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Solving Linear Optimization Problems Using the Solver Add-in. The Diet Problem

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Presentation Transcript

### Solving Linear Optimization Problems Using the Solver Add-in

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.

Hamburger Fries

Fat (g) 1018

Protein (g) 15 3

Carbohydrate (g) 32 32

Calories (kcal) 220 396

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

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

Enter labels in cells A2:A6

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

Enter labels in cells B1:E1

10 HB +18 FF

Preparing the Worksheet for Solver

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

15 HB + 3 FF

Preparing the Worksheet for Solver

32 HB + 32 FF

Preparing the Worksheet for Solver

220 HB + 396 FF

Preparing the Worksheet for Solver

15 cells B6:C6HB+ 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

Preparing the Worksheet for Solver cells B6:C6

Preparing the Worksheet for Solver

Highlight the four cells B1:C2

Preparing the Worksheet for Solver

Starting from the main menu bar click on Insert

Preparing the Worksheet for Solver

Click on Name

Preparing the Worksheet for Solver

Click on Create

Preparing the Worksheet for Solver

Make sure the Top row box is checked

Preparing the Worksheet for Solver

Click OK

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 cells B6:C6

Preparing the Worksheet for Solver

Click on cell D3

enter the formula for the objective function cells B6:C6

Preparing the Worksheet for Solver

Click on cell B3

Type =

enter the formula for the objective function cells B6:C6

Preparing the Worksheet for Solver

Click on cell B2

Type *

enter the formula for the objective function cells B6:C6

Preparing the Worksheet for Solver

Click on cell C3

Type +

enter the formula for the objective function cells B6:C6

Preparing the Worksheet for Solver

Click on cell C2

Type *

enter the formula for the objective function cells B6:C6

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 cells B6:C6

Preparing the Worksheet for Solver

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

copying down the formulas for totals cells B6:C6

Preparing the Worksheet for Solver

Copy the formula in D3 down to cell D6

copying down the formulas for totals cells B6:C6

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 totals cells B6:C6

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 totals cells B6:C6

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

Preparing the Worksheet for Solver cells B6:C6

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

Using Solver cells B6:C6

Starting from the main menu bar click on Tools

Using Solver cells B6:C6

Click on Solver

Using Solver cells B6:C6

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

Using Solver cells B6:C6

select the value of the objective function as the Target Cell

Click on the Set Target Cell: box

Using Solver cells B6:C6

select the value of the objective function as the Target Cell

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

Using Solver cells B6:C6

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

Using Solver cells B6:C6

the Changing Cells are the decision variables

Click on the By Changing Cells: box

Using Solver cells B6:C6

the Changing Cells are the decision variables

Highlight cells B2:C2 containing the decision variables

Using Solver cells B6:C6

Click on the Subject to the Constraints: box

Using Solver cells B6:C6

Using Solver cells B6:C6

select the constraints’ left-hand-sides

Click on the Cell Reference: box

Using Solver cells B6:C6

select the constraints’ left-hand-sides

Highlight cells D4:D6

Using Solver cells B6:C6

select the constraints’ left-hand-sides

Using Solver cells B6:C6

select the constraints’ types

Click on 

Using Solver cells B6:C6

select the constraints’ types

Click on >=

Using Solver cells B6:C6

select the constraints’ right-hand-sides

Click on the Constraint: box

Using Solver cells B6:C6

select the constraints’ right-hand-sides

Highlight cells E4:E6

Using Solver cells B6:C6

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

Using Solver cells B6:C6

setting the Options

Click on Options

Using Solver cells B6:C6

setting the Options

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

Using Solver cells B6:C6

setting the Options

Click OK

Using Solver cells B6:C6

executing Solver

Click on Solve

Using Solver cells B6:C6

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

Using Solver cells B6:C6

Click on Answer in the Reports box

Using Solver cells B6:C6

Click on OK

Using Solver cells B6:C6

Click on Answer Report 1 tab

Using Solver cells B6:C6

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

Using Solver cells B6:C6