Optimal Inventory Allocation for Indifferent Goods Under Dynamic Substitution

Stockout-based substitution creates complex stochastic dynamics in inventory systems even in highly symmetric settings. We study a joint assortment and inventory allocation problem in which a firm allocates a fixed total inventory of m units across n perfectly substitutable product types (e.g., colors or designs). Customers are indifferent among available types and arrive sequentially, each purchasing one unit chosen uniformly at random from the nonempty types. The sales process terminates when only one product type remains in stock. The objective is to choose the initial inventory allocation that maximizes the expected total number of sales. This seemingly simple problem is highlighted as an open problem in recent work with prior solutions only for $n=2$ and $n=3$ based on intricate, case-specific induction. This paper resolves the general case for arbitrary n and m, proving that an optimal allocation policy is balanced, meaning inventory levels for any two product types differ by at most one unit. Our proof relies on a continuous-time embedding of the discrete sales process via independent exponential clocks, and this converts inventory depletion times into independent Erlang random variables. We further illustrate the scope of the embedding method in a classical setting with a fixed number of customers whose choices follow a multinomial logit (MNL) model. In the symmetric MNL case, the embedding gives a short proof that balanced allocations are optimal, substantially simplifying prior inductive arguments. In the asymmetric MNL case, it recovers the classical fluid relaxation as a linear program and suggests a route to tighter bounds beyond the fluid approximation.

Funding: Z. Zhou was partly supported by the National Natural Science Foundation of China [Grants 72222008, 72571231]. T. Wang was partly supported by the National Natural Science Foundation of China [Grants 72221001, 72192833/72192830, 72131010] and the Research Grants Council of Hong Kong [GRF 11502225]. J. Zhang is partly sup-ported by the National Natural Science Foundation of China (NSFC) [Grant 72394361] and the Guangdong Provincial Key Laboratory of Mathematical Foundations for Artificial Intelligence [2023B1212010001].

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://orcid.org/10.1287/opre.2025.2228.