MachineReliabilityOptimizationbyGeneticAlgorithmApproach

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1 Global Journal of Researches n Engneerng: A Mechancal and Mechancs Engneerng Volume 20 Issue 2 Verson 1.0 Type: Double Blnd Peer Revewed Internatonal Research Journal Publsher: Global Journals Onlne ISSN: & Prnt ISSN: Machne Relablty Optmzaton by Genetc Algorthm Approach By Ngnass Djam Aslan Brsco, Nze Wolfgang & Doka Yamgno Serge Unversty of Ngaoundere Abstract- To defne the relablty network of a system (machne), we start wth a set of components arranged n an approprate topology (seres, parallel, or parallel-seres), choose the best terms of the rato performance / cost, and gather by lnks wth the am to combne them. Ths process requres a long tme and effort, gven the very large number of possble combnatons, whch becomes tedous for the analyst. For ths reason, t s essental to use an approprate optmzaton approach when desgnng any product. However, before tryng to optmze, t s necessary to have a relablty assessment method. The objectve of ths paper s to dsplay a meta-heurstc method, whch s sustaned on the genetc algorthm (GA) to mprove the machnes relablty. To acheve ths objectve, a methodology that conssts of presentng the functonaltes of genetc algorthms s developed. The result acheved s the proposal of a relablty network for the optmal soluton. Keywords: relablty, cost, relablty network, topology. GJRE-A Classfcaton: FOR Code: MachneRelabltyOptmzatonbyGenetcAlgorthmApproach Strctly as per the complance and regulatons of: Ngnass Djam Aslan Brsco, Nze Wolfgang & Doka Yamgno Serge. Ths s a research/revew paper, dstrbuted under the terms of the Creatve Commons Attrbuton-Noncommercal 3.0 Unported Lcense permttng all non commercal use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted.

2 Machne Relablty Optmzaton by Genetc Algorthm Approach Ngnass Djam Aslan Brsco α, Nze Wolfgang σ & Doka Yamgno Serge ρ Abstract- To defne the relablty network of a system (machne), we start wth a set of components arranged n an approprate topology (seres, parallel, or parallel-seres), choose the best terms of the rato performance / cost, and gather by lnks wth the am to combne them. Ths process requres a long tme and effort, gven the very large number of possble combnatons, whch becomes tedous for the analyst. For ths reason, t s essental to use an approprate optmzaton approach when desgnng any product. However, before tryng to optmze, t s necessary to have a relablty assessment method. The objectve of ths paper s to dsplay a meta-heurstc method, whch s sustaned on the genetc algorthm (GA) to mprove the machnes relablty. To acheve ths objectve, a methodology that conssts of presentng the functonaltes of genetc algorthms s developed. The result acheved s the proposal of a relablty network for the optmal soluton. Keywords: relablty, cost, relablty network, topology. I. Introducton T he fundamental functon of a system s to provde ts customers wth a farly economcal cost, acceptable relablty requred. These constrants requre an optmal desgn. In the engneerng context, the fundamental nterest of manufacturers s to fnd a balance between the relablty of a system and ts cost. These two factors consttute the most mportant decson varable for optmzng a system. Ths s generally manfested n mnmzng the cost under the constrant of relablty on the one hand, and n mprovng performance to meet the needs of customers under the constrant of cost on the other. In ths area, researchers have developed and mproved many methods and algorthms. The set of methods can be dvded nto two man categores: exact methods, whch guarantee to obtan an optmal soluton for problems of reasonable sze, and approxmate methods (heurstcs and meta-heurstcs), whch gve good solutons. Qualty, wthout guarantee of optmalty, but for the beneft of shorter calculaton tme. Author α: Department of Fundamental Scences and Technques of Engneer, EGCIM, Unversty of Ngaoundere, Cameroon. e-mal: ngnassbrs@yahoo.fr Author σ: Department of Mechancal Engneerng, ENSAI, Unversty of Ngaoundere, Cameroon. Author ρ: Department of Physcs, Faculty of Scences, Unversty of Ngaoundere, Cameroon. If the exact methods are based on the enumeraton, often mplctly, for the search for the set of solutons of the search space, then the approached methods rather requre random processes n the exploraton of potental solutons, and ths to deal wth the combnatoral exploson generated when usng the exact methods. In ths perspectve, we are essentally seekng n ths work to ntegrate an effcent and adapted optmzaton method to solve the problem of optmzng relablty by takng nto account the most relevant constrants. Gven the large number of confguratons that can result from a set of components, the procedure of enumeratng all the possble archtectures s no longer pleasant. Consequently, t s necessary to opt for an approxmate method, whch wll make t possble to fnd the soluton closest to the optmal soluton because t s not obvous to examne all the possbltes. In ths perspectve, we opted for the method of genetc algorthms as an optmzaton technque. It s consdered an effectve meta-heurstc method n the feld of dependablty. It s nspred by genetc bology and s based on the prncple of the search for evoluton. It does not guarantee an exact soluton, but t generates a soluton close to the optmum (Panton and Campbell [1], Levtn and Lsnansk [2]). II. The Functonng of the Genetc Algorthm a) Orgn and prncple Genetc algorthms (GA) are heurstc optmzaton algorthms based on the prncples of natural selecton and genetcs. The researcher Rechenberg [3], s the frst scentst who ntroduced evolutonary algorthms by publshng hs work ''Evoluton strateges. These algorthms are broadly nspred by Darwn's theory of evoluton publshed n Next, Holland [4] proposed the frst genetc algorthms to solve combnatoral optmzaton problems, and they were also developed by the work of Davd Goldberg publshed n 1989 (Goldberg [5] and Goldberg [6]). The am of the genetc algorthm s to brng up, from one generaton to another, the canddates (potental solutons) most suted to solvng the problem. Each generaton s made up of a defned number of ndvduals, these form a populaton, and each of them represents a pont n the search space. Each ndvdual (chromosome) has nformaton coded n the form of a 35

3 36 chan of characters that analogcally consttutes genes. Then the passage from one generaton to another s carred out based on the process of evoluton by the use of evolutonary operators lke selecton, crossng, and mutaton. Ther operatng prncple s qute smple. From an ntal populaton created at random, composed of a set of ndvduals (chromosomes), we proceed to the evaluaton of ther parent qualfcatons to hghlght the best suted, as long as the least effectve are rejected. Then, the most qualfed ndvduals are chosen by prvleged selecton by gvng them a chance to reproduce by crossng and mutatng va the two operators of crossng and mutaton. Then by relaunchng ths process several tmes, the optmal soluton can be refned by passng from one generaton to another (Dour et al. [7]). b) Descrpton of the formalsm used Clusterng s a process that parttons a set of data nto meanngful subclasses (clusters or clusters). The convergence of genetc algorthms has been demonstrated for many problems, although optmalty cannot be guaranteed. The ablty of a genetc approach to fnd the rght soluton often depends on the adequacy of the codng, the evoluton operators, and the measures of adaptaton to the problem beng addressed. The method proposed here s based on genetc algorthms (Goldberg [6]) and evolutonary strateges (Schewefel [8]). It combnes the prncple of survval of the ablest ndvduals and genetc combnatons for an eltst research mechansm. The genetc method produces new solutons (chldren) by combnng exstng solutons (parents) selected from the populaton, or by mutaton. The central dea s that parent solutons wll tend to produce superor chld solutons n terms of adaptaton so that ultmately a soluton obtaned s optmal. In ths study, we used a genetc method prevously defned by Bckng et al. [10] wth a defnton of the chromosome and the operators of selecton, combnaton, and mutaton concerned. Unlke genetc algorthms, the genetc method used s desgned to mnmze and not maxmze. Ths method, lke genetc algorthms, s not lmted by assumptons about the objectve functon and research space, such as contnuty or dfferentablty. It uses a populaton of ponts smultaneously by contrast wth usual methods usng only one pont. Genetc operators are eltstcally mprovng the search process to fnd the global optmum. There are more complcated genetc operators, but the basc operators and ther varous modfcatons can generally be appled. The choce of these operators depends on the nature of the problem and the performance requrements. The genetc algorthm that we are gong to mplement s as follows, where the process s appled to teraton k: 1. Data codng; 2. Generaton of the ntal populaton P 0 of N ndvduals; 3. Assessment of the adaptaton of all ndvduals n the populaton; 4. Selecton of a proporton of the best ndvduals (parents for the producton of new ndvduals); 5. The Crossng of all ndvduals n the populaton P k two by two wth a probablty P m, we wll have N chldren noted C k ; 6. Mutaton of all ndvduals n the populaton, we wll have N elements noted M k ; 7. Choce of the most sutable ndvduals,.e., those who optmze the objectve functon; 8. If the stop test s verfed, stop, otherwse return to step a. We wll choose, as a stop test n our mplementaton, a fnte number of teratons. It s mportant to note that the stoppng crteron can be several cycles of the algorthm (number of generatons), the average of the adaptatons of ndvduals, a convergence factor, etc. An ndvdual represents a vector of decson varable (parameters), and ts adaptaton s measured by the objectve functon. The formalsm and the genetc operators are detaled below.. Data codng The frst step s to properly defne and code the problem. That step assocates wth each pont of the search space a specfc data structure called a chromosome, whch wll characterze each ndvdual n the populaton. Ths step s consdered to be the most mportant step n GA because the success of these algorthms depends heavly on how ndvduals are coded. There are dfferent choces for codng a chromosome, ths choce beng a very mportant factor n the progress of the algorthm so t must be well suted to the problem beng addressed: Bnary codng: It s the most used codng. The chromosome s coded by a strng of bts (whch can take the value 0 or 1) contanng all the nformaton necessary to descrbe a pont n space; Mult-character codng: ths s often more natural. We are talkng about multple characters as opposed to bts. A chromosome s then represented by a seres of numbers or characters, each representng a gene; Codng n the form of a tree: ths codng n tree structure starts from a root (comprsng several parts equal to the number of ntal ndvduals), from whch one or more chldren can be derved. The tree then bulds up gradually, addng branches to each new generaton.

4 . Generaton of the ntal populaton Each chromosome s the potental result of the optmzaton problem. We defne a chromosome as a chan composed of genes, whch are the parameters (decson varables) to fnd. The value of a gene s called an allele. The possble value of an allele s an nteger or a real value. Each gene s created randomly, usng equaton 1. γ j {0;1} s chosen randomly a, a a a a a j j l j u j l j j l j u are the mnmum and maxmum lmts of the allele a j. They are chosen accordng to the problem to be treated. Each chromosome, called an ndvdual n a haplod representaton, can be wrtten: a,..., a,..., a 1 j m Wth: m s the number of genes = 1,..., N and N s the sze of the populaton (number of ndvduals). All the constrants are taken nto account n the ntal phase of populaton creaton. When an ndvdual s created, f the constrants are respected, ths ndvdual s ntegrated nto the ntal populaton; otherwse, t s not. At the start of the algorthm, the ntal populaton contans ndvduals. The length of the chromosome m and the sze of the populaton N s two of the four adjustment parameters of the genetc method.. Objectve functon and adaptaton We evaluate the dfferent solutons proposed to treat them accordng to ther relevance and to see whch the best are. For ths, we use the objectve functon. Ths functon measures the performance of each ndvdual. To be able to judge the qualty of an ndvdual and thus compare hm to others. The objectve functon of our case s to mnmze the cost whle maxmzng relablty. The evaluaton of each ndvdual n the populaton then makes t possble to make the selecton. For a system made up of n components n parallel, the relablty to be maxmzed s gven by equaton 2. n R 1 1 r 1 n s the number of components r s the relablty of the component For a parallel-seres system, the relablty to be maxmzed s gven by equaton 3. (1) (2) p n R 1 1 r (3) j 1 j1 p s the number of stages of the system r j the relablty of the j th element of the th stage n s the number of components of the th stage The cost functon to be mnmzed deducted from the work of (Gutha and Vadlaman [9]) for a parallel -seres system (machne) wth p stages s gave by equaton 4. p C C e 1 /4 C s the cost vector of the components of the chromosome p s the number of stages of the machne. v. Selecton of the most sutable ndvduals When the entre populaton s assessed at generaton t, ndvduals are ranked n ascendng order of objectve functon. Then the selecton s made. Selecton helps to statstcally dentfy the best ndvduals n a populaton and elmnate the bad ones from one generaton to the next. Ths operator also gves a chance to the bad elements because these elements can, by crossng or mutaton, generate relevant descendants compared to the optmzaton crteron. The frst N G ndvduals (the best N G ) are selected to be parents. G s the thrd settng parameter of the genetc method. G s called the generaton gap. G makes t possble to select a part of the populaton to provde suffcent genetc materal wthout decreasng the speed of convergence (Goldberg [6]). There are dfferent selecton technques: Selecton by rank: Ths selecton method always chooses the ndvduals wth the best adaptaton scores, wthout allowng chance to ntervene; Selecton by wheel: For each parent, the probablty of beng selected s proportonal to ther adaptaton to the problem (ther score by the ftness functon). Ths selecton be maged by a casno roulette wheel, on whch all the chromosomes of the populaton are placed, the place s gven to each of the chromosomes beng proportonal to ts adaptaton value. Also, the hgher an ndvdual's score, the more lkely he s to be selected. We spn the wheel as many tmes as we want ndvdual sons. The best wll be able to be drawn several tmes, and the worst never; Selecton by tournament: Two ndvduals are chosen at random, ther adaptaton functons are compared, and the best suted s selected.; Unform selecton: We are not nterested n the adaptaton value of the objectve functon,and the selecton s made n a random and unform manner (4) 37

5 38 such that each ndvdual has the same probablty P() = 1 / N as all other ndvduals, where N s the total number of ndvduals n the populaton; Eltsm: The passage from one generaton to another through the crossng and mutaton operators creates a great rsk of losng the best chromosomes. Therefore, eltsm ams to copy the best (or frst - best) chromosome (s) from the current populaton to the new populaton before proceedng to the mechansms of crossng and mutaton. Ths technque quckly mproves the soluton because t prevents the loss of the most qualfed chromosome when passng from one generaton to another. v. Crossng The selected populaton s dvded nto N / 2 couples formed randomly. Two parents, and are chosen randomly from the potental parents P 1 and P 1 ther genes are combned accordng to equaton 5. a ( k) a P a P a P j j 1 j 2 j 1 j (5) γ j s a unform random number, k = N G +1,..., N, the k th ndvdual, mode can j = 1,..., m. The newly created ndvdual s then evaluated. If ts adaptaton s better than that of the worst parent, t s ntegrated nto the populaton to tranng the next generaton. If t s not the case, we repeat the combnaton. v. Mutaton of all ndvduals n the populaton The mutaton operator s a process where a mnor change n the genetc code s appled to an ndvdual to ntroduce dversty and thus avod fallng nto local optma. Ths operator s appled wth a probablty P m. P m generally lower than that of the crossng P c. Ths probablty must be low. Otherwse the GA wll turn nto a random search. v. Choosng the best solutons Ths choce conssts n retanng the solutons whch have a lower value of the objectve functon, and puttng them n the populaton P k+1. v. Stoppng crteron The stoppng crteron s evaluated n the current populaton. If t s flled, the whole populaton has converged on the soluton. Otherwse the reproducton pattern wll be repeated. The stoppng crteron used n ths method expresses that all ndvduals have converged on the same soluton and assumes that evoluton s no longer possble, that s to say, that no better soluton can be found. The whole strategy s eltst because only the best ndvduals are selected for survval from one generaton to the next and can be the parents of new and better ndvduals. To ensure convergence of the algorthm, the parameters N and G must be adjusted wth care. The sze of the populaton N affects both the performance and effcency of the algorthm (Bckng et al. [10]). The algorthm s less effcent wth very small populaton szes. Large populaton sze may contan more nterestng solutons and dscourage premature convergence towards sub-optmal solutons, but requres more assessments per generaton, whch can lead to a low convergence rate. The generaton gap G determnes the proporton of the populaton that remans unchanged between two generatons. It s chosen to select ndvduals as severely as possble, wthout destroyng the dversty of the populaton too much. The global strategy used assumes that all the ndvduals who make up the populaton, from generaton to generaton, satsfy all the constrants. The best soluton for the latest generaton represents the soluton to the problem by the defned crtera. III. Applcaton Consder a machne made up of fve components mounted n parallel (see Fgure 1). Our goal s to defne an optmal relablty network, to ensure operaton wth mnmum relablty r mn = 0.80 whle mnmzng the cost of the structure. E C 1 C 2 C 3 Fgure 1: Structure (machne) to optmze Table 1 groups the parameters of the fve components C 1, C 2, C 3, C 4 and C 5 : C 4 C 5 Table 1: Parameters of the structure components Components Relablty Cost CC CC CC CC CC Step 1: Data codng We choose bnary codng usng a 5-bt character strng. Bts 1 through 5 represent the components of C 1 through C 5, respectvely. If a component exsts n the generated soluton, then ts F

6 correspondng bt takes the value "1"; otherwse, t takes the value "0". Step 2: Choce of GA parameters Ths choce s random whle watng to mprove t afterward. The parameters of the GA are as follows: The sze of the populaton: N = 4 ; The number of generatons: K = 10 ; The probablty of crossng: P c = 0.6 ; The probablty of mutaton: P m = Step 3: Generaton of the ntal populaton We randomly generate a populaton of 4 noted chromosomes 1, 2, 3 and 4 : 1 = [01001], 2 = [10110], 3 = [10011] and 4 = [11001]. Fgure 2 represents the relablty network of the confguraton correspondng to the chromosome 1. E C 2 C 5 Fgure 2: The relablty network correspondng to the chromosome 1. Step 4: Defnton of objectve functon The objectve functon f to optmze s defned by equaton 6. mnmze f( ) = C( ) (6) s.t r r Wth: r mn : the mnmum relablty of the structure to be optmzed; C ( ): the cost of the soluton. Step 5: Evaluaton of the relablty of each chromosome n the populaton For each chromosome generated, the relablty and the cost are evaluated respectvely by relatons 2 and 4. So: r and f ; r and f ; r and f ; r and f Step 6: Selecton of the most sutable chromosomes The best chromosomes n terms of cost n descendng order are: 1, 2, 3 and 4. We select the chromosomes 1, 2 and 3 for reproducton and we elmnate the chromosome 4 because t s the worst as far as cost s consdered. mn F Step 7: Crossng We cross a couple among the selected chromosomes wth a crossover rate of 0.6 to form new chldren. Consder the couple ( 1, 2 ): ' 1 1 ' Step 8: Mutaton We do a random draw of a sngle chromosome gene ' 1, ' 2 and 3 ; then the selected bt wll be mutated wth a mutaton probablty of Consder the second bt of the chromosome ' 2 : Step 9: Substtuton We are replacng the new populaton wth the new chromosomes. Chromosomes 3 and 4 wll s elmnated because they are the least sutable, and the chromosome 1 and 2 wll be kept.after all, they are the best qualfed among the ndvduals of the populaton. So the new populaton wll be made up of chromosomes 1, 2, ' 1 and '' 2. 1 = [01001], 2 = [10110], ' 1 = [01010] and '' 2 = [11101]. Step 10: Repeat steps 4 to 9 The algorthm stops ether after the reproducton of 16 generatons or when we notce that the soluton does not mprove after a defned number of generatons. Step 11: Optmal solutons The correspondng chromosome that optmze objectve functon s: a 10001, b 11000, c 10100, 01001, 00101, The optmum relablty and costs for each soluton are: By consderng the components selected by the genetc algorthm n the prevous optmal solutons, we generate all the possble paths (connectons) between the component of the machne. In soluton 1 ( a chromosome), we retan the components n parallel C '' ' 2 2 d e f a r and f( a) 39, r b and f( b) 40.01, r c and f( c) 40.5, r d and f( d ) 40.25, r e and f( e) 37.5, r f and f( f ) 36.25, r and f( ) g g 39

7 40 and C 5 ; n soluton 2, we retan the components C 1 and C 2 (thus defnng a path); n soluton 3, we retan the components C 1 and C 3 (defnng a path) and from end to end untl the last soluton ( g chromosome), the last path connectng the components C 3 and C 4 s shown. From where the network of optmal relablty of the machne s gven by Fgure 3. E C 1 C 5 C 2 C 3 C 4 Fgure 3: Optmal network relablty of our machne By evaluatng the overall relablty of the relablty network, we obtan a relablty of 0.88, hgher than the mnmum relablty set. Smlarly, by evaluatng the overall cost of the machne by applyng relaton 4, for the relablty network we obtan a cost of 88.75, whch s much lower than the overall cost generated by the ntal system, whch s Objectvely, the relablty network has favored cost reducton on the one hand relablty ncrease on the other. IV. Concluson Havng completed the wrtng of ths paper, whch concerns the optmzaton of the relablty of a system by the genetc algorthm, t appears that the general objectve has been acheved. Indeed, through the functonalty of genetc algorthms, mplemented on a structure wth fve components, we were able to reduce the cost and ncrease the relablty of the parallel structure. However, although the genetc algorthm s easy to mplement, t mght requre an nfnte number of teratons to the best approach of optmal soluton. Ths would make the algorthm less robust. In perspectve for ths work, t would be wse to develop or mplement another meta-heurstc that would not requre a large number of teratons. Data Avalablty The data needed to support these results are avalable n the text, specfcally n Table 1. Conflcts of Interest All authors declare that there are no conflcts of nterest regardng the publcaton of ths paper. Acknowledgments Ths work was funded by the authors of the paper and was carred out n the Department of Fundamental Scences and Engneerng Technques at the Chemcal Engneerng and Mneral Industres School at the Unversty of Ngaoundere. F References Références Referencas 1. L. Panton and J. Campbell, Genetc algorthm n optmzaton of system relablty, IEEE Trans Relab, 44:172 8, G. Levtn and A. Lsnansk, Optmzng survvablty of vulnerable seres parallel mult-state systems, Relablty Engneerng and System Safety, 79: , I. Rechenberg, Cybernetc soluton path of an expermental poblem, Lbrary Translaton 1122, Royal Arcraft Establshment, Famborough, UK, J. H. Holland, Adaptaton n natural and artfcal systems, Unversty of Mchgan press, D. Goldberg, Genetc algorthms, Addson Wesley, ISBN, , 1989a. 6. D. Goldberg, Genetc algorthms n search, optmzaton and machne learnng, Addson Wesley, 1989b. 7. S.M. Dour, S. Elbarnouss and H. Lakhbab, 'Course of Heurstc and Metaheurstc Exact Resoluton Methods', Mohammed V Unversty, Faculty of Scences of Rabat, Morocco, Mathematcs, Computer Scence and Applcatons Research Laboratory, H.P. Schewefel, Numercal Optmzaton of computer models, Edtons Wley, J. K Gutha, and R. Vadlaman, Modfed Harmony Search Appled to Relablty Optmzaton of Complex Systems, Advances n Intellgent Systems and Computng 382, F. Bckng, C. Fontex, J.P. Corrou, and I. Marc, Global optmzaton by artfcal lfe: a new technque usng genetc populaton evoluton, RAIRO-Operatons Research, vol. 28(1), 23-36, 1994.