Optimal parallel machines scheduling with availability constraints

We address a generalization of the classical multiprocessor scheduling problem with non simultaneous machine availability times, release dates, and delivery times. We develop new lower and upper bounds as well as a branching strategy which is based on a representation of a schedule as a permutation of jobs. We show that embedding a semi-preemptive lower bound based on max-flow computations in a branch-and-bound algorithm yields very promising performance. Computational experiments demonstrate that randomly generated instances with up to 700 jobs and 20 machines are solved within moderate CPU time. Moreover, the versatility of the proposed approach is assessed through its ability to solve large instances of two important particular cases P,NC"i"n"c||C"m"a"x and P|r"j,q"j|C"m"a"x.

[1]  Jacques Carlier,et al.  An Exact Method for Solving the Multi-Processor Flow-Shop , 2000, RAIRO Oper. Res..

[2]  Mohamed Haouari,et al.  An improved max-flow-based lower bound for minimizing maximum lateness on identical parallel machines , 2003, Oper. Res. Lett..

[3]  J. Erschler,et al.  Ordonnancement de tâches sous contraintes: une approche énergetique , 1992 .

[4]  Philippe Baptiste,et al.  Solving hybrid flow shop problem using energetic reasoning and global operations , 2001 .

[5]  Philippe Baptiste,et al.  Tight LP bounds for resource constrained project scheduling , 2004, OR Spectr..

[6]  David Pisinger,et al.  Dynamic Programming on the Word RAM , 2003, Algorithmica.

[7]  Eric Pinson,et al.  Jackson's Pseudo Preemptive Schedule for the Pm/ri, qi/Cmax scheduling problem , 1998, Ann. Oper. Res..

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

[9]  Chung-Yee Lee,et al.  Parallel machines scheduling with nonsimultaneous machine available time , 1991, Discret. Appl. Math..

[10]  Jacques Carlier,et al.  Scheduling jobs with release dates and tails on identical machines to minimize the makespan , 1987 .

[11]  Günter Schmidt,et al.  Scheduling with limited machine availability , 2000, Eur. J. Oper. Res..

[12]  Abdelkader Lahrichi,et al.  Ordonnancements. La notion de «parties obligatoires» et son application aux problèmes cumulatifs , 1982 .

[13]  H. Kellerer Algorithms for multiprocessor scheduling with machine release times , 1998 .

[14]  W. A. Horn Some simple scheduling algorithms , 1974 .

[15]  Chung-Yee Lee,et al.  A note on "parallel machine scheduling with non-simultaneous machine available time" , 2000, Discret. Appl. Math..

[16]  Paolo Toth,et al.  An exact algorithm for the subset sum problem , 2002, Eur. J. Oper. Res..

[17]  S. Webster A general lower bound for the makespan problem , 1996 .

[18]  Eugene L. Lawler,et al.  Preemptive scheduling of uniform machines subject to release dates : (preprint) , 1979 .

[19]  Philippe Baptiste,et al.  Satisfiability tests and time‐bound adjustmentsfor cumulative scheduling problems , 1999, Ann. Oper. Res..

[20]  J. Carlier,et al.  Une méthode arborescente pour résoudre les problèmes cumulatifs , 1991 .

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

[22]  Paolo Toth,et al.  Knapsack Problems: Algorithms and Computer Implementations , 1990 .

[23]  Mohamed Haouari,et al.  Minimizing makespan on parallel machines subject to release dates and delivery times , 2002 .

[24]  Hans Kellerer,et al.  An efficient fully polynomial approximation scheme for the Subset-Sum Problem , 2003, J. Comput. Syst. Sci..