Research Article Generalized Fuzzy Bonferroni Harmonic Mean Operators and Their Applications in Group Decision Making

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1 Joural of Applied Mathematics Volume 203 Article ID pages Research Article Geeralized Fuzzy Boferroi Harmoic Mea Operators ad Their Applicatios i Group Decisio Makig Ji Ha Park ad Eu Ji Park Departmet of Applied Mathematics Pukyog Natioal Uiversity Busa Republic of Korea Correspodece should be addressed to Ji Ha Park; ihpark@pku.ac.kr Received 9 May 203; Accepted 7 August 203 Academic Editor: Deg-Feg Li Copyright 203 J. H. Park ad E. J. Park. This is a ope access article distributed uder the Creative Commos Attributio Licese which permits urestricted use distributio ad reproductio i ay medium provided the origial work is properly cited. The Boferroi mea BM) operator is a importat aggregatio techique which reflects the correlatios of aggregated argumets. Based o the BM ad harmoic mea operators H. Su ad M. Su 202) developed the fuzzy Boferroi harmoic mea FBHM) ad fuzzy ordered Boferroi harmoic mea FOBHM) operators. I this paper we study desirable properties of these operators ad exted them by cosiderig the correlatios of ay three aggregated argumets istead of ay two to develop geeralized fuzzy weighted Boferroi harmoic mea GFWBHM) operator ad geeralized fuzzy ordered weighted Boferroi harmoic mea GFOWBHM) operator. I particular all these operators ca be reduced to aggregate iterval or real umbers. The based o the GFWBHM ad GFOWBHM operators we preset a approach to multiple attribute group decisio makig ad illustrate it with a practical example.. Itroductio Multiple attribute group decisio makig MAGDM) is the commo pheomeo i moder life which is to select the optimal alteratives) from several alteratives or to get their rakig by aggregatig the performaces of each alterative uder several attributes i which the aggregatio techiques play a importat role. Cosiderig the relatioships amog theaggregatedargumetswecaclassifytheaggregatio techiques ito two categories: the oes which cosider the aggregated argumets depedetly ad the others which cosider the aggregated argumets idepedetly. For the first category the well-kow ordered weighted averagig OWA) operator 2 is the represetative o the basis of which a lot of geeralizatios have bee developed such as the ordered weighted geometric OWG) operator 3 5 the ordered weighted harmoic mea OWHM) operator 6 the cotiuous ordered weighted averagig C-OWA) operator 7 the cotiuous ordered weighted geometric C-OWG) operator 8. The secod category ca reduce to two subcategories: the first subcategory focuses o chagig the weight vector of the aggregatio operators such as the Choquet itegral-based aggregatio operators 9 i which the correlatios of the aggregated argumets are measured subectively by the decisio makers ad the represetatives of aother subcategory are the power averagig PA) operator 0 ad the power geometric PG) operator both of whichallowtheaggregatedargumetstosupporteachother i aggregatio process o the basis of which the weighted vector is determied. The secod subcategory focuses o the aggregated argumets such as the Boferroi mea BM) operator 2. Yager 3 provided a iterpretatio of BM operator as ivolvig a product of each argumet with the average of the other argumets a combied averagig ad adig operator. Beliakov et al. 4 presetedacomposed aggregatio techique called the geeralized Boferroi mea GBM) operator which models the average of the couctive expressios ad the average of remaiig. I fact they exteded the BM operator by cosiderig the correlatios of ay three aggregated argumets istead of aytwo.howeverbothbmoperatoradthegbmoperator igore some aggregatio iformatio ad the weight vector of the aggregated argumets. To overcome this drawback Xia et al. 5 developed the geeralized weighted Boferroi mea GWBM) operator as the weighted versio of the GBM operator. Based o the GBM operator ad geometric

2 2 Joural of Applied Mathematics mea operator they also developed the geeralized Boferoi geometric mea GWBGM) operator. The fudametal characteristic of the GWBM operator is that it focuses o the group opiios while the GWBGM operator gives more importace to the idividual opiios. Because of the usefuless of the aggregatio techiques which reflect the correlatios of argumets most of them have bee exteded to fuzzy ituitioistic fuzzy or hesitat fuzzy eviromet Harmoic mea is the reciprocal of arithmetic mea of reciprocal which is a coservative average to be used to provide for aggregatio lyig betwee the max ad mi operatorsadiswidelyusedasatooltoaggregatecetral tedecy data 2. I the existig literature the harmoic mea is geerally cosidered as a fusio techique of umerical data iformatio. However i may situatios the iput argumets take the form of triagular fuzzy umbers because of time pressure lack of kowledge ad people s limited expertise related with problem domai. Therefore how to aggregate fuzzy data by usig the harmoic mea? is a iterestig research topic ad is worth payig attetio to. So Xu 2 developed the fuzzy harmoic mea operators such as fuzzy weighted harmoic mea FWHM) operator fuzzy ordered weighted harmoic mea FOWHM) operator ad fuzzy hybrid harmoic mea FHHM) operator ad applied them to MAGDM. Wei 22 developed fuzzy iduced ordered weighted harmoic mea FIOWHM) operator ad the based o the FWHM ad FIOWHM operators preseted theapproachtomagdm.h.suadm.su23 further applied the BM operator to fuzzy eviromet itroduced the fuzzy Boferroi harmoic mea FBHM) operator ad the fuzzy ordered Boferroi harmoic mea FOBHM) operator ad applied the FOBHM operator to multiple attribute decisio makig. I this paper we will develop some ew harmoic aggregatio operators icludig the geeralized fuzzy weighted Boferroi harmoic mea GFWBHM) operator ad geeralized fuzzy ordered weighted Boferroi harmoic mea GFOWBHM) operator ad apply them to MAGDM. I order to do this the remaider of this paper is arraged i followig sectios. Sectio 2 first reviews some aggregatio operators icludig the BM GBM ad GWBM operators. The some basic cocepts related to triagular fuzzy umbers ad some operatioal laws of triagular fuzzy umbers are itroduced. The desirable properties of the FBHM ad FOBHM operators are discussed. We exted them by cosiderig the correlatios of ay three aggregated argumets istead of ay two to develop geeralized fuzzy weighted Boferroi harmoic mea GFWBHM) operator ad geeralized fuzzy ordered weighted Boferroi harmoic mea GFOWBHM) operator. I particular all these operators ca be reduced to aggregate iterval or real umbers. Sectio 3 presets a approach to MAGDM based o the GFWBHM ad GFOWBHM operators. Sectio4 illustrates the preseted approach with a practical example verifies ad shows the advatages of the preseted approach ad makes a comparative study to the existig approaches. Sectio5 eds the paper with some cocludig remarks. 2. Geeralized Fuzzy Boferroi Harmoic Mea Operators The Boferroi mea operator was iitially proposed by Boferroi 2 ad was also ivestigated itesively by Yager 3. Defiitio. Let p q 0 ad let a i i 2...) be a collectio of oegative umbers. If BM pq a a 2...a ) a p i ) aq ) i i /p+q) the BM pq is called the Boferroi mea BM) operator. ) Beliakov et al. 4 further exteded the BM operator by cosiderig the correlatios of ay three aggregated argumets istead of ay two. Defiitio 2. Let p q r 0 ad let a i i 2...) be a collectio of oegative umbers. If GBM pqr a a 2...a ) ) 2) ik i k a p i aq ar k ) /p+q+r) the GBM pqr is called the geeralized Boferroi mea GBM) operator. I particular if r0 the the GBM operator reduces to the BM operator. However it is oted that both BM operator ad the GBM operator do ot cosider the situatio that i or k or i kadtheweightvectorofthe aggregated argumets is ot also cosidered. To overcome this drawback Xia et al. 5 defied the weighted versio of the GBM operator. Defiitio 3. Let p q r 0 ad let a i i 2...) be a collectio of oegative umbers with the weight vector ww w 2...w ) T such that w i >0 i 2... ad i w i.if GWBM pqr a a 2...a ) ik w i w w k a p i aq ar k ) /p+q+r) the GWBM pqr is called the geeralized weighted Boferroi mea GWBM) operator. Some special cases ca be obtaied as the chage of the parameters as follows. 2) 3)

3 Joural of Applied Mathematics 3 ) If r 0 the the GWBM operator reduces to the followig: GWBM pq0 a a 2...a ) ik i i w i w w k a p i aq ) /p+q) w i w a p i aq k w k ) w i w a p i aq ) /p+q) /p+q) which is the weighted Boferroi mea WBM) operator. 2) If q0ad r0 the the GWBM operator reduces to the followig: GWBM p00 a a 2...a ) ik i i w i a p i w i w w k a p i ) /p w k w i a p i ) /p w k ) which is the geeralized weighted averagig operator. Furthermore i this case let us look at the GWBM operator for some special cases of p. ) If p the GWBM operator reduces to the weighted averagig WA) operator. 2) If p 0 the the GWBM operator reduces to the weighted geometric WG) operator. 3) If p + the the GWBM operator reduces to the max operator. The previous aggregatio techiques ca oly deal with the situatio that the argumets are represeted by the exact umerical values but are ivalid if the aggregatio iformatio is give i other forms such as triagular fuzzy umber 24 which is a widely used tool to deal with ucertaity ad fuzziess described as follows. Defiitio 4 see 24). A triagular fuzzy umber a ca be defied by a triplet a L a M a U.Themembershipfuctio μ a x) is defied as 0 x<a L ; /p x a L { μ a x) a M a L al x a M ; x a U a M a U am x a U ; { { 0 x>a U 4) 5) 6) where a U a M a L 0 a L ada U stad for the lower ad upper values of a respectivelyada M stads for the modal value 24. I particular if ay two of a L a M ad a U are equal the a reduces to a iterval umber; if all a L a M ada U areequalthe a reduces to a real umber. For coveiece we let Ω be the set of all triagular fuzzy umbers. Let a a L a M a U ad b b L b M b U be two triagular fuzzy umbers the some operatioal laws defied as follows 24: ) a + b a L a M a U +b L b M b U a L +b L a M + b M a U +b U ; 2) λ a λa L a M a U λa L λa M λa U ; 3) a b a L a M a U b L b M b U a L b L a M b M a U b U ; 4) / a /a L a M a U /a U /a M /a L. I order to compare two triagular fuzzy umbers Xu 2 provided the followig defiitio. Defiitio 5. Let a a L a M a U ad let b b L b M b U be two triagular fuzzy umbers; the the degree of possibility of a b is defied as follows: p a b) b M a L δmax { max a M a L +b M 0)0} bl b U a M + δ) max { max a U a M +b U 0)0} bm which satisfies the followig properties: 0 p a b) p a a) 0.5 p a b) + p b a). δ 0 7) Here δ reflects the decisio maker s risk-bearig attitude. If δ > 0.5 the the decisio maker is risk lover; if δ0.5 the the decisio maker is eutral to risk; if δ<0.5 the the decisio maker is risk avertor. I the followig we will give a simple procedure for rakig of the triagular fuzzy umbers a i i 2...). First by usig 7) we compare each a i with all the a 2...); for simplicity let p i p a i a )adthewe develop a possibility matrix 25 26as 8) p p 2 p p 2 p 22 p 2 P ) 9). p p 2 p where p i 0 p i +p i p ii /2 i2...

4 4 Joural of Applied Mathematics Summig all elemets i each lie of matrix P wehave p i p i i 2... The i accordace with the values of p i i 2...)werakthe a i i 2...)i descedig order. To aggregate the triagular fuzzy correlated iformatio basedothebmadweightedharmoicmeaoperators H. Su ad M. Su 23 developed the fuzzy Boferroi harmoic mea operator. Because this operator cosiders the weight vector of the aggregated argumets we redefie this operator as fuzzy weighted Boferroi harmoic mea operator. Defiitio 6 see 23). Let a i a L i am i a U i i 2...) be a collectio of triagular fuzzy umbers let w w w 2...w ) T be the weight vector of a i i2...)where w i idicates the importace degree of a i satisfyigw i >0 i2...ad i w i.if FWBHM pq a a 2... a ) i w iw / a p i a q ))/p+q) i w iw /ai L)p a L)q )) /p+q) i w iw /a M i ) p a M )q )) /p+q) i w iw /a U i ) p a U )q )) /p+q) 0) where p q 0 the FWBHM pq is called the fuzzy weighted Boferroi harmoic mea FWBHM) operator. I particular if w //.../) T the the FWBHM operator reduces to the followig: FBHM pq a a 2... a ) / 2 ) i / ap i a q ))/p+q) ) i w i/ai L)p )) /p i w i/ai M ) p )) /p i w i/ai U ) p )) /p 2) which we call the fuzzy weighted geeralized harmoic mea FWGHM) operator. O the basis of the operatioal laws of triagular fuzzy umbers the FWBHM operator has the followig properties. Theorem 7. Let p q 0 adlet a i a L i am i a U i i 2...)be a collectio of triagular fuzzy umbers ad the followig are valid. ) Idempotecy. If all a i i2...)are equal that is a i aforallithe FWBHM pq a a 2... a ) a. 3) 2) Boudedess. a FWBHM pq a a 2... a ) a + where a mi i {a L i } mi i{a M i } mi i {a U i } ad a + max i {a L i } max i{a M i } max i {a U i }. 3) Commutativity. Let a i a L i a M i a U i i 2...)be a collectio of triagular fuzzy umbers ad the FWBHM pq a a 2... a ) 4) FWBHM pq a a 2... a ) where a a 2... a ) is ay permutatio of a a 2... a ). Proof. Sice 2) ca be prove easily we prove ) ad 3) as follows. ) Sice a i awehave which we call the fuzzy Boferroi harmoic mea FBHM) operator. I additio a special case ca obtaied as the chage of parameter. If q0 the the FWBHM operator reduces to the followig: FWBHM p0 a a 2... a ) i w iw / a p i ))/p i w i/ a p i ) w ) /p i w i/ a p i ))/p FWBHM pq a a 2... a ) i w iw / a p a q )) /p+q) i w iw / a p+q )) /p+q) a i w i w ) /p+q) a. 5)

5 Joural of Applied Mathematics 5 3) Sice a a 2... a ) is ay permutatio of a a 2... a )the FWBHM pq a a 2... a ) i w iw / a p i a q ))/p+q) i w iw / a i )p a )q )) /p+q) FWBHM pq a a 2... a ). 6) I particular if the triagular fuzzy umbers a i a L i a M i a U i i 2...) reduce to the iterval umbers a i a L i au i i 2...) the the FWBHM operator 0) reduces to the ucertai weighted Boferroi harmoic mea UWBHM) operator as follows: UWBHM pq a a 2... a ) i w iw / a p i a q ))/p+q) i w iw /ai L)p a L)q )) /p+q) i w iw /a U i ) p a U )q )) /p+q). 7) If w //.../) T the the UWBHM operator reduces to the ucertai Boferroi harmoic mea UBHM) operator as follows: UBHM pq a a 2... a ) / 2 ) i / ap i a q ))/p+q) / 2 ) i /al i )p a L)q )) /p+q) 8) I this case if w //.../) T the the WBHM operator reduces to the Boferroi harmoic mea BHM) operator: BHM pq a a 2...a ) / 2 ) i /ap i aq ))/p+q). 20) Example 8. Give a collectio of triagular fuzzy umbers: a a a a 4 3 5letw ) T be the weight vector of a i i 234); the by FWBHM operator 0) let pq2) we have FWBHM 22 a a 2 a 3 a 4 ) 4 i w iw /ai L)2 a L)2 )) /4 4 i w iw /a M i ) 2 a M )2 )) /4 4 i w iw /a U i ) 2 a U )2 )) / ) Based o the OWA ad FWBHM operators ad Defiitio 5 we defie fuzzy ordered weighted Boferroi harmoic mea FOWBHM) operator as follows. Defiitio 9. Let a i a L i am i a U i i 2...) be a collectio of triagular fuzzy umbers. For p q 0 a fuzzy ordered weighted Boferroi harmoic mea FOWBHM) operator of dimesio is a mappig FOWBHM pq :Ω Ω that has a associated vector ωω ω 2...ω ) T such that ω i 0ad i ω i.furthermore FOWBHM pq a a 2... a ) i w iw / a p σi) aq σ) ))/p+q) / 2 ) i /au i ) p a U )q )) /p+q) If a L i a U i a i foralli the the UWBHM operator 7) reduces to the weighted Boferroi harmoic mea WBHM) operator as follows: WBHM pq a a 2...a ) i w iw /a p i aq ))/p+q).. 9) i w iw /aσi) L )p a σ) L )q )) i w iw /a M σi) )p a M σ) )q )) i w iw /a U σi) )p a U σ) )q )) /p+q) /p+q) /p+q) 22)

6 6 Joural of Applied Mathematics where a σi) a L σi) am σi) au σi) i 2...)adσ) σ2)... σ)) is a permutatio of 2...) such that a σi ) a σi) for all i. However if there is a tie betwee a i ad a thewe replace each of a i ad a by their average a i + a )/2 i process of aggregatio. If k itemsaretiedthewereplace these by k replicas of their average. The weightig vector ω ω ω 2...ω ) T ca be determied by usig some weights determiig methods like the ormal distributio based method; see for more details. If ω //.../) T the the FOWBHM operator reduces to the FBHM operator; i additio if q0 the the FOWBHM operator reduces to the followig: FOWBHM p0 a a 2... a ) i w iw / a p σi) ))/p i w i/ a p σi) ) w ) /p i w i/ a p σi) ))/p i w i/aσi) L )p )) /p i w i/aσi) M )p )) /p i w i/a U σi) )p )) /p 23) which we call the fuzzy ordered weighted geeralized harmoic mea FOWGHM) operator. I particular if the triagular fuzzy umbers a i a L i am i a U i i 2...) reduce to the iterval umbers a i a L i au i i 2...) the the FOWBHM operator reduces to the ucertai ordered weighted Boferroi harmoic mea UOWBHM) operator as follows: UOWBHM pq a i a 2... a ) i ω iω / a p σi) aq σ) ))/p+q) i w iw /aσi) L )p a σ) L )q )) i w iw /a U σi) )p a U σ) )q )) /p+q) /p+q) 24) where a σi) a L σi) au σi) adσ) σ2)... σ)) is a permutatio of 2...)such that a σi ) a σi) for all i. If there is a tie betwee a i ad a thewereplaceeachof a i ad a by their average a i + a )/2 i process of aggregatio. If k items are tied the we replace these by k replicas of their average. If a L i a U i a i foralli the the UOWBHM operator reduces to the ordered weighted Boferroi harmoic mea OWBHM) operator as follows: OWBHM pq a a 2...a ) i ω iω /b p i bq ))/p+q) 25) where b i is the ith largest of a i i 2...). The OWBHM operator 25)hassomespecialcases. ) If ω0...0) T the OWBHM pq a a 2...a )max {a i }b. 26) 2) If ω00...) T the OWBHM pq a a 2...a )mi {a i }b. 27) 3) If ω//.../) T the OWBHM pq a a 2...a ) / 2 ) i /bp i bq ))/p+q) / 2 ) i /ap i aq ))/p+q) BHM pq a a 2...a ). 28) Example 0. Let a a a a ad a be a collectio of triagular fuzzy umbers. To rak these triagular fuzzy umbers we first compare each triagular fuzzy umber a i with all triagular fuzzy umbers a 2345) by usig 7) without loss of geerality set δ 0.5); let p i p a i a ) i ) the we utilize these possibility degrees to costruct the followig matrix Pp i ) 5 5 : P ). 29) Summig all elemets i each lie of matrix Pwehave p p p ) p p ad the we rak the triagular fuzzy umbers a i i ) i descedig order i accordace with the values of p i i2345)as follows: a σ) a 4 a σ2) a 5 a σ3) a 3) a σ4) a 3 a σ5) a 2.

7 Joural of Applied Mathematics 7 Suppose that the weightig vector of the FOWBHM operator is ω ) T derived by the ormal distributio based method 27) ad the by 22) let pq2) we get FOWBHM 22 a a 2 a 3 a 4 a 5 ) 5 i ω iω /aσi) L )2 a σ) L )2 )) 5 i ω iω /a M σi) )2 a M σ) )2 )) 5 i ω iω /a U σi) )2 a U σ) )2 )) /4 /4 /4. 32) Both FWBHM ad FOWBHM operators however ca oly deal with the situatio i which there are correlatios betwee ay two aggregated argumets but ot the situatio i which there exist coectios amog ay three aggregated argumets. To solve this issue motivated by Defiitio3 we defie the followig. Defiitio. Let a i a L i am i a U i i 2...) be a collectio of triagular fuzzy umbers ad let w w w 2...w ) T be the weight vector of a i i 2...) where w i idicates the importace degree of a i satisfyig w i >0 i2...ad i w i.forp q r 0if GFWBHM pqr a a 2... a ) ik w iw w k / a p i a q ar k ))/p+q+r) ik w iw w k /ai L)p a L)q ak L)r )) /p+q+r) ik w iw w k /a M i ) p a M )q a M k )r )) /p+q+r) ik w iw w k /ai U ) p a U)q a U k )r )) /p+q+r) 33) the GFWBHM pqr is called geeralized fuzzy weighted Boferroi harmoic mea GFWBHM) operator. I particular if w //.../) T the the GFWBHM operator reduces to the followig: GFBHM pqr a a 2... a ) / 3 ) ik / ap i a q ar k ))/p+q+r) 34) which we call the geeralized fuzzy Boferroi harmoic mea GFBHM) operator. I additio some special cases ca be obtaied as the chage of parameters. ) If r0 the the GFWBHM operator reduces to GFWBHM pq0 a a 2... a ) ik w iw w k / a p i a q ))/p+q) k w k) i w iw / a p i a q ))/p+q) i w iw / a p i a q ))/p+q) 35) which is the FWBHM operator. 2) If q 0 ad r 0 the the GFWBHM operator reduces to GFWBHM p00 a a 2... a ) ik w iw w k / a p i ))/p w ) k w k) i w i/ a p i ))/p i w i/ a p i ))/p 36) whichisfwghmoperator.ithiscaseifp the FWGHM operator reduces to FWHM operator. Similar to the FWBHM operator the GFWBHM operator has the followig properties. Theorem 2. Let p q r 0 adlet a i a L i am i a U i i 2...)be a collectio of triagular fuzzy umbers ad the followig are valid. ) Idempotecy. If all a i i2...)are equal that is a i aforallithe GFWBHM pqr a a 2... a ) a. 37) 2) Boudedess. a GFWBHM pqr a a 2... a ) a + where a mi i {a L i } mi i{a M i } mi i {a U i } ad a + max i {a L i } max i{a M i } max i {a U i }. a + max i {a L i }max i {a M i }max i {a U i }. 38)

8 8 Joural of Applied Mathematics 3) Commutativity. Let a i a L i a M i a U i i2...)be a collectio of triagular fuzzy umbers ad the GFWBHM pqr a a 2... a ) 39) GFWBHM pqr a a 2... a ) where a a 2... a ) is ay permutatio of a a 2... a ). I particular if the triagular fuzzy umbers a i a L i am i a U i i 2...) reduce to the iterval umbers a i a L i au i i 2...) the the GFWBHM operator 24) reduces to the geeralized ucertai weighted Boferroi harmoic mea GUWBHM) operator as follows: GUWBHM pqr a a 2... a ) ik w iw w k / a p i a q ar k ))/p+q+r) ik w iw w k /ai L)p a L)q ak L)r )) /p+q+r) ik w iw w k /ai U ) p a U)q a U k )r )) /p+q+r) 40) If w//.../) T the the GUWBHM operator reduces to the geeralized ucertai Boferroi harmoic mea GUBHM): GUBHM pqr a a 2... a ) / 3 ) ik / ap i a q ar k ))/p+q+r) / 3 ) ik /al i )p a L)q ak L)r )) /p+q+r) / 3 ) ik /au i ) p a U)q a U k )r )) /p+q+r) 4) Furthermore if a L i a U i a i foralli the the GUWBHM operator reduces to the geeralized weighted Boferroi harmoic mea GWBHM) operator: GWBHM pqr a a 2...a ) ik w iw w k /a p i aq ar k ))/p+q+r). 42) I this case if pad qr0thegwbhmoperator reduces to the weighted harmoic mea WHM) operator... Example 3. Cosider the four triagular fuzzy umbers a i ad their weight vector w give i Example 8. Thebythe GFWBHM operator 33) without of geeralizatio let p qr3) we have GFWBHM 333 a a 2 a 3 a 4 ) 4 ik w iw w k /ai L)3 a L)3 ak L)3 )) /9 4 ik w iw w k /a M i ) 3 a M )3 a M k )3 )) /9 4 ik w iw w k /a U i ) 3 a U )3 a U k )3 )) / ) Defiitio 4. Let a i a L i am i a U i i 2...) be a collectio of triagular fuzzy umbers. For p q r 0 a geeralized fuzzy ordered weighted Boferroi harmoic mea GFOWBHM) operator of dimesio is a mappig GFOWBHM pqr :Ω Ω that has a associated vector ω ω ω 2...ω ) T such that ω i 0 ad i ω i. Furthermore GFOWBHM pqr a a 2... a ) ik ω iω ω k / a p σi) aq σ) ar σk) ))/p+q+r) ik ω iω ω k /aσi) L )p a σ) L )q aσk) L )r )) ik ω iω ω k /a M σi) )p a M σ) )q a M σk) )r )) ik ω iω ω k /a U σi) )p a U σ) )q a U σk) )r )) /p+q+r) /p+q+r) /p+q+r) 44) where a σi) a L σi) am σi) au σi) i 2...)adσ) σ2)...σ)) is a permutatio of 2...) such that a σi ) a σi) for all i. However if there is a tie betwee a i ad a thewereplace each of a i ad a by their average a i + a )/2 i process of aggregatio. If k items are tied the we replace these by k replicas of their average.

9 Joural of Applied Mathematics 9 If ω //.../) T the the GFOWBHM operator reduces to the GFBHM operator. Moreover some specialcasescabeobtaiedasthechageofparameters.if r0 the the GFOWBHM operator reduces to FOWBHM operator; if r 0 ad q 0 the GFOWBHM operator reduces to FOWGHM operator. I particular if the triagular fuzzy umbers a i a L i am i a U i i 2...)reduce to the iterval umbers a i a L i au i i 2...) the the GFOWBHM operator reduces to the geeralized ucertai ordered weighted Boferroi harmoic mea GUOWBHM) operator: GUOWBHM pqr a i a 2... a ) ik ω iω ω k / a p σi) aq σ) ar σk) ))/p+q+r) ik ω iω ω k /aσi) L )p a σ) L )q aσk) L )r )) ik ω iω ω k /a U σi) )p a U σ) )q a U σk) )r )) /p+q+r) /p+q+r) 45) where a σi) a L σi) au σi) adσ) σ2)... σ)) is a permutatio of 2...)such that a σi ) a σi) for all i. If a L i a U i a i foralli 2... the the GUOWBHM operator reduces to the geeralized ordered weighted Boferroi harmoic mea GOWBHM) operator: GOWBHM pqr a a 2...a ) ik ω iω ω k /b p i bq br k ))/p+q+r) 46) where b i is the ith largest of a i i 2...).Ithiscaseif pad qr0 the the GOWBHM operator reduces to the ordered weighted harmoic mea OWHM) operator. The GOWBHM operator 46) has some special cases. ) If ω0...0) T the GOWBHM pqr a a 2...a ) max {a i } b. 47) 2) If ω00...) T the GOWBHM pqr a a 2...a ) mi {a i } b. 48) 3) If ω//.../) T the GOWBHM pqr a a 2...a ) / 3 ) ik /bp i bq br k ))/p+q+r) / 3 ) ik /ap i aq ar k ))/p+q+r) 49) which we call the geeralized Boferroi harmoic mea GBHM) operator. Example 5. Cosider the four triagular fuzzy umbers a i ad their weight vector w give i Example 0. Thebythe GFOWBHM operator 44) let pqr3) we have GFOWBHM 333 a a 2 a 3 a 4 a 5 ) 5 ik w iw w k /aσi) L )3 a σ) L )3 aσk) L )3 )) 5 ik w iw w k /a M σi) )3 a M σ) )3 a M σk) )3 )) 5 ik w iw w k /a U σi) )3 a U σ) )3 a U σk) )3 )) /9 /9 /9 50) I the followig sectio we will apply the developed operators to multiple attribute group decisio makig. 3. A Approach to Multiple Attribute Group Decisio Makig with Triagular Fuzzy Iformatio For a group decisio makig with triagular fuzzy iformatio let X{x x 2...x } be a discrete set of alteratives let G {G G 2...G m } be the set of m attributes whose weight vector is w w w 2...w m ) T with w i 0 ad m i w i ad let D{d d 2...d s } be the set of decisio makers whose weight vector is V V V 2...V s ) T where V k 0 ad s k V k.supposethata k) a k) i ) m is the decisio matrix where a k) i a Lk) i a Mk) i a Uk) i is a attribute value which takes the form of triagular fuzzy umber of the alterative x Xwith respect to the attribute G i G. I the followig we apply the GFWBHM ad GFOWBHM operators to group decisio makig with triagular fuzzy iformatio. Step. Normalizeeachattributevalue a k) i i the matrix A k) ito a correspodig elemet i the matrix R k) r k) i ) m r Mk) ) usig the followig formulas: r k) i r k) i r Lk) i a k) i i ak) i r Uk) i

10 0 Joural of Applied Mathematics r k) i a Lk) i auk) i a Mk) i amk) i a Uk) i alk) i for beefit attribute G i i2...m / a k) i / ak) i ) 2... k2...s /a Uk) i /alk) i ) /a Mk) i /amk) i ) /a Lk) i /auk) i ) for cost attribute G i i2...m 2... k2...s. 5) Step 2.UtilizetheGFWBHMoperator33) as follows: r k) GFWBHM pqr r k) rk) 2... rk) m ) m ihl w iw h w l / r k) i )p r k) m ihl m ihl m ihl r Lk) i r Mk) i r Uk) i w i w h w l ) p r Lk) h h )q r k) l )r )) /p+q+r) ) q r Lk) l ) r )) w i w h w l ) p r Mk) h ) q r Mk) l ) r )) w i w h w l ) p r Uk) h ) q r Uk) l ) r )) /p+q+r) /p+q+r) /p+q+r) 52) to aggregate all the elemets i the th colum of R k) adgettheoverallattributevalue r k) of the alterative x correspodig to the decisio maker d k. Step 3. Utilize the GFOWBHM operator 44): r GFOWBHM pqr r ) r 2)... r s) ) s khl ω kω h ω l / s khl r Lσk)) r σk) ) p r σh) ) q r σl) ) r /p+q+r) )) ω k ω h ω l ) p r Lσh)) ) q r Lσl)) r )) ) /p+q+r) s khl s khl r Mσk)) r Uσk)) ω k ω h ω l ) p r Mσh)) ) q r Mσl)) ) r )) ω k ω h ω l ) p r Uσh)) ) q r Uσl)) r )) ) /p+q+r) /p+q+r) 53) to aggregate the overall attribute values r k) k 2...s) correspodig to the decisio maker d k k 2...s)ad get the collective overall attribute value r where r σk)) r Lσk)) r Mσk)) r Uσk)) is the kth largest of the weighted data ad r k) r k) sv k r k) k 2...s) ω ω ω 2...ω s ) T is the weightig vector of the GFOWBHM operator with ω k 0ad s k ω k. Step 4. Compareeach r with all r i i 2...)by usig 7) ad let p i p r i r )adthecostructthepossibility matrix Pp i ) wherep i 0 p i +p i p ii 0.5 i2... Summig all elemets i each lie of matrix P wehavep i p i i 2... ad the reorder r 2...) i descedig order i accordace with the values of p 2...). Step 5.Rakallalterativesx 2...)by the rakig of r 2...) ad the select the most desirable oe. Step 6.Ed. 4. Example Illustratios I this sectio we use a multiple attribute group decisio makig problem of determiig what kid of aircoditioig systems should be istalled i a library adopted from 2 30) to illustrate the proposed approach. A city is plaig to build a muicipal library. Oe of the problems facig the city developmet commissioer is to determie what kid of air-coditioig systems should be istalled i the library. The cotractor offers five feasible alteratives which might be adapted to the physical structure ofthelibrary.thealterativesx 2345) are to be evaluated usig triagular fuzzy umbers by the three decisio makers d k k 2 3) whose weight vector is V ) T ) uder three maor impacts: ecoomic fuctioal ad operatioal. Two moetary attributes ad six omoetary attributes i.e. G :owigcost$/ft 2 ) G 2 :operatigcost$/ft 2 ) G 3 :performace ) G 4 :oise level Db) G 5 : maitaiability ) G 6 : reliability %) G 7 : flexibility ) G 8 :safety ) where uit is from 0 to scale three attributes G G 2 adg 4 are cost attributes ad the other five attributes are beefit attributes ad suppose that the weight vector of the attributes G i i 2...8) is w ) T )emerged from three impacts is Tables 2ad3. Ithefollowigweutilizethedecisioprocedureto select the best air-coditioig system.

11 Joural of Applied Mathematics Table : Triagular fuzzy umber decisio matrix A ). x x 2 x 3 x 4 x 5 G G G G G G G G Table 2: Triagular fuzzy umber decisio matrix A 2). x x 2 x 3 x 4 x 5 G G G G G G G G Table 3: Triagular fuzzy umber decisio matrix A 3). x x 2 x 3 x 4 x 5 G G G G G G G G Step.Byusig5) we ormalize each attribute value a k) i i the matrices A k) k23)ito the correspodig elemet i the matrices R k) r i ) 8 5 k23)tables4 5ad6). Step 2.UtilizetheGFWBHMoperator52)letpqr3) to aggregate all elemets i the th colum R k) ad get the overall attribute value r k) : r ) r ) r ) r ) r ) r 2) r 2) r 2) r 2) r 2) r 3) r 3) r 3) r 3) r 3) )

12 2 Joural of Applied Mathematics Table 4: Normalized triagular fuzzy umber decisio matrix R ). x x 2 x 3 x 4 x 5 G G G G G G G G Table 5: Normalized triagular fuzzy umber decisio matrix R 2). x x 2 x 3 x 4 x 5 G G G G G G G G Table 6: Normalized triagular fuzzy umber decisio matrix R 3). x x 2 x 3 x 4 x 5 G G G G G G G G Step 3. Utilize the GFOWBHM operator 53) supposethat its weight vector is ω ) T determied by usig the ormal distributio based method 27letδ 0.5 ad pqr3)toaggregatetheoverallattributevalue r k) k23) correspodig to the decisio maker d k k 2 3)adgetthecollectiveoverallattributevalue r : r r r r r ) Step 4. Compareeach r with all r i i 2345)by usig 7)withoutlossofgeeralitysetδ 0.5) ad let p i p r i r ) ad the costruct a possibility matrix: P ) 56) Summig all elemets i each lie of matrix Pwehave p p 2.7 p p p ) ad the reorder r 2345)i descedig order i accordace with the values of p 2345): r 4 > r 3 > r 5 > r > r 2. 58) Step 5. Rakallthealterativesx 2345) by the rakig of r 2345): x 4 x 3 x 5 x x 2 59) ad thus the most desirable alterative is x 4. From the previous aalysis the results obtaied by the proposed approach are very similar to the oes obtaied Xu s approach 2 but our approach is more flexible tha that of Xu 2 because it ca provide the decisio makers more choices as parameters are assiged differet values.

13 Joural of Applied Mathematics 3 Table 7: Compariso of the proposed approach with other approaches. Xu s approach 2 Wei s approach22 Su ad Su s approach 23 Proposed approach Problem type MAGDM MAGDM MADM MAGDM Applicatio area Air-coditioig system Air-coditioig system Ivestmet of moey EPR system selectio selectio selectio Decisio iformatio Triagular fuzzy decisio Triagular fuzzy Triagular fuzzy decisio Triagular fuzzy decisio matrix decisio matrix matrix matrix Solutio method Aggregatio stage FWHM operator FIOWHM operator GFWBHM operator Exploitatio stage FHHM operator FWHM operator FOBHM operator GFOWBHM operator Rakig stage Complemetary matrix Complemetary matrix Possibility matrix Possibility matrix Fial decisio Rakig of alteratives Rakig of alteratives Rakig of alteratives Rakig of alteratives 5. Compariso of the Proposed Approach with Other Approaches I this sectio we compare the proposed approach with other approaches. The approaches to be compared here are the approaches proposed by Xu 2 Wei 22 ad H. Su ad M. Su 23 respectively. Each of methods has its advatages ad disadvatages ad oe of them ca always perform better tha the others i ay situatios. It perfectly depeds o how we look at thigs ad ot o how they are themselves. The differeces i four approaches are the followig. ) The H. Su ad M. Su s approach is oly suitable for solvig multiple attribute decisio makig MADM) while the proposed approach ad Xu s ad Wei s approaches are suitable for solvig MAGDM because the approaches provide the aggregatio stage i aggregatio process. 2) The Xu s ad Wei s approaches have simple computatio process tha the proposed approach ad H. Su ad M. Su s approach while the proposed approach ad H. Su ad M. Su s approach are more flexible tha Xu s ad Wei s approaches because these ca provide the decisio makers more choices as parameters are assiged differet values. 3) The Wei s approach uses the weights of decisio makers as the order iducig variables i aggregatio stage while other approaches use the weights of decisio makers to determie the order positios of the overall attribute values i exploitatio stage. Others of relative compariso with Xu s Wei s ad H. Su ad M. Su s approaches are show i Table Coclusios I this paper we have exteded the GWBM operator to the triagular fuzzy eviromet ad developed the fuzzy harmoic aggregatio operators icludig the FWBHM ad GFWBHM operators. Based o the these operators ad Yager s OWA operator we have developed the FOWBHM operator ad the GFOWBHM operator respectively ad discussed their properties ad special cases. It has bee poited out that if all the iput fuzzy data reduce to the iterval or umerical data the the GFWBHM operator reduces to the GUWBHM operator ad GWBHM operator respectively; the GFOWBHM operator reduces to the GUOWBHM operator ad GOWBHM operator respectively. I these situatios the WHM resp. OWHM) operator is the special case of the GWBHM resp. GOWBHM) operator. Based o the GFWBHM ad GFOWBHM operators we have developedaapproachtomultipleattributegroupdecisio makig with triagular fuzzy iformatio ad have also applied the proposed approach to the problem of determiig what kid of air-coditioig systems should be istalled i the library. Furthermore the compariso of the proposed approach with other existig approaches is preseted. The merit of the proposed approach is that it is more flexible tha the classical oes because it ca provide the decisio makersmorechoicesasparametersareassigeddifferet values. Apparetly the proposed aggregatio techiques ad decisio makig method ca also exteded to the itervalvalued triagular fuzzy eviromet. Ackowledgmet This work was supported by a Research Grat of Pukyog Natioal Uiversity 203). Refereces R. R. Yager O ordered weighted averagig aggregatio operators i multicriteria decisiomakig IEEE Trasactios o Systems Ma ad Cyberetics vol.8o.pp R. R. Yager Families of OWA operators Fuzzy Sets ad Systemsvol.59o.2pp F. Chiclaa F. Herrera ad E. Herrera-Viedma The ordered weighted geometric operator: properties ad applicatio i Proceedigs of the 8th Iteratioal Coferece o Iformatio Processig ad Maagemet of Ucertaity i Kowledge-Based Systems pp Madrid Spai 2000.

14 4 Joural of Applied Mathematics 4 Z. S. Xu ad Q. L. Da The ordered weighted geometric averagig operators Iteratioal Joural of Itelliget Systems vol.7o.7pp Z.S.XuadQ.L.Da Aoverviewofoperatorsforaggregatig iformatio Iteratioal Joural of Itelliget Systems vol. 8 o. 9 pp H. Y. Che C. L. Liu ad Z. H. She Iduced ordered weighted harmoic averagig IOWHA) operator ad its applicatio to combiatio forecastig method Chiese Joural of Maagemet Sciece vol. 2 pp R. R. Yager OWA aggregatio over a cotiuous iterval argumet with applicatios to decisio makig IEEE Trasactios o Systems Ma ad Cyberetics Bvol.34o.5pp R.R.YageradZ.Xu Thecotiuousorderedweighted geometric operator ad its applicatio to decisio makig Fuzzy Sets ad Systemsvol.57o.0pp R. R. Yager Choquet aggregatio usig order iducig variables Iteratioal Joural of Ucertaity Fuzziess ad Kowledge-Based Systemsvol.2o.pp R. R. Yager The power average operator IEEE Trasactios o Systems Ma ad Cyberetics Avol.3o.6pp Z. Xu ad R. R. Yager Power-geometric operators ad their use i group decisio makig IEEE Trasactios o Fuzzy Systems vol. 8 o. pp C. Boferroi Sulle medie multiple di poteze Bollettio dell Uioe Matematica Italiaa vol.5o.3-4pp R.R.Yager OgeeralizedBoferroimeaoperatorsfor multi-criteria aggregatio Iteratioal Joural of Approximate Reasoigvol.50o.8pp G. Beliakov S. James J. Mordelová T. Rückschlossová ad R. R. Yager Geeralized Boferroi mea operators i multicriteria aggregatio Fuzzy Sets ad Systemsvol.6o.7pp M.XiaZ.XuadB.Zhu Geeralizedituitioisticfuzzy Boferroi meas Iteratioal Joural of Itelliget Systems vol. 27 o. pp C. Ta ad X. Che Ituitioistic fuzzy Choquet itegral operator for multi-criteria decisio makig Expert Systems with Applicatiosvol.37o.pp Z. Xu Choquet itegrals of weighted ituitioistic fuzzy iformatio Iformatio Sciecesvol.80o. 5pp Z. Xu ad R. R. Yager Ituitioistic fuzzy boferroi meas IEEE Trasactios o Systems Ma ad Cyberetics B vol.4 o. 2 pp D. Yu Y. Wu ad W. Zhou Geeralized hesitat fuzzy boferroi mea ad its applicatio i multi-criteria group decisio makig Joural of Iformatio ad Computatioal Sciecevol.9o.2pp Z. Xu Approaches to multiple attribute group decisio makig based o ituitioistic fuzzy power aggregatio operators Kowledge-Based Systemsvol.24o.6pp Z. Xu Fuzzy harmoic mea operators Iteratioal Joural of Itelliget Systemsvol.24o.2pp G.-W. Wei FIOWHM operator ad its applicatio to multiple attribute group decisio makig Expert Systems with Applicatiosvol.38o.4pp H. Su ad M. Su Geeralized Boferroi harmoic mea operators ad their applicatio to multiple attribute decisio makig Joural of Computatioal Iformatio Systems vol.8 o.4pp P. J. M. va Laarhove ad W. Pedrycz A fuzzy extesio of Saaty s priority theory Fuzzy Sets ad Systems vol. o. 3 pp F. Chiclaa F. Herrera ad E. Herrera-Viedma Itegratig multiplicative preferece relatios i a multipurpose decisiomakig model based o fuzzy preferece relatios Fuzzy Sets ad Systemsvol.22o.2pp Z. S. Xu ad Q. L. Da The ucertai OWA operator Iteratioal Joural of Itelliget Systems vol.7o.6pp Z. Xu A overview of methods for determiig OWA weights Iteratioal Joural of Itelliget Systems vol.20o.8pp R. R. Yager Cetered OWA operators Soft Computigvol. o.7pp X. Liu ad S. Ha Oress ad parameterized RIM quatifier aggregatio with OWA operators: a summary Iteratioal Joural of Approximate Reasoigvol.48o.pp K. Yoo The propagatio of errors i multiple-attribute decisio aalysis: a practical approach Joural of the Operatioal Research Societyvol.40pp

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