An Algorithm for Multistage Dynamic Networks with Random Arc Capacities, with an Application to Dynamic Fleet Management

Cheung, Raymond K.; Powell, Warren B.

doi:10.1287/opre.44.6.951

Cited by 100 publications

(48 citation statements)

References 12 publications

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“…Cheung and Powell (1996) and Kunnumkal and Topaloglu (2008b) show similar results in fleet management and network revenue management with customer choice settings. Adelman and Mersereau (2008) show a similar result for weakly coupled dynamic programs over an infinite horizon, but their proof simplifies significantly when one considers the finite horizon case.…”

Section: Proposition 2 If the Lagrange Multipliers Are Nonnegative Tmentioning

confidence: 61%

“…They call these dynamic programs as weakly coupled and show that such dynamic programs are particularly suitable for Lagrangian relaxation ideas. Karmarkar (1981), Cheung and Powell (1996), Castanon (1997) and Kunnumkal and Topaloglu (2008a) apply Lagrangian relaxation ideas to the dynamic programs arising in multi-location inventory allocation, fleet management, sensor management and multi-echelon inventory distribution settings.…”

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confidence: 99%

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Computing Time-Dependent Bid Prices in Network Revenue Management Problems

Kunnumkal

Topaloğlu

2010

Transportation Science

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We propose a new method to compute bid prices in network revenue management problems. The novel aspect of our method is that it naturally provides dynamic bid prices that depend on how much time is left until departure. We show that our method provides an upper bound on the optimal total expected revenue and this upper bound is tighter than the one provided by the widely known deterministic linear programming approach. Furthermore, it is possible to use the bid prices computed by our method as a starting point in a dynamic programming decomposition-like idea to decompose the network revenue management problem by the flight legs and to obtain dynamic and capacity-dependent bid prices. Our computational experiments indicate that the proposed method improves on many standard benchmarks.

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Section: Proposition 2 If the Lagrange Multipliers Are Nonnegative Tmentioning

confidence: 61%

mentioning

confidence: 99%

Computing Time-Dependent Bid Prices in Network Revenue Management Problems

Kunnumkal

Topaloğlu

2010

Transportation Science

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“…Given the large size of the problem, there is a strong desire to take advantage of the underlying network structure. Prior e orts at solving this problem (Frantzeskakis and Powell 1990, Powell 1988, and Cheung and Powell 1996 have focused on replacing the expected recourse function in Equation (1) with a piecewise linear, separable functionQ(S), which can be written aŝ…”

Section: The Basic Ideamentioning

confidence: 99%

Shape – A Stochastic Hybrid Approximation Procedure for Two-Stage Stochastic Programs

Cheung

Powell

2000

Operations Research

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We consider the problem of approximating the expected recourse function for two-stage stochastic programs. Our problem is motivated by applications that have special structure, such as an underlying network that allows reasonable approximations to the expected recourse function to be developed. In this paper, we show how these approximations can be improved by combining them with sample gradient information from the true recourse function. For the case of strictly convex nonlinear approximations, we prove convergence for this hybrid approximation. The method is attractive for practical reasons because it retains the structure of the approximation.A common problem in operations research is the challenge of making a decision now in a way that minimizes the expectation of costs in the future that depend on random events. For example, we may have the situation of determining how much product to ship from plants to warehouses from which we then satisfy demands at di erent retailers. We must decide how much product to ship to each warehouse before we know the retail demand. Once retail demands are known, we are able to optimize shipping patterns between warehouses and retailers.This problem, and many like it, can be posed as two-stage stochastic programs. The decision made now (in stage 1) determines what state we are in when we have to solve the problem in stage 2. If we could exactly capture the structure of the expected cost function (or recourse function) for stage 2, we would be able to make optimal decisions now. The di culty is that in most cases, the structure of this expected cost function is too complex.There is an extensive literature on two-stage stochastic programming problems, which is nicely summarized in several recent books (Infanger 1994 andKall andWallace 1994). General solution methods include scenario optimization (e.g., Rockafellar and Wets 1991), stochastic gradient techniques (e.g., Ermoliev 1983, Ruszczynski 1980, Benders decomposition and its variants (e.g., Van Slyke and Wets 1969, Birge 1985, and Higle and Sen 1991, sample path optimization (Robinson 1996), and other approximation techniques (e.g., Beale et al. 1980). These techniques are, for the most part, very general and are not designed speciÿcally to take advantage of approximations that may produce good but not optimal solutions.In this paper, we propose a new algorithm called SHAPE (successive hybrid approximation procedure) that combines an initial nonlinear approximation with iteratively sampled stochastic gradient information. The initial nonlinear approximation can exploit problem structure, while the stochastic gradient information, which is easy to obtain for most problems, tunes the approximation. We describe the algorithm, provide a small numerical illustration, and prove convergence.Section 1 provides a formal problem statement and introduces the basic idea behind the algorithm. Section 2 presents a method we call the stochastic hybrid approximation procedure (SHAPE) for solving two-stage stochastic programs with recourse. ...

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“…A major strength of the technique was that it produced nonlinear functions of the value of vehicles in the future. The ideas behind this research produced a series of articles that approximated the recourse function by using the structure of the underlying recourse function (Powell & Frantzeskakis (1992), Frantzeskakis & Powell (1990), Powell & Cheung (1994b), Powell & Cheung (1994a), Cheung & Powell (1996)). These papers introduced concepts such as nodal recourse (the recourse is to allocate flow over different links out of a node), tree recourse (the recourse is to optimize flows over a tree) and other restricted recourse strategies aimed at approximating more complex problems.…”

Section: Experimental Results For Multistage Problemsmentioning

confidence: 99%

Stochastic Programming in Transportation and Logistics

Powell

Topaloğlu

2003

Handbooks in Operations Research and Management Science

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Freight transportation is characterized by highly dynamic information processes: customers call in orders over time to move freight; the movement of freight over long distances is subject to random delays; equipment failures require last minute changes; and decisions are not always executed in the field according to plan. The high-dimensionality of the decisions involved has made transportation a natural application for the techniques of mathematical programming, but the challenge of modeling dynamic information processes has limited their success. In this chapter, we explore the use of concepts from stochastic programming in the context of resource allocation problems that arise in freight transportation. Since transportation problems are often quite large, we focus on the degree to which some techniques exploit the natural structure of these problems. Experimental work in the context of these applications is quite limited, so we highlight the techniques that appear to be the most promising.

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An Algorithm for Multistage Dynamic Networks with Random Arc Capacities, with an Application to Dynamic Fleet Management

Cited by 100 publications

References 12 publications

Computing Time-Dependent Bid Prices in Network Revenue Management Problems

Computing Time-Dependent Bid Prices in Network Revenue Management Problems

Shape – A Stochastic Hybrid Approximation Procedure for Two-Stage Stochastic Programs

Stochastic Programming in Transportation and Logistics

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