We consider the periodic review dynamic pricing and inventory control problem with fixed ordering cost. Demand is random and price dependent, and unsatisfied demand is backlogged. With complete demand information, the celebrated [Formula: see text] policy is proved to be optimal, where s and S are the reorder point and orderupto level for ordering strategy, and [Formula: see text], a function of onhand inventory level, characterizes the pricing strategy. In this paper, we consider incomplete demand information and develop online learning algorithms whose average profit approaches that of the optimal [Formula: see text] with a tight [Formula: see text] regret rate. A number of salient features differentiate our work from the existing online learning researches in the operations management (OM) literature. First, computing the optimal [Formula: see text] policy requires solving a dynamic programming (DP) over multiple periods involving unknown quantities, which is different from the majority of learning problems in OM that only require solving singleperiod optimization questions. It is hence challenging to establish stability results through DP recursions, which we accomplish by proving uniform convergence of the profittogo function. The necessity of analyzing actiondependent state transition over multiple periods resembles the reinforcement learning question, considerably more difficult than existing bandit learning algorithms. Second, the pricing function [Formula: see text] is of infinite dimension, and approaching it is much more challenging than approaching a finite number of parameters as seen in existing researches. The demandprice relationship is estimated based on upper confidence bound, but the confidence interval cannot be explicitly calculated due to the complexity of the DP recursion. Finally, because of the multiperiod nature of [Formula: see text] policies the actual distribution of the randomness in demand plays an important role in determining the optimal pricing strategy [Formula: see text], which is unknown to the learner a priori. In this paper, the demand randomness is approximated by an empirical distribution constructed using dependent samples, and a novel Wasserstein metricbased argument is employed to prove convergence of the empirical distribution. This paper was accepted by J. George Shanthikumar, big data analytics.
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Learning in Structured MDPs with Convex Cost Functions: Improved Regret Bounds for Inventory Management
We consider a stochastic inventory control problem under censored demand, lost sales, and positive lead times. This is a fundamental problem in inventory management, with significant literature establishing near optimality of a simple class of policies called “basestock policies” as well as the convexity of longrun average cost under those policies. We consider a relatively less studied problem of designing a learning algorithm for this problem when the underlying demand distribution is unknown. The goal is to bound the regret of the algorithm when compared with the best basestock policy. Our main contribution is a learning algorithm with a regret bound of [Formula: see text] for the inventory control problem. Here, [Formula: see text] is the fixed and known lead time, and D is an unknown parameter of the demand distribution described roughly as the expected number of time steps needed to generate enough demand to deplete one unit of inventory. Notably, our regret bounds depend linearly on L, which significantly improves the previously bestknown regret bounds for this problem where the dependence on L was exponential. Our techniques utilize the convexity of the longrun average cost and a newly derived bound on the “bias” of basestock policies to establish an almost black box connection between the problem of learning in Markov decision processes (MDPs) with these properties and the stochastic convex bandit problem. The techniques presented here may be of independent interest for other settings that involve large structured MDPs but with convex asymptotic average cost functions.
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 Award ID(s):
 1846792
 NSFPAR ID:
 10374181
 Date Published:
 Journal Name:
 Operations Research
 Volume:
 70
 Issue:
 3
 ISSN:
 0030364X
 Page Range / eLocation ID:
 1646 to 1664
 Format(s):
 Medium: X
 Sponsoring Org:
 National Science Foundation
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