A Mean Deviation Based Method for Intuitionistic Fuzzy Multiple Attribute Decision Making

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1 00 Iteratoal Coferece o Artfcal Itellgece ad Coputatoal Itellgece A Mea Devato Based Method for Itutostc Fuzzy Multple Attrbute Decso Makg Yeu Xu Busess School HoHa Uversty Nag, Jagsu 0098, P R Cha xuyeoh@63co Abstract The a of ths paper s to develop a ethod to detere the weghts of attrbutes obectvely uder tutostc fuzzy evroet Based o the ea devato, we establsh a optzato odel whch the forato about attrbute weghts s copletely ukow By solvg the odel, we get a sple ad exact forula whch ca be used to detere the attrbute weghts After that, we utlze the tutostc fuzzy weghted average (IFWA operator to aggregate the gve tutostc fuzzy forato correspodg to each alteratve, ad the select the ost desrable alteratve accordg to the score fucto ad accuracy fucto Fally, a practcal exaple s gve to verfy the developed ethod ad to deostrate ts practcalty ad effectveess Keywords-Itutostc fuzzy set; ultple attrbute decso akg; ea devato; I INTRODUCTION Itutostc fuzzy sets(ifs troduced by Ataassov[, ] have bee foud to be well suted to dealg wth vagueess IFS characterzed by a ebershp fucto ad a o-ebershp fucto, s a exteso of Zadeh s fuzzy set[3] whose basc copoet s oly a ebershp fucto Sce ts appearace, the IFS have receved ore ad ore atteto ad appled t to the feld of decso akg Gau ad Buehrer[4] preseted the cocept of vague sets Burllo ad Bustce[5] showed that the oto of vague sets cocdes wth that of tutostc fuzzy sets Based o vague sets, Che ad Ta[6], ad Hog ad Cho [7] utlzed the u ad axu operatos to develop soe approxate techque for hadlg ultattrbute decso akg probles uder fuzzy evroet Szdt ad Kacprzyk [8] proposed soe soluto cocepts such as the tutostc fuzzy core ad cosesus wer group decso akg wth tutostc (dvdual ad socal fuzzy preferece relatos, ad proposed a ethod to aggregate the dvdual tutostc fuzzy preferece relatos to a socal fuzzy preferece relato o the bass of fuzzy aorty equated wth a fuzzy lgustc quatfer L ad Cheg[9], Lag ad Sh[0], Huag ad Yag[], ad Wag ad X[] troduced soe slarty easures of tutostc fuzzy sets ad appled the to patter recogto Xu ad Yager[3] developed soe aggregato operators, such as the tutostc fuzzy weghted geoetrc (IFWG operator, the tutostc fuzzy ordered weghted geoetrc(ifowg operator, the tutostc fuzzy hybrd geoetrc (IFHG operator to ultple attrbute group decso akg wth tutostc fuzzy forato Xu[4] developed the tutostc fuzzy ordered weghted averagg (IFOWA operator, ad the tutostc fuzzy hybrd averagg (IFHA operator However, whe usg these operators, the assocated weghtg vector s ore or less detered subectvely ad the decso akg forato tself s ot take to cosderato suffcetly All of the above ethods wll be usutable for dealg wth such stuatos Therefore, t s ecessary to develop a ethod for deterg the weghts obectvely of the ultple attrbute decso akg probles uder tutostc fuzzy evroet I ths paper, we focus our atteto o developg a ethod obectvely aed ea devato ethod to detere the attrbute weghts uder the codto that the attrbute weghts are copletely ukow, ad the attrbute values are takg the for of tutostc fuzzy ubers, to overcoe the above ltatos To do so, the rest of the paper s orgazed as follows I Secto, we troduce soe basc cocepts of tutostc fuzzy sets I Secto 3, we establsh a optzato odel based o the ea devato ethod By solvg ths odel, a sple ad exact forula s derved to detere the attrbute weghts We utlze the tutostc fuzzy weghted averagg (IFWA operator to aggregate the tutostc fuzzy forato correspodg to each alteratve, ad the rak the alteratves ad select the ost desrable oe(s accordg to the score fucto ad accuracy fucto I Secto 4, a practcal exaple s used to llustrate the developed odels I Secto 5, we coclude the paper ad gve soe rearks II PRELIMINARIES I the followg, we troduce soe basc cocepts related to tutostc fuzzy sets I[], Ataassov troduced a geeralzed fuzzy set called tutostc fuzzy set, show as follows Defto A IFS X s gve by A = { < x, μa( x, va( x > x X} ( whch s characterzed by a ebershp fucto μ A : X [0,] ad a o-ebershp fucto va : X [0,], wth the codto /0 $ IEEE DOI 009/AICI0044

2 0 μa( x + va( x, x X where the ubers μ A( x ad v ( A x represet, respectvely, the degree of ebershp ad the degree of o-ebershp of the eleet x to the set A Defto For each IFS A X, f π A( x = μa( x va( x, x X ( s called the deteracy degree or hestato degree of x to A Especally, f π A( x = μa( x va( x = 0, x X (3 The, the tutostc fuzzy set A s reduced to a coo fuzzy set[3] For coveece, we call = ( μ a tutostc fuzzy uber(ifn([5], where μ [0,] [0,], ad μ + v Defto 3[3] Let = ( μ be a tutostc fuzzy uber, a score fucto S of a tutostc fuzzy uber ca be represeted as follows: S( = μ v (4 where S( [,] For a IFN = ( μ, t s clear that f the devato betwee μ ad v gets greater, whch eas the value μ gets bgger ad the value v gets saller, the the IFN gets greater Defto 4[3] Let = ( μ be a tutostc fuzzy uber, a accuracy fucto H to evaluate the degree of accuracy of the tutostc fuzzy uber ca be represeted as follows: H ( = μ + v (5 where H ( [0,] The larger the value of H (, the hgher the degree of accuracy of the degree of ebershp of the IFN Xu[3] troduced a order relato betwee two tutostc fuzzy ubers the followg Defto 5 Let = ( μ ad β = ( μ β be two tutostc fuzzy ubers, S( = μ v ad S( β = μβ v β be the scores of ad β, respectvely, ad let H ( = μ + v ad H ( β = μ β + v β be the accuracy degrees of ad β, the If S( < S( β, the s saller tha β, deoted by < β If S( = S( β, the ( If H( = H( β, the ad β represet the sae forato, e, μ = μβ, v = vβ, deoted by = β ; ( If H( < H( β, the s saller tha β, deoted by < β To aggregate tutostc preferece forato, Xu [39] defed the followg operatos Defto 6[6] Let = ( μ ad β = ( μ β be two tutostc fuzzy ubers, the ( + β = ( μ + μβ μ μβ, v vβ ; ( β = ( μ μβ, v + vβ v vβ ; λ λ (3 λ = ( ( μ, λ > 0; (4 λ = ( μ λ, ( v λ, λ > 0 Defto 7[5] Let = ( μ, β = ( μ β be two tutostc fuzzy ubers, the we call d( β, = β = ( μ μβ + v vβ (6 the devato betwee ad β III MEAN DEVIATION METHOD The ultple-attrbute decso-akg probles uder study ca be descrbed detal as follows Let X = { x, x,, x } ( be a dscrete set of feasble alteratves, U = { u, u,, u } be a fte set of attrbutes For each alteratve x X, the decso aker gves hs/her preferece value r wth respect to attrbute u U, where r takes the for of tutostc fuzzy r = μ v, μ [0,], v [0,], ad =, =,,,, the all the ubers, that s (, μ + v,,,, preferece values of the alteratves cossts the decso atrx R = ( r Defto 8[4] Let R = ( r be the tutostc fuzzy decso atrx, r = ( r, r,, r be the vector of attrbute values correspodg to the alteratve x, =,,,, the we call z( w = IF WA w( r, r,, r = wr + wr + + wr w w = ( μ, ( v (7 = = the overall value of the alteratve x, where w= ( w, w,, w T s the weghtg vector of attrbutes I the stuato where the forato about attrbute weghts s copletely kow, e, each attrbute weght ca be provded by the expert wth crsp uercal value, we ca aggregate all the weghted attrbute values correspodg to each alteratve to a overall oe by usg (7 Based o the overall attrbute values z ( w of the alteratves x ( =,,,, we ca rak all these alteratves ad the select the ost desrable oe(s The greater z ( w, the better the alteratve x wll be However, ths paper, we cosder the attrbute weght forato about the attrbute s copletely ukow, thus, we eed to detere the attrbute weght frstly 3

3 The ea devato ethod s proposed by Wag[7] to deal wth MADM probles wth uercal forato The authors of ths paper[8] also used ths ethod to deal wth the lgustc group ultple attrbute decso akg probles, whch the forato about the attrbute weghts are copletely ukow ad the attrbutes values are the fors of lgustc varables Its a deal s as follows For the MADM probles, we eed to copare the collectve preferece values to rak the alteratves, the larger the rakg value z ( w, the better the correspodg alteratve x s If the perforace values of each alteratve have lttle dffereces uder a attrbute, t shows that such a attrbute plays a sall portat role the prorty procedure Cotrarwse, f soe attrbute akes the perforace values aog all the alteratves have obvous dffereces, such a attrbute plays a portat role choosg the best alteratve So to the vew of sortg the alteratves, f oe attrbute has slar attrbute values across alteratves, t should be assged a sall weght; otherwse, the attrbute whch akes larger devatos should be assged a bgger weght, spte of the degree of ts ow portace Especally, f all avalable alteratves score about equally wth respect to a gve attrbute, the such a attrbute wll be udged uportat by ost experts I other word, such a attrbute should be assged a very sall weght Wag[7] suggests that zero should be assged to the attrbute of ths kd The dfferece of attrbute values ca be easured usg ea devato I the followg, we wll propose the ea devato ethod to deal wth the group decso akg proble uder tutostc fuzzy evroet For the attrbute u, the ea devato of alteratve x to all the other alteratves ca be expressed as follows: V = w r r = w d( r, r, =,,, t = t= = where r = rt = ( μt, ( vt t= t= t= (8 deotes the ea value of the attrbute u, dr r ( μ (, = + ( μt + v ( vt deotes the devato of t= t= ea value r to the attrbute value r of the alteratve x for the attrbute u So V deotes the ea devato for the attrbute u Based o the aforeetoed aalyss, we have to choose the weght vector w to axze all the ea devato values for all the attrbutes To do so, we ca costruct the odel as follows: Let (M- ax F( w = V = w d( r, r = = = (9 st w =, w 0 (0 = = dr (, r ( = The, the above odel ca be trasfored to the followg odel (M- ax Fw ( = w = (M- st w =, w 0, =,,, = To solve the above odel, we costruct the Lagrage fucto Lw (, λ = w + λ w = ( = where λ s the Lagrage ultpler Sce both fuctos F( w ad Lwλ (, are dfferetable for w, =,,, dfferetatg ( wth respect to w, =,,, ad settg the partal dervatves equal to zero, we get the followg set of equatos: L = + λw = 0, =,,, (3 w L = w = 0 (4 λ = Solvg ths odel, we get w = (5 ( = Thus (5 s the extree pot of odel(m- By oralzg w to let the su of w, =,,, be a ut, we have w w = =, =,,, (6 w = = As a ater of fact, represets the ea devato of all alteratves for the attrbute u Because the larger, the ore portat the attrbute u s, Eq(6 s obtaed drectly by usg each dvde the su of The theoretc foudato of ths ethod s based o forato 4

4 theory, that s, the attrbute provdg ore forato should be assged a bgger weght Based o the above odels, we develop a practcal ethod for solvg the ultple attrbute decso akg probles, whch the forato about attrbute weghts s copletely ukow, ad the attrbute values take the for of tutostc fuzzy values The ethod volves the followg steps: Step For each alteratve x X, the decso aker gves hs/her preferece value r wth respect to attrbute u U, where r takes the for of tutostc fuzzy ubers, that s r = ( μ, v, μ [0,], v [0,], ad μ + v, =,,,, =,,,, the all the preferece values of the alteratves cossts the decso atrx R = ( r Step If the forato about the attrbute weghts s copletely ukow, we solve the odel (M- to obta the optal weghtg vector w = ( w, w,, w T Step 3 Utlze the weghtg vector w = ( w, w,, w T ad by (7, we ca obta the overall values z ( w ( =,,, of the alteratves x ( =,,, Step 4 Calculate the scores Sz ( of the overall tutostc fuzzy preferece value z ( w ( =,,, to rak all the alteratves x ( =,,, ad the to select the best oe(s(f there s o dfferece betwee two scores Sz ( ad Sz (, the we eed to calculate the accuracy degrees H ( z ad H ( z of the overall tutostc fuzzy values z ad z, respectvely, ad the rak the alteratves x ad x accordace wth the accuracy degrees H ( z ad H ( z Step 5 Rak all the alteratves x ( =,,, ad select the best oe(s accordace wth the Sz ( ad H ( z ( =,,, Step 6 Ed IV ILLUSTRATIVE EXAMPLE I ths secto, we dscuss a proble cocerg wth a aufacturg copay, searchg the best global suppler for oe of ts ost crtcal parts used asseblg process (adapted fro[9] The attrbutes whch are cosdered here selecto of fve potetal global supplers x ( =,,5 are ( u : Overall cost of the product; ( u : Qualty of the product; (3 u 3 : Servce perforace of suppler; (4 u 4 : Suppler s profle; ad (5 u 5 : Rsk factor The expert represets the characterstcs of the potetal global supplers x ( =,,5 by the IFNs r (, =,,,5 wth respect to the attrbutes u ( =,,,5, lst Table (e tutostc fuzzy decso atrx R = ( r 5 5 TABLE I INTUITIONISTIC FUZZY DECISION MATRIX R=( R 5 5 u u u 3 u 4 u 5 x (04,05 (05,0 (06,0 (08,0 (07,03 x (06,0 (07,0 (03,04 (05,0 (08,0 x 3 (07,03 (08,0 (05,05 (03,0 (06,03 x 4 (03,04 (07,0 (06,0 (04,03 (09,0 x 5 (08,0 (03,04 (04,05 (07,0 (05,0 Step Assue the weghtg vector of the attrbute s copletely ukow, by applyg (6, we get the optal weghtg vector T w =(0358,0945,058,0994,0545 Step Utlze the weghtg vector w = ( w, w,, w5 T ad (7 to calculate the overall values z ( w ( =,,,5 of the alteratves x ( =,,,5 z ( w = (067,030, z ( w = (0599,003, z ( w = (0668,0495, z ( w = (063,076, 3 4 z ( w = (05960, Step 3 Utlze (4 to calculate the score of scores Sz ( of the overall tutostc fuzzy preferece values z ( w ( =,,,5 Sz ( = 03869, Sz ( = 03968, Sz ( 3 = 03673, Sz ( 4 = 04497, Sz ( 5 = 0359 thus Sz ( Sz ( Sz ( Sz ( Sz ( Step 4 Utlze the scores Sz ( ( =,,,5 to rak the alteratves x ( =,,,5 x4 x x x3 x5 ad the the ost desrable global suppler s x 4 V CONCLUSIONS I ths paper, we study the ultple attrbute decso akg probles, whch the forato about attrbute weghts s copletely ukow ad the attrbute values are expressed tutostc fuzzy ubers(ifns I order to get the optal attrbute weghts, we establsh a optzato odel based o the ea devato ethod By solvg the odel, we get a sple ad exact forula whch ca be used to detere the attrbute weghts After that, we utlze the tutostc fuzzy weghted average (IFWA operator to aggregate the gve tutostc fuzzy ubers decso forato, ad the select the ost desrable alteratve accordg to the score fucto ad accuracy fucto Fally, a practcal exaple s gve to verfy the developed ethod ad to deostrate ts practcalty ad effectveess Ad also, the ethod ca be exteded to the group tutostc fuzzy decso akg easly 5

5 ACKNOWLEDGMENT Ths work was supported by Hoha Uversty "the Fudaetal Research Fuds for the Cetral Uverstes (009B0454 " REFERENCES [] K T Ataassov, "Itutostc fuzzy sets," Fuzzy Sets ad Systes, vol 0, pp 87-96, 986 [] K T Ataassov, Itutostc Fuzzy Sets Hedelberg: Sprger-Verlag, 999 [3] L A Zadeh, "Fuzzy sets," Iforato ad Cotrol, vol 8, pp , 965 [4] W L Gau ad D J Buehrer, "Vague sets," IEEE Trasactos o Systes, Ma, ad Cyberetcs, vol 3, pp 60-64, 993 [5] P Burllo ad H Buste, "Vague sets are tutostc fuzzy sets," Fuzzy Sets ad Systes, vol 79, pp , 996 [6] S M Che ad J M Ta, "Hadlg ultcrtera fuzzy decso-akg probles based o vague set theory," Fuzzy Sets ad Systes, vol 67, pp 63-7, 994 [7] D H Hog ad C H Cho, "Multcrtera fuzzy decso-akg probles based o vague set theory," Fuzzy Sets ad Systes, vol 4, pp 03-3, 000 [8] E Szdt ad J Kacprzyk, "Dstaces betwee tutostc fuzzy sets," Fuzzy Sets ad Systes, vol 4, pp , 000 [9] D F L ad C T Cheg, "New slarty easures of tutostc fuzzy sets ad applcato to patter recogtos," Patter Recogto Letters, vol 3, pp -5, 00 [0] Z Lag ad P Sh, "Slarty easures o tutostc fuzzy sets," Patter Recogto Letters, vol 4, pp , 003 [] W L Hug ad M S Yag, "Slarty easures of tutostc fuzzy sets based o Hausdorff dstace," Patter Recogto Letters, vol 5, pp 603-6, 004 [] W Q Wag ad X L X, "Dstace easure betwee tutostc fuzzy sets," Patter Recogto Letters, vol 6, pp , 005 [3] Z S Xu ad R R Yager, "Soe geoetrc aggregato operators based o tutostc fuzzy sets," Iteratoal Joural of Geeral Systes, vol 35, pp , 006 [4] Z S Xu, "Itutostc fuzzy aggregato operators," IEEE Trasactos o Fuzzy Systes, vol 5, pp 79-87, 007 [5] Z S Xu, "Models for ultple attrbute decso akg wth tutostc fuzzy forato," Iteratoal Joural of Ucertaty, Fuzzess ad Kowledge-Based Systes, vol 5, pp 85-97, 007 [6] Z S Xu, "Itutostc preferece relatos ad ther applcato group decso akg," Iforato Sceces, vol 77, pp , 007 [7] Y M Wag, "A ethod based o stadard ad ea devatos for deterg the weght coeffcets of ultple attrbutes ad ts applcatos," Matheatcal Statstcs ad Maageet, vol, pp -6, 003 [8] Y J Xu ad Q L Da, "Stadard ad ea devato ethods for lgustc group decso akg ad ther applcatos," Expert Systes wth Applcatos, vol 37, pp , 009 [9] Z S Xu, "Ucerta lgustc aggregato operators based approach to ultple attrbute group decso akg uder ucerta lgustc evroet," Iforato Sceces, vol 68, pp 7-84, 004 6