Answer to Question #150563 in Operations Research for kc jimeneZ

Question

A manufacturer of shoes makes two kinds of shoes: running shoes and basketball shoes. Cutting machines and sewing machines are used to produce the shoes. Each type of shoes requires 3 minutes per pair on the cutting machine. The running shoes require 2 minutes of sewing per pair, and the basketball shoes require 4 minutes of sewing per pair. Only one operator is assigned to each machine and each process is available for only 8 hours per day. If the profit is P1,700 for each pair of running shoes and P2,000 for each pair of basketball shoes, how many pairs of each type should be produced so that the profit will be maximized?

Answer 1

This is a problem of linear programming.

Let the manufacturer makes x pairs of running shoes and y pairs of basketball shoes per day.

So objective function is

Maximize Z = 1700x + 2000y

Subject to the conditions

3x + 3y ≤ 8*60 i.e x + y ≤ 160

2x + 4y ≤ 8*60 i.e x + 2y ≤ 240

x ≥ 0, y ≥ 0

Drawing the graph of constraints we get

Quadrilateral OABC is the feasible solution region of the problem.

Let us apply corner method to find optimal solution.

Corner points Value of Z

O ( 0, 0). 0

A( 160,0) 1700*160

B( 80,80) 1700*80+2000*80

C(0, 120) 2000*120

Rewriting the table we get

Corner points. Value of Z

O ( 0, 0). 0

A( 160,0) 272000

B( 80,80) 296000

C(0, 120) 240000

Maximum value of Z is P 296000 and it is attained when x = 80, y = 80

So the manufacturer will produce 80 pairs of running shoes and 80 pairs of basketball shoes to get a maximum profit of P 296000