A Lower Bound for the Job Insertion Problem

This note deals with the job insertion problem in job-shop scheduling: Given a feasible schedule of n jobs and a new job which is not scheduled, the problem is to find a feasible insertion of the new job into the schedule which minimises the makespan. Since the problem is NP-hard, a relaxation method is proposed to compute a strong lower bound. Conditions under which the relaxation provides us with the makespan of the optimal insertion are derived. After the analysis of the polytope of feasible insertions, a polynomial time procedure is proposed to solve the relaxed problem. Our results are based on the theory of perfect graphs and elements of polyhedral theory.

[1]  Edward G. Coffman,et al.  Computer and job-shop scheduling theory , 1976 .

[2]  Y. Sotskov,et al.  The complexity of shop-scheduling problems with two or three jobs , 1991 .

[3]  William J. Cook,et al.  A Computational Study of the Job-Shop Scheduling Problem , 1991, INFORMS Journal on Computing.

[4]  Peter Brucker,et al.  Tabu-search for the multi-mode job-shop problem , 1998 .

[5]  Dell'AmicoMauro,et al.  Applying tabu search to the job-shop scheduling problem , 1993 .

[6]  V. Chvátal.  On certain polytopes associated with graphs , 1975 .

[7]  Mauro Dell'Amico,et al.  Applying tabu search to the job-shop scheduling problem , 1993, Ann. Oper. Res..

[8]  Luca Maria Gambardella,et al.  Effective Neighborhood Functions for the Flexible Job Shop Problem , 1998 .

[9]  R. Storer,et al.  New search spaces for sequencing problems with application to job shop scheduling , 1992 .

[10]  R. Möhring.  Algorithmic graph theory and perfect graphs , 1986 .

[11]  Frank Werner,et al.  On the application of insertion techniques for job shop problems with setup times , 1999, RAIRO Oper. Res..

[12]  M. Golumbic.  Algorithmic Graph Theory and Perfect Graphs (Annals of Discrete Mathematics, Vol 57) , 2004 .

[13]  E. Nowicki,et al.  A Fast Taboo Search Algorithm for the Job Shop Problem , 1996 .

[15]  Peter L. Hammer,et al.  Discrete Applied Mathematics , 1993 .

[16]  Tamás Kis.  Insertion techniques for job shop scheduling , 2001 .

[17]  Frank Werner,et al.  Insertion Techniques for the Heuristic Solution of the Job Shop Problem , 1995, Discret. Appl. Math..

[18]  Stéphane Dauzère-Pérès,et al.  An integrated approach for modeling and solving the general multiprocessor job-shop scheduling problem using tabu search , 1997, Ann. Oper. Res..

[19]  Stéphane Dauzère-Pérès,et al.  Multi-resource shop scheduling with resource flexibility , 1998, Eur. J. Oper. Res..

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

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

[22]  E. Balas,et al.  The one-machine problem with delayed precedence constraints and its use in job shop scheduling , 1995 .

[23]  Christian Artigues,et al.  A polynomial activity insertion algorithm in a multi-resource schedule with cumulative constraints and multiple modes , 2000, Eur. J. Oper. Res..

[24]  Jan Karel Lenstra,et al.  Job Shop Scheduling by Local Search , 1996, INFORMS J. Comput..