Table of Contents. Best-Case Lower Bounds in a Group Sequence for the Job Shop Problem. Group Sequencing. Introduction IFAC 2008 WC

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1 Table of Contents Best-Case Lower Bounds in a Group Sequence for the Job Shop Problem 1 Introduction Group Sequencing Guillaume Pinot Nasser Mebarki 3 The Best-Case Completion Time of an Operation IRCCyN UMR CNRS 97 Nantes, France firstname.lastname@irccyn.ec-nantes.fr IFAC 008 WC Lower Bounds Experiments Conclusion Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 1/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence / Introduction Group Sequencing Group sequencing: is a scheduling method; describes a set of schedules; guarantees a minimal quality corresponding to the worst case. A best-case evaluation of a group sequence could be interesting. Group sequencing: provides sequential flexibility during the execution of the schedule; guarantees a minimal quality corresponding to the worst case. To manage sequential flexibility, usage of groups of permutable operations. Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence / Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence /

2 Example: a Job Shop Problem i: the index of the operations, Γ (i): the set of the predecessors of O i, m i : the resource needed by O i, p i : the processing time needed by O i. A Job Shop Problem i Γ (i) {1} {} {} {} {7} {8} m i p i A Solution to This Problem 7 1, 8 3, Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 7/ 9 Execution of the Example The Group Sequence 1, , The Corresponding Semi-Active Schedules Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 8/ Why is Group Sequencing Interesting? Why is group sequencing interesting? predictive reactive method; flexibility on sequences; widely studied in the last twenty years: [Erschler and Roubellat, 1989, Wu et al., 1999, Artigues et al., 00] no need to model the uncertainties; the method is able to absorb some uncertainties: [Wu et al., 1999, Esswein, 003, Pinot et al., 007]; evaluation of the group sequence in the worst case in polynomial time for minmax regular objectives as C max and L max. The best-case evaluation of a group sequence could be usefull. Algorithms Intuitive Formulation θ i = max χ i = θ i + p i Improved Formulation θ i = max θ i χ i χ i = θ i + p i ( ) r i, max χ j, max χ j g (i) j Γ j (i) ( ) r i, γ g (i), max χ j Γ j (i) = C max of 1 r i C max, O i g l,k, r i = θ i Lower bound of the starting time of O i Lower bound of the completion time of O i Lower bound of the completion time of g l,k Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 9/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 11/

3 Example The Problem i Γ (i) m i p i g(i) 1 1 g 1,1 {1} g,1 3 3 g 1,1 {3} g,1 1 g 1, {} 1 g, The Group Sequence 1, 3, Intuitive Formulation Improved Formulation Optimal Solution Simple Lower Bound It can be used directly to compute a lower bound of the group sequence: LB(L max ) = max L i (χ i ) = max(χ i d i ) O i O i LB(C max ) = max g l,k (Natural LB) Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 1/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 1/ Makespan Lower Bound Classical job-shop lower bound: one-machine-problem relaxation [Carlier, 198] on each machine. The one-machine-problem relaxation require some tools: a head for each operations: θ i ; a tail for each operations: a reversed θ i. For group sequencing the relaxation is done on groups instead of machines (more subproblems, but smaller). Solving the one-machine problems: using Jackson Preemptive Schedule: JPS OMP LB; using the exact Carlier s algorithm [Carlier, 198]: Optimal OMP LB. Gaps Instances : la01 to la0 from [Lawrence, 198]. For each instances, we generate a group sequence with a known optimal makespan[brucker et al., 199]; a very high flexibility [Esswein, 003]. Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 1/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 17/

4 Other results Conclusion Computation times: time(optimal OMP LB) time(jps OMP LB) time(natural LB) In an exact method using these lower bounds: 10 time(exact(optimal OMP LB)) time(exact(jps OMP LB)) We have proposed: different lower-bound tools; lower bounds. They can be used directly: more complet description of a group sequence in its globality; its usage in a decision support system gives additional information to the operator. These tools can also be usefull in: heuristics; exact methods. Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 18/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 0/ Thank You Bibliography I Artigues, C., Billaut, J.-C., and Esswein, C. (00). Maximization of solution flexibility for robust shop scheduling. European Journal of Operational Research, 1(): Thank you for your attention. Brucker, P., Jurisch, B., and Sievers, B. (199). A branch and bound algorithm for the job-shop scheduling problem. Discrete Applied Mathematics, 9(1-3): Carlier, J. (198). The one-machine sequencing problem. European Journal of Operational Research, 11(1): 7. Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 1/ Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence /

5 Bibliography II Erschler, J. and Roubellat, F. (1989). An approach for real time scheduling for activities with time and resource constraints. In Slowinski, R. and Weglarz, J., editors, Advances in project scheduling. Elsevier. Esswein, C. (003). Un apport de flexibilité séquentielle pour l ordonnancement robuste. Thèse de doctorat, Université François Rabelais Tours. Lawrence, S. (198). Resource constrained project scheduling: an experimental investigation of heuristic scheduling techniques (supplement). Technical report, Graduate School of Industrial Administration, Carnegie-Mellon University, Pittsburgh, Pennsylvania. Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence 3/ Bibliography III Pinot, G., Cardin, O., and Mebarki, N. (007). A study on the group sequencing method in regards with transportation in an industrial FMS. In Proceedings of the IEEE SMC 007 International Conference. Wu, S. D., Byeon, E.-S., and Storer, R. H. (1999). A graph-theoretic decomposition of the job shop scheduling problem to achieve scheduling robustness. Operations Research, 7(1): Guillaume Pinot, Nasser Mebarki Best-Case Lower Bounds in a Group Sequence /