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A Fuzzy Economic Replacement Decision Model Prakash R Chavan

Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 pp Fuzzy Economc Replacement Decson Moel Praash R Chavan ssstant Professor Department of Statstcs Smt. asturba
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Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 pp Fuzzy Economc Replacement Decson Moel Praash R Chavan ssstant Professor Department of Statstcs Smt. asturba Walchan Collage (rt & Sc) Sangl Maharashtra INDI. bstract: In every fel of our real lfe stuatons we eal wth a replacement problem when some tems such as machnes mecal equpment mltary tan electrc bulb etc. or worers nee to replace ue to ther ecrease effcency falure or brea own. To get a more realstc vew of a replacement problem here we conser that the analytc herarchy process (HP) for economc Replacement of component. In ths Paper we use Fuzzy HP metho for solvng replacement problem n fuzzy envronment. Lngustc values are use to assess the ratngs an the weghts for ey components. These lngustc ratngs can be expresse n trapezoal fuzzy numbers. Euclan stance metho s use to calculate the stance between two trapezoal numbers. Fnally a closeness coeffcent of each alternatve s efne to etermne the ranng orer of all alternatves (ey Components). Introucton The stanar replacement polcy s a basc an well-nown polcy n mantenance optmzaton. It s concerne wth the queston whch component s replace frstly. System servce lfe can be extene f a sutable mantenance polcy has been aopte. ccorng to the contents mantenance s classfe n to two types preventve mantenance (PM) an correctve mantenance (CM). The former concerns wth the actvtes e.g. austng the operaton parameters reparng or replacng elements before the system brea-own etc. The latter eals wth the necessary repar or replacement of component as t fal. The avantage of PM s that the system can always be ept n an avalable conton as the stuaton nees. However the costs are sometmes much hgher as the replacement of component s taen [57]. In ths paper the mantenance s consere only for replacng elements no matter whch of PM or CM s chosen. Ths stuaton occurs frequently n the mantenance of some proft-orente system for example vehcles machne tool systems etc. The stues mentone above eals wth the PM polcy about the component whch has alreay been ece. For system the choce of ey components whch shoul be replace preventvely s another ssue. The man purpose of ths paper s to gve the replacement Orers of ey components n a system. The selecton of ey components follows fve crtera once the ey components have been chosen the orer of replacement s ece base on the nalytc Herarchy Process (HP) []. From the lterature t can be conclue that n replacement polcy the classcal concept of Optmalty may not always be the most approprate polcy. Over all speang we conclue that Corresponng ress: Research rtcle replacement polcy nvolve several an fferent types crtera combnaton of fferent ecson moels group ecson mang an varous forms of uncertanty. It s ffcult to fn out the replacement orers of ey components n a system. In essental the replacement of ey components problem s a group ecson-mang uner multple crtera. Uner many contons crsp ata are naequate to moel real-lfe stuatons snce human ugments nclung preferences are often vague an cannot estmate hs orers wth an exact numercal value. more realstc approach may be to use lngustc assessments nstea of numercal values. In other wors the ratngs an the weghts of the crtera n the problem are assesse by means of lngustc varables. Conserng the fuzzness n the ecson ata an group ecson mang process lngustc varables are use to assess the weghts of all crtera an the ratngs of each alternatve wth respect to each crteron. We can convert the ecson matrx nto a fuzzy ecson matrx an construct a weghtenormalze fuzzy ecson matrx once the ecsonmaer s fuzzy ratngs have been poole. In the concept of HP we gve the orers of ey components then Euclan stance metho s apple to calculate the stance between two fuzzy ratngs. In ths paper we use Fuzzy HP metho for solvng replacement problem n fuzzy envronment. Lngustc values are use to assess the ratngs an the weghts for ey components. These lngustc ratngs can be expresse n trapezoal fuzzy numbers. Euclan stance metho s use to calculate the stance between two trapezoal numbers. Fnally a closeness coeffcent of each alternatve s efne to etermne the ranng orer of all alternatves (ey Components). The analytc herarchy process (HP) s a structure technque for organzng an analyzng complex ecsons. Base on mathematcs an psychology t was evelope by Thomas L. Saaty n the 970s an has been extensvely stue an refne snce then. It has partcular applcaton n group ecson mang an s use aroun the worl n a we varety of ecson stuaton n fels such as government busness nustry healthcare an eucaton. Rather than prescrbng a correct ecson the HP helps ecson maers fn one that best suts ther goal an ther unerstanng of the problem. It proves a comprehensve an ratonal framewor for structurng a Copyrght 0 Statperson Publcatons Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 Mhatre R. G. Sananse S. L. ecson problem for representng an quantfyng ts elements for relatng those elements to overall goals an for evaluatng alternatve solutons. Users of the HP frst ecompose ther ecson problem nto a herarchy of more easly comprehene sub-problems each of whch can be analyze nepenently. The elements of the herarchy can relate to any aspect of the ecson problem tangble or ntangble carefully measure or roughly estmate well- or poorly-unerstoo anythng at all that apples to the ecson at han. Once the herarchy s bult the ecson maers systematcally evaluate ts varous elements by comparng them to one another two at a tme wth respect to ther mpact on an element above them n the herarchy. In mang the comparsons the ecson maers can use concrete ata about the elements but they typcally use ther ugments about the elements' relatve meanng an mportance. It s the essence of the HP that human ugments an not ust the unerlyng nformaton can be use n performng the evaluatons. The HP converts these evaluatons to numercal values that can be processe an compare over the entre range of the problem. numercal weght or prorty erve for each element of the herarchy allowng verse an often ncommensurable elements to be compare to one another n a ratonal an consstent way. Ths capablty stngushes the HP from other ecson mang technques. In the fnal step of the process numercal prortes are calculate for each of the ecson alternatves. These numbers represent the alternatves' relatve ablty to acheve the ecson goal so they allow a straghtforwar conseraton of the varous courses of acton. In ths Paper we use Fuzzy HP metho for solvng replacement problem n fuzzy envronment. Table : Matrx relatonshp between Functons an Components Functons Components C C C.. C m m postve trapezoal fuzzy number (PTFN) n% can be efne as efne as: 0 x n x / ( n ) n x n µ n% ( x ) = n x n x / ( n ) n x n 0 x n Lngustc values are use to assess the ratngs an the weghts for ey components. These lngustc ratngs can be expresse n trapezoal fuzzy numbers. Euclan stance metho s use to calculate the stance between two trapezoal numbers. Fnally a closeness coeffcent of each alternatve s efne to etermne the ranng orer of all alternatves (ey Components). The stanar replacement polcy s a basc an well-nown polcy n mantenance optmzaton. It s concerne wth the queston whch component s replace frstly. The mantenance s consere only for replacng elements no matter whch of PM or CM s chosen. Ths stuaton occurs frequently n the mantenance of some proft-orente system for example Vehcles machne tool system etc.. Components for Economc Replacement moel System performance can be ept as goo as possble f great care s taen n ts mantenance urng ts operaton. Mean-whle the lfe cycle of the system s extene an the effcency promote. To acheve ths goal the manner of how to mantan the system n a normal conton becomes mportant. Thus tang perocal replacement for some components n a system shoul be consere. The selecton of ey components an the replacement prorty/orers are presente n [57]. system conssts of many subsystems each carres out a specfc functon a typcal example of the machnery relatonshp between functons an components can be represente by a matrx wth elements enotng relaton exsts an 0 otherwse see table. t s better to esgn a system when t fals all components fal smultaneously. But ths s very ffcult to acheve for the real system thus replacement of component shoul be taen. () n n n n The membershp functon µ ( x) s n% Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue Page 86 Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 pp Euclean stance metho % an Let m = ( m ) m m m n % = n n n n be two trapezoal fuzzy numbers. Then the stance between them can be calculate by usng the Euclean stance metho as; v ( m % n % ) = / ( m ) + ( m ) + ( m ) + ( m ) () The Euclean stance metho s an effectve an smple metho to calculate the stance between trapezoal fuzzy numbers. ccorng to the Euclean stance metho two trapezoal fuzzy numbers m% an m % n % = 0. n% are entcal f an only f v µ ( x) n% Propose Methoology systematc approach to exten the HP s propose to solve the replacement of ey component n mechancal system uner fuzzy envronment. In ths paper the mportant weghts of varous crtera (ey components) an the ratngs of qualtatve crtera are consere as lngustc varables. Because lngustc assessments merely approxmate the subectve ugment of ecson-maers we can conser lnear trapezoal membershp functons to be aequate for capturng the vagueness of these lngustc assessments []. These lngustc varables can be expresse n postve trapezoal fuzzy numbers [Lngustc varables for mportance weght- Low Very Low Meum Meum Low Hgh Meum Hgh Very Hgh] [Lngustc varables for ratng- Poor Very poor Meum Poor Far Meum Goo goo Very Goo]. The mportance of weght of each crteron can be by ether rectly assgnng or nrectly usng parwse comparson. Fg.6.: Trapezoal fuzzy number n% In fact replacement of ey components s a group ecson mang problem whch may be escrbe by means of the followng sets: () a set of ecson-maers calle E = { D D...D } ; () a set of m possble functons calle = {... m }; () a set of n crtera C = { C C...C n } wth whch functon performances are measure; (v) a set of performance ratngs of ( =... m) wth respect to crtera C ( =... n) calle { } X = x =... m =... n. ssume that a ecson group has ecson maers an the fuzzy ratng of each ecson-maer D ( =...) can be represente as a postve trapezoal fuzzy number R % ( =...) wth membershp functonµ ( x ) R %. goo aggregaton metho shoul be consere the range of fuzzy ratng of each ecson-maer. It means that the ranges of all ecson-maers fuzzy ratngs. Let the fuzzy ratngs of all ecson-maers be trapezoal fuzzy numbers R % = ( a b c ) =.... Then the aggregate fuzzy ratng can be efne as; R % = abc =... () Copyrght 0 Statperson Publcatons Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 Mhatre R. G. Sananse S. L. Where a = mn ( a ) b = / b c = / c = = = max ( ) Let the fuzzy ratng an mportant weght of the th ecson maer be x % = ( a b c ) an w% = ( w w w w ) =... m an =... n respectvely. Hence the aggregate fuzzy ratngs ( x% ) of alternatves wth respect to each crteron can be calculate as: x % = a b c () Where; a = mn a = max ( ) ( ) b = / b = c = / c = The aggregate fuzzy weghts ( w% of each crteron ) can be calculate as: w % = w w w w (5) Where; w = mn a w = / b w = / c = = ( ) w = max s state above a replacement of ey components problem can be concsely expresse n matrx format as follows; x% x %... x %... x% n x% x %... x %... x% n D % = x% x %... x %... x% n x% x %... x %... x% m m m m n w % = w % w % w % w % Where; x = a b c w % = ( w w w w ) % an =... m an =... n can be approxmate by postve trapezoal fuzzy numbers. To avo complexty of mathematcal operatons n a process the lnear scale transformaton s use here to transform the varous crtera scales n to comparable scales. The set of crtera can be ve nto beneft crtera (the larger the ratng the greater the performance) an cost crtera (the smaller the ratng the greater the performance). Therefore the normalze fuzzy-ecson matrx can be represente as R % = r% (6) m n a b c r % = a - a - a - a - r % = c a + = m a x ( ) - a = m n ( a ) The normalzaton metho mentone above s esgne to preserve the property n whch the elements % are stanarze (normalze) trapezoal r fuzzy numbers. Conserng the fferent mportance of each crteron the weghte normalze fuzzy ecson matrx s constructe as V % = v % m n =... m =... n (7) Where v % = r % (.) w ccorng to the weghte normalze fuzzy ecson matrx normalze postve trapezoal fuzzy numbers can also approxmate the elements v%. Then the + fuzzy postve-eal soluton (FPIS ) an fuzzy - negatve-eal soluton (FNIS ) can be efne as = ( v % v % v % v% ) (8) = ( v % v % v % v% ) (9) Where + v = max ( v ) % an - v = m n v % =... m an =... n The stance of each alternatve (component) from + - an can be currently calculate as + n + =... m = v v v % = (0) - = v ( v v % ) =... n = () Where v numbers... s the stance between two fuzzy 5 Numercal Example We conser an example motvate by a reallfe system to emonstrate the practcal use of propose soluton. Power loom system n a formng machne conssts of fve components to carry out fve functons. fter prelmnary screenng three functons Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue Page 88 Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 pp ( ) reman for further evoluton. commttee of three ecson-maers D D an D has been forme to select the most sutable ey component. Fve beneft crtera are consere: () Motor ol (C ) () Ppe (C ) () Release valve (C) () Loom Motor (C ) (5) Motor Belt (C5) The propose metho s currently apple to solve ths problem the computatonal proceure of whch s summarze as follows: Step : Three ecson-maers use the lngustc weghtng varables to assess the mportance of the crtera. The mportance weghts of the crtera etermne by these three ecson-maers are shown n table. Step : three ecson-maers use the lngustc ratng varables to evaluate the ratngs of functons wth respect to each component. The ratngs of the fve functons by the ecson-maers uner the Varous crtera. Table : Importance weght of crtera from three ecsonmaers Decson-maers Crtera D D D C H VH VH C VH H H C VH H VH C VH H VH C 5 H VH H Step : The lngustc evaluaton shows n tables 6. an 6. are converte nto trapezoal fuzzy numbers to construct the fuzzy-ecson matrx an etermne the fuzzy weght of each crteron as n table. Step : The normalze fuzzy-ecson matrx s constructe as n table Table: Ratngs of the fve Functons by ecson-maers uner Varous Crtera. Crtera Functons Decson-maers MG MG MG G G G VG VG G C C C C C 5 MG MG VG G G VG G VG VG VG G VG VG G VG G G G G G G VG VG G G G VG VG G VG G VG G VG G VG Table : Prorty wth respect to C C C C C 5 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ccorng to the Decson Table of Prorty the replacement orers of ey Components are: 6 Conclung Remars extenson verson of HP n a fuzzy envronment s Fuzzy Economc Replacement problem ahere propose n ths Paper. to uncertan an mprecse ata an fuzzy set theory s ccorng to the Prorty wth respect to the aequate to eal wth them. In a replacement ecson crtera an alternatves we can etermne not only the process the use of lngustc varables n replacement ranng orer but also the assessment status of all ey ecson problems s hghly benefcal when components. Sgnfcantly the propose metho performance values cannot be expresse by means of proves more obectve nformaton for Economc numercal values. Due to the ecson-maers replacement of ey component n a system. experence feel an subectve estmates often appear n the replacement of ey component n a system an Copyrght 0 Statperson Publcatons Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue 0 Mhatre R. G. Sananse S. L. References. Bellman R. E. an Zaeh L.. Decson mang n a fuzzy envronment Management Scence. 7 () Fshburn P. C. (967). Utlty theory for Decson mang Wley New Yor.. Jarn Banevc an Mals. Optmal replacement polcy an the structure of software for contonbase mantenance. Journal of Qualty n Mantenance Engneerng () elly.. (997). Mantenance strategy. Butterworth Hene-mann. Lonon.U. 5. Macone.. Wess.E. TPM: planne an autonomous mantenance: brgng the gap between practce an research Proucton an Operatons Management. 7() Mann L. Saxena. napp.g. Statstcal-base or conton-base preventve mantenance. Journal of Qualty n Mantenance Engneerng. () Mler an star. Executve ecson wth Operatons Research Prentce Hall Englewoo olffs New Jersey 969. Internatonal Journal of Statsta an Mathemata ISSN: E-ISSN: Volume 6 Issue Page 90
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