Linear Programming

Question

A manufacturer produces a product which is either bottled or canned. Daily Machine capacity is at 320 man-hours. The bottled product need 2hours per unit, while each unit of canned product requires 1 and 1/2 hours. The manufacturer is committed to supply a customer at least 60 canst everyday. If the objective is to maximize profit, how many units of bottled and canned products should be produced?

Answer 1

Firstly, you are missing a vital piece of information; we cannot calculate profit if we don't know the sales prices. We'll proceed anyway.

We need to build a system of linear inequalities here, graph them, test the corner points of the enclosed region, and see which one maximizes the profit. (This is the outline to all linear programming problems.)

We must first define our variables. Let x be the number of bottles produced and y be the number of cans produced.

Let us now convert between English and math:

"The bottled product need 2hours per unit" means the man-hours for the bottles is 2x.

"each unit of canned product requires 1 and 1/2 hours" means the man-hours for cans is 1.5y.

"Daily Machine capacity is at 320 man-hours" means, adding the man-hours above, 2x + 1.5y <= 320.

"The manufacturer is committed to supply a customer at least 60 canst [sic] everyday" is pretty clear: y>=60.

So our system of inequalities is: (considering you can't make a negative number of products)

x >= 0

y >= 0

2x + 1.5y <= 320

y >= 60

Graph these and test the profit equation with those values of x and y. The largest one is the maximum profit.