Question

Consider the following problem faced by a Dietician: A diet needs to be created that contains not less than 2213 calories, not more than 75 grams of protein, not less than 16 grams of carbohydrates and not less than 13 grams of fat. Also, the diet should have minimal cost. In addition the diet should include at least 1.1 Units of fish and at least 0.7 cup of milk. The diet will consist of the six different foods: Bread, Milk, Cheese, Fish, Potato and Yogurt. The following table lists the cost per unit of each item and the number of grams of protein, fat, and carbohydrates per unit as well as the calories that will be supplied by each food unit. Milk 23 9 7 16.7 178 Fish 81 8 8 0 205 Bread Cheese P Cost, (cents)8 6 3 15 25 7 15 0.4 182 Potato 15 1.3 0.1 30.6 114 Yogurt 29 11.2 3 25 342 Protein, g Fat, g Carbohydrates,g Calories, Ca176 Find a diet that meets the requirements above while keeping the cost at an absolute minimum. Make sure that your Food Units are all Integers

Answer 1

Let’s create a linear programming model and solve it on excel solver.

Here our decisions are the amount of bread, milk, cheese, potato, fish and yogurt to consume. We can denote them as X1, X2, X3, X4, X5, and X6.

The objective is to minimize the cost. Thus our objective function is

Minimize 8X1 + 23X2 + 25X3 + 15X4 + 81X5 + 29X6

The constraints are

176X1 + 178X2 + 182X3 + 114X4 + 205X5 + 342X6 >= 2213 (Calorie constraint)

6X1 + 9X2 + 7X3 + 1.3X4 + 8X5 + 11.2X6 <= 75 (protein constraint)

15X1 + 16.7X2 + 0.4X3 + 30.6X4 + 0X5 + 25X6 >= 16 (carbohydrate constraint)

3X1 + 7X2 + 15X3 + 0.1X4 + 8X5 + 3X6 >= 13 (fat constraint)

X5 >= 1.1 (fish constraint)

X2 >= 0.7 (milk constraint)

X1, X2, X3, X4, X5, X6 >= (non-negative constraint)

The solver model is shown below

The formulas are shown below

The solver parameter is shown below

SUM(B18:G18) Solver Parameters G HK Bread Milk Cheese Potato Fish Yogurt 2 Cost 3 Protein 4 Fat 5 Carb 6 Calories 23 25 15 1.3 0.1 81 Set Objective: To: OMax By Changing Variable Cells: sis 11.2 Mi Oyalue Or: 15 25 15 16.7 0.430.6 176 178 182 114205 342 BS9.SGS9 Subject to the Constraints: X1 X2 X3 X4 X5 X6 SBS9:5GS9-integer S11 SKS11 SIS12SKS12 9 Dec variables 10 11 Calorie const0 12 Protein const 0 13 Carb const 14 Fat const 15 Fish const 16 Milk const 17 18 Cost 19 20 21 0 Total Limit 0 2213 0 < 75 0 16 0 13 Reset All 0 0 0 0 Load/Save 0 0.7 Make Unconstrained Variables Non-Negative Select a Solving Method: Simplex LP Options 0 0 0 0 Solving Method Select the GRG Nonlinear engine for Solver Problems that are smooth nonlinear. Select the LP Simplex engine for linear Solver Problems, and select the Evolutionary engine for Solver problems that are non-smooth. 23 24 Sheet450Sheet451 Sheet452 Sheet453 Sheet454 囲ロ凹 115%

The result is shown below. The total cost is 292 cents

Each items are

Bread = 6

Milk = 1

Cheese = 0

Potato = 2

Fish = 2

Yogurt = 1