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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution. Philippe LACOMME, Mohand LARABI Nikolay TCHERNEV LIMOS (UMR CNRS 6158), Clermont Ferrand, France IUP « Management et gestion des entreprises ».

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Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

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  1. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Philippe LACOMME, Mohand LARABI Nikolay TCHERNEV LIMOS (UMR CNRS 6158), Clermont Ferrand, France IUP « Management et gestion des entreprises »

  2. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Plan • Plan • Introduction • Algorithm based framework • Computational evaluation • Conclusions and further works IESM 2009, MONTREAL – CANADA, May 13 TR 2

  3. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV FMS definition IESM 2009, MONTREAL – CANADA, May 13 TR 3

  4. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV • AGV system Guide path layout Automated Guided Vehicles IESM 2009, MONTREAL – CANADA, May 13 TR 4

  5. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV Flexible machines IESM 2009, MONTREAL – CANADA, May 13 TR 5

  6. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV Flexible cells IESM 2009, MONTREAL – CANADA, May 13 TR 6

  7. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV Input/Output buffers IESM 2009, MONTREAL – CANADA, May 13 TR 7

  8. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction AGV operating (1/2) IESM 2009, MONTREAL – CANADA, May 13 TR 8

  9. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction AGV operating (2/2) • There are two types of vehicle trips: • the first type of loaded vehicle trips ; • the second one is the empty vehicle trips. IESM 2009, MONTREAL – CANADA, May 13 TR 9

  10. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (1/5) • Problem definition • The scheduling problem under study can be defined in the following general form: • Given a particular FMS with several vehicles and a set of jobs, • the objective is to determine the starting and completion times of operations for each job on each machine • and the vehicle trips between machines according to makespan or mean completion time minimization. IESM 2009, MONTREAL – CANADA, May 13 TR 10

  11. Empty trip Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (2/5) • Problem definition : Example of solution IESM 2009, MONTREAL – CANADA, May 13 TR 11

  12. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (3/5) • Problem definition : Complexity • Combined problem of: • scheduling problem of the form(n jobs, M machines, G general job shop, Cmax makespan), a well known NP-hard problem (Lenstra and Rinnooy Kan 1978); • a generic Vehicle Scheduling Problem (VSP) which is NP-hard problem (Orloff 1976). IESM 2009, MONTREAL – CANADA, May 13 TR 12

  13. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (4/5) • Problem definition : Assumptions in the literature • All jobs are assumed to be available at the beginning of the scheduling period. • The routing of each job types is available before making scheduling decisions. • All jobs enter and leave the system through the load and unload stations. • It is assumed that there is sufficient input/output buffer space at each machine and at the load/unload stations, i.e. the limited buffer capacity is not considered. • Vehicles move along predetermined shortest paths, with the assumption of no delay due to the congestion. • Machine failures are ignored. • Limitations on the jobs simultaneously allowed in the shop are ignored. IESM 2009, MONTREAL – CANADA, May 13 TR 13

  14. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (5/5) • Under these hypotheses the problem can be without doubt modelled as a job shop with several transport robots. • notation introduced by Knust 1999 • J indicates a job shop, • R indicates that we have a limited number of identical vehicles (robots) and all jobs can be transported by any of the robots. • indicates that we have job-independent, but machine-dependant transportation times. • indicates that we have machine-dependant empty moving time. • The objective function to minimize is the makespan . IESM 2009, MONTREAL – CANADA, May 13 TR 14

  15. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework General template • General template IESM 2009, MONTREAL – CANADA, May 13 TR 15

  16. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph definition (1/3) • Non oriented disjunctive graph • consists of: • Vm : a set of vertices containing all machine operations; • Vt : a set of vertices containing all transport operations; • C : representing precedence constraints in the same job; • Dm : containing all machine disjunctions; • Dr : containing all transport disjunctions. IESM 2009, MONTREAL – CANADA, May 13 TR 16

  17. 5 7 M2 M3 M1 4 0 4 0 5 1 M3 M4 M1 0 0 3 5 5 M5 M1 M3 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph definition (2/3) J1 J2 * J3 IESM 2009, MONTREAL – CANADA, May 13 TR 17

  18. Machine disjunction problem Robot disjunction problem Robot assignment problem Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph definition (2/2) 5 8 M2 r1 M3 r2 M1 4 0 4 0 5 1 M3 r1 M4 M1 0 * 0 3 5 5 M5 M1 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 18

  19. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph definition (3/3) • To obtain an oriented disjunctive graph we must : • define a job sequence on machines ; • define an assignment of robots to each transport operation ; • define a precedence (order) to transport operations assigned to one robot. Using two vectors: MTS which defines Machine and Transport Selections OA which defines Operation Assignments to each robot IESM 2009, MONTREAL – CANADA, May 13 TR 19

  20. m1 m1 m3 m2 m3 m5 m3 m4 m1 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 5 5 5 4 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph orientation (1/2) MTS Transport operations Transport operations 5 7 M2 M3 M1 4 0 4 0 5 1 M3 M4 M1 0 * 0 7 0 3 2 5 2 5 M5 M1 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 20

  21. 3 r1 r1 r3 r2 r2 r3 tr11 tr21 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkDisjunctive graph orientation (2/2) tr11 tr21 tr31 tr12 tr22 tr32 MTS 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Machine operations Machine operations Machine operations 2 5 3 7 M2 r1 M3 r2 M1 4 0 7 5 5 5 4 2 4 3 0 5 1 M3 r1 M4 r2 M1 0 * 0 0 3 2 5 2 5 M5 r3 M1 r3 M3 OA IESM 2009, MONTREAL – CANADA, May 13 TR 21

  22. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Graph evaluation and Critical Path Makespan =24 0 7 9 14 17 2 5 3 7 M2 r1 M3 r2 M1 4 0 7 5 5 3 5 4 0 16 20 23 14 24 2 4 3 0 5 1 M3 r1 M4 r2 M1 0 * 0 5 7 12 14 0 3 2 5 2 5 M5 r3 M1 r3 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 22

  23. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Memetic algorithm • begin • npi := 0 ; // current iteration number • ni := 0 ; // number of successive • unproductive iteration • Repeat • SelectSolution (P1,P2) • C := Crossover(P1,P2) • LocalSearch(C) with probability pm • InsertSolution(Pop,C) • Sort(Pop) • If (npi=np) • Restart(Pop,p) • End If • Until (stopCriterion). • End IESM 2009, MONTREAL – CANADA, May 13 TR 23

  24. m2 m3 m5 tr11 tr21 tr31 m3 m4 m1 tr12 tr22 tr32 m1 m1 m3 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 MTS OA r1 r1 r3 r2 r2 r3 tr11 tr21 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Chromosome Chromosome is a representation of a solution Makespan = 24 IESM 2009, MONTREAL – CANADA, May 13 TR 24

  25. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (1/5) • For one iteration: • Change one machine disjunction orientation (in the critical path) OR • Change one robot disjunction orientation (in the critical path) OR • Change one robot assignment. IESM 2009, MONTREAL – CANADA, May 13 TR 25

  26. Change transport disjunction m1 m1 m3 m2 m3 m5 m3 m4 m1 1 2 3 1 2 3 1 2 3 1 2 3 1 2 3 Robot block 4 7 r1 r1 r3 r2 r2 r3 tr11 tr21 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (2/5) tr11 tr21 tr31 tr12 tr22 tr32 MTS OA 0 7 9 14 17 2 5 3 7 M2 r1 M3 r2 M1 4 0 5 5 3 5 4 0 16 20 23 14 24 2 4 3 0 5 1 M3 r1 M4 r2 M1 0 * 0 5 7 12 14 0 3 2 5 2 5 M5 r3 M1 r3 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 26

  27. Change machine disjunction m1 m1 m3 m2 m3 m5 m3 m4 m1 1 2 3 1 2 3 2 1 3 1 2 3 1 2 3 Machine block Makespan =23 4 r1 r1 r3 r2 r2 r3 tr21 tr11 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (3/5) tr21 tr11 tr31 tr12 tr22 tr32 MTS OA 0 8 10 15 18 2 5 3 7 M2 r1 M3 r2 M1 4 0 3 5 5 3 5 0 7 18 22 5 23 2 4 3 0 5 1 M3 r1 M4 r2 M1 0 * 0 5 7 12 15 0 3 2 5 2 5 M5 r3 M1 r3 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 27

  28. m1 m1 m3 m2 m3 m5 m3 m4 m1 1 2 3 1 2 3 2 1 3 1 2 3 1 2 3 3 4 r1 r1 r3 r2 r2 r3 tr21 tr11 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (4/5) tr21 tr11 tr31 tr12 tr22 tr32 MTS r3 OA Change robot assignement 0 8 10 15 18 2 5 3 7 M2 r1 M3 r2 M1 4 0 5 5 3 5 0 7 18 22 5 23 2 4 3 0 5 1 M3 M4 r2 M1 0 r3 r1 * 0 5 7 12 15 0 3 2 5 2 5 M5 r3 M1 r3 M3 IESM 2009, MONTREAL – CANADA, May 13 TR 28

  29. m1 m1 m3 m2 m3 m5 m3 m4 m1 1 2 3 1 2 3 2 1 3 1 2 3 1 2 3 4 4 r1 r1 r3 r2 r2 r3 tr21 tr11 tr31 tr12 tr22 tr32 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (5/5) tr21 tr11 tr31 tr12 tr22 tr32 MTS r3 OA Change robot assignement 0 7 9 14 17 2 5 3 7 M2 r1 M3 r2 M1 4 0 5 5 3 5 0 7 17 21 5 22 2 4 3 0 5 1 M3 M4 r2 M1 0 r3 r1 * 0 9 11 16 18 0 3 2 5 2 5 M5 r3 M1 r3 M3 New transport disjunction is added IESM 2009, MONTREAL – CANADA, May 13 TR 29

  30. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Instances • Two types of experiments have been done using well known benchmarks in the literatures. • The first type of experiments concerns instances of: • Hurink J. and Knust S., "Tabu search algorithms for job-shop problems with a single transport robot", European Journal of Operational Research, Vol. 162 (1), pp. 99-111, 2005. • The second one with two identical robots from: • Bilge, U. and G. Ulusoy, 1995, A Time Window Approach to Simultaneous Scheduling of Machines and Material Handling System in an FMS, Operations Research, 43(6), 1058-1070. IESM 2009, MONTREAL – CANADA, May 13 TR 30

  31. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (1/4) • Experiments on job-shop with one single robot on Hurink and Knust instances based on well-known 6x6 and 10x10 instances: • J.F. Muth, G.L. Thompson, Industrial Scheduling, Prentice • Hall, Englewood Cliffs, NJ, 1963. IESM 2009, MONTREAL – CANADA, May 13 TR 31

  32. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (2/4) • Experiments on Bilge & Ülusoy (1995) 40 instances • 4 machines, 2 vehicles • 10 jobsets, • 5 - 8 jobs, 13 - 23 operations • 4 different structures for FMS IESM 2009, MONTREAL – CANADA, May 13 TR 32

  33. M1 M2 M3 M4 LU Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (3/4) • Exemple of FMS structure IESM 2009, MONTREAL – CANADA, May 13 TR 33

  34. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (4/4) IESM 2009, MONTREAL – CANADA, May 13 TR 34

  35. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Conclusion and further worksConclusion • Step forwards the generalization of the disjunctive graph model including several robots; • Memetic algorithm based approach for a generalization of the job-shop problem; • Specific properties are derived from the longest path to generate neighbourhoods; IESM 2009, MONTREAL – CANADA, May 13 TR 35

  36. Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Conclusion and further works Further works • Additional constraints; • Axact methods; • Larger instances; IESM 2009, MONTREAL – CANADA, May 13 TR 36

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