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Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost

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  • José R. Correa
  • Maurice Queyranne

Abstract

In the last decade, there has been much progress in understanding scheduling problems in which selfish jobs aim to minimize their individual completion time. Most of this work has focused on makespan minimization as social objective. In contrast, we consider as social cost the total weighted completion time, that is, the sum of the agent costs, a standard definition of welfare in economics. In our setting, jobs are processed on restricted uniform parallel machines, where each machine has a speed and is only capable of processing a subset of jobs; a job's cost is its weighted completion time; and each machine sequences its jobs in weighted shortest processing time (WSPT) order. Whereas for the makespan social cost the price of anarchy is not bounded by a constant in most environments, we show that for our minsum social objective the price of anarchy is bounded above by a small constant, independent of the instance. Specifically, we show that the price of anarchy is exactly 2 for the class of unit jobs, unit speed instances where the finite processing time values define the edge set of a forest with the machines as nodes. For the general case of mixed job strategies and restricted uniform machines, we prove that the price of anarchy equals 4. From a classical machine scheduling perspective, our results establish the same constant performance guarantees for WSPT list scheduling. © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2012

Suggested Citation

  • José R. Correa & Maurice Queyranne, 2012. "Efficiency of equilibria in restricted uniform machine scheduling with total weighted completion time as social cost," Naval Research Logistics (NRL), John Wiley & Sons, vol. 59(5), pages 384-395, August.
  • Handle: RePEc:wly:navres:v:59:y:2012:i:5:p:384-395
    DOI: 10.1002/nav.21497
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    1. Wayne E. Smith, 1956. "Various optimizers for single‐stage production," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 3(1‐2), pages 59-66, March.
    2. Dimitris Bertsimas & José Niño-Mora, 1996. "Conservation Laws, Extended Polymatroids and Multiarmed Bandit Problems; A Polyhedral Approach to Indexable Systems," Mathematics of Operations Research, INFORMS, vol. 21(2), pages 257-306, May.
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    1. Felipe T. Muñoz & Rodrigo Linfati, 2024. "Bounding the Price of Anarchy of Weighted Shortest Processing Time Policy on Uniform Parallel Machines," Mathematics, MDPI, vol. 12(14), pages 1-12, July.
    2. Cole, Richard & Correa, Jose & Gkatzelis, Vasillis & Mirrokni, Vahab & Olver, Neil, 2015. "Decentralized utilitarian mechanisms for scheduling games," LSE Research Online Documents on Economics 103081, London School of Economics and Political Science, LSE Library.
    3. Herbert Hamers & Flip Klijn & Marco Slikker, 2019. "Implementation of optimal schedules in outsourcing with identical suppliers," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 89(2), pages 173-187, April.
    4. Yossi Azar & Lisa Fleischer & Kamal Jain & Vahab Mirrokni & Zoya Svitkina, 2015. "Optimal Coordination Mechanisms for Unrelated Machine Scheduling," Operations Research, INFORMS, vol. 63(3), pages 489-500, June.
    5. Braat, Jac & Hamers, Herbert & Klijn, Flip & Slikker, Marco, 2019. "A selfish allocation heuristic in scheduling: Equilibrium and inefficiency bound analysis," European Journal of Operational Research, Elsevier, vol. 273(2), pages 634-645.
    6. Varun Gupta & Benjamin Moseley & Marc Uetz & Qiaomin Xie, 2020. "Greed Works—Online Algorithms for Unrelated Machine Stochastic Scheduling," Mathematics of Operations Research, INFORMS, vol. 45(2), pages 497-516, May.

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