Asymptotically Optimal Inventory Control for Assemble-to-Order Systems

Abstract

We consider assemble-to-order (ATO) inventory systems with a general bill of materials and general deterministic lead times. Unsatisfied demands are always backlogged. We apply a four-step asymptotic framework to develop inventory policies for minimizing the long-run average expected total inventory cost. Our approach features a multistage stochastic program (SP) to establish a lower bound on the inventory cost and determine parameter values for inventory control. Our replenishment policy deviates from the conventional constant base stock policies to accommodate nonidentical lead times. Our component allocation policy differentiates demands based on backlog costs, bill of materials, and component availabilities. We prove that our policy is asymptotically optimal on the diffusion scale, that is, as the longest lead time grows, the percentage difference between the average cost under our policy and its lower bound converges to zero. In developing these results, we formulate a broad stochastic tracking model and prove general convergence results from which the asymptotic optimality of our policy follows as specialized corollaries.

A Survey of Recent Progress in the Asymptotic Analysis of Inventory Systems

“…Similar to the development that leads to Lemma 6, Reiman and Wang (2012), Reiman et al. (2018) establish a lower bound on the expected cost at time t , which is also a lower bound on the average inventory cost of the ATO system.…”

“…Also as in the identical lead time case, Reiman et al. (2018) again reformulate the relevant optimization problem in terms of certain observable shortages, and again develop policies to keep the inventory positions and backlog levels near their ideal target levels. The targets are updated by applying new demand inputs and re‐solving the stochastic program of Lemma 7.…”

“…Existence and uniqueness of such a solution is addressed in Reiman and Wang (2015), Reiman et al. (2018), and for simplicity of discussion here let us simply assume existence and uniqueness. For a given policy π , let us define .…”

“…Instead of formally re‐defining the associated problem, we simply note that the formal problem definition is essentially the same, with the caveat that now one associates a given policy π with two on‐hand component inventory processes, one each for those components with lead time , respectively, with the associated defining dynamical inventory equations reflecting these different lead times (see Reiman et al. (2018) for details).…”

“…Due to the additional complexity introduced by non‐identical lead times, Reiman et al. (2018) introduce several additional methodological innovations. First, to show that all relevant quantities are “asymptotically close” to the idealized processes given by the re‐solved stochastic programs, the authors formulate a general “stochastic tracking model” within which one can seamlessly demonstrate all relevant convergences (as opposed to requiring separate notations, tools, and analyses for each required convergence).…”