Some Aggregation Operators with Intuitionistic Trapezoid Fuzzy Linguistic Information and their Applications to Multi-Attribute Group Decision Making

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1 Appl. Math. If. Sc. 8 No (2014) 2427 Appled Mathematcs & Iformato Sceces A Iteratoal Joural Some Aggregato Operators wth Itutostc Trapezod Fuzzy Lgustc Iformato ad ther Applcatos to Mult-Attrbute Group Decso Mag Yabg Ju 1 Shaghog Yag 1 ad Ahua Wag 2 1 School of Maagemet ad Ecoomcs Bejg Isttute of Techology Bejg Cha 2 Graduate School of Educato Peg Uversty Bejg Cha Receved: 10 Sep Revsed: 8 Dec Accepted: 9 Dec Publshed ole: 1 Sep Abstract: We study the mult-attrbute group decso mag (MAGDM) problems whch the attrbute values provded by the decso maers tae the form of tutostc trapezod fuzzy lgustc (ITrFL) formato cosderg the ucertaty ad accuracy of put argumets. Some ew aggregato operators called tutostc trapezod fuzzy lgustc weghted average (ITrFLWA) tutostc trapezod fuzzy lgustc ordered weghted average (ITrFLOWA) ad tutostc trapezod fuzzy lgustc hybrd weghted average (ITrFLHWA) operators are proposed at frst. The we study some desrable propertes of the proposed operators such as mootocty dempotecy commutatvty ad boudedess. Next two ovel approaches based o the proposed operators are developed to solve MAGDM problems wth tutostc trapezod fuzzy lgustc formato. Fally a llustratve example of emergecy logstcs suppler selecto s provded to verfy the feasblty of the proposed approaches. Keywords: Mult-attrbute group decso mag Itutostc trapezod fuzzy lgustc (ITrFL) formato Itutostc trapezod fuzzy lgustc aggregato operators 1 Itroducto Mult-attrbute group decso mag (MAGDM) problem s oe of the most mportat parts of decso theory. Due to the creasg complexty of the socoecoomc evromet ad the lac of owledge or data about the group decso mag problems doma the attrbutes volved the decso problems are ot always expressed as crsp umbers ad some of them are more sutable to be deoted by fuzzy umbers such as terval umber lgustc varable tutostc fuzzy umber etc. The fuzzy set theory (FS) tally troduced by Zadeh [1] s a good method to research MAGDM problems. Sce Zadeh troduced fuzzy set theory to deal wth vague problems aother useful tool called lgustc varables [2] are utlzed to express a decso maer s preferece formato over objects process of decso mag uder ucerta or vague evromets. It maes evaluato by meas of lgustc terms whch descrbes qualtatve lgustc formato from extremely poor to extremely good ad have bee prove more practcally ad flexbly tha FS. So far a umber of MAGDM approaches have bee proposed for dealg wth lgustc assessmet formato. Herrera et al. [3] preseted a cosesus model group decso mag uder lgustc assessmets. Herrera ad Martłez [4] developed 2-tuple lgustc approach whch composed a lgustc term ad a real umber. L et al. [5] proposed the defto of terval-valued 2-tuple lgustc approach. Zhag [6 7] preseted some terval-valued 2-tuple lgustc aggregato operators. Xu [8] proposed ucerta lgustc varables whch are the exteso of lgustc varables. Recetly a ew method called trapezod fuzzy lgustc varables receved lots of atteto from researchers sce t had bee proposed by Xu [9] whch s the geeralzed form of ucerta lgustc varables essece but more sutable for processg vague formato. However the fuzzy set s used to character the fuzzess just by membershp degree. O the bass of the fuzzy set theory Ataassov [10 11] proposed the tutostc fuzzy set characterzed by a membershp Correspodg author e-mal: juyb@bt.edu.c Natural Sceces Publshg Cor.

2 2428 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... fucto ad a o-membershp fucto. Obvously the tutostc fuzzy set ca descrbe ad character the fuzzy essece of the objectve world more exqustely ad t has receved more ad more atteto sce ts appearace [12 17]. I the real world decso maers usually caot completely express ther opos by a lgustc varable from a predefed lgustc term set or a tutostc fuzzy umber dvdually. Sometmes they ca express the formato by combg lgustc varables ad tutostc fuzzy set. Based o tutostc fuzzy set ad lgustc varables Wag ad L [18] proposed the cocept of tutostc lgustc set t ca overcome the defects for tutostc fuzzy set whch ca oly roughly represet crtera s membershp ad omembershp to a partcular cocept such as good ad bad etc. ad for lgustc varables whch usually mples that membershp degree s 1 ad the o-membershp degree ad hestato degree of decso maers caot be expressed [19]. Lu ad J [20] defed the tutostc ucerta lgustc varables based o ucerta lgustc varables ad tutostc fuzzy sets. Furthermore Lu [21] defed the terval tutostc ucerta lgustc varables based o ucerta lgustc varables ad terval-valued tutostc fuzzy sets (IVIFS). I order to process ucerta ad accuracy formato more effcecy ad precsely t s ecessary to mae a further study o the exteded form of the tutostc ucerta lgustc varables by combg trapezod fuzzy lgustc varables ad tutostc fuzzy set. For example we ca evaluate the trasportato rs of the emergecy logstcs suppler by the lgustc set S = {s 0 (extremely low); s 1 (very low); s 2 (low); s 3 (medum); s 4 (hgh); s 5 (very hgh); s 6 (extremely hgh)}. Perhaps we ca use the trapezod fuzzy lgustc [s α s β s θ s τ ] (0 α β θ τ 6) to descrbe the evaluato result but ths s ot accurate because t merely provdes a lgustc rage. I ths stuato we ca use a tutostc trapezod fuzzy lgustc (ITrFL) [s α s β s θ s τ ](uv) to descrbe the trasport rs by gvg the membershp degree u ad o-membershp degree v to [s α s β s θ s τ ]. Ths s the motvato of our study. As a fact ITrFL avods the formato dstorto ad losg the decso mag process ad overcomes the shortcomgs of the tutostc lgustc varables ad the tutostc ucerta lgustc varables. I ths paper a ovel cocept called tutostc trapezod fuzzy lgustc varable whch combes the trapezod fuzzy lgustc varables ad tutostc fuzzy set s proposed. The some ew operators for aggregatg tutostc trapezod fuzzy lgustc formato are proposed such as tutostc trapezod fuzzy lgustc weghted average (ITrFLWA) tutostc trapezod fuzzy lgustc ordered weghted average (ITrFLOWA) ad tutostc trapezod fuzzy lgustc hybrd weghted average (ITrFLHWA) operators. Furthermore two ovel methods to solve the MAGDM problems whch the attrbute weghts tae the form of real umbers attrbute values tae the form of tutostc trapezod fuzzy lgustc formato are developed based o the proposed operators. Fally a umercal example of emergecy logstcs suppler selecto s gve to llustrate the applcatos of the developed methods. The rest of ths paper s orgazed as follows. I Secto 2 some basc deftos of trapezod fuzzy lgustc varables ad tutostc lgustc umbers are revewed ad tutostc trapezod fuzzy lgustc varables s defed as well as operatos comparso ad dstace formula. I Secto 3 we propose some ew operators for aggregatg tutostc trapezod fuzzy lgustc formato ad study several desrable propertes of these operators. I Secto 4 two ovel approaches for MAGDM based o the tutostc trapezod fuzzy lgustc formato are developed. I Secto 5 a umercal example of emergecy logstcs suppler selecto s gve to llustrate the applcatos of the developed methods. The paper s cocluded Secto 6. 2 Prelmary 2.1 Trapezod fuzzy lgustc varables A lgustc set s defed as a fte ad completely ordered dscrete term set S = {s 0 s 1...s l 1 } where l s the odd value. For example whe l = 7 the lgustc term set S ca be defed as follows: S = {s 0 (extremely low); s 1 (very low); s 2 (low); s 3 (medum); s 4 (hgh); s 5 (very hgh); s 6 (extremely hgh)} Defto 2.1. [9] Let S={s θ s 0 s θ s l 1 θ [0l 1]} whch s the cotuous form of lgustc set S. s α s β s θ s τ are four lgustc terms S ad s 0 s α s β s θ s τ s l 1 f 0 α β θ τ l 1 the the trapezod fuzzy lgustc varable s defed as s = [s α s β s θ s τ ] ad S deotes a set of trapezod fuzzy lgustc varables. I partcular f ay two of αβθτ are equal the s s reduced to a tragular fuzzy lgustc varable; f ay three of α β θ τ are equal the s s reduced to a ucerta lgustc varable. 2.2 Itutostc lgustc umbers Based o tutostc fuzzy set ad lgustc term set Wag ad L [18] preseted the exteso form of the lgustc set.e. tutostc lgustc set whch s show as follows. Defto 2.2. [18] A tutostc lgustc set A X ca be defed as A={ x[s θ(x) (u(x)v(x))] x X} (1) Natural Sceces Publshg Cor.

3 Appl. Math. If. Sc. 8 No (2014) / where s θ(x) S u(x) [01] v(x) [01] ad u(x)+v(x) 1 x X. s θ(x) s a lgustc term u(x) represets the membershp degree of the elemet x to lgustc term s θ(x) whle v(x) represets the o-membershp degree of the elemet x to lgustc term s θ(x). Let π(x) = 1 u(x) v(x)π(x) [01] x X the π(x) s called a hestacy degree of x to lgustc term s θ(x). 2.3 Itutostc trapezod fuzzy lgustc umbers Defto 2.3. [22] A tutostc trapezod fuzzy lgustc set Ã X ca be defed as Ã={ x[[s α(x) s β(x) s θ(x) s τ(x) ](u(x)v(x))] x X} (2) where s α(x) s β(x) s θ(x) s τ(x) S u(x) [01] v(x) [0 1] ad u(x) + v(x) 1 x X. [s α(x) s β(x) s θ(x) s τ(x) ] s a trapezod fuzzy lgustc term u(x) represets the membershp degree of the elemet x to lgustc term [s α(x) s β(x) s θ(x) s τ(x) ] whle v(x) represets the o-membershp degree of the elemet x to lgustc term [s α(x) s β(x) s θ(x) s τ(x) ]. Let π(x)=1 u(x) v(x)π(x) [01] x X the π(x) s called a hestacy degree of x to lgustc term [s α(x) s β(x) s θ(x) s τ(x) ]. I Eq. (2) [s α(x) s β(x) s θ(x) s τ(x) ](u(x)v(x)) s a tutostc trapezod fuzzy lgustc umber (ITrFLN). Obvously Ã ca be vewed as a collecto of tutostc trapezod fuzzy lgustc umbers (ITrFLNs). For coveece ã = [s α(ã) s β(ã) s θ(ã) s τ(ã) ](u(ã)v(ã)) s used to represet a ITrFLN. Defto 2.4. [22] Let ã = [s α(ã )s β(ã )s θ(ã )s τ(ã )](u(ã )v(ã )) ad ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) be two ITrFLNs ad λ 0 the the operatos of ITrFLNs ca be defed as Eqs. (3-6). Theorem 2.1. Let ã = [s α(ã )s β(ã )s θ(ã )s τ(ã )](u(ã )v(ã )) ad ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) be two ITrFLNs ad λλ λ j 0 the we ca obta the followg rules. 1.ã ã j = ã j ã 2.ã ã j = ã j ã 3.λ(ã ã j )=λã λã j 4.λ ã λ j ã =(λ + λ j )ã 5.ã λ ã λ j = ã λ +λ j 6.ã λ ã j λ =(ã ã j ) λ j Defto 2.5. [22] Let ã = [s α(ã )s β(ã )s θ(ã )s τ(ã )](u(ã )v(ã )) be a ITrFLN the expected fucto E(ã ) ad the accuracy fucto H(ã ) of ã are defed as Eqs. (7-8) respectvely. Assume that ã ad ã j are two ITrFLNs they ca be compared by the followg rules: 1.If E(ã )>E(ã j ) the ã > ã j ; 2.If E(ã )=E(ã j ) the f H(ã )>H(ã j ) the ã > ã j ; f H(ã )=H(ã j ) the ã = ã j ; f H(ã )<H(ã j ) the ã < ã j ; Defto 2.6. [22] Let ã = [s α(ã )s β(ã )s θ(ã )s τ(ã )](u(ã )v(ã )) ad ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) be two ITrFLNs the ormalzed Hammg dstace betwee ã ad ã j s defed as Eq. (9). 3 Some aggregato operators based o tutostc trapezod fuzzy lgustc umbers Based o the ordered weghted average (OWA) operator [23] we defe three ew operators such as tutostc trapezod fuzzy lgustc weghted average (ITrFLWA) operator tutostc trapezod fuzzy lgustc ordered weghted average (ITrFLOWA) operator ad tutostc trapezod fuzzy lgustc hybrd weghted average (ITrFLHWA) operator what follows. Defto 3.1. Let ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) ( j = ) be a collecto of the ITrFLNs. The tutostc trapezod fuzzy lgustc weghted average (ITrFLWA) operator ca be defed as follows ad ITrFLWA: Ω Ω: ITrFLWA(ã 1 ã 2...ã )= ω j ã j (10) where Ω s the set of all tutostc trapezod fuzzy lgustc umbers ad ω = (ω 1 ω 2...ω ) T s the weght vector of ã j ( j = 12...) such that ω [01] ω j = 1. Especally f ω = ( )T the the ITrFLWA operator wll be smplfed to a tutostc trapezod fuzzy lgustc average (ITrFLA) operator. Theorem 3.1. Let ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) ( j = 12...) be a collecto of the ITrFLNs. Eq. (10) ca be trasformed to Eq. (11) whch s stll a ITrFLN. Theorem 3.1 ca be prove by mathematcal ducto. The steps the proof are as follows: Proof. 1.Whe =1 obvously Eq. (11) s correct. Natural Sceces Publshg Cor.

4 2430 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... ã ã j = [s α(ã )+α(ã j )s β(ã )+β(ã j )s θ(ã )+θ(ã j )s τ(ã )+τ(ã )](u(ã )+u(ã j ) u(ã )u(ã j )v(ã )v(ã j )) (3) ã ã j = [s α(ã ) α(ã j )s β(ã ) β(ã j )s θ(ã ) θ(ã j )s τ(ã ) τ(ã )](u(ã )u(ã j )v(ã )+v(ã j ) v(ã )v(ã j )) (4) λã = [s λ α(ã )s λ β(ã )s λ θ(ã )s λ τ(ã )]u(ã ) λ )(v(ã ) λ ) (5) d(ã ã j )= 1 2(l 1) ã λ = [s (α(ã )) λs (β(ã )) λs (θ(ã )) λs (τ(ã )) λ]((u(ã ) λ 1 v(ã ) λ )) (6) E(ã )= 1+u(ã ) v(ã ) s 2 α(ã )+β(ã )+θ(ã )+τ(ã )/4 = s (α(ã )+β(ã )+θ(ã )+τ(ã )) (1+u(ã ) v(ã ))/8 (7) H(ã )=(u(ã )+v(ã )) s α(ã )+β(ã )+θ(ã )+τ(ã )/4 = s (α(ã )+β(ã )+θ(ã )+τ(ã )) (u(ã )+v(ã ))/4 (8) ( (1+u(ã ) v(ã )) α(ã )+β(ã )+θ(ã )+τ(ã ) 4 (1+u(ã j ) v(ã j )) α(ã j)+β(ã j )+θ(ã j )+τ(ã j ) 4 ) (9) ITrFLWA(ã 1 ã 2...ã )= [s ω j α(ã j )s ω j β(ã j )s ω j θ(ã j )s ω j τ(ã j )] u(ã j ))) ω j (11) 2.Whe =2 sce ω 1 ã 1 = [s ω1 α(ã 1 )s ω1 β(ã 1 )s ω1 θ(ã 1 )s ω1 τ(ã 1 )] u(ã 1 ))) ω 1 (v(ã 1 )) ω 1 ω 1 ã 2 = [s ω2 α(ã 2 )s ω1 β(ã 2 )s ω1 θ(ã 2 )s ω2 τ(ã 2 )] thus u(ã 2 ))) ω 2 (v(ã 2 )) ω 2 ITrFLWA(ã 1 ã 2 ) = ω 1 ã 1 + ω 2 ã 2 = ( [s ω1 α(ã 1 )s ω1 β(ã 1 )s ω1 θ(ã 1 )s ω1 τ(ã 1 )] u(ã 1 ))) ω 1 (v(ã 1 )) ω 1 ) +( [s ω2 α(ã 2 )s ω2 β(ã 2 )s ω2 θ(ã 2 )s ω2 τ(ã 2 )] u(ã 2 ))) ω 2 (v(ã 2 )) ω 2 ) = [s 2 ω j α(ã j ) s 2 ω j β(ã j ) s 2 ω j θ(ã j ) s 2 ω j τ(ã j ) ] 2 u(ã j ))) ω j 2 3.Suppose that whe = Eq. (11) s correct.e. ITrFLWA(ã 1 ã 2...ã ) = [s ω j α(ã j ) s ω j β(ã j ) s ω j θ(ã j ) s ω j τ(ã j ) ] u(ã j ))) ω j the whe =+ 1 we have ITrFLWA(ã 1 ã 2...ã ã +1 ) = [s ω j α(ã j ) s ω j β(ã j ) s ω j θ(ã j ) s ω j τ(ã j ) ] u(ã j ))) ω j +( [s ω+1 α(ã +1 )s ω+1 β(ã +1 ) s ω+1 θ(ã +1 )s ω+1 τ(ã +1 )] u(ã +1 ))) ω +1 (v(ã +1 )) ω +1 ) = [s +1 ω j α(ã j ) s +1 ω j β(ã j ) s +1 ω j θ(ã j ) s +1 ω j τ(ã j ) ] +1 u(ã j ))) ω j +1 therefore whe =+1 Eq. (11) s correct as well. Thus Eq. (11) s correct for all. Theorem 3.2. (Mootocty) Let (ã 1 ã 2...ã ) ad (ã 1 ã 2...ã ) be two collectos of ITrFLNs. For all j= f ã j ã j the ITrFLWA(ã 1ã 2...ã ) ITrFLWA(ã 1 ã 2...ã ) (12) Natural Sceces Publshg Cor.

5 Appl. Math. If. Sc. 8 No (2014) / Proof. Accordg to the defto of ITrFLWA Eq. (11) we have ITrFLWA(ã 1ã 2...ã ) = [s ω j α(ã j ) s ω j β(ã j ) s ω j θ(ã j ) s ω j τ(ã j ) ] u(ã j))) ω j ITrFLWA(ã 1ã 2...ã ) = [s ω j α(ã j )s ω j β(ã j ) s ω j θ(ã j )s ω j τ(ã j )] u(ã j ))) ω j (v(ã j)) ω j ad ther expected values ca be calculated by Eq. (7) E(ITrFLWA(ã 1ã 2...ã ))= s ( ω j α(ã j )+ ω j β(ã j )+ ω j θ(ã j )+ ω j τ(ã j )) (1+ u(ã j ))ω j ) )(v(ã j ))ω j )/8 E(ITrFLWA(ã 1 ã 2...ã ))= s ( ω j α(ã j )+ ω j β(ã j )+ ω j θ(ã j )+ ω j τ(ã j )) (1+ u(ã j)) ω j ) )(v(ã j)) ω j )/8 sce ã j ã j the E(ITrFLWA(ã 1 ã 2...ã )) E(ITrFLWA(ã 1 ã 2...ã )) thus ITrFLWA(ã 1 ã 2...ã ) ITrFLWA(ã 1 ã 2...ã ) Theorem 3.3. (Idempotecy) Let ã j = ã for all j= the Proof. Sce ã j = ã thus ITrFLWA(ã 1 ã 2...ã )=ã (13) ITrFLWA(ã 1 ã 2...ã ) = [s ω j α(ã j )s ω j β(ã j ) s ω j θ(ã j )s ω j τ(ã j )] u(ã j ))) ω j = [s ω j α(ã)s ω j β(ã) s ω j θ(ã)s ω j τ(ã)] u(ã))) ω j (v(ã)) ω j = [s α(ã) s β(ã) s θ(ã) s τ(ã) ](u(ã)v(ã)) = ã Theorem 3.4. (Boudedess) Let ã m = m 1 j {ã j} ad ã max = max 1 j {ã j} the ã m ITrFLWA(ã 1 ã 2...ã ) ã max (14) Proof. Sce ã m ã ã max the ω j ã m ω j ã j ω j ã max thus ã m ITrFLWA(ã 1 ã 2...ã ) ã max Defto 3.2. Let ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) ( j = ) be a collecto of the ITrFLNs. The tutostc trapezod fuzzy lgustc ordered weghted average (ITrFLOWA) operator ca be defed as follows ad ITrFLOWA: Ω Ω: ITrFLOWA(ã 1 ã 2...ã )= w j ã σ( j) (15) where Ω s the set of all tutostc trapezod fuzzy lgustc umbers ad w = (w 1 w 2...w ) T s a assocated weght vector wth ITrFLOWA such that w j [01] ( j = 12...) w j = 1. (σ(1)σ(2)...σ()) s a permutato of (12...) such that ã σ( j 1) ã σ( j) for all j = w j s decded oly by the j-th posto the aggregato process. Therefore w ca also be called the posto-weghted vector. Accordg to the method of determg posto-weghted vector proposed [24] w ca be calculated by the combato umber. The calculato formula s as follows: w +1 = c 1 2 1(= ) (16) where the combato umber c 1 ca be deoted as c 1 = ( 1)!!( 1)! Theorem 3.5. (Mootocty) Let (ã 1 ã 2...ã ) ad (ã 1 ã 2...ã ) be two collectos of ITrFLNs. For all j= f ã j ã j the ITrFLOWA(ã 1ã 2...ã ) ITrFLOWA(ã 1 ã 2...ã ) (17) Theorem 3.6. (Idempotecy) Let ã j = ã for all j= the ITrFLOWA(ã 1 ã 2...ã )= w j ã j = ã (18) Theorem 3.7. (Boudedess) Let ã m = m 1 j {ã j} ad ã max = max 1 j {ã j} the ã m ITrFLOWA(ã 1 ã 2...ã ) ã max (19) Theorems ca be easly prove smlar to Theorems so the proofs are omtted. Natural Sceces Publshg Cor.

6 2432 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... Theorem 3.8. (Commutatvty) If (ã 1 ã 2...ã ) s ay permutato of(ã 1 ã 2...ã ) the ITrFLOWA(ã 1 ã 2...ã )=ITrFLOWA(ã 1ã 2...ã ) (20) Proof. Sce (ã 1 ã 2...ã ) s ay permutato of (ã 1 ã 2...ã ) the 1 w j α(ã σ( j) ) = w j β(ã σ( j) ) = w j θ(ã σ( j) ) = w j τ(ã σ( j) ) = u(ã σ( j) )) w j = 1 (v(ã σ( j) )) w j = w j α(ã σ( j) ) w j β(ã σ( j) ) w j θ(ã σ( j) ) w j τ(ã σ( j) ) u(ã σ( j) ))w j (v(ã σ( j) ))w j thus ITrFLOWA(ã 1 ã 2...ã )=ITrFLOWA(ã 1 ã 2...ã ). Defto 3.3. Let ã j = [s α(ã j )s β(ã j )s θ(ã j )s τ(ã j )](u(ã j )v(ã j )) ( j = ) be a collecto of the ITrFLNs. The tutostc trapezod fuzzy lgustc hybrd weghted average (ITrFLHWA) operator ca be defed as follows ad ITrFLHWA: Ω Ω: ITrFLHWA(ã 1 ã 2...ã )= w j bσ( j) (21) where Ω s the set of all tutostc trapezod fuzzy lgustc umbers w = (w 1 w 2...w ) T s a assocated weght vector wth ITrFLHWA such that w j [01] ad w j = 1. b j = ω j ã j j = ( b σ(1) b σ(2)... b σ() ) s a permutato of ( b 1 b 2... b ) such that bσ( j 1) b σ( j) for j = ω = (ω 1 ω 2...ω ) T s the weght vector of ã j ( j = 12...) ω j [01] ω j = 1 ad s the balacg coeffcet. Accordg to the operatos of ITrFLNs Eq. (21) ca be trasformed to the followg form. ITrFLHWA(ã 1 ã 2...ã ) (22) = [s w j α( b σ( j) ) s w j β( b σ( j) ) s w j θ( b σ( j) ) s w j τ( b σ( j) ) ] u( b σ( j) ))) w j (v( b σ( j) )) w j Two specal cases of the ITrFLHWA operator are as follows: 1.Whe w = ( )T the ITrFLHWA operator reduces to the ITrFLWA operator Eq. (10). 2.Whe ω = ( )T the ITrFLHWA operator reduces to the ITrFLOWA operator Eq. (15). 4 Two approaches for mult-attrbute group decso mag based o tutostc trapezod fuzzy lgustc aggregato operators For the mult-attrbute group decso mag problems ths paper weghts of both the attrbute ad the decso maers tae the form of real umbers ad the attrbute preferece values tae the form of tutostc trapezod fuzzy lgustc varables. Therefore we shall develop approaches to group decso mag based o tutostc trapezod fuzzy lgustc preferece formato. Let A = {A 1 A 2...A m } be a dscrete set of alteratves C={C 1 C 2...C } be a set of attrbutes ad ω = (ω 1 ω 2...ω ) T be the weght vector of the attrbutes such that ω j [01] ( j = 12...) ad ω j = 1. DM = {DM 1 DM 2...DM } s the set of decso maers. R f = [ã f ( f = 12...) s the j ] m decso matrx gve by the decso maer DM f where ã f j = [s f α(ã )s j β(ã f )s j θ(ã f )s j τ(ã f )](u(ã f j )v(ã f j )) j deotes the evaluato o alteratve A wth respect to attrbute C j gve by the decso maer DM f ad t taes the form of tutostc trapezod fuzzy lgustc umber. Approach 1 If the weght vector of the decso maers s ow ad defed as w = (w 1 w 2...w ) T we select the best alteratve by the followg steps. Step 1: For each decso maer DM f ( f = 12...) utlze the weght vector of the attrbutes ω = (ω 1 ω 2...ω ) T ad ITrFLWA operator Eq. (10) to calculate the comprehesve attrbute preferece value ã f of each alteratve determed by the f -th decso maer respectvely. ã f = ITrFLWA(ã f 1 ãf 2...ãf )= w j ã f j (=12...m) (23) Step 2: For each alteratve calculate the collectve overall preferece values Z of each alteratve determed by all decso maers by Eq. (24). Z = f=1 w f ã f (=12...m) (24) Step 3: Utlze the expected fucto Eq. (7) to calculate the expected values E(Z ) (=12...m) of the Natural Sceces Publshg Cor.

7 Appl. Math. If. Sc. 8 No (2014) / collectve overall preferece values Z. If there s o dfferece betwee two expected values E(Z ) ad E(Z j ) the we eed to calculate the accuracy values H(Z ) ad H(Z ) of value Z ad Z j by Eq. (8). Step 4: Ra these alteratves ad select the best oe by the values of E(Z ) ad H(Z ). Approach 2 If the weght vector of the decso maers s uow we select the best alteratve by the followg steps. Step 1: See Step 1 Approach 1. Step 2: Calculate the weght vector w = (w 1 w 2...w ) T of the decso maer DM f ( f = 12...) by Eq. (16) at frst. The utlze ITrFLOWA operator to calculate Z by Eq. (25) Z = f=1 σ( f) w f ã (=12...m) (25) where (ã σ(1) ã σ(2)...ã σ() ) s a permutato of (ã 1 ã2...ã f 1) σ( f) ) such that ãσ( ã ( f = 23...). Step 3: See Step 3 Approach 1. Step 4: See Step 4 Approach 1. 5 A llustratve example A serous publc health evet happes oe cty of Cha recetly emergecy maagemet ceter of the govermet wats to select a most approprate emergecy logstcs suppler order to mplemet rescue actvtes mmedately reducg the ecoomc losses ad rescue cost maxmally. There are fve supplers the cty: {A 1 A 2 A 3 A 4 A 5 }. The emergecy maagemet ceter must mae a decso accordg to the followg four attrbutes wth the attrbute weght vector ω =( ) T : (1) C 1 s the emergecy resource supply capacty; (2) C 2 s the trasportato speed; (3) C 3 s the trasportato dstace; (4) C 4 s the trasportato rs. Four experts (DM 1 DM 2 DM 3 DM 4 ) are vted to evaluate the supplers wth respect to each attrbute by usg the predefed lgustc term set S = {s 0 (extremely low); s 1 (very low); s 2 (low); s 3 (medum); s 4 (hgh); s 5 (very hgh); s 6 (extremely hgh)} ad the weght vector of the decso maers s w = ( ) T. The tutostc trapezod fuzzy lgustc decso matrces R f = [ã f j ] 5 4 ( f = ) are costructed as show [Tables 1-4]. Step 1: Utlze the weght vector of the attrbutes ω = ( ) T to calculate the comprehesve attrbute preferece values ã f ( = 12345; f = 1234) by Eq. (23) whch are show the followg respectvely. ã 1 1 = [s 1.8 s 3.4 s 4.5 s 5.5 ]( ); ã 1 2 = [s 1.3 s 2.3 s 4.4 s 5.4 ]( ); ã 1 3 = [s 0.8 s 2.2 s 3.9 s 5.1 ]( ); ã 1 4 = [s 1.4 s 2.8 s 3.8 s 5.1 ]( ); ã 1 5 = [s 1.5s 2.5 s 4.5 s 5.5 ]( ); ã 2 1 = [s 1.1 s 2.1 s 4.4 s 5.4 ]( ); ã 2 2 = [s 2.1 s 3.2 s 4.5 s 5.7 ]( ); ã 2 3 = [s 0.7 s 1.7 s 4.6 s 5.6 ]( ); ã 2 4 = [s 1.5 s 3.3 s 4.6 s 5.6 ]( ); ã 2 5 = [s 1.8s 3.1 s 4.3 s 5.3 ]( ); ã 3 1 = [s 1.4s 2.4 s 4.4 s 5.7 ]( ); ã 3 2 = [s 1.5s 3.1 s 4.7 s 5.9 ]( ); ã 3 3 = [s 1.6s 2.7 s 4.1 s 5.5 ]( ); ã 3 4 = [s 1.7s 2.7 s 4.1 s 5.1 ]( ); ã 3 5 = [s 1.7s 2.9 s 4.0 s 5.4 ]( ); ã 4 1 = [s 1.5 s 2.5 s 4.5 s 5.5 ]( ); ã 4 2 = [s 1.6 s 2.8 s 4.2 s 5.5 ]( ); ã 4 3 = [s 1.2 s 1.8 s 4.5 s 5.5 ]( ); ã 4 4 = [s 1.8 s 2.8 s 4.4 s 5.5 ]( ); ã 4 5 = [s 1.4s 2.7 s 3.0 s 5.3 ]( ). Step 2: Calculate the collectve overall preferece value Z of each alteratve by Eq. (24). The collectve overall preferece values Z are show the followg. Z 1 = [s s s s ]( ); Z 2 = [s s s s ]( ); Z 3 = [s s s s ]( ); Z 4 = [s s s s ]( ); Z 5 = [s s s s ]( ). Step 3: Utlze the expected fucto Eq. (7) to calculate the expected values E(Z ) ( = 12345) of collectve overall preferece values Z E(Z 1 )=s ; E(Z 2 )=s ; E(Z 3 )=s ; E(Z 4 )=s ; E(Z 5 )=s Step 4: Ra these alteratves ad select the best oe by the value of E(Z ). A 2 A 1 A 5 A 4 A 3 Therefore the best alteratve s A 2. If the weght vector of the decso maers s uow we select the best alteratve by the followg steps. Step 1: See Step 1 above example. Step 2: If the weght vector of the decso maers s uow t s ecessary to calculate the weght vector Natural Sceces Publshg Cor.

8 2434 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... Table 1: Itutostc trapezod fuzzy lgustc decso matrx R 1 gve by DM 1 Alteratves C 1 C 2 C 3 C 4 A 1 [s 2 s 3 s 4 s 5 ](0.60.3) [s 2 s 4 s 5 s 6 ](0.70.3) [s 1 s 3 s 4 s 5 ](0.50.4) [s 2 s 3 s 5 s 6 ](0.70.2) A 2 [s 1 s 2 s 4 s 5 ](0.70.3) [s 2 s 3 s 5 s 6 ](0.60.3) [s 0 s 1 s 4 s 5 ](0.70.1) [s 2 s 3 s 4 s 5 ](0.60.3) A 3 [s 0 s 2 s 4 s 5 ](0.70.2) [s 1 s 2 s 4 s 5 ](0.50.3) [s 1 s 2 s 3 s 5 ](0.80.1) [s 2 s 4 s 5 s 6 ](0.70.2) A 4 [s 1 s 2 s 3 s 5 ](0.50.2) [s 1 s 3 s 4 s 5 ](0.70.1) [s 2 s 3 s 4 s 5 ](0.60.3) [s 3 s 4 s 5 s 6 ](0.50.4) A 5 [s 2 s 3 s 5 s 6 ](0.60.2) [s 2 s 3 s 4 s 5 ](0.70.3) [s 0 s 1 s 5 s 6 ](0.50.4) [s 1 s 2 s 4 s 5 ](0.70.1) Table 2: Itutostc trapezod fuzzy lgustc decso matrx R 2 gve by DM 2 Alteratves C 1 C 2 C 3 C 4 A 1 [s 0 s 1 s 5 s 6 ](0.70.2) [s 1 s 2 s 4 s 5 ](0.60.4) [s 2 s 3 s 4 s 5 ](0.50.5) [s 3 s 4 s 5 s 6 ](0.40.2) A 2 [s 2 s 3 s 5 s 6 ](0.50.3) [s 3 s 4 s 5 s 6 ](0.50.4) [s 1 s 2 s 3 s 5 ](0.80.1) [s 1 s 3 s 4 s 5 ](0.50.5) A 3 [s 0 s 1 s 4 s 5 ](0.60.2) [s 0 s 1 s 5 s 6 ](0.70.2) [s 3 s 4 s 5 s 6 ](0.60.3) [s 1 s 2 s 4 s 5 ](0.70.2) A 4 [s 1 s 3 s 4 s 5 ](0.60.4) [s 2 s 4 s 5 s 6 ](0.80.2) [s 2 s 3 s 5 s 6 ](0.60.2) [s 0 s 2 s 4 s 5 ](0.60.3) A 5 [s 2 s 4 s 5 s 6 ](0.70.3) [s 2 s 3 s 4 s 5 ](0.60.3) [s 1 s 2 s 4 s 5 ](0.70.1) [s 2 s 3 s 4 s 5 ](0.60.4) Table 3: Itutostc trapezod fuzzy lgustc decso matrx R 3 gve by DM 3 Alteratves C 1 C 2 C 3 C 4 A 1 [s 0 s 1 s 4 s 6 ](0.70.1) [s 3 s 4 s 5 s 6 ](0.60.1) [s 1 s 2 s 4 s 5 ](0.70.2) [s 0 s 1 s 4 s 5 ](0.60.2) A 2 [s 2 s 4 s 5 s 6 ](0.80.2) [s 2 s 3 s 5 s 6 ](0.60.2) [s 0 s 2 s 4 s 6 ](0.60.3) [s 1 s 3 s 4 s 5 ](0.60.1) A 3 [s 2 s 3 s 4 s 5 ](0.60.3) [s 1 s 2 s 4 s 6 ](0.70.1) [s 2 s 3 s 4 s 5 ](0.60.4) [s 2 s 4 s 5 s 6 ](0.70.3) A 4 [s 1 s 2 s 4 s 5 ](0.60.2) [s 2 s 3 s 4 s 5 ](0.50.5) [s 3 s 4 s 5 s 6 ](0.50.2) [s 0 s 1 s 3 s 4 ](0.70.2) A 5 [s 3 s 4 s 5 s 6 ](0.50.4) [s 1 s 2 s 3 s 5 ](0.80.1) [s 1 s 3 s 4 s 5 ](0.50.5) [s 2 s 3 s 5 s 6 ](0.50.3) Table 4: Itutostc trapezod fuzzy lgustc decso matrx R 4 gve by DM 4 Alteratves C 1 C 2 C 3 C 4 A 1 [s 1 s 2 s 4 s 5 ](0.70.2) [s 3 s 4 s 5 s 6 ](0.60.3) [s 0 s 1 s 4 s 5 ](0.60.2) [s 0 s 1 s 5 s 6 ](0.70.2) A 2 [s 2 s 3 s 4 s 6 ](0.60.2) [s 1 s 2 s 4 s 5 ](0.70.1) [s 2 s 4 s 5 s 6 ](0.70.3) [s 2 s 3 s 4 s 5 ](0.60.3) A 3 [s 0 s 2 s 4 s 5 ](0.60.3) [s 2 s 3 s 5 s 6 ](0.60.2) [s 1 s 3 s 4 s 5 ](0.50.4) [s 2 s 4 s 5 s 6 ](0.80.2) A 4 [s 3 s 4 s 5 s 6 ](0.70.2) [s 2 s 3 s 4 s 5 ](0.50.5) [s 0 s 1 s 5 s 6 ](0.70.2) [s 1 s 2 s 3 s 5 ](0.60.4) A 5 [s 1 s 3 s 4 s 5 ](0.50.5) [s 1 s 2 s 3 s 5 ](0.80.1) [s 2 s 3 s 5 s 6 ](0.50.3) [s 3 s 4 s 5 s 6 ](0.50.4) w = (w 1 w 2 w 3 w 4 ) T of decso maers by Eq. (16) at frst. w 1 = c = w 1 = c = w 1 = c = w 1 = c3 3 = The calculate the collectve overall preferece value Z 1 by Eq. (25) ad the results are show the followg. Z 1 = [s s s s ]( ); Z 2 = [s s s s ]( ); Z 3 = [s s s s ]( ); Z 4 = [s s s s ]( ); Z 5 = [s s s s ]( ). Step 3: Calculate the expected values E(Z ) = (12345) of collectve overall preferece values Z by Eq. (7). E(Z 1 )=s ; E(Z 2 )=s ; E(Z 3 )=s ; E(Z 4 )=s ; E(Z 5 )=s Step 4: Ra these alteratves ad select the best oe by the value of E(Z ). A 2 A 1 A 5 A 4 A 3 Therefore the best alteratve s A 2 as well. 6 Coclusos I ths paper we study the mult-attrbute group decso mag (MAGDM) problems whch the attrbute values provded by the decso maers tae the form of tutostc trapezod fuzzy lgustc formato. We have developed some ew tutostc trapezod fuzzy lgustc aggregato operators such as tutostc trapezod fuzzy lgustc weghted average (ITrFLWA) operator tutostc trapezod fuzzy lgustc ordered weghted average (ITrFLOWA) operator ad tutostc trapezod fuzzy lgustc hybrd weghted average (ITrFLHWA) operator frstly. The we have studed some desred propertes of the developed operators such as mootocty commutatvty dempotecy ad Natural Sceces Publshg Cor.

9 Appl. Math. If. Sc. 8 No (2014) / boudedess. Moreover we have developed two approaches to deal wth mult-attrbute group decso mag problems uder tutostc trapezod fuzzy lgustc formato. If the weght vector of the decso maers s ow we develop a approach based o the ITrFLWA operator to aggregate all attrbute preferece values to comprehesve attrbute preferece values ad the derve the collectve overall evaluato values of each alteratve. O the other had f the weght vector of the decso maers s uow we develop aother approach based o the ITrFLOWA to aggregate all the comprehesve attrbute preferece values to collectve overall preferece values ad derve the overall evaluato values of each alteratve. Fally a llustratve example has bee gve to show the developed method. Apparetly our approaches are straghtforward ad have less loss of formato both theoretcally ad practcally ad ca be appled to hadle MAGDM problems a flexble ad objectve maer uder tutostc trapezod fuzzy lgustc evromet. It s worth otg that addto to emergecy logstcs suppler selecto problem the proposed methodes are equally applcable to other maagemet decso problems. I the future we shall cotue worg the exteso ad applcato of the developed operators. Acowledgemet Ths research s supported by Program for New Cetury Excellet Talets Uversty (NCET ) Natural Scece Foudato of Cha (No ) ad Bejg Mucpal Natural Scece Foudato (No ). Refereces [1] L. A. Zadeh Fuzzy Sets Iformato ad Cotrol (1965). [2] L. A. Zadeh The cocept of a lgustc varable ad ts applcato to approxmate reasog Iformato sceces (1975). [3] F. Herrera E. Herrera-Vedma ad J. L. Verdegay A model of cosesus group decso mag uder lgustc assessmets Fuzzy Sets ad Systems (1996). [4] F. Herrera ad L. Martłez A 2-Tuple Fuzzy Lgustc Represetato Model for Computg wth Words IEEE Trasactos o Fuzzy Systems (2000). [5] J. L J. B. La ad Y. H. L Mult-attrbute group decso-mag method based o the aggregato operators of terval 2-tuple lgustc formato Joural of Jl Normal Uversty (2009). [6] Z. S. Xu Ucerta lgustc aggregato operators based approach to multple attrbute group decso mag uder ucerta lgustc evromet Iformato Sceces (2004). [7] H. M. Zhag The multattrbute group decso mag method based o aggregato operators wth terval-valued 2-tuple lgustc formato Mathematcal ad Computer Modellg (2012). [8] H. M. Zhag Some terval-valued 2-tuple lgustc aggregato operators ad applcato multattrbute group decso mag Appled Mathematcal Modellg (2013). [9] Z. S. Xu A Approach Based o Smlarty Measure to Multple Attrbute Decso Mag wth Trapezod Fuzzy Lgustc Varables Fuzzy Systems ad Kowledge Dscovery (2005). [10] K. T. Ataassov Itutostc fuzzy sets Fuzzy Sets ad Systems (1986). [11] K. T. Ataassov Itutostc fuzzy sets: theory ad applcatos Hedelberg: Physca-Verlag (1999). [12] R. Y. Che A problem-solvg approach to product desg usg decso tree ducto based o tutostc fuzzy Europea Joural of Operatoal Research (2009). [13] Z. S. Xu Some smlarty measures of tutostc fuzzy sets ad ther applcatos to multple attrbute decso mag Fuzzy Optmzato ad Decso Mag (2007). [14] Z. S. Xu Itutostc fuzzy herarchcal clusterg algorthms Joural of Systems Egeerg ad Electrocs (2009). [15] D. F. L Some measures of dssmlarty tutostc fuzzy structures Joural of Computer ad System Sceces (2004). [16] D. F. L Multattrbute decso mag models ad methods usg tutostc fuzzy sets Joural of Computer ad System Sceces (2005). [17] M. M. Xa Z. S. Xu ad B. Zhu Some ssues o tutostc fuzzy aggregato operators based o Archmedea t-coorm ad t-orm Kowledge- Based Systems (2012). [18] J. Q. Wag ad J. J. L The mult-crtera group decso mag method based o mult-graularty tutostc two sematcs Scece & Techology Iformato (2009). [19] P. D. Lu Some geeralzed depedet aggregato operators wth tutostc lgustc umbers ad ther applcato to group decso mag Joural of Computer ad System Sceces (2013). [20] P. D. Lu ad F. J Methods for aggregatg tutostc ucerta lgustc varables ad ther applcato to group decso mag Iformato Sceces (2012). [21] P. D. Lu Some geometrc aggregato operators based o terval tutostc ucerta lgustc varables ad ther applcato to group decso mag Appled Mathematcal Modellg (2013). Natural Sceces Publshg Cor.

2436 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... [22] Y.B. Ju ad S.H.

port (2013). [23] R. R. Yager O ordered weghted averagg aggregato operators multcrtera decso mag IEEE Trasactos o Systems Ma ad Cyberetcs-Part B 18 183-190 (1988). [24] Y. Wag ad Z. S. Xu A ew method of gvg OWA weghts Mathematcs Practce ad Theory 38 51-61 (2008).

Hs ma research terests are decso mag uder ucerta evromet strategc maagemet ad huma resource maagemet Yabg Ju s a assocate professor of Departmet of Maagemet Scece ad Egeerg School of Maagemet ad

10 2436 Y. Ju et. al. : Some Aggregato Operators wth Itutostc Trapezod... [22] Y.B. Ju ad S.H. Yag Some aggregato operators wth tutostc trapezod fuzzy lgustc umbers ad ther applcato to multple attrbute group decso mag Techcal Report (2013). [23] R. R. Yager O ordered weghted averagg aggregato operators multcrtera decso mag IEEE Trasactos o Systems Ma ad Cyberetcs-Part B (1988). [24] Y. Wag ad Z. S. Xu A ew method of gvg OWA weghts Mathematcs Practce ad Theory (2008). Shaghog Yag s a PhD studet of Departmet of Maagemet Scece ad Egeerg School of Maagemet ad Ecoomcs Bejg Isttute of Techology. Hs ma research terests are decso mag uder ucerta evromet strategc maagemet ad huma resource maagemet Yabg Ju s a assocate professor of Departmet of Maagemet Scece ad Egeerg School of Maagemet ad Ecoomcs Bejg Isttute of Techology. Hs research terests are decso mag uder ucerta evromet system evaluato techology ad dscrete evet system smulato. He receved hs PhD degree from Behag Uversty Ahua Wag s a Assocate Professor of Departmet of Educatoal Techology Graduate School of Educato Peg Uversty. Her research terests are formato techology educato educato resource sharg ad educatoal decso mag. She graduated from Departmet of Computer Scece Peg Uversty 2002 recevg PhD degree. Natural Sceces Publshg Cor.

An Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Research on scheme evaluation method of automation mechatronic systems

An Indian Journal FULL PAPER ABSTRACT KEYWORDS. Trade Science Inc. Research on scheme evaluation method of automation mechatronic systems [ype text] [ype text] [ype text] ISSN : 0974-7435 Volume 0 Issue 6 Boechology 204 Ida Joural FULL PPER BIJ, 0(6, 204 [927-9275] Research o scheme evaluato method of automato mechatroc systems BSRC Che