Abstract
We show that them-machine open shop problem in which all operations have unit processing times can be polynomially transformed to a special preemptive scheduling problem onm identical parallel machines. Many results published recently as well as some new results are derived by using this transformation. The new results include solutions of open problems mentioned in a recent paper by Kubiak et al. p]A similar relationship is derived between no-wait open shop problems with unit time operations andm-machine problems with jobs having unit processing times.
Similar content being viewed by others
References
Bräsel H (1990) Lateinische Rechtecke und Maschinenbelegung, Dissertation B, TU Magdeburg
Bräsel H, Kluge D, Werner F (1991) A polynomial algorithm for then¦m¦O¦t ij =1;tree¦C max problem, to appear in EJOR
Bräsel H, Kluge D, Werner F (1991a) A polynomial algorithm for an open shop problem with unit processing times and tree constraints, Preprint, Technische Universität Otto von Guericke, Magdeburg
Bräsel H, Kluge D, Werner F (1992) A polynomial time algorithm for a 2-machine open shop problem with precedence constraints and unit processing times, Preprint, Technische Universität Otto von Guericke, Magdeburg
Brucker P, Garey MR, Johnson DS (1977) Scheduling equal-length tasks under treelike precedence constrains to minimize maximum lateness. Math. Oper. Res. 2:275–284
Coffman EG, Graham RL (1972) Optimal Scheduling for two-processor systems. Acta Informatika 1:200–213
Gabow HN, Kariv O (1982) Algorithms for edge coloring bipartite graphs and multigraphs. SIAM J. Comput. 11:117–129
Garey MR, Johnson DS (1976) Scheduling tasks with nonuniform deadlines on two processors. J. Assoc. Comput. Mach. 23:461–467
Garey MR, Johnson DS (1977) Two-processor scheduling with start-times and deadlines. SIAM J. Comput. 6:416–426
Gonzalez T (1982) Unit execution time shop problems. Math. Oper. Res. 7:57–66
Gonzalez T, Johnson DS (1980) A new algorithm for preemptive scheduling of trees. J. Assoc. Comput. Mach. 27:287–312
Graham RE, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1979) Optimization and approximation in deterministic sequencing and scheduling: a survey. Ann. Discrete Math. 5:287–326
Horn WA (1974) Some simple scheduling algorithms. Naval Res. Logist. Quart. 21:177–185
Hu TC (1961) Parallel sequencing and assembly line problems. Oper. Res. 9:841–848
Jackson JR (1955) Scheduling a production line to minimize maximum tardiness, Research Report 43, Management Science Research Project, University of California, Los Angeles
Karp RM (1972) Reducibility among combinatorial problems. In: R.E. Miller, J.W. Thatcher, Complexity of Computer Computations, Plenum Press, New York, pp. 85–103
Kubiak W, Sriskandarajah C, Zaras K (1991) A note on the complexity of openshop scheduling problems. INFOR 29:284–294
Labetoulle J, Lawler EL, Lenstra JK, Rinnooy Kan AHG (1984) Preemptive scheduling of uniform machines subject to release dates. In: WR Pulleyblank (ed.), Progress in combinatorial optimization, Academic Press, New York, pp. 245–261
Lawler EL (1976) Sequencing to minimize the weighted number of tardy jobs. RAIRO Rech. Opér. 10:5 Suppl. 27–33
Lawler EL (1982) Preemptive scheduling of precedence-constrained jobs on parallel machines. In: MAH Dempster, JK Lenstra, AHG Rinnooy Kan, Deterministic and stochastic scheduling, Reidel, Dordrecht, pp. 101–123
Lenstra JK, Rinnooy Kan AHG (1978) Complexity of scheduling under precedence constraints. Oper. Res. 26:22–35
Liu CY, Bulfin RL (1988) Scheduling open shops with unit execution times to minimize functions of due dates. Oper. Res. 36:553–559
McNaughton R (1959) Scheduling with deadlines and loss functions. Management Sci. 6:1–12
Monma CL (1979) The two machine maximum flow-time with series parallel precedence constraints: an algorithm and extensions. Oper. Res. 25:792–798
Muntz RR, Coffman EG (1969) Optimal preemptive scheduling on two processor systems. IEEE Trans. Computers C-18:1014–1020
Muntz RR, Coffman EG (1970) Preemptive scheduling of real time tasks on multiprocessor systems. J. Assoc. Comput. Mach. 17:324–338
Sethi R (1976) Algorithms for minimal length schedules. In: JR Coffman, Computer and job-shop scheduling theory. Wiley, New York, pp. 51–99
Papadimitriou CH, Steiglitz K (1982) Combinatorial optimization, Algorithms and complexity, Prentice-Hall
Simons B, Warmuth MK (1989) A fast algorithm for multiprocessor scheduling of unit length jobs. SIAM J. Comput., to appear
Tanaev VS, Sotskov YN, Strusewich VA (1989) Theory of scheduling-multistage systems, Moscow, Nauka (in Russian)
Tautenhahn T (1991) Minimizing maximal lateness for open shop witht ij =1 and release times, Preprint, Technische Universität, Magdeburg
Ullman JD (1975) NP-Complete scheduling problems. J. Comput. System Sci. 10:384–393
Author information
Authors and Affiliations
Additional information
This work was supported by Deutsche Forschungsgemeinschaft (Project JoPTAG).
Rights and permissions
About this article
Cite this article
Brucker, P., Jurisch, B. & Jurisch, M. Open shop problems with unit time operations. ZOR - Methods and Models of Operations Research 37, 59–73 (1993). https://doi.org/10.1007/BF01415528
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF01415528