Harmonic Mean Operators for Aggregating Linguistic Information

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1 Fourth Iteratoal Coferece o Natural Computato Harmoc Mea Operator for Aggregatg Lgutc Iformato Zehu Xu College of Ecoomc ad Maagemet Southeat Uverty, Nag, Jagu 0096, Cha E-mal: xu_zehu@63.et Abtract Harmoc mea wdely ued to aggregate cetral tedecy data, whch uually expreed exact umercal value. I th paper, we vetgate the tuato where the put data are gve the form of lgutc label, ad develop ome lgutc harmoc mea aggregato operator, uch a the lgutc weghted harmoc mea (LWHM operator, the lgutc ordered weghted harmoc mea (LOWHM operator, ad the lgutc hybrd harmoc mea (LHHM operator for aggregatg lgutc formato. Some example are gve to llutrate the developed operator.. Itroducto Over the lat decade, may operator have bee developed for aggregatg umercal data formato [], where the weghted averagg operator, the ordered weghted averageg operator, the weghted harmoc mea operator, the weghted geometrc mea operator, ad the ordered weghted geometrc mea operator are fve of the mot commo aggregato operator the formato fuo lterature [-7]. The weghted averagg operator [] ad the weghted geometrc mea operator [5] are the aggregato techque whch take to accout all the gve umercal data together wth ther weght. The ordered weghted averagg (OWA operator [3] wa troduced to provde a method for aggregatg everal put that le betwee the max ad m operator, whoe fudametal apect the re-orderg tep, that, the OWA operator frt reorder all the gve data decedg order, ad the aggregate the reordered data together wth the weght of ther poto, the ordered weghted geometrc mea operator [6,7] baed o the OWA operator ad o the geometrc mea, whch ha ome mlar charactertc wth the OWA operator. The weghted harmoc mea operator [4] a coervatve average, whch wdely ued to aggregate cetral tedecy data. I the real-lfe world, there are may tuato, uch a electg applcato for dfferet kd of cholarhp ad electg proect for dfferet kd of fudg polce, ad evaluatg the peed, comfort or deg for dfferet kd of car, whch a more realtc approach may be to ue lgutc aemet tead of umercal value [8]. Some author [8-7] exteded the weghted averagg operator, the ordered weghted averageg operator, the weghted geometrc mea operator, ad the ordered weghted geometrc mea operator to accommodate the tuato where the put data are expreed lgutc label, ad appled the exteded operator for olvg varou deco makg problem uder lgutc evromet. I th paper, we exted the well-kow harmoc mea to accommodate lgutc tuato, ad develop ome lgutc harmoc mea aggregato operator, uch a the lgutc weghted harmoc mea (LWHM operator, the lgutc ordered weghted harmoc mea (LOWHM operator, ad the lgutc hybrd harmoc mea (LHHM operator for aggregatg lgutc formato. The charactertc of thee developed operator are alo aalyzed.. Prelmare For coveece, we frt recall ome bac oto ad operato related to lgutc formato, whch wll be ued th paper: Lgutc label are a bac tool ued to decrbe the qualtatve apect of a problem. I [8], Xu defed S { t,...,,,,..., t} a a lgutc label et wth odd cardalty. Ay label,, repreet a poble value for a lgutc varable, ad t requred that the lgutc label et hould atfy the followg charactertc: > β ff > β ; There the recprocal operator: rec( β uch that β. Epecally, rec(, where the md /08 $ IEEE DOI 0.09/ICNC

2 lgutc label repreet a aemet of dfferece, ad wth the ret of the lgutc label beg placed recprocally aroud t. I partcular, t ad t dcate the lower ad upper lmt of the lgutc label S, repectvely, t a potve teger, ad the cardalty of S t. For example, a et of e lgutc label could be a follow: S { oe, very low, low, lghtly low, medum, lghtly hgh, 3 hgh, 4 5 very hgh, perfect} Xu [85] exteded the dcrete lgutc label et S to a cotuou label et S { [ q, q]} o a to preerve all the gve formato, where q ( q > t a uffcetly large potve teger. If S, the termed a orgal lgutc label, otherwe, termed a vrtual lgutc label. I geeral, the vrtual lgutc label ca oly appear calculato. If, β S, λ [0,], the ther operatoal law were defed a follow [8]: β ; + β λ λ ; 3 (. 3. Lgutc harmoc mea aggregato operator Baed o the well-kow harmoc mea [4], the followg, we develop ome operator for aggregatg lgutc formato: Defto. Let LWHM : S S, ad (,,..., be a collecto of lgutc label. If LWHM (,,..., ( w( w( w ( ( the LWHM called a lgutc weghted harmoc mea (LWHM operator, where w ( w, w,..., w T the weght vector aocated wth the lgutc label (,,...,, w 0,,,...,, ad w. Baed o the operatoal law of the lgutc label, we ca traform ( to the followg: LWHM (,,..., ( w( w ( w ( ( w w w ( w w w ( w where. a By Defto, we have the followg cocluo for ome pecal cae: If w, w 0,, the LWHM (,,..., (3 If w (,,..., T, the the LWHM operator reduced to the lgutc harmoc mea (LHM operator: LWHM (,,..., where ( w( w ( w ( ( ( ( ( LHM (,,..., (4. a Moreover, the LWHM operator a bouded operator whch provde for aggregato lyg betwee the lgutc max operator ad the lgutc m operator: m{ } LWHM (,,..., max{ } (5 I fact, by the operatoal law of the lgutc label, we have LWHM (,,..., ( w( w( w ( ( w(max{ } (max{ } (max{ } w w ( ( (max{ } w w w ad max{ } LWHM (,,..., 05

3 ( w( w( w ( ( (m{ } (m{ } w (m{ } w w ( ( (m{ } w w w m{ } thu, (5 hold. The LWHM operator aggregate all the gve lgutc label together wth ther aocated weght. It a coervatve average, whch very utable to be ued a a tool to aggregate cetral tedecy data. Example. Gve a collecto of fve lgutc label:, 3,,, 4, let w (0.3, 0.,0.,0.3,0. T be the weght vector of (,,3,4,5, the by (, we have LWHM(,,,, ( w( w( w3( w4( w5( ( 0.3 ( 0. ( 3 0. ( ( 0. ( Motvated by the ordered weghted averagg (OWA operator [3], here we defe a lgutc ordered weghted harmoc mea (LOWHM operator: Defto. A lgutc ordered weghted harmoc mea (LOWHM operator of dmeo a mappg LOWHM : S S, that ha a aocated vector ω ( ω, ω,..., ω T uch that ω 0,,,...,, ad ω. Moreover, LOWHM (,,..., ( ω ( σ ( ω ( σ ( ω ( σ ( (6 where (,,..., be a collecto of lgutc label. ( σ (, σ (,..., σ ( a permutato of (,,..., uch that, for all. σ ( σ( Baed o the operatoal law of the lgutc label, (6 ca be further traformed to the followg: LOWHM (,,..., (7 ω where. σ ( The aocated vector ω ( ω, ω,..., ω T ca be determed by ug ome weght determg method, for example, O Haga [9] developed a procedure to geerate the weght that have a predefed degree of ore ad maxmze the etropy of the weght. Flev ad Yager [0] developed two procedure, baed o the expoetal moothg, to obta the weght. Yager [] troduced a bac ut-terval mootoc (BUM fucto baed approach to determg the weght. Xu [] made a urvey of the weght determg method ad the developed a ormal dtrbuto baed method, whch ca releve the fluece of the ufar argumet by agg low weght to thoe fale or baed oe. By Defto, we have If ω (,0,...,0 T, the LOWHM (,,..., max{ } (8 If ω (0,0,..., T σ(, the LOWHM (,,..., m{ } (9 σ( 3 If ω 0, ω 0,, the LOWHM (,,..., σ( (0 4 If ω (,,..., T, the the LOWHM operator reduced to the LHM operator: LOWHM (,,..., LHM (,,..., ( Furthermore, the LOWHA operator ha ome derable properte mlar to thoe of the OWA operator [3]: (Idempotecy: Let (,,..., be a collecto of lgutc label, f all (,,..., are equal,.e.,, for all, the LOWHM (,,..., ( (Boudary: Let (,,..., be a collecto of lgutc label, the m{ } LOWHM (,,..., max{ } (3 σ( σ( * * (,,..., ad 3 (Mootocty: Let (,, *..., be two collecto of lgutc label, f *, for all, the * * * LOWHM (,,..., LOWHM (,,..., (4 06

4 4 (Commutatvty: Let (,,..., be a collecto of lgutc label, the LOWHM LOWHM (5 ' ' ' (,,..., (,,..., ' ' ' where (,,..., ay permutato of (,,...,. The LOWHM operator alo provde a techque for fug the gve lgutc label that le betwee the max ad m lgutc aggregato operator. Smlar to the Yager OWA operator, the fudametal apect of the LOWHM operator the re-orderg tep, that, the LOWHM operator frt reorder all the gve lgutc label decedg order, ad the aggregate the reordered lgutc label together wth the weght of ther poto. Example. Gve a collecto of four lgutc label: 3, 4, 3, 4. We frt rak the lgutc label (,,3,4 decedg order:, ( 3 σ ( 3 4 ( 3 σ, σ ( 4 4 σ Let ω (0.55,0.345,0.345,0.55 T be the aocated vector of the LOWHM operator derved from the ormal dtrbuto baed method [], the by (6, we have LOWHM (,,, 3 4 ( ω ( σ( ω ( σ( ω3 ( σ( 3 ω4 ( σ( 4 ( 0.55 ( ( ( 0.55 ( I what follow, we troduce aother lgutc harmoc mea aggregato operator lgutc hybrd harmoc mea (LHHM operator, whch baed o the LWHM ad LOWHM operator: Defto 3. A lgutc hybrd harmoc mea (LHHM operator of dmeo a mappg LHHM : S S, whch ha a aocated vector ω ( ω, ω,..., ω T wth ω 0,,,...,, ad ω, uch that LHHM (,,..., ( ω ( σ ( ω ( σ ( ω ( σ ( (6 where σ ( the th larget of the weghted lgutc label ( w,,,...,, w ( w, w,..., w T the weght vector of the lgutc label (,,..., w,,,...,, ad, wth 0 w. Epecally, f w (,,..., T, the,,,...,, th cae, the LHHM operator reduced to the LOWHM operator; f ω (,,..., T, the the LHHM operator reduced to the LWHM operator. Obvouly, the LHHM operator geeralze the LOWHM ad LWHM operator, ad reflect the mportace of both the gve lgutc label ad ther ordered poto. Example 3. let 4,, 3 3,, 4 5 4, ad be a collecto of x 6 lgutc label, ad let w (0.0,0.5,0.0,0.0,0.0,0.5 T be the weght vector of (,,...,6, the the weghted lgutc label ( 6w,,,..., 6 are a follow: , , , , We rak the weghted lgutc label decedg order: ( (,,...,6,, σ, ( σ σ ( 4.4 ( σ ( 3 3.8, σ σ ( Let ω (0.09,0.7,0.4,0.4,0.7,0.09 T be the aocated vector of the LHHM operator derved from the ormal dtrbuto baed method [9], the by (6, we have LHHM (,,..., 6 ( ω ( σ ( ω ( σ ( ω6 ( σ ( 6 ( 0.09 ( ( ( ( ( (

5 4. Cocluo Harmoc mea oe of the mot commoly-ued meaure of cetral tedecy. Coder that, real-lfe tuato, the put data are uually expreed lgutc label tead of umercal value, th paper we have exteded the tradtoal harmoc mea to deal wth lgutc formato. We have developed ome lgutc harmoc mea aggregato operator, cludg the lgutc weghted harmoc mea (LWHM operator, the lgutc ordered weghted harmoc mea (LOWHM operator, ad the lgutc hybrd harmoc mea (LHHM operator for aggregatg lgutc formato. Moreover, we have aalyzed the charactertc of the developed operator,.e., the LWHM operator drectly aggregate all the gve lgutc label together wth ther aocated weght; the LOWHM operator frt reorder all the gve lgutc label decedg order, ad the aggregate the reordered lgutc label together wth the weght of ther poto, the fudametal apect of the LOWHM operator the re-orderg tep; whle the LHHM operator geeralze both the LWHM ad LOWHM operator, ad emphaze the mportace of both the gve lgutc label ad ther ordered poto. I the future reearch, thee lgutc harmoc mea aggregato operator ca be appled may feld uch a deco makg, medcal dago, data mg, ad oft computg, etc. Ackowledgemet The work wa upported by the Natoal Natural Scece Foudato of Cha (No ad the Natoal Scece Fud for Dtguhed Youg Scholar of Cha (No Referece [] Z.S. Xu ad Q.L. Da, A overvew of operator for aggregatg formato, Iteratoal Joural of Itellget Sytem, 003, 8: [] J.C. Haray, Cardal welfare, dvdualtc ethc, ad terperoal comparo of utlty, Joural of Poltcal Ecoomy, 955, 63: [3] R.R. Yager, O ordered weghted averagg aggregato operator mult-crtera deco makg, IEEE Traacto o Sytem, Ma, ad Cyberetc, 988, 8: [4] P.S. Bulle, D.S. Mtrovć, ad P.M. Vać, Mea ad Ther Iequalte, Redel, Dordrecht, 988. [5] T.L. Saaty, The Aalytc Herarchy Proce, McGraw-Hll, New York, 980. [6] F. Herrera, E. Herrera-Vedma, ad F. Chclaa, Multpero deco-makg baed o multplcatve preferece relato, Europea Joural of Operatoal Reearch, 00, 9: [7] Z.S. Xu ad Q.L. Da, The ordered weghted geometrc averagg operator, Iteratoal Joural of Itellget Sytem, 00, 7: [8] Z.S. Xu, Lgutc aggregato operator: a overvew, I: Fuzzy Set ad Ther Exteo: Repreetato, Aggregato ad Model (Edtor: H. Butce, F. Herrera, ad J. Motero, Sprger-Verlag, 007, pp [9] M. Delgado, J.L. Verdegay, ad M.A. Vla, O aggregato operator of lgutc label, Iteratoal Joural of Itellget Sytem, 993, 8: [0] G. Bordoga, M. Fedrzz, ad G. Pa, A lgutc modelg of coeu group deco makg baed o OWA operator, IEEE Traacto o Sytem, Ma, ad Cyberetc-Part A, 997, 7: 6-3. [] F. Herrera ad E. Herrera-Vedma, Aggregato operator for lgutc weghted formato, IEEE Traacto o Sytem, Ma, ad Cyberetc-Part A, 997, 7: [] F. Herrera ad L. Martíez, A -tuple fuzzy lgutc repreetato model for computg wthword, IEEE Traacto o Fuzzy Sytem, 000, 8: [3] F. Herrera ad L.A. Martíez, A model baed o lgutc -tuple for dealg wth multgraular herarchcal lgutc cotext mult-expert deco makg, IEEE Traacto o Sytem, Ma, ad Cyberetc-Part B, 00, 3: [4] V. Torra, Aggregato of lgutc label whe ematc baed o atoym, Iteratoal Joural ofitellget Sytem, 00, 6: [5] Z.S. Xu, A method baed o lgutc aggregato operator for group deco makg wth lgutc preferece relato, Iformato Scece, 004, 66: [6] Z.S. Xu, Devato meaure of lgutc preferece relato group deco makg, Omega, 005, 33: [7] Z.S. Xu, Group deco makg baed o multple type of lgutc preferece relato, Iformato Scece, 008, 78: [8] Z.S. Xu, EOWA ad EOWG operator for aggregatg lgutc label baed olgutc preferece relato, Iteratoal Joural of Ucertaty, Fuzze ad Kowledge-Baed Sytem, 004, : [9] M. O Haga, Aggregatg template rule atecedet real-tme expert ytem wth fuzzy et logc, I: Proc d Aual IEEE Alomar Coferece o Sgal, Sytem ad Computer, IEEE ad Maple Pre, Pacfc Grove, CA, 988, pp [0] D.P. Flev ad R.R. Yager, O the ue of obtag OWA operator weght, Fuzzy Set ad Sytem, 998, 94: [] R.R. Yager, OWA aggregato over a cotuou terval argumet wth applcato to deco makg, IEEE Traacto o Sytem, Ma, ad Cyberetc-Part B, 004, 34: [] Z.S. Xu, A overvew of method for determg OWA weght, Iteratoal Joural of Itellget Sytem, 005, 0: