Abstract
The article focuses on effective single-machine scheduling algorithms. We consider the optimization of a functionf defined on feasible permutations assuming that the functionf induces certain job interchange relations. The interchange relations include “job insertion,” interchange of symbol chains, and the mutually complementary properties of interchange and embedding. Some new nontraditional problem formulations are considered together with the corresponding methods of solution.
Similar content being viewed by others
Literature cited
V. Ya. Burdyuk and V. N. Reva, “An optimization method for functionals on permutations subject to constraints,” Kibernetika, No. 1, 99–103 (1980).
V. G. Vizing, “Schedules observing job due dates,” Kibernetika, No. 1, 128–135 (1981).
V. S. Gordon and E. P. Chegolina, “Single-stage scheduling problem with a tree-ordered job set,” Izv. Akad. Nauk BSSR, Fiz.-Mat. Ser., No. 4, 36–40 (1977).
V. S. Gordon and Ya. M. Shafranskii, “Optimal sequencing for series-parallel constraints,” Dokl. Akad. Nauk BSSR, No. 3, 244–247 (1978).
V. S. Gordon and Ya. M. Shafranskii, “Function minimization on the set of permutations of partially ordered elements,” Vestsi Akad. Nauk BSSR, Fiz.-Mat. Ser., No. 2, 122–124 (1979).
A. M. Danil'chenko and A. V. Panishev, “Dichotomic search for a solution of one sequencing problem,” Kibernetika, No. 2, 118–121 (1980).
V. V. Kotsubo, “Scheduling on a single machine with job start and due times,” Vestsi Akad. Nauk BSSR, Fiz.-Mat. Ser., No. 3 50–55 (1970).
N. B. Lebedinskaya, “Minimizing the maximum deviation for preemptive jobs,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,80, 117–124 (1978).
N. B. Lebedinskaya, “Minimizing the maximum penalty for preemptive jobs,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst,102, 61–67 (1980).
A. V. Maksimenkov, “Scheduling of computer resources with minimal disproportions,” Avtomat. Vychisl. Tekh., No. 4, 69–75 (1978).
A. V. Maksimenkov, “Scheduling with constraints on resource consumption rate,” Kibernetika, No. 6, 91–95 (1979).
V. S. Tanaev, “Optimization of functions recursively defined on permutation sets,” Vestsi Akad. Nauk BSSR, Fiz.-Mat. Ser., No. 3, 27–30 (1977).
V. S. Tanaev and V. V. Shkurba, Introduction to the Theory of Scheduling [in Russian], Moscow (1975).
K. V. Shakhbazyan, “On structured solutions in scheduling problems,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,80, 239–248 (1978).
K. V. Shakhbazyan, “Sequencing a structured set of jobs to minimize the total penalty,” Zap, Nauchn. Sem. Leningr. Otd. Mat. Inst.90, 229–264 (1979).
K. V. Shakhbazyan, “Structured solutions in scheduling theory,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,102, 139–146 (1980).
K. V. Shakhbazyan, “On scheduling problems of type η¦1 ∑ ci (t),” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.102, 147–155 (1980).
K. V. Shakhbazyan and N. B. Lebedinskaya, “Minimizing the total penalty in parallel sequencies of n independent jobs,” Dokl. Akad. Nauk SSSR,237, No. 4, 790–792 (1977).
K. V. Shakhbazyan and N. B. Lebedinskaya, “Optimal preemptive schedules for independent jobs in a N-server queuing system,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,70, 205–231 (1977).
K. V. Shakhbazyan and N. B. Lebedinskaya, “Structural solutions for scheduling problems,” Preprint Leningr. Otd. Mat. Inst., E-3-81, Leningrad (1981).
H. M. Abdel-Wahab and T. Kameda, “Scheduling to minimizing maximum cumulative cost subject to series-parallel precedence constraints,” Oper. Res.,26, 141–158 (1978).
D. L. Adolphson, “Single-machine job sequencing with precedence constraints,” SIAM. J. Comput.,6, No. 1, 40–54 (1977).
K. R. Baker, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, “Preemptive scheduling of a single machine to minimize maximum cost subject to release dates and precedence constraints,” Tech. Rep. 802810, Erasmus Univ., Rotterdam.
R. Bellman, “Mathematical aspects of scheduling theory,” J. Soc. Industr. Appl. Math.,4, No. 3, 168–205 (1956),
P. Bratley, M. Florian, and P. Robillard, “Scheduling with earliest start and due date constraints,” Nav. Res. Logist. Q.,18, 511–517 (1971).
R. W. Conway, W. L. Maxwell, and L. W. Miller, Theory of Scheduling, Addison-Wesley, Reading Mass. (1967).
M. R. Garey, D. S. Johnson, B. B. Simons, and R. E. Tarjan, “Scheduling unit-time tasks with arbitrary release times and deadlines,” J. Comput. System Sci. (forthcoming).
R. L. Graham, E. L. Lawler, J. K. Lenstra, and A. H. G. Rinnooy Kan, “Optimization and approximation is deterministic sequencing and scheduling: a survey,” Ann. Discrete Math.,5, 287–326 (1979).
W. Horn, “Some simple scheduling algorithms,” Naval Res. Logist, Q.,21, 177–185 (1974).
S. M. Johnson, “Optimal two-and three-stage production schedules with setup times included,” Naval Res. Logist. Q.,1, 61–68 (1954).
P. Kas, “On a special type job sequencing problem,” Survey Math. Programming, Proc. 9th Inst. Math. Programming Symp., Budapest, Vol. 2 (1976), pp. 279–283.
E. L. Lawler, “Optimal sequencing of a single machine subject to precedence constraints,” Manag. Sci.,19, 544–546 (1973).
E. L. Lawler, “Optimal sequencing of jobs subject to series-parallel precedence constraints,” Math. Cent. Afd. Math. Beslisk. B.W.N., No. 54 (1975).
E. L. Lawler, “Sequencing problems with series-parallel precedence constraints,” Proc. Conf. on Combinatorial Optimization, Urbino, Italy (1979).
E. L. Lawler and J. Labetoulle, “On preemptive scheduling of unrelated parallel processors by linear programming,” J. ACM,25, 612–619 (1978).
E. L. Lawler and B. D. Sivazlian, “Minimization of time-varying costs in single-machine scheduling,” Oper. Res.,26, 263–269 (1978).
J. K. Lenstra, Sequencing by Enumerative Methods, Math. Centre Tracts 69, Math. Centrum, Amsterdam.
J. K. Lenstra and A. H. G. Rinnooy Kan, “Complexity results for scheduling chain on a single machine,” Econometric Institute, Erasmum Univ., Rotterdam, Rep. 7907/0 (1979).
J. K. Lenstra, A. H. G. Rinnooy Kan, and P. Brucker, “Complexity of machine scheduling problems,” Ann. Discrete Math.,1, 343–362 (1977).
C. L. Monma, “The two-machine maximum flow time problem with series-parallel precedence constraints: an algorithm and extensions,” Oper. Res.,27, 792–798 (1979).
C. L. Monma, “Sequencing to minimize the maximum job cost,” Oper. Res.,28, 942–951 (1980).
C. L. Monma and J. Sidney, “Sequencing with series-parallel precedence constraints,” Math. Oper. Res.,4, 215–224 (1979).
A. H. G. Rinnooy Kan, Machine Scheduling Problems: Classification, Complexity and Computations, Nijholf, The Hague (1976).
M. E. Rothkopf, “Scheduling independent tasks on parallel processors,” Manag. Sci.,12, 437–447 (1966).
J. B. Sidney, “Optimal single-machine scheduling with earliness and tardiness penalties,” Oper. Res.,25, 62–69 (1977).
B. Simons, “A fast algorithm for single processor scheduling,” 19th Annual Symp. on Foundations of Computer Sci. IEEE Computer Soc., Long Beach. Calif. (1978), pp. 246–252.
W. E. Smith, “Various optimizers for single-stage production,” Naval Res. Logistic Quart.,3, 59–66 (1956).
J. D. Ullman, “NP-complete scheduling problems,” J. Comput. Syst. Sci.,10, 384–393 (1975).
Additional information
Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 111, pp. 195–217, 1981.
Rights and permissions
About this article
Cite this article
Shakhbazyan, K.V., Lebedinskaya, N.B. Effective optimization methods for single-machine scheduling (Survey). J Math Sci 24, 133–148 (1984). https://doi.org/10.1007/BF01230275
Issue Date:
DOI: https://doi.org/10.1007/BF01230275