Abstract
We propose two modeling approaches to solve single machine preemptive scheduling problems with tardiness related objectives. Employing the conventional time-indexed formulation, we first build a model that explicitly identifies completion times of jobs with varying release times, due dates, and processing times. The second model adopts the aggregate planning view and eliminates binary constraints. Under this approach, each job is seen as a unit demand while its due date is mapped to a period where this unit is demanded. With this mapping, the periodic job allocation decisions are transformed into periodic production decisions that are measured by fraction of demand. Consequently, instead of explicit representation of the job completion times, this model tracks the amounts of production completed and backlogged via inventory and shortage variables and conservation of units constraints. We establish that the latter model provides tighter bounds and demonstrate that it provides a more efficient platform for optimization via computational analysis employing four commonly used tardiness related criteria and a case study from a real life application. Numerical computations reveal that aggregate planning view becomes more dominant in terms of computational performance as the problem size grows.
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References
Adamu, M. O., & Adewumi, A. O. (2014). A survey of single machine scheduling to minimize wieighted number of tardy jobs. Journal of Industrial and Management Optimization, 10(1), 219–241.
Baptiste, P. (1999). An O\((n^4)\) algorithm for preemptive scheduling of a single machine to minimize the number of late jobs. Operations Research Letters, 24, 175–180.
Batsyn, M., Goldengorin, B., Sukov, P., & Pardalos, P. M. (2013). Models, Algorithms, and technologies for network analysis. In Springer proceedings in mathematics & statistics, chap Lower and upper bounds for the preemptive single machine scheduling problem with equal processing times (vol. 59, pp. 11–27). Springer.
Belouadah, H., Posner, M. E., & Chris, C. N. (1992). Scheduling with release dates on a single machine to minimize total weighted completion time. Discrete Applied Mathematics, 36(3), 213–231.
Berghman, L., & Spieksma, F. C. R. (2015). Valid inequalities for a time-indexed formulation. Operations Research Letters, 43(3), 268–272.
Boland, N., Clement, R., & Waterer, H. (2016). A bucket indexed formulation for nonpreemptive single machine scheduling problems. INFORMS Journal of Computing, 28(1), 14–30.
Bulbul, K., Kaminsky, P., & Yano, C. (2007). Preemption in single machine earliness/tardiness scheduling. Journal of Scheduling, 10, 271–292.
Graham, R. L., Lawler, E. L., Lenstra, J. K., & Kan, A. H. G. (1979). Optimization and approximation in deterministic sequencing and scheduling: A survey. Annals of Discrete Mathematics, 5, 287–326.
Hariri, A., & Potts, C. N. (1983). An algorithm for single machine sequencing with release dates to minimize total weighted completion time. Discrete Applied Mathematics, 5(1), 99–109.
Hendel, Y., Runge, N., & Sourd, F. (2009). The one-machine just-in-time scheduling problem with preemption. Discrete Optimization, 6, 10–22.
Jaramillo, F., & Erkoc, M. (2017). Minimizing total weighted tardiness and overtime costs for single machine preemptive scheduling. Computers & Industrial Engineering, 107, 109–119.
Karp, R. M. (1972). Reducibility among combinatorial problems. In Complexity of computer computations (pp. 85–103). Springer.
Koulamas, C. (2010). The single-machine total tardiness scheduling problem: Review and extensions. European Journal of Operational Research, 202(1), 1–7.
Labetoulle, J., Lawler, E. L., Lenstra, J., & Rinnooy, K. A. (1984). Progress in combinatorial optimization (pp. 245–261). NewYork: Academic Press.
Lawler, E. L. (1990). A dynamic programming algorithm for preemptive scheduling of a single machine to minimize the number of late jobs. Annals of Operations Research, 26(1–4), 125–133.
Lawler, E. M. (1994). Knapsack-like scheduling problems, the Moore-Hodgson algorithm and the ’tower of sets’ property. Mathematical and Computer Modelling, 20(2), 91–106.
Lenstra, J. K., Kan, A., & Brucker, P. (1977). Complexity of machine scheduling problems. Annals of Discrete Mathematics, 1, 343–362.
Leung, J. Y. T., Li, H., Pinedo, M., & Zhang, J. (2007). Minimizing total weighted completion time when scheduling orders in a flexible environment with uniform machines. Information Pr, 103, 119–129.
Li, G. (1997). Single machine earliness and tardiness scheduling. European Journal of Operational Research, 96(3), 546–558.
Liao, C. J., & Cheng, C. C. (2007). A variable neighborhood search for minimizing single machine weighted earliness and tardiness with common due date. Computers & Industrial Engineering, 52(4), 404–413.
Liaw, C. F. (1999). A branch-and-bound algorithm for the single machine earliness and tardiness scheduling problem. Computers & Operations Research, 26(7), 679–693.
McNaughton, R. (1959). Scheduling with deadlines and loss functions. Management Science, 6(1), 1–12.
M’Hallah, R., & Bulfin, R. L. (2007). Minimizing the weighted number of tardy jobs on a single machine with release days. European Journal of Operational Research, 176(2), 727–744.
Pessoa, A., Uchoa, E., de Aragao, M. P., & Rodrigues, R. (2010). Exact algorithm over an arc-time-indexed formulation for parallel machine scheduling problems. Mathematical Programming Comput, 2, 259–290.
Schulz, A. S., & Skutella, M. (2002). The power of \(\alpha \) -points in preemptive single machine scheduling. Journal of Scheduling, 5, 121–133.
Sen, T., Sulek, J., & Dileepan, P. (2003). Static scheduling research to minimize wieghted and unweighted tardiness: A state-of-the-art survey. International Journal of Production Economics, 83, 1–12.
Sitters, R. (2010). Competitive analysis of preemptive single-machine scheduling. Operations Research Letters, 38, 585–588.
Sousa, J. P., & Wolsey, L. A. (1992). A time indexed formulation of non-preemptive single machine scheduling problems. Mathematical Programming, 54, 353–367.
Tian, Z., Ng, C. T., & Cheng, T. C. E. (2006). An O\((n^2)\) algorithm for scheduling equal-length preemptive jobs on a simgle machine to minimize total tardiness. Journal of Scheduling, 9(4), 343–364.
Tian, Z., Ng, C. T., & Cheng, T. C. E. (2009). Preemptive scheduling of jobs with agreeable due date on a single machine to minimize total tardiness. Operations Research Letters, 37, 368–374.
Vanhoucke, M., Demeulemeester, E., & Herroelen, W. (2001). An exact procedure for the resource-constrained weighted earliness-tardiness project scheduling problem. Annals of Operations Research, 102(1–4), 179–196.
Wang, G., Sun, H., & Chu, C. (2005). Preemptive scheduling with availability constraints to minimize total weighted completion times. Annals of Operations Research, 133, 183–192.
Yang, B., Geunes, J., & O’Brien, W. (2004). A heuristic approach for minimizing weighted tardiness and overtime costs in single resource scheduling. Computers and Operations Research, 31, 1273–1301.
Yano, C. A., & Kim, Y. D. (1991). Algorithms for a class of single-machine weighted tardiness and earliness problems. European Journal of Operational Research, 52(2), 167–178.
Yuan, J., & Lin, Y. (2005). Single machine preemptive scheduling with fixed jobs to minimize tardiness related criteria. European Journal of Operational Research, 164(3), 851–855.
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Jaramillo, F., Keles, B. & Erkoc, M. Modeling single machine preemptive scheduling problems for computational efficiency. Ann Oper Res 285, 197–222 (2020). https://doi.org/10.1007/s10479-019-03298-9
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DOI: https://doi.org/10.1007/s10479-019-03298-9