Methods for strategic decision-making problems with immediate probabilities in intuitionistic fuzzy setting

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Scietia Iraica E (2012) 19 (6) 1936 1946 Sharif Uiversity of Techology Scietia Iraica Trasactios E: Idustrial Egieerig www.sciecedirect.

Diagoal 690 08034 Barceloa Spai Received 6 March 2012; revised 16 May 2012; accepted 8 July 2012 KEYWORDS Strategic decisio-makig; Ituitioistic fuzzy values; Iterval-valued ituitioistic fuzzy values;

1 Scietia Iraica E (2012) 19 (6) Sharif Uiversity of Techology Scietia Iraica Trasactios E: Idustrial Egieerig Methods for strategic decisio-makig problems with immediate probabilities i ituitioistic fuzzy settig G.W. Wei a J.M. Merigó b a Departmet of Ecoomics ad Maagemet Chogqig Uiversity of Arts ad Scieces Chogqig PR Chia b Departmet of Busiess Admiistratio Uiversity of Barceloa Av. Diagoal Barceloa Spai Received 6 March 2012; revised 16 May 2012; accepted 8 July 2012 KEYWORDS Strategic decisio-makig; Ituitioistic fuzzy values; Iterval-valued ituitioistic fuzzy values; Operatioal laws; Probabilistic aggregatig operators. Abstract With respect to decisio-makig problems by usig probabilities immediate probabilities ad iformatio that ca be represeted with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio some ew decisio aalysis are proposed. Firstly some operatioal laws score fuctio ad accuracy fuctio of ituitioistic fuzzy values or iterval-valued ituitioistic fuzzy values are itroduced. The we have developed some ew probability aggregatio operators with ituitioistic fuzzy iformatio: Probability Ituitioistic Fuzzy Weighted Average (P-IFWA) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Average (IP-IFOWA) operator Probability Ituitioistic Fuzzy Ordered Weighted Average (P-IFOWA) operator Probability Ituitioistic Fuzzy Weighted Geometric (P-IFWG) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Geometric (IP-IFOWG) operator ad Probability Ituitioistic Fuzzy Ordered Weighted Geometric (P-IFOWG) operator ad applied these operators to decisio-makig problem about the selectio of the optimal productio strategy. Furthermore we shall exted the developed models ad procedures to solve probabilistic decisiomakig problems with iterval-valued ituitioistic fuzzy iformatio. Fially we give a illustrative example about selectio of strategies to verify the developed approach ad to demostrate its feasibility ad practicality Sharif Uiversity of Techology. Productio ad hostig by Elsevier B.V. Ope access uder CC BY-NC-ND licese. 1. Itroductio Ataassov [12] itroduced the cocept of Ituitioistic Fuzzy Set (IFS) characterized by a membership fuctio ad a o-membership fuctio which is a geeralizatio of the cocept of fuzzy set [3] whose basic compoet is oly a membership fuctio. The ituitioistic fuzzy set has received more ad more attetio sice its appearace [4 18]. Later Ataassov ad Gargov [1920] further itroduced the Iterval-Valued Ituitioistic Fuzzy Set (IVIFS) which is a geeralizatio of the IFS. The fudametal characteristic of the IVIFS is that the values Correspodig author. Tel.: ; fax: addresses: weiguiwu@163.com (G.W. Wei) merigo@ub.edu (J.M. Merigó). Peer review uder resposibility of Sharif Uiversity of Techology. of its membership fuctio ad o-membership fuctio are itervals rather tha exact umbers. Bustice ad Burillo [21] defied the cocepts of correlatio ad correlatio coefficiet of IVIFSs ad developed two decompositio theorems of the correlatio of IVIFSs. Hog ad Choi [22] geeralized the cocepts of correlatio ad correlatio coefficiet of IVIFSs i a geeral probability space. Hug ad Wu [23] proposed a method to calculate the correlatio coefficiet of IVIFSs by meas of cetroid. Grzegorzewski [24] proposed some distaces betwee ituitioistic fuzzy sets ad/or iterval-valued fuzzy sets based o the Hausdorff metric. Xu ad Che [25 27] developed some iformatio aggregatio operators with itervalvalued ituitioistic fuzzy iformatio. Xu ad Yager [28] ivestigated the dyamic ituitioistic fuzzy multiple attribute decisio-makig problems ad developed some dyamic aggregatio operators with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio. Wei [29] developed some dyamic geometric aggregatio operators with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio. Wei [30] developed some iduced geometric Sharif Uiversity of Techology. Productio ad hostig by Elsevier B.V. Ope access uder CC BY-NC-ND licese. doi: /.sciet

2 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) aggregatio operators with iterval-valued ituitioistic fuzzy iformatio. Li [31] proposed the liear programmig method for MADM with iterval-valued ituitioistic fuzzy sets. Ye [32] developed multicriteria fuzzy decisio-makig method based o a ovel accuracy fuctio uder iterval-valued ituitioistic fuzzy eviromet. Ye [33] exteded TOPSIS method with iterval-valued ituitioistic fuzzy umbers for virtual eterprise parter selectio. Xu [34] used the Choquet itegral to propose some correlated aggregatig operators for aggregatig ituitioistic fuzzy values or iterval-valued ituitioistic fuzzy values with correlative weights. Aother iterestig approach is the use of probabilistic iformatio i the aalysis because we are able to itroduce obectivity i our studies. I the literature we fid a wide rage of models that use probabilistic iformatio such as the immediate probabilities [35 37] ad the Dempster Shafer theory of evidece [3839]. Recetly Merigó [4041] has suggested the Probabilistic OWA (POWA) operator. It is a aggregatio operator that uifies the probability with the OWA operator cosiderig the degree of importace that each cocept has i the aggregatio. Moreover he has also suggested the Probabilistic Weighted Average (PWA) [42]. Its mai advatage is that it uifies the probability ad the weighted average i the same formulatio cosiderig how relevat they are i the aggregatio. I this paper we ivestigate the probabilistic decisiomakig problems with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio; some ew probabilistic decisio-makig methods are developed. Firstly some operatioal laws score fuctio ad accuracy fuctio of ituitioistic fuzzy values or iterval-valued ituitioistic fuzzy values are itroduced. The we have developed some ew probability aggregatio operators with ituitioistic fuzzy iformatio: Probability Ituitioistic Fuzzy Weighted Average (P-IFWA) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Average (IP-IFOWA) operator Probability Ituitioistic Fuzzy Ordered Weighted Average (P-IFOWA) operator Probability Ituitioistic Fuzzy Weighted Geometric (P-IFWG) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Geometric (IP-IFOWG) operator ad Probability Ituitioistic Fuzzy Ordered Weighted Geometric (P-IFOWG) operator. Furthermore we shall exted the developed models ad procedures to solve probabilistic decisiomakig problems with iterval-valued ituitioistic fuzzy iformatio. Fially some illustrative examples about the selectio of the optimal productio strategy have bee give to show the developed method. I order to do so the remaider of this paper is set out as follows. I the ext sectio we itroduce some basic cocepts related to ituitioistic fuzzy sets ad probabilistic aggregatig operators. I Sectio 3 we propose some ew probability aggregatio operators with ituitioistic fuzzy iformatio. I Sectio 4 we propose some ew probability aggregatio operators with iterval-valued ituitioistic fuzzy iformatio. I Sectio 5 a illustrative example about the selectio of the optimal productio strategy is poited out. I Sectio 6 we coclude the paper ad give some remarks. 2. Prelimiaries I the followig we shall itroduce some basic cocepts related to ituitioistic fuzzy sets ad immediate probabilities Ituitioistic fuzzy sets Defiitio 1. Let X be a uiverse of discourse the a fuzzy set is defied as: A { x µ A (x) x X } (1) which is characterized by a membership fuctio µ A : X [0 1] where µ A (x) deotes the degree of membership of the elemet x to the set A [3]. Ataassov [12] exteded the fuzzy set to the IFS show as follows: Defiitio 2. A IFS A i X is give by: A { x µ A (x) ν A (x) x X } (2) where µ A : X [0 1] ad ν A : X [0 1] with the coditio: 0 µ A (x) + ν A (x) 1 x X. The umbers µ A (x) ad ν A (x) represet respectively the membership degree ad o-membership degree of the elemetxto the set A [12]. Defiitio 3. For each IFS A i X if: π A (x) µ A (x) ν A (x) x X (3) the π A (x) is called the degree of idetermiacy of x to A [12]. Defiitio 4. Let ã (µ ν) be a ituitioistic fuzzy value; a score fuctio S of a ituitioistic fuzzy value ca be represeted as follows [34]: S ã µ ν S ã [ 1 1] (4) Defiitio 5. Let ã (µ ν) be a ituitioistic fuzzy value; a accuracy fuctio H of a ituitioistic fuzzy value ca be represeted as follows [43]: H ã µ + ν H ã [0 1] (5) to evaluate the degree of accuracy of the ituitioistic fuzzy value ã (µ ν) where H ã [0 1]. The larger the value of H ã the more the degree of accuracy of the ituitioistic fuzzy value ã is. As preseted above the score fuctio S ad the accuracy fuctio H are respectively defied as the differece ad the sum of the membership fuctio µ A (x) ad the omembership fuctio ν A (x). Hog ad Choi [44] showed that the relatio betwee the score fuctio S ad the accuracy fuctio H is similar to the relatio betwee mea ad variace i statistics. Based o the score fuctio S ad the accuracy fuctio H i the followig Xu [910] give a order relatio betwee two ituitioistic fuzzy values which is defied as follows. Defiitio 6. Let ã 1 (µ 1 ν 1 ) ad ã 2 (µ 2 ν 2 ) be two ituitioistic fuzzy values s ã 1 µν 1 ad s ã 2 µ2 ν 2 be the scores of ã ad b respectively ad let H ã1 µ1 + ν 1 ad H ã 2 µ2 + ν 2 be the accuracy degrees of ã ad b respectively the if S ã < S b the ã is smaller tha b ã deoted by ã < b; if S S b the:

3 1938 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) (1) if H ã H b the ã ad b represet the same iformatio deoted by ã b; (2) if H ã < H b ã is smaller tha b deoted by ã < b. Aother approach for uifyig probabilities ad OWAs i the same formulatio is the Probabilistic OWA (POWA) operator [4041]. Its mai advatage is that it is able to iclude both cocepts cosiderig the degree of importace of each case i the problem Probabilistic aggregatig operators Merigó [4042] developed the Probabilistic Weighted Average (PWA) operator which is a aggregatio operator that uifies the probability ad the weighted average i the same formulatio cosiderig the degree of importace that each cocept has i the aggregatio. Defiitio 7. A PWA operator of dimesio is a mappig PWA: R R such that: PWA (a 1 a 2... a ) v a (6) where ω (ω 1 ω 2... ω ) T is the weight vector of a ( ) ad ω > 0 ω 1 ad a probabilistic weight p > 0 p 1 v βp + ( β) ω with β [0 1] ad v is the weight that uifies probabilities ad WAs i the same formulatio the PWA is called the Probabilistic Weighted Average (PWA) operator. The PWA operator is mootoic bouded ad idempotet [4042]. Several authors [35 37] proposed the Immediate Probability (IP) which tries to iclude the decisio maker s attitude i a probabilistic decisio-makig problem. The mai advatage is that it is easy to apply it to almost all the probabilistic problems studied before such as decisio-makig problems actuarial scieces ad statistics. Because the probabilistic iformatio is obective but ucertai we caot the guaratee that the expected result is the result that will happe i the future. If we are i the situatios of ucertaity (risk eviromets) each decisio maker will have differet attitudes towards the same problem. I order to develop the aalysis we use i the same formulatio the weights of the OWA operator ad the probabilistic iformatio ad propose the Immediate Probability OWA (IP-OWA) operator. It ca be defied as follows. Defiitio 8. A IP-OWA operator of dimesio is a mappig IP-OWA: R R that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore: IP-OWA (a 1 a 2... a ) ˆp a σ () (7)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i p is the associated probability of α σ () ad ˆp w p w [35 37]. p It is worth poitig out that IP-OWA operator is a good approach for uifyig probabilities ad OWAs i some particular situatios. But it is ot always useful especially i situatios where we wat to give more importace to the probabilities or to the OWA operators. I order to show why this uificatio does ot seem to be a fial model we could also cosider other ways of represetig ˆp. For example we could w also use ˆp +p or other similar approaches. (w +p ) Defiitio 9. A POWA operator of dimesio is a mappig PWA: R R that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1 accordig to the followig formula: POWA (a 1 a 2... a ) ˆp a σ () (8)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i with i1 p i 1 ad p i [0 1] ˆp βp + ( β) w with β [0 1] ad p is the associated probability of α σ () [4041]. 3. Probabilistic aggregatig operators with ituitioistic fuzzy iformatio The WA [45] ad OWA operators [46 48] however have usually bee used i the situatios where the iput argumets are the exact values. Xu [10] exteded the WA ad OWA operators to accommodate the situatios where the iput argumets are ituitioistic fuzzy iformatio. Defiitio 10. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values ad let IFWA: Q Q ; if: IFWA ω ã1 ã 2... ã ω ã ω µ ν ω (9) where ω (ω 1 ω 2... ω ) T be the weight vector of ã ( ) ad ω > 0 ω 1 the IFWA is called the Ituitioistic Fuzzy Weighted Averagig (IFWA) operator [10]. Defiitio 11. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A Ituitioistic Fuzzy Ordered Weighted Averagig (IFOWA) operator of dimesio is a mappig IFOWA: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore: IFOWA w ã1 ã 2... ã ã σ () w w µσ () ν w σ () (10)... ) such that α σ ( 1) α σ () for all [10]. Similarly the PWA operator IPOWA operator ad POWA operator have usually bee used i the situatios where the iput argumets are the exact values. I the followig we shall exted these operators to accommodate the situatios where the iput argumets are ituitioistic fuzzy iformatio. These operators iclude: Probability Ituitioistic Fuzzy Weighted Average (P-IFWA) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Average

4 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) (IP-IFOWA) operator ad Probability Ituitioistic Fuzzy Ordered Weighted Average (P-IFOWA) operator. The P-IFWA is a aggregatio operator that uifies the probability ad the weighted average i the same formulatio cosiderig the degree of importace that each cocept has i the aggregatio ad usig ucertai iformatio represeted with ituitioistic fuzzy values. Defiitio 12. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A P-IFWA operator of dimesio is a mappig P-IFWA: Q Q such that: P-IFWAˆν ã1 ã 2... ã ˆν ã ˆν µ ν ˆν (11) where ω (ω 1 ω 2... ω ) T is the weight vector of a ( ) ad ω > 0 ω 1 ad a probabilistic weight p > 0 p 1 ˆv βp + ( β) ω with β [0 1] ad ˆv is the weight that uifies probabilities ad IFWAs i the same formulatio. Especially if β 0 the the P-IFWA operator is reduced to the IFWA operator; if β 1 it becomes the Ituitioistic Fuzzy Probabilistic Aggregatio (IFPA). Note that the P-IFWA is mootoic bouded ad idempotet. It is mootoic because if ã i u i for all ã i the P IFWA ã 1 ã 2... ã P-IFWA (u 1 u 2... u ). It is bouded because the P-IFWA aggregatio is delimitated by the miimum ad the maximum that is Mi{ã i } P-IFWA(ã 1... ã ) Max{ã i }. It is idempotet because if ã i ã for all ã i the P-IFWA (ã 1... ã ) ã. The IP-IFOWA operator is a aggregatio operator that uses probabilities ad OWAs i the same formulatio ad iformatio represeted with ituitioistic fuzzy values. Defiitio 13. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A IP-IFOWA operator of dimesio is a mappig IP-IFOWA: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore IP-IFOWAˆp ã1 ã 2... ã ˆp ã σ () ˆp µσ () ν ˆp σ () (12)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i p is the associated probability of α σ () ad ˆp w p w. p It is worth poitig out that IP-IFOWA operator is a good approach for uifyig probabilities ad IFOWAs i some particular situatios. But it is ot always useful especially i the situatios where we wat to give more importace to the probabilities or to the IFOWA operators. I order to show why this uificatio does ot seem to be a fial model we could also cosider other ways of represetig ˆp. For example we could w also use ˆp +p or other similar approaches. (w +p ) The P-IFOWA operator uifies the probability ad the OWA operator i the same formulatio cosiderig the degree of importace of each cocept i the aggregatio. It also uses iformatio represeted i the form of ituitioistic fuzzy values. Defiitio 14. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A P-IFOWA operator of dimesio is a mappig P-IFOWA: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore P-IFOWA p ã1 ã 2... ã p ã σ () p p µσ () νσ () (13)... ) such that α σ ( 1) α σ () for all 2... each ã i has associated a probability p i with i1 p i 1 ad p i [0 1] p βp + ( β) w with β [0 1] ad p is the associated probability of α σ (). Especially if β 0 the the P-IFOWA operator is reduced to the IFOWA operator; ad if β 1 it becomes the Ituitioistic Fuzzy Probabilistic Aggregatio (IFPA). The P-IFOWA operator is also bouded idempotet ad mootoic. Furthermore Xu ad Yager [9] exteded the WG ad OWG operators [47] to accommodate the situatios where the iput argumets are ituitioistic fuzzy iformatio. Defiitio 15. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values ad let IFWG: Q Q if: ω IFWG ω ã1 ã 2... ã ã µ ω ω ν (14) where ω (ω 1 ω 2... ω ) T be the weight vector of ã ( ) ad ω > 0 ω 1 the IFWG is called the Ituitioistic Fuzzy Weighted Geometric (IFWG) operator [9]. Defiitio 16. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A Ituitioistic Fuzzy Ordered Weighted Geometric (IFOWG) operator of dimesio is a mappig IFOWG: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore IFOWG w ã1 ã 2... ã µ w ãσ () w σ () w νσ () (15)... ) such that α σ ( 1) α σ () for all 2... [9]. I the followig we shall propose some probabilistic geometric aggregatig operators to accommodate the situatios where the iput argumets are ituitioistic fuzzy iformatio. These operators iclude: Probability Ituitioistic Fuzzy Weighted Geometric (P-IFWG) operator Immediate Probability

5 1940 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Ituitioistic Fuzzy Ordered Weighted Geometric (IP-IFOWG) operator ad Probability Ituitioistic Fuzzy Ordered Weighted Geometric (P-IFOWG) operator. Defiitio 17. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A P-IFWG operator of dimesio is a mappig P-IFWG: Q Q such that: P-IFWGˆν ã1 ã 2... ã µˆν ã ˆν ˆν ν (16) where ω (ω 1 ω 2... ω ) T is the weight vector of ã ( ) ad ω > 0 ω 1 ad a probabilistic weight p > 0 p 1 ˆv βp + ( β) ω with β [0 1] ad ˆv is the weight that uifies probabilities ad IFWGs i the same formulatio. Especially if β 0 the the P-IFWG operator is reduced to IFWG operator. Defiitio 18. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A IP-IFOWG operator of dimesio is a mappig IP-IFOWG: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore: IP-IFOWGˆp ã1 ã 2... ã µˆp ãσ () ˆp σ () ˆp νσ () (17)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i p is the associated probability of α σ () ad ˆp w p w. p It is worth poitig out that IP-IFOWG operator is a good approach for uifyig probabilities ad IFOWGs i some particular situatios. But it is ot always useful especially i the situatios where we wat to give more importace to the probabilities or to the IFOWG operators. I order to show why this uificatio does ot seem to be a fial model we could also cosider other ways of represetig ˆp. For example we could w also use ˆp +p or other similar approaches. (w +p ) Defiitio 19. Let ã µ ν ( ) be a collectio of ituitioistic fuzzy values. A P-IFOWG operator of dimesio is a mappig P-IFOWG: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore P-IFOWG p ã1 ã 2... ã µ p ãσ () p σ () p νσ () (18)... ) such that α σ ( 1) α σ () for all 2... each ã i has associated a probability p i with i1 p i 1 ad p i [0 1] p βp + ( β) w with β [0 1] ad p is the associated probability of α σ (). Especially if β 0 the the P-IFOWG operator is reduces to the IFOWG operator. 4. Extesios Ataassov ad Gargov [1920] further itroduced the Iterval-Valued Ituitioistic Fuzzy Set (IVIFS) which is a geeralizatio of the IFS. The fudametal characteristic of the IVIFS is that the values of its membership fuctio ad omembership fuctio are itervals rather tha exact umbers. Defiitio 20. Let X be a uiverse of discourse. A IVIFS Ã over X is a obect havig the form [1920]: Ã { x µ A (x) ν A (x) x X } (19) where µ A (x) [0 1] ad ν A (x) [0 1] are iterval umbers ad: 0 sup ( µ A (x)) + sup ( ν A (x)) 1 x X. For coveiece let µ A (x) [a b] ν A (x) [c d] so Ã ([a b] [c d]). Defiitio 21. Let ã ([a b] [c d]) be a itervalvalued ituitioistic fuzzy umber. A score fuctio S of a iterval-valued ituitioistic fuzzy value ca be represeted as follows [25 27]: S ã a c + b d S ã [ 1 1]. (20) 2 Defiitio 22. Let ã ([a b] [c d]) be a iterval-valued ituitioistic fuzzy umber. A accuracy fuctio H of a iterval-valued ituitioistic fuzzy value ca be represeted as follows [25 27]: H ã a + b + c + d H ã [0 1] (21) 2 to evaluate the degree of accuracy of the iterval-valued ituitioistic fuzzy value ã ([a b] [c d]) where H ã [0 1]. The larger the value of H ã the more the degree of accuracy of the iterval-valued ituitioistic fuzzy value ã is. Based o the score fuctio Sad the accuracy fuctio H i the followig Xu [25 27] give a order relatio betwee two iterval-valued ituitioistic fuzzy values which is defied as follows. Defiitio 23. Let ã 1 ([a 1 b 1 ] [c 1 d 1 ]) ad ã 2 ([a 2 b 2 ] [c 2 d 2 ]) be two iterval-valued ituitioistic fuzzy values s ã 1 a c 1 +b d 1 ad s ã 2 2 a 2 c 2 +b 2 d 2 be the 2 scores of ã ad b respectively ad let H ã1 a 1 +c 1 +b 1 +d 1 2 ad H ã 2 a 2 +c 2 +b 2 +d 2 be the accuracy degrees of ã ad b 2 respectively the if S ã < S b the ã is smaller tha b ã deoted by ã < b; if S S b the: (1) if H ã H b the ã ad b represet the same iformatio deoted by ã b; (2) if H ã < H b ã is smaller tha b deoted by ã < b. Xu ad Che [25] developed the Iterval-Valued Ituitioistic Fuzzy Weighted Averagig (IVIFWA) operator ad

6 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Averagig (IVIFOWA) operator. Defiitio 24. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values ad let IVIFWA: Q Q if: IVIFWA ω ã1 ã 2... ã ω ã ω a c ω d ω ω b (22) where ω (ω 1 ω 2... ω ) T is the weight vector of ã ( ) ad ω > 0 ω 1 the IVIFWA is called the Iterval-Valued Ituitioistic Fuzzy Weighted Averagig (IVIFWA) operator [25]. Defiitio 25. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Averagig (IVIFOWA) operator of dimesio is a mappig IVIFOWA: Q Q that has a associated vector w (w 1 w 2... w ) T such that w > 0 ad w 1 Furthermore: IVIFOWA w ã1 ã 2... ã w ã σ () c w w aσ () σ () d w σ () w bσ () (23)... ) such that α σ ( 1) α σ () for all 2... [25]. I the followig we shall propose some ew probability aggregatig operators with iterval-valued ituitioistic fuzzy iformatio: probability Iterval-Valued Ituitioistic Fuzzy Weighted Average (P-IVIFWA) operator immediate probability Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Average (IP-IVIFOWA) operator ad Probability Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Average (P-IVIFOWA) operator. Their mai advatage is that they uify the probability with the weighted average or with the OWA operator i a ucertai eviromet that ca be assessed with iterval-valued ituitioistic fuzzy iformatio. Defiitio 26. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A P- IVIFWA operator of dimesio is a mappig P-IVIFWA: Q Q such that: P-IVIFWA ν ã1 ã 2... ã ν ã c ν ν a d ν ν b (24) where ω (ω 1 ω 2... ω ) T is the weight vector of a ( ) ad ω > 0 ω 1 ad a probabilistic weight p > 0 p 1 v βp + ( β) ω with β [0 1] ad v is the weight that uifies probabilities ad IVIFWA i the same formulatio. Especially if β 0 the the P-IVIFWA operators are reduced to the IVIFWA operator. Ad if β 1 it becomes the Iterval-Valued Ituitioistic Fuzzy Probabilistic Aggregatio (IVIFPA). Defiitio 27. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A IP-IVIFOWA operator of dimesio is a mappig IP-IVIFOWA: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1 Furthermore IP-IVIFOWAˆp ã1 ã 2... ã ˆp ã σ () c ˆp ˆp aσ () σ () dˆp σ () ˆp bσ () (25)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i p is the associated probability of α σ () ad ˆp w p w w. We could also use ˆp +p p or other similar approaches. (w +p ) Defiitio 28. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A P-IVIFOWA operator of dimesio is a mappig P-IVIFOWA: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1 Furthermore P-IVIFOWA p ã1 ã 2... ã p ã c p p a d p p b (26)... ) such that α σ ( 1) α σ () for all 2... each ã i has associated a probability p i with i1 p i 1 ad p i [0 1] p βp + ( β) w with β [0 1] ad p is the associated probability of α σ (). Especially if β 0 the the P-IVIFOWA operators are reduced to the IVIFOWA operator; ad if β 1 it becomes the IVIFPA. Xu ad Che [26] developed the Iterval-Valued Ituitioistic Fuzzy Weighted Geometric (IVIFWG) operator ad Iterval- Valued Ituitioistic Fuzzy Ordered Weighted Geometric (IVIFOWG) operator.

7 1942 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Defiitio 29. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values ad let IVIFWG: Q Q if: IVIFWG ω ã1 ã 2... ã a ω b ω ω c ã ω ω d (27) where ω (ω 1 ω 2... ω ) T is the weight vector of ã ( ) ad ω > 0 ω 1 the IVIFWG is called the Iterval-Valued Ituitioistic Fuzzy Weighted Geometric (IVIFWG) operator [26]. Defiitio 30. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Geometric (IVIFOWG) operator of dimesio is a mappig IVIFOWG: Q Q that has a associated vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore IVIFOWG w ã1 ã 2... ã a w σ () b w σ () w cσ () ãσ () w w dσ () (28)... ) such that α σ ( 1) α σ () for all 2... [26]. I the followig we shall propose some ew probability geometric aggregatig operators with iterval-valued ituitioistic fuzzy iformatio: Probability Iterval-Valued Ituitioistic Fuzzy Weighted Geometric (P-IVIFWG) operator Immediate Probability Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Geometric (IP-IVIFOWG) operator ad Probability Iterval-Valued Ituitioistic Fuzzy Ordered Weighted Geometric (P-IVIFOWG) operator. Defiitio 31. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A P- IVIFWG operator of dimesio is a mappig P-IVIFWG: Q Q such that; P-IVIFWG ν ã1 ã 2... ã a ν b ν ν c ã ν ν d (29) where ω (ω 1 ω 2... ω ) T is the weight vector of a ( ) ad ω > 0 ω 1 ad a probabilistic weight p > 0 p 1 v βp + ( β) ω with β [0 1] ad v is the weight that uifies probabilities ad IVIFWGs i the same formulatio. Especially if β 0 the the P-IVIFWG operator is reduced to the IVIFWG operator. Defiitio 32. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values a IP- IVIFOWG operator of dimesio is a mappig IP-IVIFOWG: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore: ˆp IP-IVIFOWGˆp ã1 ã 2... ã ãσ () aˆp σ () bˆp σ () ˆp cσ () ˆp dσ () (30)... ) such that α σ ( 1) α σ () for all 2... each a i has associated a probability p i p is the associated probability of α σ () ad ˆp w p w w. We could also use ˆp +p p or other similar approaches. (w +p ) Defiitio 33. Let ã a b c d ( ) be a collectio of iterval-valued ituitioistic fuzzy values. A P- IVIFOWG operator of dimesio is a mappig P-IVIFOWG: Q Q that has a associated weight vector w (w 1 w 2... w ) T such that w > 0 ad w 1. Furthermore. p P-IVIFOWG p ã1 ã 2... ã ãσ () a p b p p c p d (31)... ) so α σ ( 1) α σ () for all 2... each ã i has associated a probability p i with i1 p i 1 ad p i [0 1] p βp + ( β) w with β [0 1] ad p is the associated probability of α σ (). Especially if β 0 the the P-IVIFOWG operators are reduced to the IVIFOWG operator. 5. Illustrative example I this sectio we shall aalyse a decisio-makig problem about the selectio of the optimal productio strategy [37]. Assume a compay wats to create a ew product ad they are aalyzig the optimal target i order to obtai the highest beefits. After aalyzig the market they cosider five possible strategies to follow: (1) A1: Creatig a ew product orieted to the rich customers; (2) A2: Creatig a ew product orieted to the mid-level customers; (3) A3: Creatig a ew product orieted to the low-level customers;

8 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Table 1: Available ituitioistic fuzzy iformatio about the strategies. S1 S2 S3 S4 S5 A 1 ( ) ( ) ( ) ( ) ( ) A 2 ( ) ( ) ( ) ( ) ( ) A 3 ( ) ( ) ( ) ( ) ( ) A 4 ( ) ( ) ( ) ( ) ( ) A 5 ( ) ( ) ( ) ( ) ( ) Table 2: Immediate probabilities of the problem. S1 S2 S3 S4 S5 IP IP IP IP IP (4) A4: Creatig a ew product adapted to all customers; (5) A5: Not creatig ay products. After careful review of the iformatio the decisio maker establishes the followig geeral iformatio about the productio strategy. He has summarized the iformatio of the strategies i five geeral characteristics: (1) S1: Beefits i the short term; (2) S2: Beefits i the mid term; (3) S3: Beefits i the log term; (4) S4: Risk of the productio strategy; (5) S5: Other factors. The five possible strategies A i (i ) are to be evaluated usig the ituitioistic fuzzy umbers by the decisio makers uder the above five geeral characteristics ad costruct the decisio matrix as show i Table 1. The we utilize the approach developed to get the most desirable alterative(s). Step 1. With respect to this problem the experts i the compay fid probabilistic iformatio give as follows: P ( ). They assume that the WA that represets the degree of importace of each characteristics is: ω ( ). Note that the probabilistic iformatio has a importace of 40% ad the WA a importace of 60%. Furthermore the policy of the compay is to be very pessimistic i order to obtai safety results wheever the future results are ot clear. Therefore they decide Table 5: Orderig of the alterative. Orderig IFWA A 2 > A 4 > A 5 > A 3 > A 1 IFOWA A 2 > A 4 > A 5 > A 3 > A 1 P-IFWA A 2 > A 4 > A 5 > A 3 > A 1 P-IFOWA A 2 > A 4 > A 5 > A 3 > A 1 IP-IFOWA A 2 > A 4 > A 5 > A 3 > A 1 IFWG A 2 > A 5 > A 3 > A 4 > A 1 IFOWG A 2 > A 5 > A 3 > A 4 > A 1 P-IFWG A 2 > A 4 > A 5 > A 3 > A 1 P-IFOWG A 2 > A 4 > A 5 > A 3 > A 1 IP-IFOWG A 2 > A 4 > A 5 > A 3 > A 1 to maipulate the probabilities by usig the followig OWA weightig vector: w ( ). As we ca see this weightig vector seems to be pessimistic because it gives higher importace to the lowest result used i the last weight. Note that the compay will use immediate probabilities i order to assess this problem. The results foud i the immediate probabilities by usig the above probabilities ad weights are show i Table 2. Step 2. We use the arithmetic aggregatig operator to aggregate the decisio iformatio i Table 1 ad probabilistic iformatio. The results are show i Table 3. These arithmetic aggregatig operators iclude the IFWA operator IFOWA P-IFWA operator P-IFOWA ad IP-IFOWA operator. Step 3. We use the geometric aggregatig operator to aggregate the decisio iformatio i Table 1 ad probabilistic iformatio. The results are show i Table 4. These geometric aggregatig operators iclude the IFWG operator IFOWG P-IFWG operator P-IFOWG ad IP-IFOWG operator. Step 4. Accordig to the differet aggregatig operators ad score fuctios the orderig of the alteratives is show i Table 5. Note that > meas preferred to. As we ca see depedig o the aggregatio operators used the orderig of the strategies is slightly differet. Therefore depedig o the aggregatio operators used the results may lead to differet decisios. However all of the operators produce the same best strategy A2. If the five possible strategies A i (i ) are to be evaluated usig the iterval-valued ituitioistic fuzzy umbers by the decisio makers uder the above five geeral characteristics ad costruct the decisio matrix as show i Table 6. Table 3: Ituitioistic fuzzy aggregated results (β 0.40). IFWA IFOWA P-IFWA P-IFOWA IP-IFOWA A 1 ( ) ( ) ( ) ( ) ( ) A 2 ( ) ( ) ( ) ( ) ( ) A 3 ( ) ( ) ( ) ( ) ( ) A 4 ( ) ( ) ( ) ( ) ( ) A 5 ( ) ( ) ( ) ( ) ( ) Table 4: Ituitioistic fuzzy aggregated results (β 0.40). IFWG IFOWG P-IFWG P-IFOWG IP-IFOWG A 1 ( ) ( ) ( ) ( ) ( ) A 2 ( ) ( ) ( ) ( ) ( ) A 3 ( ) ( ) ( ) ( ) ( ) A 4 ( ) ( ) ( ) ( ) ( ) A 5 ( ) ( ) ( ) ( ) ( )

9 1944 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Table 6: Available iterval-valued ituitioistic fuzzy iformatio about the strategies. S1 S2 S3 S4 S5 A 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 7: Immediate probabilities of the problem. S1 S2 S3 S4 S5 IP IP IP IP IP I the followig we utilize aother approach developed to get the most desirable alterative(s). Step 1. Similarly the results foud i the immediate probabilities by usig the above probabilities ad weights are show i Table 7. Step 2. We use the arithmetic aggregatig operator to aggregate the decisio iformatio i Table 6 ad probabilistic iformatio. The results are show i Table 8. The arithmetic aggregatig operator icludes the IVIFWA operator IVIFOWA P-IVIFWA operator P-IVIFOWA ad IP-IVIFOWA operator. Step 3. We use the geometric aggregatig operator to aggregate the decisio iformatio i Table 6 ad probabilistic iformatio. The results are show i Table 9. These geometric aggregatig operators iclude the IVIFWG operator IVIFOWG P-IVIFWG operator P-IVIFOWG ad IP-IVIFOWG operator. Step 4. Accordig to the differet aggregatig operators ad score fuctios the orderig of the alteratives is show i Table 10. Note that > meas preferred to. As we ca see i this example all the operators produce the same best strategy A2. 6. Coclusio The traditioal probability aggregatio operators are geerally suitable for aggregatig the iformatio takig the form of umerical values ad yet they will fail i dealig with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio. I this paper with respect to probabilistic decisio-makig problems with ituitioistic fuzzy iformatio or iterval-valued ituitioistic fuzzy iformatio some ew probabilistic decisio-makig aalysis methods are developed. Firstly some operatioal laws score fuctio ad accuracy fuctio of ituitioistic fuzzy values or iterval-valued ituitioistic fuzzy values are itroduced. The we have developed some ew probability aggregatio operators with ituitioistic fuzzy iformatio: Probability Ituitioistic Fuzzy Weighted Average (P-IFWA) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Average (IP-IFOWA) operator Probability Ituitioistic Fuzzy Ordered Weighted Average (P-IFOWA) operator Probability Ituitioistic Fuzzy Weighted Geometric (P-IFWG) operator Immediate Probability Ituitioistic Fuzzy Ordered Weighted Geometric (IP-IFOWG) operator ad Probability Ituitioistic Fuzzy Ordered Weighted Geometric (P-IFOWG) operator ad applied these operators to decisio-makig problem about the selectio of the optimal productio strategy. Furthermore we shall exted the developed models ad procedures to solve probabilistic decisiomakig problems with iterval-valued ituitioistic fuzzy iformatio. Fially some illustrative examples have bee give to show the developed method. I the future we shall cotiue workig i the extesio ad applicatio of the developed operators to other domais. Ackowledgmets The work was supported by the Natioal Natural Sciece Foudatio of Chia uder Grat No Natural Sciece Foudatio Proect of CQ CSTC of the People s Republic of Chia (No. CSTC2011BA0035) the Humaities ad Social Scieces Foudatio of Miistry of Educatio of the People s Republic of Chia uder Grat No. 11XJC No. 10XJA Table 8: Iterval-valued ituitioistic fuzzy aggregated results (β 0.40). IVIFWA IVIFOWA P-IVIFWA P-IVIFOWA IP-IVIFOWA A 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) Table 9: Iterval-valued ituitioistic fuzzy aggregated results (β 0.40). IVIFWG IVIFOWG P-IVIFWG P-IVIFOWG IP-IVIFOWG A 1 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 2 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 3 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 4 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) A 5 ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ]) ([ ] [ ])

10 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Table 10: Orderig of the alterative. Orderig IVIFWA A 2 > A 4 > A 5 > A 3 > A 1 IVIFOWA A 2 > A 4 > A 5 > A 3 > A 1 P-IVIFWA A 2 > A 4 > A 5 > A 3 > A 1 P-IVIFOWA A 2 > A 4 > A 5 > A 3 > A 1 IP-IVIFOWA A 2 > A 4 > A 5 > A 3 > A 1 IVIFWG A 2 > A 4 > A 5 > A 3 > A 1 IVIFOWG A 2 > A 4 > A 5 > A 3 > A 1 P-IVIFWG A 2 > A 4 > A 5 > A 3 > A 1 P-IVIFOWG A 2 > A 4 > A 5 > A 3 > A 1 IP-IVIFOWG A 2 > A 4 > A 5 > A 3 > A 1 ad the Chia Postdoctoral Sciece Foudatio uder Grat This research was also supported by the Sciece ad Techology Research Foudatio of Chogqig Educatio Commissio uder Grat KJ Refereces [1] Ataassov K. Ituitioistic fuzzy sets Fuzzy Sets ad Systems 20 pp (1986). [2] Ataassov K. More o ituitioistic fuzzy sets Fuzzy Sets ad Systems 33 pp (1989). [3] Zadeh L.A. Fuzzy sets Iformatio ad Cotrol 8 pp (1965). [4] Gau W.L. ad Buehrer D.J. Vague sets IEEE Trasactios o Systems Ma ad Cyberetics 23(2) pp (1993). 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Multi-perso multi-attribute decisio makig models uder ituitioistic fuzzy eviromet Fuzzy Optimizatio ad Decisio Makig 6(3) pp (2007). [13] Xu Z.S. Models for multiple attribute decisio-makig with ituitioistic fuzzy iformatio Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systems 15(3) pp (2007). [14] Li D.F. Extesio of the LINMAP for multiattribute decisio makig uder Ataassov s ituitioistic fuzzy eviromet Fuzzy Optimizatio ad Decisio Makig 7(1) pp (2008). [15] Li D.F. Wag Y.C. Liu S. ad Sha F. Fractioal programmig methodology for multi-attribute group decisio makig usig IFS Applied Soft Computig 9 pp (2009). [16] Li D.F. Che G.H. ad Huag Z.G. Liear programmig method for multiattribute group decisio makig usig IF sets Iformatio Scieces 180(9) pp (2010). [17] Wei G.W. Maximizig deviatio method for multiple attribute decisio makig i ituitioistic fuzzy settig Kowledge-Based Systems 21(8) pp (2008). [18] Wei G.W. 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A approach to group decisio makig based o iterval-valued ituitioistic udgmet matrices System Egieer-Theory & Practice 27(4) pp (2007). [26] Xu Z.S. ad Che J. O geometric aggregatio over iterval-valued ituitioistic fuzzy iformatio Fourth Iteratioal Coferece o Fuzzy Systems ad Kowledge Discovery FSKD pp (2007). [27] Xu Z.S. Methods for aggregatig iterval-valued ituitioistic fuzzy iformatio ad their applicatio to decisio makig Cotrol ad Decisio 22(2) pp (2007). [28] Xu Z.S. ad Yager R.R. Dyamic ituitioistic fuzzy multi-attribute decisio makig Iteratioal Joural of Approximate Reasoig 48(1) pp (2008). [29] Wei G.W. Some geometric aggregatio fuctios ad their applicatio to dyamic multiple attribute decisio makig i ituitioistic fuzzy settig Iteratioal Joural of Ucertaity Fuzziess ad Kowledge- Based Systems 17(2) pp (2009). [30] Wei G.W. Some iduced geometric aggregatio operators with ituitioistic fuzzy iformatio ad their applicatio to group decisio makig Applied Soft Computig 10(2) pp (2010). [31] Li D.F. Liear programmig method for MADM with iterval-valued ituitioistic fuzzy sets Expert Systems with Applicatios 37(8) pp (2010). [32] Ye J. Multicriteria fuzzy decisio-makig method based o a ovel accuracy fuctio uder iterval-valued ituitioistic fuzzy eviromet Expert Systems with Applicatios 36(3) pp (2009). [33] Ye F. A exteded TOPSIS method with iterval-valued ituitioistic fuzzy umbers for virtual eterprise parter selectio Expert Systems with Applicatios 37(10) pp (2010). [34] Xu Z.S. Choquet itegrals of weighted ituitioistic fuzzy iformatio Iformatio Scieces 180 pp (2010). [35] Egema K.J. Filev D.P. ad Yager R.R. Modellig decisio makig usig immediate probabilities Iteratioal Joural of Geeral Systems 24 pp (1996). [36] Merigó J.M. Fuzzy decisio makig with immediate probabilities Computers & Idustrial Egieerig 58(4) pp (2010). [37] Yager R.R. Egema K.J. ad Filev D.P. O the cocept of immediate probabilities Iteratioal Joural of Itelliget Systems 10 pp (1995). [38] Merigó J.M. ad Casaovas M. Iduced aggregatio operators i decisio makig with the Dempster-Shafer belief structure Iteratioal Joural of Itelliget Systems 24(8) pp (2009). [39] Merigó J.M. Casaovas M. ad Martíez L. Liguistic aggregatio operators for liguistic decisio makig based o the Dempster-Shafer theory of evidece Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systems 18(3) pp (2010). [40] Merigó J.M. New extesios to the OWA operators ad their applicatio i decisio makig Ph.D. Thesis Departmet of Busiess Admiistratio Uiversity of Barceloa Spai (i Spaish) (2008). [41] Merigó J.M. Probabilistic decisio makig with the OWA operator ad its applicatio i ivestmet maagemet Proceedigs of the IFSA-EUSFLAT Iteratioal Coferece Lisbo Portugal pp (2009). [42] Merigó J.M. The probabilistic weighted average operator ad its applicatio i decisio makig I Operatios Systems Research & Security of Iformatio G.E. Lasker ad P. Hruza Eds. 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11 1946 G.W. Wei J.M. Merigó / Scietia Iraica Trasactios E: Idustrial Egieerig 19 (2012) Guiwu Wei has a M.S. ad a Ph.D. degree i applied mathematics from Southwest Petroleum Uiversity Busiess Admiistratio from school of Ecoomics ad Maagemet at Southwest Jiaotog Uiversity Chia respectively. He is a Associate Professor i the Departmet of Ecoomics ad Maagemet at Chogqig Uiversity of Arts ad Scieces. He has published more tha 90 papers i ourals books ad coferece proceedigs icludig ourals such as Expert Systems with Applicatios Applied Soft Computig Kowledge ad Iformatio Systems Kowledge-based Systems Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systems Iteratioal Joural of Computatioal Itelligece Systems ad Iformatio: A Iteratioal Joural. He has published oe book. He has participated i several scietific committees ad serves as a reviewer i a wide rage of ourals icludig Computers & Idustrial Egieerig Iteratioal Joural of Iformatio Techology ad Decisio Makig Kowledge-based Systems Iformatio Scieces Iteratioal Joural of Computatioal Itelligece Systems ad Europea Joural of Operatioal Research. He is curretly iterested i Aggregatio Operators Decisio Makig ad Computig with Words. José M. Merigó has a M.S. ad a Ph.D. degree i Busiess Admiistratio from Uiversity of Barceloa Spai. His Ph.D. received the Extraordiary Award from the Uiversity of Barceloa. He also holds a Bachelor Degree i Ecoomics from Lud Uiversity Swede. He is a Assistat Professor i the Departmet of Busiess Admiistratio at the Uiversity of Barceloa. He has published more tha 100 papers i ourals books ad coferece proceedigs icludig ourals such as Iformatio Scieces Iteratioal Joural of Itelliget Systems Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systems Cyberetics & Systems ad Computers & Idustrial Egieerig. He has published four books icludig oe edited with World Scietific Computatioal Itelligece i Busiess ad Ecoomics. He is o the editorial board of several ourals icludig the Joural of Advaced Research o Fuzzy ad Ucertai Systems ad the ISTP Trasactios of Systems & Cyberetics. He has participated i several scietific committees ad serves as a reviewer i a wide rage of ourals icludig IEEE Trasactio o Fuzzy Systems Iformatio Scieces ad Europea Joural of Operatioal Research. He is curretly iterested i Aggregatio Operators Decisio Makig ad Ucertaity.

Interval Intuitionistic Trapezoidal Fuzzy Prioritized Aggregating Operators and their Application to Multiple Attribute Decision Making

Interval Intuitionistic Trapezoidal Fuzzy Prioritized Aggregating Operators and their Application to Multiple Attribute Decision Making Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Aggregatig Operators ad their Applicatio to Multiple Attribute Decisio Makig Xia-Pig Jiag Chogqig Uiversity of Arts ad Scieces Chia cqmaagemet@163.com