A time indexed formulation of non-preemptive single machine scheduling problems

We consider the formulation of non-preemptive single machine scheduling problems using time-indexed variables. This approach leads to very large models, but gives better lower bounds than other mixed integer programming formulations. We derive a variety of valid inequalities, and show the role of constraint aggregation and the knapsack problem with generalised upper bound constraints as a way of generating such inequalities. A cutting plane/branch-and-bound algorithm based on these inequalities has been implemented. Computational experience on small problems with 20/30 jobs and various constraints and objective functions is presented.

[1]  Egon Balas,et al.  The Shifting Bottleneck Procedure for Job Shop Scheduling , 1988 .

[2]  J. Carlier The one-machine sequencing problem , 1982 .

[3]  J. Sousa,et al.  Time Indexed Formulations Of Non-Preemptive Single-Machine Schduling Problems , 1989 .

[4]  Laurence A. Wolsey,et al.  Valid inequalities for 0-1 knapsacks and mips with generalised upper bound constraints , 1990, Discret. Appl. Math..

[5]  J. Patterson,et al.  Scheduling a Project to Maximize Its Present Value: A Zero-One Programming Approach , 1977 .

[6]  F. Brian Talbot,et al.  Resource-Constrained Project Scheduling with Time-Resource Tradeoffs: The Nonpreemptive Case , 1982 .

[7]  L. V. Wassenhove,et al.  An algorithm for single machine sequencing with deadlines to minimize total weighted completion time , 1983 .

[8]  Egon Balas,et al.  Facets of the knapsack polytope , 1975, Math. Program..

[9]  Martin E. Dyer,et al.  Formulating the single machine sequencing problem with release dates as a mixed integer program , 1990, Discret. Appl. Math..

[10]  Philip M. Wolfe,et al.  Multiproject Scheduling with Limited Resources: A Zero-One Programming Approach , 1969 .

[11]  E. L. Lawler,et al.  Recent Results in the Theory of Machine Scheduling , 1982, ISMP.

[12]  Marc E. Posner,et al.  Minimizing Weighted Completion Times with Deadlines , 1985, Oper. Res..

[13]  E. H. Bowman THE SCHEDULE-SEQUENCING PROBLEM* , 1959 .

[14]  S. Bansal Single machine scheduling to minimize weighted sum of completion times with secondary criterion -- A branch and bound approach , 1980 .

[15]  J. H. Patterson,et al.  An Algorithm for a general class of precedence and resource constrained scheduling problems , 1989 .

[16]  Stéphane Dauzère-Pérès The one-machine sequencing problem with dependent jobs , 1993 .

[17]  Jan Karel Lenstra,et al.  Complexity of machine scheduling problems , 1975 .

[18]  Graham K. Rand Teaching operational research to first year undergraduates: Experience at Lancaster , 1980 .

[19]  T. S. Abdul-Razaq,et al.  Dynamic Programming State-Space Relaxation for Single-Machine Scheduling , 1988 .

[20]  Jan Karel Lenstra,et al.  Recent developments in deterministic sequencing and scheduling: a survey : (preprint) , 1981 .

[21]  Chris N. Potts,et al.  An algorithm for single machine sequencing with release dates to minimize total weighted completion time , 1983, Discret. Appl. Math..

[22]  D. A. Wismer,et al.  A mixed integer programming model for scheduling orders in a steel mill , 1974 .

[23]  Maurice Queyranne,et al.  The Time-Dependent Traveling Salesman Problem and Its Application to the Tardiness Problem in One-Machine Scheduling , 1978, Oper. Res..

[24]  B. J. Lageweg,et al.  Minimizing Total Costs in One-Machine Scheduling , 1975, Oper. Res..

[25]  J. M. Tamarit,et al.  Project scheduling with resource constraints: A branch and bound approach , 1987 .