Solving Linear Optimization Problems Using the Solver Add-in

HamburgerFries Fat (g) Protein (g) 15 3 Carbohydrate (g) Calories (kcal) 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. How many servings of hamburger and fries would Julia need to eat to satisfy her daily diet requirements? Start with a problem to be formulated

constraints 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 function 10 HB + 18 FF decision variables nonnegativity constraints 32 HB + 32 FF >= 256 Carbohydrate constraint 220 HB FF >= 1980 Calories constraint Formulate the Model as a Linear Programming Problem Of course HB >= 0 and FF >= 0

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

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 objective function in cells B3:C3 10 HB + 18 FF

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 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 >= HB + 32 FF >= HB FF >= 1980

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

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

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 Preparing the Worksheet for Solver

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

Click OK 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 name FF and is currently blank name the cells for the two decision variables Preparing the Worksheet for Solver

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

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

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

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

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

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

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

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 Preparing the Worksheet for Solver Click on cell D4 to check your formula formula copied in D4 Protein constraint LHS 15 HB + 3 FF current value for the formula entered in D4

copying down the formulas for totals Preparing the Worksheet for Solver Click on cell D5 to check your formula formula copied in D5 Carbohydrate constraint LHS 32 HB + 32 FF current value for the formula entered in D5

copying down the formulas for totals Preparing the Worksheet for Solver Click on cell D6 to check your formula formula copied in D6 Calories constraint LHS 220 HB FF current value for the formula entered in D6

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 Starting from the main menu bar click on Tools

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

invoking Solver Add-In from the Tools menu Using Solver 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 Click on the Set Target Cell: box

select the value of the objective function as the Target Cell Using Solver Click on cell D3 which contains the function we want to optimize

indicate if the Target Cell is to be minimized/maximized Using Solver 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 Click on the By Changing Cells: box

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

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

add Constraints Using Solver Click on Add

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

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

select the constraints’ left-hand-sides Using Solver

select the constraints’ types Using Solver Click on 

select the constraints’ types Using Solver Click on >=

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

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

NOTE: Since the three constraints in the Diet Problem are of type >= they can be added all at once. Click on OK select the constraints’ right-hand-sides Using Solver

setting the Options Using Solver Click on Options

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

setting the Options Using Solver Click OK

executing Solver Using Solver Click on Solve

obtaining solutions from the worksheet Using Solver 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

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

obtaining an Answer Report Using Solver Click on OK

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

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