Developing simulation based optimization mechanism for a ...scientiairanica.sharif.edu/article_4461_3559eaf8ac5d8abc19631501ff5… · time. In fact, it implements condition based

Scientia Iranica E (2018) 25(5), 2788{2806

Sharif University of TechnologyScientia Iranica

Transactions E: Industrial Engineeringhttp://scientiairanica.sharif.edu

Developing simulation based optimization mechanismfor a novel stochastic reliability centered maintenanceproblem

S.H.A. Rahmati, A. Ahmadi�, and B. Karimi

Department of Industrial Engineering and Management Systems, Amirkabir University of Technology, Tehran, Iran.

Received 11 June 2016; received in revised form 4 April 2017; accepted 14 August 2017

KEYWORDSReliability-CenteredMaintenance (RCM);Stochastic productionmodel;Condition-BasedMaintenance (CBM);Shocking mechanism;Biogeography BasedOptimization (BBO).

Abstract. This research investigates joint scheduling of maintenance and productionplanning. This novel integrated problem takes bene�t of Reliability-Centered Maintenance(RCM) for monitoring and managing maintenance function of a stochastic complexproduction-planning problem, namely, Flexible Job Shop scheduling Problem (FJSP). Thedeveloped RCM works based on stochastic shocking of machines during their processtime. In fact, it implements condition based maintenance approach regulated accordingto stochastic reliability concept. Comparison of the system reliability with critical levelsdetermines the failure status of the machines. It activates two main types of reaction calledpreventive and corrective maintenance. Considering breakdown of the system betweeninspection intervals makes the proposed model more realistic. Moreover, maintenanceactivity times and their duration are considered stochastically. Because of the highcomplexity level of this joint system, Simulation-Based Optimization (SBO) approach isproposed for solving the problem. This SBO searches the feasible area through GeneticAlgorithm (GA) and Biogeography Based Optimization (BBO) algorithm. Di�erent testproblems, statistical methods, and novel visualizations are used to discuss the problem andthe algorithm, explicitly.

© 2018 Sharif University of Technology. All rights reserved.

1. Introduction

Production plans and the maintenance activities arejoint concepts in real world. However, most of theproduction and scheduling problems assume all timesmachine availability [1]. In contrast to this assumption,real world problems face many situations in whichmachines break down or need maintenance. [2]. More-over, ine�cient maintenance can cause one third ofmaintenance costs being wasted due to unnecessary

*. Corresponding author. Tel.: +98 21 64545394E-mail address: [email protected] (A. Ahmadi)

doi: 10.24200/sci.2017.4461

or improper maintenance activities [3]. Nonetheless,maintenance issues are not a considerable portion of theliterature on production and manufacturing problems.

On the other hand, maintenance and reliabilityhave a signi�cant share of the literature on modelingand optimization [4-9]. Thus, taking bene�t of this op-portunity to realize and reinforce production-planningproblems is of interest. One of these opportunitiesis a method called Reliability Centered Maintenance(RCM). Actually, the main goal of this paper isconsistent introducing of RCM to production problembecause of its importance in real environment. RCMhas various industrial applications in the maintenanceand reliability literature, including power distributionsystems, subsea pulpiness, steel plants, chemical indus-

S.H.A. Rahmati et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2788{2806 2789

try, transportation, water distribution, and concretebridge decks inspection [10-15].

RCM functionally controls the systems to reacha desired level by monitoring their reliability [16].Moreover, it prioritizes maintenance activities by rank-ing the failures according to their e�ects on systemreliability. In fact, RCM continuously monitors thereliability of the system and determines type of the re-quired maintenance activities according to the levels ofreliability [16]. Condition-Based Maintenance (CBM),in many cases, conducts the task of monitoring. CBMdevelopment owes to the recent emerging technologiessuch as radio frequency identi�cation (RFID), Micro-Electro-Mechanical System (MEMS), wireless tele-communication, and Product Embedded InformationDevices (PEID) [17]. In the next subsection, the litera-ture on joint scheduling of maintenance and productionplanning is reviewed.

1.1. Integration of maintenance and generalproduction problems

Graves and Lee [18] developed a single-machinescheduling problem. They assumed certain intervalsfor maintenance activities. Lee [19] studied the two-machine ow shop scheduling problem under availabil-ity constraint and developed dynamic programmingalgorithm and heuristic solutions. Lee and Chen [20]considered a scheduling model for parallel machines inwhich jobs could be maintained only once during theplanning horizon. They also assumed two strategies:machines could be maintained simultaneously or sepa-rately. Schmidt [21] reviewed deterministic schedulingproblems with availability constraints. Espinouse etal. [22] and Cheng and Liu [23] investigated a two-machine ow-shop problem in a no-wait environmentwith availability constraint. Liao and Chen [24] con-sidered several maintenance periods in their single-machine scheduling problem, which minimized themaximum tardiness of jobs.

Aggoune [25] in a ow shop problem consideredtwo variants of the non-preemptive jobs. Allaoui andArtiba [26] integrated hybrid ow shop schedulingproblem and maintenance constraints minimizing the ow time. Cassady and Kutanoglu [27] proposed amixed model of single-machine model with periodicor preventive maintenance, which was followed bySortrakul t al. [28]. Liao et al. [29] developed atwo-parallel-machines problem considering preventivemaintenance. Mauguiere et al. [30] studied unavail-ability in job-shop scheduling problem and single-machine model. Allaoui and Artiba [31] investigatedone-machine ow shop with availability constraints.Lin and Liao [32] studied hybrid parallel machineproblem and maintenance a�airs. Ruiz et al. [33]studied a permutation ow shop problem with pre-ventive maintenance. Chen [34] implemented exible

and periodic maintenance in his models. Liao andSheen [35] considered parallel machine scheduling withavailability and eligibility constraints, simultaneously.Berrichi et al. [36] studied parallel machines focusingon makespan and unavailability, simultaneously. Zribiet al. [37] integrated job-shop scheduling problem withavailability constraints.

Naderi et al. [38] scheduled a sequence-dependentsetup time job-shop with preventive maintenance. Mel-louli et al. [39] developed an integrated parallel ma-chine scheduling problem with preventive maintenance.Chen [40] studied a single machine with several mainte-nance periods and minimized the maximum tardinessof jobs. Mati [41] focused on the integration of job-shop scheduling problem and availability constraints.Pan et al. [42] considered variable maintenance timesubjected to machine degradation to make their singlemachine compatible with preventive maintenance. Lowet al. [43] considered single machine with periodicmaintenance. Safari et al. [44] developed CBM for ow shop scheduling problem. They did not developmathematical model and only simulated the concept.Moreover, their simulation did not assume the pos-sibility of breakdown between inspection times. BenAli et al. [45] proposed a multi-objective job shopproblem that optimized maintenance cost in additionto makespan. Ramezanian and Saidi-Mehrabad [46]developed parallel machine with rework process. Zhouet al. [47] proposed a multi-component system underchanging job shop with preventive maintenance con-sideration. Ozkok [48] investigated hull structure pro-duction process in a �xed-position shipyard companywith machine breakdown consideration.

Chouikhi et al. [49] integrated a single-unit systemwith CBM and optimized the cost of maintenanceand inspection time by determining the optimal in-spection. They assumed that both corrective andpreventive maintenance actions were perfect, whichmeans after such actions, the system became as goodas the new one. Besides, they assumed that durationsof inspection, corrective maintenance, and preventivemaintenance could be negligible. Kim and Ozturkoglu[50] developed a joint scheduling of single machineproblem with multiple preventive maintenances. Theyproposed ant colony optimization and particle swarmoptimization in order to solve this problem. Yinget al. [51] introduced di�erent SMPSs consideringmaintenance activity between two sequential jobs.Lin et al. [52] evaluated reliability of a multistateFLexible FSSP with stochastic capacity. Huang andYu [53] developed a two-stage multiprocessor FSSPwith maintenance and clean production aims. Cui andLu [54] investigated exible maintenance in SMPS andsolved their problem through the Earliest Release Date-Longest Processing Time (ERD-LPT), and Branch andBound (B&B) algorithm.

2790 S.H.A. Rahmati et al./Scientia Iranica, Transactions E: Industrial Engineering 25 (2018) 2788{2806

1.2. Integration of maintenance and FlexibleJob Shop scheduling Problem (FJSP)

Flexible job shop scheduling problem is a popularand complex exible manufacturing problem [55,56].In classical FJSP, most researches assume that allmachines are available during their working process.Both areas of the optimization problems, i.e., modeldevelopment [57-61] and solving method extension [62-76], can be found in the classical literature on FJSP.Demir and Isleyen [77] performed a comprehensive eval-uation of the various mathematical models presentedfor the FJSP.

Zribi and Borne [78] assumed unavailability of ma-chines due to preventive maintenance. Gao et al. [79]proposed preventive maintenance for FJSP in whichthe period of maintenance tasks was non-�xed andshould be determined during the scheduling procedure.Wang and Yu [80] developed FJSP considering mainte-nance activities either exible in a time window or �xedbeforehand. Moradi et al. [81] integrated FJSP andpreventive maintenance by optimizing unavailabilityand makespan. Mokhtari and Dadgar [82] introduceda joint FJSP and PM model that assumed the failurerates were time varying. In their model, the durationof PM activities was �xed. Ahmadi et al. [83] studiedrandom machine breakdown in FJSP with simulationconsiderations. The related important studies aresummarized in Table 1.

1.3. Gap analysisAccording to the literature, a rare portion of theproduction studies is devoted to FJSP, CBM, andRCM. Therefore, this research reinforces FJSP prob-lem through RCM concept. Real world assumptions,rarely considered in the literature, are assumed in thedeveloped RCM. For instance, breakdown possibility isassumed between inspection intervals. Also, this studyconsiders maintenance occurrence and duration time

stochastically. In addition, it stochastically assumes re-covery level of the system after preventive maintenance.Moreover, we use both types of maintenance strategies,called Corrective Maintenance (CM) and PreventiveMaintenance (PM). CBM is used to detect the levelof reliability [84].

The structure of the paper is as follows. Section2 presents the related literature review of the problem.Section 3 discusses the elements of the proposed jointproblem. The simulation-based approach related to theproposed RCM is developed in Section 4. Section 5presents the proposed problem and its solving method-ology through numerical examples. Finally, Section 7concludes the paper.

2. Preliminaries of the developed jointproblem

The considered production problem is a stochasticversion of the simple FJSP. FJPS has two tasks,namely, allocating operations to machines and deter-mining the sequence of allocated operations to eachmachine [72,79]. Simple FJSP consists of n jobs, J(Ji; i 2 f1; 2; :::; ng); each job, i (J1; :::; Jn), includes nioperations, O(Oij ; j 2 f1; 2; :::; nig), that are processedon m machines, M(Mk; k 2 f1; 2; :::;mg). The FJSPobjective function of this paper is makespan (Cmax)given below:

Cmax = maxfCkjk = 1; :::; ng; (1)

where Ck denotes complementation time of machinek [74].

Figure 1 illustrates the FJSP example with 3 jobs,4 machines, and 9 operations. This �gure includes a ta-ble and a related Gant chart. In the table, the numberspresent the processing times of operations on machinesin addition to their sets of capable machines. The

Figure 1. The machine capability table and Gant chart of a related feasible solution.


Table 1. Literature review of the integration of scheduling and maintenance.

Ref. # Year Scheduling types Objectives Types of maintenance Solvingmethodologies

CM PM CBM RCM Meta-heuristics Exact SBOGrave andLee [18]

1999 Single machine CmaxLateness

* DP

Cassady andKutanoglu [27]

2005 Single machine TWCT * Heuristic TE

Sortrakulet al. [28]

2005 Single machine TWCT * GA

Mauguiereet al. [30]

2005 Single machine & job shop Cmax * B&B

Chen [34] 2008 Single machine Cmax * HeuristicChen [40] 2009 Single machine Cmax * Heuristic B&B

Pan et al. [42] 2010 Single machine MWT * HeuristicLow et al. [43] 2010 Single machine Cmax * Heuristic

Kim andOzturkoglu [50]

2013 Single machineCmaxTCT

TWCT* GA

Ying et al. [51] 2016 Single machine T, ML, TFT, MT * Heuristic

Cui & Lu [54] 2017 Single machine * B&B, ERD-LPT

Lin andLiao [32]

2007 Parallel machine Cmax * Heuristic

Liao andSheen [35]

2008 Parallel machine Cmax * BSA

Berrichiet al. [36]

2009 Parallel machine CmaxUnavailability

* NSGAII

Mellouliet al. [39]

2009 Parallel machine TCT * DP, B&B

Lee [19] 1999 Flow shop Cmax * Heuristic DP

Espinouseet al. [19]

2001 Flow shop Cmax * Heuristic

Cheng andLiu [20]

2003 Flow shop Cmax * Heuristic

Aggoune [25] 2004 Flow shop Cmax * TS GA

Allaoui andArtiba [26]

2004 Flow shop Flow time * Heuristic *

Ruiz et al. [33] 2007 Flow shop Cmax * Random, NEH,SA, GA, ACO

Safariet al. [44]

2010 Flow shop Cmax * * * SA-TS *

Naderi et al. [38] 2009 Flexible ow shop Cmax * AIS, GA

Huang &Yu [50]

2016 Flow shop Cmax * PSO, ACO


Table 1. Literature review of the integration of scheduling and maintenance (continued).

Ref. # Year Scheduling types Objectives Types of maintenance Solvingmethodologies

CM PM CBM RCM Meta-heuristics Exact SBOZribi et al. [34] 2008 Job shop Cmax * Heuristic GA *

Mati [38] 2010 Job shop Cmax * Heuristic

Ben Ali [45] 2011 Job shop CmaxCost

* MOEA

Zhou et al. [47] 2012 Job shop CmaxCost

* DP

Zribi andBorne [78]

2005 FJSP Cmax * Hybrid GA

Gao et al. [72] 2006 FJSPCmaxTWLCWL

* GA

Wang andYu [73]

2010 FJSP Cmax

Moradi et al. [74] 2015 FJSP CmaxUnavailability

* NSGAII

Mokhtari, andDadgar [82]

2015 FJSP Cmax * SA *

Ahmadi et al. [83] 2016 FJSP CmaxStability

* NSGAII NRGA

This Study FJSP Cmax * * * * BBO & GA *

symbol ìnf ' implies that the machine cannot operatethe corresponding operation. The Gant chart depictscombination of the sequence and the assignment for asample solution.

This research realizes the basic FJSP production-planning problem through considering the real stochas-tic nature of the maintenance function. The mainconcept of the proposed approaches is RCM. RCMdetermines and classi�es the failure modes and triesto keep the reliability of the system in a level that theoccurrence of these modes is prevented [16]. In fact, itmonitors the system status predictively to recognizethe mode and do the required quali�ed actions inconsequence [85-87].

The monitoring mechanism of the proposed RCMis based on the CBM approach. CBM determines themaintenance activities according to the actual condi-tion of the systems [85]. In addition, the developedRCM mimics the shocking process [86] that degradesthe considered reliability function of the machines,stochastically. In other words, CBM monitors thereliability degradation caused by stochastic shockingprocess. Simultaneously, it predicts and determinesthe appropriate maintenance actions according to thereliability status of the machines [16,85]. The failuresconsidered in the research are of both types of CM and

PM. Now, in case the reliability status falls beneaththe �rst critical threshold, L, CBM suggests to havePM, and if it gets inferior to failure rate LL, a failureor breakdown occurs [87].

Figure 2 illustrates reliability deteriorating andfailure modes, schematically. This �gure plots the man-ner of reliability from two aspects. In the upper part,it introduces the stochastic variables of the problem,while in the lower part, on a generally similar �gure, itfocuses on the maintenance activities according to thestate of reliability. The S values in the �gure denotethe shock times that reduce machine reliability withinsimulation process. This example encompasses sevenshocks, i.e., S1 to S7, presented on the horizontal axis.The M values, i.e., M1 and M2, denote the time of thejth maintenance activity on the machine.

After shocks S1 to S3, reliability of the machineis still higher than L. Therefore, the machine does notrequire maintenance activity. Then, the fourth stochas-tic shock (S4) decreases the reliability of machineto the preventive maintenance bound L. Therefore,on the inspection time of 2T , the PM maintenanceactivity is recognized. The PM maintenance activityrecovers and improves the degradation level in M1.The machine works at this level of reliability untilS5 occurs. Since the reliability level of machine after


Figure 2. The maintenance activities due to the degradation level.

Figure 3. The proposed reliability modi�cation model.

shock S5 is higher than L; no maintenance activityis required. However, S6 degrades the machine toeven less than LL; thus, corrective maintenance shouldbe done. This corrective maintenance has two maindistinctive di�erences with PM, namely 1) happeningbetween the inspection intervals that cause breakdownof the machines, and 2) improving the reliability to anew machine reliability level or reliability zero in M2.

In Figure 2, the number represents the stochasticevent types that occur during the working process ofthe machine as follows.

Number 1 is a stochastic variable that denotesmachine reliability level (Relm) or rel and follows ex-ponential distribution with parameter (RL � Exp(�)).In fact, this number is a function of degradation ofmachine at each time (Dm(t)) according to the functionin Eq. (2). In this equation, �0 and �1 are reliabilitydeterioration rates and weighted average of criticallevels, i.e., DM = (L+ 4�LL)=5. Machine degradation(DLm) or Dm(t) follows exponential distribution withparameter (DLm � Exp(�)). It should be noticed thatin the equations of this paper, DLm and Dm(t) denotemachine degradation and RLm and Relm(t) denotemachine reliability.

Relm(t) =e��0Dm(t)

1 + e�1(Dm(t)�DM) : (2)

Number 2 denotes PM Duration (PMD) andit follows lognormal distribution (PMD � lognormal(�PM ; �PM )).

Number 3 represents the improving or recoverylevel through PM (RLPM) activity, calculated throughEq. (3), and it follows lognormal distribution (RLPM�log normal(�PM 0 ; �PM 0) ).

Relnew = Relold +RLPM ;

LL < Relold � L: (3)

Number 4 denotes the CM Duration (CMD)and it follows lognormal distribution (CMD �log normal (�CM ; �CM ).

Number 5 represents the improving or Recov-ery Level through CM (RLCM) activity, calculatedthrough Eq. (4), that either entirely removes thereliability of machine or makes it one.

Relnew = Relold +RLCM ; Relold � LL: (4)

Number 6 denotes the stochastic time between twoshocks (TBS) and it follows an exponential distribution(TBS � Exp(�)).

Figure 3 illustrates a brief explanation of theexplained reliability modi�cation process.


Figure 4. The proposed solution habitat (solution) vector of the example in Figure 1.

Figure 5. The proposed hybrid SSV operator of the habitat.

3. Simulation-Based Optimization (SBO)algorithm

The proposed SBO has two main elements, namely,optimization algorithm and simulation process. Twodi�erent meta-heuristic algorithms, namely, GA andBBO, conduct the optimization algorithm. Accord-ingly, this section is classi�ed into three parts. The �rsttwo parts introduce the mentioned elements, respec-tively, and the third one integrates the whole elementsand operators with each other.

3.1. Optimization algorithm of the SBOBefore developing the optimization algorithms, sepa-rately, let us explain them, comparatively. GA andBBO, as population-based algorithms, have many sim-ilarities. Both algorithms include a set of individuals,called chromosomes and habitats, respectively. The�tness values of the individuals are called �tness andHigh Suitability Index (HSI), respectively. Otherdetailed comparisons of the algorithms are providedin [84].

3.1.1. The BBO algorithmBBO mimics the migration term of biogeography sci-ence [88,89]. The solution or habitat structure in thispaper is a vector equal in length to the number of op-erations or total number of operations (TNOP). Eachcell of this vector is an ordered pair in which the upperobject is the operation name and the lower object isthe assigned machine to that operation. Moreover, the�rst row of the solution structure shows the sequenceof operations for operating on machines. Figure 4illustrates a sample of solution structure related to theGant chart of Figure 1.

BBO implements di�erent strategies in its mu-tation operator. In Sequencing Sub-Vector (SSV), itapplies a hybrid strategy, including swap, reversion,and insertion, through a random process, as shown inFigure 5.

For the assignment sub-vector (MASV), BBOperforms through machine changing from the capabletable of each operation as in Figure 6.

For executing the migration, in sequencing part,permutation operator conducts the migration as in


Figure 6. MASV operator of BBO.

Figure 7. Proposed migration of sequencing.

Figure 8. Proposed migration of assignment.

Figure 7, and in assignment part, mask operator playsthe role as in Figure 8.

3.1.2. The Genetic Algorithm (GA) operatorsGA implements reproduction, mutation, and crossoveras the conductive operators for searching the searchspace. Reproduction operator copies a set of elitechromosomes to the next generation [90].

3.2. The simulation agent of the algorithmAs mentioned in the developed scheduling model, theproposed FJSP contains di�erent stochastic compo-nents, such as RL, PMD, RLPM, CMD, RLCM, orTBS, to encompass a realistic version of the RCM.These variables change the states of the solutionsdynamically.

SBO, as a powerful tool of optimization, isinvolved in almost every aspect of stochastic pro-gramming [84]. Two general classes of stochasticoptimization problems exist in the literature, namely,the parametric (static) and the control (dynamic)ones. The static optimization includes a set of static

parameters for all states. However, in the controloptimization, solutions change according to dynamicstates [84]. Here, because of the stochastic natureof problem, dynamic strategy controls the simulationprocess. Figure 9 plots the general structure of theproposed SBO.

The input to Figure 8 is a solution from theoptimization process and its output is the simulatedversion of the objective function. This SBO conducts aloop of simulation runs (Numsim) to obtain averageand standard deviation of solutions for reporting amore robust solution. In this owchart, dt regulatessample time of the simulation. Moreover, V T andLV T denote predetermined length between visit timesand the obtained last visit time, respectively. Besides,the terms IJSfjg(i), IJFfjg(i), and IMB(m) inFigure 10 to Figure 12 are binary logical variablesthat represent ìs operation j of job i started,' ìsoperation j of job i �nished,' and ìs machine m busy,'respectively.

The reliability updating function of Figure 10determines the level of reliability for machines and the


Figure 9. The overall owchart of the proposed simulation part of SBO.

maintenance decision. Figure 11 includes the logic ofthe maintenance decision determination.

According to schedules, machine and job statusdetermination functions are activated as given in Fig-ure 12 and Figure 13, respectively. These functions de-termine the start and �nish status of jobs plus the busi-ness status of machines at each moment of simulation.

The job status function includes the shockingtime determination functions. Figure 14 illustrates theproposed shocking logic. SBO at these shock timesupdates the reliability level of machines during theoperating times for the related assigned operations.Certainly, they have impact on the types of themaintenance decisions according to the reliability levelobtained after the shock times.

4. Computational results

This section provides us with the numerical examples

of the problem to have a detailed view of the developedstochastic problem and the simulation based algo-rithms. The general information of these test problemsis provided in Section 2 and their detailed descriptionsare in a �le, called RCM, placed in ResearcheGate siteof the �rst two authors. In this section, the proposedSBO is compared with Genetic Algorithm (GA).

4.1. Parameter tuningParameters of the algorithms are tuned throughTaguchi method [91].

Tables 2 and 3 show the determined levels ofparameters of BBO and GA.

4.2. Outputs of the algorithmsTables 4 and 5 present the outputs of the algorithmsfor the developed stochastic problem for GA and BBO,respectively. Moreover, these tables include the resultsof the algorithms for simple version of the problem as


Figure 10. Proposed reliability updating function.

Figure 11. The proposed maintenance decision function.

Figure 12. The proposed machine status determination function.


Figure 13. The proposed job status determination function.

Figure 14. The proposed shock creation function.

Table 2. The factor levels of BBO.

A:Iteration size

B:Population size

C:Mutation rate

D:E rate

A:I rate

10 10 0.1 0.8 0.8

30 30 0.2 1 1

50 50 0.3 1.2 1.2

Table 3. The factor levels of GA.

A:Iteration size

B:Population size

C:Crossover rate

D:Mutation rate

10 10 0.5 0.1

30 30 0.6 0.2

50 50 0.7 0.3


Table 4. Outputs of the algorithms for test problems.

Problem # GA BBOCmaxMean Time 1 CmaxSTD CmaxMean Time 1 CmaxSTD

FJSP1 108 440.02 0 108 873.81 0.0011FJSP2 193 924.09 15.202 173.5 1907.36 98.005FJSP3 93 683.67 0 93 1429.61 0FJSP4 146.75 728 13.22 141.05 1365.07 16.97FJSP5 167.7 1255.8 25.243 212.35 2871.43 30.759FJSP6 76 690.4 0 76 1222.08 0FJSP7 261.9 1277.74 4.596 265 2347.27 21.637FJSP8 169.2 1197.9 12.727 183.1 2200 3.889FJSP9 266.95 2702 30.193 252.8 5244.42 47.16FJSP10 204.05 2496.35 8.414 203.09 4550.73 21.99Average 168.655 1239.597 10.959 170.789 2401.178 26.14111

Table 5. Outputs of the algorithms for test problems.

Problem # GA BBOCmax Time 2 Cmax Time 2

FJSP1 108 2.17 108 8.36FJSP2 149 2.65 147 14.03FJSP3 93 2.67 93 13.57FJSP4 133 2.73 118 14.79FJSP5 154 3.66 146 19.09FJSP6 76 6.02 76 19.15FJSP7 203 6.82 194 20.51FJSP8 139 7.45 120 20.07FJSP9 167 11.37 167 28.72FJSP10 154 11.7 139 30.64Average 137.6 5.724 130.8 18.893

a lower bound validation. The lower bound model isthe simple version of the FJSP with any stochasticparameter or maintenance consideration. Obviously,in such situation, both Cmax and execution time of thealgorithm present lower bound values for the developedstochastic problem. The simple problem does notencounter PM, CM, or breakdown. Moreover, it doesnot need inspection. Therefore, Cmax values are only

dependent on the main operations and are in the worstcase equal to the stochastic version. In terms ofexecution time, low time is required for processing onlysome operations in comparison with the case in whichdi�erent maintenance components are also insertedbesides the operations.

In each table, for the main developed problem,because of the stochastic nature of the problems, eachtest problem is run several times and the average(CmaxMean), standard deviation of Cmax (CmaxSTD)values, and average execution times (Time) are re-ported. In the simple model part of the tables, Di�1 isdi�erence value of Cmax in stochastic model and simplelower bound model (i.e., Di�1 = CmaxMean � Cmax).Similarly, Di�2 shows di�erence of time values of themodels (i.e., Di�2 = Time 1� Time 2).

In both Tables 4 and 5, the last columns representthe average values of the columns. Since Cmax,standard deviation, and time objective functions are allto be minimized, the smallest values are the best ones.

Figure 15 compares the algorithms regardingthree metrics of average Cmax (CmaxMean), averagetime, and average standard deviation for the obtainedsimulated solutions. As it is clear, GA is betterthan BBO only in time metric. Figure 16 carriesout the comparison of the obtained outputs from the

Figure 15. Comparison of algorithms for the stochastic problem with maintenance considerations.


Figure 16. Comparison of algorithms for the simple problem without maintenance considerations.

Figure 17. Comparison of algorithms regarding their obtained lower bounds.

algorithms for the deterministic version or the lowerbound problem.

As can be seen in Figure 17, algorithms do nothave di�erence on Cmax. Besides, although they havethe same trend in time, the vertical dimensions of theoutputs of algorithms are di�erent.

Tables 6 and 7 conduct the statistical tests for thesimple and stochastic versions. In fact, they prove thatthe algorithms in terms of Cmax are non-dominatedand in terms of time, GA is superior.

Figures 18 compares the convergence plots of

GA and BBO for the stochastic and simple problemsregarding the mentioned metrics. Moreover, thereal-time novel reliability monitoring illustration ispresented in Figure 19 for problem FJSP9. GAis used for drawing these �gures. This developedand innovative �gure illustrates the developedreliability-centered maintenance approach in detail. Inthis �gures, whenever a task is assigned to a machine,its reliability decreases during the task operation.Then, according to the mentioned logic behind thePM and CM, suitable maintenance reaction is taken.

Table 6. T-test for comparing GA and BBO regarding the metrics of Table 4.

Metric name P -value Description

Cmax (CmaxMean) 0.943 They are not considerably di�erentTime 1 0.044 GA outperforms BBOStandard deviation (CmaxSTD ) 0.150 They are not considerably di�erent

Table 7. T-test for comparing GA and BBO regarding the metrics of Table 5.

Metric name P -value Description

Cmax 0.926 They are not considerably di�erent

Time 2 0.000 GA outperforms BBO


Figure 18. Comparison of convergence plots of GA andBBO for three problems.

4.3. Discussion

As mentioned in Figure 2, our RCM problem assumestwo determining levels, i.e., L and LL. These levels aretuned to 0.81 and 0.11, respectively. According to this�gure, 6 stochastic components are considered in theproposed RCM to make it realistic. These componentsand variables are also shown in Figure 20 for the mainselected problem of FJSP9. In fact, this �gure is sameas Figure 19, but in reliability part, it only reports theoutputs of Machine 2 for presentation simplicity.

Number 1 or RL and number 6 or TBM in Figure20 depict a set of reliability degradations and set ofshocks, respectively, due to activation of operation 1.1on Machine 2. However, since the values of thesevariables are very small, the associated values arepresented all together for a speci�c operation. RL isregulated according to the function in Figures 3 and10. Shock times of TBM are generated according toFigure 14. Besides, the (3) values show the e�ect of PM(RLPM) on the reliability level of machine and theycause PM with duration denoted by number 2. The PMoccurs when the degradation level goes less than the Llevel at the inspection times or before them. Inspectiontimes are presented in Gant chart part of the �gure.CM recovery levels (RLCM) and their durations arepointed by numbers 5 and 4, respectively. CM happenswhen the reliability level violates LL level. Activationof PM or CM and their durations are denoted by themaintenance decision function given in Figure 11. Inthe Gant chart part of the �gure, machines and jobs

Figure 19. Real-time reliability level according to the Gant chart evolution for FJSP9.


Figure 20. Description of the main results of the outputs for FJSP9.

are scheduled through Figures 12 and 13, respectively.Figure 9 manages the whole simulation task.

Numbers 7 and 8 in this �gure show the wastedtime according to the maintenance requirement rec-ognized with the autonomous detection engine of thesimulation algorithm. It means that during the periodsshown by numbers 7 and 8, operations O3;3 andO4;4 have been started, respectively, since they weredegraded in the reliability �gure. However, since theirreliability levels have become less than LL and L,respectively, they require CM and PM. Therefore, theirmain operations are interrupted and the maintenanceoperations are started. Certainly, since the jobs are notresumable in our problem, they are started from thebeginning after their maintenance activities. To sumup, these �gures prove that the designed algorithm cancontrol the process autonomously.

5. Conclusion

This research focused on the maintenance consider-ation in production problems. A stochastic FJSPwas developed by considering a modern maintenancesystem called RCM. This autonomous RCM monitoredreliability level permanently and decided which main-tenance activity should be done. Since the devel-oped problem needed real-time checking of stochasticevents, it was so complicated. Therefore, two SBOmechanisms, namely, GA and BBO, were developedto conduct the optimization problem. The requiredmain and sub functions of the proposed algorithms weredescribed in detail with su�cient examples. According

to the results, the proposed RCM took bene�t fromits considered CBM concept properly. Moreover, ithandled the considered assumptions and constraintsduring the optimization process completely. Moreover,di�erent innovative and novel visualization techniquesillustrated the proposed logics of the stochastic prob-lem explicitly. Future work following this researchmay control the cost term of the maintenance withina multi-objective problem or develop other stochastictechniques, based on decomposition, to handle thesame problem.

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Biographies

Seyed Habib Rahmati received the BSc and MScdegrees in Industrial Engineering in 2007 and 2010,respectively, from Qazvin Islamic Azad University(QIAU). Now, he is PhD Candidate in IndustrialEngineering at Amirkabir University of Technology. Hejoined the Qazvin Islamic Azad University (QIAU) in2012 as faculty member of the Department of Industrialand Mechanical Engineering. His research interests arein stochastic optimization, maintenance and reliabilitymodels, scheduling, supply chain management, andbusiness intelligence.

Abbas Ahmadi received the BSc degree in IndustrialEngineering in 2000 from Amirkabir University of


Technology, MSc degree in Industrial Engineering in2002 from Iran University of Science and Technology,and PhD degree in Systems Design Engineering in 2008from University of Waterloo. He joined AmirkabirUniversity of Technology, Iran, in 2009, where he isat present Professor in the Department of IndustrialEngineering and Management Systems. Dr. Ahmadi'sresearch interests are in supply chain management,business intelligence, swarm intelligence, computa-tional intelligence, data and information management,system analysis and design, and cooperative intelligentsystems. He has authored and co-authored several pa-pers in journals and conference proceedings, chapters inbooks, and numerous technical and industrial project

reports. Under his supervision, several students havecompleted their degrees.

Behrooz Karimi is Professor in the Departmentof Industrial Engineering and Management Systemsat Amirkabir University of Technology. He receivedhis BSc degree in Industrial Engineering in 1990 fromAmirkabir University of Technology, MSc degree inIndustrial Engineering in 1994 from Iran University ofScience and Technology, and PhD degree in IndustrialEngineering in 2001 from Amirkabir University of Tech-nology. Dr. Karimi's research interests are in supplychain management, computational intelligence, meta-heuristic, inventory control, and production planning.

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Developing simulation based optimization mechanism for a ...scientiairanica.sharif.edu/article_4461_3559eaf8ac5d8abc19631501ff5… · time. In fact, it implements condition based