Fixed-Dimensional Stochastic Dynamic Programs: An Approximation Scheme and an Inventory Application

Chen, Wei; Dawande, Milind; Janakiraman, Ganesh

doi:10.1287/opre.2013.1239

Cited by 28 publications

(18 citation statements)

References 24 publications

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“…Using a very different approach, Halman et al provide an approximate dynamic programming algorithm that, combined with ideas from discrete convexity, yields a so-called fully polynomial-time approximation scheme for various related inventory control problems [17,18]. These techniques were recently extended to lost-sales models with positive lead times (as considered in this paper) by Chen et al [6], who provide a pseudo-polynomial-time additive approximation algorithm. Namely, under a suitable encoding scheme, an algorithm is presented that, for any ǫ > 0, returns a policy whose performance differs additively from that of the optimal policy by at most ǫ, in time which is polynomial in ǫ −1 if the overall encoding length of the problem is held fixed while ǫ is varied, and otherwise is pseudo-polynomial in the overall encoding length (which grows with the lead time L); we refer the reader to [6] for details.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times

Goldberg

Katz-Rogozhnikov²,

Lu³

et al. 2016

Mathematics of OR

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Lost sales inventory models with large lead times, which arise in many practical settings, are notoriously difficult to optimize due to the curse of dimensionality. In this paper we show that when lead times are large, a very simple constant-order policy, first studied by Reiman [39], performs nearly optimally. The main insight of our work is that when the lead time is very large, such a significant amount of randomness is injected into the system between when an order for more inventory is placed and when the order is received, that "being smart" algorithmically provides almost no benefit. Our main proof technique combines a novel coupling for suprema of random walks with arguments from queueing theory.

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Section: Introductionmentioning

confidence: 99%

Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times

Goldberg

Katz-Rogozhnikov²,

Lu³

et al. 2016

Mathematics of OR

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“…To deal with these curses, [5] studies fixed-dimensional stochastic dynamic programs with discrete state and action spaces over a finite horizon. Assuming that the cost-to-go functions are discrete L -convex (classes of discrete convex functions are discussed later in this subsection), [5] proposes a pseudopolynomialtime approximation scheme that satisfies an arbitrary pre-specified additive error guarantee. The proposed approximation algorithm is a generalization of the explicit enumeration algorithm.…”

Section: Relevance To Existing Literaturementioning

confidence: 99%

“…The main differences between our paper and [5] are: (i) [5] considers discrete state and action spaces (as opposed to continuous); (ii) it considers fixed dimensional (as opposed to one-dimensional) state spaces; (iii) it gives additive (as opposed to relative) error approximation; (iv) the running time of the approximation algorithm in [5] is pseudopolynomial (as opposed to polynomial) in the binary size of the input. Both [5] and the current paper are based on generalization of the technique of K-approximation sets and functions. Another relevant work in dealing with the curses of dimensionality in options pricing and optimal stopping is [6].…”

Section: Relevance To Existing Literaturementioning

confidence: 99%

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Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Halman¹,

Nannicini²

2019

SIAM J. Optim.

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We propose a Fully Polynomial-Time Approximation Scheme (FPTAS) for stochastic dynamic programs with multidimensional action, scalar state, convex costs and linear state transition function. The action spaces are polyhedral and described by parametric linear programs. This type of problems finds applications in the area of optimal planning under uncertainty, and can be thought of as the problem of optimally managing a single non-discrete resource over a finite time horizon. We show that under a value oracle model for the cost functions this result for one-dimensional state space is "best possible", because a similar dynamic programming model with two-dimensional state space does not admit a PTAS.The FPTAS relies on the solution of polynomial-sized linear programs to recursively compute an approximation of the value function at each stage. Our paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action spaces and with vectors of continuous random variables with bounded support. These results bring us one step closer to overcoming the curse of dimensionality of dynamic programming.

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“…Another recent line of research yields efficient approximation algorithms for any fixed lead time by carefully approximating the associated dynamic programs (cf. Halman et al (2009), Halman, Orlin andSimchi-Levi (2012), Chen, Dawande and Janakiraman (2014)). Despite this progress, the aforementioned work leaves open the problem of deriving efficient algorithms with arbitrarily small error, when the lead time is large.…”

Section: Introductionmentioning

confidence: 99%

Optimality Gap of Constant-Order Policies Decays Exponentially in the Lead Time for Lost Sales Models

Xin

Goldberg

2016

Operations Research

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Inventory models with lost sales and large lead times have traditionally been considered intractable due to the curse of dimensionality. Recently, Goldberg and co-authors laid the foundations for a new approach to solving these models, by proving that as the lead time grows large, a simple constant-order policy is asymptotically optimal. However, the bounds proven there require the lead time to be very large before the constant-order policy becomes effective, in contrast to the good numerical performance demonstrated by Zipkin even for small lead time values. In this work, we prove that for the infinite-horizon variant of the same lost sales problem, the optimality gap of the same constant-order policy actually converges exponentially fast to zero, with the optimality gap decaying to zero at least as fast as the exponential rate of convergence of the expected waiting time in a related single-server queue to its steady-state value. We also derive simple and explicit bounds for the optimality gap, and demonstrate good numerical performance across a wide range of parameter values for the special case of exponentially distributed demand. Our main proof technique combines convexity arguments with ideas from queueing theory.

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Fixed-Dimensional Stochastic Dynamic Programs: An Approximation Scheme and an Inventory Application

Cited by 28 publications

References 24 publications

Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times

Asymptotic Optimality of Constant-Order Policies for Lost Sales Inventory Models with Large Lead Times

Toward Breaking the Curse of Dimensionality: An FPTAS for Stochastic Dynamic Programs with Multidimensional Actions and Scalar States

Optimality Gap of Constant-Order Policies Decays Exponentially in the Lead Time for Lost Sales Models

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