Compromise Ratio Method for Decision Making under Fuzzy Environment using Fuzzy Distance Measure

Transcription

1 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Coprose ato Method for Decso Makg uder Fuzzy Evroet usg Fuzzy Dstace Measure Debashree Guha, Deba Chakraborty Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 Abstract The a of ths paper s to adopt a coprose rato (C ethodology for fuzzy ult-attrbute sgle-expert decso akg proble. I ths paper, the ratg of each alteratve has bee descrbed by lgustc ters, whch ca be expressed as tragular fuzzy ubers. The coprose rato ethod for fuzzy ultattrbute sgle expert decso akg has bee cosdered here by takg the rakg dex based o the cocept that the chose alteratve should be as close as possble to the deal soluto ad as far away as possble fro the egatve-deal soluto sultaeously. Fro logcal pot of vew, the dstace betwee two tragular fuzzy ubers also s a fuzzy uber, ot a crsp value. Therefore a fuzzy dstace easure, whch s tself a fuzzy uber, has bee used here to calculate the dfferece betwee two tragular fuzzy ubers. Now ths paper, wth the help of ths fuzzy dstace easure, t has bee show that the coprose rato s a fuzzy uber ad ths eases the proble of the decso aker to take the decso. The coputato prcple ad the procedure of the coprose rato ethod have bee descrbed detal ths paper. A coparatve aalyss of the coprose rato ethod prevously proposed [] ad the ewly adopted ethod have bee llustrated wth two uercal exaples. Keywords Coprose rato ethod, Fuzzy ult-attrbute sgle-expert decso akg, Fuzzy uber, gustc varable I. INTODUCTION AKING decso s udoubtedly oe of the ost M fudaetal actvtes of hua begs. Mult Attrbute Decso Makg (MADM probles have a portat part real lfe stuatos. Sce MADM has foud acceptace areas of operato research ad aageet studes, dfferet ethodologes have bee created for akg decso. But the applcato of the dfferet ethods s coplex ad fuzzy ature. I recet tes, wth the help of coputers, the decso akg ethods have foud great acceptace all areas of decso akg process. Especally, the last few years, wth coputers becog coected to every feld of lfe, the applcatos of varous ethodologes Mauscrpt receved July 6, 7. Dr. Deba Chakraborty s wth the Departet of Matheatcs, Ida Isttute of Techology, Kharagpur, Kharagpur 73, Ida (e-al: deba@aths.tkgp.eret. Debashree Guha, s wth the Departet of Matheatcs, Ida Isttute of Techology, Kharagpur, Kharagpur 73, Ida (correspodg author to provde phoe: (e-al: deb7@yahoo.co.. for MADM have becoe easer for the decso aker. The a cocept of the MADM proble s to fd the best opto aog all feasble alteratves based o ultple attrbutes both qualttatve ad quattatve. There are varous ethods that exst by whch we ca deal wth MADM proble. Techque for Order Preferece by Slarty to Ideal Soluto (TOPSIS s oe of the kow classcal MADM ethods, developed by Hwag ad Yoo [7]. The TOPSIS ethod s based upo the dea that the chose alteratve has shortest dstace fro the postve deal soluto ad farthest dstace fro the egatve deal soluto. But the classcal TOPSIS ethod the weghts of the attrbutes ad ratg of the alteratves are gve crsp values. Uder ay codtos, decso akg probles, crsp data are suffcet to odel real lfe stuatos [8]. Decso aker s respose to the dfferet alteratves ad also prefereces to the varous attrbutes ay be soetes expressed lgustc varables [9,,,, 3]. Therefore fuzzy set theory s used to deal wth MADM proble. May researchers have used fuzzy set theory decso akg whe fuzzess preset hua udget. Based o the slarty easure proposed by Che [5], the lteratures [3, 6, 8, 9], a rakg strategy s developed for the subects. A web-based decso-support-syste based o fuzzy set approach s pleeted [] that tegrates the subectve ad obectve forato for the evaluatg the grade of ourals. Uder these crcustaces, the TOPSIS ethod was exteded for group decso akg probles uder fuzzy evroet [4]. Coprose rato ethod for fuzzy Mult Attrbute Decso Makg ethod was troduced by Deg-Feg 6 []. The basc prcple of the coprose rato ethod s that the chose alteratve should have the closest dstace fro the postve deal soluto ad the farthest dstace fro the egatve deal soluto. But t s ot possble real lfe stuato that oe partcular alteratve satsfy both these codtos sultaeously. So, the questo here arses, that how the decso s to be ade, uder such kd of crcustaces. I the paper [] relatve portace has bee gve to both of these dstaces. A coprose rato ethodology to solve fuzzy ult attrbute group decso akg probles has bee developed the paper []. I the process of coprose rato ethod, lgustc varables have bee used to capture fuzzess decso akg forato Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 54 scholar.waset.org/37-689/77

2 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 ad decso akg process by eas of a fuzzy decso atrx. Now, ths paper, Coprose ato ethod, troduced by Deg-Feg [], has bee odfed. Dstace easures have portat role Coprose rato ethod. But paper [] the dstace easure betwee two precse ubers has bee used, whch gve us a crsp value. But a logcal cosequece defg a fuzzy dstace easure for geeralzed fuzzy ubers s that dstace betwee two precse ubers should also be a precse (.e. fuzzy uber. Also the coprose rato, troduced paper [], gve crsp value. Ths leads to a proble for the decso aker, as for two dfferet alteratves the C ca gve the sae value. So ths paper, a ew fuzzy dstace easure, troduced by Chakraborty ad Chakraborty [], has bee used. So the C s a fuzzy uber here. Also, for rakg the alteratves, the rakg ethod proposed the paper [6] has bee used. I secto II, otato of fuzzy uber has bee gve. Ad also the fuzzy dstace easure ad the rakg ethod have bee dscussed ths secto. The basc prcple ad procedure of the ethodology has bee gve secto III. I secto IV, the proposed ethod has bee llustrated wth two uercal exaples. A coparatve aalyss betwee the two ethods has also bee gve ths secto. Fally a short cocluso has bee gve secto V. II. PEIMINAIES A Notato of fuzzy uber I 965 Zadeh [7] frst troduced the fuzzy set for dealg wth vagueess type of ucertaty. A fuzzy set A defed o the uverse X whch s characterzed by a ebershp μ : X,. The support of A, say fucto such that A [ ] supp ( A s defed by the set x X μ ( x { / > } ad the level set of A leads to a set such { x X / μ x } for all [, ]. that ( A B Geeralzed fuzzy uber (GFN A geeralzed fuzzy uber A, covetoally represeted by A = ( a, a ; β, γ.e. (left pot., rght pot, left spread, rght spread,s a oralzed covex fuzzy subset o the real le f ( Supp ( A s a closed ad bouded terval.e. a β, a + γ ; [ ] ( μ A s a upper se cotuous fucto. ( a β < a a + γ ; ad (v the ebershp fucto s of the followg for: f ( x for x [ a β, a ] μ ( [, ] x = for x a a h( x for x [ a, a + γ ] Where f(x ad h(x are the ootoc creasg ad decreasg fuctos respectvely. C -type fuzzy uber A = a, a ; β, γ s sad to be -type f there exsts A GFN ( referece fuctos (for left, (for rght ad sealers β >, γ > wth the ebershp fucto of the for a x f a β x a β μ ( x f a x a = x a f a x a + γ γ Where, for (x ad (x dfferet fuctos ay be chose. For exaple, (x = ax (,-x p wth p> or (x= e -x (Zera 996. I partcular, f a = a = the A s wrtte as(, β, γ. The forula of a opposte fuzzy uber s ( β γ =( β γ,,,,. D Tragular fuzzy uber (TFN A -type fuzzy uber A s sad to be a tragular fuzzy uber (TFN deoted by = (, β, γ f ts ebershp TFN fucto s of the followg for μ ( x x f β x β = x f x + γ γ E Dstace betwee geeralzed fuzzy ubers I ths secto we take the pot of vew that the dstace betwee two fuzzy ubers should tself be fuzzy. et us cosder two GFNs as A = a, a ; β, γ ad A = a, a ; β, γ. ( ( 3 4 Therefore -cut of A ad A represets followg two tervals respectvely [ ] [ = A ( (, A ] ad [ ] [ = A ( ( [ ], A ] for all,. It s clear that dstace betwee two tervals ca be easured by takg ther dfferece. So here the terval- [ A, A ] ad dfferece operato for the tervals ( ( [ A (, A ( ] has bee used to forulate the fuzzy dstace betwee A ad A. Now, the dstace betwee [ A ] ad [ A ] for all [,] s oe of the followg ether ( a [ ] [ ] Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 55 scholar.waset.org/37-689/77

3 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 A ( + A ( A ( + A ( f or ( b [ ] [ ] A ( + A ( A ( + A ( f < I order to cosder both the otatos together a dcator varable η s troduced such that η([ A ] [ A ] + η ([ A ] [ A ] ( = [ d, d ] ( for η = after coparg fro ( A ( + A ( A ( + A ( f A ( + A ( A ( + A ( f < d = [ A ( A ( + A ( A ( ] + η [ A ( A ( ]ad d = [ A ( A ( + A ( A ( ] + η [ A ( A ( ] Therefore, the fuzzy dstace betwee ad s defed by (, (, ;, d = d d θ σ ( = = Where θ = d ax = { d d }, ad σ = d d d = F akg Method Ths subsecto gves a short descrpto of the rakg ethod proposed by [6]. Here cetrod pot of a fuzzy uber has bee deoted by x o the horzotal axs ad y o the vertcal axs. The cetrod pot ( x, y for a fuzzy uber A (subsecto.. has bee defed as x( = y( = a a a + xf ( x dx + xdx + xh( x dx a β a a a a a + f ( xdx + dx+ hxdx ( a β a a ( yh [ y f ( y] dy ( [ h y f ( y] dy where f(x ad h(x are the left ad rght ebershp fuctos of fuzzy uber A respectvely. f ( y ad h ( y are the verse fuctos of f(x ad h(x respectvely. γ γ The area betwee the cetrod pot ( x, y ad the orgal pot (, of the fuzzy uber A s defed as area( A = x. y. The area ( A has bee used for rakg the alteratves. For ay two dfferet fuzzy ubers A ad A f area( A = area( A the A = A, f area ( A > area ( A the A > A. Fally, f area ( A < area ( A the A < A. III. FUZZY COMPOMISE ATIO METHOD (FCM FO MADM (SINGE EXPET I ths paper, the followg MA (Sgle Expert DM fuzzy evroet has bee dscussed. Suppose there exst possble alteratves s, s,..., s fro whch the decso aker has to choose o the bass of attrbutes c, c,..., c, both qualtatve ad quattatve. Now here the attrbutes set C has bee dvded to two subsets C ad C where C* (k=, s the subset of beeft attrbutes & cost attrbutes respectvely. Furtherore C = C C ad C C =Φ. Here t has bee assued that the attrbutes have equal weghts. et us suppose that the ratg of alteratve s (=,, o attrbute (=,, as gve by the decso aker be f = ( ;, β. Hece, a FMA Sgle expert DM proble has bee cocsely expressed atrx forat as follows: f f Y = ( f f f Ths s referred to as fuzzy decso atrces, whch s usually used to represet the FMA Sgle expert DM proble. Sce the attrbutes ay be easured dfferet ways, the decso atrx Y eeds to be oralzed. The lear scale trasforato has bee used here to trasfor the varous attrbute scale to a coparable scale [5]. After oralzato we get β r = ;, for c C ax ax ax d d d ad a a. β a. ;, ( a.( + β.( r = for c C β ;, ( a = ax ax ax d d d Where ax d = ax{ + β f = ( ;, β }ad < < = = β < < a { f ( ;, } Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 56 scholar.waset.org/37-689/77

4 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 Deote r = ( σ ; ξ, ν The oralzato ethod etoed above s to preserve the property that the rage of a oralzed tragular fuzzy uber belogs to the closed terval [, ]. The the fuzzy decso atrx Y = ( f ca be trasfored to oralzed fuzzy decso atrx: r r = ( r r r Obvously, all r ( σ ; ξ, ν = are oralzed postve tragular fuzzy ubers ad ther rages belog to the closed terval [, ]. The, the fuzzy postve deal soluto s + ad the fuzzy egatve deal soluto s have bee defed, whose a + = a + a + a + weghted oralzed fuzzy vectors are (, ad a ( a, a a respectvely, where a + = (,, = a =,, =. ad ( Now ths paper dfferece betwee each alteratve s (=, ad the postve deal soluto ad the fuzzy egatve deal soluto has bee easured by usg equato (as follows respectvely: + + = Ds (, s dr (, a = Ds (, s dr (, a Now the saller Ds (, s +, the better s. Therefore, rak the alteratve s (=,, by Ds (, s + creasg order. A alteratve s satsfyg * Ds (, s + = { Ds (, s + } * < < should be the best coprose soluto whch has the shortest dstace fro the postve deal soluto. However shortest dstace fro the postve deal soluto ay ot always autoatcally ply axu dstace fro the egatve deal soluto. Slarly whe the decso aker raks the alteratves wth respect to the egatve deal soluto; the t s clear that bgger the value of Ds (, s, the better s. So rak the alteratves s (=,. by Ds (, s decreasg order. A alteratve s ** satsfyg Ds (, s ** = ax{ Ds (, s } should be the best coprose soluto, < < whch has the farthest dstace fro the egatve deal soluto. (3 But every stuato t ay ot happe that s = s. So * ** here also a coprose rato for every alteratve s (=,, s calculated as Ds + + ( Ds (, s Ds (, s Ds ( ε( S = ε + ( ε Ds + + ( Ds ( Ds ( Ds ( + + ( s = ax{ ( s, s } where, < < + + < < < < < < ( s { D( s, s } ( s ax{ D( s, s } ( s { D( s, s } S Here, t s cosdered that ε [,] dcates the atttudal factor of the decso aker. Whe ε =, the decso aker gves ore portace to the dstace fro the postve deal soluto. Whe ε =, the decso aker s the terested the dstace fro the egatve deal soluto. Ad equally portace to both the dstaces Ds (, s + ad Ds (, s wll be gve whe ε = /. Also t s assued that f ε >.5, the, the decso aker s based above the fuzzy postve deal soluto. The dex ε ( s easures the extet of coprose of the proxty of the alteratve s to the postve deal soluto s + ad ts dstace fro the egatve deal soluto s. The bgger ε ( s s the better s. The preferece order of the alteratve s (=, s geerated accordg to ε ( s. A alteratve s whch has the best coprose level betwee the dstace fro the postve deal soluto s + ad the dstace fro the egatve deal soluto s, satsfyg ε( s = ax{ ε( s } should be the best < < coprose soluto. Now ths ewly developed ethod we wll get ε ( s tself as a fuzzy uber. So here how decso aker copare ε ( s to each other. I ths regards, we wll apply the rakg ethod proposed the paper [6] ad usg ths ethod we choose the alteratve s. IV. NUMEICA EXAMPE Exaple : et us assue that a reputed aageet copay wats to hre a perso as the geeral aager. After the wrtte test, the experts coduct a tervew ad GD. The expert wll, the, take the fal decso based upo the followg crtera: C. M.B.A degree fro a well-kow aageet sttute. C. Oral coucato skll (4 Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 57 scholar.waset.org/37-689/77

5 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 C3. Presece of d ad the capacty to hadle crtcal stuatos C4. Group work ad leadg power. C5. Persoalty Now here t s assued that based o the above fve crtera the expert wll take the tervew of each caddate. The expert use lgustc ters the akg of hs expert coets, whch are expressed ters of tragular fuzzy uber (wthout loosg ts precseess.. A PC-aded evaluato procedure [4] ay be cosdered. It helps the decso aker to express ther opo. To desg a PC-based forato syste t eeds to have a tal kowledgebase (KB that store the kowledge about a doa represeted ache-processable for. The PC-aded procedure should be user-fredly so that as ad whe DM feels tal kowledge base (KB could be updated by corporatg ore optos havg ore ters. The optos havg dfferet ter dfferetals help the DM to express the resposes cofortably. et us cosder the doa as [,] o whch DM s resposes are to be explaed; e.g., a saple of KB cosstg of four optos [3] s cosdered here gve below: Opto ( ters Not-satsfactory/satsfactory (NS/S, Opto (3 ters Not-satsfactory/satsfactory/good (NS/S/G Opto 3 (4 ters Notsatsfactory/satsfactory/good/excellet (NS/S/G/EX Opto 4 (5 ters No-ert/poor/satsfactory/good/excellet (NM/P/S/G/EX Actually here the doa [,] has bee fuzzly parttoed to dfferet uber of optos. As the ubers of optos are creased, the doa wll be parttoed to ore uber of tervals. Now the ecessty of takg dfferet optos kowledgebase s that, f the DM s ot satsfed wth the ter fro opto, the opportuty s to be gve to h to choose further frutful opto. A algorth s desged here to trasfor the fuzzy ters of dfferet optos to correspodg tragular fuzzy ubers as follows: Algorth. Step : Cosder kth opto that cossts of (k+ terdfferetals. Step : Suppose for the th ter where =, (k+, the spread s equal to (/k ad ceter say, s coputed as: for = = for, 3 k for = k + Step 3: (boudary codto If the left or/ad rght pot of a fuzzy uber s/are outsde the doa [,], the left ad rght pots would be autoatcally replaced by ad, respectvely. Therefore, the outputs of the algorth for varous optos are represeted as fuzzy ubers as follows: : opto opto opto 3 opto 4 No-ert (,, 5 Poor (, 5, 5 No-sats. (,, (,, 5 (,, Sats (,, (,5, (,33.3,66.7 (5,5,75 Good - (5,, (33.3, 66.7, (5, 75, Excellet - - (66.7,, (75,, A hua expert s lgustc expresso s a outcoe of a reacto hs d toward a partcular query. I fact, a expert (let us assue that a expert s ot at all aware of what s called fuzzy atheatcs ay be ore cofdet ad spotaeous expressg hs opo oe lgustc ter or oe lgustc phrase (gve dfferet optos wth hs ow way of percepto, reasog ad expresso. Depedg o the DM s satsfacto level, the sae lgustc ter all of the sets of ter dfferetals that carres ultplcty of eag s placed to varous postos. Oe terpretato ca be draw here: the ter satsfactory opto 4 s relatvely ore precse that that opto 3 where t s ostly precse opto. I vew of ths, the DM selects the sae ter fro dfferet optos to ake respose ore eagful. Thus a respose atrx s obtaed as follows TABE I: Expert s lgustc resposes fro KB havg dfferet optos Caddates s s s 3 Crtera C S fro opto S fro opto3 S fro opto4 C NS fro opto 3 S fro opto 4 NS fro opto C3 G fro opto EX fro opto G fro opto3 C4 G fro opto 4 NS fro opto NS fro opto 4 C5 Ex fro opto3 Ex fro opto4 S fro opto Now fro the above table we ca wrte the fuzzy decso atrx the followg table: TABE II: The fuzzy decso atrx of three caddates: s s s 3 C (5, 5, 5 (33.3, 33.3, 33.3 (5,5,5 C (,, 33.3 (5, 5, 5 (,, C3 (,, (, 5, (66.7, 33.3, 33.3 C4 (75, 5, 5 (,, (5, 5, 5 C5 (, 33.3, (, 5, (5, 5, 5 TABE III: The oralzed fuzzy decso atrx of three caddates: s s s 3 C (.5,.5,.5 (.33,.33,.34 (.5,.5,.5 C (,,.33 (.5,.5,.5 (,,. C3 (.,.,. (.,.5,. (.67,.34,.33 C4 (.75,.5,.5 (,,. (.5,.5,.5 C5 (.,.33, (.,.5, (.5,.5,.5 Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 58 scholar.waset.org/37-689/77

6 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 TABE IV: akg results obtaed by the odfed coprose rato ethod (usg equato 3 Caddates s s s 3 Ds (, s + (.75,.875,.4 (3.7,.96,.665 (.83,.665,.96 akg order s > s3 > s Ds (, s (3.5,.4,375 (.83,.665,.96 (.7,.795,.9 akg order s > s > s3 Hece here the decso aker wll be cofuso. So here we calculate ε ( s for =,, 3. by equato (4. Ad here the expert s atttude s specfed by ε =.6 Here the value of the fuzzy uber ε ( s s wrtte (for ε =.6 s s s 3 ε ( s (,, (-.674,.44,.5 (-.979,.44,.948 akg order s > s > s3 The coprose soluto obtaed by the odfed coprose rato ethod s the alteratve s. Exaple : Our ext a s to show, wth help of ths exaple, the shortcogs of the Deg Feg s ethodology ad the advatages of proposed fuzzy coprose rato ethod (FCM. et us cosder a proble wth a gve crtero ad two alteratves s ad s aog whch the expert have to choose the best opto. The expert gves hs expert s coets lgustcally ad wthout ay loss of geeralty ths resposes are expressed ters of tragular fuzzy ubers as follows: s = (.6;.3,.65 TFN s = (.55;.55,.45 TFN Decso results obtaed by the coprose rato ethod troduced by Deg- Feg [] ad fuzzy coprose rato ethod are gve the followg table. Here, the coprose rato ethodology (CM, to calculate the dstace betwee two tragular fuzzy ubers, the dstace easure proposed by Che [4] has bee used. Table V. akg results obtaed by Deg-Feg s coprose rato ethod ad the fuzzy coprose rato ethod: Caddates s s akg order CM Ds (, s s > s Ds (, s.6.7 s > s ε ( s.5.5 # FCM Ds (, s + (.4;.5,.8 TFN (.45;.5,.75 TFN s > s Ds (, s (.4;.5,.8 TFN (.45;.75,.5 TFN s > s ε ( s (;, (;, s > s The expert s atttude s specfed by ε =.5 # eas that wth help of the Deg Feg s coprose rato ethod, the expert ca ot coe to a cocluso. Uder ths kd of crcustaces, the proposed fuzzy coprose rato ethod wll gve better result. I ths way, fro the above exaple t s proved that soe cases the odfed fuzzy coprose rato ethod wll gve better result. Here the best alteratve s s. The coprose rato ethod [] troduces a aggregatg fucto for rakg equato (, whch reflects the extet that the alteratve s (=,... closes to the postve deal soluto s + ad s far away fro the egatve deal soluto s. Now the coprose rato ethod [] we choose the alteratve s (=,,, for whch ε ( s has the axu value.but for ay two alteratves s (=k, ε ( s ca gve the sae value, the ths creates a very bg proble to the decso aker ( as show the exaple. Now uder ths pot of vew, we use here the fuzzy dstace easure troduced by Chakraborty ad Chakraborty []. Usg ths dstace easure the ewly developed ethod we get ε ( s tself as fuzzy uber ad the easly decso aker usg the rakg ethod [6] ca coe to a cocluso. I ths way we ca overcoe the shortcogs of coprose rato ethod []. Also paper [], to easure the dstace fro postve ad egatve deal soluto, Mkowsk dstace (or p etrc s used. But ths dstace ethod bascally copute crsp dstace values for partcular fuzzy ubers, ot for geeralzed fuzzy ubers. But t s very atural questo that, f the ubers tself are ot kow exactly, how the dstace betwee the ca be a exact value. I ths regard, the fuzzy dstace easure proposed by Chakraborty ad Chakraborty [], s used here. V. CONCUSION Sce atttude of a expert has a ecessary part decso akg actvtes ad also the rakg or orderg of the caddates chage due to the atttude of the decso aker. I ths regard the odfed fuzzy coprose rato ethod wth the atttudal factor wll play a portat role decso akg actvtes. Fuzzy Multple Attrbute Group Decso Makg s future work of ths paper. Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 59 scholar.waset.org/37-689/77

7 World Acadey of Scece, Egeerg ad Techology Iteratoal Joural of Matheatcal ad Coputatoal Sceces Vol:, No:, 7 Iteratoal Scece Idex, Matheatcal ad Coputatoal Sceces Vol:, No:, 7 waset.org/publcato/77 EFEENCES [] Deg- Feg, Coprose rato ethod for fuzzy ultattrbute group decso akg, Appled Soft Coputg vol. 7, pp , 7. [] C. Chakraborty, D. Chakraborty, A theoretcal developet o fuzzy dstace easure for fuzzy ubers, Matheatcal ad Coputer Modelg, vol. 43, pp. 54-6, 6. [3] D. Chakraborty, Structural quatzato of vagueess lgustc expert opos a evaluato prograe, Fuzzy Sets Systes. Vol. 9, pp. 7 86,. [4] C. Chakraborty, D. Chakraborty, A decso schee based o OWA operator for a evaluato.prograe, a approxate reasog approach, Appled Soft coputg, vol. 5, pp , 4. [5] D., Fuzzy, Multobectve May Perso Decso akg gae ad Gaes, Natoal defese IdustyPress.Beg.3. [6] Y- M. Wag, J-B. Yag, D-. Xu, K-S. Ch, O the cetrods of fuzzy ubers, Fuzzy Sets ad Systes, vol. 57, pp , 6. [7] C..Hwag. K.Yoo, Multple Attrbutes Decso Makg Methods ad Applcatos, Sprger, Berl. Hedelberg. 98. [8] S.J.Che. C..Hwag, Fuzzy Multple Attrbute Decso Makg: Methods ad Applcatos, Sprger-Verlag Berl, 99. [9].-C. Wag, S.-J.Chuu, Group decso-akg usg a fuzzy lgustc approach for evaluatg the flexblty a aufacturg syste, Europea Joural of.operatoal. esearch, vol. 54, pp , 4. [].E.Bella,.A.Zadeh, Decso akg a fuzzy Evroet, Maageet Scece, vol. 7, pp. 4-64, 97. [] M.Delgado, J..Verdegay, M.A.Vla, gustc decso-akg odels, It. J. Itellget Syste, vol. 7, pp ,99. [] F.Herrera, E.Herrera-Veda, J..Verdagay, A odel of Cosesus group decso akg uder lgustc assessets, Fuzzy Sets ad Systes, vol. 78, pp , 996. [3].A.Zadeh, The cocept of a lgustc varables ad tsapplcato to approxate reasog, Ifor-Scece, vol. 8, pp (I, pp (II, 975. [4] C.T. Che, Exteso of the TOPSIS for group decso akg uder fuzzy evroet, Fuzzy Sets ad systes, vol. 4, pp. -9,. [5] S.M. Che, A ew approach to hadlg fuzzy decso- akg probles, IEEE Tras. Systes Ma Cyberatcs, vol. 8, pp. - 6, 988. [6] D. Chakraborty, Estato of aggregatve rsk software developet: a approxate reasog approach, : Proceedgs of the Coferece Fuzzy Set Theory ad ts Matheatcal Aspects ad Applcatos, BHU, Varaas, Deceber 6 8,, Fuzzy Set theory Math.Aspects Applcato, Alled Publshers, pp [7] C.Chakraborty, D.Chakraborty, Approxate reasog wth OWA operator a evaluato schee, Cobatoral ad Coputatoal Matheatcs, Narosa Pub, New Delh, 4, pp [8]. Bswas, A applcato of fuzzy sets studets Evaluato, Fuzzy Sets Systes, vol. 74, pp , 995. [9] D. Zhou, J. Ma, E. Turba, N. Bollou, A fuzzy set approach to the evaluato of oural grades, Fuzzy Sets Syst. Vol. 3, pp ,.. Iteratoal Scholarly ad Scetfc esearch & Iovato ( 7 6 scholar.waset.org/37-689/77