Skip to main content
SpringerLink
Log in

Scheduling with time-dependent discrepancy times

  • Published:
Journal of Scheduling Aims and scope Submit manuscript

Abstract

In time-dependent scheduling, various processing time functions are studied, yet absolute value functions have surprisingly been omitted from the discussion. Such a processing time function increases linearly with a job’s discrepancy from its ideal midtime. The objective is to find a schedule that minimizes the makespan, introducing the discrepancy time minimization problem. This single-machine scheduling problem with time-dependent processing times is motivated by optimization of walking times on a car assembly line. Its decision version is NP hard, as we show by reduction of the even–odd partition problem. For the variant with known start time, we develop several heuristics. Further insights form lower bounds and dominance rules for a branch-and-bound search. Numerical experiments show the performance of our algorithms on problem instances of up to 60 jobs. For the variant with common ideal midtime and flexible start time, we present a polynomial-time algorithm.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13

Similar content being viewed by others

References

  • Ahn, B. H., & Shin, J. Y. (1991). Vehicle-routeing with time windows and time-varying congestion. The Journal of the Operational Research Society, 42(5), 393–400.

    Article  Google Scholar 

  • Alidaee, B., & Womer, N. K. (1999). Scheduling with time dependent processing times: Review and extensions. The Journal of the Operational Research Society, 50(7), 711–720.

    Article  Google Scholar 

  • Andrés, C., Miralles, C., & Pastor, R. (2008). Balancing and scheduling tasks in assembly lines with sequence-dependent setup times. European Journal of Operational Research, 187(3), 1212–1223.

    Article  Google Scholar 

  • Bautista, J., & Pereira, J. (2007). Ant algorithms for a time and space constrained assembly line balancing problem. European Journal of Operational Research, 177(3), 2016–2032.

    Article  Google Scholar 

  • Biskup, D. (2008). A state-of-the-art review on scheduling with learning effects. European Journal of Operational Research, 188(2), 315–329.

    Article  Google Scholar 

  • Boysen, N., Fliedner, M., & Scholl, A. (2007). A classification of assembly line balancing problems. European Journal of Operational Research, 183(2), 674–693.

    Article  Google Scholar 

  • Boysen, N., Emde, S., Hoeck, M., & Kauderer, M. (2015). Part logistics in the automotive industry: Decision problems, literature review and research agenda. European Journal of Operational Research, 242(1), 107–120.

    Article  Google Scholar 

  • Browne, S., & Yechiali, U. (1990). Scheduling deteriorating jobs on a single processor. Operations Research, 38(3), 495–498.

    Article  Google Scholar 

  • Bukchin, Y., & Meller, R. D. (2005). A space allocation algorithm for assembly line components. IIE Transactions, 37(1), 51–61.

    Article  Google Scholar 

  • Bülbül, K., Kaminsky, P., & Yano, C. (2007). Preemption in single machine earliness/tardiness scheduling. Journal of Scheduling, 10(4–5), 271–292.

    Article  Google Scholar 

  • Cai, J. Y., Cai, P., & Zhu, Y. (1998). On a scheduling problem of time deteriorating jobs. Journal of Complexity, 14(2), 190–209.

    Article  Google Scholar 

  • Cheng, T. C. E., Ding, Q., Kovalyov, M. Y., Bachman, A., & Janiak, A. (2003). Scheduling jobs with piecewise linear decreasing processing times. Naval Research Logistics, 50(6), 531–554.

    Article  Google Scholar 

  • Cheng, T. C. E., Ding, Q., & Lin, B. M. T. (2004). A concise survey of scheduling with time-dependent processing times. European Journal of Operational Research, 152(1), 1–13.

    Article  Google Scholar 

  • Chica, M., Cordón, Ó., Damas, S., & Bautista, J. (2012). Multiobjective memetic algorithms for time and space assembly line balancing. Engineering Applications of Artificial Intelligence, 25(2), 254–273.

    Article  Google Scholar 

  • Du, J., & Leung, J. Y. T. (1990). Minimizing total tardiness on one machine is NP-hard. Mathematics of Operations Research, 15(3), 483–495.

    Article  Google Scholar 

  • Garey, M. R., Tarjan, R. E., & Wilfong, G. T. (1988). One-processor scheduling with symmetric earliness and tardiness penalties. Mathematics of Operations Research, 13(2), 330–348.

    Article  Google Scholar 

  • Gawiejnowicz, S. (2008). Time-dependent scheduling. Monographs in theoretical computer science. Berlin: Springer.

    Google Scholar 

  • Hall, N. G., Kubiak, W., & Sethi, S. P. (1991). Earliness-tardiness scheduling problems, II: Deviation of completion times about a restrictive common due date. Operations Research, 39(5), 847–856.

    Article  Google Scholar 

  • Held, M., & Karp, R. M. (1962). A dynamic programming approach to sequencing problems. Journal of the Society for Industrial and Applied Mathematics, 10(1), 196–210.

    Article  Google Scholar 

  • Janiak, A., Krysiak, T., & Trela, R. (2011). Scheduling problems with learning and ageing effects: A survey. Decision Making in Manufacturing and Services, 5(1–2), 19–36.

    Google Scholar 

  • Ji, M., & Cheng, T. C. E. (2007). An FPTAS for scheduling jobs with piecewise linear decreasing processing times to minimize makespan. Information Processing Letters, 102(2–3), 41–47.

    Article  Google Scholar 

  • Kacem, I. (2010). Fully polynomial time approximation scheme for the total weighted tardiness minimization with a common due date. Discrete Applied Mathematics, 158(9), 1035–1040.

    Article  Google Scholar 

  • Kanet, J. J. (1981). Minimizing the average deviation of job completion times about a common due date. Naval Research Logistics Quarterly, 28(4), 643–651.

    Article  Google Scholar 

  • Keha, A. B., Khowala, K., & Fowler, J. W. (2009). Mixed integer programming formulations for single machine scheduling problems. Computers and Industrial Engineering, 56(1), 357–367.

    Article  Google Scholar 

  • Kellerer, H., & Strusevich, V. A. (2006). A fully polynomial approximation scheme for the single machine weighted total tardiness problem with a common due date. Theoretical Computer Science, 369(1–3), 230–238.

    Article  Google Scholar 

  • Klampfl, E., Gusikhin, O., & Rossi, G. (2006). Optimization of workcell layouts in a mixed-model assembly line environment. International Journal of Flexible Manufacturing Systems, 17(4), 277–299.

    Google Scholar 

  • Kononov, A. V. (1997). On schedules of a single machine jobs with processing times nonlinear in time. In A. D. Korshunov (Ed.), Operations research and discrete analysis, vol. 391 (pp. 109–122). Dordrecht: Springer.

    Chapter  Google Scholar 

  • Kovalyov, M. Y., & Kubiak, W. (1998). A fully polynomial approximation scheme for minimizing makespan of deteriorating jobs. Journal of Heuristics, 3(4), 287–297.

    Article  Google Scholar 

  • Kubiak, W., & van de Velde, S. L. (1998). Scheduling deteriorating jobs to minimize makespan. Naval Research Logistics, 45(5), 511–523.

    Article  Google Scholar 

  • Kunnathur, A. S., & Gupta, S. K. (1990). Minimizing the makespan with late start penalties added to processing times in a single facility scheduling problem. European Journal of Operational Research, 47(1), 56–64.

    Article  Google Scholar 

  • Lawler, E. L. (1977). A “pseudopolynomial” algorithm for sequencing jobs to minimize total tardiness. Annals of Discrete Mathematics, 1, 331–342.

    Article  Google Scholar 

  • Lawler, E. L., & Moore, J. M. (1969). A functional equation and its application to resource allocation and sequencing problems. Management Science, 16(1), 77–84.

    Article  Google Scholar 

  • Lee, C. Y., & Vairaktarakis, G. L. (1993). Complexity of single machine hierarchical scheduling: A survey. In P. M. Pardalos (Ed.), Complexity in numerical optimization (pp. 269–298). River Edge, NJ: World Scientific.

    Chapter  Google Scholar 

  • Malandraki, C., & Daskin, M. S. (1992). Time dependent vehicle routing problems: Formulations, properties and heuristic algorithms. Transportation Science, 26(3), 185–200.

    Article  Google Scholar 

  • Martino, L., & Pastor, R. (2010). Heuristic procedures for solving the general assembly line balancing problem with setups. International Journal of Production Research, 48(6), 1787–1804.

    Article  Google Scholar 

  • Picard, J. C., & Queyranne, M. (1978). The time-dependent traveling salesman problem and its application to the tardiness problem in one-machine scheduling. Operations Research, 26(1), 86–110.

    Article  Google Scholar 

  • Scholl, A., Boysen, N., & Fliedner, M. (2013). The assembly line balancing and scheduling problem with sequence-dependent setup times: Problem extension, model formulation and efficient heuristics. OR Spectrum, 35(1), 291–320.

    Article  Google Scholar 

  • Sourd, F. (2009). New exact algorithms for one-machine earliness-tardiness scheduling. INFORMS Journal on Computing, 21(1), 167–175.

    Article  Google Scholar 

  • Sourd, F., & Kedad-Sidhoum, S. (2008). A faster branch-and-bound algorithm for the earliness-tardiness scheduling problem. Journal of Scheduling, 11(1), 49–58.

    Article  Google Scholar 

  • Tanaka, S., Fujikuma, S., & Araki, M. (2009). An exact algorithm for single-machine scheduling without machine idle time. Journal of Scheduling, 12(6), 575–593.

    Article  Google Scholar 

  • Wan, L., & Yuan, J. (2013). Single-machine scheduling to minimize the total earliness and tardiness is strongly NP-hard. Operations Research Letters, 41(4), 363–365.

    Article  Google Scholar 

  • Wu, C. C., Shiau, Y. R., Lee, L. H., & Lee, W. C. (2009). Scheduling deteriorating jobs to minimize the makespan on a single machine. The International Journal of Advanced Manufacturing Technology, 44(11–12), 1230–1236.

    Article  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Augsburg, Augsburg, Germany

    Florian Jaehn & Helmut A. Sedding

Authors
  1. Florian Jaehn
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Helmut A. Sedding
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Helmut A. Sedding.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jaehn, F., Sedding, H.A. Scheduling with time-dependent discrepancy times. J Sched 19, 737–757 (2016). https://doi.org/10.1007/s10951-016-0472-2

Download citation

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10951-016-0472-2

Keywords

Search

Navigation