Question

A small manufacturer produces two kinds of good, A and B, for which demand exceeds capacity. The production costs for A and B are 6$ and 3$, respectively, each, and the corresponding selling prices are $7 and $4. In addition, the transport costs are 20 cents and 30 cents for each good of type A and B, respectively. The conditions of a bank loan limit the manufacturer to maximum weekly production costs of $2700 and maximum weekly transport costs of $210. How should the manufacturer arrange production to maximize profit?

Answer 1

Let 'a' be the no.of units of good A &
'b' be the no. of units of good B

Profit = 7a - (6a + .2a) + 4b - (3b + .3b)
= .8a + .7b

Since, we are trying to maximize our profit, our constraints would be as follows :

(1) a<=0, b<=0,
(2) 6a+3b >= 2700
(3) .2a + .3b >= 210

Both our corner points on solving equations 1,2 & 3 are (0,700) & (450,0)
But, the profit is maximized at (150,600) [Satisfies 1,2 & 3]

[Substtuting the points in the profit function]
(.8*0 + .7*700) = 490
(.8*450 + .7*0) = 360
(.8*150 + .7*600) = 540

Hence, by producing 150 units of A and 600 units of B, max weekly production costs are equal to $2700 and max weekly transport costs are equal to $210 and profit is also maximum at this point.