Simultaneous scheduling of machines and automated guided vehicles

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 »

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

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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Type of system under study: FMS based on AGV system Guide path layout Automated Guided Vehicles 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 Flexible machines 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 cells 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 Input/Output buffers IESM 2009, MONTREAL – CANADA, May 13 TR 7

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

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

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

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

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, Cmax 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). IESM 2009, MONTREAL – CANADA, May 13 TR 12

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Introduction Problem definition (4/5) Problem definition : Assumptions in the literature q q q q 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

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 q q 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. q indicates that we have job-independent, but machinedependant transportation times. q 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

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

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: 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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/3) J 1 M 2 7 M 3 5 M 1 4 0 J 2 J 3 0 0 0 M 3 M 5 5 5 IESM 2009, MONTREAL – CANADA, May 13 M 4 M 1 4 5 M 1 M 3 1 * 3 TR 17

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/2) Machine disjunction problem Robot disjunction problem M 2 8 r 1 M 3 5 r 2 M 1 4 0 0 M 3 M 5 5 r 1 5 M 4 M 1 4 5 M 1 M 3 1 * 3 Robot assignment problem 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 (3/3) To obtain an oriented disjunctive graph we must : q define a job sequence on machines ; q define an assignment of robots to each transport operation ; q 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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (1/2) m 2 m 3 m 5 MTS 1 2 3 m 1 m 3 m 4 m 1 1 2 3 1 Transport operations M 2 0 0 M 3 0 M 5 5 5 IESM 2009, MONTREAL – CANADA, May 13 5 M 3 5 0 1 3 2 3 Transport operations 7 0 2 5 M 4 2 M 1 7 M 1 5 4 M 1 2 4 M 3 1 * 3 TR 20

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (2/2) tr 11 tr 21 tr 31 MTS 1 2 3 Machine operations M 2 7 0 0 0 r 1 M 3 0 M 5 5 5 tr 12 tr 22 tr 32 1 2 Machine operations 2 r 3 2 2 1 5 M 3 5 r 1 3 5 M 4 M 1 2 r 2 5 r 2 r 3 2 3 M 1 5 3 2 Machine operations 3 3 4 1 3 4 4 M 1 M 3 1 * 3 OA r 1 r 3 r 2 r 3 tr 11 tr 21 tr 31 tr 12 tr 22 tr 32 IESM 2009, MONTREAL – CANADA, May 13 TR 21

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

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 Select. Solution (P 1, P 2) C : = Crossover(P 1, P 2) Local. Search(C) with probability pm Insert. Solution(Pop, C) Sort(Pop) If (npi=np) Restart(Pop, p) End If Until (stop. Criterion). End IESM 2009, MONTREAL – CANADA, May 13 TR 23

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Chromosome is a representation of a solution m 2 m 3 m 5 tr 11 tr 21 tr 31 m 3 m 4 m 1 tr 12 tr 22 tr 32 m 1 m 3 MTS 1 2 3 OA r 1 r 3 r 2 r 3 1 2 3 tr 11 tr 21 tr 31 tr 12 tr 22 tr 32 Makespan = 24 IESM 2009, MONTREAL – CANADA, May 13 TR 24

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (1/5) For one iteration: q Change one machine disjunction orientation (in the critical path) OR q Change one robot assignment. 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 (2/5) Change transport disjunction m 2 m 3 m 5 tr 11 tr 21 tr 31 m 3 m 4 m 1 tr 12 tr 22 tr 32 m 1 m 3 MTS OA 1 2 3 r 1 r 3 r 2 r 3 1 2 1 3 2 3 Robot block tr 11 tr 21 tr 31 tr 12 tr 22 tr 32 0 M 2 7 0 0 2 r 1 74 0 0 7 0 M 3 0 M 5 5 5 9 M 3 5 14 r 1 5 r 3 IESM 2009, MONTREAL – CANADA, May 13 2 2 5 5 16 M 4 7 M 1 14 r 2 3 3 4 5 17 M 1 5 20 r 2 12 r 3 3 2 4 4 23 M 1 14 M 3 24 1 * 3 TR 26

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (3/5) Change machine disjunction m 2 m 3 m 5 tr 21 tr 11 tr 31 m 3 m 4 m 1 tr 12 tr 22 tr 32 m 1 m 3 MTS OA 1 2 3 2 1 3 1 2 1 3 2 Machine block r 1 r 3 r 2 r 3 3 Makespan =23 tr 21 tr 11 tr 31 tr 12 tr 22 tr 32 0 M 2 7 0 0 2 r 1 3 0 0 8 0 M 3 0 M 5 5 5 10 M 3 5 5 r 1 5 r 3 IESM 2009, MONTREAL – CANADA, May 13 2 2 5 5 7 M 4 7 M 1 15 r 2 3 3 4 5 18 M 1 5 18 r 2 12 r 3 3 2 4 4 22 M 1 15 M 3 23 1 * 3 TR 27

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (4/5) m 2 m 3 m 5 tr 21 tr 11 tr 31 m 3 m 4 m 1 tr 12 tr 22 tr 32 m 1 m 3 MTS OA 1 2 3 2 1 3 1 2 3 r 1 r 3 r 2 r 3 1 2 1 3 2 3 Change robot assignement tr 21 tr 11 tr 31 tr 12 tr 22 tr 32 0 M 2 7 0 0 2 r 1 3 0 0 8 0 M 3 0 M 5 5 5 10 M 3 5 5 r 3 r 1 5 r 3 IESM 2009, MONTREAL – CANADA, May 13 2 2 5 5 7 M 4 7 M 1 15 r 2 3 3 4 5 18 M 1 5 18 r 2 12 r 3 3 2 4 4 22 M 1 15 M 3 23 1 * 3 TR 28

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (5/5) m 2 m 3 m 5 tr 21 tr 11 tr 31 m 3 m 4 m 1 tr 12 tr 22 tr 32 m 1 m 3 MTS OA 1 2 3 2 1 3 1 2 3 r 1 r 3 r 2 r 3 1 2 1 3 2 3 Change robot assignement tr 21 tr 11 tr 31 tr 12 tr 22 tr 32 0 M 2 7 7 2 r 1 0 0 M 3 5 5 r 3 r 1 2 5 M 3 5 0 0 9 5 7 M 4 14 r 2 3 3 4 17 M 1 5 17 r 2 3 4 4 21 M 1 22 1 * 4 0 0 M 5 5 9 r 3 2 11 M 1 5 16 r 3 2 18 M 3 3 New transport disjunction is added IESM 2009, MONTREAL – CANADA, May 13 TR 29

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

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 6 x 6 and 10 x 10 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 Knust 13, 40 16, 16 IESM 2009, MONTREAL – CANADA, May 13 14, 22 16, 63 Our method 13, 33 TR 31

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 q 4 machines, 2 vehicles q 10 jobsets, q 5 - 8 jobs, 13 - 23 operations q 4 different structures for FMS IESM 2009, MONTREAL – CANADA, May 13 TR 32

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation Experimental results (3/4) Exemple of FMS structure M 1 M 2 M 3 M 4 LU IESM 2009, MONTREAL – CANADA, May 13 TR 33

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

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

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