What is the maximum profit in this word problem?

Question

A company makes color televisions. It produces a bargin set that sells for $100 profit and a deluxe set that sells for $150 profit. On the assembly line the bargain set requires 3 hours, the deluxe set takes 5 hours. The Cabinet shop spends 1 hour on the cabinet for the bargain set and 3 hours for the deluxe set. Both sets require 2 hours of time for testing and packing. The company has available 3900 hours for assembly line, 2100 hours in the cabinet shop and 2200 hours in testing and packaging. How many sets of each type should the company produce to make a maximum profit? what is the maximum profit?

Answer 1

The explanation depends on what class this is for. If you're studying Linear Optimization, I would say to use the Simplex Method. But this problem could also appear in an algebra class, so I'll try to answer it at that level. Both methods produce the same answer in the end.

Let b stand for the number of bargain sets to be made; let d stand for the number of deluxe sets.

Since the bargain set takes 3 hours and the deluxe set 5, and since there are 3900 assembly line hours available, we can write the inequality 3b+5d<=3900. (If one bargain set "costs" 3 hours of assembly line work, then b of them cost 3b hours.)

Since the bargain set takes 1 hour in the cabinet shop and the deluxe set takes 3, and since there are 2100 cabinet hours available, we can write the inequality 1b+3d<=2100.

Since both sets require 2 hours for testing and we have 2200 testing hours available, we can write the inequality 2b+2d<=2200.

Also, common sense tells us that b>=0 and d>=0 (you can't make a negative number of TV's).

Graph all five of these inequalities in order to find the "feasible region", which tells us all of the production options available to the company. I can't draw it for you, but it's a pentagon with vertices (0,0), (0, 700), (300,600), (800, 300), and (1100,0). My points are in the form (b,d).

For reasons I won't get into, maximum profit happens at one of these vertices. Find profit at each vertex and pick the best one.

(0,0): profit 0.
(0, 700): profit 100(0) + 150(700) = 105000.
(300,600): profit 100(300)+150(600) = 120000.
(800, 300): profit 100(800)+150(300) = 125000.
(1100,0): profit 100(1100)+150(0) = 110000.

Looks like 800 bargain sets and 300 deluxe sets get the maximum profit of $125,000.

Answer 2

a graphical solution is available at
http://www.geocities.com/regform20052007/mathpro14/1.html