Optimal Use and Replenishment of Two Substitutable Raw Materials in a Stochastic Capacitated Make-to-Order Production System

How should a firm make joint production and replenishment decisions in a make-to-order setting where it can use either of two kinds of raw materials (or their mixture) to produce an end product using a shared production line with stochastic capacity? The problem is motivated by the practice of producing the final end product using two types of substitutable raw materials in the energy and manufacturing industries. In such settings, one of the two kinds of raw materials is operationally more efficient (i.e. it has a higher conversion rate to the end product) whereas the other is cheaper. Moreover, the availability of the raw materials is also stochastic due to, among other things, delivery uncertainty. We formulate the problem into a stochastic dynamic program and develop a novel approximate optimization approach to analyze and characterize the structure of the optimal policy. We show that a Use-down-to/Balancing Production Policy and modified Order-up-to Ordering Policy is optimal. Since using a brute-force approach to compute the optimal policy is time-consuming due to the curse of dimensionality of the dynamic program, we use the structural properties of the optimal policy to develop an efficient algorithm to compute the optimal policy. Although the optimal policy is complex, our numerical study shows that solving for it using the algorithm we develop is feasible, and it significantly outperforms two straw heuristic policies that could be used in practice. Our numerical results reveal the significant cost-saving opportunity of using the more sophisticated optimal policy, and it also highlights that, by leveraging the structure of the optimal policy, it can be computationally feasible to implement the optimal policy