A Fuzzy Cost-based FMEA Model

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1 Proceedigs of the 00 Iteratioal oferece o Idustrial Egieerig ad Operatios Maagemet Dhaka, agladesh, Jauary 0, 00 Fuzzy ost-based FME Model fshi Jamshidi Departmet of Idustrial Egieerig Payame Noor iversity, Tehra, Ira Reza aradara Kazemzadeh Departmet of Idustrial Egieerig Tarbiat Modarres iversity Tehra, Ira bstract Nowadays, improvig ad upgradig quality of products ad services is oe the mai reaso for outgoig the competitors ad peetratio ito markets. This article is about the deficiecy of the traditioal FME, ad itroduces a ew method based o estimated costs fuzzy FME ad utility values. I the proposed method a ew fuzzy RPN is defied. pair wise compariso amog Severity, Occurrece ad Detectio by the HP method has bee doe to obtai a ew fuzzy membership fuctio. I the proposed method the limited budget of compay for improvig activities is also cosidered. Fially, the case study shows this ew membership fuctio calculate actual costs due to failures, ca better prioritize failure modes, ca improve more potetial failures tha Dog method ad this is the most importat superiority our ew method. Keywords FME, Fuzzy Logic, tility Theory, Risk Priority Idex, HP.. Itroductio FME is a systematically method for idetifyig ad prevetig the occurrece of potetial failures i the product desig, product ad productio process. More tha 0 years of FME method i the world passes ad this method first time was used i erospace Idustries NS. Despite the abudat use of FME, this method is faced with restrictios. First; determie the exact probability of failure is difficult or impossible. May of the iformatio i this way are ordial. Secod; i FME method ifluece parameters (occurrece, severity, detectio) ofte cosider with a same weight. Third; i FME method abutmet betwee scores is t clear. For example a perso may cosiders a severity score, ad other oe 6 or 7. Fourth; I FME method i prioritized fails about costs ad profitability does ot speak ad be igored. Fifth; Some times by differet O, D, S the same RPNs are obtaied that will cause cofusio i priority. Sixth; Most of fuzzy FME methods, apply if-the methods but this method eeds expert kowledge. Seveth; ustomer discussio i FME is a very pale ad partly opiios, demads ad customer security is cosidered that after multiplyig the two quatities of occurrece ad detectio its effect reduce. Very importat efforts i FME literature for overcome the iadequacies of the traditioal RPN is doe. For example fuzzy logic widely is used for FME. hag ad Lee () used fuzzy liguistic terms such as very low, low, to evaluate O, S, D ad utilized grey relatioal aalysis to determie the risk priorities of potetial causes []. owles, Pelaez ()described a fuzzy logic based approach for prioritizig failures i a system FME, which uses fuzzy liguistic terms to describe O, S, D ad the risks of failures[]. ased o the above described fuzzy logic approach, Xu ad Zhu (00) developed a fuzzy FME assessmet expert system for diesel egie s gas turbocharger [] ad hi, ha, ad Yag(i press) developed a fuzzy FME based product desig system called EPDS-[]. uildig a fuzzy if the rule base is thought to be tedious ad critical to fuzzy FME. raglia et al. (00) proposed a risk fuctio which allows fuzzy if the rules to be geerated i a automatic way [6]. Garcia ad Melo(00) a data evelopmet aalysis method (DE) for FME suggested[] ad accordig to Fuzzy probability model developed by Lertworasirikul ad Nuttle (00) to determie the rakig idicators betwee fail modes used[]. Kai ad Lim(006) argued that it might be ot true to assume fuzzy if the rules to be certai ad of equal importace [7]. They therefore proposed the use of weighted fuzzy productio rules i fuzzy iferece system of FME, which allows a global weight to be attached to each if the rule. Recetly also Yag ad hi (00) have preseted a paper that i this paper they treat the risk factors O, S ad D as fuzzy variables ad evaluate them usig fuzzy liguistic terms ad fuzzy ratigs []. I view of fact this subject that oe of the major disadvatages of FME is ot cosiderig the cost i calculatig RPN, but there are few research published sice 00. Seu ad Ishii (00) some of disadvatage the traditioal FME 0

2 idetified ad ew method for FME based o cost life course itroduced that risks i terms of cost to measure[]. Dog (00) used utility theory ad fuzzy membership fuctios for the assessmet of severity, occurrece, ad detectio [] ad armigai (00) preseted a ew itegrated approach, amed priority-cost FME (P-FME), i order to exceed limits of FME[0]. So this paper is about solvig disadvatages of FME method, presetig a ew fuzzy method based o cost, cosiderig firm budget as a limit ad assessig weight for each RPN factors ad each expert.. T ad FT-based FME ad New approach Sice the traditioal FME uses ordial umbers to rak the severity, occurrece, ad detectio of failure modes, it caot provide a estimatio of the cost due to failure sice the cost of a failure mode raked 0 is ot always te times of a failure mode raked. Sice the ultimate goal of FME is to reduce the cost due to failure, the cost due to failure modes should be the objective for decisio-makig. The expected cost E() due to a failure mode ca be expressed as: E( ) fmpfm( Pd ) () where fm is the cost due to a failure mode, pfm is the probability of this failure mode ad pd is the probability that this failure will be detected. Sice the severity, occurrece ad detectio of a failure mode determies the failure cost, they ca be regarded as cost drivers i the utility theory (T). tility theory is a attempt to ifer subjective value, or utility, from choices. I this case, each cost driver is raked from to 0. ost values are coverted ito utility values by dividig the cost value of the highest level for each cost driver, i.e.[] i i () 0 I FT, the utility values are expressed by membership fuctios istead of real umbers. osider severity, it is raked from to 0. The cost value for level i give by egieer j is deoted as s, i 0, j, where is the umber of egieers. s s () s 0, j fter this trasformatio, utility values are betwee 0 ad. The cost ad utility values for detectio ca be derived i the same approach as severity. The evaluatio of occurrece is differet from that of severity ad detectio, sice the probability of failure is give, as show i Table I. These probability values are coverted to the utility values as: o log () where po is the probability that a failure mode occurs. fter this coversio, the utility values for occurrece are betwee 0 ad. I FT method the resultig RPI is fuzzy ad is expressed by the membership fuctio istead of a geeral utility value as: RPI ssoodd µ ( RPI ) [ µ ( s ) µ ( o ) µ ( d )] () I this paper, triagular membership fuctios are used. For a triagular membership fuctio, the miimum ad maximum utility values give by the egieers form the two bottom poits, ad the average of the utility values form the top poit, i.e. u max(j ) m j Po l mi( j ) (6) The membership grades are 0 for utility values L ad ad for utility value M. This is based o the assumptio that amog the utility values give by the egieers, the average of the values is more likely to deote the actual failure cost tha the miimum value or maximum value[6,]. For a specific failure mode, each egieer determies the cost values si for the severity ad the rakig of severity. The utility values are obtaied usig Eq. () ad the membership fuctio for severity is derived usig Eq. (6). I the same approach, the membership fuctios for the occurrece ad detectio ca be derived. ut the process has may disadvatages. Such as it takes log time for performace ad requires too much calculatio for gettig to utility values. lso each three RPN factors have the same importace. For example if egieers take part i FME team, sice for each RPN factors, there are 0 ratig ad for RPN factors(s,o,d), therefore 00 calculatio are eeded to achievemet for cost values ad 0 calculatio for utility values. I order to solve this disadvatage, we should describe s, o, d i away that the FME method ca be performed easily by users ad reached to desirable results. I this study i order to provide a ew FME model i fuzzy mode ad cosiderig cost idex, to reach a completely ew fuzzy membership fuctio o, s ad d are defied ad modified as follows. j 06

3 o d d d s s s i,,, ; Egieers umber; j,,, (Failure modes); Po The failure evet probability. eefits ad dvatages of the New Fuctios to the Previous Dog Fuctios. I fuctios proposed, by usig relatios (7), we will get ormalized umbers. ecause these relatios exactly are the same with relatio () for ormalizatio i tables comparig umbers i HP method.. Dog process method is very time cosumig ad eeds log computig for achievig utility values. For example if egieers exist i FME team, by cosiderig for each RPN factors, there is 0 rak ad cost values is eeded for detectio ad severity factors. Therefore eed to be calculated 00 umbers ( 0 00) ad for utility values because of for each RPN factors, there is 0 rak ad utility values is eeded for detectio ad severity ad occurrece factors, so eed to calculate 0 umbers for determiig utility values ( 0 00 for detectio ad severity factors ad 0 umbers for occurrece factor). I Dog method we have to calculate 0 umber, while usig relatio 7 oly eed to calculate 0 umbers (00 umbers for detectio ad severity cost values ad 0 umbers for calculatig occurrece utility values).. Work with proposed ew fuctios is very easy ad there is o eed to draw detectio ad severity utility values diagram. s a result, usig s ad o ad d proposed i this article, ew fuzzy membership fuctio will be follows form. s d µ ( RPI ) µ ( ) ( ) ( ) µ µ s d It should be metioed that the egieers based o their experiece ad kowledge, we ca get particular weight which this weight should be betwee zero ad oe ad total weights for all egieers, should be oe. For example if egieers exist i FME team, their weights ca be 0., 0.0, 0.0, 0., 0. respectively that total weights for them is oe. ccordig to weight (W) for each egieer ad cosiderig their kowledge level i the cost calculatio for RPN factors ew fuzzy fuctio (0) ca to be defied as follows: s µ ( RPI ) W ( ( ) ( ) ( µ µ µ s d d )) r m ase study shows that work with the ew fuzzy membership fuctio is very simple ad requires o log ad complex calculatio. s i part of the traditioal FME disadvatages metioed, oe of the major disadvatages of the traditioal FME is allocate equal weight to each idicator of the RPN that it has criticized by a lot of authors such as a Gilchrist [0]. Now i order to overcome this limitatio i traditioal FME method, the HP process i order to obtai a more reliable idicator i this research i fuzzy mode is used. Therefore cosiderig cost ad time criteria a pair of compariso betwee idices (detectio, occurrece, severity) is carried out. So by usig a questioaire we did these comparisos i Tractor ompay Tabriz-Ira i quality cotrol sectio to related HP matrix derive. The results of the questioaire are summarized i Table. a () Table : Rakig of occurrece a (7) (0) 07

4 Table : ssiged weights Idex Weight Detectio 0.0 Severity 0. Occurrece 0.6 Now after fidig related weights to each RPN factors ad by cosiderig (0) formula, we ca defie ew RPI as follows: ( ) s d µ ( RPI ) W (0. ( ) ( ) 0.0 ( )) µ µ + µ s d. ase study s a case study ew created fuzzy membership fuctio is expaded based o Dog example to clear its effects i prioritizatio. I Dog method failure mode by traditioal FME ad FME based o FT have evaluated respectively. Table : Traditioal FME for failure mode [] Table shows that for these three failure modes, the traditioal FME gives the equal RPN while the FTbased FME gives differet RPI s. Thus, the failure modes ca be better prioritized usig the FT-based FME. Whe the FT-based FME is used, the cost due to failure ca be assiged. ost ad utility values for severity ad detectio of failure mode by egieers i figure (), (),(),() is show. Figure : ost value for severity Figure : tility value for severity 0

5 Figure : ost value for Detectio Figure : tility value for Detectio tility values for the occurrece idex through Table () are obtaied. It should be metioed that these values are the same for each egieer. Detectio, Occurrece, Severity rakig are show i Table. Table : Rakig ad utility values for severity, occurrece, ad detectio Failure mode Table : Risk idices after defuzzificatio (RPI) risk priority idex (OM method) fter defuzzificatio, the RPI s of failure modes - are show i Table. Now ew created fuzzy membership fuctio is expaded based o previous example to clear its effects i prioritizatio. The for 0

6 testig the proposed method the umbers related to the previous example ad just three Figures (), () ad table () are used. Table 6: Rakig of severity, occurrece, ad detectio Failure mode egieer W rakig (s) cost S S S W S rakig O W O rakig ost(di) d d d W d s 6 i S s 7 7 i d s 6 0 i S s i d s i i S s i d Table 6 compares the ew fuzzy membership fuctio calculatigs with those of Dog method. I the third colum ( W ) for each egieers as a iovatio, a particular weight betwee zero ad oe is cosidered that the total value of them is equal to oe. The each of these weights i the calculatio of each fuzzy membership fuctio (detectio, severity, ad occurrece) are used. I the fourth colum (rak), cosidered raks for each failure mode by each egieers, for the severity factor is writte. O the base of these raks, usig, figures ad table, related cost (i the fifth colum) are obtaied. ccordig to s obtaied, fuzzy utility values by ew method i the sixth colum is preseted that covey Superiority of this method i compariso with Dog method i terms of comfort calculatios. To calculate fuzzy utility values for severity factor, it oly eeds to divide related cost for each egieer to the total costs obtaied from five egieers. The calculatios i colum 7 up to last colum are similar to those of colums oe to six. ased o the relatioship () ad fuzzy membership Fuctio obtaied through the table (6), RPI of each failure mode is obtaied (Figure ).. M(RPI) RPI Figure : The membership fuctio for RPI 0

7 Table 7: Risk idices after defuzzificatio Failure mode (RPI) The membership fuctio for the RPI eeds to be defuzzified to obtai the RPI value. We have used OM method i this study. I the OM method, the average of the miimum utility value ad the maximum utility value is cosidered to be the expected RPI. The results i Table (7) shows that priority failures, will be respectively ad ad. While priorities i the first example were ad ad. The advatage of the ew priority will be discussed at the ed of sectio (-) (We will show that i ew priority we ca improve more failure modes tha Dogs method). - Optimal mix of failure selectio Not all failures ca always be repaired or avoided due to corrective actios high costs. Oly a specific mix of failure modes ca be modified accordig to a specific budget provided by the firm. Therefore we eed a algorithm able to fid the optimal mix of faults to be repaired as well as to obtai the highest sum of their.i., imposig aboud o the cost that the firm is iclie to ivest i these problems. sig the simple simplex algorithm we ca obtai the followig relatios [0]: max j X j. I. j j X j cos t j budget Xj { 0, } with j is a geeric fault, xj is a variable that ca assume the value 0 if the actio is ot developed ad if the actio is developed,.i. is the ritical Idex of j-th fault. The budget is the available budget for the corrective actios. For example if firm budget to be cosidered 0 currecy, First the cost of each failure modes should be calculated usig the Table (6). The based o obtaied costs for each failure modes ad cosiderig limited budget for improvig potetial failure modes, Fial decisio o the selectio of failure modes for corrective actio occurs. Estimated cost of each failure modes are as follows: - The cost of failure mode : y cosiderig that i OM method (defuzzificatio method) RPN values of each failure modes obtai through ad, So costs of failure modes based o OM method ca be estimated. y cosiderig the relatios l u obtaied i the previous chapter ad Table (6) we have: Table : Risk idices after defuzzificatio ad ost of actio Failure mode (RPI) ost of actio(currecy) ad d are the osts related to the miimum of W i the table (6) ad S ad s,, are the osts related to the miimum of d s, d, W i the same table (osts for the occurrece factor is ot

8 cosidered, because i calculatig the utility value for this idex, there is o eed to calculate costs for occurrece factor ad utility values for this idex directly obtai through log ). Therefore, the total cost for will be equal (+) ad the total cost for l will be equal 7(+). Now based o OM method we u should calculate mea of these cost ie 7 ad that it will be. For other failure modes, costs are calculated i the same form. Now based o obtaied costs for each failure mode(table ) ad cosiderig limited budget (0 currecy), fial decisio o the selectio of failure modes for corrective actio occurs. We will choose ad failure modes for corrective actio. While without cosiderig limited budget, all three failures were amedable. Therefore, cosiderig limited budget is oe of the most importat factors i choosig failure modes for improve. lso ew priority shows that i ew method ad failure modes are selected for corrective actios, while i Dog method oly failure mode ca be chose for corrective actio. It meas that i ew method we ca improve more potetial failures tha Dog method ad this is most importat Superiority our ew method. 6. oclusio The same way that was expressed i itroductio, Traditioal FME uses RPN to prioritize failure modes. Sice the three idices used for RPN calculatio are ordial variables, thus producig these three descriptive variables ca ot be defie real costs due to failures. Therefore, i this research a ew fuzzy membership fuctio for RPN with regard to cost riteria were defied that advatage of the ew fuzzy membership fuctio is that the cost criteria is icluded ad also the level of experiece ad commets o each egieers has bee cosidered ad each RPN factors have a specific importat to each other based o cost ad time criteria. I additio, this method icludes expert kowledge ad if this people ot be available, this way ca use their kowledge. Therefore, it does t have limitatio related to the availability of a strog team i the old method. Fially, the case study shows this ew membership fuctio calculate actual costs due to failures ad ca better prioritize failure modes. Thus this method provides effective ad coveiet tool for failure aalysis ad improves FME implemetatio i failure ad risk aalysis for desig ad maufacturig productio ad assembly lies. Refereces. Dog,., 00, Failure Mode ad Effects alysis ased o Fuzzy tility ost Estimatio-Departmet of Mechaical Egieerig- urti iversity of Techology- ustralia.. raglia, M., Frosolii, M., ad Motaari, R., 00, Fuzzy TOPSIS pproach for Failure Mode, Effects ad riticality alysis, Quality ad Reliability Egieerig Iteratioal, vol., o., pp... hi, K. S., ha,., ad Yag, J.., 00, Developmet of a fuzzy FME based product desig system, Iteratioal Joural of dvaced Maufacturig Techology, vol. 6, pp. 6.. Xu, K., Tag, L.., Xie, M., Ho, S.L., ad Zhu, M.L., 00, Fuzzy assessmet of FME for egie systems, Reliability Egieerig ad System Safety, vol. 7, pp. 7.. Wag, Y.-M., hi, K.-S., Poo, G. K. K., ad Yag, J.-., 00, Risk evaluatio i failure mode ad effects aalysis usig fuzzy weighted geometric mea, Expert Systems with pplicatios, vol. 6, o., pp raglia, M., ad Frosolii, M., 00, Fuzzy criticality assessmet model for failure modes ad effects aalysis, Iteratioal Joural of Quality ad Reliability Maagemet, vol. 0, o., pp Tay, K. M., ad Lim,. P., 006, Fuzzy FME with a guided rules reductio system for prioritizatio of failure, Iteratioal Joural of Quality ad Reliability Maagemet, vol., o., pp Garcia, P..., Schirru, R., Frutuoso, P. F., ad Melo, E., 00, fuzzy data evelopmet aalysis approach for FME, Progress i Nuclear Eergy, vol. 6, o., pp. 7.. Rhee, S. J., ad Ishii, K., 00, sig cost based FME to ehace reliability ad serviceability, dvaced Egieerig Iformatics Joural, vol. 7, pp armigai, G., 00, itegrated structural framework to cost-based FME: The priority-cost FME, Reliability Egieerig ad System Safety, vol., o., pp hag,. L., Wei,.., ad Lee, Y. H.,, Failure mode ad effects aalysis usig fuzzy method ad grey theory, Kyberetes, vol., pp owles, J.., ad Pela ez,. E.,, Fuzzy logic prioritizatio of failures i a system failure mode, effects ad criticality aalysis, Reliability Egieerig ad System Safety, vol. 0, pp. 0.. Lertworasirikul, S., Fag, S.., Joies, J.., ad Nuttle, H. L. W., 00, Fuzzy data evelopmet aalysis (DE): a possibility approach, Fuzzy Sets ad Systems, vol., pp. 7.. Pillay,., ad Wag, J., 00, Modified failure mode ad effects aalysis usig approximate reasoig, Reliability Egieerig ad System Safety, vol. 7, pp. 6. O Po

A Fuzzy Cost-based FMEA Model

A Fuzzy Cost-based FMEA Model Proceeigs of the 00 Iteratioal oferece o Iustrial Egieerig a Operatios Maagemet Dhaka, aglaesh, Jauary 0, 00 Fuzzy ost-base FME Moel fshi Jamshii Departmet of Iustrial Egieerig Payame Noor iversity, Tehra,