Abstract
Single machine scheduling (SMS) problems seem to have received substantial attention because of several reasons. These type of problems are important both because of their own intrinsic value, as well as their role as building blocks for more generalized and complex problems. In a multi-processor environment single processor schedules may be used in bottlenecks, or to organize task assignment to an expensive processor; sometimes an entire production line may be treated as a single processor for scheduling purposes. Also, compared to multiple processor scheduling, SMS problems are mathematically more tractable. Hence more problem classes can be solved in polynomial time, and a larger variety of model parameters, such as various types of cost functions, or an introduction of change-over cost, can be analyzed. Single processor problems are thus of rather fundamental character and allow for some insight and development of ideas when treating more general scheduling problems.
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References
H. H. Ali, J. S. Deogun, A polynomial algorithm to find the jump number of interval orders, Preprint, Univ. of Nebraska Lincoln, 1990.
D. Adolphson, T. C. Hu, Optimal linear ordering, SIAM J. Appl. Math. 25, 1973, 403–423.
U. Bagchi, R. H. Ahmadi, An improved lower bound for minimizing weighted completion times with deadlines, Oper. Res. 35, 1987, 311–313.
S. P. Bansal, Single machine scheduling to minimize weighted sum of completion times with secondary criterion — a branch-and-bound approach, European J. Oper. Res. 5, 1980, 177–181.
J. Bruno, P. Downey, Complexity of task sequencing with deadlines, set-up times and changeover costs, SIAM J. Comput. 7, 1978, 393–404.
P. Bratley, M. Florian, P. Robillard, Scheduling with earliest start and due date constraints, Naval Res. Logist. Quart. 18, 1971, 511–517.
P. Bratley, M. Florian, P. Robillard, On sequencing with earliest starts and due dates with application to computing bounds for the (n/m/G/Fmax) problem, Naval Res. Logist. Quart. 20, 1973, 57–67.
V. Bouchitte, M. Habib, The calculation of invariants of ordered sets, in: I. Rival (ed.), Algorithms and Order, Kluwer, Dordrecht, 1989, 231–279.
P. C. Bagga, K. R. Kalra, Single machine scheduling problem with quadratic functions of completion time — a modified approach, J. Inform. Optim. Sci. 2, 1981, 103–108.
J. Błazewicz, Scheduling dependent tasks with different arrival times to meet deadlines, in: E. Gelenbe, H. Beilner (eds.), Modelling and Performance Evaluation of Computer Systems, North Holland, Amsterdam, 1976, 57–65.
K. R. Baker, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, Preemptive scheduling of a single machine to minimize maximum cost subject to release dates and precedence constraints, Oper. Res. 31, 1983, 381–386.
H. Buer, R. H. Möhring, A fast algorithm for the decomposition of graphs and posets, Math. Oper. Res. 8, 1983, 170–184.
L. Bianco, S. Ricciardelli, Scheduling of a single machine to minimize total weighted completion time subject to release dates, Naval Res. Logist. Quarterly. 29, 1982, 151–167.
K. R. Baker, Z.-S. Su, Sequencing with due dates and early start times to minimize maximum tardiness, Naval Res. Logist. Quart. 21, 1974, 171–176.
K. R. Baker, L. Schrage, Dynamic programming solution for sequencing problems with precedence constraints, Oper. Res. 26, 1978, 444–449.
K. R. Baker, L. Schrage, Finding an optimal sequence by dynamic programming: An extension to precedence related tasks, Oper. Res. 26, 1978, 111–120.
R. N. Burns, G. Steiner, Single machine scheduling with series-parallel precedence constraints, Oper. Res. 29, 1981, 1195–1207.
R. N. Burns, Scheduling to minimize the weighted sum of completion times with secondary criteria, Naval Res. Logist. Quart. 23, 1976, 25–129.
G. R. Bitran, H. H. Yanasse, Computational complexity of the capacitated lot size problem, Management Sci. 28, 1982, 1174–1186.
J. Carlier, The one-machine sequencing problem, European J. Oper. Res. 11, 1982, 42–47.
M. Chein, P. Martin, Sur le nombre de sauts d’une foret, C. R. Acad. Sc. Paris 275, serie A, 1972, 159–161.
R. W. Conway, W. L. Maxwell, L. W. Miller, Theory of Scheduling, Addison-Wesley, Reading, Mass., 1967.
E. G. Coffman Jr. (ed.), Scheduling in Computer and Job Shop Systems, J. Wiley, New York, 1976.
S. Chand, H. Schneeberger, A note on the single-machine scheduling problem with minimum weighted completion time and maximum allowable tardiness, Naval Res. Logist. Quart. 33, 1986, 551–557.
M. I. Dessouky, J. S. Deogun, Sequencing jobs with unequal ready times to minimize mean flow time, SIAM J. Comput. 10, 1981, 192–202.
W. C. Driscoll, H. Emmons, Scheduling production on one machine with changeover costs, AIIE Trans. 9, 1977, 388–395.
J. Du, J. Y.-T. Leung, Minimizing total tardiness on one machine is NP-hard, Math. Oper. Res. 15, 1990, 483–495.
J. Erschler, G. Fontan, C. Merce, F. Roubellat, A new dominance concept in scheduling n jobs on a single machine with ready times and due dates, Oper. Res. 31, 1983, 114–127.
H. Emmons, One machine sequencing to minimize mean flow time with minimum number tardy, Naval Res. Logist. Quart. 22, 1975, 585–592.
M. H. El-Zahar, I. Rival, Greedy linear extensions to minimize jumps, Discrete Appl. Math. 11, 1985, 143–156.
P. C. Fishburn, Interval Orders and Interval Graphs, J. Wiley, New York, 1985.
M. L. Fisher, A. M. Krieger, Analysis of a linearization heuristic for single machine scheduling to maximize profit, Math. Programming 28, 1984, 218–225.
M. Florian, J. K. Lenstra, A. H. G. Rinnooy Kan, Deterministic production planning: algorithms and complexity, Management Sci. 26, 1980, 669–679.
U. Faigle, R. Schrader, A setup heuristic for interval orders, Oper. Res. Lett. 4, 1985, 185–188.
U. Faigle, R. Schrader, Interval orders without odd crowns are defect optimal, Report 85382-OR, University of Bonn, 1985.
M. Florian, P. Trepant, G. McMahon, An implicit enumeration algorithm for the machine sequencing problem, Management Sci. 17, 1971, B782–B792.
M. R. Garey, D. S. Johnson, Scheduling tasks with non-uniform deadlines on two processors, J. Assoc. Comput. Mach. 23, 1976, 461–467.
M. R. Garey, D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman, San Francisco, 1979.
M. R. Garey, D. S. Johnson, B. B. Simons, R. E. Tarjan, Scheduling unit-time tasks with arbitrary release times and deadlines, SIAM J. Comput. 10, 1981, 256–269.
S. K. Gupta, J. Kyparisis, Single machine scheduling research, OMEGA Internat. J. Management Sci. 15, 1987, 207–227.
G. V. Gens, E. V. Levner, Approximation algorithm for some scheduling problems, Engrg. Cybernetics 6, 1978, 38–46.
G. V. Gens, E. V. Levner, Fast approximation algorithm for job sequencing with deadlines, Discrete Appl. Math. 3, 1981, 313–318.
A. Gascon, R. C. Leachman, A dynamic programming solution to the dynamic, multi-item, single-machine scheduling problem, Oper. Res. 36, 1988, 50–56.
C. R. Glassey, Minimum changeover scheduling of several products on one machine, Oper. Res. 16, 1968, 342–352.
R. L. Graham, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, Optimization and approximation in deterministic sequencing and scheduling: a survey, Ann. Discrete Math. 5, 1979, 287–326.
M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.
S. K. Gupta, T. Sen, Minimizing the range of lateness on a single machine, Engrg. Costs Production Economics 7, 1983, 187–194.
S. K. Gupta, T. Sen, Minimizing the range of lateness on a single machine, J. Oper. Res. Soc. 35, 1984, 853–857.
M. R. Garey, R. E. Tarjan, G.T. Wilfong, One-processor scheduling with earliness and tardiness penalties, Math. Oper. Res. 13, 1988, 330–348.
T. C. Hu, Y. S. Kuo, F. Ruskey, Some optimum algorithms for scheduling problems with changeover costs, Oper. Res. 35, 1987, 94–99.
N. G. Hall, W. Kubiak, S. P. Sethi, Earliness-tardiness scheduling problems, II: Deviation of completion times about a restictive commen due date, Oper. Res. 39, 1991, 847–856.
W. A. Horn, Single-machine job sequencing with tree-like precedence ordering and linear delay penalties, SIAM J. Appl. Math. 23, 1972, 189–202.
W. A. Horn, Some simple scheduling algorithms, Naval Res. Logist. Quart. 21, 1974, 177–185.
L. A. Hall, D. B. Shmoys, Jackson’s rule for one-machine scheduling: Making a good heuristic better, Department of Mathematics, Massachusetts Institute of Technology, Cambridge, 1988.
T. Ichimori, H. Ishii, T. Nishida, Algorithm for one machine job sequencing with precedence constraints, J. Oper. Res. Soc. Japan 24, 1981, 159–169.
O. H. Ibarra, C. E. Kim, Approximation algorithms for certain scheduling problems, Math. Oper. Res. 3, 1978, 197–204.
J. R. Jackson, Scheduling a production line to minimize maximum tardiness, Research Report 43, Management Sci. Res. Project, UCLA, 1955.
J. J. Kanet, Minimizing the average deviation of job completion times about a common due date, Naval Res. Logist. Quart. 28, 1981, 643–651.
R. M. Karp, Reducibility among combinatorial problems, in: R. E. Miller, J. W. Thatcher (eds.), Complexity of Computer Computations, Plenum Press, New York, 1972, 85–103.
H. Kise, T. Ibaraki, H. Mine, A solvable case of a one-machine scheduling problem with ready and due times, Oper. Res. 26, 1978, 121–126.
H. Kise, T. Ibaraki, H. Mine, Performance analysis of six approximation algorithms for the one-machine maximum lateness scheduling problem with ready times, J. Oper. Res. Soc. Japan 22, 1979, 205–224.
K. R. Kalra, K. Khurana, Single machine scheduling to minimize waiting cost with secondary criterion, J. Math. Sci. 16–18, 1981–1983, 9–15.
W. Kubiak, S. Lou, S. Sethi, Equivalence of mean flow time problems and mean absolute deviation problems, Oper. Res. Lett. 9, 1990, 371–374.
W. Kubiak, Completion time variance minimization on a single machine is difficult, Oper. Res. Lett. 14, 1993, 49–59.
W. Kubiak, New results on the completion time varaince minimization, Discrete Appl. Math. 58, 1995, 157–168.
E. L. Lawler, On scheduling problems with deferral costs, Management Sci. 11, 1964, 280–288.
E. L. Lawler, Optimal sequencing of a single machine subject to precedence constraints, Management Sci. 19, 1973, 544–546.
E. L. Lawler, Sequencing to minimize the weighted number of tardy jobs, RAIRO Rech. Opér. 10, 1976, Suppl. 27–33.
E. L. Lawler, A ‘pseudopolynomial’ algorithm for sequencing jobs to minimize total tardiness, Ann. Discrete Math. 1, 1977, 331–342.
E. L. Lawler, Sequencing jobs to minimize total weighted completion time subject to precedence constraints, Ann. Discrete Math. 2, 1978, 75–90.
E. L. Lawler, Sequencing a single machine to minimize the number of late jobs, Preprint, Computer Science Division, University of California, Berkeley, 1982.
E. L. Lawler, Recent results in the theory of machine scheduling, in: A. Bachern, M. Grötschel, B. Korte (eds.), Mathematical Programming: The State of the Art, Springer, Berlin, 1983, 202–234.
R. E. Larson, M. I. Dessouky, Heuristic procedures for the single machine problem to minimize maximum lateness, AIIE Trans. 10, 1978, 176–183.
R. E. Larson, M. I. Dessouky, R. E. Devor, A forward-backward procedure for the single machine problem to minimize maximum lateness, IIE Trans. 17, 1985, 252–260.
J. K. Lenstra, Sequencing by Enumerative Methods, Mathematical Centre Tract 69, Mathematisch Centrum, Amsterdam, 1977.
J. Labetoulle, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, Preemptive scheduling of uniform machines subject to release dates, in: W. R. Pulleyblank (ed.), Progress in Combinatorial Optimization, Academic Press, New York, 1984, 245–261.
B. J. Lageweg, J. K. Lenstra, A. H. G. Rinnooy Kan, Minimizing maximum lateness on one machine: Computational experience and some applications, Statist. Neerlandica 30, 1976, 25–41.
E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, Recent development in deterministic sequencing and scheduling: a survey, in: M. A. H. Dempster, J. K. Lenstra, A. H. G Rinnooy Kan (eds.), Deterministic and Stochastic Scheduling, Reidel, Dordrecht. 1982, 35–73.
E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, D. B. Shmoys, Sequencing and scheduling: Algorithms and complexity, in: S. C. Graves, A. H. G. Rinnooy Kan, P. H. Zipkin (eds.), Handbook in Operations Research and Management Science, Vol. 4: Logistics of Production and Inventory, Elsevier, Amsterdam, 1993.
E. L. Lawler, J. M. Moore, A functional equation and its application to resource allocation and sequencing problems, Management Sci. 16, 1969, 77–84.
J. K. Lenstra, A. H. G. Rinnooy Kan, Towards a better algorithm for the job-shop scheduling problem — I, Report BN 22, 1973, Mathematisch Centrum, Amsterdam.
J. K. Lenstra, A. H. G. Rinnooy Kan, Complexity of scheduling under precedence constraints, Oper. Res. 26, 1978, 22–35.
J. K. Lenstra, A. H. G. Rinnooy Kan, Complexity results for scheduling chains on a single machine, European J. Oper. Res. 4, 1980, 270–275.
J. K. Lenstra, A. H. G. Rinnooy Kan, P. Brucker, Complexity of machine scheduling problems, Ann. Discrete Math. 1, 1977, 343–362.
J. Y-T. Leung, G. H. Young, Minimizing total tardiness on a single machine with precedence constraints, ORSA J. Comput, to appear.
R. McNaughton, Scheduling with deadlines and loss functions, Management Sci. 6, 1959, 1–12.
G. B. McMahon, M. Florian, On scheduling with ready times and due dates to minimize maximum lateness, Oper. Res. 23, 1975, 475–482.
S. Mitsumori, Optimal production scheduling of multicommodity in flow line, IEEE Trans. Syst. Man Cybernet. CMC-2, 1972, 486–493.
S. Miyazaki, One machine scheduling problem with dual criteria, J. Oper. Res. Soc. Japan 24, 1981, 37–51.
R. H. Möhring, Computationally tractable classes of ordered sets, in: I. Rival (ed.), Algorithms and Order, Kluwer, Dordrecht, 1989, 105–193.
C. L. Monma, Linear-time algorithms for scheduling on parallel processors, Oper. Res. 30, 1982, 116–124.
J. M. Moore, An n job, one machine sequencing algorithm for minimizing the number of late jobs, Management Sci. 15, 1968, 102–109.
R. H. Möhrig, F. J. Radermacher, Generalized results on the polynomiality of certain weighted sum scheduling problems, Methods of Oper. Res. 49, 1985, 405–417.
C. L. Monma, J. B. Sidney, Optimal sequencing via modular decomposition: Characterization of sequencing functions, Math. Oper. Res. 12, 1987, 22–31.
J. H. Muller, J. Spinrad, Incremental modular decomposition, J. Assoc. Cornput. Mach. 36, 1989, 1–19.
T. L. Magnanti, R. Vachani, A strong cutting plane algorithm for production scheduling with changeover costs, Oper. Res. 38, 1990, 456–473.
M. E. Posner, Minimizing weighted completion times with deadlines,Oper. Res. 33, 1985, 562–574.
C. N. Potts, An algorithm for the single machine sequencing problem with precedence constraints, Math. Programming Study 13, 1980, 78–87.
C. N. Potts, Analysis of a heuristic for one machine sequencing with release dates and delivery times, Oper. Res. 28, 1980, 1436–1441.
C. N. Potts, A Lagrangian based branch and bound algorithm for a single machine sequencing with precedence constraints to minimize total weighted completion time, Management Sci. 31, 1985, 1300–1311.
W. R. Pulleyblank, On minimizing setups in precedence constrained scheduling, Report 81105-OR, University of Bonn, 1975.
C. N. Potts, L. N. van Wassenhove, An algorithm for single machine sequencing with deadlines to minimize total weighted completion time, European J. Oper. Res. 12, 1983, 379–387.
M. Raghavachari, A V-shape property of optimal schedule of jobs about a common due date, European J. Oper. Res. 23, 1986, 401–402.
F. M. E. Raiszadeh, P. Dileepan, T. Sen, A single machine bicriterion scheduling problem and an optimizing branch-and-bound procedure, J. Inform. Optim. Sci. 8, 1987, 311–321.
A. H. G. Rinnooy Kan, B. J. Lageweg, J. K. Lenstra, Minimizing total costs in one-machine scheduling, Oper. Res. 23, 1975, 908–927.
I. Rival, N. Zaguiga, Constructing greedy linear extensions by interchanging chains, Order 3, 1986, 107–121.
S. Sahni, Algorithms for scheduling independent tasks, J. Assoc. Comput. Mach. 23, 1976, 116–127.
L. E. Schrage, Obtaining optimal solutions to resource constrained network scheduling problems, AIIE Systems Engineering Conference, Phoenix, Arizona, 1971.
L. E. Schrage, The multiproduct lot scheduling problem. in: M. A. H. Dempster, J. K. Lenstra, A. H. G Rinnooy Kan (eds.), Deterministic and Stochastic Scheduling, Reidel, Dordrecht, 1982.
G. Schmidt, Minimizing changeover costs on a single machine, in: W. Bühler, F. Feichtinger, F.-J. Radermacher, P. Feichtinger (eds.), DGOR Proceedings 90, Vol. 1, Springer, 1992, 425–432.
J. B. Sidney, An extension of Moore’s due date algorithm, in: S. E. Elmaghraby (ed.), Symposium on the Theory of Scheduling and Its Applications, Springer, Berlin, 1973, 393–398.
J. B. Sidney, Decomposition algorithms for single-machine sequencing with precedence relations and deferral costs, Oper. Res. 23, 1975, 283–298.
B. Simons, A fast algorithm for single processor scheduling, Proc. 19th Annual IEEE Symp. Foundations of Computer Science, 1978, 50–53.
W. E. Smith, Various optimizers for single-stage production, Naval Res. Logist. Quart. 3, 1956, 59–66.
J. B. Sidney, G. Steiner, Optimal sequencing by modular decomposition: polynomial algorithms, Oper. Res. 34, 1986, 606–612.
W. Townsend, The single machine problem with quadratic penalty function of completion times: A branch and bound solution, Management Sci. 24, 1978, 530–534.
F. J. Villarreal, R. L. Bulfin, Scheduling a single machine to minimize the weighted number of tardy jobs, AIIE Trans. 15, 1983, 337–343.
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Błażewicz, J., Ecker, K.H., Pesch, E., Schmidt, G., Węglarz, J. (1996). Scheduling on One Processor. In: Scheduling Computer and Manufacturing Processes. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-03217-6_4
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