Question

Andrew, who is ill, takes vitamin pills. Each day, he must take at least 16 units of myra X, at least 5 units of myra Y, and at least 20 units of myra Z. He can choose between tablets and capsules. Each tablet contains 8 units of myra X, 1 unit of myra Y, and 2 units of myra Z, while each capsule contains 2 units of myra X, 1 unit of myra Y and 7 units of myra Z. The cost per tablet is Php15, and Php30 per capsule. How many tablets and capsules should he buy in order to minimize his cost?​

Answer 1

z (min) = 105

x₁ = 3

x₂ = 2

Explanation:

Let´s call x₁ (number of tables) and x₂ ( number of capsules)

assuming it is only possible to get a whole table or capsule we have:

Table x₁ 8 u myra X 1 u of myra Y and 2 u of myra Z

Capsule x₂ 2 " 1 " and 7 u "

Objective function

z = 15* x₁ + 30* x₂ to minimize

First constraint ( Myra X )

8*x₁ + 2*x₂ ≥ 16

Second constraint ( Myra Y )

x₁ + x₂ ≥ 5

Third constraint ( Myra Z )

2*x₁ + 7*x₂ ≥ 20

General constraints x₁ ≥ 0 x₂ ≥ 0 both integers

Then the model is:

z = 15 * x₁ + 30 * x₂ to minimize

Subject to

8*x₁ + 2*x₂ ≥ 16

x₁ + x₂ ≥ 5

2*x₁ + 7*x₂ ≥ 20

x₁ ≥ 0 x₂ ≥ 0 both integers

As the constraints, all are of the form ≥ we have to subtract surplus variables sₐ and add artificial variables Aₐ

sₐ ≥ 0 Aₐ ≥ 0

The first table is

z x₁ x₂ s₁ s₂ s₃ A₁ A₂ A₃ = Cte

1 - 15 - 30 0 0 0 M M M = 16

0 8 2 -1 0 0 1 0 0 = 5

0 1 1 0 -1 0 0 1 0 = 20

0 2 7 0 0 -1 0 0 1 = 20

Using AtoZmax and after 6 iterations we find

z (min) = 105

x₁ = 3

x₂ = 2