Skip to main content
SpringerLink
Log in

Complexity of shop-scheduling problems with fixed number of jobs: a survey

  • Original Article
  • Published:
Mathematical Methods of Operations Research Aims and scope Submit manuscript

Abstract

The paper surveys the complexity results for job shop, flow shop, open shop and mixed shop scheduling problems when the number n of jobs is fixed while the number r of operations per job is not restricted. In such cases, the asymptotical complexity of scheduling algorithms depends on the number m of machines for a flow shop and an open shop problem, and on the numbers m and r for a job shop problem. It is shown that almost all shop-scheduling problems with two jobs can be solved in polynomial time for any regular criterion, while those with three jobs are NP-hard. The only exceptions are the two-job, m-machine mixed shop problem without operation preemptions (which is NP-hard for any non-trivial regular criterion) and the n-job, m-machine open shop problem with allowed operation preemptions (which is polynomially solvable for minimizing makespan).

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Akers SB (1956) A graphical approach to production scheduling problems. Oper Res 4:244–245

    Google Scholar 

  • Akers SB, Friedman J (1956) A non-numerical approach to production scheduling problems. Oper Res 3:429–442

    Google Scholar 

  • Aho AV, Hopcroft JE, Ullman JD (1974) The design and analysis of computer algorithms. Addison-Wesley, Reading

    MATH  Google Scholar 

  • Brucker P (1988) An efficient algorithm for the job-shop problem with two jobs. Computing 40:353–359

    Article  MATH  MathSciNet  Google Scholar 

  • Brucker P (1994) A polynomial algorithm for the two machine job-shop scheduling problem with a fixed number of jobs. Oper Res Spekt 16:5–7

    Article  MATH  MathSciNet  Google Scholar 

  • Brucker P, Jurisch B, Meyer W (1989) Geometric methods for solving the job-shop scheduling problem. Preprint Heft 127, Fachbereich Mathematik/Informatik, Universität Osnabrück

  • Brucker P, Kravchenko SA, Sotskov YuN (1994) On the complexity of two machine job-shop scheduling with regular objective functions. Oper Res Spect 16:5–7

    Article  MATH  Google Scholar 

  • Brucker P, Kravchenko SA, Sotskov YuN (1999) Preemptive job-shop scheduling problems with a fixed number of jobs. Math Methods Oper Res 49:41–76

    MATH  MathSciNet  Google Scholar 

  • Brucker P, Schlie R (1990) Job-shop scheduling with multipurpose machines. Computing, 45:369–375

    Article  MATH  MathSciNet  Google Scholar 

  • Gonzalez T, Sahni A (1976) Open shop scheduling to minimize finish time. J Associat Comput Mach 23:665–679

    MATH  MathSciNet  Google Scholar 

  • Hardgrave WH, Nemhauser GL (1963) A geometric model and graphical algorithm for a sequencing problem. Oper Res 11:889–900

    MATH  Google Scholar 

  • Johnson SM (1954) Optimal two- and three-stage production schedules with setup times included. Naval Res Logist Quart 1:61–68

    Article  Google Scholar 

  • Kravchenko SA, Sotskov YuN (1996) Optimal makespan schedule for three jobs on two machines. Math Methods Oper Res 43:233–238

    Article  MATH  MathSciNet  Google Scholar 

  • Masuda T, Ishii H, Nishida T (1985) The mixed shop scheduling problem. Discr Appl Math 11:175–186

    Article  MATH  MathSciNet  Google Scholar 

  • Servach VV (1983) On the Akers–Friedman problem. Upravljajemye Syst Novosibirsk 23:70–81 (in Russian)

    Google Scholar 

  • Shakhlevich NV, Strusevich VA (1990) Scheduling two jobs in a multi-machine open shop to minimize an arbitrary regular penalty function. Report 9125/A, Erasmus University Rotterdam, The Netherlands

  • Shakhlevich NV, Sotskov YuN (1994) Scheduling two jobs with fixed and nonfixed routines. Computing 52:17–30

    Article  MATH  MathSciNet  Google Scholar 

  • Shakhlevich NV, Sotskov YuN, Werner F (1999) Shop-scheduling problems with fixed and non-fixed machine orders of jobs. Ann Oper Res 92:281–304

    Article  MATH  MathSciNet  Google Scholar 

  • Shakhlevich NV, Sotskov YuN, Werner F (2000) Complexity of mixed shop scheduling problems: A survey. Euro J Oper Res 120:343–351

    Article  MATH  MathSciNet  Google Scholar 

  • Sotskov YuN (1985) Optimal scheduling of two jobs with a regular criterion. Institute of Engineering Cybernetics of the Academy of Sciences of BSSR, Minsk, pp. 86–95 (in Russian)

  • Sotskov YuN (1989) Complexity of scheduling with fixed number of jobs. Dokl Akad Nauk BSSR. 33:488–491 (in Russian)

    MATH  MathSciNet  Google Scholar 

  • Sotskov YuN (1990) Complexity of optimal scheduling of three jobs Kibernetika N 5:74–78 (in Russian)

  • Sotskov YuN (1991) The complexity of shop-scheduling problems with two or three jobs. Euro J Oper Res 53:326–336

    Article  MATH  Google Scholar 

  • Sotskov YuN, Shakhlevich NV (1990) NP-hardness of optimal scheduling three jobs. Vestsi Akad Navuk BSSR Ser Fiz-Mat Navuk 4:96–101 (in Russian)

    MathSciNet  Google Scholar 

  • Sotskov YuN, Shakhlevich NV (1995) NP-hardness of shop-scheduling problems with three jobs. Discr Appl Mathe 59:237–266

    Article  MATH  MathSciNet  Google Scholar 

  • Strusevich VA (1986) On the possibility of constructing optimal makespan schedules for multi-stage systems with nonfixed routes. Vestsi Akad Navuk BSSR Ser Fiz-Mat Navuk 6:43–48 (in Russian)

    MathSciNet  Google Scholar 

  • Szwarc W (1960) Solution of the Akers-Friedman scheduling problem. Oper Res 8:782–788

    Article  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Osnabrück, Fachbereich Mathematik, Osnabrück, 49069, Germany

    Peter Brucker

  2. United Institute of Informatics Problems of the National Academy of Sciences of Belarus, Surganov St. 6, Minsk, 220012, Belarus

    Yu N. Sotskov

  3. Otto-von-Guericke-Universität, Fakultät für Mathematik, PSF 4120, Magdeburg, 39016, Germany

    Frank Werner

Authors
  1. Peter Brucker
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yu N. Sotskov
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Frank Werner
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Peter Brucker.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Brucker, P., Sotskov, Y.N. & Werner, F. Complexity of shop-scheduling problems with fixed number of jobs: a survey. Math Meth Oper Res 65, 461–481 (2007). https://doi.org/10.1007/s00186-006-0127-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00186-006-0127-8

Keywords

Search

Navigation