A firm has two factories.One factory has the cost function c1(y1)= 2y21 + 90 and the other has the cost function c2(y2)= 6y22 +40.If the firm wishes to produce a total of 32 units as cheaply as possible, how many units will be produced in the second factory?
A)7
B)2
C)8
D)14
E)None of the above.

Question

Quizplus · Accepted Answer

To minimize the total cost, we need to choose the factory with lower cost per unit. 
For factory 1: 
c1(y1) = F10 + 2y21F1S1 + 90F1 
= 90F1 + 2y21F1S1 + F10 
= 90F1 + 2(16-y2)F1S1 + F10 
= 90F1 + 32F1S1 - 2y2F1 + F10

For factory 2: 
c2(y2) = F20 + 6y22F2S2 + 40F2 
= 40F2 + 6y22F2S2 + F20 
= 40F2 + 6(16- y1)F2S2 + F20 
= 40F2 + 96F2S2 - 6y1F2 + F20

Total cost: 
C = c1(y1) + c2(y2) 
= (90F1 + 32F1S1 - 2y2F1 + F10) + (40F2 + 96F2S2 - 6y1F2 + F20) 
= (90F1 + F10) + (40F2 + F20) + 32F1S1 + 96F2S2 - 2y2F1 - 6y1F2

To produce 32 units in total, we need to have y1 + y2 = 32. 
Substituting y1 = 32 - y2 into the total cost equation:

C = (90F1 + F10) + (40F2 + F20) + 32F1S1 + 96F2S2 - 2y2F1 - 6(32-y2)F2 
= (90F1 + F10 + 1920F2 - 192F2 - 192y2) + 32F1S1 + 96F2S2 - 6y2F2

For the total cost to be minimized, we need to minimize the expression: 
90F1 + F10 + 1920F2 - 192F2 - 192y2 
= 90F1 + 1728F2 - 192y2 + F10

Since F1 = F2 = 1, we only need to minimize -192y2 + 90 + 1728 = -192(y2-9) + 1818. 
This is minimized when y2 = 8. 
Therefore, the best choice is to produce 8 units in factory 2.

A Firm Has Two Factories