Skip to main content
Log in

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

  • Published:
Journal of Scheduling Aims and scope Submit manuscript


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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or Ebook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others


  • Adhau, S., Mittal, M. L., & Mittal, A. (2012). A multi-agent system for distributed multi-project scheduling: An auction-based negotiation approach. Engineering Applications of Artificial Intelligence, 25(8), 1738–1751.

    Article  Google Scholar 

  • Agnetis, A., Mirchandani, P. B., Pacciarelli, D., & Pacifici, A. (2004). Scheduling problems with two competing agents. Operations Research, 52, 229–242.

    Article  Google Scholar 

  • Ahuja, R. K., Magnanti, T. L., & Orlin, J. B. (1993). Network flows. Upper Saddle River, NJ: Prentice-Hall.

    Google Scholar 

  • Bachelet, B., & Mahey, P. (2003). Minimum convex-cost tension problems on series-parallel graphs. RAIRO Operations Research, 37–4, 221–234.

    Article  Google Scholar 

  • Brânzei, R., Ferrari, G., Fragnelli, V., & Tijs, S. (2002). Two approaches to the problem of sharing delay costs in joint projects. Annals of Operations Research, 109, 359–374.

    Article  Google Scholar 

  • Ciurea, E., & Ciupalâ, L. (2004). Sequential and parallel algorithms for minimum flows. Journal of Applied Mathematics and Computing, 15, 53–75.

    Article  Google Scholar 

  • Confessore, G., Giordani, S., & Rismondo, S. (2007). A market-based multi-agent system model for decentralized multi-project scheduling. Annals of Operations Research, 150, 115–135.

    Article  Google Scholar 

  • Demeulemeester, E. L., & Herroelen, W. S. (2002). Project scheduling—a research handbook. Dordrecht: Kluwer Academic Publishers.

    Google Scholar 

  • Estévez-Fernández, A. (2012). A game theoretical approach to sharing penalties and rewards in projects. European Journal of Operational Research, 216(3), 647–657.

    Article  Google Scholar 

  • Evaristo, R., & van Fenema, P. C. (1999). A typology of project management: Emergence and evolution of new forms. International Journal of Project Management, 17(5), 275–281.

    Article  Google Scholar 

  • Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: W. H. Freeman & Co.

    Google Scholar 

  • Homberger, J. (2012). A (mu, lambda)-coordination mechanism for agent-based multi-project scheduling. OR Spectrum, 34(1), 107–132.

    Article  Google Scholar 

  • Knotts, G., & Dror, M. (2003). Agent-based project scheduling: Computational study of large problems. IIE Transactions, 35(2), 143–159.

    Article  Google Scholar 

  • Knotts, G., Dror, M., & Hartman, B. C. (2000). Agent-based project scheduling. IIE Transactions, 32(5), 387–401.

    Google Scholar 

  • Lau, J. S. K., Huang, G. Q., Mak, K. L., & Liang, L. (2005a). Distributed project scheduling with information sharing in supply chains: Part I an agent-based negotiation model. International Journal of Production Research, 43(22), 4813–4838.

    Article  Google Scholar 

  • Lau, J. S. K., Huang, G. Q., Mak, K. L., & Liang, L. (2005b). Distributed project scheduling with information sharing in supply theoretical analysis and computational study. International Journal of Production Research, 43(23), 4899–4927.

  • Leung, J. Y.-T., Pinedo, M. L., & Wan, G. (2010). Competitive two-agent scheduling and its applications. Operations Research, 58, 458–469.

    Article  Google Scholar 

  • Li, Z., & Zhou, H. (2012). Coordination mechanism of economic lot and delivery scheduling problem. Industrial Engineering Journal, 15, 18–22.

    Google Scholar 

  • Phillips, S., & Dessouky, M. I. (1977). Solving the project time/cost tradeoff problem using the minimal cut concept. Management Science, 24(4), 393–400.

  • Wang, L., Zhan, D., Nie, L., & Xu, X. (2011). Research framework for decentralized multi-project scheduling problem. In 2011 International Conference on Information Science and Technology, ICIST 2011, pp. 802–806.

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Cyril Briand.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Agnetis, A., Briand, C., Billaut, JC. et al. Nash equilibria for the multi-agent project scheduling problem with controllable processing times. J Sched 18, 15–27 (2015).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: