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

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 It’s free and can be downloaded by following the instructions provided in the following link (copy and paste into your browswer):

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: 350X X2 } 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. Objective Function Equation Cell
Solver Parameters
Objective Function Equation Cell
Reserved Cells for Optimal Solution
R.H.S. of Constraints
L.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 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.
Model 1 Model 2 Model 3
Number ordered 3,000 2,
Hours of wiring/unit
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.

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 = } model 3
Resource Constraints
2M M2 + 3M3 <= 10,000 } wiring
1M M2 + 1M3 <= 5,000 } harnessing
Nonnegativity Conditions
M1, M2, M3, B1, B2, B3 >= 0

31. Implementing the Model
See file Fig3-17.xls
See pp 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: .0865X X2+ .10X X X5+ .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 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: 21X X X16 +
35X X X26 +
55X X X36

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 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