Pasquale's Problem

Pasquale is a self-employed machinist who sells two products: P and Q. The market will accept up to 100 Ps and 100 Qs per week. P sells for $200 and Q sells for $195. The raw materials for each part cost $50. It takes him 18.5 minutes to build a P and 16.5 minutes to build a Q. Pasquale's supplier also sells partially-assembled raw materials for P and Q. The partially-assembled raw materials save him 3 minutes per P and 3.5 minutes per Q, but they also cost more: an additional $25 per P and $29 per Q. Given that Pasquale works at most 40 hours per week, what is Pasquale's maximum weekly profit? For simplicity, assume that the quantity of Ps and Qs he produces each week can be specified by a rational number.

Calculators are allowed, but solving this without a computer is quite a challenge.

Here's a hint (rot13): Ubj zhpu jbhyq Cnfdhnyr or noyr gb znxr vs ur qvqa'g unir gur bcgvba bs hfvat cnegvnyyl-nffrzoyrq enj zngrevnyf?

This problem can be solved quite easily with a linear programming solver, such as the one at http://www.zweigmedia.com/RealWorld/simplex.html. Let p and q be the number of Ps and Qs produced each week using only raw materials, and let p' and q' be the number of Ps and Qs produced each week using partially-assembled raw materials.

Maximize z = 150p + 125p'+ 145q + 116q' subject to 
18.5p + 16.5q + 15.5p' + 13q' <= 2400
p + p' <= 100
q + q' <= 100

Optimal Solution: z = 20581.1; p = 40.5405, p' = 0, q = 100, q' = 0

So the most Pasquale can make is $20581.10/week. He can do this by avoiding the partially-assembled raw materials entirely, making as many Qs as the market will accept (100), and spending the remainder of his time on Ps (40.5405).

Click here to view the solution

This puzzle was inspired by a paper by Balakrishnan [1].

[1] Balakrishnan, Jaydeep. The theory of constraints and the make-or-buy decision. Journal of Supply Chain Management, January 1, 2005 .