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Optimal Policies for a Two‐Product Inventory System under a Flexible Substitution Scheme
Author(s) -
Xu He,
Yao David D.,
Zheng Shaohui
Publication year - 2016
Publication title -
production and operations management
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.279
H-Index - 110
eISSN - 1937-5956
pISSN - 1059-1478
DOI - 10.1111/poms.12536
Subject(s) - substitution (logic) , product (mathematics) , function (biology) , substitution method , submodular set function , dynamic programming , substitution effect , economics , mathematical optimization , transformation (genetics) , selection (genetic algorithm) , scheme (mathematics) , computer science , microeconomics , mathematics , telecommunications , biochemistry , chemistry , mathematical analysis , geometry , evolutionary biology , artificial intelligence , gene , biology , programming language
We study a two‐product inventory model that allows substitution. Both products can be used to supply demand over a selling season of N periods, with a one‐time replenishment opportunity at the beginning of the season. A substitution could be offered even when the demanded product is available. The substitution rule is flexible in the sense that the seller can choose whether or not to offer substitution and at what price or discount level, and the customer may or may not accept the offer, with the acceptance probability being a decreasing function of the substitution price. The decisions are the replenishment quantities at the beginning of the season, and the dynamic substitution‐pricing policy in each period of the season. Using a stochastic dynamic programming approach, we present a complete solution to the problem. Furthermore, we show that the objective function is concave and submodular in the inventory levels—structural properties that facilitate the solution procedure and help identify threshold policies for the optimal substitution/pricing decisions. Finally, with a state transformation, we also show that the objective function is L ♮ ‐concave, which allows us to derive similar structural properties of the optimal policy for multiple‐season problems.