Theory of constraints and the combinatorial complexity of the product-mix decision

The theory of constraints (TOC) proposes that, when production is bounded by a single bottleneck, the best product mix heuristic is to select products based on their ratio of throughput per constraint use. This, however, is not true for cases when production is limited to integer quantities of final products. Four facts that go against current thought in the TOC literature are demonstrated in this paper. For example, there are cases in which the optimum product mix includes products with the lowest product margin and the lowest ratio of throughput per constraint time, simultaneously violating the margin heuristic and the TOC-derived heuristic. Such failures are due to the non-polynomial completeness (NP-completeness) of the product-mix decision problem, also demonstrated here.