Research Article Frank Aggregation Operators for Triangular Interval Type-2 Fuzzy Set and Its Application in Multiple Attribute Group Decision Making

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1 Hidawi Publishig Corporatio Joural of pplied Mathematics rticle ID pages Research rticle ra ggregatio Operators for Triagular Iterval Type-2 uzzy Set ad Its pplicatio i Multiple ttribute Group Decisio Maig Jidog Qi ad Xiwag Liu School of Ecoomics ad Maagemet Southeast Uiversity Naig Jiagsu Chia Correspodece should be addressed to Xiwag Liu; xwliu@seu.edu.c Received 18 ebruary 2014; ccepted 28 Jue 2014; Published 2 September 2014 cademic Editor: Hug-Yua Chug Copyright 2014 J. Qi ad X. Liu. This is a ope access article distributed uder the Creative Commos ttributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial wor is properly cited. This paper ivestigates a approach to multiple attribute group decisio-maig (MGDM problems i which the idividual assessmets are i the form of triagle iterval type-2 fuzzy umbers (TIT2Ns. irstly some ra operatio laws of triagle iterval type-2 fuzzy set (TIT2S are defied. Secodly some ra aggregatio operators such as the triagle iterval type-2 fuzzy ra weighted averagig (TIT2W operator ad the triagle iterval type-2 fuzzy ra weighted geometric (TIT2WG operator are developed for aggregatio TIT2Ns. urthermore some desirable properties of the two aggregatio operators are aalyzed i detail. ially a approach based o TIT2W (or TIT2WG operator to solve MGDM is developed. illustrative example about supplier selectio is provided to illustrate the developed procedures. The results demostrate the practicality ad effectiveess of our ew method. 1. Itroductio uzzy multiple attribute group decisio maig (MGDM is a hot area of research i fuzzy decisio aalysis theory. It ivolves the process to select the best alterative(s from the set of feasible alteratives with respect to multiple attributes i which the decisio iformatio usually with fuzziess orucertaityisprovidedbyagroupofdecisiomaes uder fuzzy eviromet. I recet years may techiques have bee developed to solve MGDM problems 1 8. Li ad Yag 1 developed a liear programmig techique for multidimesioal aalysis of prefereces i multiple attribute group decisio maig uder fuzzy eviromets. Wag ad Lee 2 geeralized TOPSIS method for MGDM based o high ad low operators which satisfy the partial orderig relatio o fuzzy umbers. Xu ad Che 3 establisheda iteractive model for MGDM i which the iformatio about attribute weights is partly ow. Li ad Wu 4 preseted a fuzzy DEMTEL method for group decisio maig to gather group ideas ad aalyzed the cause-effect relatioship of complex problems i fuzzy eviromet. Guha ad Charaborty 5 developed a MGDM techique cosiderig the degrees of cofidece of experts opiios ad aggregated the decisio iformatio by usig similarity. Hatami-Marbii ad Tavaa 6 proposed a alterative fuzzy outraig method by extedig the ELECTRE I method to tae ito accout the ucertai imprecise ad liguistic assessmets provided by a group of decisio maers. Mousavi et al. 7 utilized Mote Carlo simulatio techique to derive a stochastic approach for hadlig MGDM ad applieditirisselectioproblemiahighwayproect.zhao et al. 8 utilized the fuzzy prioritized operators to develop some models for triagular MGDM i which both the attributesaddecisiomaesareidifferetprioritylevel. Type-2 fuzzy set (T2S was first proposed by Zadeh 9 which ca be viewed as a effective extesio of traditio type-1 fuzzy set (T1S. The membership fuctio of T2S which is characterized by primary fuctio secodary fuctio ad a footprit of ucertaity (OU. It is the ew third dimesio of T2S ad the footprit of ucertaity ca reflect more additioal degrees of freedom that mae it more capable of hadlig imprecisio ad imperfect iformatio i real-world applicatio Hece T2S are receivig more ad more attetios from researchers ad have bee

2 2 Joural of pplied Mathematics successfully developed both i theoretical ad i practical aspects However because the computatioal complexity of geeral T2S is too high it is usually difficult to use i real practice situatios 22. To overcome the limitatio of T2S iterval type-2 fuzzy set (IT2S which is also called iterval-valued fuzzy set (IVS ad cotais membership values that are crisp iterval from zero to oe is proposed 23 ad has bee the most useful tool for modelig ucertaity ad complexity i may domais I particular may approaches have bee preseted i the field of multiple attribute group decisio maig (MGDM. Che ad Lee 31 presetedaewmethod based o arithmetic operatios for hadig multiple attribute hierarchical group decisio-maig problems. Che ad Lee 32 further developed a iterval type-2 TOPSIS method to solve MGDM ad gave some practical examples to illustrate the procedure of the proposed method. Che et al. 33 preseted a raig formula for IT2S iformatio ad established a framewor based o raig method for hadlig MGDM with iterval type-2 fuzzy iformatio. Hu et al. 34 preseted a approach based o possibility degree for iterval type-2 fuzzy multiple criteria group decisio maig ad established a maximizig deviatio optimal model for determiig the attribute weights i which the weight iformatio is partly ow. Wag et al. 35 establishedsome weight optimal models to determie the weights of attributes ad utilized the iterval type-2 weighted averagig (IT2W operator for hadlig MGDM uder iterval type-2 fuzzy eviromet. Che 36 preseted a siged-distace-based method for determiig the obective importace of criteria ad hadlig fuzzy multiple criteria group decisio-maig problems i a flexible ad itelliget way. Che et al. 37 ivestigated exteded QULILEX method for group decisio maig i which the attribute values tae the form of IT2S ad further applied it i medical decisio maig. Baležetis ad Zeg 38 further exteded MULTIMOOR method with IT2S ad applied it i the field of huma resource maagemet ad performace maagemet. Chiao 39 itroduced the cocepts of triagular iterval type- 2 fuzzy set ad developed the aalytic hierarchy process (HP method based o logarithmic regressio fuctio for hadlig MGDM problem. Nga 40 established the probabilistic liguistic framewor with IT2S MGDM problems ad some arithmetic operatios such as uio ad itersectio betwee the iterval type-2 fuzzy liguistic umbers were also defied. Che 41 developed a ew liear assigmet method based o siged distaces for hadlig MGDM problems with IT2S iformatio ad applied it i the selectio of ladfill site. Z. M. Zhag ad S. H. Zhag 42 proposed the cocept of trapezoidal iterval type-2 fuzzy soft set ad further developed a MGDM approach uder iterval type-2 fuzzy eviromet. or MGDM problems a essetial importat step is how to aggregate the decisio-maig iformatio i differet formats with aggregatio operators. s a importat tool for iformatio fusio aggregatio operators are widely used i various fuzzy set (S such as T1S ituitioistic fuzzy set (IS ad hesitat fuzzy set (HS. However up till ow the researches o type-2 fuzzy aggregatio operators are very rare. Wu ad Medel 43 preseted the cocept of liguistic weightedaverage(lwiwhichtheattributesadthe weights both are i the form of IT2S the algorithm was provided ad verified the LW operator ca be decomposed ito fidig two fuzzy weighted average (W. Liu et al. 44 preseted a aalytical solutio method for the W; the method had some good mathematical aalysis properties which ca be used for iterval type-2 aggregatio process. Zhou et al proposed the type-2 OW operator to aggregate the liguistic opiios or prefereces i huma decisio-maig modelig by T2S ad a direct method to aggregate IT2S by type-2 OW operator was provided. Liu et al developedsomeaggregatiooperators suchas weighted aggregatio operators geometric aggregatio operators ad hybrid harmoic averagig operators to hadle MGDM problems with iterval-valued trapezoidal fuzzy umbers. ll the above iterval type-2 fuzzy aggregatio operators are based o alpha-cut decompositio theorem 9; the mai differece betwee the existig iterval type-2 fuzzy aggregatio operators ad other S aggregatio operators is that the iterval type-2 fuzzy aggregatio operators are derived by fuzzy extesio priciple while the other fuzzy aggregatio operators are obtaied by triagle orms s the fudametal of iformatio fusio triagle orms play a importat role ad have bee successfully used to derive various fuzzy aggregatio operators However util ow there is o research about iterval type-2 aggregatio operator based o triagle orm operatios. I order to fill this gap we will utilize the ra triagle orms to develop some desirable iterval type-2 fuzzy aggregatio operators i this paper. ra triagle orms 55 which is the oly oe type triagle orm satisfyig the compatibility. Sice the ra triagle orms ivolve the parameter this ca provide more flexibility ad robustess i the process of iformatio fusio admaeitmoreadequatetomodelpracticaldecisiomaig problems tha other triagle orms 56. Calvo et al. 57 study the fuctioal equatios of ra ad lsia for two classes of commutative associative ad icreasig biary operators. Yager 58 itroduced the extesios operatios to the lattice of closed iterval-valued fuzzy set based o ra t-orms ad gave ecessary ad sufficiet coditios such that these operatios ca form a complete algebraic structure. Casaovas ad Torres 59 ivestigated the additive geeratig fuctio of ra t-orms ad established a framewor with ra t-orms i approximate reasoig. Saroci 60 made a compariso betwee the ra t-orms ad the Hamacher t-orms ad proved that two differet torms form the same family. However the previous research wors almost focus o the basic mathematical costruct ad properties of ra t-orms; the researches of its applicatios are very rare especially i aggregatio ad decisio maig. Todatewehaveotseeayresearchesoaggregatio operators based o ra t-orms for decisio-maig problems. Therefore it is meaigful to study aggregatio operators based o ra triagle orms operatios ad their applicatio i MGDM uder triagle iterval type-2 fuzzy eviromet.

3 Joural of pplied Mathematics 3 The remaider of this paper is show as follows. I Sectio 2 we briefly itroduce some basic cocepts of T2S triagle iterval type-2 fuzzy set ad ra triagle orms. I Sectio 3 weestablishtheraoperatioallawsof triagle iterval type-2 fuzzy set. I Sectio 4 wepreset triagle iterval type-2 fuzzy ra weighted averagig (TIT2W operator ad triagle iterval type-2 fuzzy ra geometric average (TIT2WG operator ad also discuss some desirable properties i detail. I Sectio 5 we develop a approach based o the proposed ra aggregatio operators to MGDM uder the triagle iterval type-2 fuzzy eviromet. practical example is provided to illustrate the procedure of our method i Sectio 6. ially we ed the paper with some coclusios ad poit out future researches i Sectio Prelimiaries I this sectio we will briefly review some basic cocepts of type-2 fuzzy set ad ra triagular orms Type-2 uzzy Set Defiitio 1 (see 23. type-2 fuzzy set i the uiverse of discourse X ca be represeted by a type-2 membership fuctio μ (x u as follows: ={((x u μ (x u x X u J x 0 1} (1 where J x deotes a iterval i 0 1. Moreoverthetype-2 fuzzy set ca also be expressed as the followig form: μ = (x u = x X u J x (x u x X ( u Jx μ (x u /u (2 x where J x 0 1 is the primary membership at x ad u Jx μ (x u/u idicates the secod membership at x. or discreet situatios is replaced by. Defiitio 2 (see 23. Let be a type-2 fuzzy set i the uiverse of discourse X represeted by a type-2 membership fuctio μ (x u. Ifallμ (x u = 1 the is called a iterval type-2 fuzzy set (IT2S. iterval type-2 fuzzy set caberegardedasaspecialcaseofthetype-2fuzzysetwhich is defied as 1 = x X u J x (x u = x X ( u Jx 1/u. (3 x It is obvious that the iterval type-2 fuzzy set defied o X is completely determied by the primary membership which is called the footprit of ucertaity (OU ad the OU ca be expressed as follows: OU ( = J x = {(x u u J x 0 1}. (4 x X x X 2.2. Triagular Iterval Type-2 uzzy Set. Because the operatios o iterval type-2 fuzzy sets are very complex accordig to the decompositio theorem 23 the iterval type-2 fuzzy sets are usually tae i some simplified formatios i applicatios. Here we follow the results of Chiao 39 which adopted triagular iterval type-2 fuzzy set for solvig MGDM problems. I 39Chiaoproposedthecoceptoftriagulariterval type-2 fuzzy set i which the upper membership fuctio (UM ad the lower membership fuctio (LM are represeted by triagular fuzzy umber. Let = (l l m r r be a triagular iterval type-2 fuzzy set defied o X whichisshowiigure1 where l l m r r are the referece poits of the triagular iterval type-2 fuzzy set satisfyig 0 l l m r r 1. The upper ad lower membership fuctios of are defied as x l { m l l x<m UM (x = 1 x=m { x r { m r m x<r x l l x<m LM (x = { m l 1 x=m x r { m { m r x<r. The footprit of ucertaity (OU of the membership fuctio of is show as the shaded regio i igure 1. Obviously if l = l r = r theum (x = LM (x for all x X ad the triagular iterval type-2 fuzzy set reduces to the triagular type-1 fuzzy set. If X is a set i which the elemets are all real umbers the a triagular iterval type-2 fuzzy set i X is called a triagular iterval type-2 fuzzy umber (TIT2N. Che et al. 33 proposed the raig formula for iterval type-2 fuzzy umber as follows. Defiitio 3 (see 33. Let = (l l m r r be a triagular iterval type-2 fuzzy umber i igure 1; the raig value Ra( of the triagular iterval type-2 fuzzy umber is defied as follows: Ra ( = ( l + r 2 (5 +1 l + l +r + r +4m. 8 (6 ThelargertheraigvalueRa( the greater the TIT2N ra Triagular Norms. ra operatios iclude the ra product ad ra sum which are examples of triagle

4 4 Joural of pplied Mathematics μ 1 0 l l m r r x igure 1: triagular iterval type-2 fuzzy set. orms ad triagle coorms respectively. ra product is a t-orm ad ra sum is a t-coorm ad the mathematical forms are defied as follows 55: a b= (λ1 a 1(λ 1 b 1 λ>1 (a b a b= (λa 1(λ b 1 λ>1 (a b It is poited out that the ra product ad ra sum have the followig property 61: (a b + (a b = a + b (a b a à + (a b a =1. Based o the limit theory we ca easily prove the iterestig results as follows 61. (7 (8 (1 If thea b a+b ab a b ab ad the ra product ad ra sum are reduced to the probability product ad probability sum. (2 If λ thea b mi(a + b 1 a b max(0a+b 1 ad the ra product ad ra sum are reduced to the Luasiewicz product ad Luasiewicz sum. 3. ra Operatios of Triagular Iterval Type-2 uzzy Numbers Based o ra t-corm ad ra t-coorm previously metioed we will establish the basic operatio laws of triagular iterval type-2 fuzzy umbers i this sectio. Defiitio 4. Let 1 ad 2 be three TIT2Ns ad λ>1; the we defie their operatio laws as follows. (1 dditio operatio: 1 2 =( 2 (λ1 l 1 2 (λ1 l 1 2 (λ1 m 1 2 (λ1 r 1 2 (λ1 r 1. (2 Multiplicatio operatio: 1 2 =( 2 (λl 1 2 (λl 1 2 (λm 1 2 (λr 1 2 (λr 1. (3 Multiplicatio by a ordiary umber: =( (λ1 l 1 1 ( (λ1 r 1 ( 1 (λ1 m 1 ( 1 (9 (10

5 Joural of pplied Mathematics 5 (4 Power operatio: (λ1 r 1 1 ( (λ1 r 1 (. 1 =( (λl 1 1 ( (λr 1 ( 1 (λm 1 1 ( (11 ( Triagular Iterval Type-2 uzzy ra Weighted veragig Operator Defiitio 6. Let = (l l m r r ( = be a collectio of TIT2Ns ad let TIT2W: Ω Ω;if TIT2W ω ( =ω 1 1 ω 2 2 ω (13 the the fuctio TIT2W is called a triagular iterval type-2 fuzzy ra weighted averagig (TIT2W operator where ω = (ω 1 ω 2...ω T is the weight vector of ( = 12...ω 0ad ω =1.Iparticular if ω = (1/1/...1/ T the the TIT2W operator is reducedtoatriagularitervaltype-2fuzzyraarithmetic averagig (TIT2 operator of dimesio whichis defied as follows: (λr 1 1 ( (λr 1 (. 1 Theorem 5. Let 1 ad 2 be three TIT2Ns ad 1 2 >0; the oe has (1 1 2 = 2 1 ; (2 ( 1 2 = 1 2 ; (3 1 2 =( ; (4 ( 1 2 = 1 ( 2. TIT2 ω ( = 1 ( 1 2. (14 Based o the ra operatio laws from Theorem 5 we ca derive the followig Theorem 7 which shows that the W of TIT2 fuzzy sets is also a TIT2 fuzzy set. Theorem 7. Let = (l l m r r ( = beacollectiooftit2ns;thetheiraggregated value by TIT2W operator is still a TIT2N ad TIT2W ω ( =( (λ 1 l 1 ω It is easy to prove the formulas i Theorem 5; thus they are omitted i here. 4. ra ggregatio Operators for Triagular Iterval Type-2 uzzy Numbers I this sectio we will develop the ra operator uder triagular iterval type-2 fuzzy eviromet ad propose the triagular iterval type-2 fuzzy ra weighted averagig (TIT2W operator ad the triagular iterval type- 2 fuzzy ra weighted geometric (TIT2WG operator basedoratriagularormsrespectively. (λ 1 l ω 1 (λ 1 m 1 ω (λ 1 r 1 ω (λ 1 r ω 1 (15

6 6 Joural of pplied Mathematics where ω=(ω 1 ω 2...ω T is the weight vector of ( = ω >0ad ω =1. Proof. Here we prove Theorem 7 by usig the mathematical iductio method as follows. (1 or = 2 based o the ra operatioal laws of TIT2Ns we have ω 1 1 =( (λ 1 l 1 1 ω 1 ( ω 1 1 (λ 1 l ω ( ω 1 1 ω1 (λ1 m 1 1 ( ω 1 1 (λ 1 r 1 1 ω 1 ( ω 1 1 ω1 (λ1 r 1 1 ( ω 1 1 ω 2 2 =( (λ 1 l 2 1 ω 2 ( ω 2 1 (λ 1 l ω ( ω 2 1 ω2 (λ1 m 2 1 ( ω 2 1 (λ 1 r 2 1 ω 2 ( ω 2 1 ω2 (λ1 r 2 1 ( ω. 2 1 I accordace with Defiitio 2weobtai (16 TIT2W ω ( 1 2 =ω 1 1 ω 2 2 =( 2 (1+(λ1 l (λ1 ( 1 ω /( ω (1+(λ1 l (λ1 ( 1 ω /( ω (λ1 ( (1+(λ1 m 1 ω /( ω (λ1 ( (1+(λ1 r 1 ω /( ω (λ1 ( (1+(λ1 r 1ω /( ω 1 1

7 Joural of pplied Mathematics 7 =( 2 (λ1 l 1 ω 2 ω (λ1 l 1 ω ( ω 1 log 1+ω 2 1 λ ( ω 1+ω (λ1 m 1 ( ω 1+ω (λ1 r 1 ω 2 ω (λ1 r 1 ( ω 1 log 1+ω 2 1 λ ( ω 1+ω 2 1 =( 2 (λ 1 l 1 ω 1 logλ 2 (λ 1 l ω 1 2 (λ 1 m 1 ω 2 (λ 1 r 1 ω 1 logλ 2 (λ 1 r ω 1. (17 If (15holdsfor=thatis TIT2W ω ( =( (λ 1 l 1 ω (λ 1 l ω 1 (λ 1 m 1 ω (λ 1 r 1 ω (18 (λ 1 r 1 ω (λ 1 r ω 1 ( (λ 1 l +1 1 ω +1 ( ω +1 1 (λ 1 l +1 1 ω +1 ( ω +1 1 (λ1 m +1 1 ω+1 ( ω +1 1 (λ 1 r ω 1 the if = +1 based o the operatioal laws of the TIT2Ns we ca deduce that TIT2W ω ( =( (λ 1 l 1 ω (λ 1 l ω 1 (λ 1 m 1 ω (λ 1 r +1 1 ω +1 ( ω +1 1 (λ1 r +1 1 ω+1 ( ω +1 1 =( +1 (λ1 l 1 ω ( +1 ω 1 +1 (λ1 l 1 ω ( +1 ω 1 ω +1 (λ1 m 1 ( +1 ω 1

8 8 Joural of pplied Mathematics =( +1 (λ1 r 1 ω +1 ( +1 ω 1 +1 (λ1 r 1 ω ( +1 ω 1 (λ 1 l 1 ω (λ 1 l ω 1 (λ 1 m 1 ω (λ 1 r 1 ω (λ 1 r ω 1. (19 That is (15 holds for=+1. Therefore (15 holds for all which completes the proof of Theorem 7. I what follows we ca easily prove that the TIT2W operator has the idempotecy boudary ad other properties that ordiary aggregatio operators usually have. Theorem 8 (idempotecy. Let = (l l m r r ( = 12...be a collectio of TIT2Ns where 0 l l m r r 1.Ifall ( = 12...are equal that is = forallthe TIT2W ω ( =. (20 Proof. By usig Theorem 7wehave TIT2W ω ( =( (λ 1 l 1 ω (λ 1 l 1 ω (λ 1 m 1 ω 1 (λ 1 r 1 ω (λ 1 r 1 ω = ( λ 1 l λ 1 l λ 1 m 1 λ 1 r λ 1 r = (1 (1 l 1 (1 l 1 (1 m 1 (1 r 1 (1 r =(l l m r r =. Thus the proof is completed. (21 Theorem 9 (boudary. Let = (l l m r r ( = be a collectio of TIT2Ns where 0 l l m r r 1adlet The + ={ max {l } max {l } max {m } max {r } max ={ mi {l } mi Proof. Sice mi {r } {r }} {l } mi {r }}. mi {m } (22 TIT2W ω ( (23 mi {l } l max {l } mi {l } l max {l } mi {m } m max {m } for all theitfollowsthat ad the mi {r } r max {r } mi {r } r max {r } (24 1 max {l } 1 l 1 mi {l } (25 (λ 1 max {l } 1 ω (λ 1 l 1 ω (λ 1 mi {l } 1 ω. (26

9 Joural of pplied Mathematics 9 The it follows that (λ 1 max {l } 1 ω (λ 1 l 1 ω (λ 1 mi {l } 1 ω λ 1 max {l } 1 max 1 mi {l } (λ 1 l 1 ω logλ λ 1 mi {l } {l } mi {l } 1 log λ (λ 1 l 1 ω (λ 1 l 1 ω Let TIT2W ω ( = = (l l m r r ; thebyusigtheraigvalueformulaof TIT2N we have Ra ( = ( l + r 2 +1 l + l +r + r +4m 8 max (l +max (r ( 2 +1 (max + max = Ra ( + Ra ( = ( l + r 2 (l +max (l +max (r (r +4max (m ( l + l +r + r +4m 8 mi (l +mi (r ( 2 +1 max {l }. Similarly we have mi {l } 1 log λ max {l } mi {m } 1 log λ max {m } mi {r } 1 log λ max {r } mi {r } 1 log λ max {r }. (λ 1 l 1 ω (λ 1 m 1 ω (λ 1 r 1 ω (λ 1 r 1 ω (27 (28 Therefore (mi (l +mi (l +mi (r +mi = Ra (. (r +4mi (m (8 1 (29 TIT2W ω ( (30 which completes the proof of Theorem 9. Theorem 10. Let = (l l m r r ( = be a collectio of TIT2Ns where 0 l l m r r 1. If +1 = (l +1 l +1 m +1 r +1 r +1 isalsoatit2noxthe TIT2W ω ( = TIT2W ω ( (31

10 10 Joural of pplied Mathematics Proof. Sice +1 i={+1} (λ 1 l i 1 =( i={+1} (λ 1 l i 1 i={+1} (λ1 m i 1 i={+1} (λ 1 r 1 i={+1} (λ 1 r 1 (32 accordig to Theorem 7wehave TIT2 ω ( 1 =( (λ 1 ( (1+ i={+1} (λ1 l i 1/( 1ω (λ 1 ( (1+ i={+1} (λ1 l i 1/( 1ω (λ 1 ( (1+ i={+1} (λ1 m i 1/( 1 ω (λ 1 ( (1+ i={+1} (λ1 r i 1/( 1ω (λ 1 ( (1+ i={+1} (λ1 r i 1/( 1 ω =( (λ1 l 1 ω 1 l (λ +1 1 ω ω (λ1 l (λ 1 1 l ω +1 1 (1+ 1 log λ (1+ ω (λ1 m 1 (λ 1 m +1 1 ω (λ1 r 1 ω 1 r (λ +1 1 ω ω (λ1 r 1 (λ 1 r +1 1 ω (1+ 1 log λ (1+ =( (λ1 l 1 ω 1 l (λ +1 1 ω (λ1 l 1 (λ 1 l log λ

11 Joural of pplied Mathematics 11 ω (λ1 m 1 (λ 1 m +1 1 (λ1 r 1 ω 1 r (λ +1 1 ω (λ1 r 1 (λ 1 r log λ. (33 O the other had based o Theorem 7 ad the operatioal laws of TIT2Ns we have TIT2 ω ( =( (λ 1 l 1 ω 1 logλ (λ 1 m 1 ω (λ 1 r 1 ω 1 logλ (l +1 l +1 m +1 r +1 r +1 (λ 1 l ω 1 (λ 1 r ω 1 =( (λ1 l 1 ω 1 l (λ +1 1 ω (λ1 l 1 (λ 1 l log λ ω (λ1 m 1 (λ 1 m +1 1 (λ1 r 1 ω 1 r (λ +1 1 ω (λ1 r 1 (λ 1 r log λ. (34 Therefore we have TIT2W ω ( Theorem 11. Let = (l l m r r ( = 12...be a collectio of TIT2Ns where 0 l l m r r 1.Ifr>0the = TIT2W ω ( which completes the proof of Theorem 10. (35 TIT2W ω (r 1 r 2...r =r TIT2W ω ( (36

12 12 Joural of pplied Mathematics Proof. Based o the ra operatio laws of TIT2Ns we have r =( (λ 1 l 1 r (λ 1 l r 1 r ( r 1 ( r 1 (λ1 m 1 ( r 1 (λ 1 r 1 r (λ 1 r r 1 ( r 1 ( r 1 TIT2 ω (r 1 r 2...r =( (1+ (λ 1 ( (1+(λ1 l 1 r /( r 1 1ω (λ 1 ( (1+(λ1 m 1 r /( r 1 1ω (λ 1 ( (1+(λ1 r 1 r /( r 1 1ω (λ 1 ( (1+(λ1 l 1 r /( 1ω r 1 (λ 1 ( (1+(λ1 r 1 r /( r 1 1ω =( (λ1 l 1 rω rω (λ1 l 1 rω ( r 1 ( r 1 (λ1 m 1 ( r 1 (λ1 r 1 rω rω (λ1 r 1 ( r 1 ( r 1. (37 lso sice r TIT2W ω ( =r ( (λ 1 l 1 ω 1 logλ (λ 1 m 1 ω (λ 1 r 1 ω 1 logλ (λ 1 l ω 1 (λ 1 r ω 1

13 Joural of pplied Mathematics 13 =( 1 log λ (1+ (λ 1 (1 log (λ1 l r λ (1+ 1 ω 1 ( r 1 (1+ (λ 1 ( (1+ (λ1 m 1 ω r 1 ( r 1 (λ 1 (1 log (λ1 r r λ (1+ 1 ω 1 (1+ ( r 1 (1+ r (λ 1 ( (1+ (λ1 l 1 ω 1 ( r 1 (λ 1 ( (1+ (λ1 r 1 ω r 1 ( r 1 =( (λ1 l 1 rω rω (λ1 l 1 rω (1+ ( r 1 (1+ ( r 1 (1+ (λ1 m 1 ( r 1 (λ1 r 1 rω rω (λ1 r 1 ( r 1 ( r 1 (38 therefore we have TIT2W ω (r 1 r 2...r Thus the proof is completed. =r TIT2W ω ( (39 Theorem 12. Let = (l l m r r ( = be a collectio of TIT2Ns. If r > 0 +1 = (l +1 l +1 m +1 r +1 r +1 is a TIT2N o Xthe TIT2W ω (r 1 +1 r r +1 =r TIT2W ω ( (40 Proof. ccordig to Theorems 10 ad 11wecaeasilyprove Theorem 12. Theorem 13. Let = (l l m r r ( = 12...be a collectio of TIT2Ns ad assume that λ>1. s λ 1 the TIT2 operator approaches the followig limit: lim TIT2W ω ( =( 1 1 (1 l ω 1 (1 l ω (1 m ω 1 (1 r ω. (1 r ω 1 (41 Proof. rom Theorem 7wehave lim TIT2W ω ( =( lim (1 (λ 1 l 1 ω lim (1 (λ 1 l ω 1 lim (1 (λ 1 m ω 1

14 14 Joural of pplied Mathematics lim (1 (λ 1 r 1 ω lim (1 (λ 1 r ω 1. (42 Sice the five referece poits have the same mathematical form so we tae the first referece poit as a example; that is lim (1 (λ 1 l 1 ω =1 (1 l ω. (43 s λ 1 the (λ1 l 1 ω 0 by logarithmic trasform ad the rule of equivalet ifiitesimal replacemet: (λ 1 l 1 ω l (λ1 l 1 ω (λ1 l 1 ω =. l λ l λ (44 BasedoTaylor sexpasioformulawehave λ 1 l (1 l =1+(1 l l λ+ 2 (l λ 2 +. (45 lso sice λ>1thelλ>0 λ 1 l =1+(1 l l λ+ O(l λaditfollowsthat Therefore λ 1 l 1 (1 l l λ. (46 (λ 1 l 1 ω (1 l ω l λ (λ1 l 1 ω l λ Thewehave (1 l ω. lim (1 (λ 1 l 1 ω = lim (1 l (λ1 l 1 ω l λ (λ1 l 1 ω =1 lim l λ =1 Similarly we have (1 l ω. lim (1 (λ 1 l ω 1 =1 (1 l ω lim (1 (λ 1 m ω 1 =1 (1 m ω lim (1 (λ 1 r 1 ω =1 (1 r ω lim (1 (λ 1 r ω 1 (47 (48 (λ 1 l 1 ω ((1 l l λ ω (λ 1 l 1 ω (1 l ω (l λ ω (λ 1 l 1 ω (1 l ω (l λ ω (λ 1 l 1 ω (1 l ω (l λ ω =1 (1 r ω which completes the proof of Theorem 13. (49 Theorem 14. Let = (l l m r r ( = be a collectio of TIT2Ns ad assume that

15 Joural of pplied Mathematics 15 λ > 1. s λ + the TIT2W operator approaches the followig limit: lim TIT2W ω ( λ =( ω l ω l ω m ω r Proof. By Theorem 7wehave lim TIT2W ω ( ω r. =( lim (1 (λ 1 l 1 ω lim (1 (λ 1 l ω 1 lim (1 (λ 1 m ω 1 lim (1 (λ 1 r 1 ω (50 lim (1 (λ 1 r ω 1. (51 Sice the five referece poits have the same mathematical form so we tae the first referece poit as a example; that is lim (1 (λ 1 l 1 ω = ω l. λ + s λ + by logarithmic trasform we have (52 Based o L Hospital s rule it follows that lim (1 (λ 1 l 1 ω λ + l (λ1 l 1 ω =1 lim λ + l λ =1 lim λ + ( =1 lim = =1 λ + ω l. Similarly we have (λ1 l 1 ω 1+ (λ1 l 1 ω ( λ l ω (1 l ( 1 1 λ 1 l 1 λ (λ1 l 1 ω 1+ (λ1 l 1 ω ( ω (1 l λ1 l ω (1 l λ 1 l 1 lim (1 (λ 1 l ω 1 = ω l λ lim (1 (λ 1 m ω 1 = ω m λ lim (1 (λ 1 r 1 ω = ω r λ lim (1 (λ 1 r ω 1 = λ which completes the proof of Theorem 14. ω r (54 (55 lim (1 (λ 1 l 1 ω λ + l (λ1 l 1 ω =1 lim λ + l λ. (53 Theorem 15. or give argumets ( = 12...ad λ (1+ the TIT2W operator is mootoically decreasig with respect to λ. Proof. I order to verify that the TIT2W operator is mootoically decreasig with respect to λ we should oly prove that every referece poit fuctio is mootoically

16 16 Joural of pplied Mathematics decreasig with respect to λ. Due to the space limit we tae the first referece poit fuctio as a example. Let T l (λ = 1 (λ1 l 1 ω adtaig the derivative of T l (λ with respect to λweobtai dt l (λ dλ (λ1 l 1 w ( (w (1 l /(λ λ l = (λ1 l 1 w. l λ Sice λ>1 0 l 1wecaeasilyprovethat (56 dt l (λ <0. (57 dλ Hece T l (λ is mootoically decreasig with respect to λ which implies that TIT2W operator is mootoically decreasig with respect to λ. Thustheproofiscompleted. Theorem 16. Let = (l l m r r ( = be a collectio of TIT2Ns for ay λ (1+ ; the sup {TIT2W ω ( } 4.2. Triagular Iterval Type-2 uzzy ra Weighted Geometric Operator Defiitio 17. Let = (l l m r r ( = be a collectio of TIT2Ns ad let TIT2WG: Ω Ω;if TIT2WG ω ( = ω 1 1 ω 2 2 ω (59 the the fuctio TIT2WG is called a triagular iterval type-2 fuzzy ra weighted geometric (TIT2WG operator where ω = (ω 1 ω 2...ω T is the weight vector of ( = 12...ω 0ad ω =1.Iparticular if ω = (1/1/...1/ T the the TIT2WG operator is reduced to a triagular iterval type-2 fuzzy ra geometric averagig (TIT2G operator of dimesio whichis defied as follows: TIT2G ω ( =( 1 2 1/. (60 Based o the ra operatio laws from Theorem 5 we ca derive Theorem 18. Theorem 18. Let = (l l m r r ( = beacollectiooftit2ns;thetheiraggregated value by TIT2WG operator is still a TIT2N ad TIT2WG ω ( =( (1 l ω 1 (1 l ω (1 m ω (1 r ω 1 (1 r ω if {TIT2W ω ( } =( ω l ω l ω m ω r ω r. Proof. BasedoTheorems13 15theresultisobvious. (58 =( (λ l 1 ω (λ l ω 1 (λ m 1 ω (λ r 1 ω (λ r ω 1 (61 where ω=(ω 1 ω 2...ω T is the weight vector of ( = ω >0ad ω =1. Proof. The proof of Theorem 18 is similar to Theorem 7.

17 Joural of pplied Mathematics 17 Theorem 19 (idempotecy. Let = (l l m r r ( = 12...be a collectio of TIT2Ns where 0 l l m r r 1.Ifall (2... are equal that is = forallthe Proof. The proof of Theorem 22 is similar to Theorem 11. Theorem 23. Let = (l l m r r ( = be a collectio of TIT2Ns. If r > 0 +1 = (l +1 l +1 m +1 r +1 r +1 is a TIT2N o Xthe TIT2WG ω ( =. (62 Proof. The proof of Theorem 19 is similar to Theorem 8. TIT2WG ω ( r 1 +1 r r +1 Theorem 20 (boudary. Let = (l l m r r ( = 12...be a collectio of TIT2Ns where 0 l l m r r 1adlet The + =( max {l } max {l } max {m } max {r } max {r } =( mi {l } mi {l } mi {m } mi {r } mi {r }. (63 TIT2WG ω ( (64 Proof. The proof of Theorem 20 is similar to Theorem 9. Theorem 21. Let = (l l m r r ( = be a collectio of TIT2Ns where 0 l l m r r 1. If +1 = (l +1 l +1 m +1 r +1 r +1 is a triagular iterval type- 2 fuzzy umber o Xthe TIT2WG ω ( = TIT2WG ω ( Proof. The proof of Theorem 21 is similar to Theorem 10. (65 Theorem 22. Let = (l l m r r ( = 12...be a collectio of TIT2Ns where 0 l l m r r 1.Ifr>0the TIT2WG ω ( r 1 r 2... r (66 =(TIT2WG ω ( r. =(TIT2WG ω ( r +1. Proof. The proof of Theorem 23 is similar to Theorem 12. (67 Theorem 24. Let = (l l m r r ( = 12...be a collectio of TIT2Ns ad assume that λ>1. s the TIT2WG operator approaches the followig limit: lim TIT2WG ω ( =( (l ω (l ω (m ω (r ω (r ω. Proof. The proof of Theorem 24 is similar to Theorem 13. (68 Theorem 25. Let = (l l m r r ( = be a collectio of TIT2Ns ad assume that λ> 1. sλ + the TIT2WG operator approaches the followig limit: lim TIT2WG ω ( λ =( ω l ω l ω m ω r ω r. Proof. The proof of Theorem 25 is similar to Theorem 14. (69 Theorem 26. or give argumets ( = 12...ad λ (1+ the TIT2WG operator is mootoically icreasig with respect to λ. Proof. The proof of Theorem 26 is similar to Theorem 15.

18 18 Joural of pplied Mathematics Theorem 27. Let = (l l m r r ( = be a collectio of TIT2Ns for ay λ (1+ ; the sup {TIT2WG ω ( } =( ω l ω r ω l ω r ω m if {TIT2WG ω ( } ω =( l ω l ω l ω l. m ω Proof. TheproofofTheorem27 is similar to Theorem 16. ( The Relatioship betwee the TIT2W Operator ad TIT2WG Operator Theorem 28. Let = (l l m r r ( = 12...be a collectio of TIT2Ns ad λ (1+ ;the TIT2W ω ( TIT2WG ω ( Proof. BasedoTheorem12wehave The it follows that if {TIT2W ω ( } =( ω l ω r ω l ω r. TIT2W ω ( ( ω l ω r ω l ω r. ω m ω m (71 (72 (73 rom Theorem 27wehave The sup {TIT2WG ω ( } =( ω l ω r ω l ω r. TIT2WG ω ( ( ω l ω r ω l ω r. ω m ω m (74 (75 Therefore accordig to the trasfer of iequality we ca easily deduce that TIT2W ω ( The proof is completed. TIT2WG ω ( (76 Theorem 28 shows that the values obtaied by the TIT2WG operator are ot bigger tha the oes obtaied by the TIT2W operator for ay λ>1. 5. pproach to Multiple ttribute Decisio Maig uder the Triagular Iterval Type-2 uzzy Eviromet I this sectio we ivestigate the group decisio-maig problems uder the triagular iterval type-2 fuzzy evirometbasedotheproposedraaggregatiooperatorsi Sectio 4. irst we describe the MGDM problems with triagular iterval type-2 fuzzy iformatio i this paper. Let = ( m be the set of alteratives let D = (D 1 D 2...D p be the set of decisio maers (DMs ad let e = (e 1 e 2...e p be the weight vector of DMs where e 0 = 12...pad p e = 1.Let C = (C 1 C 2...C be a set of attributes ad let ω = (ω 1 ω 2...ω be a set of weight vectors of them satisfyig ω 0 1 ad ω = 1.Let ( = ( a ( i m be a triagular iterval type-2 fuzzy decisio matrix where a ( i is measured by TIT2Ns which is provided by the decisio maer D for the alterative i with respect to attribute C. I the followig we will use the TIT2W (or TIT2WG operator to develop a approach to MGDM with triagular iterval type-2 fuzzy iformatio which ivolves the followig steps.

19 Joural of pplied Mathematics 19 Step 1. Normalize the decisio-maig iformatio matrix ( =( a ( i m. I geeral attributes are divided ito two types; oe is beefit attribute (the bigger the better ad the other is cost attribute (the smaller the better; i order to maitai the cosistecy of the attribute values we should trasform the decisio matrix ( = ( a ( i m ito the ormalizatio matrix R ( =( r ( i m uless all the attributes are the same type. I this paper we use the followig formula to obtai the ormalized decisio matrix R ( =( r ( i m : r ( i ={ a( i for beefit attribute C ( a ( i c for cost attribute C where ( a ( i c is the complemet of a ( i. Step 2. Utilize the TIT2W operator r i = TIT2W ( r (1 i r(2 i... r(p i i=12...m or the TIT2WG operator 2... r i = TIT2WG ( r (1 i r(2 i... r(p i i=12...m 2... (77 (78 (79 to aggregate all the idividual triagular iterval type-2 fuzzy decisio matrix R ( =( r ( i m ito the collective triagular iterval type-2 fuzzy decisio matrix R =( r i m. Step 3. ggregate the triagular iterval type-2 fuzzy values r i for each alterative i (i = 12...m by usig the TIT2W (or TIT2WG operator: or r i = TIT2W ( r i1 r i2... r i i=12...m (80 r i = TIT2WG ( r i1 r i2... r i i=12...m. (81 Step 4. Calculate the score values S( r i (i=12...mof the overall preferece values r i (i=12...m. Step 5. Ra all the alteratives i (i=12...mad select the best oe(s i accordace with S( r i.thegreaterthescore values S( r i are the better the alteratives i (i=12...m will be. Step 6. Ed. 6. Illustrative Example I this sectio a illustrative example for MGDM is provided as a demostratio of the applicatio based o the proposed triagular iterval type-2 fuzzy ra aggregatio operator i real-world practice situatios. Let us cosider a high-tech compay which aims to select a supplier of USB coectors (adapted from 42. There are four potetial suppliers i (i = 1234 that have bee desigated for further evaluatio ad assume that there are four attributes to be cosidered i the evaluatio process: (1 C 1 : fiacial; (2 C 2 : performace; (3 C 3 : techical capacity; ad (4 C 4 : orgaizatioal culture ad strategy ad ω = ( T is the weight vector of them. The decisio committee cotais three decisio maers D ( = icludig egieerig expert fiacial expert ad quality cotrol expert whose weight vector is e = ( T.ThedecisiomaersD ( = express the attribute values of the potetial supplier i (i = with respect to C i (i = 1234 by TIT2Ns respectivelywhicharelisteditables1 2ad3. Based o the TIT2W operator the decisio-maig steps are show as follows. Step 1. Cosider that all the attributes are beefit type; therefore the decisio matrices do ot eed ormalizatio; therefore R ( = ( =( a ( i 4 4 =( r ( i 4 4. Step 2. Utilize the TIT2W operator to aggregate all the idividual triagular iterval type-2 fuzzy decisio matrix R ( =( r ( i 4 4 itothecollectivetriagularitervaltype-2 fuzzy decisio matrix R = ( r i 4 4 adwithoutthelossof geerality we let λ=2; theresultisshowitable4. Step 3. Utilize the TIT2W operator to aggregate all the preferece values r i ( = 1234 i the ith lie of R ad the we ca get the overall preferece values r i (i=1234 as follows: r 1 = ( r 2 = ( r 3 = ( r 4 = ( (82 Step 4. CalculatethescorevaluesS( r i (i = of the overall preferece values r i (i=1234respectively: S( r 1 = S ( r 2 = S( r 3 = S ( r 4 = (83 Step 5. Ra all the alteratives i (i = 1234ad select the best oe(s i accordace with S( r i. Sice therefore we have S ( r 2 >S( r 1 >S( r 3 >S( r 4 ( (85 where the symbol meas superiorto. Thusthebest supplier is 2.

20 20 Joural of pplied Mathematics Table 1: Triagular iterval type-2 fuzzy decisio matrix R (1. C 1 C 2 C 3 C 4 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 2: Triagular iterval type-2 fuzzy decisio matrix R (2. C 1 C 2 C 3 C 4 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 3: Triagular iterval type-2 fuzzy decisio matrix R (3. C 1 C 2 C 3 C 4 1 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 4: Triagular iterval type-2 fuzzy decisio matrix R. C 1 C 2 C 3 C 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 5: Triagular iterval type-2 fuzzy decisio matrix R. C 1 C 2 C 3 C 4 ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Table 6: Score values obtaied by the TIT2W operator ad the raigs of alteratives Raig λ= λ= λ= λ= λ

21 Joural of pplied Mathematics 21 Table 7: Score values obtaied by the TIT2WG operator ad the raigs of alteratives Raig λ= λ= λ= λ= λ Based o the TIT2WG operator i order to select the most desirable supplier we ca also develop a method for hadlig MGDM problems with triagular iterval type-2 fuzzy iformatio which ca be described as follows. Step 1.ItisthesameasStep1. Step 2.UtilizetheTIT2WGoperatortoaggregateallthe idividual triagular iterval type-2 fuzzy decisio matrix R ( =( r ( i 4 4 itothecollectivetriagularitervaltype-2 fuzzy decisio matrix R = ( r i 4 4 adwithoutthelossof geerality we let λ=2; theresultisshow i Table5. Step 3.UtilizetheTIT2WGoperatortoaggregateallthe preferece values r i ( = 1234 i the ith lie of R ad the we ca get the overall preferece values r i (i=1234 as follows: r 1 = ( r 2 = ( r 3 = ( r 4 = ( (86 Step 4.CalculatethescorevaluesS( r i (i = of the overall preferece values r i (i=1234respectively: S( r 1 = S ( r 2 = S( r 3 = S ( r 4 = (87 Step 5.Raallthealteratives i (i = 1234ad select the best oe(s i accordace with S( r i. Sice therefore we have S ( r 2 >S( r 1 >S( r 3 >S( r 4 ( (89 where the symbol meas superiorto. Thusthebest supplier is 2. It is clear to see that they are the same raig results for two methods based o TIT2W operator ad the TIT2WG operator. I order to reflect the ifluece with the Table 8: Comparisos with type-2 OW ad type-2 W. ggregatio operator Type-2 OW operator Type-2 W operator The proposed operators Computatioal complexity lexible Order of alteratives High Noe Very high Noe Low High differet value of parameter λwe use differet parameter λ i our proposed methods to ra the alteratives. The raig results are show i Tables 6 ad 7. s we ca see from Tables 6 ad 7theaggregatioscore values based o TIT2W ad TIT2WG operators with differet parameter γ are differet but the raig orders of the alteratives are the same i this example. The decisio maers ca choose the appropriate value i accordace with their prefereces. By further aalyzig Tables 6 ad 7 we ca easily see that the score values obtaied by the TIT2W operator became smaller as the parameter λ icreases for the same aggregatio elemets but the TIT2WG operator became greater as the parameter λ icreases for the same aggregatio elemets. urthermore the values obtaied by the TIT2W operator are always larger tha the oes obtaied by the TIT2WG operator for the same value of the parameter λ ad the same aggregatio elemets. I order to verify the validity of our methods we mae a compariso betwee our proposed operators with the type-2 OW operator 46 ad the type-2 fuzzy weighted averagig operator 62 for solvig multiple attribute group decisio maig with iterval type-2 fuzzy iformatio. Due to the space limit we give the results directly. The raig results are show i Table 8. rom Table 8 we ca see that our methods ca obtai thesameraigresultswiththetype-2owoperator proposed by Zhou et al. 46 ad type-2 W operator proposed by Liu ad Wag 62 i this example. This fact verifies that the TIT2W ad TIT2WG operators we proposed are reasoable ad valid for iterval type-2 fuzzy decisio-maig problems. Compared with the type-2 OW operator ad the type-2 W operator the mai advatage of TIT2W ad TIT2WG operators we proposed is that it ca reduce the cost of computatioal complexity ad ehace the reliability of the aggregatio result uder fuzzy eviromet. urthermore our proposed methods iclude

22 22 Joural of pplied Mathematics a parameter which ca adust the aggregate value based o the real decisio eeds ad reflect the decisio maer s ris prefereces. Therefore the beefit is that the proposed operators are more geeral ad more flexible. ccordig to the comparisos ad aalysis above our proposed aggregatio operators are better tha the other two methods. 7. Coclusios This paper puts forward a ew method to solve MGDM with triagle iterval type-2 fuzzy iformatio. We have defied ra operatio laws of iterval type-2 fuzzy set ad by usig ra triagular orms some ew aggregatio operators icludig the TIT2W ad TIT2WG operators are developed. The properties of these operators have bee discussed i detail. Moreover we have applied the TIT2W ad TIT2WG operators for hadlig MGDM problems uder the triagle iterval type-2 fuzzy eviromet. ially a illustrative example about supplier selectio has bee provided to demostrate how to apply the proposed procedure. I further research we will cotiue our worig ad exted the developed aggregatio operators to the fields of data miig 63 techology evaluatio 64 computig with words 65 ad cluster aalysis 66. Coflict of Iterests The authors declare that there is o coflict of iterests regardig the publicatio of this paper. cowledgmets The wor is supported by the Natioal Natural Sciece oudatio of Chia (NSC uder Proects ad Ph.D. Program oudatio of Chiese Miistry of Educatio ad the Scietific Research ad Iovatio Proect for College Graduates of Jiagsu Provice CXZZ Refereces 1 D. Li ad J. Yag uzzy liear programmig techique for multiattribute group decisio maig i fuzzy eviromets Iformatio Scieces vol. 158 pp Y. J. Wag ad H. S. Lee Geeralizig TOPSIS for fuzzy multiple-criteria group decisio-maig Computers ad Mathematics with pplicatios vol.53o.11pp Z. Xu ad J. Che iteractive method for fuzzy multiple attribute group decisio maig Iformatio Scieces vol. 177 o. 1 pp C.J.LiadW.W.Wu causalaalyticalmethodforgroup decisio-maig uder fuzzy eviromet Expert Systems with pplicatiosvol.34o.1pp D. Guha ad D. Charaborty uzzy multi attribute group decisio maig method to achieve cosesus uder the cosideratio of degrees of cofidece of experts opiios Computers ad Idustrial Egieerig vol.60o.4pp Hatami-Marbii ad M. Tavaa extesio of the Electre I method for group decisio-maig uder a fuzzy eviromet Omegavol.39 o.4pp S.M.Mousavi.JolaiadR.Tavaoli-Moghaddam uzzy stochastic multi-attribute group decisio-maig approach for selectio problems Group Decisio ad Negotiatiovol.22o. 2 pp X. Zhao R. Li ad G. Wei uzzy prioritized operators ad their applicatio to multiple attribute group decisio maig pplied Mathematical Modellig vol.37o.7pp L.. Zadeh The cocept of a liguistic variable ad its applicatio to approximate reasoig-i Iformatio Scieces vol. 8 o. 3 pp J. M. Medel Computig with words ad its relatioships with fuzzistics Iformatio Scieces vol. 177 o. 4 pp I. B. Turse Type 2 represetatio ad reasoig for CWW uzzy Sets ad Systemsvol.127o.1pp N. N. Kari ad J. M. Medel Cetroid of a type-2 fuzzy set Iformatio Sciecesvol.132o.1 4pp N. N. Kari ad J. M. Medel Operatios o type-2 fuzzy sets uzzy Sets ad Systemsvol.122o.2pp Liu efficiet cetroid type-reductio strategy for geeral type-2 fuzzy logic system Iformatio Sciecesvol.178o.9 pp J. M. Medel Type-2 fuzzy sets ad systems: a overview IEEE Computatioal Itelligece Magazievol.2o.1pp D. Wu ad J. M. Medel Ehaced Kari-Medel algorithms IEEE Trasactios o uzzy Systems vol.17o.4pp D. Y. Zhai ad J. M. Medel Ehaced cetroid-flow algorithm for computig the cetroid of geeral type-2 fuzzy sets IEEE Trasactios o uzzy Systemsvol.20o.5pp J.M.Medel.LiuadD.Zhai α-plae represetatio for type-2 fuzzy sets: theory ad applicatios IEEE Trasactios o uzzy Systemsvol.17o.5pp C.HwagM.YagadW.Hug Osimilarityiclusio measure ad etropy betwee type-2 fuzzy sets Iteratioal Joural of Ucertaity uzziess ad Kowledge-Based Systems vol.20o.3pp H. B. Mitchell Patter recogitio usig type-ii fuzzy sets Iformatio Sciecesvol.170o.2 4pp H. R. Tizhoosh Image thresholdig usig type II fuzzy sets Patter Recogitiovol.38o.12pp J. isbett J. T. Ricard ad D. Morgethaler Multivariate modelig ad type-2 fuzzy sets uzzy Sets ad Systems vol. 163pp J.M.MedelR.I.Johad.Liu Itervaltype-2fuzzylogic systems made simple IEEE Trasactios o uzzy Systemsvol. 14o.6pp H. Zhou ad H. Yig method for derivig the aalytical structure of a broad class of typical iterval type-2 mamdai fuzzy cotrollers IEEE Trasactios o uzzy Systems vol.21 o.3pp

23 Joural of pplied Mathematics bbadi L. Nezli ad D. Bouhetala oliear voltage cotroller based o iterval type 2 fuzzy logic cotrol system for multimachie power systems Iteratioal Joural of Electrical Power ad Eergy Systemsvol.45o.1pp Y. Maldoado O. Castillo ad P. Meli Particle swarm optimizatio of iterval type-2 fuzzy systems for PG applicatios pplied Soft Computig Jouralvol.13o.1pp S.T.Wag.L.ChugY.Y.LiD.W.HuadX.S.Wu ew gaussia oise filter based o iterval type-2 fuzzy logic systems Soft Computig vol. 9 o. 5 pp B. Choi ad. Chug-Hoo Rhee Iterval type-2 fuzzy membership fuctio geeratio methods for patter recogitio Iformatio Sciecesvol.179o.13pp D. ay O. Kula ad B. Heso Coceptual desig evaluatio usig iterval type-2 fuzzy iformatio axiom Computers i Idustry vol. 62 o. 2 pp J. Hwag H. Kwa ad G. Par daptive iterval type-2 fuzzy slidig mode cotrol for uow chaotic system Noliear Dyamicsvol.63o.3pp S. Che ad L. Lee uzzy multiple criteria hierarchical group decisio-maig based o iterval type-2 fuzzy sets IEEE Trasactios o Systems Ma ad Cyberetics : Systems ad Humasvol.40o.5pp S. M. Che ad L. W. Lee uzzy multiple attributes group decisio-maig based o the iterval type-2 TOPSIS method Expert Systems with pplicatios vol.37o.4pp S.M.CheM.W.YagL.W.LeeadS.W.Yag uzzy multiple attributes group decisio-maig based o raig iterval type-2 fuzzy sets Expert Systems with pplicatiosvol. 39 o. 5 pp J.H.HuY.ZhagX.H.CheadY.M.Liu Multi-criteria decisio maig method based o possibility degree of iterval type-2 fuzzy umber Kowledge-Based Systemsvol.43pp W. Wag X. Liu ad Y. Qi Multi-attribute group decisio maig models uder iterval type-2 fuzzy eviromet Kowledge-Based Systemsvol.30pp T. Che siged-distace-based approach to importace assessmet ad multi-criteria group decisio aalysis based o iterval type-2 fuzzy set Kowledge ad Iformatio Systems vol.35o.1pp T. Che C. Chag ad J. R. 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24 24 Joural of pplied Mathematics 60 P. Saroci Domiatio i the families of ra ad Hamacher t-orms Kyberetia vol. 41 o. 3 pp W. Wag ad H. He Research o flexible probability logic operator based o ra T/S orms cta Electroica Siica vol.37o.5pp X. Liu ad Y. Wag aalytical solutio method for the geeralized fuzzy weighted average problem Iteratioal Joural of Ucertaity uzziess ad Kowledge-Based Systems vol.21o.3pp C. Qiu J. Xiao L. Yu L. Ha ad M. N. Iqbal modified iterval type-2 fuzzy C-meas algorithm with applicatio i MR image segmetatio Patter Recogitio Letters vol. 34 o. 12 pp T. Dereli ad K. ltu Techology evaluatio through the use of iterval type-2 fuzzy sets ad systems Computers ad Idustrial Egieerigvol.65o.4pp D. Wu ad J. M. Medel Computig with words for hierarchical decisio maig applied to evaluatig a weapo system IEEE Trasactios o uzzy Systemsvol.18o.3pp H. Le Capitaie ad C. rélicot cluster-validity idex combiig a overlap measure ad a separatio measure based o fuzzy-aggregatio operators IEEE Trasactios o uzzy Systems vol. 19 o. 3 pp

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