Question

The boy scouts are preparing trail mix for an upcoming backpacking trip. They are planning to use almonds, raisins, and M&Ms for the trail mix. Each 16oz bag should have a total ingredient cost $10.00 or less. Due to their level of activity on the trail, at least 50% of total calories should come from carbs. And one of the moms insists that at most 50% of total calories can come from M&Ms. The boys have collected some nutrionial and cost information per ounce of each ingredient. They would like to create a mix which maximizes the total calories provided by each bag of trail mix within the above restrictons. How should they create their recipe? What will be the calorie content of the bags of trail mix? Will there be anything leftover?

almonds|raisinsM&Ms carb calories per ounce 20 80 80 total calories 160 80 140 per ounce cost per ounce (S) 0.250.80 0.50

Answer 1

Let’s first formulate the problem

Maximize 160X1 +80X2 +140X3

Subject to

20X1 + 80X2 + 80X3 >= 0.5*(160X1 +80X2 +140X3)

140X3 <= 0.5*(160X1 +80X2 +140X3)

0.5X1 + 0.25X2 + 0.8X3 <= 10

X1, X2, X3 >= 0

Next, let’s solve the problem on excel solver. The output and the result is shown below.

| Х ﹀ .fi SUMPRODUCT(ВЗ:D3,SBS6:SDS6) 2 Carb calories 3 Total calories 160 4 Cost Almonds Raisins M&M 80 80 0.25 80 140 0.8 Max calories 3200 20 Solver Results 0.5 Solver found a solution. All Constraints and optimality conditions are satisfied 6 Dec var 11.4286 17.1429 Sensitivity Limits Keep Solver Solution O Restore Original Values Return to Solver Parameters Dialog 8 Constraint 1 9 Constraint 2 10 Constraint 3 1600 1600 10 20 80 0 0.25 80 1600 0 10 Outline Reports 0.5 0.8 Save Scenario Solver found a solution. All Constraints and optimality conditions are 12 13 14 15 16 17 18 19 When the GRG engine is used, Solver has found at least a local optimal solution. When Simplex LP is used, this means Solver has found a global optimal solution Sheet1049 Sheet1050 Sheet1051 Sheet1052 Sheet1053 Sheet1054 + 145%

They should create their recipe with 11.42 ounces of Almonds and 17.14 ounces of raisins. The calorie content will be 3200. M&Ms will be completely unutilized and will be left over completely.