Some Studies on Multistage Decision Making Under Fuzzy Dynamic Programming

P.K.Parida, S.K.Sahoo, K.C.Sahoo
International Journal of Logic and Computation (IJLP), Volume (1): Issue (1) 52
Some Studies on Multistage Decision Making Under
Fuzzy Dynamic Programming
Prashanta Kumar Parida prashantaparida_77@rediffmail.com
Department of mathematics,
Eastern Academy of Science and Technology
Bhubaneswar, Orissa, India.
S.K.Sahoo sahoosk1@rediffmail.com
Institute of Mathematics & Applications,
Bhubaneswar-751003, Orissa, India.
K.C.Sahoo kishorechandra_sahoo@rediffmail.com
Department of mathematics,
Eastern Academy of Science and Technology
Bhubaneswar, Orissa, India.
Abstract
Studies has been made in this paper, on multistage decision problem, fuzzy dynamic
programming (DP). Fuzzy dynamic programming is a promising tool for dealing with
multistage decision making and optimization problems under fuzziness. The cases of
deterministic, stochastic, and fuzzy state transitions and of the fixed and specified,
implicity given, fuzzy and infinite times, termination times are analyzed.
Keywords: Multistage decision process, multistage decision making under fuzziness, multistage
optimization problem under fuzziness, fuzzy dynamic programming.
1. INTRODUCTION
Multistage decision problems usually arise when decisions are made in the sequential manner over time
where earlier decisions may affect the feasibility and performance of later decisions. Dynamic
programming approach is used in such multistage decision problems which are based on Bellman’s
Principle of Optimality.
The common elements of dynamic programming models include decision stages, a set of possible states
in each stage, transitions from states in one stage to states in the next, value functions that measure the
best possible objective values that can be achieved starting from each state, and finally the recursive
relationships between value functions and different states.
A decision made at a given stage, and at a given state, induces a change in the output of the process.
The performance of the process is measured over some planning horizon, and is expressed by an
aggregate of partial scores that express the performance of the particular stage decisions. An optimal
sequence of decisions or controls at the consecutive stages over the planning horizon is then sought.
Dynamic programming is a powerful formal tool for dealing with a large spectrum of multistage decision
making and control problems. In mid 1950’s (cf. Bellman, 1957), dynamic programming has become a
standard tool in many fields including operation research, control theory and engineering, engineering,
computer science, etc.

Dynamic programming has been one of the earliest general techniques to which fuzzy sets theory has
been applied (cf. Chang, 1969; Bellman and Zadeh 1970, Esogbue and Ramesh, 1970).
2. MULTISTAGE DECISION MAKING PROCESS
The multistage decision making process can be separated into a number of sequential steps, or stages,
which is completed in one or more ways. The options for completing stages are known as decisions. The
condition of the process at a given stage is known as state at that stage; each decision effects a transition
from the current state to a state associated with the next stage.
The multistage decision making process is finite if there are only a finite number of stages in the process
and a finite number of states associated with each stage. Many multistage decision processes have
returns (cost or benefits) associated with each decision, and these returns may very with both the stage
and state of the process. The multistage decision process is deterministic if the outcome of each decision
is known exactly.
Mathematical Program
The mathematical program
optimize ( )∑=
=
n
i
ii xfZ
1
subject to bx
n
i
i ≤∑=1
(1)
0≥ix ,
ni ,......,2,1=
in which
( )ii xf ,
ni ,......,2,1= are known (non-linear) functions of a single variable and b is a non-
negative integer.
2.1. PRINCIPLE OF OPTIMALITY IN DYNAMIC PROGRAMMING
Dynamic programming is an approach for optimizing multistage decision making processes. It is based on
Bellman’s principle of optimality. An optimal policy has the property that, “regardless of the decisions
taken to enter a particular state in a particular stage, the remaining decisions must constitute an optimal
policy for leaving that state”. This principle, begin with the last stage of an n-stage process and determine
for each state the best policy for leaving that state and completing the process, assuming that all
preceding stages have been completed.
Let ≡u the state variable, whose values specify the states
( )≡umj
optimum return from completing the process beginning at stage
j in state u
( )≡ud j
decision taken at stage j that achieves
( )umj
The entries corresponding to the last stage of the process,
( )umn and
( )ud n are generally
straightforward to compute. The remaining entries are obtained recursively; i.e. the entries for
j th stage
( )1,......,2,1 −= nj are determined as functions of the entries for the
( )1+j th stage.
The dynamic programming approach is well suited processes modeled by system Eq.(1) in which each
decision pays off separately, independent of other decisions. For system Eq.(1), the value of
( )umn for
bu ,......,2,1,0= are given by
( ) ( ){ }xfoptimumum n
ux
n
0 ≤≤
=
(2)

The recursion formula is
( ) ( ) ( ){ }xumxfoptimumum jj
ux
j −+= +
≤≤
1
0 (3)
for 1,.......,2,1 −−= nnj . In Eq.(2), the decision variable x (which is denoted nx in Eq.(1)) runs
through integral values, as x ( )jx≡
in Eq.(3). That value of x which yields the optimum in Eq.(2) is
taken as
( )udn and that value of x which yields the optimum in Eq.(3) is taken as
( )ud j
. If more than
one values of x yields either optimum, arbitrarily choose one as the optimal decision. The optimal
solution to program Eq.(1) is ( )bmZ 1
*
= the optimal return from completing the process beginning at
state1 with b units available for allocation. With
*
Z determined, the optimal decisions
**
2
*
1 ,......,, nxxx
are found sequentially from
( )
( )
( )
( ) 








−−−−=
−−=
−=
=
−.......
..................
*
1
*
2
*
1
*
*
2
*
13
*
3
*
12
*
2
1
*
1
nnn xxxbdx
xxbdx
xbdx
bdx
(4) 2.2. DYNAMIC PROGRAMMING WITH DISCOUNTING
If money earns interest at the rate of i per period, an amount ( )np due n periods in the future has the
present (or discounted) value
( ) ( )npp n
α=0 , where i+
≡
1
1
α
(5)
The multistage decision making processes, the stages represent time periods and the objective is to
optimize a monetary quantity. The solution by dynamic programming, the recurrence formula for
( )umj
,
the best return beginning in stage j and state u , involves terms of the form
( )ym cj+
, the best return
beginning in the stage cj + (c time periods after stage j ) and state y . If
( )ym cj+
is multiplied by
c
α ,
where α is the above-defined discount factor, then
( )ym cj+
is discounted to its present value at the
beginning of stage
j . So
( )um1 will be discounted to the beginning of stage 1, which is the start of the
process.
2.3. STOCHASTIC MULTISTAGE DECISION PROCESSES
The multistage decision making process is stochastic if the return associated with at least one decision in
the process is random. This randomness generally two ways: either the states are uniquely determined
by the decisions but the returns associated with one or more states are uncertain or the returns are
uniquely determined by the states arising from one or more decisions are uncertain.
If the probability distributions governing the random events are known and if the number of stages and
the number of states are finite, then the dynamic programming approach is useful for optimizing a
stochastic multistage decision making process. The general procedure is to optimize the expected value
of the return. In those cases, where the randomness occurs exclusively in the returns associated with the
states and not in the states arising from the decisions, this procedure has the effect of transforming a
stochastic process into a deterministic one.

3. MULTISTAGE DECISION MAKING UNDER FUZZINESS
3.1. DECISION MAKING UNDER FUZZINESS
The basic elements of [19] general approach are: a fuzzy goal G in X , a fuzzy constraint C in X ,
and a fuzzy decision D in X ; X is a (non-fuzzy) space of options. The fuzzy relation
( ) ( ) ( )xxx GCD µµµ ∧= , Xx ∈∀ (6)
which expresses the “goodness” of an Xx ∈ as a solution to the decision making problem. Considered,
1 for definitely good (perfect) to 0 for definitely bad (unacceptable), through all intermediate values. The
“ ( )baba ,min=∧ ” operation is commonly used and is assumed throughout this paper though many
other operations are also employed ([11] and [12]).
For an optimal (non-fuzzy) solution, an Xx ∈ such that
( ) ( ) ( ) ( )( )xxxx GC
Xx
D
Xx
D µµµµ ∧==
∈∈
supsup*
(7)
is a natural (but not the only possible [11]) choice.
The multiple fuzzy constraints and fuzzy goals is defined in different spaces. Suppose that the fuzzy
constraint C is defined as a fuzzy set in { }xX = the fuzzy goal G is defined as a fuzzy set in { }yY =
and a function YXf →: , ( )xfy = is known. X and Y may be sets of decisions and their
outcomes, respectively.
Now, the induced fuzzy goal, both G′ and C are defined as
( ) ( )[ ]xfx GG µµ =′
for each Xx ∈ (8)
and the (min-type) fuzzy decision is
( ) ( ) ( ) ( )[ ] ( )xxfxxx CGCGD µµµµµ ∧=∧= ′
, for each Xx ∈ (9)
The n fuzzy constraints defined in X , nCCC ,......,, 21 , m fuzzy goals defined in Y , mGGG ,......,, 21 ,
and a function ( )xfy = , then the (min-type) fuzzy decision is
( ) ( )[ ] ( )[ ] ( ) ( )xxxfxfx nm CCGD µµµµµ ∧∧∧∧∧= ........ 11G
for each Xx ∈ (10)
Now we proceed the multistage decision making case within the above general framework.
3.2. MULTISTAGE DECISION MAKING (CONTROL) UNDER FUZZINESS
By [11], [12] and [19], we assume, the dynamics of the deterministic dynamic system under control is
described by a state transition equation
( )ttt uxfx ,1 =+ , ,.......1,0=t
(11)
where { } { }ntt sssxXxx ,......,,, 211 ==∈+ are the states at time (stage) t and 1+t , respectively
and { } { }mcccuUu ,........,, 21==∈ is the decision (control or input) at t ; X and U , are
assumed finite.

0
C 1
C 2
C 1−N
C
0u 1u 2u 1−Nu
0x 1x 2x ...... 1−Nx Nx
0
G 1
G 2
G 1−N
G N
G
0=t 1=t 2=t 1−= Nt Nt =
FIGURE 1: A General Framework For Multistage Decision Making Under Fuzziness
The multistage decision making (control) under fuzziness may be depicted in Fig.1. We start from an
initial state at stage (time) 0=t , 0x make a decision 0u attain a state, at time 1=t , 1x make a
decision 1u ,…… Finally, being at 1−= Nt in 1−Nx , we apply 1−Nu and attain the final state Nx .
The state transitions are given Eq.(11), the consecutive decision tu are subjected to fuzzy constraints
t
C
and on the states 1+tx fuzzy goals
1+t
G are imposed,
.1,.......,1,0 −= Nt The multistage decision
making process is evaluated by the fuzzy decision which is assumed a decomposable fuzzy set in
XUXU ×××× ....... , is
( ) ( ) ( ) ( ) ( )NGNCGCND xuxuxuuu NN µµµµµ ∧∧∧∧= −− − 1100110 110 ..........,.....,,
( ) ( )[ ]1
1
0
1 +
−
=
+∧∧= tGtC
N
t
xu tt µµ
(12)
where N is some termination time is fixed. The basic case, the optimal sequence of
decisions *
1
*
1
*
0 ,.....,, −Nuuu , such that
( ) ( ) ( ) ( ) ( )[ ]NGNCGCuuu
ND xuxuxuuu NN
N
µµµµµ ∧∧∧∧= −− −
−
110
,...,,
0
*
1
*
1
*
0 110
110
..........max,.....,,
( ) ( )[ ]1
1
0,...,,
1
110
max +
−
=
+
−
∧∧= tGtC
N
tuuu
xu tt
N
µµ
(13)
For simplicity, a fuzzy goal, ( )NG
xNµ is only imposed on the final state Nx . Then the
fuzzy decision is
( ) ( ) ( ) ( ) ( )NGNCGCND xuxuxuuu NN µµµµµ ∧∧∧∧= −− − 1100110 110 ..........,.....,,
(14)
and the problem is to find *
1
*
1
*
System
under
control
S
System
under
control
S
System
under
control
S

( ) ( ) ( ) ( ) ( )[ ]NGNCGCuuu
ND xuxuxuuu NN
N
µµµµµ ∧∧∧∧= −− −
−
110
,...,,
0
*
1
*
1
*
0 110
110
..........max,.....,,
(15)
This general problem formulation may be extended, mainly with respect to ([11] and
[12]):
● The type of termination time: fixed and specified in advance, fuzzy implicitly given
infinite.
● The type of the dynamic system: deterministic, stochastic, fuzzy and fuzzy
stochastic.
● The type of objective function: cost minimization, profit/benefit maximization and
fuzzy
criterion set based satisfactory degree maximization, and virtually all cases a
dynamic-
programming-type algorithm can be devised.
4. FUZZY DYNAMIC PROGRAMMING FOR THE CASE OF A FIXED AND
SPECIFIED TERMINATION TIME
The case of a fixed and specified (in advance) termination time is basic, and provides a suitable point of
departure for extensions. Here we discussed a deterministic, stochastic, fuzzy dynamic system, and fuzzy
criterion set dynamic program.
4.1. THE CASE OF A DETERMINISTIC DYNAMIC SYSTEM
A deterministic system is described by its state transition Eq.(11), i.e. ( )ttt uxfx ,1 =+ ;
{ }ntt sssXxx ,......,,, 211 =∈+ ; { }mt cccUu ,........,, 21=∈ ; 1,......,1,0 −= Nt , Xx ∈0 is the initial
state, and ∞<N is a fixed and specified termination time. The fuzzy constraints are
( ) ( )10 10 .,,......... −− NCC
uu Nµµ and the fuzzy goal is ( )NG
xNµ . The fuzzy decision is
( ) ( ) ( ) ( )NGNCCND xuuxuuu NN µµµµ ∧∧∧= −− − 100110 10 ..........,.....,, (16)
and to find
*
1
*
1
*
( ) ( ) ( ) ( )[ ]NGNCCuuu
ND xuuxuuu NN
N
µµµµ ∧∧∧= −− −
−
10
,...,,
0
*
1
*
1
*
0 10
110
..........max,.....,, (17)
Clearly, the last two terms are ( ) ( )( )111 ,1 −−− ∧− NNGNC
uxfu NN µµ depend on 1−Nu , hence Eq.(17) can
be rewritten as
( ) ( ) ( ) ( )[ ]NGNCCuuu
ND xuuxuuu NN
N
µµµµ ∧∧∧= −− −
−
10
,...,,
0
*
1
*
1
*
0 10
110
..........max,.....,,
( ) ( ) ( ) ( )( )( )


 ∧∧∧∧= −−−− −
−
−
−
11120
,...,,
,max.....max 1
1
20
110
NNGNCu
NCCuuu
uxfuuu NN
N
N
N
µµµµ (18)

On repeating this backward iteration for 021 ,.....,, uuu NN −− , we obtain the set of dynamic programming
recurrence equations
( ) ( ) ( )[ ]
( )



==
∧=
−−+−
+−−− +−−
−
−
Niuxfx
xux
iNiNiN
iNGiNCu
iNG iNiN
iN
iN
,........,2,1,,
max
1
11µµµ
(19)
where ()⋅−iN
G
µ is a fuzzy goal at iNt −= induced by a fuzzy goal at 1+−= iNt .
An optimal sequence of decisions sought,
*
1
*
1
*
0 ,.....,, −Nuuu is given by the successive maximizing values
of iNu − in Eq.(19). It is convenient to represent the solution,
*
tu , by an optimal policy UXat →:*
, such
that ( )ttt xau **
= , 1,........,1,0 −= Nt , i.e., relating an optimal decision to the current state.
4.2. THE CASE OF A STOCHASTIC DYNAMIC SYSTEM
The stochastic system is assumed to be a Markov chain whose temporal evolution is described by a
conditional probability ( )ttt uxxP ,1+ such that { }ntt sssXxx ,......,,, 211 =∈+ ,
{ }mt cccUu ,........,, 21=∈ ; Xx ∈0 is an initial state, 1,......,1,0 −= Nt and ∞<N is a fixed
and specified termination time. Now the following two problem formulations:
● due to [19]: find an optimal sequence of decisions
*
1
*
1
*
0 ,.....,, −Nuuu to maximize the
probability of attainment of the fuzzy goal, subject to the fuzzy constraints, i.e.
( ) ( ) ( ) ( )[ ]NGNCCuuu
ND xuuxuuu NN
N
µµµµ ∧∧∧= −− −
−
10
,...,,
0
*
1
*
1
*
0 10
110
..........max,.....,, (20)
where the fuzzy goal is viewed to be a fuzzy event in X whose (non-fuzzy) probability is [17],
( ) ( ) ( )∑∈
−−=
Xx
NGNNNNG
N
NN xuxxPxE µµ ., 11 (21)
● due to [14]: find an optimal sequence of decisions
*
1
*
1
*
0 ,.....,, −Nuuu to maximize the
expectation of the fuzzy decisions membership function, i.e.
( ) ( ) ( ) ( )[ ]NGNCCuuu
ND xuuExuuu NN
N
µµµµ ∧∧∧= −− −
−
10
,...,,
0
*
1
*
1
*
0 10
110
..........max,.....,, (22)
These formulations are clearly not equivalent.
Bellman and Zadeh’s approach [19]
Since in Eq.(20), ( ) ( )[ ]111 ,1 −−− ∧− NNGNC
uxfEu NN µµ depend only on 1−Nu , the next two right-most
terms depend only on 2−Nu etc., the structure of Eq.(20) is essentially the same as that of Eq.(18), and
the set of fuzzy dynamic programming recurrence equation is
( ) ( ) ( )[ ]
( ) ( ) ( )




==
∧=
∑∈
+−−−+−+−
+−−−
+−
+−+−
+−−
−
−
NixuxxPxE
xEux
Xx
iNGNNiNiNG
iNGiNcu
iNG
iN
iNiN
iNiN
iN
iN
,....,2,1;,
max
1
11
1
11111
1
µµ
µµµ
(23)

and we consecutively obtain
*
iNu − or optimal policies
*
iNa − , such that ( )iNiNiN xau −−− = **
,
Ni ,.......,2,1= .
Kacprzyk and Staniewski’s approach [14]
To solve problem Eq.(22), first introduce a sequence of functions [ ]1,0:
1
→×
=
UXXh
i
j
i and
[ ]1,0:
1
1
→×
+
=
UXXg
j
l
j ; Ni ,......,2.1,0= ; 1,.......,2,1 −= Nj , such that
( ) ( ) ( ) ( )
( ) ( ) ( )
( ) ( )
( ) ( )












=
=
=
∧∧∧=
−
=
+
−−−
∑
−
00000
010
1
010
0101010
,max
.............................................................................................................
,........,,max,........,,
,.,........,,,........,,
............................................................................................................
,.....,.....,,.........,
0
10
uxgxh
uuxguuxh
uxsPuushuuxg
xuuuuuuxh
u
kkk
u
kkk
n
i
kkikikkkk
NDNCCNNN
k
N µµµ
(24)
The consecutive decisions and states are juuu ,........,, 10 and jxxx ,........,, 10 , respectively, then jg is
the expected value of ( )0xD ⋅µ provided that the next decisions are optimal, i.e.
*
1
*
2
*
1 ,........,, −++ Njj uuu . It
can be shown ([11], [12] and [14]), that there exist functions UUXX
k
j
k →×
=1
:ω , such that
( ) ( )( )101010 ,........,,,,........,,,........,, −−− = kkkkkkkkk uuxuxguuxh ωω .
Then, an optimal policy sought,
*
ta , 1,.......,2,1,0 −= Nt is given by
( )
( ) ( )( )
( ) ( )( )( )






=
=
=
−−−−−− ,......,.......,,.......,,,......,,
.........................................
,,
0
*
020
*
211110
*
1
0
*
01110
*
1
00
*
0
xaxxaxxxxa
xaxxxa
xa
NNNNNN ω
ω
ω
(25)
It is depends not only on the current state but also on the trajectory.
4.3. THE CASE OF A FUZZY DYNAMIC SYSTEM
In this case the system is fuzzy and its dynamics is governed by a state transitions equation
( )ttt UXFX ,1 =+ , ,.........2,1,0=t (26)

where tX , 1+tX are fuzzy states at time (stage) t and 1+t , and tU is a fuzzy decision at t
characterized by their membership functions ( )tX xt
µ , ( )11 ++ tX xt
µ and ( )tU ut
µ , respectively.
Eq.(26) is equivalent to a conditioned fuzzy set ( )tttX uxxt
,11 ++
µ ([11] and [12]).
Baldwin and Pilsworth [13] proposed a dynamic programming scheme. First, for each
1,,.........2,1,0 −= Nt a fuzzy relation ( ) ( ) ( )11 1, ++ +∧= tGtCttR
xuxu ttt µµµ is constructed. The
degree to which tU and 1+tX satisfy
t
C and
1+t
G is
( )( ) ( ) ( )( ) ( ) ( )( )


 ∧∧∧= ++++ +
+
+
1111 1
1
1
maxmax,,, tGtX
x
tCtU
u
tttRtT xxuuxxuu t
t
t
t
t
t
t µµµµµµ (27)
The fuzzy decision is
( )010 ,........, XUU ND −µ
( ) ( )( ) ( ) ( )( ) ( ) ( )( )


 ∧∧∧∧∧∧= −− −
−
−
NGNX
x
NCNU
uCU
u
xxuuuu N
N
N
N
N
N
µµµµµµ maxmax.......max 1100 1
1
1
0
0
0
(28)
and an optimal sequence of fuzzy decisions
*
1
*
0 ,........, −NUU , such that
( ) ( )010
,.....,
0
*
1
*
0 ,........,max,........,
10
XUUXUU ND
UU
ND
N
−−
−
= µµ
( ) ( )( )
( ) ( )
( ) ( )( ) 















∧
∧∧
∧∧∧=
−− −
−
−− NGNX
xX
NCNU
uUCU
uU xx
uu
uu
N
N
NN
N
N
NN µµ
µµ
µµ
maxmax
maxmax.......maxmax
11
00
1
1
11
0
0
00
(29)
Hence the set of dynamic programming recurrence equation is
( ) ( ) ( )[ ]
( ) ( ) ( )( ) ( )
( ) ( ) ( )( ) ( )









−=
∧



 ∧=



 ∧∧=
∧=
−−−+−−+−
+−−−−
−+−−
−−
+−
+−−
−
−−
−
1,........,2,1
,maxmax
maxmax
max
11
1
1
1
1
1
1
Ni
xuxxux
XuuX
xxX
iNXiNiNiNXiNU
ux
iNX
iNGiNCiNU
uU
iNG
NGNX
x
NG
iNiNiN
NiN
iN
iNiN
iN
NiN
iN
N
N
N
N
µµµµ
µµµµ
µµµ
(30)
They redefine the problem formulation in terms of the reference fuzzy states and fuzzy decisions to finally
make Eq.(30) solvable ([11], [12] and [13]).
5. FUZZY DYNAMIC PROGRAMMING FOR THE CASE OF A FUZZY TERMINATION
TIME
In many real-world problems it may be more adequate (sufficient) to assume a fuzzy termination time as
more or less 5 years, a couple of days, ten years or so [7].

Let { }NkkkR ,.......,1,,1,......,1,0 +−= be the set of decision making states. At each Rt ∈ , a fuzzy
constraint ( )tC
utµ , and a fuzzy goal ( )vG
xvµ , Rv∈ is imposed on the final state. The fuzzy
termination time is given by ( )vTµ , Rv ∈ , which is a termination time v .
The fuzzy decision
( ) ( ) ( ) ( ) ( )vGTvCCvD xvuuxuuu vv µµµµµ ...........,.....,, 100110 10 ∧∧∧= −− − (31)
and to find an optimal termination time
*
v and an optimal sequence of decisions
*
1
*
1
*
0 ,.....,, −vuuu such that
( ) ( ) ( ) ( ) ( )[ ]vGTvCCuuu
vD xvuuxuuu vv
N
µµµµµ ...........max,.....,, 10
,...,,
0
*
1
*
1
*
0 10
110
∧∧∧= −− −
−
(32)
5.1. THE CASE OF A DETERMINISTIC DYNAMIC SYSTEM
Eq.(32) was formulated and solved by Kacprzyk ([7] and [8]). Then, Stein [23] presented a
computationally more efficient model and solution.
Kacprzyk’s approach
In Kacprzyk’s ([7] and [9]) formulation the set of possible termination times is
( ){ } { } RNkkvRv T ⊆+=>∈ ,.......,1,0: µ , hence an optimal sequence of decisions is
*
1
**
1
*
1
*
0 ,....,,,.....,, −− vkk uuuuu . The part
*
1
**
1 *,....,, −− vkk uuu is determined by solving
( ) ( ) ( )[ ]
( )






−+=+−=
=
∧=
−−+−
+−−− +−−
−
−
1,......,1,;1,......,2,1
,
,max,
1
11
Nkkvivi
uxfx
vxuvx
iviviv
ivGivCV
ivG iviv
iv
iv µµµ
(33)
where ( ) ( ) ( )vGTvG
xvvx vv µµµ ., = . An optimal termination time
*
v , then found by the maximizing
v in
( ) ( )vxx kGv
kG kk ,max 11 11 −− −− = µµ (34)
The part
*
2
*
1
*
0 ,.....,, −Kuuu is then determined by solving
( ) ( ) ( )[ ]
( )



−==
∧=
−−−−−
−−−+− −−−
−−
+−
1,.......,2,1;,
max
11
11 1
1
1
kiuxfx
xux
ikikik
ikGikCu
ikG ikik
ik
ik µµµ
(35)
Stein’s approach
Stein [23] presented a computationally more efficient dynamic programming approach. At 1−= Nt ,
{ }1,......,2,1 −∈ Ni , and attain
( ) ( ) ( )iNGTiNG
xiNx iNiN −− −− −= µµµ . or apply iNu −
and attain ( ) ( )11 +−− +−− ∧ iNGiNC
xu iNiN µµ .
The set of recurrence equation is therefore

( ) ( ) ( ) ( )[ ]
( )



==
∧=
−−+−
+−−−− +−−
−
−
−
Niuxfx
xuxx
iNiNiN
iNGiNCu
iNGiNG iNiN
iN
iN
iN
,.......,2,1;,
max
1
11µµµµ
(36)
and an optimal termination time is such a iNt −= at which terminating decision *
1*
−v
u , occurs, i.e.
when
( ) ( ) ( )[ ]11max +−−− +−−
−
− ∧> iNGiNCu
iNG
xux iNiN
iN
iN µµµ
(37)
5.2. THE CASE OF A STOCHASTIC DYNAMIC SYSTEM
This stochastic dynamic system was first formulated and solved in Kacprzyk ([8] and [9]) by combining the
section 4.2 and 5.1. An optimal termination time
*
v and an optimal sequence of decisions
*
1
*
1
*
0 ,.....,, −vuuu such that
( ) ( ) ( ) ( )[ ]vGvCCuuuv
vD xEuuxuuu vv
v
µµµµ ∧∧∧= −− −
−
10
,...,,,
0
*
1
*
1
*
0 10
110
..........max,.....,, (38)
where ( ) ( ) ( )vGTvG
xvx vv µµµ .= ; and ( ){ } { }Nkkvv T ,......,1,0: +=>µ . As in section 4.2, we
determine
*
v and
*
1
**
1 *,....,, −− vkk uuu by solving
( ) ( ) ( )[ ]
( ) ( ) ( )




=×=
∧=
+−
∈
−−+−+−
+−−−
+−
+−
+−
+−−
−
−
∑ NixuxxPxE
xEux
ivG
Xx
vvivivG
ivGivCu
ivG
iv
iv
iv
iviv
iv
iv
,.....,2,1;,
max
11111
1
1
1
1
1
µµ
µµµ
(39)
and
*
v is given by the maximizing v in
( ) ( )vxx kGv
kG kk ,max 11 11 −− −− = µµ (40)
The remaining part
*
0
*
3
*
2 ,.....,, uuu kk −− is obtained by solving
( ) ( ) ( )[ ]
( ) ( ) ( )




−=×=
∧=
∑∈
−−−−−−−−
−−−−−
−
−−−
−−−
−−
−−
1,.......,2,1;,
max
111
11
1
1
1
1
kixuxxPxE
xEux
Xx
ikGikikikikG
ikGikCu
ikG
ik
ikik
ikik
ik
ik
µµ
µµµ
(41)
In the later Stein’s [23], formulation the problem is solved by the following set of recurrence equations
( ) ( ) ( ) ( )[ ]
( ) ( ) ( )( )




=×=
∧∧=
∑∈
−−−−+−+−
+−−−−
−
+−+−
+−−
−
−
−
NiuxfuxxPxE
xEuxx
Xx
iNiNGiNiNiNiNG
iNGiNCu
iNGiNG
iN
iNiN
iNiN
iN
iN
iN
,.......,2,1;,,
max
11
1
11
1
µµ
µµµµ
(42)
where
*
v occurs when
( ) ( ) ( )[ ]11max +−−− +−−
−
− ∧> iNGiNCu
iNG
xEux iNiN
iN
iN µµµ (43)

5.3. THE CASE OF A FUZZY DYNAMIC SYSTEM
In this case, fix some (finite and relatively small) number of reference fuzzy states (and possibly
decisions), and obtain an auxiliary approximate system whose state transitions are of deterministic
system type ([12] and [16]). Then, Stein’s [23] approach can be employed.
6. MULTISTAGE DECISION MAKING (CONTROL) WITH AN IMPLICIT SPECIFIED
TERMINATION TIME
Now the process terminates when the state enters for the first time a termination set of states
{ } XsssW npp ⊂= ++ ,......,, 21 . We determine an optimal sequence of decisions
*
1
*
1
*
0 ,........,, −N
uuu , such that
( )0
*
1
*
1
*
0 ,........,, xuuu ND −
µ ( ) ( ) ( )]..........[max 1100
,...,, 110
NGNNCC
uuu
xxuxu N
N
µµµ ∧∧∧= −−
−
(44)
where Xxxx N
∈−110 ,.....,, W , and WxN
∈
The solution of Eq.(44) may proceed by using:
● Bellman and Zadeh’s [19] iterative approach,
● Komolov’s et al. [22] graph theoretic approach and
● Kacprzyk’s ([5] and [8]) branch and bound approach.
7. MULTISTAGE DECISION MAKING (CONTROL) WITH AN INFINITE
TERMINATION TIME
In all the problems considered the solution process required some iteration over consecutive stages. This
may be justified the number of stages is not too high, and when the process itself exhibits a sufficient
variability over time. In the fuzzy setting, the multistage decision making (control) problem with an infinite
termination time was first formulated and solved by Kacprzyk and Staniewski ([15] and [16]).
For the deterministic dynamic system Eq.(11), the fuzzy decision is
( ) ( ) ( ) ( ) ( ) ..........,......., 211100010 ∧∧∧∧= xxuxxuxuu GCGCD µµµµµ
( ) ( )[ ]1
0
lim +
=∞→
∧∧= tGttC
N
tN
xxu µµ (45)
and to find an optimal stationary strategy ( ),........., ***
aaa =∞
, such that
( ) ( )00
*
max xaxa D
a
D ∞∞
∞
= µµ
( )( ) ( )[ ]1
0
limmax +
=∞→
∧∧=
∞
tGttC
N
tNa
xxxa µµ (46)
Eq.(46) may be solved in a finite number of steps by using a policy iteration algorithm whose essence is a
step-by-step improvement of stationary policies. A policy iteration type algorithm was also proposed for
the stochastic system by [16].

8. COMPUTATIONAL COMPLEXITY
The development of efficient algorithms for processing various aspects of fuzzy dynamic programs is an
important research area in dynamic programming. In this section, we will summarize main results of the
computational complexity analysis provided by (1997). Esogbue [2] using two algorithms of Kacprzyk [7]
and Stein [23] for the fuzzy termination time discussed.
Esogbue showed that the dynamic programming approach presented by Kacprzyk requires ( ) 2/1+NN
iterations while the one proposed by Stein [23] requires only N and it is computationally more efficient.
Let, the time and space complexity analysis for both.
Here, consisting of n equations and 1−Nu controls each assuming m values, and time
kNt ,.......,1−= , the total number of operations involved is
( ) ( ) ( ) ( )12..11222 −−++−++−−− mnkkNnktmmkmtn (47)
Let tkm == , then Eq.(47) is of order ( )3
2 kO .
The basic dynamic programming formulation of the problem proposed by Stein [23] is the same expect for
the structure of the recurrence equations that require N only iterations of the optimizing process as
opposed to ( ) 2/1+NN required is Kacprzyk’s algorithm.
The space and the time complexities are of order ( )nO and ( )mnO , respectively. Essentially, the time
complexity is of order ( )( )( )kNmO −− 12 . This is of order ( )2
kO . The total storage demand is
( )( ) ( )122/21 −+++−+− knnkNkNn . This is order ( )kO . It was pointed out, the difference
between the earliest and the latest possible termination times is an important factor in the computationally
burden of this optimization process.
So, the model proposed by Stein [23] is computationally more efficient, taking ( )kO memory spaces in
( )2
kO operations, as opposed to ( )2
kO memory spaces in ( )3
kO operations in Kacprzyk’s (1977)
model. The superiority is exhibited both from the space and time complexity considerations.
The case of stochastic system, it was shown that the computational burden and storage requirements are
identical for both algorithms. If in the dynamic programming formulation, the objective is to maximize the
probability of attainment of the fuzzy goal G subject to non fuzzy constraints.
9. CONCLUSION
We have presented a brief exposition of main aspects of fuzzy dynamic programming, including main
problem classes and major applications in a variety of fields. We have studied dynamic programming
problems in fuzzy environments. Some basic problems have been studied and development of fuzzy
dynamic programming is given. It is shown that the fuzzy dynamic programming may be a promising tool
for dealing with multistage decision making and optimization problems under fuzziness. For other surveys
of fuzzy dynamic programming and its applications, we refer the reader to [1] and then the fundamental
presentation by [6] and [11].
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[1] A.O. Esogbue and R.E. Bellman. “Fuzzy dynamic programming and its extensions”. TIMS/
Studies in the Management Sciences, 20, 147-167, 1984.
[2] A.O. Esogbue and V. Ramesh. “Dynamic Programming and fuzzy allocation processes”.
Technical Memo No. 202, Dept. of Operation Research Case Western University, Cleveland,
OH., (1970).

[3] G. Conrnuejds and R. Tutuncu. “Optimization Methods in Finance”. Cambride University Press,
(2007).
[4] H.-J. Zimmermann. “Fuzzy Sets, Decision Makings, and Experts Systems”. Kluwer, Dordrecht
(1987).
[5] J. Kacprzyk. “A branch-and-bound algorithm for the multistage control of a non fuzzy system in a
fuzzy environment”. Control and Cybernetices, 7, 51-64, 1978b.
[6] J. Kacprzyk. “A generalization of fuzzy multistage decision making and control via linguistic
quantifiers”. Int. J. of Control, 38, 1249-1270, 1978b.
[7] J. Kacprzyk. “Control of non-fuzzy system in a fuzzy environment with fuzzy termination time”.
Systems Sciences, 3, 325-341, 1977.
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Systems Science, 4, 291-300, 1978b.
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[19] R.E. Bellman and L.A. Zadeh. “Decision-making in a fuzzy environment”. Management Sci., 17,
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