On Certain Greedoid Polyhedra, Partially Indexable Scheduling Problems, and Extended Restless Bandit Allocation Indices

UPF Economics & Business Working Paper No. 456

29 Pages Posted: 31 May 2000

See all articles by José Niño Mora

José Niño Mora

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences

Date Written: April 5, 2000

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 Ni?o-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.

Keywords: Stochastic scheduling, restless bandits, greedoids, polyhedral methods, conservation laws, achievable region.

JEL Classification: C60, C61

Suggested Citation

Niño Mora, José, On Certain Greedoid Polyhedra, Partially Indexable Scheduling Problems, and Extended Restless Bandit Allocation Indices (April 5, 2000). UPF Economics & Business Working Paper No. 456, Available at SSRN: https://ssrn.com/abstract=230871 or http://dx.doi.org/10.2139/ssrn.230871

José Niño Mora (Contact Author)

Universitat Pompeu Fabra - Faculty of Economic and Business Sciences ( email )

Ramon Trias Fargas 25-27
Barcelona, 08005
Spain
(34-3) 542 26 73 (Phone)
(34-3) 542 17 46 (Fax)

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