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Journal of Scheduling

, Volume 18, Issue 2, pp 131–145 | Cite as

Scheduling multi-colour print jobs with sequence-dependent setup times

  • A. P. Burger
  • C. G. Jacobs
  • J. H. van VuurenEmail author
  • S. E. Visagie
Article

Abstract

In this paper, a scheduling problem is considered which arises in the packaging industry. Plastic and foil wrappers used for packaging candy bars, crisps and other snacks typically require overlay printing with multiple colours. Printing machines used for this purpose usually accommodate a small number of colours which are loaded into a magazine simultaneously. If two consecutively scheduled print jobs require significantly different colour overlays, then substantial down times are incurred during the transition from the former magazine colour configuration to the latter, because ink cartridges corresponding to colours not required for the latter job have to be cleaned after completion of the former job. The durations of these down times are therefore sequence dependent (the washing and refilling time is a function of the number of colours in which two consecutive printing jobs differ). It is consequently desirable to schedule print jobs so that the accumulated down times associated with all magazine colour transitions are as short as possible for each printing machine. We show that an instance of this scheduling problem can be modelled as the well-known tool switching problem, which is tractable for small instances only. The problem can, however, be solved rather effectively in heuristic fashion by decomposing it into two subproblems: a job grouping problem (which can be modelled as a unicost set covering problem) and a group sequencing problem (which is a generalisation of the celebrated travelling salesman problem). We solve the colour print scheduling problem both exactly and heuristically for small, randomly generated test problem instances, studying the trade-off between the time efficiency and solution quality of the two approaches. Finally, we apply both solution approaches to real problem data obtained from a printing company in the South African Western Cape as a special case study.

Keywords

Job grouping Job scheduling Job sequencing Tool switching 

Mathematics Subject Classification

90B35 05A18 05C15 

Notes

Acknowledgments

Work towards this paper was supported financially by the South African National Research Foundation under Grant GUN 2072999.

References

  1. Allahverdi, A., Gupta, J. N. D., & Aldowaisan, T. (1999). A review of scheduling research involving setup considerations. Omega International Journal of Management Science, 27, 219–239.CrossRefGoogle Scholar
  2. Allahverdi, A., Ng, C. T., Cheng, T. C. E., & Kovalyov, M. Y. (2008). A survey of scheduling problems with setup times or costs. European Journal of Operational Research, 187, 985–1032.CrossRefGoogle Scholar
  3. Avci, S., & Akturk, M. S. (1996). Tool magazine arrangement and operations sequencing on CNC machines. Computers and Operations Research, 23, 1069–1081.CrossRefGoogle Scholar
  4. Balakrishnan, N., & Chakravarty, A. K. (2001). Opportunistic retooling of a flexible machine subject to failure. Naval Research Logistics, 48, 79–97.CrossRefGoogle Scholar
  5. Balas, E. (1983). A class of location, distribution and scheduling problems: Modeling and solution methods. In R. Gray & L. Yuanzhang (Eds.), Proceedings of the Chinese–US Symposium on systems analysis. New York: Wiley.Google Scholar
  6. Ball, M. O., Magnanti, T. L., Monma, C. L., & Nemhauser, G. L. (1995). Network models. In G. L. Nemhauser & A. H. G. Rinnoy Kan (Eds.), Handbooks in operations research and management science (Vol. 7). Amsterdam: Elsevier.Google Scholar
  7. Bard, J. F. (1988). A heuristic for minimizing the number of tool switches on a flexible machine. IIE Transactions, 20, 382–391.CrossRefGoogle Scholar
  8. Belady, L. A. (1966). A study of replacement algorithms for virtual storage computers. IBM Systems Journal, 5, 78–101.CrossRefGoogle Scholar
  9. Blazewicz, J., & Finke, G. (1994). Scheduling with resource management in manufacturing systems. European Journal of Operational Research, 76, 1–14.CrossRefGoogle Scholar
  10. Ceria, S., Nobili, P., & Sassano, A. (1997). Set covering problem. In M. Dell’Amico, F. Maffioli, & S. Martello (Eds.), Annotated bibliographies in combinatorial optimization (pp. 415–428). Chichester: Wiley.Google Scholar
  11. Crama, Y., Kolen, A. W. J., Oerlemans, A. G., & Speksma, F. C. R. (1994). Minimizing the number of tool switches on a flexible machine. International Journal of Flexible Manufacturing Systems, 6, 33–54.CrossRefGoogle Scholar
  12. Crama, Y., & Oerlemans, A. G. (1994). A column generation apporach to job grouping for flexible manufacturing systems. Europena Journal of Operational Research, 78, 58–80.CrossRefGoogle Scholar
  13. Crama, Y. (1997). Combinatorial optimization models for production scheduling in automated manufacturing systems. European Journal of Operational Research, 99, 136–153.CrossRefGoogle Scholar
  14. Crama, Y., van de Klundert, J., & Spieksma, F. C. R. (2002). Production planning problems in printed circuit board assembly. Discrete Applied Mathematics, 123, 339–361.CrossRefGoogle Scholar
  15. Dantzig, G. B., Fulkerson, D. R., & Johnson, S. M. (1954). Solution of a large-scale traveling salesman problem. Operations Research, 2, 393–410.Google Scholar
  16. Follonier, J.-P. (1994). Minimization of the number of tool switches on a flexible machine. Belgian Journal of Operations Research, Statistics and Computer Science, 34, 55–72.Google Scholar
  17. Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. New York: Freeman.Google Scholar
  18. Ghiani, G., Grieco, A., & Guerriero, E. (2010). Solving the job sequencing and tool switching problem as a nonlinear least cost hamiltonian cycle problem. Networks, 55, 379–385.CrossRefGoogle Scholar
  19. Gray, A. E., Seidmann, A., & Stecke, K. E. (1993). A synthesis of decision models for tool management in automated manufacturing. Management Science, 39, 549–567.CrossRefGoogle Scholar
  20. Gutin, G., & Punnen, A. P. (2002). The travelling salesman problem and its variations. Combinatorial optimization series (Vol. 12). New York: Springer.Google Scholar
  21. Hertz, A., & Widmer, M. (1993). An improved tabu search approach for solving the job shop scheduling problem with tooling constraints. Discrete Applied Mathematics, 65, 319–345.CrossRefGoogle Scholar
  22. Hertz, A., Laporte, G., Mittaz, M., & Stecke, K. E. (1998). Heuristics for minimizing tool switches when scheduling part types on a flexible machine. IIE Transactions, 30, 689–694.Google Scholar
  23. International Business Machines. (2013). CPLEX Optimizer, October 17th 2013. http://www-01.ibm.com/software/commerce/optimization/cplex-optimizer/
  24. Kiran, A. S., & Krason, R. J. (1988). Automated tooling in a flexible manufacturing system. Industrial Engineering, 20, 52–57.Google Scholar
  25. Knuutila, T., Hirvikorpi, M., Johnsson, M., & Nevalainen, O. (2002). Grouping PCB assembly jobs with typed component feeder units. Technical Report 460. Finland: Turku Centre for Computer Science.Google Scholar
  26. Laporte, G., Salazar-González, J. J., & Semet, F. (2004). Exact algorithms for the job sequencing and tool switching problem. IIE Transactions, 36, 37–45.CrossRefGoogle Scholar
  27. McGeoch, L. A., & Sleator, D. D. (1991). A strongly competitive randomized paging algorithm. Algorithmica, 6, 816–825.Google Scholar
  28. Nemhauser, G. L., Trotter, L. E., & Nauss, R. M. (1972). Set partitioning and chain decomposition. Management Science, 20(22), 1413–1423.Google Scholar
  29. Oerlemans, A. G. (1992). Production planning for flexible manufacturing systems. PhD Dissertation, University of Limburg, Maastricht.Google Scholar
  30. Privault, P., & Finke, G. (2000).k-Server problems with bulk requests: An application to tool switching in manufacturing. Annals of Operations Research, 96, 255–269.Google Scholar
  31. Rosen, K. H. (2000). Handbook of discrete and combinatorial mathematics. Boca Raton, FL: CRC Press.Google Scholar
  32. Salonen, K., Smed, J., Johnsson, M., & Nevalainen, O. (2006). Grouping and sequencing PCB assembly jobs with minimum feeder setups. Robotics and Computer-integrated Manufacturing, 22, 297–305.CrossRefGoogle Scholar
  33. Sodhi, M. S., Askin, R. G., & Sen, S. (1994). Multiperiod tool and production assignment in flexible manufacturing systems. International Journal of Production Research, 32, 1281–1294.CrossRefGoogle Scholar
  34. Tang, C. S., & Denardo, E. V. (1988). Models arising from a flexible manufacturing machine Part I: Minimization of the number of tool switches. Operations Research, 36, 767–777.CrossRefGoogle Scholar
  35. van der Veen, J. A. A., Woeginger, G. J., & Zhang, S. (1998). Sequencing jobs that require common resources on a single machine: A solvable case of the TSP. Mathematical Programming, 82, 235–254.Google Scholar
  36. Weisstein, E. W. (2003). CRC concise encyclopedia of mathematics (2nd ed.). Boca Raton, FL: Chapman & Hall/CRC.Google Scholar
  37. Yildirim, M. B., Duman, E., Krishna, K., & Senniappan, K. (2007). Parallel machine scheduling with load balancing and sequence dependent setup times. International Journal of Operations Research, 4(1), 42–49.Google Scholar

Copyright information

© Springer Science+Business Media New York 2014

Authors and Affiliations

  • A. P. Burger
    • 1
  • C. G. Jacobs
    • 1
  • J. H. van Vuuren
    • 2
    Email author
  • S. E. Visagie
    • 1
  1. 1.Department of LogisticsStellenbosch UniversityMatielandSouth Africa
  2. 2.Department of Industrial EngineeringStellenbosch UniversityMatielandSouth Africa

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