Distance and Similarity Measures for Intuitionistic Hesitant Fuzzy Sets

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1 Iteratoal Coferece o Artfcal Itellgece: Techologes ad Applcatos (ICAITA 206) Dstace ad Smlarty Measures for Itutostc Hestat Fuzzy Sets Xumg Che,2*, Jgmg L,2, L Qa ad Xade Hu School of Iformato Egeerg, Ahu Xhua Uversty, Hefe , P. R. Cha 2 School of Maagemet, Hefe Uversty of Techology, Hefe 23000, P. R. Cha * Correspodg author ad formato processg capactes, the recommeders caot provde ther recommedatos wth a sgle umercal value, a marg of error, some possblty dstrbuto o the possble values, several possble umercal values, several possble terval umbers, but several possble tutostc fuzzy umbers. For example, to get a reasoable recommedato result, a recommedato orgazato, whch cotas a lot of recommeders, s requred to estmate the degree that a tem satsfes a attrbute. Suppose there s a case: some recommeders provde {(, )}, ad the others provde {(, )}, ad these two parts caot persuade each other to chage ther opos. We ca easly see that such case caot be dealt wth by fuzzy sets, hestat fuzzy sets, ad ther extesos, such as terval-valued fuzzy sets, tutostc fuzzy sets, terval-valued tutostc fuzzy sets, type 2 fuzzy sets, terval-valued hestat fuzzy sets [3]. Thus, t s very ecessary to troduce a ew exteso of hestat fuzzy sets to address ths ssue. The am of ths paper s to preset the oto of tutostc hestat fuzzy set (IFS), whch exteds the hestat fuzzy set to tutostc fuzzy evromets ad permts the membershp of a elemet to be a set of several possble tutostc fuzzy umbers. Thus, tutostc hestat fuzzy set s a very useful tool to deal wth the stuatos whch the recommeders hestate betwee several possble tutostc fuzzy umbers to assess the degree to whch a tem satsfes a attrbute. I the prevous example, the degree to whch the alteratve satsfes the attrbute ca be represeted by a tutostc hestat fuzzy set {(, ), (, )}. Abstract As geeralze of the hestat fuzzy sets, tutostc hestat fuzzy sets (IHFSs), whch permts a membershps degree ad a o-membershp degree of a elemet to a gve set, ca be cosdered as a useful tool to express ucerta formato the huma decso makg process. Based o the tradtoal dstace measures, such as Eucldea dstace, Hammg dstace, Hausdorff dstace, ad other geeralzed dstace measures, ths paper, a varety of dstace measures for tutostc hestat fuzzy sets are proposed, based o whch the correspodg smlarty measures ca be obtaed. We vestgate the coectos of the aforemetoed dstace measures ad further develop a umber of tutostc hestat ordered weghted dstace measures. They ca allevate the fluece of uduly large (or small) devatos o the aggregato results by assgg them low (or hgh) weghts. A umercal example s provded to llustrate these dstace ad smlarty measures. Keywords- tutostc hestat fuzzy sets; dstace measures; smlarty measures; multple attrbute decso makg I. INTRODUCTION The cocept of tutostc fuzzy set (IFS) was troduced by Ataassov [] to geeralze the cocept of Zadeh s fuzzy set [2]. Each elemet a IFS s expressed by a ordered par, ad each ordered par s characterzed by a membershp degree ad a o-membershp degree. The sum of the membershp degree ad the o-membershp degree of each ordered par s less tha or equal to oe. Sce t was frst troduced 986, the IFS theory has bee wdely vestgated ad appled to a varety of felds e.g. mathematcal programmg,[3] decso makg,[4-6] patter recogto,[7-9] mage processg, [0] grey relatoal aalyss, [6] etc. A growg umber of studes focus o the dstace measure ad the smlarty measure for HFSs [4] ad some extesos of HFS [5,6]. Dstace measures are fudametally mportat varous felds such as decso makg[7-8], rsk vestmet [9], ad patter recogto[20]. Hestat fuzzy sets (HFSs), a exteso of tradtoal fuzzy sets, ca address these stuatos. HFSs were frst troduced by Torra ad Narukawa [], ad they permt the membershp degrees of a elemet to be a set of several possble values betwee zero ad oe. HFSs are hghly useful resolvg stuatos where people hestate whe provdg ther prefereces the decso-makg process, ad they have bee a subect of great terest to researchers. Recetly, Rodríguez et al. [2] preseted a overvew ad dscussed future treds for HFSs. The aforemetoed measures, however, caot be used to deal wth the dstace measures ad smlarty measures betwee tutostc hestat fuzzy sets. Due to the fact that hestacy s a very commo problem huma decso makg process as metoed earler, t s ecessary to develop some dstace measures ad smlarty measures for tutostc hestat fuzzy sets. I order to do so, the remader of ths paper s set out as follows. I the ext secto, we troduce some basc cocepts related to hestat fuzzy sets ad tutostc hestat fuzzy sets. I Secto 3, we propose some dstace measures ad smlarty measures for tutostc hestat fuzzy sets. I Secto 4, a llustratve However, the process of some practcal group recommedato, sometmes, due to the tme pressure ad lack of kowledge or data or the recommeders lmted atteto 206. The authors - Publshed by Atlats Press 82

2 example s poted out. I Secto 5, cocludes the paper wth some remarks. II. PRELIMINARIES A. Itutostc Fuzzy Sets Defto ([]) Let X be a uverse of dscourse, a IFS A X s defed as: A { x, u A( x), va( x) x X} where the fuctos u A (x) ad v A (x) deote the degrees of membershp ad o-membershp of the elemet xx to the set A, respectvely, wth the codto: 0 u A (x), 0 v A (x), 0 u A (x)+ v A (x). =( u +v ) s amed as a tutostc fuzzy value (IFV). B. Hestat Fuzzy Sets Defto 2 ([]) Let X be a referece set, a hestat fuzzy set (HFS) A o X s defed terms of a fucto h A (x) whe appled to X returs a fte subset of [0, ],.e., A { x, ha( x) x X} where h A (x) s a set of some dfferet values [0, ], represetg the possble membershp degrees of the elemet xx to A. For coveece, we call h A (x) a hestat fuzzy elemet (HFE) [4]. III. DISTANCE AND SIMILARITY MEASURES FOR INTUITIONISTIC HESITANT FUZZY SETS It has bee kow that the group recommedato systems, t s somewhat dffcult for the recommeders to assg exact value for a membershp degree ad a omembershp degree of certa elemet to A, but maybe hestat from oe tutostc fuzzy value to aother. It meas that t s very ecessary to troduce the cocept of tutostc hestat fuzzy set (IHFS). Defto 3 Let X be a fxed set; a tutostc hestat fuzzy set (IHFS) o X s gve terms of a fucto that whe appled to X returs a subset of D. To be easly uderstood, we express the IHFS by a mathematcal symbol, ad defe as: A { x, h ( x ) x X,,2,, } A where h A ( x ) s a set of some dfferet tutostc fuzzy values D, deotg the possble membershp degree ad omembershp degree of the elemet xx to the set A ~. where ( ) ( ), ( ) ( ), h x h x h x h x h ( x ) h ( x ) For coveece, we call h ( x ), h ( x ) A% A% A% A% A% A% A% A% a tutostc hestat fuzzy elemet (IHFE). It s oted that the umber of values dfferet IHFEs may be dfferet. To compute the dstace ad smlarty betwee two IHFSs, Whe the legth of two IHFSs are ot equal, oe ca make them havg the same umber of elemets through addg some elemets to the IHFE whch has less umber of elemets. I terms of the pessmstc prcple, the smallest elemet wll be added whle the opposte case, the optmstc prcple may be adopted. I the preset work, we use the former case. Ths dea has bee successfully appled to dstace ad smlarty measures for HFSs [4]. Drawg o the well-kow Hammg dstace ad the Hausdorff dstace, we preset the followg weghted dstace measures for tutostc hestat fuzzy sets. Assume that the weght of the elemet x X s w (,2, L, ) wth w [0,],ad w, the we get some geeralzed tutostc hestat fuzzy weghted dstace. Defto 4 The geeralzed tutostc hestat fuzzy weghted ormalzed Hammg dstace betwee A % ad B % are defed as follows: l x d ( A, B) w h ( x ) h ( x ) h ( x ) h ( x) GIHFWNHD A( ) B( ) A( ) B( ) 2lx The geeralzed tutostc hestat fuzzy weghted ormalzed Hausdorff dstace betwee A ~ ad B ~ are defed as follows: d A B w h x h x h x h x GIHFWNHD (, ) max ( ) ( ) B ( ) ( ), ( ) ( ) B ( ) ( ) A A 83

3 Combg the above equatos, we defe a geeralzed hybrd tutostc hestat fuzzy weghted ormalzed dstace as follows: dghihfwnd ( A, B) w l x h ( x ( ) ) h ( )( ) ( ) ( ) ( )( ) A B x h x h B x A 4lx max ha ( ) ( x ) hb ( ) ( x ), ha ( ) ( x ) hb ( ) ( x ) 2 Based o the relatoshp betwee smlarty measures ad dstaces measures, the weghted smlarty measures of tutostc hestat fuzzy set are defed as follows: Defto 5 The geeralzed tutostc hestat fuzzy weghted ormalzed Hammg smlarty measure betwee A ~ ad B ~ are defed as follows: s ( A, B) d GIHFWNHSM GIHFWNHD The geeralzed tutostc hestat fuzzy weghted ormalzed Hausdorff smlarty measure betwee A ~ ad B ~ are defed as follows: s GIHFWNSM ( AB, ) d GIHFWNHD( AB, ) Combg the above equatos, we defe a geeralzed hybrd tutostc hestat fuzzy weghted ormalzed smlarty measure as follows: sghihfwnsm ( A, B) dghihfwnd ( A, B) IV. AN APPROACH BASED ON SIMILARITY MEASURES TO MULTI-CRITERIA DECISION MAKING WITH IHFSS Mult-crtera decso makg, whch ca be characterzed terms of a process of choosg or selectg suffcetly good alteratve(s) (or course(s)) from a set of alteratves to atta a goal (or goals), ofte happes our daly lfe. For example, choosg a car to buy, or selectg a electroc product from amazo or ebay. A mult-crtera decso makg problem wth IHFS formato ca be terpreted as follows: Suppose that a decso maker s asked to evaluate a set of alteratves X={x,x 2,,x } wth respect to several crtera c (,2, L, m). The crtera have a weghtg vector m w={w,w 2,,w m }, where 0 ad. The w w decso maker mght feel much easer ad are more wllg to gve ther assessmets by provdg some IHFS formato. Example. Cosder a move recommedato system. To better recommed dfferet types of moves o the market, we calculate ther dstace ad smlarty accordg to four attrbutes: story (x ), actg (x 2 ), vsuals (x 3 ) ad drecto (x 4 ). The weghg vector of these four attrbutes s w=(5,,, ). Suppose that a group teds to gve ratgs o fve moves M, M 2, M 3, M 4. Gve the recommeders who make such a recommedato have dfferet backgrouds ad levels of kowledge, terests ad hobbes, sklls, experece ad persoalty, etc., ths could lead to a dfferece the recommedato formato. To clearly reflect the dffereces of the opos of dfferet recommeders, the data of recommedato formato are represeted by the IHFSs ad lsted Table. TABLE I. THE DATA OF RECOMMENDATION INFORMATION x x 2 x 3 x 4 M {(,),(,)} {(,)} {(,)} {(,)} M 2 {(,0.),(,)} {(,),(,)} {(,),(,)} {(,)} M 3 {(0.9,0.),(,0.)} {(,0.),(,)} {(,0.)} {(0.9,0.)} M 4 {(,)} {(,)} {(,),(,)} {(,)} M 5 {(,)} {(,)} {(,),(,)} {(0.,)} Therefore, we defe each deal IHFE h (, 0), (,2, L, ) the deal tem M { x, h x X}, (,2, L, ). We use the geeralzed tutostc hestat fuzzy weghted ormalzed Hammg smlarty measure, the geeralzed tutostc hestat ormalzed Hausdorff smlarty measure, ad the geeralzed hybrd tutostc hestat weghted ormalzed smlarty measure to calculate the devatos betwee each alteratve ad the deal alteratve, the we get the rakgs of these alteratves, whch are lsted Fgure 3, respectvely, ad varato of smlarty measure values wth respect to the parameter λ. We ca rak the moves accordace wth the geeralzed tutostc hestat fuzzy weghted ormalzed 84

4 Hammg smlarty measure: M 3 >M 2 >M >M 4 > M 5, The move M 3 s the best choce accordg to ths smlarty measure. Usg the geeralzed tutostc hestat ormalzed Hausdorff smlarty measure, or the geeralzed hybrd tutostc hestat weghted ormalzed smlarty measure, we ca get the same cocluso. We fd that the rakgs are dfferet as the parameter λ (whch ca be cosdered as the decso makers rsk atttude) chages, cosequetly, the proposed dstace measures ca provde the decso makers more choces as the dfferet values of the parameter are gve accordg to the decso makers rsk atttudes. Smlarty 0.9 M V. CONCLUSION As geeralze of the hestat fuzzy sets, tutostc hestat fuzzy sets, whch permts the membershps of a elemet to a gve set havg a few dfferet tutostc fuzzy sets rather tha real umbers, ca be cosdered as a powerful tool to express ucerta formato the huma decso makg process. Based o the tradtoal Hammg dstace, Eucldea dstace, Hausdorff dstace, ad geeralzed dstace, ths paper, we propose a varety of dstace measures for tutostc hestat fuzzy sets, based o whch the correspodg smlarty measures ca be obtaed. We vestgate the coectos of the aforemetoed dstace measures. They ca allevate the fluece of uduly large (or small) devatos o the aggregato results by assgg them low (or hgh) weghts. Fally, we shall preset a umercal example to show potetal evaluato of emergg techology commercalzato wth tutostc hestat fuzzy formato order to llustrate the method proposed ths paper. I the future, we shall cotue workg the applcato of the tutostc hestat fuzzy multple attrbute decso makg to other domas FIGURE I. VARIATION OF THE s GIHFOWNSM WITH RESPECT TO THE PARAMETER ACKNOWLEDGMENT The authors would lke to thak the aoymous revewers for ther valuable suggestos as to how to mprove ths paper. Ths work was supported by the Natural Scece Foudato of the Ahu Educatoal Commttee of Cha (Grat Nos. KJ205A300, KJ204A00). Smlarty s FIGURE II. VARIATION OF THE GIHFOWNHSM WITH RESPECT TO THE PARAMETER Smlarty s FIGURE III. VARIATION OF THE GHIHFOWNSM WITH RESPECT TO THE PARAMETER M M REFERENCES [] K. T. Ataassov, Itutostc fuzzy sets, Fuzzy Sets Syst. 20() (986) [2] L.A. Zadeh, Fuzzy Sets If. Cotrol 8 (965) [3] D. F. L, Mathematcal-programmg approach to matrx games wth payoffs represeted by Ataassov's terval-valued tutostc fuzzy sets, IEEE Tras. Fuzzy Syst. 8(6) (200) [4] G.W. We, X.F. Zhao, Some duced correlated aggregatg operators wth tutostc fuzzy formato ad ther applcato to multple attrbute group decso makg, Expert Syst. Appl. 39 (2) (202) [5] Z.L. Yue, Y.Y. Ja, G.D. Ye, A approach for multple attrbute group decso makg based o tutostc fuzzy formato, It. J. Ucertaty Fuzzess Kowl. Based Syst. 7 (3) (2009) [6] G.W. We, Gray relatoal aalyss method for tutostc fuzzy multple attrbute decso makg, Expert Syst. Appl. 38 (9) (20) [7] D.F. L, C.T. Cheg, New smlarty measures of tutostc fuzzy sets ad applcato to patter recogtos, Patter Recogt. Lett. 23 (2002) [8] S.K. De, R. Bswas, A.R. Roy, A applcato of tutostc fuzzy sets medcal dagoss, Fuzzy Sets ad Systems 7 (200) [9] I.K. Vlachos, G.D. Sergads, Itutostc fuzzy formato applcatos to patter recogto, Patter Recogto Letters 28 (2007) [0] T. Chara, Itutostc fuzzy segmetato of medcal mages, IEEE Tras. Bomed. Eg. 7(6) (200) [] V. Torra, Hestat fuzzy sets, It. J. Itell. Syst. 25 (200) [2] R.M. Rodríguez, L. Martíez, V. Torra, Z.S. Xu, F. Herrera, Hestat fuzzy sets: state of the art ad future drectos, It. J. Itell. Syst. 29 (204)

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