Spreadsheet Modeling & Decision AnalysisA Practical .docx

Spreadsheet Modeling
& Decision Analysis
A Practical Introduction to Management Science
5th edition
Cliff T. Ragsdale
© 2007 South-Western College Publishing
Modeling and Solving LP Problems in a Spreadsheet
Chapter 3
IntroductionSolving LP problems graphically is only possible
when there are two decision variablesFew real-world LP have
only two decision variablesFortunately, we can now use
spreadsheets to solve LP problems
Spreadsheet SolversThe company that makes the Solver in
Excel, Lotus 1-2-3, and Quattro Pro is Frontline Systems, Inc.
Check out their web site:
http://www.solver.comOther packages for solving MP problems:

AMPLLINDO
CPLEXMPSX
The Steps in Implementing an LP Model in a Spreadsheet
1.Organize the data for the model on the spreadsheet.
2.Reserve separate cells in the spreadsheet for each decision
variable in the model.
3.Create a formula in a cell in the spreadsheet that corresponds
to the objective function.
4.For each constraint, create a formula in a separate cell in the
spreadsheet that corresponds to the left-hand side (LHS) of the
constraint.
Let’s Implement a Model for the
Blue Ridge Hot Tubs Example...
MAX: 350X1 + 300X2} profit
S.T.:1X1 + 1X2 <= 200} pumps
9X1 + 6X2 <= 1566} labor
12X1 + 16X2 <= 2880} tubing
X1, X2 >= 0} nonnegativity
Implementing the Model
See file Fig3-1.xls

How Solver Views the ModelTarget cell - the cell in the
spreadsheet that represents the objective functionChanging cells
- the cells in the spreadsheet representing the decision
variablesConstraint cells - the cells in the spreadsheet
representing the LHS formulas on the constraints
Let’s go back to Excel and see how Solver works...
Goals For Spreadsheet DesignCommunication - A spreadsheet's
primary business purpose is communicating information to
managers. Reliability - The output a spreadsheet generates
should be correct and consistent.Auditability - A manager
should be able to retrace the steps followed to generate the
different outputs from the model in order to understand and
verify results.Modifiability - A well-designed spreadsheet
should be easy to change or enhance in order to meet dynamic
user requirements.
Spreadsheet Design Guidelines - IOrganize the data, then build
the model around the data.Do not embed numeric constants in
formulas.Things which are logically related should be
physically related.Use formulas that can be copied.Column/rows
totals should be close to the columns/rows being totaled.

Spreadsheet Design Guidelines - IIThe English-reading eye
scans left to right, top to bottom.Use color, shading, borders
and protection to distinguish changeable parameters from other
model elements.Use text boxes and cell notes to document
various elements of the model.
Make vs. Buy Decisions:
The Electro-Poly CorporationElectro-Poly is a leading maker of
slip-rings.A $750,000 order has just been received. The
company has 10,000 hours of wiring capacity and 5,000 hours
of harnessing capacity.
Model 1 Model 2Model 3
Number ordered3,0002,000900
Hours of wiring/unit21.53
Hours of harnessing/unit121
Cost to Make$50$83$130
Cost to Buy$61$97$145
Defining the Decision Variables
M1 = Number of model 1 slip rings to make in-house
B1 = Number of model 1 slip rings to buy from competitor

Defining the Objective Function
Minimize the total cost of filling the order.
MIN:50M1+ 83M2+ 130M3+ 61B1+ 97B2+ 145B3
Defining the ConstraintsDemand Constraints
M1 + B1 = 3,000} model 1
M2 + B2 = 2,000} model 2
M3 + B3 = 900} model 3Resource Constraints
2M1 + 1.5M2 + 3M3 <= 10,000 } wiring
1M1 + 2.0M2 + 1M3 <= 5,000 } harnessingNonnegativity
Conditions
M1, M2, M3, B1, B2, B3 >= 0
See file Fig3-17.xls
An Investment Problem:
Retirement Planning Services, Inc.A client wishes to invest
$750,000 in the following bonds.
Years to
CompanyReturn MaturityRating
Acme Chemical8.65%111-Excellent

DynaStar9.50%103-Good
Eagle Vision10.00%64-Fair
Micro Modeling8.75%101-Excellent
OptiPro9.25%73-Good
Sabre Systems9.00%132-Very Good
Investment RestrictionsNo more than 25% can be invested in
any single company.At least 50% should be invested in long-
term bonds (maturing in 10+ years).No more than 35% can be
invested in DynaStar, Eagle Vision, and OptiPro.
X1 = amount of money to invest in Acme Chemical
X2 = amount of money to invest in DynaStar
X3 = amount of money to invest in Eagle Vision
X4 = amount of money to invest in MicroModeling
X5 = amount of money to invest in OptiPro
X6 = amount of money to invest in Sabre Systems
Maximize the total
annual investment return:

MAX: .0865X1+ .095X2+ .10X3+ .0875X4+ .0925X5+ .09X6
Defining the ConstraintsTotal amount is invested
X1 + X2 + X3 + X4 + X5 + X6 = 750,000 No more than 25% in
any one investment
Xi <= 187,500, for all i 50% long term investment restriction.
X1 + X2 + X4 + X6 >= 375,00035% Restriction on DynaStar,
Eagle Vision, and OptiPro.
X2 + X3 + X5 <= 262,500Nonnegativity conditions
Xi >= 0 for all i
A Transportation Problem: Tropicsun
Mt. Dora
1
Eustis
2
Clermont
3

Ocala
4
Orlando
5
Leesburg
6
Distances (in miles)
Capacity
Supply
275,000
400,000
300,000
225,000
600,000
200,000
Groves
Processing
Plants
21
50
40
35
30
22
55

25
20
Xij = # of bushels shipped from node i to node j
Specifically, the nine decision variables are:
X14 = # of bushels shipped from Mt. Dora (node 1) to Ocala
(node 4)
X15 = # of bushels shipped from Mt. Dora (node 1) to Orlando
(node 5)
X16 = # of bushels shipped from Mt. Dora (node 1) to Leesburg
(node 6)
X24 = # of bushels shipped from Eustis (node 2) to Ocala (node
4)
X25 = # of bushels shipped from Eustis (node 2) to Orlando
(node 5)
X26 = # of bushels shipped from Eustis (node 2) to Leesburg
(node 6)
X34 = # of bushels shipped from Clermont (node 3) to Ocala
(node 4)
X35 = # of bushels shipped from Clermont (node 3) to Orlando
(node 5)
X36 = # of bushels shipped from Clermont (node 3) to Leesburg
(node 6)
Minimize the total number of bushel-miles.
MIN:21X14 + 50X15 + 40X16 +

35X24 + 30X25 + 22X26 +
55X34 + 20X35 + 25X36
Defining the ConstraintsCapacity constraints
X14 + X24 + X34 <= 200,000} Ocala
X15 + X25 + X35 <= 600,000} Orlando
X16 + X26 + X36 <= 225,000} LeesburgSupply constraints
X14 + X15 + X16 = 275,000} Mt. Dora
X24 + X25 + X26 = 400,000} Eustis
X34 + X35 + X36 = 300,000} ClermontNonnegativity
conditions
Xij >= 0 for all i and j
A Blending Problem:
The Agri-Pro CompanyAgri-Pro has received an order for 8,000
pounds of chicken feed to be mixed from the following
feeds.The order must contain at least 20% corn, 15% grain, and
15% minerals.
NutrientFeed 1Feed 2 Feed 3Feed 4
Corn30%5%20%10%
Grain10%3%15%10%
Minerals20%20%20%30%
Cost per pound$0.25$0.30$0.32$0.15

Percent of Nutrient in
X1 = pounds of feed 1 to use in the mix
MIN: 0.25X1 + 0.30X2 + 0.32X3 + 0.15X4
Defining the ConstraintsProduce 8,000 pounds of feed
X1 + X2 + X3 + X4 = 8,000Mix consists of at least 20% corn
(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8000 >= 0.2Mix consists of
at least 15% grain
(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8000 >= 0.15Mix consists of
at least 15% minerals
(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8000 >= 0.15Nonnegativity
conditions
X1, X2, X3, X4 >= 0
A Comment About ScalingNotice the coefficient for X2 in the

‘corn’ constraint is 0.05/8000 = 0.00000625As Solver runs,
intermediate calculations are made that make coefficients larger
or smaller.Storage problems may force the computer to use
approximations of the actual numbers.Such ‘scaling’ problems
sometimes prevents Solver from being able to solve the problem
accurately.Most problems can be formulated in a way to
minimize scaling errors...
Re-Defining the Decision Variables
X1 = thousands of pounds of feed 1 to use in the mix
Re-Defining the
Objective Function
MIN: 250X1 + 300X2 + 320X3 + 150X4
Re-Defining the ConstraintsProduce 8,000 pounds of feed
X1 + X2 + X3 + X4 = 8Mix consists of at least 20% corn
(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8 >= 0.2Mix consists of at
least 15% grain
(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8 >= 0.15Mix consists of at
least 15% minerals
(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8 >= 0.15Nonnegativity

conditions
X1, X2, X3, X4 >= 0
Scaling: Before and AfterBefore:
Largest constraint coefficient was 8,000
Smallest constraint coefficient was
0.05/8 = 0.00000625.After:
Largest constraint coefficient is 8
Smallest constraint coefficient is
0.05/8 = 0.00625.The problem is now more evenly scaled!
The Assume Linear Model OptionThe Solver Options dialog box
has an option labeled “Assume Linear Model”. This option
makes Solver perform some tests to verify that your model is in
fact linear. These test are not 100% accurate & may fail as a
result of a poorly scaled model.If Solver tells you a model isn’t
linear when you know it is, try solving it again. If that doesn’t
work, try re-scaling your model.
A Production Planning Problem:

The Upton CorporationUpton is planning the production of their
heavy-duty air compressors for the next 6 months.Beginning
inventory = 2,750 units Safety stock = 1,500 unitsUnit carrying
cost = 1.5% of unit production costMaximum warehouse
capacity = 6,000 units
12 3456
Unit Production Cost$240$250$265$285$280$260
Units Demanded1,0004,5006,0005,5003,5004,000
Maximum Production4,0003,5004,0004,5004,0003,500
Minimum Production2,0001,7502,0002,2502,0001,750
Month
Pi = number of units to produce in month i, i=1 to 6
Bi = beginning inventory month i, i=1 to 6
Minimize the total cost production
& inventory costs.
MIN:240P1+250P2+265P3+285P4+280P5+260P6
+ 3.6(B1+B2)/2 + 3.75(B2+B3)/2 + 3.98(B3+B4)/2
+ 4.28(B4+B5)/2 + 4.20(B5+ B6)/2 + 3.9(B6+B7)/2
Note: The beginning inventory in any month is the same as the
ending inventory in the previous month.

Defining the Constraints - IProduction levels
2,000 <= P1 <= 4,000 } month 1
1,750 <= P2 <= 3,500 } month 2
2,000 <= P3 <= 4,000 } month 3
2,250 <= P4 <= 4,500 } month 4
2,000 <= P5 <= 4,000 } month 5
1,750 <= P6 <= 3,500 } month 6
Defining the Constraints - IIEnding Inventory (EI = BI + P - D)
1,500 < B1 + P1 - 1,000 < 6,000 } month 1
1,500 < B2 + P2 - 4,500 < 6,000 } month 2
1,500 < B3 + P3 - 6,000 < 6,000 } month 3
1,500 < B4 + P4 - 5,500 < 6,000 } month 4
1,500 < B5 + P5 - 3,500 < 6,000 } month 5
1,500 < B6 + P6 - 4,000 < 6,000 } month 6
Defining the Constraints - IIIBeginning Balances
B1 = 2750
B2 = B1 + P1 - 1,000
B3 = B2 + P2 - 4,500
B4 = B3 + P3 - 6,000
B5 = B4 + P4 - 5,500
B6 = B5 + P5 - 3,500
B7 = B6 + P6 - 4,000
Notice that the Bi can be computed directly from the Pi.
Therefore, only the Pi need to be identified as changing cells.

A Multi-Period Cash Flow Problem:
The Taco-Viva Sinking Fund - ITaco-Viva needs a sinking fund
to pay $800,000 in building costs for a new restaurant in the
next 6 months. Payments of $250,000 are due at the end of
months 2 and 4, and a final payment of $300,000 is due at the
end of month 6.The following investments may be used.
InvestmentAvailable in MonthMonths to MaturityYield at
Maturity
A1, 2, 3, 4, 5, 611.8%
B1, 3, 523.5%
C1, 435.8%
D1611.0%
Summary of Possible Cash Flows
Investment1234567
A1-11.018
B1-1<_____>1.035
C1-1 <_____><_____> 1.058
D1-1 <_____> <_____> <_____> <_____> <_____> 1.11
A2-11.018
A3-11.018
B3-1 <_____> 1.035
A4-11.018
C4-1 <_____> <_____> 1.058

A5-11.018
B5-1 <_____> 1.035
A6-11.018
Req’d Payments $0$0$250$0$250$0$300
(in $1,000s)
Cash Inflow/Outflow at the Beginning of Month
Ai = amount (in $1,000s) placed in investment A at the
beginning of month i=1, 2, 3, 4, 5, 6
Bi = amount (in $1,000s) placed in investment B at the
beginning of month i=1, 3, 5
Ci = amount (in $1,000s) placed in investment C at the
beginning of month i=1, 4
Di = amount (in $1,000s) placed in investment D at the
beginning of month i=1
Minimize the total cash invested in month 1.
MIN: A1 + B1 + C1 + D1
Defining the ConstraintsCash Flow Constraints
1.018A1 – 1A2 = 0 } month 2
1.035B1 + 1.018A2 – 1A3 – 1B3 = 250 } month 3
1.058C1 + 1.018A3 – 1A4 – 1C4 = 0 } month 4

1.035B3 + 1.018A4 – 1A5 – 1B5 = 250 } month 5
1.018A5 –1A6 = 0 } month 6
1.11D1 + 1.058C4 + 1.035B5 + 1.018A6 = 300 } month
7Nonnegativity Conditions
Ai, Bi, Ci, Di >= 0, for all i
Risk Management:
The Taco-Viva Sinking Fund - IIAssume the CFO has assigned
the following risk ratings to each investment on a scale from 1
to 10 (10 = max risk)
InvestmentRisk Rating
A1
B3
C8
D6The CFO wants the weighted average risk to not exceed 5.
Defining the ConstraintsRisk Constraints
1A1 + 3B1 + 8C1 + 6D1
< 5
A1 + B1 + C1 + D1
} month 1

1A2 + 3B1 + 8C1 + 6D1
< 5
A2 + B1 + C1 + D1
} month 2
1A3 + 3B3 + 8C1 + 6D1
< 5
A3 + B3 + C1 + D1
} month 3
1A4 + 3B3 + 8C4 + 6D1
< 5
A4 + B3 + C4 + D1
} month 4
1A5 + 3B5 + 8C4 + 6D1
< 5
A5 + B5 + C4 + D1
} month 5
1A6 + 3B5 + 8C4 + 6D1
< 5
A6 + B5 + C4 + D1
} month 6
An Alternate Version of the Risk ConstraintsEquivalent Risk
Constraints
-4A1 – 2B1 + 3C1 + 1D1 < 0 } month 1
-2B1 + 3C1 + 1D1 – 4A2 < 0 } month 2
3C1 + 1D1 – 4A3 – 2B3 < 0 } month 3
1D1 – 2B3 – 4A4 + 3C4 < 0 } month 4
1D1 + 3C4 – 4A5 – 2B5 < 0 } month 5

1D1 + 3C4 – 2B5 – 4A6 < 0 } month 6
Note that each coefficient is equal to the risk factor for the
investment minus 5 (the max. allowable weighted average risk).
Data Envelopment Analysis (DEA):
Steak & BurgerSteak & Burger needs to evaluate the
performance (efficiency) of 12 units. Outputs for each unit (Oij)
include measures of: Profit, Customer Satisfaction, and
CleanlinessInputs for each unit (Iij) include: Labor Hours, and
Operating CostsThe “Efficiency” of unit i is defined as
follows:
Weighted sum of unit i’s outputs
Weighted sum of unit i’s inputs
=
wj = weight assigned to output j
vj = weight assigned to input j
A separate LP is solved for each unit, allowing each unit to
select the best possible weights for itself.

Maximize the weighted output for unit i :
MAX:
Defining the ConstraintsEfficiency cannot exceed 100% for any
unit
Sum of weighted inputs for unit i must equal 1
Nonnegativity Conditions
wj, vj >= 0, for all j
Important Point
When using DEA, output variables should be expressed on a
scale where “more is better” and input variables should be
expressed on a scale where “less is better”.

Analyzing The
Solution
End of Chapter 3
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Module1------- Slip Ring -------Number toModel 1Model
2Model 3Make000Buy000Cost toMake$50$83$130Total
CostBuy$61$97$145$0# Available000#
Needed3,0002,000900Hours
RequiredUsedAvailableWiring2.01.53.0010,000Harnessing1.02.
01.005,000
&A
Page &P

Electro-Poly Corporation
Minimize: E11
By changing: B6:D7
Subject to: B13:D13>=B14:D14
E17:E18<=F17:F18
B6:D7>=0
Variable cell
Variable cell
Variable cell
Variable cell
Variable cell
Variable cell
Set cell
Constraint cell
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Constraint cell

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