Journal of Combinatorial Optimization

, Volume 12, Issue 4, pp 387–394

A note on the complexity of the problem of two-agent scheduling on a single machine

  • C. T. Ng
  • T. C. E. Cheng
  • J. J. Yuan


We consider a two-agent scheduling problem on a single machine, where the objective is to minimize the total completion time of the first agent with the restriction that the number of tardy jobs of the second agent cannot exceed a given number. It is reported in the literature that the complexity of this problem is still open. We show in this paper that this problem is NP-hard under high multiplicity encoding and can be solved in pseudo-polynomial time under binary encoding. When the first agent's objective is to minimize the total weighted completion time, we show that the problem is strongly NP-hard even when the number of tardy jobs of the second agent is restricted to be zero.


Production/scheduling Multi-agent deterministic sequencing Games/group decisions Cooperative sequencing 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Agnetis A, Mirchandani PB, Pacciarelli D, Pacifici A (2004) Scheduling problems with two competing agents. Oper Res 52:229–242MathSciNetCrossRefGoogle Scholar
  2. Clifford JJ, Posner ME (2001) Parallel scheduling with high multiplicity. Mathematical Programming, Ser. A, 89:359–383MATHMathSciNetCrossRefGoogle Scholar
  3. Garey MR, Johnson DS (1979) Computers and intractability: a guide to the theory of NP-completeness. Freeman, San Francisco, CAMATHGoogle Scholar
  4. Hochbaum DS, Shamir R (1991) Strongly polynomial algorithms for the high multiplicity scheduling problem. Oper Res 39:648–653MATHCrossRefGoogle Scholar
  5. Lawler EL (1977) A pseudopolynomial algorithm for sequencing jobs to minimize total tardiness. Annals Discr Math, 1:331–342MATHMathSciNetCrossRefGoogle Scholar
  6. Moore JM (1968) An n job, one machine sequencing algorithm for minimizing the number of late jobs. Manag Sci 15:102–109MATHGoogle Scholar
  7. Rinnooy Kan AHG (1976) Machine scheduling problems. Martinus Nijhoff, The HagueGoogle Scholar

Copyright information

© Springer Science + Business Media, LCC 2006

Authors and Affiliations

  • C. T. Ng
    • 1
  • T. C. E. Cheng
    • 1
  • J. J. Yuan
    • 2
  1. 1.Department of LogisticsThe Hong Kong Polytechnic University, Hung HomKowloonPeople”s Republic of China
  2. 2.Department of MathematicsZhengzhou UniversityZhengzhouPeople”s Republic of China

Personalised recommendations