Research ArticleModelling and Solving Rescheduling Problems in DynamicPermutation Flow Shop Environments

Pablo Valledor1 Alberto Gomez 2 Paolo Priore2 and Javier Puente2

1ArcelorMittal Inc Global RampD Asturias Gijon Spain2University of Oviedo Department of Business Administration Polytechnic School of Engineering 33203 Gijon Spain

Correspondence should be addressed to Alberto Gomez albertogomezuniovies

Received 7 December 2019 Revised 25 May 2020 Accepted 18 June 2020 Published 24 July 2020

Academic Editor Roberto Natella

Copyright copy 2020 Pablo Valledor et alis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

e aim of this paper is to analyse model and solve the rescheduling problem in dynamic permutation flow shop environmentswhile considering several criteria to optimize Searching optimal solutions in multiobjective optimization problems may bedifficult as these objectives are expressing different concepts and are not directly comparable us it is not possible to reduce theproblem to a single-objective optimization and a set of efficient (nondominated) solutions a so-called Pareto front must befound Moreover in manufacturing environments disruptive changes usually emerge in scheduling problems such as machinebreakdowns or the arrival of new jobs causing a need for fast schedule adaptation In this paper a mathematical model for thistype of problem is proposed and a restarted iterated Pareto greedy (RIPG) metaheuristic is used to find the optimal Pareto frontTo demonstrate the appropriateness of this approach the algorithm is applied to a benchmark specifically designed in this studyconsidering three objective functions (makespan total weighted tardiness and steadiness) and three classes of disruptions(appearance of new jobs machine faults and changes in operational times) Experimental studies indicate the proposed approachcan effectively solve rescheduling tasks in a multiobjective environment

1 Introduction

Scheduling in production systems addresses the problemof sequencing the manufacturing of a series of jobsassigned to different machines in a production environ-ment subject to certain requirements Since 1950s thesecomplex problems of (NP-hard) type as shown by Gareyet al [1] have been studied in depth by the scientificcommunity

Flow shop systems seek the optimal scheduling of ldquonrdquojobs j1 j2 jn in ldquomrdquo machines m1 m2 mm Eachjob consists of ldquomrdquo tasks the ith task being processed by themachine mi In each specific period of time every machineprocesses a single taskus to finish a job on the machine ithis machine must be available and the job on machineldquomminus 1rdquo should have been fully processed

When job scheduling is identical in each machine thereis a permutation flow shop problem (PFSP) which usuallyassumes no precedence among different job tasks or

interruptions on them PFSP is the problem tackled in thispaper as it has many real-world applications although otherenvironments may appear such as the blocking flow shopproblem [2] or the integrated planning and schedulingproduction system [3 4]

Due to the combinatorial nature of the problem offactorial order the number of schedules increases to n

Multiobjective optimization seeks to recognize the bestadvantageous solution by simultaneously analysing multiplediscrepant objectives (for example cost quality or time)Every objective can be measured in different units or havedifferent meanings making them incomparable Conse-quently they do not allow a possible optimal solution for allcriteria to be obtained While finding efficient solutions(known as nondominated) the final objective would be toobtain the group of nondominated solutions also calledPareto-optimal solutions making up the Pareto frontSubsequently the decision maker will select the solution thatbetter meets the business needs

HindawiComplexityVolume 2020 Article ID 2862186 17 pageshttpsdoiorg10115520202862186

Multiobjective optimization problems need to usemetrics which allow comparing the performance efficiencyof the Pareto-Front obtained with every algorithm Based onthe multiobjective optimization literature the followingmetrics have been selected

(i) Hypervolume [5](ii) Unary epsilon indicator [6ndash8]

e methods used to solve multiobjective optimizationproblems are usually based on scalar [9] or metaheuristictechniques [10] Regarding scalar methods many strategiescan be used to set up the objective weighting conventional(CWA) dynamic (DWA) random (RWA) and bang-bangweighted aggregation (BWA) All of them try to convert amultiobjective problem into a single one by modifying theweights of the objective functions at each process step inorder to find the Pareto-front solution set Other scalarmethods are the epsilon-constraint goal programming andlexicographical order

Concerning metaheuristic methods there are techniquessupported by evolutionary algorithms [11ndash17] particleswarm optimization systems [18] tabu search [19] simu-lated annealing [20ndash22] ant colonies [23 24] greedy al-gorithms [25] or local search methods [26]

Most of the papers on these methods study biobjectiveproblems which are more easily represented graphically andwhose results are easier to analysee limitations of currentalgorithms are usually analysed to solve problems with agreater number of objective functions and subsequentlydesign new techniques which allow the specific problems tobe efficiently tackled [27 28]

is paper is structured in the following way Firstrescheduling schemes studied in the literature are com-mented Once the problemrsquos mathematical formulation isdefined as an integer linear programming model therescheduling architecture is presented to solve the problembased on a predictive-reactive strategy Next the RIPGmetaheuristic is described and validated on Dubois-LacosteInstances in a static biobjective environment After cali-brating the parameters the RIPG metaheuristic is applied toa benchmark that was specifically designed for this paperFinally comparative results are obtained and the discussionand the main conclusions of the paper are shown

2 Rescheduling Systems

In classical scheduling problems an initial set of known jobsmust be scheduled considering the available machines in thesystem and the existing technological constraints is static(also so-called offline) approach assumes that the jobs to besequenced and the configuration of the system are known inadvance and constant in time In real production envi-ronments there may be new job arrivals as well as certainevents not known beforehand that induce variations re-quiring a new schedule such as breakdowns or preventivemaintenance activities in machinery or changes in thepriority of jobs ese conditions make it necessary to in-corporate dynamism in the scheduling systems through thereorganization of jobs as the static approach is not optimal

anymore Rescheduling is the process of actualizing thecurrent scheduling to adapt to the disruptions that mayappear e basic phases of this process are the following

(i) e rescheduling strategy to be used defining whennecessary the need for a new production schedule

(ii) e designated reprogramming policy specifyingthe time and manner of carrying out the resched-uling process

(iii) e designed technique to update the originalschedule

ere are mainly three rescheduling strategies describedin the literature predictive-reactive proactive (or robust)and dynamic [29ndash32] among which the predictive-reactiveone was selected in this paper Furthermore three policiesare usually used to complete the rescheduling of the jobswithin this approach event-driven periodic and hybrid[33] of which the second one was chosen ere are alsothree main procedures available to actualize a non-feasibleschedule due to the occurrence of disruptions on the pro-duction system right-shift rescheduling [34 35] partialregeneration [34] and complete regeneration [36 37] thelast of them being used in this paper

Metrics normally employed in evaluating the reprog-ramming performance can be based on efficiency (for ex-ample makespan mean flow time or total weightedtardiness) or robustness (for example the systemrsquos averagestability as proposed by Pfeiffer et al [33])

In the literature there are hardly any papers which coverrescheduling in dynamic flow shop environments withpermutation and multiple objective functions Itayef et al[38] and Liefooghe et al [39] developed algorithms focusedon the search for nondominated solutions in a PFSP en-vironment with a proactive-reactive strategy without usingthe objective function weighting method Itayef et al [38]used the Multiobjective Simulated-Annealing Algorithm(MOSA) while Liefooghe et al [39] proposed metaheuristicsbased on the hypervolume known as the Indicator-BasedEvolutionary Algorithm (IBEA) Liu et al [40] proposed amodified iterated greedy algorithm (eIG_Rep) introducingan effective escape mechanism from a local optimum toconsider the arrival of new orders in a PFSP

Likewise the literature does not abound whenrescheduling analysis is extended to environments which aredifferent from the PFSP environment Xiong et al [41]adapted the NSGA-II Nondominated Sorting Genetic Al-gorithm [12] for a stochastic flexible job shop problem(FJSP) Iima [42] developed a multiobjective genetic algo-rithm to minimize the schedulersquos total tardiness and stabilityin an environment with parallel machines and variations injob delivery dates Zhang et al [43] proposed the applicationof a multiobjective evolutionary algorithm based on de-composition (MOEAD) to solve a hybrid flow shoprescheduling problem with 2 objectives (makespan andstability as the number of jobs assigned to different ma-chines versus the original schedule) and random machinebreakdowns He et al [44] implemented the NSGA-III al-gorithm [27] to solve the rescheduling problem of rush

2 Complexity

orders in a hybrid flow shop system considering three ob-jectives makespan total transportation time and sequencestability In reviews on rescheduling techniques by Vieiraet al [45] and Ouelhadj and Petrovic [46] the predictive-reactive strategy is identified as the one most frequentlyused

3 Proposed Solution

First having found no references in the literature amathematical formulation for solving this type of problem asan integer linear programming model is proposed emodel tries to optimize three objective functions makespantotal weighted tardiness (TWT) and stability in order toincrease productivity in (1) the production environment (byreducing the makespan) (2) the customer service (byminimizing TWT) and (3) the schedule stability in therescheduling process when faced with different suddendisruptions

Subsequently the architecture for the proposedrescheduling system its implementation and the appliedRIPG method are described RIPG is a non-population-based algorithm recently developed by Ciavotta et al [25]and applied to a biobjective PFSP environment In thispaper the algorithm was used in a different context havingbeen adapted to the dynamic rescheduling architecturedeveloped and used to optimize the three objective functionsdefined makespan total weighted tardiness and stability

31 Mathematical Model Before presenting the mathe-matical model formulation the scientific literature dealingwith uncertainty and multiobjective optimization ap-proaches will be briefly discussed Razmi et al [47] presenteda mathematical model that included uncertainty throughfuzzy processing times that is to say a stochastic model inwhich processing times are not input data for the model butrandom variables with a known probability distribution Formultiobjective environments Itayef et al [38] formulated abiobjective mathematical model to address the arrival of newjobs and thus minimize the maximum completion time forthe tasks and improve the schedule stability In such amodeltwo independent sets of jobs are considered new arrivals tothe system and jobs which were already scheduled in theprior rescheduling period

Fattahi and Fallahi [48] formulated a dynamic FJSPwhich considered the starting time of the reschedulingperiod and the rescheduling time (RT) trying to optimizetwo objectives makespan and stability In order to model theFJSP system restrictions on operations precedence wereincluded (this type of constraint is not present in PFSPproblems)

Ramezanian et al [49] proposed a mathematical modelto formulate a biobjective PFSP with the aim of minimizingthe linear combination of the stock jobs cost and the latereleases cost including the starting time of the jobs at themachines (release time) e thesis written by ornblad[50] on mathematical formulations in FJSP explained theimportance of analysing the objective functions when

developing the problem-solving method For this analysisdifferent formulations were proposed amongst which theso-called time-indexed model based on the temporal dis-cretization of the scheduling period stood out When dif-ferent formulations were evaluated taking into considerationthe objectives of makespan and the average between thetardiness and the completeness time for the jobs a greatdifference in performance was observed depending uponthe objective to be reached

Yenisey and Yagmahan [51] considered a static multi-objective PFSP formulation which includes the analysis ofthree time types waiting times (for jobs at themachines) jobpreprocessing times (before jobs are processed by the sys-tem-ready times) and processing times Li et al [52] pro-posed a mathematical formulation for the PFSPrescheduling problem that includes recovery times forbroken machines with the aim of minimizing a bi-criteriafunction while weighting the makespan and the systemstability (calculated as the number of jobs that have a releasetime which is different from the planned release time cal-culated within the prior rescheduling point) Finally Yuanand Yin [53] developed a fuzzy multiobjective local search-based decomposition (FMOLSD) algorithm to solve a bio-bjective problem to minimize makespan and total flow timewhile considering fuzzy processing times for the jobsscheduled in a permutation flow shop problem

e mathematical formulation to solve the problem isdetailed in the following is formulation is based on thepreviously mentioned approaches and incorporates newcharacteristics Because this is a PFSP each job must accesseach machine following the same processing order that is tosay from the first to the last machine In the problem thefollowing considerations are considered

(1) Overlapping is not permitted in consecutive oper-ations for the same job that is a new operation in ajob cannot begin until the preceding operation forthat job is completed

(2) Job pre-emption is not permitted that is to say eachjob must wait until the preceding job has finished inorder to be processed on the same machine

(3) ere are infinite buffers (intermediate storage areas)between two machines

(4) All the jobs that have begun to be processed in thesystem must maintain their scheduling order whenany type of disruption arises

(5) All jobs interrupted due to a disruption must remainon their assigned machine with the pending jobsbeing delayed if necessary Jobs not impacted by thedisruption so not delayed will go on with theirnormal process

(6) Rescheduling is carried out periodically thereforetwo types of jobs exist in the system those which arealready being processed whose schedule order can-not vary and those whose order can vary (new jobsand jobs which have not yet begun to be processed)It is important to consider the jobs that are beingprocessed because depending upon the disruptions

Complexity 3

which arise (for example machine breakdowns) acertain minimum starting time will be established forthe next incoming job in the system (in order to beprocessed by the first machine)

(7) Machine breakdowns can arise at any time and notonly when the machines are busy

(8) If a machine breakdown arises when a job is beingprocessed the job will restart processing at the pointwhere the interruption occurred

(9) When a variation in the processing time of a jobarises the new processing time is fixed until the nextprocessing time variation occurs on the same job

As it is a predictive-reactive rescheduling process theMixed-Integer Linear Programming (MILP) model can berun at each rescheduling point meaning that the past dis-ruptions (new jobs processing time variations and machinebreakdowns) that have happened during each reschedulingperiod are received as inputs in the modelose disruptionsimpact the next schedule erefore there are no predicteddisruptions considered in this formulation

e problem is defined by three finite sets

(i) J the set of new jobs to be scheduled from 1 n(ii) M the set of machines from 1 m(iii) O the set of operations for each job at each machine

from 1 m

Here Ji is the job at ith position of the permutatione parameters used in the model are as follows

(i) pij the processing time for job Ji euro J on machine j(ii) di the delivery date for job Ji euro J(iii) wi the priority (weight) of job i with respect to the

rest of the jobs(iv) rli (release time) once a new job has arrived at the

system the release time is the time necessary forjob i to be available in order to be processed againby the production environment

(v) rtj (ready time) the time in which machine j isready to process a new job is input data vectorrepresents the initial situation of the machinesbefore undertaking the new scheduling It is cal-culated based on the current time the disruptionswhich have arisen and the scheduling schemecalculated at the previous rescheduling point It isan important factor as it cannot be supposed thatall the machines are initially available

(vi) RT current temporal instant is represents thecurrent point of rescheduling in the productionenvironment

(vii) To take into account the schedule stability crite-rion the number of unprocessed jobs in the baseschedule (nnot processed) needs to be defined in-cluding new arrivals to the system the startingtime for the jobs at each machine in the baseschedule (Sbaselineij ) the current rescheduling in-stant (RT) and a scale constant (scale 10 in

accordance with the value used in Pfeiffer et al[33] used to apply a quadratic penalty method tothe starting time variation in the base schedulewith respect to RT)

(viii) BJstart a breakdown instant for machine j

(ix) BJend recovery instant from the breakdown of

machine j(x) BigM large constant (BigM⟶infin)

In terms of decision variables the following wereconsidered

(i) Binary variables which represent the position ofeach of the jobs in the planned schedule

forall i j isin J xij f(x) 1 if Ji is at j position

0 otherwise1113896 (1)

(ii) Ci completion time for the job at position i(iii) Cij completion time for the job at position i on

machine j(iv) Sij starting time of job i on machine j(v) Binary variables that indicate the circumstances

under which a machine breakdown has occurred(three situations are modelled)

(a) forall i isin Jand j isinM yij1 it takes the value 1 if theoperation Oij finishes before the breakdownoccurs that is to say the time the operationtakes to finish is less than the machinersquosbreakdown starting point

(b) forall i isin Jand j isinM yij2 it takes the value 1 if theoperation Oij is not yet finished at the momentwhen the breakdown occurs that is to say thetime the operation takes to finish overlaps withthe machinersquos breakdown starting point

(c) forall i isin Jand j isinM yij3 it takes the value 1 if theoperation Oij has a starting time which is afterthe moment when the machine breaks down

(vi) Cmax TWT STB these variables correspond to themaximum completion time for the tasks the totalweighted tardiness and the schedule stabilityrespectively

e mathematical model for the MILP formulation ofthe permutation flow shop rescheduling problem was basedon the following equations

e objectives of the problem were to minimize themakespan the total weighted tardiness (TWT) and thestability of the schedule to be planned (STB) obtaining theoptimal Pareto frontier (PF)

min PF Cmax TWT STB( 1113857 (2)

e makespan was calculated as the maximum com-pletion time for all the jobs and it corresponds to thecompletion time for the last scheduled job that is the timewhen the last task in the last machine finishes

Cmax maxi1n

Ci(π)( 1113857 Cn (3)

4 Complexity

e total completion time for each job Ci is defined bythe following equation according to the completion time inthe last machine M

forall i isin J Ci CijM (4)

where M is the set of machines in the systemMore specifically the completion time for each job i on

each machine j takes into consideration the followingcomponents

(i) e time at which the job i can begin to be processedby machine j is time depends on the following

(a) e completion time of job i on the previousmachine (Cijminus1)

(b) e completion time of the previous job locatedat position (iminus 1) on the current machine j(Ciminus1j)

(c) e maximum of these terms will be the instantof time in which job i can begin to be processedby machine j (Sij)

(ii) e processing time for job i on machine j (pij)(iii) Time of breakdown onmachine jwhen job i is being

processed (BJend minus BJ

start)

When calculating the completion time along with ma-chine breakdowns the following equations are modelledconsidering the three possible breakdown situations

If the operation finishes before the breakdown appears

foralli isin Jandforallj isinM Cij + 1 minus yij11113872 1113873 middot BigMge Sij + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM Cij minus 1 minus yij11113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj

(5)

If the breakdown appears while the operation Oij isbeing processed it is necessary for the job to wait until themachine is ready again in order to continue its processingforalli isin Jandforallj isinM

Cij + 1 minus yij21113872 1113873 middot BigMge Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

foralli isin Jandforallj isinM

Cij minus 1 minus yij21113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

(6)

Finally if the operation Oij has not yet begun the workmust wait until the machine breakdown has finished

foralli isin Jandforallj isinM

Cij + 1 minus yij31113872 1113873 middot BigMge Max Sij BJend1113872 11138731113873 + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM

Cij minus 1 minus yij31113872 1113873 middot BigMle Max Sij BJend1113872 1113873 + 1113944

n

k1xki middot pkj

(7)

Although the above equations contain the max function(that is not linear) they can be expressed in a linearized wayusing a big-M plus binary variables representation beingautomatically transformed by most of the commercialsolvers and respecting theMILP formulation of the problem

If a machine breakdown appears only one of these threesituations can arise this can be ensured by including thefollowing equation

foralli isin Jandforallj isinM 11139443

k1yijk 1 (8)

e starting time for each job i on each machine j de-pends upon the following equation

foralli isin Jand j isinM Sij Max Cijminus1 Ciminus1j1113872 1113873 (9)

In order to define the starting times in the MILP modelthe following equations are used

Operation Oij cannot begin until the operation for theprevious job has finished on the same machine

foralligt 1 isin Jandforallj isinM Sij geCiminus1j (10)

and operation Oij cannot begin until the previous operationfor the same job has finished

foralli isin Jandforalljgt 1 isinM Sij geCijminus1 (11)

e starting time for job i on the first machine must begreater than the overall release time as this is the momentwhen the job is ready to begin to be processed

foralli isin J Si1 ge rli (12)

Because not all the machines are initially available inorder to be able to process the next job it is important to takeinto consideration the fact that the starting time for the jobscannot begin until the machine is available that is to say

foralli isin Jandforallj isinM Sij ge rtj (13)

In case there are delaysmdashthat is when a job completiontime is greater than its delivery datemdashthe total weightedtardiness (TWT) is calculated as the weighted sum of dif-ferences between the total completion time for each job andits delivery date according to their assigned priority weights(wi)

TWT 1113944

n

i1Max Cim minus di 0( 1113857 middot wi (14)

For modelling purposes in the MILP problem the fol-lowing equations were included

TWTge 1113944n

i1Cim minus di( 1113857 middot wi

TWTge 0

(15)

e third objective function considered in the problem isthe stability (STB) with respect to the base schedule cal-culated in the previous period (baseline) Stability isexpressed as the average of the absolute differences of

Complexity 5

unprocessed job starting times adding a penalty whichdepends upon the variation in the starting time for the baseschedule regarding the current RT Unprocessed tasks aredefined as those jobs which have previously been scheduledat the prior point of rescheduling and which have not yetbegun to be processed by the system at the current RT

STB 1

nnot processedmiddot 1113944

n

i1Si1 minus S

baselinei1

11138681113868111386811138681113868

11138681113868111386811138681113868 +scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(16)

e absolute value function used in calculating thestability objective is nondifferentiable in addition to beingnonlinear Nevertheless it can be expressed in an integerlinear way as follows

STB 1

nnot processedmiddot 1113944

n

i1DeltaStart Timei +

scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(17)

Two new binary variables d1j and d2j were introduced aswell as a large constant BigM and the following equationsfor MILP formulationforallj isin J 0le Sj1 le BigM

forallj isin J 0leDeltaStart Timej minus Sj1 minus Sbaselinej11113872 1113873le 2 middot BigM middot d2j

forallj isin J 0leDeltaStart Timej minus Sbaselinej1 minus Sj11113872 1113873le 2 middot BigM middot d1j

forallj isin J d1j + d2j 1

(18)

If d1 1 then delta at the starting time would take thevalue Sj1 minus Sbaselinej1 otherwise if d2 1 it would take thevalue Sbaselinej1 minus Sj1

Finally the model ensures that a job must be in onesingle position and that only a single job can exist at anyposition of the schedule is is done through the followingequations

foralli isin J 1113944n

j1xij 1

forallj isin J 1113944

n

i1xij 1

(19)

32 Rescheduling Architecture e rescheduling strategyused was based on the predictive-reactive approach thus abaseline schedule was first generated and then updatedaccording to the disruptions that arise in system A periodicrescheduling policy was used through time windows Finallythe rescheduling method consisted of a complete resched-uling based on heuristicmetaheuristic algorithms (asrescheduling method we proposed RIPG algorithm de-scribed in detail in the next section) RIPG is selected be-cause its efficient behaviour in static multiobjectivepermutation flow shop problems [25] makes it a promisingtechnique for the dynamic environment under study eproposed rescheduling architecture is shown in Figure 1

Initially the baseline schedule was generated accordingto both the flow shop system involved and the jobs to bescheduled en a synchronous control was establish-edmdashevery T units of timemdashin order to perform arescheduling process when disruptions arise in the systemobtaining the Pareto-front output from the solutions foundin each executed rescheduling interval It is important tospecify an appropriate period based on the dimensions ofthe problem for each rescheduling process In this paperthis period was calculated as the maximum completiontime for jobs (Cmax) divided by the total number ofrescheduling points to be executed on the system (we haveconsidered 5 rescheduling points as it is an appropriatenumber to evaluate the architecture adaptation capacityand does not excessively increase the number of executionsin the system)

Likewise at each point of rescheduling it is necessary toselect a representative solution for the calculated Paretofront e selected solution will be used as a baselineschedule to be employed for each new rescheduling efundamental importance of this base schedule lies in its usein evaluating the stability objective function To calculate therepresentative solution for the Pareto front a knee pointmethod was used [54ndash56] A detailed description of the kneepoint method used to calculate this most representativesolution can be found in Valledor et al [57]

33 Restarted Iterated Greedy Algorithm e implementa-tion of the developed RIPG algorithm was based on thedesign proposed by Ciavotta et al [25] eir application ofthe technique was explained using a biobjective PFSP andits results on Taillard Instances were described comparingthe RIPG technique with 17 algorithms including MOSAand Multiobjective Tabu Search (MOTS) Ciavotta et al[25] described the RIPG as a state-of-the-art method be-cause of its excellent results when compared with the bestapproaches previously presented in the literature eRIPG algorithm is an extension of the IG (Iterated Greedy)technique proposed by Ruiz and Stutzle [58] ese tech-niques belong to the stochastic local search (SLS) categorye RIPG technique consists of 5 phases as shown inFigure 2

Phase 1 First an Initial Solution Set (ISS) is determinedbased upon the execution of different simple heuristicsMore specifically NEH heuristics [59 60] are usedeach one being applied to obtain acceptable solutionswith different objective functions For each of the twosolutions obtained via heuristics the greedy phase(described below in phase three) is applied generatingan initial set of nondominated solutions (working set)Phase 2 Second the optimization loop begins with theprocess of selecting a solution via the ModifiedCrowding Distance Assignment (MCDA) method inorder to improve the working set during the greedyphase e MCDA process receives the working set asan input parameter and calculates the distance to eachone of the solutions e selected solution is the one

6 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

orders in a hybrid flow shop system considering three ob-jectives makespan total transportation time and sequencestability In reviews on rescheduling techniques by Vieiraet al [45] and Ouelhadj and Petrovic [46] the predictive-reactive strategy is identified as the one most frequentlyused

3 Proposed Solution

First having found no references in the literature amathematical formulation for solving this type of problem asan integer linear programming model is proposed emodel tries to optimize three objective functions makespantotal weighted tardiness (TWT) and stability in order toincrease productivity in (1) the production environment (byreducing the makespan) (2) the customer service (byminimizing TWT) and (3) the schedule stability in therescheduling process when faced with different suddendisruptions

Subsequently the architecture for the proposedrescheduling system its implementation and the appliedRIPG method are described RIPG is a non-population-based algorithm recently developed by Ciavotta et al [25]and applied to a biobjective PFSP environment In thispaper the algorithm was used in a different context havingbeen adapted to the dynamic rescheduling architecturedeveloped and used to optimize the three objective functionsdefined makespan total weighted tardiness and stability

31 Mathematical Model Before presenting the mathe-matical model formulation the scientific literature dealingwith uncertainty and multiobjective optimization ap-proaches will be briefly discussed Razmi et al [47] presenteda mathematical model that included uncertainty throughfuzzy processing times that is to say a stochastic model inwhich processing times are not input data for the model butrandom variables with a known probability distribution Formultiobjective environments Itayef et al [38] formulated abiobjective mathematical model to address the arrival of newjobs and thus minimize the maximum completion time forthe tasks and improve the schedule stability In such amodeltwo independent sets of jobs are considered new arrivals tothe system and jobs which were already scheduled in theprior rescheduling period

Fattahi and Fallahi [48] formulated a dynamic FJSPwhich considered the starting time of the reschedulingperiod and the rescheduling time (RT) trying to optimizetwo objectives makespan and stability In order to model theFJSP system restrictions on operations precedence wereincluded (this type of constraint is not present in PFSPproblems)

Ramezanian et al [49] proposed a mathematical modelto formulate a biobjective PFSP with the aim of minimizingthe linear combination of the stock jobs cost and the latereleases cost including the starting time of the jobs at themachines (release time) e thesis written by ornblad[50] on mathematical formulations in FJSP explained theimportance of analysing the objective functions when

developing the problem-solving method For this analysisdifferent formulations were proposed amongst which theso-called time-indexed model based on the temporal dis-cretization of the scheduling period stood out When dif-ferent formulations were evaluated taking into considerationthe objectives of makespan and the average between thetardiness and the completeness time for the jobs a greatdifference in performance was observed depending uponthe objective to be reached

Yenisey and Yagmahan [51] considered a static multi-objective PFSP formulation which includes the analysis ofthree time types waiting times (for jobs at themachines) jobpreprocessing times (before jobs are processed by the sys-tem-ready times) and processing times Li et al [52] pro-posed a mathematical formulation for the PFSPrescheduling problem that includes recovery times forbroken machines with the aim of minimizing a bi-criteriafunction while weighting the makespan and the systemstability (calculated as the number of jobs that have a releasetime which is different from the planned release time cal-culated within the prior rescheduling point) Finally Yuanand Yin [53] developed a fuzzy multiobjective local search-based decomposition (FMOLSD) algorithm to solve a bio-bjective problem to minimize makespan and total flow timewhile considering fuzzy processing times for the jobsscheduled in a permutation flow shop problem

e mathematical formulation to solve the problem isdetailed in the following is formulation is based on thepreviously mentioned approaches and incorporates newcharacteristics Because this is a PFSP each job must accesseach machine following the same processing order that is tosay from the first to the last machine In the problem thefollowing considerations are considered

(1) Overlapping is not permitted in consecutive oper-ations for the same job that is a new operation in ajob cannot begin until the preceding operation forthat job is completed

(2) Job pre-emption is not permitted that is to say eachjob must wait until the preceding job has finished inorder to be processed on the same machine

(3) ere are infinite buffers (intermediate storage areas)between two machines

(4) All the jobs that have begun to be processed in thesystem must maintain their scheduling order whenany type of disruption arises

(5) All jobs interrupted due to a disruption must remainon their assigned machine with the pending jobsbeing delayed if necessary Jobs not impacted by thedisruption so not delayed will go on with theirnormal process

(6) Rescheduling is carried out periodically thereforetwo types of jobs exist in the system those which arealready being processed whose schedule order can-not vary and those whose order can vary (new jobsand jobs which have not yet begun to be processed)It is important to consider the jobs that are beingprocessed because depending upon the disruptions

Complexity 3

which arise (for example machine breakdowns) acertain minimum starting time will be established forthe next incoming job in the system (in order to beprocessed by the first machine)

(7) Machine breakdowns can arise at any time and notonly when the machines are busy

(8) If a machine breakdown arises when a job is beingprocessed the job will restart processing at the pointwhere the interruption occurred

(9) When a variation in the processing time of a jobarises the new processing time is fixed until the nextprocessing time variation occurs on the same job

As it is a predictive-reactive rescheduling process theMixed-Integer Linear Programming (MILP) model can berun at each rescheduling point meaning that the past dis-ruptions (new jobs processing time variations and machinebreakdowns) that have happened during each reschedulingperiod are received as inputs in the modelose disruptionsimpact the next schedule erefore there are no predicteddisruptions considered in this formulation

e problem is defined by three finite sets

(i) J the set of new jobs to be scheduled from 1 n(ii) M the set of machines from 1 m(iii) O the set of operations for each job at each machine

from 1 m

Here Ji is the job at ith position of the permutatione parameters used in the model are as follows

(i) pij the processing time for job Ji euro J on machine j(ii) di the delivery date for job Ji euro J(iii) wi the priority (weight) of job i with respect to the

rest of the jobs(iv) rli (release time) once a new job has arrived at the

system the release time is the time necessary forjob i to be available in order to be processed againby the production environment

(v) rtj (ready time) the time in which machine j isready to process a new job is input data vectorrepresents the initial situation of the machinesbefore undertaking the new scheduling It is cal-culated based on the current time the disruptionswhich have arisen and the scheduling schemecalculated at the previous rescheduling point It isan important factor as it cannot be supposed thatall the machines are initially available

(vi) RT current temporal instant is represents thecurrent point of rescheduling in the productionenvironment

(vii) To take into account the schedule stability crite-rion the number of unprocessed jobs in the baseschedule (nnot processed) needs to be defined in-cluding new arrivals to the system the startingtime for the jobs at each machine in the baseschedule (Sbaselineij ) the current rescheduling in-stant (RT) and a scale constant (scale 10 in

accordance with the value used in Pfeiffer et al[33] used to apply a quadratic penalty method tothe starting time variation in the base schedulewith respect to RT)

(viii) BJstart a breakdown instant for machine j

(ix) BJend recovery instant from the breakdown of

machine j(x) BigM large constant (BigM⟶infin)

In terms of decision variables the following wereconsidered

(i) Binary variables which represent the position ofeach of the jobs in the planned schedule

forall i j isin J xij f(x) 1 if Ji is at j position

0 otherwise1113896 (1)

(ii) Ci completion time for the job at position i(iii) Cij completion time for the job at position i on

machine j(iv) Sij starting time of job i on machine j(v) Binary variables that indicate the circumstances

under which a machine breakdown has occurred(three situations are modelled)

(a) forall i isin Jand j isinM yij1 it takes the value 1 if theoperation Oij finishes before the breakdownoccurs that is to say the time the operationtakes to finish is less than the machinersquosbreakdown starting point

(b) forall i isin Jand j isinM yij2 it takes the value 1 if theoperation Oij is not yet finished at the momentwhen the breakdown occurs that is to say thetime the operation takes to finish overlaps withthe machinersquos breakdown starting point

(c) forall i isin Jand j isinM yij3 it takes the value 1 if theoperation Oij has a starting time which is afterthe moment when the machine breaks down

(vi) Cmax TWT STB these variables correspond to themaximum completion time for the tasks the totalweighted tardiness and the schedule stabilityrespectively

e mathematical model for the MILP formulation ofthe permutation flow shop rescheduling problem was basedon the following equations

e objectives of the problem were to minimize themakespan the total weighted tardiness (TWT) and thestability of the schedule to be planned (STB) obtaining theoptimal Pareto frontier (PF)

min PF Cmax TWT STB( 1113857 (2)

e makespan was calculated as the maximum com-pletion time for all the jobs and it corresponds to thecompletion time for the last scheduled job that is the timewhen the last task in the last machine finishes

Cmax maxi1n

Ci(π)( 1113857 Cn (3)

4 Complexity

e total completion time for each job Ci is defined bythe following equation according to the completion time inthe last machine M

forall i isin J Ci CijM (4)

where M is the set of machines in the systemMore specifically the completion time for each job i on

each machine j takes into consideration the followingcomponents

(i) e time at which the job i can begin to be processedby machine j is time depends on the following

(a) e completion time of job i on the previousmachine (Cijminus1)

(b) e completion time of the previous job locatedat position (iminus 1) on the current machine j(Ciminus1j)

(c) e maximum of these terms will be the instantof time in which job i can begin to be processedby machine j (Sij)

(ii) e processing time for job i on machine j (pij)(iii) Time of breakdown onmachine jwhen job i is being

processed (BJend minus BJ

start)

When calculating the completion time along with ma-chine breakdowns the following equations are modelledconsidering the three possible breakdown situations

If the operation finishes before the breakdown appears

foralli isin Jandforallj isinM Cij + 1 minus yij11113872 1113873 middot BigMge Sij + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM Cij minus 1 minus yij11113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj

(5)

If the breakdown appears while the operation Oij isbeing processed it is necessary for the job to wait until themachine is ready again in order to continue its processingforalli isin Jandforallj isinM

Cij + 1 minus yij21113872 1113873 middot BigMge Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

foralli isin Jandforallj isinM

Cij minus 1 minus yij21113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

(6)

Finally if the operation Oij has not yet begun the workmust wait until the machine breakdown has finished

foralli isin Jandforallj isinM

Cij + 1 minus yij31113872 1113873 middot BigMge Max Sij BJend1113872 11138731113873 + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM

Cij minus 1 minus yij31113872 1113873 middot BigMle Max Sij BJend1113872 1113873 + 1113944

n

k1xki middot pkj

(7)

Although the above equations contain the max function(that is not linear) they can be expressed in a linearized wayusing a big-M plus binary variables representation beingautomatically transformed by most of the commercialsolvers and respecting theMILP formulation of the problem

If a machine breakdown appears only one of these threesituations can arise this can be ensured by including thefollowing equation

foralli isin Jandforallj isinM 11139443

k1yijk 1 (8)

e starting time for each job i on each machine j de-pends upon the following equation

foralli isin Jand j isinM Sij Max Cijminus1 Ciminus1j1113872 1113873 (9)

In order to define the starting times in the MILP modelthe following equations are used

Operation Oij cannot begin until the operation for theprevious job has finished on the same machine

foralligt 1 isin Jandforallj isinM Sij geCiminus1j (10)

and operation Oij cannot begin until the previous operationfor the same job has finished

foralli isin Jandforalljgt 1 isinM Sij geCijminus1 (11)

e starting time for job i on the first machine must begreater than the overall release time as this is the momentwhen the job is ready to begin to be processed

foralli isin J Si1 ge rli (12)

Because not all the machines are initially available inorder to be able to process the next job it is important to takeinto consideration the fact that the starting time for the jobscannot begin until the machine is available that is to say

foralli isin Jandforallj isinM Sij ge rtj (13)

In case there are delaysmdashthat is when a job completiontime is greater than its delivery datemdashthe total weightedtardiness (TWT) is calculated as the weighted sum of dif-ferences between the total completion time for each job andits delivery date according to their assigned priority weights(wi)

TWT 1113944

n

i1Max Cim minus di 0( 1113857 middot wi (14)

For modelling purposes in the MILP problem the fol-lowing equations were included

TWTge 1113944n

i1Cim minus di( 1113857 middot wi

TWTge 0

(15)

e third objective function considered in the problem isthe stability (STB) with respect to the base schedule cal-culated in the previous period (baseline) Stability isexpressed as the average of the absolute differences of

Complexity 5

unprocessed job starting times adding a penalty whichdepends upon the variation in the starting time for the baseschedule regarding the current RT Unprocessed tasks aredefined as those jobs which have previously been scheduledat the prior point of rescheduling and which have not yetbegun to be processed by the system at the current RT

STB 1

nnot processedmiddot 1113944

n

i1Si1 minus S

baselinei1

11138681113868111386811138681113868

11138681113868111386811138681113868 +scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(16)

e absolute value function used in calculating thestability objective is nondifferentiable in addition to beingnonlinear Nevertheless it can be expressed in an integerlinear way as follows

STB 1

nnot processedmiddot 1113944

n

i1DeltaStart Timei +

scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(17)

Two new binary variables d1j and d2j were introduced aswell as a large constant BigM and the following equationsfor MILP formulationforallj isin J 0le Sj1 le BigM

forallj isin J 0leDeltaStart Timej minus Sj1 minus Sbaselinej11113872 1113873le 2 middot BigM middot d2j

forallj isin J 0leDeltaStart Timej minus Sbaselinej1 minus Sj11113872 1113873le 2 middot BigM middot d1j

forallj isin J d1j + d2j 1

(18)

If d1 1 then delta at the starting time would take thevalue Sj1 minus Sbaselinej1 otherwise if d2 1 it would take thevalue Sbaselinej1 minus Sj1

Finally the model ensures that a job must be in onesingle position and that only a single job can exist at anyposition of the schedule is is done through the followingequations

foralli isin J 1113944n

j1xij 1

forallj isin J 1113944

n

i1xij 1

(19)

32 Rescheduling Architecture e rescheduling strategyused was based on the predictive-reactive approach thus abaseline schedule was first generated and then updatedaccording to the disruptions that arise in system A periodicrescheduling policy was used through time windows Finallythe rescheduling method consisted of a complete resched-uling based on heuristicmetaheuristic algorithms (asrescheduling method we proposed RIPG algorithm de-scribed in detail in the next section) RIPG is selected be-cause its efficient behaviour in static multiobjectivepermutation flow shop problems [25] makes it a promisingtechnique for the dynamic environment under study eproposed rescheduling architecture is shown in Figure 1

Initially the baseline schedule was generated accordingto both the flow shop system involved and the jobs to bescheduled en a synchronous control was establish-edmdashevery T units of timemdashin order to perform arescheduling process when disruptions arise in the systemobtaining the Pareto-front output from the solutions foundin each executed rescheduling interval It is important tospecify an appropriate period based on the dimensions ofthe problem for each rescheduling process In this paperthis period was calculated as the maximum completiontime for jobs (Cmax) divided by the total number ofrescheduling points to be executed on the system (we haveconsidered 5 rescheduling points as it is an appropriatenumber to evaluate the architecture adaptation capacityand does not excessively increase the number of executionsin the system)

Likewise at each point of rescheduling it is necessary toselect a representative solution for the calculated Paretofront e selected solution will be used as a baselineschedule to be employed for each new rescheduling efundamental importance of this base schedule lies in its usein evaluating the stability objective function To calculate therepresentative solution for the Pareto front a knee pointmethod was used [54ndash56] A detailed description of the kneepoint method used to calculate this most representativesolution can be found in Valledor et al [57]

33 Restarted Iterated Greedy Algorithm e implementa-tion of the developed RIPG algorithm was based on thedesign proposed by Ciavotta et al [25] eir application ofthe technique was explained using a biobjective PFSP andits results on Taillard Instances were described comparingthe RIPG technique with 17 algorithms including MOSAand Multiobjective Tabu Search (MOTS) Ciavotta et al[25] described the RIPG as a state-of-the-art method be-cause of its excellent results when compared with the bestapproaches previously presented in the literature eRIPG algorithm is an extension of the IG (Iterated Greedy)technique proposed by Ruiz and Stutzle [58] ese tech-niques belong to the stochastic local search (SLS) categorye RIPG technique consists of 5 phases as shown inFigure 2

Phase 1 First an Initial Solution Set (ISS) is determinedbased upon the execution of different simple heuristicsMore specifically NEH heuristics [59 60] are usedeach one being applied to obtain acceptable solutionswith different objective functions For each of the twosolutions obtained via heuristics the greedy phase(described below in phase three) is applied generatingan initial set of nondominated solutions (working set)Phase 2 Second the optimization loop begins with theprocess of selecting a solution via the ModifiedCrowding Distance Assignment (MCDA) method inorder to improve the working set during the greedyphase e MCDA process receives the working set asan input parameter and calculates the distance to eachone of the solutions e selected solution is the one

6 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

which arise (for example machine breakdowns) acertain minimum starting time will be established forthe next incoming job in the system (in order to beprocessed by the first machine)

(7) Machine breakdowns can arise at any time and notonly when the machines are busy

(8) If a machine breakdown arises when a job is beingprocessed the job will restart processing at the pointwhere the interruption occurred

(9) When a variation in the processing time of a jobarises the new processing time is fixed until the nextprocessing time variation occurs on the same job

As it is a predictive-reactive rescheduling process theMixed-Integer Linear Programming (MILP) model can berun at each rescheduling point meaning that the past dis-ruptions (new jobs processing time variations and machinebreakdowns) that have happened during each reschedulingperiod are received as inputs in the modelose disruptionsimpact the next schedule erefore there are no predicteddisruptions considered in this formulation

e problem is defined by three finite sets

(i) J the set of new jobs to be scheduled from 1 n(ii) M the set of machines from 1 m(iii) O the set of operations for each job at each machine

from 1 m

Here Ji is the job at ith position of the permutatione parameters used in the model are as follows

(i) pij the processing time for job Ji euro J on machine j(ii) di the delivery date for job Ji euro J(iii) wi the priority (weight) of job i with respect to the

rest of the jobs(iv) rli (release time) once a new job has arrived at the

system the release time is the time necessary forjob i to be available in order to be processed againby the production environment

(v) rtj (ready time) the time in which machine j isready to process a new job is input data vectorrepresents the initial situation of the machinesbefore undertaking the new scheduling It is cal-culated based on the current time the disruptionswhich have arisen and the scheduling schemecalculated at the previous rescheduling point It isan important factor as it cannot be supposed thatall the machines are initially available

(vi) RT current temporal instant is represents thecurrent point of rescheduling in the productionenvironment

(vii) To take into account the schedule stability crite-rion the number of unprocessed jobs in the baseschedule (nnot processed) needs to be defined in-cluding new arrivals to the system the startingtime for the jobs at each machine in the baseschedule (Sbaselineij ) the current rescheduling in-stant (RT) and a scale constant (scale 10 in

accordance with the value used in Pfeiffer et al[33] used to apply a quadratic penalty method tothe starting time variation in the base schedulewith respect to RT)

(viii) BJstart a breakdown instant for machine j

(ix) BJend recovery instant from the breakdown of

machine j(x) BigM large constant (BigM⟶infin)

In terms of decision variables the following wereconsidered

(i) Binary variables which represent the position ofeach of the jobs in the planned schedule

forall i j isin J xij f(x) 1 if Ji is at j position

0 otherwise1113896 (1)

(ii) Ci completion time for the job at position i(iii) Cij completion time for the job at position i on

machine j(iv) Sij starting time of job i on machine j(v) Binary variables that indicate the circumstances

under which a machine breakdown has occurred(three situations are modelled)

(a) forall i isin Jand j isinM yij1 it takes the value 1 if theoperation Oij finishes before the breakdownoccurs that is to say the time the operationtakes to finish is less than the machinersquosbreakdown starting point

(b) forall i isin Jand j isinM yij2 it takes the value 1 if theoperation Oij is not yet finished at the momentwhen the breakdown occurs that is to say thetime the operation takes to finish overlaps withthe machinersquos breakdown starting point

(c) forall i isin Jand j isinM yij3 it takes the value 1 if theoperation Oij has a starting time which is afterthe moment when the machine breaks down

(vi) Cmax TWT STB these variables correspond to themaximum completion time for the tasks the totalweighted tardiness and the schedule stabilityrespectively

e mathematical model for the MILP formulation ofthe permutation flow shop rescheduling problem was basedon the following equations

e objectives of the problem were to minimize themakespan the total weighted tardiness (TWT) and thestability of the schedule to be planned (STB) obtaining theoptimal Pareto frontier (PF)

min PF Cmax TWT STB( 1113857 (2)

e makespan was calculated as the maximum com-pletion time for all the jobs and it corresponds to thecompletion time for the last scheduled job that is the timewhen the last task in the last machine finishes

Cmax maxi1n

Ci(π)( 1113857 Cn (3)

4 Complexity

e total completion time for each job Ci is defined bythe following equation according to the completion time inthe last machine M

forall i isin J Ci CijM (4)

where M is the set of machines in the systemMore specifically the completion time for each job i on

each machine j takes into consideration the followingcomponents

(i) e time at which the job i can begin to be processedby machine j is time depends on the following

(a) e completion time of job i on the previousmachine (Cijminus1)

(b) e completion time of the previous job locatedat position (iminus 1) on the current machine j(Ciminus1j)

(c) e maximum of these terms will be the instantof time in which job i can begin to be processedby machine j (Sij)

(ii) e processing time for job i on machine j (pij)(iii) Time of breakdown onmachine jwhen job i is being

processed (BJend minus BJ

start)

When calculating the completion time along with ma-chine breakdowns the following equations are modelledconsidering the three possible breakdown situations

If the operation finishes before the breakdown appears

foralli isin Jandforallj isinM Cij + 1 minus yij11113872 1113873 middot BigMge Sij + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM Cij minus 1 minus yij11113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj

(5)

If the breakdown appears while the operation Oij isbeing processed it is necessary for the job to wait until themachine is ready again in order to continue its processingforalli isin Jandforallj isinM

Cij + 1 minus yij21113872 1113873 middot BigMge Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

foralli isin Jandforallj isinM

Cij minus 1 minus yij21113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

(6)

Finally if the operation Oij has not yet begun the workmust wait until the machine breakdown has finished

foralli isin Jandforallj isinM

Cij + 1 minus yij31113872 1113873 middot BigMge Max Sij BJend1113872 11138731113873 + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM

Cij minus 1 minus yij31113872 1113873 middot BigMle Max Sij BJend1113872 1113873 + 1113944

n

k1xki middot pkj

(7)

Although the above equations contain the max function(that is not linear) they can be expressed in a linearized wayusing a big-M plus binary variables representation beingautomatically transformed by most of the commercialsolvers and respecting theMILP formulation of the problem

If a machine breakdown appears only one of these threesituations can arise this can be ensured by including thefollowing equation

foralli isin Jandforallj isinM 11139443

k1yijk 1 (8)

e starting time for each job i on each machine j de-pends upon the following equation

foralli isin Jand j isinM Sij Max Cijminus1 Ciminus1j1113872 1113873 (9)

In order to define the starting times in the MILP modelthe following equations are used

Operation Oij cannot begin until the operation for theprevious job has finished on the same machine

foralligt 1 isin Jandforallj isinM Sij geCiminus1j (10)

and operation Oij cannot begin until the previous operationfor the same job has finished

foralli isin Jandforalljgt 1 isinM Sij geCijminus1 (11)

e starting time for job i on the first machine must begreater than the overall release time as this is the momentwhen the job is ready to begin to be processed

foralli isin J Si1 ge rli (12)

Because not all the machines are initially available inorder to be able to process the next job it is important to takeinto consideration the fact that the starting time for the jobscannot begin until the machine is available that is to say

foralli isin Jandforallj isinM Sij ge rtj (13)

In case there are delaysmdashthat is when a job completiontime is greater than its delivery datemdashthe total weightedtardiness (TWT) is calculated as the weighted sum of dif-ferences between the total completion time for each job andits delivery date according to their assigned priority weights(wi)

TWT 1113944

n

i1Max Cim minus di 0( 1113857 middot wi (14)

For modelling purposes in the MILP problem the fol-lowing equations were included

TWTge 1113944n

i1Cim minus di( 1113857 middot wi

TWTge 0

(15)

e third objective function considered in the problem isthe stability (STB) with respect to the base schedule cal-culated in the previous period (baseline) Stability isexpressed as the average of the absolute differences of

Complexity 5

unprocessed job starting times adding a penalty whichdepends upon the variation in the starting time for the baseschedule regarding the current RT Unprocessed tasks aredefined as those jobs which have previously been scheduledat the prior point of rescheduling and which have not yetbegun to be processed by the system at the current RT

STB 1

nnot processedmiddot 1113944

n

i1Si1 minus S

baselinei1

11138681113868111386811138681113868

11138681113868111386811138681113868 +scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(16)

e absolute value function used in calculating thestability objective is nondifferentiable in addition to beingnonlinear Nevertheless it can be expressed in an integerlinear way as follows

STB 1

nnot processedmiddot 1113944

n

i1DeltaStart Timei +

scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(17)

Two new binary variables d1j and d2j were introduced aswell as a large constant BigM and the following equationsfor MILP formulationforallj isin J 0le Sj1 le BigM

forallj isin J 0leDeltaStart Timej minus Sj1 minus Sbaselinej11113872 1113873le 2 middot BigM middot d2j

forallj isin J 0leDeltaStart Timej minus Sbaselinej1 minus Sj11113872 1113873le 2 middot BigM middot d1j

forallj isin J d1j + d2j 1

(18)

If d1 1 then delta at the starting time would take thevalue Sj1 minus Sbaselinej1 otherwise if d2 1 it would take thevalue Sbaselinej1 minus Sj1

Finally the model ensures that a job must be in onesingle position and that only a single job can exist at anyposition of the schedule is is done through the followingequations

foralli isin J 1113944n

j1xij 1

forallj isin J 1113944

n

i1xij 1

(19)

32 Rescheduling Architecture e rescheduling strategyused was based on the predictive-reactive approach thus abaseline schedule was first generated and then updatedaccording to the disruptions that arise in system A periodicrescheduling policy was used through time windows Finallythe rescheduling method consisted of a complete resched-uling based on heuristicmetaheuristic algorithms (asrescheduling method we proposed RIPG algorithm de-scribed in detail in the next section) RIPG is selected be-cause its efficient behaviour in static multiobjectivepermutation flow shop problems [25] makes it a promisingtechnique for the dynamic environment under study eproposed rescheduling architecture is shown in Figure 1

Initially the baseline schedule was generated accordingto both the flow shop system involved and the jobs to bescheduled en a synchronous control was establish-edmdashevery T units of timemdashin order to perform arescheduling process when disruptions arise in the systemobtaining the Pareto-front output from the solutions foundin each executed rescheduling interval It is important tospecify an appropriate period based on the dimensions ofthe problem for each rescheduling process In this paperthis period was calculated as the maximum completiontime for jobs (Cmax) divided by the total number ofrescheduling points to be executed on the system (we haveconsidered 5 rescheduling points as it is an appropriatenumber to evaluate the architecture adaptation capacityand does not excessively increase the number of executionsin the system)

Likewise at each point of rescheduling it is necessary toselect a representative solution for the calculated Paretofront e selected solution will be used as a baselineschedule to be employed for each new rescheduling efundamental importance of this base schedule lies in its usein evaluating the stability objective function To calculate therepresentative solution for the Pareto front a knee pointmethod was used [54ndash56] A detailed description of the kneepoint method used to calculate this most representativesolution can be found in Valledor et al [57]

33 Restarted Iterated Greedy Algorithm e implementa-tion of the developed RIPG algorithm was based on thedesign proposed by Ciavotta et al [25] eir application ofthe technique was explained using a biobjective PFSP andits results on Taillard Instances were described comparingthe RIPG technique with 17 algorithms including MOSAand Multiobjective Tabu Search (MOTS) Ciavotta et al[25] described the RIPG as a state-of-the-art method be-cause of its excellent results when compared with the bestapproaches previously presented in the literature eRIPG algorithm is an extension of the IG (Iterated Greedy)technique proposed by Ruiz and Stutzle [58] ese tech-niques belong to the stochastic local search (SLS) categorye RIPG technique consists of 5 phases as shown inFigure 2

Phase 1 First an Initial Solution Set (ISS) is determinedbased upon the execution of different simple heuristicsMore specifically NEH heuristics [59 60] are usedeach one being applied to obtain acceptable solutionswith different objective functions For each of the twosolutions obtained via heuristics the greedy phase(described below in phase three) is applied generatingan initial set of nondominated solutions (working set)Phase 2 Second the optimization loop begins with theprocess of selecting a solution via the ModifiedCrowding Distance Assignment (MCDA) method inorder to improve the working set during the greedyphase e MCDA process receives the working set asan input parameter and calculates the distance to eachone of the solutions e selected solution is the one

6 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

e total completion time for each job Ci is defined bythe following equation according to the completion time inthe last machine M

forall i isin J Ci CijM (4)

where M is the set of machines in the systemMore specifically the completion time for each job i on

each machine j takes into consideration the followingcomponents

(i) e time at which the job i can begin to be processedby machine j is time depends on the following

(a) e completion time of job i on the previousmachine (Cijminus1)

(b) e completion time of the previous job locatedat position (iminus 1) on the current machine j(Ciminus1j)

(c) e maximum of these terms will be the instantof time in which job i can begin to be processedby machine j (Sij)

(ii) e processing time for job i on machine j (pij)(iii) Time of breakdown onmachine jwhen job i is being

processed (BJend minus BJ

start)

When calculating the completion time along with ma-chine breakdowns the following equations are modelledconsidering the three possible breakdown situations

If the operation finishes before the breakdown appears

foralli isin Jandforallj isinM Cij + 1 minus yij11113872 1113873 middot BigMge Sij + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM Cij minus 1 minus yij11113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj

(5)

If the breakdown appears while the operation Oij isbeing processed it is necessary for the job to wait until themachine is ready again in order to continue its processingforalli isin Jandforallj isinM

Cij + 1 minus yij21113872 1113873 middot BigMge Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

foralli isin Jandforallj isinM

Cij minus 1 minus yij21113872 1113873 middot BigMle Sij + 1113944n

k1xki middot pkj + B

Jend minus B

Jstart1113872 1113873

(6)

Finally if the operation Oij has not yet begun the workmust wait until the machine breakdown has finished

foralli isin Jandforallj isinM

Cij + 1 minus yij31113872 1113873 middot BigMge Max Sij BJend1113872 11138731113873 + 1113944

n

k1xki middot pkj

foralli isin Jandforallj isinM

Cij minus 1 minus yij31113872 1113873 middot BigMle Max Sij BJend1113872 1113873 + 1113944

n

k1xki middot pkj

(7)

Although the above equations contain the max function(that is not linear) they can be expressed in a linearized wayusing a big-M plus binary variables representation beingautomatically transformed by most of the commercialsolvers and respecting theMILP formulation of the problem

If a machine breakdown appears only one of these threesituations can arise this can be ensured by including thefollowing equation

foralli isin Jandforallj isinM 11139443

k1yijk 1 (8)

e starting time for each job i on each machine j de-pends upon the following equation

foralli isin Jand j isinM Sij Max Cijminus1 Ciminus1j1113872 1113873 (9)

In order to define the starting times in the MILP modelthe following equations are used

Operation Oij cannot begin until the operation for theprevious job has finished on the same machine

foralligt 1 isin Jandforallj isinM Sij geCiminus1j (10)

and operation Oij cannot begin until the previous operationfor the same job has finished

foralli isin Jandforalljgt 1 isinM Sij geCijminus1 (11)

e starting time for job i on the first machine must begreater than the overall release time as this is the momentwhen the job is ready to begin to be processed

foralli isin J Si1 ge rli (12)

Because not all the machines are initially available inorder to be able to process the next job it is important to takeinto consideration the fact that the starting time for the jobscannot begin until the machine is available that is to say

foralli isin Jandforallj isinM Sij ge rtj (13)

In case there are delaysmdashthat is when a job completiontime is greater than its delivery datemdashthe total weightedtardiness (TWT) is calculated as the weighted sum of dif-ferences between the total completion time for each job andits delivery date according to their assigned priority weights(wi)

TWT 1113944

n

i1Max Cim minus di 0( 1113857 middot wi (14)

For modelling purposes in the MILP problem the fol-lowing equations were included

TWTge 1113944n

i1Cim minus di( 1113857 middot wi

TWTge 0

(15)

e third objective function considered in the problem isthe stability (STB) with respect to the base schedule cal-culated in the previous period (baseline) Stability isexpressed as the average of the absolute differences of

Complexity 5

unprocessed job starting times adding a penalty whichdepends upon the variation in the starting time for the baseschedule regarding the current RT Unprocessed tasks aredefined as those jobs which have previously been scheduledat the prior point of rescheduling and which have not yetbegun to be processed by the system at the current RT

STB 1

nnot processedmiddot 1113944

n

i1Si1 minus S

baselinei1

11138681113868111386811138681113868

11138681113868111386811138681113868 +scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(16)

e absolute value function used in calculating thestability objective is nondifferentiable in addition to beingnonlinear Nevertheless it can be expressed in an integerlinear way as follows

STB 1

nnot processedmiddot 1113944

n

i1DeltaStart Timei +

scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(17)

Two new binary variables d1j and d2j were introduced aswell as a large constant BigM and the following equationsfor MILP formulationforallj isin J 0le Sj1 le BigM

forallj isin J 0leDeltaStart Timej minus Sj1 minus Sbaselinej11113872 1113873le 2 middot BigM middot d2j

forallj isin J 0leDeltaStart Timej minus Sbaselinej1 minus Sj11113872 1113873le 2 middot BigM middot d1j

forallj isin J d1j + d2j 1

(18)

If d1 1 then delta at the starting time would take thevalue Sj1 minus Sbaselinej1 otherwise if d2 1 it would take thevalue Sbaselinej1 minus Sj1

Finally the model ensures that a job must be in onesingle position and that only a single job can exist at anyposition of the schedule is is done through the followingequations

foralli isin J 1113944n

j1xij 1

forallj isin J 1113944

n

i1xij 1

(19)

32 Rescheduling Architecture e rescheduling strategyused was based on the predictive-reactive approach thus abaseline schedule was first generated and then updatedaccording to the disruptions that arise in system A periodicrescheduling policy was used through time windows Finallythe rescheduling method consisted of a complete resched-uling based on heuristicmetaheuristic algorithms (asrescheduling method we proposed RIPG algorithm de-scribed in detail in the next section) RIPG is selected be-cause its efficient behaviour in static multiobjectivepermutation flow shop problems [25] makes it a promisingtechnique for the dynamic environment under study eproposed rescheduling architecture is shown in Figure 1

Initially the baseline schedule was generated accordingto both the flow shop system involved and the jobs to bescheduled en a synchronous control was establish-edmdashevery T units of timemdashin order to perform arescheduling process when disruptions arise in the systemobtaining the Pareto-front output from the solutions foundin each executed rescheduling interval It is important tospecify an appropriate period based on the dimensions ofthe problem for each rescheduling process In this paperthis period was calculated as the maximum completiontime for jobs (Cmax) divided by the total number ofrescheduling points to be executed on the system (we haveconsidered 5 rescheduling points as it is an appropriatenumber to evaluate the architecture adaptation capacityand does not excessively increase the number of executionsin the system)

Likewise at each point of rescheduling it is necessary toselect a representative solution for the calculated Paretofront e selected solution will be used as a baselineschedule to be employed for each new rescheduling efundamental importance of this base schedule lies in its usein evaluating the stability objective function To calculate therepresentative solution for the Pareto front a knee pointmethod was used [54ndash56] A detailed description of the kneepoint method used to calculate this most representativesolution can be found in Valledor et al [57]

33 Restarted Iterated Greedy Algorithm e implementa-tion of the developed RIPG algorithm was based on thedesign proposed by Ciavotta et al [25] eir application ofthe technique was explained using a biobjective PFSP andits results on Taillard Instances were described comparingthe RIPG technique with 17 algorithms including MOSAand Multiobjective Tabu Search (MOTS) Ciavotta et al[25] described the RIPG as a state-of-the-art method be-cause of its excellent results when compared with the bestapproaches previously presented in the literature eRIPG algorithm is an extension of the IG (Iterated Greedy)technique proposed by Ruiz and Stutzle [58] ese tech-niques belong to the stochastic local search (SLS) categorye RIPG technique consists of 5 phases as shown inFigure 2

Phase 1 First an Initial Solution Set (ISS) is determinedbased upon the execution of different simple heuristicsMore specifically NEH heuristics [59 60] are usedeach one being applied to obtain acceptable solutionswith different objective functions For each of the twosolutions obtained via heuristics the greedy phase(described below in phase three) is applied generatingan initial set of nondominated solutions (working set)Phase 2 Second the optimization loop begins with theprocess of selecting a solution via the ModifiedCrowding Distance Assignment (MCDA) method inorder to improve the working set during the greedyphase e MCDA process receives the working set asan input parameter and calculates the distance to eachone of the solutions e selected solution is the one

6 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

unprocessed job starting times adding a penalty whichdepends upon the variation in the starting time for the baseschedule regarding the current RT Unprocessed tasks aredefined as those jobs which have previously been scheduledat the prior point of rescheduling and which have not yetbegun to be processed by the system at the current RT

STB 1

nnot processedmiddot 1113944

n

i1Si1 minus S

baselinei1

11138681113868111386811138681113868

11138681113868111386811138681113868 +scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(16)

e absolute value function used in calculating thestability objective is nondifferentiable in addition to beingnonlinear Nevertheless it can be expressed in an integerlinear way as follows

STB 1

nnot processedmiddot 1113944

n

i1DeltaStart Timei +

scale

Sbaselinei1 minus RT1113969⎛⎜⎜⎜⎝ ⎞⎟⎟⎟⎠

(17)

Two new binary variables d1j and d2j were introduced aswell as a large constant BigM and the following equationsfor MILP formulationforallj isin J 0le Sj1 le BigM

forallj isin J 0leDeltaStart Timej minus Sj1 minus Sbaselinej11113872 1113873le 2 middot BigM middot d2j

forallj isin J 0leDeltaStart Timej minus Sbaselinej1 minus Sj11113872 1113873le 2 middot BigM middot d1j

forallj isin J d1j + d2j 1

(18)

If d1 1 then delta at the starting time would take thevalue Sj1 minus Sbaselinej1 otherwise if d2 1 it would take thevalue Sbaselinej1 minus Sj1

Finally the model ensures that a job must be in onesingle position and that only a single job can exist at anyposition of the schedule is is done through the followingequations

foralli isin J 1113944n

j1xij 1

forallj isin J 1113944

n

i1xij 1

(19)

32 Rescheduling Architecture e rescheduling strategyused was based on the predictive-reactive approach thus abaseline schedule was first generated and then updatedaccording to the disruptions that arise in system A periodicrescheduling policy was used through time windows Finallythe rescheduling method consisted of a complete resched-uling based on heuristicmetaheuristic algorithms (asrescheduling method we proposed RIPG algorithm de-scribed in detail in the next section) RIPG is selected be-cause its efficient behaviour in static multiobjectivepermutation flow shop problems [25] makes it a promisingtechnique for the dynamic environment under study eproposed rescheduling architecture is shown in Figure 1

Initially the baseline schedule was generated accordingto both the flow shop system involved and the jobs to bescheduled en a synchronous control was establish-edmdashevery T units of timemdashin order to perform arescheduling process when disruptions arise in the systemobtaining the Pareto-front output from the solutions foundin each executed rescheduling interval It is important tospecify an appropriate period based on the dimensions ofthe problem for each rescheduling process In this paperthis period was calculated as the maximum completiontime for jobs (Cmax) divided by the total number ofrescheduling points to be executed on the system (we haveconsidered 5 rescheduling points as it is an appropriatenumber to evaluate the architecture adaptation capacityand does not excessively increase the number of executionsin the system)

Likewise at each point of rescheduling it is necessary toselect a representative solution for the calculated Paretofront e selected solution will be used as a baselineschedule to be employed for each new rescheduling efundamental importance of this base schedule lies in its usein evaluating the stability objective function To calculate therepresentative solution for the Pareto front a knee pointmethod was used [54ndash56] A detailed description of the kneepoint method used to calculate this most representativesolution can be found in Valledor et al [57]

33 Restarted Iterated Greedy Algorithm e implementa-tion of the developed RIPG algorithm was based on thedesign proposed by Ciavotta et al [25] eir application ofthe technique was explained using a biobjective PFSP andits results on Taillard Instances were described comparingthe RIPG technique with 17 algorithms including MOSAand Multiobjective Tabu Search (MOTS) Ciavotta et al[25] described the RIPG as a state-of-the-art method be-cause of its excellent results when compared with the bestapproaches previously presented in the literature eRIPG algorithm is an extension of the IG (Iterated Greedy)technique proposed by Ruiz and Stutzle [58] ese tech-niques belong to the stochastic local search (SLS) categorye RIPG technique consists of 5 phases as shown inFigure 2

Phase 1 First an Initial Solution Set (ISS) is determinedbased upon the execution of different simple heuristicsMore specifically NEH heuristics [59 60] are usedeach one being applied to obtain acceptable solutionswith different objective functions For each of the twosolutions obtained via heuristics the greedy phase(described below in phase three) is applied generatingan initial set of nondominated solutions (working set)Phase 2 Second the optimization loop begins with theprocess of selecting a solution via the ModifiedCrowding Distance Assignment (MCDA) method inorder to improve the working set during the greedyphase e MCDA process receives the working set asan input parameter and calculates the distance to eachone of the solutions e selected solution is the one

6 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

with the greatest modified crowding distance (MCD)value which coincides with solutions in sparselypopulated search spaces e main difference between

MCDA and Crowding Distance Assignment (CDA)proposed originally in the NSGA-II [12] is that the firstmethod considers the number of times that a solution

Synchronous control(every T units of time)

Evaluation of everyobjective function for

each solution

Disruptions

Periodic control ofdisruptions

Baselineschedulegenerator

Rescheduling

New schedule(baseline)

Baselineschedule

Jobs to scheduleFlow shop environment definition

Figure 1 Rescheduling architecture

Restart working set

No

No

Start RIPG

Solution working set initialization(NEH amp Rajendran heuristics + greedy phase)

Yes

Modified Crowding Distance Assignmentselection

Greedy phase1 Destruction of ldquokrdquo contiguous positions in the sequence

2 Construction phase

Greedy phase initial solution selection usingModified Crowding Distance Assignment operation

Local search(Insertion nsel jobs in nneigh neighbours)

Return Pareto front

Externalsolutionarchive

Working set restarting

YesStopping criterion

Figure 2 Flow diagram for the RIPG technique

Complexity 7

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

has been previously selected giving it less opportunityof being reselectedPhase 3 As the next step the greedy phase is executede selected solution is modified by applying both adestruction operation to eliminate a sub-schedule ofjobs from the solution and a greedy process known as aconstruction operation to insert each of the eliminatedjobs in one of the possible schedule positions In thisway a solution pool is createdPhase 4 Next once again by the MCDA method asolution is selected from the solution pool generatedduring the greedy phase and a local search method isapplied for improvement As local search we appliedthe same classical approach as proposed by Ciavottaet al [25] where nsel jobs are randomly selected theneach of them is eliminated from the schedule andreinserted in nneigh consecutive positions half of whichprecede the initial position of the selected job and theother half follow it Once all the movements are un-dertaken for the selected jobs the generated solutionsare subsequently evaluated and the dominated solu-tions are eliminated e nondominated solutions areadded to the working setPhase 5 Finally a restarting mechanism is imple-mented on the set of solutions to prevent the algorithmfrom reaching only a local optimal solution and causinga stagnation effect e mechanism is simple and re-liable it consists of storing the set of nondominatedsolutions in the working set in an external file andsubsequently randomly restarting the working setisrestart process is launched if the size of the working setthat is the number of nondominated solutions founduntil that time does not vary in a pre-set number ofiterations

e RIPG algorithm requires the configuration of 5input parameters the size of the set of solutions (workingset) k (the number of consecutive elements eliminatedduring the destruction operation of the greedy phase) nsel(the number of elements randomly selected in the localsearch phase) nneigh (the number of consecutive positions inwhich a job is reinserted in the local search phase) and therestart threshold (the maximum number of iterationsallowed without modifying the size of nondominated so-lution set once this limit is surpassed the working setautomatically restarts)

4 Results and Discussion

41 Experimental Design e lack of a common scheme toassess rescheduling problems in a PFSP leads to the gen-eration of databases that permit to work with diverse dis-ruptions in the production environment (appearance of newjobs machine failures etc) and evaluate a multiobjectiveoptimization problem e main database to assess PFSPswas established by Taillard [61] whose instances were ini-tially generated to be applied into single-objective envi-ronments minimization of the completion time of the jobs

Subsequently these instances were modified to incorporate adue date to each job similarly to Minella et al [62] andHasija and Rajendran [63] allowing the rescheduling systemto be assessed under different objective functions such astotal weighted tardiness (TWT) or maximum delay of jobs[64]

In the proposed rescheduling system after the originalschedule comes into effect different disruptions may occurover time NEH heuristic was used [59 65] to achieve theoriginal schedule in order to optimize themakespan Minellaet al [62] also used it to have a feasible original scheduleAdditionally in our generated benchmark the disruptionsto be considered and the way in which they occur are de-fined Although most rescheduling systems focus on onetype of disruption in this paper we use three machine faultsnew job arrivals and variable processing times

Machine faults were defined through an exponentialdistribution defined by the level of failure reached by thesystem the mean time to repair (MTTR) and the mean timebetween faults (MTBF) e breakdown level (Ag) per-centage of machine failure time was estimated by means ofthe following expression

Ag MTTR

MTTR + MTBF (20)

We have followed the machine faults parameterizationproposed by Adibi et al [66] and Mason et al [67] Re-garding the arrivals of new jobs an exponential distributionwas also adopted with an arrival rate based on the quotientbetween the degree of use of the system and the averageoperating time [68] Concerning the deviation of the jobprocessing times they were randomly generated (1probability) with a variation factor of 30 in line with theparameters established by Swaminathan et al [69] with asymmetric triangular distribution

From the 120 instances of Taillard another 120rescheduling instances were created and from them a subsetof 50 was selected to be evaluated with the RIPG algorithme reason for not evaluating all 120 instances is the highexecution time required us for each of these instances 5rescheduling points were executed during which the systemwas updated according to the events in the flow shop en-vironment and therefore the technique was executed 5times per instance e 50 instances used were randomlyselected from the set of 120 Taillard Instances excludingthose of greater size (with 500 jobs) because it is not usual towork with such a high number of jobs in environments withmore than one objective function At the website Instances ampSolutions the following files have been included

(i) Training instances used with the Irace Package(ii) Evaluation instances(iii) Pareto-front solutions from the RIPG for each one

of the 10 repetitions undertaken for each evaluationinstance

42 Validation of the RIPG Algorithm on Dubois-LacosteInstances in a Static Biobjective Environment Since we have

8 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

not found a benchmark for the dynamic three-objectiveproblem in the literature we adapted the algorithm to astatic biobjective approach and used the multiobjectiveperformance metrics proposed by Dubois-Lacoste et al [26]as a way to validate the correct behaviour of the algorithme aim of Dubois-Lacoste et al was minimizing both themaximum completion time for the tasks and the totalweighted tardiness e reason for selecting this environ-ment is the ability to compare the results of the appliedalgorithm with the best results obtained in the literaturethrough a Two-Phase Local Search + Pareto Local Search(TP+ PLS) hybrid algorithm

In our experiments the RIPG metaheuristic used classicparametrization as proposed in the literature [25] andrecommended for biobjective environments (see Table 1)

Each experiment was repeated 5 times (independently)in order to evaluate the multiobjective metrics of hyper-volume and the unary epsilon indicator (I1ε ) on each of theDubois-Lacoste et al instances [26] Based upon a prelim-inary statistical study with the Wilcoxon test the RIPGalgorithm was inferred to behave in a similar way to theTP+PLS algorithm in a static biobjective environment ascan be seen in the box plot in Figure 3 Similar results areobtained with the unary multiplicative epsilon indicator ascan be seen in Figure 4

In Figure 5 a comparison between the Pareto frontsobtained by the RIPG algorithm and the best results found inthe literature is shown [26] for some of the TaillardInstances

e RIPG algorithm approaches the best Pareto frontsobtained in the literature following the same trends withoutexcessively deviating from the reference nondominatedsolutions New nondominated solutions have even beendetected in some cases as can be seen in instances ta018 andta024

43 Parameter Calibration As classical parameters in staticenvironments could not adjust properly to dynamic sce-narios we have run a parameter calibration of RIPG in ourrescheduling multiobjective environment e hardwareenvironment for experimentation consisted of a machinewith a Quad-Core Intel Xeon X5675 processor 307GHzand 16GB of RAMmemory e operating system used wasWindows 7 Enterprise Edition 64 bit architecture e

software was developed on the NET platform with the Cprogramming language using the Microsoft Visual Studio2010 Development Framework

In order to define the stopping criterion we analyseddifferent approaches from the literature [58 70ndash74] Mostuse a maximum execution time calculated depending on thesize of the problem that is defined by the number of jobs andmachines that the system has available as the stoppingcriterion For this study the time limit was calculated withthe following equation t n times m2 times 100 where t is the timeinmilliseconds n is the number of jobs andm is the numberof machines e above expression was used to calculate the

Table 1 Classical parametrization for the RIPG algorithm

Parameter Default valueSize of the set of job solutions after the restart phase (working set) 100k (number of consecutive elements eliminated in the greedy phasersquos destructionoperation) 5

nsel (number of elements selected randomly in the local search phase) Dynamicparameter dependent on the number of jobs in the instance to be solved

nsel ncount if ncountle nlowast 05else nsel nlowast 05

n is the number of jobs and ncount is the number of timesthat a solution has been selected

nneigh (consecutive positions where a job is reinserted in the local search phase) 5Restart threshold (maximum number of iterations allowed without modifyingthe size of the nondominated solutions set Once this value is surpassed theworking set randomly restarts)

2lowast n (where n is the number of jobs)

Hyp

ervo

lum

e

10

08

06

04

02

RIPG TP + PLS

Figure 3 Box plot showing hypervolume in a biobjectiveenvironment

Una

ry ep

silon

indi

cato

r

20

18

16

14

12

10

RIPG TP + PLS

Figure 4 Box plot showing the unary epsilon indicator in abiobjective environment

Complexity 9

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

execution time because both the number of jobs and thenumber of machines played a role in the difficulty of theproblem to be solved In addition it allowed the use of morecalculation time in problems with greater complexity

Likewise the automatic configuration of the parametersdefined by the RIPG algorithm was proposed based on anextension of the iterated F-Race method and implemented

in the Irace tool [75] To accomplish this hypervolumewas defined as a metric to evaluate the configurationschemes as this is the most common criterion in mul-tiobjective problems To select the training instances acomplexity analysis was used depending on the resultsprovided in the literature regarding Taillard Instances[26]

0

5000

10000

15000

20000

1250 1300 1350 1400Makespan

Proposed metaheuristicTP + PLS

Tota

l wei

ghte

d ta

rdin

ess

(a)

Proposed metaheuristicTP + PLS

02000400060008000

1000012000

1340 1360 1380 1400 1420 1440Makespan

Tota

l wei

ghte

d ta

rdin

ess

(b)

Proposed metaheuristicTP + PLS

0

2000

4000

6000

8000

10000

1540 1560 1580 1600 1620Makespan

Tota

l wei

ghte

d ta

rdin

ess

(c)

Proposed metaheuristicTP + PLS

0

200

400

600

800

1000

2220 2225 2230 2235 2240Makespan

Tota

l wei

ghte

d ta

rdin

ess

(d)

Proposed metaheuristicTP + PLS

130000135000140000145000150000155000160000

2700 2750 2800 2850Makespan

Tota

l wei

ghte

d ta

rdin

ess

(e)

Proposed metaheuristicTP + PLS

020000400006000080000

100000120000140000160000180000

3100 3200 3300 3400Makespan

Tota

l wei

ghte

d ta

rdin

ess

(f )

Proposed metaheuristicTP + PLS

0200000400000600000800000

10000001200000

6300 6500 6700 6900 7100

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(g)

Proposed metaheuristicTP + PLS

0100000020000003000000400000050000006000000

10600 10700 10800 10900 11000 11100 11200

Tota

l wei

ghte

d ta

rdin

ess

Makespan

(h)

Figure 5 Comparison of Pareto frontiers between the RIPG metaheuristic and the TP+PLS algorithm in a static biobjective flow shopenvironment (a) Problem instance ta001 (b) problem instance ta002 (c) problem instance ta018 (d) problem instance ta024 (e) probleminstance ta031 (f ) problem instance ta047 (g) problem instance ta084 and (h) problem instance ta100

10 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

As representative instances a total of 6 problems wereselected (ta02 ta031 ta047 ta061 ta080 and ta024) inwhich two have low complexity two have medium com-plexity and two are of high complexity in terms of solvingdifficulty e criterion chosen to evaluate the complexity ofan instance was based on a trade-off between the number ofnondominated solutions found in the literature for bio-bjective static problems (makespan and total weightedtardiness) combined with the relative error found in mono-objective problems considering the makespan criterion einstances used in the calibration phase were not subse-quently used for the evaluation of the algorithm in order toavoid overfitting

After analysing the Irace results the RIPG algorithmparametersrsquo configurations did not have a great impact onthe results therefore this algorithm is robust in terms of

parameter variation In Table 2 the results of the optimalIrace automatic configuration for the RIPG algorithm areshown

44 Performance of the RIPG Algorithm in MultiobjectiveDynamic Environments With the aim of verifying theperformance of the RIPG metaheuristic it was executed onthe 50 benchmark problems which were developed specif-ically in this study for a flow shop rescheduling environmente RIPG algorithm was executed 10 times (independently)for each of the experiments in order to carry out a reliablestatistical analysis As it was impossible to compare the RIPGalgorithm with other metaheuristics in the proposed envi-ronment the results obtained from the simulations in 4selected instances are presented below In Figures 6ndash9 the

Table 2 Automatic RIPG configuration via Irace

Parameter Value Datatype

Range establishedfor Irace

Size of the set of job solutions after the restart phase (workingset) 124 Integer [50 200]

k (number of consecutive elements eliminated in the greedyphasersquos destruction operation) 5 Integer [1 10]

nsel (number of elements selected randomly in the local searchphase) Dynamic parameter dependent on the number of jobsin the instance to be solved

nsel ncount if ncountle nlowast 01else nsel nlowast 01

n is the number of jobs and ncount is thenumber of times that a solution has been

selected

Decimal [01 1]

nneigh (consecutive positions where a job is reinserted in thelocal search phase) 8 Integer [1 12]

Restart threshold (maximum number of iterations allowedwithout modifying the size of the nondominated solutions setOnce this value is surpassed the working set randomlyrestarts)

3lowast n (where n is the number of jobs) Integer [1 4]

Dynamic Pareto front in TA_FSRP_20 times 5 times 1

1400 1600 1800Makespan

2000 2200 minus2000 2000 4000 60

00 8000

1000

0

Total weighted tardiness1200

014

000

1600

0

Stab

ility

0

050100150200250300350400

minus50

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 6 RIPG Pareto front in TA_20_5_1

Complexity 11

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

Pareto fronts obtained in the 4 chosen instances are shownFive Pareto fronts are shown from each instance corre-sponding to the five generated at each point of reschedulingWe can see how the number of nondominated solutionsfound goes down as the problemrsquos size increases

All the Pareto fronts obtained from the RIPG algorithmcan be viewed at Instances amp Solutions httpswwwdropbox

comsl4mqu5p17wov5s6Multiobjective Rescheduling-Instances26Solutionszipdl0

Finally with the aim of viewing the rescheduling pro-cesses when disruptions appear a small instance was se-lected (TA_20_5_3) to show the evolution of themanufacturing schedule It should be noted that as this is asmall instance machine breakdowns did not occur

Dynamic Pareto front in TA_FSRP_50 times 5 times 4

30003500 4000 4500

Makespan5000

5500 6000220000

240000

Total weighted tardiness260000280000

300000

Stab

ility

200

300

400

500

600

700

800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 8 RIPG Pareto front in TA_50_5_4

Dynamic Pareto front in TA_FSRP_20 times 10 times 3

15002000

2500

Makespan3000

35004000minus5000

05000

Total weighted tardiness1000015000

20000

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 7 RIPG Pareto front in TA_20_10_3

12 Complexity

however the rest of the disruptions did appear (see Table 3)In Figure 10 the schedule of the jobs at the different points ofrescheduling is shown J16 is the first job in the initial baseschedule NJX identifies new jobs e base schedule ispresented (t 0) as well as 5 rescheduling runs (at times

t 232 t 464 t 696 t 928 and t 1160) Due to theappearance of new jobs and variations in job processingtimes the schedule was modified Figure 11 shows the RIPGPareto front for each rescheduling point in the instanceTA_20_5_3

Table 3 Disruptions in TA_20_5_3

Time Disruptionst1 232 7 new jobs 2 variations in processing timet2 464 3 new jobs 1 variation in processing time (job 9)t3 696 4 new jobs 4 variations in processing timet4 928 5 new jobs 3 variations in processing timet5 1160 4 new jobs 2 variations in processing time

Dynamic Pareto front in TA_FSRP_50 times 20 times 1

4000 4500 5000 5500 6000

Makespan70006500

7500 8000 8500

5000

060

000

7000

080

000

9000

010

0000

4000

0

Total weighted tardiness

3000

020

000

1000

0

Stab

ility

2001000

300400500600700800

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 9 RIPG Pareto front in TA_50_20_1

Initial sequence J16 J3 J20 J18 J7 J1 J12 J10 J5 J2 J9 J4 J19 J14 J17 J6 J13 J11 J8 J15t0 = 0

NJ3 J10 J5 J9 J17 J4 J14 NJ5 J15 J11 J6 NJ7 J13 J19 J8 NJ2 NJ1 NJ6 NJ4 J2t1 = 232

J11 J19 J9 J4 J17 NJ3 J15 J6 J14 J13 NJ8 NJ9 NJ1 NJ2 NJ10 J8 NJ4 NJ6t2 = 464

J6 J13 J14 J15 NJ3 NJ8 NJ1 NJ2 NJ10 NJ13 NJ4 NJ11 J8 NJ6 NJ9 NJ14 NJ12t3 = 696

NJ8 NJ3 NJ1 NJ12 NJ13 NJ2 NJ4 NJ11 J8 NJ9 NJ14 NJ17 NJ10 NJ6 NJ18 NJ15 NJ19 NJ16t4 = 928

NJ21 NJ2 NJ11 J8 NJ9 NJ14 NJ17 NJ10 NJ16 NJ23 NJ6 NJ19 NJ15 NJ18 NJ22 NJ20t5 = 1160

Makespan = 1840 TWT = 32310 stability = 31

Makespan = 2150 TWT = 15872 stability = 9

Makespan = 2323 TWT = 11816 stability = 21

Makespan = 1159=gtrescheduling interval = 232

Makespan = 1445 TWT = 37885 stability = 158

Makespan = 1608 TWT = 32326 stability = 94

Figure 10 Schedule of jobs at the different rescheduling points in TA_20_5_3

Complexity 13

5 Conclusions and Future Lines of Research

is research proposes several relevant contributions Firsta newMILPmathematical formulation has been proposed torepresent the dynamic triobjective rescheduling problem tobe solved Likewise a set of problems (benchmark instances)has been defined to model dynamic rescheduling environ-ments including the specification of different disruptionswhich can arise in a production environment Additionallya rescheduling architecture has been designed and imple-mented based on a predictive-reactive strategy with periodicrescheduling Moreover the RIPG metaheuristic has beenapplied and its proper operation in dynamic reschedulingsystems verified

To validate the proposed approach and having found noreferences in the literature a benchmark based on a staticbiobjective environment was used It is important to note thatRIPG algorithm approaches the best Pareto frontiers obtainedin the literature following the same trends without exces-sively deviating from the reference nondominated solutionsset in a static biobjective environment New nondominatedsolutions have even been detected in some cases as can beseen in instances ta018 and ta024 Likewise RIPG is veryrobust because any parametrization of the algorithm yieldssimilar results in terms of the median value As it has beenimpossible to compare the RIPG algorithm with othermetaheuristics in the proposed environment due to the lackof techniques applied in such problems Pareto frontiersobtained for several instances of problems of different sizeshave been shown e number of nondominated solutionsfound goes down as the problemrsquos size increases

For future lines of research it may be useful to delvedeeper into analysing the influence of the base schedule

(baseline) used to evaluate the stability objective function onthe algorithmrsquos results In addition although this paper hasanalysed all the experiments using 5 rescheduling points thedynamic system could be evaluated for a variable number ofrescheduling points between 0 and 1000 as Sabuncuogluand Karabuk [76] considered in their work Likewise theRIPG metaheuristic could be adjusted to dynamic situationsin problems known asmany-objective optimization in whichthe number of objective functions is greater than threeLastly another existing metaheuristic could be adapted toresolve this problem so that the results may be comparedwith those obtained through RIPG

Data Availability

Instances and Solutions Dataset referred in this paper can beaccessed at httpsdoiorg10176324p4jcwdwpt2 (Men-deley Website)

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] M R Garey D S Johnson and R Sethi ldquoe complexity offlowshop and Jobshop schedulingrdquo Mathematics of Opera-tions Research vol 1 no 2 pp 117ndash129 1976

[2] G Lebbar A El Barkany A Jabri and I El Abbassi ldquoHybridmetaheuristics for solving the blocking flowshop schedulingproblemrdquo International Journal of Engineering Research inAfrica vol 36 pp 124ndash136 2018

[3] Z I M Hassani A El Barkany A Jabri I El Abbassi andM Darcherif ldquoNew approach to integrate planning and

Dynamic Pareto front in TA_FSRP_20 times 5 times 3

1200 1400 1600 1800 2000Makespan

2200 2400 2600

2000

015

000

3000

035

000

4000

045

000

5000

0

2500

0

Total weig

hted tar

diness

1000

050

00

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 11 RIPG Pareto front in TA_20_5_3

14 Complexity

5 Conclusions and Future Lines of Research

is research proposes several relevant contributions Firsta newMILPmathematical formulation has been proposed torepresent the dynamic triobjective rescheduling problem tobe solved Likewise a set of problems (benchmark instances)has been defined to model dynamic rescheduling environ-ments including the specification of different disruptionswhich can arise in a production environment Additionallya rescheduling architecture has been designed and imple-mented based on a predictive-reactive strategy with periodicrescheduling Moreover the RIPG metaheuristic has beenapplied and its proper operation in dynamic reschedulingsystems verified

To validate the proposed approach and having found noreferences in the literature a benchmark based on a staticbiobjective environment was used It is important to note thatRIPG algorithm approaches the best Pareto frontiers obtainedin the literature following the same trends without exces-sively deviating from the reference nondominated solutionsset in a static biobjective environment New nondominatedsolutions have even been detected in some cases as can beseen in instances ta018 and ta024 Likewise RIPG is veryrobust because any parametrization of the algorithm yieldssimilar results in terms of the median value As it has beenimpossible to compare the RIPG algorithm with othermetaheuristics in the proposed environment due to the lackof techniques applied in such problems Pareto frontiersobtained for several instances of problems of different sizeshave been shown e number of nondominated solutionsfound goes down as the problemrsquos size increases

For future lines of research it may be useful to delvedeeper into analysing the influence of the base schedule

(baseline) used to evaluate the stability objective function onthe algorithmrsquos results In addition although this paper hasanalysed all the experiments using 5 rescheduling points thedynamic system could be evaluated for a variable number ofrescheduling points between 0 and 1000 as Sabuncuogluand Karabuk [76] considered in their work Likewise theRIPG metaheuristic could be adjusted to dynamic situationsin problems known asmany-objective optimization in whichthe number of objective functions is greater than threeLastly another existing metaheuristic could be adapted toresolve this problem so that the results may be comparedwith those obtained through RIPG

Data Availability

Instances and Solutions Dataset referred in this paper can beaccessed at httpsdoiorg10176324p4jcwdwpt2 (Men-deley Website)

Conflicts of Interest

e authors declare that they have no conflicts of interest

References

[1] M R Garey D S Johnson and R Sethi ldquoe complexity offlowshop and Jobshop schedulingrdquo Mathematics of Opera-tions Research vol 1 no 2 pp 117ndash129 1976

[2] G Lebbar A El Barkany A Jabri and I El Abbassi ldquoHybridmetaheuristics for solving the blocking flowshop schedulingproblemrdquo International Journal of Engineering Research inAfrica vol 36 pp 124ndash136 2018

[3] Z I M Hassani A El Barkany A Jabri I El Abbassi andM Darcherif ldquoNew approach to integrate planning and

Dynamic Pareto front in TA_FSRP_20 times 5 times 3

1200 1400 1600 1800 2000Makespan

2200 2400 2600

2000

015

000

3000

035

000

4000

045

000

5000

0

2500

0

Total weig

hted tar

diness

1000

050

00

Stab

ility

minus100

0

100

200

300

400

500

Rescheduling point 1Rescheduling point 2Rescheduling point 3

Rescheduling point 4Rescheduling point 5

Figure 11 RIPG Pareto front in TA_20_5_3

14 Complexity

scheduling of production system heuristic resolutionrdquo In-ternational Journal of Engineering Research in Africa vol 39pp 156ndash169 2018

[4] Z I M Hassani A El Barkany I El Abbassi A Jabri andM Darcherif ldquoNew model of planning and scheduling forjob-shop production system with energy considerationrdquoManagement and Production Engineering Review vol 10no 1 pp 89ndash97 2019

[5] E Zitzler and L iele ldquoMultiobjective evolutionary algo-rithms a comparative case study and the strength Paretoapproachrdquo IEEE Transactions on Evolutionary Computationvol 3 no 4 pp 257ndash271 1999

[6] J Knowles E iele and E Zitzler ldquoA tutorial on the per-formance assessment of stochastic multiobjective optimizersrdquoComputer Engineering and Networks Laboratory (TIK) ETHZurich Zurich Switzerland 2006

[7] E Zitzler L iele M Laumanns C M Fonseca andV G Fonseca da ldquoPerformance assessment of multiobjectiveoptimizers an analysis and reviewrdquo IEEE Transactions onEvolutionary Computation vol 7 no 2 pp 117ndash132 2003

[8] E Zitzler D Brockhoff and L iele ldquoe hypervolumeindicator revisited on the design of Pareto-compliant indi-cators via weighted integrationrdquo in Evolutionary Multi-Cri-terion Optimization S Obayashi K Deb C PoloniT Hiroyasu and T Murata Eds Springer Berlin Heidel-berg Germany pp 862ndash876 2007

[9] G Lebbar I El Abbassi A Jabri A El Barkany andM Darcherif ldquoMulti-criteria blocking flowshop schedulingproblems formulation and performance analysisrdquo Advancesin Production Engineering amp Management vol 13 no 2pp 136ndash146 2018

[10] GMinella R Ruiz andM Ciavotta ldquoA review and evaluationof multiobjective algorithms for the flowshop schedulingproblemrdquo INFORMS Journal on Computing vol 20 no 3pp 451ndash471 2008

[11] H Amirian and R Sahraeian ldquoMulti-objective differentialevolution for the flow shop scheduling problem with amodified learning effectrdquo International Journal of Engineering- Transactions C Aspects vol 27 no 9 p 1395 2014

[12] K Deb A Pratap S Agarwal and T Meyarivan ldquoA fast andElitist multiobjective genetic algorithm NSGA-IIrdquo IEEETransactions on Evolutionary Computation vol 6 no 2pp 182ndash197 2002

[13] H Ishibuchi and T Murata ldquoMulti-objective genetic localsearch algorithmrdquo in Proceedings of IEEE InternationalConference on Evolutionary Computation pp 119ndash124Nagoya Japan May 1996

[14] N Karimi and H Davoudpour ldquoA high performing meta-heuristic for multi-objective flowshop scheduling problemrdquoComputers amp Operations Research vol 52 pp 149ndash156 2014

[15] G Lebbar I El Abbassi A El Barkany A Jabri andM Darcherif ldquoSolving the multi objective flow shop sched-uling problems using an improved NSGA-IIrdquo InternationalJournal of Operations and Quantitative Management vol 24no 3 pp 211ndash230 2018a

[16] B-B Li and L Wang ldquoA hybrid quantum-inspired geneticalgorithm for multiobjective flow shop schedulingrdquo IEEETransactions on Systems Man and Cybernetics Part B (Cy-bernetics) vol 37 no 3 pp 576ndash591 2007

[17] T Pasupathy C Rajendran and R K Suresh ldquoA multi-objective genetic algorithm for scheduling in flow shops tominimize the makespan and total flow time of jobsrdquo Inter-national Journal of Advanced Manufacturing Technologyvol 27 pp 804ndash815 2006

[18] D Y Sha and H-H Lin ldquoA particle swarm optimization formulti-objective flowshop schedulingrdquo International Journal ofAdvanced Manufacturing Technology vol 45 no 7-8pp 749ndash758 2009

[19] M A Allouche ldquoManagerrsquos preferences modeling withinmulti-criteria flowshop scheduling problem a metaheuristicapproachrdquo International Journal of Business Research andManagement vol 1 no 2 pp 33ndash45 2010

[20] A Jabri A El Barkany and A El khalfi ldquoMulti-pass turningoperation process optimization using Hybrid Genetic-Sim-ulated Annealing algorithmrdquo Modelling and Simulation inEngineering vol 2017 Article ID 1940635 10 pages 2017

[21] S-W Lin and K-C Ying ldquoMinimizing makespan and totalflowtime in permutation flowshops by a bi-objective multi-start simulated-annealing algorithmrdquo Computers amp Opera-tions Research vol 40 no 6 pp 1625ndash1647 2013

[22] T K Varadharajan and C Rajendran ldquoA multi-objectivesimulated-annealing algorithm for scheduling in flowshops tominimize the makespan and total flowtime of jobsrdquo EuropeanJournal of Operational Research vol 167 no 3 pp 772ndash7952005

[23] M Gravel W L Price and C Gagne ldquoScheduling continuouscasting of aluminum using a multiple objective ant colonyoptimization metaheuristicrdquo European Journal of OperationalResearch vol 143 no 1 pp 218ndash229 2002

[24] A Rabanimotlagh ldquoAn efficient ant colony optimizationalgorithm for multiobjective flow shop scheduling problemrdquoInternational Journal of Mechanical Aerospace IndustrialMechatronic and Manufacturing Engineering vol 5 no 3pp 598ndash604 2011

[25] M Ciavotta G Minella and R Ruiz ldquoMulti-objective se-quence dependent setup times permutation flowshop a newalgorithm and a comprehensive studyrdquo European Journal ofOperational Research vol 227 no 2 pp 301ndash313 2013

[26] J Dubois-Lacoste M Lopez-Ibantildeez and T Stutzle ldquoA hybridTP + PLS algorithm for bi-objective flow-shop schedulingproblemsrdquo Computers amp Operations Research vol 38 no 8pp 1219ndash1236 2011

[27] K Deb and H Jain ldquoAn evolutionary many-objective opti-mization algorithm using reference-point-based non-dominated sorting approach Part I solving problems withbox constraintsrdquo IEEE Transactions on Evolutionary Com-putation vol 18 no 4 pp 577ndash601 2014

[28] Y YuanH Xu andBWang ldquoAn improvedNSGA-III procedurefor evolutionary many-objective optimizationrdquo in Proceedings ofthe 2014 Annual Conference on Genetic and Evolutionary Com-putation pp 661ndash668 New York NY USA 2014

[29] J Kuster D Jannach and G Friedrich ldquoApplying localrescheduling in response to schedule disruptionsrdquo Annals ofOperations Research vol 180 no 1 pp 265ndash282 2008

[30] K Lee F Zheng and M L Pinedo ldquoOnline scheduling ofordered flow shopsrdquo European Journal of Operational Re-search vol 272 no 1 pp 50ndash60 2019

[31] W Liu Y Jin and M Price ldquoNew scheduling algorithms anddigital tool for dynamic permutation flowshop with newlyarrived orderrdquo International Journal of Production Researchvol 55 no 11 pp 3234ndash3248 2017

[32] Z Zakaria and S Petrovic ldquoGenetic algorithms for match-uprescheduling of the flexible manufacturing systemsrdquo Com-puters amp Industrial Engineering vol 62 no 2 pp 670ndash6862012

[33] A Pfeiffer B Kadar and L Monostori ldquoStability-orientedevaluation of rescheduling strategies by using simulationrdquoComputers in Industry vol 58 no 7 pp 630ndash643 2007

Complexity 15

[34] R J Abumaizar and J A Svestka ldquoRescheduling job shopsunder random disruptionsrdquo International Journal of Pro-duction Research vol 35 no 7 pp 2065ndash2082 1997

[35] S F Smith ldquoReactive scheduling systemsrdquo in IntelligentScheduling Systems D E Brown and W T Scherer EdsSpringer Boston MA USA 1995

[36] L K Church and R Uzsoy ldquoAnalysis of periodic and event-driven rescheduling policies in dynamic shopsrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 3pp 153ndash163 1992

[37] G E Vieira J W Herrmann and E Lin ldquoAnalytical modelsto predict the performance of a single-machine system underperiodic and event-driven rescheduling strategiesrdquo Interna-tional Journal of Production Research vol 38 no 8pp 1899ndash1915 2000

[38] A B Itayef T Loukil and J Teghem ldquoRescheduling a per-mutation flowshop problems under the arrival a new set ofjobsrdquo in Proceedings of the International Conference onComputers Industrial Engineering pp 188ndash192 TroyesFrance July 2009

[39] A Liefooghe M Basseur J Humeau L Jourdan andE-G Talbi ldquoOn optimizing a bi-objective flowshop sched-uling problem in an uncertain environmentrdquo Computers ampMathematics with Applications vol 64 no 12 pp 3747ndash37622012

[40] W Liu Y Jin andM Price ldquoNewmeta-heuristic for dynamicscheduling in permutation flowshop with new order arrivalrdquoe International Journal of Advanced Manufacturing Tech-nology vol 98 pp 1817ndash1830 2018

[41] J Xiong L Xing and Y Chen ldquoRobust scheduling for multi-objective flexible job-shop problems with random machinebreakdownsrdquo International Journal of Production Economicsvol 141 no 1 pp 112ndash126 2013

[42] H Iima ldquoGenetic algorithm approach to multiobjectiverescheduling on parallel machinesrdquo IFAC Proceedings Vol-umes vol 38 no 1 pp 139ndash144 2005

[43] B Zhang Q Pan L Gao and Y Zhao ldquoMOEAD for multi-objective hybrid flowshop rescheduling problemrdquo in Pro-ceedings of the ASME 2018 13th International ManufacturingScience and Engineering Conference College Station TXUSA June 2018

[44] X He S Dong and N Zhao ldquoResearch on rush order in-sertion rescheduling problem under hybrid flow shop basedon NSGA-IIIrdquo International Journal of Production Researchvol 58 no 4 pp 1161ndash1177 2019

[45] G E Vieira J W Herrmann and E Lin ldquoReschedulingmanufacturing systems a framework of strategies policiesand methodsrdquo Journal of Scheduling vol 6 no 1 pp 39ndash622003

[46] D Ouelhadj and S Petrovic ldquoA survey of dynamic schedulingin manufacturing systemsrdquo Journal of Scheduling vol 12no 4 pp 417ndash431 2009

[47] J Razmi R T Moghaddam and M Saffari ldquoA mathematicalmodel for a flow shop scheduling problem with fuzzy pro-cessing timesrdquo Journal of Industrial Engineering vol 3pp 39ndash44 2009

[48] P Fattahi and A Fallahi ldquoDynamic scheduling in flexible jobshop systems by considering simultaneously efficiency andstabilityrdquo CIRP Journal of Manufacturing Science and Tech-nology vol 2 no 2 pp 114ndash123 2010

[49] R Ramezanian M B Aryanezhad and M Heydari ldquoAmathematical programming model for flow shop schedulingproblems for considering Just in time productionrdquo Interna-tional Journal of Industrial Engineering and Production

Research vol 21 no 2 2010 httpswwwresearchgatenetpublication49591605_A_Mathematical_Programming_Model_for_Flow_Shop_Scheduling_Problems_for_Considering_Just_in_Time_Production

[50] KornbladMathematical Optimization in Flexible Job ShopScheduling Modelling Analysis and Case Studies ChalmersUniversity of Technology and University of GothenburgGoteborg Sweden 2013

[51] M M Yenisey and B Yagmahan ldquoMulti-objective permu-tation flow shop scheduling problem literature reviewclassification and current trendsrdquo Omega vol 45 pp 119ndash135 2014

[52] J Li Q Pan and K Mao ldquoA discrete teaching-learning-basedoptimisation algorithm for realistic flowshop reschedulingproblemsrdquo Engineering Applications of Artificial Intelligencevol 37 pp 279ndash292 2015

[53] F Yuan and M Yin ldquoA novel fuzzy model for multi-objectivepermutation flow shop scheduling problem with fuzzy pro-cessing timerdquo Advances in Mechanical Engineering vol 11no 4 pp 1ndash9 2019

[54] J Bukchin and M Masin ldquoMulti-objective lot splitting for asingle product m-machine flowshop linerdquo IIE Transactionsvol 36 no 2 pp 191ndash202 2004

[55] P M Chaudhari D R V Dharaskar and D V M akareldquoComputing the most significant solution from Pareto frontobtained in multi-objective evolutionaryrdquo InternationalJournal of Advanced Computer Science and Applicationsvol 1 no 4 pp 63ndash68 2010

[56] H A Taboada and D W Coit ldquoData clustering of solutionsfor multiple objective system reliability optimization prob-lemsrdquoQuality Technology ampQuantitative Management vol 4no 2 pp 191ndash210 2007

[57] P Valledor A Gomez P Priore and J Puente ldquoSolvingmulti-objective rescheduling problems in dynamic permu-tation flow shop environments with disruptionsrdquo Interna-tional Journal of Production Research vol 56 no 19pp 6363ndash6377 2018

[58] R Ruiz and T Stutzle ldquoA simple and effective iterated greedyalgorithm for the permutation flowshop scheduling problemrdquoEuropean Journal of Operational Research vol 177 no 3pp 2033ndash2049 2007

[59] M Nawaz E E Enscore and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983

[60] C Rajendran ldquoHeuristics for scheduling in flowshop withmultiple objectivesrdquo European Journal of Operational Re-search vol 82 no 3 pp 540ndash555 1995

[61] E Taillard ldquoBenchmarks for basic scheduling problemsrdquoEuropean Journal of Operational Research vol 64 no 2pp 278ndash285 1993

[62] G Minella R Ruiz and M Ciavotta ldquoRestarted iteratedPareto Greedy algorithm for multi-objective flowshopscheduling problemsrdquo Computers amp Operations Researchvol 38 no 11 pp 1521ndash1533 2011

[63] S Hasija and C Rajendran ldquoScheduling in flowshops tominimize total tardiness of jobsrdquo International Journal ofProduction Research vol 42 no 11 pp 2289ndash2301 2004

[64] P Perez-Gonzalez and J M Framinan ldquoScheduling permu-tation flowshops with initial availability constraint analysis ofsolutions and constructive heuristicsrdquo Computers amp Opera-tions Research vol 36 no 10 pp 2866ndash2876 2009

[65] E Taillard ldquoSome efficient heuristic methods for the flow shopsequencing problemrdquo European Journal of Operational Re-search vol 47 no 1 pp 65ndash74 1990

16 Complexity

[34] R J Abumaizar and J A Svestka ldquoRescheduling job shopsunder random disruptionsrdquo International Journal of Pro-duction Research vol 35 no 7 pp 2065ndash2082 1997

[35] S F Smith ldquoReactive scheduling systemsrdquo in IntelligentScheduling Systems D E Brown and W T Scherer EdsSpringer Boston MA USA 1995

[36] L K Church and R Uzsoy ldquoAnalysis of periodic and event-driven rescheduling policies in dynamic shopsrdquo InternationalJournal of Computer Integrated Manufacturing vol 5 no 3pp 153ndash163 1992

[37] G E Vieira J W Herrmann and E Lin ldquoAnalytical modelsto predict the performance of a single-machine system underperiodic and event-driven rescheduling strategiesrdquo Interna-tional Journal of Production Research vol 38 no 8pp 1899ndash1915 2000

[38] A B Itayef T Loukil and J Teghem ldquoRescheduling a per-mutation flowshop problems under the arrival a new set ofjobsrdquo in Proceedings of the International Conference onComputers Industrial Engineering pp 188ndash192 TroyesFrance July 2009

[39] A Liefooghe M Basseur J Humeau L Jourdan andE-G Talbi ldquoOn optimizing a bi-objective flowshop sched-uling problem in an uncertain environmentrdquo Computers ampMathematics with Applications vol 64 no 12 pp 3747ndash37622012

[40] W Liu Y Jin andM Price ldquoNewmeta-heuristic for dynamicscheduling in permutation flowshop with new order arrivalrdquoe International Journal of Advanced Manufacturing Tech-nology vol 98 pp 1817ndash1830 2018

[41] J Xiong L Xing and Y Chen ldquoRobust scheduling for multi-objective flexible job-shop problems with random machinebreakdownsrdquo International Journal of Production Economicsvol 141 no 1 pp 112ndash126 2013

[42] H Iima ldquoGenetic algorithm approach to multiobjectiverescheduling on parallel machinesrdquo IFAC Proceedings Vol-umes vol 38 no 1 pp 139ndash144 2005

[43] B Zhang Q Pan L Gao and Y Zhao ldquoMOEAD for multi-objective hybrid flowshop rescheduling problemrdquo in Pro-ceedings of the ASME 2018 13th International ManufacturingScience and Engineering Conference College Station TXUSA June 2018

[44] X He S Dong and N Zhao ldquoResearch on rush order in-sertion rescheduling problem under hybrid flow shop basedon NSGA-IIIrdquo International Journal of Production Researchvol 58 no 4 pp 1161ndash1177 2019

[45] G E Vieira J W Herrmann and E Lin ldquoReschedulingmanufacturing systems a framework of strategies policiesand methodsrdquo Journal of Scheduling vol 6 no 1 pp 39ndash622003

[46] D Ouelhadj and S Petrovic ldquoA survey of dynamic schedulingin manufacturing systemsrdquo Journal of Scheduling vol 12no 4 pp 417ndash431 2009

[47] J Razmi R T Moghaddam and M Saffari ldquoA mathematicalmodel for a flow shop scheduling problem with fuzzy pro-cessing timesrdquo Journal of Industrial Engineering vol 3pp 39ndash44 2009

[48] P Fattahi and A Fallahi ldquoDynamic scheduling in flexible jobshop systems by considering simultaneously efficiency andstabilityrdquo CIRP Journal of Manufacturing Science and Tech-nology vol 2 no 2 pp 114ndash123 2010

[49] R Ramezanian M B Aryanezhad and M Heydari ldquoAmathematical programming model for flow shop schedulingproblems for considering Just in time productionrdquo Interna-tional Journal of Industrial Engineering and Production

Research vol 21 no 2 2010 httpswwwresearchgatenetpublication49591605_A_Mathematical_Programming_Model_for_Flow_Shop_Scheduling_Problems_for_Considering_Just_in_Time_Production

[50] KornbladMathematical Optimization in Flexible Job ShopScheduling Modelling Analysis and Case Studies ChalmersUniversity of Technology and University of GothenburgGoteborg Sweden 2013

[51] M M Yenisey and B Yagmahan ldquoMulti-objective permu-tation flow shop scheduling problem literature reviewclassification and current trendsrdquo Omega vol 45 pp 119ndash135 2014

[52] J Li Q Pan and K Mao ldquoA discrete teaching-learning-basedoptimisation algorithm for realistic flowshop reschedulingproblemsrdquo Engineering Applications of Artificial Intelligencevol 37 pp 279ndash292 2015

[53] F Yuan and M Yin ldquoA novel fuzzy model for multi-objectivepermutation flow shop scheduling problem with fuzzy pro-cessing timerdquo Advances in Mechanical Engineering vol 11no 4 pp 1ndash9 2019

[54] J Bukchin and M Masin ldquoMulti-objective lot splitting for asingle product m-machine flowshop linerdquo IIE Transactionsvol 36 no 2 pp 191ndash202 2004

[55] P M Chaudhari D R V Dharaskar and D V M akareldquoComputing the most significant solution from Pareto frontobtained in multi-objective evolutionaryrdquo InternationalJournal of Advanced Computer Science and Applicationsvol 1 no 4 pp 63ndash68 2010

[56] H A Taboada and D W Coit ldquoData clustering of solutionsfor multiple objective system reliability optimization prob-lemsrdquoQuality Technology ampQuantitative Management vol 4no 2 pp 191ndash210 2007

[57] P Valledor A Gomez P Priore and J Puente ldquoSolvingmulti-objective rescheduling problems in dynamic permu-tation flow shop environments with disruptionsrdquo Interna-tional Journal of Production Research vol 56 no 19pp 6363ndash6377 2018

[58] R Ruiz and T Stutzle ldquoA simple and effective iterated greedyalgorithm for the permutation flowshop scheduling problemrdquoEuropean Journal of Operational Research vol 177 no 3pp 2033ndash2049 2007

[59] M Nawaz E E Enscore and I Ham ldquoA heuristic algorithmfor the m-machine n-job flow-shop sequencing problemrdquoOmega vol 11 no 1 pp 91ndash95 1983

[60] C Rajendran ldquoHeuristics for scheduling in flowshop withmultiple objectivesrdquo European Journal of Operational Re-search vol 82 no 3 pp 540ndash555 1995

[61] E Taillard ldquoBenchmarks for basic scheduling problemsrdquoEuropean Journal of Operational Research vol 64 no 2pp 278ndash285 1993

[62] G Minella R Ruiz and M Ciavotta ldquoRestarted iteratedPareto Greedy algorithm for multi-objective flowshopscheduling problemsrdquo Computers amp Operations Researchvol 38 no 11 pp 1521ndash1533 2011

[63] S Hasija and C Rajendran ldquoScheduling in flowshops tominimize total tardiness of jobsrdquo International Journal ofProduction Research vol 42 no 11 pp 2289ndash2301 2004

[64] P Perez-Gonzalez and J M Framinan ldquoScheduling permu-tation flowshops with initial availability constraint analysis ofsolutions and constructive heuristicsrdquo Computers amp Opera-tions Research vol 36 no 10 pp 2866ndash2876 2009

[65] E Taillard ldquoSome efficient heuristic methods for the flow shopsequencing problemrdquo European Journal of Operational Re-search vol 47 no 1 pp 65ndash74 1990

16 Complexity

[66] M A Adibi M Zandieh and M Amiri ldquoMulti-objectivescheduling of dynamic job shop using variable neighborhoodsearchrdquo Expert Systems with Applications vol 37 no 1pp 282ndash287 2010

[67] S J Mason S Jin and C MWessels ldquoRescheduling strategiesfor minimizing total weighted tardiness in complex jobshopsrdquo International Journal of Production Research vol 42no 3 pp 613ndash628 2004

[68] L Tang W Liu and J Liu ldquoA neural network model andalgorithm for the hybrid flow shop scheduling problem in adynamic environmentrdquo Journal of Intelligent Manufacturingvol 16 no 3 pp 361ndash370 2005

[69] R Swaminathan M E Pfund J W Fowler S J Mason andA Keha ldquoImpact of permutation enforcement when mini-mizing total weighted tardiness in dynamic flowshops withuncertain processing timesrdquo Computers amp Operations Re-search vol 34 no 10 pp 3055ndash3068 2007

[70] A Abdelhadi and L H Mouss ldquoAn efficient hybrid approachbased on multi agent system and emergence method for theintegration of systematic preventive maintenance policies inhybrid flow-shop scheduling to minimize makespanrdquo Journalof Mechanical Engineering Research vol 5 no 6 pp 112ndash1222013

[71] A Costa F A Cappadonna and S Fichera ldquoA hybridmetaheuristic approach for minimizing the total flow time inA flow shop sequence dependent group scheduling problemrdquoAlgorithms vol 7 no 3 pp 376ndash396 2014

[72] S Hatami R Ruiz and C A Romano ldquoTwo simple con-structive algorithms for the distributed assembly permutationflowshop scheduling problemrdquo in Managing ComplexityC Hernandez A Lopez-Paredes and J M Perez-Rıos EdsSpringer Cham Switzerland 2014

[73] D L de Souza F S Lobato and R Gedraite ldquoA comparativestudy using bio-inspired optimization methods applied tocontrollers tuningrdquo Frontiers in Advanced Control Systemvol 7 2012 httpwwwintechopencombooksfrontiers-in-advanced-control-systemsa-comparative-study-using-bio-inspired-optimization-methods-applied-to-controllers-tuning

[74] E Vallada and R Ruiz ldquoGenetic algorithms with pathrelinking for the minimum tardiness permutation flowshopproblemrdquo Omega vol 38 no 1-2 pp 57ndash67 2010

[75] M Lopez-Ibantildeez J Dubois-Lacoste L Perez CaceresM Birattari and T Stutzle ldquoe irace package iterated racingfor automatic algorithm configurationrdquo Operations ResearchPerspectives vol 3 pp 43ndash58 2016

[76] I Sabuncuoglu and S Karabuk ldquoRescheduling frequency inan FMS with uncertain processing times and unreliablemachinesrdquo Journal of Manufacturing Systems vol 18 no 4pp 268ndash283 1999

[77] P Czyzzak and A Jaszkiewicz ldquoPareto simulatedannealingmdasha metaheuristic technique for multiple-objectivecombinatorial optimizationrdquo Journal of Multi-Criteria Deci-sion Analysis vol 7 no 1 pp 34ndash47 1998

[78] H Ishibuchi T Yoshida and T Murata ldquoBalance betweengenetic search and local search in memetic algorithms formultiobjective permutation flowshop schedulingrdquo IEEETransactions on Evolutionary Computation vol 7 no 2pp 204ndash223 2003

Complexity 17