Chapter 3

Modeling and Solving LP

Problems in a Spreadsheet

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-1

Introduction
Solving LP problems graphically is only
possible when there are two decision
variables
Few real-world LP have only two
decision variables

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-2

Spreadsheet Solvers

The company that makes the Solver in

Excel, Lotus 1-2-3, and Quattro Pro is
Frontline Systems, Inc.
web site:
http://www.frontsys.com

Other packages for solving MP problems:

AMPL
CPLEX

LINDO
MPSX

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-3

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 to
represent 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 lefthand side (LHS) of the constraint.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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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

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-1.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-6

How Solver Views the Model

Target cell - the cell in the spreadsheet
that represents the objective function
Changing cells - the cells in the
spreadsheet representing the decision
variables
Constraint cells - the cells in the
spreadsheet representing the LHS
formulas on the constraints

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-7

Goals For Spreadsheet Design

Communication - A spreadsheet's primary business

purpose is that of 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 the model and verify results.

Modifiability - A well-designed spreadsheet should be

easy to change or enhance in order to meet dynamic user
requirements.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-8

Spreadsheet Design Guidelines

Organize 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.
The 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.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Make vs. Buy Decisions:

The Electro-Poly Corporation

Electro-Poly is a leading maker of slip-rings.

A $750,000 order has just been received.
Model 1

Model 2

Model 3

3,000

2,000

900

Hours of wiring/unit

1.5

Hours of harnessing/unit

Cost to Make

$50

$83

$130

Cost to Buy

$61

$97

$145

Number ordered

The company has 10,000 hours of wiring capacity and

5,000 hours of harnessing capacity.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Decision Variables

M1 = Number of model 1 slip rings to make in-house
M2 = Number of model 2 slip rings to make in-house
M3 = Number of model 3 slip rings to make in-house
B1 = Number of model 1 slip rings to buy from competitor
B2 = Number of model 2 slip rings to buy from competitor
B3 = Number of model 3 slip rings to buy from competitor

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Objective Function

Minimize the total cost of filling the order.
MIN: 50M1 + 83M2 + 130M3 + 61B1 + 97B2 + 145B3

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Constraints

Demand Constraints
M1 + B1 = 3,000 } model 1
M2 + B2 = 2,000 } model 2
M 3 + B3 =

900 } model 3

Resource Constraints
2M1 + 1.5M2 + 3M3 <= 10,000 } wiring
1M1 + 2.0M2 + 1M3 <= 5,000 } harnessing

Nonnegativity Conditions
M1, M2, M3, B1, B2, B3 >= 0

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-17.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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An Investment Problem:
Retirement Planning Services, Inc.

A client wishes to invest $750,000 in the

following bonds.
Return

Years to
Maturity

Rating

Acme Chemical

8.65%

11

1-Excellent

DynaStar

9.50%

10

3-Good

Eagle Vision

10.00%

4-Fair

Micro Modeling

8.75%

10

1-Excellent

OptiPro

9.25%

3-Good

Sabre Systems

9.00%

13

2-Very Good

Company

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Investment Restrictions
No 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.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-16

Defining the Decision Variables

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

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Objective Function

Maximize the total annual investment return.
MAX: .0865X1 + .095X2 + .10X3 + .0875X4 + .0925X5 + .09X6

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Constraints

Total 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 >= 375,000
35% Restriction on DynaStar, Eagle Vision, and
OptiPro.
X2 + X3 + X5 <= 262,500

Nonnegativity conditions
Xi >= 0 for all i
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-20.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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A Transportation Problem:
Tropicsun
Supply
275,000

Groves

Distances (in miles)

Processing
Plants
Capacity

21

Mt. Dora

Ocala

50

200,000

40

35

400,000

30

Eustis

Orlando

22

600,000

55

300,000

20

Clermont

Leesburg

25

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

225,000

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Defining the Decision Variables

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)
XSpreadsheet
bushels shipped from Clermont (node 3) to Leesburg (node 6)
36 = # of
Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.
2-22

Defining the Objective Function

Minimize the total number of bushel-miles.
MIN: 21X14 + 50X15 + 40X16 +
35X24 + 30X25 + 22X26 +
55X34 + 20X35 + 25X36

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Constraints

Capacity constraints
X14 + X24 + X34 <= 200,000

} Ocala

X15 + X25 + X35 <= 600,000

} Orlando

X16 + X26 + X36 <= 225,000

} Leesburg

Supply constraints
X14 + X15 + X16 = 275,000 } Mt. Dora
X24 + X25 + X26 = 400,000 } Eustis
X34 + X35 + X36 = 300,000 } Clermont

Nonnegativity conditions
Xij >= 0 for all i and j

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-24.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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A Blending Problem:
The Agri-Pro Company

Agri-Pro has received an order for 8,000 pounds of

chicken feed to be mixed from the following feeds.
Percent of Nutrient in
Nutrient

Feed 1

Feed 2

Feed 3

Feed 4

Corn

30%

5%

20%

10%

Grain

10%

3%

15%

10%

Minerals

20%

20%

20%

30%

Cost per pound

$0.25

$0.30

$0.32

$0.15

The order must contain at least 20% corn, 15%

grain, and 15% minerals.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Decision Variables

X1 = pounds of feed 1 to use in the mix
X2 = pounds of feed 2 to use in the mix
X3 = pounds of feed 3 to use in the mix
X4 = pounds of feed 4 to use in the mix

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Objective Function

Minimize the total cost of filling the order.
MIN: 0.25X1 + 0.30X2 + 0.32X3 + 0.15X4

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Constraints

Produce 8,000 pounds of feed

X1 + X2 + X3 + X4 = 8,000

Mix consists of at least 20% corn

(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8000 >= 0.2

Mix consists of at least 15% grain

(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8000 >= 0.15

Mix consists of at least 15% minerals

(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8000 >= 0.15

Nonnegativity conditions
X1, X2, X3, X4 >= 0

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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A Comment About Scaling

Notice that the coefficient for X2 in the corn

constraint is 0.05/8000 = 0.00000625
As Solver solves our problem, intermediate
calculations must be done that make coefficients
large 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...

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-30

Re-Defining the Decision

Variables
X1 = thousands of pounds of feed 1 to use in the mix
X2 = thousands of pounds of feed 2 to use in the mix
X3 = thousands of pounds of feed 3 to use in the mix
X4 = thousands of pounds of feed 4 to use in the mix

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-31

Re-Defining the Objective

Function
Minimize the total cost of filling the order.
MIN: 250X1 + 300X2 + 320X3 + 150X4

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-32

Re-Defining the Constraints

Produce 8,000 pounds of feed

X1 + X2 + X3 + X4 = 8

Mix consists of at least 20% corn

(0.3X1 + 0.5X2 + 0.2X3 + 0.1X4)/8 >= 0.2

Mix consists of at least 15% grain

(0.1X1 + 0.3X2 + 0.15X3 + 0.1X4)/8 >= 0.15

Mix consists of at least 15% minerals

(0.2X1 + 0.2X2 + 0.2X3 + 0.3X4)/8 >= 0.15

Nonnegativity conditions
X1, X2, X3, X4 >= 0

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-33

A Comment About Scaling

Earlier the largest coefficient in the
constraints was 8,000 and the smallest
is 0.05/8 = 0.00000625.
Now the largest coefficient in the
constraints is 8 and the smallest is
0.05/8 = 0.00625.
The problem is now more evenly scaled.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-34

The Assume Linear Model Option

The

Solver Options dialog box has an option

labeled Assume Linear Model.
When you select this option Solver performs some
tests to verify that your model is in fact linear.
These test are not 100% accurate & often fail as a
result of a poorly scaled model.
If Solver tells you a model isnt linear when you
know it is, try solving it again. If that doesnt work,
try re-scaling your model.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-35

Implementing the Model

See file Fig3-33.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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A Production Planning Problem:

The Upton Corporation

Upton is planning the production of their heavy-duty air

compressors for the next 6 months.
Month
1

Unit Production Cost

$240

$250

$265

$285

$280

$260

Units Demanded

1,000

4,500

6,000

5,500 3,500

4,000

Maximum Production 4,000

3,500

4,000

4,500 4,000

3,500

Minimum Production

1,750

2,000

2,250 2,000

1,750

2,000

Beginning inventory = 2,750 units

Safety stock = 1,500 units
Unit carrying cost = 1.5% of unit production cost
Maximum warehouse capacity = 6,000 units
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Decision Variables

Pi = number of units to produce in month i, i=1 to 6
Bi = beginning inventory month i, i=1 to 6

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Objective Function

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.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Constraints

Production 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

Ending 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

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

1,500 <= B6 + P6 - 4,000 <= 6,000 } month 6

2-40

Defining the Constraints

(contd)

Beginning 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.
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-31.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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A Multi-Period Cash Flow Problem:

The Taco-Viva Sinking Fund - I

Taco-Viva needs to establish 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.

Investment
A
B
C
D

Available in Month Months to Maturity Yield at Maturity

1, 2, 3, 4, 5, 6
1
1.8%
1, 3, 5
2
3.5%
1, 4
3
5.8%
1
6
11.0%

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Summary of Possible Cash Flows

Investment
A1
B1
C1
D1
A2
A3
B3
A4
C4
A5
B5
A6

Cash Inflow/Outflow at the Beginning of Month

1
2
3
4
5
6
7
-1
1.018
-1 <_____> 1.035
-1
<_____> <_____> 1.058
-1
<_____> <_____> <_____> <_____> <_____> 1.11
-1
1.018
-1
1.018
-1 <_____> 1.035
-1
1.018
-1
<_____> <_____> 1.058
-1
1.018
-1
<_____> 1.035
-1
1.018

Reqd Payments $0
(in $1,000s)

$0

$250

$0

$250

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

$0

$300
2-44

Defining the Decision Variables

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

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Defining the Objective Function

Minimize the total cash invested in month 1.
MIN: A1

+ B1 + C1 + D 1

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-46

Defining the Constraints

Cash 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 7

Nonnegativity Conditions
Ai, Bi, Ci, Di >= 0, for all i

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-35.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Risk Management:
The Taco-Viva Sinking Fund - II

Assume the CFO has assigned the following risk ratings to

each investment on a scale from 1 to 10 (10 = max risk)
Investment

Risk Rating

The CFO wants the weighted average risk to not exceed 5.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-49

Defining the Constraints

Risk Constraints

1A1 + 3B1 + 8C1 + 6D1

A1 + B1 + C1 + D1
1A2 + 3B1 + 8C1 + 6D1
A2 + B1 + C1 + D1
1A3 + 3B3 + 8C1 + 6D1
A3 + B3 + C1 + D1
1A4 + 3B3 + 8C4 + 6D1
A4 + B3 + C4 + D1
1A5 + 3B5 + 8C4 + 6D1
A5 + B5 + C4 + D1
1A6 + 3B5 + 8C4 + 6D1
A6 + B5 + C4 + D1

<= 5

} month 1

<= 5

} month 2

<= 5

} month 3

<= 5

} month 4

<= 5

} month 5

<= 5

} month 6

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-50

An Alternate Version of the Risk Constraints

Equivalent 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 maximum allowable weighted average risk).
Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Implementing the Model

See file Fig3-38.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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Data Envelopment Analysis (DEA):

Steak & Burger

Steak & Burger needs to evaluate the performance (efficiency) of

12 units.
Outputs for each unit (Oij) include measures of: Profit, Customer
Satisfaction, and Cleanliness
Inputs for each unit (Iij) include: Labor Hours, and Operating
Costs
The Efficiency of unit i is defined as follows:
nO

Weighted sum of unit is outputs

Weighted sum of unit is inputs

nO

Oij w j
j 1
nI

I v
=
j 1

ij j

Oij w j
j 1
nI

I ij v j
j 1

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-53

Defining the Decision Variables

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.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-54

Defining the Objective Function

Maximize the weighted output for unit i :
nO

MAX: Oij w j
nO

Oij w j
j 1

j 1

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-55

Defining the Constraints

Efficiency cannot exceed 100% for any unit

nO

nI

j 1

j 1

Okj w j I kj v j , k 1 to the number of units

Sum of weighted inputs for unit i must equal 1

nO nI

nI

I ij v j 1
j 1 j 1

I ij v j 1
j 1

Nonnegativity Conditions
wj, vj >= 0, for all j

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-56

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.

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

2-57

Implementing the Model

See file Fig3-41.xls

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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End of Chapter 3

Spreadsheet Modeling and Decision Analysis, 3e, by Cliff Ragsdale. 2001 South-Western/Thomson Learning.

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