The P&Q Problem

The P&Q problem was presented by Eliyahu M. Goldratt in [1]. The idea is to find the best product mix with a maximum net profit by varying the numbers of products and under given circumstances. A solution is feasible as long as one resource is not overloaded. More details are described in the Details section.


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The P&Q problem was presented by Goldratt [1]. It aims to permit discussion—by means of a simple and easy example—of the differences between the assumptions made by the theory of constraints (TOC) and other management approaches, especially those that use some form of cost allocation. In this Demonstration, the goal is to make use of this example and of scenarios derived from it to discuss the feasibility of optimal solutions in light of the TOC.
In this problem, a specific enterprise manufactures and sells two types of products, and . For that, four resources are required: , , and . Sales prices, costs of raw materials—assumed here as the only total variable cost (TVC)—and the demand for each product, as well as the unit processing time (in minutes) required for each resource, are shown in the image. Throughput (T) is calculated by Sales minus TVC.
These resources are paid to operate for 2400 minutes/week, and the enterprise's total costs and expenses (operational expenses, OE), except for those associated with raw materials, correspond to $6000/week. The question is how many and must be produced and sold so that the enterprise's profit is maximized. One can readily see that, in contrast to the other resources, resource does not present enough capacity to meet the weekly and demand. For this, 3000 minutes would be necessary, or 125% of available capacity. It is necessary to decide on a production mix that would maximize the profit of this enterprise.
Of course, the problem can be solved mathematically in an easy way. In [2] is a link to a Wolfram Mathematica notebook with the solution, but first, you can check your intuition by playing with the simulation or by making your own calculation.
The optimal solution for checking your own results: net profit = $300, , .

[1] E. M. Goldratt, The Haystack Syndrome: Sifting Information Out of the Data Ocean, Croton-on-Hudson, NY: North River Press Inc., 1990.
[2] J. Kanz. "P&Q Solution for Mathematica."
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