Optimal Inventory Allocation for Indifferent Goods Under Dynamic Substitution

• 成果类型: Article
• 署名作者: Zhou, Ziyan; Wang, Tong; Zhang, Tingwei
• 署名单位: Shanghai Jiao Tong University; City University of Hong Kong; The Chinese University of Hong Kong, Shenzhen
• 刊物名称: OPERATIONS RESEARCH
• ISSN/ISSBN: 0030-364X
• DOI: 10.1287/opre.2025.2228
• 发表日期: 2026
• 关键词: assortment planning problem

摘要

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, casespecific 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.