Answer to Question #283179 in Operations Research for Abateneh

Question

A Kelifa Business Centre plans to sell two models of an item at costs of $350 and $400. The $350 model yields a profit of $85 and the $400 model yields a profit of $90. The total demand per month for the two models will not exceed 150. Find the number of units of each model that should be stocked each month in order to maximize the profit. Assume the merchant can invest no more than $56,000 for inventory of these items.

Formulate Linear Programming problems (LPP)

Find the optimal solution by Graphic method

Answer 1

A.

maximize the profit:

$z=85x+90y$

subject to:

$x+y\le 150$

$350x+400y\le 56000$

where x, y are numbers of units of models

B.

corner points (Extreme Points):

for $x+y= 150$ :

$y=0\implies x=150$

for $350x+400y= 56000$ :

$x=0\implies y=140$

for intersection of $x+y= 150$ and $350x+400y= 56000$ :

$x=150-y$

$7(150-y)+8y=1120$

$y=1120-7\cdot150=70$

$x=150-7=80$

Objective function value at Extreme Points:

$z(150,0)=12750$

$z(80,70)=13100$

$z(0,140)=12600$

The maximum value of the objective function Z=13100 occurs at the extreme point (80,70).

Hence, the optimal solution to the given LP problem is : x=80, y=70 and max Z=13100.