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A modified shifting bottleneck heuristic anddisjunctive graph for job shop scheduling problemswith transportation constraintsQiao Zhanga, Hervé Maniera & Marie-Ange Maniera IRTES-SET, Université de Technologie de Belfort-Montbéliard, Belfort Cedex, France.Published online: 14 Aug 2013.

To cite this article: Qiao Zhang, Hervé Manier & Marie-Ange Manier (2014) A modified shifting bottleneck heuristic anddisjunctive graph for job shop scheduling problems with transportation constraints, International Journal of ProductionResearch, 52:4, 985-1002, DOI: 10.1080/00207543.2013.828164

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International Journal of Production Research, 2014Vol. 52, No. 4, 985–1002, http://dx.doi.org/10.1080/00207543.2013.828164

A modified shifting bottleneck heuristic and disjunctive graph for job shop scheduling problemswith transportation constraints

Qiao Zhang, Hervé Manier and Marie-Ange Manier∗

IRTES-SET, Université de Technologie de Belfort-Montbéliard, Belfort Cedex, France

(Received 29 June 2012; final version received 16 July 2013)

In this paper, we consider job shop scheduling problems with transportation constraints and bounded processing times. We usea modified disjunctive graph to represent the whole characteristics and constraints of such considered problems. Comparedwith classical disjunctive graph, it contains not only processing nodes, but also transportation and storage nodes. There arealso positive and negative arcs for bounded processing time constraints, transportation times and minimum and maximumallowed storage times before and after each processing task. The objective is to minimise makespan. A feasible solution formakespan is found, if its associated graph contains no positive cycle. A modified shifting bottleneck procedure is used tosolve the studied job shop problems which are represented by disjunctive graphs. It is coupled with a heuristic for assigningand sequencing transportation tasks iteratively. To validate our approach, several types of benchmarks with fixed or boundedprocessing times are tested, corresponding to flexible manufacturing systems, robotic cells and surface treatment facilities.Computational results show that the modified disjunctive graph and the proposed method are efficient and can deal withvarious cases.

Keywords: job shop scheduling; transportation; bounded processing times; disjunctive graph; shifting bottleneck procedure

1. Introduction

During the last decades, more and more complex manufacturing systems have been used for various kinds of production,for example: flexible manufacturing systems (FMSs), robotic cells (RCs) and surface treatment facilities (STFs). In thesesystems, job shop scheduling problems with transportation constraints are needed to be solved, but constraints may varyaccording to the system and production environment. Various kinds of main characteristics may be encountered:

• flexibility: in fact two kinds of flexibility may be considered. The first one corresponds to the fact that one machine canperform several operations of a given processing sequence. In this case, we talk about re-entrant job shop problem. Inthe second case, several processing resources (respectively, transportation resources) can perform the same processingtask (respectively, transportation task). Then an assignment problem needs to be considered additionally to thesequencing one. Here, we talk about flexible job shop scheduling problem (FJSSP).

• buffer associated with machines: it can be allowed or prohibited between any two processing resources;• no wait constraint: this one may exist or not. In the first case, once one product enters the system, it can be either on

a machine fulfilling its treatment or on a transport resource moving to the next destination;• processing times: they may be fixed in advanced. But in some cases, they may be bounded by maximum finite values

and even some times unbounded (maximum processing time is +∞). This constraint ensures product quality.

In the systems and associated problems that we study, a combination of several (or all) characteristics may exist. Our goal isto propose a method which is able to solve such problem whatever the combination is.

Scheduling problems in these systems have received a great deal of attention because reducing lead time is always a veryimportant goal for industry. Job shop scheduling with transportation constraints is NP-hard and only very special cases maybe solved polynomially (Knust 1999). We can distinguish two types of such problems: with fixed processing durations andunlimited storage; with bounded/fixed processing times but no buffer exists.

The first type can be found in FMS and some RCs. Bilge and Ulusoy (1995) considered a job shop problem withtwo transport resources, fixed processing times, fixed assignment of processing tasks and four topologies of transporta-tion layout. A time window approach is proposed to schedule machines and material handling system. Hurink and Knust(2005) proposed a two-stage local search for solving a job shop problem with a single robot, fixed processing dura-tions and fixed assignment. Lacomme, Larabi, and Tchernev (2007) applied a disjunctive graph to represent the problem

∗Corresponding author. Email: marie-ange.manier@utbm.fr

© 2013 Taylor & Francis

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proposed by Bilge and Ulusoy (1995) with two transportation resources, and a memetic algorithm is developed to solve it.Deroussi, Gourgand, and Tchernev (2008) studied the same problem by using a simple but efficient meta-heuristic approach.Deroussi and Norre (2010) studied flexible job shop problems with transportation with two possible resources for eachtask. Instances are extensions of benchmarks of Bilge and Ulusoy (1995). A dynamic heuristic is proposed to solve them.Other researchers (like Moslehi and Mahnam 2011; Lee et al. 2012) studied flexible job shop problems by using artificialintelligence tools. However most of the time, transportation constraints are not considered.

The second type of problems exists mostly in systems where no buffer is allowed (RC and STF). Crama et al. (2000)classified problems in RCs into two types: the first one is referred to unbounded processing time windows (Sethi et al. 1992;Crama and Van de Klundert 1997a; Hall, Kamounb, and Sriskandarajah 1998), and the second one is known as zero-widthprocessing time windows or no-wait (Levner, Meyzin, and Ptuskin 1998). Crama and Van de Klundert (1997b) andCrama et al. (2000) analysed the complexity of finding cyclic schedules in robotic flowshop cells. Didem Batur,Ekin Karasan, and Selim Akturk (2012) studied multi-part jobs cyclic scheduling problem in a RC with two flexible machinesand one robot. Processing times are adjustable for all parts. A 2-stage heuristic algorithm is constructed to minimise thecycle time. STF is a system composed of tanks containing chemical or electrolytic baths, in which hoists are responsible fortransportation tasks. Processing times are limited by minimum and maximum durations. Hoist scheduling problems (HSPs) areconsidered for such systems (Bloch et al. 2008). Manier and Bloch (2003) classified problems in STF into four types: CHSP(cyclic HSP), PHSP (predictive HSP), DHSP (dynamic HSP) and RHSP (reactive HSP). Like for RC scheduling problems,cyclic instances are often the most studied ones. Phillips and Unger (1976) studied a 1-degree cyclic HSP(CHSP) with a singlehoist.Abranch and bound-based mixed integer linear programming was applied to solve it. Che, Chu, and Chu (2002) appliedan exact method “branch and bound” to find multi-degree cycle schedules of CHSP with one hoist. Subai, Baptiste, and Niel(2006) have taken into account environmental constraints in HSPs while maximising the throughput. A multi part-type CHSPproblem was treated by Mateo, Companys, and Bautista (2002). El Amraoui et al. (2011) extended this problem for morecomplex lines and also to find n-degree cycles. Manier and Lamrous (2008) used an evolutionary approach to solve cyclicHSPs with several hoists. The encoding is based on the representation of empty hoists’moves. Most of the studied problems inRCs and STFs are cyclic ones in which types of parts are limited. Bloch, Varnier, and Baptiste (1999) studied the predictiveHSP by using a disjunctive graph representation and a modified shifting bottleneck procedure. But the number of hoistswas unlimited.

Until now, few researches consider a general model which can incorporate various kinds of problems encountered inthose different systems. For this goal, we consider a FJSSP with transportation constraints involved by critical handlingresources, in which the processing durations may be limited by lower and upper bounds, and with null or unlimited capacityof buffers. This model is a generalisation of the classical job shop scheduling problem and FJSSP, which allows an operationto be processed by any resource from a given set. Moreover, processing times are bounded and transport constraints cannotbe neglected. Thus problems with fixed processing times or/and fixed assignment can be considered as special cases for thisgeneral model. One interest of our general method is to handle more possible realistic industrial cases. As an example, wemay solve scheduling problems for various kinds of production demands on a same given flexible set of resources. Also,in the production process, some industries use various kinds of shops. They often apply dedicated methods to optimise theperformance for each one. With our model, we could consider the various shops as a single one, and then provide a globaloptimisation of the production.

In a previous work, we elaborated a hybrid method of genetic algorithm and tabu search for this kind of problems(Zhang, Manier, and Manier 2012a). However, it is less efficient for HSP which requires zero storage strictly. In this paper,we use a modified disjunctive graph representation so as to model the general model. This graph is detailed and compared withclassical graph representation in Section 2. A modified shifting bottleneck procedure is applied to solve job shop problemsrepresented by the disjunctive graph (Section 3). Various types of instances are tested and results are shown and analysed inSection 4 including a comparison with the best results found in the literature. The conclusion and perspectives are given inSection 5.

2. Disjunctive graph representation

In the literature, many researchers have applied disjunctive graphs for various job shop scheduling problems.Blazewicz, Pesch, and Sterna (2000) used a disjunctive graph for machine representation. Hurink and Knust (2005) andLacomme, Larabi, and Tchernev (2007) applied disjunctive graph models containing nodes of processing and transport tasksfor a system with one or several transportation resources. The disjunctive graphs used in those three studies suit only forfixed processing times. Then they cannot model HSPs. For such problems, Bloch, Varnier, and Baptiste (1999) proposed amodified disjunctive graph by adding negative arcs for representing maximum processing times. It contains processing nodesand can represent predictive problems in STFs. But, it cannot represent schedules on transport resources. We have modified

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Figure 1. Conjunction among basic elements in proposal disjunctive graph.

this disjunctive graph by adding transport nodes (Zhang, Manier, and Manier 2011). This graph can easily model HSPs forboth processing and transport resources. However, only solutions without storage can be represented by this graph. So itdoes not suit for other types of problems (FJSSP for FMS and RC). Until now and to our knowledge, no model exists torepresent various kinds of scheduling problems with both transport and bounded processing times, which can be found inthe three identified systems (FMS, RC and STF). In this section, we describe the considered general problem and a modifieddisjunctive graph representation.

2.1 General FJSSP

The considered general problem contains: a set of N independent jobs J = {J1, J2, . . . , JN }, a set of m processing resources(machines) M = {M1, M2, . . . , Mm} and a set of r identical transport resources (robots) R = R1, R2, . . . , Rr . Each job Ji

consists of ni totally ordered processing tasks Oi1, Oi2, . . . , Oini . Each operation Oi j can be performed without preemptionby any machine chosen from a set of processing machines Ui j ⊆ M . The input and output storage activities before and

after an operation Oi j are S−i j and S+

i j , respectively. The processing time for Oi j is within a time window[

p−i j , p+

i j

]and is

machine-independent. When Oi j is finished, a transportation task Ti j needs to be processed by one and only one transportresource. Its minimal associated travel time is vi j . The maximal transportation time on a transportation resource is noted Z3.Transportation times are machine-dependent. Empty travel times may differ from loaded ones between any two machinesMk to Ml . The capacity of buffers can be null, limited or unlimited, depending on the associated system requirement. In ourstudy, we consider null or unlimited capacity. The maximum allowed waiting time for each input and output storage activitiesare, respectively, noted as Z1 and Z2.

The objective is to minimise the makespan Cmax while respecting processing time constraints, travel time constraints,and/or storage requirement, and/or no wait constraint. For the proposed model, following assumptions are made:

• resource failures are ignored;• all resources can handle at most one job at a time;• each task can only be assigned to one and only one resource;• all resources are available at time zero;• release times of jobs are zero;• each job consists of its fixed processing sequence;• once a task is performed on a resource (machine/transport resource), it cannot be interrupted until it is finished (no

preemption is allowed);• once a machine finishes any task, it becomes available for another operation.

The mathematical model associated with this problem, that we call general FJSSP with transportation, can be found inZhang, Manier, and Manier (2012a).

2.2 A modified disjunctive graph

We define a disjunctive graph G = V ∪ C ∪ D with V = Vm ∪ Vt ∪ Vs , D = Dm ∪ Dt . Vm denotes all processing tasks ofjobs and two dummy nodes: a source and a sink nodes. Vt is the set of nodes of all transportation tasks and Vs denotes all inputand output storages. A set of conjunctive arcs C represents precedence constraints in the same job (see Figure 1). Any jobfollows alternatively a set of waiting/processing/waiting/transportation tasks. Conjunctive arcs consist of positive/negativearcs to model minimum/maximum limits on tasks’ durations (processing, storage and transportation tasks).

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Table 1. Various values according to constraints.

Characteristics Constraints Usually encountered systems Z1 p+i j Z2 Z3

Processing times Bounded RC, STF > p−i j

Fixed FMS, RC = p−i j

Unbounded FMS, STF, RC = +∞Buffers No wait RC, STF = 0 = 0 = vi j

No storage RC, STF = 0 = 0 = vi jStorage allowed FMS, RC = +∞ = +∞ = vi j

C also includes arcs between each processing/transport node and its next output/input buffer. Positive and negative valueson these arcs correspond to processing time limits or loaded travel times. These added negative arcs are useful to bound timesof tasks. Values on negative arcs vary according to the associated constraints. The minimum waiting time in each bufferis zero, while the maximum waiting time depends on the characteristics of the considered system. Table 1 shows variouspossibilities for those values according to the different types of systems and associated constraints. In STF and RC, whereno buffer is allowed Z1 = Z2 = 0. Otherwise, in FMS, Z1 = Z2 = ∞. This flexibility allows the graph to represent variouskinds of problems. In STF, processing times are bounded, then positive and negative values between a processing node andits output buffer equal to p−

i j and p+i j , respectively. In other systems, where processing times are fixed p+

i j = p−i j . Finally,

Dm and Dt are sets of disjunctions connecting non-sequenced tasks that require the same resource (machine or transportresource). The procedure used to arbitrate disjunctions is described in the next section.

Note that this disjunctive graph cannot represent assignment problem. It is constructed once an assignment is chosen.In this paper, we generate assignments randomly.

2.3 Arbitration of disjunctions

A solution is modelled by the modified disjunctive graph, once all disjunctions in Dm and Dt are arbitrated. The orientationof disjunction is inspired by Bloch, Varnier, and Baptiste (1999) and Zhang, Manier, and Manier (2011) for a more simpledisjunctive graph dedicated to STFs. Figures 2 and 3 shows the way of orientating processing and transport disjunctions. Thearbitration of disjunctions in this disjunctive graph is different from the classical way. In this one, we can arbitrate directlyfrom one node to another one, for example, by adding arcs Oi j → Okl or/and Tkl → Ti j . The values of arcs correspond tothe real associated times (known fixed ones) of Oi j or/and Tkl . But with bounded durations, we cannot know the real timein advance. In Figure 2, the arbitration “Oi j before Okl ” is introduced by adding the arc S+

i j → Okl . The value on this arc iszero, which means that Okl can start right after Oi j . In Figure 3, in the same way, the order “Tkl before Ti j ” is represented bythe arc S−

kl+1 → Ti j . This arc is valued by the empty travel time from the destination machine of Tkl to the starting machineof Ti j . It gives the robot enough time to move between two successive transport tasks.

Compared with classical disjunctive graphs and their arbitration procedure, the proposed graph and orientation can modelthe scheduling on both machines and transport resources while considering bounded processing times and loaded and emptytravel times. Furthermore, the existence of storage activity nodes allows us to represent solutions with or without storage.

The modified disjunctive graph can represent more solutions than a simple graph like in Bloch, Varnier, and Baptiste(1999). To illustrate this difference, let us consider a small example to show the representations of a solution. This example

Figure 2. Orientation of the disjunction between two processing tasks.

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Figure 3. Orientation of the disjunction between two transportation tasks.

Table 2. Data for a job shop problem.

job i Processing tasks

Oi1 Oi2 Oi3 Oi4

1 M0 : [0, ∞] M1 : [8, ∞] M2 : [16, ∞] M4 : [12, ∞]2 M0 : [0, ∞] M1 : [20, ∞] M3 : [10, ∞] M2 : [18, ∞]3 M0 : [0, ∞] M4 : [14, ∞] M2 : [18, ∞]

consists of 3 jobs, 5 machines (including a loading machine M0) and unlimited buffers (see data in Table 2). The loaded andempty transportation times are 1. A possible machine scheduling is shown as below:

M0 : O11 → O21 → O31M1 : O12 → O22M2 : O13 → O24M3 : O23M4 : O32 → O14

This feasible schedule is represented in the simple graph in Figure 4, and in the modified graph in Figure 5, respectively.We can see that there is a positive circuit (in bold dotted line: O14 → O24 → end of J2 → O33 → O14) in this simplegraph even without adding transport resource schedules. This means that a feasible solution can be detected as a non-feasiblesolution by a simple graph. This is because the transportation tasks are not represented individually by nodes in the graph.Indeed, in order to integrate the transport constraints, the transport durations are rather added to the processing time in theassociated arcs. In the modified graph, the same machine schedule is represented without positive circuit. We conclude thatthe modified graph can represent more feasible solutions than a simple one. On the other hand, a simple graph is based onunlimited number of robots. Then it may be unable to detect if one solution is not feasible in terms of robot capacity. Ourgraph can address such problem. Other graphs in the literature can also do it but with other hypotheses (unlimited storage,fixed times, ...).

A complete feasible solution (for the same example in Table 2, with 2 robots) is represented by the proposed graphin Figure A1 in Appendix A. This feasible solution integrating the previous machine sequencing and the following robotsequencing :

R1 : T11 → T31 → T22 → T13R2 : T12 → T21 → T32 → T23

The evaluation of a solution represented by the modified graph can be made by using Dantzig’s algorithm(Gondran and Minoux 1983). A valid solution’s makespan is the longest path without positive circuit.

To conclude, the graph we use here can be considered as an extension of more simple graphs (Bloch, Varnier, and Baptiste1999; Hurink and Knust 2005; Lacomme, Larabi, and Tchernev 2007). None of them consider both processing, transport,storage activities and bounded times. Of course, this involves more nodes and arcs, and then more time for the evaluationstep.

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Figure 4. A simple graph representation for machine sequences.

Figure 5. Our proposed graph representation for machine sequences.

3. A modified Shifting Bottleneck heuristic (MSB)

A Shifting Bottleneck heuristic (SBH), first presented by Adams, Balas, and Zawack (1988), is used to find feasible schedulesfor job shop problems. It solves iteratively a single machine sub-problem for all machines that are not scheduled yet whilefinding the bottleneck resource. The bottleneck resource is the machine which mostly impacts the performance (makespan).Its determination is done by a branch and bound procedure and is based on processing times, release dates and due dates oftasks. Release dates and due dates are calculated by the associated disjunctive graph during each iteration. Then the minimumlateness of each machine is determined by using a branch and bound technique to find the machine schedule which reducesmaximum lateness for the associated job. Once all machines’ maximum latenesses are determined, the one with the largestmaximum lateness is chosen as the bottleneck machine and arcs corresponding to its schedule are arbitrated in the graph.If there are several machines with the same maximum lateness, any one can be determined for the bottleneck. It continuesuntil all machines are scheduled.

In our modified procedure, robots are also considered as additional resources. The procedure starts with a set of resources(labelled SR) whose sequences are already fixed. In our case, those resources are often loading stations whose input sequencesof jobs are given (generated randomly). The release date of a task corresponds to the length of a longest path from the startingnode to its associate node in graph. Finding the longest path is made by using Dantzig’s algorithm (Gondran and Minoux1983), which can also detect positive cycles. Classically, duedates of each task are obtained by subtracting the remainingprocessing times of the jobs from the makespan. In our case, a task’s duedate equals to the difference between Cmax and thelength of the longest path from the considered node to the ending node. Because of the bounded processing time constraints,

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Algorithm 1. Heuristic to assign and schedule transportation tasks in graph G

Dantzig(G), obtain the longest path from the starting node to every node in G.Calculate the earliest finish time for each transport resource.From the smallest to the biggest, order all transportation tasks Ti j based on their release times (equal to the length of the longest pathfrom the starting node to each task’s node).Value = current longest path from the starting node to every node in G.repeat

for each task Ti j in order doif Ti j is not assigned yet then

assign this task to the transport resource which can execute it at the earliestupdate the finish time of the chosen resource.break;

end if

end for{arbitration of disjunction between the new assigned task Ti j and other assigned tasks on the same resource}while disjunctions exist among assigned tasks on the resource of the new assigned task do

choose one couple of tasks among the ones which have the smallest release date, arbitrate its disjunctionif the added arc can update the second task’s release date in the chosen couple then

if Dantzig (G) = −1 thendelete the added arc, add the inverse one.if Dantzig (G) = −1 then

Value = −1return Value

end ifend ifrecalculate the finish time for each resource and reorder all transportation tasks.

end ifend whileif all tasks are assigned and no more disjunction needs to be oriented then

Value = Cmax of Gbreak;

end if

until Value ≥ −1return Value

not all schedules are feasible ones. If the schedule of the bottleneck resource found by the branch and bound procedure isnot valid, then a simple repair is applied. It consists in permuting any pair of tasks in this sequence. This simple repair isnot always efficient when there is a big amount of tasks on a resource, especially in STF and for robots which have manytransportation tasks to perform.

To overcome this shortcoming, we develop a new heuristic for assigning and sequencing transportation tasks one byone, while taking into account current release dates (see Algorithm 1). At each step, it chooses the task with the smallestrelease date and assigns it to the robot with the earliest available date. Once a new task is assigned, new disjunctions must bearbitrated. The disjunctions will be arbitrated in an increasing order of already assigned tasks (considering the earliest releasedate in the sequence of the considered robot). For each arbitrated disjunction, the associated arc in the graph is oriented.The new graph obtained is then verified and evaluated by Dantzig’s algorithm. The procedure is repeated until all tasks areassigned and no disjunction remains. Then it returns the value of Cmax for the last evaluated graph G. But the algorithmmay also stop when infeasible solutions are identified during arbitration, i.e. if the graph contains a positive cycle. Then theDantzig’s algorithm returns −1.

Figure 6 shows the combination of the proposed heuristic and the modified shifting bottleneck procedure (MSB) withrepair for transportation assignment and schedules. First, we use the shifting bottleneck procedure to find feasible schedulesfor all machines (Zhang, Manier, and Manier 2011); second, we use our new heuristic (Algorithm 1) to find feasible schedulesfor transport resources. This heuristic orders and assigns tasks step by step by recalculating release dates once a new arcis added. It is then more realistic than the original way (determining assignments in advance and applying EDD rules forfinding schedules for each sub-problem). In fact, this algorithm may also be used for scheduling machines and not onlyrobots. But the proposal heuristic is time consuming (i.e. the complexity of each longest path computing is O(n3)). The

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Figure 6. The global framework of the modified shifting bottleneck procedure.

shifting bottleneck with simple repair can find feasible machine schedules using much less time. That is why we apply thisheuristic only for transport tasks.

4. Computational results

4.1 Instances

In order to validate the modified disjunctive graph model and methodology, four classes of instances corresponding to variousproduction systems are tested. Each class can be seen as a special case of our general model.

• The first class contains instances proposed by Hurink and Knust (2005). They are extensions of job shop problemsproposed by Muth and Thompson (1963). These instances are characterised by a single-available machine for eachprocessing task, a single-transport resource and fixed processing times. So, no assignment problem needs to beconsidered.

• The second class contains instances of Bilge and Ulusoy for FMS in Ulusoy and Bilge (1993) and Bilge and Ulusoy(1995). There are one available machine for each processing task, fixed processing times and two transport resources.So, the assignment of transport tasks needs to be considered.

• The third class is an extension of FJSP (flexible job shop scheduling problem) with transportation and was proposedby Deroussi et al. in Deroussi and Norre (2010). It is characterised by: each processing task can be processed onone machine among two available ones, fixed processing times and two transport resources. There are assignmentproblems for both processing and transport tasks.

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Table 3. Cmax for the first-class instances (Hurink and Knust 2005).

Inst. TS MSB Gap% CPU

P01 D1 d1 87 156 79.31 10.68P01 D1 t1 81 93 14.81 10.46P01 T2 t1 74 93 25.68 10.98P01 T3 t0 92 95 3.26 10.12P01 D3 d1 216 224 3.70 17.69P01 D2 d1 148 158 6.76 17.71Average 22.23 12.94

• The fourth class contains two types of HSP instances. They are characterised by: a single available machine for eachprocessing task, bounded processing times, no storage allowed between two operations or during transportation taskand one or several transport resource(s). Constraints of no wait and no buffers must be considered.

• The first type of instance was proposed by Mateo and Companys (2007). These instances contain two typesof jobs which enter alternatively the production line, with various transportation times (hoist speeds). All jobsfollow the same processing sequence but with different processing time windows. We used those instancesto solve job shop problems. But Mateo and Companys (2007) used them to solve a 2-degree cyclic HSP. Theauthors searched the optimal cycle time T of a repetitive hoist moving sequence for a single hoist. Becausethere are few benchmarks adapted to our considered job shop environment, we extend these instances intopredictive ones with 5 and 21 jobs (respectively, 3 and 11 jobs of type one, 2 and 10 jobs of type two) and 1or 2 robots. Then, we leave the cyclic environment to solve job shop ones. Based on a cycle time T , we candeduce a Cmax value for N jobs, as: Cmaxcyc = �N/2 × T + t t1. t t1 is the time spent on the line by thelast job (supposed of type 1). It equals the sum of processing and transportation times for this job.

• The second type of HSP instances was proposed by Paul, Bierwirth, and Kopfer (2010). In these instances,40 jobs must be treated. They are composed of four types of jobs with various rates. The total number ofprocessing tasks varies from 263 to 290. There are 18 machines, a single-transportation resource and twoloading and unloading stations.

• The fifth class is composed of ten instances. We extended those ones from the third class by introducing boundedprocessing times (Zhang, Manier, and Manier 2012a). The minimal durations p−

i j correspond to the fixed times ofthe third class. Each maximal duration p+

i j is determined by 1.2 × p−i j . Then, those instances contain two machines

for each processing task, bounded processing times, with unlimited buffer and 2 robots.

4.2 Results

Results are obtained by running our algorithm 1000 times with random assignment, if there exists assignment problems,and the best results are shown in the following tables. Programs are written in C++ language, and are tested on a AMDPhenom X2 2.8 GHz. CPU times vary according to different kinds of instances: from several minutes for small instances,to one hour for bigger instances (HSP instances). Mean CPU times (in minutes, labelled CPU) with our MSB approach areprovided for each instance in Tables 3, 4, 5(a) and (b), 6 and 9. For the bigger size instances (Tables 7 and 8), the results areobtained after running our MSB algorithm during one hour.

Tables 3–7 compare the makespan (Cmax) obtained with our method and the best ones found in literature and the previousone we proposed in Zhang, Manier, and Manier (2012a). In those tables, our algorithm is labelled MSB. The gap betweenMSB and any reference method (Ref) is calculated as: G Ref = (M SB − Ref )/Ref (%). The reference methods used forthe four classes of instances are respectively:

• TS: Tabu Search developped by Hurink and Knust (2005), for the first-class instances in Table 3.• GAA, SALS, MA: genetic algorithm used by Abdelmaguid et al. (2004), simulated annealing local search proposed

by Deroussi, Gourgand, and Tchernev (2008) and memetic algorithm applied by Larabi (2010) for the second-classinstances in Table 4.

• ILS: Iterated local search proposed by Deroussi and Norre (2010) for the third-class instances in Tables 5(a) and (b).Table 5(a) provides results in terms of exit time of the last job of the system, instead of in terms of makespan.

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Table 4. Cmax for FMS instances (Bilge and Ulusoy 1995).

Inst. SALS GAA MA MSB CPU Gap %

GS AL S GG AA G M A

Ex11 96 96 96 97 4.03 1.04 1.04 1.04Ex12 82 82 82 82 3.74 0.00 0.00 0.00Ex13 84 84 84 88 3.78 4.76 4.76 4.76Ex14 103 103 103 110 3.07 6.80 6.80 6.80Ex21 100 102 100 109 5.90 9.00 6.86 9.00Ex22 76 76 76 80 4.76 5.26 5.26 5.26Ex23 86 86 86 87 4.57 1.16 1.16 1.16Ex24 108 108 108 126 4.83 16.67 16.67 16.67Ex31 99 99 99 110 5.64 11.11 11.11 11.11Ex32 85 85 85 87 6.17 2.35 2.35 2.35Ex33 86 86 86 89 5.85 3.49 3.49 3.49Ex34 111 111 111 136 5.48 22.52 22.52 22.52Ex41 112 112 112 131 7.13 16.96 16.96 16.96Ex42 87 88 87 96 7.14 10.34 9.09 10.34Ex43 89 89 89 99 7.12 11.24 11.24 11.24Ex44 121 126 121 132 6.46 9.09 4.76 9.09Ex51 87 87 87 90 3.25 3.45 3.45 3.45Ex52 69 69 69 73 4.48 5.80 5.80 5.80Ex53 74 74 74 76 4.22 2.70 2.70 2.70Ex54 96 96 96 99 3.25 3.13 3.13 3.13Ex61 118 118 118 123 7.55 4.24 4.24 4.24Ex62 98 98 98 104 6.05 6.12 6.12 6.12Ex63 103 104 103 106 6.61 2.91 1.92 2.91Ex64 120 120 120 140 7.13 16.67 16.67 16.67Ex71 111 115 111 122 13.11 9.91 6.09 9.91Ex72 79 79 79 86 9.77 8.86 8.86 8.86Ex73 83 86 83 91 11.45 9.64 5.81 9.64Ex74 126 127 126 149 10.55 18.25 17.32 18.25Ex81 161 161 161 161 9.71 0.00 0.00 0.00Ex82 151 151 151 151 9.57 0.00 0.00 0.00Ex83 153 153 153 153 9.11 0.00 0.00 0.00Ex84 163 163 163 172 8.47 5.52 5.52 5.52Ex91 116 118 116 123 4.84 6.03 4.24 6.03Ex92 102 104 102 107 5.37 4.90 2.88 4.90Ex93 105 106 105 107 5.58 1.90 0.94 1.90Ex94 120 122 120 125 4.57 4.17 2.46 4.17Ex101 147 147 146 163 8.14 10.88 10.88 11.64Ex102 135 136 135 147 8.38 8.89 8.09 8.89Ex103 138 141 137 149 9.17 7.97 5.67 8.76Ex104 159 159 157 189 7.84 18.87 18.87 20.38Average 6.60 7.32 6.64 7.39

• GATS: a simple genetic algorithm and tabu search procedure developped by Zhang, Manier, and Manier (2012a),for the third and fifth-class instances in Tables 5(a), (b) and 9. Table 5(a) provides results in terms of exit time of thelast job of the system, instead of in terms of makespan.

• BB: Branch and Bound used by Mateo and Companys (2007) dedicated to cyclic HSP in STF;• ATW: a heuristic proposed by Paul, Bierwirth, and Kopfer (2010) to solve a predictive HSP.

For the first-class instances, MSB is not more efficient than the reference method (gap between 3.26% and 79.31%,mean gap equals 22.25%) (Table 3). This can be explained as we use a method which is not dedicated to this kind ofproblems, and only one part of the algorithm is used (as those instances contain no assignment problem and only fixedprocessing times).

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Table 5. Results for FJSP instances (Deroussi and Norre 2010).

(a) Exit time criterionInst. ILS MSB Gap% CPU

FJSP1 160 176 10 18.35FJSP2 138 138 0 12.25FJSP3 142 152 7.04 13.67FJSP4 138 148 7.25 13.53FJSP5 112 116 3.57 8.07FJSP6 158 166 5.06 13.09FJSP7 150 166 10.67 23.35FJSP8 197 204 3.55 16.00FJSP9 166 176 6.02 10.86FJSP10 188 206 9.57 15.39Average 6.27 14.46

(b) Cmax criterionInst. GATS MSB Gap% CPU

FJSP1 144 156 8.33 9.13FJSP2 118 124 5.08 6.82FJSP3 124 140 12.90 7.63FJSP4 124 132 6.45 7.20FJSP5 94 96 2.13 4.56FJSP6 144 148 2.78 6.68FJSP7 124 132 6.45 10.22FJSP8 180 191 6.11 11.34FJSP9 150 154 2.67 7.57FJSP10 178 192 7.87 9.49Average 6.08 8.06

For FMS instances, SALS, GAA and MA remain better than MSB. We found the same makespan for 10% of instances.The gap is less than 5% in, respectively, 45%, 47.5% and 42.5% of the forty instances, and the biggest gap is about 22%. Theaverage gaps are, respectively, 7.32%, 6.64% and 7.39%, and the standard deviation is 5.79% for SALS. Our results are stillsatisfying as they are rather near the best ones of dedicated methods. Moreover, for the methods in reference, results partlydepend on the initial solutions whose generation is not always detailed in the literature. So the analysis of the differencesobtained is difficult to complete. For all the instances Exi4, (i ∈ [1, 10]), the makespan in Deroussi, Gourgand, and Tchernev(2008), Larabi (2010) and our results are longer than those for other instances, because of long travel times among machines.

For FJSP instances, two types of criterion are compared: makespan and exit time of the system (Table 5). For exit timecriterion, MSB provides a mean gap equal to 6.27% and obtains the same value for one instance (Table 5(a)). Gaps areno more than 11% and the standard deviation is 3.35%. For the makespan criterion, MSB provides a mean gap of 6.08%(Table 5(b)). These instances contain both assignment and sequencing problems. Compared with the dedicated method, ourapproach only uses randomly generated assignment, but still provides comparable results. We can observe that the meanCPU time varies according to the criterion considered. Indeed, the difference of running times between makespan and exittime is 79.26% in average. It can be explained as two tasks are added by job (one transportation and one fictitious processingtask) to handle exit time criterion.

Tables 6 and 7 provide results for STF. In those tables, the column Cmaxcyc corresponds to a cyclic single hoist problem, asexplained in Subsection 4.1. We compare our results with the cyclic solutions (Gcyc = (Cmax1hoist − Cmaxcyc)/Cmaxcyc).We also compare the results we have obtained for one and two hoists (gain = (Cmax2hoist − Cmax1hoist )/Cmax1hoist ).Finally, in Table 6, we show the improvement brought by our method MSB, compared with the algorithm GATS wepreviously developped (GG AT S = (Cmax1hoist − G AT S)/G AT S).

Table 6 shows our results for instances with five jobs and one or two hoists. In single-hoist cases, our Cmax is less thanCmaxcyc for 96.7% of the instances (29 out of 30). This means that MSB provides better results than those obtained with a2-degree cycle time (average improvement: 12.57%). Indeed, in cyclic problems, periodic schedules are searched in a steadystate hypothesis. Such solutions may be no longer optimal ones while considering transient states. In the job shop problem,transient states at the beginning and at the end cannot be neglected. This can explain that we can improve the results provided

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Table 6. Results for instances of Mateo and Companys (2007) with five jobs, one and two hoist(s).

Inst. Cmaxcyc GATS MSB Gap% CPU MSB gain CPU %

1hoist Gcyc GG AT S 2hoist

501 1312 1099 1120 −14.63 1.91 2.08 795 −29.02 1.94502 1079 914 962 −10.84 5.25 2.78 723 −24.84 1.43503 1327 1422 1232 −7.16 −13.36 3.29 714 −42.05 2.98504 1092 924 943 −13.64 2.06 3.40 647 −31.39 1.93505 1281 1262 1105 −13.74 −12.44 2.45 697 −36.92 1.47506 1201 1040 1083 −9.83 4.13 3.62 727 −32.87 3.02507 1313 1315 1233 −6.09 −6.24 1.57 708 −42.58 2.95508 1050 1018 890 −15.24 −12.57 1.91 612 −31.24 1.35509 1065 1082 914 −14.18 −15.53 4.28 642 −29.76 2.03510 1051 1028 949 −9.71 −7.68 4.97 625 −34.14 2.11511 1472 1131 1107 −24.80 −2.12 2.69 765 −30.89 1.71512 980 949 837 −14.59 −11.80 3.21 684 −18.28 2.36513 1191 1151 1022 −14.19 −11.21 1.62 741 −27.50 2.36514 1246 1025 1046 −16.05 2.05 1.69 774 −26.00 1.36515 939 987 943 0.43 −4.46 2.21 587 −37.75 2.02516 1392 1486 1351 −2.95 −9.08 5.30 756 −44.04 2.43517 1376 1145 1201 −12.72 4.89 2.57 760 −36.72 1.78518 1167 1103 1007 −13.71 −8.70 3.70 725 −28.00 1.66519 1335 1245 1189 −10.94 −4.50 3.93 825 −30.61 1.96520 939 972 895 −4.69 −7.92 2.53 556 −37.88 1.47521 1340 1106 1078 −19.55 −2.53 3.42 728 −32.47 2.53522 2034 1622 1754 −13.77 8.14 2.98 1107 −36.89 1.63523 1638 1630 1550 −5.37 −4.91 4.48 929 −40.06 3.11524 1129 1011 970 −14.08 −4.06 3.70 701 −27.73 1.73525 1830 1616 1457 −20.38 −9.84 5.52 958 −34.25 1.60526 1760 1438 1428 −18.86 −0.70 2.86 969 −32.14 2.34527 1326 1111 1137 −14.25 2.34 3.79 805 −29.20 2.34528 1153 1026 1029 −10.75 0.29 4.16 692 −32.75 2.23529 1341 1119 1145 −14.62 2.32 2.81 865 −24.45 1.65530 1782 1579 1495 −16.11 −5.32 4.18 979 −34.52 2.34Average −12.57 −4.05 3.26 −32.56 2.06

by Mateo and Companys (2007). Moreover, those authors considered that jobs of two types enter the line alternatively.Our method does not fix any entering order of jobs. For instance 515, for which MSB does not provide a better solution, thegap is very small (0.43%). Figures B1 and B2 in the Appendix B detail the solutions obtained for instance 504 and 5 jobs.Figure B1 represents the optimal solution for the associated cyclic problem (Cmaxcyc = 1092) with alternative entering orderof jobs (of type 1 and 2). Figure B2 shows our results with MSB (Cmax = 943) where no constraint is fixed on the enteringsequence for the general job shop problem. For this obtained solution, the entering sequence is J2 → J5 → J3 → J1 → J4(type 2-1-1-1-2), whereas it is J1 → J2 → J3 → J4 → J5 (type 1-2-1-2-1) in the cyclic sequence. In Figure B1, we canobserve that for the cyclic solution, the processing durations in one tank are the same for jobs of the same type. It is a constraintof the cyclic problem which does not exist in a general job shop environment (see Figure B2). Another difference betweenthese two solutions is that the processing tasks are better distributed in the periodic solution. With our MSB approach, thosetasks are more concentrated and are shorter at the beginning of the schedule, which involves more waiting times and longerdurations for latter tasks. We also compare our method with the algorithm GATS we previously developed. MSB improvesthe Cmax for nearly 70% of instances (20 out of 30), with an average gap −4.05%.

In a second time, the gain when adding one hoist, is studied (gain = (Cmax2hoist − Cmax1hoist )/Cmax1hoist ). MSBgives better results for two hoists (mean improvement about 32.5%). It is not surprising that adding one hoist can improvegreatly the productivity, whatever the method is. But those results may be used as reference ones in further researches. Themean CPU time is smaller for two hoists than for one hoist (average gain 30.30%). This can be explained by the fact thatthere are more feasible solutions in two hoist cases. Then MSB can find a feasible one more easily to stop the procedure.

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Table 7. Results for instances of Mateo and Companys (2007) with 21 jobs, 1 and 2 hoist(s).

Inst. Cmaxcyc MSB Gap% MSB Gain %1hoist Gcyc 2hoist

501 4256 4251 −0.12 2900 −31.78502 3631 3606 −0.69 2736 −24.13503 5239 5265 0.50 3535 −32.86504 3868 4237 9.54 3082 −27.26505 4521 4932 9.09 3929 −20.34506 4161 4105 −1.35 3285 −19.98507 5041 5172 2.60 3782 −26.88508 3714 3607 −2.88 2704 −25.03509 3762 3769 0.19 2922 −22.47510 4035 3844 −4.73 2921 −24.01511 4928 4343 −11.87 3901 −10.18512 3060 3262 6.60 2774 −14.96513 4199 4769 13.57 3665 −23.15514 4262 3991 −6.36 3831 −4.01515 3539 3768 6.47 2532 −32.80516 5344 4994 −6.55 3742 −25.07517 4448 4485 0.83 3573 −20.33518 4111 3955 −3.79 3004 −24.05519 4703 4401 −6.42 3208 −27.11520 3539 3289 −7.06 3040 −7.57521 4476 4144 −7.42 3653 −11.85522 6786 6652 −1.97 5259 −20.94523 6246 5732 −8.23 4570 −20.27524 3713 3635 −2.10 2757 −24.15525 6198 5926 −4.39 4696 −20.76526 5856 4830 −17.52 4669 −3.33527 4430 4203 −5.12 3720 −11.49528 3801 3682 −3.13 2957 −19.69529 4405 4111 −6.67 3156 −23.23530 6246 5204 −16.68 5072 −2.54Average −2.52 −20.07

MSB runs faster for instances of Mateo and Companys (2007) than for other ones with similar size and for the samenumber of transportation resources (first-, second- and third-class instances). It is because no assignment problem exists, andvisiting order of machines are identical for jobs in Mateo and Companys (2007). Thus, machines can be scheduled by thepropagation of arcs determined by the entering sequence of jobs.

Instances with five jobs and five tanks may be considered as small size cases. To test the performance of MSB in largersize problems, we kept the same data but introduced 21 jobs. The results were globally less interesting than for five jobsproblems (Table 7). For 70% of instances, Cmax is less than Cmaxcyc (average gain: 5.96%). For nine instances, Cmaxcyc isbetter (average gap 5.49%). The average gap for the 30 instances is −2.52%, which means we have globally better results.Note that the influence of transient states (upstream and downstream) is stronger for small size instances (five jobs) than fora great number of jobs (21). This may partially explains why MSB is slightly less efficient compared to the optimal cyclicsolutions. Besides, with MSB, we solved the whole instances, whereas with our previous method GATS, it was difficult tofind feasible solutions in reasonable time.

MSB provides better average results than ATW in Paul, Bierwirth, and Kopfer (2010) for 78.57% instance (Table 8), withimprovements from 1.11% to 20.39% (average improvement: 6.45%). We solve these instances in a job shop environmentwith release dates of jobs equal to zero. While in Paul, Bierwirth, and Kopfer (2010), jobs enter the system with differentrelease dates which are not given.

For the fifth-class instances (Table 9), compared with our previous methods GATS, we got better results for two instances,and the same makespan for four instances. The average gap is 2.34% with great variation from −4.84% to 14.52%. GATSsolves assignment problems for both machines and robots whereas MSB assigns tasks to robots but takes into account

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Table 8. Results for the fourth-class instances of Paul, Bierwirth, and Kopfer (2010).

Inst. Ref. Cmax average Cmax of MSB

CAT W Std. deviation Best AverageCM SB Std. deviation GapAT W (%)

1 28.69 1.72 22.56 23.71 1.23 −17.382 27.94 2.63 24.27 24.10 1.02 −13.753 27.76 2.63 21.10 23.11 1.12 −16.744 27.60 3.27 21.56 25.65 2.13 −7.085 27.91 3.23 20.60 22.21 1.34 −20.396 26.73 2.76 22.34 22.30 1.01 −16.557 27.37 3.03 22.79 25.31 1.29 −7.528 27.69 2.73 24.01 27.38 2.02 −1.119 26.88 2.83 22.04 24.22 1.87 −9.8810 27.07 3.05 26.11 27.89 1.05 3.0311 27.74 3.48 24.78 26.83 1.69 −3.3012 26.49 3.00 23.35 25.95 1.83 −2.0413 27.27 3.22 26.44 28.80 1.13 5.5914 26.63 1.99 28.18 31.12 1.43 16.87Average 2.83 1.44 −6.45

Table 9. Makespan for the fifth-class instances.

Inst. GATS MSB Gap% CPU

FJSP1 144 144 0.00 13.55FJSP2 118 116 −1.69 9.13FJSP3 124 142 14.52 11.26FJSP4 124 118 −4.84 10.94FJSP5 94 94 0.00 5.08FJSP6 144 150 4.17 12.19FJSP7 124 124 0.00 14.21FJSP8 180 182 1.11 17.23FJSP9 150 150 0.00 11.11FJSP10 178 196 10.11 15.23Average 2.34 11.99

randomly generated machine assignment. Then, if this one is not good, the results may be less interesting than with GATS.Nevertheless, MSB succeeds in finding at least as good makespan as GATS for 6 instances out of 10.

Finally, in this paper, we have shown the ability of our disjunctive graph to represent not only problems encounteredin various types of systems, but also to represent more feasible solutions than conventional graphs. The above results andanalyses show that the presented modified shifting bottleneck procedure using this disjunctive graph is a self-sufficient methodto solve effectively and efficiently the studied problems, with interesting performance compared with those in the literature,and without the support of other tools. This approach can also be advantageously integrated in a hybrid procedure, for exampleto improve the performance of GATS. Zhang, Manier, and Manier (2012b) explain how to combine those heuristics, and thefirst obtained results emphasise the overall performance of such a possible combination. Nevertheless, the resulting hybridalgorithm is even heavier and time consuming. Moreover, it involves a modification of our presented shifting bottleneckprocedure, in particular in the loop related to the arbitration of arcs for the same transportation resource.

5. Conclusion

In this paper, various kinds of FJSSP with transportation are considered. For this purpose, we defined a generic problemwhich gathers all characteristics identified in each system (bounded processing times, transportation constraints, ...), knowingthat each individual problem is known as NP-hard one. Then we use a modified disjunctive graph adapted to represent thisgeneric FJSSP with transport once assignments are generated. In particular, it contains nodes for processing, transportationand storage tasks. It also includes negative arcs to incorporate constraints of bounded processing time, transportation time and

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limited storage time. To solve the various problems modelled by this general graph, a modified shifting bottleneck procedurewith repair is used. A new heuristic is coupled with this procedure. It assigns and sequences transportation tasks step by stepwhile arbitrating disjunctions. Various tests on benchmarks of the literature are performed. We also validated our method onnew instances which correspond the most to our generic problem. Computational results show that the modified disjunctivegraph model and algorithm are globally efficient. Indeed, even if they do not improve the best results obtained with dedicatedmethods, they are generally not so far from those ones while being able to deal with various kinds of problems. Finally,compared with our previous work GATS (Zhang, Manier, and Manier 2012a), MSB partly solves assignment problem, but itprovides better results for bigger size and very constrained instances (no storage and bounded times). We can conclude thatthis new method is a self-sufficient one and provides a good compromise between best results, evaluation time and flexibility(ability to address several kinds of complex job shop scheduling problems with transportation).

In our future work, the proposal method may be improved in several directions, upstream and downstream of MSB:upstream, we can couple it with other approaches for initial assignment problem; downstream, we can also envisage tocombine our method with other local search techniques to improve the feasible solution found by the modified shiftingbottleneck heuristic. It will be interesting to extend the proposed heuristic to solve both machine and transportation resourcesassignment. But necessary reduction technology must be applied to reduce the evaluation time.

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Appendix A. An example of oriented disjunctive graph

Figure A1. Oriented disjunctive graph for a complete solution of example in Table 2 .

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