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. IESM 2009, MONTREAL – CANADA, May 13 TR 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

3. IESM 2009, MONTREAL – CANADA, May 13 TR 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

4. IESM 2009, MONTREAL – CANADA, May 13 TR 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

5. IESM 2009, MONTREAL – CANADA, May 13 TR 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

6. IESM 2009, MONTREAL – CANADA, May 13 TR 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

7. IESM 2009, MONTREAL – CANADA, May 13 TR 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

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

9. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

10. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

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

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

13. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

14. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

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

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

17. IESM 2009, MONTREAL – CANADA, May 13 TR 17 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/3) * M2 7 M3 5 M1 4 0 0 0 0 M3M4M1 5 4 1 M5M1M3 5 5 3 J1 J2 J3

18. IESM 2009, MONTREAL – CANADA, May 13 TR 18 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/2) 0 * M2r1M3r2M1 8 5 M3r1M4M1 5 4 M5M1M3 5 5 4 0 0 0 1 3 Machine disjunction problem Robot assignment problem Robot disjunction problem

19. IESM 2009, MONTREAL – CANADA, May 13 TR 19 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive 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: MTSwhich defines Machine and Transport Selections OAwhich defines Operation Assignments to each robot

20. IESM 2009, MONTREAL – CANADA, May 13 TR 20 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (1/2) 0 * M2M3M1 7 5 M3M4M1 5 4 M5M1M3 07 5 252 4 0 0 0 1 3 4 5 5 5 MTS 123123 1 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3 Transport operations Transport operations

21. IESM 2009, MONTREAL – CANADA, May 13 TR 21 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (2/2) 0 * M2r1M3r2M1 7 253 M3r1M4r2M1 5 243 M5r3M1r3M3 0 5 252 4 0 0 0 1 3 7 3 4 5 5 5 MTS 121233121233 tr11tr21tr31tr12tr22tr32 123 OA r1 tr11 r1 tr21 r3 tr31 Machine operations r2 tr12 r2 tr22 r3 tr32

22. IESM 2009, MONTREAL – CANADA, May 13 TR 22 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Graph evaluation and Critical Path 0 * M2r1M3r2M1 0791417 7 253 M3r1M4r2M1 0 14 162023 5 243 M5r3M1r3M3 0571214 5 252 4 24 0 0 0 1 3 7 34 5 5 5 Makespan =24

23. IESM 2009, MONTREAL – CANADA, May 13 TR 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

24. IESM 2009, MONTREAL – CANADA, May 13 TR 24 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Chromosome 1231231 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3tr11tr21tr31tr12tr22tr32 OA r1 tr11 r1 tr21 r3 tr31 r2 tr12 r2 tr22 r3 tr32 MTS Chromosome is a representation of a solution Makespan = 24

25. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

26. IESM 2009, MONTREAL – CANADA, May 13 TR 26 74 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (2/5) * M2r1M3r2M1 0791417 7 253 r1M4r2M1 14 162023 243 r3M1r3M3 571214 252 4 24 0 0 M3 0 5 M5 0 5 0 0 1 3 34 5 5 5 MTS 123123 1 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3 r1 tr11 r1 tr21 r3 tr31 r2 tr12 r2 tr22 r3 tr32 OA tr11tr21tr31tr12tr22tr32 Change transport disjunction Robot block

27. IESM 2009, MONTREAL – CANADA, May 13 TR 27 0 * M2r1M3r2M1 08101518 7 253 M3r1M4r2M1 0 5 71822 5 243 M5r3M1r3M3 0571215 5 252 4 0 0 0 1 3 3 3 4 5 5 5 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (3/5) MTS 213123 1 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3 r1 tr21 r1 tr11 r3 tr31 r2 tr12 r2 tr22 r3 tr32 OA tr21tr11tr31tr12tr22tr32 23 Makespan =23 Machine block Change machine disjunction

28. IESM 2009, MONTREAL – CANADA, May 13 TR 28 r1 tr11 r3 0 * M2r1M3r2M1 08101518 7 253 M3 r1 M4r2M1 0 5 71822 5 243 M5r3M1r3M3 0571215 5 252 4 0 0 0 1 3 3 3 4 5 5 5 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (4/5) MTS 213123 1 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3 r1 tr21 r3 tr31 r2 tr12 r2 tr22 r3 tr32 OA tr21tr11tr31tr12tr22tr32 23 Change robot assignement r3

29. IESM 2009, MONTREAL – CANADA, May 13 TR 29 4 r1 tr11 r3 0 * M2r1M3r2M1 0791417 7 253 M3 r1 M4r2M1 0 5 71721 5 243 M5r3M1r3M3 09111618 5 252 4 0 0 0 1 3 3 4 5 5 5 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (5/5) MTS 213123 1 m2 2 m3 3 m5 1 m3 2 m4 3 m1 1 2 3 m3 r1 tr21 r3 tr31 r2 tr12 r2 tr22 r3 tr32 OA tr21tr11tr31tr12tr22tr32 22 Change robot assignement r3 New transport disjunction is added

30. IESM 2009, MONTREAL – CANADA, May 13 TR 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.

31. IESM 2009, MONTREAL – CANADA, May 13 TR 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. Deviation in percentage from the best solution found by each method to lower bound proposed by Hurink and Knust Four methods proposed by Hurink and KnustOur method 13,4016,1614,2216,6313,33

32. IESM 2009, MONTREAL – CANADA, May 13 TR 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

33. IESM 2009, MONTREAL – CANADA, May 13 TR 33 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (3/4) Exemple of FMS structure M1M2M3M4 LU

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

35. IESM 2009, MONTREAL – CANADA, May 13 TR 35 Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Conclusion and further works Conclusion  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;

36. IESM 2009, MONTREAL – CANADA, May 13 TR 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;