Answer to Question #248702 in Operations Research for Unique

Question

A pharmacy has determined that a healthy person should receive 70 units of proteins, 100 units of carbohydrates and 20 units of fat daily. If the store carries the six types of health food with their ingredients as shown in the table below, what blend of foods satisfies the requirements at minimum cost to the pharmacy? Make a mathematical model for the given problem

Foods Protein units Carbohydrates units Fat units Cost per unit

A 20 50 4 2

B 30 30 9 3

C 40 20 11 5

D 40 25 10 6

E 45 50 9 8

F 30 20 10 8

Answer 1

minimize cost:

$z=2x_1+3x_2+5x_3+6x_4+8x_5+8x_6$

subject to:

$20x_1+30x_2+40x_3+40x_4+45x_5+30x_6\ge70$ : amount of protein

$50x_1+30x_2+20x_3+25x_4+50x_5+20x_6\ge100$ : amount of carbohydrate

$4x_1+9x_2+11x_3+10x_4+9x_5+10x_6\ge20$ : amount of fat

where x₁, x₂, x₃, x₄, x₅, x₆ are units of 6 foods

solution using Simplex method:

After introducing artificial variables:

Min Z=2x₁+3x₂+5x₃+6x₄+8x₅+8x₆+0S₁+0S₂+0S₃+MA₁+MA₂+MA₃

subject to

20x₁+30x₂+40x₃+40x₄+45x₅+30x₆-S₁+A₁$\ge$ 70

50x₁+30x₂+20x₃+25x₄+50x₅+20x₆-S₂+A₂$\ge$ 100

4x₁+9x₂+11x₃+10x₄+9x₅+10x₆-S₃+A₃=20

and 

x₁,x₂,x₃,x₄,x₅,x₆,A₁,A₂,A₃≥0

Positive maximum Z_j-C_j is 104M-8 and its column index is 5. So, the entering variable is x₅.

Minimum ratio is 1.5556 and its row index is 1. So, the leaving basis variable is A₁.

The pivot element is 45.

Entering =x₅, Departing =A₁, Key Element =45

Positive maximum Z_j-C_j is 27.7778M+1.5556 and its column index is 1. So, the entering variable is x₁.

Minimum ratio is 0.8 and its row index is 2. So, the leaving basis variable is A₂.

The pivot element is 27.7778.

Entering =x₁, Departing =A₂, Key Element =27.7778

Positive maximum Z_j-C_j is 4M-1.92 and its column index is 6. So, the entering variable is x₆.

Minimum ratio is 1.3636 and its row index is 1. So, the leaving basis variable is x₅.

 The pivot element is 0.88.

Entering =x₆, Departing =x₅, Key Element =0.88

Positive maximum Z_j-C_j is 0.3818M-0.3273 and its column index is 7. So, the entering variable is S₁.

Minimum ratio is 1.4286 and its row index is 3. So, the leaving basis variable is A₃.

The pivot element is 0.3818.

Entering =S₁, Departing =A₃, Key Element =0.3818

Positive maximum Z_j-C_j is 3.8571 and its column index is 3. So, the entering variable is x₃.

Minimum ratio is 1.2766 and its row index is 1. So, the leaving basis variable is x₆.

The pivot element is 1.119.

Entering =x₃, Departing =x₆, Key Element =1.119

Positive maximum Z_j-C_j is 1.1489 and its column index is 2. So, the entering variable is x₂.

Minimum ratio is 1.8182 and its row index is 1. So, the leaving basis variable is x₃.

The pivot element is 0.7021.

Entering =x₂, Departing =x₃, Key Element =0.7021

Since all Z_j-C_j≤0

Hence, optimal solution is arrived with value of variables as :

x₁=0.9091,x₂=1.8182,x₃=0,x₄=0,x₅=0,x₆=0

Min Z=7.2727