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 ABSTRACT: I this paper, we ivestigate the iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig (MADM problems i which the attributes are i differet priority level. Motivated by the ideal of prioritized aggregatio operators bu R.R. Yager, we developed some prioritized aggregatio operators for aggregatig iterval ituitioistic trapezoidal fuzzy iformatio, ad the apply them to develop some models for iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig (MADM problems i which the attributes are i differet priority level. Fially, a practical example about talet itroductio is give to verify the developed approaches ad to demostrate its practicality ad effectiveess. Keywords: Multiple Attribute Decisio Makig (MADM, Iterval Ituitioistic Trapezoidal Fuzzy Iformatio, Prioritized Aggregatio Operators, Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Weighted Average (IITFPWA Operator, Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Weighted Geometric (IITFPWG Operator Received: 2 April 2018, Revised 29 April 2018, Accepted 8 May 2018 DOI: 2018 DLINE. All Rights Reserved 1. Itroductio Ataassov[1-3]itroduced the cocept of ituitioistic fuzzy set(ifs, which is a geeralizatio of the cocept of fuzzy set[4]. Ataassov ad Gargov[5] itroduced the cocept of iterval-valued ituitioistic fuzzy sets (IVIFSs as a further geeralizatio of that of IFSs, as well as of IVFSs. Ataassov[6] defied some operatioal laws of the IVIFSs. The ituitioistic fuzzy set ad iterval-valued ituitioistic fuzzy sets has received more ad more attetio sice its appearace[7-26]. Shu, Cheg ad Chag[27] gave the defiitio ad operatioal laws of ituitioistic triagular fuzzy umber ad proposed a algorithm of the ituitioistic fuzzy fault-tree aalysis. Wag[28] gave the defiitio of ituitioistic trapezoidal fuzzy umber ad iterval ituitioistic trapezoidal fuzzy umber. Wag ad Zhag [29] gave the defiitio of expected values of ituitioistic trapezoidal fuzzy umber ad proposed the programmig method of multi-criteria decisio-makig based o ituitioistic trapezoidal fuzzy umber with icomplete certai iformatio. Wag ad Zhag[30] developed the Hammig distace of ituitioistic trapezoidal fuzzy umbers ad ituitioistic trapezoidal fuzzy weighted arithmetic averagig (ITFWAA operator, the proposed multi-criteria decisio-makig 102 Joural of Itelliget Computig Volume 9 Number 3 September 2018
method with icomplete certai iformatio based o ituitioistic trapezoidal fuzzy umber. Wa[31] defied the score fuctio ad accuracy fuctio of a iterval ituitioistic trapezoidal fuzzy umber ad proposed the iterval ituitioistic trapezoidal fuzzy weighted average (IITFWA operator ad the iterval ituitioistic trapezoidal fuzzy weighted geometric(iitfwg operator. Wei[32] ivestigated the multiple attribute group decisio makig (MAGDM problems i which both the attribute weights ad the expert weights takes the form of real umbers, attribute values takes the form of iterval ituitioistic trapezoidal fuzzy umbers. The some ew aggregatio operators icludig iterval ituitioistic trapezoidal fuzzy ordered weighted geometric (IITFOWG operator ad iterval ituitioistic trapezoidal fuzzy hybrid geometric (IITFHG operator, are proposed ad some desirable properties of these operators are studied, such as commutativity, idempotecy ad mootoicity. The, a IITFWG ad IITFHG operators-based approach is developed to solve the MAGDM problems i which both the attribute weights ad the expert weights take the form of real umbers, attribute values take the form of iterval ituitioistic trapezoidal fuzzy umbers. Fially, some illustrative examples are give to verify the developed approach ad to demostrate its practicality ad effectiveess. From above aalysis, we ca see that iterval ituitioistic trapezoidal fuzzy set is a very useful tool to deal with ucertaity. More ad more multiple attribute decisio makig theories ad methods uder iterval ituitioistic trapezoidal fuzzy eviromet have bee developed. Curret methods are uder the assumptio that the attributes are at the same priority level. However, i real ad practical multiple attribute decisio makig problem, the attributes have differet priority level commoly. To overcome this drawback, motivated by the ideal of prioritized aggregatio operators [33-34], i this paper, we propose some iterval ituitioistic trapezoidal fuzzy prioritized aggregatio operators: iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator ad iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (IITFPWG operator. The promiet characteristic of these proposed operators is that they take ito accout prioritizatio amog the attributes. The, we have utilized these operators to develop some approaches to solve the iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig problems i which the attributes are i differet priority level. To do so, the remaider of this paper is set out as follows. I the ext sectio, we itroduce some basic cocepts related to iterval ituitioistic trapezoidal fuzzy umbers ad some operatioal laws of iterval ituitioistic trapezoidal fuzzy umbers. I Sectio 3 we have developed some iterval ituitioistic trapezoidal fuzzy prioritized aggregatio operators: iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator ad iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (ITTFPWG operator ad studied some desirable properties of the proposed operator. The promiet characteristic of these proposed operators is that they take ito accout prioritizatio amog the attributes. I Sectio 4, we have developed apply these operators to develop some models for iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig (MADM problems i which the attributes are i differet priority level. I Sectio 5, a practical example about talet itroductio is give to verify the developed approach ad to demostrate its practicality ad effectiveess. I Sectio 6, we coclude the paper ad give some remarks. 2. Prelimiaries 2.1 Iterval Ituitioistic Trapezoidal Fuzzy Numbers Set I the followig, we shall itroduce some basic cocepts related to ituitioistic trapezoidal fuzzy umbers ad iterval ituitioistic trapezoidal fuzzy umbers. Defiitio 1[28]. Let ã is a ituitioistic trapezoidal fuzzy umber, its membership fuctio is: x a b a μ ã, a < x < b; μ ã (x = μ ã, b < x < c; d x d c μ ã, c < x < d; 0, others (1 Its No-membership fuctio is: Joural of Itelliget Computig Volume 9 Number 3 September 2018 103
ν ã (x = b x + ν ã (x a 1, a b a 1 < x < b; 1 ν ã, b < x < c; x c + ν ã (d 1 x, c d 1 c < x < d ; 1 0, others. (2 Where 0 < μ ã < 1; 0 <ν ã < 1; ad μ ã + ν ã < 1; a, b, c, d R. The ã = <([a, b, c, d]; μ, ([a 1, b, c, d 1 ]; ν > is called a ituitioistic trapezoidal fuzzy umber. For coveiece, let ã = ([a, b, c, d]; μ, ν. If (x [0, 1] ad (x [0, 1] are iterval umbers, ad 0 < sup ( (x + sup ( (x < 1, x X, for coveiece, let (x =, (x =, The ã = ([a, b, c, d]; = ([a, b, c, d]; is called a iterval ituitioistic trapezoidal fuzzy umber[31]. Defiitio 2[31]. 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 ituitioistic trapezoidal fuzzy umber, ad λ > 0, the (1 ã 1 ã 2 = ([a 1 a 2, b 1 b 2, c 1 c 2, d 1 d 2 ]; Defiitio 3. Let ã = ([a, b, c, d]; be a iterval ituitioistic trapezoidal fuzzy umber, a score fuctio S of a iterval ituitioistic trapezoidal fuzzy umber ca be represeted as follows: S (ã 1 = a+ b + c + d 4 4, S (ã [ 1, 1]. (3 Defiitio 4[31]. Let ã = ([a, b, c, d]; be a iterval ituitioistic trapezoidal fuzzy umber, a accuracy fuctio H of a iterval ituitioistic trapezoidal fuzzy umber ca be represeted as follows: H (ã = a+ b + c + d 4 2, H (ã [0, 1]. (4 to evaluate the degree of accuracy of the iterval ituitioistic trapezoidal fuzzy umber ã, where H(ã [0, 1]. The larger the value of H (ã, the more the degree of accuracy of the iterval ituitioistic trapezoidal fuzzy umber ã. 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 ad the o-membership fuctio. Based o the score fuctio S ad the accuracy fuctio H, i the followig, Wa[31] give a order relatio betwee two iterval ituitioistic trapezoidal fuzzy umber, which is defied a follows: 104 Joural of Itelliget Computig Volume 9 Number 3 September 2018
Defiitio 5. 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 ituitioistic trapezoidal fuzzy umber, s (ã 1 ad s (ã 2 be the scores of ã ad b, respectively, ad let H(ã 1 ad H(ã 2 be the accuracy degrees of ã ad b, respectively, the if S (ã < S(b, the ã is smaller tha b, deoted by ã < b; if S(ã < S(b, the if, the ã ad b represet the same iformatio, deoted by ã = b; (2 if ã is smaller tha b, deoted by ã < [10-11]. 2.2 Prioritized Average (PA Operator The prioritized average (PA operator was origially itroduced by Yager [33], which was defied as follows: Defiitio 6[33]. Let G = {G 1, G 2,..., G } be a collectio of attribute ad that there is a prioritizatio betwee the attribute expressed by the liear orderig G 1 G 2 G 3... G, idicate attribute G has a higher priority tha G k, if < k. The value G (x is the performace of ay alterative x uder attribute G, ad satisfies G (x [0,1]. If where w = PA (G i (x = w G (x, T = - 1 G k (x ( = 2,...,, T 1 = 1. The PA is called the prioritized average (PA operator. 3. Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Aggregatio Operators (5 3.1. Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Weighted Average (IITFPWA Operator The prioritized average [33] operators, however, have usually bee used i situatios where the iput argumets are the exact values. We shall exted the PA operators to accommodate the situatios where the iput argumets are iterval ituitioistic trapezoidal fuzzy iformatio. I this Sectio, we shall ivestigate the PA operator uder iterval ituitioistic trapezoidal fuzzy eviromets. Based o Defiitio 6, we give the defiitio of the iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator as follows: Defiitio 7. Let ã ( =1, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, the we defie the iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator as follows: IITFPWA(ã 1,..., ã T 1 = = 1 T 2 ã 1 ã 2... ã = = 1 T ã (6 where ad s (ã is the score values of ã ( =1, 2,...,. Based o operatios of the iterval ituitioistic trapezoidal fuzzy iformatio described i Sectio 2.1, We ca drive the Theorem 1. Theorem 1. Let ã ( =1, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, the their aggregated value by usig the IITPWA operator is also a iterval tuitioistic trapezoidal fuzzy umber, ad IITFPWA(ã 1,..., ã = T 1 ã 1 ã 2... ã = T = 1 T 2 T ã Joural of Itelliget Computig Volume 9 Number 3 September 2018 105
= a b c d,,, T 1 (1 μ, 1 (1 μ ν ν (7, - 1 where = s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã ( = 2,...,. It ca be easily proved that the IITFPWA operator has the followig properties. Theorem 2. (Idempotecy Let ã (, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, where - 1 =, the s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã i ( =1, 2,...,. If all ã ( =1, 2,..., are equal, i.e. ã = ã for all IITFPWA(ã 1,..., ã = ã Theorem 3. (Boudedess Let ã (, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, where = - 1 s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã ( =1, 2,...,., ad let (8 The ã = mi ã, ã + = max ã ã < IITFPWA(ã 1,..., ã < ã + Theorem 4. (Mootoicity Let ã (, 2,..., ad ã (, 2,..., be two set of iterval ituitioistic trapezoidal fuzzy umbers, - 1 where = s (ã T = - 1 s (ã (, 2,...,, T 1 = T 1 = 1, s (ã is the is the score values of ã (, 2,...,, s (ã is the score values of ã (, 2,...,, if ã < ã, for all, the IITFPWA(ã 1,..., ã < IITFPWA(ã 1, ã 2,..., ã (9 3.2. Iterval Ituitioistic Trapezoidal Fuzzy Prioritized Weighted Geometric (IITFPWG Operator Based o the IITFPWA operator ad the geometric mea, here we defie a iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (IITFPWG operator: Defiitio 8. Let ã (, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, the we defie the iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (IITFPWG operator as follows: IITFPWG(ã 1,..., ã 106 Joural of Itelliget Computig Volume 9 Number 3 September 2018
- 1 T 2 T T = ã 1T1 ã 2 T... ã = where = s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã ( =1, 2,...,. ã (10 Based o operatios of the iterval ituitioistic trapezoidal fuzzy umbers described i Sectio 2, we ca drive the Theorem 5. Theorem 5. Let ã ( =1, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, the their aggregated value by usig the IITPWA operator is also a iterval ituitioistic trapezoidal fuzzy umber, ad IITFPWG(ã 1,..., ã T 2 T = T = ã 1T1 ã 2 T... ã ã = ã 1 (a T, (b, (c, (d (11 (μ T 1, (μ 1 (ν, 1 (1 ν - 1 where = s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã ( =1, 2,...,. It ca be easily proved that the IITFPWG operator has the followig properties. Theorem 6. (Idempotecy Let be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, where ad is the score values of. If all are equal, i.e. for all, the (12 Theorem 7. (Boudedess Let (ã (, 2,..., be a collectio of iterval ituitioistic trapezoidal fuzzy umbers, where (13 Joural of Itelliget Computig Volume 9 Number 3 September 2018 107
- 1 = s (ã ( = 2,...,, T 1 = 1 ad s (ã is the score values of ã ( =1, 2,..., ad let the ã = mi ã, ã + = max ã ã < IITFPWG (ã 1,..., ã < ã + (13 Theorem 8. (Mootoicity Let ã (, 2,..., ad ã (, 2,..., be two set of iterval ituitioistic trapezoidal fuzzy umbers, - 1-1 where = s (ã, T s (ã (, 2,...,, T 1 = T 1 = 1, s (ã is the score values of ã (, 2,..., if ã < ã for all, = the IITFPWG (ã (14 1,..., ã < IITFPWA(ã 1, ã 2,..., ã 4. A Approach to Multiple Attribute Decisio Makig with Iterval Ituitioistic Trapezoidal Fuzzy Iformatio I this sectio, we shall utilize the iterval ituitioistic trapezoidal prioritized aggregatio operators to multiple attribute decisio makig with iterval ituitioistic trapezoidal fuzzy iformatio. For a multiple attribute decisio makig problems with iterval ituitioistic trapezoidal fuzzy iformatio, let X = {X 1, X 2,..., X m } be a discrete set of alteratives, Let G = {G 1, G 2,..., G } be a collectio of attribute ad that there is a prioritizatio betwee the attribute expressed by the liear orderig G 1 G 2 G 3... G, idicate attribute G has a higher priority tha G s. If < s the decisio makers provide several values for the alterative A i uder the attribute G with aoymity, these values ca be cosidered as a iterval ituitioistic trapezoidal. I the case where two decisio makers provide the same value, the the value emerges oly oce i. Suppose that is the iterval ituitioistic trapezoidal fuzzy decisio matrix, where idicates the degree that the alterative A i satisfies the attribute G give by the decisio maker, idicates the degree that the alterative A i does t satisfy the attribute G give by the decisio maker,, i = 1, 2,... m,, 2,.... The, we utilize the IITFPWA (or IITFPWG operator to develop a approach to multiple attribute decisio makig problems with iterval ituitioistic trapezoidal fuzzy iformatio, which ca be described as followig: Step 1. Calculate the values of T i ( i = 1, 2,... m,, 2,... as follows - 1 T i = s (h i, (1, 2,... m,, 2,... λ = 1 T i1 = 1, i = 1, 2,...m (15 (16 Step 2. Aggregate all iterval ituitioistic trapezoidal fuzzy umbers h i (, 2,... by usig the iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator: 108 Joural of Itelliget Computig Volume 9 Number 3 September 2018
(17 i = 1, 2,...m. Or the iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (IITFPWG operator: (18 Step 3. Calculate the scores S(h i (i = 1, 2,...m of the overall iterval ituitioistic trapezoidal fuzzy preferece values h i (i = 1, 2,...m to rak all the alteratives A i (i = 1, 2,...m ad the to select the best oe(s. Step 4. Rak all the alteratives A i (i = 1, 2,...m ad select the best oe(s i accordace with S(h i (i = 1, 2,...m.. Step 5. Ed. 5. Numerical Example I order to stregthe academic educatio, promote the buildig of teachig body, the school of maagemet i a Chiese uiversity wats to itroduce oversea outstadig teachers (adapted from [35]. This itroductio has bee raised great attetio from the school, uiversity presidet, dea of maagemet school ad huma resource officer sets up the pael of decisio makers which will take the whole resposibility for this itroductio. They made strict evaluatio for 5 cadidates X i (i = 1, 2, 3, 4, 5 accordig to the followig four attributes: G1 is the morality; G2 is the research capability; G3 is the teachig skill; G4 is the educatio backgroud Uiversity presidet have the absolute priority for decisio makig, dea of the maagemet school comes ext. Besides, this itroductio will be i strict accordace with the priciple of combie ability with political itegrity. The prioritizatio relatioship for the criteria is as below, G 1 G 2 G 3... G. The five possible cadidates X i (i = 1, 2, 3, 4, 5 are to be evaluated usig the iterval ituitioistic trapezoidal fuzzy values by the three decisio makers uder the above four attributes, ad costruct the decisio matrix as listed i the followig matrices as follows: Joural of Itelliget Computig Volume 9 Number 3 September 2018 109
[(0.5, 0.6, 0.7, 0.8];[ 0.4, 0.5];[0.3, 0.4];[0.1, 0.2, 0.3, 0.4];[ 0.4, 0.5];[0.1, 0.2] [(0.6, 0.7, 0.8, 0.9];[0.2, 0.7];[0.2, 0.3];[0.5, 0.6, 0.7, 0.8];[0.3, 0.6];[0.2, 0.4] [(0.1, 0.2, 0.4, 0.5];[ 0.3, 0.4];[0.1, 0.2];[0.2, 0.3, 0.5, 0.6];[ 0.3, 0.5];[0.1, 0.4] [(0.3, 0.4, 0.5, 0.6];[0.3, 0.6][0.2, 0.4];[0.1, 0.3, 0.4, 0.5];[0.4, 0.6];[0.2, 0.3] [(0.2, 0.3, 0.4, 0.5];[ 0.4, 0.7];[ 0.1, 0.3];[0.3, 0.4, 0.5, 0.6];[0.5, 0.6];[0.3, 0.4] [(0.5, 0.6, 0.8, 0.9];[ 0.3, 0.6];[ 0.2, 0.3];[0.4, 0.5, 0.6, 0.7];[ 0.3, 0.7];[0.1, 0.3] [(0.4, 0.5, 0.7, 0.8];[ 0.4, 0.7];[ 0.1, 0.2];[0.5, 0.6, 0.7, 0.9];[ 0.5, 0.8];[0.1, 0.2] [0.5, 0.6, 0.7, 0.8];[ 0.2, 0.6];[ 0.2, 0.3];[0.3, 0.5, 0.7, 0.9];[ 0.2, 0.4];[0.1, 0.5] [0.1, 0.3, 0.5, 0.7];[ 0.3, 0.6];[ 0.1, 0.4];[0.6, 0.7, 0.8, 0.9];[ 0.3, 0.7];[0.1, 0.2] [0.2, 0.3, 0.4, 0.5];[ 0.2, 0.5];[ 0.3, 0.4];[0.5, 0.6, 0.7, 0.8];[ 0.5, 0.6];[0.2, 0.4] The, i order to select the most desirable cadidate, we utilize the IITFPWA operator to develop a approach to multiple attribute decisio makig problems with iterval ituitioistic trapezoidal fuzzy iformatio, which ca be described as followig: Step 1. Utilize (19-(20 to calculate the values of T i (i = 1, 2,...m, = 2,... as follows: 1.0000 0.3575 0.0581 0.0244 1.0000 0.4500 0.1682 0.0706 T i = 1.0000 0.1800 0.0414 0.0155 1.0000 0.2588 0.0526 0.0126 1.0000 0.2363 0.0638 0.0112 Step 2. Aggregate all iterval ituitioistic trapezoidal fuzzy umbers by usig the iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator to derive the overall iterval ituitioistic trapezoidal fuzzy umbers of the cadidates A i. ([0.3990, 0.4990, 0.6030, 0.7030];[0.5882, 0.4975];[0.2268, 0.3259] ([0.5493, 0.6493, 0.7592, 0.8634];[0.3937, 0.6588];[0.2023, 0.2855] ([0.1304, 0.2317, 0.4283, 0.5296];[0.3790, 0.4101];[0.1047, 0.2238] ([0.2558, 0.3793, 0.4833, 0.5873];[0.4555, 0.5921]; [0.2042, 0.3555] ([0.2206, 0.3206, 0.4206, 0.5206]; [0.4961, 0.6678];[0.1326, 0.3167] Step 3. Calculate the scores of the overall iterval ituitioistic trapezoidal fuzzy umbers of the cadidates A i : 110 Joural of Itelliget Computig Volume 9 Number 3 September 2018
Step 4. Rak all the cadidates A i (1, 2, 3, 4, 5 i accordace with the scores of the overall iterval ituitioistic trapezoidal fuzzy umbers, ad thus the most desirable cadidate is A 2. Based o the IITFPWG operator, the, i order to select the most desirable cadidate, we ca develop a approach to multiple attribute decisio makig problems with iterval ituitioistic trapezoidal fuzzy iformatio, which ca be described as followig: Step 12. See Step 1. Step 22. Aggregate all iterval ituitioistic trapezoidal fuzzy umbers usig the iterval ituitioistic trapezoidal fuzzy prioritized weighted geometric (IITFPWG operator to derive the overall iterval ituitioistic trapezoidal fuzzy umbers of the cadidates A i. ([0.3340, 0.4544, 0.5688, 0.6752]; [0.3908, 0.4926]; [0.2654, 0.3464] ([0.5448, 0.6455, 0.7576, 0.8620]; [0.2372, 0.6390]; [0.2067,0.3073] ([0.1184, 0.2226, 0.4240, 0.5254]; [0.2885, 0.4037]; [ 0.1121, 0.2373] ([0.2332, 0.3758, 0.4808, 0.5848]; [0.3105, 0.5846]; [0.2082, 0.3699] ([0.2168, 0.3178, 0.4184, 0.5188]; [0.3979, 0.6397]; [0.1649, 0.3201] Step 3. Calculate the scores of the overall iterval ituitioistic trapezoidal fuzzy umbers of the cadidates A i : Step 4. Rak all the cadidates A i (1, 2, 3, 4, 5 i accordace with the scores of the overall iterval ituitioistic trapezoidal fuzzy umbers, ad thus the most desirable cadidate is A 2. I this sectio, we have proposed two approaches to solve the iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig problems i which the attributes are i differet priority level. From the above aalysis, we ca see that the mai advatages of the proposed operators ad approaches over the traditioal iterval ituitioistic trapezoidal fuzzy operators ad approaches are ot oly due to the fact that our operators accommodate the iterval ituitioistic trapezoidal fuzzy eviromet but also due to the cosideratio of the prioritizatio amog the attributes, which makes it more feasible ad practical. 6. Coclusio I this paper, we ivestigate the iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig (MADM problem i which the attributes are i differet priority level. The, motivated by the ideal of prioritized aggregatio operators[33], we have developed some prioritized aggregatio operators for aggregatig iterval ituitioistic trapezoidal fuzzy iformatio: iterval ituitioistic trapezoidal fuzzy prioritized weighted average (IITFPWA operator ad iterval ituitioistic trapezoidal fuzzy umbers prioritized weighted geometric (ITFPWG operator. The promiet characteristic of these proposed operators is that they take ito accout prioritizatio amog the attributes. The, we have utilized these operators to develop some approaches to solve the iterval ituitioistic trapezoidal fuzzy multiple attribute decisio makig problems i which the attributes are i differet priority level. Fially, a practical example about talet itroductio is give to verify the developed approach ad to Joural of Itelliget Computig Volume 9 Number 3 September 2018 111
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