Question

A health clinic dietician is planning a meal consisting of three foods whose ingredients are summarized as follows: One Unit ofFood I Food II Food IIIUnits of Protein 10 15 20Units of Carbohydrates 1 2 1Units of Iron 4 8 11Calories 80 120 100The dietician wishes to determine the number of units of each food to use to create a balanced meal containing at least 40 units of protein, 6 units of carbohydrates, and 12 units of iron, and with as few calories as possible. Use solver to find how many units of each food should be used in order to minimize calories.

Answer 1

z (min) = 360 x₁ = x₃ = 0 x₂ = 3

Explanation:

Protein Carbohydrates Iron calories

Food 1 (x₁) 10 1 4 80

Food 2 (x₂) 15 2 8 120

Food 3 (x₃) 20 1 11 100

Requirements 40 6 12

From the table we get

Objective Function z :

z = 80*x₁ + 120*x₂ + 100*x₃ to minimize

Subjet to:

Constraint 1. at least 40 U of protein

10*x₁ + 15*x₂ + 20*x₃ ≥ 40

Constraint 2. at least 6 U of carbohydrates

1*x₁ + 2*x₂ + 1*x₃ ≥ 6

Constraint 3. at least 12 U of Iron

4*x₁ + 8*x₂ + 11*x₃ ≥ 12

General constraints:

x₁ ≥ 0 x₂ ≥ 0 x₃ ≥ 0 all integers

With the help of an on-line solver after 6 iterations the optimal solution is:

z (min) = 360 x₁ = x₃ = 0 x₂ = 3