Decision Science Letters

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1 Decson Scence Letters (5) 89 Contents lsts avalable at GrowngScence Decson Scence Letters homepage: Mult-objectve avalablty-redundancy allocaton problem for a system wth reparable and non-reparable components Hossen Zoulfaghar a*, Al Zenal Hamadan a and Mostafa Aboue Ardakan b a Department of Industral and Systems Engneerng, Isfahan Unversty of Technology, Isfahan, Iran b Department of Industral Engneerng, Daneshpajoohan Hgher Educaton Insttute, Isfahan, Iran C H R O N I C L E A B S T R A C T Artcle hstory: Receved October 9, Receved n revsed format: March, 5 Accepted Aprl, 5 Avalable onlne Aprl 5 5 Keywords: System avalablty and relablty Mult-objectve optmzaton Redundancy allocaton Reparable and non-reparable components Genetc algorthm Relablty s one of the most mportant characterstcs of the electrcal and mechancal systems wth applcatons n the space communcaton ndustres, nternet networks, telecommuncaton systems, power generaton systems, and productve facltes. What adds to the mportance of relablty n these systems are system complcatons, nature of compettve markets, and ncreasng producton costs due to falures. Ths paper nvestgates avalablty optmzaton of a system usng both reparable and non-reparable components, smultaneously. The avalablty-redundancy allocaton problems nvolve the determnaton of component avalablty (.e., lfe tme and repar tme of the components) and the redundancy levels that produce maxmum system avalablty. These problems are often subject to some constrants on ther components such as cost, weght, and volume. To maxmze the avalablty and to mnmze the total cost of the system, a new Mxed Integer Nonlnear Programmng (MINLP) model s presented. To solve the proposed model, an mproved verson of the genetc algorthm s desgned as an effcent meta-heurstc algorthm. Fnally, n order to verfy the effcency of the proposed algorthm, a numercal example of a system s presented that conssts of both reparable and non-reparable components. 5 Growng Scence Ltd. All rghts reserved.. Introducton Relablty optmzaton s an mportant topc that has attracted the attenton of many researchers. In relablty optmzaton, the am s to desgn a system structure that acheves a hgher level of relablty at mnmum budget and recourses. In order to mprove the relablty of a specfc system, one of the followng approaches may be adopted: a) ncreasng the relablty of each component n the system, b) usng the redundant components n parallel, c) combnng (a) and (b) above, and d) reassgnment of nterchangeable components (Kuo & Prasad, ). * Correspondng author. Tel:(+98)9558 E-mal address: hamadan@cc.ut.ac.r (H. Zoulfaghar) 5 Growng Scence Ltd. All rghts reserved. do:.567/j.dsl.5..7

2 9 Many dfferent types of the relablty optmzaton problems have been encountered and nvestgated. However, the two common types nclude the Redundancy Allocaton Problem (RAP) and the Relablty-Redundancy Allocaton Problem (RRAP) (Kuo & Prasad, ). In RAP, there are dscrete component choces wth predefned characterstcs such as relablty, cost and weght, where the goal s to fnd the optmal number of redundances n each subsystem n order to maxmze system relablty consderng some system constrants. The Relablty-Redundancy Allocaton Problem (RRAP) s formulated as a mxed-nteger non-lnear programmng problem. It s a more complcated verson compared wth the RAP snce the relablty and, therefore, other related specfcatons of the components are not predetermned and consdered as decson varables. The purpose of RRAP s to maxmze system relablty by selectng component relablty and component redundancy levels, whch forms a dffcult but realstc optmzaton problem. Component specfcatons such as cost, weght, and volume are defned as ncreasng non-lnear functons of component relablty (Cot, ). Because of the complex nature of ths problem, most classcal mathematcal methods have faled to yeld optmal or near optmal solutons for ths problem (Coelho, 9). All prevous studes of RRAP consdered the system as a non-reparable one. In ths paper, we consdered, for the frst tme, RRAP when the system conssts of both reparable and non-reparable components. Therefore, the problem may be renamed as the Avalablty-Redundancy Allocaton Problem (ARAP). Generally, the relablty of a system (or a component) s the probablty that t wll adequately perform ts specfed purpose for a specfed perod of tme under specfed envronmental condtons (Barlow & Proschan, 98) and avalablty s defned as the probablty that a system s n ts ntended functonal condton and, therefore, capable of beng used n a stated envronment (Hamadan, 98). The man dfference between these two concepts s that avalablty s used for reparable components whle relablty s a functonal ndex for the non-reparable ones. Furthermore, the term relablty can be used for the frst falure of a reparable component whle the term avalablty s used for the entre lfe of reparable components. A system usually conssts of a number of subsystems n whch each subsystem uses several components n parallel. These components are placed n each subsystem accordng to the system applcaton requrements; and each component has ts own predefned avalablty (relablty), weght, volume, and cost, whch should be consdered n desgnng the system and n determnng ts optmzaton condtons. Optmzaton of such a system can be turned nto a mult-objectve problem due to the presence of several and, sometmes, conflctng objectves such as maxmzng avalablty (or relablty), mnmzng system cost, weght, and volume. Snce RAP and RRAP belong to the NP-hard class of optmzaton problems (Chern, 99; Ha & Kuo, 6) they are generally too dffcult and tme-consumng to solve usng tradtonal optmzaton methods. More specfcally when the problem sze s large, most classcal mathematcal methods have faled to handle these optmzaton problems properly (Soltan, ). Dfferent methods have been developed for solvng the RAPs. Exact optmzaton methods lke dynamc programmng (Fyffe et al., 968; Ng & Sancho, ), nteger programmng (Msra & Sharma, 99), Lagrangean multplers (Msra, 97), and varous types of the meta-heurstc algorthms such as genetc algorthm (Hamadan & Khorshd, ; Ardakan & Hamadan, ; Ardakan et al., 5), ant-colony optmzaton (Cha & Smth, ), Immune algorthm (Chen & You, ), the surrogate constrant method (Onsh et al., 7), varable neghborhood search (VNS) algorthms (Lang & Chen, 7), Tabu search (TS) algorthm (Ouzneb et al., 8), and partcle swarm optmzaton (Bej et al., ; Wu et al., ; Garg, ) have been used for maxmzng system relablty. For solvng the RRAPs, numerous meta-heurstc algorthms such as genetc algorthm (Zoulfaghar et al., ; Ardakan & Hamadan, ), artfcal bee colony algorthm (Yeh & Hseh, ), Partcle

3 H. Zoulfaghar et al. / Decson Scence Letters (5) 9 Swarm Optmzaton (Coelho, 9), harmony search (Zou et al., ; Wang & L, ; Zou et al., ), cuckoo search algorthm (Valan, ; Valan et al., ), and mperalst compettve algorthm (Afonso et al., ) have been wdely employed over the past decade. As already mentoned, ths paper deals wth the RRAPs whose man dfference from smlar problems les n the assumpton that the system nvolves both reparable and non-reparable components; hence t must be consdered as ARAP. To solve ths problem, a new mxed nteger non-lnear programmng model s ntroduced and solved by an mproved verson of Genetc Algorthm (GA). To solve the proposed mathematcal model and to show the capablty of the proposed GA n handlng the problem, a modfed problem from the lterature s consdered and the GA results are compared wth those of the Improved Partcle Swarm Optmzaton (IPSO) algorthm (Wu et al., ) as one of the best algorthms reported n the lterature. In contrast to the studes conducted on relablty optmzaton, fewer studes have been devoted to the avalablty allocaton and optmzaton to nvestgate the optmal falure and repar rates of each component n a system amed at maxmzng (or mnmzng) the objectves. In most cases, the problem of avalablty allocaton and optmzaton can be defned as a mult-objectve optmzaton problem, whch ams to maxmze system avalablty and mnmze tts cost (Elegbede & Adjallah, ). Levtn and Lsnansk () ntroduced a model n whch the cost of desgnng the system s fxed and ts purpose s to optmze system avalablty. Also, Zo and Bazzo () presented an analyss on level dagrams of Pareto front for the redundancy allocaton problem. Ther ams were to maxmze system avalablty and mnmze the cost and weght of the whole system. Chang and Chen (7) proposed a new mult-objectve genetc algorthm, namely the smulated annealng based on a multobjectve genetc algorthm (MOGA), to resolve the avalablty allocaton and optmzaton problems of a reparable system. In the exstng lterature, a number of researchers have nvestgated the theoretcal problems of avalablty modelng (Srvasvata & Fahm, 988; Zhao, 99; Lee, ; Cao et al., ; Sercola, 999; Ma et al., ; Dewnter, ). The present artcle ams at both avalablty and relablty allocaton and avalablty optmzaton. The rest of the paper s organzed as follows. In Secton, the structure of the problem s presented and the proposed model s establshed. A descrpton of the proposed genetc algorthm s provded n Secton and a numercal example for the problem s gven Secton. Fnally, the paper concludes wth results, conclusons, and some suggestons for future study.. Problem defnton and the proposed model Seres-parallel systems are used here as a well-known system structure for descrbng the proposed model. The common structure of a seres-parallel system s llustrated n Fg..,, M, , j, j M, j, K, K M, K m Fg.. General structure of seres-parallel system.

4 9 Wthout loss of generalty, t s supposed that all the components n each subsystem are dentcal (the components have the same relablty and avalablty). In general, one has two objectves for these types of systems: maxmzng system avalablty and mnmzng system cost (Elegbede & Adjallah, ). As already mentoned n the prevous sectons, most studes on redundancy allocaton problems have consdered optmzaton of system relablty and t has been supposed that all the components are nonreparable, at the cost of neglectng the avalablty and mantanablty of components. Furthermore, n cases where system avalablty s consdered, t has been assumed that the system conssts of only reparable components. Ths s whle n real world condtons, there are a few systems that are desgned to use ether reparable or non-reparable components. In fact, most complcated systems consst of both reparable and non-reparable subsystems. Examples nclude systems composed of electronc and mechancal sectons such as automoble motor systems, and arplane systems, producton systems, where the electronc sectons consst of non-reparable components whle the mechancal sectons have reparable ones. Recently, Zoulfaghar et al. () presented a mathematcal model for such a system consstng of both reparable and non-reparable components. They developed the model for the case n whch the relablty or avalablty of the components are pre-determned. As an extenson to that study, a new model s developed n ths paper to consder the problem for the case n whch the relablty and avalablty of the components are not gven n advance but they are consdered as decson varables. Ths problem can be called the avalablty-redundancy allocaton problem. It s, therefore, assumed that some subsystems use non-reparable components whle others have reparable ones. In ths case, t wll not be possble to use the relablty formulaton for the objectve functon snce some of the components are reparable. It follows then that the modelng should be accomplshed n such a way that the objectve functon s consdered for maxmzng system avalablty. In the next subsectons, notatons and relatons used n the proposed mxed nteger non-lnear programmng model and the mathematcal representaton of the model are presented.. Notaton ( ) Asys t Avalablty of the system at tme t, C sys Cost of the system, R ( t) Relablty of the component used n the subsystem at tme t, A ( t) Avalablty of the component used n the subsystem at tme t, Av ( t) Avalablty of the subsystem at tme t, s Number of total subsystems, s Number of subsystems that nclude non-reparable components, A Set of subsystems ncludng reparable components, R Set of subsystems ncludng non-reparable components, n Number of components n subsystem, α, β Constants representng the physcal characterstc of each component n subsystem, T Operatng tme durng whch the component must not fal, a, b, q and p Real numbers used to calculate the cost of reparable component n subsystem, E( X), EY ( ) Average lfe tme and repar tme of reparable components,

5 H. Zoulfaghar et al. / Decson Scence Letters (5) 9 w, wv Real numbers used to calculate the weght and volume of components n subsystem, V Maxmum permtted volume of the system, W Maxmum permtted weght of the system, N Maxmum permtted number of components n subsystem, P Mnmum permtted number of components n subsystem L, U Lower and upper bounds of relablty n subsystem, R R L, U Lower and upper bounds of avalablty n subsystem. A A. The mathematcal model In ths secton, a b-objectve mathematcal model s developed and presented for the problem. The proposed model s as follows: n ( ( t ) ) f : Max Asys () t = Max Av () n n = Max ( ( R() t ) ) ( ( A( t) ) ) R A () β n p q T s f : Mn Csys = α n + e + n( a + b ) R ln R A E( X) E( Y) () n s wne W () s.t: ( A R) ( A R) wv n V p n N, ( A R) n Z +, ( A R) LR R U, R R L A U, A A A In ths model, Eq. () denotes the frst objectve functon for maxmzng system avalablty. The functon s a multplcaton of two parts: the frst maxmzes the relablty of non-reparable subsystems whle the second maxmzes the avalablty of the subsystems wth reparable components. Eq. () s the second objectve functon whch s related to the total cost of the system. Ths functon also conssts s of two parts: the frst one calculates the cost of reparable components. In ths part, e accounts for the nterconnectng hardware, α and β are the constants representng the physcal characterstc of each component n subsystem R, and T s the operatng tme durng whch the component must not fal (Dhngra, 99). The second part calculates the cost of reparable components, where a, b, and q are postve real numbers, whle p s negatve, ( A) (Chang & Chen,7). Eq. () shows the constrant on maxmum weght, whle Eq. () ndcates the constrant of maxmum volume for the system. Constrant (5) s related to the maxmum and mnmum numbers of permtted components n each subsystem and constrant (6) shows the condtons of the decson varables. Constrant (7) denotes the doman of relablty for non-reparable components and constrant (8) denotes the doman of n () (5) (6) (7) (8)

6 9 avalablty for reparable components. For the purposes of system avalablty analyss and calculaton, we assume that the lfe tme and repar tme are exponental dstrbutons. Therefore: n f : Max Asys ( t) = Max ( ( Av ( t)) ) f t( ) λt n λ µ + λ µ n = ( ( e ) ) ( ( e ) λ + µ λ + µ R A β n : T s ( p q MnC ( ) ( ) ) sys = α n + e + n a λ + b µ R λt A (9) () subject to n () s wne W ( A R) ( A R) wv n V p n N, ( A R), ( A R) λ U, ( A R) n Z + L L λ λ µ µ µ U, A Chern (99) proved that the redundancy allocaton problem n ts smplest form of seres system was an NP-hard problem. Therefore, the proposed model whch s more complcated than the model gven by Chern would also be NP-hard. In order to maxmze the objectve functon of ths model, t s reasonable to use a meta-heurstc method such as GA as used n ths paper.. The proposed genetc algorthm Genetc algorthm (GA) was frst proposed by Holland (975) and has been one of the most applcable meta-heurstc methods for solvng combnatoral optmzaton problems over the past three decades. Generally, GA s employed for solvng models wth one objectve functon. In ths paper, ths algorthm s used to solve the proposed b-objectve model. In the frst step, theε -constraned method s used to determne the optmzed value for the second objectve functon (cost). Ths value s then used n the problem constrants, and the sngle-objectve problem s solved by the genetc algorthm. Snce the value obtaned for the second objectve functon s the mnmum value of the system cost, by the optmal value for the frst objectve functon can also be obtaned by the gradual ncrease of ths value n the constrant. It s clear that whenever the cost ncreases, due to releases n the added constrant, avalablty of the problem should be better than before. Ths trend produces dfferent solutons for the problem for dfferent levels of costs and avalablty, whch makes the decson makers able to select approprate solutons by consderng dfferent crtera. Below s presented a complete descrpton of the proposed genetc algorthm.. Chromosome defnton For the proposed GA, each chromosome ncludes s genes where the frst row presents the number of components used n subsystems, the second row presents the lfe tmes of the components n each subsystem and the thrd row shows the repar tmes of the components n the reparable subsystems. These genes are randomly produced at gven ntervals. In other words, the value for each gene should () () () (5) (6)

7 H. Zoulfaghar et al. / Decson Scence Letters (5) 95 be verfed n constrants (), (5), and (6). Fg. represents the chromosome structure consdered for ths problem. Components level Lfe tme n n λ λ Non-Reparable n s λ s n s λ s n s + λ s + Repar Tme µ s + Reparable n s + λ s + µ s + ns n s λs λ s µ s µ s Fg.. Chromosome (soluton representaton). Ftness functon As mentoned above, to solve the proposed b-objectve model by GA, the second objectve s consdered as a constrant so that the model becomes a sngle-objectve one. Ftness functon (ff) s the value of the frst objectve functon (system avalablty) plus the penalty for constrant volaton. In other words, the problem constrants are added to the objectve functon n such a way that f one soluton goes beyond the constrants, a relatvely large amount of penalty s added to the objectve functon. Ths penalty keeps the feasblty of the fnal soluton whle t also provdes the search n the nfeasble space of the problem. The search n the nfeasble space leads to an approprate dversty for the genetc algorthm.. Intal populaton In order to produce an ntal populaton, Pop chromosomes are randomly generated. In ths paper, populaton sze (Pop) s equal to. Ths number of populaton has been used n prevous studes such as Safar () and Debb et al. (). Safar () states that n problems wth a bg soluton space, the number of prmary populaton should be more than.. Selecton In order to select the requred chromosomes n the crossover operaton, the followng steps need to be taken. The ftness functon (ff) s calculated for all the exstng chromosomes (Pop) n the present populaton. Then, from Pop present chromosomes, k chromosomes are randomly selected and sorted based on ff. The chromosome wth the largest ftness functons (avalablty-penaltes) s selected as the parent for generatng a new populaton. Ths process wll be repeated Pop tmes untl Pop parents are fnally selected for the crossover and mutaton operators..5 Crossover Crossover takes place at a certan rate. Usng the crossover operaton, sx offsprng are generated from each two parents. The two parents and the sx offsprng create eght chromosomes and the two premer chromosomes based on ff are selected to transfer to the next generaton. As a result, there wll be Pop populaton at the end of the crossover operatons. In order to produce these sx offsprng from the two selected parents, the followng steps are taken: Step : Two random numbers (, ) s selected from s + to s. m m are selected such that m s n the nterval ( to s ) and m

8 96 Step : The genes n the nterval ( to m ) for each parent are exchanged to produce two offsprng. Step : The genes n the nterval ( s + to m ) for each parent are exchanged to produce two other offsprng. Step : The genes n the nterval ( to m ) and those n the nterval ( s + to m ) for each parent are exchanged at the same tme to produce the last two offsprng. Ths knd of crossover leads to n the enhanced capablty of the algorthm for fndng better solutons. Here, an llustratve example s used to explan the crossover operaton. Suppose that for the case n Fg., there s a system wth 6 subsystems, of whch nclude non-reparable components and nclude ' reparable ones whle: m=, m = ; n 5 =,,...,6 ;. λ. =,,...,6 ;. µ.5 =,5, 6. Now, there are eght chromosomes and we should select two superor ones. For ths selecton, ff s calculated for all the eght chromosomes and they are compared wth each other. Fnally, two chromosomes wth the hghest value of ff are selected..6 Mutaton The mutaton operator s also used at a certan rate whch s less than that of the crossover operator. The man purpose of applyng the mutaton operator s to ncrease dversty and to avod trappng n the local optmzaton. In ths operator, one offsprng s randomly selected from among two chromosomes produced by the crossover operator. Non-reparable reparable Frst Parent Second Parent Frst Chld Second Chld Thrd Chld Fourth Chld Ffth Chld Sxth Chld Fg.. Crossover Operaton

9 H. Zoulfaghar et al. / Decson Scence Letters (5) 97 ' ' ' ' Two random numbers ( m, m ) are consdered such that m s selected from ( to s ) and m s selected from ( s + to s ) and the values of these two genes are exchanged. Then, ff s calculated for the muted offsprng and compared wth the value for ff of the pre-mutaton chromosome. If the value for ff of the new offsprng s greater than that of the prevous one, t wll then be replaced by the newly generated offsprng. Otherwse, the prevous offsprng remans as the superor ones. For example, suppose that n ' ' Fg., the offsprng has been selected for mutaton, m = and m = 5. Fg. represents the mutaton operator for these random values. Chld before mutaton Non-reparable reparable..7.. Chld after mutaton Fg.. Mutaton Operaton Stoppng crtera The GA process wll contnue untl a predefned number of teratons (Gen). In ths paper, the number of teratons s set equal to 5 generatons. 5. A numercal example Ths part of the paper ncludes an example whose data s a combnaton of those appled n Zou et al. () and Chang and Chen (7). In ths example, the system ncludes subsystems where subsystems to 5 have non-reparable components whle subsystems 6 to have reparable components. Maxmum allowable weght and volume for the system are (unts of weght) and 8 (unts of volume), respectvely. Maxmum and mnmum numbers of allowable components n each subsystem have been consdered as 5 and, respectvely. Other detals are presented n Table. To solve ths problem, the mproved genetc algorthm proposed n ths paper has been used. The mproved GA desgned here has been coded by MATLAB software and run on a computer wth G of RAM. In ths paper, some prelmnary experments were used and the crossover and mutaton rates were set to.9 and., respectvely. Also, the populaton sze and maxmum generatons were taken to be and 5, respectvely. Table detals of problem nonreparable Reparable w wv α 5 β a b Sub Sub Sub Sub Sub Sub Sub Sub Sub Sub p q L λ U λ L µ U µ

10 98 In order to show the capablty of the genetc algorthm, the problem has also been solved by the Improved Partcle Swarm Optmzaton (IPSO) algorthm proposed n Wu et al. (), whch s consdered as one of the best algorthms n RAP so far. They demonstrated that IPSO s an algorthm wth a great capablty for solvng these problems. Therefore, ths algorthm was selected as sutable for makng comparsons. Also, for the IPSO, the populaton sze was selected to be PS =, maxmal number of teratons was set at K=5, and the mutaton probablty to p m =.5. The two algorthms were run tmes for each value of cost movng from to 5 and the results were presented n Table. In ths Table, SD represents standard devaton whch s based on the convergng values of the objectve functon. SD s expressed as follows: SD = f f ( k ) (7) k = where, f k s the k th convergng value of the objectve functon, and f represents the average value (medan) of the objectve functon. Based on the four crtera (Best, Worst, Medan, and SD) n Table, the mproved GA proposed here outperformed IPSO for all groups of cost values. The outperformance of the mproved GA s due to the use made of the crossover and mutaton operatons desgned here. These results also show that the avalablty of the system ncreases wth ncreasng cost. The solutons thus obtaned are shown n Table and the detaled soluton for the cost value of s presented n Table. Table Comparson of results for example between GA and IPSO Soluton number Cost Algorthm Best Worst Medan SD GA e- IPSO e-5 5 GA e-5 IPSO e- GA e-5 IPSO e-5 5 GA e- IPSO e- 5 GA e-5 IPSO e- 6 5 GA e-5 IPSO e- 7 GA e- IPSO e- 8 5 GA e-5 IPSO e- Fg. 5 shows the convergence of the objectve functon value n each generaton. Ths soluton belongs to one of the teratons for a cost equal to. The near-optmal soluton (objectve functon value = ) was acheved after approxmately 5 generatons. Table Detals of soluton number wth cost= Sub. Sub. Sub. Sub. Sub.5 Sub.6 Sub.7 Sub.8 Sub.9 Sub. n λ µ

11 H. Zoulfaghar et al. / Decson Scence Letters (5) 99 Objectve functon value generaton Fg. 5. Objectve functon value convergence The Pareto front results are llustrated n Fg. 6 by usng the medan of avalablty. Ths Fgure shows that for each value of cost, the avalablty obtaned by GA s better than that obtaned by IPSO. Avalablty Medan GA IPSO Cost Fg. 6. Pareto front of results for GA and IPSO These results also demonstrated that the convergence and stablty of the proposed GA are better than those of the IPSO algorthm. Fg. 7 shows that n 7 out of 8 cases, the value of SD for GA was smaller than that for IPSO. The precson of the genetc algorthm s also observed to be hgher than that of the IPSO algorthm. These ndcate that the proposed GA s a robust optmzaton algorthm. SD 9.E- 8.E- 7.E- 6.E- 5.E-.E-.E-.E-.E-.E Soluton Number GA IPSO Fg. 7. Compare stablty algorthms

12 6. Summary and Conclusons In redundancy allocaton problems (RAPs), t s commonly assumed that the system conssts of ether only reparable or non-reparable components. As an extenson to ths assumpton, a system consstng of both reparable and non-reparable components was consdered n ths paper and a new mathematcal model was developed for the system. The problem has been formulated as a nonlnear nteger programmng model subject to a number of gven constrants. Snce the RAPs belong to the NP-hard class of optmzaton problems, t s not easy to solve the proposed model n real cases, especally for large systems. Therefore, meta-heurstc methods are suggested for solvng such a hard and complex problem. In ths paper, an mproved genetc algorthm (GA) was developed as an effectve metaheurstc algorthm for solvng the RAP. The results obtaned by the genetc algorthm showed the satsfactory and approprate avalablty of the system. In addton, the precson of the genetc algorthm was shown to be hgh when compared wth one of the best algorthms reported n the lterature. For future work, the authors are nvestgatng the extenson of the proposed model by ntroducng fuzzy numbers. References Ardakan, M. A., & Hamadan, A. Z. (). Relablty redundancy allocaton problem wth coldstandby redundancy strategy. Smulaton Modellng Practce and Theory,, 7-8. Ardakan, M. A., & Hamadan, A. Z. (). Relablty optmzaton of seres parallel systems wth mxed redundancy strategy n subsystems. Relablty Engneerng & System Safety,, -9. Ardakan, M. A., Hamadan, A. Z., & Alnaghan, M. (5). Optmzng b-objectve redundancy allocaton problem wth a mxed redundancy strategy. ISA transactons, 55, 6 8. Afonso, L. D., Maran, V. C., & dos Santos Coelho, L. (). Modfed mperalst compettve algorthm based on attracton and repulson concepts for relablty-redundancy optmzaton. Expert Systems wth Applcatons, (9), Barlow RE, Proschan F. (98).Statstcal theory of relablty and lfe testng probablty model. Slver sprng. Bej, N., Jarbou, B., Eddaly, M., & Chabchoub, H. (). A hybrd partcle swarm optmzaton algorthm for the redundancy allocaton problem. Journal of Computatonal Scence, (), Cao, Y., Sun, H., Trved, K. S., & Han, J. J. (). System avalablty wth non-exponentally dstrbuted outages. Relablty, IEEE Transactons on, 5(), Chen, T. C., & You, P. S. (5). Immune algorthms-based approach for redundant relablty problems wth multple component choces. Computers n Industry, 56(), Chern, M. S. (99). On the computatonal complexty of relablty redundancy allocaton n a seres system. Operatons Research Letters, (5), 9-5. Lang, Y. C., & Smth, A. E. (). An ant colony optmzaton algorthm for the redundancy allocaton problem (RAP). Relablty, IEEE Transactons on,5(), 7-. Chang, C. H., & Chen, L. H. (7). Avalablty allocaton and mult-objectve optmzaton for parallel seres systems. European journal of operatonal research, 8(), -. dos Santos Coelho, L. (9). An effcent partcle swarm approach for mxed-nteger programmng n relablty redundancy optmzaton applcatons.relablty Engneerng & System Safety, 9(), Cot, D. W. (). Maxmzaton of system relablty wth a choce of redundancy strateges. IIE transactons, 5(6), Deb, K., Pratap, A., Agarwal, S., & Meyarvan, T. A. M. T. (). A fast and eltst mult-objectve genetc algorthm: NSGA-II. Evolutonary Computaton, IEEE Transactons on, 6(), DeWnter, F. A., Paes, R., Vermaas, R., & Glks, C. (). Maxmzng large drve avalablty. Industry Applcatons Magazne, IEEE, 8(), Dhngra, A. K. (99). Optmal apportonment of relablty and redundancy n seres systems under multple objectves. Relablty, IEEE Transactons on,(),

13 H. Zoulfaghar et al. / Decson Scence Letters (5) Elegbede, C., & Adjallah, K. (). Avalablty allocaton to reparable systems wth genetc algorthms: a mult-objectve formulaton. Relablty Engneerng & System Safety, 8(), 9-. Fyffe, D. E., Hnes, W. W., & Lee, N. K. (968). System relablty allocaton and a computatonal algorthm. Relablty, IEEE Transactons on, 7(), Garg, H. (). Performance analyss of complex reparable ndustral systems usng PSO and fuzzy confdence nterval based methodology. ISA transactons, 5(), 7-8. Ha, C., & Kuo, W. (6). Relablty redundancy allocaton: An mproved realzaton for non-convex nonlnear programmng problems. European Journal of Operatonal Research, 7(), -8. Hamadan, A. Z., & Khorshd, H. A. (). System relablty optmzaton usng tme value of money. The Internatonal Journal of Advanced Manufacturng Technology, 66(-), Hamadan A.Z. (98). Avalablty and Relablty Modelng, Ph.D. thess, Unversty of Bradford, England. Holland JH. (975). Adaptaton n natural and artfcal systems. Mchgan: Unversty of Mchgan Press. Kuo, W., & Prasad, V. R. (). An annotated overvew of system-relablty optmzaton. Relablty, IEEE Transactons on, 9(), Lee, K. W. (). Stochastc models for random-request avalablty. Relablty, IEEE Transactons on, 9(), 8-8. Levtn, G., & Lsnansk, A. (). A new approach to solvng problems of mult state system relablty optmzaton. Qualty and relablty engneerng nternatonal, 7(), 9-. Lang, Y. C., & Chen, Y. C. (7). Redundancy allocaton of seres-parallel systems usng a varable neghborhood search algorthm. Relablty Engneerng & System Safety, 9(), -. Ma, Y., Han, J. J., & Trved, K. S. (). Composte performance and avalablty analyss of wreless communcaton networks. Vehcular Technology, IEEE Transactons on, 5(5), 6-. Msra, K. B., & Sharma, U. (99). An effcent algorthm to solve nteger-programmng problems arsng n system-relablty desgn. Relablty, IEEE Transactons on, (), 8-9. Msra, K. B. (97). Relablty optmzaton of a seres-parallel system. Relablty, IEEE Transactons on, (), -8. Ng, K. Y., & Sancho, N. G. F. (). A hybrd dynamc programmng/depth-frst search algorthm, wth an applcaton to redundancy allocaton. IIE Transactons, (), Onsh, J., Kmura, S., James, R. J., & Nakagawa, Y. (7). Solvng the redundancy allocaton problem wth a mx of components usng the mproved surrogate constrant method. Relablty, IEEE Transactons on, 56(), 9-. Ouzneb, M., Nourelfath, M., & Gendreau, M. (8). Tabu search for the redundancy allocaton problem of homogenous seres parallel mult-state systems. Relablty Engneerng & System Safety, 9(8), Safar, J. (). Mult-objectve relablty optmzaton of seres-parallel systems wth a choce of redundancy strateges. Relablty Engneerng & System Safety, 8, -. Sercola, B. (999). Avalablty analyss of reparable computer systems and statonary detecton. Computers, IEEE Transactons on, 8(), Soltan, R. (). Relablty optmzaton of bnary state non-reparable systems: A state of the art survey. Internatonal Journal of Industral Engneerng Computatons, 5(), 9-6. Srvastava, V. K., & Fahm, A. (988). k-out-of-m system avalablty wth mnmum-cost allocaton spares. Relablty, IEEE Transactons on, 7(), Valan, E., & Valan, E. (). A cuckoo search algorthm by Lévy flghts for solvng relablty redundancy allocaton problems. Engneerng Optmzaton,5(), Valan, E., Tavakol, S., Mohanna, S., & Hagh, A. (). Improved cuckoo search for relablty optmzaton problems. Computers & Industral Engneerng, 6(), Wang, L., & L, L. P. (). A co-evolutonary dfferental evoluton wth harmony search for relablty redundancy optmzaton. Expert Systems wth Applcatons, 9(5), Wu, P., Gao, L., Zou, D., & L, S. (). An mproved partcle swarm optmzaton algorthm for relablty problems. ISA transactons, 5(), 7-8.

14 Yeh, W. C., & Hseh, T. J. (). Solvng relablty redundancy allocaton problems usng an artfcal bee colony algorthm. Computers & Operatons Research, 8(), Zhao, M. (99). Avalablty for reparable components and seres systems. Relablty, IEEE Transactons on, (), 9-. Zo, E., & Bazzo, R. (). Level Dagrams analyss of Pareto Front for mult-objectve system redundancy allocaton. Relablty Engneerng & System Safety, 96(5), Zou, D., Gao, L., Wu, J., L, S., & L, Y. (). A novel global harmony search algorthm for relablty problems. Computers & Industral Engneerng, 58(), 7-6. Zou, D., Gao, L., L, S., & Wu, J. (). An effectve global harmony search algorthm for relablty problems. Expert Systems wth Applcatons, 8(), Zoulfaghar, H., Hamadan, A. Z., & Ardakan, M. A. (). B-objectve redundancy allocaton problem for a system wth mxed reparable and non-reparable components. ISA transactons, 5(), 7-.