DOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA. Col.lecció d Economia

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1 DOCUMENTS DE TREBALL DE LA FACULTAT D ECONOMIA I EMPRESA Col.lecció d Ecoomia E09/232 The iduced 2-tuple liguistic geeralized OWA operator ad its applicatio i liguistic decisio makig José M. Merigó Lidahl Aa M. Gil-Lafuete Adreça correspodècia: Departamet d Ecoomia i Orgaització d Empreses Facultat d Ecoomia i Empresa Uiversitat de Barceloa Edifici Pricipal, Torre 2 3a plata Av. Diagoal 690, Barceloa, Spai Telephoe: Fax: merigo@ub.edu; amgil@ub.edu

2 Abstract: We preset the iduced 2-tuple liguistic geeralized ordered weighted averagig (2-TILGOWA) operator. This ew aggregatio operator exteds previous approaches by usig geeralized meas, order-iducig variables i the reorderig of the argumets ad liguistic iformatio represeted with the 2-tuple liguistic approach. Its mai advatage is that it icludes a wide rage of liguistic aggregatio operators. Thus, its aalyses ca be see from differet perspectives ad we obtai a much more complete picture of the situatio cosidered ad are able to select the alterative that best fits with with our iterests or beliefs. We further geeralize the operator by usig quasi-arithmetic meas, ad obtai the Quasi-2-TILOWA operator. We coclude this paper by aalysig the applicability of this ew approach i a decisio-makig problem cocerig product maagemet. Keywords: 2-tuple liguistic aggregatio operator; 2-tuple liguistic OWA operator; Liguistic geeralized mea; Liguistic decisio makig. JEL Classificatio: C44, C49, D81, D89. Resume: Se preseta el operador de media poderada ordeada geeralizada ligüística de 2 tuplas iducida (2-TILGOWA). Es u uevo operador de agregació que extiede los ateriores modelos a través de utilizar medias geeralizadas, variables de ordeació iducidas e iformació ligüística represetada mediate el modelo de las 2 tuplas ligüísticas. Su pricipal vetaa se ecuetra e la posibilidad de icluir a u gra úmero de operadores de agregació ligüísticos como casos particulares. Por eso, el aálisis puede ser visto desde diferetes perspectivas de forma que se obtiee ua visió más completa del problema cosiderado y seleccioar la alterativa que parece estar e mayor cocordacia co uestros itereses o creecias. A cotiuació se desarrolla ua geeralizació mayor a través de utilizar medias cuasi-aritméticas, obteiédose el operador Quasi-2-TILOWA. El trabao fializa aalizado la aplicabilidad del uevo modelo e u problema de toma de decisioes sobre gestió de la producció.

3 1. Itroductio The ordered weighted averagig (OWA) operator (Yager 1988) is a well-kow aggregatio operator for fusig umerical iformatio ad for decisio makig problems (Ah ad Park 2008; Aloso et al. 2008; Beliakov 2005; Beliakov et al. 2007; Caós y Lier 2008; Chiclaa et al. 2007; Emrouzead 2008; Liu 2008, 2009; Merigó ad Gil-Lafuete 2008a, 2008b, 2008c; Wag 2008; Xu 2005, 2008a, 2008b; Yager 1993, 1996, 2007a, 2008; Yager ad Kacprzyk 1997; ad Zarghami et al. 2008). However, situatios might arise i which the iformatio available is vague or imprecise ad we are uable to aalyze it usig umerical values. I such istaces, we require a alterative method such as a qualitative approach based o liguistic assessmets (Zadeh 1975). The literature describes various types of OWA operators usig liguistic iformatio (Boissoe 1982; Bustice et al. 2008; Herrera ad Herrera-Viedma, 1997; Herrera et al. 1995, 2008; Herrera ad Martíez 2000a, 2000b, 2001; Xu 2004, 2007, 2008; Yager 2007b; ad Zadeh 1975). I this paper, we adopt the 2-tuple liguistic OWA (2-TLOWA) operator ad its extesios (Herrera ad Martíez 2000a; Wag ad Hao 2006), ad OWA operator based o the 2-tuple liguistic represetatio model itroduced by Herrera ad Martíez (2000a). A iterestig extesio of this is the iduced OWA operator (Yager ad Filev 1999). This uses a more geeral formulatio i the reorderig process of the argumets by usig order-iducig variables. Its applicatio meas we are able to deal with more complex situatios that are ot depedet o the values of the argumets, that is, o the degree of optimism. Sice its itroductio the IOWA operator has received cosiderable attetio, see, for example, Chiclaa et al. (2004, 2007); Merigó ad Gil-Lafuete (2008b); ad Yager (2003). Further iterestig extesios of the OWA operator are the geeralizatios that use geeralized meas (Dyckhoff ad Pedrycz 1984) ad quasi-arithmetic meas, kow respectively as the geeralized OWA (GOWA) operator 3

4 (Karayiais 2000; Yager 2004) ad the Quasi-OWA operator (Beliakov 2005; Beliakov et al. 2007; Calvo et al. 2002; ad Fodor et al. 1995). They geeralize a wide rage of aggregatio operators such as the average, the OWA ad the ordered weighted geometric (OWG) operator (Herrera et al. 2003). Recetly, Merigó ad Gil-Lafuete (2008b) have suggested a geeralizatio of the IOWA operator by usig geeralized meas. This operator, kow as the iduced geeralized OWA (IGOWA) operator, geeralizes a wide rage of aggregatio operators such as the OWA ad the IOWA operator. Note that a further geeralizatio is possible by usig quasi-arithmetic meas (Quasi-IOWA operator). Takig this geeralizatio oe step further, i this paper we preset the iduced 2-tuple liguistic geeralized OWA (ILGOWA) operator. This represets a extesio of the IGOWA operator for those cases i which the iformatio available is assessed with liguistic variables i the form of the 2- tuple liguistic approach. I this way, we ca geeralize a wide rage of 2-tuple liguistic aggregatio operators icludig the 2-tuple iduced liguistic OWA (2-TILOWA), the 2-TLOWA, the 2-tuple liguistic weighted average (2- TLWA), the 2-tuple liguistic geeralized mea (2-TLGM), the 2-tuple liguistic weighted geeralized mea (2-TLWGM) ad the 2-tuple liguistic GOWA (2-TLGOWA), amog others. The mai advatage of this operator is that it icludes a wide rage of specific cases which eables us to cosider may differet situatios ad select the oe that best fits with our iterests. We also preset a further geeralizatio of the 2-TILGOWA operator - the Quasi-2-TILOWA operator - by usig quasi-arithmetic meas. Note that while various approaches have bee developed for dealig with liguistic iformatio (Boissoe 1982; Herrera ad Herrera-Viedma, 1997; Herrera et al. 1995, 2008; Herrera ad Martíez 2000a, 2000b, 2001; Wag ad Hao 2006; Xu 2004, 2007, 2008; ad Zadeh 1975), i this paper we focus o the ideas of Herrera ad Martíez (2000a, 2000b, 2001) ad compute with words (CWW) directly. It 4

5 should be oted, therefore, that this geeralizatio ca be see as a iitial step i the process of geeralizig the 2-TLOWA operator with geeralized meas ad quasi-arithmetic meas, sice further geeralizatios usig other liguistic models are possible. I our discussio of the applicability of the 2-TILGOWA operator, we are able to show that it is applicable i a wide rage of situatios. Ad we preset a specific applicatio of this ew approach i a liguistic decisio makig problem cocerig product maagemet. We focus, i particular, o the selectio of productio strategies. The mai coclusio we draw whe usig the 2-TILGOWA operator is that decisios ca vary depedig o the specific case used. Therefore, with the 2-TILGOWA operator, the decisio maker obtais a more complete view of the problem ad will select the decisio that is i closest accordace with his iterests. The rest of this paper is orgaized as follows. Sectio 2 presets some basic cocepts about the 2-tuple liguistic represetatio model, the 2-TLOWA operator ad the IGOWA operator. I Sectio 3, we preset the 2-TILGOWA operator ad study some of its mai properties. I Sectio 4 we aalyze a wide rage of families of 2-TILGOWA operators distiguishig betwee the weightig vector W ad the parameter λ. Sectio 5 itroduces the Quasi-ILOWA operator ad Sectio 6 presets a applicatio of the ew approach i a liguistic decisio makig problem. Sectio 7 brigs the paper to a close by summarizig its mai coclusios. 2. Prelimiaries I this sectio, we briefly review the 2-tuple liguistic approach, the 2-tuple liguistic OWA ad the iduced geeralized OWA operator. 5

6 2.1. The 2-tuple liguistic represetatio model We are used to workig i a quatitative settig, where the iformatio is expressed by meas of umerical values. However, may aspects of the real world caot be assessed i a quatitative maer ad we must work i a qualitative form, i.e., with vague or imprecise kowledge. I such istaces, a better approach might be provided by the use of liguistic assessmets rather tha umerical values. The liguistic approach represets qualitative aspects as liguistic values by meas of liguistic variables (Zadeh 1975). I adoptig this approach, we eed to select the appropriate liguistic descriptors for the term set ad its sematics. Oe way to geerate the liguistic term set ivolves directly supplyig the term set by cosiderig all the terms distributed o a scale alog which a total order is defied (Herrera ad Herrera- Viedma 1997). For example, a set of seve terms S could be give as follows: S = {s 1 = N, s 2 = VL, s 3 = L, s 4 = M, s 5 = H, s 6 = VH, s 7 = P} where N = Noe, VL = Very low, L = Low, M = Medium, H = High, VH = Very high, P = Perfect. Typically, i such cases, the liguistic term set should have the followig characteristics: A egatio operator: eg(s i ) = s such that = g+1 i. Be ordered: s i s if ad oly if i. Max operator: max(s i, s ) = s i if s i s. Mi operator: mi(s i, s ) = s i if s i s. Various approaches have bee forwarded for dealig with liguistic iformatio such as Boissoe (1982); Herrera ad Herrera-Viedma (1997); Herrera et al. (1995, 2008); Herrera ad Martíez (2000a, 2000b, 2001); Wag 6

7 ad Hao (2006); Xu (2004, 2007, 2008); Yager (2007b) ad Zadeh (1975). I this paper, we adopt the approach suggested by Herrera ad Martíez (2000a, 2000b, 2001). They developed a fuzzy liguistic represetatio model, which represets liguistic iformatio by usig a pair of values that they refer to as 2- tuple, (s, α), where s is a liguistic label ad α is a umerical value represetig the value of the symbolic traslatio. With this model, it is possible to udertake CWW processes without ay loss of iformatio, thereby overcomig oe of the mai limitatios of earlier liguistic computatioal models (Boissoe 1982; Herrera et al ad Zadeh 1975). Defiitio 1. Let β be the result of a aggregatio of the idexes of a set of labels assessed i the liguistic label set S = {s 0, s 1,, s g }, i.e., the result of a symbolic aggregatio operatio. β [0, g], where g + 1 is the cardiality of S. Let i = roud (β) ad α = β i be two values, such that, i [0, g] ad α [ 0.5, 0.5), the α is kow as the symbolic traslatio. Note that the 2-tuple (s i, α) that expresses the equivalet iformatio to β is obtaied with the followig fuctio: Δ : [0, g] S [ 0.5, 0.5), Δ(β) = s i α = β i i = roud( β ), α [ 0.5,0.5). (1) where roud is the usual roud operatio, s i has the closest idex label to β ad α is the value of the symbolic traslatio. For further iformatio o the 2-tuple liguistic represetatio model, see (Herrera ad Martíez 2000a, 2000b, 2001). 7

8 2.2. The 2-tuple liguistic OWA operator The 2-tuple liguistic OWA (2-TLOWA) operator is a liguistic aggregatio operator that uses the 2-tuple liguistic represetatio model i the OWA operator. It ca be defied as follows: Defiitio 2. Let Ŝ be the set of the 2-tuples. A 2-TLOWA operator of dimesio is a mappig f: Ŝ Ŝ, which has a associated weightig vector W such that w [0, 1] ad = 1w = 1, the: f ((s 1, α 1 ), (s 2, α 2 ),, (s, α )) = Δ ( w β * ) (2) = 1 * where β is the th largest of the 2-tuples (s i, α i ). Note that it is possible to distiguish betwee descedig (2-TDLOWA) ad ascedig (2-TALOWA) orders. Note also that the weights of these operators are related by w = w* +1, where w is the th weight of the 2-TDLOWA (or 2- TLOWA) operator ad w* +1 the th weight of the 2-TALOWA operator. Followig Herrera ad Herrera-Viedma (1997), we ca refer to the ascedig order as the iverse 2-TLOWA operator. By usig a differet weightig vector W, it is possible to study a wide rage of families of 2-TLOWA operators icludig the olympic-2-tlowa, the S-2- TOLWA, cetered-2-tolwa, etc. For further iformatio, see, for example, Merigó ad Gil-Lafuete 2008b; Xu 2005; or Yager

9 2.3. The iduced geeralized OWA operator The iduced geeralized OWA (IGOWA) operator is a extesio of the GOWA operator, with the differece that the reorderig step of the IGOWA operator is ot defied by the values of the argumets a i, but rather by order iducig variables u i, where the ordered positio of the argumets a i depeds upo the values of the u i. It ca be defied as follows: Defiitio 3. A IGOWA operator of dimesio is a mappig IGOWA: R R defied by a associated weightig vector W of dimesio such that the sum of the weights is 1 ad w [0, 1], a set of order-iducig variables u i, ad a parameter λ (, ), accordig to the followig formula: IGOWA( u 1,a 1, u 2,a 2,, u,a ) = 1/ λ λ b w (3) = 1 where (b 1,, b ) is (a 1, a 2,, a ) reordered i decreasig order of the values of the u i, the u i are the order-iducig variables, ad a i are the argumet variables. Note that it is possible to geeralize the IGOWA operator further by usig quasi-arithmetic meas. The, we obtai the Quasi-IOWA operator, which ca be defied as follows: Defiitio 4. A Quasi-IOWA operator of dimesio is a mappig QIOWA: R R defied by a associated weightig vector W of dimesio such that the sum of the weights is 1 ad w [0, 1], ad by a strictly mootoic cotiuous fuctio g(b), as follows: 1 QIOWA( u 1,a 1, u 2,a 2,, u,a ) = g ( ( )) w g b (4) = 1 9

10 where the b are the argumet values a i of the Quasi-IOWA pairs u i, a i ordered i decreasig order of their u i values. As we ca see, the differece betwee the IGOWA ad the Quasi-IOWA, is that we replace b λ with a geeral cotiuous strictly mootoic fuctio g(b). 3. The iduced liguistic geeralized OWA operator The 2-TILGOWA operator is a extesio of the OWA operator that uses liguistic assessmets, geeralized meas ad order-iducig variables i the reorderig of argumets. By usig liguistic iformatio assessed by the 2-tuple liguistic represetatio model, we are able to represet ucertaity more completely without losig ay iformatio i the computig process. By usig geeralized meas, we ca geeralize a wide rage of mea operators icludig the arithmetic mea, the geometric mea ad the quadratic mea. Ad by usig order-iducig variables, we obtai a more geeral formulatio of the reorderig process that ca deal with more complex situatios that are ot oly depedet o the values of the argumets. The 2-TILGOWA operator provides a parameterized family of 2-tuple liguistic aggregatio operators that icludes the 2-TILOWA operator, the 2-TLOWA, the 2-tuple liguistic maximum, the 2- tuple liguistic miimum ad the 2-tuple liguistic average (2-TLA), amog others. It ca be defied as follows: Defiitio 5. Let Ŝ be the set of the 2-tuples. A 2-TILGOWA operator of dimesio is a mappig f: Ŝ Ŝ, which has a associated weightig vector W such that w [0, 1] ad = 1w = 1, the: 10

11 f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 1/ λ λ w β (5) Δ = 1 where β are the argumet values (s i, α i ) of the 2-TILGOWA triplets (u i, s i, α i ) ordered i decreasig order of their u i, ad λ is a parameter such that λ (, ). Remark 1: Note that if λ 0, we ca oly use positive umbers R +, i order to obtai cosistet results. Remark 2: From a geeralized perspective of the reorderig step, it is possible to distiguish betwee the descedig 2-TILGOWA (2-TDILGOWA) operator ad the ascedig 2-TILGOWA (2-TAILGOWA) operator. The weights of these operators are related by w = w* +1, where w is the th weight of the 2- TDILGOWA ad w* +1 the th weight of the 2-TAILGOWA operator. Remark 3: If B is a vector correspodig to the ordered argumets s β λ, we shall call this the ordered argumet vector ad W T is the traspose of the weightig vector, the, the 2-TILGOWA operator ca be expressed as: f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = ( W T B) 1/ λ (6) Remark 4: Note that if the weightig vector is ot ormalized, i.e., W = = 1w 1, the, the 2-TILGOWA operator ca be expressed as: 11

12 f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 1 W = 1 w s λ β 1/ λ (7) The 2-TILGOWA operator is a mea or averagig operator. This is a reflectio of the fact that the operator is commutative, mootoic, bouded ad idempotet. These properties ca be demostrated with the followig theorems. Theorem 1 (Commutativity). Assume f is the 2-TILGOWA operator, the f ((u 1, s 1, α 1 ),, (u, s, α )) = f ((u 1, s 1, α 1 ),, (u, s, α )) (8) where ((u 1, s 1, α 1 ),, (u, s, α )) is ay permutatio of the argumets ((u 1, s 1, α 1 ),, (u, s, α )). Proof. Let f ((u 1, s 1, α 1 ),, (u, s, α )) = 1/ λ λ w β (9) Δ = 1 f ((u 1, s 1, α 1 ),, (u, s, α )) = Δ = 1 1/ λ λ X w (10) Sice ((u 1, s 1, α 1 ),, (u, s, α )) is a permutatio of ((u 1, s 1, α 1 ),, (u, s, α )), we have β = X, for all, ad the f ((u 1, s 1, α 1 ),, (u, s, α )) = f ((u 1, s 1, α 1 ),, (u, s, α )) 12

13 Theorem 2 (Mootoicity). Assume f is the 2-TILGOWA operator, if β i X i, for all i, the f ((u 1, s 1, α 1 ),, (u, s, α )) f ((u 1, s 1, α 1 ),, (u, s, α )) (11) Proof. Let f ((u 1, s 1, α 1 ),, (u, s, α )) = 1/ λ λ w β (12) Δ = 1 f ((u 1, s 1, α 1 ),, (u, s, α )) = = 1 1/ λ λ s χ w (13) Sice β i X i, for all i, it follows that, β X, ad the f ((u 1, s 1, α 1 ),, (u, s, α )) f ((u 1, s 1, α 1 ),, (u, s, α )) Theorem 3 (Bouded). Assume f is the 2-TILGOWA operator, the mi{(s i, α i )} f ((u 1, s 1, α 1 ),, (u, s, α )) max{(s i, α i )} (14) Proof. Let max{(s i, α i )} = c, ad mi{(s i, α i )} = d, the f ((u 1, s 1, α 1 ),, (u, s, α )) = 1/ λ λ Δ β w = 1 1/ λ λ w c = = 1 1/ λ λ c w (15) = 1 13

14 f ((u 1, s 1, α 1 ),, (u, s, α )) = 1/ λ λ Δ β w = 1 1/ λ λ w d = = 1 1/ λ λ d w (16) = 1 Sice = 1w = 1, we obtai f ((u 1, s 1, α 1 ),, (u, s, α )) c (17) f ((u 1, s 1, α 1 ),, (u, s, α )) d (18) Therefore, mi{(s i, α i )} f ((u 1, s 1, α 1 ),, (u, s, α )) max{(s i, α i )} Theorem 4 (Idempotecy). Assume f is the 2-TILGOWA operator, if (s i, α i ) = (s k, α k ), for all i, the f ((u 1, s 1, α 1 ),, (u, s, α )) = (s k, α k ) (19) Proof. Sice s αi = s α, for all i, we have f ((u 1, s 1, α 1 ),, (u, s, α )) = 1/ λ λ Δ β w = = 1 1/ λ λ Δ β w = = 1 λ Δ β = 1 1/ λ w (20) 14

15 Sice = 1w = 1, we obtai f ((u 1, s 1, α 1 ),, (u, s, α )) = (s k, α k ) Remark 5: Aother iterestig poit to cosider is the differet measures available for characterizig the weightig vector. For example, we could cosider the etropy of dispersio (Yager 1988), the divergece of W (Yager 2002) or the balace operator (Yager 1996). The etropy of dispersio is defied as follows: H(W) = w l( w ) (21) = 1 For the balace operator, we have: BAL (W) = w = 1 1 (22) Ad for the divergece of W: 2 DIV(W) = w ( W ) = 1 1 α (23) Note that i this case, it is also possible to distiguish betwee descedig ad ascedig orders. Remark 6: A iterestig poit whe aalyzig iduced liguistic aggregatio operators is the problem of ties i the reorderig step. To solve this problem, we recommed followig the method developed by Yager ad Filev (1999) 15

16 whereby they replace each argumet of the tied IOWA pair by its average. For the 2-TILGOWA operator, istead of usig the arithmetic mea, we replace each argumet of the tied 2-TILGOWA pair by its 2-TLGM depedig o the parameter of λ. Remark 7: As explaied i Yager ad Filev (1999) for the IOWA operator, we should ote that the values used for the order-iducig variables of the IGOWA operator, ca be draw from ay space that has a liear orderig. Thus, it is possible to use differet kids of attributes for the order-iducig variables; specifically, we ca mix umbers with words i the aggregatios (Yager ad Filev 1999). 4. Families of 2-TILGOWA operators I this sectio, we preset a wide rage of particular cases of 2-TILGOWA operators. We distiguish betwee the parameter λ ad the weightig vector W Aalysig the parameter λ If we aalyze differet values of the parameter λ, we obtai aother group of particular cases icludig the usual 2-TILOWA, the 2-TILOWG, the 2- TILOWHA ad the 2-TILOWQA operator. Note that we ca distiguish betwee descedig ad ascedig orders i each of these cases. Remark 8: Whe λ = 1, we obtai the 2-TILOWA operator. f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = w = 1 β (24) 16

17 Note that if w = 1/, for all i, we obtai the 2-TLA. If the ordered positio of u i = i, for all i, the 2-TLWA. Ad if u i =, for all i, the, we obtai the 2- TLOWA. Remark 9: Whe λ = 2, we obtai the 2-TILOWQA operator. f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = = 1 1/ 2 2 w β (25) If w = 1/, for all i, we obtai the 2-tuple liguistic quadratic average (2- TLQA). If the ordered positio of u i = i, for all i, the 2-tuple liguistic weighted quadratic average (2-TLWQA). Ad if u i =, for all i, the, we obtai the 2- tuple liguistic ordered weighted quadratic averagig (2-TLOWQA) operator. Remark 10: Whe λ = 0, we obtai the 2-TILOWG operator. f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = β (26) = 1 w If w = 1/, for all i, we obtai the 2-tuple liguistic geometric average (2- TLGA) ad if the ordered positio of u i = i, for all i, the 2-tuple liguistic weighted geometric average (2-TLWGA). If u i =, for all i, the, we obtai the 2-tuple liguistic ordered weighted geometric averagig (2-TLOWGA) operator. Remark 11: Whe λ = 1, we obtai the 2-TILOWHA operator. f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 1 = 1 w β (27) 17

18 Note that if w = 1/, for all i, we obtai the 2-tuple liguistic harmoic mea (2-TLHM) ad if the ordered positio of u i = i, for all i, the 2-tuple liguistic weighted harmoic mea (2-TLWHM). If u i =, for all i, the, we obtai the 2- tuple liguistic ordered weighted harmoic averagig (2-TLOWHA) operator. Note that we could aalyze other families by usig differet values i the parameter λ. Note also that it is possible to study these families idividually i a similar way to that reported i Sectios 3 ad Aalysig the weightig vector W By choosig a differet maifestatio of the weightig vector i the 2- TILGOWA operator, we are able to obtai differet types of aggregatio operators. For example, we ca obtai the 2-tuple liguistic maximum, the 2- tuple liguistic miimum, the 2-tuple liguistic geeralized mea (2-TLGM), the 2-TLWGM ad the 2-TLGOWA operator. Remark 12: The 2-tuple liguistic maximum is obtaied if w 1 = 1 ad w = 0, for all 1. The 2-tuple liguistic miimum is obtaied if w = 1 ad w = 0, for all. More geerally, if w k = 1 ad w = 0, for all k, we obtai the step-2- TILGOWA. The 2-TLGM is foud whe w = 1/, for all i. The 2-TLWGM is obtaied whe the ordered positio of i is the same as. Fially, the 2- TLGOWA is foud if the ordered positio of u i is the same as the ordered positio of the values of the a i. Remark 13: The 2-tuple liguistic media ca also be used as 2-TILGOWA operators. If is odd we assig w ( + 1)/2 = 1 ad w * = 0 for all others. If is eve the we assig, for example, w /2 = w (/2) + 1 = 0.5 ad w * = 0 for all others. 18

19 Remark 14: The olympic-2-tilgowa is foud whe w 1 = w = 0, ad for all others w * = 1/( 2). Note that if = 3 or = 4, the olympic-2-tilgowa is trasformed i the media-2-tilgowa ad if m = 2 ad k = 2, the widow-2-tilgowa is trasformed i the olympic-2-tilgowa. Remark 15: Followig (Liu 2009), it is possible to develop a geeral form of the olympic-2-tilgowa operator cosiderig that w = 0 for = 1, 2,, k,, 1,, k + 1; ad for all others w * = 1/( 2k), where k < /2. Note that if k = 1, the, this geeral form becomes the usual olympic-2-tilgowa. If k = ( 1)/2, the, this geeral form becomes the media-2-tilgowa aggregatio. Remark 16: Note that it is also possible to develop the cotrary case of the geeral olympic-2-tilgowa operator. I this case, w = (1/2k) for = 1, 2,, k,, 1,, k + 1; ad w = 0, for all others, where k < /2. Note that if k = 1, the, we obtai the cotrary case of the media-2-tilgowa. Remark 17: A further type of aggregatio that could be used is the E-Z 2- TILGOWA weights based o the E-Z IGOWA weights (Merigó ad Gil- Lafuete, 2008b). I this case, we should distiguish betwee two classes. I the first class, we assig w * = (1/q) for * = 1 to q ad w * = 0 for * > q, ad i the secod class, we assig w * = 0 for * = 1 to q ad w * = (1/q) for * = q + 1 to. Remark 18: The widow-2-tilgowa is foud whe w * = 1/m for k * k + m 1 ad w * = 0 for * > k + m ad * < k. Note that k ad m must be positive itegers such that k + m 1. 19

20 Remark 19: A further family of liguistic aggregatio operator that could be used is the cetered-2-tilgowa operator, based o the OWA versio (Yager, 2007a). We ca defie a 2-TILGOWA operator as a cetered aggregatio operator if it is symmetric, strogly decayig ad iclusive. It is symmetric if w = w + 1. It is strogly decayig whe i < ( + 1)/2 the w i < w ad whe i > ( + 1)/2 the w i < w. It is iclusive if w > 0. Note that it is possible to cosider a softeig of the secod coditio by usig w i w istead of w i < w. (softly decayig cetered-2-tilgowa operator). Aother particular situatio of the cetered-lgowa operator appears if we remove the third coditio (o-iclusive cetered-2-tilgowa operator). Remark 20: Aother iterestig family is the S-2-TILGOWA operator based o the S-OWA operator (Yager 1993). It ca be subdivided i three classes, the orlike, the adlike ad the geeralized S-2-TILGOWA operator. The geeralized S-2-TILGOWA operator is obtaied whe w 1 = (1/)(1 (α + β)) + α, w = (1/)(1 (α + β)) + β, ad w = (1/)(1 (α + β)) for = 2 to 1 where α, β [0, 1] ad α + β 1. Note that if α = 0, the geeralized S-2- TILGOWA operator becomes the adlike S-2-TILGOWA operator ad if β = 0, it becomes the orlike S-2-TILGOWA operator. Also ote that if α + β = 1, we obtai the 2-tuple iduced liguistic geeralized Hurwicz criteria. Remark 21: Aother iterestig family is the omootoic-2-tilgowa operator that follows the ideas of (Yager, 1999). It is foud whe at least oe of the weights w is lower tha 0 ad = 1w = 1. Note that a key aspect of this 20

21 operator is that it does ot always accomplish the mootoicity property. Therefore, this operator is ot strictly a particular case of the 2-TILGOWA ad ca be see as a differet type of aggregatio operator. Remark 22: Usig a similar methodology, may other families of 2-TILGOWA weights could be similarly developed as have bee reported i may studies for the OWA operator, icludig Ah ad Park (2008); Beliakov (2005); Beliakov et al. (2007); Chiclaa et al. (2007); Emrouzead (2008); Liu (2008, 2009); Merigó ad Gil-Lafuete (2008a, 2008b, 2008c); Xu (2005); ad Yager (1993, 1996, 2007a). Remark 23: Note that it is relatively straightforward to apply these methods to the 2-TILGOWA operator as the weights are ot affected by the liguistic iformatio. Obviously, more complex aalyses might be udertake i which the weights are also liguistic variables, but i this paper we do ot tackle this problem. 5. Quasi-2-TILOWA operators As explaied i Beliakov (2005), a further geeralizatio of the GOWA operator is possible usig quasi-arithmetic meas. Adoptig the same methodology, we ca suggest a similar geeralizatio of the 2-TILGOWA operator by usig these quasi-arithmetic meas. We call this geeralizatio the Quasi-2-TILOWA operator. The, we obtai a more geeral formulatio of the reorderig process by usig order iducig variables ad this is able to deal with more complex situatios. The Quasi-2-TILOWA operator ca be defied as follows: 21

22 Defiitio 6. Let Ŝ be the set of the 2-tuples. A Quasi-2-TILOWA operator of dimesio is a mappig f: Ŝ Ŝ that has a associated weightig vector W of dimesio such that the sum of the weights is 1 ad w [0,1], the: = f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = g w g( s ) 1 1 β (28) where s β is the th largest of the s αi. As we ca see, we replace s λ β with a geeral cotiuous strictly mootoe fuctio g(s β ). I this case, the weights of the ascedig ad descedig versios are also related by w = w* +1, where w is the th weight of the Quasi-2- TDILOWA ad w* +1 the th weight of the Quasi-2-TAILOWA operator. Remark 24: As explaied i the case of the 2-TILGOWA, if the weightig vector is ot ormalized, i.e., W = operator ca be expressed as: = 1w 1, the, the Quasi-2-TILOWA W 1 1 f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = g w g( s ) = 1 β (29) Remark 25: Note that all the properties ad particular cases commeted i the 2-TILGOWA operator are also icluded i this geeralizatio. For example, we could study differet families of Quasi-2-TILOWA operators such as the Quasi- 2-TLA, the Quasi-2-TLWA, the Quasi-S-2-TILOWA, the Quasi-olympic-2- TILOWA, the Quasi-cetered-2-TILOWA, etc. Remark 26: Note also that the Quasi-2-TILOWA operator icludes may other cases that are ot icluded i the 2-TILGOWA such as the trigoometric 2-22

23 TILOWA, the radical 2-TILOWA, the expoetial 2-TILOWA, etc. These aggregatios follow the same methodology as the OWA versio (Beliakov et al. 2007) with the differece that ow we are usig liguistic iformatio i the problem. For the radical 2-TILOWA operator, we obtai: f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 1 1/ s = 1 β w γ log γ (30) For the trigoometric 2-TILOWA operator we form the followig equatios: f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 2 arcsi w π = π si sβ 1 2 (31) f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 2 arccos w π = π cos sβ 1 2 (32) f ((u 1, s 1, α 1 ), (u 2, s 2, α 2 ),, (u, s, α )) = 2 arcta w π = π ta sβ 1 2 (33) Ad for the expoetial 2-TILOWA, we obtai: the 2-TILOWA if γ = 1. log γ = 1 w γ sβ, if γ 1; ad 23

24 6. Applicatio i liguistic decisio makig May differet applicatios are possible usig the 2-TILGOWA operator. I priciple, it ca be applied to similar situatios to those described whe cosiderig the OWA operator. Moreover, it has a rage of other applicatios that ca be developed i may differet fields, icludig: Decisio theory Statistics Ecoomics Busiess decisio makig Mathematics Physics Clearly, i each field, may differet applicatios are also possible. For example, i busiess decisio makig we might cosider fiacial problems (Merigó ad Gil-Lafuete, 2007), huma resource maagemet, strategic maagemet or product maagemet, amog others. Below, we focus o a applicatio of the 2-TILGOWA operator to a busiess decisio-makig problem. Specifically, we aalyze a product maagemet problem i which a compay seeks to pla its productio strategy for the forthcomig year. Let us assume they cosider five alteratives: A 1 = Create a ew product for high-icome customers. A 2 = Create a ew product for medium-icome customers. A 3 = Create a ew product for low-icome customers. A 4 = Create a ew product suitable for all customers. A 5 = Do ot create a ew product. 24

25 As the eviromet is highly ucertai, the eterprise s experts are uable to draw o umerical iformatio i coductig their aalysis. Rather, they have to rely o liguistic iformatio assessed usig the 2-tuple liguistic represetatio model. The results of these liguistic values are as follows. Note that i this example the experts use a set of seve terms S as follows: S = {s 1 = N, s 2 = VL, s 3 = L, s 4 = M, s 5 = H, s 6 = VH, s 7 = P} where N = Noe, VL = Very low, L = Low, M = Medium, H = High, VH = Very high, P = Perfect. We ext aalyze the results obtaied by usig differet types of 2-TILGOWA operators i order to see the rage of differet results i lie with the attitude adopted by the compay i the face of ucertaity. I this example, we cosider the 2-tuple liguistic maximum, the 2-tuple liguistic miimum, the 2-TLA, the 2-TLGA, the 2-TLQA, the 2-TLWA, the 2-TLOWA operator, the 2-TILOWA, the 2-TILOWG, the 2-TILOWQA operator, the media-2-tilowa ad the olympic-2-tilowa. I evaluatig these strategies, the experts cosider the key factor as beig the firm s ecoomic situatio over the forthcomig year. Followig careful aalysis, they cosider five potetial scearios: S 1 = Very bad, S 2 = Bad, S 3 = Regular, S 4 = Good, S 5 = Very good. The expected liguistic results depedig o situatio N i ad alterative A k are show i Table 1. Note that the results are liguistic values represeted with the 2-tuple liguistic approach. 25

26 Table 1: Liguistic payoff matrix N 1 N 2 N 3 N 4 N 5 A 1 (s 4, 0.5) (s 3, 0.1) (s 3, 0) (s 4, 0.4) (s 6, 0.2) A 2 (s 4, 0) (s 5, 0.2) (s 2, 0.3) (s 4, 0.2) (s 5, 0.1) A 3 (s 2, 0.2) (s 3, 0.3) (s 3, 0.5) (s 5, 0) (s 6, 0.1) A 4 (s 3, 0) (s 5, 0.3) (s 5, 0) (s 4, 0.4) (s 3, 0.2) A 5 (s 5, 0) (s 4, 0) (s 4, 0.2) (s 3, 0.5) (s 2, 0.4) I this example, we assume that the experts assume the followig weightig vector for all the cases: W = (0.1, 0.2, 0.3, 0.4, 0.5). Note that this weightig vector is used as a weighted average i the 2-TLWA, while for the rest it is used to represet the attitudial character of the eterprise. Note that the attitudial character of the compay is particularly complex sice it ivolves the opiios of the differet members sittig o the board of directors. Thus, the compay s experts use order-iducig variables to represet the attitudial character. The results are show i Table 2. Table 2: Order-iducig variables N 1 N 2 N 3 N 4 N 5 A A A A A

27 This iformatio ca the be aggregated i order to take a decisio. The results are show i Tables 3 ad 4. Table 3: Aggregate liguistic results 1 Max Mi LA LGA LQA LWA A 1 (s 6, 0.2) (s 3, 0) (s 4, 0) (s 4, 0.13) (s 4, 0.13) (s 4, 0.13) A 2 (s 5, 0.2) (s 2, 0.3) (s 4, 0) (s 4, 0.17) (s 4, 0.18) (s 4, 0.09) A 3 (s 6, 0.1) (s 2, 0.2) (s 4, 0.22) (s 3, 0.46) (s 4, 0.06) (s 4, 0.19) A 4 (s 5, 0) (s 3, 0) (s 4, 0.1) (s 4, 0.19) (s 4, 0.02) (s 4, 0.08) A 5 (s 5, 0) (s 2, 0.4) (s 4, 0.26) (s 4, 0.37) (s 4, 0.17) (s 3, 0.48) Table 4: Aggregate liguistic results 2 LOWA ILOWA ILOWG ILOWQA Media Olympic A 1 (s 4, 0.28) (s 4, 0.15) (s 4, 0.03) (s 4, 0.26) (s 3, 0.1) (s 4, 0.16) A 2 (s 4, 0.35) (s 4, 0.07) (s 4, 0.33) (s 4, 0.11) (s 5, 0.2) (s 4, 0.37) A 3 (s 4, 0.63) (s 4, 0.19) (s 4, 0.1) (s 4, 0.43) (s 3, 0.5) (s 4, 0.27) A 4 (s 4, 0.3) (s 4, 0.04) (s 4, 0.16) (s 4, 0.01) (s 5, 0) (s 4, 0.3) A 5 (s 3, 0.48) (s 4, 0) (s 4, 0.09) (s 4, 0.07) (s 4, 0.2) (s 4, 0.24) As we see, differet results are obtaied depedig o the liguistic aggregatio operator used ad, cosequetly, the decisio maker ca take differet decisios. Note that more specific istaces of the LGOWA operator could be cosidered i the aalysis such as those described above i the previous sectios. 27

28 A further iterestig issue ivolves establishig a orderig for the productio strategies. Note that this is particularly useful whe we wish to cosider more tha oe productio strategy i the aalysis. The results are show i Table 5. Table 5: Orderig of the productio strategies Orderig Orderig Max A 3 A 1 A 2 A 4 =A 5 2-TLOWA A 1 A 4 A 2 A 5 A 3 Mi A 1 =A 4 A 5 A 3 A 2 2-TILOWA A 3 A 1 A 5 A 4 A 2 2-TLA A 1 =A 2 A 4 A 3 A 5 2-TILOWG A 1 A 5 A 3 A 4 A 2 2-TLGA A 1 A 4 A 2 A 5 A 3 2-TILOWQA A 3 A 1 A 2 A 5 A 4 2-TLQA A 2 A 1 A 3 A 4 A 5 Media A 2 A 4 A 5 A 3 A 1 2-TLWA A 3 A 1 A 2 A 4 A 5 Olympic A 4 A 1 A 5 A 3 A 2 As we ca see, depedig o the liguistic aggregatio operator used, the orderig of the productio strategies differs. Thus, the decisio maker ca the cosider a wide rage of scearios ad select the specific case that best fits with his iterests. 7. Coclusios I this article we have preseted the iduced 2-tuple liguistic geeralized OWA operator, a aggregatio operator that uses geeralized meas, order-iducig variables i its reorderig of argumets ad ucertai iformatio assessed with the 2-tuple liguistic represetatio model. We have demostrated that this operator ca be of great use because it geeralizes a wide rage of liguistic aggregatio operators icludig the 2-TLA, the 2-TLWA, the 2-TLOWA, the 2-28

29 TILOWA, the 2-TLWGM ad the 2-TLGOWA, amog others. The mai advatage of this operator is that it makes it possible to cosider a wide rage of results depedig o the particular type of 2-TILGOWA operator beig used. I this way, the same problem ca be viewed from a rage of perspectives ad the solutio that best fits our iterests ca be selected. I a additioal step, we have further geeralized the 2-TILGOWA operator usig quasi-arithmetic meas. This we have called the Quasi-2-TILOWA operator. I this case, the mai advatage is that the more geeral formulatio provided allows us to cosider may situatios that are ot icluded i the 2- TILGOWA. We have also discussed here the applicability of this ew approach. We have demostrated that it is applicable i a wide rage of fields icludig decisio theory, statistics, ecoomics, etc. We have preseted a applicatio i a decisio makig problem cocerig product maagemet. The mai advatage of usig the 2-TILGOWA operator is that the decisio maker obtais a more complete view of the problem because he is able to cosider a wide rage of situatios ad select the oe that best fits with his iterests. I our future research, we wish to exted this approach to other situatios that ca be assessed by applyig liguistic approaches. Our primary motivatio is that we believe the computig with words process of the 2-tuple liguistic approach eeds to be improved so as to make it more efficiet. Moreover, we also wish to examie other decisio-makig applicatios i fields such as fiacial maagemet ad huma resource selectio. 29

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