Yl1J7 mmflu'Y11,'illnl111U"1 1 - 2 1oif'j":;tJ::'Yl1' 15 nlm:1-1m LUu<1!u
' . . .q 4 J ~ .... <!I .q
m11u1m1Hmn'llu1un 5 mmn 5.l tlM'l'l 5.1.1

• •~ <:I <=> cl ~ I
!'j0~'YI 5.1.2 fll11!fl'jl~fi!fl'j0'1JlE.l~l'WH'UU Shortest Route
Lfl~il'liltMlUbLlJlJ Shortest Route bU.LlTI!lT'.iLbflUbJ'Vlll<il tJiJLUl'Vlm tJ~'11fi'bjJflil nl".i'Vll
'l~tJ~'YllJ~~u~iw11um'lL~il:J-Jlm'il1nl 'Vl.L<il'Vl~J11Efofinl'Vl.L<il'Vl~~~ilt!JJmfl''lil'li1mm l<il tJ'il~'Y11m1' .1'i'ill'lb11Lft.L'Yll1'illmb'Vli1~o/i'u'V111 (Source Node) bULl1bb'Vli11l.imtJ'V111 (Sink Node)
' .vl10UHTI 5.2 Lft.L'Vll1!ll'lL~.L'Vll~'llltJWll'l~'Vl111bilil1Yl1 7 bi'.lil1 l<i1tJilbfl.L'Vll1":i~'Vl111bilil1iif1
11tJ!~batJ;iii11m'Wi'I 5.2
17
15 6 5
[O, S] 6
3
10
4
4
'il1nm'W~ 5.2 'il~b~:1J1'i'il1":iru11<iltJn1'Vlu;i11X'l<il 1 LUU'J<i1o/i'um1L~:1Jo/i'u'ii~111s-bJi'n1'lcll.i.,~Yi'JJ
<il":illll1".i (permanent labeled) flil [O, SJ l'1tJ~iii1b<l'llbb'rnflfl 0 'Vl:l-Jltl~J ".i~tJ~'Vll1'illff'<ilb~:IJo/i'u111
'
ti1'il<i1tXu iii1bi'11~ 2 fifl'il<il~riflu'Vlu1il<i1tXu 1u~d'il<il 1 fifl'ilrn~:J-Jo/i'u iii1tXuiii1b<1'11bb1nfifl o iii1b<1'11YI'I 'I • q q
2 i'iil S bb'Vl.L'il<ilb'l:J-Jo/i'u (Starting Node)
'
m1fim1mVilLlfu'Vll1~~u~fl<ii m m1llfi1u1ru'Vilbl"i'iii1d
iu~ 1 1'i'il11m1'illff"<i1o/i'umJlu~dfifl'il<il~ 1 11X1'i'ilnru1'il<il<fi111 ~fltJ'VllJ'ii<i1o/i'u'V111 ~1li q q .,, q
m:J-111m~u'V1111l.iiiJ'J<iltXu1;'i'l<iltJb:i-io/1€11~1u'l<il5u1 'il1nm'W'il~i'.i'J<i1~ 2 "'"~'l<il~ 3 l<iltl~'J<il~ 2
''"~'l<il~ 3 'ii~L-ii'iii1Ll1'1IRb!Jil'n1'lcliiiJd [15, 11 Lb!~ [10, 11 iii1ilt!11'l<il~ 2 <if1Ll1'll [15, 11 <if1Ll1'llbL1n
' ' '~ ~ .flil 15 'Vl:l-lltl11".i~tl~'Vll1'ill!l'J<il<il.L'Vll1 ('J<ilfl 1) mLl'J'J<ilYI 2 il1~tJ~'V111 15 <if1L<1'llYI 2 fiil 1 fifl'l<il
rifluV1u1fiil'J<il'11 1 dJuo/i'u

Yl'i117ll1"<'1rit11::Vfu"<'1ri 2 6Lill:'il<'1ri 3 •fi<im::tJ::m"ri~uri11<'1'i11n<11<'1iiu1t1i'i"'il<'1ritl7:;Vfu~ ~ q q q q
"171'il::bi1u11 ~<'1ri 3 i'.J7rn::'Yl1"~urilj'<'1flil 10 1>1-:iiiu~<'1ri 3 'il::tJ7::Vfu"171m'l7flil ('l-:ib<'luYu)
(permamently labeled) il-:im'1ri 5.3 1'41u~<'1ri 2 i'i"1:iJtJ7::nu1~11•llm<'iu'Yl1-:iri~urilj'<'1 1>1-:iiiu~<'1ri
2 ~-:ii'.Jm7tJ;::Yfu<ii':!l~'lm11 (temporary labeled) ~'1 [15,1J ~"111i.11;ntlw1~n1l'1rn<'iu'Yl1-:iL'Vl:i.iri
~urilj'<'1ltliJ"~<'1,Xu li1'-:ibb11<'1"H'l:um'1ri 5.3
[15, 1]
17
15 6 5
[O, S] 6
1
3
10
4
2
4
[10, 1] ('Vlt.i1tJ: nll16aJ<i11)
' '
<"I ,j "-' ""'.flll'l'fl 5.3 nn 1~nJJm111n111~'fl 3
~uri 2 Lubbwil1::~u'iJ::Yl'ii1;ru1b<'iu'Yl1-:iri,~eii.i'il1ffi1<'1ri1h::Vf1J<i111m17L'Vl:i.i (fl<i'il<'1ri 3)• •
'il1nrnl'1ri 5.3 •71'ii::Yl'ii11ru1•<'iu'Yl1-:ion<i-:i~<'1ri 3 11.liJ"~(i)ri 2 (3-2) ~'1-i!;::tJ::'Y11-:i•'Yi1nu 3 ''"::
'il1n'il<'1ri 3 1t1iJ-:i'il<'1ri 5 (3-5) ~-:i1-i!;::tJ::'Yl1'1b'Yi1nu 4 1>1-:iiiu•<'iu'Y11-:i'll1n'il<'1ri 3 1tii'i-:i'il<'1ri 2 •'Yi1nu'I q q q
10 + 3 = 13 bbl1l:bfl'll'Yll'1'il1n~<'1ri 3 1tii'fo~<'1ri 5 •'Yi1nu 10 + 4 = 14 1>1-:iiiu m'ii::tl1::Vfum1m11
LV1:i.i~ei'il<'1ri 2 l'ruli'rn,rcl [13, 3J bb'YIU<J1rn~i.iriil1::tJ::m-:i~uri11<'1~t1 [15, 11 ~-:i'VlmtJii-:i'i1 1::tJ::'Y11-:i
'illn'il<'16~i.i.i'u ('ii11~ 1) miJ'1"l111ri 2 il1::tJ::m-:iri~uri11111fi<i 13 "t111tJ"'1rl<iuV1u1d~ei..,111ri 3 rl1u..,<'1riq q q q i; 4 ~
5 'il::ilm1t11::Vfu<i111~1m11~t1 [14, 3] li1'-:ibb11111-:i11'l'Wl1'!"lri 5.4

15
10
[13, 3]
2
3
[10, 1]
6
4
17
5
6
4
4
2
[14, 3]
~u~ 3 '!1nmw~ 5.4 'j'1~11m·m11l'Jlff1J'1~ 2 11~'1'151.11 1'1i1'11.1m~"1il'j'1~ 4 bb'1t'j'1~
7 b~urrn~~u~"1'1'lln'l'1~ 1 11.l!'.1~•'1~ 2 $if] 13 $11ilub~U'Yll~'l1n'l'1~ 1 1u!'.i1'l'1~ 4 t'11.1N1U'l'1q q ~ ~ ~ q
~ 2 b'Vi1nu 13 + 6 = 19 bb'1tbl.'fum~1u~~ii~lJ'1'lln'j'1 1 1u"l'1~ 7 b'Vi1nu 13 + 11 = 30 b~il
Vl'l1•ru1•1nm'l"l~ 5.5 bi.l'J1.1rnYl1.1u'j'11l•tnum1~1m11"1il'j'1~ 4, 5 bb<lt 7 bbfl'l b';-l'Jtbi'iilni.l•~nu
'11•1•'1~ 5 b~a1-;11n1oiib~um~~~u~1'1$ia 14 $11iiu111•::1l•::nu'11•1m1•~•'1~ 5 $ia [14, 3]
' ' '
~~bbi.1'1111'lumw~ 5.5
[13, 3] [30, 21
17
15 6 5
[19, 2]
6
3
4
10
4
4
[10, 1] [14, 3]

. ' '
o/iufi 4 'Jlnm'V'lfi 5.5 'l~'ill'l1111.m'1urrn 5-4 bbi1~ 5-6 bftU'Yl1.:J'l1n~111fi 1 t-i1w1111 5 11.l!'.fo
'1111~ 4 'l~bUUbftU'Yll.:J 14 + 4 = 18 zj.:ibUUbftum.:i~Aun-iib'1u'Yl1.:i'l1n'l<11~ 1 N1U'l<11 2 11lt:i.:i'l<11~ 4q q q q
zj.:il'.h'1um.:i = 19 ~.:i,:Cu,~1il!l.:is:im1tlw.-i1~'l<11 4 bUU 118, 5] <t1wo111~ 6 'l~1'6,!1um.:i'l1n'l<11~ 1
' ' '
N1U'l<11 5 11lt:i.:i'l<11~ 6 'l~dJubftU'Yll.:J 14 + 2 = 16 bfttt'Yl1.:i~Au~'1<11 rn ~uit~!l'l<11~ 6 ~.:i,:Cu,111~ 6q q q q q
,~t1.,~Yi1Jm1m1'l'~!l 116, 5l ~.:ibb'1111.:i1'1'1M1'V'1~ 5.6
[13, 3] [30, 2]
17
15 6 5
[O, SJ 6
3
10
4
2
4 [16, 5]
110, 1] [14, 3]
~u~ 5 'l1nm'V'I~ 5.6 ~'l1'l'lll1bftU'Y11.:J 5-4 bbi1~ 6-7 bftU'Yl1.:J'l1n'l111fi 1 N1U'l<11 6 11.ltr.:i
' '
~111~ 7 'l~blirnfiu'Yll.:i 16 + 6 = 22 zj.:ibUUbfttt'Yl1.:i~Aun-)1bftU'Yll.:J'lln'j<11~ 1 t-i1u~111 2 11lt:i.:i<j<11~ 7 zj.:i
s:ib'1u'Yl1.:i = 30 ~.:i,:Cu,~lil!l.:iilmww.-11~~111 7 dlu 122, 6] b~!l'ill'lT>rn1'l1nm'V'I~ 5.7 bmEJrnfiEJlJ
'l<111l1~Yi'1J<i111~1m11~!l'l<11~ 4, 7 bbii'1n1'l~bli!ln1.l'l'~Yl1J<1l'l'l'l<11~ 4 ,d!l.:i'l1nL'6bfttt'Yl1.:i~Au~"'<11~!lq q q ~.i
18 ~.:i,:Cum'l~1l.,~Yi1J<il'l'lm11~~111~ 4 ~!l 118, 5] ~.:Jbbi1<11.:it-1'1M1'V'I~ 5.7

5-12 n7'fSu•1'nzmtff1V"imrutifl::-n1-N'"1n1m11ffel1iUw1u.
[13, 3] [22, 6]
2
17
15 6 5
[O, SJ [18, 5] 6
3
4
10
4
2
4 [16, 5]
[10, 1] [14, 3]
" ' '
oifufi 6 'illn!ll'wfi 5.7 ~'il1<J1;i,rnl1um' 4-7 '""~ 6-7 •Kum,,nn'ililfi 1 Nl'U'ilil 4 1uE'.i'• •
"lil~ 7 'il~blJubKurm 18 + 5 = 23 ol1ub'1urrn'il1n'jil~ 1 ;,l1u'jil~ 6 1uE'.i~'jil~ 7 'il~blJubKurrn
16 + 6 = 22 %~blJubi1'urrn~~un11 ~~iium'il~u1~Yium1m11'lil~ 7 ~eJ [22, 6] ~~m1il~11i.u
fll'l'i'Yl 5.8
[13, 3] [22, 6]
2
17
15 6 5
[O, SJ 118, 5] 6
1
3
4
4
2
4 [16, 5]
110, 1] 114, 3]
' '
"" t1 OJ .,.
ftlVfll 5.8 fll1 ~~'l'l1J1l~l!J11~~'1'1'l 7

' 9-< ) '
m-;'11u1rnmLli'um~rifiui'i<11i1'illn"JlilYl 1 fi~"JlilYl 7 '~loiiL'1U'Yll~ 1-3-5-6-7 ~~bbffil~L"1
' .. l t
ii "'l ..... <:II ""'
~VI !l!'U'llH'lll!'U'm!Wlllfl~Vl'll 1 1:::v:::111~
(Olll!:IJl'l1)
2 1-3-2 13
3 1 - 3 10
4 1-3-5-4 18
5 1 - 3 - 5 14
6 1-3-5-6 16
7 1-3-5-6-7 22
~~iiu ~~mm•11ff1t114i"11m-;'1m1rnm Lli'um1~~u~'!Iiim m•11n11"1~1d
iu~ 1 tl•~Yfum1m1•"Jlil1i'um1$iiJ [O, SI liltii'.inn111uillVl"Jlil~ 1 bUU"Jlilb~:1-11lu
iu~ 2 f1'ill'>illl"Jlil"111'1 ~mm•m~ill-lfllJ"Jlil~tl-;~Yfulil'>lm1•~oi1<11i1'11i'.l"JlillilUl~ liltJ
ril'VIUil1VlbUU"JlilU'J~YllJll'11~1m11rlilU liltJll'i4i'1bi1'JJfflj!'1n1:1rn"lu1~LffilJ liltJ~Wl1bl1'JJbb'Jn'lu1~bffilJ
'VlmtJti~ 'J~tJ~'Yll~'illn"Jlilb~:1-11lu1tlti1"liliiu WJ1bi1'JJ~ 2 $iil"Jlil~mriilU'VIUl"Jliliiu liltJtln~~n'il~~l-1
'
'illn"JilYJ 1 rlilU
iu~ 3 •1n-iru~ 2 b~iln''liltl'J~YllJlil'Jl~1m11~Hr.ii'u'Yl1~~~u~'!lil ~~'l~L71ln"Jliliiu11Ll'iu
'illilti'>~Yfum1m1-;
'
iu~ 4 i'l1'YJn"Jlillu~1tJ11u ilm-;tl-;~Yfu<ii'Jlm1•fl'JlJ'YJn"llilLLiK1 m•'11u1rn•~~~il1811i.J
f1'>1J'Yln'lil1Vili!Jun'1uhl'l111u-iru~ 2
' ' I 9' > ! < V I I
2' ..., =-=I !!J A..., ,,,,i ""'- !!J .di A ~ V A ' AA ell
'JJilff1Ln<il 1llfll'J'VllbffU'Yll~'YlffU'Yl'!lil'illn"Jlilb'>l-l<ilUflil"Jlil'Yl 1 'JlU1U'JJU<ililU'YIVl111:!-ln'JillJ'Vl'Jil
1i'il~'l11m•V1l~f1~~iiu•~•'Yi1nu«i1u1u N - 1 (liltJ~ N flil «i1u1U"Jlil<i11:1-1Wl1mh~il 7 "Jlil ~1iiu
N - 1 flil 7 - 1 = 6 bbilil~11'il~1i'mm~~~Ub'YilnlJ 6 WlJ ~1'J~L"1'11<ilillJ)
. ' ',.... d. ~ a .::1. v '_I -1""' ,.......,,....,..
'11ll~'illflflfllfl!'ll!l'llll!l'!~!'JlMfl 5.L2 Ull1 rn'JV111fjlJl'lfl'ilfl1'J:IJ 5.1.2
' ' . .!I d. ' .d. .q ~ <=l
!llU'll1fll'>flfl111'11'111Dfl 5 fill'llfl 5.1 m1~fl 5.1.2

• •
<!I "" "" d <!I '
!'Hl'.l'fl 5.1.3 fll':i1!fl':il~'ti!fl':i8'lllf.J'.lllH!1J'U
Minimal Spanning Tree
bf1~£J'li1m11.muu Minimal Spanning Tree dJu1ll'm·:mni'.:lt1J,.,l1<111.1ilbll1,.,m1.1~fi1.1'blJi'iB
m1mbfl'U'Yll1~b~1l:i.ibflU'Yll~'YJff'l<il1u'li1m1u1<111.11'1-i'ilbflU'Y111-;1;i.i~~U~')<il
~16VH~ s.3 m1111!'!1mmu!1L1h'i1mnu~.!111 T<111.11t"mrnf1bil!1~tl1:;,.,iJ<11~')<11
22 26
6
13
27 18 24
20 33
18
5
32
49
(,,,u1!l: bl.Im)
•1nmv;Y 5.9 1Vlt11m1111bb~um111;immf1bUl11Vllt1'i1"l<11"11~1 i!f1~u 7 "l<il 1<11!11t°mtJ
.. '
bmi'll1Yi~uYiGJ<11
iwmm1um1.i1u1rnt11h'lii';id
iu~ 1 b~iln"l<il1<111 fl1'1'lu'li1m1ubllu"l<ilb~:i.iliiu
iu~ 2 b~eimfiu'Y111•1n"l<ilb~:i.11ii'm'lu"l<11~1nl1~GJ<il ~1blJUbflU'Yl11~~U~GJ<11 (%1m•,.,m1.1'i1
nm '111'6''1111.1 1:;tJ:;'Y111J
iu~ 3 •1n"l<il~!Jm~m.ifo1bflU'Yll1bbi11 'li1•11rn1b~iln"l<11~!'i11l-11"1!Jm~ei;i.iTm~iltj1ni1~<11
~~m1b~il:i.Jfo1,1i11mn~GJ<11 b~ein"l<il,X'Ubb!1:;b~ei;i.illl1"l<il{(ufiu"l<il~!Jm~ei;i.iTmbbi11
iu~ 4 1'1141tuiu~ 3 •un-:hYJn~<11•:;!Jrn~ei:i.iTm (n11:;1.1:;'Y111b'Yi1nu1uiu~ 3 1~ein"l<11b<11R
10l,v;11:;m1il-;:;1.1:;'Y111b'Yi1nu•:;1"1.i1<11eiu~ilmnn-:h 1 .i1<11iJU)

' '
"Jlnm'WYi 5.9 ilEJ1J'11ilEJ1JYi~Yi'l'1
1. "J1m11'W~ 5.9 liiEln'lil~ 1 dju"lill~:J.J~u "J1mXul~EJ:J.JLm"lil~EJ~Lniln1J"lil~ 1 mn~
'l'1iiiEl"lil~ 2 n1J"lil~ 3 "lil~ 2 1-if1~1J~m,,Yi1n1J 13 "lil~ 3 1-if,i'um,,Yi1n1J 1s (ii,i:iu '~EJ:J.JTm"lil
Yi 1 fl1J"l'1Yi 2 (ii,m'WYi s.10
22 26
6
27 18 24
33
5
18
32
49
2. "J1ni:iu"J1n"lil~l~EJ:J.JlmLlil1fiEJ"lil~ 1 lli'l~"lil~ 2 '0"J11Ul1"lil~ti~hi1~t;Jm~m-i~EJ~
Lnii'~ijilfiEJ~il~ 1 11.l~il~ 3 lli'l~~il~ 2 11.l~il~ 3 '0"J11Ul1llii'1"J~1~u1~11~il1~EJi.JLtJ,{i'il1ll"J~lUU
"Jil~ 1 11.lti'"il~ 3 1'W11~1-if,i'uvrnAu~l1ilfiEJ 18 (ii"1l1il~11'lum'W~ 5.11. . .
26
6
18 2
33
32
49

3. 'illff'Jil~L:iE1:1-1Lu~LL<i'1fier'Jil~ 2 bil1~'ilil~ 3 'W'il11ru,1'3il~t'.fo1l11iil~m~B:1-1~mJ1n.i'~ftilq q q q 'II 'II •
fiEI '3il~ 4 'ilil~ 5 bbl1~'3il~ 7 'ii~LoiibftU'Yll~bYhnu 22, 27, 30 bbl1~ 49 <iil:J-11ii1iilu~Bv;'ill'l[ll,lbb.i'1'3~
LliBn::iE1:1-111.i~'ilil~ 2 1u~il~ 4 L~E1~'31n'loii1"1u'Yl1~~~u~ftil ~~bifti1~11'lum~~ 5.12
' ' '
26
6
18 24
33
32
49
4. 'illff"il~i~ernll'J~Liii'1fi!l"lil~ 4 mi~'ilil~ 3 v;'ilnrn1'ilil~i:i~1l11iil1m1:i!l:1-1~!ltJ'ln.i'~ftil~ q q q 'II 'II q
fierilil~ 5 'il11~ 6 t11v;'311rn1'31n'ilii~ 4 1tJ<Jii~ 5 "'"~'ii~ 6 ,~'loii1"1um~1·vhnu 18 "'"~ 26 <i11:1-1q q q q q
iii1iilu "'"~A1v;<J11rn1<J1fl<Jii~ 3 1tJi:i~<Jii~ 5 "'"~'ii~ 7 ,~'loii1"1u'Yl1~1·vhnu 30 bbl1~ 49 <i11:1-11ii1iilu
' ' '
~~~U<J~b:ij!J:J-ilm")11~ 4 nlJ")11~ 5 b~'ll~loiJbftU'Yll~~~U~tJ11 = 18 ~~bbfti1~11'lum~~ 5.13
22 26
27 18 24
33
30 32
49

5. 'illff'Ji1~b~!li.Jtu~bb~'l~!l'jil~ 5 bLrn~~i1~ 3 ~'ill'Jllil~i1~8~tlJ1.'i'IJm~!lJ.JYrn~!l~LnK~
l1ilfiei'ilil~ 6 bbl'l~'ilil~ 7 t'l1~'i11-;ru1'il1n'ilil~ 5 ttJ'ilil~ e LLl'l~"Ji1~ 7 "~'loifblru'Y11~b'Yi1nu 24 bbl'!~ 32q q q q q q
"11;iii1'1u ii1u•lfu'Yl1~'i11n~i1~ s 1tJ'jil~ e •'Yi1nu 2e bbl'l~'il1n~il~ 3 1tJ'lil~ 7 b'Yi1nu 49 '1~~u 'ii~
b~!li.Jfo~'jil~ 5 nU'jili1 6 b'V'i';l~LoifblfU'Yll~~~u~f!il~!l 24 '1~Lbl1il~1i'lwi1'V'i~ 5.14
6
33
6. "Jlf1'ilil~b~!lJ.JLm1L~1fi!l<Jil~ 6 ii~ll'iilil~ii~1lJ1.'i'11m~!lJ.J5n 1 <Jilfiei<Jil~ 7 'il~~'J1-;ru1q q q 9J q q
LL~1 1lfu'Yll~~mm-;m~ili.Jtti~m'jil~ 7 1.'i'fiil"Jlf1~il~ 3 1tJ'jil~ 7 ~ii~ 5 1tJ~'jil~ 7 bb'1~'jil~ 6
lU'jil~ 7 "J~LoifbliU'Yll~L'Yi1nu 49, 32 bb'1~ 33 ;lli.Jii1'1u '1~~.b'<~L~ili.Jtm'jil~ 5 1tJii~~il~ 7 b'V'i';l~
'lolfdiu'Yl1~~~u~f!il = 32 'il~L~u1.'i'1111n~i1)m~!li.Jfo~111J.Jil'YJf1'j'1LbK1l11tJ1#1'1~m'V'i~ 5.15
22
13
18 24
18
32

• •"'! <: "" d "'! I
!':ifl~'fl 5.1.4 fl1':i1!fl':i1~'tt!fl':i0'1l1£1~1'UH'IJ'IJ Maximum Flow
b'1"l!l'll1tJ-:numJlJ Maximum Flow bUu11lm·mni1lJ.!Vll11'1tJ:i:ibU1VllJ1rn.;zj!lVll~l'U1um11vrn
~1111<114i'mn~lll'l (•°liu ~1u1u111tJu#i~~~!l~uu11uu 1fhrnuil:11hnh il1iiu vi"lm•ni1~'ii~'Ylw1i!l)
'illmJl'lqf'U'Yll~ (Source Node) ttlti~'ill'ltlmtJ'Yll~ (Sink Node) 1urnn~rilVl'Ul'l l'U':i:;JJlJ'llm Maximum
' '
Flow ~lbUU!ltl1~ri~~'il~qr!l~Yl'ill':illil'11ll-li1ll-ll':ill'Um':iLV1'1 (flow capacities) 'illn''Jl'lqr'U'Yll~ltl
8~~1'1tl<01tJ'Yll~~mmw1viio14i'mn~ll'1
~101.h~~ 5.4 h~niui!l ii'ubbvi~vi<l~qrmm1'ii~ili:iiu1Vi'mn~il'1Ltlti~u71ffi~l'l~lVIU1tJi!lii'u
~1b<l'll~!ltlum{ium~ (arcs) ~!ltl7mrnil:1:iiu (wi.l1tJ ~'Ubbn'1'1!lu<li!l~1Tl-!~) •°liu•
5 3
'il1nm'l'lV1mtJm1l-l11 tl'hnrnm1'llmb~1iiu'il1n'ill'l~ 1 1tiu~'il'1~ 2 14i'mn~fll'l~!l s.ooo
' ' '
bbnflfl!lUl'i!l~11l-l~Lu'llrn~~ti1'ii~mu'il1n'ill'l~ 2 1tiu~'ill'l~ 1 .J11.11um11viiomn~fll'l~'ii~14i'~!l 3.ooo
' ' '

~heidH~ 5.5 m'l1111~1u1um'l'lJUll,J1:i'.iu•nn"l(l}iifu'Yl11~ieY'j(I}~
7 /1111l(1}11Ji'hm1~~ 5.16
3 0
0 2 0
5 2 3 1
0 7 0 7
3
0
nBU~'ii~i'.lm'l~lu1ru.1111~1u1u~mm'ln1111'11~mn~'j(I} <1bmJ11'1JBlVfV1"1"ru.1'lu.,1t1'1~1lit1(1}
'IJB1m'l1Vlmii'urrn';lln"l'1~ 1-3-6-7
i'i1V1"1"ru.1m'l1Vl'1'illf1"l'1~ 3 1uii1"l(I}~ 6
7 0
Vlmt1'111J.111iJ~mru.m'l'lJUll1';11f1"l(I}~ 3 1uii1"l(I}~ 6 1Ji'mn~'j(l}~il 7,ooo 1m'1'1BuiilB
411J.1, 'lu'lJru.~~n1il1m'iu"1n"l(I}~ 6 1uii1"l(I}~ 3 iiu1l-ill1m'lmh1~
n11'1';1l'lU1.l';llmflu'Y11,m1J.1mm'lf1'lum'l1Vll1';11n';l'1~ 3 1u'il'1~ 6 1~ll,ll'1b'Yi1nu 6,000~ q 'lJ ~
1m11'1Bu.iB411J.1, 1111~'luYi1uB11$imnun11'1'il1'lru.1iiBum'iu"1n"(I}~ 6 1uii1"(1}~ 3 i'.lm1J.1l1J.11'ln
' '
'lum'lLVl11nr:'iu1~1'Yi1nu 6,ooo •m'1mm.iB411J.1"'llu1$imnu bbl(1},1~/11111~
6 6
lii~iiumTI'i';ll':lUl.l'lJil,'iTim'l Maximum Flow i'.l~U'1il'Ull1U1~ /11'1
~u~ 1 11111i'u'Y111<J1n"l(I}•~i<<ii'u 1uifo~(lllJmtJ'Yl11 1(1}t11'1<J1'lru.1m1J.1mm'ln'lum'l1Vl'1~
11Ju1u1~ (iifB~i'.Jfi1mnn11 0) 'lJil~ 1i'm1.i11~1Ru'Y11' n11l-illll-ll'lf1VllbRU'Yll'<Jln'il(l}iifU'Yl1'1uii'"(I}
' '
umt1'Yll~~ll1m'ln1111'11~bbl(1}~111~'11'1BU~rnm~llJ-111~1

iu~ 2 n1m1:1-1mmo11'lumoLn1;1umfiu'Yl11'llil1LL'iil1~"l<il1JUbflU'Y111~uill'J~ij'1lmb'iil1~bflU
m~ (Pf) ~b~il1Yil1n~u~ 1 Lr;il'Jr11mob~:1-1-ii1u1um11'1,rn'lu~11'J~1utr;il'Ju1L1.h1:1-1nu-ii1u1u~fi1b~:1-1
'OJ1n'OJr;iliru'Y111Ltltf~,r;itlml'J'Yl11. .
iu~ 3 'OJ1mflu'Y11~~b~iln'lu~u~ 1 u1'11-iim1umol1'1l1 (Pf) L1.J11ufi1~ililfl'OJ1mfiurrn
1:;n111"lr;i'ii1~'1 umfiu'Yl11'0J1n"lr;iliru'Yll~Ltltf1'jr;itlmtJ'Yll~ bbl1~ii1Ltlu1nnubfl'U'1'111{founfiu1u
"ii1mm@lmnu (Pf) Lbli'1{founfiuLtlr11iu~ 1
'OJ1fl<if1€1U11~ 5.4 LL'1<il1fll'lfi1'U1lli<il1:1-i~WllilULJl<iiit
iu~ 1 b~ilmflu'Y11~ 1-3-6-7 81-ol1u1uni-;L~~" (Pf) ~mm111fi~tl1:iiu1Jlmn~l1'1<illmfiu•
'Yl1~d,ri1nu 6,ooo bbfll1l1ilu'iiil~1t:1-1~ (•flu'Yl1~ 1-3) bflU'Yl11 1-3 1a-Jmm-:mfi~w1u1Ji5mbli'1 tl1mrn
~il1ln'OJ1mfiu'Yl1~ 1-3, 3-6 bbl1~ 6-7 111111~Ltl•ri1nu 6,ooo bm1111ilu 'il~bn~mri1nu o, 1oo LLl1~
1,ooo bbfll1l1ilu<111:1-1ti1iilu tl1mrntl1:iiu~fi~ifilunfi'u'luYi.i'Vrn 7-6-3-1 R<il~b~mri1nu 6,ooo
bbfll1l11l'U Jl1m'liunuiil~'b"'11L1'lum'V1~ 5.17
3 0
0 2 0
6
5 2 3
£)6
6 1
6
ff :r
5
0
0
' .d. -~ feO.J ii
!lWl'l 5.17 U1"11l1011'11'11ll~UllJ'llUll!ll'll'lN 1-3-6-7
~u~ 2 V1'0J11rn1b~eirnfiu'Yl11'lna-J$'iJ 1-2-5-7 '11-ol1u1um11n11 (Pf) ~mm111fi~tl1:iiu1JJ
mn~!1'1<111mfiu'Yl1~dbri1nu 3,ooo bm1111eiu<iiei~1T:1-11 (bfl'U'Yll~ 2-5) i1~1iuu1mrntl1:iiu~fi11JJ,~:1-1
~u5n.3,ooo bm111;1eiu b~eiu1Ltl11:1-1nutl1mrn~fi11Jltu~u~ 1 -ol1u1u 6,ooo bbfll1l11l'U 'il~11:1-11JJ
fl~~u 9,ooo bm1111eiu -ol1u1um11n11uubfl'U'Yl11 1-2-5-7 mtJnfi~m1tlfo<if1bl1'll1J'ULfl'U'Yl1~bb'1'1~1JJ
l'Ulll'11~ 5.18

0 3 5z ,lj
-8"
2
1
2 6
),3' 2 3 1
9
0 6 1 1
3
' ..,. _l':l r o .... !ll
11Wn1 5.18 lJ'llJlll!fll1'11Hl~llllJll!l'll!llll'fll~ 1-2-5-7
~u~ 3 'l'.1"J1011rn~flmG'fum~1111:1-Jiliil 1-2-3-5-7 rhm1lV1'1 (Pf) ~mm-;11ci~~1iru1'1!:1-11n~
<t1?1mmG'fu·vrnd•rhnu 2.000 •m'1'11lU (•G'fum~ 1-2 111'fo 2-3) •~1lil111.h1i.Jnutl'hnru~ri~1liln~u
~flu~ 1 "'"~ 2 l'fi'1 "'~01i.J1'1f,if~~u 11,000 Lm'1'11l'il 'ii1mum'll'Vl'11Jm'1um~ 1-2-3-5-7 m!J'Vlil~
m-;tl1'u~1L'1'1Jl'l11?1~1'1f~~mY<~ 5.19
0 3 5
5
:z 20 2,Z 11
4 1 6
Oz 2; .g
1
11
0 6 1 6 1
5 3
' .e:!. -~ I o <V !V
lllWl'I 5.19 lJ'llJlll!fll~'llllll~llllllllJ'll!llll'fll~ 1-2-3-5-7

%u~ 4 v:i>Jl':H:111bi1!lmftu'Y11'l'11!-iiiei 1-4-6-7 .-11m"Jl'lli.1 (Pf) ~C1m•11;;hJ1~u1#1mn~
6j'1il1:1-J•ftuY11,ihrhnu 1,000 •mmrnu (bftuY11' 6-7) .~!lu111h1:1-Jnu'll~mrn~<i''1n%uil!lu~ 1, 2
bbi'I~ 3 bbii'1>J~"l1l-llWJ~~~u 12,000 bbni'li'l!lU ~lU1Un1".il'11111JUbftU'Y11' 1-4-6-7 llltJ'lliii',m"l'll'luif1
bi'l'llbbM1~1#Jil,m1'1~ 5.20
0 3 3
5 0 2
0 4 1 1
.fJ' 7
12 ------;.., 1
0 6
3
1 6
6
.r
0
,8 3 fr
4 1
1 )3
' .mwli s.20 tl1mwm"i'lnJi:i~ll11¥iillmil'um~ 1-4-6-7
%u~ 5 1Ai•11rn1•i1!lmftu'Y11'l'11!-i?i!l 1-4-6-5-7 .11m•l'111'1 (Pf) ~mm111<1,J1~u1#mn~
C'1lll:J-Jiflu'Yllib·hnu 1,000 bm-.imm (bfluY11' 6-5) b~!lu11'll.,1:1-Jnu'll~mrn~<i"1n%uil!lu~ 1,
'
2, 3 mi~ 4 bbfl1 •~"iw1Jf,;r~~u 13,000 bbnl1!1!lu ~imum"il'lli'!Umflurm 1-4-6-5-7 mtJ'lliii'~m"i
u·l'uii'1bl1'llbbe'1'1Jr'1',m1'1~ 521
0
13 ------;..,
0
0 3
5 0
4 1
6 1
3
3
' .
2
;;r
2
.nmli 5.21 'll':lmwm"i'llllli~ll11¥immil'ii'l'rn 1-4-6-5-7
7

.fru~ 6 Vl"J1'n111Lfium.:iLV1ii$ieJ 1-4-6-3-5-7 .-i1m-;1vil'I (Pf) ~mmwfi.:i1<1'mn~aoimlcJ,
LflU'Y11.JdL'Yi1nu 1,000 LLnl'll'leJU (LflUl'll.J 3-5) L~eJ'i.i111h1lcJtllJmlcJ1rn~fi.JDDn"l1n.fruwieJu~ 1 ~.J 5
" " 1LL'11 "J~'>1lcJ1<1'Yi.:ifiu 14,000 LLnl'll'IDU '11u1um-; Vll'IUmflum.:i 1-4-6-3-5-7 mt.1vi'1'.:im-;1J~'ulii1
Ll'l'lJLLC!O'l.JL<il~.Jlll'W~ 5.22
0 3
5 0
3 :z: 14
0
7
0 4 % 2 5 0
14
0 6 -1:' JiJ 0
3
2
3
' .m'l'ln 5.22 tl11nru111'l'll1Ju~il1iJ1Ju1Jtii'1mN 1-4-6-3-5-7
"llnfll'WLL'l"n (.nw;~ 516) Lbl'l~fll'W'J~Yllt.J (fll'W~ 5.22) <if1Dth.:im'>LVll'llJULflWYll.J 3-6
u~mrnL~lcJLL'l"nm1lcJmm-;riLum'>lVil'! LLl'l~u~mrnm-;LVll'l'1~.:i'J'1Yl1mLCf'1.JL<il<if.:id
u~mrnm'lLVl'1'1~.JLL'ln
7 0
2 5
i:iuLLaoi.:i1vr,Vlw11m1lcJmlcJ11ri1um11V1l'lumfiu'Yl1.:i 3-6 <'Yi1nu 5,ooo LLmrneJuiiieJ~11lcJ.:i
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11 n'ilm':i:i.Jlf11j<ll (critical activities) 111tJ zj.Jbft'.l'Yll.J'lnri<11d~u~·vmlm.:im':i'i1~41mmlJ'J:i.I1Vfn'ilm-;:;.i
l!il.J'l bl1~<ii'lll:i.lri1'1'1'.lilb'll11 fl1ii1oii'1'i1~<l.:ii:.il11'1ll vrb1l11'll1l.Jlm.:im-;ii1'lf11t11i1 tJ <l1rnft'.l'Yll.J~'.l~li.i
1'lf,{fu'Yll.J'in '1'11 n'ilm':i:i.JlJ'.lbft'.l'Yll.Jdmmw1'11ii1oii'11~'111:;.inm~ilVI ~'.l'llil.Jbbl!i l1~n'iln-;w
fl11rll'l.l1ill'l1111l:ll'UO~1fl1~fll1
m11'im1ru1111n m'llil.J1'11.Jm':iil~u<11eium-;l'i1u1ru1111nmei ~ 4 ~'.l'llil'.l~'11fl'J!Wl.:id
0 ~-· ~ ~ ~ • '
1. fll~i'llJ1W !U'llH'llJl (Forward Pass Computation) "l~:i.lf11'lm'.l'llli'Vllnm'llil.Jbb'lll1~
n'ilm'l-:i.i'lJEl.J1'1':i.Jf111 2 fi11iiei nmb~:i.i41rn~'l~!jil (ES) bbl1l::b1'1lbbfr'lbi1~"lb~'l~!jil (EF) til tJiJ
'VIi'in rnru.;I'1m1ru'Vil'1m ~:;.i41ub~'l~'lil'llt1.:i bbllil1 ~n"lnT>:;.i1~Wi.:id
t- c;; c;; ,,.
EF = b'll11bbl1'lbi11"lb'l'l'Yl!jil
t = nm1um-;1'11n'iln->1:i.i
1il tJ~ EF = ES + t

ES EF
~
"Jlnm'l'l n"Jfl"i';l-1 A flll-115rn~l-l~'UYl1L#i'Yfuf b~m"J1rnuun"JnTirn~l-l~'U'llil'Lm,m-; )(,,:(',
ESA = 0
tA = 5
EFA = 0+5
= 5
'lum-;V11nm b~l-l~ub11~!j'1 (ES) 'llil,Wlifl:;n<Jm-;mXuilni:i1um-;V11L#f)l,d
nlJ'llil,nwvnrnrn~l-l~'Ub~1~lj'1 (ES) nmb~l-l~Ub~1~lj'1'llil~n"Jfl"i'll-ll'1'1 (ES) "J~b'Yi1nul'i1~3-11n
' ' '
fi!l'1"J1nl'i1'llei~nmbb~1bl1~"Jb'l:11'iii'1 (EF) 'llmn"Jm'l"-Jfiei~rfou1>1ii1
b~il~"J1-;nn;nnm'l'l~ 5.26 l'11Jm-;u1nl:J'llil,m•V1lb'li11b~l-l~m11~lj~'.bm'loif m-;'11u1rn
1>11b'lfll (ES) bbfl:;b'lfll (EF) "J:;f11u1rn11Hh~V1ii:1 "Jln"l~b~l-l~u1tii'f'"l~'!~Yfl!J'llil~fi"Jm5l-I ("Jln
-ii11J:i'lil1t1m~'ll11:1'lil'lJeNm'l'l) )l~m'l'l~ 5.27
ES EF
A 0 5
5
c H
Start
ES EF 4 12
B 0 6
6
' . ' '
ilWITI 5.27 !1lll!~ll~!!lhfiq~ (ES) !!ll~!1lll!!a1m'lll!hfi11~ (EF) 'llMO'ilfl1'>11 A !!ll~O'ilfl1111 B

~
l-lnm
' ' '
nl'lW'1l'Hllb1inb7:i.i<liub~1Yifl<11 (ES) firhrnw B
ES
8
= 0
tB = 6
EF =
B
0 + 6
= 6 '9'1l1111'11
~~
c'Ylf1'1nd'l:i.I
ES =
c
5
tc = 4
EF =
c 5 + 4
= 9 '9'1l1111'11
. '
61JB'lfi"<ln11:J.J H Uu1J~~B.:i'IN-c.J1aru1f1-<un11;wYltltJri€lu
EF
~
:i.llf1'Yl'1<11
'
rnn ESH
ESH
i
=
=
=
9
9
12
EFH = 21 '9'1J1111'11
nl'lW'1l''lilllfll ES bbfl~ EF 'll8~n'1rnW H bb'1<11~1Ji'lUJll'V'i~ 5.28
A 0 5
5 EFc ESH
/' '-.,, i i
c 5 9 H 9
Start
EFB 4 12
'-· J
B 0 6
6
•
21

111,~H'il"1m"Hl~'ill'illilll ES b6'1~fll EF '])fl,fi'ilrn'il-l till'J1i111Anm'lfilU'llli11Hh~111ib
'ilUmU'YJnfi'ilm'il-l'ilUbbli'lL'11'il ~'n"1i'i,~wJJEMlm,m'ld'linm 26 iltl1i11V: 111''11li1#i'lwi1'r'I~ 5.29
E 5 6 F 6 10
1 4
"A 0 5 D 5 8 G 10 24
5 3 14
/' i
c 5 9 H 9 21 I 24 26
Start Finish
4 12 2
..
B 0 6
6
2. 111'lft1'111!l!li11unii'u (Backward Pass Computation) Lrifll'ilm'ifi1u1rn1t1-if1,111u1'1JEJ~'!i11'J
•1 ""' C 0 = 2 1 C 0 D D 21 ..... D
~lU'OJU111'11 ES LL'1~ EF '])fl''YJnn'ilm'il-ll'1'i'i1'1ULL'1'l n>J~'Yllm'imU1llil'J!l'l-hn'1U 'il1nm'l'11U'llli
1t1-if1~111ii:1 L'il'Yl'i1u-i1L'1'i~m•d'lin'11i'i,~u 26 iltl1i11V: L'il'il~L~l-lfi1mrn1111'11 LF LL'1~ LS Lill'.I
' '~Viii' A V ii .<>!.
L LS = nmL'il-lilU'lll'Yl<jil
" ~ ~
LF = nmL<l'i"J'lll'Yl<jil
t = L'l'11
L~W~ LS = LF - t
~LS LF
fi'ilm'il-l I d'Jufi<Jm'il-l<jilYlll'.l'lJEJ~L'1'i~m1d m1fi1u1rn1'i!lun<i'u >J~Yi'ill'illilfi>Jrn'il-l
'1i!Ylll'J {ie:iunAu1t1-if1,111ii:1 ('illn'JJ11i'J!llti'Yll,oif1 l'.li'.iEJ'lJ e:i~m'r'I) Lil l'.lri1111uil1iX LF 'll!l' fi'ilnnl-l
'
'11i1Ylll'.l'il~Lvi1nunm•1l-l~1m,m•L'1~'ilL'1l-l!l 1111EJi'.i'11Lvi1i'ufi1mn~'1il'llfl"1mLLli1L'11'ilL11~,;il'll!l'
' ' '
fi'ilm'il-l~~u<jil'll!l~L'1"m' (EF)

LFI = 26
tr = 2
LSI = 26 - 2
24 iill~n~
1um7fi1u1rnt:femnflu•f11l'vrn1m•~:1-14iu~'6'1~lf01
w1i"~ihm77:1-1m41tm!Jm7fi1u1rn1111$1~d
c;; !l; .,,.
ni;i'llmmi'mrnrn<17<J'lll'Yl'l01 (LF)
mn..ii1•<1~oii1~'1'1'll!l'n"rnd:1-1lff] <J~i'.ifi1•'Yi1nufi1~ustJ~ijli1'il1nfi1'lls,nmc4:1-11fuoii1~'1ii1 (LS)
n"~"ii''m17"l'llil,tm,m1~•'Yi1nu 26 ii1l011~ m1~fi1 EF1 = 26
$1~iiufi1 LF1 l1fi1 = 26 iillm~
=2
= LF1 - t
= 26 - 2
=24
~~~ufi1 LSI !Jfi1 = 24 i'1!011~ ,
~ .., "" .<>I ..,. ..., f!<v ""' "" ""' &I =
<1l'117lJl1'0Jl1d7l-I A 'OJ~l.lffl1l-ll1l-l'V'lUlll1lJn'OJl1d1l-l'Ylli11l-ll-11 3 ll'OJ!lddl-1 '1il ll'OJllTl'l-1 E, D, C
l01ti~fi1 LSE = 5, LS0 = 7, LSC = 8 fi1 LFA 'OJ::W'OJ1drn1'0J1nfi1 LS 'llil~,i(~ 3 n'OJl1ddl-l~!Jfi1ueJtJ~
'f11 1u~d;:ieJ LSE = 5 $(~~u LFA = 5
" '
$1~iiufin'ilrnd:1-1 A
LFA = 5
t = 5
LSA=5-5
= 0 ii1l011~

E 5 6 F 6 10
1 5 6 4 6 10
"'A 0 5 D 5 8 G 10 24
5 0 5 3 7 10 14 10 24
?<'
i
c 5 9 .H
'
9 21 I 24 26
Start ' Finish
4 8 12 12 12 24 2 24 26
.,
B 0 6
6 6 12
3, m1111nmti1w~mi11ll (Slack) mt1111ii'~<J1n~1~'11irn;urn:rnm~tm~m•~~•1nm•f11uirn
11Joih~111u1 bmtm•f11u1rnlfElunii'ubbl11 b•1'11i.Jl'm'11u1rn1111nmiih•El~Yib111i1Ellltl'JJEl~bblli'1tn•nT'liJ•
Slack = LS - ES
~
LF - EF111•[) =
,;
5.30 4f1Eltl1~ii•m-rn c"Jlnl1l'l'lYI
LSC 8
ESC = 5
Slack = 8 - 5
= 3 ilu111~
111lJltJ'111lJ11 n"Jn"l'llJ C i'.inmfi1'l1Hb111i1eJeJ~ 3 i'i.Jm~ ~~'11m•rni1ii1oif11~ 3 ifi.Jm~
L11t1~1i.Jii~N'1mt'Y11Jlim1m11iJ'JJEl~Lf1•~m•~•t1'i11Vi'tm~m.,dii1oif1iiu111mt1i1~11ii•m·rn c 1i.J
1'liii•n'1iJ1nl1'1
1u'JJrnt~n<Jn'1iJ E
LSE = 5
EFE = 5
Slack = 5 - 5
= 0

Lbfi!l,11n<Jnn:J.J E hi:i'.inmclT>il,,'Vl~ilfl~ ii',iXun'ilm"l:J.Jd'l.1l11m1lti1!i1<li11Jl L'l'l11dil
fi<Jn1'l:J.Jdti1!i1<lf1<Jtfl' Nii1'1tJ'l1:J.J<iim1fn"i1:J.J'llil~1'1'l,nl'l ~'OJttillVilf11,m1d'l<lfnmLfiWlll 26
i1iiJ'11V1 bbfi'1~11n'lm·rn E d'.iun'ilmw'lnl.j'1 (n'lfl':i'l:J.J'lnl.1'1 Slack = o Lfi:J.Jfl)
m1fi1u1ruV11fi1 Slack 'llfl"L<liiitn'JnTl:J.J cr1JJ1Jllli',<11Tll'~ 5.1
Earliest Latest Earliest Latest Slack Critical
Activity Start Start Finish Finish (LS-ES) Path?
A
B
c
D
E
F
G
H
I
(ES) (LS) (EF)
0 0 5
0 6 6
5 8 9
5 7 8
5 5 6
6 6 10
10 10 24
9 12 21
24 24 26
4. m1m1ifirnHifl']P1 .mtJ'Vlll,"J1n~1Jlfi1u1rumfi1
4.1 m"lfi1u1ru1J.Joii1~V1Ulb~fl'Vllfi1 ES, EF
4.2 m"lfi1mrut'iemn!lrnY'lilm'11 LS, LF
(LF)
5
12
12
10
6
10
24
24
26
4.3 m"lfilU1rum nmcl15ei"'VI~ilei~'11fl'bb<liiitn'Jm'l:>J
0 Yes
6
3
2
0 Yes
0 Yes
0 Yes
3
0 Yes
%u<11flu<iieilD'J~1'i1m'l~"Jl'ltlll'Vll bft'U'Yll,'lnl.j<ll 'OJlfl!ll"il'~ 5.1 lVl~'Jl'ltlllll Slack
Column 11n<Jm"i:J.Jl!l:i'.i.11 Slack = 0 fi'lfl"i'>:J-J,;(U'OJ~bllun"Jn'l"i:J.Jlnl.j<11 (critical activities) 'IJeJ'
1.i.,,m., ~'iJ'l~nmJ1uJ11tJn"OJm"i:>J A, E, F, G ""~ 1 ~~fi'lm'l:J.Jlli'~nli11<J~iJ"~nflrn~fl:J.J1tJ,nu
d'lubftWYrn A-E-F-G-l b"ilb"itJff.h bft'U'Yll~'lnl.j!l (critical path) (~~bft'U'Yll~'lnl.j!ld<J~blJUbftU'Yll'~
'l<li'l~tJ~nm"i1:J.J'llei'n"m":J.J<ii1''1 ~mn~'J!liirnei~)
<Jlflfll'llbfl'll~Vlbf11il'liltMlU PERT/CPM <Jln<ii'1mh~~ 5.5 ~1J'l'Vll'l1fl'l~nl'li'fl:J.ll"il!lillJ
fi1bl1:J.J~~ 4 fi1l1:J.J cr1JJ1Jlfiil
1. nm~~'Vl:J.Joi~1m~m'l"J~bbi1'1Lfi~"OJ fil!lfl1J~1Jlfifl'l<linm~~~u 26 fiiJmV1
2. nmb~:J.J1lubbl1~nm~ucroi'llfl'bb'iiii~fi<Jm"i:J.J'llfl~1'1-;-~m'ldfi1!leiufi~Jlfifl 'lVf~<Jl"itlllfi'lD• •
l~<Jlfl!ll"il~~ 5.1

3. fi'Jn'l'll.J1~n'.i1~~dJun;,in'l'll.J"ini;i<il ~'il~efi<J~vh1Vlbi~'il<i11l.Jn1viu<11 •1m ,Ylmlium'l
'l1J1h~nwi1L•n~m'lftbi~'1Yiuml.Jn1viu<11nm fl1<il<l1J~L.i'~<J n;,innl.J A, E, F, G ""'°'~ I bUU
n'Jn'l'll.J"infj<il
4. fi'Jn'l'll.JL<11u1~~1i.J1'lifi'iilm'll.l"infj<il ~ill.Jl'Hw11l11oif11~.rn~mm'lnli1oif11~~1u~1i.J
rl~e.i;;,Yi11~;;i.,~m.,dii1oif1 '11<ilil1J~1.i'~il1Vi~'iil1'lrn11.l;,i1n"1l'l1~~ 5.1 ~!ln'iilm'll.J B, c. D """~
H •llun'iilm•l.J~1i.i"infJ"1
~ " JI .. ,; ~ .._. -·~~~
'l'li:Hlllilt!il'llll'U0'111i!l'l~!'lM'l'l 5.2.2 m:n !lJ'lAu{l'lJl'lilJil'l'l:IJ 5.2.2
' ' ' '
1 ~ I <':: ":I. ~ ""
l1Ull11ll'lrlil'lll'l'lll11l'l151'10ll'l1 5.2 !'lM'll 5.2.2
1um~mru~1J'i:m'lL'1'l~m'lm'il'il~l'lm'l~'"'"rn11um'ld~Tm~m'lLVibi~'ilL~1~un11n1wu<11
Ll/11.Jnl'lbi'iilnn'ilm'll-11Jl~n;,im'll-lL'.nl'l1'<11b1mm'l'Y'il~l'.1ViLi~'ilb~1~'.~'il~rl~N1'Ll/11J'l1l.J"1ilb1in'l1l-I
'llmLm~m'l ~~;jlblJu'il~~il~l'lm'lbYll.11'1'1-wmrnrn~mh~bYll.J~U b'li'U '1'.~l'. '11il1~mn m'lLoifb'14il~
l'1m;;i41J~..;JrmYl:1-1~u Llluefiu Iii~~UN1J'lVll'lTm~m'l'il~efiml'lm•~'iill'll.i1lif<11lluhL1i'fomfh.1u1m4fl~
•
'llil~nm~ii<11;o,~n1J'111oif'ii11J~LYl:1-1~u1i'lrnfl1Jm'l~'ill'll.i1 (time-cost trade-offs) lif~'iil~flIT1J11J<i11l-I
'
1if1€181~Yi 5.7
i'i'10th~~ 5.7 n'ilnl'lbbvl~Vlrt~1~l'lm'l..;J<11Yi1L'1'l~nl'llUm'lli11~'lni,r11viqjL'14f]~..;)m ~~bU'.
t;;i.,~m'l~n'iilm'l""vi~d1.l~<11Vi1~'.bllu1i'l~;j1mm<Ju'l~1J~L1m'llil~m•lil'~~'lni,r11viqj WI~~'nm~
1-ii'lµmw11~'lni,r1'llil~bb<liii~n'iilm'l:1-1 ~1J'lvi1.,tm~m.,d~~mm'lnm11Jnm'llfl~bb<liii~fi'iilm'l:1-1L'.m'l
li11~'lni,r11viqj b'14il~~m'llil~h~~rn1.i'EJ81~bbU'.lilU Wi~bbil'l~'llmi~b5l'J<11Li'lii~d

A
B
c
D
E
' ' '
m'l1Jrnfl'im'llil·:lbfl'fo1~n-ffi 1
m'llhmeirn~m~n'l'fi 1
' ' '
m'l1Jrnfl'im'llil~bfl'im~n-ffi 2
mw•mmJLfl4m~rn~ 2
nl'll'l~i1il!J'l~UU
A
7
Start
c
6
B
3
D
3
A
c
B,D
E
2
7
3
6
3
2
Finish
'illnml'lfi 5.31 m:J.1Tm11i1bi11.1~11.1L1Joih~111ifo~m1mh ES, EF bb<i~l1i1bi11.1~11.1iiei1.1nli'rn~ei
vmh LS, LF nmill"lil~'llil~bb'li<i~n'ilmw bb<i~1111Lfi1.1m~'lnriG111~ei Lfiu·vrn A-B-E 1~1.1lm~m"l
.tf,~1-iinm"l1:J.Jilf~~1.1 12 'i'u 'l11J<l~bB1J~bbrW1~11'11.1rn'l1~~ 5.2
~ , 0 "' ""'
m'lWl'I 5.2 fll'l111F11!1lll ES, LS, EF, LF !1llll!l'l0~!1llZ!lllfl':i'l:ll1fllJl'I
(ES) (LS) (EF) (LF)
A 0 0 7 7 0 Yes
B 7 7 10 10 0 Yes
c 0 1 6 7 1
D 6 7 9 10 1
E 10 10 12 12 0 Yes

CPM
>J1nm"iiiln1't1m"i"i"1"1~~'li1tJ,1urlmi11m••li1 ~u'lm"ilfl"i,m"i"1m"it:nhihbb1J1J PERT/
b-ii1m1.;tlumn1~LbNULb'1~mlJfl:1-1Lfl"i,m•14il91tJ L~:l-1~~bLliim"liiln1't1bb'1~"iLfl"il~~"iltJ'1~,5tJ91•
'lleNLfl•~m• m"l'1fi,'li1tJ,1u LL'1~m"i'iLm1~~'li1m1u 1'1lnmLLfi1L'17>J'll1l'Lfl•,m• 1'11b.9'U'Yll''imy11
bb'1~nmR1•1l''ll1l~bLlii'1~n>JmwLbli1 bu1Jl,n"irn~u'l1'11"iLfl':l,m"i1ll'J'J~l'Jm"irfl91iiuh1um"i'191"i~tJ~
nmm"l<1i1Luu,1u1'1'l1lnm••li1L'11>J'll1l~Tm,m"iM •'liu 'innrfl1mh,ri 5.7 Lfl"i,m"i1um"iUl"i~7n1't1•
b1'1qjbfl41l,"imh,~1uri>J1m~:1-11-ifnm11:1-1~'~u 12 1u th~u'lm'l"'ll1l'h,~1mLvi~d<ifo~n1'rn9inm
'lum1ui-p'..'n1't1l1'1f;J) Lfl41l~"in"ih~~1u 'lVimihinm•Yl tJ~ 1o -Ju L~1l'lVi1fl"i~m1d•'17••71n11
ri1'1'1U91•~:1-1 el'um•Li'l1l~m>J1mw1iw'1"11~'1 L'liu ufo1JltJriiifo~nn11~iiu.11L71~un11ri11'1U9i m1
111l'nui11].!1'1lfl11:1-1li1-if1'lum"i"i91fl,191')~1J ·rn"'I vi1'lVi~u'lm"i'i1dJumh'~~ri>J~<if1l~m1Jf):J.!'lVi
Lfl1~ni1•<17••71~un-)1ri11'1u9i•1'1i
mw91'l"~tJ~L1m'll1l'1fl1'niw1>Jvi114il91tJm1•~:J.11117~mn1'Yll~'1frn••.,~~rn'lVinun•n"l1:J.Jri
"int:]91 L'liu nwvi1,1uli1,nm m••~:1-11117~mmrn,mh~ •'liu Lfl4il~l'11l Lfl41l~"in1 ••lii•~l'l.!1'l.jf
<l1m~:1-1~u'111tJ el'ufi~~mvi1'lVi<ifu111ul9itJ"i1:1-1'll1l~Lfl1~m1'1~~un11L~:1-1rlri11'1u9111 4i'~iXu ~1J'l1'1l'l"' . .>J~<if1l~rfl91fl'Uh time-cost trade-offs 1~1'111~nmri'191'1~1'11•1nm•d~Tm~mT11'1:1-1.!1nu<ifu111u
1'1'l1l'11'loif<l1tJri<ifil~Lilm~:1-1~u1'1'l1l1~ • '
m1d~Lfl1~m• (project crashing) dJi{'inm•lumw1'lVi1rn~nm'll1l~Lfl"i~m1~u'1~ L91tJ
mw91•1m'll1l'n>Jm'w"inq9lwd~,,,'l<1mnn111'1i'l~n•n11:1-1'1~'lViifotJn11nmun~
L91tJm"id~Lfl"i~m1~~<l1Li'lu<ifml'J·ii1l~'14i'~"11l1ud
1. nm<1i1Luu~rn<111:1-1un~ (normal time: t ) 1"11lnmriu1~mru11luiu911l'Unl'l"11~LLN'Un !"-' I ! ....
2. nmlii1Luu~1u1Jcl1~d~791 (crash time: t ) 1"11lnmfifl'ufl'19!Yl'J~d~n•nTI:1-1iiu'l 1'11
C I Ji t q
3. '11'[.jf<J1tJun~ (normal cost c ) fi1l'11'[.jf'111tJYiLn9i'llULl'i1ln'Jm1:1-1l'Jm1ii1Luu~1u911:1-1n
4. '11'loif<l1m'l'~791 (crash cost: c ) 1"1€1.!1'l-if'ii1tJriLn9i~m~m-i~n•m1:1-1iXu'l 'lVil'17'JL91tJ
a; d c
L':l1Ylij91
-li1l:1-1'14i'~nli11mrn~11lum11~ri 5.3
•

Time(days) Total Cost Maximum Crash
Reduction Cost Per day
Activity Normal Crash Normal(C) Crash (C) in Time
( C -CJ
n c
(Mi)
K ~ _,__n
' M.
'
A 7 4 $500 $800 3 $100
B 3 2 200 350 1 150
c 6 4 500 900 2 200
D 3 1 200 500 2 150
E 2 1 300 550 1 250
-- --$1,700 $3,100
1unTrnfh.1mfi!Jufi1'loif'ii1!J'lunT;d~'1u'J~l'i1nTl.il'i.r1rnfi1'loif'ii1!J'lum'lb1,lliEJ'Vltl,·m.brn1m
(crash cost per time period) 'll1l,n'Jnnwlliw1 btrh.1ubfl!Junu iii,d
c - cc n
T -T
n c
'
'J1n'll1'l1,fi 5.3 <ii'1eim,m1rn'li.J A nmlin~~1l 7 "Ju fi1Loif'ii1!Jlin~~fo 500 lJl'YI n1d'
rnn rnn~d''~~1l 4 1ufi1'loii'ii1!J'lum'ld,~1l 800 rn'YI <ii,,Xu '11'loii'ii1LI'lum'ld,,1ulli1l'Vltl,rnn
'll1l'fi'rn·:m A
=
=
800 - 500
7-4
300
3
100 lJl'Yl/lU

800
650
500
Maximum Possible
Crash Operation
4 5.5
Normal Operation
7
<: f G} 91 T ~ I """
mvm 5.32 rn.nuu::;m1'll'ill!l1um11'i~fl'ilm11J A
<nnm'l'ri;j 5.32 rnrnrn~f11'l.if'ii1tJmn'ilm'i>i A Lfimn'ilm'lm@im U~l:i-il:'l).Jl'i1JlWl"111
'
n'iln11.i A il<ilL.if'i1tJl't-1.nl'ib1~~1UJ~111~il~ln11n'ilnTrn5u '<i1 Liiui1fil~ilm1Lti~mJLYtJlJnlJ
n'ilm'i>i5u L'lfU n'iln'l'l).J B '1'i1m1LmtJ1JLYJtJ1JnU 'illnm'Y'I~ 5.31 Li1UYll~'lnt:ililflil A-B-E b"1~
'il1n-ifol;!11'lu<i1111~~ 5.3 'il~l'i'il11ru1d~n'ilm'l>i'l<ii .,~11111~ A, B, E L~il~'il1milun<mTrn'lnt:ilil
LLiitLriill'i'ill'lUl1'11'l.if'ii1tJ'lum1d~.iei111'lt~111u1tJnmLLii'1 n'iln'i'i).J A 'il~i1'11'l.if'ii1tJUiltJ~l)il LiltJ
<J~d~nm'lun'ilm'l.id 2 1u 'il1m&i.i 7 1u LVIRil 5 1u L"1tJi1'11'l.if'ii1rn~.i~u~n 200 rnYI L"1tJi1'11
'l.if'ii1tJ'l1).J (1,700 + 200 = 1,900) Lllu 1,900 rnYI 1m~m.,d'il~L<l~'illill).Jnm~i1fil~m'l i'iil 10 1u
nnd~L<i11~m'l ,~,;ril~ d~~n'ilm'l.i'lnt:i<ii~il~'lumtJ~l'll'lnt:ililL'Yi1~'.l L'Y'l'l1~1111nd~'lu
fi'iln'l'l>il:iJ"int:i'lln'il~l:i-iLn11N!llVlL1i11LLii'1 L<l~'il'lJil~llil'l~nl'i1111~l~ LLRrnl'l!1~tm~m'inl:iJ<i1LlJU
<ifil~b1~11nn<Jm-1.i"inf:llil LLl'i"J~LRilnL1~L:'Y'll~1un'iln'l'l>i1nt:i<ii~i1<11u'YJUnl'ib1~IiiilVIU1tJL1m~l~ljii
LLR~1:i-id'im1mLil1111mtJ~<ifil~m'> uiln'il1nd Lriiln'ilm'l:i.1111'lt~'l ~!Jm1~LLii'1 rn~mrnm'i1Lfi"1m'l
Lti~tJumJM'lJil~mtJ~rn'lnt:i111fiililm m1u1&1).J~1:i-i1nt:i'llLtl~tJU>i1Lllumm1u1nt:i111'1~NR'l'il'lVILn111
n'iln'i'l).J'lnt:i111l111:i-i ~'il~<ifB~!Jn'Ull-lll'i'ill'iUll~'ii~d~1ti~1tJ i;'l~~U 'il~Li'.iu1~1lm'ld~Lm~m'iLlJU
m~1J1'.lm'l~Lllu'l~1J1J~'il~<ifil~m~'l'i1ilth~LUU~u111ilU
v "' .!! .. .,; ~ ''-' -·~ ~~'lllH'illfll'lfl'lfl!Wl'llllfT>::;!1lHl'I 5.2.3 !!il1 lll':i"1U!)1Jl'lflll!l1':ilJ 5.2.3
, ' ' .q <=t ' .... <=t Al <=t
!ll!!U1fl111'1fl'!!l'l1ll1!ll'I 5 l'llllll'I 5.2 !18~l'I 5.2.3

'nn<ibf:lth"~ 5.6 bli1.i.lm"n1TI1mm"11:1ri1'Vlu01"1~u~nmnT'1'11bilwllmw1i;o~n'ilm"ij.J1Jleith~
bbU'Uel'U (nm~bbU'Uf:l'U) botl'U n'ln"i"ij.J A Loiinm 5 i11l011~ fi'ln"l"lj.J B 1oiinm 6 illl011~ bU'U<iJ'U
b'ciei"'l1nTm"m"d~u1V11"1i11l-;~11um-;rn"lum"i'11bUu"1ul.11rieiu ~"mmwr1'Vlu01nm1unl''i111buu
n'iln'i'i:W<i!1"'1 1Ji'mh"oif01b'l'U (i(u$imb<iii0~n'iln'i'i:W'J~i'.ifi1nm~oif01b'lUb'i'lU"fllb~m) bb'iilumru~
bliuTm"m"l'Vll-i'l ~~u11111-;1i.ib'1U'11bUUm"lmrieiu'Vl1mliu1m"nT'iL'Vll-1Yi1'lVi'tl-imm-;1:11l'i~mrn
nm~1oii'lum"11'i1n'ilm'ij.J<ii1"'1 1Jim.h"bbuoifoi;o"11l1Ji b~ei"'l1ni'.im1:wLi.ibbU'UeJUb~mnunm1um-;
'11buun'ilm'l:w<ii1"'1 ~"ll-;~mrnnm'llei"bb.i;o~fr•m-;:wd]u 3 fi1ri"d
' '..:J • "" a '1 v a ;>l 61 v.... ..., r:
1. -;~u~nm'Vl1'11n'ilm"i:wbil"i'lb"1b'i1'Vllj"1 (optimistic time) b'llilf!!i0n1'trn a
2. 'i~ti~nm~Yi1n'ilm'i:wbil~'JlJi'oii1~il01 (pessimistic time) 1oiiilrui:i'nm'ii b
' .
3. 'i~ti~nm~Yi1n'i1m-;:w1Ji'bliui:humn (most likely time) 1oiiilrui:i'n1'tru m
b~ei"'il1nm'ibb-;inLb'il"m1:wu1'l~Lliu'll€1"b1m'11buu"1u'llei"Lb<ii;;,~n'ilm-;:w<ii1"'1 blim•uubu<if1
(beta distritution) rl"bbilOI"11'1.umy;~ 5.33 rl"iiunl''l<i1U11:1.bfllb>t~tJ'lleJ"nmlii1bU'.1"1Wllel"bb.i;o~
n'ilm'il.11Ji'illn~m
a+4m+b
=
6

a m
a
a
t
rnn
t
rnn
m,t
rnn
b
m
b
d_ I £1"j .:t 1'
ill'l'l'fl 5.33 l'lli!!llfl!!ll~i'nllJlllll~!uU'lJ!M!1l;ll!l1"Vl!1JlJ!IJAl (beta distribution)

' '
Y111um1l-ll'll'l1ll bl'l~eiu'llei'•1mbl:.11l'J'llei'fi"mw
1'11eir.h~~ 5,8 u'llffi~Ni1"1i'lufi1fj"1ff1'lm1:1-1bbvi,.,,i'.t,li!il1Am:iin'llei,Yl-!,1u'i~l'Jbbi1n~%Ju1
Ni1"1ifrwn"lJi',5ueiLl'l"l,m11um1eiein!lu4il'llul'1l'll-i fiml'l~<Nl!Jl'l~wlfu1>1L~A1t.1 •Au!l1Vf~u'l'111u'llffi
Ni1"1 ~1>1i'luh1um1Ni1"1i'lu#h'Jfu1>1l'll-i!l!ln~"1m1>1lul'1~1 '7!lu L1>1t.1'J~1Ji'ifo'1u!l"l1f.l,1um1~n1'!'1
1'111:1-1dJuh11Ji''ll!l'L'11,m"ldn!lu'J~~1•ilu,1u'J'l' L1>11'Jl111 m1n8l'J01'll!l'Tm,m1~,d
A
B
c
D
E
F
G
H
I
J
., ""' ., If
n1'm%J'Uln1"li:lfilflbb'U'!JNil"111fl'YI
n1"l11"bN'Un1"ll~l'J"1illl'l
m111'"Nu"lll'.lil~•Bl'.ll'll'l1'1Ji'1u'lmrn1i;rmNl1ll
VIV .,,,.,e;
nl"lli"ll,ll'Ubi'U!JNilllllfl'YI
nlTil~biNUibWUW!Jl!l-!1'i'fl1
n1111,ibNUm"l!J1~mrnn11'1uYJU
m"ll'l1>1fl81J1um111,!lul"i1Lullm1>1m"~u~
.. . 'n1"li11"l1'Jll"111'l'l1Ul11Jj"lfl
., . ~ t ,;
m1'l01Y1111m1unmm1ll,"llfllbbi1m11v.mrnrnl'.l!ll'l'llll'J
m~1>1'Yi111m1u~u')1>1Ylll'J'llml'11~m.,,l
A
A
A
c
D
B,E
H
F, C, I

c F
-Routing Cost Estimates
A D G J
~ Ho ~ Ho Finish
Prepare Design Prototype Testing Final Report
/'
"' !'
E H I
Start f------io Ho
Marketing Brochure Market Survey Pricing and Forecast
'·
~B
Plan Market Research
'"'5. f ' ">. ii q I
ill"1'11 5.34 'lllDHll PERT im~!ll'Hl!l!lllt!fll!'ll:IJ
'il1nmw'll1m1u PERT L'ii!l~'il1nL'l'i~m'iddJu1m~m•l'1'1l-J~n"Jn'i'ii.Jt'.i~1l-1L'1tJvi1mn!lu
lii'~,Xu nm'll!l~w1iil::n'ilnni.i~~i'll'i1nm~1i.Jm,(u!luoifilb"JULv:itJ~l'i1L~tJ1 w1i'i1::i'll'i1nm 3 1'11 (~!l
1'11 a, m, b) 4i'~LLftil~1i'lu'111'i1~~ 5.4
4 f I ""
l'll'lH'l'I 5.4 l'll!11ll a, m, b 'llM!!l'lll~!lllllnll
Activity Optimistic Most Probable Pessimistic
(a) (m) (b)
A 4 5 12
B 1 1.5 5
c 2 3 4
D 3 4 11
E 2 3 4
F 1.5 2 2.5
G 1.5 3 4.5
H 2.5 3.5 7.5
I 1.5 2 2.5
J 1 2 3

bbll11lnu'll<N bblli11~n'ilm1:i.i1'1'111n~il1
ifi''l!ltll~b'liU n'illl"lW A
t
e
te(A)
a2
cr2
A
=
=
=
=
=
=
=
=
a+4m+b
6
a + 4(5) + 12
6
36
6
6 '11l'11Vl
(b ~ ar
[ 12; 4r
t:r
1.78
iii~~rr:nmiinmb1<~1'J'll!l~bblli11~n'ilrn1:i.i bb11~'11m1:i.ibb1..l1i.J11uil~uu<!:i.i:i.i~~1u~11 b'lm'll~~
w1i11~n'ilm1:i.il1n1'm'ilrnb'il~bb1Jrnu<ii1 (beta distribution) iii1iltJ1~n'ilm1:i.i A bl1mnfi1nmb1<l'il'J
bb<i~fi1m1:i.ibbll•ll•1ubbii1 m7bb'ilrnb'il~'lJil~n'ilrn1:i.i A ~LDum7bb'ilrnb'il~bburnu<ii1 bb'1'1~11'l>l'lu
'
mv;Y 5.35

Most Probable Expected (t)
Optimistic Pessimistic
...
0 4 5 6 12 rnn (Rll>nVi)
Activity Time (in Weeks)
1u'li1u1D'b~mnu m1fi1u1mm.i1b1mb»~l'Jbbiii~.11m11Jbblh1h1w111D,bb'liiii~n'ilm11Jmm111
fim1m1<il Ji''bb!W1,11'lum11'~ 5.5
. .
m1NTI s.s nnmtl1n111m~!J1111~tlim11JmhtlnwuMtttla~ii~n-;-;IJ
Expected Time
Activity (weeks) Variance
A 6 1.78
B 2 0.44
c 3 0.11
D 5 1.78
E 3 0.11
F 2 O.Q3
G 3 0.25
H 4 0.69
I 2 0.03
J 2 0.11
Illl'J"v1!'(,'illn~1<iii1nl'lL~tJ'U'lill'J~l'U bbli'1i1nl'lfil'U1m1'11.11b1'11b%~[J Lbiii~.11'111IJ lbU'llh'l'U
bbli'1 ~u.ieim'il~'li1m11'11biiwm,"im1<11 1111tJ1-ii-li'ei:i;Jlilnm b>1~1.1'il1nm11,~'li1m11u{f1~1'1u1b-rlei"v11.i1
ES. EF bblil~'li1m1fi1mmifoun!'lrn-rlei"v1111 LS. LF 'll1l,bb"11ii~n'ilm11J Ji'~btl1111,11'lu.n11"l~ 5.36

c 6 9 F 9 11
3 10 13 2 13 15
A 0 6 D 6 11 G 11 14 J 15 17
6 0 6 5 7 12 3
Finish
12 15 2 15 17
/' 7'
Start
E 6 9 H 9 13 I 13 15
3 6 9 4 9 13 2 13 15
B 0 2
2 7 9
,.i "" I
illW'fl 5.36 mn ES, EF, LS, LF 'llll~fl~m1:mn~i
'innmYrYi 5.36 1imllueim'l~ni,;1m1:wd'Ju1u1iff,nnL'il'l,m'lih:;,,i'1,st1'JmtJ1m1m 17
ffifumii bi'iu'Yl1,1nl]<il1'11'tm~m.,d $h:ni'iu'Yl1' A·E·H+J 111~bbil'il~11'lum'l1~~ 5.6
~ I <> ..... ""
~n:n~'fl 5.6 flWll1!11!1lll ES, LS, EF, LF !1llllll1ll~!!ll~fl'ilm1111fll)fl
(ES) (LS) (EF) (LF)
A 0 0 6 6 0 Yes
B 0 7 2 9 7
c 6 10 9 13 4
D 6 7 11 12 1
E 6 6 9 9 0 Yes
F 9 13 11 15 4
G 11 12 14 15 1
H 9 9 13 13 0 Yes
I 13 13 15 15 0 Yes
J 15 15 17 17 0 Yes

"i11nm•1~~ 5.6 bbl1il~nmbb~'lbl11"ii'lJD~1fl•~m.-hii~'l·!hn.1~nm'lum-,iiwilu,1u 17
51Jil1Vi bbiili~D~"il1niim11J1:i.ibbilueiui~mn1.1nmm•oi1iuu~1u'lJ1l,n"ilm•iJiil1~'1 "1~iim'f1m•oi1iuu
'lU'lJD,fi"ilnnlJtilmiWfl~DtJ1,ri~n"ilm•lJlllrj<iibl11"ii-iflfl11rnrn;:i,~tJ (t ) n"ii~''11lVlfil'V1U>1bb~'lbl1~"il
' ' e
'!JD~1fl"m•illi1LJubbUii1~1iJ&'i'1t.1 'V1'iDmh11&'111m1rni:ltJ~b1JU'lJD,n"ilm•iJ'lnrj<il (standard devia-
tion) "il~iiNii1'1Dm•ri1'1'1'Uilbb~'lbl11"il'lJ€1'lfl"m'
m•oi1bU'U~l'U'lill'J,l'Ubb1J1J PERT ~'oilbUU,1'>1'11brit.1"1J'UlJl<il•jl'U'lJD'lm'm' (project
•j 2' .... &:!.
standard deviation) bilil''U
'
=
~ ""- IV "" ,ol! V
blJD~"illilll"illflbl1'UY11'1llrj<ilflD bll'U A-E-H-1-J
m-,,,,1'11britJ'buum<il•jl'UllllJl•llfi1u1ru1&'ii,d
'11fl'lllJbbll•ll•'l'U'lJ€1'dfaYrn'lnrJ<il
if= 2 2 2 2 2
CJA + O"E + O"H + O"I + O"J
= 1.78 + 0.11 + 0.69 + 0.03 + 0.11
:::: 2.72
()" = {c;2
= ;;;;
= 1.65
n1N'1.1'im•'lJt1,1.1'ii:imiv1,d<ift1,m•'lV!m-,fim11m1lJbUU1iJ1&'i''lJmlfl"m''lVibl11wnt.1'lu 20
•
51JmVib~€1'lVi'Yf'Un1.1ri1'1'1Uilm•Ni1<ii1u'11~1<if11'bbNU11 "ii ~iim1:wii1"il~bUU'V1'it1i'.itm,m•mrn'.fou
i'i'lt.1'lil'lum-,'1'11tfl"m'ii'n911'lVibb~'lbl11"il'lunm~ril'V1'Uil
"illllll:J.Jm•llnGimmJ"lU (standard normal equation)
' '
L1fl1fir11'VlUWI - b1~1!.%RtJ
z =
S.D.
20 - 17
=
1.65
= 1.82

'
"J111<ill"l1~m·m•mb"J~btul.Jlln~ (normal distribution) fl1'1'l11JUH~bliu'fl Z = 1.82
b'Yilnl.J 0.5 + 0.465 = 0.9656 #l~bbMl~li'lWl1'W~ 5.37
' .n<i1114111llfomt1 96.56 •ll1l'hlilw1'ifi•~Yi1tm~nwi'.t11'1"lmtJLu 20 ii'lll'l1~ #l~bbt'lil~11'lu
d
il1'W1'1 5.37 - 5.38
cr =1.65 ii'lll'll~
/
17
m~~ 5.37 m~U~lll!MUflfl'lJ!l~fa~~m~
CT=1.65 P(T $ 20)
' '
T = 20
z 20 - 17 = 1.82
1.65
~ ' "' ' ~ 0 • 1 ~-· 'm~TI 5.38 fnl!J'l!l~~!Utl'lJM!fl'HflUTI'lll!ll'~~mD tlt1l'll 20 ll'u"111'1
"Jlni11IJ'"~141•1nm1>iiu1ru6111.l. '
1. L'1"l~m1d•~L-ifnm~1ruu~1u11'1m>i~tJ 17 ii'lll'l1~
2. 1eim'1ffim~m1d•~••ii'1Lt11"lmtJLu 20 ii'tJm~ 811 96.56 %
3. fi•m11J'lnri1118il A-E-H+J ~~n1fi•m11J#l~n<i11•'11•1i1n11ri1111ul'l'il~'i111Vi1'11~m.,
'"~'-if11u411tJ
4. fi'ilm"llJYl"ll.J'lnri118il fi'il11TIIJ B, C, D, F, G
Ll'ltJllrn11~•~<i1-if114i'Lliu 7, 4, 1, 4, 1 ii'lll'l1~11111J~1#1l.J #!~iiu~l.J111111m'ilt111J1"lbl~~
'Vlliv.mm'il1nn•n111J~ll.J'lnri1111tJ'li1tJ1u~1wim~n•n111J~'lnrill

L~€l1~ 5.3.1 LL'U1~'1L~trn'fu/f1LL1.11.1m4'1mN
~il~fi 5.3.2 m•'l.J•ttJn.l'loii'/f1LLu1.1m4mi'V'J
~ei1Yi 5.3.3 lii1L•1.11.1m4'1ei'V'JmtJ1<ilL~eiu1'll"l"l'Jm'W
1. /f1lll.11.Im4'1ei'V'J Ll'ium•'ilm1tVi'Wq~n..,..,l.l'llei1/f1Ll'l.J'i~ftuh firn."11L~€l~<Jtri1mtJ
1111ei'Wmmrn'Wq~nnl-11ueiu1'11w11ei1/f1LL'!J..,iiw1 1'11'.J~m.,ri1u1tJ1111ei'Wmn•rn
ii'~nt111 <Jtm4itJoirei:1;11;1ftmut1ui:l<J'jUU'llei1/f1LL'l.J• LrntLLl.IY1~n.Wm1l.lu1<JtLU'U'lleJ1
m•LU~l'J'UlLU"''"i;i1utLUU-iiei:!;!"1um'i'iLmltM'Wl]~nnl.l'llei1/f1LL'l.J•ii'1nt111
2. /f1mJum4'1ei'Y1 mm•i;iu11'1.J'!JntJm'ilifnui:ItJ!VI1n1';1ii'1fiuh 'llll.IVIU1~m~
11•n<J<1i1''1 1~L•n m•'Wrnmcl1111'11u'1•ei''llm'1 m•'Wrnmclm1l.lu1<J~Ll'i1.1'llrn
L'1Tu,'1imLlltJt1f,1u1i.J1~ m•'Wrnmrnm1l.IU1<J~LU1.1'11ei1m•Ln'llvidjtJ! fll™mmrn
fflll.IU1<J~dlu1unnL'!J~l'JULL'!JM<i!1Llviu,,mYl~'Wrnml.IU1'1il
'
3. /f1L<1.11.1m4'1ei'Y1mtJ1<iideiul'll<1)'1Wll'W Ltlum1l.IUl<JtLU'U'lleJ,ftmut/i1<L'l.J'i~ft'Uh
,,; " ' ~. ,~ d ~ !'1 ~ ·~ ~ '
;1rn:r1Ll.lm1mmu Lu m'i~l'J~Vl'U' 111•muu'i~l'Jtnm l'J11<JtL'lJ1j LUftm1~'11'111 m
m1l.lu1<J~Ltlu'Jlei,ftmu~lii1LL'!J..,,iu'1 <Jtl'l.!1'11~ ~1m1l.lu1<JtLtlu'Jlei,ftm1.1~<1i1''1
6l .... Ji '1 ,x ,.., ,,,:,. 1"
mftm1t'1"'11'U<J~ <l.l'lJ'Uil~nmlmun•l.l'll'U
v -· J1V1Qu'l:::tr'1fl
L~eifin1'11'111l'U~
1. eill1.11mLu1~'1L~tJ1n1.1/f1LL1.11.1m4'1ei'V'J1~
2. 'l.J•~sin'll"li'lii1LLuumi•m'Y1nui:l'l!1111m,11•n<J1"1
3. eillu1tJlii1LL1.1um4'1ei'V'JmtJL<ii'L~eiu1'11'1)l'ltJm'W1"1

m-,~i?llluh'llil'NlJ'lvi1'iiin'l~i'.lm1;;.i1:iJbbuu1Jub1i1mb~m1im~'~1bllum.h'~'~~lJ"iV111'l~lif1l,• •
i1m'ib'll11'J;;.im'i11,mm11<i1,V1u1 b~1l'i1l,{lJfi1V1{um1;;.i1;;.ibuu1Ju~m'bni?l~u111i'lum1'11buu'iu
11;;.i,if"'im111bni?l~m.illiw1 ~iim'ibli~l'JUbbli11'lli<iiilili?lrn11 J(,~um'i11'bbNU11<11,V1u1 1:iJ11'l~bUU
m'i11,bbNUrm1hum'i<ilflli?I m'iNl'il nl'ib'.lu V111l'Yl'i'111mnwtt1'ltJ 'l~'li11'Jri11Vim-,4fi?illuh'll1l'
NlJ"l'111-,biJu1llm.h,iili'i~R'Yllim111mn~'~u 1<ill'J~NlJ"iV11'imm-,i:iu14l1bblJ1Jm{f11JW (Markov Model)
• •
mlii1imf1~1l,i'.J1l'llU<il'll;i,1um'i11,bbNuii1,V1u11unT'.1'11bui."1uJ!,nii11141
4l1bblJlJJ.11{f11JW 14i{um1i'i@m1~u1<iltJUnf1rn'ilf115'i1{•1m{m'i!tJ~1l ii'~bi?l'i m{f11JW (Andrei
Markov) 14i'b5U1lbb'U1~i?lLUm'i1llilJ1tJbbl1~'Yi1ml'J111i:iW!m1J.1'll1l'1J'l}fllf1'll1l'n1'li LM1'liu~lli11bbl1nrif,
'lln~u14innu11111ifmJ1,n-'h,'l111'1um'ibbn't'lluruV11rml'lfin~ 'l51'l'YIm b>'1~bf1,m"15<ii{ 11;;.i~,. . ..
m'ibbnUqJV11m'i~i11lluhm,Vlul~'Yll'1l'in'Jlli1''1 b'liu m1'Vi1u1tJ'W!lWim1;;.i'l11M~1J71nf11u4!1um1;;.i
'l~-l'nfin~lli1l'il'l1llufi1bbl1tnl'lbli~l'Jmbli'1~1li'ildlllu#h5u m1'ibm1~Vlii1ubbD,'ilflli?I m1'Vi1u1l'J11lm5
~ bf1~1l,{n'l"'Vi1,1u1:iJ141 m1'Vi1u1tl'l'<'l Win11;;.im1'll1'i~V1d'll1l'jnV1d m1'Viiull'Jnl'ibli~l'JUbbli~'
'1lbb'llU,'ll1l'l.Jf1flln'l d'Ju41u
m1'hm1~Vim{f11JW (Markov Analysis) bl'Ju'iiim1•:S'li'imrn~b~m1f1l,n1Jm-,'lbf1'i1~Vi
111i:iWlmw1uu'J-oi1iu'll1l'4l"bli'iLi?i'l b~1l~'l~'Vi1mtJ'W!lWim'iJ.1lU1l'Ulf1<il'll1l,411bbll'i~U'l 1<ill'J'liim1
~b~tl11f1J,nlJm1;;.iu1'J~bUU (probabilities) b~1l'li11'J1um14li11lluh
KflMW~1m~tr:1J:1J~~1u'lltM~1111111i11inu'l'l
nn1oii411bblJum1111lwb~ilnl'>'Wmn'intm1;;.iu1'JnUu1um-,d'li?iilmu~'l~4imll1rn1llJ<1i1 tJ
1J'~li'itn1llJbb~~5J.IJ.IW!jl'UJl,r;i1J11id
1. 'il~W'B~mm'ltl!Hlfl!!'il~i:l'orn~ (State) 'll!l'l.J!1!1'1'1111l~'~ri1fl'i'in1'!1141 UqJ'111Lum14li11lluh
1'.)nlJf1!'lll'l~4f1l,mm1m~l.Jilmu~1<11 4i11J81,biu m1i'in1'!1111i:iWim'iJ.1nl'>~1lllufr1 2 wmll,iff1 ?'iil
'il'i1lluf11 A bb~~m1llufi1 B 'l~bbD,biJu 2 5mu~?'i1l~1l'il'i1lluff1 A V111l~1l'il'llllufi1 B
1i?ItJnl'ibb'lrnb"ilmi:~~ilJ.1J.1W!j1uJ1,d ,
1.1 lJ!1'1~Vl11lll,firi1fi,ijjnm'l~4f1l,1lULuilmu~Li?iilmu~'llU' •'liu 'l~4ieNbUU'1nff1' . .'lleM A Vl11l B 1lLl1'Li111lLl1''ll;i,brl1~u'J~biJW"Jnff1'l11l',)('51J,'il'1lluff1Lu'llrn~b~mnu1:iJ1<11
1.2 1iJi'.lm1bli~tJUbhli'1,1fm;Jl1~Ujl'Ub~mnuli1~mmb~~~1u1u5mu~ b'liu t'hi'in1'!1
'll'llRUfll A tllJ<ildlR'Ufll B bb~1n'l~1JJiim'ibli~tJ'UbU'U'ildlllufi1 c Vl11l<ii'ilR'Ufll D Vl11l1l-ii'.lm'i
b~J.ibUU 3 ilmu~ ?'iil m1b1iujnfl1'il'lllluf11 A m'ibUUjnfl1'll1l,'il'i1Rufi1 B bb~tnl'nUujnfl1

2. ;i:::w1HUl'l11lJU1'il::ai'.lu1unl'nfi~l;flll'll::: (State Probability) LUU'il"ltJU (~ n = O) m1:;;'.1
'iJ:::LiJu1um7Lnilamu:::'iJ:::i'Li~tJUbb'Lil1~1t1"11:;;m;;in<n LiltJLVinmLui:l<J<Jtimllu n = o Ji~iXu
'
m:;;Tmfi1u1m1111m1:;;'.1<J:::iliu'llei~amu:::'11~'1 1J1Ji~••'1'li1~n;;i1nil1t1 n = 1, 2, 3, ... n
~10vN~ s.9 1ui:l<J"lliu~nfl1~ei"1'l"lliufl1 A Llimhu1u 60 L'Liei7i~u<i'ibbl1:::~ei"1'l"lliufl1 B
bUWihu1u 40 btlei{i~u<i'i
Ji~iXu amu:::1u7nJ1Jdi'l 2 amu::: m7fi1111uilm1:;;u1<J:::LUu'llei~amu:::LU7tlJJ"Jtb~tJn11
d'Junrn"1ei7m1:;;u1"JtbUU'JJ!l~amu::: (Vector of State Probaility)
~~ < ' !'1 ~ ~
ivi n(i) = nrn"1mm1:;;m<JtbJJWJ1ei~amu::: bU~'lill'I i
= (n1, n2' n3 ,..., nn)
LiltJ~ n = <l1u1wJ1ei~amu:::
It ' ~ 61 .d
(n
1
, n
2
, n
3
,..., n ) = nrn"1!l"'lm1:;;m<J:::biJu'llei~amu:::m~1ill'I o, 1, 2, ..., n
' n,,
<J1n<ii1m.h~'il 5.9 m7:Jleiliufi1'1Jei~~nf11"1"'llliuf11 A bbl1:::imliufl1 B mm"'lm:iitJu
1ul'Linrn"1dm1:;;u1<J:::LUu'JJei~amu:::1u7tlJJ'IJei~lli:l'il"lliu (n = O) 1Jl$iei
n0
= (0.60 0.40)
uein,1ndt'11iifmm"'l .i11:;;u1<J:::Lliu1um7Ln"1amu::: m am1rn~<ii1 (steady state proba-
bility) oiiei:J.Jm~mnum1:;;u1<J:::d'Ju1um"'lLn"1amu::: n = o 1a.J<l1Llluiifei~i.i1m1oif,~eim"'lfi1u1m1u
• <
nnvw1mm
3. 1'111!.lUl'il:::!i'.IU'lJ0~fl17!tl~t:JU!!'lfo~i:mrn::: (Transition Probability) m1:;;'.l1,:::bUU'll!l·Jn17
L'Li~tJmL'Lil1~amu::: $iei m1:;;u1,:::Lliu1um••n"1LVl"Jm'm1ueim.i'1L~mnum•.i~ei~1uamu:::L~:;;
vi~eimdbtl~tJUbb'Lil1~amu:::1via.!
a:;;:;;o/i'<i1u<J1nm1:;;uv•:::bUU'llei~m"'lLti~tJUbb'Li;;i~amu:::$1ei ,:::iifei~1i.J•ti~tJUbb'Lil1~ill1!lilm•
"'
r!mni:l(J!1111L"1i:l(J!1111vid~vi~eim<Jmh11Jl11 <J:::1oirm1:;;u1<Jnllu'llei~m••tl~tJuamu:::'J!lilL@imnu1u
m"'lrni,;1i:l(J!1111L"1i:l(J!V11V1d~ i'l.11.i~~La:;;ei1a.JL'Li~tJULL'Lil1~m:;;rnn
vimmVl<IJLum,rni,;1i:l(J!V11Lili:l(J!vi1111d~ <J:::iifei~i'lm1:;;u1'11nllu'llei~m7Ltl~tJULL'Lil1~amu:::
<ii1ei~1~mwil1'ii:::dJwim~m7Ltl~tJubb'Lil1~amu::: (transition probability) 'll!l~flufi1'171
liufl1 A bbl1::: B ''"'"~11'lu"1171~m1:;;u1'ii:::diu'lleJ~m"tl~t.JUbb'Lil1~amu:::1JIJJ~d
m1:;;u1'11:::•llwirn~m••ti~tJm•1Jl1~flufi1
~!lil71ilufi'1 A
~rnmlluf'1 B
0.80
0.30
0.20
0.70

)(1rn'llL.bm11'Limm.b!l.b~ 1 <Jn#i'1~Lfl1l~!liil.bfll'il':lliil.bfll A 'dlfom<i 80% ~-~~!l
iil.bfll'il':ll A 1inL.bfl~'li!l11i LLl'l~l'.iT!lmft 20% ~-~•1i~im11i~!liil.bfll'il':ll B
'i!1.b)(1Ll'l'llLm•m.b!l.b~ 2 <Jnf11~Lfl1l~!l B •~l'.il!lmft 30% ~-~•1i~1l.b11i~!lfil.bfll'il':ll A
Lbl'lrnnf11~~!l!l.bfll'il':ll B l'.iT!lmef 70% ~-~~!l!l.b#i'1m1 B 1inli!lL1i
•
!l;:(, m1:wu1<J~LU.b'llil'f11':lL1i~8.bLblil'l,e!11.b~ LbYl.b~"J~LLC!fl'lwili'll!l,<ill':ll' ill'OJ'OJ~LbMn
11l.b11iLL:wY11nin1Ji ~"l!l~l"l1ld1Lb:wY11nim1:wu1<J~LlJ.b'll!l,m':ll1i~tJ.bLbDl'I' (Matrix of Transi-
•
tion Probabilities) )(1Li'l'll~€ltjL.bLL:W1'11nim1:wu1•~•U.b'll!l,fll':lbD~1lmL1il'l"'fon11 '11m1:wu1<J~
LU.b'll1l'm':lLli~1l.bLbD1M)(,Lbe!fl'11ii',d
P11 P12 P13 pln
P21 P22 P23 P2n
p =
p
mn
P = ••:wY11ni'll1l,fl11:wu1<J~•lJ.b'll!l,m1•1i~tJ.bbblil'I'
P = m1i;u1<J~LlJ.bL.bm1•1i~tJ.b'OJ1nl1m.b~~ i L.btl<J~1f.b11l•lJ.bftm.b~~ i L.bilmfl<il!]
= 1, 2, 3, ..., m
= 1, 2, 3, ..., n
<J1n<if1!ltJ1,•~:w m1:wu1<J~bU.b'llmm1•tl~tJ.bLbUl'l,ftm.b~'~'~tJ.bt.b':llibluuLb:w1'11nilJJii',d•
<ii':ll!l.bfll A 0.80 0.20
m1!l.bfll B 0.30 0.70

,, v
'llfltl'.:JllWI
1. m1:1.1ii1"il~blluhmumwub~11'l1:1.1nu"il~<ii11~1if',vi1nlJ 1 b!'f:J-111 (bimbmu11u~ 1 "il~1if1
b'Yi1nlJ 0.80 + 0.20 = 1.oo m11u11u~ 2 ,~1if1b'Yi1nlJ o.3o + 70 = 1.oo)
2. m1:1.1ii1"il~•llu1u,bmMmilb~11'l1:1.1nm•ii'11aJ-ii1•llu<ii11~1if1bvi1n11 1 (•'liu bbml1111:1.1i~ 1
~v·~ A .,,: ~~·~ ,j
'ilnl1lb'Yl1n11 0.80 + 0.30 = uo 'll~ ;<1 bbl'l~bbtl11'f111:Wil'YI 2 "il~bl1lb'Ylln11 0.20 + 0.70 = 0.90 'lf~ ;r1)
. ' '<V d: <:!:!. <:!:I. <::!. ii 1-' _j">."-"=>."'>
11a~;i1nt1fl'!ll!'llf>'llll:n'l~!w~n 5.3.1 mn u'll'lufllJl'lfl'ilm'l:1.1 5.3.1
' . . '
1 d: I "" ~ <:!:!. .c:S.
'J.ll!'U1fll'lftfl'!l111'1fJ!Jfl 5 ~Hl'Ufl 5.3 !'l!Mfl 5.3.1
vd !U ""' ""' ..., t:
Vl'Ul'Yl'Yl1~1111'.lnl'l!1l'l1111 nl'lNl'l11 nl'lb~'.I bbl'l~'Yl'l'V'it.lln'l:l.l'ij'tltJ
m'lbbnUf.1.!11111111tJ1-if<ii1bblllJm~flE1'W i'iv m'lfi1u111wnm1:J-1ii1"il~bllu~'l~111J"il~11~1umnut
>11~'1 1u111-11fl!1 1111tJ"il~fi11-11m1if1"il1nm:nu~1uu'il~iiubbfitbb:W'Yl'in4m1:1.1ii1"il~bllu'Yl1~m'lb1.J~tJubbl.Jfl~
mnu~
ri1111u1111VI P = bb:W'Yl'in-Bm1:1.1ii1'ilnUW111l~m'lb1.J~tJ'.lbb1.Jl'l~'J111ff1tJ
n(n) = m1:J-1U1'il~bU'.l'll<i~m'lbn1111'fmu~>11~'1 1u~1111~ n, n=O, 1, 2,...
n(n+l) = m1:1.1ii1'il~bU'.l'Yl1~m'lbn1111'fmu~>11~'1 1u~1111~ n+1
1111 tJ n(O) = m1:1.1iHnllu'Yl1~m'lbn111!'fmu~LU!11lUb~:J-J<iiu
o/1111th~m'l1111m1:1.1ii1'il~bllu1ul'fmu~>11~'1
n(l) = n(O) . P
n(2) = n(l) . P
n(n+l) = n(n) • P
v I ~ o !U ' J ""-V 1 V d:I V
ll1fl!JHfl 5.10 111~1J1'.lbbVl~Vl'.l~:l.l'l1'.I Minimart <l'.J 2 'llU f!El11'.I A-mart bbl'l~ B·mart '1'11~
m'lbnlJ'l111'l1:W-if<i:wl'll'lm'h'V'11Jl1L'.IU'il"ilU'.ll'lm'f1 A-mart i'lm.il.J'l~mm 60% l'i1ul'ln'11 B-mart SJ'ii "' q 'lJ 'lJ 'IJ
mJl.J'l~mm 40% '.IE1n'il1ndti,'V'l1Jll'V'l"~n'l'l:W'll1l'l'ln'11 70% ~il'illn~1u A-mart [i,.,~~<i"il1n1i'1u. ., .' ~ ' v.,::i "!""- "lo l.<!I JV Ii) <=I V "-' A JV
A-mart bbfl~iln 30% bufltJ'.lbu'll1l'il1n'l1'.I B-mart bU'llt1l~'YllJnfl1 B-mart tJ'fl~'ll1l"i11n'l1'.I B-mart
~bfltJ~il'illn'i1u 85% b1.J~tJu11.J~11"il1n'i1u A-mart

1ltnn'Yl"lllJ111u5n 2 lloih.JVIUlR1UIVU.Jnl"lil"1il'lJ1l.Jiii.J 2 -l'1wi1~1llm'Yi1h
1!lfii 'illnh'YI t<ir11l t.i1.Jmm"lt:i"l~1j1l.J.i1h~n1llJ'lJ1l.J<if1tblJlJm~'11l1'11"1r,,d
1. flmu~'ll1l.JU!j!Vl1l1 2 '1mu~fi1l
1.1 -l'1u A-mart
1.2 -l'1u B-mart
2. m1l1Ul'il~1uu1um"l1n?lflmu~ rn n;o1il'il<jiiu (n = o) fi1l
P (A-mart) = 0.60
P (B-mart) = 0.40
m1i1'1-il'il~Lllu1um"lhn?lirnrn~'ll1l.J-l'1uiif, 2 m m"ln1~tm1uiu'll1l.Jnn1011l~m1i1'1-il'il~1liu
'llil.Jiifm'U~ (Vector of State Probabilities) l'il.J'l 1ili'Jr.Jd
n(O) = [O 60 0.40]
3. 11l1'Yl~n.j'.m1~u1'i1~1llu'll1l.Jm"l1u~uuL1u;o.Jf!mu~
A-mart B-mart
A-mart 0.70 0.30
p =
B-mart 0.15 0.85
= 7!(0). p
=
' 'A <~
11mu1l'U'YI 1 ;;irn '1il~ll'YI 1
(60 .40) ( 701 =
l15
[0.60 0.40] (0.70 0.30J
0.15 0.85
(60) (70) + (.40) (15)
= 0.42 + 0.06
= 0.48
R1'UbbU.Jilfl?l'll1l.J-l'lu A-mart ~'Ull 1 (n=1) = .48 (n = 1)
(.60 .40) (301 =
l.85 ~
(.60) (.30) + (40) (85)
0.18 + 0.34
~ 0.52

= nil) • P
=
[0.48 0 52] (.70) =
.15 =
[0.48 0.52] (.70
.15
.30)
.85
(0.48) (0.70) + (0.52) (0.15)
0.336 + 0.078
= 0.414
. ',.J ' • "' ,.,~
bbYl~1Ubbrn<i'l!11il'llil''i1U A-mart ~Uu'I 2 (n = 2) = 0.414
[O 48 0 52] (30) =
l85 =
(0.48) (0.30) + (0 52) (0 85)
0.144 + 0.442
= 0.586
fl1UbbU~<i'l!11il'lJil' B-mart ~utl~ 2 (n = 2) = 0.586
<11,i'.iuri1UbbU,<i'l!11il'lJil~ A-mart mi~ B-mart briil~Utl~ 2 mm'lm:iitJu1uiu'lm,nmiiid
1ii'i'iei [0.414 0 586]
A"JtiUH~ 5.11 ~ilm111nn:J.1~~1111u1tJ1l'l'i0"v>1Yil'.leiriei'l1tJLV1!) 3 'lltJ i'iil A, B lbi1~ c l'.lm1il
'lJltJ1u1li1<J~uui'iei 150 ff1urn'I 200 ffrnrnl'I Lli1~ 150 ff1mJ1'I mmh<ifu LiltJ~']~n'l'i:J.J'lJiN
N'u11nf1 ilm'lbu~tJUbbui1~luLiltJi'.im1:J.Jul"J~Lflu'lJ ei~m'lbu~tJUbbu i1"'Hnu~m'l~il1l'l'lff~Yil'.leifiili'f,
•
3 'll!.J
•"" # ..... ,,.
:J.l'l1!.Ji1~b!l!.Jilil'U
'111:!.IUl"J ~lflU'lJ !)~nl'ibU ~tJUbbUi1~!1mu~
,~u A
A 0.80
B 0.50
c 0.10
B c
0.10 0.10
0.90 0.05
0.05 0.85

11hh 'il1n1ihmh~~ 5.11 ffll.ll'lT11::'41l~l"ith:;nE1u'll1l~;'i1bb1Jum1'11l'VJ1~;1~d
1. '1tmi:;'llE1~i:l'!Jvnd!l 3 <!mu:: ~8
1.1 ~-.j1wt.i1dY11;''vrYii'.leinei A
1.2 ~-.j1111U11'JlY11;''vfYii'.leinei B
t.3 ~-.j1111't.i11'JLY11;'''WYii'.lE1n1l c
2. m1l.IU1'il::b1Ju'him1bnlil'1mu:: ru, rnni:l'ii~iiu (n = 0)
d
flEl
w-.j1111u11'J1Y11#'WYii'.lE1n1l A
150
0.30= =
• 500
w-.j1111u1l'J1Y11;'''WYii'.lE1n1l B
200
0.40= =
• 500
~-.i1111u1dY11;'''Wrii'.leili1l c
150
0.30= - =
500
1.00
mll-IUl'ill:bUu'llEl~ w-.l1111u1l'J1Y11;'''WYii'.leinei;'i~mh1'11m1i;ib~ l'Julu1l.i'IJE1~b1m9l84'i11nm1l.I
• •
't.i1'J::blJu'lJE1~'1mu::o/11~1 1~;1~d
7t(O) = [O 30 0.40 0 30]
3. bbl.IY1~nim1:l-IU1'J::blJu'lJei~m1bl.i~l'Jmbl.i'1~'1mu::
A B c
A
[ oeo
0.10
'"]p = B 0.50 0.90 0.05
c 0.10 0.05 0.85
m1fi1u1ru,1111m1l.IU1'J:;blJu1um1bfiil '1mu::ll~ 1 (n =
n(1) = n(O) • P
['"
0.10
orn]= I0.30 0.40 o 30] 0.50 0.90 0.05
0.10 0.05 0.85
1)

nl1jb1Xl~b111l1 [O 30 0;40 0 30] bb<l~bbl-IVl'in.Ji P 1-i1L~.1,d
~ ·~
"1:11'1-l!J'UVI 1 jllil'l<ill-li1VI 1
[0.30 0.40 0 30]
r0.80
l
0.50
0.10
= (0 30)(0 80) + (0 40)(0 05) + (0 30)(0 10)
= 0.24 + 0.02 + 0.03
= 0.29
:• cbumJ,'ilm<il'll!l,~'11VIUltJlV111>'f'Y'l~l1!Jti!l A ~uu~ 1 (n = 1) = 0.29
,J ~ &J.
bb'11'U!J'UVI 1 jllil'lill-111VI 2
0.101
[0.30 0.40 0.30] 0.90 = (0.30)(0.10) + (0.40)(0 90) + (0.30)(0 05)
0.05
= 0.030 + 0.360 + 0.015
= 0.405
•• ri1ubbli,<ilm<il'll!l,~'11V1UltJlV111>'f'Y'l~l1!lti!l B ~uu~ 1 {n = 1) = 0.405
'~ ·~ml1'U!J'UVI 1 jlliMll-11lVI 3
I
0.101
[0.30 0.40 0.30] 0.05
0.85
= (030)(0.10) + (0.40)(0.05) + (0.30)(0.85)
= 0.030 + 0.020 + 0.255
= 0.305
ri1mbli,<ilm<il'll!l,~'11V1u1ulV11l>'f'Y'l~l1!lti!l c ~uu~ 1 (n = 1) = 0.305

rn-:;.-i1u1ruvi1m1;;iu1'iinlJu1urn-:;Ln"1•m1u~t1~ 2 (n = 2)
n(2) = n(l) . P
[
0.80 0.10 0.10]
= [0.29 0.405 0305] 0.50 0.90 0.05
0.10 0.05 0.85
fll':l'JlliL'"lrnlllEJ7 [029 0.405 0.305] LLS~LL;;J'Yl'ini p 'Yi1t~(i(~d
' 'a <a
LLll1UelU'YI 1 'Jlli<'f91;;Jfl'YI 1
0.80
[029 0.405 0.305] 0.50 = (0 29)(0.80) + (0.405)(0 05) + (0 305)(0.10)
0.10
= 0.2320 + 0.0202 + 0.0305
= 0.2827
li1ULLU~91'1191'1Jel~N'il1VIU1tJL'Yl'ifl1'fYlrleinei A ~ull~ 2 (n = 2) = 0.2827
•' '
d ·~mnue:iu'YI 1 fJlli'191;;Jfl'YI 2
0.10
[O 29 0.405 0.305] 0.90 = (029)(0 10) + (0.405)(0.90) + (0.305)(0 05)
0.05
= 0.0290 + 0.3645 + 0.0153
= 0.4088
li1ULLU~91!1191'llel~N'il1VIU11Jl'Yl•f'i~Yrleinei B ~uli~ 2 (n = 2) = 0.4088
•' '~ <a
LL11U'1U'YI 3 'Jfil<'f91;;Jfl'YI 3
0.10
[O 29 0.405 0.305] 0.05 = (029)(0.10) + (0.405)(0.05) + (0.305)(0.85)
0.85
= 0.0290 + 0.0203 + 0.2593
= 0.3085
'11ULLU~lil119l'llfl~~'illVIUltJL'Yl'ifl~Yrleinei c ~uli~ 2 (n = 2) = 0.3085
(i(~~u'11uLLU~lll"W1'llei~N''il1viu1tJL'Yl•ff~Yrleinmii~ 3 -:;1tJ L~ei~uli~ 2 '11m-:;rn~tJulu•ll• •
nrnlllei71~~ei [0.2827 0.4088 0.3085]

<u1n<i11eJ!i11Yi 5.1o <u~~u11 m1J-1u1<u~d'iu'lleJ11jnfll1um1~eJfiu.'11<u1n'l'ru A-mart <urnl'1<i~
11lb~eJl'J'l briei111'1nmb~l-l~U LWllrn~~m1J-1Ul'J~bfJU'llfl11Jnff11um1~eifiu.'11<u1n-i'1u B-mart 'J~
' d.. Ji .,J ,,.'. Ji' ,,g .., '4 2.1 l ' ~ ' .& .., .::
fleJ!J'l b~l-l'llUbl-leJ111'1b1mml-l'llU "UUl11l'flll1tf11il1'Jnl'1'1lmll-1Ul"U~bUU'11VIU1 bb<lt111'1b11'1"UlflU
1~1' • f1 11.-GJ x=!V ~ ,,.,,,,. &
ufll'11ll-IUl"U~blJU'lleJ1<lflfll mm':i'lfeJl'fUfll'Jlfl'JlU A-mart bb<l~ B-mart 'Jtl-lfllb1'1l-IYl!l111'1 'lf·l'lt
. '
•711mi1d11•fJum1J-1u1<u~•fJu ill l'fm1t'11<il1111"leim1J-1u1<ut•fJt1l'fmut1rn~l'Jl1 vi"leim1J-1Ul'il~bfJu
ill l'fmUti1)<ll'Jlll~
. . .<I.I 4. .d, <!:!: ""' v ~-1 -'""'""""'""'
'11!l~'illllftll'.lll!U61111l'l'J~l'J6~'11 5.3.2 11!11 IU'!OlU{)lJ!lfHlll'!'!U 5.3.2
1 4. ' "" ,,; J; ,,,;,
Ull'lfllll'>ffll'.111'111-1111'11 5 !161-1'11 5.3 l'!ll~YI 5.3.2
' ',,. "" "" ,,!j "" ' .fl 61 ""'
l'fmU~'11YIVl':ieJl'fmU~i1);;; l'JJll~Vl'JeJ l'fmU~'Jrn~ 1Jl1 fl fl l'fmU~YI mll-IUl'iltbut! bUfll'Jbfl l'1
l'fmut (state probability) hJi1ni-ni.J~1JUbbUMV1'i11'illfl~N1U1t1111mtJ'l 111'1bl<it'il~b"ltJnm1J-1
Ul'iltlfJuLum1bnl'1l'fm~d11d'lt1m1J-1u1'il~bfJULUl'fm~f11~ (steady state probability) vi"leil'fmut
'11<if1111"lm7vn11d'lm~eiu1'll rn "j11i1)!11Jlll~ (equilibrium condition)
'i11n<i11ei1J11~ 5.10 <ut~u11m1J-1u1'iintlu'l1ei1i:Jn.'111um1~eiifu.'i'1'i11n-:f1u A-mart 'il~i111
<i1Lllb~1l1J'l briei1111nmb~l-l~t1 LWllillt~m1J-1Ul'iltbfJWnei1i:Jn.'111um1~eiftt1fll'il1n-:f1u B-mart 'ilt
fl eJ l'J'l l~l-l~Ubriei1111b1!11b~l-l~U 'ilU~1l'fJll1tf11<il1'il~1iflfll m ll-IUl'iltbUUfllV1ii1 bb<lt11 11il11'illnd
1ufi1m1J-1u1'iltbfJU'lleJ1<infr11um1~eifiu.'i'1'i11n-:f1u A-mart bbi1~ B-mart <u~i1'11b~l-1Ylfl111'1 ~1'ilt. '
b711n<i1d11d'Jum1J-1u1<utbfJu rn l'fm1~.i1<il1111"lmn1J-1u1•tbfJt1l'fmut1ttJ~tn1 111"leim1J-1Ul'il~bfJU
ill '1b11Uti1)i.11Jl11~ Ji'1Lb'111111lu1111'Jl1~ 5.7

U'1 l.l'lll'U~1'm A-mart •!Hll'U~~m B-mart
0 0.6 0.4
1 0.48 0.52
2 0.414 0.586
3 0.3777 0.6223
4 0.357735 0.642265
5 0.3467542 0.6532457
6 0.3407147 0.659285
7 0.337392 0.6626066
8 0.3350671 0.6644332
9 0.3342097 0.6652883
10 0.3337397 0.6657579
11 0.3334811 0.6665189
12 0.3334145 0.6665855
13 0.3333779 0.6666221
14 0.3333578 0.6666422
15 0.3333467 0.6666533
16 0.3333405 0.6666595
17 0.3333354 0.6666646
18 0.3333343 0.66666557
19 0.3333338 0.6666662
20 0.3333335 0.6666665
21 0.3333333 0.6666667
22 0.3333333 0.6666667
' '
<nnmTJ-ifi 5.7 m1i.;ii1'il~LliwmN-i'1u A-mart LL'1t-i'1u B-mart Liim-ii'1~-i1<;1Yi 21 Lliwiiu
. ' 'l,.1 JU 1 9'1 JU I>) ,._ ""' JU V JU ,,i 1 .c1 A
bubb1"1'"11i.JU1'il~b1JWJJ1l-1l;jnm mnl''l'lf1li1U¥11'illn'l1U A-mart Lb'1~"l1U B-mart "Jrn¥i1¥1-11'1 'lf-1
,,,. , % , a 1 a ..., ""' , a; ..oi , .;;;
L"Hmmu11L1ium1i.;m'il~L1iu rn ilm1~f!-J<;11·wrnm1:wu1"J~L1iuilmu~"l~CJ~m1 wrnm1i.Jm"J~L1iu
rn mnu~~1"rn11"W

•
n(n) = n(n-1) • P
n = n. P
't"I "" 1.!Urn 'ililil'1tJmv;vi.,-1l11murn~'YI 'ii~ il
' ' •
n (~1ilVIUl) = n (~1ilu) P
n(n) = n(n-1) P
~
n.PVl71l n =
• d
ii'~ifa 'illn'IJ1l:i;j'1<il11lEll~'YI 5.10
lilv~
'illn (1) 030n1
'illn (2) -0.30n
1
'illn (4)
[n1 n2l
n1
n2
n1
n2
n1 + n2
- 0.15n
2
+ 0.15n
2
0.30n
1
= [n1 n2] [0.70 0.30 J
0.15 0.85
= m1;;;u1'il~LD'U'll1l~'1n~1~1l'illn-i'1u A-mart
•
= m1;;;u1'il~LD'U'll1l~'1n~1~fl'illn-i'1u B-mart
= 0.70n
1
+ 0.15n
2
= 0.30n
1
+ 0.85n
2
= 1
= 0
0
= 0.15 n2
0.15
•
(1)
(2)
..... (3)
. (4)
.. (5)
n1 = 0.30 n2
1
=
2
n2

u1.-i1
1
LLYl'U.-ill'U (3)
' = - 11
2 2
1
2112 + 112 = 1
3112 = 2
2
112 =
3
2
l'U (3)112 =
3
2
1111 + =
3
111 1 . 2
=
3
1
=
3
' .l'U"l~tJ~tn1 fl11:1-1U1"J~LlJufi11n#i'l'J~'ilil'llil~"J1n~1u A-mart lLl1~ B-mart flil 0.33 lbl1~
0.67 m:1-1li14i'u'Vl'lm~1.1ueJ~1u~tl'lm~nm11eJ4'llil~m1:1-1u1"J~Lilu1.i'fieJ [O 33 o 67]
oifoifom11 t'f1u1"11m1:1-1u1"J~LlJu'llei~4mu~ ill 'liiw11'1'1 Ltlf,Jillnrn1:1-1Y1'ln-ifm1:1-1u1"J~d'iu
'lleJ~m'>Lti~tJmLtlii~ Nl1ii'l'11l~1#1,~ll.-i1L'Yi1num1:1-1u1"J~LlJu'JJeJ~l4mu~ ill "l111il'1tllll'l'1
lii1eJtJ1~
[0.33 0 67]
[0.33 0.67]
[
0.70
0.15
[
0.70 l0.15
d v ird
Llt11Uil'UYI 1 ~illnrn1111:1-111Y1 2
[0.33 0 67]
[
0.30 l0.85
0.30]
0.85
= (0.33)(0 70) + (0.67)(0.15)
= 0.231 + 0.1005
= 0.3315
= 0.33
= (0 33)(0 30) + (0.67)(0 85)
= 0.099 + 0.5695
0.6685
= 0.67

f11ubbli~'ll'ill'll'!Jm'.i'1u~~ 2 .~1.1u1uiunrn'llei1 [O 33 0.67]
zj~•Dum'l'll11'ilfleJU11 N'1~'Wtl rn "lliHiJ'1W11l"HJrnlfei~'Y1'!eJta.J
-;nn4hei~1~~ 5.11 mo.J1u1rnmm1:1-1u1'il~•Du1u~n~rn~'1~~ rn "l'llli)'1Wl1'WL1iii~d
•
n(~1<i1'Y1u1J = n(~1<i1u)P
n(n) = n(n-1)P
n = nP
(n1 n2 n3) = (n1 n2 n3) [ o.80
0.05
0.10
0.10
0.90
0.05
0.10]
0.05
0.85
1~ n1
= m1:1-1u1'il~bDu'llei~~<i1'Y1u11.il'!'l"lfl'WVil'.iei!iei A
n2
= m1:1-1u1'il~•Dwnei~~<i1'Y1U11.IL'Yl'lfl'WViiiei!iei B
n3
= m1:1-1u1'il~•Du'llei~~<i1'Y1U1tJL'Yl'lfl'WViiiei!iei C
n
1
= 0.80n
1
+ 0.05n
2
+ 0.10n
3
n
2
= 0.10n1
+ 0.90n2 + 0.05n3
n
3
= 0.10n
1
+ 0.05n2
+ 0.85n
3
n1 + n2
+ n3
= 1
'il1n (1) 0.20n
1
- 0.05n
2
- 0.10n
3
= o
'il1n (2) 0.10n
1
+ 0.10n
2
- 0.05n
3
= O
'il1n (3) 0.10n
1
- 0.05n
2
+ 0.15n
3
= O
(6) - (7) o15n
2
- 0.20n
3
= 0
= 0.20n
3
= 0.20 n
0.15 3
4
=-n3 3
(5) x 2, 0.40n
1
- 0.10n
2
- 0.20n
3
= o
(6) + (8) 0.30n
1
- 025n
3
= O
= 0.25n
3
= 0.25 n
0.30 3
5
= -n
6 3
(1)
. (2)
(3)
.. (4)
(5)
(6)
.. (7)
(9)

bb'YIUfll (8), (9) lU (4)
N'~<ii>i1viu1tJ1m·#V<Yil1eiiir.i A
5
=• 19
N~<ii>llVIU1tJ1'Yl7'1Y<Yil1eirieJ B
8
=• 19
N~<il>ilVIUltJl'Yl7#t'V<Yil1eiriei c 6
=• 19
19
-n = 1
6 3
n3
6
= 19
5 6
n1 = -x
6 19
5
= 19
n2
4 x __()_= -
3 19
8
= 19
= 0.265
= 0.420
= 0.315

Anderson, David R. Sweeney, Dennis J. and Williams, Thomas A. Quantitative Methods for Business.
gth ed. Ohio: South - Western College Publishing. 2001.
Levin, Richard I and Rubin, David S. Statistics for Management. 7th ed. New Jersey: Prentice Hall
Render, Barry and Heizer, Jay. Principles of Operation Management with Tutorials. 2nd ed. New
Jersey: Prentice Hall International, Inc., 1997.
Render, Barry and Stair, Ralph M. Jr. Quantitative Analysis for Management. 5th ed. New Jersey:
Prentice Hall International. Inc., 1997.
Stevenson, William J. Production Operation Management. 5th ed. Boston: Irwin McGraw-Hill, 1997.
Taylor lll, Barnard W. Introduction To Management Science. gth ed. New Jersey: Prentice Hall

illfll'l'W1fl
0 z
Area or
Probability
Entries in the following table give tile area under tile curve between the mean and z standard de-
viations nbove the mean. FQrexarnplet for z = t.25 the area under the curve between the mean
and zis 0.3944.
' 0.00 0.01 0.02 0.03 0.04 o.os 0.06 0.67 0.08 0.09
o.o 0.0000 0.0040 0.0080 0~0120 0.0160 0.0199 0.0239 0.0279 0.0319 0.0359
o.t 0.0398 0.0438 0.0478 0.0517 0.0557 0.0596 0.0636 0.0675 0.0714 0.0753
0.2 0.0793 0.0832 0.0871 0.0910 0.0948 0.0987 0.(026 0.1064 0.1103 O.l 141
0.3 0.!179 0.1217 0.1255 0.1293 0.1331 0.1368 0.1406 0.1443 0.1480 0.1517
0.4 0.1554 0.1591 0.1628 0.1664 O.J700 0.1736 0.1772 0.1808 0.1844 0.1879
0.5 0.1915 0.1950 0.1985 0.2019 0,2054 0.2088 0.2123 0.2157 0.2190 0.2224
0.6 0.2257 0.2291 0.2324 0.2357 0.2389 0.2422 0.2454 0.2486 0.2518 0.2549
0.7 0.2580 0.2612 0.2642 0.2673 0.2704 0.2734 0.2764 0.2794 0.2823 0.2852
11.8 0.2881 0.2910 0.2939 0.2967 0.2995 0.3023 0.3051 0.3078 0.3106 0.3133
0.9 0.3159 0.3186 0.3212 0.3238 0.3264 0.3289 0.3315 0.3340 0.3365 0.3389
1.0 0.34)3 0.343& 0.3461 0.3485 0.3508 0.3531 0.3554 0.3577 0.3599 0.3621
1.1 0.3643 0.3665 0.3686 0.3708 0.3729 0.3749 0.3770 0.3790 0.3810 0.3830
L2 0.3849 0.3869 0.3888 0.3907 03925 0.3944 0.3962 0.3980 0.3997 0.4015
1.3 0.4032 0.4049 0.4066 0.4082 0.4099 0.4ll5 0.4131 0.4147 0.4162 0.4177
l.4 0.4192 0.4207 0.4222 0.4236 0.4251 0.4265 0.4279 0.4292 0.4306 0.4319
1.5 0.4332 0.4345 0.4357 0.4370 0.4382 0.4394 0.4406 0.4418 0.4429 0.4441
1.6 0.4452 0.4463 0.4474 0.4484 0.4495 0.4505 0.4515 0.4525 0.4535 0.4545
L7 0.4554 0.4564 0.4573 0.4582 0.4591 0.4599 0.4608 0.4616 0.4625 0.<Ui33
1.8 0.4641 0.4649 0.4656 -0.4664 0.467! 0.4678 0.4686 0.4693 0.4ti99 0.4706
1.9 0.4713 0.47!9 0.4726 0.4732 0.4738 0.4744 0.4750 0.4756 0.4761 0.4767
2.0 0.4772 0.4778 0.4783 0.4788 0.4793 0.4798 0.4803 0.4808 0.4812 0.4817
2.1 0.4821 0.4826 0.4830 0.4834 0.4838 0.4842 0.4846 0.4850 0.4854 0.4857
.u 0.4861 0.4864 0.4868 0.4871 0.4875 0.4878 0.4881 0.4884 0.4887 0.4890
2.3 0.4893 0.4896 0.4898 0.4901 0.4904 0.4906 0.4909 0.49ll 0.4913 0.4916
2.4 0.4918 0.4920 0.4922 0.4925 0.4927 0.4929 0.493 l 0.4932 0.4934 0.4936
2.5 0.4938 0.4940 0.4941 0.4943 0.4945 0.4946 0.4948 0.4949 0.4951 0.4952
2.6 0.4953 0.4955 0.4956 0.4957 0.4959 0.4960 0.4961 0.4962 0.4963 0.4964
2.7 0.496S 0.4966 0.4967 0.4968 0.4969 0.4970 0.497! 0.4972 0.4973 0.4974
2.8 0.4974 0.4975 0.4976 0.4977 0.4977 0.4978 0.4979 0.4979 0.4980 0.4931
" 0.4981 0.4982 0.4982 0.4983 0.4984 0.4,984 -0.4985 0.4985 0.4986 0.4986
JJI 0.4986 0.4987 0.4987 0.4988 0.4988 0.4989 0.4989 0.4989 0.4990 0.4990

·6. "''Yi'l-.!'JU'l'l
4a
1'"
fl1u11ii-3
il1u11il~
Yiii&£i~tl~utl1'3
11".'lJ., 'ji1.~J. u·,.::;~.r;<
"iD ;fi~"fGl"i~:'ll".i~.~-~1:::"~-,~·f :'rn'->r:·:''Jri)' ·1:11·~·ti>cr-: 1'1
~<11':~'Yl~J-,~tl~~1·31~tJ::;··;,.J';1i71't
C:.1.l
;ffl.lJ., r-,1.B.A. Ph.D. (Hu.sines::; Adn1inktra!ion)
Pt.D. (E:-.:.,,<nsrx :.g i'<1an'lver:1ent)
o/1J:j;-,:Y1;1f' 1~::;-s~~, eo:,•i•li '1.1n .••··..ll lli11'""'"'"· ··•··• ,,111lfln
0.1

'I "'
'11'l-l1fffl 6
,, Jl
!flllfl'J,:Jl'lHl'l'il
d
lileJU'YI 6.1 Gl1bb1J1Jbbm<ileJtJ
6.1.1 '1mtru:::rf1Ltl'llei,1:::urnbm<ileitJ
= ..., .., b .... ~ .... ....
6.1.2 '1:1-1:1-ilil:JlU '1llJ'1ntlru bb'1:::vi;;inm1mi;iqJ'lleJ'lil1bb1Jrncm<ileitJ
••.... ' .... ..., .=I
6.1.3 meirn,mbblJrnbm<ileJtJ'llUY<U:JlU
6.2.1 m1'loii'm1~1'1ei,'1mum1cl
6.2.2 ifi1b'1'llzj:1-1ilum1~1'1ei''1mum1cl
6.2.3 4fu<11eium1'loii'm1~1'1ei,'1mum1cl
6.2.4 ifi1mh,m1tl1i:!Jn#im1~1'1ei,'1mum1cl
1. Gl1bb1J1Jbbm<ileitJ 1h:::neiuJi'1tJei~.1t11:::neiu~u31u $iei m1mfi, l'i1u'lVfu1m1 bb'11:
cb!l1<ileJtJ 1:::urnbm<ileitJmm1;;rv1rnViu1Jiri'11t1'lu'il11iitl1~~11u ~''~bnl'l~Ubi'l:J-Jei
'~eim1!Jifiei,m1u1m1l'.Imnn11~'1 m1:1-1mm1D'llei'511.1'lVfu1m1 bl'!tJrf11t11~uu
bbm<ilfltJ1J'lmfl1JJi'1tJb<il'l,'1-'l,~U:JlU 4 bb1J1J bb'11:,~'1l-J:l-l~:JlUluG1'ntlru~ii1~'l flu
bl'!tJ~'fi1Iii eiu~1ifi,1nm11b<il'll~~'l~1J1J bbm<ilfltJ'~1i.J'l'l1'11lilfllJbVlm~~'ll'I~' b'liu
Gl1bb1J1J1Jl,1J'll:ITT'YI
2. nn~l'1eN '1mum1cld'Jut11:::1miiiil1V1'lu m11 b<il'll~~U(]JVll~'1'1uoiiuoifou'u
1i.Jm'1b<i111:::~Ji'1tJGf1 bb1J1J<ilrnlilm'11iiflJl nT'i'loii'm1~1"fl' '1mum1cl':::m~itJ
Gf1b'1'llzj1Jmrhm'l'YI111'1fl' bYlfl'1-i'1~Yl~n'iium1bb'nbb'~mll-JUTilntiu'llfl~Gf1bbtl1rj1J
b~fl0nmVIU1tJ~ 6 'rnb.11 ii:n$nti1mm1D
1. bbi'llil~m1'lb<il'lll:~Gf1 bb1Jrnbm<ilfltJ'lu'1ntlru:::lii1~'l 1Ji
2. tl1i:tJrnti'loii'm1~1'1eN'1mum1clnuiJqJvi1llufl1'"'rn~ei •~urncmi;ieitJ bb'1l:nl'l
'lb<i111:::~m1'1~'VJULJi'

•
""llHl'l-fl'J 6.1
b~B~~ 6.1.1 fj'ni,rru~rf1Ltl'lJB~"i~1J!JbbmG'leitJ
~<M~ 6.1.2 fl:w:wili.:i1u #qii'ni,rrn bbl1~111fj'nm"ifi1fi't]J'lJll,<ii1bb1J!Jbbm'1lltJ
' ' !!.' v&!I. A <V I "-' 'V 4
b"ill,111 6.1.3 mllm~<mbb!J!JbbmfllltJ'lJ'il'W'iljl'il
1. i'ni,rru~rf1Ltl'lJei,.,~1J!JbbmG'IBtJ i.h:::nll1J #11tJll,f11.l-,~neiJJ~'il!ll'il ~ll m"imii'
fhu1Viu~m., bbl1~bbm'1BtJ 1111tJrf11u1'1-,,fl-i'1~~u.:i1u'lJei~-,~1J!JbbmG'leitJilei~ 4 bb!JIJ
1#i'bbri (n) "i~1Ju111i,11ei~111i,%u111eiu ('iJ) "irn1J111mt1iei,-111ii~%u.illu ('1) "i~JJIJ
111ii,11ei~-VlmtJ%u.ieiu bbi1~ (~) "i~1J1JV1mtJiei~-111mt1%u111eJu '11'ileJ1J~L#i',nn<ii1bb1J1J
bbm fill mi~1iJ1i'11'illl1JbVlm~~'llil bb.i'il~1Vii'ni,rru~b%'Wl~'lJll,m"iihbi'.lumnll~
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bbmfieJtJ~rllVl'UlilLiltJ Kendall-Lee ~ei (a/b/c/)(d/e/f) zj,bbl'l111'ii'<ii1bb!J!JbbmG'llltJ
1uflmum-,cl.!1''1 bbl1~5t]Ji'ni,rcl~1-ifbb111u.-i1e!llili"11''1 1u<ii1bb!J!Jbbmflll!.I 1#1bbri
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~ei~1u"l~1JJJL111rn%~tJ nm"illG'lll tJ1Ubbmflei!.I1111rn%~tJ nm~eiaj1'il"i~IJJJh m%~tJ
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bbi'.i.iJ11n'111.J1,iXuilm1m;in1rm11~ bbc;i11m1Lnilbbm<i1181i.J~1bllu1ie:Mmill'Jd11u1'JJ~'mh1 LuVlml'J<i~'
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m15lsimi'1 (N'fou'imo) b-ii1bCm';liJ-'uu'im1blJu'11u1umn Slm>ll'il:::bfl1wn1mrim11ib1.J• • •
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"il:oili'nJo1ru:;ri1i.JbbUUelU n!i11fiil i'nJo1rurn1wii1m'i'uu'Jm-;d.lu1'll1lu1~fi3-1 elTI1J1tJL.i1'1tJnl'J'l'6-;'llbbUU' .m'l'bb"ilnbb"il~m1l-IU1"il:oii'lu ~~iiV1mmbuu'ii'1tJnu bb1i~'li'numnr:j'1i!l1l m:;u1um-;il1'1J!l~ (Poisson
Process) bbfl:Ob4m'IJm "e'i111'l'lm'l'm'i'umm-;" ii'luihbfl'lJbbl1'1~'11muN'i-ii1'i'uu'Jm-;'V!~ilflnii'111'1mil.~tJ• •
~i-ih'i'uu'Jrn-;1im'ltl~'V!ti1m1m <il11lUl~b'liu jnii'1b-il1:J.J1'lh'l':ofi11luii'11'1mil.~tJ 2 1'1'U1i1l'UTYl dluJfu
bbfl:o'lum'l'i-iii'i'uu'Jm.,,Xuflnii'1'Vl~ilN''i'uu'Jm"l'l:;l1111n ~n'J'Jl-lribb111n1i1~nu11.l'lurn~l'l~~flnil1n'li.Jilm1J-1~ ~ ul ~
eloirmrimniil'l'U'>1l1'1ilmi'lu'ii1u1'Ul-11n bb1i'lurn~l1mum'>ruJnii'1l11m'Jti1l'1'Yl'U'Jill'11ltJ1Jf il1ilu1~i'Jiu
m-,i-ii'1bbml'11lm~1lN1nb'.Ju'lu11'1.,~m-,b-luN1n'11lnii1'tJ-Ab'11'1'1J1l~1Jl~llUll'1l'>~m~'1U'Yl'>'>f1m'> bl'Jui1iu
1?1~iXu bYlilmi:J.J~1tJ'lum-,'lim1:0Vi'l':ournbml'1iltJ <J:Ol1J-1J-1~-i1jnii'1"il:O'>!ll'11ltJL'Ubbml'1eJtJ"ilun11"il:o1Jrw
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1Jl~'l':;uuN''i'uu"im-,'Vl~1lflnil11lu'lu~bzymfiu <il11Ju1~b'liu nl'lb-ii1bbmb~1l~1l~1L'1tJl11Titi1'V'l d'Jui1iu
• • •
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l1'11noib.WtJ~ 20 ;'iu iilu<1iu iii1ljnii'11rn<iimb~'ljn.l11'1uL'VllJ'il:o•-ii1~-,:;uubbml'11ltJ1i.J1Jf<1i1l~il1ln11l,1n
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bfl4ei~i'.iei vi'i'ei,i1,;;iei,eiLi1,1.l1:::neiunu ii'1eiLi1wliu v;irn1u~bfllUl"1il~'llmTiu1m1 •1h,ii'1lf,ll-J vi'i'ei
v;irn1u'l1uci1tJllu#i'1v;~ei;ieiumclm1'll'Uci1tJ bUu<li'u N'liiiu::lm1mm1fld'm11Vfu::lm11"1'ilufi,f1.i11• •
"'""1flm1111'it,b1'i1iiu i11N'm1.ifu::lm1i'.iflmw,n11fl.i11m1u::lm1n'iltln1lrn1nnurhil' 1un1rn'll q 'ii ~
b'littd'ilt<li'ei,iJm1b~;i~1u1urtVfu::lm1bvleib~l-1fl1111m1'tVfu::lm1 i11:1:::iitllb~'lbbfl'lfleitJ'iltilm1;irn1
b'Jiil-l~ttl4iltJ'l 'kHl'll"1~ln11 t1iln'il1nd'tm11u'tVlu::lm1li'<ifil,'11u,fi,~m 1tlutJurn11Viu'lmobbfl'lfliltJ
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~il mrfou-1rnJ'lm1rieJu (first come first serve) ~"uuiubb1J1J~L.ifnumn~'l11 bl<lilum,oimum1cl
€ll'J<ifil~l,i(,1u~mnvi'1,1uu'lm1fiilUbY<1ltiJm1;i~1bl'JUd,~'JU ii11eith'b'JiU m1"111'il1n1'tlflUL.if
qrn{wim'bbv;111~ bUu<iiu~'~b~m.ifei'nuri1u'tVfu'lm15nu1:::m1111d,1#1bbri nm'tVfu::lm1 ~'liltJrf111.l
lb~'lnm'tViu::lm1il1vi1uim#i'1vi'i'ilN'1uu::lm1bb<li11:::'1u•:::w11n<li1~nueiein1u m1;i1i.JbbUUilU'llil'nm1Vi• •
JJ'lm1•:::m:::'J1tJ1u'1n1'trnn~'ri:1-1 m b1umrn i:::uu~'1'11~1u#l1mf14ei,~mnm1Vfu::lm1'iltf1'~'Yi1nu•
vm"1il1vi1m1n#i'1bb<li11:::flu 111eiLi11l'liu m'liJ"1'Jf1Nl'll111l1ltJL.ifbfl4ei,~m bUuriiu nmliiiu::lm1mm"lfl• •
~-:::eilill1tJl~11tJ'!N'n-ifum1bl'Jn'"'"ni;iul'J:::dlul11tJeieinm'tu;jU'llil'f1ioifi~~""11~fi,u1:::ll111rrmv;
'llmN'liiiu'lm1 ~il fl'i111m11Viu::im11riibbri'11u1uim#i'111lm1:1~uYi~rii1uu'lm1bb~1bO!~'JmtJ'tuvi'it,viu1tJ• •
nm b'litt m1mm1fl1iiiu'lm1jmi'11#i'lufl'i111 5 flU/~1Tm bUu<li'u
•~ ,, ..2. !ft-:i<Hl'-:iNVl'IJ~l'IJ'llfl.:J-:i%1J1J!!tl1fHl!'J
1"1tJvf11u 1ff''"~1,~u..1u'Jlei~1:::urnbmfleJtJileiLi#l1tJnu 4 bbuu~1lm;i.i'n1'trnt'llei,ri1u~ .611'""' """ d' .J!; .&!:: .J
bVllJ1f111flil (n) 1t1J1JVIM'lJil,'Vll,-VltlnU1liltl ('ll) 1t1J1JVlmtJ'liil,'Vll,-Vltl,'llU"1ilU (fl) 1t1J1J111U'
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~1U1U'liil~'lVl1Jim"l'lUbbll1G'l8!Jbb<ll::~lU'lU~Uo/JilU'1:::U~Uilnn~~lU1U'VIU1l'J'lVl1JimTI1i;im'h'V:::
<ifel~~lU1ti'1Un"i:::Yi'~m"i'lVl1Jinl"lb'1i'1~U'1l-11.J'inl "110i'J1~'1Jm'il:'.1J1Jbbtl'lG'lil!Jbb1J1J'Vlrt~'li8~-'Vlrt~~WililU
b'liu ~·yhm'i1t11i,rm6~i1b~l'VIUi~'Vf1Jim'iblNl'J~G'!Ub&'im bbwlrha:ib~l'VIUl~lVf1Jim'i'Vlml'JG'!Un'V:::nml'J
~ ' & ~
blJU'il'.:1J1Jbbt11G'leJ!JbbU1J'Vlml'J'lrn~-'VIU~'IJUo/JilU
.510ND1l'JLtl1'1rnb1'1'YI6'~il~1Jm'i~ni,r1 N'111m~n'V:::<ifei~'iil'luViil~~rnil'lVia:im'ib"ltm~ilbbwi'l::;
• •
1'1U11'lu'i:::11111~~,:i~.,eii:iu 1'1mrn~m'l'Yi1m1m1<J11~mtJ'lu~ubi~'ei~.lu (b'liu ~~J1111un 1'<11m1l-!ilu
l<liiio/J) nilU'1:::b-if11'11Jbb'<1'YI6inmi1:::'1J€1~bbll1G'lill'Jl uin1'1'l.b:::d<J:::bi'JUbb1J1JVlrt~"lfil~-Vlml'J~Uo/1€1U bbwif11i'.J
bb'l"l'Yl6bb~:::'<1mrn<lVlml'JG'IU'i:::D1Jbbll1G'lill'J'l:::bi'lubb1J1JV1ml'J'liei~-111ml'J~Uo/JilU'lu'll'.i111<Ji~m<J1'11JbbmG'l01'J
~bbo/Jmh~Ltl'11ntm~'11'1~~u;[1U~~ 4 bb1J1J~mh1m l1mJ1~b'liu m'Va:i'VlmmbmG'lill'J 'Vl~1tJ'lim1Jim'i
ii~~rVlu'luVf1~'1TI1A1llu.'11~a:i1A1ii:rn1mn1J b'.lU'VI~l l'JG'!Ubrn:::a:imnG'lill'Jbi1<11~u'lubbwi~:::'liil~'lVi1Jim1 bb~:::
a v .., v Jf ..., ' da1 ' "" !:: d i.>v
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uim'imJ-J1w'1ii'rn'i'uboii1~uuim'i<J1n'Vlu1l'J'lVi1Jim'i bbwl0i'J1~hn1111l-I 'YlfJ1'1'DbbmG'lill'Jbdil~<ifumm'i11
Ul:i.Jl'.ibG'l'ill::~'i:::UrnbmG'lill'J~i'.Ji(ni,rru::;wJ1~'1 tlUL<if bb~ll'il'.:1J1Jbbbl1G'lill'J,:(Wl <J:::i'.Jm1!-1'1ii'U-if1J~ilU
mn
9.1 .0, 61 ~ d
3. '110ffJ':l'Vl'ill':!Wl !llll1':!1!A':ll:::tt':l:::lJ1J!lfJ1A01J
'YlfJ1"Dbbbl1G'lill'J'1:::'lVi.-i1111il1J'lu~tl'lm~iin1'1'l.b:::b%1A11:::'11il~m.,i;l1bUU~1uil~•'liu 1'11'"'~l'J'llil~~1mu
~n.'11'lu1:::uubbll1G'lil tJ bb<l:::l'i1b%~l'J'llil~nm'iilG'lill'J ~,mi(wfi'llil~m'i1bG'l"il:::~'i:::uubbbl1'1il l'Jl'flm"ll1Ulltl
•
88nbb1J1J"j::;1J1Jbbll1G'IEltJ1<1f11<1f0~m"l'lVia:i~lU1U'VIU1l'J'lViuim"l~VIU1tJ
il~Yi'l#i'rnh1mbbwl<1i'ubbR111 m11•m1rn1:::uubbmG'IEIl'J'Y111#i'lu111ml'Jiini,rrn:::bb<l:::'YlfJ1"DbbmG'IEItJ
n'lVi.-i1111E11J'IJE1~m'i1•m1:::~'i:::uubbmG'IEll'J1<1f'lU'Vlm l'Jiini,rru::;b'liunu ~uim"l'1:::#i'il~b~iln1i11<11iini,rru::;
'"''Wl:::'lm~m1i;l1'uu~1u'lVib'Vll-lll::l'll-ltl1J"i:::1Jmbm'1ilm•wi<lrnuu "11E1 i'J1~b'Jiu bb'YIU~'V:::'l".1f1l11'<11'Y1m 2
1'11 il'bbl'l<il~'lu1l1m.h~~ 6.1 N'ui1111'im"Jn1111u<11111'11 '"'~l'J'lJiN-1fill'J<l:::nm~ili'.l11~'llil~'Vlu1l'J'lViuim'i
• •
'li.im'i•iiu 40 btiil1b<'Bw11 'Vl1il5nrnm'Vlrt~~ui'Vl11m'1n1V1u111~1u1u1J~'l<11'llmG'lu~1u~il~'lu'i:::um•m
G'leil'J'll-im'imnn11 5 G'IU #11tJm1l-!u1<J:::bi'lu 90 btlil1,JBwil b~ilrll'VIU<111J11'<11il~nri11 ~ui'Vll"il'llm"i!l
,.;tjl-1b&il-1dilUl'11 .~mnu'Vl'i-h1l'J1Jim'i1<1f5n#l11'J ii'ni,rrn:::b%1A11:::'1Jil~m'ii;l1buu~1u~ul'Jl-!'l4tlim1'i1•m1:::~
1:::uubbm'1ill'J 1<1f,,ri
G'l1ll-IU1'1:::bi'IU~'VIU1tJ'lViuim"l'V:::EILill~•' '' .fJ "" ""'-. • di 61
G'l1ll-IU1'1:::bu'U'Yl'1:::l-/'11U1UG'l'U'11U1UVl'U~ bU"i:::uu
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1'11b%~m1m~il~'lu.,::;uuwiil'1u

4. ffrnu1111ll-.~ui'il1~u11i'u11t]Jll11101Aeu
il);JJ'Vll~.~m.ifeJ~nu•:::'IJUbbll1'11ll'J<J:::1iJ1.i'l«.il"1illJb'VIi-11:;~fj>l <ii~b'liu <ii'"lbb'IJ'IJnl"l11.J"lbbn''m
btil~bftU bblli'il:::11Xiln~run%11n:::'lleJ~m"l<'ilbUUm"l'lleNbbll1'11ltJ (the queuing system's operating
characteristics) iln~rul:b%1111:;'llmm"l<?i1biiunT'i'ill:eJIT'IJ1tJl'<!1m"lvh:i1u'lleJ~"l:::'IJ'IJbtll1'11ltJ'lU''iU'llil'
•
nm"lilfiill'J1'1l'Jb%~1'J'llel~<Jn.i'1 bbfl:::·fatJ!1:::'llel'nm~rl'JUl1Xu"Jm"l'il:::ei~11~ 1'1tJrf'JbUU);jJ'Vllb~mfi'u
"lt'IJ'IJtbll1'1'1tJ'il:::t~m.if'1~nu.i1mi-1~11,:;/iei~~'1'1J"lm"l11.,:::iliuL11 mh1;.iei ,:;1.if'VIU'll'J1~u"im•~
'VIU'ltJV1~m111fa1:::V1ii'mu"lm"i'ii:::<ifo~ilri1il~m•l'<i°i"1b'Vill11 n1<if1btU"l4i''1iiluh0ei"l:::"1um•'l1Xu"lm•<if1bb'IJU
~ftfi~~u,:;/imei~1u~'llil~fi'lli-Jiffi.1'lifu1h:::V111,.,:;ilium•11Xu"lm·m11:;<if1btU"i5'il'l ~b~m.ifei~ mm.vi
~'l.jfum"i<if'1R'ilh114i'1tturntmfimmuul'1'il:::l.Jnbl'ieinn'i!:;~~¥i1b%~tJ/i'il'Yj'il"l'li-J (total expected
cost)
m1i-i#i.i'lifun1'1tJrf11u'lm,<if1bbU"l<ii'1R'ilh "it<iium"i11Xu"lm"ibbl1t¥i1'"~tJ/i'il'YJ'il"i'li.Jbbft>l'ei~
LW11111~ 6.3 'ill:b~'il1.i"J1¥i1t%~tJ'llil~<if'il'Yj'il'l'li.JbUUNm'li.i'llil~<ifu'Yj'il 2 'lJU>l ;.ieJ (n) ¥i1'loii'ii1tJ'lum"i'l~
u"lm"l bb<it ('ll) ¥i1'loii'ii1tJ'lum"i"iilfiiltJ
4.1 l'i11'li''11111im11lti'11~1111 (Service Cost) ilbbU'l1'11''lum"ifl>l 3 bb'il'l1'11~#111'Jn'il $lei
n. fi'1'il1nfi1'loii'ii1tJ'lum•u"lm"i1'1tJ"1"l' <if'leJt.!1~•'liu il1ilfiu~1u 2 fi'il (tWl'il~'il:::bU'il
1 fiU) 'llU.lltJR'il.i'l'i!ln"itllJ"ld'Ylfl~b.iflm'ileJ11~vJl'lTI 1'11'loii'ii1tJl'ilnl'l'IJ"lm"l<J:::ft~~'ilb'Viln'IJfi1-.l1~'llil'' .fi'il,1Ufi'il~ 2 ttiu<ifu
'll. fl>l'illnmwey);JJtflm%-w1:::nm~'VIU'ltJ11Xu"lm"leJ~11~ t1/l"il:::t~eJ'l:::iluu"lm"ij'~~u
nm~'VIU'ltJu"lm"ieJ~11~n,:;ey,~umi.i1u#11tJ
fi. fi'1'il1nfi11i1tJ1eJmft'lleJ'61m11~'llil~'VIU'ltJ'IJ"lm"i <ifu'Yjufi1bfltJ1eJmft'll€Mm"l1iJ~>lft"i'l
'l1X'Vltl1tJu"lm'l~'il1u'loii'lun'ilrn"ii-im"lNi°i"15w1 e:n<1'ii:::'.1m'loiitt'Ylufi1-.l1~'llei~V1il1tJu"lm•
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4.2 ~'ll'IJ'll'l0ri01J bbil'11'111:i.J61:i.J1~irr1:::vi11,,::;#iunnu'im1bbli:::'11b%~tl'llil'nl'1il'1ilt1<J:::
mjluilni,rru,::;Nni'.hi bbllinilill<J1<iJ-ifil<!1tlb~mi'u'1fu"'U1il'1iltilliilVIU1tinm'iilflil ti <;t::;,Xut11:::b~U. ' "
#f'1n'i11<J::;1:iJl'iilt11<ii·fom1n'i11li''lum1".hm1:::~'il::lJ'IJ bbmflilt1 '1i'ulJU"lilflilt1bUul'i1tl1:::mru,'lleM
1iulJU~~llJbl'!tl"llt11~'11il'1pfi<J b~illn4l1Vl:i.J;im1:i.Jil'1'YIU bbfl::;ililn<J1nbbmflilt1bY<'il:::'11il'1ilmu
bfiu1ti '1i'u'Yludm<J bbl1'1'li'ri1h~vi1t11t1bUU'1~'m11 vi1ilill<Jftl::Yliluli,ri11•~vi1t11tiei81,m11 il1
'
1:::lJ'IJbbfl1flilmfi;i~wi1t1'luviii1t1Ij1fi<Jbm b'Jiu b~eJflu,1u'1ei'1ei{u~uri1utl1millJ1!1t1u.l<J1mbNUn
il::1vi'i 1iu'YIU"ji)flil ti'lumrud<J::<l::Ylilu1fu'YIU'llil'Nli'111111'<~"'f1I bl'!tl1ti'llil'1flU'11U~ilti'lu"l::vi11,m1q q "ql g{i::I 'II
1ilflil t1 Uiln<J1nd1iulJU1ilflilti'lunl'i"ibf111:::~rn'fl~' <J:::tl'i::nilu<il1ti.;(,b1m~'iilil~'lumnflilt1bb'11::
nm1u'i:::vi11,m•-l'JJu'im1lln~1t1
~10th~~ 6.1 ~~;im;t-:;,,mtl•:::nil'IJ"l!1t1u"1bb'Vi,vii,ri1;;1,1(;i5uh11<J:::'1i'il''l'61ilm1ub{i1viii1~
'lusbu'lViu'imdbiln~Ubfl~il'1i'.1il~flu'l,<J:;villVf$1l'i1b%~t11fu'Y.)U71:i.J<1h'j';i 'irnubbfl'lflilt1~1bfl71:tVl1u
1f1EJ81'1d m<J<J::bi'Jumru (n) vi1il ('ll) 'lwi1Y<~ 6.2m'imli"i1u'lVfJJ'im•'llil,AU'11ubi'Ju1t11;it1zj:i.J1;it1
b%~mvi1i'u 12 '1Ulliil~11:i.J'1 (V11il 1 flULUlJfl 5mfl) L'<i'1V1ii1~'lu'11u'lViu'im'i mm;n'lViu'im'iflU,lU
'lum'iLUrnL'1:::fiuLfl~mi'.lil'l-6L1'111;im%~t1 3.75 mfl (V11il 16 flUlliil~11m) bb<i::nm~'lVfJJ'im1n•i'Ju1t1
1;it1rj:i.J ~~;im'i<W1'i11{11,b{i1V1ii1~•~ilmtliJu&i'11u'lmhu1Vfu'im'i 'lVii'iT~1,~11:i.J,1i::: 12 rn'YI bb1i::
!'11bb'i,'llmflmlU~'iililajlubbmAilt1i'.l!'i1bYi1i'u 16 lJ1'Yllliil~11m ~~;im~::'l1ii;j<1!'11#1,n'i11mtl'i::mrn
l;jlifll'lleNNli'llmY<~~llJbfltl1t1b~ilf1U'11U1fil,'iililaj'lUbbil1fliltJ .;;'1nm~'l'1tlUnl'i'iilflilti'lUbb!l'lfliltlbb!il::
nm~'l'1tlum'ibunvi1ilfiULfl~il,i'.Jm.;im1:i.Jil~'lunm~il~11,'llil'1'1U'11u ~'1nfiil b1m~il1~1'11,1um:i.J
tln&i
~~'1m'it'6'Ylfli,r~mnflilt1m'l'1flum'iri1viu;ifi1'"~tl'llm'il1u1u'1u,1u'lw;::uubb!l1'1ilt1 ('il1u1u
fl1'"1U~<J~lUbbmbb<J::~il~'lurl1ulJ'illl'i)'lu 3~JilJ fiil viit,V1u1t1'1J'im1 ft€1,VIU1tlmf1l'i bb611;<l:i.JVIU1tl
mm'i nT'l"ibfl'ill;~'il;lJlJbb!l'lfliltltu1f1iltil,d<Jl;l'6~1ii'i'l'rnmit'llf!l<flil~~'<J:;1#'n'i11.Jil1tilUb~il'1~ 6.1.3
bbllilu~d<J::'llilbbl1'1'1N!ii1'Y<rrm•1b'1'il:::~b~il'l'6illirnt1m11l;i5uh'll<J'1N~'1m'ih"1mb'Vi,d
1
2
3
•
3.0
0.8727
0.7647

..... , $' ""' • • •;~~liJfll-:i 111J-:ifll-:i l'l'Uf!U1l'im; l'l'Uf!'U-:i<llHJ!J* l'l'Ufll,!;11J
1 (12) (1) =12 (16) (3) =48 60.00
2 (12) (2) =24 (16) (08727) =13.96 37.96
3 (12) (3) =36 (16) (0 7647) =12.24 48.24
*·:n:i.irnnu'irn:i
m1nrlBu1'1181~'11h~<i11.11t111111nn'VJ fh1:::11l~i.Jrn1'VJ'1'11tJnurn"I~ 6.3 1111:i.i@i1BtJ1~<J:::Y1u)1
ti%111m':lh~~11.1'11':lb!iBn1'l1~b!iein~ 2 ~%111Hfil'Vli.brnnm':lbv:ltJ~ 2 'VIU'ltJ b"1':11:::1Vi'11<i11.1'Yl1.1':l1:i.i~1<1<11
. ' '
11.1~ltil-ifeim':lfl<J1':ln.neitj 2 ti':l:::b~ufiei ti':l:::m':l~'Vlit~<111:i.irn11~1-if1um':l1b'1':11:::V1'i.1.1~dilwt.bmti1.1
~1Tm ~~<11m':lmbtil'1~1Viil'11bti1.11u1<ii'l:111tJm':l'Jtl.l~11.111.1~1hMm':ll'i1~11.11u 1 1m-if1111 <f:i.i:i.i~1Viil'11
whnu s ~11m •:::11i'N1"1lUbb.il'1:::'Yl1~blieinWi~d (1) 480 lJ1'Yl.iei1u (2) 303.68 lJ1'Y1'1ei1u •m::: (3) 385.92
rn'Yll1iei1u ~~!'.1~'1~1ViN"1m':l@i111i1.1hb'Vlj'jeirn~J,J ti':l:::m~<f!l~b~mnum':lbliein1-ii.i1b%~tJ'IJei~~11.111.1
'11.1~11.111.1-i-:::uumn'1ei mb'YIU~"l:;1-ii.11 '"~tJ'IJ!l~nm~!l~1u-,:;uuJ,,11~11.1'lm<iiu111.1':l!lfieitJ 1i'u~<Ji~bbfi1
-o;nli!ln1-ii'111111/1<J:;1ViNl'1'11Ub'Vlj'j!lunu mll'lfi!l n11.if'11b%~tJ'IJ!l~nm~eitj11.':l;lJlJ ~<tr'ilm':lh~~11.I
-o;:;<i(!l~1-if'11b%~tJ'IJ!l~nl'>':l!lfi!ltJ'IJm'11.1~11.1u'11"1:::'11.1~mli~umfieitJ @11eJtJ1~b'li1.1 <f:i.i:i.i~-:i1bU1.1'Y11~bli!ln
~ 1 n1il'11.1~11.1mli~bb[)'Jfl!JtJL<11m%~tJ 12'11.111iei~11J,J~ bbl'1:::'11.1~1'.l,bb8"1:::'1U"l:::'l-iinmlU'J":::lJlJL'ilm%~tJ
1 ~'JLJ,J~ 1i,:;,Xu,:::il'1'U~l'.I, (12x1) = 3 '1'.l,!l~,'.l,':l;lJlJ n1 A bb'Yl'U.i1b%~tJ1i''il-rln1'>1J1li~ (mean arrival
rate) w bb'YIU'11m~m1m~!ltjlU':l:::uu bb"1::: L bb'Yl'U'11b%~tJ'll!l~~11.11u'1u~11.11u-,:::uu 1i,:;,X1.1-o;:::1~111
L = t..w ~~bli1.1-ifeJ<f1t1~'11fi!l!L1.1'Yltil'1-amn'1m.JWi'~•:::1.lnGi11'lu-,1tJl'1:::blll'J11111ieJ1ti
' . ' '
1 ..... 1 .:::! d, <!! .:::!
'Ul!'U1ill-:iH!llfl'll'U1!l'l'l 6 ll<l'U'l'l 6.1 110~'1'1 6.1.1

ri€lu~,~~rn±l"il!Jfi~b5!J'1<fi1bblJlJbbt11'1il!Jliiil1u~u'lu~ullu,:::<ifil"'l11m1:wb<ii1L'lnu<l:w:w@i31u
bbfi~ii'l!'1n1'.'lcllii1"'l ~Hr'lm1ri1"Dm11"il!Jnilu4i'"liiil1ud
'..:::.. .d. QJ QJ
1. irl-l:J.JIll!}l 'l!lflffJfl'IJll1!!'IJrnm1~ol'l
'YlrJ1'1Dbbm"el!J1;1!Jm~"rnm~ul'1!Jill'1!J<l:w:w@i31u'Y1m!JU"iml"ielUb~m<iiel"nlJel"<iftl"lmellJ
~u3m'llel"bbm"el!J m"ibll~!JUbbtifi"":w:w@i31um"ll.,:::m"in'Id'lll-lfiIiiilm"i'ibfl"il~1X"i:::lJlJbbm"il!Jftl'Y1'lu
'luvh~el"d':::n~11b%'jlfl:;cff1bblJlJbbmflelmflil"<iiu <l:w:w@i31ulii1"'l ~b~m-n'el"l:l41"d~el
1.1 fll".il!'ilfl!!'il~'IJ!Hfll"imi:l~ (Distribution of Arrivals)
'11 bblJlJbbf.Ylflel!J-1'1bi'lucff1bblJlJ~ b~monel"fi'lJ'111:WUN:::bi'lu (probabilistic model) ,if"d
b'Y'l"il::<11il1ull"i::<rnmm"lil1'llel"n"itlJ1Unl"i!Jn"i1:Wiltj'lu<ff1bblJlJ'luj1~~ bi'Ju/11bbU'l~:W /11 bbU"i~:W
''Y1~1d'l:::l:JYl"n-h'um"ibb'nbb'~m1:will'I:::b:UUb%·m:::cff1,ii~nfi1m"imli~bbfi:::nm'lVflJ~m"iliil~nb:Uu/11bbU'l
~:w '11mJ1~b'llu ~1u1u'Jnfi'1~mli~iiiel'Y1ii~'Y1u1m1m (b'llu 1iiel~1lm) 'lu~1ucfim-1:wbbvi~'Y1rt~ m'lbn'1~u
l '1!J~:Wbblii'I:::l:Ji1~n-h'um"ibb'ilnbb'il~m 1:WU1'1::;blJUb:I.·vn:;cfi1
0 " ' ' ,J'lv ~ ,, ~ Ji ' _J v -" <v
'l1u1um"imrn'ilel'Y1U1m1m rn <lmu'YI 'Y1lJ"im"i'Y1U~'l m'lm'1'l1Ub'1!J'1:W'i11:Wfin1'1rn:::.,..~n'llu
m·m'ilrnl'il~m1:wul'ln:Liu'1n1"rn:::ii11~'l bblii~iJn'l::'l'1lJnu1<1i'l:'1!Jrf11ui!lel nl'lbb'ilrnb'il~u1•1m~ (Poisson
Distribution) <l:w:w@ijlU4i'~n~111;l'lum"i<lUlJ'1'.lU'l1nm"l~n1'.'11bil~ll"i:::-ln~m!J~U '1n1"rn:::'11iljj
·im~m•u'lrnb'il~uulJiJ1'll'el~~el n1~m1um1mli~iiiel~1l:w~l:lm"llb'ilnbb'il~bblJlJi:l1'll'el~;''1!Jil<i111bQ~tJ~~
bb'Vlu;''1tJ A. cfi~~u nm~eltj"i:::w:h~m1mli~ (time between arrivals) 'l:::l:lm"lbb'ilnbb'il~bblJlJblini-
11lbbUUb'il!Jfi (exponential distribution) ;''1tJfllb%~tJ ~ Q:;~u fllel'il'rnQ~tJm1:w1li~!iiel~11:w~b'Yhfi'lJ
2 .-iu nm~iltj1:::'Y111~m"imii~'l~b'Yi1nlJ ~ ~1l:W~'Y11el 30 u1fl m1:wfi:w~uli'4l~rni11'11t11;141~d
i'J'l'llei-:1
'11bQ~!J = A.
A= 2 '1Uliiil~'l1m
' ..: 1
'11bQl1!J = l
1
= 112 ~11:w~l
= 30 mfi

• v ~
1.2 lll1Wil0!!'1~'1J0~!1llll'll1J-:llll1 (Distribution of Service Times)
b1m1«1J~m11um:::1J1um1'll<Mbbmflfl tJm'll'.1nl'l"bb"nbb'l~ffll:J.JUl'll::d'iu1Ji'u111mtJitlbb1J1J
w1i~iin'l:::l1:1-J:1-J@inu $lei 1«l'.lm7bb'lrnL•~bb1Jrn5n.flt1..um:StJ01 •1n~1<iinl111m•~mn1Jm1:1-Jii'm'iun
1:::11111~ m1bb<Jnbb"J~bb1J1JU1'llfl~bb<1:::bbuub5n.fltlbbum:Stm ii'1nm'l«1J7m1l'.lm1bb<Jnbb"J~bb1J1JbBn.flt1-
bbUub:iltJ<1bbii'1 el<il"il'l«u7m1 (service rate) ~~bb'Ylu<ii1tJ µ 'l:::l'.lm7bb"Jrnb'l~bb1J1JU1'llfl~
1.3 1~!UUU'lJM!!trlfl!l1l (Queue Discipline)
#1~~1<iiflTIJltJmbbii'111 1:::butJ1J'llfl~bbmfleil'J bUUn{.]m'l~<ii'11#1u~n.11~•nil'fou7m7 'lu
/11bb1J1J'l"l::1J1JbbmflfltJ iin<J:::l1m1@i'l«01n.111<ii"luu7m7bb1J1Jmrfou1<ii"luu7m1riflu ii'1!1:1-J:1-J@i.-1u. ~
If~nli11i11<iil1ei<iiflii'mnufl11:J.JbUU'l7~ n<J:::¢!ei~$1<i1'1u/f1bb1J1JmnfleitJ5U'l i1'n1'frnl::'Wl@irnw'llfl~~n.11
~m'lm:::'Yluliieim7~w,'11#i'um1'l«u~m'lb'W'll:::~n.11~ei~'lubbmfleJtJm<JV1:J.J<ilm1:1-Jei<il'YIULbii'1eiein'l1n
bbfl1flBtJ1ti (reneging)
l.4 V-:l~'lflfl-:iil1nvi1m~1iJ;J1nA (Infinite and Finite Calling Population)
n~:1-Jti1:::'Jl1n11U'i'l11'1-DbbmfleiEJ bUUbb'Vlli~~m'llmm1m1i~mn<iieiEJ n~:J.Jtl1:::'Jl1ml'.11<iiiif~bb1J1J
-ol1n<i1-oliu1u••01:::1iJ-oJ1n<i1-ol1u1u#l~nli11bbii'11u•~ei~~ 6.1.1 'lum'l'ibfl11:::Vibbmflei1.1/11bburn•mfleitJ
dlm¢fu ifn-.,:::l1m.;@i11tl1:::'Jl1mi1'llUl<il1iJ-oJ1n<il bb<il"lumru~tl7:::'ll1rnl'.1'1Jm<il<l1fl<ilfl';ll::'Wrn'l'.iu1<iivi11ti
'lum1:mtlu'l7~L'liuflu
1.5 fldl:IJ1lld'll0~!!1.l"Jfl01lfi;110A!m~1iJ;J1fi'A (Infinite and Finite Queue Length)
bbil11/11bb1J1Jbbmfl<JtJb<iltJyf11111iin-.,:::l1:J.J:1-J@il«m1:1-Jm1'1J<J~bbmfleJtJi:J1Ji'liJ-oJ1n<ilbb<il"lum1:1-J
•tlu•7~ni11'1•tlm'liuiXubl1:1-Jei111 m•l'.lrnru~~u~i1flt.i<l1n<il'Yi11«bbmfleitJ<iiei~nn-ol1n<ilm1:1-Jm1<111:J.J111'll I I 'i) 'IJ
i - ' < V i l & A ~OJ .:t ' " " ' ""'y 1 ,..,.""°
<il'll'J <mflm~ b'llU 'WU'Yl'ilfl<il'l"rJl1l'Vl'l"lJ1rJ1J'l"'l"1'.jn'i'11eJfleJtJn11'11U11l'JVI'j"eJ1JTI1'.jnl1ufll'IJeJ~Vll'l"rJbbVI~VIU~;;J
<liu1u<l1n<il 'lum1'l••m:::Vibbm<ilfltJ'lumrud<J:::i1/11bbu1J~bb<11nii1~•1nbbm<ilfltJ~i'.Jm1:J.Jm11iJ-oJ1n<i1
QJ <=S IV q!V.,,.
1.6 0A1llll-:l:l.lltH!!tl~0ll-:ll l'llU-:llll1 (Arrival Rate Versus Service Rate)
b<iltJvi11tibbii'1 'l"l::1J1Jbbf11<ilfltJ~~n1'fl'll::fl~1JUl1:J.J:J.J@i;JlU'llfl~l1m1:::fl~/11 (steady-state queuing
system) ~~m1:J.Jl1l:J.JT:m'lum11«u1m1V1~eie1<1111m11«1J1m1l<ilm1:J.J-.,:::~~n11€i<i111m1mwu1m1
i],..:;,Xum1:J.Jm1'llei~bbm<ileitJ"J:::m1~u.4fltJ'l <Ju1i.Jmm1nbi1rj11m1:::<il~<if1 m1:1-Jii':1-JYi'ut1:::11111~
e1<1111m1mn~ eJ<il"il1«u1m1bb<1:::m1:J.Jm1•%~tJ'IJfl~bbm<ilfltJ bbl1<il~fl~Lwi1'W~ 6.6 t11€i<i1"Jlm1mn~
'il:::<iiei~iifltJn11€1<il"il'l«u1m1 bbii'1€i<il"Jlfi1u'IJfl~eJm1m1:1-nn~iiflel<il'lll«u1m'l""J:::i'.J'11ii<JtJn11 1 bb<1:::
b~ei<i1€i<i111fi1u#l~nli11l'.1'11bi1'lnii' 1 <il11:J.Jtn1b%~tJ'IJfl~bbm<ileim1ni1'lni''11mXm'i (infinity) 'lu
m'W~ 6.4 '11 'A Lbl1<i1~'11b%~tJ'IJei~e1<il'llm1mfi~ bb<1::: µ bbV1u<i1b%~tJ'IJei~;J<il11'l«mm1 'il:::ii'~m1111#111
Lrleie1<i111fi1u'IJfl~ 'A liifl µ l'.1<11mnn11 0.7 m1:J.Jm1m~tJ'IJeJ~bbmfleJtJ"":::bv1:1-J~ueitJ1,TI<ilL11

. '
00
lQ)
£
s
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
(Arrival/service rate ratio - A/µ)
QJ <V c!at at v OJ QJ
2. l:TtJ!tiIll!ti.!'l'l!!ltf1'110~ fl'IJ~1l!'IJ'U!Iti1fl0 ti
00
i1qJ~n1'rrn~l~ti1on€Mnu~1LL1J1J Lbm'18 ti~<Jia~ri1fl1ll-J Lonll<Jrim1wrn1'rlYlt]1'i"D mnil€l ti <ii€lltl
1h~nmJ#i'1tJ ifqJ~n1'rclmm3mbLYIU~1LL1J1JLLmil8 tJ LL<l~ifqJ~n,,;rnLLYJUfilf!n?'i'llm~1 LL1J1Jllr11il€l tJ
11.J' <1..1 tf ""'
2.1 <r'!!1,1nH111m Illl !!rn1!fl'llIll11111m1mi'l!l11
Kendall - Lee 1#i'ri1111u'1ifqJ~n1'rrni.J11wi-31m~mLYJU~1LL1JrnLmflmmuu<i11~'11'1t.J1-if~1
e'Jnm 6~1L1mo/ieinu'11i.J!11'1'lJ1uiuLL'lJ'lJ'1~d
(a!b/c) (d/e/f)
t.m~
&
a = nTm"ilnLL"il~'lleMBil'llm1mrn
b = nl1LL"ilnLL"il~'lleJ~elil'llnl1lif'lJ'lm1
c = "ollU1UVIU11J1Viu'lm1
d =
d
~L'lJ 1J'lJ'll8~bLmil<JtJ
e = "ollU1U'J~<jil~€laj1U1~'lJ'lJL#J
f = 'llmil'llei~u1~'ll1rn
ifqJ~n1'rrn~1-ifLLYIUnl1LL'rnl'1~U1l;LJlYJ<ill~'1 ii~~d ~€)
M = n11LL"ilnLL'1~ Lb'lJ'lJU1'll€l~
..!
D = nl1LL'1nLL'1~LL'lJ1JLLU'UeJ'U, il~'YI
<
Ek = nl1LL"ilnLL'1~LL'lJ1JL€l€l'i"LL'1~ (Erlang)
G = nl1bL"ilnLb"il~vf1LUfi~i.J-;;J1L'l'il~L"ill~'J~

lilulii''lil81,'llil'lii''lbb1J1Jm11'1iltJ~ull'iUhiJoi,d
(1) (M/M/1) (FIFO/co/oo) 'VlmtJ!i, <i'i'lbb1J1Jbbrl'l'1iltJbb1J1J'liil'1Viu1nT'l'.iiil,b@im '111:1-Jl'.ll'l
um.iiltJ1l-i<i1nm•11:;ilti"l:;'ll1m1l-i~1nil~1u1u ~,.1'.lultluuu~v.rniiu1~Yi'11t11u~'lliltl"l:::~11u <i'f1il81'
b'liU ~lUill'Vll"lill:J.J~'~il'JiJ.i-l'1'1Ub@im nl"lb4i'1bb1'lb~il"lil0lh%b'.'lu.;1!'lu.11ill:J.JbbNUn11h''l LUVll'-
i1"l"lr<i'lU.11 dJu~u
(2) (M/M/1) (FIFO/n/oo) 'VlmtJli' <il'lbb1J1Jbb1'l'1iltJbb1J1J'liil'1Viu1m"l'liil'b@im ml:J.JtJl'l
LLm.iilu~1nii ••11:::t11:::'ll1m1l-i~1nil'ilm1u <il1mJw11il,"l:::urn•m'1ilutl1:;L111'1d 1#1mi 1rnul1'1"lfi'v.Yi
il1:J.Jil1urn1u~rhV1uil-l'umu~1'1"lb4i'1m"51wiu n mu '11u~ilzjLu call waiting ':::ilbRmbr<i1'1vrw,
~'n~il m"l<iim•mmmtilu':::1Jn'ii1nilH~1u1uV1rt, '11u~Lnu,:;1Jnlii''1mm1iln111
(3) (M/M/S) (FIFO/oo/oo) 'VlmtJli' @11b61Jrnbm'1iltJ~il"liil,LVfu1m"lmmd1 1 'liil'
m1:J.Jm1LLm'1ilu1l-i'11i'il LL11:;tl1:;'ll1n11l-i'11nil~m1u <il1mJ1' L"liu m"lb4i'1mn"lilIJ.!1nniluV11'il'Y!1
1l1n11:J.Jnuv;urn1u1h:::~1b'11Ublilil1rruim1~'V1mtJ 'Vl1'ilm11il'lf11:;~11oifu1m11'11,.i1u'lleJ'woiiu~• •
1nuu~~efieJ~m1Loif1'11~.J1u d]uefiu
(4) (M/G/1 )(FIFO/oo/oo) V1:J.J1uli' <i'i1bb1J1Jbbm.iilmbuu•1fo'1Viu7m1"liil'L~m nm1Vi
u7m"lilm1u'ijrnb"••uuL'1'1 mwm1••m.iiltJ1iJ~1nil••11:::ilt11:;'ll1m1l-i~1nil~m1u
(5) (M/M/R)(FIFO/K /K) 'VlmtJli' @11bb1J1Jbb1'l'1iltJ~il'liil,LVl1J7m"l:J.Jlnn11 1 'lim
m1:J.Jl'.ll1bb1'l'1iltJ'ii1nilbbi1ru"l:;'lflm'ii1nil'<l1u1u 'Vl1m1tJn'll Machine Repair Man Model mrnd
~rn'11V11il~wu7m11l-i1~m'ij1nmm1rn:::b'Vll'lilu"1,1f1bbuu5u'l @11!ltJ1'6'Jiu b'1~il,~n"l<ii1''1 ~1oif,1u
1uh"1mL11:;1!lwu7nwiieJ:J.Jli11'ilzj Llluefiu
(6) (M/M/oo) (FIFO/oo/oo) 'VlmtJli" lii1bb1J1Jbbtn'1iltJ~'ii1wm"liil"LViu1m"lil1l-i'<l1nil
'11u1u m1:J.Jm1um.ieJu1a.hi1n'1 •rn:::t11:::·1nm1l-i'<l1nil'<l1uiu @11iltJ1"•m1.ieJut11n111'1d 1#i'••ri
!1Gi''1m11u~1Uill'VIT'lbb1J1J Fast Food 'Vl1'il~1Uill'V11"l~LVfu"im"lbb1J1Jl_J'/.Jb'i!JG°i blluefiu
<V V J9'ljj I """~ <V
2.2 <lt)!illl'lltu'l'I !'lll!'l'l'llfll<llJl'I !'lll'l1!1lJ1J!!IJ11'101l
ftilJ1ni,;cl~1oif1um1n1V1uil~i11i:i~"11''l 1ulif1LL1J1Jbbtn.iiltJ tl1:::nilu#'ltJ
A. = [}1m1m1m-l'uu1m1 V1:J.J1tJli' 'iim1u~nrl11'1rn'i<~tJ~:J.J1-l'uu1m1<iieJV1rt,
V1u1rn1m
µ = 1ilil"llm1'lVi11"im"l 'VlmtJli' '11u1u~nrl1tilrn'i<~tJ~L#i'-l'uu'lm1bb~'lbi11<ij<iJeJ
" ''VIU,'VIU'ltJb'l'11
L = '11u1u11n.11~mJ'lumn.iiltJtilrn'i<~tJq • •
L = 'ii1U'lU<Jnrf1~ilzjLWil::1J1Jbbrl'lililtJliltJb%~tJ (~"~n1ii',"lilililtJbb'1:::~n1Gi'~
w =q
w =
-l'uu'lm"l)
nm"lilililtJ1umn.iiltJtilrn'i<~tJ
n111~iJtJLu1:;u11tilrn'i:I~tJ•

Po
p
n
p
=
=
=
il1l:WUl'OJ:;blJ'U~'j:;lJlJ"Jdl~bU11 (VIU11'JLVllJ~nl'ifltj11~)
.-i11:wii1"ln1iu~'ll:;ii.-iu n .-iu'lwJ:;lJlJ
ffo1'11w11ei~nm~rtlVilJ~m-;~ei~rh~1u
3. o"J::anu Birth-Death
V11h'lla~m14'fnjj1-;:;lJlJ mn.-iei tJ L<"ibbrl m'l'ib.-i-;1:;'im.J1.-i11:wii1'il:a1iu'llei~m-;ii,;,m'1•
'ii1mu'1i1~'1 'lu1:;lJlJm11.-ial'JV11ei1:;lJlJlJ1m'l ~~:i:in'l:;b~l'l'ilbb1'l'U#i'11'J P t,m n Vlml'lii~ ~m1u,;,m'1n •
V11ei~1lJlJ1m'l'lu-;:;lJlJ 'lum-;frnjj111 ru nml'1'11amff~'l:;lJlJ'OJ:;i'.iiJnfi1eitj'lu1:;lJlJ~1u1u n .-iui:iu
mmwv:J,11ru1b1imm11:w'llei~bV11?Jm1ru'ii1~'1 ~m,bfi'1~u1<"1#1~d
P = (1) mrn~b~:wii,;,nfi1mJ n-1 .-iu bbA'1ii.-iu'lV1i.ib.if1m'lu1:;lJlJti11'JL:i.iii,;,nfi1bn1n 'll 'll 'IJ
eianL'll'il1rn:;lJlJ rh'lVibV1i'imri1nlJ n .-iu
+ (2) mrn~b~:wi'.iiJnfi1eitj n+1 .-iu bbA'1i'.1iJnfi1~1lJlJ1m1bff~'!lein,1n-;:;lJu111
bb1":;1:iJii.-iu'lV1:i.ib.if1:w1ril'lY!bV11i1mri1nlJ n .-iu
+ (3) mrn~b~:Wi'.1iJnfi1eitj n l'l'U bb.1'1i'.iiJnfll~1lJu1m'lbff~"l!leJn'illn';:;lJu1u
~-l'ei:wr'ilJi'.iiJnfi1'lV1l-ib.if1mYmiiirh'lVibV1i'imri1nlJ n .-iu
+ (4) mrn~b~:Wi'.iiJn.1'1eitj n il'U bb.1'11:iJiijnfll!l!lnLU"Jlfl'l:;lJlJbbl1:'.;1iJii.-iulVl:i.i
b.if1m 1'11'lVibV1i'imri1nlJ n .-iu
'OJ1nmrn~bfi'1:iliJnfi1 n .-iu'lu'l:;lJlJ ru nmh'l i:iumm'lti'jf<illm'iiei1<1111ilil'l1m-;m1ulJ1m-;
(A) blJ'U Poisson Distribution (nm1:;V111~m-;mi~blJ'U Exponential) bbl1:;ilil'llnl'l'lVilJ1m'l (µ)
briu Poisson Distribution (nm'lum'l1uu1nTib1iu Exponential) bb.1'1 bri!l'l:;lJueitJ'luffm1:;
•"b<lfll'l'J (Steady State) 'OJ:;bfi'1 Flow Balance Equations #f~{l
t ' 1i ""' .,,,, 2" 61
i1tJ Pn = ml:W'Ul'OJ:;b 'i1'1'll;:WiJnm n .-ium-;:;uu
= ii'lildlm-;:w11'uu~m'l'll!l~iJn.1'1'lmJ briai'.iiJn.1'1!ltjbb.1'1 n .-iu'lu-;:;lJlJ
,µn = ii'lildlnl'llVllJ1m-;brieii'.iiJn.1'1 n ~u'lu-;:;lJlJ
t v ~.Iv v ~ v d ' .J v ~ o o o o ' v
i11'J1'11 bu!lil'llnl'im'llJlJ'lnl'l:Wn'l:;:wm<i~1'1 'U'ill'l!l "'o = "'i = "'
2
= ...= "' <l1'ileJildlnl'l
,., x I n
'lVilJ1m1iium'il'IJU!ltjnlJ~1u1u0VilJ1m'lfi:il 1!1eitJ1~b'liu 'lu1:;lJufi:ilrtlVflJ1m1 3 <iu bb<il,;,:;.-iuii
eJ'1"'Jlnl'llYllJ1m-; 5<i'U/~1t:w~

J. "' .....
ba.Jili.<~nm 1 '1'U µ1 = 5
b~ilil~n.11 2 '1'U µ2
= 10 bl';Ti:;'l:;lif~Vlu~m• 2 '1'U
b~ilil~n.11 3 '1'U µ3 = 15 bl';Ti:;,:;t-if~Vlu~m• 3 '1'U
,o!l # ij..I I ..tifij ..... =
bb~:;ba.<ila.<~n'11mnm1 3 '1'U µ4 =µ5
=µ6 =...= µn = 15 bl';Tl~'l~a.<~ bVllJ''mld
bV'lm 3 '1'U
'lln Flow Balance Equation iiu 'll:bi'.lu1#f-J1 bJ1il n = 0 : P
1
'll:b°lilnU 0 bb~:; µ n'l:;n- n
b°li1nu o b'iiunu ~~iiu ,:;1#1
P1µ1 = Po"-o
"-o= -·P
µ1 0
= =
= =
p "-o"-1"'2···"'n-1=
n
µl . µ2 µ3 ..."-n
c A.o"-1"'2··· "'n-1=
n µ! · µ2 µ3···"-n
p = CnPon
·P
0
·P
0
·P0
~ ,o!l ,,,i,J A ii ij ij..I
...:..P = 1 ba.<ei k = d1i.<YJnn>ru'Ylm'lbrnn'lJ'Ub'1
J
P0
+P1
+P2
+ ... Pk=1
Po +Clo+ Clo+··· CkPo= 1
k
P
0
( 1 + 2: cl = 1
j=1 J

1
(1 +LC)
i=1 J
~'1u~d ti'n:::urnbm'1!l tJ0T1~1-,i;i1Vll'.J~n.'fwli1:J.J1eitj1wo::;uu1.i'mh'1iJ-;hnil ~l'lJ€1' P0
1
00
(1 +LC)
j=1 J
w1iti'1-,:::uul'.l~n.'f1eiaj1u-,:::rnJ1.i'!l~1,~1nil ~1'lJ!l' P0
n•::;'11u1rn1.i'il1m;iil.,.if1~.ru bri!l~1
v • ~
k = 'lJ!l'illrn•m-;;n'lJ!l~-;;::;rnJbbm'1!ltJ
1J<mm-;;'1m1rn P
0
duu11bliu11ilb~:J.J.i'u'lJ!l~lii1bb1Jrnbmfl!ltJ m-;;'11u1rn~1m1:J.Ju1'il:::bliu
f!u'l •:::1'1•1n~m111mbb~1 fl!l P = c P
0
(b<:wn:::rnru~l'.Jm-;;bb"JnU'il~b,lJ1Ji:l1'l!!l~) bb11:::m-;;'11u1rnn n
• ' .... 41.
L bbl1::; Lq 'il::::J.Jl'illTlnl';i'11U1"hillillilYl1~ 'el
L =
k
L n • Pn (k = oo brim::;urnUUbb1JlJ1iJ~1nil~lU1UlU-;;::;uu)
n=O
k
Lq = L (n-s) · Pn (s ='ii1mu~Vlu~m-;;)
n=s+1
w =
1
brn::; Wq =
'A
Lq
'A
111[]1'1~bbtl1il!ltJ'il::;'l-hmmi'nm-;;i'i'1'il1n Birth-Death Process dtuv:i'iljUlblJU';jil';inl'"ifllU1"h
1111~1 ill:J.Jtl-;;nrrn'lJei~bbmileimbil11::;tl-;;::;b.r11111ii~bb'1il'1m~m~ 6.1.3
4. m-a1h::a'Jfllll"'l'll~11mm1i;i11'10rt
1um~tl~ifq]m-;;tl-,::;1jrn'1'1:ii111ri1'1~mn'1€1tJi'.11ilJtl-;;::;0~'1 2ti'J:::m-;;fl€1
ti'J:::m-;;,b-;;n blium-;;0·1h~il·mmJbbmileim~ei1oifbb111u0mum-;;m'll!l'nlTI!lfl!ltJLiltJl'.11'<iifJtl'J::;m.1
4i1Mm-;;1111~10fiqj.J1~1 ~loifbliulii11il~·mm-;;ohbUWll!l~-;;::;uubbmil!ltJ,XU'l

i.hrnl';;~crei' bUum1'1N<J11rn1bl1einG1'1bblJlJbbmm1d1iitJmWfl'.l.if1ll;J'1~1crfi@l~bb!'flil'N<im1'11buu
'lleN-;:;uu~,dbvlmh:::Tmru"lum-;m:inbblJu-;:::uu1Viu7m-; 'l!'bUUVi1L<Ji11fi'llJ'll1l'm-;tJ-;:;~rn"l1.if'l:::uu
mnfieitJ m'lGi'lilfiuh 1um1ri11'1Ulil1:::i1um'lLViu7m1<Jn~m.ifei,nufi1'l.if<i1tJ~<Jni'ilil~u 1.lmi ~1-
'l.ji<J1l'JLUn11'1'1'J'U;t,'il1:b~lJ~W'1llJ'iilU1U'll1l'0 VllJ7nl'l1'1:;1l'111:1-i!'fl:l-ll'l'1'll1l''l1:lJlJm·iVllJ7fll'l nu
~1'l.jf<J1tJ'lum111Jm1l'.!1'1~fo~1)llJbi'!tJl1lm!'f ;t,<J1:'11i1'1,i;'l1-;:::uubbm'1eitJil'ii1u1u0Viu7m1b~:1.J~u
1'1:;1l'l:::lJlJi'.l'111:1.Jb11'lum-;'lViu7m'lmn~u '11'l.ji~1tJn:1-1<J:::ilw,~iiu<i1''j!JmT1N.i1:1-1~bbM1'mbbii'1'lu
m"I'<~ 6.3 ;t,~"llil~l'1lil'll1l'~1'l.ji<J1m1:1-1 ~u7m'l1'1~ei~~ilihu1<J'lum1'1lilfiuhcr1m'l11ri11'1ulil'ii1u1u
W'l.Viu7m'l1'1~eim1:1-1cr1m1ii'll1l,'l1:lJlJ~b1'1m:::cr:1-1~crlil1'1. '
.i1:1-1~n'111bbii'1'lm41l,~ 6.1.2 11'11bburnbm'11ltJil1'1ml'.l~tlbblJlJ b-dei~<J1n1l,.1tJ1rn1lu~h''l
'll1l''l:;urnm1'1mJ'l!,tJ1rn1lu.1'1l'.lnzj:1.JtJ1:::·1nn1 bb'11'11ll'J bb<i:::0Viu7m'lil1'1ml'.ltJ-;rn1'l 'lm4m~
6.1.3 d<J:::bb!'flil,m'l'ib'1"1:::,.::;<ii1bblJlJbbm'11ll'J~u~u!l1u ~l'.ltl'l:::'ll1m1.J'ii1n1ii'ii1u1u bb<i:::buu1u.i1:1-1
crm.;@l!Jmm1:1-111i,ilm'lbb<Jmb<J'bblJUU1'll1l' ;t,biJu~tlbblJlJfll'lbb<Jnbb<J,~itl'.!:1-1numn lliltJrnn'lViu7m'l
ilm'lbb<Jmb'il,bburniini°Ttlbuub'lll'.l<i L.lbbrl
1. Gi'1bblJrnbm'11lmbuu'liil~'lViu7m'l'Yi1l'b$1m m1:1-1m1bbll1'11ll'J1.J'ii1nlilbbm~iltJ1:::'l11m1.J'ii1nlil
'ii1u1u 1'1~1l<ii1bblJlJbbl1'1ill'JbblJlJ (M/M/1) (FIFO/oo/oo)
2. o/11bburnbmflilmbuu'liil''lViu7m'l'Yi1l'b$m m1:1-1m1bbmflill'J'ii1nlilbb<i:::tJ1:::'ll1m1.Jil1nlil
iJ1u1u 1'1~1l<ii1bblJlJbbll1flill'JbblJlJ (M/M/1) (FIFO/n /oo)
3. <ii1bburnbm'11ltJ~l'.l'li1l''lViu7m1mnn11 1 til' m1:1-1m1bbmf11ltJ1.J'ii1nlil •rn:::tJ1:;'Ji1m
hiiJ1nlil'iilU1U1'1~1Jo/11bblJlJbbl1flill'JbblJlJ (MIMIS) (FIFQ/oo/oo)

rv ' 6l V -<::: ' ..=:t :tJI 'o rv """-1
1. ll1!!1J1J!!t11fl01'.l1!1J1J'lfO'l!'111.J'.i!ll'.i'lHM!llffJ fl11i1£J111!tl1fl01'1 !il111fllll!ll~il1J'.i~'lf1fl'.i
iiill1nV1'illtt1'U tt1oil1111J1J!!t11flell'.ll!ll1J (MIM/1) (FIFO/oo/oo)
lihlbuulbmmitJ'.i'.IJlbuudbuulihlburnbmfleJmbuu'liei.:irn.:ib~tn%u~mnu~AlubVlu1#1Yi11:U. ~
'lu%'l<11'.IJ1:;.j1-Ju i1:i-J:i.i~jl'U'llei.:i<i'f1bbuuLLmfleitJi:Ubbuud:U1rneirnihtJ
1.1 :LJ1:;'ll1mloii11uu:;m-:;1<?1tJi'.hh,vi1,.,:;vd1'b1m'lum-:;boii11uu:;m.,"""n"""'bbuuban.fl:U-
•um%tm
1.2 nm'lum-:;'lViu"lm1l'lm1LL•mlb"J'lburnlurn5n.fl:Ub'U'Ub%tJ'I
1.3 1~U1Jbbll1fleJtJbbUU'li€MYll,U"inl'ib~l'J1'Yl~€1ch'ULVfu:;m;b~m %u<11eim~m
1.4 'lViu"lm1<111:1-J~1ilumrieiu'lViu"lm1rieiu
1.5 n~:i-J:LJ1:;'ll1ml'J.j1u1u1~<i1n<?1
1.6 .bbll1fl!ltJtJ11l1i"l~.j1fi"1
1.7 .ilm1b>1~tJ'llei.:im1boii11uu"lm1l'lfi1umm11il<?1'llb>1~tJ'lleJ'm1'lVi'u"lm1 (A<µ)
<i'f1bbuu~u31u~'1!l"1fl!i'ei.:ii1:1-1:1-1~31uoii1.:illui.J1meiu#11tJ 'lj"1'1:1-Jm1 s '1:1-1m1 ~.:ibl'iufi1'1i'i~'llei.:i
' .A Q v ._, ""-
bbl1fl!JtJ'Jllillfl'lJ "1.:Jib
(1) m1:i.iui<i:;•l'Ju~<i:;1~l'lN'1uu"lm1mJ'lu1:;uu ~.:i'YlmCJti,NlViu"lm111,,1u• • •
A.
Po = 1 - -
µ
(2) fl1l:J.JUl<i:;•lJu~l'JN'1uu"lrn1 n f!ueit.i'lu1:;uu• •
P = Po+ P1 + P2 + ··· + P~ n ~ n
(4) .j1u1u<jnii'11<?1m>1~tJ~ll1.J1u1:;uu
A.
L =
µ-A.
(5) .j1u1u<jnii'11<?1m>1~tJ~ei~l'Ubbm.-ieit1
L =q µ(µ.A)

gznL;gL;tngriLµn11.nLNJJ;llvl·v
9znL;gL;tngriLµn11.mNJJ;1tv1-s
vz11,L;vLL;11,},lflLlltbl1.tbL~lJ11,lv·z 'li.,,.',..,,
8z11,L;8L!ftbgflL!-'Jtbl1.tbLNJ);lvl·1
rl'{tll!trmLtlllrtl!lnu@!!mLLt
"'tr;~Ji>"
~/1hCll'lJCll11~LVi~jptbQ1JPliltblil,LlJ~~L.!ftLgflL!-'J
nll,'l'LLWJ;Mlt~lJ~(lfM.LLJ~rtx1v,llJU.lil,,l(l~l.!ftLgriq.mrM.LLJrtrt~l.'l'Ll,r.i.ngllljpl.LLJ'l'LUWl./1111'.,J!>
llllll~rll~Jlo1,lf<rlg1)il1,ll'~(',~~11,l·~l.!ftLg
riqm11,l.!ftLgmµn11,mrnJ;1M~L~:ir.i,rnl.!ftLg~J!>Ull.Lb'~rt~n!!'~lJ!-tl~ClJ!-Ulr.~ttle,~/Vilrtoo1mltt:ll.J1
Lf<tl~'l'LQ1Ill~l.!ftLgrilfltbll.Cl~mClL;Cll.ltilmg1~ClJtlll.rt)1'nLWJ;blvlCl[$1flClt.<Cll.tLLJniA.n;prn1l)1'/Vilrt09
'''
~l~lf<llk-JP'l'Lt.<~ti!'l11l.!f'l'LgriLµn11.nlWJ;bMrn1e,~1ltilml,lll.tbJtvJillriLµn11,utLm~n0vl.!ftLg
rll!-l'l'Ll/,'l'LLWJ;blvl)!o1,l!}Cl~'l'L~'l'Lt-)lll.lil,,lltllil'l'Lwtt~1!,LLJ~rt)i,1,l.llJll.lil,,l)ii1,ll'~r.l.!f'l'LgrlL!-l'l'Ll/,l.!ftLgrlL!-l'l'LI!.
nLWJ;bM'l'LlnLJ?tt~1L!}tiri&mn111]Ju~1mm~r.11ur.11i,LLJ($l.!ftLgriqm11,i,Lunui,µltil1~ti11~/ln11,ui,rn11,
lttl.lLJ~l)IL~l/l,~Cll/,fllr.:lWl.lLJl~~tttt~rn1l)1'1ltilml,rlll.l'll!.'l'LJt8Jilll.!fngmµn11,ui.lttl8:lr.e,~1m~'t1
''
rllill'!'Lt.<8llWlr.rnl.!f'l'LgtL11.l.!ftLgrll!-l'l'Ll!.'l',lWJ;bM!:J:li'll.!f'l',g~jpLJ1µrn1('.,*Cll''l',)tz'1Ullt~l.J1rl~'t1fllill
~J!>ulµL.!fng~)lltt:lr.~/ln11,ui,/11,Cl!'10Vi1:l!.J1~Ll'/1~f.llil~Cl~m~m@lllrne,m1.Vi,J;l~rt~Cll!.l.!fng~J!>UlµlllJ1
lll~J1lm.wi,1;1gu1l.!fng~f.ll'1!.l.LU!:J~µ"~fi11i,1;1gu1l.LLJ~lVitir111,~uCJCl~f.lVi£l~f1z·9~~L~Bt~
rl
'{
=d
m~ll'l.LLJ~rt)ilWµrnunlf.lli>J?Cl~l/I,(d)l.LLJ~rt)il1.nlf.l~Cl11.m~z.,,.~.LLJLl.lil,,l(8)
£Z-9_.111.!,LLJ'f'LLIJEN.3~L,c~,!-Lu:::l!lnfi13~lunnn11l.J!1
('{-rl)rlb
--=M
'{
'{-rl
l
rl@0@!,µfl~'t1fllilll'll1(l)
=M
rtrt:ii,ni.,l'rnµri~'tmllllL'll1(9)

m1fi1mmV11<11'1n~.!1~'1 11111.J'lim.Jm1~m~nci11J.J1bbi1'1 "l::bbM1'111UbbU1m1~mn,4il,
~ 6.1.3 ~i'Vl1u'l·1m'1u~b'Vlliil'JJEJ'<ii'1EJ1.i1,<f,nlium1<ii'"1i1uh'lubd'llil,1i'U'YJU~"l::bn"1~U~"l:;iluu1m•
'11''1 nuL.it1oJ'i'.lV1u1~<ii'"1i1Uh"l:;<il"1i1UL"lbliilnm1ri1'VIU"1rl"111LVi'u1m'l~LVi'1i'u'YJU~1~')"1
'Vlli'mnm'l1'lum'lbliilnYl1'bliiln $il bliilnYll' bliiln~S11i'uvm'l1;;.iL"1m%~t1~1~61.i 1i'mm. ' '
1h:;nmJ.11t1 iilu'l'Jum1'llurht1i1uG"l1 bbfl::iilu'YJUnl'lbilm1m'JJil,~urn1wiiu'l11
ri1'VIU"1LVl E(T
0
) = fl1'11"1'V11' (Expected Value) 'll€M1i'U'YJUL"11'J11J.J
' .., ~ d .... ..... ' ,,!!
E(W
0
) = mm"1'Vl1~'JJil,<11U'l'JUb61m1m'JJm'V'lUrn1u'lll.J'l!1<11il'1U
E(S
0
) = fi1'11"1'Vl1,'JJil,iifU'YJULUm"l'JJUrlll'Ji1UG"i1.JiJ$u
L"11'J~ E(T ) = E(W ) + E(S )
c c c ' '/.I
m1fi1mmV11.11m.i'Vl1','JJil,iifU'YJUL.im1J.J'llil,'Vrnbli1mYi 1fi1u1m1.1il,i(
1. <i1u1mm.11m.iV11'~'JJei1iiiu'YJUbilm1m'JJil'~urn1wiiu111.!EJ$u E(W
1) iilu'YJUnldbilti
2-<x , 'l.I c ,
nm'JJil,Y«UrnTu-iiu111u'JJUil~nunmVi'l.tlu•:;uunci11$iil 'li1,nm<ii'1bb'1wi1uG"l1m~1lnil1"Jum:;1'11
'JJuri1t1i1uG"l1bbi1'1b61~"l nm'"~ti~EJl'J"lu1:;uu<i1u1m1"1'11n~<111
~ ..:
w11'111bfliln1'1 1: =
~ ..:
1'111bfliln1'1 2: w2 =
~ ~
w31'11"fliln1'1 3: =
~ ..:
1'111bfliln1'1 4 : w4 =
w =
1
3-2
1
4-2
1
5-2
1
6-2
=
=
=
=
1
µ-A.
1~11;;.i~
112 ~1LJ.J1'1111il 30 mfl
113 ~1l;;.i1V11EJ 20 mYi
114 ~i1J.J1'1111il 15 mYi
iiimmm1lill'Jnfl1'JJmv;urn1u-iiu111<1iEJ$iu~uil1.lnmilu1u111 nm~ilti'lu'l:ouu bbfl::'11bb&1' . .
'llil1'V'<Urn1u-iiu111~-l11,1uil~ 'lu~d"l::b'fon~1t1'111'11bilm1m

li1'Yl1U'Yll~b!ie:Jn~ 1 "Jll'llUff~mft~iiu.'11~1n~~b'>~!.I 2 ~U/~1ti-1~ ~~,Xu 1unm 1 $'Ju (8
~11i-1~) <i:;il"J11ml'i-:iiiu.'11~1n~~m~!.I 2x8=16 ~u/~1ti-1, "Jl'llUfl~bblii!1:;~u<i:;bilm1m 1 ~1li-J' (W)
• ' 1
ilu~'bbliimli,<ium:;n,'lluri1!.liiu.'11bb{{1'"~' 1wih''1m~~rn111vrum1u'1iu"l114i'fl~bftl'I'1mii~d'Ju
i,Ji1fi1m1miiEJ'Yl1rn..h:;i-11rn 100 m'Yl/~1ti-1, ~,,Xu fi1q;i1~'Yl-'i'~'lleJ,4fUYJUnl"Jbil!.lnm'llil,.,u'1Jrn11mm"Jll
'1iu1rn1>l~~d
=
'' ..... '""* di.al
mfll~'Yl1,'ll!l,mb5rn1m'llil~'Yll""fln'YI 1
= if11mm"Jm x ~iu1u~1t:1-1~ x nmLu"J:;uu x fi1bft!.l'Yll!.llii!l~1Ti-1,
= 2 x 8 x 1 x 100 lJl'YI = 1,600 1Jl'YI
2. '11u1rn'Yl1fi1q;i1~'Yl1'~'ll!l,4fu'Ylum"J'llUff~liu.'111iie:JmTii1,1un:;nm~$iu (8 ~1ti-1~) E(S )
' c1
'11u1rn11>1~,d
fi1{)1,'V'lilm1u'lluci1!.liiu.'111 '1ULUm"J'Yi1,1unmm,$iu = 50 m'YI x s ~11i-1, = 400 Ul'YI
li1'Yl1um~b~iln~ 1 fi1{)1~'V'liln:nu'lluci1!.11lu.'11 3 '1ULUm"J'Y11,1u'li1~nmnm~$iu = 400 lJl'YI
x 3 l'IU = 1,200 lJl'YI
3. '11u1rn'Yl1fi11'11~'Yl1~'llfl~4i'u'Ylu1~!.l"J1i-J'll!l~'Yl1~b~fln~ 1 E(T ) ~~d
' c
E(Tc1) = E(Wc1) + E(Scl)
= 1,600 + 1,200 = 2,800 lJl'YI
1u'Yl1umb~mnu <i:;'11u1rn4i'UYJUnl"J"Jilfl!l!.lbbi1:;4i'uYJUm"J'l1Uci1!.l1iu.'11li1'Yl1u'Yl1~'~!ln5u'l
11>1~~;ll"Jl~~ 6.2
m~1li0ft
1~~'Uftl1
h'u~m1
1. µ = 3
2. µ = 4
3. µ = 5
4. µ = 6
''U'lltll!l
3
4
5
6
d
!llll!l
~eg1u
1~'U'U
1
W=--
µ-A
1 'll.:1-1.
1/2 'll.:1-1.
113 'll.:1-1.
1/4 'll.:1-1.
' ..1'101'1'1!
SA= 16 A'u
16
16
16
16
' ~
i'lll'llff!11~
' d
'll!Mi'll!ll'a!1lll
9i01'lu1u
ii'vm 100
'lll'll/~11m
E(WJ
1,600
800
533.33
400
''U0~ftl1'U'1!1Jl!l
9i0flu 1uii'i'l11
400 'Ul'll/i'l'I!
E(Tc)
1,200
1,600
2,000
2,400
v
'110~i'l'l!'IJ'I!
1vmnu
E(Tc) =
E(WJ +E(Sc)
2,800
2,533.33
2,800

"J1n«11'n~~ 6.2 "l~L~u11m'iLliilnm~Li:1ein~ 2 ~ei .n'v;urn1u'11uti1tJi1uff1 4 '1u h'W"Jl:;LVI
<ifu'Ylu'i1:1<L'1m%~tJ.iei~u~1'1'1 m'i1.n'v;urnm'11uli1tJi1uff1 4 '1u ~~d'iu'11mi1~mm:;'1:i<~'1'1bbi1~
' ' '
1l"J~1'1tJi1~Gj'1 uem"l1ndmi'1 •~mti'itJrnYitJu"J:;1'111~'Yl1~hliein~ 1 nlJ'Yll~hliein~ 4 bb.11 bb~11,,~1vr
<ifu'Ylu'i1mrhnu 2,soo m'YIL'Yhnu bh"1'Yl1,Lliein~ 4 "J:;Li1m1m1u.,~uuueitJn11 "1~iiu'Yl1~Lliein~ 4
'
~~~tl'i:;i1'Yl'ilmv;~n11'Yl1~bliein~ 1
"J1rn:;"1u1Vfu'im'iLLi1:;q;1m'11'1l~'11!l~<iiu11uLiltJ'i1:i<,,.~tJ~~1u1ru1itlui11'i1'~ 6.2 mm'ir;i
lt1:1<1L:iitJun"Jl'l'lbb'1"1~m1:1<i':i<Yiutl.~"1~mv;~ 6.5
•"1U11U'lil'1iltJ
o~-----~-------- c;:;liiumw'im'i
2. Ill'l!!'IJ1J!ltl'll'l0!JI!'IJ'IJoH0~'l-tt'IJ~Ill'i,/M !~rl'l l'l'll lJrJl'l!!ii'lfl0riiJl ftVl!!l.l::'l.h:::'lll fl11iiiJl ftVI
0 "' v
'ill'!fl'l.J 'tt'i0Vl'l!!'IJ'IJ!!ii'lfl0!JI!'IJ'IJ (M!M/1) (FIFOin/oo)
'1:i<:i<~jm'11ei~<if1LblJlJLLmmi tJ'lllbbJJud b1'1~eiunu'1:1<:1<~31u'11ei~<if1bblJlJLLm'1i'ltJLhuu'liei~'Yll~
u'im'iL~m"1~n~11mbb.111i1~<ifu11n1iei tJm'1u1iei'1:1<:1<~31uL~mnrnm'1'1l!l~bbm'1!ltJ ~~<if1LLuu1urnm
dLLm'1iltJ"l~lJn"51ni1~1rndi'l~ ri1i'~'111:1<i1lm'ir;i1um'i1Vlu'im'i1'1'ieim1:1<LijLli:i<h'1lil~ijn.1'1~"~~ei~
Li1rn1m'iil'1!ltJ 1ii'l'ii1ni1h1'16'i1~mm'i:;'Ylu"1eim1uu1"J:;Lliu'1l€1,'ii1muiJnff11u"J:;uu m1:1<m11'1tJb%~tJ
'11il"hm'1i'lrnh"1~'11m'i:;uu nm'i!l'1i'ltJLi1rn%~tJ1uLLm'1i'ltJ bb"1:;nmei~lu'i~lJLi1rn%~tJ
ll11'1'U'1LVf~lU1'Uljnfl1~!l~l'U'i~lJrnLm'1i'ltJ~L.fuJLnu M '1U 'lj'1'1:1<m'i'11!l~<i11bblJlJLLm'1!ltJ
• •a 9il "-'A
'i:!lJlJ'U"J:;b1JUi1~U

=
A.
1- -
µ
[
A.Jm+11- -
µ
~ po [A.µ]nP = 611111-l'u n ~ Mn
..1.l-11l n = 0, 1, 2, ..., M
(3) m1:1-1il1,~1llu~il~ni'1ueil'.ln11V1~mvhfiu k f!uei i:,i1m-:;uu
L
A.
µ
= - - -
A.
1- -
µ
Lq = L + P0
- 1
(6) nm1'1rn'1.~l'.l~eitj1umrmeil'.l
wq
Lq
w = w 1
q µ
(M + 1)[~]M+l
l-[~]M+l
(8) el'1"llfl1111f~1u'llei~ohu1Viu':im1 (pl 111~eifi'1Gi1unm~~Viu':imd'Y11~1u
A.
p =
µ

i;i'·;rnclH~ 6.3 Call Center bbvi1V1i1'.1l'.l~n.'111'Yl"S?f'l1YiL-ii1m1'1m>l~tJ~'lLi<111~ 10 <iu ll1m1Jei-
b"Sb'llffl 1 <iu ~1L'1m»~tJ"l~L.jfnm1h~mrn 5 mYl<iiB~n#i'1w1immu vnn Call Center iltjl'l1tJ~1V1i<'1
5 tjl'lltJ nfi11¥iB 6'11m-.nLVfll~n#i'1"SBl'l1u1#i' 4 tjl'l1u V11nl.jf<iw~1 5 tjl'llmb~'l ~n#i'1~L'Yl"l b-ii1m
111i1-J'J1nd<J~Ll-il'l1m1n1oif11~uu1#1 BtJlfl'Yl"SlUilLBml'l~LmtlmJL'lld<J~1l-ii1,dJmvhL'1 bb'1~LBml'!~
"n.111'Yl.,.mbL~'l1-ii11~uu1i.i1'11'Yi1nub'Y111.,
•
<J1nm1.-i1u1ru.'llll-l"S~1J1JbLm<iBmb1J1J (M/M/1) (FIFO/n/co) "l~L#i'
P
0
= 0.25
pl = 0.21
p3 = 0.17
p4 = 0.12
3. ft'111um101A0rJ~i'i'li0~1'ti'u'-im'l'i11flfl11 1 '110~ A1iimn1101A0£Ji3ii11nl'l 11l,'l:;1.h:;'lf1fl'l'
i3ii11nl'l~lt!11-l'l'l10ft'111U1J!!01flflfJ!!lJjJ (MIMIS) (FIFO/co/co)
.,~urnbm<iBu~ll'liB1LVi'u'im'l'mnni1 1 'liEi,~'11rn~unumn~iJ'1L#i'bbri.,~uu~'1'liEJ'1vr
u'im"S'll,J'llmunu bW1rnm<iilullbil1mLLmb~tJ'l "Snuuum"S1Vfu'imlbU'UbbUuL<i"SmriEiu1'11uu'im1
riEJu 'liil'1Vfu'imlbL<iiii~'liB'"~l'.15ll11nT:;1Vfu'im"Sb'Y11nuL'1ul'.im"SLL"lnbb"l-J'll'lll-!Ul"l~bUULLuub~u1nu
lULLJlJm1i-11n,JLU'Ubb1JlJ~l-ILL'1~bU'Ui5ft"S~<iiEJn'U L'11,Jl'l-'1,JJ~1JlJLbll'l'11l[JbblJUd'J~bU'Uil,JJ11'11~ 6.2 ('1)
Lub~il'~ 6.1.1 l'll-ll-lo/li;;m'llEJ,Jiil1LLurn1m<iilu111mu'l!EJ,m1iuvi'u•nmfi1lu~'V1l-l'1b'Vlil1lunul'll-ll-lo/li;;1u
" "' "
'llmil1LLuu'liEJ,'Yll,Jb~tJ'l~uvi'uJ'1ubbuu1l-i<l1n'1m1l-Im1'll1J,Jbbtn'llEJuL'1uil'l!'1l'll-lm"l<ii1-J'I il,d
(1) m1l-IU1'J~L1iu~"~1l-ill~1uu'im11l~Lu1~uu
(A.iµ)n
n!
p
n (A./µ)n
s!sn-s
=
1
s-1 (/1./µ)n (A./µ)s
2: - - + - - - -
n=o n! s1(1 - A.ISµ)
..:
P
0
, bl-IEJ n:;:; S
..P0, bl-lil n > S

(4)
(5)
(6)
(7)
p =
A.
Sµ
'iii1u1ui,jn.'111.-im;;i,~1.1~ei~LUbb11'1iltJ
L
po (/.,,/µ)Sp
=
8!(1 - p)2q
'iii1u1ucin.'111.-im;;i,~1.1~1lt.11w:mJu
• •
A.
L = Lq+ -
µ
b1m1.-imu~tJ~1l~1umnmw
L
w =
_g_
q A.
rnn1.-im;;i,~1.1~ei~1u-;~uu
1
w = w +q µ
;l10aN~ 6.4 m-11m'ibbvi~'Vlrt~~ei~m-;n1111uwi'iiim1u'llei~1Vlu"im-;biJn'ii1m'.Ju'11V1.lfui,jn.'111u
'iii1u1u~b111m~fti-1 btiEJ~'lln'Yll~l!Ulfil'il'lu11.1mtJ~il~m11Vim1i-ift~mm1.-ib~1Lum1u"im1i,jn~ilbUU
ii1firu Jr~i:iu m1i-1biltJ1111tJ~bn.-i•1nm11il'11l8'll1l~'1n.'11~.-ibUUi-1ci'11 1oo m'Yl/411m 1.-itJ~eJ'il'll. . .
'11~1~'!Nurn1ubvi1nu 75 m'Y11411m m1m1'1iu"im1'll1l~i.Jn.'11bliu11ieit.i1~rji-1bb'1~bliu!i&11~'1ieinu t.-i1.1
bu~tm'11'l~l'1'1nf'i1m1'1iu"im1 40 'il8/~11i-i~ el''ll11m11Vlu"im1bllmbumiln.fllibum:fitJ'1 '!Nurn1u141
•
nmli1~mrn 2mfi1um-;LVlu'lm-;'1nf'i1w1i'1~1ltJ N'u'l'V111'l11l~l!Ulm1bbvi~dl'JufomtJ'l1'1nf'i1•~~1l~• • •
1i-h1l'1iltJUlmfiu111n11 1 u1fi TIUlm'i~~~1l~f1•11rn1'l1•~<if.-i'iii1u1u'll1l~1Vlu"im1'iii1u1mvi1Lwi~~·~

rdei~'>nn5<i111m-;-m1'~ A.= 40 'll!J/~11:w~ il~1mmr-J15il'llm'l1Vfu7m'l µ = 30 'llW~11:w~
(:w1'J1n 60 ul'fl + 2 uiri/'1u) 1umrudt'f1~111'lViil'V'liirnmu1m'lb'IAit.1~'1m~m'"~himm'lnu7nn
C1nfi'11Jf11u ;;'mnit~b'liudbbm'18t.1'J~m1b~:w~m~•m'l bbiid(;]u~N'u71111'lilufornu11C1n.i'1him'l
' ' .'l8'1<iuuiubfiu1i.Jn11 1 uiri mTI111'ii1mu'li0~'lViu7m1'J~Ji'0~i'.imnn11 1 'limbbUUEJU w1i'ii1mu'li0~
LVfu7m1m1J'.ib'Yi1l111~~bVl:wl~"l:W~fl91 bUUb~El~~~lJ7Vll"l'J~<ifil-1fl'J11rn1b'V'l"ll~'J~b~m~il~fllJ<ifWIJU~
~ }; v
'il~bni11'lf'Ui111U
11:w:w~1Vi111',i;0nrv1ilm'lqf01l1uhil#f,d
"m'bliiln'YI 1 ~111LVfii'liil~LVf1J7m'l 2'liil' (S = 2)
'Yll,bliiln~ 2 ~011Vli'.i'liil'LVf1J7m1 3'liil~ (S = 3)
m"liiln~ 3 ~011Vli'.i'liil'1Vlu'Jm1 4'lieN (S = 4)
111;;'rnnrn'l1~1flum1qf1111uh lii0 b!iEJmrnb!iEJn~iinm1um11EJ'1EJUl111m11~u~1~ri111 bbi1~
bUU'Yll~b!iEln~LVf'11'111111111,onEJ,<ifUYJU~lfli11 <ifu11w11il''l~lJuil 2 'lfU111 Wiil <ifu11um11EJ'1ilmbi1~<ifu11u
m11Vlu1m'l briil'ii1u1u'liil'1Vlu7m"lb~:w~u <ilu11um1'lil'18U'J~C1111m bbi;i<ilu11um'l1Vfu7m'l'J~
b~:w~u <ilu11u1111m1:wmm1n'11mrn1Jr#r,d
MinimizeE(T) = E(Wc) + E(Sc)
E(Sc) liiEJ ~1'11011111'-:ionm<ilu11um11Vlu7m'l
E(Wc) '11u1rn'J1n~1b1lu1111u1um'l"lEJ'1il!J/~11m x eJil"llm'lmn-:i x nmil~LUbbf.Yl'1ilU
1111m'Yi1nu 100A.wq
E(Sc) '11u1rn'J1n5<ii'll'11~1-:i'V'liirn1U/'1u x 'ii1u1u'liEJ-:iLVf1J7m'l1111m'Yi1nu 758
briilbb'Ylu'11LU~il'li;i1-:i'l ~n!i11mbbil'1'J~1J1'1111l'i~#l-:iiiJ1'l1-:i~ 6.3

~ ' """"'m ~rn!lnftlil'!Jl'I S=2 S=3 S=4
Po 0.200 0.254 0.262
Lq 1.067 •m 0.145 m1 0.015 •m
L 2.400 !lU 1.478 !lU 1.348 !lU
Wq 0.0267 'JJlJ. 0.0036 'lllJ. 0.0004 'lllJ.
vi'lei i.602 u1Yi 111'iei 0.216 uiYi
A A
wm 0.024 Ul'YI
w 0.0600 'JJlJ. 0.0369 'lljj, 0.0337 'lljj,
111'iei 3.6 u1Yi 111'lei 2.214 uiYi 111'lei 2.022 uiYi
'J1n4iB:1-1ri'11i1fl@iLu<i11"ll~~ 6.3 i1lm"i111-i1mm<i1•%~1.11iiB-ii'1t:1-1~'llil~'11~u111um"i"i<lfl<l!J ~u'Ylu
. ' '
m"i1Viu'lm"i ""ri~~u11u11:1-1 ii''"""!l~LU<ill'i'l~~ 6.4
fi11'1101tt'l~fi!lii'1Tm ' . ~
Alftl.itt'l~fi0ii'1Tm'ilM
m~1li!l111u111'li1m1.i
'11lWl'f'llml'U
• A!l~1T:i.i~ •
r 1ii ""'
'IJMl'l'll'IJ'Ufl1'l'l!l'1!JO l'l'll'IJ'lln"
'If!)~ til.J'lfll'l
E(Wc)=lOOAWq E(SJ = 75S E(T) = E(Wcl + E(Scl
100(40)(.0267)
75(2) = 150 1Jl1'1S=2 256.8 Ul'YI
= 106.8 1Jl111
100(40)(.0036)
8=3 75(3) = 225 1J1'YI 123941Jl1'1 I= 14.41Jl111
100(40)(.0006) 301.6 1Jl'YI
8=4 75(4) = 300 1J1''I
= 1.6 1Jl1'1
ii'~ti:u t'i1'1i1'l1'i'rnH1nuforn1.1~ ~u'l1111"in1111u!l11'111i-J<ifo~nn1Vlrim'i'nBu1mnu 1 u1Yi
• •
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"1l'll'Vrni11u Operations Research rf1111 m.h~hihn~ 4i1bb1J1Jbb!l1'11ltJ~fl'111m,i/~Vl~11 "J~t.b111
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11-;~Sluu'lJVllLum~li~iiiib'liud 1111~<J<Jn1111~111~~11ii<J 1i'b111"1tt"1m~161il~Slmum-;rn 1i'lm1d,~~161<J~
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1.z ilqJV11~<if1l~m"lbbil'1:Wm:J.J1"ln'l-ii<if1bb:1J:1Jlilrulil.rl<11i1.fl<if b'liu 111bbUrnbmlil1ltJ~l'.l&i'mtn•~
1m'lrnllublil~EJ'li1tJ
1.3 fll'i"'Ylillfl!l~ n:IJ"l~:IJ:IJVl"J!l n"l~:IJ1Ufll"l'1~ l'.i<ifUYJU~~ b'liU nn'Ylillfl1l~l-if">~:IJl.Jblil"JeJ'lil tJ
- ''11l:J.J~1blil1l"l
1.4 m"l'Ylillfl1l~nu"l~uu111ieJm~u1um':l,1~ m''l-ifnmu11-1,fiuff)1~'~1i11ltJNfl&i'~t1.<if b'liu
m"l$lf11'r1Nflm~'Yl:IJ'lJiNl111~lflnfou'lu1lU11i1"14il~VIUl ,~<ifm'l-iib1fll'UlUU:IJUn11,~b~UNfl&i'~u
1.5 m'i"'Ylillfl1l~nu':l~:1J:1JV1~1lm~u1um"l,1~m,l'.lm1i-iaj~mn b'liu rii1b6uu~u.11f!~mliEJ~l'.l
m"l~~~1lVlflltJ~ii!u (multiple stocking points) VlieJl'.lm':lbn:IJ'luvimdnil~~uf11
v 0 11,.1 qg.o Q .J
2. 'UO'illfWl'UO~fll':i !'llfll':i'illtlO~tl'tllllfll':itll
r:~"~nTru~~rn111i'm"l'ii1fleJ~<1mum':lm,~il1bdtJ'lJi1111mm.hrn1"l bbiilnl'.l-ifo'ii1nlilrn~1h~m"l
2.1 m"l'l-iinn'ii1<1ei~;;11;nunY1nl:J.JlbbflDqJVll'il~'lVil'i1"11l:1J~1:iJ'l'lil'i1<i11l:IJbVlm~fl:J.J~')ill (non-
optimum solution) bYm~m':l'ii1<1md'lVibb!ilbv:iml'i1"11l:IJ~bVlm~<li-ibbfl~UlY<!lhbri1i:iu bbfl~l'il<i11l:IJ~1<if
1ll'il!ieJ111:i.Jiilil'lll-1b~m"1•~'Yll~lilrn<i1'11MI~ b'V<'jl~l'il<i11l:IJ~1<ifm'ilbU~tJUbbUfl~111 b~1l'¥hfll'jbbflDqJVll
• • J
<if1tJm':l'ii1<1eJ~<1mum"lruVimtJ'l lil1'~ l'i1'11l:1Jbb'1fl~lil1~,~1:i.J,r11nu•Miil1~'1nm'i"'l-ii111bb:1J:1Jlilru"1m<i"1~
5u'l b'liu 111bbuum.,11hbbm:J.J,'ii~b<lu1:i.J11'~"nilqJV11~lil~~n"11i-in'~1<ifl'i1"11l:1J,vii-11~~'lill,,,hffml1:J.J1l
2.2 m"l1<ifm~~m':l'ii1<1eJ~<imum':lru~@!,:(U'il:;<ifeJ~'l-iinmbbfl:;'11'l-il''ii1tJ<;l~
2.3 m':l'ii1;;w~<1mum':lru1-il'l<ifnuilqJVllb%~1~LfJUL~El~'l111 ilqJV11~~'V1m~num':l'l-il'm'i"'ii1fl!l~
6' 1" .91 .J d 1" v 1' ' 6' '"' 6'.d
<imum•ru'il~l'l!l~ buuilqJVll'YI mm'lJ!l~num1i-1 :J.JbbUU!lU'lJEJ,ll•1ni)m"lru Vl•!lbVl<iJfll"lru'Yl<11i-11w
•1u':l1i-1.ifm~mb<i~'lViNfl'Yll'l1tWi1!11
2.4 m':l'ii1<1il~<imum':lll1Vil'i1"1eJumu'l<if<imum':lru!il1~'1 ~~u1'V11':lmm"lnu11111i•~':ilmb<i~
bl.i~tJ:1JbYitJm1'iil111111m1nT;ru~'V1m~<1:J.J~l1ill bb.i1:i.J111i-1Tm'lVibbu1V11,V1im1<1tJ'YITI~u11l.i;;fam1m':lru~' ' .<ifEJ,m'i"1<if

~" 0 '
3. 'IJ01J!'IJll'IJ0'1fll':i !'llfll':i'il1i:l0'1trtil 'UIll'HU
m"l"51'1fl'f!ll1.lm"lcl1Jnl-i1mH'lum"lm'ii:l11JV11.!1''1 1<>i'l. 2 il'n1'tru:;~fl Ul1J'Vl1rf111l'Yl1'
1'YlmmMr~mtJm''1 u61:;i:ll]JV11rf1111~'1'1rn~u'lu*1'il'l~'
3.1 i:lruV11rf11um~1'Ylmii11'1<il1mtJm'l'11iiluri m"l~n1'!'1m"lLL'l'1'l'rn:;'J11.J'llil~il.lil1'1 m"l~n1'!'1
m"lbfi~il.I~'llil;fl'lt111'1'lu"j:;mlJ m"lLbn~il<i:i.Jm"lL'll'L~u i:ll]J'VllL'Vlrhdmm"l11V11.i1i;ifl~11ii1'11tJ'loiim"l
"51'1fl~'Yll~f1ru'ilm'1'il1 LL'111lm"lJi~nfi11m'l'loiinmmuLL61:;~uL1llia~<i1'loii'ii1tJ'1~ ":;,:(u~~vru1u'loiim"l
•
'ii1'1a~'1mum"lill ,:;'lVl.i1'ilfl1J~i1'11liltJll"l:;mruL'Yi1i:iu LL.InmbL'1:;<i1'loii'ii1tJ'lum"j'111"11'ila1JuatJn-:i1
m"i'loiiJ11bblJlJfiru'ililfl'il1
3.z i:ll]JV11rf1111~'1'1rn~u'lu*1i;i"~~~'11m"lrnLni:l11J1·n11ii1iltJ'loiim"j-;j1"a~<imum"lcl1JJmi
3.2.1 m"l'ii1'1a~f!mum"lrni:ll]J'V11mw~'Vll"l!J'il'11'Vlrn"l:i.J L'Jiu m"lilflrnbuurn:;u1um"l
'Yll'Lfii'l "l:;uumu');i.;llufi'1'1'b'Vli'lei mwilrnLlJlJ"l:;uuLLmfieitJ m"lailrnLlJlJ"l:;uu'll.1'1' Lliu<ilu
3.2.2 m"j-;j161ei,'1mum1cl'llfl'i:l11JV11m'1l"'n'brn:;b!fWjf11f!'il.J L'liu m1~n1'!'1'¥111~rn"l:1-1
'llfl'~~i'i;i LL'1:;~u~Lnf1 m1~n1'!'1'1m'l'1m"lLL'li~-iiu'llil~'ilmil m"l~n1'!'1 ~'1n1:;'Yl1J'lleNm"l'1il '11 L'.l.I
d'Ju4fa
3.2.3 m"l<ill'1fl~ '1mum"lrn'll<i~i:l11JV11 'Yl1~ '1'1bJ~n"l"lmb'1:;i'~ fi:J-J L'liu m"l~n1'!'1nn
bU~tJ.ILLU61~'ll<i~'11u1ull"l:;'ll1rn'll<i~L'1n m"l~n1'1'1'1'1r]~rn"i:J-J'll<J~il.lL.l'll.llJ'VI Lliu<ilu
3.2.4 m"l'ii161<i~f!mum"irn"l:;uu*1LL'l'1ri!°i b'liu m"l~n1'!'1"l:;uum"l'Vl:i.Jm'l tJ.l'llfl~ll1~'il
m"i'ii1'1fl~m"lti1~1.l'llil~f!:J-Jfl~ m<;~n1'!'11:;V1'11~m"lti1~1.l'llil~i'.lmbl1:;Lrf1 bUulifu
3.2.5 m"i'ii1'1il~'1mum"lrn'i:um"l'Yil'1~il"l1:J-J
' t ' •
1 d ' .::I. .::I. ""' ""
t!!!tl1fll1ffil'lflttt!11'Jfl 6 1'10'1J'l'I 6.2 !10~'1'1 6.2.1
'

'lum-:;'l1im1'111iiil16'm1.m1·mf 'Yi1m1V1'1iiil1<ii'1 bblJUlil tJfll"'l~a.J<il1mh1bYlil'lVllJi<il1bii'll<i11
V1rt1,1nVl1n'iium1bb>Jmb>J1m1'1filii <i11rn'llJ11ni111ll~1il~"l:;V1-:l11 o bbii:; 1 (L:iJna.J~1 o bbii:; 1) <J:;
b'itJn<i11bii'lldi1 <ii'1bii'llfll-J (random number) 'lurn1'1~1>J~b'itJn1um"J~LJim~1<ii'1bii'llria.Jtti1m"lria.J
' ' '
<ii'18Lll1bblJ1Jl-J8'1.llii'11.flii (Monte Carlo Sampling) 1ll<Jfll'll1LJl11 bVl'1U'1nl"J~a.J<ii'1ilLll1bblJ1Jl-J8'1.llii'11.flii
1:iJ'l'lidlum"liilnmfll"l'illii816'fll'l.lfll"Jlli"i.'I.I<111:ii'I.lb8'1bbwi bU'UbVI'1U'11'111m1mil"J:;bum~1flru<IIfll'1<111~
'li11'.J'lum"l'il1iiil1'1mum"lrn
nn'l1im1'111ii€1111mum"lrn il"l'"b~a.J1fua.JH1nm"J~'l.l'1l1 bbii:;1'JtJ"lnu'1li"'lmtj1:;V1111
'11'1"'lla.Jliin<i~1~ 2 l'1tJiin1{1'.J 2 rll'l.l~il 18'1.l U'lbb:J.J'l.llf (Von Neumann) bbii:;~mu (Ulan) 11u1{1'.J
Jf1nl'l11 blium1'111ii81'1!11'1.lmmb~l'.J1rllJ'lllJ1'1.lfll"JVIii€1j.J<if1'llil1il1¥11'1U1l'l"Jil'l.l VIfi1>J1n,X'I.I~n1:iJ'I.Il'l.l
m1'l1im'1u'1m"'l~:J.J<ii'18t.J11bblJU:J.Jil'illiim.flii1Ji'llmtJil1ililfl1t1'lu1'1n111mn~u
VIfin m1~uj1u'llil'I bVI'1U'1m"l~a.J<ii'1il u1'1 bbUU:J.Jilulii mftiim<J1n m"lfi1 mWJm1rn'll81m1
Vl:J.J'l.l'llil11'1fiil"'lbiil'1Yi (roulette wheels) bbiil;fll"lbli'l.lLY<1Utiil'l.lfll"JY<Uu~ll:J)ilnil'Ihn L'l.lfll"JVll-J'l.111fiil
' . '
bbwim:fl~'l<J:;nil'lVibfiil<il1bii'll~a.JV1rl1fl1wi8Lli<J:;bbl1il1'lVibi'.iufi,m"J'li<il..ii'll~:J.J'lum"l'il1iitJ1'1mum"Jrn
ilu1,~11'.J
9'1ri~H~ 6.s m"l'il1ii8111mum1clei1.1.,1f;'l1il1~u.11 eit1.,1f;'llrn'ifb8uwiilfiliil1v1~~1u.'11il1"V1u
bb'li'IV1ii1llfim1-m:;bll'l.l<ii'1bbli"J~a.J~L:iJwimii81 ~1ilfll"Jbb<Jmb>J1'111:J.Ju1>J:;bllu il1d
u ,. uu ,1J 11~1w101r i;im • i'l u ,m1mrn1~1 'IJ'll~~'l tr~fl
x p (x)
10 0.2
11 0.4
12 0.2
13 0.1
14 0.1
<J1nWJ1"Jl11i11iiiu P (x) UVl'l.lml:J.JUl<J:;dlu~<J:;i'.leili111f;'ll8'l'ifb8ubri1nu x b'14Nwiilfiliil1v1
1<11111.l-:;:;6',fi'll il'lm"l'il1~il'l!1mum"lrnuuu:J.J8'1.lliim.flii~8 m"li'l~11il1 bbli"lrl:J.J ('lu~;l~8 €111!1'1f;'ll €1'1
' ' '
'ifb8uwiilfillm v1) Lil1'.Jfll-:;~a.J<1118 u11>J1nfll"lbb<Jnbb<J'Im1:J.Jul>J:;bU'l.l ei11'1'1f;'ll€1'l'ifb8uwi€1fillil1v1mm"lt1
~x>...J , .fi l ~v 'l .J
'1"ll'l'll'l.ILill'.J'1l-J'lnm1wilmb<J'lml:J.J'l.ll>J:;blJ'l.l P (x) ilLifll"lVllj'l.111ffil il'lbbl1il1 W11Y<VI 6.6

40%
x ~ 11
bri•:n~ii'eJ'VI1.j1lilaj~ri1uL1l'llil,,,iei rl'l~Llii'1JliA,.l'lliJ'~bilw1ieiifli"11Vi (biu nl'VI1.j11luri1u~nu
~u~ 40% n'l~'lViri1'll8' x bri1nu 11 bl'l~m'iieiifli"11Vi dJu<i!u) brimi1m"lV1:J-lm'ti'8V1mLl'l l'l~'n'l~Llil
l'l'lll-J~A~AJ.J'llei~1)11'1~.I'llei~~bilu'iieiAli111Vl ~~'l~:i1l1'nm>b~'lnii'bflmm1bb'lmb'l'm1J.Jul'l~bUU P(x)
m~mum1611:f1,ri1'llil~ x 'l1nm1b~1mri1m1bb'lmb'l,m1J.Ju1'lnuu P(x) bUu/11eitJ1,V1rt~'llil~b'Vll'lUl'l
m'i'rjl-J/f1ilLll~bbUUl-JiJU@i1'11-fli'I lU'Vll~lii)iJ@ib"lleJl'l'l~1~<i(il'rllnl"l'Vll-JU1~ii'eJ'l~~'l bb'ii'l~l'1f'11dl~bi'l'll
rji-;bmu ~~"~11iini111'lu111.16'1~b51.1"1'iiei1u
~ ,, .vr:i~r:n:mi::;'.U1l!fllit!:IJ
b'Vll'lUl'lnl"l'loif/11611l'1~m~u1um1rjl-J (a random process generator) 'l~'loii'Y1d.ifum1bb'lmb'l~
A~1J.J, F(x) 'llei~m1bb'lmb'l,fi1u1'l~bliu, P(x) 'lum1A'l'1~ri1/11bbU"lrjJ.J, x '11J.1/11iJtJ1,~bbii'1Y1~n'1ium1
bb'lnLb'l~!l~f.'l-J, F(x) bb!l11~8ajhw1111~~ 6.5

iJ t1 !II a
~ <r~ll'll0~~Ja'U ll'JU!Ul'il~IU'U fll1!!'ilnl!'lH
l'ialf'u"ni 'l10~~u<r~fi' <r~<r!J
x P(x) F(x)
10 0.2 0.2
11 0.4 0.6
12 0.2 0.8
13 0.1 0.9
14 0.1 1.0
m-,,,,rn,,Ni1:;1'f:i.11Jnn1'lllu'11w1h' (O, 1) bb'1:;bmum1:i.iu1'1:;bliu~~l.li1,,),:;:iJfi1rvhnu 'lll~il
um.1n--)1 x (b'liu F(12) = P(x ,,; 12) = 0.8) m11~ F(x) bbi1'1,1Ull1'1'<~ 6.71Vift',brn1111m1:i.im1'llil'
b1i'u'1-,,1ubbu1#,1ubblii'1~~u ,:;:iJfi1•vi1nufi1 P(xl 'llil,'11 x bblii'1~fi-, b'liu ~~u~'ij°'1bftU'1.,,1ubbu1#,
~:!1'11 x mnn-11 14 ,:;:iJ'i'111:i.im1•~:i.iii'u~ o.9 bb'1111i~uij°'1~ 1.00 'li1'~'ni111i1il'1'i'lii'il'num1:i.i
u1'1::•liu'llil,~1ii1,f1'llil'~'~mvi1nu 14 b'i'l~El' P(xl = 0.101mi1uil'b~tnnu•l1u'1"'1mm1#,~il ~b'lllUil
'11x•vi1nu13 ,:;i1~Yi'Elufi1m1:i.iu1'1::•limvi1nu 01 (o e - o.8 = 01) bb01::tu'Y11uil''~mnu m1:i.iu1
'1:;bliu P(x) '11'llli'u x = 12 '1:;'11111<1i''11rni1u'1-;,tubbwl#,~:ilm1:i.im1b~:J.1ii'u~ o.s 11i~Uij°'1~ 0.8 'lll~il
P(x) = 0.20
F(x)
1.0
0.9
;i; 0.8
"'" 0.7
"'"7
<;; 0.6
c= 0.5
~,.. 0.4r
c::
0.3
0.2
0.1
10 11 12 13 14 x

.,:;,Xu m1"'rnb,.:ii1:;'1:wn;;i.:i x '~1h~nmJef'1m'lil11'llil.:i'li1.:i"11.:i'l ~i1D111'1liD.:ir'ilJflUl'!.:J.1w1i11~
>i1tl1ilb1'lll(i1u1uV11'.i.:i r il.-i1D•i1:;V1-J1.:i o bb1~ 1 nni1'1'1.:i~u1'1tJn1~1J1um1fi1J bbli1'Yi1m1'111'11 r d111n1J 'lJ q 'lJ
€1~'lu'li1.:i'l'1lUbbnU~.:J'llil.:Jm'W~ 6.7 mfi,~'YmlJ11qiJM'1 (x) lJUbbnUU€1ULJi b'liu r = 0.76 111m1~'lu
• • d
'li1.:i o.60 ,; F(x) ,; o.so ':;1Ji11 '11 x il.i1b'Yhn1J 12 dJuefiu il.:iiiu1'1tJm1bii1ilrnihm'llrj1J~um<l'1viu.:i
m'l:;mm1!lV11'11'11€l'11lUl'!.:J.1 (x) lJUbbnUUi!JULefi
m~1num1i1'1'1.:i<ll1bbu1rj1J1'1tJ'l-ii•'Yl<F1U<f1;.J€Juii!<F1111:11n1:;'Yi11Jref'1 tJiJ€l bb"1Lu'Yll.:ill~liiii~n,:;
'loil'"D;.1Vi1b111d'lu4Ubb1n'lli!l.:Jm~1num1'~.;rD.:J •11;;intihb1'llrj1J~umriDu 'lum11(i111D.:ii1mum'iclef'1tJ
t "" ' """ 2.1 <v ""' ii o6l al!'
U'ibbn'i!J<f1D;.J'W1bl11B'l'~;.im'ii1'il.:Jl1Jl'i1.:J"11b1'llrj!JbblJ1Jb'YltJ;.J (pseudorandom numbers) 'JIU bbl11 bU'YlU
'ii~'loilm'i1.:i<l'1b1'llrj;.i~u'iln1Jm.flun1'1Nu1rnm.:iviil1tJd (1111'i1.:J~ n-2) 1111'i1.:i<l'1b1'llrj;.iil.:imh1iJ'irnmJ
cJl1m11'lJ 2 viKn il.11~.:ibb"1 oo il.:i 99 L111tJ11~"l'1'YlflutJ;.ivii1<ll1b11'll11 '~'i1u1efl11<ll1b11'lJ<il1.:i'l bvii11dil.-i1
bU~tJUbbU1.:J1u111ml'1nm
'lilm'i'l-ifo/11m'llrj;.i 'llfll1ll'il.:J'lun1'1Ni.nn'Yi11Ji'VlmtJ'iil bb<il'l~bbi1'1.:J'lVi•i1u•V1m'lilvii'.i.:iil.:id
d v ,.) v l a~ ¥} '[ ~d ,.) Ji
(1) rnDnmt'1'll"JIJl111bb1n '1'1 m111 m1111'il.:i b"1tJ11lfll'i"JIJ'JJUIJ1
(2) 'l1n1111'il.:i'lVibl1Dn<ll1•11'llil'11umh.:iilmJrnbNUm'ibi:1Bn~ri1V1u11111'6bilimu
fll'ibl1Dno/11L1'llrj1J<l'1 bb'in m'll~IJ1iu1mml'i'VIl11J1111 bbli1'loii'U1~11.:i1Ju111111.:itii1b1'llrj1J b~D1Jr
o/11t1'llfl!Jo/11"'inm6bli1 'lnii1i!lno/11t1'llfl;.Jil'1111L'1tJri11'm1116blJ1JbtNWlli!l.:Jfl1'itii1€lrnbuui!Ju t'liu tii1i!ln
. '
tii1il'11u~D~'luw:rnWimnu<il€l1ut~i!ltJ'l 'luvi;.i1116bm bbli1t~;.icfi'ml1i!ln<ll1L1'll'lub6rnil1111u vi~m~i!l1,'i
tii1t<>'llrj1J<ll?•.t'lnm6bli1'lVit~i!lno/11b<>'llil1111u,1ni11111JntWimnu1ub~i!ltJ'l 'uV1;.J?1'11111Jniiu 6bli1~ucfi'u
l'!'11Jnil'1111 m1ti:1€lntli'1t11'llil'11u'l1n111111.:i<l'1t11'llrj1J1;Jm1m~l111'1-ii11J<ll1t11'JJ'l111tii1 b1'lJV11'.i.:it'W'il~m'
'il~ll i'l,'lo/11b1'll~1;J'l'li'Jj'1<ll1b1'JJ€l1.i1.:ibbYl'iJ~.:J
vl10VH~ 6.6 fll'>'loii<ll1b1'llrj;.J'lum11(i11il.:Ji1!11Ufll">clqui1.:i.l'l1€l.:J~b8U l'!;.J;.iii!111i'€l.:Jfll'il(il1€l.:J
l'!mum1cl€lui1.:i'1'llD.:i¢1t8mliunm 15 fill'11~ '1n11ll'il.:Jtii1 b1'llfi1Jri1V1u111<l'1 b1'11ft!J o/11b6'irnlJu 39q 'lJ q q
bti1'1tii1Bn<ll1b11'llrj;.i<l'1il1111u'lui1'11JntWimnu-ommlJ 15 l(i1u1u 'il~'lVi.-i1tii1 b61l1rj1J (quM'1'lJD.:J~L1lu) ~
'1€liWlli€l.:JrlU il.:116'1"1.:JL.1111111.:J~ 6.6

....." cl.:t1:! O!lll'fl r ~ilmt'i (x)
1 39 11
2 73 12
3 72 12
4 75 12
5 37 11
6 02 10
7 87 13
8 98 14
9 10 10
10 47 11
11 93 14
12 21 11
13 95 14
14 97 14
15 69 12
<iil'il~~ 6.6 bbMl~Nl1m">~ll1B~i.'1mum">ru <il1'1~.-'1'll11~~b~u<il11fil1'11~b1Ju-,:;1J:;nmmu 15
fii.Jm~ 111li.'1~.-'m~1J<i111fil1'11~ill'l1bri1n'IJ 12.06 bfl4B~ (181/15 = 12 06J Nl1ii'wi5~1~•nnm">~ll1B~
'
i.'lmummd m'ilu111lbtl'hJ'IJL'i11J'IJn'IJfi1fl1'1'1111~~'11u1rn•nnw~n-Wum-,bL'ilnbb'il~m1~ii1'ilnllu
X.
l
E(x)
n
E(x) = LPX
i=1 1 1
= (0 2)(10) + (04)(10) + (0.2)(12) + (0 1)(13) + (0 1)(14)
.:
= 11.5 bfl'iB'1

eilior~.l<iieiillim111'1i?lm;::i,~mvhnu 12 06 ,.,~ei~ ~d'luN'1'1V<TI'il1nm1'51'1B,'1mum1clilfil
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<iieinu (interaction) 'IJB,i111bli1flm111<i1d 'il~l'.iN'1LVf4iei,'Y11rn1V1"1'1il,mn'1~'~u:Ylei1VfN'1llV<TI'IJeJ,rn1
'
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" ~
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m1ri1V1uil1VlliruY.JUbD'Ubmll'Y'l"lum"l1(i)N'1~'1'1TI m11nini;i1mwrim1~.:i~e:il'!ufi'1briilfi.:i'l(i)~.J~il 11si:;
ni;imrn-Yib~tni'i'u1i1111.l1wi1.:i'l 1'1lu eiuii.:i.l '1l1.:inm1ill'lu.11n.:il:iei b1si:;~u.11i11e:i.:i (buffer stock)
nwiill '1e:i.:il'!mum"lrnriilmrn1um"l~-o;i:;'l'lm m :i.Jtini±l'V'ffliiin''i"l:i.Jiil .J'l 'iiil.:im:;u1um1V1"!01:;uu'lu
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rn1'Y11.:i1u'lJil.:im:;u1um"lVl"le:i1:;uu~ri1~.:itlni±18~1~e:i81.:ioif(i)b"Ju m.,-<i1u.:ii'i.:iil.,-1n1'YIm "11.:i'l 11:l-Ji'i.:i
m:i'.i(jji1un'lle:i.:i~1m1:;Vi1m<fl111uu~~,:;l'!1m1n'lViieJ:i,isi~ilJrnJ;-:;btJ'liu"l~
3. m·rn0fl!!'Uum"i'flAfl0<i
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bb'ilnbb"J.Jf111:l-J~ bbsi:;m1a.J'1-i1"J:;blJu4i'.Jbbli91.Jlulil1"1.Jii.J<ii.Jd ~' (x) i'1~1b>1~mvhi'u 4. 50
fll'UM (x)
.;
' 'J ' 'J'111lJll '11llJ'lllll~! jj fl11lJ'll11l~! 'llll'~ll'IJ
2 10 0.10 0.10
3 15 0.15 0.25
4 30 0.30 0.55
5 35 0.35 0.90
6 10 0.10 1.00
100

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fiu<i'1'll11?1l'.im'Vi1nu 100 rn'YlliieJffu (C ) m'llL'lrnb"J~m1:!-lu11>Jt>Uu'llei~eJucr~.'i'lJeJ~-1JmmuliieJ'1u1?11Vi• s •
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t .,,. v r ~ d:I.
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(ililvi1ii y)
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~i.liml P(x)
0.2
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0.2
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01
' i'lfl111Jlll'il~l 'II
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• •
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LL;n$iei 00 LLfl'lL~ein<if'lbfl'llrjl-l!.'f11LULU'111:J.JnL~l'J'lflUliieivlu~eil'l'1 ·lwi'hm.i11d<J::;'l11 rl LL'YIU<if'lLl'l'llrj:J.J
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d
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1
2
3
4
5
6
7
8
9
10
20
6
0
25
7
30
55
119
104
88
0
0
40
0
40
40
80
0
0
0
20
6
40
25
47
70
135
119
104
88
00
49
41
94
87
33
67
44
63
25
14
15
15
18
17
15
16
15
16
15
ii-ufl'1
fl~~M~0
..Jm~
6
0
25
7
30
55
119
104
88
73
46
80
61
89
47
(y)
2
3
3
3
2
+c
'
+cs
150 30 0
~T
'
180
150
150
150
150
0 900 1050
125
35
150
275
595
520
440
365
0
0
0
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0
0
0
0
275
185
300
275
595
520
440
365

m-;'l111m<!mum-;ru'l11l1uqJmd1h:;miuJhl'J 4 ~wmm <ii1d
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fiu.11'1~bViflilb:I~l'JwJ1lfl'Liill'lll"cifbYi1fi'u 418.50 lJl'YI L,I,'Yl11U,jif~,;(,l,'l:;lif1l1'Yilf1l'l'Y1'1i1!J1'llU1U!.llf1
!II g, !II ' 1' '
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'11b'llli'lei11eii1llil1~1Yi1nu 418.50 lJl'YI d,llubv:lm.111l1'1m.h1ri1111d11ri1ifm-w'llnllum-;'Yl'1i'l1l1~ri1111u'1
61 V ::: 2' IV 6J 21 ~ IS'<=! 6' JI. G V
b'lll Q = 40 bbl'l:; R = 30 1<:;uumlllei1m'lb'lllf1l'l'lli'l1l1fl'mum-;rnmn11Jfl'i.Jl,l'lrnmn'llun•rnei1
bU~l'J'Ubb'L1111.;1 Q bbi1:; R i.V!l:l.;1"111'1 nuei1lnlllbbi11bliein'11 Q bbl'l:; R ~lVlNi1'l1l-l'll1l1<iiu'l'lufiu.11
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• •
bi'leini.i'm-;'11111l1fl'mum1cl-;:;1Jusu.11f11b1111'ei ~11>1-;1:;~bYim bbiiiileiufi11111'1 ~<ii1l1m-;1111ll (b'liu
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'11mliei~~1~fl'ilmi.Vi' Nl'li'i'WTI'llnl'Li'lbbrni.i AB: OM bbfl'il~1l!.li.wi1-w~ 6.9
' .

Simulation/Inventory Simulation
***Input Data***
Ordering Cost (Baht/order)
Holding Cost (Baht/unit)
Shortage Cost (Baht/unit)
Business Days (-/year)
Min order 0 (units)
Max order 0 (units)
Min R (units)
Max R (units)
Initial inventory (units)
Demand Prob. Cumulative Prob.
14.00 0.2 0.2
15.00 0.4 0.6
16.00 0.2 0.8
17.00 0.1 0.9
18.00 0.1 1.0
***Program Output***
150.00
5.00
100.00
365.00
30.00
35.00
30.00
35.00
20.00
Lead Time
2.00
3.00
4.00
Minimum Annual Total Inventory Cost : 3,924,699.00
Minimum Order Quantity
Minimum Reorder Point
***End of Output***
: 35
: 31
~ ~ , ~-·mvrn 6.9 N1ltl'ftHll1fl!lJ1!!111lJ AB: QM
Prob. Cumulative Prob.
0.6 0.6
0.3 0.9
0.1 1.0
N111'i-w£lwi1-w~ 6.9 d'JuN<i'J1nm1<(1111mrrmum1oo(i']rni1111u(i']r1-nei~ o tc11::: R bblllrnh~fiu
bv:iffV11.fiu'ljui1u.i'1f!~b111liei11:1-1'11:1-1<if1ei!.i1~dn1111u(i']11-1' o :ilri1ei~lu'li1~ 30 bbi1~ 35111tl1ti '"'~r1 Rei~
1u'li1~ 30 bbi1~ 35 111tl1ti 'Jt:L'1Ni1'111i<ii~d
<fiu'ljui1u.i'1f!~l111~ei11:1-1~1<j•wi0tlbvi1nu 3,924,699.oo rnVI
mmrurn-;~~~8~1'1llbvllfi1J 35 fl'.•
!.11mru"l(i']~~~ei~1'l(i']bvi1nu 31 l"iu

i;l1!ldH~ 6.8 m'l'il1ii1J~rimum"lru"l::urntmfl1l1.I 1lmfl1"ltbvi~111rt~1Vfu"lm"l1Jrn·'f1ttuuoif1J"m
t1f11JJ1.jfu"Jm'l (a drive in teller window) ~,;;111m'l'll1l~!i'Ulfll'l"1~ni111ifi1l~m'lflmn"ll:lJlJf11'lLVllJ"im'l
l1?11.1'11u~i'l~'l::1.1::nm"l1lfl1l1.l'll1l~1Jn'11 ttii~1i111'llm'l1Vfu"lm'l'll1l~V<iim1ul!U1f11.,,;'liil~1Vfu"lm'l'li1~nm
'l::11111~mw1l'~'ll1l~1Jn'11tt.iii::flUbb'1~"l~1.1~nii11Vfu"im"ltbr111~1<11<1111.1m'lbb<Jnbt"l~m1uu1<J~dJu"1~d
' ' ~
' 11 ' 11 ' ~
''IJ1~!11.ll1~'1'11l~fll1l.ll~~ fl1ll.l'lllll~! u fl1ll.IUHI~! '1-11.1~1.ll.I '111~'1JMVl1!1.l'IJ'!l.I
x p (x) F(x) rl
1 0.1 0.1 oo- 09
2 0.3 0.4 10- 39
3 0.3 0.7 40- 69
4 0.2 0.9 70- 89
5 0.1 1.0 go- gg
~ v ~
' 11 R11l.lu11J~1111-11.1~1.1" ' ~
'1~V~!11.ll 'l'ilJ1fll1 fl1ll.l'llll~! 'I-I '111~'110~1'11!1.l'IJ'!l.I
y P(y) F(y) r2
2 0.4 0.4 oo- 39
3 0.4 0.8 40--+ 79
4 0.2 1.0 so- 99
'11m'll'J'l-1'1/"1bb"in (r
1
) "~1.jftbmii1~')111t!luo/11t~l-1Wfutbli'1e:i1u1i1l'i'1111u1umrn~irn'u1u1'1w1111
b~1l1.1'1 "l~1.'f111tii'll~l-1'11tb"lrnliu 77 i11111w'11bii'll~l-l'l/<11~rim (r
2
l "~1.jjri111l-ln~111n<J1n.jf1rn11111u111'11bii'll
~l-lb~l-JWl'U~bbm'11~')"1bbli'1e:i1uo/11!'i'1111u1'Ui1"1l-lllb~mnu1um~1il~1J'U~1l1.1'1 "Jl:1.'flihb1'/J~l-1'11bb'lrn!iu
•..., ;; • If 0 ..... ii.> """" ""
28 N'1'1'i11l'll1l~m"l"J1iiil~r!mum'lrui11111"l1Jljflfll 20 flu l-1"1~'U

• fi'11ti'Ut{3-l
~""'
,;1.:Jl1i:ll r~n:n !1i:l1 'j:i:£1::::11in ':f::::U::::t1Q1 iii'11•~<i" 'l::::u::::11-a1 11tn 1::::a::::rJa1
~
'i::::1rl1"n11 • Q • . . 1if11"in1"i mn1il '16.:JijftA'l
""" '1 3.JlU:J t".ilJ~t.I ';!"6"Uf).:J 'Jl't"l'WfHll'h!
'2
u1ti.:i (x) 'Ujfll'j •mnm 0ti-.h.:i (y) 1m:1111
1 77 4 4 4 omi1 4 u1i1 28 2 mi1 6 2 mi1
2 94 5 9 9 0 3 24 2 11 2
3 30 2 11 11 0 0 00 2 13 2
4 05 1 12 13 1 0 92 4 17 5
5 39 2 14 17 3 0 07 2 19 5
6 28 2 16 19 3 0 18 2 21 5
7 10 2 18 21 3 0 93 4 25 7
8 99 5 23 25 2 0 99 4 29 6
9 00 1 24 29 5 0 87 4 33 9
10 27 2 26 33 7 0 33 2 35 9
11 12 2 28 35 7 0 69 3 38 10
12 73 4 32 38 6 0 97 4 42 10
13 73 4 36 42 6 0 28 2 44 8
14 99 5 41 44 3 0 39 2 46 5
15 12 2 43 46 3 0 87 4 50 7
16 49 3 46 50 4 0 06 2 52 6
17 99 5 51 52 1 0 26 2 54 3
18 57 3 54 54 0 0 97 4 58 4
19 94 5 59 59 0 1 79 3 62 3
20 82 4 63 63 0 1 79 3 66 3
54 9 57 111
• •V G ff 2JiJ<=IVd
'llWl11Jum-;>J11"ei~ffmum-;ru-,:;uubbm.-ieiU'lll~l'IU:W 11~u
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ii~bilu 4 u1i1 ,.~tiui,Jmi'1.-iu~V1tl~mii~.sim1m~ 4 (irufinlnumwmiiunm~~'~mm'hinumfi
b~:w111b4eiU'l) ljn.11.-iu~Vltl~>J~b4i1'fou'lm-;Vfuflbb1"~nmmwm'lleMm-;1Vfu'lm-;'1l~ri1V1u11.;1b~:w1lu
t2.1.d • ..., 2' ..., & 1n2'<=1 , u ~ .d ....
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eitJ11~nniruflnl'iilu 4 mi1
• •

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) 'l~141~1b.,'lltjlJbllu 28 ~~'l~'lVi''1T'i~tJ~nm1Viu1nTmi1nu 2
u1Yi bri€Ju1nT'i~uijoi"~ nm'll€J~mwmct'l1n1JJ'l~Qniiu'fin11ctn.,1b'Yi1nu 6 ~~bUUN.,mn'llEl~nm
b~lJ4iuu1m"i (mnct 4) nu"i~tJ~nm'lViu'im"i (2 u1n) "i~tJ~nm"i1lJct~nflw1uct'Vlrt~ct€J~lu"i~uu
ii1u1rn'l1rn~tJ~nm'l€Jf1€JtJ'll€J~~nfl1 (O mn) mrn~tJ~nmct1uu1m"i (2 mn) 'l~141Nm'il'1nb'Yi1nu
2mri
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ll~~lii!l~~lillluhfi€1 d€JuL'llb~lJ<iiu ""~~€Ju1-n~uijlil b~!luL'llb'Vlchd'l~iJN.,'li!lNm'il'1TI'll!l~m"i<llmM
ffmum'lrnlil1lJ~1€11.11~d m.;lJGi'lVib~BuL'lJb~lJliiu11•~uu!ltJ'luffm'111~•JJfi11iJiJ.,nfi1Bti'lu"i~urn"tJ• • •
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•
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Yi~~
'lu'Yi1u€1~<&itnnu ~1€1l'i1-:idn1V1ulil .~BuL'lJ~uffoi11111iJIii!l~~'l1•rn1nnmfin€J~"nfl1• •
tllil<Jlnfiuct 20 bllu<iiu1JJ bb'lim1<l1cimffmunl''irn'l~~u""ri€J~nfi1'1uct 20 €J!ln<J1n•~uu1JJ m"i
<l1C1€1-:Jffmum"irn'lJ€J~"i~1J1Jnl"i'lViu1nl''i'll€1-:J1lUlfll"ibbvi-:Jdvi1b.Wrn'lw1b~nmiJ"i~lJlnl 1 ~1w~ (~nfl1
liluct 20 <Jln1JJctnm 66 uTYi) ~~LlJbY1rnl'1€Jliieml''iLV1'"b'N.,i'il'1Tiy€1~'luffm1~'1-:i~1 <l1bllu<ii€1-:i'Yi1m•
'Yllilij8-:J'lVimn<i~~~u ~-:im<J'l.n1JJ"ibbn"ilJ'1€JlJ~'lblil€11i1tJ1um•'Yllil"1N14flil.,€Jlil1u'Yi1m"i (8 ~11:>1~)
Nijfil'1nct14f..,1nrn"i·rw1cimd'l~'lVifffiGictu1ffuL'l'VlmtJ.11 •iu .,,~tJ~nm"i€J<i€1m;:i,~tJ'lJ€J~cin#l1...ici~<iu•
ll'11b'Yi1nu 54120 V1~m'Yi1nu 2.7 uTYi 'lVl#-:imlil11'l~il~nfl1«~VllJlil 6 <iu <J1n«~'VllJlil 20 '1u1iJ14i<i'ltJ
nm"i!l<i€JmcitJ ""~ll~nfl114 fu <iil'l~bi'ltJnm"i€Jf1€1tJlubb11'1€11'.1 u€Jn'l1nd~-:iff1:>11"inff1JJi'in1'frn~m"i
"i€1'1€1tJ'll!l-:J~m'11Ji'loitJ'l-iflill"il~m"ibb'lrnb'l~mil.I~ li!~d
~~11~rrnn1lfl1lt1 .,; .,;
v Q
mnm fl 1m1111~1m
'll1l~~flfll (lHll)
0 6 6
1 2 8
2 1 9
3 5 14
4 1 15
5 1 16
6 2 18
7 2 20

<;nn.jfeJ:;,m1um11~.jf1~ifiu N~ilm'i"llUlfll'i"lbvi~dm'ilrnfri1ili111h~mrn-l'm.16'~ 50 'llil~'1nfi1
• • •
1fa11oil'nm'i"ilfliltJUmm1Tvi'%mrhi'1.1 2 mfi vi'%elilbi'.ilm-l'iltJ'1~ 20 'llil1jnfl11!el11oil'nm·:rnfliltJ~~w;i
6 mfi~u1u Uiln'il1nd~~i1m'i"'llil~llmfl1'i"bbvi~dm'ililm1:;,i<iuh•~rm1.111 1'eltJ1"~b'Yi1Lfl'llil~n;;i1~
QJ ' ' .... r: • r: llJ ' ...,
'V1UmlU'llil~llmfl1'i"'J~ilfJ11~ 'illnN'1'1'V11l'llel~nl'Nl<im<imum'i"rn'llil1~nm 20 flU 'V1l.111 'V1Urn1u
'J~il!.1111 9mfi •1nrnrnY1vi:;,ii1 57 mfi vi'%ilfiflblJu7mm~ 15.8 'llil~nm~~Vi:;.Jfl
•
'i"::tJ::nmLi1m<>1~1J~~nf11bb<il1"rnu'J~ifiil~1-ii'tu.,~1.11.1ilfi1b'ihi'1.1 111120 vi'%mrhill.I 5.5 mfi
ilu1~hn'11:;.i ~n1'11'ii1u1u 6 'flU ~1l-11fil~bl'lm1m'i"ilLUbbll1fliltJ •nl'lm1milrJ1U'l::1.11.11i1m<>t~mv:im
(2 + 2 + 2 + 4 + 3 + 3) vi'%eJ 16 mfi + 6 flu= 2.66 mfi LU'Yll~nii'1.1i'u <1nl'11'i11wm 14 flu~ifiil~•
bl'lm1m'i"ilLUbbll1flil1J •~1oil'nmel~LU'i"~l.ll.I1flm<>t~tJ (111-16 )V1'%el 95 mfl/14 flU = 6.7 mfi
mh11i1tJ<i,U iin/;in~1•ni1u11m'i"Loil'm'l'il1iimm11um'i"ruilonel~~bl'iui'li1'1'i"~~mm'lrwi1u1rn
'11fi'n~rn:;b<>1'V1l~'llil~m'i"'11bii,u~1u'llm'i"::1.11.11Wlvimnvimm11m'i"1i'li1bbl.l1.1'Yl11'flrni1'11<1'115w1
1'110aN~ 6.9 1.11ij'YINlii1b'fl4<M11itvJw1bbvi1vili~ 1!el~m'l1b'fl'i"1::v1m'l<i~'Ylu il1wi'l.lm'i"Nli'1
l'YIEJiu1vii.J ililn'il1V1UltJLUm'i"Nlii1l'YIEJiu1vii.Jd ifiel~1"~'YJULU<fi'U'YJUfl1~ 'i11u1u 1,~00,000 l.111'1 'i"l'fll
'i11vi'l.i1tJ'llil1"l'YIEJfl1'i"il'i"1fl11l-ibi'iu 400 m'Yl<ilm'fl4el~ 1.11miifiil~m'i"1Vf'i11u1um'i"'lll1JilU11~11ulJbb'i"n
'J~ifia~1l-1~1n11'li1.J:;.i'YJU (break-even point) 'llil~m'i"Ni!ii1 UllJVll'llil~l.11ij'Ylbbvi~dMiel m1:;,i1l-i
bb'l.iuelu'llel~<flu'YJUbbU'i"iitubb<i::'lll<i~fl"'llel~"l'YIEJiU1vil-1 bbNun"l~tJ.jfeli.i1""l1Jl11i'uY!UbbU'i"iiJmifi1iltj'l::vi11~
200 ii~ 220 m'YI bb'1~<!U<i~fl"'llel~"l'YIEJ1.i"'11.1m'i"flli1V1mtJ11•::~ueltji'uui,jfi1m'llil~~bb'li~-iiut'i11.11mi
.ibb'li~i1a1.1111Jlmm'i"i1moiYfufi LUUbb'i"n'ii1u1utJai1'llltJ'llel11.11mi,ntlu 4,000 b'fl4el~ 5,000 b'fl4el~
•
vi'%a 6,000 b'fl4il~ bb.it'i11.11mi~bb'li1l'l1Wfi1il1JL<filu'i"lbb'i"n flli1lltJili1'llltJ'J::blJu 6,ooo bfl4il~ 7,000 b'fl4el~
vi'%a s.ooo bfl4a~ ~~d u7mimi111ilm1:;,id'iu1u1JJ 60% ~1.11mi~bb'li~•::iluJ.jfi1mi1au1ifi'lulibb'i"n
'J1nh'Yl~N~oim'i"'llmu1mi1fil~m'i"'Yl'i"11Jf111:;.Jb~tJ''llel1nl":i<i~'YIULUm'i"Ni!ii11'YltJ~u1vii.Jd vi'%a'll q ~ ~
m•rnh15n1uiitJvitt~11 N'~oim'l1imnnm1um1m.l1'J~blJU~tJelfl'llltJ'llel~"l'YltJLUUbb":in •::i'ifi1•'Yi1nu
. '
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bbni:lllJvi11WI ''W":il~i:lllJmd i'ilif1bbU'i"VlmtJlif1""~"'.i"::lil1i'im1:;,i;;1:;,iYi'uili'u ueln•1nd lif1bbUW1,lil1i'i
'111i.Jbb'1-iuilu~ueltji'1.1fi1m1:;,i'l.i1'J~blJu mi~lii'1bbU":i<il1~'1 fii'i-iieii.iii'Yll~Wf1u<ifi~~bflml"lel.ielm":i'il1iiil1
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•1nm-;;'i11;;iel1<imum-;;ru1u~di'ia ri11-;; ~,m:;,i1.,mb1ub~tJum1'1'Jbb<iil~m1:;.i~'llel~tJili1'll1tJ~n"flfi:;.i
' '
'YIU ;;o::i:iu1"1111.J-;;::mfl"'llel~m":i'il1<1el~<imum'lrudi'ia m'i"V11Y11fii'ubb<ii)~m-;;bb'Jnbb'il~m1:;,i'l-i1•::blJu'llil~
' '
ri1h1u'i"lbb":in
<i:;,im'i"~U!JlU'llil~ri11~'il~LJi'J1nm'i"'il1vi'l.i1tJ"l'YIEJ i'iil
ri1h = (":il'fll'llltJ -1i'U'YJUbbU'i"NUl'iilVIU11J) x ('!um.I") - ii'U'YJU'fl~~

2.- d. ! E d l l ' V <>OJ"VV
lil'U'VJUm'VIb1"~'1<11l'llltJ<ll!JVIU1tJlJ<lllb'YllnlJ 1,000,000lJl'VI WA~ 400 lJl'VI <llllJfllillJ'11Vl'llJ'llU'VJU
v 6 - d , .d ~ J ~ .!' ~ "1 ~ ~v • 6
bb'lhNUbb'1~'llJ'1,'1'llm'l'Vl1,i lJ<lllblJ'1tJUbblJ'1' blJ 'l"'J~'1'1''ll'U bil biltJ b'll<111b'1'llrjlJ nl'l'Jl'1!l,'1ll1Unl'lni
1ul:lb1Jmd mm1m~l'JudJuiJ,,iu1.lli'"'"il'1'W11~~ 6.10
bliemi'hmJ<JrjlJ
bYliD'Vll1i'u11um.hi'.iu
•iiiDnif1bb1hrjrnYl!lm
u{imm11i''1!lu'llm~bb'li~
,11!lnlii1bb1J<JrjlJ•Ylavn
<ilJ1.'1,fi'lla~'i'Vl1,i
>J1nm~~ 6.1o m1'ii1'1!l,'1mum1clb~lJ1i'u.11tJn1'ib~!ln#1b'1'llrjlJ bv:l!lvi1~u11uu1J1i'.iu 'Vl.1''
'il1niiu1Vi•~!ln#1b'1'llrj:iJ#1~<l!l' ,v:J!lvi11Jt]n'imt~lil!llJ'll!l'u'i*-1~bb'1khilV11!l1iJ bb'1n~!ln~1b'1'llrji.J
#1~'1llJby:j!l'Vll'llJ'1,.i'll!l~'l'Vl1,i 'il1niiu1Vifi1mruri1hLliltJ1-ii'1:iJm'iri1l'i-iiwlfu m~~ 6.1o 1ViN'1'1'~rr
'i11nm1'ii1'1!l,'1mum1ci ~!l ri1hvi~'fi1 ,v:J!l1Vimm1nvi1.Y:bn-ifu m1u>Jmb'il~'ll<nri1l:i>J~.1!l,'1'i1m'i
'ii1m1,'1mum'icl~1muvimtJ'l '1~~ ,Ji11l1Vit-1N'1'1'~rrmmwrn~!l bbnm'ib~tJun111'1bb'1'1,m'ibb'Jmb'J'
'll!l,ri1h L'1tJvi111Jbb.1'1 m'i'l'i1m'l'Vlil'1!l,1iJ~1n11100 '1~'
lUn1'lrilVIUil.i1#1b'1'llrji.JlVfnuo/fU'VJUbblhNU tit]mml~n !llJ'JlnlJ'i*-1~bb'li~ bb'1~'lum.'1J!J'
1'VltJ51m1mi11>li',d ~ei 1umm'll!l~~u'VlubblJ'ii'.iu b~!l,'il1n1fu,,,ubb1J1r:iui1'11mh~vi11, 200 ii'~ q ·~ ~
220 lJl'VI tl1'1i.Ji.J&i ril'VIUil1Vi1i'u'VJUbblJ'lr:iuil.11dJu 200 1Jl'VI 205 m'VI 210 1Jl'VI 215 m'VI bb'1~ 220 1Jl'VI
~1tJ'11llJUl'J~bllmvil'l nu u~iiu 'J~'1lm'in1-iiili1b'1'llrjmwrnV1&1rn~m by:j!lhb'VIUfll1i'u11m•1i1i'.iu
bbl'i'1~'11#1"'"il~LWi11'1~~ 6.7

0-1
2-3
4-5
6-7
8-9
200
205
210
215
220
'11'!-j1''1Jl.J{jn~mlWlilB'IJ'lJB~'IJ~zjwiJ~ b~B~'>l1niim1mliu1JJ111 60% ri,::iii.J{jn1mTWfilB'IJ
'>!1n.-ibb'JJ~"":: 40% ri'IJ11:frifib•'ll~'il::hiiii.JnmmTWlilB'IJ ,.:;,Xumm1tiri1.ju.i<ii'1•'1'lJ'4i,J '11'!-j1'1Jl.Jnn1m'l! ~ ~ q 'W
TWlilil'IJ'>!1n~•b'll~1.i'<ii~"'".i~'lu111-n~ri 6.s
0-5
6-9
ftlj.j1''1J'lD0~.'lJB~1'YI~ b~il~'il1neiiJ0~.ii.-J1dJu 4,000 bfl4B~ 5,000 bfl4B~ j.j~iJ 6,000 bi14B~
d'.iBiim11WfilB'IJ'>lln'IJ1jj'Ylzjbb'll~ 6bt1::eiiJ0~.ii.-i1dJu 6,000 bi14B~ 7,000 bfl4B~ '}j~il s,ooo bfl4B~
b~BhJiiiJ{jn~m1Wfi1Bmnn'IJ1jj'YIzjbb'JJ~ i'l1ri1v11-1.i'lVi .-11m1i,JUl'iltblJu'lJ!l~'li.Jn-,clbb'fit1::"11u1uii
'11bvi1ffuj.ji,Ji1 ?iilii1Bm0Bti1~t1:: i'lum1bnoi mmm1n'loif<ii'1bt1'lJ~i,J 1 <ii'1 i!IB 1, 2bb'1:: 3bb'Ylueii.J0~fi
ri,::bnil~u'lubb'fit1::rnru <ii'~bb<lil~'lwm-n~ri 6.9
1 6,000 4,000
2 7,000 5,000
3 8,000 6,000

<J1niii1..uu~1"m~ff-i'1~~ubbl1:::iii1bl1'JJrji-i~n1viu<ii b"J1ff1i<T'i!.l~1"il~bVl'Jm"cl 1<iiu1-ifm.,
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vi~ei BASIC 11J1-11lu V11t1m'1'l-i{[JJ.,1mwrl11~'1~1.Jb~t1rn'l'ii1'1mi.'im1-1rn'lru~i'.J'l-iin1-1t1tJ1,bbY<~V1'11!.I biu
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Gould, F.G., G.D. Eppen, and C.P. Schmidt. Introductory Management Science. 4th ed. Englewood
Cliffs: Prentice - Hall, 1993.
Moore, L.J., S.M. Lee, and B. W. Taylor, III Management Science. 4th ed. Boston: Allyn and Bacon,
1993.
Render, Barry, Stair Jr., Ralph and Hanna, Michael E. Quantitative Analysis for Management.
8th ed. Upper Saddle River, New Jersey: Prentice-Hall, 2003.
Taylor Ill, Bernard W. Management Science. 6thed. Upper Saddle River, New Jersey: Prentice-
Hall, 1999.
Winston, Wayne L. Operations Research: Applications and Algorithms. 3'ded. Belmont, Wadsworth
Publishing Company, 1994.

rmH~ ft·l P(O) for Multichannel Posission/ Exponential Queuing Process: Probabilltity ofZero in System
R 2 3 • s 6 7 a 9 10 15
0.02 0.96079 0.94177 0.92312 0.90484 0.66692 0.86936 0.85215 063527 081873 0.74082
0.04 0.92308 0.86892 0.85215 0.81873 0 78663 0.75578 0.72615 0.69768 0.67032 0.54881
0.06 0.88679 0.83526 0.78663 0.74082 069768 0.65705 0.61878 0.58275 0.54881 0.40657
0.08 0.85185 0.78659 0.72615 0.67032 0.61878 0.57121 0.52729 0.48675 0.44983 0.30119
0.10 0.81816 0.74074 0.67031 0.60653 0.54881 0.49659 0..44933 0.40657 0.3'5766 0.22313
0.12 0.78571 0.69753 0.61876 0.54881 0.48675 0.43171 0.38289 0.33960 0.30119 0.16530
0.14 0.75439 0.65679 0.57116 0.49657 0.43171 0.37531 0.72628 0.28365 0.24860 0.12246
0.16 0.72414 0.61838 0.52720 0.44931 0.38289 0.32628 0.27804 0.23693 0.20190 0.09072
0.18 0.89492 0.58214 0.48660 0.40653 0.33959 0.28365 023693 0.19790 0.16530 006721
0.20 0.88667 054795 0.44910 0.36782 0.30118 0.24659 0.20189 0.16830 0.13534 0.04979
0.22 0.63934 051567 0.41445 0.33277 0.26711 0.21437 0.17204 0.13807 0.11080 0.03688
024 0.61200 0.48519 0.38244 0.30105 0.23688 0.18636 0.14660 0.11532 0.09072 0.02732
0.26 0.58730 0.45640 0.35284 0.27233 0:21007 Q.16200 0.12492 009632 0.07427 0.02024
0.28 0.56250 0.42918 0.32548 0.24633 0.18628 0.14082 0 10645 0.06045 0.06081 001500
0.30 0.53846 0.40346 0.30017 0.22277 0.16517 0 12241 0.09070 0.06720 0.04978 0.01111
0.32 0.51515 0.37913 0.27676 0.20144 0.14644 0.10639 0.07728 0.05612 0.0407B 0.00823
0.34 0.49254 0.35610 0.25510 0.18211 0.12981 0 09247 0.06584 0.04687 0.03337 0.00610
036 0.47059 0.33431 0.23505 0.16460 0.11505 0.08035 005609 0.03915 0.02732 0.00452
0.38 0.44928 0.31367 0.21649 0.14872 0.10195 0.06981 0.04778 0.03269 0.02236 0.00335
0.40 042857 0.29412 0.19929 0.13433 0.09032 006065 0.04069 0.02729 0.01830 0.00248
0.42 0.40845 0.27559 0.1B336 0.12128 0.07998 0.05267 0.03465 0.-02279 0.01498 0.00184
0.44 0.38889 0.25802 0. 16860 0.10944 0.07080 0.04573 0.02950 0.01902 0.01225 0.00136
0.46 0.38986 0.24135 0.15491 0.09870 0.06265 003968 0.02511 0.01587 0.01003 0.00101
0.48 035135 0.22554 0.14221 0.08895 0.05540 003442 002136 0.01324 0.00826 0.00075
0.50 033333 0.2"1053 013043 0.08010 0.04898 002984 0.018~6 0.01104 0.00871 0.00055
0.52 0.31579 0.19627 o. t 1951 0.07207 004323 0 02586 001544 0.00920 0.00546 0.00041
054 029870 0.18273 0.10936 0.06477 0 03814 002239 0.01311 0.00767 0.00446 0.00030
0.56 0.28205 0.16986 0.09994 005814 0.03362 0.01936 001113 o.ocis38 0.00356 0.00022
0.58 0.26582 0.15762 0.09119 0.05212 0.02959 0.01673 0.00943 0.00531 0.00298 0.00017
0.80 0.25000 0.14599 0.08306 0.04665 0.02601 0.01443 0.00799 0.00441 0.00243 0.00012
0.62 0.23457 0.13491 0.07550 0.0<1167 0.02282 0.01243 D.00675 0.00386 0.00198 o:OOOils
0.64 0.21951 0.12438 0.06847 0.03715 0.01999 001069 0.00570 0.00303 0.00161 0.00007
0.66 0.20482 0.11435 0.06194 0.03304 0.01746 0.00918 0.00480 0.00251 0.00131 0.00005
066 0.1904B 0.10479 0.05587 0.02930 0.01522 0.00786 0 00404 Oll0207 0.00106 0.00004
0.70 0.17647 0.09569 0.05021 0.02590 0.01322 0.00670 0 0033'5 0.00170 0.00085 0.00003
0.72 0.16279 0.08702 0.04495 0.02280 0.01144 0.00570 0.00283 0.00140 0.00069 0.00002
0.74 0.14943 0.0"7675 0.Q40'06 001999 Q,00986 0.00483 0.00235 -0.00114 000055 0.00001
0.76 0.13636 0.07087 0.00550 0.01743 0.00846 0.00407 000195 000093 000044 0.00001
078 0.12360 0.06335 0.03125 o.01s10 0.00721 0.00341 000160 0.00075 0.00038 0.00001
080 0.11111 005618 0.02730 0.01299 0.00610 0.00284 0.00131 0.00060 0.00028 0.00001
0.82 0.09890 004933 0.02362 0.01106- 000511 0.0023" 0.00106 0.00048 0.00022 0.00000
064 008696 004280 0.02019 000931 000423 0.00190 0.00085 0.00038 0.00017 0.00000
0.86 0.07527 0.03656 0.01700 0.00772 0.00345 0.00.153 0.00067 0.00029 0.00013 0.00000
a.es 006383 0.03060 0.01403 0.00627 0.00276 0.00120 0.00052 0.00022 0.00010 0.00000
0.90 0.05263 0.02491 0.01126 0.00496 0.00215 0.00092 0.00039 0.00017 0.00007 0.00000
0.92 0.04167 0.0~947 0.00687 0.00377 D.00161 0.00068 000028 0.00012 0.00005 0.00000
0.94 0.03093 0.01427 0.00627 0.00268 0.00113 0.00047 0.00019 0.00006 o.00003 0.00000
0.96 0.02041 0.00930 0.00403 0.00170 0.00070 0.00029 0.00012 0.00005 0.00002 0.00000
0.98 0.01010 0.00454 0.00194 0.00081 0.00033 0.00013 000005 0.00002 000001 000000

39 65 76 45 45 19 90 69 64 61 20 26 36 31 62 58 24 97 14 97 95 06 70 99 00
73 71 23 70 90 6597601211 31 56 34 19 19 47 83 75 51 33 30 62 38 20 46
7.2 20 47 33 84 51 67 47 97 19 98 40 07 17 66 23 05 09 51 80 5978115249
75 17 25 69 17 17 95 21 78 58 24 33 45 77 48 69 81 84 09 29 93 22 70 45 80
37 48 79 88 74 63 52 06 34 30 01 31 60 10 27 35 07 79 71 53 28 99 52 01 41
02 89 08 16 94 85 53 83 29 95 56 27 09 24 43 21 78 55 09 82 72 61 88 73 61
87 18 15 70 07 37 79 49 12 38 48 13 93 55 96 41 92 45 71 51 09 18 25 58 94
98 83 71 70 15 89 09 39 59 24 00 06 41 41 20 14 36 59 25 47 54 45 17 24 89
10 08 58 07 04 76 62 16 46 68 58 76 17 14 86 59 53 11 52 21 66 04 18 72 87
47 90 56 37 31 71 82 13 50 41 27 55 10 24 92 28 04 67 53 44 95 23 00 84 47
93 05 31 03 07 34 18 04 52 35 74 13 39 35 22 68 95 23 92 35 36 63 70 35 33
21 89 11 47 99 11 20 99 45 18 76 51 94 84 86 13 79 93 37 55 98 16 04 41 67
95 16 94 06 97 27 37 83 28 71 79 57 95 13 91 09 61 67 25 21 56 20 11 32 44
97 08 31 55 73 10 65 81 92 59 77 31 61 95 46 20 44 90 32 64 26 99 76 75 63
69 26 88 86 13 59 71 74 17 32 48 38 75 93 29 73 37 32 04 05 60 82 29 20 25
~1 47 10 25 03 87 63 93 95 17 81 83 83 04 49 77 45 85 50 51 79 88 01 97 30
91 94 14 63 62 08 61 74 51 69 92 79 43 89 79 29 18 94 51 23 14 85 11 47 23
80 06 54 18 47 08 52 85 08 40 48 40 35 94 22 72 85 71 08 86 50 03 42 99 36
67 72 77 63 99 B9 85 84 46 06 64 71 06 21 66 89 37 20 70 01 61 65 70 22 12
59 40 24 13 75 42 29 72 23 19 06 94 76 10 08 81 30 15 39 14 81 83 17 16 33
63 62 06 34 41 79 53 36 02 95 94 61 09 43 62 20 21 14 6B 86 84 95 48 46 45
78 47 23 53 90 79 93 96 38 63 34 B5 52 05 09 85 43 01 72 73 14 93 87 81 40
87 68 62 15 43 97 48 72 66 48 53 16 71 13 81 59 97 50 99 52 24 62 20 42 31
47 60 92 10 77 26 97 05 73 51 8B 46 38 03 58 72 68 49 29 31 75 70 16 08 24
56 88 87 59 41 06 87 37 78 48 65 BB 69 58 39 88 02 B4 27 B3 85 81 56 39 38
22 17 68 65 84 87 02 22 57 51 68 61J 80 95 44 11 29 01 95 BO 49 34 35 86 47
19.36 27 59 46 39 77 32 77 09 79 57 92 36 59 89 74 19 B2 15 08 5B 94 34 74
16 77 23 02 77 28 08 24 25 93 2245448411 87 BO 61 65 31 09 71 91 74 25
78 43 76 71 61 97 67 63 99 61 80 45 67 93 82 59 73 19 85 23 53 33 65 97 21
03 28 28 26 08 69 30 16 09 05 53 58 47 70 93 66 56 45 65 79 45 56 20 19 47
04 31 17 21 56 33 73 99 19 87 26 72 39 27 67 53 77 57 68 93 60 61 97 22 61
61 06 98 03 91 87 14 77 43 96 43 00 65 98 50 45 60 33 01 07 98 99 46 50 47
23 68 35 26 00 99 53 93 61 28 52 70 05 48 34 56 65 05 61 86 90 92 10 70 80
15 39 25 70 99 93 86 52 77 65 15 33 59 05 28 22 87 26 07 47 86 96 98 29 06
58 71 96 30 24 rn 46 23 34 27 85 13 99 24 44 49 18 09 79 49 74 16 32 23 02
93 22 53 64 39 07 10 63 76 35 87 03 04 79 88 0.8 13 13 85 51 55 34 57 72 69
78 76 58 54 74 92 38 70 96 92 52 06 79 79 46 82 63 18 27 44 69 66 92 19 09
61 81 31 96 82 00 57 25 60 59 46 72 60 18 77 55 66 12 62 11 08 99 55 64 57
42 88 07 10 05 24 98 65 63 21 47 21 61 88 32 27 so 30 21 60 10 92 35 36 12
77 94 30 05 39 28 10 99 00 27 12 73 73 99 12 49 99 57 94 82 96 88 57 17 91

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m1N~<11 bbl1~m'll-leJ1JVlmtmu blJu<iiu
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'l1~bbN'Um'lN~'11 bbl1~m~;i'Yi1~utl1:omrn
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1v ""' "' $1 t::I .... E"l !IJ iJ ""'.. 1 ~
3. fl1'l 'Iffl1Ulfll'll'l1'1-1'll!J~i;jI'If111'111l)lIll'I'll:::Ill'lJ1 lUel~'illnf1l'llUU~'lellJ~ll'1::::wu'l:::flJm'lrubu
' . ..:i; .... ' ,:!;I <'II
l'leJ~UU'l eltJ1~'1f1'lJ~
4. m1fl'll11w1'il1111~'iJ1;l'lflNilflllru'l'i1m~''>il'lllN~lliiru'l'l~nel~ 41:::m:$iel 1:::tJ:::llUv'1-t1 "~
l~lJtll l'il'lt)l~lJ1llll:w~ ll'1:::llf1~1 f11n'ilm'lLi'liJm'l'Wmrnrub'S~mmrueltJl~l~l'J'llU'Yjrn:::tJ:::'lJel~'l~'il'l
'il'lllm<JYi1'l~Y<mn"illif1'1Y<m'11<il 'J1nmY<~ 7.1 lli''1~1~.~un~'lJrn:::~1~,.,'if1111Nlilliiru<Vi1i"1:::tJ:::l'il'ltJ
.~lJtllblli:W~bli'1 vi'lei~l~tJn11d'Ju1:::tJ:::~:w<ii1 'illf1,XU'il:::•llW~'1lU~tJU'llel~tJel'1'llltJ ~~tJel'1'llltJ'il:::l~:W~'1
~~•.jJ1rj1:::tJ:::lln~1 ~"l'1•1i~tJudm-tl•'111:::~•'E~iJ~mru1:!-immw1•'1"i'1:::1'11<il ~'l"lmrnru<i'lel"1-iim1

YImmrnb'TI~fllli1111"1L4i1·.1hEJ <ii1fl81~1l'in'Jm'inill1TMmm'i'lf'ii ~l'.im'iYImmrn11e'l'ii'llm'ibiilul'llff~mq q q 'll
'ill1il'il1"ifl~~'l'il~:1-1<ii1bb~11iJrmuifo'loifm'i1"!tnrnrnL'R~mmrnilaj~~fl111111e'l'ii'llnl'ibiilul'ii'J~b~:1-1.r;u8n~~
Lii1111m~m-,nill171~mm-,'ll111.r;u:1-11mnmEJ bb~1'lu~l)111'J~bii11111qJm1iJmm-,11'll1l'J1"1111i.J111 'Yi1liXJi'in'J
tl'i~f!unum1:1-1'lm11;1
rnn
5. m1if1n11V1ml'I 1fid,~1"1ifili.Jl1'Jlnm'iftl'i1'Jm1:1-14'1111b~'U'llm11n;f1 111'iilN~fl11111111~11'J~• • •
dJujn;f1'llil~A'm"'lui;im111Li11111mtJ1111l'Jnl'iililrnbuu11ilum:1-1 i1:1-1rn1'!rn 111'iilV1'ill1ilui;im111 'Yi1'lVll41
iiil:i-111b~mnubbwum'i~il'llil~11m'fl ~~u111.l'loii'lum'i1"1rnmrntJfl'il'llltJ b611~if~mm-,11u1111'loii'lum•
• •
t1-TI.Jt11~w11i;iifrn.ri'ii11fl'il'J'Unl'i11~b6W'UVi'WJmwi1'llifrn'Yl'l111iJ14
6. 11l1V1l:li'l'l (Delphi Method) u'iWnb6'i"'U.l 14i'Vi'WJm'lfi6~1111Y'J,r;u'lull fl.I'!. 1950 'ltidnn
Vi''IJJ'Ul;'Umb~fl'lvtl411uWl1U'i"~lEJ'11U'Jlnm'ib611rntl~l'J'Uifm;111'luntj:1-1 'llill~b~mnumm'imbnUqJ~l
m1:1-1oifiibbU~'llil~ntj:1-1141 'ltib111111Y'J'J~tl•~nilu.11l'J<irn~1Jflfll1~bUuntj:1-1'llm~bzjl'J1'111qJ"iii1u1u111ii~ 1111tJ
Yi'11t1l1"iii1u1u 5-1o fl'U ~~bb'ii11~<iw<J~1m1'ilflufi1mi.J 'lViirm;111 bbl1~1"1tnmrn~~~<i'fil~m• 111fi~"11ni:iu
b~lVlill~"'il~'Yilm'i'i11J'i"1:1-14ifl:l-ll16611~'111"1mmrn'JimCibzjm'lflnIb6'iil1~fl'U:l-ll:l-lfl1J'lVIV11111it1mi:1-1111'iilN<ii'ii~'Uh'IJ 'IJ ~ti q 11.1
~~ifll:1-1ii'ii1~'1 ~11u'i1:1-1md"'il~u1mf!'lli66'1~11tJ~1unii'u1t1if~e-'.Jbzjm'lf1n 'bb'iiii~flulliltJ1i-ibil'ilbWl'J11Luu'll ~ 'I.I ~1,:1
ifm;111111'iilwiim'i1"!mmrn'llil~~bzjl'J1'11lqjfl'Ulii "'illniim;1iibVl6hd'lVi~bzjtl1'11l!;JJ66'ii'1~fl'Unfl1J1tiVl1J1'11'U bb1~
'Yi1m'i1"!mn-ntl111iJbb~1l'.im'lbnu11u11:1-111iff1t1 bbii~ci~nfiu'lVIV1"'ii1•rn1~~6uum~mum~~1nufl~1~d
111ii1milu "'il'Um~Yi'~1"1'111"1mmrn~LU'U:l-liilbiln'j;uV1"lu~l1ii liitJmmmmtl~ui;iilu1.'iiii~d• •
1. Lfiiln~bzjm'lf1qJ"iii1mu111ii~ ~~m'i"'il~'lVILL'ii11~<iul1m1:1-1im1mzjm'lf1qJ'lubb'ii11~m'JJ1~bb"1n

2. "i'JlJ'l"'J:J-1fi1YHJ1n"ici'inn~btiil'J1'l1qjbb'lifi:;<ilu ~'11ll"JloiibblJlJ'1€llJm:J-J'i,j7e:i5b:J-JMi<i1<iie:inlJ~
L'iftn'lllqj
3. "1uwmm:;li,;n~JJ1u1V1~,o/lm'l1qJ11n<iluv.~<i:J-JnlJ1Vlv.mmru11,j;;J
4. Yl1~1~'U'ililU 2 3 <Jun-:h<J:;Ll"INl1n1"iV1tnmrn~uT/1€lh bbi11ii'1Nl1'!<i1Yl1!JlUti'1~Lollm'l1qj
''Hh<i1l1L'V'lmm•ti1VI~·ollu1'l1qJLL<iifl:;<ilu1'111JJu•:;lu'l:fu<J1n<i111:wl""~u•:;l'flJm'l"rn'llil-1~
,ollm'll1umu5u'l l<ilEJ~<ii1,;~1u1:W~1d'Ju~fo,;e:itJy;~e:J:J-11,ju1nu'i,j7ei~-.i'nnu Y111Vlmm•til1<i1m1:w'1i<ilwff,;• • •
ii1ulJ<il<i1'1l1'111'1bbl1~ti,;L11<i11e:im'11Vlv.iirn1u1u•~l"ilJ~1n-fo<ll:wh~<J~1Vl'1ie:i~<i1Liiu~Ll'lum1:w<J1,;l<il!J1:W•
<ile:i'1Lm'1n~1 ',j'l"mm,;hr;fl<il
11i~'1iml1EJ~lVIN'1'1i1 bbl1~'111'1i..J18J'1n'J1m"i'INmmcil<ilEJnzjl.l11'l"'l":J-1<i11 iltl1,;L'l"R"11:J-I m-;-
1Yl"l<ill.l'Ul<ill.IlUU<J<JUU Y111Vlm-,~ilftl"i"i'J<ilL~'Jd,;~u ~,;t'11m•n'li1Ufl<ilnm~1'1i'l.um-;-v.mmcl"'1'11•
11iL<i1'1L'V'lilul.l1'1i'lum1v.mmrn1~EJ~m1 L'liu m-;-y;mn•cl!Je:J<i1'111!J1u 5 ll'1i1,;'i,jii:1 Lvle:i1'1i'l.u
m-;-o/J<iilUh11,JbbNWi1~,Jn1"iNii"1 1,j7€Jn1"iV1mmrnl"imLYI<i1lUl'1~l1"J~l1m-;-LU~UUbbU'1,JlUilU1<i1"1Ll'JU
eitl1-:i1•
n1WWU'j'11Tin1"JY<tnn"lcib'R'1fjtw11'1NLvl€l~"J:;lVlfl1y;mn"lci<J1nm"iloii1Tim"JY<tnmrnb'R'1fjtw11V1
l1m1:w1Jn<.111l'11ni1Li'imm1:w<J'1' ~-:il1nni1dllim•<ii1-:i'l m1'1i'l.umw1lJu1'm™mmrn W!-:i~
1. m"i-ii'<ilr11n•~lJ1um1v.mn•rnlVILl'lum•n;J1U •'liu 1un•rn~l1m11'1iv.iirn1uY11nn
V1mmclUil<i1'111UYlnb~ilU bvle:i1Vlifuh11v.iirn1ui'.Jm-;-d0
lJ"i'llJ"i'J:J-1'1ieJ:J-1'1fo!J1Tin1"lb&!J'lflUYln<i1~'1 ~'1• • •
-ii'<i1Yl1:J-11"1"i'11U1um-;-'11LUUn1"iV1mn1ru'1nVlv;un,J1U'Jl1!J b'liu YinL~€lU"J~U1bblJlJ'1illJm:J-1Lui11-;-1<Jo(l) I q E
A 0 • < A • ~ "1 .,, " A ' " • I~ • I ~,, < ~ Ji
<il'l1:J-l<il(ilL'i,jU'Jlil'1jnmn~l.lb<ill.I Yl1 b',j b<il'11€ll;J'1'li'1ftll.11'l"mLm1~'1,jb1,Jfl11:J-lbu'1!JUbbul1'1 L<i1€ltn'1'li<i1L"JU'JIU
• 6) 'l-'' r: ii' JI.
LL'1~1'11 L',jmv;mmru1Jn"1il'1:J-11n'llu
2. m•mlJ<ill.lfi1v;mmcl e:J~i'.i'i,ju1~'INmmrnm"i"J~<ile:i-:i1'1lwm•tlneim:w1i,;1ifm•v.mmcl• •
•rn:;L~ilr11m•v;mmclLLil1m•<J:;l"e:i-:ii'.im•o/i<illilll.l<il.!1~Ln<il~u"1' ,vle:i,:;1'11.iJfomfi!JlJ t11JJtl•-:i "":;• •
<i1'llJ<il:WY<mn"iru1Vlun~'fo-:id-:i~u1u<i1~'1<iiil1ti o/11e:itl1-:iL'liu N'11 mn"iclLriilo/i<ilm:J-1<i1NmL~1Y<lJ11 l1q '!.< '!! '!!
v;mmm!Jil<il'll1rnnu<i111:J-J<J'1-:i•t'1l.le:i Wl-:ii:ium1v.mmrn<il~-:i<iiil1ti ~v..mmrnm"i"l~Y11m1t11JJ.i1
v.mmru"l1,j:J-J'lVl'1<i1'1'1<J1n~m<i111 ',j7e:ii'.im-;-V1<J1"iru11Vl'1:;L5U<il"lillJ<ililrn.nnri-:i~u
3. m•'1~""'1J'1h 1mJ1Vl•1-:i1''1mi~v.mmm~l]n<iim
4. m•1'1inzjl.I m-;-v.mn"iOO<i1tJ1'1inzj:w~Lo/ll'J1'll1qJ'l,jft11J<ilU ifn<J~1Vl.!1~1]n<iiei-:imnn11m-;-

'liim1'1'1Ulfl'illit'ii~tA:irn,u (Quantitative Forecasting Methods) LllUm"l'l~im"l'Yll~i1!'iiii
m1::flrn<ilflli1<il~ 1Lfl"l1::Vi<ifEJ:>1fl'lu!l~<ilLLf1::1"hm'l'V'lmmruw1<ilm"lnbluEJmfl<il 'lmn'inm'l'll!l~m"l
. '
'V'lmn"lruL'il~tl'imm~u!iEJ11<i1EJ:>1fll'.ht1LLUUEJ~~~111m"l'V'lmmnfoui fl<ii'ltl LLUU'llEJ~<iiEJ:!-1fl~u T<iitJ'I.I 'll 'II 'II 'll
mm'lfl'iilLLUn1im'l'l'l1Jln"lillL'il~mmmEJ€lnd'Ju 2 ~tiLLlJlJ /;)€) <ii'1LL1JlJ€l1tn'llJL'lfllLLfl::<ii'1LL1JlJLLi1<il~
m1:>1ft:>1WUW€l~<ii'1Lbti"l <ii'1LL1JlJ€l1tn'l":>lb'lfll (time-series models) 'lu111ibtJ~ 7 d'il::bi1UEJl~ 5 1i /;)EJ
' ' '
1. 1i~w;<.iiml'liiEJuYi (Moving Averages)
2. 'lim'l'll~'rn'forn3n~tiLUUb:fftJfl (Exponential Smoothing)
3. Double Exponential Smoothing
4. Winter's Three-Parameter Trend and Seasonality Method
5. 1im'lV!lbbWJTit:!-1 (Trend)
<ii'1uuu€l1tmm1m tllum"l'V'lmmM<iltJ'l-iiolim1:>1:>1iiiJ"1u':i1EJi110limllu'lll~fi'11u'llm€l~<il 111'i!l
nl:'l111inutJ111i'l~11Lllum'l'V'ImmruEJmfl<il1<iltJ'l-iiolim;1fl'luEJ~<i1'ii1u1u111il~ L'liii t.'f1iifo~m'l'l'lmmrutJEJ<il
'llltJi'tl<i11Vi111ul'llEJ~Ll'l4EJ~iJ1umm0il'il::'loiitJEJ<il'llltJ'llm1'tl<i11111"luB~<ii~N1i1:>11'llmLfl4EJ~t11umm0:>11
111nl'l'l'l1Jln'lll nl'l'l'l1Jln'Jtll't<iitJ'l4i<ii1bLlJlJEl'ltn'l:!-1nm~~i1:>i:>iiii11~LLlJlJ'llEl~<ii€li;Jfl~<1i1Litil:>il'lU€l~<il'il::
LllumwEJnEJml'l<ilLt11::~~l11m'l'V'l1Jlmilliiuhi1EJUlfl<illiJSJEJ::hLli~tJULLtiMtDu1inLLUU
m"l'V'l1Jln"lm'T•w'l4i<ii1LL'YIU'll€l~El'ltm:!-1nm<J::Llliim'l'V'ImmrnEJi11fl<il"Jln<ifEJi;Jfl'lUEJ~<iitvh~u
1<iltJL:iJi1uh<ii'1tLti"llJU'l u11::nm"J::mJ~tllu'lf1~'.i1~t1'il'l nu L'im<il1rn.i1<ii'ultl L'lfil 'lltJfttl<i11Vi 'llm~Elil
'lltJL<il"lmi1 m'l::'lltJU tllwiiu
'·' - < "' • 'l 4 '"' J€l~flu'l::n€llJ'll€l~€l1tn'l:>lb'lfll m"l1Lfl'll::Vl!l1tn'l":>ib'lfl1Luilm'lwiln'll!ll;Jfl U€J(IJ<i1€l€lnLu'U'11U
ti'lmEJu.!1~'1 u11::l11m'll'11(1Jfl::LuEJu1f!<i1 1(1J!J!litm:>1rnni'.l 4 1Nf;iJ5::nEJu1i1m'liii'it1 um1U:>i 11~m11
1{l'<ifm LLfl::m'lNUbLU~L:iJLLUU€lU
1. tm11u:>1 (Trend) tllum'lLLi1(1J~Yifl'Yll~'llt1~'1it1:>111i1i'ni;.rn::m"ltfl~EJu11111'lJEJ~<ifB:>111i'.l
• •
i'ni;.m::~BtJ'l ~u111'iEJ~EltJ'1 fl~ 'lu'lf1~nm~N1i1:>11
" ·' • - Ji ~ d ~ ,,/ ~ Ji 4 'l2. l]~mfl (Seasonality) LuU~uLL1JlJ'll€l~'ll€li;Jflm<il'll'U'l!lVl'l€l:>1m'lLfl!1€l'UbVl'l'llUVl'l€lfl~ u
l11uEJ~L~mnu'luLL"111::'lf1~nm'llB~1u l'iiJ(IJ1Vi L~EJu 111'iEJLmmi1 ~~oliEJ:>1fliin111Nfln"l::'Ylu<J1nm'l
•
Lti~tJULL1ifl~l]~m11 t'lfu oif€li;Jflm"l'll1m~EJ111m1
3. 1D'<ifm (Cycles) i'iB ~LLUU'llEJ~<iiBi;JflLnOl~u~1'lu'lf1~111mtJ'l U ~~iin•::Slm1:>1!'i:>1wu~nu
1D'<ifn'l'll€l~Jfifi"JLLfl::Lfl~n'I LLfl::m1:iJ'Yl'llU11LL'111~'lf1~'ll€l~1'D'<ifm•::ih::tJ::nmmuLYi1l<ii

4. m'J~'Ubbli,,y'li.Jm.iu!lu (Irregular Movement m!J Random Variations) iiiJ m-;d''1
Vltin1wifm.m~~1:iJihliLL1J1J~LLUu!lui16'1LVlil:WTil1nl1mum·rn1~N'1lln~~L:iJ611mw<i1'1m"'lrn"t~<i1~Vlu1. . "
•'liu m-;Ln'1m1~1Yi1:w 'ilm'ilsi bbsi~m"'lVll'J'1~1ull-,~Vi1~'ll!l~'V11Xrnm
t
1 2 3 4 5 6 7 s 9 10 11 12 (tl)
nfn
(fl) ~n~ruz1.bu':l1U~
'OJ El V'l YHl if! '1 Ell '<il fl ¥1 'V'lb) ff fl 81 ~ El (H1J~nVi)
n~1
(•) iin1'ru~q'i)m~
1 2 3 4 5 6 7 8 9 10 1112 13 14 (ti)
n;;:i1
('1) in,,;~'i!)%n'
1 2 3 4 5 6 7 s 9 o11 12 3 4 (u)
1.1?n
('1) in,,,rumwl1uu"~'*'

1f1LLlJlJil'lfn'l":Wnmmm'm<il1u1ru'11m1:w1fil~m"iih1.'11 (demand) 1iil2~LLuuii11tJfiuqiJ LLUU
•
'Jill (multiplicative model) mi:muumn (additive model) iii~t
9!1111J1J~l,U
il11111J1J1J'Jfl
'11~mrnclm1:w1fil~m"i = T + s + c + I
Lilfl T qiJ '11LLU11u:w
s qil '1ih)~mii
c qfl '11'llml[J~rn
qil m.'11h~nflU'llmm'lNULLiJ7Vil1JLLuuflu
m~~ 7.3 -~bbf'1~.iffl:Wil'llEJ~m1:;,i1lmm'lftU.11~i1LLU11ill.JrviJ.J~UlU 4 i'J~NlU:;Jl bbi1~.iffl:Wi1i1
• •
0 1 2 3 4
fl~1~hlhn:w 1um"i<il1u1ru'11~mmcl:i'.in<i~'Li.lu1'11ni->NuLLlJ'lL'll~~:;,iL.if1m<il1u1ru Lum<i1n
dJunl''lmn~<i~V11'11~'1iJn~i:iu i;'f~,Xuei~f1iJ.,~nEJJJ~~•V1~m'i'lm'11LLmtuJ.J h)~rnii LLii~1[J~m
lU'lltJi1~L5tJ'1'llmm'lLL'1n.ifm;Ji1il'lfn'l":Wnm<i~n~11luL~fl~~ 7.2.4
1f1LLJJ1Jbbf'1~m1:wif:w'liiui'llfl~1f16bl.J'i (casual model) dJum.,~mmru'1'1tJYi<i1"lru1m1:;,i
if:w'liiui'll1N<ii1LLlJ"i <i1n~<ii1LLlJ"i<ii1'Vlrt~ (dependent variable) LLD'l'1l:W<ii1LLD"l5n<ii1V1~ (independent
variable) 1miutJ:w1.ifmnqfi m"ilbfl"il~Vim"ill'1llilm'll~Liiu (linear regression analysis) ~~-~ilm"i
Yi<i1.,ru11i~<ii1LLiJ~i1~'Ylif~iilliil~~~"~ri1m.,~mmcl •'liu rn'l~mmrntJil'1'llltJLfl~miJ'lumm'1•~i1

ili1bb1J1'llm.j1u1uii1uwn "ll'1liiu4i'1'llil-J~wii-:iiiu bbi0MU1J1~mrum1'1-:ibei'l:J.Jm"l'lJ1tJ~i1-:iN10m~'Ylu'1il
tJil'1'llltJ•1m-:ifi'lm1
1ui11u"lltJ10nEitJ'1'llil-:i'iilm1fi1u1ru'lJiJ-:im1Y<mmrnb'B-:iu~mru<Jmsil11ub~m'1il1u
11a~m11:lmrnd0muT>::1~M~ 7.1.3 1d{1 Tti1wtii)lliiil11111w 7.1.3
1iminm'l:lmnmlw~ 7 l'l!l'll~ 7.1 1~0~~ 7.1.3

'
"'~auri 7.2
b'l°fl~Yi 7.2.1
L'l°mfi 7.2.2
' '
' ' '' ""' ,,,. ""nTl1'1l'fllb%fltM1fl1l'l.1'1
m-;;:Ll-fub'l°tJu•llni'l:llmu•'llvil
b'l'ei~Yi 7.2.3 Double Exponential Smoothing
L'l°mYi 7.2.4 Double Exponential Smoothing: Holt's Two-Parameter Method
b'l'ei~Yi 7.2.5 Winter's Three-Parameter Trend and Seasonality Method
•'l°ei~Yi 7.2.6 m-m1..mT™1
•'l°ei~Yi 7.2.7 m7bbilnoiifll;)flffl-!n'lm1m
1. m-m1fi1L%~m.-i~eiu~•Vlm::<t~nunTrnmrnm"lurnm~.jjeil;)fl~l1~rn;.ru::hi~ull'l'eifl~
eitJ1~'l1ilb~1 bbfl::'1.i.il1ilrn;.m::'llm11~mfl 11im-m1.-i1'"~m.-i~eiu~ bbU~eieimliu11ifi1
< I I I l I ~
""' ""' .,. ' ' ' .,. ."l "" ' ~ ..,
b%fitJbilfl1lU'Yl1ltJl~~ltJ bbfll:'fllb%fitJbilfl1lU'Ylbb1JlJ1:11~UlllUn
2. nl'lU'frn'l°tJUbBn.aiJbUUb'JltJfl bDU'iTinl'l°l'Itlln'lUb~bll:J.n::filJ'ifol;)fl~flflU.jj'l~hJ
bU~tJubbllm bbfl::l-if.jjeil;JmWm 1 ~1il m°l'lmrnru"l~iltJlVf.-i1.-i~~ a ii1Vl-fum'llVf
•
u1lliinffu.jj'eil;)fl
3. Double Exponential Smoothing bliu'i'iim'l°l'ltJln'lUbfllVl'llJ.jjfll;)fl~l'Jifni;.ru::
bbU1tu~Ji1tJ Til1.1l1m'lrllllUilfllil~~ 2 fll $iei a bbfl:: y
4. Holt's Two-Parameter Method bliu'i'iiVltt~'llfl~m'l°l'ltJln'lUb~L-iifiu.jj'm.rn~l'Jv
~ni;.m::bbmL™1,;a~bftu<il'l~ bbfl::1.i'lVim1~ii1f'iqJfiu.jj'eil;)flli1'Jil~1n~'Jil
5. Winter's Three-Parameter Trend and Seasonality Method 1-iilum'l°l'lmn'lrn
.jJei~fl~l1iini;.m::"m1™1bbfl::l11i'Yl~fl'llfl~n ilnlflv .,.
6. m'lVl1fi1•m1Tii~ bl'Ju'i1im'l"l'lmmrn.jjeil;)fl~l1ilni;.rurnu11u~1iltJ1-ii'l1in1ii'~<tei~
!V ..J. lU !V jl-- x
U1ltJ'Yl'Jil'1'll~b'1UbbU1 bU~'llU
11.r !V .fl 6'ol 4 j1'
7. 'll1ll;)fl1l1-!n'l~nmmm'lmbtJn'll1ll;)flbuU 4 m.-iu'l::neiu .-iei bbU1 b~ 11~mzi
bbfl::m'le:iUbbll'lYi~i.ibbuueiu

v -· '191QIJ':i~ff,:jfl
b~eii!ni,r1'ililU~ 7.2 "llJbbll1 i:ni!m,;1mm-,n
1. V11'11wmm1Mbt.1'lim-,V11'11b<l~m'il~ilu~1'1
2. V11'11wmmni'11t.1'lim-,iJ1rn~t.1rn~n.£1:iJbum'6t.161L'1
3. V11'11wmmm'11t.1'li Double Exponential Smoothing 1'1
4. V11'11wmmrn'11t.1'li Holt's Two-Parameter Method 1'1
' If~ """"
5. 'Vllfl1Wmn-;m1111t.11u Winter's Three-Parameter Trend and Seasonality
Method 1'1
6. V11'11wmmrn'111Jm-;m'11bm11~1'1
7. ei'ilrn!!m-;bb'ilnoiieii,:'1eiitmi-inm1'1

'ilim>1111ri1bQ~ti•.-i~eiu~ blJu'ilim"l'V'immT.1~~1ti~'flil'll!l'ii1bblJUeiiinwnm$1ei 'ili'11,,.!]ti
bfl~eiu~'iml'l~1oii,~ei~ni,;ru.::'llm.jjm.Jf11iJ141~m-.~u1111eimeitJ1'n'"'~1 ""~1l-i~~ni,;ru:o'll8,n'1mf1 'iil. .,,
m"l'V'imrnru'1'1tim"lu1.jjm.Jf1LUil~"1~N1m11"il1u1u111rt,m'Yi1mtieim'1"11'1ti"1"i' filbQ~mfl~eiu~:i'.inL-if•
1um"l'V'imrncl-.~ti~~u •'liu m"i'l'imn..clbliu-.1rn~iJ'"1Vi1111ei"llm~8u '11bi:l~m'1~8u~~utia.J1-ifnu~ei
' ' ' ' ' ' 2'
'11b%llm'1!1eiurieitJ1,~1tJ (simple moving average) bb11:;'11bi:1i'im'1l1eiuribbuun1,u1111iin (weight moving
average)
' ' '0 2-' al I ""2,.o AI.,;:~ tot f I Aal! .d .d ,.J. ! I
1. l'll!lllll'.l!flllll'llflll!JH~l!J 11l'11bQ6'Wb'1f1ilU'Yl€ltJ1''1tJ 'J~Ul'llill;jf1'Ylfnl'jilmU'YIU'Yl'llill;jf1'YI
•n1Yi'l'" 11n.-i~'1ummmrnru 11ilti~ij"1"l1um-.fim1rn~,d
At-1 + At-2 + At-3 + ··· + At-n
n
=
' 'o 2' #I o l do I d
'
At-1
At-2' At-3 bbf1~ At-n
i11fl1.h~fi 1.1
=
=
=
'J1u1U'llil''llili;Jf1V1"l1l'llmwir1,nm'Ylmm1111.-i1bi:lf1ti
'11'l1'~'fil'l~ULu'lf1,nm t-1
f11'J1,~bnil~ULU'lf1"1m t-2, t-3, ... Uf1~ t-n
N°{lilm"lh,,mNl1"1•.-i4ei,iJwmmmr1A,111rt,1'i1m-.'V1mrnclm1a.Jiif8,m-.w;i11~•~8u4i'1ti'ili
•' ' ' '( <=!. ,d. ""- I ' ,=i, ..,j
'11bQf1!Jb'1f1ilU'YliltJ1"1tJUlllJ 3 bil1JU "11a.J"11"il''YI 7.1

.. m v ,,,. 4
!l'l!ll-I !Jlflffl1llJl'l!l~nl1'il':i~ (!!111H)
' ,fll'i'l1Jlfljfl!
' ' '
t A, F, (fl1!>1lirn'1~ilui1 3 !iiilu)
!Jfl'>Wl!J 100
~ <
fllJl1l'i'IU!l 120
~
130lJUlfllJ
•mmm 160 (100 + 120 + 130)/3 = 116.67
V'lfj1'tl11'1!J 190 (120 + 130 + 160)/3 = 136.67
i1l)UltJU 230 (130 + 160 + 190)/3 160
f1jf1/;jl'1lJ 260 (160 + 190 + 230)/3 193.33
11~'/llfl!J 300 (190 + 230 + 260)/3 226.67
flutJl!JU 280 (230 + 260 + 300)/3 = 263.33
"Jl'llfllJ 180 (260 + 300 + 280)/3 = 280
'i'lf)MmtJU 160 (300 + 280 + 180)/3 253.33
UU11'1!J (280 + 180 + 160)/3 206.67
<J111$11ill'l1~~ 7.1 "l~bl<l<il~1VIL~ui'~'lilm'l.i1u1m1<1ltJ1.iffi1L%~m'1~ilU~'llm-ifm.il'l 3 Liliilu
(3-month moving average) L'llu LiliilU!:i.11"ll'JUfllV'lmmru'l~L4i<J1nfi1L%~1'J'lliNfll"l'l~ 3 Lili<Jun!lui-iil
A +A +A.
F =
&.i.lfl. n.'V'l. ;i..J.fl.
3IJ.J. ti.
100 + 120 + 130
3
= 116.67
qllV'll'Jlmru'llil~ LiliilUV'lf)1"111'1:i.J<J:;U1'1iil:1;jl'l.11<J'l~li1ij<ilflil La1iJUL:i.11"ltJUm1.if.i1mm LLm:$i<1l
-ifil:1;J!1'11<J'l~~Ln1~'l<1li-iil 1iiilU:i.Jm1.i:i.Jililn1u$i~d
=
=
A +A. +An.w. 3-.l.fL 1J.J_t1_
3
120 + 130 + 160
3
= 136.67

=
280 + 180 + 160
3
= 206.67
' ' ' ' ' '
LL'101~~11m7V'lmrnclmJ1J'111>1i'11.M1!'1EJuYI 3 ft't1011VI LL11~'11L>1i'1rnfli'1EJuYI g ft't1011VI ~1,,,fo
tl'tfon'ti f'l'Jl)J~fl~fll'iil~~
1 800
2 1,400
3 1,000
4 1,500
5 1,500
6 1,300
7 1,800
8 1,700
9 1,300
10 1,700
11 1.700
12 1,500
13 2,300
14 2,300
15 2,000
16 1,700
17 1,800
18 2,200
19 1,900
20 2,400
21 2.400
22 2.600
23 2,000
24 2,500
25 2,600
26 2,200
27 2,200
28 2,500
29 2.400
30 2,100
A1't'U.llfl'i~
3Mweek
Moving average
1,067
1,300
1,333
1,433
1,533
1,600
1,600
1,567
1,567
1,633
1,833
2,033
2,200
2,000
1,833
1,900
1,967
2,167
2,233
2,467
2,333
2,367
2,367
2,433
2,333
2,300
2,367
' 'fll'VU'.l1fl'it'U
9-week
Moving average
1,367
1,467
1,500
1,556
1,644
1,733
1,811
1,800
1,811
1,911
1,933
2,011
2,111
2,144
2,111
2,167
2,267
2,311
2,311
2,378
2,378

'illn'ill"ll~~ 7.2 Ulrll''J~bbfl~rllV<1Jln'rnhrn'Jimin11W ml'<~ 7.4 ~~'n1.J"itJmfitJ1JlVlbi1U
m1i-1bb'ilrnh~1~·w)1~ri1''l~nuri1v;mm-ii10~bb1J1J ri1b%~rnfl~ilu~ 3 #i.J1111Vi bbfl~rilb%~rnfl~ilu~ g #J.JmVi
J)~,~bViu1~11 rllb%~rnfl~ilU~ 9 #JJ1111Vi '~'ilil1JlUil~'fiilmob1.J~tJUbbD'1~m1i-ilifo~m1,'.b1~-if1n'11 bb'fi
'~'lVlri1~i1m1i-1b"itJu (smooth) mnnllbb1J1Jrllb%~rnfl~ilu~ 3 iifJ.JmVi
2,800
2,400
,..r
<= 2,000'7
OJ
•i;;
1,600;;;
r ,,......('"
~ 1,200
r
;;;
800
IS
400
4 8 12 16 20 24 28 32
m'l'i~ 7,4 u11mm11!l1vrniiauri1~~~i'miiwa1111cl?i'11J'lliri1m~ml'lg<il-l~<idH~11J
uuu 3 ITT.J~mi1rn~ 9 !ltfomi
! I ! I 51/
,d, =A<=I A I l -=! d. <=I ! ~ ._,
ilrnll'Ylb.,tJn1lfllb%fltJbflflilUYI bb1J1Jl1~Ul'j,jUn
#J.JmVi
2. ri1m~mri~tlu~uuutl1~J111il11 ri1b%~mfl~ilu~bburnh~~1~un d'.iu'li5m1v;mrnru~i1
m1'lVim1i-11l1'1QJj,j"imi1~~l~un (weight) nu·lfo:1Ji1'Jl11!"11~'1 w11nr;h~nu 1111tJ-iiil:1;Jrlli1ij111utJa.J'lV1~1~un
mn~!Jl1ibbfl1-iim;1!1bn1tf1111JJ '~i1~11'jUn!1111UiltJmLtl<i11i-1li1ii'u i;im'lum-,rim1rni'.1Jl~d
.:
Ftba.Jil
w1
w2
wn
n
=
=
=
=
=
ri1v;mrncl~'61~nm t
~1~un~'lVlnuri1<J'l~~•n111~u'lu'61~nm 1-1
~1wun~'lVlnuri1"J'l~~'n111~u'lu'61~nm t-2
tl1~un~'l>i'nuri1<J'l~~bn111~u'lwih~nm t-n
~1u1u'61~nm0~1'ji-J111~'l1i'lum1v;mrnrubm~i1b1ilu1'lrh

• •
~
~
i:<i1'>1:i.J'111l~'11t1V1iinVi~V1:i.Jil<J:;<if1J~b'l'i1nu 1
LiltJrf1111utJa.i1Viil1V1iin w1
"i!:;Lci1Vinu-ii1li;Ji1<i1'lil w2
,:;1mVinu-iieii;Ji1ililm<J1n-iieii;Ji1<i111<il
v d ,:,, .... ~ .... ..,.,,... ,,J,.. ,,%
ufl:;nii:1-11•'>1JtJ'1 1:'Uilei ulVIUn•:0:1-1mueitJmb:l-l1J'llm;Jmfll'll.I
v ' "'
m!luHn 7.3
i11~~ilm-,h~~1ui:<li<11•'14ei~11~umn1.-i vhn1'>'Y>ltJ1rnillm1:1-1iiiei~n1">1i'1!.i"iufi1•>1~!.1•'1~eiu~
••uuci1~i!1V1iin'll<N-iieii;Ji1 3 b~eiu~Nl.1:1-11L•m1Vi'11i!1V1iiniii~i1
.urnull
0.50
0.33
0.17
A _J i-'
2 blililU'Jbbtn
3 b~ilUfibbi'1

~
tl"lmwm1iil'11Mm~117~ • '!W!nl flWWlfl~IU
' ' ' •
t At F, (fllt>tilmfl~eiufimJuci1"u1111iln 3 t~eiu)
imolfllJ 100
~ ' 120fJlJ111'l.hTI
A
130lJ.hlfllJ
0.5(1~0)+0.33(120)+017(100) = 121.6t!Jl,fltJ.h 160
'l']1"11lfllJ 190 0.5(160)+0.33(130)+0.17(120) = 143.3
l1).hltJ.h 230 0 5(190)+0.33(160)+0 17(130) =169.9
mnlJlfllJ 260 0.5(230)+0.33(190)+0.17(160) = 204 9
il"mflii 300 0 5(260)+0 33(230)+017(190) = 238.2
ifamtJu 280 0.5(300)+0.33(260)+0.17(230) = 274.9
~fllfl~ 180 0 5(280)+0.33(300)+0.17(260) = 283.2
'l']flijf11 tJ.h 160 05(180)+0.33(280)+017(300) = 233.4
iim1fllJ 0.5(160)+0.33(180)+0.17(280) = 187
'lln"1T'i1"~ 7.3 'l~tt'1fl"1Viti'.iullf)llm-;.-lm1rn1'1u1% '11t%~mfl~eiu~tt1Jrnh"il1V1ii:n'IJei"1ieii.J'1
• •
3 tllieiu t'llu tllieiutmnuu '11Ym1mrn"l4i,1n'l1.1V1ii:m,Jrnfiufi1'l7" 3 tlli1lu~ei
= 0.5(130) + 0 33(120) + 0.17(100)
= 121.6
' (;' di. 0 ~ ._, d.c1 ' d .... ' "" ~ ~ ....
fl1'lmmrn'llmtfl€JU'l']1"1Wi1l-J 'l~UlU1VIUf1'Yll-Jfl1l-Jlf1'Vl')'1~Ulf11Jfll'l':i"t'11lUtl-J1"11'JU UlVIUn
' ,., ' ' !!.-
,,;:,,;: I OJ I ""' 4 ,,;: o V """"" I £1 d, V I = .:I( <V ft OJ ..:!
'YI:J.,Jflli1l"m ~Ulf11Jfll'li"tfl1lUlJUlfll-J ttil~UlVIUf1'YIl-Jfl1U€ll'J'YI')fl~Ulf11Jfll'li" t'1€lUfll-JJl1'IUjj fl"U
F = 0.5(160) + 0.33(130) + 0.17(120)
'V>l.fL
' '~mm1l'lmmrnt'11lU'l'lt]1"m>il-l = 143.3
1uwhm.h"~ 7.1 ""~<1f1mh"~ 7.3 d'Ju1i1ll-lil'li'1t~mnu '~ti'.iu1;1111uffmum1rndm11~1l. '
'lmmrnttuu'11t%~mfl~1lu~tt1J1Jri1"il1V1ii:n 1'11'!1Vim1l-Jct1fil1!fi'u1irn;Jil<i1')'1mnn-i1 ,~1ViNl1m-;
'lmrncl~1nii't~l'J".i1'7"mnn11m1'lmrnrnmJu.11t%~tJtfl~1lu~ei~1"~1u tt"1~"~ct1filJ~1lmit~ein
.i1il1V1ii:nm1i'.1'11t~1L'1 btil~m-;i'.l"ii1u1u1irn;1'1 (n) t~lLfl~~tVlm~'1l-J

nTni1eimili!1v11Xn 1um,i.Jlju<1l<i'fei,1-n'i.h:;num;ru •m:;m;V1fl<1eiu111m1.1fr¥, L1'lei1111'11Yi
•111m:;<1lJn1J'lj'1-ii'm;JAmnriiJ'1 1'11.1liei"11lJn{;]ii'1'1 1iJ11 oii'<ii;JA1uei~"1riLYl,N1u1iJLi'.Jwii1iJ,~rict1'1QJ
rin'111ei:;1;;J:;Lnfl~u1ueimfl"1 ~'1Vf.i1il1wunmnrin'1• •
~ .. Ji .. .; v ,_, -·~~~
11tM'11flftfl:1J11u0min1:::1·nNn 7.2.1 mn rn>lil11{Jllllflllm11J 7.2.1
1uuu1m1fln:1J1mha~ 7 llll'i-1~ 7.2 !~M~ 7.2.1
1u1li'llei~'11'"~1.1Lfl~eiuriei!J1,~1mm:;'11L%~mfl~eiu~LLurni1,J:1V1iin l1-ii'<iLfi1.1~~'1mmcl'iii:;
<i'ml1-ifom11uei~"1ri~m'iiei~ei!l~1wmmnl'1<i<llJm; ~~Yi1(Vf,fim1mLLA:;'111-n'~11.1n~ m;1mnm;iJ'fu• • •
~1.1rn~n"1ti.Jmm'li1.1A (Exponential Smoothing 'Vl7ei Single Exponential Smoothing) Lum;
l'1mmru'1:;<1:;wnrnl'1;1d:iJ<i'ei~•Ru-iieii;Jiiluei~"1Lllu~1mumn 11ii:;'lnm;iJ'fu17!!rn~n"1tiJmuL'lim1 1liu
1TIV11i~riu1.1lJ1oti'lummmmcl ~'LllUL'Vlfluflri1Vlm1:1cJct1'1QJnuoii'eii;JA'11iJ'1mn~iJ'1
O:;Jm;1um11'i1u1mfi11'1mn;ruli1~d
I F, = F,_1 + a (At-1 - FM)
1l11l F, = .11'1rnmm"lw1h~rnn t
F = '111'1mmm"lw1h~nmri t-1t-1
At-i = '11'iil'i~~1n'1~u1u'll1~nm~ t-1
a = fi1fl~~Yi11VfL71.1u (smoothing constant) 0,;; a,;; 1
'ii11n<1:1cJm;n<i111Jl5nm.h,'Vlrt~11 mmmmclct1111'l'u'li1~nm1fl'l ~1l m;u1.111'1mmruLriei
~1'1rimi'1 miJ~u<K11.1'11i'.l'1'1m'1'1m~m1'1mrnm"lu~1'1riLL'11 il1m11'1mmru1riD~1'1riLLii'1l1'11mn
n11'11'iil'i~ "1~~u'111'1mmm"l111:iJ'iii:;lniJ'fu1V1~1ii~mLllu a L'Yi1'll<i~m1lJN'1'1m'1 LLA:;i'11m™mmcl
1rim1'1~LLii"ll1fil'u1l1.1n-:i1'11'iil'i~ '111'1mn;m"l111:!-l'iii:;lJniJ~u1Vl,]'~uLllu a L'Yi1'll<i,m1lJN'1'1m'1 ~'
il~m"11&111 l11Y11.1~'11 3 '11L'Yi1~urio/iei~l1m;~'1LRu-ii1li;Jl1~1l '111'1mn;m"lu'li1~nm<i1fj'1 (F
1
)
-iiei:1cJn.11'iii'l~ri,n'1~u<i1n'1 (A ) •m:;.11 a 1'11.1ii'11iJ1uTI1fi'iil :iin1oti'lu'li1~1:;11111' .05 -50 ~~'11 a 'iii:;! ~ t-1 q
""' v .,. v If ""' G g_, A
lJn'1'iil11m1'iil1niin:i,;m:;'lJmNA"1Jlm'YILLA:;m1lJfl'1L'VIU'll<i~ r;jlJjV!lj

'
rnh111111m'™mmcl1!1t11llm'll.l':iut'it1ut5n.fll.I•um'llm1 6bYJfi"J"i~6bii'1~€Jm'l'INmn'lcl1i1LI
15'116%~t16.i~eiu~6b1J1Jtl1~~11'1itn ~l'.lm1i.iil~i~t1ilmn~u ~~mm'ltifij<JUlVf•~ul>l~d
"Jlntll-lnl'l Ft = Ft-l + a(At-l - Ft_1) .. (1)
b'lit1uL1'1;;J11fd'Ju Ft = aAt-l + (1-a) Ft-l
bb1'1U Ft 1!1t1 aA,_1
+ (1-a) F,_1
"~'lurri.im'l (1) 'VJffl t
,~111 F, = aA,_
1
+ (1-a) [aA,_2
+ (1-a)F,_2
1
= aA,_1
+ a(1-a) A,_2
+ (1-a)
2
[aA,_3
+ (1-a) Ft_3
]
= aAt-l + a(1-a) A,_2
+ a(1-a)
2
At_3
+ a(1-a)
3
A,_4
+ (1-a)
4
[aAt_5
+ (1-a) F,_5
1
. V l - ' 1 1 2 3
~1nrri.im'>'lll'ilu~m111 w1
= ~, w~ = a(1-a), W3
= <;7<1-a) , w4
= a(1-a) , . . ,
'll~,~•~u11111'11u11'1iln<J~l'.lmnfi<J'1 fi-ifm;1'1li1<jil ""~'11u11'1iln<Jrnil'1~ (1-a) 'VJn'li1,nmfi
d'lu€1$iil illl-1ill'll'~ 7.4 '~''"ii''lVimm'lm-if1hli,biu1~ild
'
.;
flll'l~'ll
smoothing constant
(a)
a= 0.1
a= o.5
'v ' ..
l'l1!ltJNl'l 7 .4
''111~!1lll
.' . v
l'llU111Ufl
J111Ufl~l.101l111'1U!!111ll~,/1~!1lll
t t-1 t-2 t-3 t-4
a a(l-a) a(l-aJ
2
a(l-a)8
a(l-a)4
0.1 0.09 0.081 0.073 0.066
05 0.25 0.125 0.063 0.031
f11~~ilm'l11ivhm".i1Nmn".irn'1111 Lu6~€1ul'.lm.imhiNmmclm1i.i<ifo1m".i6'14iJ~l.l':fummf!
'lJil1b~iJU6l.J1'tlLIU6vilnlJ 120 6'14€l~ W1lJ".iln~1ltrlei~m~€lU6l-11'tl1.JU <fll'lJlLI"J"i,'lJ€l~i~€lU6l.J1'tltJU l'.j 160
6'1~€l1 ~~'1m".i11Khi!'iein<f11.i1~ smoothing constant diu 0.50 n1".i1Nmmcl.i11i.i#lei1m'l6~€lU
1N'11'tJl1'1l.J~ll'1itilt16b1'1U.i1LUrfl-lm".iil~tt
=
=
=
Ft-1 + a (A,_1 - F,)
120 + 0.5 (160 - 120)
'
140 6'1'iei1

mmrn 500
F1
= F,_1
+ a(A,_1
- F,_,)
400 a= 0.1, o.3, o.5
300
200 a= 0.1
100
J F M A M J J A S 0 :J D J F M AM J J A S 0 N DJ F ~~f.l'U
m.Ji,<ii'urnTI'lmrnrn 1urnrufi1i-Jj'.j·iie1:1;j!1'1l1'1mrnru.1U<l&i"1m1vr 'ltf11TI'Imrnrn'l~b~l-l<ii'u
'l1rn1111~ 2 dJu<ii'u1iJ 11111'Jri11'1u111'11v..mrnrn'llm'l1111~ 2 •l'i1nufi1,~-:i'llei-:i'l1111~ 1 i1'1iii'lmh~~ 7.5
l'l10rh~ri 7.5 h'l'l1ue..ii1"1Lfl4ei,iJwm rn.rj'.j-iJ!l:l;J!1'11eJ'll'JeJ111'llll'JJi,w;i1,&i!lUi.Jrn1fll-lfllJ1,J-11'1-1l'JU
.. ..!~0il fl0~'Ulfl (11'W~)
l-ln"llfll-1 100
• <
120nl-!Jll1'1Uli
'd
130l-!Ulfll-1
bl-11'!-ll'JU 160
<fi'<l'If11TI'l l'Jln"llll'J!l111'llll'Jb~!JU1'1t]1'tmfll-1~1l'J'iTim"liJ-J'ub'JtllJb~n.fl:JJbLU.lb'JltJ!'l 1111l'Jfil'VIU111fi1
a = 0.5
1uil1ei ~1'1ii'ti-i1<i'lVf'ii11'1mrnrn'llm b~!lu1111m ~'l<ii'mb~l-l<ii'ul1111'Jril'VIU11iLVf'ii11'1mrnrn'll!l'I
' ' .di .... I! ' .... "" ... ... ""' 1.J ' I{"-' ( ,<!! ""
1,111eJUf1l-lm1'1Umrnnul'J1Ji11'llll'J'llmbi11eiUi.Jn"llfll-1 fl!l 1oo bm!l'I bb!'l~b"ll-1"1U1'11'111'1l'Jln-mb"1'1bb"1bi11!lUi.JU1fll-1
•llwiiu1iJ i1,d

:;nni':J.Jm-;; Fl = F a(A -F )
H H H
<A d
F + a(A - F )fllf'j mrnmLilil'U:J.J.lfl:J.J F. =:i.J.fl. n.'Vll. n.'lfll. n:v.i.
= 100 + 0.5 (120 - 100)
= 100 + 10
110
d
= L'1'J1l'
<A
fllf'j tJln"lllibli11l'Ub:J.11'11tJ'U = 110 + 0.5 (130 -110)
= 120 Lfl~m
<A
fllf'jtJlff'llliLil1J'U'V'j']1'flllfl:J.J = 120 + 0.5 (160 - 120)
= 140 Lfl~fl'
fln!iiif>flAl a timm~lf~ 'lim-;;tl:;rn'forn5n.fri:tlL'Um:iitJfiifeJ,m-;;fi1 a ~bvimnl:J.J ~'"~
'Yi11Vlfi1'V'jmrnm"lnll'L~tJ,nufl'll:J.JLD'U<J~~ t'l1eitli1~.i1~nmu~fiflu.jf1~fl~~ fi1 a "~ilfi1u1ltJ u11it'l1
eJtli'~.iilm-;;Lvi:J.J~'UVl'iflfilf~eJti1~-;1ilb~1 L'llu 1'ufl1tl-;~bl1'YILL'i1~'U"J~1VIfi1 a :J.Jl LY11l<m :J.Jm'J
•
Lti~tJuutlm1J!YrumJ1~hlil'l1:J.Jm~<J~'V'jtJlmm"lVi'nniif1l~mn~filNu~1~~iiffl~Yi1u1tJfll a ~LVlm~f:J.J• • •
nfl'U ~~il'lim-;;1um-;;muf):J.Jfi1 a 11i1tml'JW<J1-;;m1fi1m1:J.JflmilLfl~1l'U'llfl~m-;;f'jmrnrn (forecast er-
ror) Lri1l'Ylilffl~li'fi1 a 1111~'1 $i~d
,; 'fl1 mfllllfl!flllf>ll'Uf>~mwmi 11~ru
=
A -F
t t
n
= .Le1=1 t
Lilfl et $if] fl'll:J.Jflmilbfl~fl'U'llfl,nl'J'V'jtJlrnrnrllVIW'li'l~L'lm t
1u~d"~LL<'1~'lin171111fi1m1:J.Jflmiiibfl~fl'U'llfl~m1'Y'lmrnrnJl'ltJ'lim71111fi1Lu~tJ'llei~'hmrim-
' '({ """"' I A ,"! 0 V
blJ'Ui':J.Jj7m bbf~'lun171111fl1LuftJm1:J.Jflf1flLflMl'Uml'M<fl~
1) fllL:~tJ'llml'l1ubflmLlJ'Ui1:J.Jjorn (Mean Absolute Deviation; MAD) mTJilfl'll:J.Jflmiil
Lf1~flu'llfl~m-;;'V'jmrnrniii~vi:J.Jiii'llivii1~ $11) m-;;1-iifi1Lu~tJ'llei~l'l1iurimrnui1:J.J1.J"rn (MAD) fi11-11m1iiitJ
' .o I ,<i V ({ ~ O ~ OJ -=i.
m'J1-llf11Nf"l1:J.J'llmm1:J.Jf1miilbflfflUft:J.Jj'Jlli (absolute value) Vl1'Jli1'ltl"J1U1U'llfll;jfli1~'U
MAD =
L lm1:1-1f1miilbf1~fl'UI
n
A
MAD
~1letl·wm =
n

'
w11JaNn 7.6
~<rllnl"lNlLJ Nfl"1<ifei~m"l'Vlllileium"l1'6'i'fim"lwmn"lrti<lhu'i:fim,11''ub"i!JuL~n.flumub:if!J9
'11111''um"lwmmciu~mru.m1:w<if<Mm1!1ufi1'JJEJ~h~~1u~~l1'6eii,il1'11'J~~ 9 b~EJu "11:W"11'l1~~ 7.5
bbl9:m1m1bm!JrnYi1.1u.11m1:w'1mllb'1~ilu'J1nm'll'6'11 a ~11-1.11-1. 2'11 ~il a= 0.10 bb9~ a= 0.50
1l11.11ub~<JU:Wn'lll'l:w1Jillm"lwmmcim1:w<if<J~m,fiufl1li'b'Yi1nu 175 viu1!J
. 'f]:W1111"Ull
i'Jm'1:w
bi.JlfltJU
Y<)1'rl11fl:W
il!JU1tl'U
mn!Jlfli<
l'l~Vllfl:W
. 'f:J.l.Jfll'lfl...J.15
iJUlflaJ
i:!Jl~f)tl'j,l,
Yl~1'!fll'1>J
~fJ'UltlU
"'"ll'"'"ftJ11'1>J
180
168
159
175
190
205
180
182
180
168
159
175
190
205
180
182
q ' , , ~
m1Hn 7.5 11mm111n.m1111111 'If a 0.10 irn:: a 0.50
I J 6} V
11mmmru•111 a= 0.10
175
176 = 175.00 + .10(180 -175)
175
173
173
175
178
178
' '
I J ~ :V
!ll'l'Wlfl1!l!*!'lf a = 0.50
175
178 = 175.00 + 50(180 -175)
173
166
170
180
193
186
q ' ,,
11111~n 7.6 111 MAD m!l a 0.10 1m:: a 0.50
175
176
175
173
173
175
178
178
a= 0.10
i1so-1751
j168·1761
MAD
MAD = 84 = 10.50
8
=5
=8
16
2
17
30
2
4
= 84
175
178
173
166
170
180
193
186
a· o.5o
jlB0-1751
1!68·178j
MAD
MAD = 100 = 12.50
8
=5
=10
14
9
20
25
13
4
100

<Jln<ill'll~~ 7.6 1'11 MAD bilil a = 0.10 i11'11iieitJffl1'11 MAD b~il a = 0.50 f!'jl.JL.111
a= 0.10 blJu'11~b'Vl:J.11~l1l-liif1Vl1'u.l11u1.jflunT>'V'lmm~1nn11 a= o.50
' '
2) fllb>1titJm1l-l<11m<11b<11tieiufi1i'~f!il~ (Mean Squared Error; MSE) .Jwr1tium1l-1<11m111
b<11~eiufi1i'~f!ei~ (MSE) blJui:in'lilwd~1unT>1<i1mll-1<11m<i1b<11~eiu'llei~nl'o'V'lmmrn MSE fiei1'11c>1~EJ
'llmm1l-l<11m<i1b<11tieiuunfi1~~fiil~
MSE
MSE =
'v ' d
l'l'JOl'IN'l'I 7.7
L (mll-l<ilm<i1b<il~ilu)2
n
n 2
Let=1 t
n
<J1n-ifeim11wibmh~~ 7.6 bbf~<ill"ll~~ 7.6 .l1m,1u1ru'11 MSE <illl-l<ill'll~~ 7.7
•
a • 0.10 a • 0.50
;j.Jn";llfl;j.J 180 175 25 175 25
" 'f))JlllV'lU'D 168 176 64 178 100
iiu1f13.J 159 175 256 173 196
Wit1t!U 175 173 4 166 81
'l'lffj'lifllfJ;j.J 190 173 289 170 400
i1tJU1!'J'U 205 175 900 180 625
mn!Jl~~ 180 178 4 193 169
~-lVl1'fl3.J 182 1'18 16 186 16
~'fl'S"13.J"IJ€!-l l}Jfl".113.J"Ilfl.:J
{'fl11:i.Jfl~f1(f)Lfl~'i.)'U)2
= 1,558 (fl1l;j.Jfl'flli;Hfl~D'U)2
= 1,612
MSE =
1588
MSE =
1,612
8 8
= 194.75 = 201.5
<J1nm1cu'itJrni1tJu'11 MSE b~il a = 0.10 bbf~ a = 0.50 1h1nJJ111'11 MSE b~il a = 0.10
i'.ifi1iimm11 bbM1~11'11 a= 0.10 ,1Ju1'11~b'Vlm~fil-ln111umm11u1.jflum-mmmrn
. ' '...,. ~ 4 a .q " '_I -l""'v,c,,""'
'll'<Hlllflrlll'lfl!ll01'111!1"l~!~O~'l'I 7.2.2 uin m~Yiu{IUYlfl'ilfl111J 7.2.2
' . ' ., d: ' ""' "" ."! q
lll!ll111l~Yll11fl'llll11'lfl 7 YIOll'l'I 7.2 !~O~fl 7.2.2

• •
!j0.;in 7.2.3 Double Exponential Smoothing
1.-iulin~Lrie:rvi1nT;mmrncl1~u~~uifn•~1i.J1ili'l:vim1:mhii11JnJJri1m.n1ui.i (Trend) LL<lid'.l!l
vm1rncl1rn~ll1unm~~ulU<J~L~J.I'lVim1i.ii114i'11Jn1JrllLLU1Lui.imn~u "lilm™mrncl~l'lm1~<Jl'lllil
!i~!l~illi'lm!llJ'll!l~bLU'lLUi.i'lum.,,,..mrnruLLlJlJU-J'rn'lurn~n.frTliLUUL:iimli L'lun'i1 Double Exponen-
tial Smoothing ~~l'lm·:ni1111Ullrll1"l'llfo11J!l~ 2 o/11 ~!l ri1 a (Alpha) m1~ Y (Gamma) T.-iu1ilful
<1~m~'l4tlum.,,,..mrncl Jr~d
..LJ.l!l
•
FT,
s,
T,
A,
a
FT, = S +Tt-1 t-1
s, = FT, + a(A, - FT,)
T, = T,_1
+ Y(FT, - FT,_1
- T,)
= ri1V<mmrn"lmh~mn t
= ri1V<mmru<J1nm'l'i11'lVil'lU1J (smoothed forecast) 'lmh~rnn t
= rllU'l~mlli'll!l~LLU1L~ (trend estimate) 'lmh~nm t
= ri1<J'l~'lu'!!1~nm t
= rll'1~~~villVlL'lU1J'l~Vlll~.if!ll;ji9fi'1Jrll1"mn'lru
y = ri1m~~vi1'lVJL'lU1J'l~1111l~LLU1L~-'l~nuri11h~mm'llmLLU1LUJ.1
rll a LL9~ Y l'lrll!l~'l~Vl'il~ 0 - 1
oU'il!llfl'illll':iftl'il1ill
""' ! V c ! ff 1' A<>I
<l~~ll11ll!l~nl'l'11U1lli'111"Ulf1'>Dlll1mIT Double Exponential Smoothing
i11111-J'uf!1JlllVl~ 7 <J~villll~~Ull!lUJi~d
1. rllVIUllrll a bb9~ Y
2. "11u1mri1 s6
<L"9~ T
6
rieJu~~4imL4i.i4iuV<mmclJi~w1iL~!lu~ 1 d'.Jwliu1u
3. .Jm1mri1V<mmclilu.-i1Vi~ 7 FT7
= S6+ T6
4. m'l"11u1mri1V<mmru'll!l~i'lli.-i1Vi<lieJLli vi1m.,"11mm s7
LL9~ T
7
<Jln)1111
s7 = FT7 + a(A7 - FT7)
T
7
== T
6
+ Y(FT7
- FT
6
- T
6
)

v ' ""
ffJ01'JH'l17.8
fll11Lli'l"llu'Yl4 <ii8'1m'l"'1mnwlmw1'll111~1'1K<11~.JLlJui{ufil'lJU<i11'1ii.J'llil'1n'Jm'l" LYlil'J~1~
'l1'1LLN'Unl'l"'llm tJrilK'1nl'>N11<ii Lbfl~~ii L<il~tll-l'Yl1'1'1m mihwl'um-,,'il Lu'1-1'111-1 fll11Lliillu'Yl4Lzj1111iiil~fl, .
tJil<i1'llltJLut1&l<11'lt1.J 6 L&lilu~N1um-!iu'J~LlJwil1LL'Yl'Ulum'l"m<11fl~LUtJ8il'llltJluil'Ulfl<il1~ LLfl~1~L~iln
1-iiL'Ylf!Ufl Double Exponential Smoothing 1um71'1mrnrntJilil'lllm~eiu~ 7 lliltJL~'11 a = 0.2,
.. l'J!lVl'IJll'J (~IW1'11)l>101.1
(t) (A,)
1 130
2 136
3 134
4 140
5 146
6 150
"""' '1 llfll'Hl1'1l1W
21 f( a: d ~2' ' '
<11il.Jrn'l"V1mrnmtJeilil'll1mililu'YI 7 <nnl<il" FT
7
= s
6
+ T
6
'll.J<ilil.J'Yl"1rn1 s
6
LLfl~ T
6
neiu
'1J.J<ifil.J,'i1Luum-,~'1"1il11.ld
.,!21 ~ •"-'""'d
1. L'l"~<il'U'11'Jln'l"11tJilil'llltJ'111'1Wiilil'U'YI 1 (FT
1
) rim.1
11rn-, $iJ1~'11V1mrnrn•Yi1fiu'1111iJi;i'lJ111<J1.J'llil'1.~ilu~ 1
•
~'1UU FT = A = 130
1 1 ' '
2. 1.1-,~mt11'1l'llil'1LL'U'lliil-J (Tl) lliltJU1'1l'J"i'1~8'1L~il'U'11 6 fllJ '11'J1'1'llil'1L~il'UVi 1 mhvm
~ J._'4• < A . . , j . . S p j . o !
il'ltJ 5 'lJ'1LUWill'U'l'U'llil'1'lJ'l'1L'lfll"JlnLilil'U'YI 1 b1'1Li18'U'YI 6
Tl
A6-A1
= 5
150 - 130
= 5
= 4
3. ¥i11-11m'11 s1
'Jlnj<il'l" s = FT, + a (A, - FT,)
t
LL'Yl'i-1'11 s1 = 130 + 0.2 (130 - 130)
= 130

', ~ ~
4. 'V'ltJln'l'illtJilo/l'l!lrno/leJU'Yl 2
"Jlfljo/l'i' FTt = S +Tt·l t-1
Lb'YIU.11 FT2
= S1
+ T1
= 130 + 4
= 134
5.
', ~ ~
'V'ltJln'l'illtJ1Wl'lJ1Llbo/leJU'Yl 3
FT
3 = Sz + Tz
sz = FT
2
+ a(A
2
- FT
2
)
= 134 + 0.2 (136- 134)
= 134.40
Tz = T
1
+ Y(FT
2
- FT
1
-T
1
)
= 4 + 0.3 (134 - 130 - 4)
= 4
•
lii,iuu FT3 = 134.40 + 4
= 138.40
6. Yi1m™mrnrntJeio/l'l!1rn~eiu~ 4 "OJuii"'~eiu~ 7 1Ji'11i<~uo/leiu.if1"~uo/11i<<il1~"~ 7.8
LLl'l'1'1l'l"m"'11w'lrnlii',i~a1ud ~"":;1J'.JTvm1rnrnL~au~ 7 ~~a'1m1i;la 153.09 ('W'ulJ1'Yl) 'Y1ia 153,090
lJl'Yl
FT7 = SB+ TB
SB = 149.28
TB = 3.81
FT7 = 149.28 + 3.81
= 153.09

<:: <f ii ....Q • •
Vll~Wfl 7.8 Jl.lllflWVWlfl~ll!Vl1!111l Double Exponential Smoothmg
(t) (A,) FT, + a(A, - FT,) = st
1 130 130 + 0.2(130 - 130) = 130.00
2 136 134 + 0.2(136 - 134) = 134.40
3 134 138.40 + 0.2(134 - 138.40) = 137.52
4 140 141.64 + 0.2(140 - 14164) = 141.31
5 146 145.17 + 0.2(146 - 145.17) = 145.34
6 150 !149.io I + 0.2(150 - 149.10) = f 149.28
I
(t) (A,) T + Y(FT, - FTt·1 - T,) = T,t·1
1 130 given = 4.00
2 136 4.00 + 0.3(134 - 130 - 4.00) = 4.00
3 134 4.00 + 0 3(138.40 - 134 - 4 00) = 412
4 140 4.12 + 0.3(141.64 - 138.40 - 412) = 386
5 146 3.86 + 0.3(145.17 - 141.64 - 3 86) = 3.76
6 150 3.76 + 0.3(149.10 - 145.17 - 3 76) = ITfil~
(t) (At) s,.1 + Tt-1 = FT,
1 130 given = 130.00
2 136 130 + 4.00 = 134.00
3 134 134.40 + 4.00 = 138.40
4 140 137.52 + 4.12 = 141.64
5 146 141.31 + 3.86 = 145.17
6 150 145.34 + 3.76 = I 149.10,
7 - 149.28 + 3.81 = 153.09
t
' ' ' l
q d 1 Q <: 4 &:I
!'llll'll1fll1ftfllfll'l'll1tl'l1 7 116'11'11 7.2 [16~'11 7.2.3

' '
!~0-::in 7.2.4 Double Exponential Smoothing: Holt's
Two-Parameter Method
lTifll'l'V'Immrn'll!l' Holt Loiifiu.jf€l:l;Jilriiii:'ni,;ru::;Lb'l-h11um'Jl~bftU<11'l~ bbi1!!J'.iV1i:'flm'l'l1€J~fl1'llVf
.,..., v i i I ,,,l c<v J .JJl..o 1 ii 1 JK
m1!Jmfl'lJflU'll!l:l;Jilml'j111mm1'l111 Lm:;m1lJmfltjJ'J!!ii111ii~ L'lilt.1'1 L!JEJ'llil:l;Jil'll~'Jlfl'll!l:l;J<im'l111mmm
L4ilt.1'1 'ili'llm Holt 'il!!l'.imirilV1U11181'V'll'lli'lb<11EJ7 2 81 flil 81 a Lb<i:; Y 1111t.1iift!Jm~1-ir'tumi
y
F = s,+t,mt+m
.o'( ' fl.' ... '
<lil '1l'V'l1.llfl'llli'Vl'Jf'Nnm t+m
= ax, + (1 - a)(s,_1
+ b,)
= Y(s, - S,) + (1 - Y)b,_1
f!EJ 81fl~riri'Yi11ViL~t.1u1:;V111~.jfEJ:1;J<ifiu81'V'lmrnrn
f!EJ 81fl~riri'Yi11Vi.~t.1u-;:;V111'Lbmtu!J~"nu81t1i:;iml'llil"'m1tu!J
81 a n<i:; Y mh:;wi1" o ~" 1 t'f181L.if11ni:' 1 LLft111"111Vim1!Jih4'n1nu.ifeilJ<iri101111mnri"'111'IJ "ii:! 'IJ q ..~
1um1.-i1mrnV1181'V'lmmrn 'il:;<ifei,iimiri1V1u11181L~lJJiu'llei,81 s1
LL<i:; b
1
<ii"d
LVl S
1
= X
1
(X2 - XI) + (X4 - X3)
=
2
'
1'11tiVHn 7.9
m11.J~m1Jid1u.,-m11!J.jfEJ!J<it.1ei111'1J1t.1L<l~ei~<if111V1ru1mwti,ri111u•ei,'i'ui1111'lleiueitJ 1uL<11'l!J111riq 'IJ ., q 'IJ
v.J1'1-b!J1'V'lu11.ifEJ:1;J<iiii:'ni,;ru::;'IJEJ'LLU'l1iilJ ~'1mli'llil~ Holt's Two Parameter 'Yi1m1'V'!mrnrnt.1ei111'1J1t.1
LUEJUlfl<11 Lm:;ri1V1u1111Vi81'V'11'lli'JL<11€17 a = 0.2 bb<l:5 Y = 0.3

i~mrnr
d
!'l!l'1'lll!'l (!m!l~)
1 130
2 136
3 133
4 144
5 148
6 155
7 150
8 160
~ ' "" IV
nl'lf11'llU'il'11b'W'ilU
1~ s1 = x1
.. s1 = 130
1~ bl = x2 -x1
= 136 - 130 =6
nl'VIU'ill~ a. =0.2, y =0.3
82 = a.x2 + (1 - a.)(81 + bl)
= 0.2(136) + 0 8(130 + 6)
= 136
b2 = y(82 - 81) + (1 - y)b2-1
= 0 3(136 - 130) + 0 7(6)
= 6
F2+1 = 82 - b2(1)
= 136 + 6(1)
F3 = 142
83 = a.X3 + (1 - a.)(8
2
+ b
2
)
= 0.2(133) + 0.8(136 + 6)
= 140.2
b3 = y(8
3
- 8
2
) + (1 - y)bH
= 0.3(140.2 - 136) + 0 7(6)
= 5.46

F3+1 = S
3
+ b
3
m
= 140.2 + 5.46(1)
F4 = 145.66
s4 = 0 2(144) + 0.8(140.2 + 5.46)
= 145.33
b4 = 0.3(145.33 - 140.2) + 0 7(5.46)
= 5.36
F4+1 = 145.33 + 5.36(1)
F5 = 150.69
V11~Hn 7.9 mwm111~ul;'11'J"ili Holt
..; ,.nvi11 l'J!lVl'llll'J (!ftW~)
x, s, b, F,
1 130 130 6
2 136 136 6
3 133 140.2 5.46 142
4 144 145.33 5.36 145.66
5 148 150.15 5.20 150.69
6 155 155.28 5.14 155.35
7 150 158.34 4.52 160.42
8 160 162.29 4.35 162.86
= S
8
+ b
8
(1)
= 162.29 + 4.35(1)
= 166.64
= S
8
+ b
8
(2)
= 162.29 + 4.35(2)

'
.!1'V1rnmiil'1111Yi 11 ii'iil F8+3 = SB + b8(3)
= 162.29 + 4 35(3)
d
= 175.34 bfl':l€1~
I ff ..J A
F8+4
S
8
+ b
8
(4)fll'V1tJln'llli~111'YI 12 '1€1 =
= 162.29 + 4.35(4)
d
= 179.69 fl':fm
. ' 'Q.I 4 ~ .t:!!. &:t v 'r.I .I""' v.<:>."'>
11M'il1flftfl'!l1!'1J0l11trl'l:::t1M'l'l 7.2.4 mn rn'l"l'lufllll'lfl'ilf11W 7.2.4
' ' ' ', 4 l -=:!. ~ .t:!I, "
'IJU'IJ1f111ftfl'lfl'tl'IJ11l'l'l 7 l'l0'1J'l'l 7.2 110~'1'1 7.2.4
• •<!! "'! '
!"Jfl~'YI 7.2.5 Winters Three-Parameter Trend and
Seasonality Method
1umru~iei:1mi1li'rn:Jm:;iiu1t{(~iLl1:::l'.lii'Yl1i'V1'1'11€1~n111ml1111 tJmm-,111oiim-,'¥1rnmlli111 tJiu. .,.
Winters ~~l'.lmmT1'1u111'11'V11'lli'lL'fl€1~ 3 '11 flil a (Alpha) Y (Gamma) LL'1:; ~ (Beta)
'11'V1mmlli~'li1~nm t+m 1oiilillJi'i'mtrn11 Ft+m
F = (S + b m) I
t+m t t t-L+m
' ~ ' v ){
b11tJ~l'f~fll'l"'fll~'l 111~14
l'f~rn.,urnw~u1VIL~tJufiei
st = a Ixt + (1 - al (SH + bt-1)
t-L

x,
I, = f:l - + (1 - f:l) I
St t-L
bl'.iil L ~il •1b~m1:wm1'1Jml]~f111" b'llu ~1U1Ub~iluvi'l"il1'117mfl'lu 111
~"',~ d~""
ll1bIJU'1Jil:i;j1"71tWililUbU 1 lJ L = 12
d'Jwiiil:1;J1"7lt1L111'>mfl'lu 1 l'I L = 4
iil10aN~ 7.10 u~lffi'l'n'lfl~L.'lbnu.if!l:l;Jfltlil'11'1J1tJRU~1'llU111Wd~'1Jil~lJ~lffid'iu'JltJL'117:wlfl~1U1U
3 l'I 'V'lu11tJill1l'lJ1u'lubbl1ifl~1111-,m11ililni,;rn~~d'Ju,i1~1bu1tii:wbbfl~blluri~mfl ~~14i'lmrr'JJEJ~ Winter's
Three-Parameter Trend and Seasonality Method m'l.if'lum'>'V'ltJ1f17rnf.lill1l'lJ1tJlUilU1'1'11 b4:w
iifU'1111lftmrl1111Ui11'111'1171i'.Jbl1lil-fl.'J'lVf a= 0.1, Y= 0.15 bbfl~ f:l = 0.1
ii~ iwnrnr Uell'l'UlU
1 1 25
2 45
3 88
4 9
2 1 42
2 67
3 146
4 48
3 1 124
2 160
3 225
4 125
. ' = 2'
f117f11111U111'11b'J:W111U
m'>b4:wiiiu'lumw11VIb'fou1.'l"i11n11:wm"J.i'~d
SL = XL
m-,14:wiiiu'lum-,1111'1111~m'"1111u1111'11811171ft1U'1Jil~'11"il~~brn~.-11111,~tJ'lul'ILb'>n~~d

1il!111fl1'111111
L X
X = 2: -'
"°1 L
1u411il~1.J~•~•~u1#1111iil:J;Jmilu11tJlil1m'1Lu 1 lJ Sl 4 Lmm'1 41.Ji:iu L = 4 '1:i.im1m'l•~:i.i
v
<ilufi1u1rn
111' s4 = x4
I, =
S_
' x
12 =
x2
x
I
x3
=.j
x
14 =
x4
x
x
4 x,
= 2:-
t=1 4
b,
(XL+1 - X1) + (XL+2 - X2) + (~+3 - X) + (XL+4 - X4)
= 4L
s4 = x1
= 9
x 25+45+88+9
41.75= =4

. ' .mii11LJn1mmmrun1v'q7n'<iJ 7-39
25
11 = = 0.60
41.75
12
45
108= - - =
41.75
13
88
2.11= =41.75
14
9
0.22= =41.75
rilVl'.1'1.-1 b
4
b4
(X5 - XI) + (X6 - X2) + (X7 - X3) + (XB - X4)
=
4(4)
(42 - 25) + (67 - 45) + (146 - 88) + (48 - 9)
=
16
= 8.5
85
x5
= a -
1
- + (1 - a) (8
5
_
1
+ b
5
)
5-4
(42)
= 0.1 - + (1 - 0.1) (9 + 8.5)
0.60
= 22.75
b5 = Y(8
5
- s
4
) + (1 - Y) b
4
= 0.15 (22.75 - 9) + (1 - 015) 8.5
= 9.28
15
x5
= ~- + (1 - ~) ls 5-4
5
(42)
= 0.1 22.75 + (1 - 0 1) (0.60)
= 0.73
F5+1 = [S5 + b5 (1)] 15-4+1
= [22.75 + 9.28 (1)] 1.08
= 34.59

,J
'ill'll~'YI 7.10
.;
~1'11'1
1
2
3
4
5
6
7
8
9
10
11
12
s6
0 1(67)
= - - + (1 - 0 1) (22.75 + 9 28)
1.08
= 35.03
b6 = 0.15 (35.03 - 22 75) + (1 - 0 15) 9.28
= 9.73
I. =b
O.l(G7) + (0.9) (108)
35.03
= 1.16
F7 = [35 03 + 9.73 (1)] 2.11
= 94.44
..,,; <! 1! ""'"' •
Vil~Hl'l 7.10 fll~'l'W1ft1U1>11U1fi Wmters
U!l'1'llm (X
1
) s, b,
25
45
88
9 9 8.5
42 22.75 9.28
67 35.03 9.73
146 47.20 10.10
48 73.39 12.51
124 94.53 13.81
160 111.30 14.25
225 123.18 13.90
125 171.45 19.06
r, F,
0.60
1.08
2.11
0.22
0.72
1.16 34.59
2.21 94.44
0.26 12.61
0.78 61.85
1.19 125.67
2.17 277.47
0.31 35.64

if?uLJLJn1mmn7llin1-J~fi'V 7-41
F12+1 = [812 + b12(l)] 112-4+1
F13 = [171.45 + 19.06(1)] 0.78 = 148.6
F12+2 = [S12 + b12(2)] 112-4+2
F14 = [171.45 + 19.06(2)] 119 = 249.39
F12+3 = [S12 + b12(3)J 112-4+3
F15 = [171.45 + 1906(3)] 2.17 = 496.13
F12+4 = [S12 + b12(4)J 112-4+4
F16 = [171.45 + 19.06(4)] 0.31 = 76.78
rn'Wlmn:ml~btJ'ili Winters ii fi11'lmmrn'l~bbi.Juei11niffi1'l~'bflmL'1~u~uilurn'lTi1111u'1
fll'Vil'llfail<i-nii, 3 fll a, Y bbil~ f} ~'l~lVlfllfflla.Jfli11'1bfl~EIU'JIEl'fll'Wltlln'irntt<itJ~lj'1 b'lfU 'llfl
4f1m.h,.jj'1,<ifu'l~biiu1#111fiTvm1mrnti'1i.J1#1.-i1~1nifbl'imilufi1'l~' bbl'l'1,i1fi1 a, Y bb'1~ [) ~1oifti,1Vi
fi11i.Jmm~l'a.Jtll~l'ltlln'irn'llllfi1 a, y bbil~ r> ~1Vifi1 MAD ~llJ'1'l~Yi1LVifi11'lmmrnYi"l#liilb~Elli<l
11
a.Jlf)'Jl'.I,
. '
~ 4 A: J. .d 91 l AjAVAA
'11!l~lll!l!lll'llllt!!l'111ll'l'l~l'll'M'fl 7.2.5 U!l1 !tl'lAU{Jllllll'illl'l'llJ 7.2.5
. ' . '
Ill ~ 1 ~ "" ~ ""
!Ullt!1!ll'lft!llfl'11U1!l'fl 7 ll!lt!'fl 7.2 mM'fl 7.2.5
"I ""' l v
!~fl~fl 7.2.6 fll~'ttlH'l-f'.H'l-nJ
fll'l'llllbb'l-111Ua.J (Trend) bUU'itm'l'Vimmrnb~e:i.jj'El:l;Ji1l'.l'1ni,;~~bi1'UbbU11ila.Ji1fi11'lmmrn'il~
l'.lrn<Jb:tl~tJubb:tlii'1:tJ1ufi.iyrnL'1 'iilm·m1bm1Ltta.JbbUU~ltJ~ij'11'i<i m-:;1oil'i1im-:;rrnl'ln~11iril'1'l'le:i'
UEltJ~l''1 (least squares method) f!·~·mbu1Tiia.J<ifu<11-:;~~'U ~'l'.l~m'foni1m-:;n'1n<im'll'bifu (regres-
'
y = a+ bx
.. y fi11'lmmrn (buufobb:tl<J"11a.i)ba.JEI =
a = fi14l'1bb1l'U y
b =
~ ~
fflla.J'JIU'll<l'bl'l'.hil':i~
x = 'll1'b1m (bUu4J1bb:tl<JBl'l<J~)

b
Ixy- nxy
=
Ix2
- nx2
a = y-bx
x
LX
= n
y =
Iy
n
t?l10d1.:iii 7.11
1ull~l'i1umuM1Ji•nu-ifil:1;Ji'lm1;;.iriiEJ~m-,i1ufl111'Ji~w1i•~EJu;;.irn1<i;;.inuifm1<i;;.i•llw~1wm
12 .~eiu m;;.i111-,1~~ 7.11 <:nn-ifil:l;Ji'l'l~•~uKni,;rn~~l'.ibbu11ii;;.ib~l-l~u (rn'i'I~ 7.6) ~~1m1l'llmm">
'lb'1'll~Vlnl"Jl11lilrn:ir~bft'U11"J~ (linear regression analysis) l'Unl"J'i'Imrnrnm1;;.iriiEJ~nl"J'llil~.~il'U
Iiiil11.J rii~bbl'Wl~lU111"'1~~ 7.12
"' !'Jll"1'1ll!'J (11'1i1a) "' all"1'1ll!'J <11'li1aJ!"11l'll !Vlll'll
1 600 7 2.600
2 1,550 8 2,900
3 1,500 9 3,800
4 1,500 10 4,500
5 2,400 11 4,000
6 3,100 12 4,900

l'Jil91~ll'J 5,000 y .}
(Ul'I) 4,500
y10
y 12
r 11 y12
4,000 Yg •
3,500 YB
{° y10 Y11
Y5 y7 yg
3,000
Y5 r , •
2,500 Ya
2,000 y4
{' y6 Y7
y
.}
y5
1,500 rY2 3
1,000 y1 y3 Y4
500 {. y2
Y1
0
1 2 3 4 5 6 7 8 9 10 11 12
4 1v v v
i11Vm 7.6 fll1 'If Least Square Method !f1H!!f1-lllH
x y xy x2 Y =a+ bx
1 600 600 1 801.3
2 1,500 3,100 4 1,160.9
3 1,500 4,500 9 1,520.5
4 1,500 6,000 16 1,880.1
5 2,400 12,000 25 2,239.7
6 3,100 18,600 36 2,599.4
7 2,600 18,200 49 2,959.0
8 2,900 23,200 64 3,318.6
9 3,800 34,200 81 3,678.2
10 4,500 45,000 100 4,037.8
11 4,000 44,000 121 4,397.4
12 1.J/00 58,800 144 4,757.1
78 33,350 268,200 650

bbVJU.ilLU'J"1"ilJi'
x = 6.5 y = 2,779.17
b = 359.615 a = 441.66
•
11-:iuu y = 441.66 + 359.6x
,~eiu'iiei1u b'Jiwifei-:im"Jwmrnmb~€lmJn"ilf1~1i"1il1 <fim~flu~
13 =Y13
=441.66 + 359.6 (13) =5,116.4 "1U1tJ
. '
13) <li;iuu "11wmrnmtJ€li1'll1m~fluYi
' ' ''lf1~'1JMf'lmmmru
<il1wmn"illi~1tJ'lrr'lbfl"il:;Vim.,1111111€ltJb1l-Jbil'u111"J;id,l'Ju'111h~mrn~;il'.im1~flm111bf1~€lu€l~ ~'
ittJ~U€ln<il1wmn"illid'lu'Ji1-:i~-:iil1m"Jti'11u1rn1~-.1n'11li1ubi'.lmb1JmJl"1"iJlU (Standard Deviation; SD)
~bni1~u-.1nm111iil~m"Jtii1ti€ltJ <ii;id
SD =
SD =
~ (y - y)2
i-=1 1 1
n-2
(600-8013)
2
+ (1,550-1,160.9)
2
+ (1,500-1,502.5)
2
+ '' + (4,900-4,757.1)2
10
= 363.9
-.1n'11li1mi'.itMbUUm111"J;JlU~'11u1rn1~d-.~'Ji1tJ1Vf~wmrnmmm"Jt1f11i1fl~bu'Ji1;i'l1€l-J<il1
wmrnrn"l~Ji1tJ1~1lum1~b~€l~u11h;i'l 1111~1111"J1-J~ 7.13 L'1tJl'.iil~~~"1u11-if€l~<1l'.i~ni,r~m"im~"J1tJ
"' ."uu1f;iun~

. . .""- I J V ""I. <I.I J
1111n~11 7.13 l!tl'Pl~Al'l'WlmW w 'l"::0'1'.UA1l:IJ!'lf0!11-!VIH1
..u .. u
' ' ''j"::;p]1Jft1l!J!'lf0!11-! 'lf1~'110~Al'l'l!Jlfl1W
68% Y±SD
95% Y±2SD
99.8% Y±3SD
= 5,116.4 ± 2 (363.9)
= 5,116.4 ± 727.8
~ ' ~ '
'1<J'l":::'Vl11~ 4,388.6 rn 5,844.2 'VIU'ltJ
. . '
IV .:.i <!:I <!:! ""- ii "'ti ti~ ......~
111:M;n11t111m11-1<Jm-.r11::1·rn~n 7.2.6 mn 1 1w {)1JV1111J1111i; 7.2.6
• I 1 T
1
~ f &:! .::{ <!:! <=:I
1-!l!'l-!1fll11'!fl'lfl'tt'lfl1Jl1 7 Vl!l'l-!l1 7.2 !1Ml1 7.2.6
il1~~n'111mbbi'i111 .ifeil;Jl11l'l-!m~nmileJ~,f;iJ'l"millJ'll!l~bb'l.-b1tit~ ri~mi1 1'1;r'ln'l" bbi1:::m'l"
NUbbiJ~1i.imiueJu m&bbiln<lieiJ;Jl11l'l-!fl'l"~nm (decomposition of time series) 'VlmtJ!i~ fll'l"bbtJfl!l~,f;
1h:::nmJ"11~1 eJ!lmnn.ifm;1'11l'l-!fl'l"~nmbvj8il'l"1'ili181Jli'Vlli'wi1'1liNil~r;1h:::nmJb'Vl'11iXu b'Jiu 1'i11Vf'V1'11J
11ri~mi11<11illi'VITI'Y1'1~1niteJ tJbv:ltJ~1<11 1um~li[iii&im'l"'Vllfi1ei~,f;iJ.,:::neJ1J'll!l~bbU11it~ bbi1:::r]',lflli1
mm&i;i1'i11Jl~1tJ bbl'im'S'Vl1q;hm,f;iJ.,meil.J1'{1~miXu1'i11Jlmn~u biteJ~'il1nm'lbli~tJUbblil1~il1~1'{J~m
lj'l"ii<J:i'.in1-iinmu1u m<JdJunm'VlmtJu bm::::i'.in<J:::'1l-IYiuiJ'.nl.JmJbli~tJUbbUi1~'1J!l~ilm1:::b'1~n<J rhu
m'lV11G'i1eJ~.'iiJ.,:::neJum&NUbbU&~1l-1bbUUeJuiiu Vi11JifieJu.Ji1~mn bdeJ~<J1n1l-lmm&11'11ww1:::mm'lbn<11
b'Vl11Jm"ru~ti<11iJn&i1Jloi1~'V1it1
1<11tJrf11lim"J~<J:::v:i<J1'l"rn18~,';iJ.,:::neirn~ 4 'll!l~.ifm;1'1ei'l-!m~nmiXub'Vlm:::~<J:::1-iiif1V11'1Jm'l
'Y1mn'l"m'l:::tJ:::m1 bbl'iif1V11'1Jm"J'Y1mmrubv1eim.,ii1buu~1um~m,Nliiiibllum,'Y1mmm"i:::m;~u bbl1:::
'l:::tJ:::iJ1unm~ 1u'VrniJ[iii&i~~:i'.in1Vi'm1~if1fitj!b1<'¥11:::ei~,f;iJ"i:::nill.JbbU'llWJbbi1:::r]~ml1L'Vh.tiu ~~.tiu1u

m'wmn'l"rubriEJ'Yi1m'lwmmru.11ml'ilifa.ibbll'N:;iJw#i'1tJrn.'i1h:;n1lUb]~mt'1 byjEJ1Vf'11wmmru"lnil
,,. =,,,: %.
i'1rn;;i11m.,,tJ,'llu
briEJ.11wmmru"lu7:;tJ:;~mhmEJu#i'1tJ EJ,.'i1h:;nEJrniu11ill-Ibbt'l:;b]~m'1 ~'~~'Yi1m'l'Wmmru
':;o/frn1bm1:;V~EJ EJ'.'iiJ-;:;nmJ~' 2 ft l'lm1i.Ji1'i.Jwurrnuilth,h <J:;d'.ium1:J-Ji1':J-Jwurrbbuuu1n 1'111l
v v '
m1i.Jil:J-J'W'llmbuu~rn
m1:J-Ji1'i.J~'llTibblJ1JlJ'lfl d'.lumwi1':J-JW'llTI~ff:J-1:J-Jo/i'l1.11~'lliil]~mi1l'l<l1u1uf1'~ 1iJ11.11bb'U'lLill-I
<J:;al~l'11'lbvlll<ii<iil:J-llllW~ 7.7 iiu~EJ
'11wmmru = '11bbu11ill-I + '11~'llu11~ml'1
~'bbfl<il,<ill:J-1111,,.,fi 7.7
m1:J-Jfii.JwuTibblJUf1ru ilJum1i.Jfi:J-Jwuii~.11"u11Ui.J<Jrnnf1ru#i'1tJ'11~'liftn <ilm'1,:(u~1lv v 'II ~1 q;
'11wmmru = '11bm11ii:i.J x .11~'llub]~m'1
'11~'l1Ub]~fll'1 (Seasonal Index) '11~'lliib]~nl'1~EJ <l1mu~1.jfi.J~u-li1l:!J'1il11fl'l"i.Jb'lmbyjil1Vf
ffil<ilf1llil'fl1Jb]~fll'1"1W1 'llEJ'l'.J
m1:J-Ji1':J-Jwuubbuu~rnflilJui'WiJm'lru~ii'i<ii~um:J-JiJno/ivf1'1 1i.J ~"1w11#111 t'i1'11m'lrn<ii;;i:;m
'llil"b'll'lLU:J-Ji.Jln~u .11m1:J-JN'lltb'l.J'l<J:;mn~u<il1:J-11iJ#i'1tJ ~'bbil<ii,l#i'<i11i.Jm,_.,~ 7.8 ~,t'iu 1iim7tb1J1J
~rnd~,,11u~u1m11inul<iitJrf11iJ tb'1:;1u~U<J:;bbil<i1Wiiw1:;"lfauu~ru ~,$11EJ81,~ 7.12

nm
i.u1. n.'1. ~.'1. n.'1. ~-'1- n.'1. ~.'1. n.'1.
1uiihB!J1'~ 7.12 'IJ:;bb!1<11'li'1li nl"ibb"1noiim;1l'lm1m~nii1 Lllu'11Jimlq~ml'I bbii:;fi1Lm1tiiJJ
bbii'1u1'11Jl'liiit)~~li1bb!i~fllbb'U1tiiJJ1Jl.jfl,um-mmmm ~'blJu1li111d,~u8~1-ifnu
l'l1oriNfl 7.12
V11,!1'l"rnii1ufi1Lb'Vi,111d~iioiim;1iim"l'll18'11B"liu•~n1uwliii:;Lll"imfi•im,.,:;8:;nm 3i"l~~hum "i1~
12 L<11"iml1 Ji,<111"!1'~ 7.14 ~''ll:;Li°iu11ll11itlmJu'llB'oiim;1iiiili'VITI~ii'llm")~mii$1B 8!l<11'1ll8'ii:;~16J<111u
L<11"iml1~ 1 'llB~lJ bb!i:;OJ~fiB8'1 ,.y;~~u 'il'l-l'P'l<111u1"1"iml1~ 4 'llB'lJ uBnOJ1ntl:iubb1J1J'llm-iim;1l'I~'
""'"'1Vl,i'iuli~bb'U1tii~118!l<11'1118l'.ibbmtm,,1'1~~u1uwliii:;'J bd!l~OJ1noiim;1iil'lLY18' 3 i"l ~'hi111m"in
m<11'1~bUB,fitl.,:;nBu'llB''llJ<fm1JJ
1m:1mr~ !J!ll'l'Ul!J
•('111l1!J : 10,000 1Jl'11)
1 30
2 42
3 55
4 100
5 35
6 46
7 59
8 120
9 43
10 57
11 71
12 142

1f!~ml!~
' ' ' • ,; .; ,; •' ""' ~ ~ ~
110'1'11111 ftl!lll11!ftMU11 I'll!11l11!ftl0U11 0'1~ll!1Ul)~flll
•('HU111 : 10,000 Ul'l'I) 4 ~1'1 2 nfl
1 30 -
2 42 -
56.75
}=-----57.43 55 0.958
,__ 58
}-58.54 100 - 1.709
59
5 35 - 59.5 0.588
60
6 46 62.5 0.736
65
7 59 66.0 0.894
67
8 120 68.4 1.754
69.75
9 43 71.3 0.603
72.75
10 57 75.5 0.755
78.25
11 71
12 142
' v ..
fll'j'i'IWllVl'lf'lltjliJfll'1
1ih11LL<iin-il€1i;Jl1€11-!rni.;rnrn~€1vmh~'lluri~m<1mw1~li'liwn'>l~~ 7.13 t1Btii'l~u<ii€l'U~~d
1. •nn-if€1i;Jl1u€11''lJltil'lii'ni,i.m~dJuri~m11<il1i.J1mmfl ~~'1'11m1vn.11~'lluri~m1111'1m~i.;1fu
<i1u1m1111'11L%~m.i~€1u~ 4 L<ii"imfl ''"~1'1li'lufll'li.Jn~ 3 •'liu
t1Bu '11 56.75 "'J~€1~<i17~nm~7~111-:i1~1<il1m'1~ 2 ""~ 3
~1~ulii€1Ll.J"'J~<i1u1m'11L%~m.i~eiu~ 4 L<iiW1'1~~d
'11L%~m<i~€1'U~ 4 L<ii7mfi'lJ€1~L<ii"imfl~ 2, 3, 4 mi~ 5
30 + 42 + 55 + 100
=
4
= 56.75
42 + 55 + 100 + 35
=
4
= 58

1'1EJ 1'11 58 'Jteiaj<i1'l~rn11~'l::11111,1<i1'lm11~
L<il'li-1111 ;1EJ'li5nwii1,iilu'Jufluij'1i:f~ 121<i11m11
' ' '
3 mi:: 4 bbfl::'Jnhm'l'11mrnl'i1•%i1m<iiieiuYi 4
11mmtt~ o/11mJi,;'.(d]wii!li;J<i'lll'J1'1r,ml1~.:Jfi1mrnl'i1L%~rn'll~!lu~ 4 1'111i-11l1 Lb<iirli~!li;J<iLlJU'lll'J 12 L&i!lu
LULL<ii<i~il 'l::fi1u1rn1111'11L%~rn'll~!lu~ 12 •iii!lu
2. b'iiil,'JlMhb>b~l'JYi1Ji°lu11'1i.in~ 3 iXu dJmi1~eiaj'l::'w)1~L<il'll-1111 ~'iii'il,'Yhm'l1111fi1b:i.~tJ
b'll~i'l'U~bblJlJ 2 '1'1 b~ei1Vf'11bQ~l'J~1;'1<i1nilajlubb'ii<i::1<i1w111<i11i-111'1J-Jn~ 4
t(i)l'J '11 57.4 'Jmaj<i1'l,1<i1.,m11~ 3
=
56.75 + 58
2
= 57.4
bbfll:'J~'Yilm'l'1l'U1rnl'il b>b~l'JL'fl~i'JU~bblJlJ 2 ~1o/ld'J'Ub11~fl'U<i11l-J<i1l'll~
3. VllfllB<i1'llft1'UfJ'{Jnlfl (l1'1l-JJl~ 5) lo/ltJUll'ill'Ji'lo/l'llll'JL'U11'1l-JJl~ 2 Vll'l;'i1tJfllbQ~tJl'U11'1l-JJl~
d ,,.: o 1V j d ~.%
4 'li~'Jn'li-irnu1rnb'1'Jln •mmi'f'YI 3 '1,'U
= 55
57.4
= 0.958
'11fi'<i1'llc!1ufJ'{JnlfldmJ<1m1J-JV1mtJ1;'i·)i m1i.iiiieJ~m'lLU1<i1'lmi'f~ 3 btlu 95.8 •tidb~u4i
'llEJ~fi1t;::i.~t1'iiEJlJ bbfl~'l~'Yi1m'lfi1u1rnmfi1ii'<i1'l1ii1uri'{Jmfl1u1<i1'lm115u'l ••<1~1li11'luM1i-iil~ 5 <i11i-i
<i11'll'

1m:irnr
ii~ 1
1
2
3
4
0.958
1.709
1Jrri1~'lluri~m11~1l
lll11l-Jlfl~ 1
lll1l-nfl~ 2
lll1:Wlfl~ 3
lll1l-nfl~ 4
m':ltt1fi111'1niu:tt
=
=
=
=
ii~ 2
0.588
0.736
0.894
1.754
0.596
0.746
0.926
1.732
' v '
m0m1ir1uq~mQ
ii~ 3 '
.,;
l'll!lllla
0.603
0.588 + 0.603
= 0.596
2
0.755
0.736 + 0.755
= 0.746
2
0.958 + 0.894 = 0.926
2
7.709 + 1.754
= 1.732
2
'il1n-ii1l:1;JiWm11tJ1l'1'll1t.ii'lbb'U'lLu:wb~:w~ii m11111ri1bbi111U:W'il~1-ii'lrrm1'ibrn1~Vlrn1!1'1!11lt.1 (re-
gression analysis) lwi'hmh~d'il~l-ifrnTibfl11~Vlnl1!1'1flilt.lb~~bifiill11~ ~~bbl1'1~lll:Wlll'V1~ 7.9
1Jil?l'llltJ
.liJ11>1-,:m'1 160
140
120
100
80
60
40
20
1 2 3 4 1 2 3 4 1 2 3 4
1 2 3
' ' ' '
mwfi 7.9 11~1vluir'1~n1~!l'1'lll~ 1'11mllmfll10ufi !!Q~rUu11mti!"

m'l"Vlll~.m1-111ihimm'l"n'loii-iieJl;jfltJeJ'1'll1tJ~m;r'il~~1zyr u.Jfi1V11u1!lm'l"bl.in-iiell;JflbllJu~'loiinu
iin'il~'loii.-iw;1~ml'l~el'U~l-11Vllbl1.m1-111mi 11'.lil~'illn'il~'lVf.-11-wmmru~.-ieJ1-1-ii1~'1~~
'lu~d'il~ml'1~1um'l"11m1~Vim·m'1neJm:S~1fiu.i·;M1'1tJ'loii-iia:i,ifl'illn.-i11>1~ml'l~eiu~~.-i11-11ru.11'
uli'1'l1-1!1'1l-lli~ 4'lleJ~.i1'l"l~~ 7.141'1tJ-iieJl;Jfl'il~•~l-l'il1n1mmfl~ 3 ~~d
1mimr
x
3
4
5
6
7
8
9
10
M1Til-I 52
x
y
b
a
=
=
=
=
=
mlP1'111am11
' ' '
' .<:!, .<!!. ""
Alt!Hl!l!tllt1'J'llfl 2 11PI
y
57.4
58.5
59.5
62.5
66.0
68.4
71.3
?55
519.1
LX
=n
Ly
=n
Lxy- ilXy
2: 2 -2x -nx
110.32
=42
y-bx =
=
xy
172.2
234
297.5
375
462
547.2
641.7
75]!
3,484.6
52
=8
519.1
=8
=
2.63
6.5
64.89
9
16
25
36
49
64
81
19.!2
38.0
3,484.6 - (8)(6.5)(64.89)
380 - (8)(6 5)2
64.89 - (2 63)(6.5)
47.8

l1:wm-,il1111wm.,'V'lmmrnfi1"m1ii::w 0B
fi1•m11U:w =T =47.8 + 2.63x
n1iliil~m.,'V'ltnmrntiil1i'lll1J'llil~ll'liillJ f1ill.i1:w1l1~ 13, 14, 15 bb'1~ 16 'il~<iim1>h•itunwm:w
%u<ililiil'iil1udf1il
%u~ 1 Ul':Wm':mm1ii::wV1Jim'V'lmmrufi1•m1Lii::wL1im•'Y1Ufll x = 13, 14, 15 bb'1~ 16 <ill:W
T = 47.8 + 2.63x
T
13
= 47.8 + 2.63(13)
T
14
= 47.8 + 2.63(14)
T
15
= 47.8 + 2 63(15)
= 82.0
= 84.6
= 87.2
T
16
= 47.8 + 2.63(16) = 89.9
%u~ 2 u1fi1bbu11ii::wi;irnn1Jfi1'1'1iilfJ~m01~1111mm1'11u.iiluiliu 'il~l4ifi1'V'lmmrn~<ifil~m1
fi1'V'lmmrn = fi1•m11U:w x fi1'1'1iilri~m01
fi1'V'ltnmrn'1<il">ml1~ 13 = 82 x 0.596 = 48.87
fi1'V'ltnmrn'1<il-,m'~ 14
fi1'V'ltnmrn'1<il1m<!~ 15
fi1'V'lmmrn'1<il">m<!~ 16
.~ . ~
1'1161111'11 7.13
=
=
=
84.6 x 0.746 = 63.11
87.2 x 0.926 = 80.75
89.9 x 1.732 = 155.71
1J™1timumtJ ri11''~•N'B1!Ji:lllJV1lm">Ni1<iiwm01~dJwiun-,ffim-,1'i1muhiYiu<ililm">fl~:wilul!lufl1
<l~14i•nlJ-,11J">1:w-ii'ili,J'1tJil1i'lJ1tJ~N1W-J1 4 ti <i11:1--1<i11">1~~ 7.18 m1•m1~Vi'V'llJ11tJil1i'lJ1tJllin'l:IUl~'llil~
ri~mmm~•m11u:wil~ ~~1Ji'lm1lm-,wnn4iili,J'1il1tm:wnm•~il'V'lmmrutJil1i'lJ1tJ1u1l111ii:1

"'
l'l~>i'llll'l (BU)
m1m
ll~ 1 ii~ 2 ii~ 3 ll~ 4
1 55,220 39,875 32,180 62,377
2 57,350 64,128 38,600 66,501
3 15,445 47,653 25,020 31,404
4 24,776 43,050 51,300 36,504
5 21,408 39,359 31,790 16,888
6 17,118 10,317 32,100 18,909
7 18,028 45,194 59,832 35,500
8 19,883 46,530 30,740 51,250
9 15,796 22,105 47,800 34,443
10 53,665 41,350 73,890 68,088
11 83,269 46,024 60,202 68,175
12 72,991 41,856 55,200 61,100
~"' "1liflU IIII fl'Uflajllfl~ fl'J:tJ11ll1
<J1noiieJJ.Ji'l'llil,u"i~mumml'luoiieJJ.Ji'liltJb~ilu 1um-;1111il'Jfunilmi'IL4J.JW!u'i'i1m11111'11b%~tJ'II 'II ~I 'II
bfl~il'U~ 12''lilrlil'U1ii11luft11J.Jli~ 4 'llil,ll111'~ 7.1B 1ilti•11fi~#f<J:;dJu'11~iltjll"nm'-;~11111,b~il'U
'11LL1m1th~wi1'b~ilu~ 6 Lm~ 7 '11~ 2 eJ•i1:;11111'L~ilu 7 bbfl~ s Lllm'llult<Ju~uftilL'UftilJ.Jli~ 4 (Lu. ~ '
~d--~iJ11'11'Yl'1UtJ:i.J1~Ll'Ju-;;j1u1ubl:i.J)
111i,<J1rnXu ~'vi1m11111'11b%~mfi~ilu~ 2 b~ilu 1ii11luftil:i.Jti~ 5 bbl1:;'11u1ru.8'll11ii1u
ri'{Jmfl1uftil:i.Jti~ 6 ~,ll,.,,,,;<i1ud

55,220
57,350
15,445
bl-11'<11'.J.I, 27,776
21.408
1 17,118
18,028
19,883
fl.l,l'Jll'J.I, 15,796
53,665
83,269
72,991
39,875
64,128
47,653
43,050
39,359
2 10,317
45,194
46,530
fl.l,tJll'J.I, 22,105
41,350
46,024
41,856
32,180
38,600
25,020
51,300
31,790
3 32,100
f I t ' J f
t d. <!!. .tj I d. .!!, d.
f'll!fllla!l'lll!Hl'll l'll!lllla!l'llltl'l.!'11
12 ~1A
38,162
36,884
37,449
40,133
41,405
42,901
42,335
44,598
46,819
47,345
46,318
43,215
40,620
39,979
37,852
35,965
36,653
36,022
37,837
39,057
37,741
39,883
42,594
43,776
44,888
37,523
37,166
38,791
40,769
42,153
42,618
43,466
45,709
47,082
46,832
44,767
41,917
40,299
38,915
36,908
36,309
36,338
36,930
38,447
38,399
38,812
41,239
43,185
44,332
0.4805
0.5350
0.4072
1.3163
1.9754
1.7127
0.9174
1.4030
1.0121
0.9193
0.8792
0.2461
1.1215
1.1957
0.5989
1.1388
1.2666
1.1334
0.8370
10052
0.6446
1.2440
0.7361
0.7241

~ .1ll'JHfl 7.19 (fl!l)
59,832
30,740
fiumtJu 47,800
73,890
60,202
55,200
62,377
66,501
31.404
36,504
16,888
4 18,909
35,500
51,250
34,443
68,088
68,175
61,100
12 ~1A
47.404
49,729
50,261
49,028
47,787
46,687
44,660
46,369
45,256
44,772
45.437
45,928
46,146
48,567
49,995
49,645
48.407
47,237
45,673
45,514
45,812
45,014
45,104
45,682
1.2966
0.6329
0.9561
1.4884
1.2437
1.1686
1.3657
1.4611
0.6855
0.8109
0.3744
0.4139
111i1~'illf1Yi1~·hfl<ii'l"lfi'l'U'J~nl'1LU111"1~~ 7.19 bbl9''l u1.l1fi<il'l"lfi'l'U'J~nl'1dm1111'11m~tJ
iii1111~u.!1fi<il'l"1fi1uLuw1i'1n~eiu #l~bbl'i<il~L'U<iil">l~~ 7.20 1'11b:b~t1~1~<J~•lluwi1#l'liil']~m61'llil~wii61~
.~eiu1u.,eiuli

Month
u1
0.9174 0.8370
1.4030 1.0052
1.0121 0.6446
0.9193 1.2440
0.8792 0.7361
0.2461 0.7241
0.4805 1.1215 1.2966
0.5350 1.1957 0.6329
0.4072 05989 0.9561
1.3163 1.1388 1.4884
1.9754 1.2666 1.2437
1.7127 1.1333 1.1686
m1111ri111ii1!U':J.J
1.3657
1.4611
0.6855
0.8109
0.3744
0.4139
0.9174+0 8370+1.3657
3
14030+1.0052+1.4611
3
1.0121+0.6446+0.6855
3
0.9193+1.2440+0.8109
3
0.8792+0.7361+0.3744
3
0.2461+0.7241+0.4139
3
0.4805+1.1215+1.2966
3
0.5350+1.1957+0.6329
3
0.4072+0.5989+0.9561
3
1.3163+1.1388+1.4884
3
1.9754+1.2666+1.2437
3
1.7127+1.1333+1.1686
3
= 1.0400
= 1.2898
= 0.7808
= 0.9914
= 0.6633
= 0.4614
= 0.9662
= 0.7879
= 0.6541
= 1.3145
= 1.4952
= 1.3382
•nnoiim-1'1'V'l1JlltJ1li1lm'l-if-i11u411l1LLWltW;L~i.J~um'TI'!1<il1LLU1tW;,~14fl!im1iL<i1"1:;V;m1t11lt11ltJ
•
1u~hilti1-id,~1oiim-,1L<fl"1~Vimcmi11n1lm:;r-iL~u<11'l-i
m'l111L~mm11ui.Jmm1n1-litJ1li1lm'l-if-i11u411~LLn'il'i-i1.1 w1i1'11V11u1!im·llL<iln'iim;JmLuu~1i
nuiin'il~1i<il1L:I~tJL<f~1JU~mV11Lft'ULb'U11Ui.J L-dil-i•nn'il~1Vi'<il1'V'lmmn'i~<fl1luoii1-J<f-l~
l1.1~d,nL<fl"1~Vim'lt11ln1ltJL:;r,,fiu.i1-Jli11tJ1ioiim;J"'1nfi1L~tJL<fl~1lu~~'11u1nM'LLa11u'1i1li.Jn~
5 'IJ1l-J<ill"1-J~ 7.19 li11tJ-ii1l:l;J'1'il~~i.J'illrn~1lumnJ.Jlfli.J'IJ1l~i'J~ 1

N x y xy x•
1 7 37,523 262,661 49
2 8 37,166 297,329 64
3 9 38,791 349,115 81
4 10 40,769 407,689 100
5 11 42,153 463,686 121
6 12 42,618 511,415 144
7 13 43,466 565,063 169
8 14 45,709 639,921 196
9 15 47,082 706,227 225
10 16 46,832 749,305 256
11 17 44,767 761,031 289
12 18 41,917 754,513 324
13 19 40,299 765,690 361
14 20 38,915 778,303 400
15 21 36,908 775,078 441
16 22 36,309 798,802 484
17 23 36,338 835,763 529
18 24 36,930 886,315 576
19 25 38,447 961,183 625
20 26 38,399 998,383 676
21 27 38,812 1,047,925 729
22 28 41,239 1,154,678 784
23 29 43,185 1,252,367 841
24 30 44,332 1,329,955 900
25 31 46,146 1,430,527 961
26 32 48,567 1,554,137 1024
27 33 49,995 1,649,846 1089
28 34 49,645 1,687,924 1156
29 35 48,407 1,694,260 1225
30 36 47,237 1,700,528 1296
31 37 45,673 1,689,916 1369
32 38 45,514 1,729,538 1444
33 39 45,812 1,786,676 1521
34 40 45,014 1,800,557 1600
35 41 45,104 1,849,279 1681
36 42 45,682 1,918,662 1764
L 882 1,541,704 38,544,247 25,494

.'2:x 822
x = = = 24.5
n 36
v
Ly 1,541,704
42,825= = =n 36
n
Lxy- nxy 38,544,247 - 36(245)(42,825) 38,544,247 - 37,771,650
b = = =:2: 2 -2 25,494 - 36(24.5)(24 5) 25,494 - 21,609x -nx
772,597
202.52= =
3,815
a = y-bx =42,825 - (202.52)(24.5) =37,863.26
f!:wm"lil1111wmmmn'Wl:.!11bu1lihi ~ei
.!1bm1lii:w = T= a +bx
= 37,863.26 + 202.52x
' tr~ .,,;
'11'1"<1Jln'Titbb'11'l'Ul'I 49 = 37,863.26 + 202 52(49)
=
=
47,786.74
47,787 fi'u
.tj 1 '1 v
m11~n 7.22 fll!l'lll•'lllJ
.. .;
!Vl!lll'I ' 1vftl!llll 'lllJ (T)
49 47,787
50 47,989
51 48,192
52 48,394
53 48,597
54 48,799
55 49,002
56 49,204
57 49,407
58 49,609
59 49,812
60 50,014

'11~mmrn = '111vu1tmi x ~'li'U111Jrn'1
' ef 4 ""'
'11~tnn'ltU'1il'U1'1 49 = 47,787 x 1.040
= 49,700 fiu
bb<l:;i.i1rn-rnmmrn1~ilufi 50-60 1#1'11~mmru~~m"ll~~ 7.23
' 'lll~H 7.23 l'lWlalft~ll!
.. ,;
f'i1umhl':1.1 (T) ' v "' ' '!?ll'l'llTI 1'11'1'1f'll'J~fllll 1'11'1'11llmw
49 47,787 1.040 49,700
50 47,989 1.290 61,895
51 48,192 0.781 37,626
52 48,394 0.991 47,978
53 48,597 0.663 32,232
54 48,799 0.461 22,515
55 49,002 0.966 47,344
56 49,204 0.788 38,766
57 49,407 0.654 32,316
58 49,609 1.315 65,212
59 49,812 1.495 74,479
60 50,014 1.338 66,930
. ' '
v "' .. .. "' v ~-· -·~ ~~mM1111111111J11uemirn:::110~n 7.2.7 wn m1Yiu{JUl'lil11il'fi" 7.2.7
. ' . .q cit I q ""- <!t -"'.
1u11u1m111mn11u111n 7 l'lti'lln 7.2 11e~n 7.2.7

'
"'ll0'1-!fl 7.3
"11H'IJ'IJ!!t;Tvt~rt11t1tl't1TI'i!~'ll6~"111iil1
~
!!'ll1fWI
A d = i: "" v , ,
•·m'n 7.3.1 nT'1b'1'1:::V1rn">rw111Bm'il'b<iuilm''1t1
b~il'~ 7.3.2 fl:J.J1h:::iir1t<1V1lY:J.J'l1u-IT
' ' 'b'lm-Yi 7.3.3 i'imh:;iin1lmii,rn">ri1V1U'1
1. rn1l•m1:::V:rn-;n'1nill'Jb'S'bftUeJ81,~11'J blJuiilm-;'V'lmn-nil'lleJ,<ii1mJ1Jbb<i'1,m1:J.J
i'i:J.J'V1un'lleJ'lif1bblh 'il1nrn1fi'il1111.n!,<ii1bbl.h 1<ii1~l1.-i11:J.Ji'i:J.J'V1uiinu~'~'J:::'V'Imrnru
2. i'i:J.J1.h:::1lnt<iV1lY:J.J'11u-IT d'iwri1~bbft'1'li'm1:J.JiY:J.J'V1u-IT-;:::V1'i1,<ii1mh 2 <ii1'i1i1m1:J.J
i'i:J.J'W'unnU:J.J1ml:eJmV1tML01
3. i'imh:::llntbbvi,rn-,ri1V1u01 bDwri1~bbft'1'11<ii1bb1h1i<i-;:::m:J.J1-;nflilmm1-;btl~t1uutlm
'llm<ii1bbtl-;m:J.J1.1'~bv:lmb01
4. rn1ib.-idl:::V:rn-;n'1nflt1'V'l'1'l~ru •fluiilrn-;'V'lmrnru'llB,<ii1uuubb<i'1"m:J.Ji'i:J.J1iufi'llil'
<ii1utl-; bi1eJi1<ii'mti-;1lft-;:::mnn11 1 <ii1
' '<!IA A V<vdl
b:J.Jil.imrwmun 7.3 'illJbbm un.inmmm-;n
1. V11.i1'V'lmrnru'11t111lrn-;ib.-i'1:::V:rn-;n01nBm'S'bfium.h,~1t11.1
2. V11.i1lYmh:::iintftV1fl:J.J'W'utl.1
3. V11.11i'i:J.Jtl-;:::1ln~uvi,rn-;ri1V1U011.1
4. V11.i1'V'lmrnru.1'1t1illrn1l•m1:::V:rn-,n'1nflt1'V'l'1'l~ru1'1

m"i'V'lmmci<i11tJ<ii'lbbllllm'l'l-lm1J-Ji1J-J'W'uil'll!l-l<ii'lLL1h (causal models) dium"i'V'lmmrn~ll
m-m•1.,-rn1ii-l<ii1rn.h~l1m1J-JiiJ-JYiut'in11~-l~ri1il-l<J:::1'1mrnrn Lil111'11J'i1<ii1mh1'1i1m1J-Ji1J-JY1ut'inu II
•:::1-l1<ii1LL1hiXua-J11<ii'lum.,-1'1mrnci<if1tJ11l1wrn1-lflfi~ <ii1!lE.11-lL'Jiu ~lwmihu1111;,lllN<Wi!ltJ!l'l'llltJ
L<ll~!l~fl'llii'rn.ri t'i1l1m•n!li171-i~!lElmfrm~J-J~.l<J:;Yi11VitJ!l11'll1tJm~!l-lfl'llii'rn<Yimn~u ilui!i!lm1J-J• • •
i1J-J'W'uil<ii-lmi11ddJuLwml'lum1nu (causal relationship) Lil!l~-l111ii-lLn11~u<J:::Ll'iumLV1<11lVfLn'15n~-l• •
111i!-lma-Jm 1u<ii1!lE.11-ld•:;L'itJntJ!l'l'llltJLl'iwihmh<111J-Jm1:::~1u1uiJ1uLl'iu<ihLLti.,lifl.,-::;
<ii1bblJ11Lbfl'l-l<l111d-Ji1J-J'W'uil'llm<ii1bbti.,di111lm"i1'1mrnrn~U[Jd-J1-iinu 2 UT.;Ul'Yl4'1€J m.,-1Lm1:::Vi
(multiple regression analysis)
m"i'V'lmmrn1'1tJ'iilm.,-1L<1111:;Vim.,-ti'lti!ltJL:ii-lLftu!lu1-l~1tJd-.:::1-ii111!1nm.,L~mn1111lm"i'V'imnm
t•ml~LLu1tuJ-J (Trend) <ii1LL1i.,1111ml'lu.-i11'1mrnrn1-iii1~iln1'1cl Ym1:::<ii1LL1i"ifll1'l":::1-iii1~!1n1'tcl x
flJ-Jm.,1um.,-1'1mrnrnm.,-ti'1ti!lm:if-lLft'U!lU1-l~1tJi!i!l Y=a+ bx
Y i!i!l <ii1LLtl.,-1111J-J
x i!i!l <ii1LL1i'l"Bfl"i:;
a i!i!l ii1u<ii'ILLn.I y Vl'i!li!i!l.-il'll!l-l y Lii!l x l'.J.-i1Ll'i.l~.I~
b i!i!l .-11m1J-Jifu'llmLft.l<ll'l"-l 111'im'itJn11.-11i1J-Jtl.,-::;1'Yl~m•n'1n!ll'l~-lLl'iu.-11~
Llfl'1-ln-le'l11111m•Lt1~tJ'Ubbt1i1-l'll!l-l y .nil x ,11~tJu111 1111u1tJ
.
A10d1~n 1.14
w~11fll'l"lJ1i:i'YINii<llbLi1:::~1111'1-i1tJ b<ll~!l-lfl'llii'rn.riLbvi-lVlii-l<il!l-lfll"i1'1mmrntJ!l11'll1tJ1.IUVIUl'll!l-l• •
fltlrnrn<11nbb"1-lVf!l-l,ll'llU11111;j,-l~-lblJUNii111firno/i'll!l-l111i:i'YI byj!l~11b<ll'itJJ-JLLN.lfll'l"Nii<11 111i:i'YIL:fr!111tJ!l'I
'111tJi1<1111J-Jil'J-J'W'uiln11~1u1u~'W'nmfitJ~ri!l1171-l1111;,l ~-l141Ln11-ii'm;Ji1tJ!l11'111tJLm:::~1mu~Y1nmfitJ~
117i-llVIJ-Jl.IL'll<llrn-lL'Yl1'1d-JV11.lm ,Y-lLL"1lJ 2545-2550 l.1<111'1-l"1!lLtid•

~0'1'1!11J ~1'U1U~'i1nll'~1~1111.i
'l'Ul.
(11U10 : 1,000 !Jll'I) ' v
(11'1-!10 : 1,000 11~~)
2545 20 10
2546 30 30
2547 25 40
2548 20 20
2549 20 10
2550 35 70
1~1Ju11i1l:i,Jl1'l11J~'Oj1u1u~Yimn4fm1~~1vi:iJ %~dJulibmhli0-,~ (x) ii'1::;1'.11J"1'l111'J~•llulibmh<111:11
(y) m1~1'Jurn11"1V'lu111i'1"1.h~~0mllm1:11i1:110'ui5nu~1l f11vi1n-<51mu~Yin011~1vi:iJ•~:11~u 1'.11l"1'l111'.1
"l:::lliimluJ.J1~:11~uo'i11'J~~1mil'l11l~ri1l'i~'11l~U1ll'.l~iJ"1 (least square method) mrnl1u<11'>~n1'>'1"1'11ll'J
,,J o rr ,,J
(linear regression line) '11:11111'¥11'1 7.10 iil1::;1'!1m'>V'll'Jlfl'iffi"llfl'1:11nl'> Y =a+ bx '11:11<111"1~1'1 7.24
l'Jll"1'll11'J y
(0°U1Jll'I)
40
30
20
10
ol1u1u~0'n011~1vi.J
o 10 20 30 40 50 60 70 so x (0'uviifo)

11:W
x y xy
10 20 200
30 30 900
40 25 1,000
20 20 400
10 20 200
70 35 2,450
180 150 5,150
LX 180
x = - = - = 30
n 6
:Ly 150
y = = = 25
n 6
b
:l:xy- nxy 5,150 - (6)(30(25)
0.25= = =
:Lx2 - nX2 8,000 - (6)(30)
2
a = y- bx = 25 - (025)(30) = 17.5
u11tJbb'VIU.i1LUft:Wm'in'1mm Y =a+ bx 'J~1Jift:wm1<hw:fum1v;mmrnlii~U
Y = 17.5 + 0.25x
x•
100
900
1,600
400
100
4,900
8,000
ti'11u112551 m'1i1'J~i:J'11u1uri~mm1tJ'J~ftfi~~u1V1i.JLubw1n1~•'Vl'W'1'1m1u80,000 VI~~ Iii~
i:iutJ<J'1'llltJ'll<J~i'I 2551 'l~1Ji'J1nm'ii.l11tJbb'VIU.i1Luft:wm1 y = 17.5 + 0.25x
'~ d
UUflil tJ!l'1'llltJ = 17.5 + 0.25 (80)
= 37.5
= 37'500 Jl'VI
. ' '
v "' .. .. .. ,, ~-· -·~~~'11!M'illflftfllllllll'IM11:rl'i~t'il'M'11 7.3.1 !!a1 rn'JA1Jl)UAfl'ilfl1'i!J 7.3.1
' 1 ' t
, 4 ' """ ""' .<!! 4
llttll1fll'Jftfllll11ll111'11 7 Al'lll'l1 7.3 !11'1~'11 7.3.1

'llirwi'lLwn~Vlmw'ilt;iilmt" Lfiuil!J1"~11'J 111:inm'l4l'lum-:;·v1m1n·m'1l'ill'J'l-li<ihmhilin~ (x)
'1tnrncll'Jil'il'llll'J~"diu<ii1mh'll1;i.; (y) 'llim"l'1tnrnclmJlJ'dL~Lin Causal Forecasting Model nl"l~
1hmh x m;i.;1-:;i;iy;mrnrn<ii1"1h y 1Jl&J..V1m'l'il~uil~num1;i.;fi:i.iYiut-:;~'11111"<ii1"u-:;iii~ 2 1'iltJrn"l'lll1
"11fi;i.;ll-:;~!11'1!11'lllfi:i.iYiuil 'lll~ilm-:;'lll1'1:i.iu-:;~i11'l!"vi"m-:;n1'lllu'il
'1m.h~i11'l!ft'lllfi:i.iYiuTI (Coefficient of Correlation) 'l-iifi()J'1nmi r d'Julf11~ilWll'Jm1:i.i
fi;i.;vfai',:;'11111" y mi~ x Li14il"'lll:i.J1LI'llil,<f11 r ~d'Ju + 'lll~il - d'Jurn·mf!lil,Yif1m,'llmm1:i.1'1:i.1Yiuil "11411
b<1'll'llm r bbftlil~li":;#lum1:i.i#:i.1Yiuil11l'.i:i.i1niforn'!Nml'ii '11 r e:i~-:;:;1<111' -1 (i, +1 b'14il,'lllmtJ'llm r
'J:;11<1l'.iilunu11'14m'lllmtJ'llm b 'lu11:i.im"l y = a+bx '11 r ~d'JuoirnLftlil,11.!1'/Jil" y mi~ x 'J~bi1~il'U~
lul'UYlf1Yrn'il"l,.if1;i.;nu'lJru.:;~lf11 r ~Ll'Juu1n LLftlil,111f11'llil~ y bbfl~ x 'J~b'1~il'ULUl'UYlf1'YrnL~mnu
'11 r = -1 m1:i.ifi:i.1Yiuil,~'lll11, y "fl~ x il~'l1-1Yif1m'm'nwii1:i.inuil!J1,11;i.;l-:;cl
,.,;, ,,,:,,¥, did., '..:x
'lJrn:;'Yl011 x b'1:i.J'llU 1'11 y 'J~fllilfl~ Vl"lill:i.lil011 x fllilmm y 'J~L'1:J.J'll'U
"11 r = +1 m1:i.1'1:J.JYiui',:;vi11' y LLfl~ x mJ'luYif1'Yll'L~mnuil!J1,11:i.iu"lrn
• •J ' .., ' .., '
""' l A A ' "" A ,,.. .=I > 1
'lJrn:;'Yl'11 X L'1:i.J'lJU '11 y 'J~L'1:1-J'lJU 'lll"lilb:i.Jil011 x fllilfl'011 y 'J~fllilfl"
'11 r = 0 y LLfl~ x 1:i.Jl'.im1:i.1'1:i.1Yiuilnu
fll r = +0.2 y bbfl~ x l'.im1:i.1'1:i.1YiuilnuifotJ'luYi'1'Yll"~mnu
"11 r = -0.9 y w;i:; x l'.im1:i.ifi:i.1Yiuilnu:i.i1n'luYif1'Yll,'il"l'nwii1;i.;
m"lVllfll r m1o/i''Jlnj'il"l
r =
nl:xy - LxLy
Jll'1~ 7.11 biftlil~l{jL{ijUfilft:i.JU"l~i1'Yl!ft'lllft:J.JYiUTilU1ni,rru.:;'1h''l nu
my; (a) Lriillfll r = +1 bLft'il,'11<ii1LLll"l x LLfl~ y l'.im1;i.;#m~'uilnumil'ft:J.Jj"lll
m'1 (bl ,riil'11 r l'.i.!1mnn11 o bbfl~uilm11 1 ••fl'il"11<ii1m.hl'.im1;i.;#:i.1Yiuilnu'lu
Yl'1'Yll~b&°imnu tl1"11 r ri"b.if1'ln~ 1 :J.Jlmbft'il,11<ii1111l"liim1:i.iii':J.JYiuilnu;i.;1n
m'1 (c) L~il"11 r = o bbftlil,11<ii1mJ,fi,ftil,1:i.Jl'.im1:J.J#;i.;Yiuilnublm
!111'1 (d) 1riil'11 r = -1 bbftlil,11<if1LLU"ll'.im1:J.Jfi:J.JYiui'nu'luYi'1'Yll''il"nwihm1!J1,ft:J.Jmll
•

y y
•
•
• •
•
x
(a)r=+1 (b) o < r < 1
y y
•
• •• • •
• • • ••
(cl r = 0 (d) r = -1
v ' "'
A1eJ!JH'l1 7.15
u'l*vi~1111u11.1b'1~il~'!'lliirn-n"luo/11!l811~ 7.14 1Jivw1rnclt.1il'1'lllf.Il'1t.1~'l1'HUl'llfl~rn1u
~Yin<f~11111i.Jl'.lm1:1-1l'fa-1Viuilnuf.lill'l'lll1.1 m1~1u1rn1111.11l1:1-1t/1~~l'l~fl111l1m~·ufi,~uiln1Ji11o/11bbll11'!1
<fil1dl'.lm1:1-1l1:1-1Yiurinuil811lrnl~'1bV1ml'1"ium1~1mrn~~d
' ' 'mn~n 7.25 m1rlrn1w1Vl0mf11ll':JJ1h~llnlli:r111!m~·1.!~
x y xy x2 y2
10 20 200 100 400
30 30 900 900 900
40 25 1,000 1,600 625
20 20 400 400 400
10 20 200 100 400
70 35 2,45Q 4,900 Ll_2_§
"l'llJ 180 150 5,150 8,000 3,950

7-66 nl"i5tf!Tidtivifiinruu'1::n1~(P)n1<Jn11'9i1Liiw1u
n1:xy - 2:x1:y
r =
~n2:x2 - Lx)2] [n1:y2 - (Ly)2
]
6(5,150) - (180)(150)
=
,; 16(8,000) - (180)2
]16(3,950) - (150)2
]
3,900
=
v(15,600)(1,200)
3,900
= 4,326.66
r = 0.9
1"1.-i1if:i.itl->~ft'Yl~i1V1if:i.1Viuii (r) 0.9 ~~b-if1'ln~ 1 bbMw:i1<ii1bbti->iii~11ei~llm1:i.iif:i.iViuiinumh~
:i.i1nmi~1t1'luVi0'Yl1~b~mnu 11~iiu ~imu~VinmfftJ~A-i'l~l'Yla-im:i.11->11i1.1m'loitlum->V'Imn-,Uimm'll1tJ
1<11$1
' ' d
~jij'JTI 7.3.3 UinJi~~nfiHli'Jfll'lfll't'iiA~
u:m.h~anll11ti~fll'HlTl'l'IJA (The Coefficient of Determination) l-ifif1.1J'1ni;-cl r2 bllu
.-i1~bbi1'1~11<i11bbtl11lm~ x mm111ei1lmtJm->bti~tJubbUl1~'llm<i11m.h<il1:i.i y 1#i~bWtJ~l'1 ~~.-11 r2
'J~
'li1tJl~eifil.J1tJm1:i.iif:i.iViuif1~11~<i11bbU'> x bbl1~ y1#iei~wif'1b'JU:i.Jln~ut'i1.-i1 r2lJ.-i1'1~bbi1'1~11<111bbU'l'iii1->~
x 111m111eifil.J1tJm->bU~tJUbbUl1~'llm<if1bbU'><ill:i.J y 1"1~ b'liu !11 r2
= 0.9 bbi1'1~11<ii1bbU'> x ei1lmtJ
f11->bU~tJUbbUl1~'llm.-i1 y 1"11'~ 90% <hu5n 10% 'llmm->bU~tJUbbtlm1i.J111m-,11eiiimtJ1"1"11tJ<ii1bbU'>
x dbb'1tn"lbnA"l1n<i11bbti11Jw1 .-11 r2
~.1m1rn1#i,~ll.-i1ei~->~'Yl11~ o ii~ 1 !i1~1ijAi"iei o bbl1~'11'1~lJ<il
iiiei 1 !i1if:i.it11~ft'Yl~bbvi~m1ri1V1UA'J~1aJmm111uein1"111111bb1.h1l'11~bb'<i~<ii1bbU'l'<ill:i.Jilm1:i.iif:i.1Viuiifh1
'luVi0m~l<ii

1'11il'mh:::fin~<1Y1il':1-1Yiunbbl1:::1'11il'mh:::fin%bbvi~m1ri1Y1u'1 lii1~bllu1h:::1tJ'lfib!ii<Jm11'11'11m1l-I
il':1-1Yiun1::w)1~~1bL1h1l<11:::Lbl1::~1Lb1h"11l-l1um1wmmclJI1tJ'irrm-;-11"'niltJb:ii~L~u il11'11m1:1-1il':1-1Yiun
ri~mn<J:::rl~Nii1'11'1'11wmmclri~ilm1:1-1LL~utl1mn~u
1ufom.h~~ 7.15 L'1'11u1rnml'l1il'l-!1h:::fiYJ!ftY1il':1-1Yiutl'11'11 r = 0.9 ~~iiul'l1il'l-!1h::fin!LLvi'
m1ri1Y1u'1 r2 = 0.81 Y1mtJm1:1-111 81 % 'llil,m1btl~tJUbbtli1~'llmtJil'1'llltJ'11m-;-11<JlilJ1tJ1Ji<J1n~1u1u
~Yinm#tJ~l171~lYll-i
¥11arlHn 1.16
u~l'J"1bYJf1 ~1n'1 1'1u1,{i1iltirnclm-,ililnri1ifomrn{i1:1-11~1wu1tJ ~u'ivi1TI<J~u~rifo,m1
' .'ibf1"'1::Vim1:1-1il':1-1Yiun1:::Y111~<111LL1l1 2 <111 111'111oii<J1tJ1um1l11Ji,rrn1l'.iNi1'iiiltJil'1'll1tJfiu.'11'llil'u~:1-11n
Uilrn'flml'1 ~~LJibl1u-ifil:1;Ji1tJil'1'llltJbbi1::1'111.if'ii1tJ1um-;-l11J1'f111luil~"1~~·hum 4u~ii
'l'Ul. ri1i'll''ii1ahm1~hnmn l'J!Jlij'!Jll'J
(1:1U11J : 10,000 U1!1) (1:1U11J : 100,000 U1!1)
2547 48 130
2548 52 151
2549 50 150
2550 55 158
m1'11u1rn1'11il':1-1tl1:::fin!LLvi~m1ri1Y1u'1 (r2
) byjil'ii<J11rn111rn1btl~tJubbtlii~'llil'tJil>l'llltJ
fiu.'i1'l1mu7~dbn>l"OJ1n1lnliwii'llil~l'l11.if<J1tJ1um1111Ji,rrn1mnuilm'fltJ~1"' 'irrm1vnl'i1 r21Ji"OJ1nm1Y11
x y xy x2 y2
48 130 6,240 2,304 16,900
52 151 7,852 2,704 22,801
50 150 7,500 2,500 22,500
55 158 8,690 3,025 24,964
':i'll-1 205 589 30,282 10,533 87,165

r =
nl:xy - l:xl:y
4(30,282) - (205)(589)
=
,J[4(10,533) - (205)2][4(87,165) - (589)2]
r = .887
•
<ii~uu r2
= .786
1#1'11 r = .887 ~~i'l.-iwih'ln.11 bbl'l'1~11tJ<i'1'llll'Jbbl'l~fi1'l:if'ii1tJ'lum•tw1,irn1i'lm1l-lil'l-l~uilnu
<it11~l-l1mbri:::cllu1Jlul1.rvrnb~mnu
1#1'11 r2 = .786 ~~VlmtJ<nll-111 78.6% 'llil~m•btl~tJUbbUl'l~'llrntJ<il'l'llltJl'flm'l11<ifilJ1tJl#i''il1n
'11'l<ii'ii1tJ'lum•LW'ISllil V11ilnl'i111#111m•1tl~tJUbbtifl~'llil~tJill'l'llltJi'1u;;1'll<i~u~Mvi~unu'111<ii'ii1tJ'lum•
LW'ISn.hl~~ 78.6% rhu5n 21.4% iiu~w'iui:l'ii~tJ~u'l b'llu 'llfll'llltJ
A tf ~ ""'- tf A
fll 'l'J!l'l'li::11m1tl'1tl6l'l'Vt'l:lt!JW11'l6fl1111m1::11m1mirrnm'II~'Vt'l:l (Multipie Regression
Analysis) illuiilm•'l'lmmrn"lumru~i'1~1u1u<i11bbti'l5'11~VIm tJ<if1 1i'lrn'il1nmilbfl'll~Vim•11'111<itJ
<itl1.J~1tJ,XiiYi'il1•rn1'il1n<ii1bbtl•l'il'f'l~bVim<if11~tJ~.Jm'ilbllwii<i~1n<?J~1'illVi'11'1'<mmllnfl'lll-lflm'1bf~<iu
'lum•ii1bUU1j'lfi'il.-i1'1'lmmcim"J'il~<iiil~Y'i'il1'lrn1<il1bbti'll'il'f'l~~mnn11 1 <ii1 <i11<Jtlw11rntJ8'1'llltJ~~
1llu#1bbU'l<illl-li'lm1:wil':w~uilnu<il1iitl•l'il'f'l~V1mtJ<if1 b'llu 'llfll'lllmrn~.-i1'l<ii'ii1tJ'lum'lL11J1"rnli1u;;1
m•'l'lmmciJbtJ1rrm•1•m1~Vin1•11'1niltJWl1J~rn1Jl•nnm.m1•

.rill y ~il <i'i1bb1h1111~
xi ~il <ii1mh1il1">~~iilltj k <ii1 (brit1 i = 1, 2, ..k)
.... ....J ..... ....
a fitl '11.1111~1bn'U y b~il x1
= x? = ... = "k = O
b
1
, b
2
... bk ~ti fi~'l.J"i~fl'J~m-,1111111m.J 1111tJfi bi d'Ju<i1~bbftlll~lll'>bU~tJ'UbbU!W1Jt1~
<ii1bb'l.J"i1111~ Y L5lt1<ii1bb'l.J">1il1'>~ x. bU~tJul'l.J 1 VIU1tJ •rit1<i'i1bb'l.J"illl1">~llw1 ii.!1'1~~1
ii'1ft~mw11111ti tl'W'V!']rnii<ii1bbU'>llft"i~'iiilu1u:i.nn m">fi1u1rn'"~'-ii'1il~1'11b111t1~'lf1tJ w•iii'lii
<i'i1bb'l.J"illl1">:0 2 $11 mm-,11.i1u1rnl'1bt1~ 1111tJ'l-ii!1a-Jm">lli1~1 $~d
y = a + b
1
x1
+ b
2
x2
a = Y- bJx1 - bl2
(Lx~)(Lx1y) - (Lx1
x2
)(Lx2
y)
bl =
(Lx~)(Lx~) - (Lx1
x/
b2 =
(Lx~)(Lx2y) - (Lx1
x
2
)(Lx1
y)
(Lx~)(L~) - (Lx1
x/
fl1f!aHTI 7.17
lJ~~J1iiuvhr111"ilfJV1'11lJY1lJ11 tJll<11'111tJJ1iim'.il'lf'llll~lJ~iim1a-Jfi~'W'uilnlJ.!1'l-ii~1tJ'lum-,
1!141'1'rnlbbft~"ilrn'll1tJ ~~bbftlll~1'1'11tJ!1~m-;n1-,1111111lltJ'W'V!'Jlli iii~d
' •
bl'.itl y
. ~ ~
tJti1'1'111tJ'Ul~'U'W'lflll"ilfjVImlJ
<11'l-ii~1tJ'lum,,1!14:i;rn1
•,:!, I ' a v A
x
2
fitl "ilfil'llltJilltlVl'U1tJ'lltl~'Ul~'U'W'lflll"ilfjVlmlJ
<J1nm1&Jif~'W'uil1111&J!1~m,,dnci111'111
b
1
d'iutJt11'1'llltJ~b'l.J~tJ'Ul'l.J L5JeJ<i1'l-ii~1tJL'Um"i1!14:i;rn1•'1.J~tJ'Ul'l.J 1 VIU1tJ Lu'll~~"ilrn
'llltJ'1ilVIU1tJif~fi~L~~
b2
•lJutJt1111'111tJ~b'l.J~tJul'l.J •rit1-,1'11'111tJ'1ilV1U1m'l.J~tJul'l.J 1 VIU1tJ Lu'llrn~~.!1'l-ii~1tJ
'lum"i1!141'1'rnli'.i~fi~L~&J

~18ti1.:1fi 7.18
' .N11J'll11J1J~i:i'Yl'll11Ji)tlmm1uVi!l"'YI'111!l"'i'YImm111111'1'11J11 '51u1u~1f'".J!!l'll!l"Jnf11ilm1i<
#i-1Yiuiinufi'JJ'51u1ul'l~"1um'l'L'Yl'l'ff'l'l~n"Jnff1 bb11~'51u1ul'l~"1·1.m11mii111m.,.1i'"1u'llm~urr1 L'111il
nl'l'bRu4im.m'51u1u s l'l~" ii"1um'l'1~~ 7.26 b~!l'Y11m"l'l'lrnmcl'51u1u~1~"~!ll'11J14im.Yibm1~~m'l'•
" " ~ Ji1ll'IJ11-ll'llM'lf!l
4
6
6
8
6
7
5
7
1Vi y
y
4
6
6
8
6
7
5
7
Nl1Tii-I (~) 49
;Jimjf'lr~11-111,.,1n•«'l'lfiii~~11ft'1
3
6
7
9
8
7
6
8
' .d, Q Q ..... ""
l'l!l '111.11urnm'll!l
~!l '51u1u!'l~"1um'l'L'Y11fi'l'l~nu~nfl1
~!l '51u1u!'l~"1um1rt1ii111m114i~uf11
2
"i x2 xl
3 4 9
6 4 36
7 4 49
9 5 81
8 4 64
7 5 49
6 4 36
8 5 64
54 35 388
;)1ml-11'1r~11-1fll•llllil'lIll•Hl11Jft'1
4
4
4
5
4
5
4
5
x2
2 "iY ~y "ix2
16 12 16 12
16 36 24 24
16 42 24 28
25 72 40 45
16 48 24 32
25 49 35 35
16 30 20 24
25 56 35 40
155 345 218 240

y =
x1 =
x2 =
=
=
=
b2 =
=
=
a =
=
=
49
8
6.125
54
- = 6.75
8
35
8
= 4.375
(Lx;)(:l°:x
1
y) - (Lx
1
x
2
)(Lx
2
y)
(Lx~)(Lx;) - (Lx1
x/
(155)(345) - (240)(218)
(388)(155) - (240)2
0.4547
(Lx~)(Lx2y) - (Lx
1
x
2
)(Lx
1
y)
(Lx~)(Lx;) - (Lx
1
x
2
)
2
(388)(218) - (240)(345)
(388)(155) - (240)2
0.7024
Y- bix1 - bzxz
6.125 - (0.4547)(6.75) - (0 7024)(4.375)
-0.0172
<J1n'1:J.Jm-,iwmm.JY'l'J1~ru Y = a+ b1
x1
+ b2
x2
~~.au <J~1iii'1:J.Jm1
Y = -0.0172 + 0.4547x
1
+ 0.7024x
2
1wihm.h~ttt'f1i;J11'.J'JJ11'.1.;J!l~m1V1rnrnrui>i1k1~!lLu.i~~<il€111l t'111ViV1iirn1w111l'.Ilm41Y'IYin~
1jnff1~m1u 5 .i~~ bbil~~m1u.i~~Lum11'illi"1m11.n!uff1 8 .i~~
'"~1i11 '11V1mrnru~1u1uf1~~~!l = -0.0172 + o4547(5) + o7024(8)
= 7.8755
<l..i .<:! ,g ,t: ,,,,; ~ 'i'~I A,.... ~....
'trn~11111ftmrnuBmtn1::11Mn 7.3.4 mn 1u1llu{)llllfl1Jm1JJ 7.3.4
. ' ' ', 4 1 """ "" ~ ""
'UU'U1fl11ftfllfl'l1'1.111J'll 7 ll!l'U'll 7.3 !1!l~'l'I 7.3.4

•
"'~01.!'t'J 7.4
' '
b~t1-Jfi 7.4.2 n11<ild1~ff81Jq;)11f111~~1bfi~._'!"lJD._'!n11f!i.1l~b~~8u
' ' 'I .<t IV f{ I ,o! ,'9, ~<v I
'lJ<J.1fi1UblJ l'N llJU'i.1j'Jffi fll'JVll ill l>l'1 um1i.1il<ll01lil'1<JUflliNfiil~ bb<l~fll'JVIlill
~ilU'1~1Nfllb>I~U'!Jil~f!·mbi'lm blJUi'fi-1j':irn
2. m':i'il'J1"Jl'1ll.Jfl1m1i.ii11•llm'!Jmm':ll'l<>101•1'1~1lu ,Yi1l'i'i"J1':lru1'11 'llinToNm1rnrn~1oii
1l~ fi1'l'lmrnrnw1m'li1~'1n fiT•~~mnumJ•Yim101 1ll'JlU'!J1lrn'lJ'ilmu<ii.111111l1iJ
' 'A .di "" ?..I <v ,;i
bi.J1l'1rnn<i11l'UY1 7.4 "lrnbm un.inml'llm':iri
1. V11'!J'Ul01'!J1l~m1i.1ilii101bil~1lu'!J<i~m.,'l'lmmru1#1
2. 1lIDJ1UlTI!ll':i01"i1"Jl'1llJfi1m1i.ili1blitM'lJ1l~fll'Jil"1'1bil~ilUlefi'

lUnl'l~tnn'lcl<ifel~nl'lfil~tnn'lcl~'1n<ifel~bb:i.JUl'il lnli'bfll'J~fi'lJfi1,'7;imn~ff'1 "iTinl'lv •
~mmci~uOln.i1;inu '~1-Jifi1~mmci~LLOln.J1;inu1u ~~'ri1m'll'1mmcl~;i<i1el;i1n:h'illm™mmci
...i11~"ill 111'5el"i~l'6el~iXuiJ'JJm'1'1Jel'lm1J-Jflm'1•fl~elmJ1nuelmfltJ;il'1 !JeiJ-Jw1'11111el1:i.J ""~'~•iieln
Hnlimmmmrn'1'1~1-Ji.J1m1J-Jfll11'1bfl~elu~ueltJ~<j'1
L'1tJrf11um'l'Vl1'1Jm'1'1Jmfl11J-Jflm'1•fl~elu~utJJ-1L'6nmlel~ 3 :Li'l~b1w1 1'1mi m'l'Vl1'11•>1~1'J
'JJel'chmiltJ'bl.Jitfi'l-1j'lcl m'l'Vl1'11b>t~tJm1J-1flmflbfl~eluri1'1;iilel' '"'~m'l'111.11-i'eil'J11~'JJel''11L>1~l'J'JJel'
fl1uLilmmufi'J-1j'lcl . .
, di , .o:i Q,I <! • I1. 11TH!lfllHl'11'J'lllNll'1'1Jllll'M!1J'IJll'lJl'iW (Mean Absolute Deviatiou; MAD) fllU'lrn '11'J
. "0 ' ..:!! ..... E' 1V • 1V ..,,,,.
n11UlfllN'1'l'll-J'JJel'fl11),!bfl'1elUifl-Jj1!:I Vl11'1'll'J '1U'lU'JJEll;jl1'1,U
MAD
Ll.11''7~ - .i11'1mmcil
= n
d
MAD ~Vl'lel = n
..e = fl'lll-lfll11'1bfll1elU
. "n = '1U'JU'JJ€ll;jl1
2. m;mfi1111~1'Ji'l11:1Ji'1'11'111'1~6'1Jftlii'~1rM (Meau Squared Error; MSE) '11u1rn1#f<ii,d
MSE =
L (.11''7' - '11~mn'lci)2
n
L (fl11l-Jfll11'1bfl~elu)2
= n
d
MSE
Le2
V11€J =
n

. .l!V l ~ I <':!! <V d
3. fll1'11lrll161'Jll:::'U!l~l'll1'!lll'J'UM!f1'1H1Jl'J~!lJlllflJ'IJ'H;it (Mean Absolute Percent Error;
MAPE) '11u1rn1ili'1,rd
MAPE =
=
MAPE =
n
m1:1-Jrim11•ri~eiu
:2: ' A x 100
fll"J'J.:J
n
n
i11vvNll 7.19 lJ™'v!••'1u'1rn1.J ~ifi11 1iliLR1Joiiei:1;1<i1.Jil11'lJltJ<J~-:ifirnhvm1mrntJil11'lJ1rnri~ei-:i
U'l11ll'1WTfa'V1t-:ilJLL<i~1-1m0rn,;1-J111im-i~mmrn~L4imJdl'.J1'i1'lJU111'lJil.:Jfl'll:J.Jflfll11Lfl~ilUil~1.:Jl"l
' .•!.I av<"!
'llil:l;j<l:J.J11.:JU
1
2
3
4
5
6
7
8
217 215
213 216
216 215
210 214
213 211
219 214
216 217
212 216
'11 L:
l ,,. • •
(flHli:-l-fll'Yifllft'iW)
2
-3
1
-4
2
5
·1
::1
-2
1ilil'i1 MAD
:2:e=
n
22
= -
8
= 2.75
2
3
1
4
2
5
1
1
22
4
9
1
16
4
25
1
16
76
.92%
1.41
46
1.90
.94
2.28
.46
1.89
10.26 ·,;

rif1u1111n11wmn'jnfrn.Jyt'JfiI 7-75
fll MSE = 2:e2
n
76
= -
8
= 9.5
}:[Jix 10~rrnr;--:i
fll MAPE = n
10.26 %
= 8
= 1.28 %
I I I l
q ~ f ~ """ d, ~
1u11u1m~Hfl!fl'llU'lfJt1 7 ~rnun 7.4 m1~n 7.4.1
brie:Jl'i1m"iwmmutbb1:;1'1fi1~~iJ.Jm"ibbi1 -:im1i.J1.1~u<j'1b'V1mbthd bl.Im1.11'1i1-:i~1blJU~iJ-:ill
m-rn<Jnrn111Nim1"iwmmn°ibb'1n.l1-:i<J1nfi1<J7-:imnibiJ1.1bV11.1-:iL'1 11lm1V1ii-:i~1i'lum1'1"i1"Jf!i:l1Jfll
vm1mrnb~ilfllfi-:im1J-Jciuh1Vin1J~Wmmrn$le:i m11~ Tracking Signal
Tracking Signal ~il mT1''1fi1m1wmmrnl'im11.1fi1<J7-:i1'1~bV11.1-:iL'1 ~d'ium1'111<Jflil1J'11
m1J-Jih•5m (bias) •1m-:im1rim'1bf1~ilu'llmmmmmrn"lwii1-:inm~v.i1'Ul-11<Jun-:ii:1'i<Jtiu bbfi:;blJum1
rnmn-:ibbU1LUJ-J11'11mli-Jf1l11'1bfl~ilUblJU1111u'iirim-:imnn11v:'foiim.m11 L'11.1ri1mrn~1n'!'1"id$le:i
Tracking Signal (TS) =
TSFE
MAD
v

bl'lil RSFE =
=
MAD =
Nff'1:W'llil~m1:wflml1lbfl~ilU (Running Sum of Forecast Errors)
L (.i1<J7~ - '11'Wmrnru)
L l.i1<J7~ - '11'Wmrnrul
n
'11'llil~ Tracking Signal ~~1wirn141 <J~!;Jrn.1ltJf:j<J1-;rn111il~lu'llillJb'llll'llil~m-;muri:w1111il
1:i.J n1'Wu11B~umw111uil'llilrn'l1"1murii.mffl1l~111llm-;'Wmrnru~'l-iiiXuili:lqJm,j1dJu4iB~111~l1l'l-ii L'liu
m'W~ 7.12 <J~LViu11'11 Tracking Signal LnU'llillJb'llllmlJ<fl:WlJu il1f!:w:w$i11m-;mnrnrult'l-ii'lll
Lfin.fl:1hmm'litm N'Wmrnclm<J'Yll?lflm'Li·fo.11 a 'l111:i.J•
'llillJL'llllmu!1:WlJUbbn~l'i1~iXum·m~~'ll1liXU"J~1:i.Ji1~l'ilillJ.~tJ'l LL<ii N'Wmrnru<J~1iil~'Yll?ln!N' . .
.'11-11111.11~ff:wLWlJff:Wwn $iiJ.11iXu<J~<iiil~1:i.J~1n11<Ju~um1:wflm11Lfl~iluLLifL~n<J UiltJ'l 1111il~~Lnu111
<JUfl~bfltJ:wm1h:wm1:wi'.111'Lln$i111
oMAD
'~ ' ..
+
s
l'rJCtJNl'l 7.20
Tracking Signal
>--~~~~~~~~~~~"--~~
d
ill'Vnl 7.12 Tracking Signals
NlJ~111l';'llil.:JlJ'lli'YIbbffUfflJltJ 'OJ1nl1l <iiil~nl';llTNffillJ1l'lllm-;'Wmrnru~'l-iiB~L111m~ff:w1111il1:i.J• •
il1111~1m1-;'Wmrnru-ifm;1fl'lum-;1.:i~ 7.27 L11mi1111u11.i1 Tracking Signal 11~'llil1Jb'll<il ± 4 MAD

2
3
4
5
6
7
8
1
2
3
4
5
6
7
8
217
213
216
210
213
219
216
212
' 'i'lwm1m111
215
216
215
214
211
214
217
216
11im-,Wi1u1rnm'11 Tracking Signal bb<fi?l~1u.i1TJ~~ 7.28 lii~d
1m1~n 7.28 m1r1in111n1 Tracking Signal
' 'fll'Wfilfl'jt)l fllllJ RSFE
flUl~HflgtlU
217 215 2 2 2 2
213 216 -3 -1 3 5
216 215 0 6
210 214 -4 -4 4 10
213 211 2 -2 2 12
219 214 5 3 5 17
216 217 -1 2 1 18
212 216 -4 -2 4 22
MAD
20
2.5
2.0
2.5
2.4
2.8
2.6
2.8
TS= RSFE
MAD
-0.4
0
-16
-0.8
1.1
0.8
-0.7
b~1lWi111 Tracking Signal mb'iitJum1'1'1 lii~mv-;~ 7.13 'il~bilu1'111 '11v-;mmcl~~ei~1u
'JJ1JlJb'JJ'1'11Uf1:W~1lc.ibnWll1JlJb'JJ'1 ± 4 MAD bb<li?1~11'llim'lV>;tJlmcl~toiiei~d~~tJil:WWL'1 •
' .

MAD
+4
0
-4
1 2 3 4 5 6 7 8
m~~ 7.13 Tracking Signals 'IJ0W'lii'nu!l'uili ;l1ft'PJ
"-' d Ji .Ji .,,; SJ "l_ t ..... ~....
111H'il1nflmrnu<>mm'>~1·rn~n 7.4.2 mn rnw1iln11l'lfl'ilfl'>n1 7.4.2
' ' . '
1 .di f "" "" .<!{ AO:l.
ll!!ll1fll'>flnlfl11'1-11l'l'll 7 l'l<lll'll 7.4 l'>M'll 7.4.2

nmn 11ii'lftlli'!J'lfl nniffSPSS for Windows iunn?im1::i11fil!!Jfl 'l'l.llcivnmii<m'i'Ylmilv 2544
lfim1 <ifv1wilij1TI™ 1u 1i-;::mmn;::![<iJ?'lf1miiim1::i1i!ffvlfb.nruim::mr.l<iJnnnm;i1iuu-J1u i,iu1v~ 10
i<i,i1'i'YlmiltJ!jL'll'iftJm-:;mil·n'lf 2544
e],,..,, 'iu'Yl-fmv nnwmmn!irlilnnil<iJi'lui'i!rnv![m'il 'j'.11mrn~m'l'Ylm'1v 2542
Chase Aguilano Jacobs. Production and Operations Management: Manufacturing and Services. gth ed.
Irwin/McGraw-Hill, 1998.
James B. Dilworth , Operations Management. 2nd ed. McGraw-Hill, 1996.
Jay Heizer, Barry Render. Produc{jon and Operations Management: Strategic and Tactical Decision.
4th ed. Prentice Hall, 1996.
Norman Gaither and Greg Frazier. Operations Management. 9th ed. South-Western, Thomson Leaming,
2002.
Russell, RS. and Taylor III, B.W. Operation Management. Prentice Hall, 2000.
R Dan Reid and Nada R Sanders. Operations Management. John Wiley and Sons, Inc.

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