Vehicle Equipment Selection Model for Preliminary Design Using Fuzzy Multi-Criteria Decision Making Method

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1 Vehile Equipment Seletion Model for Preliminry Design Using Fuzzy Multi-Criteri Deision Mking Method Ab. Smn Abd. Kder, Jmlih Idris, Mohd Afifi Abdul Mukti, Mohd Zmni Ahmd Fulty of Mehnil Engineering, Universiti Teknologi Mlysi Mlysi Abstrt: Equipments seletion proess is one of the key plnning nd design tivities of mrine, lnd nd ir vehiles. Equipments re often seleted from set of possible lterntives. More thn one deision mkers re usully involved nd seletion of one prtiulr hoie is mde ginst set of strtegi seletion riteri. Approximte mode of deisionmking is normlly employed nd hene seletion riteri re subjetively speified. This pper presents the fuzzy pproh to the subjetive representtion of design dt nd the multi-riteri deision mking method to the tretment of subjetive riteri rting nd rnking of equipment to serve s olletive deision mking tool..0 Introdution To redue ost some systems nd equipment fitted to newly designed vehiles re of-the-shelf types. Hene designing vehiles involves the seletion of systems nd equipment suh s engine monitoring nd ontrol system for high performne r, propulsion plnts for ships nd rdio nd teleommunition systems for irrfts. They re seleted from wide rnge of hoies nd seletion is mde ginst set of seletion riteri determined by the designers. It is multi-riteri deision mking proess nd hene few importnt notes worth mentioning. Firstly, the seletion riteri normlly over the three different spets; tehnil, eonomi nd environmentl. Tehnil spet inludes elements suh s tehnology level, tehnil speifition, tehnil omplexity nd user friendliness. Eonomi spet inludes elements suh s equipment ost, mintenne ost nd mintenne level. Environmentl spet inludes elements suh s sfety stndrd, risk level nd environmentl friendliness. Therefore, it is obvious tht some riteri re subjetive in nture while some re of risp quntifible vlues. Seondly, therefore, nd espeilly t the erly stge of the design spirl, deision mking in the seletion proess n be done on n pproximtion bsis. At tht stge it is quite eptble to ept n equipment bse on its pproximte ompline to the seletion riteri determined by the designers. In other words, the equipment omplies with the seletion riteri to ertin degree only. The designers define the rnge of speifition tht represents ompline nd hene they re onsidered s hving the expert knowledge on the mtter. It is lso ommon for the designers to represent eh ompline rnge linguistilly suh s sfe, modertely sfe, very sfe nd so on. Thirdly, deision is mde in mutully exlusive mnner. A deision mde on one seletion riteri will not effet the deision to be on the next. Neither hs it been ffeted by ny deision mde erlier. This further reflets the mbiguous nture of the deision mking proess. Fuzzy sets theory is onvenient nd flexible mthemtil tool for deling with pproximtion using linguisti deription (Avineri, 2000). It hs been erlier pplied, for exmple, to the seletion of mteril hndling equipment for the mining industry (Deb et l., 2002) nd to the seletion of dvned mnufturing system for investment evlution purposes (Krsk, 200). It employs pproximte, rther thn ext, modes of resoning, nd therefore inorportes impreise, linguisti nd subjetive vlues. Fuzzy method simply mimis humn wy of expressing opinions in the simplest wy. The pper pplies fuzzy onepts to equipment seletion in the preliminry design of mrine, lnd nd ir vehiles nd hs been orgnized in the following mnner. Setion 2.0 summrises the theoretil foundtion upon whih the deision mking model, methodology nd the nlytil tools hve been bsed. Setion 3.0 deribes the proposed model nd indites the fuzzifition, fuzzy opertions nd the lgorithm behind the men opertor method. Setion 4.0 Corresponding uthor

2 diusses strength of the fuzzy multi-riteri deision mking onept being pplied, the verstility of the model developed nd the ury of the results nd highlights some problems nd possible improvement. 2.0 THEORETICAL BACKGROUND 2. Fuzzifition nd Defuzzifition Method µ A nd often denoted by A = {( x, µ () x ) x X}. µ A is generlised hrteristi funtion (the membership funtion of the fuzzy set A), x is one prtiulr element tht belongs to A, X is the universe of diourse. The onditions re µ A() x = if x is totlly in A, µ A () x = 0, if x is totlly out of A nd 0< µ A() x < if x is prtly in A. A fuzzy set is defined by funtion () x : X [ 0,] A set whose membership funtion is pieewise ontinuous is lled fuzzy number. A fuzzy number ording to the onept of fuzzy set n be represented in tringulr form s in Figure 2 (other forms re trpezoidl nd S-shped). A tringulr fuzzy number with entre my be seen s fuzzy quntity x is pproximtely equl to. A linguisti vrible n be defined s vrible whose vlues re not numbers, but words or sentenes in nturl or rtifiil lnguge (Krsk, 200). Linguisti vrible suh s lrge or smll is tken s representtion of phenomenon too omplex to be deribed using the onventionl quntittive terms. Figure Tringulr fuzzy number Figure 2 Membership funtion nd prtitioning Therefore within universe of diourse linguisti vrible represents rnge of vlues tht mke up fuzzy set. The universe of diourse n be prtitioned into s mny linguisti vribles s deemed neessry nd prtitions n overlp s shown in Figure 3. The linguisti vribles re usully defined s fuzzy sets with pproprite membership funtions (Hong nd Lee, 996). H is linguisti vrible representing prtition tht deribes ertin phenomenon with hrteristi high in the universe of diourse. In fuzzy set theory membership is mtter of degree. In the bove expression µ ( A) is defining the degree of relevnt of x to the set A. Membership of x to A is impreise or vgue nd µ ( A) is its mesure of unertinty. The fuzzy proposition is true to the degree to whih x belongs to the fuzzy set. form A () x = A symmetri tringulr fuzzy number with entre nd width α > 0 hs membership funtion of the following x if x α α. The nottion use is A=(, α) 0 otherwise The proess of ssigning membership funtions to fuzzy vribles is either intuitive or bsed on some lgorithmi or logil opertions (Krsk, 200). Intuition is simply derived from the pity of the experts to develop membership funtions through their own intelligene, experiene nd judgement (Hong nd Lee, 996; Krsk, 200). Tringulr membership funtions re hosen for pplition onsidering their intuitive representtion nd ese of omputtion (Krsk, 200). A fuzzy number n be defuzzified using the entre of grvity method. Figure 4 illustrtes the opertion of defuzzifying using suh method.

3 2.2 Fuzzy Aggregtion nd Rnking The stright forwrd method for ggregting fuzzy sets in the ontext of deision-mking uses the ggregting proedures frequently used in utility theory or multi-riteri deision theory (Rj, 999; Avineri, 2000; Krsk, 200; Deb et l., 2002]. The method onsiders men vlue between vrieties of gol using verge between the optimisti lower bound (minimum degree of membership) nd the pessimisti upper bound (the mximum degree of membership). Hene the method is lled men opertor method. This method is normlly used for nlysis involving empiril dt nd the ontinuous use of this method shows tht it is n dequte model for humn ggregtion proedures in deision environments. The proedure for the men opertor method is s below. Let there be seletion situtions where: (i) there is designer j from group of designers j = to m, (ii) there is set of lterntive equipment i for i = to n for the vehile, (iii) the equipment is to be seleted using 2-level seletion pproh ginst; first, putting weightge on k strtegi subjetive seletion riteri, nd seond, putting rting on eh lterntive for eh k for k = to p (iv) the designers will use linguisti weighing W = (VP, P, F, G, VG) for the strtegi subjetive riteri where VP = very poor, P = poor, F = fir, G = good nd VG = very good nd linguisti rting R = (VL, L, M, H, VH) where VL = very low, L = low, M = medium, H = high nd VH = very high. If two designers re involved in weighing nd rting four different equipment the rting nd rnking n be tbulted in Tble 2. R ijk nd W kj re ll fuzzy numbers (R ijk, R b ijk, R ijk ) nd (W kj, W b kj, W kj ) respetively. Aggregting within one prtiulr k gives W = [ W W 2 W 3... W m ] (W in (row 3, olumn 4) of Tble 2) m nd R = [ R ] i i, Ri, 2 Ri, 3... Ri, m ( R in (row 4, olumn 4) of Tble 2). Aggregting W nd R m i i for one prtiulr k gives F = ( W R ) where F i i i is lled the fuzzy suitbility index. Aggregting ross ll k gives Fi = [( Ri W ) ( Ri2 W 2... ( Rip Wp) ] p Tble A mtrix illustrting fuzzy ggregting opertion Rnking of fuzzy sets is bsed on extrting vrious fetures from the fuzzy sets suh s its entre of grvity. Rj (999) nd Prodnovi nd Simonovi (2002) provides good omments on the vrious methods of rnking fuzzy numbers. Rnking eqution by Chen s method (Deb et l., 2002)

4 ( δi x min) ( x mx αi) isv ( Fi) = ( γi δi) + + ( βi αi). The mximising set is 2 ( x mx x min) ( x mx x min) k ( x x min) Μ = {( x, µ M ( x)) xεr} with membership funtion. µ M ( x) = x x for x mx x x min nd ( mx min) µ ( x) = 0, otherwise. The minimising set { } k ( x x mx) M N = ( x, µ N( x)) xεr is µ N( x) = x x for ( min mx) x mx x x min nd µ ( x) = 0 N otherwise; where k represents plnner s preferene on level of risk, k = when designers re onservtive or neutrl, k = 0.5 when designers re optimist or risk tker, k = 2 when designers re pessimist or risk verse, x min = inf D (opertor inf represents infimum, whih is the globl minimum [2]), x mx = sup D (opertor sup represent supremum, whih is the globl mximum [2], D = U i=,n D i ; (i.e. the set ontining the x µ F i( x) > 0 (the elements with positive membership degree µ Fi ). elements), D i = { } 3.0 The Proposed Model T build up the model to demonstrte the pplition of the fuzzy multi-riteri deision mking let there be se where systems nd equipment to be fitted to vehile n be seleted bsed on nine seletion riteri presented in Tble 2 below. In prtie it is lso quite ommon to use nine-digit likert le of 9; being the lest preferred to deide on these seletion riteri. In ses where the seletion riteri re represented by tul tehnil speifition vlues nd figures, these figures n lso be prtitioned into severl rnges. The figures n be split into nine prtitioned to mth the likert le. Fuzzifition omprises of deiding wht re the minimum nd mximum of these figures, the number of prtitions, the figures to be represented by eh prtition, nd the linguisti nming of eh prtition. Tble 2: Equipment seletion riteri during preliminry design Tehnil Eonomi Environmentl & others Tehnology level Equipment ost Sfety stndrd Opertor skill requirement Mintenne ost Comfort mesures Mintenne skill requirement Liene nd upgrding ost Environmentl friendliness If eh seletion riteri re to be split into five eqully sped prtitions nd linguistilly nmed s VERY LOW (VL), LOW (L), MEDIUM (M), HIGH (H) nd VERY HIGH (VH), its eqully sped overlpping membership funtion will be of the form s in Figure 3 below. Observe tht eh linguisti term is representing rnge of speifition vlues or likert le vlues.

5 Figure 3: A formt of membership funtion for seletion riteri To be ble to implement the men opertor method proposed by Rj (999), Avineri (2000), Krsk (200) nd Deb(2002) two unknown re required; (i) whih is the weight of importne of the seletion riteri ( ), nd (ii) er the equipment rting for eh hoie. Thus, for nine-riteri se = wo i,, nd, er = e r, 2, 3, 4, 5, 6 7, 8, er where, in fuzzy terms wo i = (, s, ) nd e r = er er ). b ( b Referring to Figure 3 the fuzzy nd er in likert le orrespond to the designers seletion of linguisti nmes bove s in the Tble 3 below. The designers deision on nd er n be summrized in Tble 4. Tble 3: Fuzzy nd er LINGUISTIC NAME nd er VL (,, 3) L (, 3, 5) M (3, 5, 7) H (5, 7, 9) VH (7, 9, 9) Tble 4: Summry of designers hoie of nd er

6 The fuzzy suitbility index ( fsi ) for eh equipment hoie h is then lulted by multiplying the equipment rting ( er ) with wo i. The produt is fs i whih is fuzzy number ( fsi, fsib, fsi ). The net fs i " for eh h is the normlised sum of ( er X wo i ) nd divided by 9. fs i h = 9 ( ) ( er2 2 ) ( ) ( ) 3 3 er4 4 ( ) ( ) 5 5 er6 6 ( ) ( ) ( ) 7 7 er8 8 " where the lultion of normlized fuzzy suitbility index for equipment hoie- ( fsi h fsi h = 9 6 ( er ) ( ) b,, b ( er2 ) ( ) 2 b 3 2,, 2b 2 ( er3 ) ( ) 3 b 3 3,, 3b 3 ( er ) ( ) 4 4 b 4 4,, 4b 4 ( er ) ( ) 5 5 b 5 5,, 5b 5 ( er ) ( ) 6 6 b 6 6,, 6b 6 ( er7 ) ( ) 7 b 7 7,, 7b 7 ( ) ( ) ( ) ( ) er8 8 b 8 8,, 8b 8 er,, b The output will be in tringulr fuzzy form (, b, ). The lultion is repeted for ll the equipment under omprison. The fuzzy suitbility indexes re then rnked using Chen s method (Deb et l., 2002) s below where hoie rnked = 2 ( x mx ( i x x min) min b ) ( x mx i) + ( b ) ( x mx x min) + ( bi i) i i ) is

7 xmx = mximum fsi vlue from ll h x = minimum fsi vlue from ll h min The output will be in risp form suh tht 0 < hoie rnked <. 0. The best hoie would be the equipment with the lrgest Chen s (Deb et l., 2002) rnking vlue. 4.0 Diussion nd Closing The fuzzy multi-riteri deision mking onept is n estblished onept. It hs been pplied to mny res of pplition. It is verstile nd n be pplied t different level of deision mking. From the model illustrted bove it n be seen tht the tehnique ssist deision mking in mny wys. Firstly, it n serve s quntittive method to deide on qulittive vribles. Seondly, representing rnge of vlues under one linguisti term nd proessing it in fuzzy wy in tul ft is men of tking re of unertinty in deision mking. Thus, seprte exerise of estimting the unertinty I the deision mking proess is not required. It is lredy inbuilt within. Thirdly, nd the most importnt is it dopts the humn modes of resoning tht uses pproximte, impreise, linguisti, nd subjetive vlues. This mkes deision mking friendlier. The bove model n esily be modified to suit prtiulr sitution. The types of seletion riteri n be djusted nd so s the number of riteri to be used. The designers would only need to deide on the weight of importne of eh seletion riteri nd the equipment rting to eh seletion riteri deided upon. The rtings re expert knowledge nd hene the designers re lwys onsidered s the expert when published dt re not vilble. The repetitive number runhing tsk n be mde simpler using spredsheet of omputer progrm. This ould mke dt entry nd other modifitions muh fster. The finl results fter being rnked using Chen s method (Deb et l., 2002) is normlly rrnged in deending order. The highest in the order is the best equipment being onsidered. However, if the equipment, for some resons, hs to be foregone the next equipment below should be the next hoie. The opportunity forgone of not quiring the erlier n be estimted s the perentge different of the two Chen s vlue. Thus, if the first rnked equipment is of Chen s vlue nd the seond equipment in the order is of Chen s vlue, the opportunity foregone of tking the seond insted of the first is (( ) /0.7745)(00); i.e. 5.74%. the opportunity foregone would be in term of ost sving or ese of mintenne or pssenger omfort. These n be dedued from the rtings nd weight of importne put in ple during the deision mking. Results ury is not normlly fous when this onept is pplied. Otherwise it ontrdits the intention of dopting pproximtion, vgueness nd mbiguity modes of humn resoning. It is suffie to mention here tht the degree of relevnt of the whole deision mking proess depends very muh on the weight of importne nd the equipment rtings used. It n esily be rubbish-in-rubbish-out exerise depending on the ommitment put by those involved in the deision mking. However, tehnique wise, there re severl improvement possible suh s using types of membership funtion more urte then the tringulr type, using more prtitions nd using djusted non-equl overlpping membership funtion where smller prtition re use round dt presumed to be more importnt so tht those rnge of dt re not lost.

8 Referenes Ahmd, M. Z., The Applition of Fuzzy Expert System to Preliminry evelopment Plnning of Medium Size Continer Terminl, Engineering Dotorte thesis, University Teknologi Mlysi, Avineri, E., J. Prshker, nd A. Ceder, Trnsporttion Projets Seletion Proess Using Fuzzy Sets Theory. Fuzzy Sets nd Systems. 6 (): Deb, S. K., B. Bhtthryy nd S. K. Sorkhel, Mteril Hndling Equipment Seletion by Fuzzy Multi-Criteri Deision Mking Methods. Proeedings of AFSS 2002 Interntionl Conferene on Fuzzy Systems. Clutt, Indi Hong, T., P. nd C. Y. Lee, 996. Introdution of Fuzzy Rules nd membership Funtion from Trining Exmples. Fuzzy Sets nd System. 84: Krsk, E. E. nd E. Tolg, 200. Fuzzy Multi-riteri Deision-mking Proedure for Evluting Advned Mnufturing System Investments. Int. J. Prodution Eonomis. 69 (): Prodnovi, P. nd P. Simnovi, Comprison of Fuzzy Sets Rnking Methods for Implementtion in Wter Resoures Deision Mking. Cndin Journl of Civil Engineering. 29: Rj, P. A. nd D. N. Kumr, 999. Rnking Alterntives with Fuzzy Weights Using Mximizing Set nd Minimizing Set. Fuzzy Sets nd System. 05: