Behavior prediction of washing system in a paper industry using GA and fuzzy lambda–tau technique

Behavior prediction of washing system in a paper industry using GA and fuzzy lambda–tau technique

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  Behavior prediction of washing system in a paper industry using GAand fuzzy lambda–tau technique S.P. Sharma a , Dinesh Kumar b , Ajay Kumar c, ⇑ a Department of Mathematics, Indian Institute of Technology, Roorkee 247667, Uttarakhand, India b Department of Mechanical and Industrial Engineering, Indian Institute of Technology, Roorkee 247667, Uttarakhand, India c Department of Applied Mathematics, ABV-Indian Institute of Information Technology and Management Gwalior, Gwalior 474010, Madhya Pradesh, India a r t i c l e i n f o  Article history: Received 9 December 2008Received in revised form 31 August 2011Accepted 8 September 2011Available online 4 October 2011 Keywords: ReliabilityMTBFPetri netsLinguistic variablesGenetic algorithmsOptimization a b s t r a c t Availability analysis has been an important issue in the design field of any Industrial sys-tem as the system structure has become more complicated. Also, the system availability isaffected by many factors such as design, manufacturing, installation, etc., and so it may beextremely difficult to model, analyze and predict the failure behavior of the system. Thepurpose of this paper is to develop a new approach for computing various performancemeasures, namely reliability, availability, MTBF (mean time between failures), ENOF(expected number of failures), failure rate and repair time, for any industrial system. Inthe proposed approach, the failure rates and repair times of all constituent componentsare obtained using genetic algorithms and then various performance measures are com-puted using fuzzy lambda–tau methodology. Washing system, the major part of paperindustryisthesubject of study. Theinteractionsamongtheworkingcomponentsaremod-eled using Petri nets. Failure and repair rates are represented using triangular fuzzy num-bers as they allow expert opinion, linguistic variables, operating conditions, uncertaintyand imprecision in reliability information to be incorporated into system model. Basedon calculated reliability parameters, a structured framework has been developed thatmay help the maintenance engineers to analyze and predict the system behavior.   2011 Elsevier Inc. All rights reserved. 1. Introduction Thecomplexityofanindustrialsystemisgrowingdaybydayandsothejobofsystemanalystshasbecomemorechalleng-ing, as they have to study, characterize, measure and analyze the uncertain systems’ behavior, using various techniques,which require the component failure and repair pattern. But unfortunately, the data, available from the past record, areincomplete,imprecise,vagueandconflicting,thatleadstoinadequateknowledgeofbasicfailureevents.Further,age,adverseoperating conditions and the vagaries of the system, affect each unit of the system differently [1]. Therefore, it may be verydifficult to construct an accurate and complete mathematical model for the system. Thus, one comes across the problem of uncertaintyinreliabilityassessment.Forthispurpose,theprobabilisticandnon-probabilistictechniquesareused.Theprob-abilistic approaches deal with uncertainty, which is randomin nature, while the fuzzy approach deals with the uncertainty,whichisduetoimprecisionassociatedwiththecomplexityofthesystemaswellasvaguenessofhumanjudgement[2].Fuzzy methodologycandealwithimprecise,uncertaindependentinformationrelatedtosystemperformanceandprovidesabetter,moreconsistentandmathematicallymoresoundmethodforhandlinguncertaintiesindatathanconventionalmethods,such 0307-904X/$ - see front matter   2011 Elsevier Inc. All rights reserved.doi:10.1016/j.apm.2011.09.044 ⇑ Corresponding author. Tel./fax: +91 751 2449624. E-mail addresses:, (A. Kumar). URL: (A. Kumar).Applied Mathematical Modelling 36 (2012) 2614–2626 Contents lists available at SciVerse ScienceDirect Applied Mathematical Modelling journal homepage:  as Bayesian statistics [3]. These approaches are still in developing phase and often use fuzzy sets, possibility theory and lin-guistic variables. The list includes, Singer [4] developed a new methodology to find out various reliability parameters usingfuzzysetapproachandfaulttree. Thefailureratesandrepairtimeswererepresentedusingtriangularfuzzynumbers.Chengand Mon [5] used interval of confidence for analyzing the fuzzy systemreliability. Through theoretical analysis and compu-tationalresultstheyhaveshownthattheirproposedapproachismoregeneralandstraight-forwardcomparedtothatofSing-er’s. Chen [6] presented a new method for analyzing the fuzzy system reliability using fuzzy number arithmetic operations.Knezevic and Odoom [7] proposed a new methodology (fuzzy lambda–tau methodology) by making use of Petri nets (PNs)insteadoffaulttree.Fuzzysettheorywasusedtorepresentfailureandrepairdata.Themethodiscapabletoprovidethereli-abilityindicesmoreefficiently.Wangetal.[8]proposedamethodtoconstructthefailurenumberinthesuccessiveoperatingranges using fuzzy paradigm. The method can easily model repairable and non-repairable cases, and identifies the systemfailed again after repair in the next time failure sequence data. The proposed methodology is demonstrated using differenttimes of repairs for about 191 bus motors. Yadav et al. [9] presented a formal structure for capturing imprecise informationandknowledgeandutilizingitinreliabilityimprovementestimation.Acaseexamplewaspresentedtodemonstratethepro-posed approach. Jiang and Chen [10] established a basis for the reliability analysis of systems with fuzzy reliability. As anexample, a case study about the fuzzy reliability analysis of a kind of sensor used in railway systems is provided to verifythe logic of this algorithm. The computation results show that this algorithm fits the engineering experience. Tanriovenet al. [11] used fuzzy logic with Markov model to describe both transition rates and temperature-based seasonal variations,whichidentifiesmultipleweatherconditions.Theeffectivenessofthemethodologyisshownbytakinganexampleofpowersystem. ZhaoandLiu[12] investigatedtheexpectedsystemlifetime, systemlifetime, andsystemreliabilityusingfuzzysim-ulationandestimatedthesystemperformances.Somenumericalexperimentsonmulti-stagesystemandnetworksystemareprovided. Huanget al. [13] proposedanewmethodusingfuzzyarithmetics,artificialneural networkandgeneticalgorithmstodeterminethemembershipfunctionoftheestimatesoftheparametersandthereliabilityfunctionofmulti-parameterlife-time distributions. The effectiveness of the proposed method is illustrated with normal and Weibull distributions. Rao et al.[14] presented a solutionto test interval optimization problemwith uncertain/imprecise parameters withfuzzy-genetic ap-proachalongwithacaseofapplicationfromasafetysystemofIndianpressurizedheavywaterreactor.Keetal.[15]proposedaproceduretoconstructthemembershipfunctionsofthesystemcharacteristicsassumingtimestofailureandtimestorepairof the operatingand standby units to followfuzzified exponential distributions. The practicalityof the proposedapproachisillustratedbytakinga numerical example. WangandWatada[16] studiedthe redundancyallocationproblemstoa parallel-series system using fuzzy random variables. Some numerical examples are provided to illustrate the feasibility of the ap-proach and quantify its effectiveness. Komal et al. [17] proposed a genetic algorithms based lambda–tau (GABLT) techniqueusingtraditional lambda–taumethodologyandgenetic algorithm. The approachis appliedto press andwashingsysteminapaper industry and gave recommendations to improve the systemperformance. Taheri and Zarei [18] investigated Bayesiansystemreliabilityinvagueenvironments.ThemodelparametersareassumedtobevaguerandomvariableswithvaguepriordistributionsandthevagueBayesestimateofsystemreliabilityhasbeencalculated.Someexampleshavebeenshowntoclar-ify the proposedapproach. Kumar et al. [19] analyzed the reliabilityof waste clean-up manipulator using real coded geneticalgorithms and fuzzy lambda–tau methodology. Various reliability parameters have been computed and sensitivity analysishas been done with various rate of occurrence of failure.Apart fromthe reliability of repairable systems, fuzzy methodology is widely used for accessing human reliability and tofindouttherelativeimportanceofhumanfactorsaffectinghumanreliability[7,20].Bertolini[21]analyzedhumanreliability and calculated the probability of erroneous actions using a fuzzy classification system. Li et al. [22] developed a new fuzzyhuman error risk assessment methodology for determining human error risk importance. The modeling is done using fuzzylogicandacaseexampleispresentedtodemonstratetheproposedapproach. Resultsshowthatthemethodismorerealisticthan the traditional ones, and it is practicable and valuable.Among the inexact reasoning methods, fuzzy methodology (FM) acts as one of the most viable and effective tool. On theotherhandgeneticalgorithm(GA), amemberof heuristictechniques, is alsoapowerfultool andisusedmanytimesforreli-ability/availability optimization. GA performs better when the solution space to be searched, is relatively large, noisy andnonlinear.Hsiehetal.[23]utilizedgeneticalgorithmsandsolvedvariousreliabilitydesignproblems,suchasreliabilityopti-mization of series systems, series–parallel systems and complex systems. Jeang [24] suggested that computer aided simu-lation software would give the optimal design for reliability. Many methods have been developed and have been used forreliability design. However, a well-defined knowledge based method has not been found in the literature for reliability de-sign and optimization. Considering maximum system reliability and minimal total cost, Li [25] solved the related problemsby multiple fuzzy objective planning. Ramirez-Marquez and Coit [26] proposed a new heuristic approach for solving theredundancy allocation problem for multi-state series–parallel systems. Yalaoui and Chatelet [27] formulated an approxi-mated function for the reliability allocation problem in a series–parallel system. You and Chen [28] proposed an efficientheuristic approach for series–parallel redundant reliability problems. Liu et al. [29] proposed a novel state selection tech-nique for reliability evaluation of power systems. The fast sorting algorithm (FSA) has been derived to select quickly the re-quired number of the system states in descending probability order with the minimum number of computations andcomparisons. The system states can be dynamically selected and analyzed until the specified accuracy is satisfied. Juanget al. [30] proposed a new method to compute optimal values of MTBF and MTTR based on GA. A knowledge-based interac-tive decision support system was developed to assist the designers set up and to store component parameters during theintact design process of repairable series–parallel system. Azaron et al. [31] used a genetic algorithm approach to solve a S.P. Sharma et al./Applied Mathematical Modelling 36 (2012) 2614–2626   2615  multi-objective discrete reliability optimization problem in a  k -dissimilar-unit non-repairable cold-standby redundant sys-tem. The results are also compared against the results of a discrete-time approximation technique to show the efficiency of the proposed GA approach. Bhunia et al. [32] formulated the reliability optimization problem as a chance constraints basedreliability stochastic optimization problem with interval valued reliabilities of components and solved the problem by realcoded genetic algorithm. The stability of the proposed methodology have been analyzed by doing sensitivity analysis. LinsandDroguett[33]solvedredundancyallocationproblemssubjecttoimperfectrepairsbyusingmulti-objectivegeneticalgo-rithmcoupledwithdiscreteeventsimulation.Themulti-objectiveGAisvalidatedviaexampleswithanalyticalsolutionsandshows its superior performance when compared to a multi-objective ant colony algorithm.The motive of this paper is to devise a method to chalk out the performance measures of any repairable system. The de-visedmethodisanamalgamoftwotechniques,GAandfuzzylambda–taumethodology,whichcanbedescribedstepwiseas,(i) develop an optimization model of availability for the system taken, (ii) utilize GAs to obtain MTBF and MTTR, during thedesign phase, for various components of washing system in a paper industry and optimize the availability parameters. Theoptimizationmodelof availabilityandGAproceduresensurethat thecost-effectiveparametersof systemavailabilitycanbeobtained,whichhelpthesystemanalyststodeviseoptimaldesignpoliciesandrepairpolicies,and(iii)usefuzzylambda–taumethodologytocalculatevariousperformancemeasuressuchas MTBF, reliability, availability, ENOF, etc. AsPetri net (PN) isapowerfultool,itiswidelyusedinmodelingandanalysisofcomplexmanufacturingsystemsandprocessduetoitscapacityin modeling the dynamics of the system. Petri nets make use of diagraph to describe cause and effect relationship betweenconditionsandevents.PNhasastaticaswellasdynamicpart.Thestaticpartconsistsofthreeobjects:places,transitionsandarrows, while the dynamic part contains tokens only [34]. Desrochers and Al-Jaar [35], Baccelli et al. [36] and Liu and Chiou [37] demonstrated the superiority of Petri nets over fault tree analysis (FTA) and Markov chain modeling. Adamyan and He[38,39] analyzed that Petri net modeling provides the ability of assessing the quality and reliability impacts of unplannedfailuresandthesequenceofthesefailure.Petrinethastheabilitytotracksystemstatesandtransitionsbetweenthesestatesbased on some triggering requirements and this ability allows to analyze combined failure modes and to predict their po-tential severity, as well as to estimate the probability of occurrence of failure modes. Keeping these points in view, interac-tions among the working units of the washing system is modeled using Petri nets. Different cut sets are obtained usingmatrix method [37]. The results so obtained may help the system designer to formulate optimal design policy and repairpolicies. 2. Methodology  The solution methodology is divided into two phases. In the first phase, optimal values of MTBF and MTTR are obtainedusing GA and in second phase various reliability parameters are obtained using fuzzy lambda–tau methodology. The flowchart of the two phased methodology is depicted in Fig. 1 and the phases are described below:  2.1. Phase-I: Formation of the optimization model for the proposed system As stated above, the information obtained from the past record/history, is very much dependent on the system configu-ration/structure, soitis verydifficulttoanalyzethesystembehaviorcorrectly. Therefore, inorder tofindout theavailabilityparameters in a cost effective manner (optimum value), the system availability optimization model and GA are used. Thisphase can be described in the following steps:  2.1.1. List the approximate expression for system availability Theavailabilityexpressionsforaseries–parallelsystemcanbeobtainedassuming:(i)thecomponentsareoperatedinde-pendently i.e. the failure and repair characteristics of components are statistically independent, (ii) the failure rate  k i  andrepair rate  l i  are constants, (iii)  k i  l i , and (iv) separate maintenance facility is available for each component. The givenindustrial system is divided into its constituent components and based on reliability block diagram (RBD), the expressions Fig. 1.  Flow chart of proposed methodology.2616  S.P. Sharma et al./Applied Mathematical Modelling 36 (2012) 2614–2626   foravailability,failurerateandrepairrateareobtained[40].ThebasicparametersforseriesandparallelsystemareshowninTable 1. Based on the expressions in Table 1, the approximate availability expression for the system can be written as: Av  ¼  f  ð MTBF 1 ;  . . .  ; MTBF n ; MTTR  1 ;  . . .  ; MTTR  n Þ :  ð 1 Þ  2.1.2. List the expression for total system cost  The Manufacturing cost depends on product specification. If MTBF of any component is longer, the failure rate will belower, indicatingthat thecomponentis highlyreliable, leadingtoa sharpincreaseinthemanufacturingcost [25]. The MTBFof a component and manufacturing cost are related to each other and the relation can be expressed mathematically [41]: CMTBF i  ¼  a i   ð MTBF i Þ b i þ c i ;  ð 2 Þ where, CMTBF i  andMTBF i , respectively, represent themanufacturingcostandMTBFof the i thcomponent,while,  a i ,  b i  and c i are constants, representing the physical property of the  i th component and  b i  >1. The relationship is illustrated graphicallyin Fig. 2(a).The components of any system interact during operation and failure in one component may increase the failure in otheror even impair the efficiency of complete system. In order to keep the system operating, the faulty component must be re-pairedassoonas possible. Torepairthefaultycomponentintime, theexperiencedstaff will berequired. Insuchcases, hugeinvestment on equipments leads to higher repair cost, despite short repair time. Assuming a linear relationship betweenMTTR and the repairing cost of components, a lower MTTR indicates a higher repairing cost, with the relation representedmathematically [30]: CMTTR  i  ¼  a i    b i   MTTR  i ;  ð 3 Þ where, CMTTR  i  andMTTR  i , respectively, represent the repairingcost andMTTRof component of the  i thcomponent, while,  a i and  b i  are constants. The relationship is illustrated graphically in Fig. 2(b).Based on Eqs. (2) and (3), the total cost can be written as: Tc  ¼ X ni ¼ 1 ð a i   ð MTBF i Þ b i þ c i Þ þ X ni ¼ 1 ð a i    b i   MTTR  i Þ :  ð 4 Þ  2.1.3. Construct the objective function Max AvTc  ¼  F  ð MTBF 1 ;  . . .  ; MTBF n ; MTTR  1 ;  . . .  ; MTTR  n Þ :  ð 5 Þ  2.1.4. Write down the lower and upper bounds for MTBF and MTTR s : t :  LbMTBF i  6 MTBF i  6 UbMTBF i ; LbMTTR  i  6 MTTR  i  6 UbMTTR  i : ð 6 Þ  2.1.5. Solve the model The formulation in Steps (2.1.3) and (2.1.4) are combined to form an optimization model. The model can be solved usingGA. Flow chart of GA procedures is depicted in Fig. 3. The solution procedure of GA can be written simply as:(i) Generate random initial population.(ii) Evaluate the fitness of each individual in the population.(iii) Repeat,(a) Select best-ranking individuals to reproduce.  Table 1 Basic parameters of availability for series–parallel systems. Type of system ExpressionSeries configurationPAs ¼ PA 1  PA 2  PA n   1   k 1 l 1 þ k 2 l 2 þþ k n l n   k s  = k 1  + k 2  +  + k n ;  l s   k 1 þ k 2 þþ k n k 1 l 1 þ k 2 l 2 þþ k n l n Parallel configuration PA s   1   k 1  k 2  k n l 1  l 2  l n k s   k 1  k 2  k n ð l 1 þ l 2  l n Þ l 1  l 2  l n ;  l s   l 1  + l 2  +  + l n S.P. Sharma et al./Applied Mathematical Modelling 36 (2012) 2614–2626   2617  (b) Regenerate new generation through crossover and mutation (genetic operations) and give birth to offsprings.(c) Evaluate the individual fitnesses of the offspring.(d) Replace worst ranked part of population with offspring.(iv) Repeat until termination criterion is satisfied.As soon as the optimal values of MTBF and MTTR are obtained, phase two starts.  2.2. Phase-II: Calculation of various reliability parameters using fuzzy lambda–tau methodology In this phase, the optimal values of MTBF and MTTR, obtained in previous phase are used to calculate various reliabilityparameters such as fuzzy failure rate, repair time, reliability, availability, ENOF and MTBF, using fuzzy lambda–tau method-ology, so as to increase the efficiency of the methodology. The procedural steps of fuzzy lambda–tau methodology can bedescribed as follows: Fig. 2.  Relation between (a) MTBF and the manufacturing cost, (b) MTTR and the repairing cost. Start No MutationCrossoverSelectionTerminationcriterionsatisfiedEvaluation of eachindividualGeneration of initialpopulationFinal SolutionYes Fig. 3.  GA flow chart.2618  S.P. Sharma et al./Applied Mathematical Modelling 36 (2012) 2614–2626 
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