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On certain greedoid polyhedra, partially indexable scheduling problems and extended restless bandit allocation indices

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  • José Niño-Mora

Abstract

We present a polyhedral framework for establishing general structural properties on optimal solutions of stochastic scheduling problems, where multiple job classes vie for service resources: the existence of an optimal priority policy in a given family, characterized by a greedoid (whose feasible class subsets may receive higher priority), where optimal priorities are determined by class-ranking indices, under restricted linear performance objectives (partial indexability). This framework extends that of Bertsimas and Niño-Mora (1996), which explained the optimality of priority-index policies under all linear objectives (general indexability). We show that, if performance measures satisfy partial conservation laws (with respect to the greedoid), which extend previous generalized conservation laws, then the problem admits a strong LP relaxation over a so-called extended greedoid polytope, which has strong structural and algorithmic properties. We present an adaptive-greedy algorithm (which extends Klimov's) taking as input the linear objective coefficients, which (1) determines whether the optimal LP solution is achievable by a policy in the given family; and (2) if so, computes a set of class-ranking indices that characterize optimal priority policies in the family. In the special case of project scheduling, we show that, under additional conditions, the optimal indices can be computed separately for each project (index decomposition). We further apply the framework to the important restless bandit model (two-action Markov decision chains), obtaining new index policies, that extend Whittle's (1988), and simple sufficient conditions for their validity. These results highlight the power of polyhedral methods (the so-called achievable region approach) in dynamic and stochastic optimization.

Suggested Citation

  • José Niño-Mora, 2000. "On certain greedoid polyhedra, partially indexable scheduling problems and extended restless bandit allocation indices," Economics Working Papers 456, Department of Economics and Business, Universitat Pompeu Fabra.
  • Handle: RePEc:upf:upfgen:456
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    References listed on IDEAS

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    1. A. Federgruen & H. Groenevelt, 1988. "Characterization and Optimization of Achievable Performance in General Queueing Systems," Operations Research, INFORMS, vol. 36(5), pages 733-741, October.
    2. Dimitris Bertsimas & José Niño-Mora, 2000. "Restless Bandits, Linear Programming Relaxations, and a Primal-Dual Index Heuristic," Operations Research, INFORMS, vol. 48(1), pages 80-90, February.
    3. E. G. Coffman & I. Mitrani, 1980. "A Characterization of Waiting Time Performance Realizable by Single-Server Queues," Operations Research, INFORMS, vol. 28(3-part-ii), pages 810-821, June.
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    6. M. Dacre & K. Glazebrook & J. Niño‐Mora, 1999. "The achievable region approach to the optimal control of stochastic systems," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 61(4), pages 747-791.
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    More about this item

    Keywords

    Stochastic scheduling; restless bandits; greedoids; polyhedral methods; conservation laws; achievable region;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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