# Minimizing the Cost of Matched Components in Precision Manufacturing

Prof. Weber’s research on the “Optimal Matching of Random Parts,” featured in the current issue of the Journal of Mathematical Economics, investigates a firm’s optimal policy of ordering parts with randomly distributed characteristics when these inputs need to be assembled with other components whose characteristics are also uncertain.

A mechanical watch movement contains as oscillator, mounted together, a balance wheel and a hairspring whose out-of-the-box characteristics are usually too diffuse for the assembled system to be precise enough as a reliable time keeper. Therefore the parts of each category (i.e., balance wheels or hairsprings) need to be sorted into bins of similar characteristics, and subsequently they can be matched across corresponding bins. Prof. Weber’s recent results in [1] characterize the solution of the cost-minimization problem when deciding about the order quantities for each input required to build an assembled system such as a watch oscillator. They also provide an easily interpretable closed-form approximation of the optimal solution. One of the key insights by and large is that one component tends to serve as a “buffer component” for the other (e.g., hairsprings tend to be buffer components for the more expensive balance wheels). Generically this leads to asymmetric order quantities in practical applications. This research is part of an ongoing project on precision manufacturing at the Chair of Operations, Economics and Strategy (OES) at EPFL, and the current results follow the robust design of matching classes in [2], published in 2021.

**Abstract**

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.

**References**

[1] Weber, T.A. (2022) “Optimal Matching of Random Parts,” Journal of Mathematical Economics, Vol. 101, Art. 102688.

[DOI: 10.1016/j.jmateco.2022.102688; open access]

[2] Weber, T.A. (2021) “Minimum-Error Classes for Matching Parts,” Operations Research Letters, Vol. 49, No. 1, pp. 106—112.

[DOI: 10.1016/j.orl.2020.12.003; open access]