Answer to Question #205755 in Operations Research for KANE WINSTONE SENO

Question

1.) Rina needs at least 48 units of protein, 60 units of carbohydrates, and 50 units of fat each month. From each kilogram of food A, she receives 2 units of protein, 4 units of carbohydrates, and 5 units of fats. Food b contains 3 units of protein, 3 units of carbohydrates, and 2 units of fats. If food A costs Php110 per kilogram and food B costs Php 90 per kilogram. How many kilograms of each food should Rina buy each month to keep costs at a minimum?

Answer 1

minimize costs:

$z=110x_1+90x_2$

subject to:

$2x_1+3x_2\ge 48$

$4x_1+3x_2\ge 60$

$5x_1+2x_2\ge 50$

x₁, x₂ are kg of foods A and B

for Extreme Points:

for $5x_1+2x_2= 50$ :

$x_1=0\implies x_2=25$

for $2x_1+3x_2= 48$ :

$x_2=0\implies x_1=24$

intersection $5x_1+2x_2= 50$ and $4x_1+3x_2= 60$ :

$x_1=4.29,x_2=14.29$

intersection $2x_1+3x_2= 48$ and $4x_1+3x_2= 60$ :

$x_1=6,x_2=12$

Objective function values at Extreme Points:

$z(0,25)=2250$

$z(4.29,14.29)=1757.14$

$z(6,12)=1740$

$z(24,0)=2640$

The miniimum value of the objective function z=1740 occurs at the extreme point (6,12).

Hence, the optimal solution to the given LP problem is : x₁=6, x₂=12 and min z=1740.