A cutting plane algorithm for the unrelated parallel machine scheduling problem

This paper deals with the polyhedral structure of the scheduling problem R//Cmax. Strong valid inequalities are identified for fixed values of the maximum completion time and are used to build a cutting plane scheme from which an exact algorithm and an approximation algorithm are developed. The main algorithm includes a preprocessing phase to compute an upper bound with the list scheduling algorithm of Davis and Jaffe and a lower bound from the preemptive version of the problem. Computational results show that the proposed exact algorithm gives an optimal solution for almost all tested cases, within the fixed time and memory limits. For the few cases where the limit has been exceeded, rather good solutions are obtained. Computational requirements of both algorithms are reported for a collection of test problems and the quality of the schedules provided by the approximation algorithm is measured.

[1]  Alexander Schrijver,et al.  Theory of linear and integer programming , 1986, Wiley-Interscience series in discrete mathematics and optimization.

[2]  Klaus Jansen,et al.  Polynominal Time Approximation Schemes for General Multiprocessor Job Shop Scheduling , 2000, ICALP.

[3]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[4]  W. Pulleyblank Chapter V Polyhedral combinatorics , 1989 .

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

[6]  Edward G. Coffman,et al.  An Application of Bin-Packing to Multiprocessor Scheduling , 1978, SIAM J. Comput..

[7]  Eugene L. Lawler,et al.  On Preemptive Scheduling of Unrelated Parallel Processors by Linear Programming , 1978, JACM.

[8]  Michael Jünger,et al.  Practical problem solving with cutting plane algorithms in combinatorialoptimization , 1993, Combinatorial Optimization.

[9]  Jeffrey M. Jaffe,et al.  Algorithms for Scheduling Tasks on Unrelated Processors , 1981, JACM.

[10]  C. N. Potts,et al.  Analysis of a linear programming heuristic for scheduling unrelated parallel machines , 1985, Discret. Appl. Math..

[11]  N. Piersma,et al.  A local search heuristic for unrelated parallel machine scheduling with efficient neighborhood search , 1996 .

[12]  T.C.E. Cheng,et al.  A state-of-the-art review of parallel-machine scheduling research , 1990 .

[13]  Klaus Jansen,et al.  Polynomial time approximation schemes for general multiprocessor job shop scheduling , 2002, J. Algorithms.

[14]  Thomas D. Rogers,et al.  Cycle structure in discrete-density models , 1982, Discret. Appl. Math..

[15]  Z Liu,et al.  Scheduling Theory and its Applications , 1997 .

[16]  Stephen A. Cook,et al.  The complexity of theorem-proving procedures , 1971, STOC.

[17]  Giovanni Rinaldi,et al.  A Branch-and-Cut Algorithm for the Resolution of Large-Scale Symmetric Traveling Salesman Problems , 1991, SIAM Rev..

[18]  Michael A. Langston,et al.  Bounds for Multifit Scheduling on Uniform Processors , 1983, SIAM J. Comput..

[19]  Oscar H. Ibarra,et al.  Heuristic Algorithms for Scheduling Independent Tasks on Nonidentical Processors , 1977, JACM.

[20]  Eugene L. Lawler,et al.  Chapter 9 Sequencing and scheduling: Algorithms and complexity , 1993, Logistics of Production and Inventory.

[21]  Carsten Lund,et al.  Hardness of approximations , 1996 .

[22]  Steef L. van de Velde Duality-Based Algorithms for Scheduling Unrelated Parallel Machines , 1993, INFORMS J. Comput..

[23]  Jan Karel Lenstra,et al.  Approximation algorithms for scheduling unrelated parallel machines , 1987, 28th Annual Symposium on Foundations of Computer Science (sfcs 1987).

[24]  Chris N. Potts,et al.  Heuristics for scheduling unrelated parallel machines , 1991, Comput. Oper. Res..

[25]  Jan Karel Lenstra,et al.  Computing near-optimal schedules , 1995 .

[26]  Laurence A. Wolsey,et al.  Valid inequalities for mixed 0-1 programs , 1986, Discret. Appl. Math..

[27]  E.L. Lawler,et al.  Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey , 1977 .

[28]  Chris N. Potts,et al.  Unrelated parallel machine scheduling using local search , 1994 .

[29]  Laurence A. Wolsey,et al.  Integer and Combinatorial Optimization , 1988 .

[30]  Kazumiti Numata,et al.  APPROXIMATE AND EXACT ALGORITHMS FOR SCHEDULING INDEPENDENT TASKS ON UNRELATED PROCESSORS , 1988 .

[31]  Ellis Horowitz,et al.  Exact and Approximate Algorithms for Scheduling Nonidentical Processors , 1976, JACM.

[32]  W. R. Pulleyblank,et al.  Polyhedral Combinatorics , 1989, ISMP.

[33]  Paolo Toth,et al.  Exact and Approximation Algorithms for Makespan Minimization on Unrelated Parallel Machines , 1997, Discret. Appl. Math..

[34]  Graham K. Rand,et al.  Logistics of Production and Inventory , 1995 .