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Recent Results in the Theory of Machine Scheduling

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Mathematical Programming The State of the Art

Abstract

The state of the art of deterministic machine scheduling is reviewed. Emphasis is placed on efficient, i. e. polynomial-bounded, optimization algorithms. A few of the more significant NP-hardness results are highlighted, and some open problems are mentioned.

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References

  • D. L. Adolphson, “Single Machine Job Scheduling With Precedence Constraints”, SIAM J. Compute 6 (1977) 40–54.

    Article  MathSciNet  MATH  Google Scholar 

  • D. L. Adolphson, T. C. Hu, “Optimal Linear Ordering”, SIAM J. Appl. Math, 25 (1973) 403–423.

    Article  MathSciNet  MATH  Google Scholar 

  • H. M. Abdel-Wahab, T. Kameda, “Scheduling to Minimize Maximum Cumulative Cost Subject to Series Parallel Precedence Constraints”, Operations Res., 26 (1978) 141– 158.

    Google Scholar 

  • K. R. Baker, Introduction to Sequencing and Scheduling, Wiley, New York, 1974.

    Google Scholar 

  • K. R. Baker, L. E. Schräge, “Finding an Optimal Sequence by Dynamic Programming: An Extension to Precedence-Constrained Tasks”, Operations Res, 26 (1978) 111–120.

    Article  MATH  Google Scholar 

  • 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”, (1982) to appear in Operations Res.

    Google Scholar 

  • P. Brucker, M. R. Garey, D. S. Johnson, “Scheduling Equal-Length Tasks under Tree- Like Precedence Constraints to Minimize Maximum Lateness”, Math. Operations Res, 2 (1977) 275–284.

    Article  MathSciNet  MATH  Google Scholar 

  • E. G. Coffman, Jr., R. L. Graham, “Optimal Scheduling for Two-Processor Systems”, Acta Informat, 1 (1972) 200–213.

    Article  MathSciNet  Google Scholar 

  • R. W. Conway, W. L. Maxwell, L. W. Miller, Theory of Scheduling, Addison-Wesley, Reading, Mass. 1967.

    Google Scholar 

  • G. I. Davida, D. J. Linton, “A New Algorithm for the Scheduling of Tree Structured Tasks”, Proc. Conf. Inform. Sci. and Systems, Baltimore, MD., 1976.

    Google Scholar 

  • D. Dolev, “Scheduling Wide Graphs”, (1981), unpublished manuscript.

    Google Scholar 

  • H. Emmons, “One-Machine Sequencing to Minimize Certain Functions of Job Tardiness”, Operations Res, 17 (1969) 701–715.

    Article  MathSciNet  MATH  Google Scholar 

  • M. L. Fisher, “A Dual Algorithm for the One-Machine Scheduling Problem”, Math. Programming, 11 (1976) 229–251.

    Article  MathSciNet  Google Scholar 

  • M. L. Fisher, B. J. Lageweg, J. K. Lenstra, A. H. G. Rinnooy Kan, “Surrogate Duality Relaxation for Job Shop Scheduling”, Report, Mathematisch Centrum, Amsterdam, 1981.

    Google Scholar 

  • M. Fujii, T. Kasami, K. Ninomiya, “Optimal Sequencing of Two Equivalent Processors”, SIAM J. Appl. Math, 17 (1969) 784–789; Erratum, 20 (1971) 141.

    MathSciNet  MATH  Google Scholar 

  • H. N. Gabow, “An Almost-Linear Algorithm for Two-Processor Scheduling”, J. Assoc. Comput. Mach., 29 (1982 A) 766–780.

    Google Scholar 

  • H. N. Gabow, private communication, 1982 B.

    Google Scholar 

  • M. R. Garey, “Optimal Task Sequencing with Precedence Constraints”, Discrete Math, 4 (1973) 37–56.

    Article  MathSciNet  MATH  Google Scholar 

  • M. R. Garey, D. S. Johnson, “Scheduling Tasks with Nonuniform Deadlines on Two Processors”, J. Assoc. Comput. Mach, 23 (1976) 461–467.

    MathSciNet  MATH  Google Scholar 

  • M. R. Garey, D. S. Johnson, “Two-Processor Scheduling with Start-Times and Deadlines”, SIAM J. Comput, 6 (1977) 416–426.

    Article  MathSciNet  MATH  Google Scholar 

  • M. R. Garey, D. S. Johnson, R. E. Tarjan, M. Yannakakis, “Scheduling Opposing Forests”, unpublished manuscript, (1981).

    Google Scholar 

  • P. C. Gilmore, R. E. Gomory, “Sequencing a One-State Variable Machine: A Solvable Case of the Traveling Salesman Problem”, Oper. Res, 12 (1964) 655–679.

    Article  MathSciNet  MATH  Google Scholar 

  • T. Gonzalez, “Optimal Mean Finish Time Preemptive Schedules”, Technical Report 220, (1977) Computer Science Department, Pennsylvania State University.

    Google Scholar 

  • T. Gonzalez, “A Note on Open Shop Preemptive Schedules”, IEEE Trans. Computers, C-28 (1979) 782–786.

    Google Scholar 

  • T. Gonzalez, D. B. Johnson, “A New Algorithm for Preemptive Scheduling of Trees”, J. Assoc. Comput. Mach, 27 (1980) 287–312.

    MathSciNet  MATH  Google Scholar 

  • T. Gonzalez, E. L. Lawler, S Sahni, “Optimal Preemptive Scheduling of Two Unrelated Processors in Linear Time”, (1981), to appear.

    Google Scholar 

  • T. Gonzalez, S. Sahni, “Open Shop Scheduling to Minimize Finish Time”, J. Assoc. Comput. Mach, 23 (1976) 665–679.

    MathSciNet  MATH  Google Scholar 

  • T. Gonzalez, S. Sahni, “Preemptive Scheduling of Uniform Processor Systems”, J. Assoc. Comput. Mach, 25 (1978) 92–101.

    MathSciNet  MATH  Google Scholar 

  • 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.

    Article  MathSciNet  MATH  Google Scholar 

  • M. Held, R. M. Karp, “A Dynamic Programming Approach to Sequencing Problems”, SIAM J. Appl. Math, 10 (1972) 196–210.

    MathSciNet  Google Scholar 

  • W. A. Horn, “Single-Machine Job Sequencing with Treelike Precedence Ordering and Linear Dealy Penalties”, SIAM J. Appl. Math, 23 (1972) 189–202.

    Article  MathSciNet  MATH  Google Scholar 

  • W. A. Horn, “Minimizing Average Flow Time with Parallel Machines”, Oper. Res, 21 (1973) 846–847.

    Article  MATH  Google Scholar 

  • W. A. Horn, “Some Simple Scheduling Algorithms”, Naval Res. Logist. Quart, 21 (1974) 177–185.

    Article  MathSciNet  MATH  Google Scholar 

  • E. Horowitz, S. Sahni, “Exact and Approximate Algorithms for Scheduling Nonidentical Processors”, J. Assoc. Comput. Mach, 23 (1976) 317–327.

    MathSciNet  MATH  Google Scholar 

  • T. C. Hu, “Parallel Sequencing and Assembly Line Problems”, Oper. Res, 9 (1961) 841–848.

    Article  Google Scholar 

  • J. R. Jackson, “Scheduling a Production Line to Minimize Maximum Tardiness”, Research Report 43, (1955) Management Science Research Project, University of California, Los Angeles.

    Google Scholar 

  • J. R. Jackson, “An Extension of Johnson’s Results on Job Lot Scheduling”, Naval Res. Logist. Quart, 3 (1956) 201–203.

    Article  Google Scholar 

  • S. M. Johnson, “Optimal Two- and Three-Stage Production Schedules with Setup Times Included”, Naval Res. Logist. Quart, 1 (1954) 61–68.

    Article  Google Scholar 

  • S. M. Johnson, “Discussion: Sequencing n Jobs on Two Machines with Arbitrary Time Lags”, Management Sci, 5 (1958) 299–303.

    Article  Google Scholar 

  • E. P. C. Kao, M. Queyranne, “On Dynamic Programming Methods for Assembly Line Balancing”, working paper, Dept Quantitative Management Sci., University of Houston, Texas, (1980).

    Google Scholar 

  • H. Kise, T. Ibaraki, H. Mine, “A Solvable Case of the One-Machine Scheduling Problem with Ready and Due Times”, Oper. Res, 26 (1978) 121–126.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Labetoulle, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, “Preemptive Scheduling of Uniform Machines Subject to Release Dates”, Report BW 99, (1979) Mathematisch Centrum, Amsterdam.

    Google Scholar 

  • B. J. Lageweg, E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, “Computer Aided Complexity Classification of Deterministic Scheduling Problems”, Report BW 138, (1981) Mathematisch Centrum, Amsterdam.

    Google Scholar 

  • E. L. Lawler, “On Scheduling Problems with Deferral Costs”, Management Science, 11 (1964) 270–288.

    Article  MathSciNet  Google Scholar 

  • E. L. Lawler, “Optimal Sequencing of a Single Machine Subject to Precedence Constraints”, Management Sci, 19 (1973) 544–546.

    Article  MATH  Google Scholar 

  • E. L. Lawler, “Sequencing to Minimize the Weighted Number of Tardy Jobs”, RAIRO Rech. Oper, 10 suppl. (1976) 27–33.

    Google Scholar 

  • E. L. Lawler, “A Tseudopolynomial’ Algorithm for Sequencing Jobs to Minimize Total Tardiness”, Ann. Discrete Math, 1 (1977) 331–342.

    Article  MathSciNet  Google Scholar 

  • E. L. Lawler, “Sequencing Jobs to Minimize Total Weighted Completion Time Subject to Precedence Constraints”, Discrete Math., 2 (1978 A) 75–90.

    Google Scholar 

  • E. L. Lawler, “Sequencing Problems With Series Parallel Precedence Constraints”, to appear in Proc. Confer, on Combinatorial Optimization, ( N. Christofides, ed.), Urbino, Italy, 1978 B.

    Google Scholar 

  • E. L. Lawler, “Efficient Implementation of Dynamic Programming Algorithms for Sequencing Problems”, Report BW 106/79, ( 1979 A) Mathematisch Centrum, Amsterdam.

    Google Scholar 

  • E. L. Lawler, “Preemptive Scheduling of Uniform Parallel Machines to Minimize the number of Late Jobs”, Report BW 105, ( 1979 B) Mathematisch Centrum, Amsterdam.

    Google Scholar 

  • E. L. Lawler, “Preemptive Scheduling of Precedence-Constrained Jobs on Parallel Machines”, in Deterministic and Stochastic Scheduling, (M. A. H. Dempster, et al., eds.), D. Reidel, Dordrecht, Holland, 1982A, pp. 101–124.

    Google Scholar 

  • E. L. Lawler, “A Fully Polynomial Approximation Scheme for the Total Tardiness Problems”, ( 1982 B), submitted for publication.

    Google Scholar 

  • E. L. Lawler, “On Scheduling a Single Machine to Minimize the Number of Late Jobs”, ( 1982 C), submitted for publication.

    Google Scholar 

  • E. L. Lawler, J. Labetoulle, “On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming”, J. Assoc. Comput. Mach, 25 (1978) 612–619.

    MathSciNet  MATH  Google Scholar 

  • E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, ’Minimizing Maximum Lateness in a Two-Machine Open Shop, Math. Oper. Res, 6 (1981) 153–158.

    Article  MathSciNet  MATH  Google Scholar 

  • E. L. Lawler, C. U. Martel, “Scheduling Periodically Occurring Tasks on Multiple Professors”, Info. Proc. Letters, 12 (1981) 9–12.

    Article  MathSciNet  MATH  Google Scholar 

  • E. L. Lawler, C. U. Martel, “Computing ‘Polymatroidal’ Network Flows”, Math. of Operations Res., 1 (1982 A) 334–347.

    Google Scholar 

  • E. L. Lawler, C. U. Martel, “Flow Network Formulations of Polymatroid Optimization Problems”, Annals Discrete Math., 16 (1982 B) 189–200.

    Google Scholar 

  • E. L. Lawler, J. K. Lenstra, “Machine Scheduling with Precedence Constraints”, in Ordered Sets, ( I. Rival, ed.), D. Reidel, Dordrecht, Holland, 1982, pp. 655–675.

    Google Scholar 

  • E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, “Recent Developments in Deterministic Sequencing and Scheduling: A Survey”, in Deterministic Sequencing and Scheduling, (M. A. H. Dempster, et al., eds.), D. Reidel Co., Dordrecht, Holland, 1982 A, pp. 35–74.

    Google Scholar 

  • E. L. Lawler, J. K. Lenstra, A. H. G. Rinnooy Kan, “A Gift for Alexander!: At Play in the Fields of Scheduling Theory”, OPTIMA, Mathematical Programming Society Newsletter, No. 7, (1982 B).

    Google Scholar 

  • E. L. Lawler, M. G. Luby, V. V. Vazirani, “Scheduling Open Shops with Parallel Machines”, (1982), to appear in Operations Res. Letters.

    Google Scholar 

  • E. L. Lawler, J. M. Moore, “A Functional Equation and Its Application to Resource Allocation and Sequencing Problems”, Management Sci, 16 (1969) 77–84.

    Article  MATH  Google Scholar 

  • E. L. Lawler, B. D. Sivazlian, “Minimization of Time Varying Costs in Single Machine Scheduling”, Operations Res, 26 (1978) 563–569.

    Article  MathSciNet  MATH  Google Scholar 

  • J. K. Lenstra, Sequencing by Enumerative Methods, Mathematical Centre Tracts 69, Mathematisch Centrum, Amsterdam, 1977.

    Google Scholar 

  • J. K. Lenstra, A. H. G. Rinnooy Kan, “Complexity of Scheduling under Precedence Constraints”, Operations Res, 26 (1978) 22–35.

    Article  MathSciNet  MATH  Google Scholar 

  • C. Martel, “Scheduling Uniform Machines With Release Times, Deadlines and Due Times”, J. Assoc. Comput. Mach., (1981), to appear.

    Google Scholar 

  • R. McNaughton, “Scheduling With Deadlines and Loss Functions”, Management Sci, 6 (1959) 1–12.

    Article  MathSciNet  MATH  Google Scholar 

  • L. G. Mitten, “Sequencing n Jobs On Two Machines With Arbitrary Time Lags”, Management Sci, 5 (1958) 293–298.

    Article  MathSciNet  Google Scholar 

  • C. L. Monma, A. H. G. Rinnooy Kan, “Efficiently Solvable Special Cases of the Permutation Flow-Shop Problem”, Report 8105, (1981) Erasmus University, Rotterdam.

    Google Scholar 

  • C. L. Monma, J. B. Sidney, “Sequencing With Series-Parallel Precedence Constraints”, Math. Oper. Res, 4 (1979) 215–224.

    Article  MathSciNet  MATH  Google Scholar 

  • J. M. Moore, “An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs”, Management Sci, 15 (1968) 102–109.

    Article  MATH  Google Scholar 

  • J. F. Muth, G. L. Thompson, eds., Industrial Scheduling, Prentice Hall, Englewood Cliffs, N. J., 1963, p. 236.

    Google Scholar 

  • S. S. Reddi, C. V. Ramamoorthy, “On the Flow-Shop Sequencing Problem With No Wait in Process”, Oper. Res. Quart, 23 (1972) 323–331.

    Article  MATH  Google Scholar 

  • M. H. Rothkopf, “Scheduling Independent Tasks on Parallel Processors”, Management Sci, 12 (1966) 437–447.

    Article  MathSciNet  Google Scholar 

  • S. Sahni, Y. Cho, “Nearly On Line Scheduling of a Uniform Processor System With Release Times”, SIAM J. Comput, 8 (1979) 275–285.

    Article  MathSciNet  MATH  Google Scholar 

  • S. Sahni, Y. Cho, “Scheduling Independent Tasks With Due Times on a Uniform Processor System”, J. Assoc. Comput. Mach, 27 (1980) 550–563.

    MathSciNet  Google Scholar 

  • J. B. Sidney, “An Extension of Moore’s Due Date Algorithm”, in Symposium on the Theory of Scheduling and Its Applications, Lecture Notes in Economics and Mathematical Systems 86, ( S. E. Elmaghraby, ed.), Springer, Berlin, 1973, pp. 393–398.

    Google Scholar 

  • J. B. Sidney, “Decomposition Algorithms for Single-Machine Sequencing With Precedence Relations and Deferral Costs”, Oper. Res, 23 (1975) 283–298.

    Article  MathSciNet  MATH  Google Scholar 

  • J. B. Sidney, “The Two-Machine Maximum Flow Time Problem With Series Parallel Precedence Relations”, Oper. Res, 27 (1979) 782–791.

    Article  MathSciNet  MATH  Google Scholar 

  • B. Simons, “A Fast Algorithm for Single Processor Scheduling”, Proc. 19 th Annual IEEE Symp. Foundations of Computer Science, (1978) 50–53.

    Google Scholar 

  • W. E. Smith, “Various Optimizers for Single-Stage Production”, Naval Res. Logist. Quart, 3 (1956) 59–66.

    Article  MathSciNet  Google Scholar 

  • J. D. Ullman, “NP-Complete Scheduling Problems”, J. Comput. System Sci, 10 (1975) 384–393.

    Article  MathSciNet  MATH  Google Scholar 

  • J. Valdes, R. E. Tarjan, E. L. Lawler, “The Recognition of Series Parallel Digraphs”, SIAM J. Computing, 11 (1982) 298–313.

    Article  MathSciNet  MATH  Google Scholar 

  • M. K. Warmuth, “M Processor Unit-Execution-Time Scheduling Reduces to M-l Weakly Connected Components”, M. S. Thesis, (1980) Department of Computer Science, University of Colorado, Boulder.

    Google Scholar 

  • G. Weiss, M. Pinedo, “Scheduling Tasks with Exponential Service Times on Non Identical Processors to Minimize Various Functions”, J. Appl. Probab, 17 (1980) 187–202.

    Article  MathSciNet  MATH  Google Scholar 

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Lawler, E.L. (1983). Recent Results in the Theory of Machine Scheduling. In: Bachem, A., Korte, B., Grötschel, M. (eds) Mathematical Programming The State of the Art. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-68874-4_9

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  • DOI: https://doi.org/10.1007/978-3-642-68874-4_9

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