Answer to Question #292730 in Operations Research for meskele

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 the mathematical mod

Answer 1

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

Solution:

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

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