Wednesday, April 21, 2010

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?

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 .

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