Optimal Matching of Random Parts

This paper examines the minimization of the cost for an expected random production output, given an assembly of finished goods from two random inputs, matched in two categories. We describe the optimal input portfolio, first using the standard normal approximation of the binomial classification distributions, and second using a tight concave envelope instead of the exact output objective. The latter approach yields closed-form expressions for the factor demands and total costs which are linear in the expected output and which approximate the solution to the original minimum-cost matching problem for sufficiently large production batches. A key structural insight is that depending on the ratio of input prices, one of the inputs should be considered as “critical component” while the other assumes the role of a “buffer component.” As long as the cost ratio does not reach a critical threshold, which is proportional to the ratio of the grade-attainment likelihoods, the relative composition of the optimal input portfolio remains largely invariant. A numerical study confirms the practicality of the envelope approach, both as a seed for a numerical solution of the exact optimality conditions and as an approximate solution in closed-form.