Copyright © 2000, Denise Sakai Troxell. All rights reserved. (updated from © Joe Aieta and Denise Sakai Troxell – Summer 1999.) Solving Linear Optimization

Copyright © 2000, Denise Sakai Troxell. All rights reserved.(updated from © Joe Aieta and Denise Sakai Troxell – Summer 1999.)

Solving Linear

Optimization Problems

with Excel Solver

for Microsoft Excel 2000


A Linear Optimization Problema diet problem

Julia wonders if she can keep the amount of fat in her diet down and still get all the protein (45 g), carbohydrate (256 g), and calories (1,980 kcal) that she needs every day by eating fast food only. For other nutrients, such as vitamins, iron, and calcium, she will depend on pills.

She chooses her favorite fast foods: hamburger and french fries. The nutritive values per serving are given below.

How many servings of hamburger and fries would Julia need to eat to satisfy her daily diet requirements?

396220Calories (kcal)

3232Carbohydrate (g)

315Protein (g)

1810Fat (g)

FriesHamburger


Formulate the Problemas a Linear Programming problem

constraintsconstraints

that minimizes the total amount of fat

subject to the following minimum diet requirements

HB = the number of servings of hamburger FF = the number of servings of french fries

15 HB + 3 FF 45 Protein constraint

Julia is looking for

objective functionobjective function

10 HB + 18 FF

decision variablesdecision variables

nonnegativity constraints

nonnegativity constraints

32 HB + 32 FF 256 Carbohydrate constraint 220 HB + 396 FF 1980 Calories constraint

Of course HB 0 and FF 0


Preparing the Worksheet for Solverstart with a blank sheet


Preparing the Worksheet for Solverenter labels

Enter row labels in cells A2:A6

Enter row labels in cells A2:A6



Enter column labels in cells

B1:F1

Enter column labels in cells

B1:F1



Enter inequalities in

cells E4:E6

Enter inequalities in

cells E4:E6



NOTE: These labels are not essential for the use of

Solver

NOTE: These labels are not essential for the use of

Solver


Preparing the Worksheet for Solverenter constants

Enter coefficients of objective

function in cells B3:C3

Enter coefficients of objective

function in cells B3:C3

Remember…

10 HB + 18 FF Fat



Enter coefficients of left side of

constraints in cells B4:C6

Enter coefficients of left side of

constraints in cells B4:C6

15 HB + 3 FF 45 Protein

32 HB + 32 FF 256 Carbohydrate

220 HB + 396 FF 1980 Calories

Remember…



Enter right sides of

constraints in cells F4:F6

Enter right sides of

constraints in cells F4:F6




Remember…


Preparing the Worksheet for Solvername decision variables cells

Highlight cells B1:C2

Highlight cells B1:C2



Click on InsertClick on Insert



Click on NameClick on Name



Click on CreateClick on Create





Make sure the Top row

(containing the names HB and

FF) box is checked

Make sure the Top row

(containing the names HB and

FF) box is checked



Click on OKClick on OK



Cell C2 is named FF and is currently

blank

Cell C2 is named FF and is currently

blank

Cell B2 is named HB and is currently

blank

Cell B2 is named HB and is currently

blank


Preparing the Worksheet for Solverenter formula for objective function

Remember…

10 HB + 18 FF Fat

NOTE: Objective function formula in cell D3


Click on cell D3

Click on cell D3



Type in =Type in =

Remember…

10 HB + 18 FF Fat





Click on cell B3

Click on cell B3

Remember…

10 HB + 18 FF Fat





Type in *Type in *

Remember…

10 HB + 18 FF Fat





Click on cell B2

Click on cell B2

Remember…

10 HB + 18 FF Fat





Type in +Type in +

Remember…

10 HB + 18 FF Fat





Click on cell C3

Click on cell C3

Remember…

10 HB + 18 FF Fat





Type in *Type in *

Remember…

10 HB + 18 FF Fat





Click on cell C2

Click on cell C2

Remember…

10 HB + 18 FF Fat





Press EnterPress Enter

Remember…

10 HB + 18 FF Fat





Click on cell D3 to check formula

Click on cell D3 to check formula

Remember…

10 HB + 18 FF Fat

NOTE: This step is not essential for the use of

Solver


Solver


Remember…

10 HB + 18 FF Fat

NOTE: The formula is equivalent to

=B3*$B$2+C3*$C$2 . Blanks in cells HB and FF (cells B2

and C2) are considered zeros

NOTE: The formula is equivalent to

=B3*$B$2+C3*$C$2 . Blanks in cells HB and FF (cells B2

and C2) are considered zeros


Current value of formula

entered in cell D3

Current value of formula

entered in cell D3

Formula entered in cell

D3


D3


Preparing the Worksheet for Solverenter formulas for constraints




Remember…

NOTE: Formulas for the left sides of the constraints in

cells D4:D6


cells D4:D6



Click on cell D3 and grab the fill handle ++

on the lower right corner

Click on cell D3 and grab the fill handle ++

on the lower right corner




Remember…


cells D4:D6


cells D4:D6



Drag the fill handle ++ down to

cell D6

Drag the fill handle ++ down to

cell D6

NOTE: The formula in cell D3 will be copied down to

cell D6

NOTE: The formula in cell D3 will be copied down to

cell D6




Remember…


cells D4:D6


cells D4:D6






Remember…


cells D4:D6


cells D4:D6






Remember…

Click on cell D4 to check formulaClick on cell D4

to check formula


Solver


Solver






Remember…

Current value of formula entered

in cell D4


in cell D4


D4 (Protein)


D4 (Protein)

NOTE: The rows for column B and C were copied as relative references while HB and FF

remained fixed

NOTE: The rows for column B and C were copied as relative references while HB and FF

remained fixed





Remember…



to check formula


Solver


Solver






Remember…

Current value of formula entered in

cell D5

Current value of formula entered in

cell D5

Formula entered in cell D5

(Carbohydrate)

Formula entered in cell D5

(Carbohydrate)





Remember…



to check formula


Solver


Solver






Remember…


in cell D6


in cell D6

Formula entered in cell D6 (Calories)

Formula entered in cell D6 (Calories)



NOTE: These cells will be colored to indicate that they are essential for

Solver

NOTE: These cells will be colored to indicate that they are essential for

Solver

Objective function

formula in cell D3

Objective function

formula in cell D3

Decision variable

values in cells B2:C2

Decision variable

values in cells B2:C2


Using Solverinvoke Solver

Click on ToolsClick on Tools



Click on SolverClick on Solver




Using Solvercomplete the Solver Parameters dialog box

Click on cell D3Click on cell D3

NOTE: The cell displayed in the Set Target Cell: box must contain the

objective function formula (cell D3)

NOTE: The cell displayed in the Set Target Cell: box must contain the

objective function formula (cell D3)



Check the

Min: circle

Check the

Min: circle



Click on the By

Changing Cells: box

Click on the By

Changing Cells: box



NOTE: The By Changing Cells: box

must display the decision variable cells

(cells B2:C2)

NOTE: The By Changing Cells: box

must display the decision variable cells

(cells B2:C2)

Highlight cells B2:C2Highlight

cells B2:C2



NOTE: The Subject to the Constraints: box

must contain the constraints on the decision variables

NOTE: The Subject to the Constraints: box

must contain the constraints on the decision variables

Click on Add

Click on Add



Click on the Cell

Reference: box

Click on the Cell

Reference: box



Highlight cells D4:D6

Highlight cells D4:D6



Click on to select

constraint types

Click on to select

constraint types



Click on >=Click on >=



Click on the Constraint:

box

Click on the Constraint:

box



Highlight cells F4:F6

Highlight cells F4:F6




NOTE: Different type constraints must be entered

separately

NOTE: Different type constraints must be entered

separately


Using Solverset the Options

Click on OptionsClick on Options



Check the Assume Linear

Model and the Assume Non-Negative boxes

Check the Assume Linear

Model and the Assume Non-Negative boxes

NOTE: The formula (in the Target Cell D3) is

linear on the non-negative variables (HB in

B2 and FF in C2)

NOTE: The formula (in the Target Cell D3) is

linear on the non-negative variables (HB in

B2 and FF in C2)



Accept the remaining default options by clicking

on OK

Accept the remaining default options by clicking

on OK


Using Solverexecute Solver

Click on Solve

Click on Solve


Using Solverread solution

NOTE: Solver uses a method known as

SIMPLEX

NOTE: Solver uses a method known as

SIMPLEX

This solution contains Protein: 105 g Carbohydrate: 256 g Calories: 1980 kcal satisfying the constraints


Solver finds a solution HB = 6.75 servings of Hamburger FF = 1.25 servings of French fries with minimum fat content of 90 g



Using Solverobtain the Answer Report

Click on AnswerClick on Answer


Using Solverend execution



Using Solverview the Answer Report

Click on the Answer

Report tab

Click on the Answer

Report tab


Using Solverread the Answer Report

NOTE: Resize window for a complete view of the spreadsheet portion

NOTE: Resize window for a complete view of the spreadsheet portion


Using Solverread the Answer Report





Documents

Copyright © 2000, Denise Sakai Troxell. All rights reserved. (updated from © Joe Aieta and Denise Sakai Troxell – Summer 1999.) Solving Linear Optimization