This document provides instructions on how to model and solve linear programming problems in a spreadsheet using Excel's Solver tool. It begins with an introduction to LP problems and why spreadsheets are useful when there are more than two decision variables. It then explains how to access and enable the Solver add-in. The document outlines the steps to implement an LP model in a spreadsheet, including organizing the data, designating cells for decision variables and objectives/constraints. Finally, it provides examples of modeling different classic LP problems like resource allocation, transportation, and investment planning.

1. Modeling and Solving LP
Problems in a Spreadsheet
Chapter 3

2. 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 (i.e., Excel’s Solver) to
solve LP problems.

3. Spreadsheet Solvers
 We will be using Excel’s Solver to solve linear
programming problems.
 You access Solver from Excel’s Data tool bar
menu.
 If Solver is not present, click on the Office button
(in Excel 2007) or File button (in Excel 2010),
then Excel Options, followed by Add-ins, then
click on “Go” at the bottom of the window to
manage Excel’s add-ins, and finally make sure
that the check box for Solver add-In is enabled.
The next slides provide you with screen shots of
enabling Solver if it is not present.

4. Enabling Solver: Step 1
Excel 2007
Step 1: Click here

5. Enabling Solver: Step 1
Excel 2010

6. Enabling Solver: Step 2
Step 2:Click on Excel Options, then “Add-Ins”

7. Enabling Solver: Step 3
Step 3: Click on “Go”

8. Enabling Solver: Step 4
Step 4: Check the
Solver Add-In box

9. Solver for MAC
Those of you who have a MAC can still
download Solver for Macintosh Excel
2008. It’s free and can be downloaded by
following the instructions provided in the
following link (copy and paste into your
browswer):
http://www.solver.com/mac/dwnmacsolver.htm

10. Solver (continued)
 Note: there is no need to install Premium
Solver for Education (see pp. 53 of your
textbook), as the standard Solver that is
within Excel is capable of solving all
problems.
 The Simplex method is the default
algorithm that the standard version of
Excel’s Solver uses in solving LP
problems.

11. Steps in Implementing an LP
Model in a Spreadsheet
1. Organize the data for the model in 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.

12. 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

13. Using Solver
Please note that there two steps to solve
any type of a linear programming problem:
Step 1: Set the problem up in a
spreadsheet.
Step 2: Invoke Solver to enter all pertinent
parameters.
The next slide shows the first step which
entails setting the problem up in Excel.

14. Spreadsheet set up

15. Using Solver
Let’s break up the spreadsheet set up into
smaller pieces so that we do not find it
overwhelming. Please note that I’ll explain
the set up in a more generic way before
going into specifics.

16. Step 1: Designate an Area for
the Optimal Solution
We reserve any area in the
spreadsheet for the optimal
solution

17. Step 2: Create an Equation for
the Objective Function
The equation will be created in this
cell

18. Step 3: Input Each Constraint’s
Coefficients
Pump constraint is: 1X1 +1X2 <= 200, so we input the
values 1 and 1 in row 9 to reflect the coefficients of this
constraint. Same approach for the other constraints.

19. Step 4: Create an Equation for
Each Constraint’s L.H.S.
Each L.H.S. reflects consumption
of resources. We will worry about
the equation itself later on.

20. Step 5: Input Each Constraint’s
R.H.S. Value

21. Step 6: Creating the Equations
B6*B5+C6*C5
B9*B5+C9*C5
B10*B5+C10*C5 B11*B5+C11*C5

22. Solver Parameters
Objective Function
Equation Cell
Reserved Cells for
Optimal Solution
L.H.S. of
Constraints
R.H.S. of
Constraints

23. 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

24. Implementing the Model
Week 2 Multimedia contains a link to an
Excel file called Fig3-1.xls. Please open the
file to view it. Note: the first worksheet in
Fig3-1.xls,called “model”, illustrates the set
up of the problem in Excel along with all the
pertinent equations. The second worksheet,
“solution”, illustrates the optimal solution to
the problem using Solver. See pp. 48-60 of
your textbook for full details.

25. Solver (continued)
 Do not forget to always specify the
linearity and non-negativity assumptions in
Solver.
 This can be accomplished by checking the
check boxes Assume Non-Negative and
Assume Linear Model in the Solver
Options dialog box.

26. Video Clips
 Please view the video clips Sumproduct
Function and Solver Example.

27. 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.
 The company has 10,000 hours of wiring
capacity and 5,000 hours of harnessing capacity.
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

28. 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

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

30. Defining the Constraints
 Demand Constraints
M1 + B1 = 3,000 } model 1
M2 + B2 = 2,000 } model 2
M3 + 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

31. Implementing the Model
See file Fig3-17.xls
See pp. 63-67 of your textbook for full
details.

32. An Investment Problem:
Retirement Planning Services, Inc.
 A client wishes to invest $750,000 in the
following bonds.
Years to
Company Return 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

33. 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.

34. 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

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

36. Defining the Constraints
 Total amount 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

37. Implementing the Model
See file Fig3-20.xls
Also see pp. 67-71 of your textbook for full
details. Please note that I have prepared a
video clip of this problem for you to view.
The problem is set up in a more classic
and traditional way than the author’s
approach. So, please view it under the title
“The Investment Problem”.

38. 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

39. 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)
X36 = # of bushels shipped from Clermont (node 3) to Leesburg (node 6)

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

41. 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

42. Implementing the Model
See file Fig3-24.xls
Note that spreadsheet Fig3-24.xls
contains the model with all the pertinent
formulas. You need to invoke Solver and
specify your Target Cell, Changing Cells
and constraints to get the optimal solution.
See pp. 72-78 for full details.

43. Transportation Problem
(continued)
General guidelines in formulating the
transportation problem:
 When total supply = total demand, all supply
constraints will have equality signs and all
demand constraints will have equality signs.
 When total supply < total demand, all demand
constraints will have “<=“ signs and all supply
constraints will have “>=“ signs.
 When total supply > total demand, all supply
constraints will have “<=“ signs and all demand
constraints will have “>=“ signs.

44. Video Clip
 Please view the video clip “The
Transportation Problem” which will
provide you with a nice overview of how to
set up and solve a classic transportation
problem using Solver.

45. End of Chapter 3