Shortest path to nonpreemptive schedules of unit-time jobs on two identical parallel machines with minimum total completion time

Abstract.Ideal schedules reach both minimum maximum completion time and minimum total completion time of jobs. It is known that there exist computable in polynomial time ideal nonpreemptive two-machine schedules of unit-time operation jobs with equal release dates and arbitrary precedence constraints on identical parallel machines, in flow shops and open shops. In this paper, we study the possibility of extending these results to the case where release dates can be different. We establish the complexity status of P2|prec,rj,pj=1|∑Cj and F2|prec,rj,pij=1|∑Cj showing that optimal schedules for these problems can also be found in polynomial time and conjecture that all such schedules are ideal indeed. On the other hand, we show that the ideal schedules in open shops do not always exist.

[1]  V. S. Tanaev,et al.  Scheduling Theory: Multi-Stage Systems , 1994 .

[2]  Zhen Liu,et al.  Single Machine Scheduling Subject To Precedence Delays , 1996, Discret. Appl. Math..

[3]  David S. Johnson,et al.  Scheduling Tasks with Nonuniform Deadlines on Two Processors , 1976, J. ACM.

[4]  Harold N. Gabow,et al.  An Almost-Linear Algorithm for Two-Processor Scheduling , 1982, JACM.

[5]  Rolf H. Möhring,et al.  Computing the bump number is easy , 1988 .

[6]  Edward G. Coffman,et al.  Ideal preemptive schedules on two processors , 2003, Acta Informatica.

[7]  Oliver Vornberger,et al.  On Some Variants of the Bandwidth Minimization Problem , 1984, SIAM J. Comput..

[8]  Vadim G. Timkovsky,et al.  Identical parallel machines vs. unit-time shops and preemptions vs. chains in scheduling complexity , 2003, Eur. J. Oper. Res..

[9]  Jan Karel Lenstra,et al.  Complexity results for scheduling chains on a single machine : (preprint) , 1980 .

[10]  Eugene L. Lawler,et al.  Sequencing and scheduling: algorithms and complexity , 1989 .

[11]  David S. Johnson,et al.  Two-Processor Scheduling with Start-Times and Deadlines , 1977, SIAM J. Comput..

[12]  John L. Bruno,et al.  Deterministic Scheduling with Pipelined Processors , 1980, IEEE Transactions on Computers.

[13]  Ronald L. Graham,et al.  Optimal scheduling for two-processor systems , 1972, Acta Informatica.

[14]  M. Fujii,et al.  Optimal Sequencing of Two Equivalent Processors , 1969 .

[15]  Peter Brucker,et al.  Open shop problems with unit time operations , 1993, ZOR Methods Model. Oper. Res..