Int. J. Production Economics 143 (2013) 24–34

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Int. J. Production Economics

0925-52

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ijpe

Job-shop based framework for simultaneous scheduling of machines andautomated guided vehicles

Philippe Lacomme a, Mohand Larabi a,b, Nikolay Tchernev a,b,n

a Universite Blaise Pascal, LIMOS (UMR CNRS 6158), 63177 Aubi�ere Cedex, Franceb IUP (Management et gestion des entreprises), Universite d’Auvergne, 26, av. Leon Blum, Clermont Ferrand, 63000, France

a r t i c l e i n f o

Article history:

Received 28 December 2009

Accepted 13 July 2010Available online 23 July 2010

Keywords:

FMS

AGV

Job-shop

Scheduling

Memetic algorithm

73/$ - see front matter & 2010 Elsevier B.V. A

016/j.ijpe.2010.07.012

esponding author. Tel.: +33 4 73 40 77 71; fa

ail address: tchernev@isima.fr (N. Tchernev).

a b s t r a c t

This paper deals with the problem of simultaneous scheduling of machines and identical automated

guided vehicles (AGVs) which are well known difficult to solve problems. The studied problem can be

modelled as a job shop where the jobs have to be transported between machines by AGVs. This article

introduces a framework based on a disjunctive graph to modelize the joint scheduling problem and on a

memetic algorithm for machines and AGVs scheduling. The objective is to minimize the makespan.

Computational results are presented for a benchmark literature instances. New upper bounds are found,

showing the effectiveness of the presented approach.

& 2010 Elsevier B.V. All rights reserved.

1. Introduction

Automated guided vehicles (AGVs) are among various ad-vanced material handling techniques that are finding increasingapplications today. They can be interfaced to various otherproduction and storage equipment and controlled through anintelligent computer control system. Both the scheduling ofoperations on machines as well as the scheduling of AGVs areessential factors contributing to the efficiency of the overallflexible manufacturing system (FMS) (Blazevicz et al., 1991).Significant improvement of FMS performance is expected as aresult of a proper coordination of both AGV and machinescheduling problem.

The considered FMS scheduling problem has the samestructure as introduced in 1993 by Ulusoy and Bilge (1993) andsuccessively studied by Bilge and Ulusoy (1995), Ulusoy et al.(1997), Abdelmaguid et al. (2004), Lacomme et al. (2005), Reddyand Rao (2006), Deroussi et al. (2008). It is concerned with thesimultaneous scheduling of machines and AGVs in a flexiblemanufacturing environment where a set of different machinesperform different tasks and a set of identical AGVs performmaterial handling and transportation tasks between machines.The problem addressed in this paper thus falls into the classifica-tion of NP-complete combinatorial problems for which efficientoptimal solution procedures do not exist unless P ¼NP (Gane-sharajah et al., 1998). Scheduling can be achieved in a static mode(off-line scheduling) prior to execution, a dynamic mode (real-

ll rights reserved.

x: +33 4 73 40 76 39.

time or on-line scheduling) during the execution, or a combina-tion of both. This paper deals with off-line scheduling in FMS.Methods for off-line scheduling are directed at schedule genera-tion over a certain horizon period (from a day to a week), the timeduring which the schedule is not expected to require any change.This aspect of the scheduling has been studied by severalresearchers (Raman et al., 1986; Blazevicz et al., 1991; Ulusoyand Bilge, 1993; Bilge and Ulusoy, 1995; Ulusoy et al., 1997;Abdelmaguid et al., 2004; Lacomme et al., 2005; Reddy and Rao,2006; Deroussi et al., 2008, Caumond et al., 2009).

Exact methods are mainly used for the study of simple orparticular FMS, with strict assumptions. Thus, Blazevicz et al.(1991) study an FMS in which identical parallel machines are laidout in loop. Raman et al. (1986) presents a mixed integerprogramming (MIP) formulation of this problem but with theunrealistic assumption that the vehicles come back to the load/unload (LU) station after each achieved transport. The FMSwithout this assumption was later formulated in MIP by Bilgeand Ulusoy (1995). According to the authors, the resulting modelis intractable in practice, because of its nonlinearity and its size.To the best of our knowledge Caumond et al. (2009) is the lastpublication that deals with linear formulation of the FMS. Theirapproach differs from previously published publications becauseit takes into account the maximum number of jobs allowed in thesystem, limited input/output buffer capacities, empty-vehicletrips and no-move-ahead trips simultaneously. However onlyone AGV is taken into consideration as special case of the generalFMS.

Approximated methods are well adapted to study most of theFMS. Nevertheless, many works are dedicated to simplified formsof this problem. There are essentially two kinds of simplifications.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–34 25

The first ones consist in defining dispatching rules for the vehiclesand the second ones are the restriction of the material handlingsystem at only one AGV. As illustration, Lacomme et al. (2005) andSoylu et al. (2000) proposed, respectively, branch and bound andneural networks approaches for scheduling of FMS based onsingle AGV. Finally, few works are undertaken on the FMSscheduling with multiples AGVs. Ulusoy and Bilge (1993) andBilge and Ulusoy (1995) proposed an iterative method based onthe decomposition of the master problem into two subproblems:machine scheduling and vehicle scheduling. A heuristic algorithmgenerates machines schedules to solve the first subproblem.They introduced a solution heuristic based on the sliding timewindow (STW) approach to find a feasible solution to the vehiclescheduling problem given the machine schedule. Ulusoy et al.(1997) developed a genetic algorithm (GA) for solving thisproblem in which GA generates better results compared to theSTW heuristic. Abdelmaguid et al. (2004) proposed a hybridmethod composed of a genetic algorithm for the scheduling ofmachines and a greedy search heuristic algorithm for thescheduling of vehicles. This problem has been studied further byReddy and Rao (2006) for the case of multi objective optimization.They developed a GA approach that can provide a set of non-dominated solutions for the minimization of makespan, meanflow time, and mean tardiness simultaneously. Deroussi et al.(2008) described an efficient neighboring system implementedinto three different metaheuristics and a new solution represen-tation based on vehicles rather than machines. Face to thecomplexity of the joint scheduling of jobs and material handlingdevices, assumptions are made to simplify the problem. The mostcommon ones are:

�

Each job is 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. thelimited 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.

Under these hypotheses the problem can be without doubtmodelled as a job-shop with several transport robots. It isclassified as a JR9tkl,tukl9Cmax problem according to the a9b9g-notation introduced by Graham et al. (1979) and extended byKnust (1999) for transportation problems. 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. tkl

indicates that we have job-independent, but machine-dependenttransportation times. tukl indicates that we have machine-dependent empty moving time. The objective function tominimize is the makespan Cmax. The proposed modelling andsolving approach problem encompasses job-shop with one singletransport robot investigated by Hurink and Knust (2001, 2002,2005). A very similar problem is the job-shop problem withseveral identical transport robots which has been investigated byBrucker and Strotmann (2002).

In order to modelize real transportation situation, numerousstudies tend to include some additional constraints. Strusevich(1999) included the delay between the end of the processing timeon one machine and the earliest due date on the next machine for

the same job. This delay is denoted by transportation time.Constraints on the number of robots are stressed for the flow-shop in Hurink and Knust (2001) and for the job-shop in Hurinkand Knust (2002,2005). Lastly, Hurink and Knust (2005) intro-duced a highly efficient disjunctive graph for the job-shop withone single robot and derived some specific properties to a properdefinition of neighborhoods in local search procedure.

In this paper we propose a new effective framework based on adisjunctive graph to modelize the joint scheduling problem andon a memetic algorithm for jobs sequence generation onmachines, AGVs sequence generation and vehicles assignmentsto transport operations. The contribution of this paper is twofold.First, an original modelling approach based on disjunctive graph isdeveloped for the studied problem. Second, an efficient memeticalgorithm is developed for providing solutions to large instancesof the problem in short computational time. The developedframework is not restricted to the case of AGVs as it can be suitedeasily to any type of trip based material handling systems.

The remainder of the paper is organized as follows. Section 2defines the framework we promote based on a memetic algorithmsearch scheme and on a disjunctive graph. This section includes acareful description of the disjunctive graph to modelize transportoperation constraints. Section 3 describes the components of thememetic algorithm including population definition, local search,clone detection and restarts. Section 4 deals with computationalevaluation of the framework including benchmarks based on a setof instances dedicated to the single robot problem given byHurink and Knust (2005) and on set of instances dedicated to theseveral robots given by Bilge and Ulusoy (1995). Section 5concludes the paper and addresses promising directions for futureresearch.

2. Algorithm based framework

The framework is based on a memetic algorithm for sequencegeneration on both machines and robots operations in machineand transport selection (MTS) and assignment of transportoperations assignment (OA). This memetic algorithm includes apowerful local search procedure. The problem is modelled first asa non-oriented disjunctive graph. It is possible to obtain anoriented disjunctive graph where a Bellman like longest pathalgorithm permits to compute the earliest completion time of thelast operation: the makespan.

2.1. Job-shop with transportation times and several robots

The job-shop scheduling problem with transportation timesand several robots can be classified as a complex combinatorialproblem, in which a set of n jobs (J¼{J1, y, Jn}) required to beprocessed on a set of m machines (M¼{M1, y, Mm}). Jobtransports are achieved by a set of r transport robots R¼{R1, y,Rr}. An ordered set of ni operations denoted Oi,1,Oi,2, y, Oi,ni fullydefines each job Ji since one operation Oi,j refers to one machine Mj

for duration pi,j. Each machine Mj can process only one job Ji

during pi,j (without preemption) at one time and each job can beprocessed by only one machine at the same time. Between twomachine operations Oi,k and Oi,k + 1 (which refer to machine mi,k andmi,k +1) a transportation operation tp

mi,k ,mi,kþ 1is performed by robot p

which can handle at most one job at a time. For convenience,tpmi,k ,mi,kþ 1

is used to denote both a transportation operation and atransportation time. Empty transportation times vp

i,j are alsoaddressed while one robot p moves from machine Mi to Mj

without carrying a job. It is possible to assume, for each robot p,that vp

i,i ¼ 0 and tpi,jZvp

i,j. We consider that all transport times aremachine dependent and job independent.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–3426

As in the popular job-shop scheduling problem the problemof interest is restricted in by not considering buffer spacelimitation between machines, assuming that machine Mi remainsidle only when the transport operation is processed and notconsidering job transfer time from machine to buffer. Bothprocessing times needed for the jobs on the machines andtransportation times are assumed to be non-negative anddeterministic denoted as pi,j,.t

ki,j, The objective is to determine a

feasible schedule which minimizes the makespan Cmax ¼

Maxj ¼ 1,nfCjg where Cj denotes the completion time of the lastoperation Oj,nj of job Jj.

Table 1One instance of job-shop with two robots.

Processing times Transportation times for each robot r

Job 1 M3: 1 M1: 3 M2: 6 M1 M2 M3 M4

Job 2 M2: 8 M3: 5 M1: 10 M1 0 2 2 2

Job 3 M3: 5 M4: 4 M1: 9 M2 1 0 2 2

M3 2 2 0 2

M4 2 2 2 0

2.2. Proposal for a non-oriented disjunctive graph and notations

In order to model the job-shop problem with transport we usedisjunctive graph introduced by Roy and Sussmann (1964). In fact,in the disjunctive graph the fixed arcs form a set of distinctdirected paths going from the start node 0 to the finish node *, thearc weights are all positive, and the pairs of alternative arcs(called disjunctive arcs) are all in the form (i,j), (j,i) where i and j

are two operations to be processed on the same machine. Toaddress problem with several robots, it is necessary to includeassignment of robots to transport operations. As defined inLacomme et al. (2007a), the disjunctive graph G¼(Vm[Vt,C[Dm[Dr) consists of: a set of vertices Vm containing all machineoperations; a set of vertices Vt of transport operations (obtainedby an assignment of robot to each transport operation) andtwo dummy vertices 0 and *. The graph consists also of:(i) conjunctions C representing precedence constraints Oi,k-

tpmi,k ,mi,kþ 1

-Oi,kþ1, (ii) disjunctions for the machine Dm, (iii)disjunctions for the transport Dr which depends on the assign-ment of robots to transport operations Ar. For each job Ji, ni�1transport operations tp

mi,k ,mi,kþ 1are introduced including precedence

Oi,k-tpmi,k ,mi,kþ 1

-Oi,kþ1. Arcs from a machine node to a transportnode are weighted with the machine operation duration. Edgesbetween machine operations represent disjunctions between theconsidered operations since they cannot hold them simulta-neously on the same machine. In addition, to the classical set ofundirected machine disjunctions Dm (all pairs of operations whichhave to be processed on the same machine and which are notlinked by a directed path), it is necessary to consider the set ofundirected robots disjunctions (Dr). Undirected robot disjunctionscannot be represented since transport operation assignment (Ar)to one robot is not achieved. Fig. 1 represents the non-orienteddisjunctive graph linked to the data set of Table 1. To solve thescheduling problem it is necessary to turn all undirected arcs inDm [ Dr into directed ones, and to assign one robot to eachtransport operation.

0

M3 1 2

5

80

0

0

r

r

r

2

2

M1

M2 M3

M3 M4

Fig. 1. Non-oriented disjunctive graph

2.3. Solutions representation and solution feasibility

According to Hurink and Knust (2005), it is possible to defineSm a fixed machine selection which is called Machine Disjunctionsand define SR a fixed transport selection which is called TransportDisjunction. A fully oriented disjunctive graph can be obtained

using S¼ C [ Sm [ Sr called a complete selection. The challenging

problem consists in defining efficiently the complete selection S

avoiding generation of cyclic graphs. The correct definition of Sm,SR can be achieved using two vectors denoted MTS and OA. Thanksto the Bierwith’s proposal for the job-shop, it is possible to definethe machine and transport selections MTS as a vector byrepetition (Bierwith, 1995). Since each job j has nj machineoperations and nj�1 transport operations, MTS encompasses2nj�1 times the number j. The first occurrence of j in the vectorMTS refers to the first machine operation of the job, the secondoccurrence refers to the first transport operation of the job and soon. The vector OA is the transport operation assignments whereOA(i)¼p defines that the ith transport operation in MTS(i) isprocessed by robot p. For two consecutive transport operations

assigned to the same robot p the arc added from tpmi,k ,mi,kþ 1

to

tpmj,k0 ,mj,k0 þ 1

gets the weight tpmi,k ,mi,kþ 1

þvpmi,kþ 1 ,mj,k0

and tpmj,k0 ,mj,k0 þ 1

þ

vpmj,k0 þ 1 ,mi,k

if its orientation is in the opposite direction as stressed

in Fig. 2.The union S¼ C [ Sm [ Sr fully describes a solution if the

resulting oriented disjunctive graph G¼ ðVm [ Vt ,SÞ is an acyclicone. A feasible schedule can be constructed by longest pathcalculation which permits to obtain the earliest starting time ofboth machine and transport operations and fully defines a semi-active schedule with the Cmax given by the length of the longestpath from node 0 to *.

The longest path defines the set of operations for which no delayis allowed between the two dummy nodes (Cmax). Note thatunfortunately there is a great number of selections (MTS,OA) whichinduce an identical oriented disjunctive graph since it is the relativeorder of operations which is responsible of the disjunctions.

3 2

6

4 2

9

015 r

r

r

2

M2

M1

M1

for representation of the problem.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–34 27

Let us consider an instance of a job-shop problem with r¼2robots, m¼4 machines, n¼3 jobs, 9 machine operations and 6transport operations. Let us consider:

�

the following order of jobs on Machine 1: Job 1, Job 3, Job 2; onMachine 2: Job 2, Job 1; on Machine 3: Job 3, Job 1, Job 2; onMachine 4. Job 3. � the following assignment of transport robots to the transport

operations:The robot 1 is assigned in following order to: first transportof job 3, the first transport of job 1, the first transport ofjob 2, the second transport of job 1, the second transport ofjob 2;The robot 2 is assigned in following order to: the secondtransport of job 3;The corresponding schedule can be found in Fig. 3 wherenumbers in bold are earliest starting time of operations.

2.4. Solution evaluation

Evaluating a solution consists in determining a completeselection S¼ C [ Sm [ Sr , which constructs the graph linked to MTS

and OA vectors by adding the disjunctive machine arcs and

Fig. 2. Disjunction between two transport operations.

0

0

0

0 r1

r1

M2

M3

35

3

3 3

5M3

M3

M1

8

8

2

2

1

Critical path

Disjunction betwesame machine

Disjunction betwesame robot

Conjunctions bet

r1 M42

Fig. 3. The oriented d

transportation arcs in the graph G. The solution evaluationprocedure (Evaluate_A_Solution(MTS,OA,G,Cmax) is based on aBellman’s like longest path algorithm and produces the earlieststarting time of operations including the makespan Cmax (seeLacomme et al. (2007a) for details).

3. Components for the memetic algorithm

3.1. Chromosome: representation, evaluation

Many genetic algorithms use quasi-direct representations ofsolutions. A job-shop’s representation scheme addressing severalrobots constraint is addressed in Lacomme et al. (2007b): thebasic idea consists in considering Bierwirth’s sequences as achromosome. A chromosome C does not represent a solution butcan be viewed as a representation of a partial solution PS¼ C [ Sm

which is composed of a machine and transport selection MTS.

C.Father is the representation of the longest path in the graphlinked to the makespan C.Fitness. The initial population iscomposed of randomly generated solutions and the crossover isbased on the GOX crossover first introduced by Bierwirth. Clonedetection is managed using a hash function providing a clonedetection in O(1) thanks to a basic array Clone where Clone[i]¼1as soon as a chromosome i has been investigated. The key featureof the algorithm is composed of a powerful local search describedbelow.

Two properties hold in this case: (1) chromosomes can beefficiently evaluated with respect to the two vectors; (2) twochromosomes can hold to the same oriented disjunctive graph.These properties hold for many GAs dedicated to schedulingproblems. For convenience, it is possible to note C.MTS theBierwirth’s sequence linked to the machine and transportselection and C.OA the robots dedicated to the transport opera-tions. C.Fitness is defined by the evaluation procedure. Achromosome evaluation consists in a basic call to Evaluate_A_So-

lution. Evaluate_A_Solution (C.MTS, C.OA,G,C.Fitness,C.Father).It is important to denote that this representation using two

vectors (MTS and OA) provides only ‘‘feasible chromosome’’,

r1

r1

M1

M2

3

3

5 2

2

2

10

9

6

en machine operations of different jobs done by the

en transport operations of different jobs done by the

ween operations of the same job

r2 M14

9

isjunctive graph.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–3428

which corresponds to an acyclic graph. The procedure whichpermits to evaluate chromosomes is based on a longest pathalgorithm and generates semi-actives schedules only. Semi-actives schedules generation is appropriate since it is well known

Algorithm 1. Crossover procedure

that the set of semi-actives schedules contains at least oneoptimal solution.

3.2. Chromosomes generation

The chromosome generation is based on:

�

Random generation of MTS sequence; � Heuristic construction of OA for a fixed MTS sequence;

The heuristic proposed is a constructive one which consists inscheduling one operation per iteration. This heuristic is a systemgeneration schedule based on the initial proposal of Giffler andThomson in 1960. During the initialization phase, the set ofunscheduled operations U is initialized to the set of all operationsO and the set of scheduled operations S is initialized to the empty set.Note that the set O contains both machine operations and transportoperations. The main loop consists in scheduling one operation and isended when the set of unscheduled operations is empty. Theheuristic permits to generate OA for a fixed machine selection.

3.3. Proposal for crossover

Algorithm 1 presents the crossover which is based on the GOXcrossover first introduced by Bierwirth. This crossover stems fromthe OX crossover dedicated to the TSP. The main characteristic ofthis crossover is to preserve the relative order of operations. Thecrossover is managed only on MTS. Both Child.OA is generatedaccording to heuristic which computes OA for a fixed machine andtransport selection (MTS). The crossover procedure is based ontwo loops: the first one assigns job numbers which are in C1.MTSto Child.MTS. The array USE computes the total number of

occurrences of each job in the sequence. The second loop scansthe second chromosome from the first position to the last one,while inserting job number of C2.MTS since the number ofoccurrences is less than ni.

At the end of the procedure, a call to the procedureCompute_OA permits to have a full child chromosome which isevaluated by Evaluate_A_Solution.

3.4. Mutation by local search

In this section we introduce some specific properties ofproblems which can be efficiently included in neighborhoods forlocal search approach. In Van Laarhoven et al. (1992) theneighborhood is based on the permutation of two successivemachine operations on the critical path which concerns the samemachine. The block approach is due to Grabowski et al. (1996)which is based on machine-block definition. Dell’amico andTrubian (1993) first introduced a specific machine-operationapproach by exchanging one machine-operation in a block withone machine-operation at the end or at the beginning of the sameblock. Previously stressed in publications on the job-shop, blocksdefinition are strategic to obtain efficient local search scheme. Thesearch space is the set of all complete selections. At each iterationa neighboring solution is generated by moving a machineoperation or by moving a transport operation or by modifyingthe assignment of a transport operation to a robot. Let us noteFather the vector which contains one critical path in an acyclicgraph. The sequence u1,u2,u3, ..., un (where u1¼0 and un thedummy node at the end of the graph) is the sequence ofoperations (machine operations or transport operations) whichdefines the critical path. We define also: (i) a machine-block as asuccession of machine operations in the sequence which areprocessed consecutively in the critical path; (ii) robot-block as asuccession of transport operations in the sequence which isprocessed consecutively in the critical path. The theorempresented in Hurink and Knust (2005) holds and it is easy to

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–34 29

state that to obtain a new neighborhood solution with acompletion time less than or equal to the completion time ofthe current solution, either: (i) at least two transport operations ofa robot-block must be processed in the opposite order; (ii) at leasttwo machine operations of a machine-block must be processedin the opposite order. Let us note that adjusting the assignment

Algorithm 2. Local search

of one robot to one transport operation in the critical pathcan induce a new solution with completion time less than orequal to the current one. Modifications of the graph consist inmodifying MTS or OA by permutation or by modification of onevalue in a special rank. The local search procedure is given byAlgorithm 2.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–3430

3.5. Population structure and initialization

The initial population is composed of nc randomly generatedchromosomes in an array Pop sorted by decreasing fitness order.In traditional GAs, identical solutions or clones appear, introdu-cing a premature convergence to local minima. The local searchreduces cost interval in population and increases the convergencerate of the algorithm. One solution to avoid this phenomenonconsists in forbidding clones. Because exact clone detection can betime consuming, hashing techniques have been promoted forexample by Cormen et al. (1990).

Lastly an efficient hashing technique has been promoted forthe job-shop with time lags in Caumond et al. (2008): the hashfunction consists in auditioning the earliest starting time ofoperations using a modulo function avoiding large mark compu-tation. The hash function linked to a chromosome C is:

Algorithm 3. Memetic algorithm

markðCÞ ¼P

it2i ðmod KÞ where K is a large integer value, for

example 49 999.

3.6. Incremental memetic algorithm

The memetic approach takes the concept of evolution asemployed in genetic algorithms. The improvement is obtained byincorporating local search into the genetic algorithm. Evolution-ary algorithms become increasingly robust and easy to use asglobal optimization method and have received considerableamount of attention.

The well-known key features of a memetic algorithms schemeare (Moscato et al., 2007; El Fallahi et al., 2008; El Mekkawy andLiu, 2009):

�

a random generation of initial population; � application of a crossover operator to child generation; � local search with a probability to favour convergence; � periodic restart trying to avoid premature convergence.

Each iteration of the Memetic Algorithm (MA) procedure,

presented in Algorithm 3, starts by randomly selecting twosolutions P1 and P2 using binary tournament method. The rang i

of parent P1 is selected with ð2i=ncðncþ1ÞÞ, where nc is thepopulation size (see Reeves, 1993). The rank of parent P2 is drawnuniformly with a probability1=nc. One child solution C is obtainedwith a modified OX crossover presented in Section 3.3. C

undergoes with a given probability pm the local search describedin Section 3.4. The clone detection is achieved on the solutionobtained after local search. If the ended solution is a clone, theinitial solution is considered in auditioning in the population.Depending on the clone detection, the solution is included or notin the population and a new sort is managed to preserve thefitness decreasing order of the population. Since the populationsize is fixed the worst solution is discarded after each insertionthanks to the sort process. A global variable npi is used to count

the number of consecutive iterations without any improvement ofthe best solution found: restarts are managed as soon as themaximal number of iterations without improvement is reached.StopCriterion encompasses both the maximal number of iterationsas well as the lower bound of the problem.

3.7. Multi-robots framework

Due to the combinatorial nature of the problem, methodstackling several robots instances, have extreme difficulty ineffectively searching the solution space even for small scaleinstances. To navigate the search to new regions of the solutionspace, the memetic algorithm is processed iteratively with anincremental number of robots from 1 to r. The multi-robotsframework performs intensification in the search by successiveexecution of the memetic algorithm (Algorithm 4). This mechan-ism permits to transcend local minima by replacing solutions inpopulation by new chromosomes including canonical ones andrandomly generated ones.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–34 31

Algorithm 4. Incremental resolution of instances with several robots

4. Computational evaluation

Two types of experiments have been done using well knownbenchmarks in the literature. The first type of experimentsconcerns instances with one robot (Huring and Knust, 2005),and the second one with two identical robots (Bilge and Ulusoy,1995). Methods are evaluated as regards the gap between the bestsolution found and the lower bound (or the best known solution).Results are given in Tables 2–4. In this section, the followingnotations are used:

TabExp

N

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

T

M

N: total number of operations to schedule;LB: lower bound introduced by Hurink and Knust (2005);BFS: the best solution found by the framework;Dev.: deviation in percentage to LB;Avg.: average;Stop_criterion: criterion to stop algorithm execution;T: computational time (in seconds) to found BFS;n: number of jobs of the considered instance;m: number of machines of the considered instance.

le 2eriments on job-shop with one single robot on Hurink and Knust (2005) instances.

ame of instances Hurink and Knust results: deviation in percentage from

N LB UBone Dev1% UBtwo

1-P01.dat.D1_d1 66 82 87 6.10 88

1-P01.dat.D1_t1 66 77 81 5.19 83

1-P01.dat.D2_d1 66 147 148 0.68 155

1-P01.dat.D3_d1 66 213 217 1.88 219

1-P01.dat.tkl.1 66 136 137 0.74 138

1-P01.dat.T2_t1 66 71 74 4.23 76

1-P01.dat.T3_t0 66 92 92 0.00 94

1-P02.dat.D1_d1 190 880 1044 18.64 1035

1-P02.dat.D1_t0 190 880 1042 18.41 1055

1-P02.dat.D1_t1 190 880 1016 15.45 1021

1-P02.dat.D2_d1 190 892 1070 19.96 1064

1-P02.dat.D3_d1 190 906 1070 18.10 1084

1-P02.dat.D5_t2 190 1167 1325 13.54 1390

1-P02.dat.T1_t1 190 874 1006 15.10 1053

1-P02.dat.T2_t1 190 880 1015 15.34 1058

1-P02.dat.T5_t2 190 898 1102 22.72 1102

1-P02.dat.tkl.1 190 888 1086 22.30 1090

1-P02.dat.tkl.2 190 896 1028 14.73 1073

1-P02.dat. mult0.5_D1_d1 190 482 555 15.15 578

1-P02.dat. mult0.5_D1_t1 190 482 544 12.86 559

1-P02.dat. mult0.5_D2_d1 190 497 633 27.36 680

1-P02.dat. mult0.5_D2_t0 190 497 578 16.30 603

1-P02.dat. mult0.5_D2_t1 190 497 613 23.34 627

ean total deviation 13.40

The parameters used (whatever the experiments) in thememetic algorithm are as follows:

the

nm: 10// maximal number of iterations for local search;nc: 75//number of chromosomes in population;np: 2200// number of iterations before restart;pm: 0, 35//probability of local search;pr: 95//percent of chromosome replaced during restart;K: 49999//parameter used in the hash function;

All procedures are implemented under Delphi 7.0 packageand experiments were carried out on a 2.0 GHz Pentium DualCore computer under Windows XP Media Center with 2 GB ofmemory. The first benchmark is concerned with instancesbased on the Hurink and Knust’s instances with only onetransport robot (Huring and Knust, 2005). Note, only instanceswith transportation times independent of jobs have been used inthe experiments. These instances are dedicated to the single robotjob-shop problem. Extensions of these instances can be down-loaded at: www.isima.fr/� larabi. 23 instances are investigated

best solution found by each method to lower bound Framework

Dev2% UBsh Dev3% UBlg Dev4% BFS Dev5%

7.32 88 7.32 87 6.10

7.79 83 7.79 81 5.19

5.44 153 4.08 148 0.68

2.82 216 1.41 213 0.00

1.47 141 3.68 136 0.00

7.04 74 4.23 74 4.23

2.17 93 1.09 92 0.00

17.61 1013 15.11 990 12.50 1012 15.00

19.89 989 12.39 989 12.39 1017 15.57

16.02 995 13.07 989 12.39 983 11.70

19.28 1004 12.56 993 11.32 1045 17.15

19.65 1078 18.98 1072 18.32 1100 21.41

19.11 1383 18.51 1371 17.48 1361 16.62

20.48 1022 16.93 1018 16.48 978 11.90

20.23 1053 19.66 1030 17.05 993 12.84

22.72 1090 21.38 1020 13.59 1022 13.81

22.75 1061 19.48 1018 14.64 1009 13.63

19.75 1058 18.08 1014 13.17 1002 11.83

19.92 562 16.60 558 15.77 581 20.54

15.98 551 14.32 542 12.45 546 13.28

36.82 674 35.61 666 34.00 673 35.41

21.33 595 19.72 595 19.72 584 17.51

26.16 621 24.95 620 24.75 620 24.75

16.16 14.22 16.63 12.57

Table 3Experiments on job-shop with two robots on Bilge and Ulosoy (1995) instances

minimizing the exit time of the last job of the system.

Instances BKS BFS DEV% Instances BKS BFS DEV%

Ex11 114 114 0 Ex61 129 129 0

Ex12 90 90 0 Ex62 102 102 0

Ex13 98 98 0 Ex63 105 105 0

Ex14 140 140 0 Ex64 151 151 0

Ex21 116 116 0 Ex71 134 134 0

Ex22 82 82 0 Ex72 86 86 0

Ex23 89 89 0 Ex73 93 93 0

Ex24 134 134 0 Ex74 161 161 0

Ex31 121 121 0 Ex81 167 167 0

Ex32 89 89 0 Ex82 155 155 0

Ex33 96 96 0 Ex83 155 155 0

Ex34 148 148 0 Ex84 178 178 0

Ex41 138 138 0 Ex91 129 127* �1.55

Ex42 100 100 0 Ex92 106 106 0

Ex43 102 102 0 Ex93 107 107 0

Ex44 163 163 0 Ex94 149 149 0

Ex51 110 110 0 Ex101 153 153 0

Ex52 81 81 0 Ex102 139 139 0

Ex53 89 89 0 Ex103 141 139* �1.42

Ex54 134 134 0 Ex104 183 183 0

Table 4Experiments on job-shop with two robots on Bilge and Ulosoy (1995) instances

minimizing the completion time of the last job on the last machine.

Instances t/p ratio BKS BFS DEV%

EX110 0.15 126 126 0.00

EX210 0.15 148 148 0.00

EX310 0.15 150 150 0.00

EX410 0.15 119 119 0.00

EX510 0.21 102 102 0.00

EX610 0.16 186 186 0.00

EX710 0.19 137 137 0.00

EX810 0.14 292 292 0.00

EX910 0.15 176 176 0.00

EX1010 0.14 238 238 0.00

EX120 0.12 123 123 0.00

EX220 0.12 143 143 0.00

EX320 0.12 145 145 0.00

EX420 0.12 114 114 0.00

EX520 0.17 100 100 0.00

EX620 0.12 181 181 0.00

EX720 0.15 136 136 0.00

EX820 0.11 287 287 0.00

EX920 0.12 173 173 0.00

EX1020 0.11 236 236 0.00

EX130 0.13 122 122 0.00

EX230 0.13 146 146 0.00

EX330 0.13 146 146 0.00

EX430 0.13 114 114 0.00

EX530 0.18 99 99 0.00

EX630 0.14 182 182 0.00

EX730 0.17 137 137 0.00

EX830 0.13 288 288 0.00

EX930 0.13 174 174 0.00

EX1030 0.12 237 237 0.00

EX140 0.18 124 124 0.00

EX241 0.13 217 217 0.00

EX340 0.18 151 151 0.00

EX341 0.12 221 221 0.00

EX441 0.19 172 172 0.00

EX541 0.18 148 148 0.00

EX640 0.19 184 184 0.00

EX740 0.24 137 137 0.00

EX741 0.16 203 203 0.00

EX840 0.18 293 293 0.00

EX940 0.19 175 175 0.00

EX1040 0.17 240 240 0.00

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–3432

with 66 operations to schedule for the smallest one and 190operations for the largest one. Because the framework we proposeis a random search method, each instance is studied over 5 runs.Hurink and Knust (2005) provide four methods: (i) one stageapproach run within ten minutes (UBone); (ii) a two stageapproach run within ten minutes (UBtwo); (iii) a combinedapproach run within ten minutes (UBsh); (iv) a combinedapproach run within one hour (UBlg). Since the results arecompared, for this part of the computational experiment wedecided to use the same time limits as these of Hiring and Knust.Results given in Table 2 are obtained using time limit of tenminutes and show that the proposed framework has an averagedeviation of 12.57% from the lower bound provided by Hurink andKnust (2005). For five instances new BFS are obtained: threeinstances (T1-P02.dat.D3_d1, T1-P02.dat.T1_t1, T1-P02.dat.T2_t1)only new BFS are proposed and for two instances (T1-P02.dat.D1_t0, T1-P01.dat.D3_d1) the results are equal to the LB,i.e. the optimal solutions are obtained. Appendix A gives the Ganttchart of the optimal solution for instance T1-P01.dat.tkl.1. We canstate that the framework can compete with dedicated methods ofHurink and Knust (2005).

For the job-shop with two identical transport robots we usethe benchmark test suggested by Bilge and Ulusoy (1995) andused by Ulusoy et al. (1997), Abdelmaguid et al. (2004), Reddy andRao (2006) and Deroussi et al. (2008). It is composed of 40instances. All of them are made up of one load/unload station, fourmachines and two vehicles. The loading/unloading times areneglected compared to the transportation times. Empty movesand transportation moves are supposed to spend the sameamount of time. These instances are generated according to tenjob sets, combined with four different topologies for the work-shops. Topologies are assumed to be representative of existingsystems, and the job sets include between 5 and 8 jobs, andbetween 13 and 21 operations to be scheduled. The name given tothe instances is composed by the prefix ‘Ex’, followed of twonumbers representing respectively the job set and the topology.Thus, Ex12 indicates the instance generated by job set 1 andtopology 2. Results presented in Tables 2,3 and 4 are obtainedusing time limit of 2 min. In the most of the cases BFS is obtainedin few seconds. Table 3 gives the comparison of framework bestfound solutions (BFS) with the best known solutions (BKS) (seeDeroussi et al., 2008). The criterion to minimize is the exit time ofthe last job of the system taking into account the last transportoperation to the unload station. For all instances the frameworkproposed in this paper finds the best known solutions. Moreoverfor two instances new best found solutions are proposed.

The performance of the proposed framework is also tested forthe most studied in the literature criterion: minimize thecompletion time of the last job on the last machine (makespan)without taking into account the last transport operation to theunload station; this criterion is studied with the 42 problemsreported in literature by Bilge and Ulusoy (1995), Ulusoy et al.(1997), Abdelmaguid et al. (2004)and Reddy and Rao (2006) fordifferent t/p ratios (travelling time/processing time). The bestknown solutions reported in the literature are compared with thebest results obtained by the proposed method (see Table 4). Theperformance of the framework is similar to the previousexperiment. The framework finds all best known solutions infew seconds.

5. Concluding remarks

This paper is a step forwards the generalization of thedisjunctive graph model including several robots. We developeda memetic algorithm based approach for a generalization of the

Fig. A1. Gantt charts on the new optimal solution for T1-P01.dat.tkl.1. The two Gantt charts show the processing of six jobs (J1 to J6) on six machines (M1 to M6). The

schedule of each job gives the number of operation depending on the manufacturing routing. The robot schedule gives, if not empty (E), the origins machine, job number

and the destination machine.

P. Lacomme et al. / Int. J. Production Economics 143 (2013) 24–34 33

job-shop problem where additionally transportation aspects withmultiple transport resources are taken into account. An extendeddisjunctive graph is introduced to efficiently model bothproblem and solutions. Specific properties are derived from thelongest path to generate neighborhoods. The proposed modellingapproach can be used easily to solve flexible job-shop with severaltransport robots. A local search takes advantages of bothmachine-block and transport-block. The problem appears to behard to solve requiring a proper coordination of both transportrobot and machine disjunctions. The presented results prove that:

�

The framework competes, for instances with only one trans-port robot, the dedicated method of Hurink and Knust (2005)with an average deviation of 12.41 percents as regards thedeviation of 13.4. Moreover two solutions are proved to beoptimal (T1-P01.dat.D3_d1, T1-P01.dat.tkl.1). � The results, for instances with two robots, clearly show the

superiority of proposed framework. Our method finds a resultat least as good as those published so far. Moreover 2 newupper bounds are found.

This work is a step forward definition of wide ranging methodsfor job-shop with robots including the special case with one robotonly. The work pushes us into accepting a high quality results for onerobot instance and several robots instances. Our research is nowdirected on exact solutions procedures for multi-robots instances androbust resolution schemes to tackle stochastic transportation time.

Appendix A

See Fig. A1.

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