Minimum-cost ordering for selective assembly

We consider the minimization of input cost for a selective assembly system that features two random inputs and a finite number of matching classes. This setup frequently arises in high-precision manufacturing when input tolerances are not tight enough for the required output precision. We determine optimality conditions for the cost-optimal input portfolio given an expected-output target, first using a normal approximation of the multinomial binning distribution, and second employing a simple upper envelope of the output objective. We show that the relative error tends to zero as the production scale becomes sufficiently large. The envelope optimization problem also yields a closed-form solution for the cost-minimizing inputs as well as total costs, which are easy to understand for managers. A numerical study tests the practicality of the envelope approach. The latter can be used as seed for a numerical solution of the exact problem, as well as a closed-form approximation, which allows for an analysis of structural properties.