On Preference Orders for Sequencing Problems Or, What Hath Smith Wrought?

  • E. L. Lawler
Conference paper
Part of the NATO ASI Series book series (volume 82)


When in 1956 W. E. Smith proposed the ratio rule for solving the unconstrained weighted completion time problem, he suggested an abstraction in the form of a preference order. Over the years, the concept of a preference order has been much elaborated. It is now applied to abstract problems in optimal sequencing, with general precedence constraints being dealt with by the technique of modular decomposition. This paper provides a unified and self contained exposition of these results, with some new material on the treatment of contiguity constraints.


Internal Node Precedence Constraint Preference Order Decomposition Tree Parallel Component 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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  1. [1]
    Booth, K. S., and G. S. Leuker. Testing the consecutive ones Property, interval graphs, and graph planarity using PQ-tree algorithms. J. Comput. Syst. Sci., 13: 335–379, 1976.MATHCrossRefGoogle Scholar
  2. [2]
    Buer, H., and R. H. Moehring. A fast algorithm for the decomposition of graphs and posets. Math. Oper. Res., 8: 170–184, 1983.Google Scholar
  3. [3]
    Conway, R. W., W. L. Maxwell, and L. W. Miller. Theory of Scheduling, Addison-Wesley, 1967.Google Scholar
  4. [4]
    Carey, M. R. Optimal task sequencing with precedence constraints. Discrete Math., 4: 37–56, 1973.MathSciNetCrossRefGoogle Scholar
  5. [5]
    Horn, W. A. Single-Machine job sequencing with treelike precedence ordering and linear delay penalties. SIAM J. Appl. Math., 23: 189–202, 1972.MathSciNetMATHCrossRefGoogle Scholar
  6. [6]
    Korte, N., and R. H. Moehring. Transitive orientation of graphs with side constraints. Proc WG ‘85, Workshop on Graphtheoretic Concepts in Computer Science, H. Notemeier (ed.), Trauner Verlag, Linz, 143–160, 1985.Google Scholar
  7. [7]
    Lawler, E. L. Sequencing jobs to minimize total weighted completion time subject to precedence constraints. Ann. Disc. Math., 2: 75–90, 1978.MathSciNetMATHCrossRefGoogle Scholar
  8. [8]
    Lawler, E. L., and B. D. Sivazlian. Minimization of time-varying costs in single-machine sequencing. Operations Research, 26: 563–569, 1978.MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Monma, C. L. Sequencing to minimize the maximum job cost. Operations Research, 28: 942–951, 1980.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Monma, C. L., and J. B. Sidney. Sequencing with series parallel precedence constraints. Math. Oper. Res., 4: 215–224, 1984.MathSciNetCrossRefGoogle Scholar
  11. [11]
    Monma, C. L., and J. B. Sidney. Optimal sequencing via modular decomposition: Characterization of sequencing functions. Math. Oper. Res., 12: 22–31, 1987.MathSciNetCrossRefGoogle Scholar
  12. [12]
    Muller, J. H., and J. Spinrad. Incremental modular decomposition. J. Assoc. Comput. Mach., 36: 1–19, 1989.MathSciNetMATHGoogle Scholar
  13. [13]
    Rothkopf, M. E. Scheduling independent tasks on parallel processors. Management Science, 12: 437–447, 1966.MathSciNetCrossRefGoogle Scholar
  14. [14]
    Sidney, J. B. Decomposition algorithms for single-machine sequencing with precedence relations and deferral costs. Operations Research, 23: 283–298, 1975.MathSciNetMATHCrossRefGoogle Scholar
  15. [15]
    Sidney, J. B. The two-machine maximum flow time problem with series parallel precedence relations. Operations Research, 27: 782–791, 1979.MathSciNetMATHCrossRefGoogle Scholar
  16. [16]
    Sidney, J. B. A decomposition algorithm for sequencing with general precedence constraints. Math. Oper. Res., 6: 190–204, 1981.MathSciNetMATHCrossRefGoogle Scholar
  17. [17]
    Sidney, J. B., and G. Steiner. Optimal sequencing by modular decomposition: Polynomial algorithms. Oper. Res., 34: 606–612, 1986.MathSciNetMATHCrossRefGoogle Scholar
  18. [18]
    Smith, W. E. Various optimizers for single-stage production. Naval Res. Logist. Quart., 3: 59–66, 1956.MathSciNetCrossRefGoogle Scholar
  19. [19]
    Valdes, J., R. E. Tarjan, and E. L. Lawler. The recognition of series-parallel digraphs. SIAM J. Comput., 11: 298–313, 1982.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • E. L. Lawler
    • 1
  1. 1.Computer Science DivisionUniversity of CaliforniaBerkeleyUSA

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