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October  2011, 7(4): 825-848. doi: 10.3934/jimo.2011.7.825

## Single-machine scheduling with stepwise tardiness costs and release times

 1 Sabanci University, Manufacturing Systems and Industrial Engineering Program, Orhanli-Tuzla 34956 Istanbul, Turkey 2 University of Florida, Department of Industrial and Systems Engineering, Gainesville, FL 32611, United States

Received  August 2010 Revised  May 2011 Published  August 2011

We study a scheduling problem that belongs to the yard operations component of the railroad planning problems, namely the hump sequencing problem. The scheduling problem is characterized as a single-machine problem with stepwise tardiness cost objectives. This is a new scheduling criterion which is also relevant in the context of traditional machine scheduling problems. We produce complexity results that characterize some cases of the problem as pseudo-polynomially solvable. For the difficult-to-solve cases of the problem, we develop mathematical programming formulations, and propose heuristic algorithms. We test the formulations and heuristic algorithms on randomly generated single-machine scheduling problems and real-life data sets for the hump sequencing problem. Our experiments show promising results for both sets of problems.
Citation: Güvenç Şahin, Ravindra K. Ahuja. Single-machine scheduling with stepwise tardiness costs and release times. Journal of Industrial and Management Optimization, 2011, 7 (4) : 825-848. doi: 10.3934/jimo.2011.7.825
##### References:
 [1] M. van den Akker, "LP-based Solution Methods for Single-Machine Scheduling Problems," Ph.D Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1994. [2] M. van den Akker, C. A. J. Hurkens and M. W. P. Savelsbergh, Time-indexed formulations for machine scheduling problems: Column generation, INFORMS Journal on Computing, 12 (2000), 111-124. doi: 10.1287/ijoc.12.2.111.11896. [3] J. R. Birge and M. J. Maddox, Bounds on expected project tardiness, Operations Research, 43 (1995), 838-850. doi: 10.1287/opre.43.5.838. [4] P. Brucker and S. Knust, Complexity results for scheduling problems, Available from: http://www.informatik.uni-osnabrueck.de/knust/class/ (accessed 1 March 2011). [5] J. Curry and B. Peters, "Solving Parallel Machine Scheduling Problems with Stepwise Increasing Tardiness Cost Objectives," Working Paper, Department of Industrial Engineering, Texas A&M University, 2004. [6] J. Curry and B. Peters, Rescheduling parallel machines with stepwise increasing tardiness, International Journal of Production Research, 43 (2005), 3231-3246. doi: 10.1080/00207540500103953. [7] C. F. Daganzo, R. G. Dowling and R. W. Hall, Railroad classification yard throughput: The case of multistage triangular sorting, Transportation Research Part A, 17 (1983), 95-106. doi: 10.1016/0191-2607(83)90063-8. [8] C. F. Daganzo, Static blocking at railyards: Sorting implications and tracks requirements, Transportation Science, 20 (1986), 186-199. doi: 10.1287/trsc.20.3.189. [9] C. F. Daganzo, Dynamic blocking for railyards: Part I. Homogeneous traffic, Transportation Research Part B, 21 (1987a), 1-27. doi: 10.1016/0191-2615(87)90018-X. [10] C. F. Daganzo, Dynamic blocking for railyards: Part II. Heterogeneous traffic, Transportation Research Part B, 21 (1987b), 29-40. doi: 10.1016/0191-2615(87)90019-1. [11] J. Du and J. Y.-T. Leung, Minimizing total tardiness on one machine is NP-hard, Mathematics of Operations Research, 15 (1990), 483-495. doi: 10.1287/moor.15.3.483. [12] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Discrete Optimization (Proc. Adv. Res. Inst. Discrete Optimization and Systems Appl., Banff, Alta., 1977), II, Annals of Discrete Mathematics, 5 (1979), 287-326. doi: 10.1016/S0167-5060(08)70356-X. [13] L. A. Hall, A. S. Schulz, D. B. Shmoys and J. Wein, Scheduling to minimize average completion time: Off-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), 513-544. doi: 10.1287/moor.22.3.513. [14] E. R. Kraft, A Hump sequencing algorithm for real time management of train connection reliability, Journal of the Transportation Research Forum, 30 (2000), 95-115. [15] E. R. Kraft, Priority-based classification for improving connection reliability in railroad yards-part I: Integration with car scheduling, Journal of the Transportation Research Forum, 56 (2002a), 93-105. [16] E. R. Kraft, Priority-based classification for improving connection reliability in railroad yards-part II: Dynamic block to track assignment, Journal of the Transportation Research Forum, 56 (2002b), 107-119. [17] E. L. Lawler, "A 'Pseudopolynomial' Algorithm for Sequencing Jobs to Minimize Total Tardiness," Studies in Integer Programming (Proc. Workshop, Bonn, 1975), Annals of Discrete Mathematics, 1, North-Holland, Amsterdam, (1977), 331-342. doi: 10.1016/S0167-5060(08)70742-8. [18] C. D. Martland, PMAKE analysis: Predicting rail yard time distributions using probabilistic train connection standards, Transportation Science, 16 (1982), 476-506. doi: 10.1287/trsc.16.4.476. [19] E. R. Petersen, Railyard modeling: Part I. Prediction of put-through time, Transportation Science, 11 (1977a), 37-49. doi: 10.1287/trsc.11.1.37. [20] E. R. Petersen, Railyard modeling: Part II. The effect of yard facilities on congestion, Transportation Science, 11 (1977), 50-59. doi: 10.1287/trsc.11.1.50. [21] C. Phillips, C. Stein and J. Wein, Minimizing average completion time in the presence of release dates, Mathematical Programming B, 82 (1998), 199-223. doi: 10.1007/BF01585872. [22] M. W. P. Savelsbergh, R. N. Uma and J. Wein, An experimental study of LP-based approximation algorithms for scheduling problems, INFORMS Journal on Computing, 17 (2005), 123-136. doi: 10.1287/ijoc.1030.0055. [23] J. P. De Sousa and L. A. Wolsey, A time-indexed formulation of non-preemptive single-machine scheduling problems, Mathematical Programming, 54 (1992), 353-367. doi: 10.1007/BF01586059. [24] M. A. Turnquist and M. S. Daskin, Queuing models of classification and delay in railyards, Transportation Science, 16 (1982), 207-230. doi: 10.1287/trsc.16.2.207.

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##### References:
 [1] M. van den Akker, "LP-based Solution Methods for Single-Machine Scheduling Problems," Ph.D Thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1994. [2] M. van den Akker, C. A. J. Hurkens and M. W. P. Savelsbergh, Time-indexed formulations for machine scheduling problems: Column generation, INFORMS Journal on Computing, 12 (2000), 111-124. doi: 10.1287/ijoc.12.2.111.11896. [3] J. R. Birge and M. J. Maddox, Bounds on expected project tardiness, Operations Research, 43 (1995), 838-850. doi: 10.1287/opre.43.5.838. [4] P. Brucker and S. Knust, Complexity results for scheduling problems, Available from: http://www.informatik.uni-osnabrueck.de/knust/class/ (accessed 1 March 2011). [5] J. Curry and B. Peters, "Solving Parallel Machine Scheduling Problems with Stepwise Increasing Tardiness Cost Objectives," Working Paper, Department of Industrial Engineering, Texas A&M University, 2004. [6] J. Curry and B. Peters, Rescheduling parallel machines with stepwise increasing tardiness, International Journal of Production Research, 43 (2005), 3231-3246. doi: 10.1080/00207540500103953. [7] C. F. Daganzo, R. G. Dowling and R. W. Hall, Railroad classification yard throughput: The case of multistage triangular sorting, Transportation Research Part A, 17 (1983), 95-106. doi: 10.1016/0191-2607(83)90063-8. [8] C. F. Daganzo, Static blocking at railyards: Sorting implications and tracks requirements, Transportation Science, 20 (1986), 186-199. doi: 10.1287/trsc.20.3.189. [9] C. F. Daganzo, Dynamic blocking for railyards: Part I. Homogeneous traffic, Transportation Research Part B, 21 (1987a), 1-27. doi: 10.1016/0191-2615(87)90018-X. [10] C. F. Daganzo, Dynamic blocking for railyards: Part II. Heterogeneous traffic, Transportation Research Part B, 21 (1987b), 29-40. doi: 10.1016/0191-2615(87)90019-1. [11] J. Du and J. Y.-T. Leung, Minimizing total tardiness on one machine is NP-hard, Mathematics of Operations Research, 15 (1990), 483-495. doi: 10.1287/moor.15.3.483. [12] R. L. Graham, E. L. Lawler, J. K. Lenstra and A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: A survey, Discrete Optimization (Proc. Adv. Res. Inst. Discrete Optimization and Systems Appl., Banff, Alta., 1977), II, Annals of Discrete Mathematics, 5 (1979), 287-326. doi: 10.1016/S0167-5060(08)70356-X. [13] L. A. Hall, A. S. Schulz, D. B. Shmoys and J. Wein, Scheduling to minimize average completion time: Off-line and on-line approximation algorithms, Mathematics of Operations Research, 22 (1997), 513-544. doi: 10.1287/moor.22.3.513. [14] E. R. Kraft, A Hump sequencing algorithm for real time management of train connection reliability, Journal of the Transportation Research Forum, 30 (2000), 95-115. [15] E. R. Kraft, Priority-based classification for improving connection reliability in railroad yards-part I: Integration with car scheduling, Journal of the Transportation Research Forum, 56 (2002a), 93-105. [16] E. R. Kraft, Priority-based classification for improving connection reliability in railroad yards-part II: Dynamic block to track assignment, Journal of the Transportation Research Forum, 56 (2002b), 107-119. [17] E. L. Lawler, "A 'Pseudopolynomial' Algorithm for Sequencing Jobs to Minimize Total Tardiness," Studies in Integer Programming (Proc. Workshop, Bonn, 1975), Annals of Discrete Mathematics, 1, North-Holland, Amsterdam, (1977), 331-342. doi: 10.1016/S0167-5060(08)70742-8. [18] C. D. Martland, PMAKE analysis: Predicting rail yard time distributions using probabilistic train connection standards, Transportation Science, 16 (1982), 476-506. doi: 10.1287/trsc.16.4.476. [19] E. R. Petersen, Railyard modeling: Part I. Prediction of put-through time, Transportation Science, 11 (1977a), 37-49. doi: 10.1287/trsc.11.1.37. [20] E. R. Petersen, Railyard modeling: Part II. The effect of yard facilities on congestion, Transportation Science, 11 (1977), 50-59. doi: 10.1287/trsc.11.1.50. [21] C. Phillips, C. Stein and J. Wein, Minimizing average completion time in the presence of release dates, Mathematical Programming B, 82 (1998), 199-223. doi: 10.1007/BF01585872. [22] M. W. P. Savelsbergh, R. N. Uma and J. Wein, An experimental study of LP-based approximation algorithms for scheduling problems, INFORMS Journal on Computing, 17 (2005), 123-136. doi: 10.1287/ijoc.1030.0055. [23] J. P. De Sousa and L. A. Wolsey, A time-indexed formulation of non-preemptive single-machine scheduling problems, Mathematical Programming, 54 (1992), 353-367. doi: 10.1007/BF01586059. [24] M. A. Turnquist and M. S. Daskin, Queuing models of classification and delay in railyards, Transportation Science, 16 (1982), 207-230. doi: 10.1287/trsc.16.2.207.
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