# simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution

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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 ». - PowerPoint PPT PresentationTRANSCRIPT

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolutionPhilippe LACOMME, Mohand LARABINikolay TCHERNEVLIMOS (UMR CNRS 6158), Clermont Ferrand, FranceIUP Management et gestion des entreprises

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolutionPlanPlanIntroductionAlgorithm based frameworkComputational evaluationConclusions and further works

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionType of system under study: FMS based on AGVFMS definition

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionType of system under study: FMS based on AGVAGV systemGuide path layoutAutomated Guided Vehicles

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionType of system under study: FMS based on AGVFlexible machines

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionType of system under study: FMS based on AGVFlexible cells

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionType of system under study: FMS based on AGVInput/Output buffers

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionAGV operating (1/2)

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

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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionProblem definition (2/5)Problem definition : Example of solution

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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution IntroductionProblem definition (4/5)Problem definition : Assumptions in the literatureAll 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.

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

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based frameworkGeneral templateGeneral template

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.

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/3)*J1J2J3

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph definition (2/2)0*M2r1M3r2M185M3r1M4M154M5M1M355400013

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 SelectionsOAwhich defines Operation Assignments to each robot

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (1/2) 0*M2M3M175M3M4M154M5M1M3075252400013MTSTransport operationsTransport operations

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Disjunctive graph orientation (2/2) 0*M2r1M3r2M17253M3r1M4r2M15243M5r3M1r3M30525240001374555MTS121233121233tr11tr21tr31tr12tr22tr32123OAMachine operationsMachine operationsMachine operations

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Graph evaluation and Critical Path0*M2r1M3r2M107914177253M3r1M4r2M10141620235243M5r3M1r3M30571214525242400013734555Makespan =24

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Memetic algorithmbeginnpi:= 0; // current iteration numberni := 0; // number of successive unproductive iterationRepeatSelectSolution (P1,P2)C := Crossover(P1,P2)LocalSearch(C) with probability pmInsertSolution(Pop,C)Sort(Pop)If (npi=np)Restart(Pop,p)End IfUntil (stopCriterion).End

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework ChromosomeChromosome is a representation of a solutionMakespan = 24

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.

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (2/5)*M2r1M3r2M107914177253r1M4r2M114162023243r3M1r3M357121425242400M305M505001334555MTSOAtr11tr21tr31tr12tr22tr32

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (3/5)0*M2r1M3r2M1081015187253M3r1M4r2M105718225243M5r3M1r3M30571215525240001333555MTSOAtr21tr11tr31tr12tr22tr3223

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (4/5)r30*M2r1M3r2M1081015187253M3r1M4r2M105718225243M5r3M1r3M3057121552524000133555MTSOAtr21tr11tr31tr12tr22tr3223Change robot assignementr3

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Algorithm based framework Local search (5/5)r30*M2r1M3r2M107914177253M3r1M4r2M105717215243M5r3M1r3M30911161852524000133555MTSOAtr21tr11tr31tr12tr22tr3222Change robot assignementr3New transport disjunction is added

Simultaneous scheduling of machines and automated guided vehicles: graph modelling and resolution Computational evaluation InstancesTwo 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.

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

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