Journal of Scheduling

, Volume 20, Issue 5, pp 475–491 | Cite as

Finding an optimal Nash equilibrium to the multi-agent project scheduling problem

  • Cyril Briand
  • Sandra Ulrich Ngueveu
  • Přemysl Šůcha


Large projects often involve a set of contractors, each in charge of a part of the project. In this paper, we assume that every contractor is self-interested and can control the duration of his/her activities, which can be shortened up to an incompressible limit, by gathering extra resources at a given cost. In this context, the resulting project makespan depends on all the contractors’ decisions. The customer of the project is interested in a short project makespan and offers a reward, proportional to the project makespan reduction, to be shared by the contractors. In practice, either the reward sharing policy results from an upfront agreement or payments are freely allocated by the customer. Each contractor is only interested in the maximization of his/her profit and behaves accordingly. This paper addresses the problem of finding a Nash equilibrium and a sharing policy that minimize the project makespan. The aim is to help the customer to determine the duration of the activities and the reward sharing policy such that no agent has an incentive to unilaterally deviate from this solution. We show that this problem is NP-hard and how it can be modeled and solved by mixed integer linear programming. Computational analysis on large instances proves the effectiveness of our approach. Based on an empirical investigation of the influence of reward sharing policies on the project makespan, the paper provides new insight into how a project’s customer should offer rewards to the contractors.


Project scheduling Time-cost trade-off Nash equilibrium Mixed integer programming 



This work was supported by the ANR Project No. 08-BLAN-0331-02 named “ROBOCOOP” and by the Grant Agency of the Czech Republic under the Project GACR P103-16-23509S.


  1. Agnetis, A., Briand, C., Billaut, J. C., & Šůcha, P. (2015). Nash equilibria for the multi-agent project scheduling problem with controllable processing times. Journal of Scheduling, 18(1), 15–27.CrossRefGoogle Scholar
  2. Averbakh, I. (2010). Nash equilibria in competitive project scheduling. European Journal of Operational Research, 205(3), 552–556.CrossRefGoogle Scholar
  3. Bachelet, B., & Mahey, P. (2003). Minimum convex-cost tension problems on series-parallel graphs. RAIRO - Operations Research, 37, 221–234.CrossRefGoogle Scholar
  4. Bahrami, F., & Moslehi, G. (2012). Study of payment scheduling problem to achieve client–contractor agreement. The International Journal of Advanced Manufacturing Technology. doi: 10.1007/s00170-012-4023-5.
  5. Bey, R. B., Doersch, R. H., & Patterson, J. H. (1981). The net present value criterion: Its impact on project scheduling. Project Management Quarterly, 12, 35–45.Google Scholar
  6. Briand, C., Agnetis, A., & Billaut, J. C. (2012). The multiagent project scheduling problem: complexity of finding an optimal Nash equilibrium. In 13th international conference on project management and scheduling, pp. 106–109Google Scholar
  7. Dayanand, N., & Padman, R. (2001). Project contracts and payment schedules: The client’s problem. Management Science, 47, 1654–1667.CrossRefGoogle Scholar
  8. De, P., Dunne, E. J., Ghosh, J. B., & Wells, C. E. (1995). The discrete time-cost tradeoff problem revisited. European Journal of Operations Research, 81, 225–238.CrossRefGoogle Scholar
  9. Demeulemeester, E., Vanhoucke, M., & Herroelen, W. (2003). Rangen: A random network generator for activity-on-the-node networks. Journal of Scheduling, 6, 17–38.CrossRefGoogle Scholar
  10. Demeulemeester, E. L., & Herroelen, W. S. (2002). Project scheduling: A research handbook. Kluwer Academic Publishers, chap 2-1.3.4. Transforming an AoN network into an AoA network with minimal reduction complexity, pp. 44–48Google Scholar
  11. Diaby, M., Cruz, J. M., & Nsakanda, A. L. (2011). Project crashing in the presence of general non-linear activity time reduction costs. International Journal of Operational Research, 12, 318–332.Google Scholar
  12. Estévez-Fernández, A. (2012). A game theoretical approach to sharing penalties and rewards in projects. European Journal of Operational Research, 216(3), 647–657.CrossRefGoogle Scholar
  13. Farley, A. M., & Proskurowski, A. (1982). Directed maximal-cut problems. Information Processing Letters, 15(5), 238–241. doi: 10.1016/0020-0190(82)90125-9.CrossRefGoogle Scholar
  14. Hartmann, S., & Briskorn, D. (2010). A survey of variants and extensions of the resource-constrained project scheduling problem. European Journal of Operational Research, 207(1), 1–14.CrossRefGoogle Scholar
  15. Hougaard, J. L. (1990). An introduction to allocation rules. Berlin: Springer.Google Scholar
  16. Moulin, H., & Shenker, S. (1992). Serial cost sharing. Econometrica, 60, 1009–1037.CrossRefGoogle Scholar
  17. Phillips, S., & Dessouky, M. I. (1977). Solving the project time/cost tradeoff problem using the minimal cut concept. Management Science, 24, 393–400.CrossRefGoogle Scholar
  18. Sasaki, M., Campbell, J. F., Krishnamoorthy, M., & Ernst, A. T. (2014). A stackelberg hub arc location model for a competitive environment. Computers & Operations Research, 47, 27–41.CrossRefGoogle Scholar
  19. Schrijver, A. (2004). Combinatorial optimization—Polyhedra and efficiency. Berlin: Springer.Google Scholar
  20. Sprumont, Y. (2008). Nearly serial sharing methods. International Journal of Game Theory, 37, 155–184.CrossRefGoogle Scholar
  21. Szmerekovsky, J. G. (2005). The impact of contractor behavior on the client’s payment-scheduling problem. Management Science, 51, 629–640.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.CNRS, LAASToulouse Cedex 4France
  2. 2.UPS, INP, LAASUniversité de ToulouseToulouse Cedex 4France
  3. 3.Faculty of Electrical EngineeringCzech Technical University in PraguePrague 2Czech Republic

Personalised recommendations