Solving linear optimization problems using the solver add in
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Solving Linear Optimization Problems Using the Solver Add-in. The Diet Problem

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Solving Linear Optimization Problems Using the Solver Add-in

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Solving linear optimization problems using the solver add in

Solving Linear Optimization Problems Using the Solver Add-in


Faculty babson

The 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 (nutritionists would disapprove, but this example ought to be simple). She chooses her favorite fast foods: hamburger and French fries. The nutritive values per serving are given below.

HamburgerFries

Fat (g) 1018

Protein (g) 15 3

Carbohydrate (g) 32 32

Calories (kcal) 220 396

Start with a problem to be formulated

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


Faculty babson

objective function

constraints

Formulate the Model

as a Linear Programming Problem

Julia is looking for

HB= the number of servings of Hamburger

decision variables

FF = the number of servings of French Fries

that minimizes the total amount of fat

10 HB +18 FF

subject to the following minimum diet requirements

15 HB+ 3 FF>= 45 Protein constraint

32 HB+ 32 FF>= 256 Carbohydrate constraint

220 HB + 396 FF>= 1980 Calories constraint

nonnegativity constraints

Of course HB>= 0 and FF>= 0


Preparing the worksheet for solver

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants

Enter labels in cells A2:A6


Preparing the worksheet for solver1

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants

NOTE: The labels in A4:A6 name the 3 constraints in this problem

Enter labels in cells B1:E1


Preparing the worksheet for solver2

Enter coefficients of objective function in cells B3:C3

10 HB +18 FF

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants

NOTE: The labels in cells B1:C1 name the 2 variables in this problem


Preparing the worksheet for solver3

Enter coefficients of left side of Protein constraint in cells B4:C4

15 HB + 3 FF

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants


Preparing the worksheet for solver4

Enter coefficients of left side of Carbohydrate constraint in cells B5:C5

32 HB + 32 FF

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants


Preparing the worksheet for solver5

Enter coefficients of left side of Calories constraint in cells B6:C6

220 HB + 396 FF

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants


Preparing the worksheet for solver6

15 HB+ 3 FF>= 45

32 HB+ 32 FF>= 256

Enter right sides of constraint in cells E4:E6

220 HB + 396 FF>= 1980

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants


Preparing the worksheet for solver7

Preparing the Worksheet for Solver

start with blank sheet & enter labels and constants


Name the cells for the two decision variables

name the cells for the two decision variables

Preparing the Worksheet for Solver

Highlight the four cells B1:C2


Name the cells for the two decision variables1

name the cells for the two decision variables

Preparing the Worksheet for Solver

Starting from the main menu bar click on Insert


Name the cells for the two decision variables2

name the cells for the two decision variables

Preparing the Worksheet for Solver

Click on Name


Name the cells for the two decision variables3

name the cells for the two decision variables

Preparing the Worksheet for Solver

Click on Create


Name the cells for the two decision variables4

name the cells for the two decision variables

Preparing the Worksheet for Solver

Make sure the Top row box is checked


Name the cells for the two decision variables5

name the cells for the two decision variables

Preparing the Worksheet for Solver

Click OK


Name the cells for the two decision variables6

name the cells for the two decision variables

Preparing the Worksheet for Solver

Cell B2 has the name HB and is currently blank Cell C2 has the nameFF and is currently blank


Enter the formula for the objective function

enter the formula for the objective function

Preparing the Worksheet for Solver

Click on cell D3


Enter the formula for the objective function1

enter the formula for the objective function

Preparing the Worksheet for Solver

Click on cell B3

Type =


Enter the formula for the objective function2

enter the formula for the objective function

Preparing the Worksheet for Solver

Click on cell B2

Type *


Enter the formula for the objective function3

enter the formula for the objective function

Preparing the Worksheet for Solver

Click on cell C3

Type +


Enter the formula for the objective function4

enter the formula for the objective function

Preparing the Worksheet for Solver

Click on cell C2

Type *


Enter the formula for the objective function5

enter the formula for the objective function

Preparing the Worksheet for Solver

formula entered in D3 objective function (Fat content) 10 HB + 18 FF

current value for the formula entered in D3 NOTE: Blanks in HB and FF (cells B2 and C2) are considered zeros

NOTE: The formula B3*HB+C3*FF could have been typed in cell D3 directly

Enter


Copying down the formulas for totals

copying down the formulas for totals

Preparing the Worksheet for Solver

Click on cell D3 and grab the fill handle + in the lower right corner


Copying down the formulas for totals1

copying down the formulas for totals

Preparing the Worksheet for Solver

Copy the formula in D3 down to cell D6


Copying down the formulas for totals2

copying down the formulas for totals

Preparing the Worksheet for Solver

formula copied in D4 Protein constraint LHS 15 HB + 3 FF

current value for the formula entered in D4

Click on cell D4 to check your formula


Copying down the formulas for totals3

copying down the formulas for totals

Preparing the Worksheet for Solver

formula copied in D5 Carbohydrate constraint LHS 32 HB +32 FF

current value for the formula entered in D5

Click on cell D5 to check your formula


Copying down the formulas for totals4

copying down the formulas for totals

Preparing the Worksheet for Solver

formula copied in D6 Calories constraint LHS 220 HB +396 FF

current value for the formula entered in D6

Click on cell D6 to check your formula


Faculty babson

Preparing the Worksheet for Solver

copying down the formulas for totals

NOTE: In the formula, the rows for column B and C have been copied as relative references and the references to the decision variables HB (cell B2) and FF (cell C2), are fixed


Faculty babson

Using Solver

invoking Solver Add-In from the Tools menu

Starting from the main menu bar click on Tools


Faculty babson

Using Solver

invoking Solver Add-In from the Tools menu

Click on Solver


Faculty babson

Using Solver

invoking Solver Add-In from the Tools menu

NOTE: From this point on, only the spreadsheet portion of the Excel window will be displayed


Faculty babson

Using Solver

select the value of the objective function as the Target Cell

Click on the Set Target Cell: box


Faculty babson

Using Solver

select the value of the objective function as the Target Cell

Click on cell D3 which contains the function we want to optimize


Faculty babson

Using Solver

indicate if the Target Cell is to be minimized/maximized

Check the Min radio button to indicate that we want to minimize the value in the Target Cell


Faculty babson

Using Solver

the Changing Cells are the decision variables

Click on the By Changing Cells: box


Faculty babson

Using Solver

the Changing Cells are the decision variables

Highlight cells B2:C2 containing the decision variables


Faculty babson

Using Solver

add Constraints

Click on the Subject to the Constraints: box


Faculty babson

Using Solver

add Constraints

Click on Add


Faculty babson

Using Solver

select the constraints’ left-hand-sides

Click on the Cell Reference: box


Faculty babson

Using Solver

select the constraints’ left-hand-sides

Highlight cells D4:D6


Faculty babson

Using Solver

select the constraints’ left-hand-sides


Faculty babson

Using Solver

select the constraints’ types

Click on 


Faculty babson

Using Solver

select the constraints’ types

Click on >=


Faculty babson

Using Solver

select the constraints’ right-hand-sides

Click on the Constraint: box


Faculty babson

Using Solver

select the constraints’ right-hand-sides

Highlight cells E4:E6


Faculty babson

Using Solver

select the constraints’ right-hand-sides

NOTE: Since the three constraints in the Diet Problem are of type >= they can be added all at once.

Click on OK


Faculty babson

Using Solver

setting the Options

Click on Options


Faculty babson

Using Solver

setting the Options

Check Assume Linear Model and Assume Non-Negative boxes (do not modify the other Options for this problem)


Faculty babson

Using Solver

setting the Options

Click OK


Faculty babson

Using Solver

executing Solver

Click on Solve


Faculty babson

Using Solver

obtaining solutions from the worksheet

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

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

Solver uses a method known as SIMPLEX


Faculty babson

Using Solver

obtaining an Answer Report

Click on Answer in the Reports box


Faculty babson

Using Solver

obtaining an Answer Report

Click on OK


Faculty babson

Using Solver

viewing the Answer Report

Click on Answer Report 1 tab


Faculty babson

Using Solver

viewing the Answer Report

NOTE: From this point on, only the spreadsheet portion of the Excel window will be displayed


Faculty babson

Using Solver

viewing the Answer Report

with minimum fat content of 90 g

Solution HB = 6.75 servings of HamburgerFF = 1.25 servings of French fries

and with Protein: 105 g Carbohydrate: 256 g Calories: 1980 kcal satisfying the constraints

Final steps in any solution involve an interpretation of these computer-generated results


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