Journal of Scheduling

, Volume 18, Issue 1, pp 15–27 | Cite as

Nash equilibria for the multi-agent project scheduling problem with controllable processing times

  • Alessandro Agnetis
  • Cyril BriandEmail author
  • Jean-Charles Billaut
  • Přemysl Šůcha


This paper considers a project scheduling environment in which the activities of the project network are partitioned among a set of agents. Activity durations are controllable, i.e., every agent is allowed to shorten the duration of its activities, incurring a crashing cost. If the project makespan is reduced with respect to its normal value, a reward is offered to the agents and each agent receives a given ratio of the total reward. Agents want to maximize their profit. Assuming a complete knowledge of the agents’ parameters and of the activity network, this problem is modeled as a non-cooperative game and Nash equilibria are analyzed. We characterize Nash equilibria in terms of the existence of certain types of cuts on the project network. We show that finding one Nash equilibrium is easy, while finding a Nash strategy that minimizes the project makespan is NP-hard in the strong sense. The particular case where each activity belongs to a different agent is also studied and some polynomial-time algorithms are proposed for this case.


Multi-agent project scheduling  Nash equilibria Flow networks 



This work was supported by the ANR Project No. 08-BLAN-0331-02 named “ROBOCOOP.” The authors wish to acknowledge two anonymous reviewers for their remarks that significantly helped improving the paper.


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Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • Alessandro Agnetis
    • 1
  • Cyril Briand
    • 2
    • 3
    Email author
  • Jean-Charles Billaut
    • 4
  • Přemysl Šůcha
    • 2
    • 5
  1. 1.Università di SienaSienaItaly
  2. 2.CNRS, LAASToulouseFrance
  3. 3.Univ de Toulouse, UPS, LAASToulouseFrance
  4. 4.Laboratoire d’InformatiqueUniversité Francois Rabelais ToursToursFrance
  5. 5.Faculty of Electrical EngineeringCzech Technical University in PraguePrague 2Czech Republic

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