Question

3. Tigist wishes to mix two types of food C and D in such a way that the vitamin contents of the mixture contain at least 8 units of vitamin C and 11 units of vitamin D. Food C costs $ 60/kg and Food D costs $80/kg. Food C contains 3 units/kg of Vitamin A and 5 units / kg of Vitamin B while food D contains 4 units/kg of Vitamin A and 2 units/kg of vitamin B. Determine the minimum cost of the mixture by graphic model.

Answer 1

minimize cost:

$Z=60x_1+80x_2$

subject to:

$3x_1+4x_2\ge 8$ - for Vitamin A

$5x_1+2x_2\ge 11$ - for Vitamin B

where

x₁ is number of kg of food C,

x₂ is number of kg of food D

for extreme points:

$5x_1+2x_2= 11$

$x_1=0\implies x_2=5.5$

$3x_1+4x_2= 8$

$x_2=0\implies x_1=2.67$

intersection:

$x_1=(11-2x_2)/5$

$3(11-2x_2)/5+4x_2= 8$

$x_2=7/14=0.5,x_1=2$

Objective function values at extreme points:

$Z(0,5.5)=440$

$Z(2,0.5)=160$

$Z(2.67,0)=160$

The miniimum value of the objective function Z=160 occurs at 2 extreme points.

Hence, problem has multiple optimal solutions and min Z=160.