A Scheduling Problem with One Producer and the Bargaining Counterpart with Two Producers

  • Xiaobing Gan
  • Yanhong Gu
  • George L. Vairaktarakis
  • Xiaoqiang Cai
  • Quanle Chen
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4614)


First this paper considers a Common Due Window (CDW) scheduling problem of n jobs on a single machine to minimize the sum of common weighted earliness and weighted number of tardy jobs when only one manufacturer processes these jobs. Two dynamic algorithms are designed for two cases respectively and each case is proved to be ordinary NP-hard. Successively the scenario, where two manufacturers jointly process these jobs due to the insufficient production facilities or techniques of each party, is investigated. A novel dynamic programming algorithm is proposed to obtain an existing reasonable set of processing utility distributions on the bi-partition of these jobs.


Schedule Problem Optimal Schedule Dynamic Programming Algorithm Bargaining Solution Total Processing Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. 1.
    Anger, F.D., Lee, C.-Y., Martin-Vega, L.A.: Single Machine Scheduling with Tight Windows. Research Report, Department of Industrial and Systems Engineering, University of Florida, Gainesville, FloridaTransportation Science, pp. 86–16 (1986)Google Scholar
  2. 2.
    Kramer, F.-J., Lee, C.-Y.: Common Due Window Scheduling. Production and Operations Management 6, 262–275 (1997)Google Scholar
  3. 3.
    Lee, C.-Y.: Earliness-Tardiness Scheduling Problems with Constant Size of Due Date Window. Research Report, Department of Industrial and System Engineering, University of Florida, Gainesville, Florida, pp. 91–17 (1991)Google Scholar
  4. 4.
    Liman, S.D., Ramaswamy, S.: Earliness-Tardiness Scheduling Problems with a Common Delivery Window. Operations Research Letters 15, 195–203 (1994)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Liman, S.D., Panwlker, S.S., Thongmee, S.: Determination of Common Due Window Location in a Singer Machine Scheduling Problem. European Journal of Operational Research 93, 68–74 (1996)zbMATHCrossRefGoogle Scholar
  6. 6.
    Koulamas, C.: Single-Machine Scheduling with Time Window and Earliness/Tardiness Penalties. European Journal of Operational Research 91, 190–202 (1996)zbMATHCrossRefGoogle Scholar
  7. 7.
    Ventura, J.A., Weng, M.X.: Single Machine Scheduling with a Common Delivery Window. Journal of the Operational Research Society 47, 424–434 (1996)zbMATHCrossRefGoogle Scholar
  8. 8.
    Koulamas, C.: Maximizing the Weighted Number of On-Time Jobs in Single Machine Scheduling with Time Windows. Math. Comput. Modelling 25, 57–62 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Liman, S.D., Panwalkar, S.S., Thongmee, S.: Common Due Window Size and Location Determination in a Single Machine Scheduling Problem. Journal of the Operational Research Society 49, 1007–1010 (1998)zbMATHCrossRefGoogle Scholar
  10. 10.
    Liman, S.D., Panwalkar, S.S., Thongmee, S.: Scheduling Common Due Window Problems with Controllable Processing Times. Annals of Operations Research 70, 145–154 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Chen, Q.-L., Sun, S.-J.: An Earliness and Tardiness Problem in Single Machine Scheduling wiht a Common Due Window. Applied Mathematics- a Journal of Chinese universities SerA 15, 440–448 (2000)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Chen, Z.-L., Lee, C.-Y.: A Column Generation Algorithm for Parallel Machine Common Due Window Scheduling. European Journal of Operational Research 136, 512–527 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Yen, B., Wan, G.: Tabu Search for Total Weighted Earliness and Tardiness Minimizing on Single Machine with Distinct Due Windows. European Journal of Operational Research 142, 271–281 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Baker, K.R., Scudder, G.D.: Sequencing with Earliness and Tardiness Penalties: A Review. Oper. Res. 38, 22–36 (1990)zbMATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Lawler, E.L.: Fast Approximation Algorithms for Knapsack Problems. In: Proc. 18th Annual Symposium on Foundation of Computer Science, pp. 206–213. IEEE Computer Society, Long Beach, CA (1977)Google Scholar
  16. 16.
    Nash, J.: Two Person Cooperative Games. Econometrica 21, 128–140 (1953)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Muthoo, A.: Bargaining Theory with Application. Cambridge University Press, Cambridge (1999)Google Scholar
  18. 18.
    Zhang, D.: A logical Model of Nash Bargaining Solution. In: Proceeding of IJCAI, pp. 983–990 (2005)Google Scholar
  19. 19.
    Trockel, W.: Integrating the Nash Program into Mechanism Theory. Review of Economic Design 7, 27–43 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Trockel, W.: Core-equivalence for the Nash Bargaining Solution. Economic Theory 25, 255–263 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Dagan, N., Volij, O., Winter, E.: A Characterization of the Nash Bargaining Solution. Social Choice and Welfare 19, 811–823 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Touati, C., Altman, E., Galtier, J.: Generalized Nash Bargaining Solution for Bandwidth Allocation. Computer Networks (in press)Google Scholar
  23. 23.
    Nagahisa, R., Tanaka, M.: An axiomatization of the Kalai-Smorodinsky Solution when the Feasible Sets can be Finite. Social Choice and Welfare 19, 751–761 (2002)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Lahiri, S.: Axiomatic Characterization of the Nash and Kalai-Smorodinsky Solutions for Discrete Bargaining Problems. PU.M.A 14, 207–220 (2004)zbMATHGoogle Scholar
  25. 25.
    Mariotti, M.: Nash Bargaining Theory when the Number of Alternatives can be Finite. Social Choice and Welfare 15, 413–421 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Chen, Q.-L.: A New Discrete Bargaining Model on Job Partition Between Two Manufacturers. PhD Dissertation, The Chinese University of Hongkong (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Xiaobing Gan
    • 1
  • Yanhong Gu
    • 2
  • George L. Vairaktarakis
    • 3
  • Xiaoqiang Cai
    • 4
  • Quanle Chen
    • 4
  1. 1.Department of Information and System Management, Shenzhen University, Shenzhen 518060China
  2. 2.Department of Applied Mathematics, Shenzhen University, Shenzhen 518060China
  3. 3.Department of Operations, Case Western Reserve University, Cleveland, OH 44106-7235USA
  4. 4.Department of Systems Engineering and Engineering Management, The Chinese University of Hong Kong, Shatin, N.T., HK 

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