, Volume 10, Issue 4, pp 347–360 | Cite as

Soft due window assignment and scheduling of unit-time jobs on parallel machines

  • Adam JaniakEmail author
  • Wladyslaw Janiak
  • Mikhail Y. Kovalyov
  • Frank Werner
Open Access
Research Paper


We study problems of scheduling n unit-time jobs on m identical parallel machines, in which a common due window has to be assigned to all jobs. If a job is completed within the due window, then no scheduling cost incurs. Otherwise, a job-dependent earliness or tardiness cost incurs. The job completion times, the due window location and the size are integer valued decision variables. The objective is to find a job schedule as well as the location and the size of the due window such that a weighted sum or maximum of costs associated with job earliness, job tardiness and due window location and size is minimized. We establish properties of optimal solutions of these min-sum and min-max problems and reduce them to min-sum (traditional) or min-max (bottleneck) assignment problems solvable in O(n 5/m 2) and O(n 4.5log0.5 n/m 2) time, respectively. More efficient solution procedures are given for the case in which the due window size cost does not exceed the due window start time cost, the single machine case, the case of proportional earliness and tardiness costs and the case of equal earliness and tardiness costs.


Scheduling Parallel machines Earliness-tardiness Due window Unit-time jobs 

MSC classification



Open Access

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  1. Anger F, Lee CY, Martin-Vega L (1986) Single machine scheduling with tight windows. Department of Industrial and System Engineering, University of Florida, Gainsville, Res. Rep 86–16Google Scholar
  2. Biskup D, Feldmann M (2005) On scheduling around large restrictive common due windows. Eur J Oper Res 162: 740–761CrossRefGoogle Scholar
  3. Bodin L, Golden B, Assad A, Ball M (1983) Routing and scheduling of vehicles and crews: the state of the art. Comput Oper Res 10: 62–212Google Scholar
  4. Burkard R, Cela E (1999) Linear assignment problems and extensions. In: Handbook of combinatorial optimization, Supplement vol A, Kluwer, Dordrecht, pp 75–149Google Scholar
  5. Cheng T (1988) Optimal common due-date with limited completion times deviation. Comput Oper Res 15: 91–96CrossRefGoogle Scholar
  6. Chu C, Gordon V, Proth JM (2002) Due date assignment and scheduling: SLK, TWK and other due date assignment models. Prod Plan Control 13: 117–132CrossRefGoogle Scholar
  7. Gordon V, Proth JM, Chu C (2002) A survey of the state-of-the-art of common due date assignment and scheduling research. Eur J Oper Res 139: 1–25CrossRefGoogle Scholar
  8. Hashimoto H, Yagiura M, Imahori S, Ibaraki T (2010) Recent progress of local search in handling the time window constraints of the vehicle routing problem. 4OR 8: 221–238CrossRefGoogle Scholar
  9. Janiak A, Winczaszek M (2003) An optimal algorithm for a single processor scheduling problem with a common due window. In: Proceedings of 9th IEEE international conference on methods and models in automation and robotics, Miedzyzdroje, Poland, 25–28 August 2003, pp 1213–1216Google Scholar
  10. Janiak A, Kovalyov M, Marek M (2007) Soft due window assignment and scheduling on parallel machines. IEEE Trans Syst Man Cybern Part A 37: 614–620CrossRefGoogle Scholar
  11. Kacem I (2010) Fully polynomial time approximation scheme for the total weighted tardinessminimization with a common due date. Discret Appl Math 158: 1035–1040CrossRefGoogle Scholar
  12. Kanet J, Sridharan V (2000) Scheduling with inserted idle time: problem taxonomy and literature review. Oper Res 48: 99–110CrossRefGoogle Scholar
  13. Karakostas G, Kolliopoulos S, Wang J (2009) An fptas for the minimum total weightedtardiness problem with a fixed number of distinct due dates. In: Lecture notes in computer science, vol 5609. Springer, Berlin, pp 238–248. doi: 10.1007/978-3-642-02882-3_24
  14. Kellerer H, Strusevich V (2006) A fully polynomial approximation scheme for the singlemachine weighted total tardiness problem with a common due date. Theor Comput Sci 369: 230–238CrossRefGoogle Scholar
  15. Koulamas C (1997) Maximizing the weighted number of on-time jobs in a single-machine scheduling with time windows. Math Comput Model 25: 57–62CrossRefGoogle Scholar
  16. Kramer FJ, Lee CY (1994) Due window scheduling for parallel machines. Math Comput Model 20: 69–89CrossRefGoogle Scholar
  17. Kritikos M, Ioannou G (2010) The balanced cargo vehicle routing problem with time windows. Int J Prod Econ 123: 42–51CrossRefGoogle Scholar
  18. Lee CY (1991) Earliness-tardiness scheduling problems with constant size of due window. Department of Industrial and System Engineering, University of Florida, Gainsville, Res. Rep. 91–17Google Scholar
  19. Li CL, Mosheiov G, Yovel U (2008) An efficient algorithm for minimizing earliness, tardiness, and due-date costs for equal-sized jobs. Comput Oper Res 35: 3612–3619CrossRefGoogle Scholar
  20. Li X, Tian P, Leung S (2010) Vehicle routing problems with time windows and stochastic travel and service times: models and algorithm. Int J Prod Econ 125: 137–145CrossRefGoogle Scholar
  21. Liberatore F, Righini G, Salani M (2011) A column generation algorithm for the vehicle routing problem with soft time windows. 4OR 9: 49–82CrossRefGoogle Scholar
  22. Mosheiov G (2001) A due-window determination in minmax scheduling problems. INFOR 39: 107–123Google Scholar
  23. Mosheiov G, Oron D (2004) Due-window assignment with unit processing-time jobs. Naval Res Logist 51: 1005–1017CrossRefGoogle Scholar
  24. Mosheiov G, Sarig A (2010) Scheduling with a common due-window: polynomially solvable cases. Inf Sci 180: 1492–1505CrossRefGoogle Scholar
  25. Schrage L (1981) Formulation and structure of more complex/realistic routing and scheduling problems. Networks 11: 229–232CrossRefGoogle Scholar
  26. Solomon M, Desrosiers J (1988) Time window constrained routing and scheduling problems. Transp Sci 22: 1–13CrossRefGoogle Scholar
  27. Torn A, Zilinskas A (1989) Global optimization. In: (eds) Lecture notes in computer science, vol 350. Springer, HeidelbergGoogle Scholar
  28. Tuong NH, Soukhal A, Billaut JC (2010) A new dynamic programming formulation for scheduling independent tasks with common due date on parallel machines. Eur J Oper Res 202: 646–653CrossRefGoogle Scholar
  29. Wang D, Fang S, Hodgson T (1998) A fuzzy due-date bargainer for the make-to-order manufacturing systems. IEEE Trans Syst Man Cybern Part C 28: 492–497CrossRefGoogle Scholar
  30. Wang D, Fang S, Nuttle H (1999) Soft computing for multicustomer due-date bargaining. IEEE Trans Syst Man Cybern Part C 29: 566–575CrossRefGoogle Scholar
  31. Wen C, Eksioglu S, Greenwood A, Zhang S (2010) Crane scheduling in a shipbuilding environment. Int J Prod Econ 124: 40–50CrossRefGoogle Scholar
  32. Yen B, Wan G (1999) Single machine bicriteria scheduling: a survey. Int J Ind Eng Theory 10: 222–231Google Scholar
  33. Yeung W, Oguz C, Cheng T (2004) Two-stage flowshop earliness and tardiness machine scheduling involving a common due window. Int J Prod Econ 90: 421–434CrossRefGoogle Scholar

Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Adam Janiak
    • 1
    Email author
  • Wladyslaw Janiak
    • 2
  • Mikhail Y. Kovalyov
    • 3
  • Frank Werner
    • 4
  1. 1.Institute of Computer Engineering, Control and RoboticsWrocław University of TechnologyWrocławPoland
  2. 2.Faculty of Computer Science and ManagementWrocław University of TechnologyWrocławPoland
  3. 3.United Institute of Informatics ProblemsNational Academy of Sciences of BelarusMinskBelarus
  4. 4.Faculty of MathematicsOtto-von-Guericke-University MagdeburgMagdeburgGermany

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