Generalized Restless Bandits and the Knapsack Problem for Perishable Inventories

Darina Graczová () and Peter Jacko ()
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Darina Graczová: Department of Applied Mathematics and Statistics, Comenius University, 842 48 Bratislava, Slovakia
Peter Jacko: Department of Management Science, Lancaster University Management School, Lancaster, LA1 4YX, United Kingdom; and BCAM Basque Center for Applied Mathematics, 48009 Bilbao, Spain

Operations Research, 2014, vol. 62, issue 3, 696-711

Abstract: In this paper we introduce the knapsack problem for perishable inventories concerning the optimal dynamic allocation of a collection of products to a limited knapsack. The motivation for designing such a problem comes from retail revenue management, where different products often have an associated lifetime during which they can only be sold, and the managers can regularly select some products to be allocated to a limited promotion space that is expected to attract more customers than the standard shelves. Another motivation comes from scheduling of requests in modern multiserver data centers so that quality-of-service requirements given by completion deadlines are satisfied. Using the Lagrangian approach we derive an optimal index policy for the Whittle relaxation of the problem in which the knapsack capacity is used only on average. Assuming a certain structure of the optimal policy for the single-inventory control, we prove indexability and derive an efficient, linear-time algorithm for computing the index values. To the best of our knowledge, our paper is the first to provide indexability analysis of a restless bandit with bi-dimensional state (lifetime and inventory level). We illustrate that these index values are numerically close to the true index values when such a structure is not present. We test two index-based heuristics for the original, nonrelaxed problem: (1) a conventional index rule , which prescribes to order the products according to their current index values and promotes as many products as fit in the knapsack, and (2) a recently proposed index-knapsack heuristic , which employs the index values as a proxy for the price of promotion and proposes to solve a deterministic knapsack problem to select the products. By a systematic computational study we show that the performance of both heuristics is nearly optimal, and that the index-knapsack heuristic outperforms the conventional index rule.

Keywords: Markov decision processes; resource allocation; knapsack problem; index policies; Whittle index; Lagrangian relaxation (search for similar items in EconPapers)
Date: 2014
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Citations: View citations in EconPapers (6)

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