Model LP Problems in Spreadsheets

1
Cliff T. Ragsdale, Spreadsheet Modeling & Decision Analysis, Ninth Edition. © 2022 Cengage. All Rights Reserved. May not be scanned, copied or
duplicated, or posted to a publicly accessible website, in whole or in part. 1
Chapter 3
Modeling and Solving LP
Problems in a Spreadsheet
duplicated, or posted to a publicly accessible website, in whole or in part.

2
Icebreaker: Dream Job
• The class will be broken into pairs of students.
• Answer the following question:
• What was your dream job when you were little?
• What is your dream job now?

3
Chapter Objectives (1 of 2)
By the end of this chapter, you should be able to:
03.01 Formulate algebraic LP models for a variety of decision problems.
03.02 Describe and perform the steps required to implement a linear
programming (LP) model in a spreadsheet.
03.03 Apply solver software to solve LP models implemented in a spreadsheet.
03.04 Discuss the goals and guidelines of good spreadsheet design.
03.05 Contrast heuristic solutions and optimal solutions to LP problems.
03.06 Discuss the impact of scaling on LP problems and their solution.

4
Chapter Objectives (2 of 2)
By the end of this chapter, you should be able to:
03.07 Discuss the meaning and purpose of parameterized optimization.
03.08 Explain and use the following functions:
SUM( ), SUMPRODUCT( ), SUMIF( ), IF( ),OR( ), AND( ), LEFT( ),
INDEX( ), PsiOptValue( ), PsiCurrentOpt( ).

5
Introduction

6
Polling Activity
Which business discipline uses optimization problems most?
a) Accounting
b) Finance
c) Information Systems
d) Management
e) Marketing
f) Supply Chain

7
Introduction
• Solving LP problems graphically is only possible when there are two decision
variables
• Few real-world LP have only two decision variables
• Fortunately, we can now use spreadsheets to solve LP problems

8
Spreadsheet Solvers
• The company that makes the Solver in Excel, Google Sheets, Quattro Pro is
Frontline Systems, Inc.
• Check out their web site:
• http://www.solver.com
• Other packages for solving MP problems:
AMPL LINDO
CPLEX LPSolve

9
Analytic Solver
• This book comes with a 140-day trial version of Analytic Solver
• Analytic Solver includes:
• a greatly enhanced version of the Solver built into Excel
• and many other tools & features to be discussed throughout this book
• You can download Analytic Solver from:
• http://www.solver.com/student

10
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.

11
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

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Implementing the Model (1 of 9)
Fig3-1.xlsm

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Discussion Activity
This model says that X1 and X2 cannot be negative. It is intuitive that we cannot
produce a negative number of hot tubs, so why must we have these constraints?
What would happen if we did not have these constraints in place?

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How Solver Views the Model
• Objective cell - the cell in the spreadsheet that represents the objective
function
• Variable cells - the cells in the spreadsheet representing the decision variables
• Constraint cells - the cells in the spreadsheet representing the LHS formulas
on the constraints

15
Goals for Spreadsheet Design
• Communication - 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.

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Spreadsheet Design Guidelines - I
• 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.

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Spreadsheet Design Guidelines - II
• 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.

18
Make vs. Buy Decisions
The Electro-Poly Corporation

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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
Number ordered 3,000 2,000 900
Hours of wiring/unit 2 1.5 3
Hours of harnessing/unit 1 2 1
Cost to Make $50 $83 $130
Cost to Buy $61 $97 $145
• The company has 10,000 hours of wiring capacity and 5,000 hours of
harnessing capacity.

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Defining the Decision Variables (1 of 7)
M1 = Number of model 1 slip rings to make in-house
B1 = Number of model 1 slip rings to buy from competitor

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Defining the Objective Function (1 of 7)
Minimize the total cost of filling the order.
MIN: 50M1+ 83M2+ 130M3+ 61B1+ 97B2+ 145B3

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Defining the Constraints (1 of 7)
• Demand Constraints
M1 + B1 = 3,000 } model 1
M2 + B2 = 2,000 } model 2
M3 + B3 = 900 } model 2
• 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

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Fig3-19.xlsm

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Knowledge Check (1 of 2)
• In designing a spreadsheet it is a good idea to embed numeric constants in
formulas?
• True
• False

25
An Investment Problem
Retirement Planning Services, Inc.

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An Investment Problem: Retirement Planning
Services, Inc.
• A client wishes to invest $750,000 in the following bonds.
Company Return Years to Maturity Rating
ACME Chemical 8.65% 11 1-Excellent
DynaStar 9.50% 10 3-Good
Eagle Vision 10.00% 6 4-Fair
Micro Modeling 8.75% 10 1-Excellent
OptiPro 9.25% 7 3-Good
Sabre Systems 9.00% 13 2-Very Good

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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.

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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 System

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

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• 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

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Fig3-22.xlsm

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Knowledge Check (2 of 2)
• One of the goals for spreadsheet design is that the output a spreadsheet
generates should be correct and consistent. This is:
a) Auditability
b) Reliability
c) Accuracy
d) Consistency

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A Transportation Problem
Tropicsun

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A Transportation Problem: Tropicsun

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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)

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

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• 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
Xij >= 0 for all i and j

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Fig3-26.xlsm

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A Blending Problem
The Agri-Pro Company

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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.

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X1 = pounds of feed 1 to use in the mix

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MIN: 0.25X1 + 0.30X2 + 0.32X3 + 0.15X4

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• 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
X1, X2, X3, X4 >= 0

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A Comment about Scaling
• Notice the coefficient for X2 in the ‘corn’ constraint is 0.05/8000 = 0.00000625
• As 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...

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Re-Defining the Decision Variables
• X1 = thousands of pounds of feed 1 to use in the mix

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Re-Defining the Objective Function
MIN: 250X1 + 300X2 + 320X3 + 150X4

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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
X1, X2, X3, X4 >= 0

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Scaling: Before and After
• Before:
• 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!

49
Fig3-30.xlsm

50
A Production Planning
Problem
The Upton Corporation

51
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 2 3 4 5 6
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 2,000 1,750 2,000 2,250 2,000 1,750
• 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

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

53
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.

54
Defining the Constraints - I
• 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

55
Defining the Constraints - II
• 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
1,500 < B6 + P6 − 4,000 < 6,000 } month 6

56
Defining the Constraints - III
• 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.

57
Fig3-33.xlsm

58
A Multi-Period Cash Flow
Problem
The Taco-Viva Sinking Fund

59
A Multi-Period Cash Flow Problem: The Taco-
Viva Sinking Fund - I
• Taco-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.
Investment Available in Month Months to Maturity Yield at Maturity
A 1, 2, 3, 4, 5, 6 1 1.8%
B 1, 3, 5 2 3.5%
C 1, 4 3 5.8%
D 1 6 11.0%

60
Summary of Possible Cash Flows
Investment 1 2 3 4 5 6 7
A1 −1 1.1018
B1 −1 <_____
> 1.035
C1 −1 <_____
> <_____
> 1.058
D1 −1 <_____
> <_____
> <_____
> <_____
> <_____
> 1.11
A2 −1 1.018
A3 −1 1.018
B3 −1 <_____
> 1.035
A4 −1 1.018
C4 −1 <_____
> <_____
> 1.058
A5 −1 1.018
B5 −1 <_____
> 1.035
A6 −1 1.018
Req’d Payments
(in $1,000s)
$0 $0 $250 $0 $250 $0 $300

61
• 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

62
Minimize the total cash invested in month 1.
MIN: A1 + B1 + C1 + D1

63
• 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
Ai, Bi, Ci, Di >= 0, for all i

64
Fig3-37.xlsm

65
Risk Management
The Taco-Viva Sinking Fund

66
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)
Investement Risk Rating
A 1
B 3
C 8
D 6
• The CFO wants the weighted average risk to not exceed 5.

67
• Risk Constraints
1 1 1 1
1 1 1 1
1A 3B 8C 6D
month 1
A B
5 }
C D
  
 


2 1 1 1
2 1 1 1
1A 3B 8C 6D
month 2
A B C D
5 }
  
  

3 3 1 1
3 3 1 1
1A 3B 8C 6D
month 3
A B C D
5 }
  
  

4 3 4 1
4 3 4 1
1A 3B 8C 6D
month 4
A B
5 }
C D
  
 


5 5 4 1
5 5 4 1
1A 3B 8C 6D
month 5
A B
5 }
C D
  
 


6 5 4 1
6 5 4 1
1A 3B 8C 6D
month 6
A B
5 }
C D
  
 



68
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 max. allowable
weighted average risk).

69
Fig3-40.xlsm

70
Data Envelopment Analysis
(DEA)
Steak and Burger

71
Data Envelopment Analysis (DEA): Steak and
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:
Weighted sum of unit s outputs
Weighted sum of unit s inputs
= 



0
1
1
'
'
n
ij j
j
nI
ij j
j
o w
I v
i
i

72
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.

73
Maximize the weighted output for unit i:
MAX :


0
1
n
kj j
j
o w

74
• Efficiency cannot exceed 100% for any unit
 
 
 
0
1 1
to the number of uni
, 1 ts
I
n n
kj j kj j
j j
o w l v k
• Sum of weighted inputs for unit i must equal 1



1
1
I
n
ij j
j
l v
wj, vj > = 0, for all j

75
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”.

76
Fig3-43.xlsm

77
Psi Functions
• Analytic Solver includes a number of custom functions that all begin with the
letters “Psi” (short for polymorphic spreadsheet interpreter)
• When running multiple optimizations:
• PsiCurrentOpt( ) returns the integer index of the current optimization
• PsiOptValue(cell, opt #) returns the optimal value of the indicated cell for a
particular optimization (opt #)

78
Analyzing the Solution
Fig3-48.xlsm

79
Self-Assessment
How would you rate your spreadsheet skills? Novice, Intermediate, Expert,
Other?
What is the difference between a heuristic solution and an optimal solution?
Which is better?
Which of the four goals for spreadsheet design is most important? Which is most
difficult?

80
Summary (1 of 2)
Now that the lesson has ended, you should have learned how to:
• Formulate algebraic LP models for a variety of decision problems.
• Describe and perform the steps required to implement a linear programming (LP) model in a
spreadsheet.
• Apply solver software to solve LP models implemented in a spreadsheet.
• Discuss the goals and guidelines of good spreadsheet design.
• Contrast heuristic solutions and optimal solutions to LP problems.
• Discuss the impact of scaling on LP problems and their solution.

81
Summary (2 of 2)
Now that the lesson has ended, you should have learned how to:
• Discuss the meaning and purpose of parameterized optimization.
• Explain and use the following functions:
SUM( ), SUMPRODUCT( ), SUMIF( ), IF( ),OR( ), AND( ), LEFT( ), INDEX( ), PsiOptValue( ),
PsiCurrentOpt( ).

Model LP Problems in Spreadsheets

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