Stability Radius of an Optimal Schedule: A Survey and Recent Developments

  • Yuri N. Sotskov
  • Vyacheslav S. Tanaev
  • Frank Werner
Part of the Applied Optimization book series (APOP, volume 16)


The usual assumption that the processing times of the operations are known in advance is the strictest one in deterministic scheduling theory and it essentially restricts its practical aspects. Indeed, this assumption is not valid for the most real-world processes. This survey is devoted to a stability analysis of an optimal schedule which may help to extend the significance of scheduling theory for some production scheduling problems. The terms ‘stability’, ‘sensitivity’ or ‘postoptimal analysis’ are generally used for the phase of an algorithm at which a solution (or solutions) of an optimization problem has already been found, and additional calculations are performed in order to investigate how this solution depends on the problem data. We survey some recent results in the calculation of the stability radius of an optimal schedule for a general shop scheduling problem which denotes the largest quantity of independent variations of the processing times of the operations such that this schedule remains optimal. We present formulas for the calculation of the stability radius, when the objective is to minimize mean or maximum flow time. The extreme values of the stability radius are of particular importance, and these cases are considered more in detail. Moreover, computational results on the calculation of the stability radius for randomly generated job shop scheduling problems are briefly discussed. We also show that the well-known test problem with 6 jobs and 6 machines has both stable and unstable optimal makespan schedules.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    H. Bräsel, Yu. N. Sotskov, and F. Werner, Stability of a schedule minimizing mean flow time, Math. Comput. Modell. (to appear).Google Scholar
  2. [2]
    P. Chretienne, E.G. Coffman, J.K. Lenstra, and Z. Liu, (eds), 1995, Scheduling Theory and its Applications, John Wiley Sons, New York, NY.Google Scholar
  3. [3]
    H. Fisher, and M.L. Thompson, 1963, Probabilistic learning combinations of local job-shop scheduling rules, in: Industrial Scheduling; Muth, J.F. and Thompson, G.L. (eds.), Prentice-Hall, Englewood Cliffs, 225 - 251.Google Scholar
  4. [4]
    E.N. Gordeev, 1987, Algorithms of polynomial complexity for computing the stability radius in two classes of trajectory problems, U.S.S.R. Comput. Maths. Math. Phys., 27, 14 - 20.CrossRefGoogle Scholar
  5. [5]
    E.N. Gordeev, 1989, Solution stability of the shortest path problem, Discrete Math, 1, 45 - 56 (in Russian).Google Scholar
  6. [6]
    E.N. Gordeev, and V.K. Leontev, 1980, Stability in bottleneck problems, U.S.S.R. Comput. Maths. Math. Phys. 20, 275-280.Google Scholar
  7. [7]
    E.N. Gordeev, and V.K. Leontev, 1985, The complexity of the tabulation of trajectory problems, U.S.S.R. Comput. Maths. Math. Phys., 25, 199 - 201.CrossRefGoogle Scholar
  8. [8]
    E.N. Gordeev, V.K. Leontev, and I.Ch. Sigal, 1983, Computational algorithms for finding stability radius in choice problems, U.S.S.R. Comput. Maths. Math. Phys., 23, 128 - 132.CrossRefGoogle Scholar
  9. [9]
    A.W.H. Kolen, A.H.G. Rinnooy Kan, C.P.M. van Hoesel, and A.P.M. Wagelmans, 1994, Sensitivity analysis of list scheduling algorithms, Disc r. Appl. Math., 55, 145 - 162.CrossRefGoogle Scholar
  10. [10]
    S.A. Kravchenko, Yu.N. Sotskov, and F. Werner, 1995, Optimal schedules with infinitely large stability radius, Optimization, 33, 271 - 280.CrossRefGoogle Scholar
  11. [11]
    E.L. Lawler, J.K. Lenstra, A.H.G. Rinnooy Kan, and D.B. Shmoys, 1993, Sequencing and Scheduling: Algorithms and Complexity, in Logistics of Production and Inventory, Handbook in Operations Research and Management Science 4; G.C. Graves, A.H.G. Rinnooy Kan and P.H. Zipkin (eds.), North Holland, Amsterdam, 445 - 522.Google Scholar
  12. [12]
    V.K. Leontev, 1975, The stability of the traveling salesman problem, U.S.S.R. Comput. Maths. Math. Phys., 15, 199 - 213.CrossRefGoogle Scholar
  13. [13]
    V.K. Leontev, 1976, Stability in combinatorial choice problems, Soviet Mathematics Doklady, 17, 635 - 638.Google Scholar
  14. [14]
    M. Libura, 1991, Sensitivity analysis for minimum Hamiltonian path and traveling salesman problems, Discr. Appl. Math., 30, 197 - 211.CrossRefGoogle Scholar
  15. [15]
    M. Libura, On accuracy of solutions for discrete optimization problems with perturbed objective functions, Ann. Oper. Res. (submitted).Google Scholar
  16. [16]
    M. Libura, E.S. van der Poort, G. Sierksma, and J.A.A. van der Veen, 1996, Sensitivity analysis based on k-best solutions of the traveling salesman problem, Research report 96A14, University of Groningen.Google Scholar
  17. [17]
    O.I. Melnikov, 1978, Optimal schedule stability for the Bellman-Johnson problem, Vesti Akad. Navuk BSSR, Ser. Fiz.-Mat. Navuk. 6, 99-101 (in Russian).Google Scholar
  18. [18]
    R. Ramaswamy, and N. Chakravarti, 1995, Complexity of determining exact tolerances for min-sum and min-max combinatorial optimization problems, Report WPS-247/95, Indian Institute of Management, Calcutta.Google Scholar
  19. [19]
    J. Picard, and M. Queyranne, 1978, The time-dependent traveling salesman problem and its application to the tardiness problem in one machine scheduling, Oper. Res., 26, 86 - 110.CrossRefGoogle Scholar
  20. [20]
    M. Pinedo, 1995, Scheduling. Theory, Algorithms, and Systems, Prentice-Hall, Englewood Cliffs, New Jersey.Google Scholar
  21. [21]
    D.R. Shier, and G. Witzgall, 1980, Arc tolerances in shortest path and network flow problems, Networks, 10, 277 - 291.CrossRefGoogle Scholar
  22. [22]
    Yu.N. Sotskov, 1989, The stability of high-speed optimal schedules, U.S.S.R. Comput. Maths. Math. Phys., 29, 57 - 63.CrossRefGoogle Scholar
  23. [23]
    Yu.N. Sotskov, 1991, Stability of an optimal schedule, European J. Oper. Res., 55, 91 - 102.CrossRefGoogle Scholar
  24. [24]
    Yu.N. Sotskov, 1993, The stability of the approximate Boolean minimization of a linear form, Comput. Maths. Math. Phys., 33, 699 - 707.Google Scholar
  25. [25]
    Yu.N. Sotskov, N.Y. Sotskova, and F. Werner, 1996, Stability of an optimal schedule in a job shop, Preprint 15/96, Otto-von-Guericke-Universität Magdeburg, FMA.Google Scholar
  26. [26]
    Yu.N. Sotskov, and F. Werner, 1995, On the calculation of the stability radius of an optimal or an approximate schedule, Preprint 23/95, Ottovon-Guericke-Universität Magdeburg, FMA.Google Scholar
  27. [27]
    Yu.N. Sotskov, V.K. Leontev, and E.N. Gordeev, 1995, Some concepts of stability analysis in combinatorial optimization, Discr. Appl. Math., 58, 169 - 190.CrossRefGoogle Scholar
  28. [28]
    B. Sussmann, 1972, Scheduling problems with interval disjunctions, Oper. Res., 16, 165 - 178.Google Scholar
  29. [29]
    V.S. Tanaev, Yu.N. Sotskov, and V.A. Strusevich, 1994, Scheduling Theory, Multi-Stage Systems,Kluwer Academic Publishers.Google Scholar
  30. [30]
    R.E. Tarjan, 1982, Sensitivity analysis of minimum spanning trees and shortest path trees, Inform. Processing Letters, 14, 30 - 33.CrossRefGoogle Scholar
  31. [31]
    G.R. Wilson, and H.K. Jain, 1988, An approach to postoptimality and sensitivity analysis of zero-one goal programs, Naval Res. Log., 35, 73 - 84.CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 1998

Authors and Affiliations

  • Yuri N. Sotskov
    • 1
  • Vyacheslav S. Tanaev
    • 1
  • Frank Werner
    • 2
  1. 1.Institute of Engineering CyberneticsMinskBelarus
  2. 2.Fakultät für MathematikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

Personalised recommendations