Intuitionistic Fuzzy Einstein Prioritized Weighted Average Operators and their Application to Multiple Attribute Group Decision Making

Transcription

1 Appl. Math. If. Sci. 9 No Applied Mathematics & Iformatio Scieces A Iteratioal Joural Ituitioistic Fuzzy Eistei Prioritized Weighted Average Operators ad their Applicatio to Multiple Attribute Group Decisio Makig Rakumar Verma 1 ad BhuDev Sharma 1 Departmet of Applied Scieces HMR Istitute of echology ad Maagemet Guru Gobid Sigh Idraprastha Uiversity Hamidpur Delhi Idia Departmet of Mathematics Jaypee Istitute of Iformatio echology Deemed Uiversity A-10 Sector-6 Noida Uttar Pradesh Idia Received: 5 Mar. 015 Revised: 3 May 015 Accepted: 4 May 015 Published olie: 1 Nov. 015 Abstract: his study ivestigates the multiple attribute decisio makig uder ituitioistic fuzzy eviromet i which the attributes ad the decisio makers are i differet priority levels. I this paper we first propose two ew ituitioistic fuzzy aggregatio operators such as the ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator ad the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator for aggregatig ituitioistic fuzzy iformatio. hese proposed operators ca capture the prioritizatio pheomeo amog the aggregated argumets. he properties of the ew aggregatio operators are studied i details ad their special cases are examied. Furthermore based o the proposed operators a approach to deal with multiple attribute group decisio makig problems uder ituitioistic fuzzy eviromet is developed. Fially a practical example is provided to illustrate the multiple attribute group decisio makig process with ituitioistic fuzzy iformatio. Keywords: Ituitioistic fuzzy sets ituitioistic fuzzy umbers prioritized weighted average operator ituitioistic fuzzy prioritized weighted average operator ituitioistic fuzzy prioritized weighted geometric operator 1 Itroductio Ituitioistic fuzzy set IFS as a geeralized form of fuzzy set [48] was itroduced by Ataassov [1]. It is characterized by three fuctios expressig the degree of membership the degree of o-membership ad the degree of hesitacy respectively. Fuzzy sets are IFSs but the coverse is ot ecessarily true. Over the last few decades IFS theory has bee extesively ivestigated by may researchers ad applied i a variety of fields icludig decisio makig [ ][19]-[38][4 4445] ad [553] medical diagosis [518] ad patter recogitio [ ] etc. Iformatio aggregatio [15] is a importat ad useful research topic i ituitioistic fuzzy set theory which has received quite some attetio from researchers ad practitioers i the last couple of years. o aggregate ituitioistic fuzzy iformatio Xu ad Yager [3] proposed some geometric aggregatio operators such as the ituitioistic fuzzy weighted geometric IFWG operator the ituitioistic fuzzy ordered weighted geometric IFOWG operator the ituitioistic fuzzy hybrid geometric IFHG operator ad developed a applicatio of IFHG operator to multiple attribute decisio makig with ituitioistic fuzzy iformatio. Xu [33] developed some arithmetic aggregatio operators such as the ituitioistic fuzzy weighted averagig IFWA operator the ituitioistic fuzzy ordered weighted averagig IFOWA operator ad the ituitioistic fuzzy hybrid averagig IFHWA operator. Motivated by Yager [41] Zhao et al. [51] proposed the geeralized ituitioistic fuzzy weighted averagig GIFWA operator the geeralized ituitioistic fuzzy ordered weighted averagig GIFOWA operator ad the geeralized ituitioistic fuzzy hybrid averagig GIFHA operator. Further Xia ad Xu [30] developed a umber of geeralized ituitioistic fuzzy poit aggregatio operators such as the geeralized Correspodig author rkver83@gmail.com c 015 NSP Natural Scieces Publishig Cor.

2 3096 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... ituitioistic fuzzy poit weighted averagig GIFPWA operators the geeralized ituitioistic fuzzy poit ordered weighted averagig GIFPOWA operator ad the geeralized ituitioistic fuzzy poit hybrid averagig GIFPHA operator ad studied their properties with some special cases. Wei [6] proposed some iduced ituitioistic fuzzy geometric aggregatio operators ad studied their applicatios i group decisio makig uder ituitioistic fuzzy eviromet. Based o the ordered weighted distace OWD operator [38] Zeg ad Su [49] proposed the ituitioistic fuzzy ordered weighted distace IFOWD operator to aggregate the ituitioistic fuzzy iformatio. Xu ad Yager [36] ivestigated the dyamic ituitioistic fuzzy multiple-attribute decisio makig problems ad developed the dyamic ituitioistic fuzzy weighted averagig DIFWA operator to aggregate dyamic ituitioistic fuzzy iformatio. Also Wei [6] proposed the dyamic ituitioistic fuzzy weighted geometric DIFWG operator ad applied it to dyamic multiple attribute decisio makig with ituitioistic fuzzy iformatio. Recetly Wag ad Liu [19] itroduced some ew operatios o IFSs such as Eistei sum Eistei product Eistei expoetiatio etc. ad developed some ew ituitioistic fuzzy aggregatio operators such as the ituitioistic fuzzy Eistei weighted average IFEWA operator ad the ituitioistic fuzzy Eistei ordered weighted average IFEOWA operator. Wag ad Liu [0] further proposed some ew geometric ituitioistic fuzzy aggregatio operators such as the ituitioistic fuzzy Eistei weighted geometric IFEWG operator ad the ituitioistic fuzzy Eistei ordered weighted geometric IFEOWG operator. hey also established some useful properties of these operators such as commutativity idempotecy ad mootoicity ad developed a decisio makig method for solvig multiple attribute decisio makig problem uder ituitioistic fuzzy eviromet. Zhao ad Wei [50] itroduced some ituitioistic fuzzy Eistei hybrid aggregatio operators ad discussed their applicatios i multiple attribute decisio makig. Xu et al. [31] itroduced a ew aggregatio operator called iduced ituitioistic fuzzy Eistei ordered weighted averagig I-IFEOWA operator for aggregatig ituitioistic fuzzy iformatio ad studied its applicatio i multiple attribute group decisio makig. Oe more thig that may be metioed is that the above aggregatio operators for IFNs have assumed that the attributes ad the decisio makers are at same priority level. However i the real life multiple attribute group decisio makig problems attributes ad decisio makers have differet priority levels. o imbue this issue motivated by the idea of prioritized aggregatio operators [4 43] Yu [45] developed some ituitioistic fuzzy prioritized aggregatio operators such as the ituitioistic fuzzy prioritized weighted average IFPWA operator the ituitioistic fuzzy prioritized weighted geometric IFPWG operator ad proposed two approaches to solve multi-criteria group decisio makig problems uder ituitioistic fuzzy eviromet. Further Yu [46] itroduced a ew geeralized ituitioistic fuzzy prioritized geometric aggregatio operator based o Archimedea t-coorm ad t-orm. Recetly Yu [47] also proposed two ew geeralized prioritized aggregatio operators such as the geeralized ituitioistic fuzzy prioritized weighted average GIFPWA operator geeralized ituitioistic fuzzy prioritized weighted geometric GIFPWG operator ad discussed their applicatios i multi criteria decisio makig. However it seems that there is o ivestigatio o prioritized aggregatio techique usig Eistei operatios with IFNs. herefore the focus of this paper is to develop some ituitioistic fuzzy prioritized weighted average operator based o Eistei operatios. he paper is orgaized as follows: I Sectio some basic cocepts related to ituitioistic fuzzy sets Eistei operatios o ituitioistic fuzzy sets ad prioritized average operator are briefly give. I Sectio 3 we itroduce two ew prioritized weighted aggregatio operators such as the ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator ad the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator ad discuss their particular cases. Some properties of IFEPWA ad IFEPWG operators are also studied here. I Sectio 4 we develop a method for multiple attribute group decisio makig based o the proposed operators uder ituitioistic fuzzy eviromet i which the attributes ad decisio makers are i differet priority levels. I Sectio 5 fially a umerical example is preseted to illustrate the proposed approach to multiple attribute group decisio makig with ituitioistic fuzzy iformatio. Our coclusios are preseted i Sectio 6. Prelimiaries We briefly review some basic cocepts related to ituitioistic fuzzy sets ad prioritized weighted averagig operator which will be eeded i the followig aalysis. Defiitio 1. Ituitioistic Fuzzy Set [1]: A ituitioistic fuzzy set A i a discrete uiverse of discourse X x 1 x...x is give by A{µ A xν A x x X} 1 where µ A : X [01] ad ν A : X [01] with the coditio 0 µ A x+ν A x 1. For each x X the umbers µ A x ad ν A x deote the degree of membership ad degree of o-membership of x i A respectively. Further π A 1 µ A x ν A x is called the degree of hesitacy or the ituitioistic idex of x i A. For a elemet x X Xu ad Yager [3] ad Xu [33] defied the pair µ A xν A x as ituitioistic fuzzy umber IFN ad deoted it by µ ν. c 015 NSP Natural Scieces Publishig Cor.

3 Appl. Math. If. Sci. 9 No / Defiitio. Eistei Operatios o IFNs [19 0]: Let 1 µ 1 ν 1 ad µ ν be two ituitioistic fuzzy umbers ad λ > 0 the followig Eistei operatios o 1 ad are defied as µ i 1 ε 1 +µ 1+µ 1 µ ν 1 ν 1+1 ν 1 1 ν µ ii 1 ε 1 µ 1+1 µ 1 1 µ ν 1 +ν 1+ν 1 ν 1+µ iiiλ ε 1 1 λ 1 µ 1 λ 1+µ 1 λ +1 µ 1 λ νλ 1 ν 1 λ +ν λ 1 iv 1 ε λ µ λ 1 µ 1 λ 1+ν 1 λ 1 ν 1 λ +µ λ 1 1+ν 1 λ +1 µ 1 λ. Defiitio 3. Score Fuctio [4]: Let µ ν be a ituitioistic fuzzy umber the score fuctio S of a IFN is defied as follows: S µ ν S [ 11]. Defiitio 4. Accuracy Fuctio [7]: Let µ ν be a ituitioistic fuzzy umber the score fuctio H of a IFN is defied as follows: H µ + ν H [01]. 3 o rak ay two i µ i ν i i 1 Xu ad Yager [3] proposed the followig method: Defiitio 5. Rakig Method for IFNs [3]: Let 1 ad be two ituitioistic fuzzy umbers S 1 ad S be the scores of 1 ad respectively ad H 1 ad H be the accuracy values of 1 ad the iif S 1 > S the 1 is larger tha deoted by 1 >. iiif S 1 S the we check their accuracy values ad decide as follows: aif H 1 H the 1 ad represet the same iformatio deoted by 1. bhowever if H 1 >H the 1 is larger tha deoted by 1 >. It is oted that above defied score fuctio S rages from -1 to 1. Liu [14] itroduced aother score fuctio of a IFN as follows S 1+S 1+ µ ν [01]. 4 Accordig to Liu [14] if we replace the score fuctio S by S the order relatio betwee two IFNs 1 ad itroduced by Xu ad Yager [3] is also valid. he Prioritized Weighted Average PWA operator was origially itroduced by Yager [4 43] as follows: Defiitio 6. Prioritized Weighted Average PWA Operator [443]: Let G {G 1 G...G } be a collectio of attributes ad let there be a prioritizatio betwee the attributes expressed by the liear orderig G 1 G G 3 G idicatig that the attribute G has a higher priority tha G k if < k. Also let G x be the performace value of ay alterative x uder attribute G ad satisfies G x [01]. If PWAG 1 xg x...g x 1 G x 5 1 where 1 k1 G kx the PWA is called the prioritized weighted average PWA operator. I the ext sectio we ivestigate the prioritized weighted average operator uder ituitioistic fuzzy eviromet based o Eistei operatios ad propose the ituitioistic fuzzz Eistei prioritized weighted average IFEPWA operator ad the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator. 3 Ituitioistic Fuzzy Eistei Prioritized Weighted Average Operators O the basis of the defiitio of PWA operator we give the defiitio of ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator as follows: Defiitio 7. Ituitioistic Fuzzy Eistei Prioritized Weighted Average IFEPWA Operator: Give a set of ituitioistic fuzzy umbers µ ν 1... the ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator is defied as follows IFEPWA 1... ε 1 1 ε 1 1 ε 1 ε 1 ε ε ε 1 ε 6 where 1 k1 S k ad S k is the score of k µ k ν k. Next based o the Eistei operatioal laws of IFNs we have results i the followig theorems: heorem 1. Let µ ν 1... be a set of ituitioistic fuzzy umbers the the aggregated value by usig the IFEPWA operator is also a ituitioistic fuzzy umber ad IFEPWA ε 1 ε 1 ε ε ε 1 ε µ µ 11+ µ µ ν 7 1 ν ν where 1 k1 S k ad S k is the score of k µ k ν k. c 015 NSP Natural Scieces Publishig Cor.

4 3098 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... Proof: he first result follows directly from Defiitio. We prove 7 by mathematical iductio o. i First let the for 1 ad accordig to the Eistei operatioal laws of the IFNs we have 1 1 ε µ1 1 1 µ ν µ 1 1 i +1 µ ν ν 1 8 ad 1 ε i 1+ µ 1 1 µ 1 1 ν 1+ µ µ. 1 ν ν 9 Now from Defiitio 7 we have 1 IFEPWA 1 ε 1 ε 1 ε µ µ 1 1 µ µ µ µ 1 i +1 µ µ ν 1 1 ν 1 1 ν 1 1 ν ν 1 1 ν 1 1+ µ 1 11 µ 11+ µ µ ν ν ν his shows that the result 7 holds for. Next let 7 holds for k that is IFEPWA 1... k 1 k k ε 1 ε 1 k ε ε ε 1 k ε k 1 k 1 1+ µ k 1 k 1 1 µ k 1 k 11+ µ k 1 + k 11 µ k 1 k 1 k 1 ν k 1 ν k 1 + k k 1 1 ν. 11 Whe k+ 1 by the Eistei operatioal laws of the IFNs we have IFEPWA ε 1 ε 1 ε ε ε 1 ε k ε 1 ε k 1 1+ µ k 1 1+ µ 1 k 1 1 µ 1 + k 1 1 µ 1 1 k 1 1 ν k 1 ν 1 + k 1 1 ν 1+ µ 1 1 µ 1 1 ν 1+ µ 1 +1 µ 1 ν ν 1 1+ µ µ µ 1 1 µ ν 1 1 ν ν 1 1 i.e.7 holds for. herefore by mathematical iductio 7 holds for all. his completes the proof of the heorem 1. he IFEPWA operator has the followig properties: heorem. Idempotecy: Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k 3... with 1 1. Also let S k be the score of k µ k ν k. If all the ituitioistic fuzzy umbers 1... are equal i.e. µ ν the Proof: From Defiitio 7 we have IFEPWA IFEPWA 1... IFEPWA... 1 ε ε 1 ε ε ε 1 ε µ µ µ 1 11 µ 1 1 ν 1 ν ν c 015 NSP Natural Scieces Publishig Cor.

5 Appl. Math. If. Sci. 9 No / µ 1+ µ 1+ µ + 1 µ his proves the theorem. ν ν + ν µ ν. 14 Corollary 1: If µ ν 1... is a collectio of the largest IFNs i.e. 10 the IFEPWA 1... IFEPWA Proof: Corollary 1 follows directly from heorem. Corollary. No-compesatory: If 1 µ 1 ν 1 is the smallest IFN i.e the IFEPWA 1... IFEPWA Proof: Sice 1 01 the by defiitio of the score fuctio we have Further S S k 3... ad k1 So that from Equatios 17 ad 18 we have 1 S k S 1 S S 1 19 k1 0 S S ad By Defiitio 7 we have 1 IFEPWA 1... ε 1 ε 1 ε ε ε ε 1 ε 1 ε ε ε ε his corollary carries a very sigificat coclusio. Accordig to it highest priority attribute i decisio makig is ot compromised. his amout to sayig that if the highest priority attribute is ot met tha other attributes eve if they are partially/fully met will ot matter. hat is they will have o cotributio to the fial decisio. heorem 3. Mootoicity: Let µ ν ad µ ν 1... be two sets of ituitioistic fuzzy umbers 1 k1 S k 1 k1 S k Also let S k ad S k be the scores of k µ k ν k ad k µ ν k respectively. If k i.e. µ µ ad ν ν for all the IFEPWA 1... IFEPWA Proof: We kow that fx 1 x 1+x x [01] is a decreasig fuctio of x. If µ µ for all the fµ fµ i.e. 1 µ 1+µ 1 µ 1+µ for all. Now let ad 1 3 w w be the prioritized weight vectors of µ ν ad µ ν 1... such that [01] with the coditio From the result above we have hus i.e. 1 µ 1+ µ µ µ 1 µ 1 1 µ 1 1+ µ 1 1+ µ 1 1 µ µ 1 1+µ µ 1 1+ µ µ 11+ µ µ µ 1 11 µ µ µ 1 Next usig gy y y y [01] a decreasig fuctio of y if ν ν for all the gµ gµ i.e. ν ν ν ν for all. he hus ν ν 1 ν 1 ν ν ν 1 ν 1 ν c 015 NSP Natural Scieces Publishig Cor.

6 3100 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... i.e. ν 1 ν ν 1 1 ν ν 1 1 ν ν ν ν ν Note that 30 also holds eve if ν ν 0 for all. he accordig to Defiitio 5 we obtai that IFGEPWA 1... IFEPWA his proves the theorem. heorem 4. Boudedess: Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k k ad S k be the score of k µ k ν k. Also let he mi µ max ν ad + max µ mi ν. 3 IFEPWA Proof: It directly follows from heorem 3. Relatioship betwee ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator ad ituitioistic fuzzy prioritized weighted average IFPWA operator [45]: I the ext theorem we prove a relatio betwee the IFEPWA operator ad IFPWA operator proposed by Yu [45] as follows IFPWA i 1 1 µ 1 1 ν heorem 5. Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k ad S k be the score of k µ k ν k. he IFEPWA 1... IFPWA with equality if ad oly if 1 3. Proof: Usig weighted AM-GM iequality [15 41] we have 1 the 1+ µ µ 1 1+ µ 1 1+ µ µ 11+ µ µ µ µ µ 1 µ µ where the equality holds if ad oly if µ 1 µ µ 3 µ. I additio sice ν ν ν ν i1 38 we have 1 1 ν 1 ν 39 1 ν ν 1 where the equality holds if ad oly if ν 1 ν ν 3 ν. he accordig to Defiitio 5 we obtai that IFEPWA 1... IFPWA with equality if ad oly if 1 3. his proves the theorem. Special cases of IFEPWA operator: i If the priority levels of the aggregated argumets reduced to the same level the the IFEPWA operator reduces to the ituitioistic fuzzy Eistei weighted average IFEWA operator [19]: IFEPWA 1... w 1 ε 1 ε w ε ε ε w ε w w 1 1+ µ 1 1 µ w w 1 1+ µ µ 1 ν w w 1 ν + 1 ν w. 41 c 015 NSP Natural Scieces Publishig Cor.

7 Appl. Math. If. Sci. 9 No / ii If ν 1 µ for all ad the priority levels of the aggregated argumets reduced to the same level the the IFEPWA operator gives 1 1+ µ w 11 µ w IFEPWA µ w µ w 4 which we call the fuzzy Eistei weighted average IFEA operator. From the geometric perspective here we defie the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator. Defiitio 8. Ituitioistic Fuzzy Eistei Prioritized Weighted Geometric IFEPWG Operator: Give a set of ituitioistic fuzzy umbers µ ν 1... the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator is defied by IFEPWG 1... ε ε ε 1 ε ε 1 ε ε ε 1 43 where 1 k1 S k ad S k is the score of k µ k ν k. Next based o the Eistei operatioal laws of IFNs we have the followig theorem: heorem 6. Let µ ν 1... be a set of ituitioistic fuzzy umbers the usig the IFEPWG operator the aggregated value is also a ituitioistic fuzzy umber ad IFEPWG ε 1 ε ε 1 ε ε ε µ 1 µ µ ν ν ν ν 1 where 1 k1 S k ad S k is the score of k µ k ν k. Proof: It ca be proved o lies similar to that of heorem 1. Some other properties of the IFEPWG operator are proved i the followig theorems: heorem 7. Idempotecy: Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k 3... with 1 1. Also let S k be the score of k µ k ν k. If all the ituitioistic fuzzy umbers 1... are equal i.e. µ ν i the IFEPWG Proof: he proof of heorem 7 is similar to that of heorem. heorem 8. Mootoicity: Let µ ν ad µ ν 1... be two sets of ituitioistic fuzzy umbers 1 k1 S k 1 k1 S k Also let S k ad S k be the scores of k µ k ν k ad k µ ν k respectively. If k i.e. µ µ ad ν ν for all the IFEPWG 1... IFEPWG Proof: he proof of heorem 8 is similar to heorem 3. heorem 9. Boudedess: Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k ad S k be the score of k µ k ν k. Also let he mi µ maxν ad + max µ mi ν. 47 IFEPWG Proof: It directly follows from heorem 8. Relatio betwee ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator ad ituitioistic fuzzy prioritized weighted geometric IFPWG operator [45]. Based o the algebraic laws o IFNs Yu [45] defied the IFPWG operator as follows 1 1 IFPW G µ 1 1 ν We have the followig additioal theorem: heorem 10. Let µ ν 1... be a set of ituitioistic fuzzy umbers 1 k1 S k ad S k be the score of k µ k ν k. he IFPWG 1... IFEPWG with equality if ad oly if 1 3. c 015 NSP Natural Scieces Publishig Cor.

8 310 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... Proof: Usig weighted AM-GM iequality [14 41] we have or 1 µ µ µ + 1 µ µ 1 µ 51 1 µ µ 1 where the iequality holds if ad oly if µ 1 µ µ 3 µ. Additioally sice or 1+ν 1 1 ν ν ν ν ν 1 11+ν ν ν 1 11+ν ν ν where the iequality holds if ad oly if ν 1 ν ν 3 ν. he accordig to Defiitio 5 we get IFPWG 1... IFEPWG with equality if ad oly if 1 3. his proves the theorem. Special cases of IFEPWG operator: i If the priority levels of the aggregated argumets are reduced to the same level the the IFEPWG operator 43 reduces to the ituitioistic fuzzy Eistei weighted geometric IFEWG operator [0]: IFEPWG ε wi ε ε wi ε ε ε wi 1 µ w w 1 µ + 1 µ w w wi 1 1+ν 1 1 ν w w 1 1+ν ν ii If ν 1 µ 1... ad the priority levels of the aggregated argumets are reduced to the same level the the IFEPWG operator gives 1 µ w IFEPWG µ 55 w + 1 µw which we call the fuzzy Eistei weighted geometric FEWG operator. I the followig sectio we suggest a applicatio of the proposed aggregatio operators to multiple attribute group decisio makig problems with ituitioistic fuzzy iformatio ad give a illustrative example. 4 A Approach to Multiple Attribute Group Decisio Makig uder Ituitioistic Fuzzy Eviromet Let us cosider a multiple attribute group decisio makig problem ivolvig a set of optios X {X 1 X... X m } to be cosidered uder a set of attributes G {G 1 G... G } ad let there be a prioritizatio betwee the attributes expressed by the liear orderig G 1 G G idicatig attribute G has a higher priority tha G s if < s ad let D { } D 1 D... D q be the set of decisio makers ad let there be a prioritizatio betwee the decisio makers expressed by the liear orderig D 1 D D q idicatig decisio maker D η has a higher priority tha D ς if η < ς. Let A k k i m µ k i ν k i m be a ituitioistic fuzzy decisio matrix ad k i µ k i ν k i be a attribute value provided by the decisio maker D k D which is expressed i a IFN where µ k i idicates the degree that the optio X i X satisfies the attribute G G expressed by the decisio maker D k ad ν k i idicates the degree that the optio X i X does ot satisfies the attribute G G expressed by the decisio maker D k such that µ k i [0 1]ν k i [0 1]µ k i + ν k i 1 i1... m; o harmoize the data first step is to look at the attributes. hese i geeral ca be of differet types. If all the attributes G {G 1 G...G } are of the same type the the attribute values do ot eed harmoizatio. However if these ivolve differet scales ad/or uits there is eed to covert them all to the same scale ad/or uit. Just to make this poit clear let us cosider two types of attributes amely i cost type ad the ii beefit type. Cosiderig their atures a beefit attribute the bigger the values better is it ad cost attribute the smaller the values the better is it are of rather opposite type. I such cases we eed to first trasform the attribute values of cost type ito the attribute values of beefit type. So trasform the ituitioistic fuzzy decisio matrix A k k i m ito the ormalized ituitioistic fuzzy c 015 NSP Natural Scieces Publishig Cor.

9 Appl. Math. If. Sci. 9 No / decisio matrix R k r k i m by Xu ad Hu [40] where r k i usig the method give µ k i ν k i ad r k k i i for beefit attribute G C k i1...m ; 1... i for cost attribute G 57 where k C k i is the complemet of i such that k C i ν k i µ k i. With attributes harmoized ad usig the IFEPWA / IFEPWG operator we ow formulate a algorithm to solve multiple attribute group decisio makig problems with ituitioistic fuzzy iformatio: Step 1. Calculate the values of k i k 1...q as follows k 1 i γ1 k Step. Utilize IFEPWA operator: r i µ i ν i IFEPWAr 1 i r i...r q i 1 i q k1 k i q k1 q k1 ε 1+ µ k i 1+ µ k i r 1 i i ε S r γ i k3...q 58 q k1 k i 1 i ε k i q k1 k i q k1 k i q k1 k i + q k1 q k1 ν k i or the IFEPWG operator: r i µ i ν i q k1 IFEPWGr 1 i r i... r q i k i r i q i ε ε 1 µ k i 1 µ k i ν k i k i q k1 k i q k1 k i k i k i q k1 k i q k1 k i + q k1 1 i 1 ε i q k1 r k ε i i ε r i q k1 k i q k1 µ k i q k1 q k1 q k1 k i µ k i k i q k1 k i q k1 k i + q k1 1+ν k i 1+ν k i k i µ k i q k1 k i ν k i ε ε r q i q k1 k i q k1 k i k i q k1 k i q k1 k i + q k1 1 ν k i 1 ν k i k i q k1 k i ε q i ε r q i q k1 k i k i q k1 k i k i q k1 k i to aggregate all the idividual ituitioistic fuzzy decisio matrices R k k 1...q ito a r k i m collective ituitioistic fuzzy decisio matrix Rr i m i1...m; 1... Step 3. Calculate the values i i 1...m ; 1... as follows 1 i S r iν i1...m; ν1 i1 1 i1...m. 63 Step 4. Aggregate all ituitioistic fuzzy preferece values r i 1... by the IFEPWA / IFEPWG operator: or r i µ i ν i IFEPWAr i1 r i...r i i1 i i ε r i1 ε 1 i ε r i ε ε 1 i ε r i 1 i i i 1 1+ µi 1 i 1 1 µi 1 i 1 1+ i µ 1 i i i µ 1 i i r i µ i ν i 1 i 1 νi 1 i ν i 1 i i + 1 ν i 1 i i i1...m. 64 IFEPWGr i1 r i...r i ε i1 r i1 1 ε i i ε r i 1 ε i i ε ε r i 1 i 1 i 1 µi 1 i i µ 1 i i + 1 µ i 1 i i i i 1 1+νi 1 i 1 1 νi 1 i 1 1+ν i 1 i i ν i 1 i i i1...m 65 to derive the overall ituitioistic fuzzy preferece values r i i1...m of the optios X i i1... Step 5. Calculate the score values as follows: S r i 1+ µ r i ν ri i1...m. 66 Step 6. Rak all the optios X i i 1...m accordig to the score values S r i i 1...m i descedig order. he leadig X i with the highest value of S r i is the best optio. I order to demostrate the applicability of the proposed method to multiple attribute group decisio makig we c 015 NSP Natural Scieces Publishig Cor.

10 3104 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... cosider below a uiversity faculty recruitmet group decisio makig problem. Example: he departmet of mathematics i a uiversity wats to appoit outstadig mathematics teachers. he appoitmet is doe by a committee of three decisio makers Presidet D 1 Dea of Academics D ad Huma Resource Officer D 3. After prelimiary screeig five teachers X i i 1345 remai for further evaluatio. Pael of decisio makers made strict evaluatio for five teachers X i accordig to the followig four attributes: G 1 the past experiece; G the research capability; G 3 subect kowledge; G 4 the teachig skill. Durig the evaluatio process the uiversity Presidet D 1 has the absolute priority for decisio makig Dea of Academics comes ext. he prioritizatio relatioship for the attributes is as follows G 1 G G 3 G 4. he three decisio makers evaluated the cadidates X i i 1345 with respect to the attributes G 134 ad provided their evaluatio values i terms of ituitioistic fuzzy umbers ad costructed the followig three ituitioistic fuzzy decisio matrices A k k 13 see ables 1-3 a k i 5 4 able 1: Ituitioistic fuzzy decisio matrix A 1 G 1 G G 3 G 4 X X X X X able : Ituitioistic fuzzy decisio matrix A G 1 G G 3 G 4 X X X X X able 3: Ituitioistic fuzzy decisio matrix A 3 G 1 G G 3 G 4 X X X X X Based o the IFEPWA operator the steps are as follows: Step 1: Sice all the attributes G 134 are of the beefit type the the attributes values do ot eed harmoizatio therefore R k A k k i 5 4 r k i 5 4. Step : Usig Equatios i 58 ad 59 to calculate the 1 i i ad 3 i we get [ ] [ ] i i [ ] i Step 3: Usig the IFEPWA operator Equatio 60 to aggregate all the idividual decisio matrices R k k 1 3 ito the collective decisio matrix R r i µ 5 4 ri ν ri we get the followig table: 5 4 able 4: Ituitioistic fuzzy collective decisio matrix R G 1 G G 3 G 4 X X X X X Step 4: Followig Equatios i 6 ad 63 to calculate the i i1...m; 1... we get [ i ] Step 5: Aggregatig all ituitioistic fuzzy umbers r i 1... by the IFEPWA operator Equatio 64 to derive the overall ituitioistic fuzzy preferece values r i i1...m of the teachers X i we get r r r r r Step 6: Calculatig the score values S r i of the teachers X i i1...m we have S r S r S r S r S r Step 7: Rakig the teachers X i i 1345 i accordace with the score values S r i i1345 i descedig order we have X X 4 X 3 X 1 X 5. hus X is the best teacher for this appoitmet. Based o the IFEPWG operator the mai steps are as follows: Step 1 : Same as Step 1. Step : Same as Step. Step 3 : Usig IFEPWG operator Equatio 61 to aggregate all the idividual decisio matrices R k k 1 3 ito a collective decisio matrix R r i 5 4 µ ri ν ri we get the followig 5 4 table c 015 NSP Natural Scieces Publishig Cor.

11 Appl. Math. If. Sci. 9 No / able 5: Ituitioistic fuzzy collective decisio matrix R G 1 G G 3 G 4 X X X X X Step 4 : Usig Equatios i 6 ad 63 to calculate the i i1...m; 1... We get [ i ] Step 5 : Aggregate all ituitioistic fuzzy umbers r i 1... by the IFEPWG operator Equatio 65 to derive the overall ituitioistic fuzzy preferece values r i i1...m of the teacher X i we have r r r r r Step 6 : Calculatig the score values S r i of the teachers X i i1... m we have S r S r S r S r S r Step 7 : Rakig the teachers X i i 1345 i accordace with the score values S r i i1345 i descedig order we have hus the best optio is X 3. X 3 X X 4 X 5 X 1. his result is differet from the result obtaied by the IFEPWA operator because the IFEPWA operator focuses o the impact of overall data while the IFEPWG operator highlights the role of the idividual data. 5 Coclusio I this paper we ivestigate the ituitioistic fuzzy multiple attribute group decisio-makig problem i which the attributes ad decisio makers are at differet priority levels. Based o Eistei operatio some ew prioritized weighted aggregatio operators called the ituitioistic fuzzy Eistei prioritized weighted average IFEPWA operator ad the ituitioistic fuzzy Eistei prioritized weighted geometric IFEPWG operator for aggregatig ituitioistic fuzzy umbers has bee itroduced. he promiet characteristic of these proposed operators is that they take ito accout prioritizatio amog the attributes ad decisio makers. Some of their properties are ivestigated i detail. Based o proposed operators a ituitioistic fuzzy multiple attribute group decisio makig approach is developed to solve multiple attribute group decisio makig problems uder ituitioistic fuzzy eviromet. Fially a umerical example is preseted to illustrate the give approach. Coflict of Iterests he authors declare that there is o coflict of iterests regardig the publicatio of this article. Refereces [1] K.. Ataassov Ituitioistic fuzzy sets Fuzzy Sets ad Systems vol.0 o.1 pp [] F.E. Bora A itegrated ituitioistic fuzzy multi-criteria decisio makig method for facility locatio selectio Mathematical ad Computatioal Applicatios vol.16 o. pp [3] J. Chachi ad S.M. aheri A uified approach to similarity measures betwee ituitioistic fuzzy sets Iteratioal Joural of Itelliget Systems vol.8 o.7 pp [4] S.M. Che ad J.M. a Hadlig multicriteria fuzzy decisio-makig problems based o vague set theory Fuzzy Sets ad Systems vol.67 o. pp [5] S.K. De R. Biswas ad A.R. Roy A applicatio of ituitioistic fuzzy sets i medical diagosis Fuzzy Sets ad Systems vol.117 o. pp [6] A.G. Hatzimichailidis G.A. Papakostas ad V.G. Kaburlasos A ovel distace measure of ituitioistic fuzzy sets ad its applicatio to patter recogitio problems Iteratioal Joural of Itelliget Systems vol.7 o [7].H. Hog ad C.H. Choi Multicriteria fuzzy decisio makig problems based o vague set theory Fuzzy Sets ad Systems vol.114 o.1 pp [8] W.L. Hug ad M.S. Yag Similarity measures of ituitioistic fuzzy sets based o L p metric Iteratioal Joural of Approximate Reasoig vol.46 o.1 pp [9] W.L. Hug ad M.S. Yag Similarity measures of ituitioistic fuzzy sets based o Hausdorff distace Patter Recogitio Letters vol.5 o.14 pp [10] B. Li ad W. He Ituitioistic fuzzy PRI-AND ad PRI- OR aggregatio operators Iformatio Fusio vol.14 o.4 pp [11] D.F. Li ad C.. Cheg New similarity measures of ituitioistic fuzzy sets ad applicatio to patter recogitios Patter Recogitio Letters vol.3 o.1-3 pp [1] D.F. Li Multiattribute decisio makig models ad methods usig ituitioistic fuzzy sets Joural of Computer ad System Scieces vol.70 o.1 pp [13] Z.Z. Liag ad P.F. Shi Similarity measures o ituitioistic fuzzy sets Patter Recogitio Letters vol.4 o.15 pp [14] L.W. Liu Multi-criteria Decisio Makig ad Reasoig Methods i a Ituitioistic Fuzzy or Iterval-valued Fuzzy Eviromet Doctoral Dissertatio Shadog Uiversity Chia 005. c 015 NSP Natural Scieces Publishig Cor.

12 3106 R. Verma B. Sharma: Ituitioistic Fuzzy Eistei Prioritized Weighted... [15] V. orra ad Y. Narukawa Modelig Decisios: Iformatio Fusio ad Aggregatio Operators Berli: Spriger-Verlag 007. [16] R.K. Verma ad B.D. Sharma O geeralized ituitioistic fuzzy divergece relative iformatio ad their properties Joural of Ucertai Systems vol.6 o.4 pp [17] R.K. Verma ad B.D. Sharma A ew measure of iaccuracy with its applicatio to multi-criteria decisio makig uder ituitioistic fuzzy eviromet Joural of Itelliget ad Fuzzy Systems vol.7 o.4 pp DOI: /IFS [18] I.K. Vlachos ad G.D. Sergiadis Ituitioistic fuzzy iformatio-applicatio to patter recogitio Patter Recogitio Letters vol.8 o. pp [19] W. Wag ad X. Liu Ituitioistic fuzzy iformatio aggregatio usig Eistei operatios IEEE rasactios o Fuzzy Systems vol.0 o.5 pp [0] W. Wag ad X. Liu Ituitioistic fuzzy geometric aggregatio operators based o Eistei operatios Iteratioal Joural of Itelliget Systems vol.6 o.11 pp [1] J.Q. Wag ad H.Y. Zhag Multicriteria decisio-makig approach based o Ataassov s ituitioistic fuzzy sets with icomplete certai iformatio o weights IEEE rasactios o Fuzzy Systems vol.1 o.3 pp [] G.W. Wei GRA method for multiple attribute decisio makig with icomplete weight iformatio i ituitioistic fuzzy settig Kowledge Based Systems vol.3 o.3 pp [3] G.W. Wei ad J.M. Merigo Methods for strategic decisio makig problems with immediate probabilities i ituitioistic fuzzy settig Scietia Iraica vol.19 o [4] C.P. Wei P. Wag ad Y.Z. Zhag Etropy Similarity measure of iterval-valued ituitioistic fuzzy sets ad their applicatios Iformatio Scieces vol.181 o.9 pp [5] G.W. Wei Maximizig deviatio method for multiple attribute decisio makig i ituitioistic fuzzy settig Kowledge-Based System vol.1 o.8 pp [6] G.W. Wei Some geometric aggregatio fuctios ad their applicatio to dyamic multiple attribute decisio makig i the ituitioistic fuzzy settig Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systems vol.17 o. pp [7] J.Z. Wu ad Q. Zhag Multi-criteria decisio makig based o ituitioistic fuzzy weighted etropy Expert Systems with Applicatios vol.38 o.1 pp [8] M. Xia Z. Xu ad B. Zhu Geometric Boferroi meas with their applicatios i multi-criteria decisio makig Kowledge Based Systems vol.40 pp [9] M. Xia Z.S. Xu ad H. Liao Preferece relatios based o ituitioistic multiplicative iformatio IEEE rasactios o Fuzzy Systems vol.1 o.1 pp [30] M. Xia Z.S. Xu Geeralized poit operators for aggregatig ituitioistic fuzzy iformatio Iteratioal Joural of Itelliget Systems vol.5 o.11 pp [31] Y Xu Y. Li ad H. Wag he iduced ituitioistic fuzzy Eistei aggregatio ad its applicatio i group decisio makig Joural of Idustrial ad Productio Egieerig vol.30 o.1 pp [3] Z. Xu ad R.R. Yager Some geometric aggregatio operators based o ituitioistic fuzzy sets Iteratioal Joural of Geeral Systems vol.35 o.4 pp [33] Z. Xu Ituitioistic fuzzy aggregatio operators IEEE rasactios o Fuzzy Systems vol.15 o.6 pp [34] Z. Xu Ituitioistic fuzzy Multiattribute decisio makiga iteractive method IEEE rasactios o Fuzzy Systems vol.0 o.3 pp [35] Z. Xu Priority weight itervals derived from ituitioistic multiplicative preferece relatios IEEE rasactios o Fuzzy Systems vol.1 o.4 pp [36] Z. Xu ad R.R. Yager Dyamic ituitioistic fuzzy multi-attribute decisio makig Iteratioal Joural of Approximate Reasoig vol.48 o.1 pp [37] Z. Xu Ituitioistic fuzzy Boferroi meas IEEE rasactios o Systems Ma ad Cyberetics Part-B: Cyberetics vol.41 o. pp [38] Z. Xu ad J. Che Ordered weighted distace measure Joural of Systems Sciece ad Systems Egieerig vol.17 o.4 pp [39] Z. Xu ad H. Hu Proectio models for ituitioistic fuzzy multiple attribute decisio makig Iteratioal Joural of Iformatio echology ad Decisio Makig vol.9 o. pp [40] Z. Xu O cosistecy of the weighted geometric mea complex udgmet matrix i AHP Europea Joural of Operatio Research vol.16 o.3 pp [41] R.R. Yager Geeralized OWA aggregatio operators Fuzzy Optimizatio ad Decisio Makig vol.3 o.1 pp [4] R.R. Yager Prioritized aggregatio operators Iteratioal Joural of Approximate Reasoig vol.48 o.1 pp [43] R.R. Yager Prioritized OWA aggregatio Fuzzy Optimizatio ad Decisio Makig vol.8 o [44] J. Ye Fuzzy decisio makig method based o the weighted correlatio coefficiet uder ituitioistic fuzzy eviromet Europea Joural of Operatio Research vol.05 o.1 pp [45] D. Yu Ituitioistic fuzzy prioritized operators ad their applicatio i multi-criteria group decisio makig echological ad Ecoomic Developmet of Ecoomy vol.19 o.1 pp [46] D. Yu Multi-criteria decisio makig based o geeralized prioritized aggregatio operators uder ituitioistic fuzzy eviromet Iteratioal Joural of Fuzzy Systems vol.15 o.1 pp [47] D. Yu Group decisio makig based o geeralized ituitioistic fuzzy prioritized geometric operator Iteratioal Joural of Itelliget Systems vol.7 o.7 pp [48] L.A. Zadeh Fuzzy sets Iformatio ad Cotrol vol.8 o.3 pp [49] S.Z. Zeg ad W.H. Su Ituitioistic fuzzy ordered weighted distace operator Kowledge Based Systems vol.4 o.8 pp c 015 NSP Natural Scieces Publishig Cor.

13 Appl. Math. If. Sci. 9 No / [50] X. Zhao ad G. Wei Some ituitioistic fuzzy Eistei hybrid aggregatio operators ad their applicatio to multiple attribute decisio makig Kowledge Based Systems vol.37 pp [51] H. Zhao Z. Xu M.F. Ni ad S.S. Liu Geeralized aggregatio operators for ituitioistic fuzzy sets Iteratioal Joural of Itelliget Systems vol.5 o.1 pp [5] W. Zhao ad J.M. He Ituitioistic fuzzy geometric Boferroi meas ad their applicatio i multicriteria decisio makig Iteratioal Joural of Itelliget Systems vol.7 o.1 pp [53] S.F. Zhag ad S.Y. Liu A GRA-based ituitioistic fuzzy multi-criteria group decisio makig method for persoel selectio Expert Systems with Applicatios vol.38 o Rakumar Verma is Assistat Professor of Mathematics at HMR Istitute of echology ad Maagemet Delhi affiliated to G.G.S.I.P. Uiversity Delhi. He received the Ph.D. degree i Mathematical Scieces from Jaypee Istitute of Iformatio echology Deemed uiversity Noida i 014. He is referee of several iteratioal ourals i the applied mathematics fuzzy mathematics ad decisio makig. His research iterests iclude iformatio theory fuzzy iformatio measures patter recogitio multicriteria decisio-makig aalysis soft set theory etc. Bhu Dev Sharma curretly Professor of Mathematics at Jaypee Istitute of Iformatio echology Deemed Uiversity Noida Idia did his Ph.D. from Uiversity of Delhi. He was a faculty at the Uiversity of Delhi ad earlier at other places. He spet about 9 years abroad 0 years i USA ad 9 years i the West Idies as Professor of Mathematics ad i betwee as chair of the departmet. Professor Sharma has published over 150 research papers i the areas coverig Codig heory Iformatio heory Fuctioal Equatios ad Statistics. He has successfully guided 8 persos for Ph.D. i Idia ad abroad. He has also authored/co-authored 3 published books i Mathematics ad some others o Idia studies. He is the Chief Editor of Joural of Combiatorics Iformatio ad System Scieces JCISS 1976-preset ad is o the editorial boards of several other ourals. He is member of several professioal academic orgaizatios. Past Presidet of Forum for Iterdiscipliary Mathematics FIM Vice Presidet of Calcutta Mathematical Society Ramaua Math Society Vice Presidet of Academy of Discrete Mathematics ad Applicatios ADMA. After returig to Idia he is workig o establishig a Ceter of Excellece i Iterdiscipliary Mathematics CEIM i Delhi Idia. Professor Sharma has received several awards ad recogitios i Idia ad abroad. He is widely traveled that would iclude Germay ad several other coutries of Europe Brazil Russia ad Chia. He has orgaized several Iteratioal Cofereces o Math/Stat i Idia USA Europe Australia ad Chia. c 015 NSP Natural Scieces Publishig Cor.