Solving Linear Optimization Problems Using the Solver Add-in

Start with a problem to be formulated 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. Hamburger Fries Fat (g) 10 18 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?

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 objective function 10 HB + 18 FF subject to the following minimum diet requirements constraints 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 start with blank sheet & enter labels and constants Enter labels in cells A2:A6

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 Solver start with blank sheet & enter labels and constants Enter coefficients of objective function in cells B3:C3 10 HB + 18 FF NOTE: The labels in cells B1:C1 name the 2 variables in this problem

Preparing the Worksheet for Solver start with blank sheet & enter labels and constants 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 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 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 Enter right sides of constraint in cells E4:E6 15 HB + 3 FF >= 45 32 HB + 32 FF >= 256 220 HB + 396 FF >= 1980

Preparing the Worksheet for Solver start with blank sheet & enter labels and constants

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Highlight the four cells B1:C2

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Starting from the main menu bar click on Insert

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Click on Name

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Click on Create

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Make sure the Top row box is checked

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Click OK

name the cells for the two decision variables Preparing the Worksheet for Solver name the cells for the two decision variables Cell B2 has the name HB and is currently blank Cell C2 has the name FF and is currently blank

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function Click on cell D3

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function Click on cell B3 Type =

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function Click on cell B2 Type *

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function Click on cell C3 Type +

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function Click on cell C2 Type *

enter the formula for the objective function Preparing the Worksheet for Solver enter the formula for the objective function 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 Preparing the Worksheet for Solver copying down the formulas for totals Click on cell D3 and grab the fill handle + in the lower right corner

copying down the formulas for totals Preparing the Worksheet for Solver copying down the formulas for totals Copy the formula in D3 down to cell D6

copying down the formulas for totals Preparing the Worksheet for Solver copying down the formulas for totals 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 Preparing the Worksheet for Solver copying down the formulas for totals 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 Preparing the Worksheet for Solver copying down the formulas for totals 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 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

invoking Solver Add-In from the Tools menu Using Solver invoking Solver Add-In from the Tools menu Starting from the main menu bar click on Tools

invoking Solver Add-In from the Tools menu Using Solver invoking Solver Add-In from the Tools menu Click on Solver

invoking Solver Add-In from the Tools menu 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

select the value of the objective function as the Target Cell Using Solver select the value of the objective function as the Target Cell Click on the Set Target Cell: box

select the value of the objective function as the Target Cell 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

indicate if the Target Cell is to be minimized/maximized 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

the Changing Cells are the decision variables Using Solver the Changing Cells are the decision variables Click on the By Changing Cells: box

the Changing Cells are the decision variables Using Solver the Changing Cells are the decision variables Highlight cells B2:C2 containing the decision variables

Using Solver add Constraints Click on the Subject to the Constraints: box

Using Solver add Constraints Click on Add

select the constraints’ left-hand-sides Using Solver select the constraints’ left-hand-sides Click on the Cell Reference: box

select the constraints’ left-hand-sides Using Solver select the constraints’ left-hand-sides Highlight cells D4:D6

select the constraints’ left-hand-sides Using Solver select the constraints’ left-hand-sides

select the constraints’ types Using Solver select the constraints’ types Click on 

select the constraints’ types Using Solver select the constraints’ types Click on >=

select the constraints’ right-hand-sides Using Solver select the constraints’ right-hand-sides Click on the Constraint: box

select the constraints’ right-hand-sides Using Solver select the constraints’ right-hand-sides Highlight cells E4:E6

select the constraints’ right-hand-sides 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

Using Solver setting the Options Click on Options

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

Using Solver setting the Options Click OK

Using Solver executing Solver Click on Solve

obtaining solutions from the worksheet Using Solver obtaining solutions from the worksheet Solver finds a solution HB = 6.75 servings of Hamburger FF = 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

obtaining an Answer Report Using Solver obtaining an Answer Report Click on Answer in the Reports box

obtaining an Answer Report Using Solver obtaining an Answer Report Click on OK

viewing the Answer Report Using Solver viewing the Answer Report Click on Answer Report 1 tab

viewing the Answer Report Using Solver viewing the Answer Report NOTE: From this point on, only the spreadsheet portion of the Excel window will be displayed

viewing the Answer Report Using Solver viewing the Answer Report with minimum fat content of 90 g Solution HB = 6.75 servings of Hamburger FF = 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