Economic Computation and Economic Cybernetics Studies and Research, Issue 2/2017, Vol. 51

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1 Ecoomic Computatio ad Ecoomic Cyberetics Studies ad Research, Issue 2/2017, Vol. 51 Biyami YUSOFF, PhD Cadidate School of Iformatics ad Applied Mathematics Uiversity of Malaysia Tereggau, Malaysia Jose Maria MERIGÓ, PhD Departmet of Maagemet Cotrol & Iformatio Systems Uiversity of Chile, Satiago, Chile David CEBALLOS, PhD Departmet of Mathematical Ecoomics, Fiace & Actuarial Scieces Uiversity of Barceloa, Barceloa, Spai OWA-BASED AGGREGATION OPERATIONS IN MULTI-EXPERT MCDM MODEL Abstract. This paper presets a aalysis of multi-expert multi-criteria decisio makig (ME-MCDM) model based o the ordered weighted averagig (OWA) operators. Two methods of modelig the majority opiio are studied as to aggregate the experts judgmets, i which based o the iduced OWA operators. The, a overview of OWA with the iclusio of differet degrees of importace is provided for aggregatig the criteria. A alterative OWA operator with a ew weightig method is proposed which termed as alterative OWAWA (AOWAWA) operator. Some extesios of ME-MCDM model with respect to two-stage aggregatio processes are developed based o the classical ad alterative schemes. A compariso of results of differet decisio schemes the is coducted. Moreover, with respect to the alterative scheme, a further compariso is give for differet techiques i itegratig the degrees of importace. A umerical example i the selectio of ivestmet strategy is used as to exemplify the model ad for the aalysis purpose. Keywords: multi-expert MCDM; OWA operator; IOWA operator; majority cocept; weightig methods, fiacial decisio makig. JEL Classificatio: C44, D81, G11 1. Itroductio I the past, various multi-criteria decisio makig models have bee developed as tools for modelig huma decisio makig ad reasoig (see, Figueira et al., 2005). The models have bee extesively used i umerous applicatios to deal 211

2 Biyami Yusoff, Jose Maria Merigó, David Ceballos with the rakig ad selectio of optio (or alterative). I complex decisio makig problems, ormally a group of experts (or decisio makers) ivolved i which each of them offsets ad/or support the others for a exhaustive judgmet. Sice the, the expasio of such models to multi-expert MCDM (ME-MCDM) problems has become the mai focus i the literature (see, for example, i Taib et al., 2016). Cetral to the ME-MCDM problems, aggregatio process plays a crucial role i obtaiig the fial decisio, either to sythesize the criteria or to fuse the overall judgmet of experts. A overview of the mai aggregatio operators ad their properties ca be referred, for istace, i Beliakov et al. (2007) ad Grabisch et al. (2009). The weighted arithmetic mea (WA) ad the ordered weighted averagig (OWA) operators are amog the most widely used aggregatio operators i the decisio makig models. The OWA (Yager, 1988) provides a geeral class of mea-type aggregatio operators which ca be raged from two extreme cases, i.e., ad (mi) ad or (max) operators. It modifies the basic aggregatio process used i decisio makig model by applyig the cocept of fuzzy set theory, precisely, usig the fuzzy liguistic quatifiers (Zadeh, 1983) for a soft aggregatio process. I compariso to the WA which represets the degrees of importace associated with particular criteria, the weights i OWA reflect the importace or satisfactio of values with respect to orderig. By appropriately selectig the weightig vector, differet kids of relatioships betwee the criteria ca be modeled. I certai cases, the WA is ecessary i represetig the MCDM problems. For example, some experts may prefer to associate a specific weight for each criterio based o its degree of importace. Hece, cosiderig the advatages of both WA ad OWA i modelig the real applicatios, Yager (1988) the proposed the iclusio of uequal degrees of importace i OWA as a itegrated approach. Cosequetly, a umber of other techiques to deal with the same problem have bee developed. The itegratio of these weightig methods has bee formalized i two differet approaches. I the first approach, the relative weights are oly used to modify the argumet values to be aggregated, specifically without the direct itegratio with ordered weights. Examples i this category iclude the method based o max-mi ad product (Yager, 1988), fuzzy system modelig (Yager, 1998) ad hybrid weighted average (Xu ad Da, 2003). O the other had, i the secod approach, the relative weights ad ordered weights are directly itegrated as a ew set of weights, e.g., method based o liguistic quatifiers (Yager, 1996), weighted OWA (WOWA) (Torra, 1997), OWAWA (Merigó, 2012) ad immediate WA (IWA) (Llamazares, 2013). Aother importat variat of OWA is the iduced OWA (IOWA) operator (Yager ad Filev, 1999). Geerally, it is a extesio of the OWA which ivolves a pair of values, such as, the additioal parameter (order-iducig variables) used to iduce the argumet values to be aggregated. Aalogously, with respect to a group decisio makig, the majority agreemet amog experts ca be implemeted usig the IOWA operators, which sythesizes the opiios of the majority of experts. I this case, the majority opiio refers to a cosesual judgmet of 212

3 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model majority of experts who have similar opiios. I geeral, the OWA ad IOWA operators provide a more flexible model for combiig the iformatio i decisio makig problems, specifically i the complex eviromet where the attitudial character of experts is cosidered. O the basis of previous discussio, the purpose of this study is o extedig ad aalyzig the ME-MCDM model with respect to two-stage aggregatio processes, otably, the fusio of criteria ad the aggregatio of experts judgmets. Firstly, two models based o majority cocept for aggregatig the experts judgmets are reviewed. I particular, the methods as itroduced by Pasi ad Yager (2006) ad its extesio by Bordoga ad Sterlacchii (2014). Pasi ad Yager (2006) proposed the method i case of the weights betwee experts are cosidered as idetical (homogeeous group decisio makig) ad employed a support fuctio based o distace measure to compute the majority agreemet betwee experts. Besides, the support betwee experts is calculated with respect to the fial rakigs of optios which derived primarily by each expert (classical scheme). O the cotrary, Bordoga ad Sterlacchii (2014) the exteded this idea to iclude the case where the experts are assiged with differet degrees of importace (heterogeeous group decisio makig) ad utilized the similarity measure based o Mikowski OWA (MOWA) to calculate the support betwee experts. Istead of focusig o the idividual rakig o optios of each expert, they provide the similarity measure with respect to each specific criterio (alterative scheme). I this study, for the purpose of compariso, some modificatios have bee made to both methods. I specific, the extesio of Pasi- Yager method from the classical scheme to the alterative scheme has bee made. Likewise, the Bordoga-Sterlacchii method has bee modified to deal with the classical scheme. Hece, these methods with the existig origial methods are applied i the ME-MCDM model ad the a compariso as to examie the results of differet schemes is coducted. Secodly, some methods based o the itegratio of OWA ad WA for the purpose of aggregatig the criteria are preseted. I additio, a alterative OWAWA (AOWAWA) operator which combies the characteristics of IWA ad OWAWA usig the idea of geometric mea is proposed. As a compariso, the ME-MCDM model with respect to Bordoga-Sterlacchii approach o the alterative scheme is applied as to observe the results of distict weightig techiques i the aggregatio process. The outlie of this paper is as follows. I Sectio 2 the defiitios of OWA, IOWA ad MOWAD operators are preseted. I Sectio 3 the aggregatio techiques for modelig the majority opiio are discussed. The, Sectio 4 reviews the itegrated weightig methods based o WA ad OWA as well as the proposed AOWAWA operator. I Sectio 5, the geeral frameworks of ME-MCDM model based o classical ad alterative schemes are outlied. The, a umerical example i a selectio of ivestmet strategy is provided i sectio

4 Biyami Yusoff, Jose Maria Merigó, David Ceballos 2. Prelimiaries This sectio provides the defiitios ad basic cocepts related to OWA, IOWA ad MOWAD aggregatio operators that will be used throughout the study. 2.1 OWA operator Defiitio 1. (Yager, 1988). A OWA operator of dimesio is a mappig OWA: R R that has a associated weightig vector W = (w 1, w 2,, w ) of dimesio, such that w j [0,1] ad w j = 1, give by the followig formula: OWA W (a 1,, a ) = w j a σ(j) where a σ(j) is the argumet value a j beig ordered i o-icreasig order a σ(1) a σ(2) a σ(). Note that, the reorderig process makes the OWA operator is o loger a stadard liear combiatio of weighted argumets, but it is rather a piecewise liear fuctio (Beliakov ad James, 2011). Give that a fuctio Q: [0,1] [0,1] as a regular mootoically odecreasig fuzzy quatifier ad it satisfies: i) Q(0) = 0, ii) Q(1) = 1, iii) a > b implies Q(a) Q(b), the the associated OWA weights ca be derived usig this fuctio as follows (Yager, 1988): w j = Q ( j 1 ) Q (j ), j = 1,2,, (2) such that w j [0,1] ad w j = 1. The liguistic quatifier Q(Zadeh, 1983) ca be preseted i the form of Q(r) = r γ, γ > 0 with the mai characteristics such that: γ 0, the W = W, where W = (1,0,,0); γ = 1 the W = W 1/, where W 1/ = (1/, 1/,,1/); ad γ the W = W, where W = (0,0,,1). 2.2 IOWA operator Defiitio 2. (Yager ad Filev, 1999). A IOWA operator of dimesio is mappig IOWA: R R that has a associated weightig vector W such that w j [0,1] ad w j = 1, give by the followig formula: IOWA W ( u 1, a 1, u 2, a 2,, u, a ) = w j a σ(j) where a σ(j) is the argumet value of pair u j, a j of order-iducig variable u j, reordered such that u σ(1) u σ(2) u σ() ad the covetio that ifu σ(j) are (3) (1) 214

5 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model tied, i.e., u σ(j) = u σ(j+1), the, the value a σ(j) is give as their average (see, Yager ad Filev, 1999; Beliakov ad James, 2011). 2.3 Mikowski OWA distace Defiitio 3. (Merigó ad Gil-Lafuete, 2008). A MOWAD operator of dimesio is a mappig MOWAD: R R R that has a associated weightig vector W of dimesio such that = 1 with w j [0,1] ad the distace betwee w j two sets A ad B is give as follows: MOWAD W (d 1, d 2,, d ) = ( λ w j d σ(j) 1/λ ), (4) where d σ(j) is the compoet of d j beig ordered i o-icreasig order d σ(1) d σ(2) d σ() ad d j is the idividual distace betwee A ad B, such that d j = a j b j with λ is a parameter i a rage λ R {0}. By settig differet values for the orm parameter λ, some special distace measures ca be derived. For example, if λ = 1, the the Mahatta OWA distace ca be obtaied, ifλ = 2 the the Euclidea OWA distace ca be acquired, λ = the Tchebycheff OWA is derived, etc. Equivaletly, OWA ad IOWA operators ca be geeralized i the similar way (see, Merigó ad Gil- Lafuete, 2009; Merigó ad Yager, 2013; Yager, 2004).The OWA, IOWA ad MOWAD operators are all satisfyig commutative, mootoic, bouded ad idempotet properties. 3. Aggregatio Methods based o Majority Cocept I this sectio, the methods for aggregatig experts judgmets by the iclusio of majority cocept are preseted. I particular, the method by Pasi ad Yager (2006) ad its extesio by Bordoga ad Sterlacchii (2014) are studied. 3.1 Pasi-Yager approach I the followig, a brief descriptio of the metioed methods is give. Two fudametal steps i both methods are o determiig the order-iducig variable ad o derivig the associated weights of experts. The methodology used to obtai the majority opiio based o Pasi ad Yager (2006) ca be expressed as the followig. Suppose that a set of idividual opiios of hexperts (h = 1,2,, k) is give as the vector P h i = (p 1 i, p 2 i,, p k i ), i.e., with respect to each optio i, (i = 1,2,, m). For a simple otatio, P h ca be used istead ofp h i sice each optio ca be evaluated idepedetly usig the same formulatio. For a sigle optio, the similarity of each expert ca be calculated usig the support fuctio as follows: supp(p l, p h ) = { 1 if p l p h < β, 0 otherwise. (5) 215

6 Biyami Yusoff, Jose Maria Merigó, David Ceballos The support fuctio represets the similarity or dissimilarity betwee expert l with each of the other experts h, (h = 1,2,, k) (ot iclude himself/herself), such that l h. The the overall support for each idividual expert l ca be give as: k u l = supp(p l, p h ) h=1 h l where u l costitute the values of order-iducig variable U = (u σ(1),, u σ(k) ) which ordered i o-decreasig order, such that u σ(1) u σ(2) u σ(k). I cosequece, to compute the weights of the weightig vector, defie the values t l based o a adjustmet of the u l values, such that: t l = u l + 1 (icludig himself/herself:supp(p l, p l ) = 1). The t l values are i o-decreasig order, t 1 t 2 t k. O the basis of t l values, the weights are computed as follows: w l = Q(t l k) k. Q(t l k) l=1 The value Q(t l k) deotes the degree to which a give member of the cosidered set of values represets the majority. The quatifier Q with sematic most for the majority opiio of experts ca be give as follows:, (6) (7) 1 if r 0.9, Q(r) = { 2r 0.8 if 0.4 < r < 0.9, 0 if r 0.4, (8) where r = t l k. As ca be see, the weight of experts here is derived based o the arithmetic mea (AM) where each expert is cosidered as havig a equal degree of importace or trust, e.g., reflect the average of the most of the similar values..the, the fial evaluatio is determied usig the IOWA operators. Note that, here the values of order-iducig variable are reordered i o-decreasig order istead of o-icreasig order as i the origial IOWA, such i Eq. (3). This type of orderig reflects the coformity of quatifier most as to model the majority cocept (see, Pasi ad Yager, 2006) for detailed explaatio. Note also that, the quatifier Q here is a alterative represetatio of Q(r) = r γ. For represetig the majority opiio of experts, this type of quatifier will be used throughout the study. However, the vector P h i = (p 1 i, p 2 i,, p k i ), that derived after the first stage of aggregatio process shows a slight differet betwee its values due to the ormalizatio process. This coditio the leads to the values of p l p h less differetiable ad cause a difficulty i assigig a value for β. Hece, i this study, a slight modificatio to the support fuctio i Eq. (6) is suggested ad the formulatio is give as follows: 216

7 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model supp(p l, p h ) = { 1 if p l p h max p l p h < β, (9) l h 0 otherwise. where max l h p l p h is the maximum distace betwee all experts. Example 1: Suppose that a set of idividual opiio of experts is give asp h = (p 1, p 2,, p 5 ) = (0.7, 0.86, 0.76, 0.72, 0.6) with respect to each optio, A i. The, the fial majority opiio of experts ca be computed as the followig. A i E 1 E 2 E 3 E 4 E 5 E 1 E 2 E 3 E 4 E 5 h P i supp 1,h divided by supp 2,h max l p h supp 3,h h supp 4,h supp 5,h By settig β = 0.4, the overall support for each expert ca be obtaied, such as: s 1 = 3, s 2 = 1, s 3 = 3, s 4 = 2, ad s 5 = 1. I case of ties, the stricter β ca be imposed (β = 0.1,i this example), to order the p h values. The vector of orderiducig variable the ca be give as U = (u σ(1),, u σ(5) ) = (1,1, 2, 3, 3) ad the weightig vector ca be obtaied as W Maj = (w 1,, w 5 ) = (0, 0, 0.2, 0.4, 0.4). The fial majority opiio of experts ca be calculated as follows: IOWA( 1, 0.6, 1, 0.86, 2, 0.72, 3, 0.76, 3, 0.7 ) = (0 0.6) + (0 0.86) + ( ) + ( ) + ( ) = Bordoga-Sterlacchii approach I the followig, the method based o Bordoga ad Sterlacchii (2014) is preseted. Cotrary to the previous method, here the majority opiio of experts with respect to each specific criterio is cosidered. Suppose that a collectio of judgmet of hexperts is give as vector P j h = (p j 1, p j 2,, p j k ) for criterio j, (j = 1,2,, ). I this method, istead of usig the support fuctio based o distace measure, they used the Mikowski OWA-based similarity measure to obtai the Q coherece for the order-iducig variable. The Q coherece of each expert l ca be defied as follows: 217

8 Biyami Yusoff, Jose Maria Merigó, David Ceballos λ u l = Q coherece (P l, P h ) = MOWA(s 1,, s k ) = ( ω h s σ(h) k h=1 ) 1/λ, (10) where s l = s(p l, p h ) = 1 p l p h is a similarity measure betwee expert l with each of the other experts h (icludes himself), give that l h ad s σ(h) are orderig of (s 1,, s k ) i o-icreasig order (s σ(1) s σ(2) s σ(k) ). Meawhile ω h are the ordered weights with the iclusio of importace degrees of h h 1 experts t h, h = 1,2,, k, give as ω h = Q( i=1 t σ(i) ) Q( i=0 t σ(i) ), such thatω h, t h [0,1] k ad ( h=1 ω h = k h=1 t h = 1). The orm parameter λ R {0} provides a geeralizatio of the model. Here the quatifier Q(r) = r γ is employed. The OWA weights ω h will be explaied i great detail i the ext sectio. With respect to the Eq. (10), the order iducig vector ca be give as: U = (u 1,, u k ) = (Q coherece (P 1, P h ),, Q coherece (P k, P h )), (11) Moreover, Q as the geeralized quatifiers ca take ay sematics to modify the weights of experts (or trust degrees) for differet strategies. Whe Q(t h ) = t h as for (γ = 1), the Q coherece is reduced to: u l = coherece(p l, P h ) λ = ( t h s h ) k h=1 1 λ, (12) which is the Mikowski WA-based similarity measure. Formally, Q coherece ca be raged i betwee Q (t h ) for γ 0, to Q (t h )for γ. Afterwards, the weights for the IOWA operator ca be derived usig the followig formula: m h = argmi h(u 1 t 1,, u k t k ) k, (13) argmi i (u 1 t 1,, u k t k ) h=1 where m h are reordered i o-decreasig order. Aalogously, give the quatifier Q as i Eq. (8) for the majority opiio, the weightig vector W Maj = (w 1,, w k ) ca be computed as follows: w h = Q(m h) k. (14) Q(m h ) h=1 Note that, the geeral weights w h represet the quatificatio of majority of experts for the fial agreemet o each criterio, whilst the weights ω h reflect Q coherece for derivig the order-iducig values. Next, the overall aggregatio process ca be computed usig the IOWA such i Eq. (3). Similarly, the o-decreasig iputs u h, p h is implemeted as explaied 218

9 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model i previous sub-sectio. It ca be show that, the coherece fuctio Eq. (12) ca be represeted as the dual of similarity measure, which is the distace measure: coherece(p l, P h ) = ( t h (1 p l p h ) λ ) k h=1 k = 1 ( t h p l p h λ ) h=1 such that for ay p l ad p h with s(p l, p h ) [0,1], the properties: i)s(p l, p l ) = 1 (reflexive) ad, ii) s(p l, p h ) = s(p h, p l ) (symmetric) are fulfilled for each sigle value of l ad h. Aalogously, to more differetiate betwee the values ad to avoid the ties problem, a simple modificatio to the similarity measure is suggested as follows: s(p l, p h ) = 1 ( p l p h max p ), (16) l p h l h where max l h p l p h is the maximum distace betwee all experts. Correspodigly, the weights for IOWA aggregatio process Eq. (13) ca also be modified to the followig formula: m h = argmi h(u 1 t 1,, u k t k ) Max h (u 1 t 1,, u k t k ). (17) 1/λ, 1/λ (15) Example 2: Suppose that a set of opiio of experts o a sigle criterio C j is give asp j h = (p 1, p 2,, p k ) = (0.31, 0.34, 0.30, 0.28, 0.11). The majority agreemet of experts ca be calculated as follows: C j E 1 E 2 E 3 E 4 E 5 h P j t 1 t 2 t 3 t 4 t 5 U supp 1h supp 2,h s h t h supp 3,h supp 4,h supp 5,h k where U = h=1 s h t h. I this case, for Q(t h ) = t h ad by settig λ = 1, the vector of order-iducig variables ca be determied, specificallyu = (s σ(1),, s σ(5) ) = (0.21, 0.79, 0.81, 0.84, 0.85). Next, by usig the quatifier Q with sematics most for majority, the weightig vector W Maj = (w 1,, w 5 ) = 219

10 Biyami Yusoff, Jose Maria Merigó, David Ceballos (0, 0, 0.20, 0.40, 0.40) ca be obtaied. The fial majority opiio of experts ca be give as the followig: IOWA( 0.21, 0.11, 0.79, 0.34, 0.81, 0.28, 0.84, 0.30, 0.85, 0.31 ) = OWA Operators with iclusio of the Degrees of Importace I this sectio, some OWA aggregatio operators with their weightig methods are reviewed, i particular, the weightig methods based o the iclusio of WA. I additio, a alterative weightig method with its respective aggregatio operator called as alterative OWAWA operator is proposed. 4.1 Some of the existig methods Prior to the defiitio of itegrated weightig methods, the geeral defiitio of WA is give as the followig. Defiitio 4. Let V = (v 1, v 2,, v ) be a weightig vector (degrees of importace) of dimesio such that v j [0,1] ad v j = 1, the a mappig WA: R R is a weighted arithmetic mea (WA) if WA V (a 1, a 2,, a ) = v j a j. The WA satisfies mootoic, idempotet ad bouded properties, but it is ot commutative (Beliakov et al., 2007; Grabisch et al., 2009; Torra, 1997). There are a umber of methods i the literature which have bee proposed for obtaiig weights for the OWA aggregatio operators (see, Xu, 2005). Oe of them is by usig the liguistic quatifiers as defied i the prelimiaries sectio, refer to Eq. (2). Throughout the study, the OWA weightig vector Wis exclusively referred to this type of weights, specifically to be itegrated with the weightig vector, V (except for the methods i Defiitios 8 ad 9 as will be explaied later). Defiitio 5. (Yager, 1988). Let V ad W be two weightig vectors of dimesio, the a mappigowa: R R isa OWA-MP operator of dimesio if: OWA V,W (a 1, a 2,, a ) = w j a σ(j), (18) where a σ(j) is the valuea jbeig ordered i o-icreasig order a σ(1) a σ(2) a σ() such that a j = H(a j, v j ) = (v j α ) (a j ) v j α ad α is the oress measure ad α = 1 α is its complemet. This is the uified formulatio of the methods which proposed earlier i Yager (1978) ad Yager (1987), specifically based o the max-mi ad product approaches. I this study, it is deoted as OWA-MP. Notice that i the special cases: if α = 0, the it ca be reduced to a pure ad operator. Specifically, give v that a j = a j j with W = W, the a σ() is geerated, which is the smallest value of 220

11 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model a σ(j). Coversely, if α = 1, the it ca be reduced to a pure or operator. Give that a j = v j a j with W = W, the a σ(1) is geerated, which is the largest value of a σ(j). The OWA-MP operators meet mootoic ad idempotet properties, however they are ot commutative as ivolve WA. Moreover they are also ot bouded, as i the case of argumet value, a j [0,1], the modified argumet values a j are always greater tha or equal to the argumet values, a j. Defiitio 6. (Yager, 1998). Let V ad W be two weightig vectors of dimesio, the a mappigowa: R R is a OWA-FSM operator of dimesio if: OWA V,W (a 1, a 2,, a ) = w j a σ(j), (19) where a σ(j) is the value of a jbeig ordered i o-icreasig order a σ(1) a σ(2) a σ() give that a j = H(a j, v j ) = α v j + v j a j ad α = 1 α, that is the complemet of oress. This method is based o fuzzy system modelig ad is termed as OWA-FSM i this study. Notice that i the special cases: if α = 0, the it reduces to a pure ad operator. Specifically, give that a j = v j + v j a j ad w = 1, the a σ() is geerated, which is the smallest value of a σ(j). Whilst, if α = 1, the it is a pure or operator. Give that a j = v j a j ad w 1 = 1, the a σ(1) is geerated, which is the largest value of a σ(j). The OWA-FSM operators meet mootoic ad idempotet properties, but, they are ot commutative as ivolve WA. Moreover, they are also ot bouded, as i the case of a j [0,1], the a j a j. Defiitio 7. (Xu ad Da, 2003). Let V ad W be two weightig vectors of dimesio, the a mappigha: R R is a hybrid averagig operator of dimesio if: HA V,W (a 1, a 2,, a ) = w j a σ(j), (20) where a σ(j) is the argumet value a jbeig ordered i o-icreasig order a σ(1) a σ(2) a σ() give that a j = v j a j ad is the balacig coefficiet. It ca be show that whe W = (1/, 1/,,1/), the the HA operator reduces to the WA, whilst whe V = (1/, 1/,,1/), the HA operator reduces to the OWA. The HA operators meet mootoic property, however, they are either idempotet or bouded. As ca be see, the Defiitios 5-7 are based o the approach where the degrees of importace, v j are used to modify the argumet values to be aggregated. I the followig, the approaches based o the direct itegratio betwee v j ad w j are preseted. 221

12 Biyami Yusoff, Jose Maria Merigó, David Ceballos Defiitio 8. (Torra, 1997). Let V ad W be two weightig vectors of dimesio, the a mappig WOWA: R R is a weighted ordered weighted averagig (WOWA) operator of dimesio if: WOWA V,W (a 1, a 2,, a ) = ω j a σ(j), (21) where a σ(j) is the argumet value of a j beig ordered i o-icreasig order a σ(1) a σ(2) a σ() ad ω j = f( k=1 v σ(k) ) f( k=0 v σ(j) )with f beig j a mootoic o-decreasig fuctio that iterpolates the poits ((j/), j j 1 k=1 w j ) together with the poit (0,0). The fuctio f required to be a straight lie whe the poits iterpolated i this way. It ca be demostrated that whe W = (1/, 1/,,1/), the WOWA operator reduces to WA, whilst whe V = (1/, 1/,,1/), WOWA operator reduces to OWA. Moreover, they are mootoic, idempotet, ad bouded. Equivaletly, the WOWA operator ca be trasformed to the OWA operator with the iclusio of degrees of importace (Yager, 1996), if a regular mootoically o-decreasig fuzzy quatifier Q is used as the fuctio f ad it ca be defied as the followig. Defiitio 9. (Yager, 1996). LetV ad W be two weightig vectors of dimesio, the a mappig OWA: R R is a OWA operator of dimesio if : OWA V,W (a 1, a 2,, a ) = ω j a σ(j), (22) where a σ(j) is the argumet value a j beig ordered i o-icreasig order a σ(1) a σ(2) a σ() ad ω j = Q( k=1 v σ(k) ) Q( k=0 v σ(k) ) such that ω j [0,1] ad ω j = 1. Defiitio 10.(Llamazares, 2013). Let V ad W be two weightig vectors of dimesio, the a mappigiwa: R R is a immediate weighted averagig (IWA) operator of dimesio if: IWA V,W (a 1, a 2,, a ) = j j 1 π j a σ(j), (23) where a σ(j) is the argumet value a j beig ordered i o-icreasig order a σ(1) a σ(2) a σ() ad π j = w j v j / w j v j. As ca be see, the IWA is a maipulatio of immediate probability (Egema et al., 1996; Merigó, 2012; Yager et al., 1995) by usig the WA istead of the probability distributio. The IWA operators satisfy the geeralizatio properties as V = (1/, 1/,,1/), it reduces to OWA ad whe W = (1/, 1/,,1/), 222

13 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model the IWA reduces to the WA (Llamazares, 2013). The IWA operators meet mootoic, idempotet, bouded properties. Defiitio 11. (Merigó, 2012). Let V ad W be two weightig vectors of dimesio, the a mappigowawa: R R is a ordered weighted averagigweighted average (OWAWA) operator of dimesio if: OWAWA V,W (a 1, a 2,, a ) = φ j a σ(j), (24) where a σ(j) is the argumet value of a j beig ordered i o-icreasig order a σ(1) a σ(2) a σ() ad φ j = βw j + (1 β)v σ(j) with β [0,1]. OWAWA operators satisfy mootoic, idempotet, bouded properties. Moreover, the value retured by the OWAWA operator lies betwee the values retured by the WA ad OWA, ad coicides with them whe both are equal. I additio, by takig the advatages of the IWA ad the OWAWA operators, a ew weightig method ca be derived as i the ext sub-sectio. 4.2 Alterative OWAWA operator Defiitio 12.Let V ad W be two weightig vectors of dimesio, the a mappigaowawa: R R is a alterative ordered weighted averagigweighted average (AOWAWA) operator of dimesio if: AOWAWA V,W (a 1, a 2,, a ) = φ ja σ(j), (25) where a σ(j) is the argumet value of a j beig ordered i o-icreasig order a σ(1) a σ() ad φ j = (w β j v (1 β) σ(j) ) (w β j v (1 β) σ(j) ) with β [0,1], by covetio that (0 0 = 0). The AOWAWA operator are mootoic, bouded, idempotet. However, it is ot commutative because the AOWAWA operator icludes the WA. The AOWAWA operators geeralized to WA ad OWA whe β = 0 ad β = 1, respectively. Theorem 1(Mootoicity) Assume that f is the AOWAWA operator, let A = (a 1, a 2,, a ) ad B = (b 1, b 2,, b ) be two sets of argumets. If a j b j, j (1,2,, ), the: f(a 1, a 2,, a ) f(b 1, b 2,, b ). Proof. It is straightforward ad thus omitted. Theorem 2 (Idempotecy) Assume f is the AOWAWA operator, if a j = a, j (1,2,, ), the: f(a 1, a 2,, a ) = a. Proof. It is straightforward ad thus omitted. 223

14 Biyami Yusoff, Jose Maria Merigó, David Ceballos Theorem 3(Bouded) Assume f is the AOWAWA operator, the: Mi{a j } f(a 1, a 2,, a ) Max{a j } Proof. It is straightforward ad thus omitted. 5. ME-MCDM Model based o Differet Decisio Schemes I this sectio, the geeral frameworks of ME-MCDM model based o the classical ad alterative schemes are preseted. I additio to the origial methods by Pasi ad Yager (2006) ad Bordoga ad Sterlacchii (2014), some extesios have bee made as the followig. First, the majority cocept of Pasi-Yager method which is origially based o the classical scheme is exteded to the case of alterative scheme. Secodly, the Bordoga-Sterlacchii method which is based o the alterative scheme is modified to the case of the classical scheme. These methods are used for the compariso purpose i the ext sectio. The algorithms for the model are structured as i the followig. 5.1 Classical scheme Stage I: Iteral aggregatio (Local aggregatio) Step 1: First, a decisio matrix for each expert D h, h = 1,2,, k, is costructed as follows: C 1 C A h h 1 a 11 a 1 D h (26) = ( ), A h h m a m a m1 where A i idicates the optio/alterative i(i = 1,2,, m) ad C j deotes the criterio j (j = 1,2,, ). Meawhile the a h ij represets the preferece for optio A i with respect to criterio C j, such that a h ij [0,1]. Step 2: Next, determie the weightig vector for all the expert usig oe of the available methods, such as i Eqs. (18-25). Note that, i this case, the proportio of criteria to be cosidered is subject to the attitudial character of idividual experts. Hece, each expert ca provide distict decisio strategies separately. Step 3: Aggregate the judgmet matrix of each expert by the weightig vector as determied i Step 2. At this stage, each expert derives the rakig of all optios idividually. Stage II: Exteral aggregatio (Global aggregatio) With respect to the type of aggregatio methods, the cosesus measure for the majority of experts ca be calculated as follows: 224

15 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model (P-Y*) The Pasi-Yager method (Homogeeous group decisio makig): Step 4: Determie the order-iducig variable usig the Eqs. (5-6) or i the case where the argumet values are very close to each other, use the modified support fuctio such i Eq. (9). Step 5: Calculate the weightig vector which represets the majority of experts usig the Eq. (7) based o quatifier most as i Eq. (8). I this case, the weight of each expert is cosidered as equal. (B-S*) The modified versio of Bordoga-Sterlacchii method (Heterogeeous group decisio makig): Step 4: Determie the order-iducig variable usig the Eqs. (10-12) or i the case where the argumet values are very close to each other, the use the modified similarity measure such i Eq. (16). Step 5: Calculate the weightig vector usig the Eq. (14) ad Eq. (17). I this case, the weight or trust degree is associated to each expert. 5.2 Alterative scheme Stage I: Exteral aggregatio Step 1: By the similar way, a decisio matrix for each expert is costructed such i Eq. (26). The, the aggregatio based o majority cocept ca be implemeted usig oe of the followig methods: (B-S**) The Bordoga-Sterlacchii method (Heterogeeous GDM): Step 2: Determie the order-iducig variable such i Step 4(B-S*) of the classical scheme. But, istead of aggregate the opiio of experts with respect to each optio, here, the aggregatio process is coducted o each criterio. Step 3: Calculate the weightig vector such i Step5(B-S*) of the classical scheme usig the values of the order-iducig variable i the previous step. (P-Y**) The extesio of Pasi-Yager method (Homogeeous GDM): Step 2: Determie the order-iducig variable as i Step4(P-Y*) of the classical scheme. But, istead of aggregate the opiio of experts with respect to each optio, here, the aggregatio process is coducted o each criterio. Step 3: Calculate the weightig vector such i Step5(P-Y*) of the classical scheme usig the order-iducig variable derived i the previous step. Stage II: Iteral aggregatio (Global aggregatio) Step 4: Step 5: Determie the weightig vector usig oe of the methods as show i Eqs. (18-25). Fially, aggregate the judgmet matrix of the majority of experts with respect to the weightig vector derived i Step 4. Note that here, the proportio of criteria is subject to the attitudial character of the majority of experts. 225

16 Biyami Yusoff, Jose Maria Merigó, David Ceballos 6. Numerical Example I this sectio, a ivestmet selectio problem is studied where a group of experts or aalysts are assiged for the selectio of a optimal strategy. Assume that a compay plas to ivest some moey i a regio. Primarily, they cosider five possible ivestmet optios as follows: A 1 = ivest i the Europea market, A 2 =America market, A 3 =Asia market, A 4 =Africa market, A 5 = do ot ivest moey. I order to evaluate these ivestmets, the ivestor has brought together a group of experts. This group cosiders that each of ivestmet optios ca be described with the followig characteristics: C 1 = beefits i the short term, C 2 = beefits i the mid-term, C 3 = beefits i the log term, C 4 = risk of the ivestmet, C 5 = other variables. The available ivestmet strategies depedig o the characteristic C j ad the optio A i for each expert are show i Table 1. Table 1. Available ivestmet strategies of each expert, E h E 1 E 2 E 3 C 1 C 2 C 3 C 4 C 5 C 1 C 2 C 3 C 4 C 5 C 1 C 2 C 3 C 4 C E 4 E 5 C 1 C 2 C 3 C 4 C 5 C 1 C 2 C 3 C 4 C 5 A A A A A A A A A A I this study, two aalyses are coducted. First is to aalyze the effect of differet decisio schemes for the homogeeous ad heterogeeous cases. The aggregated results of aalysis are preseted i Table2. Note that, for the heterogeeous case, the weights(0.3, 0.1, 0.1, 0.4, 0.1)represet the expert E 1, E 2, E 3, E 4 ad E 5, respectively. As ca be see, there is a slight differece betwee the results that derived from both majority aggregatio approaches with respect to differet decisio schemes. The majority opiio of experts with respect to the classical scheme provides A 4, A 2, A 1, A 5 ad A 3 as the fial rakig for both methods. While the majority opiio of experts computed with respect to alterative scheme exhibits the rakig of A 4, A 1, A 5, A 2 ad A 3 (also for both 226

17 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model methods). Hece, the aggregated results demostrated the effect o differet decisio schemes i rakig the optios. ME-MCDM- PY* Table 2. The aggregated results Homogeeous case, t h = 1/ ME-MCDM- PY** Heterogeeous case, ME-MCDM- BS* t h 1/ ME-MCDM- BS** A (R3) (R2) (R3) (R2) A (R2) (R4) (R2) (R4) A (R5) (R5) (R5) (R5) A (R1) (R1) (R1) (R1) A (R4) (R3) (R4) (R3) Note: * refers to the classical scheme ad ** refers to the alterative scheme; R = rakig. Secodly, as a further aalysis, the method of ME-MCDM-BS** based o the itegratio of WA ad OWA weights is coducted. Table 3 shows the aggregated results of the model based o differet weightig techiques. Table 3. The aggregated results with respect to ME-MCDM-BS** model OWA (Q) WOWA IWA OWA- WA AOWA -WA OWA (FSM) OWA (MP) The weights v j for the criteria are give as 0.1, 0.2, 0.3, 0.3, 0.1 ad the ordered weights, w j are represeted as most (γ = 10), i.e., most of the criteria have to be satisfied. As ca be oticed, the proposed AOWAWA operator with β = 0.5idicates the similar rakig as the WOWA ad IWA methods, A 1, A 4, A 3, A 5 ad A 2. Cocurretly, the rest weightig techiques show slightly differet results. Note that i this case, the decisio strategy is subject to the attitudial character of the majority of experts. By selectig ay parameter γ to represet the liguistic HA 227

18 Biyami Yusoff, Jose Maria Merigó, David Ceballos quatifier, various decisio strategies ca be derived. Specifically for γ 0 (at least oe criteria is cosidered), γ = 1 (averagely all) ad γ (all criteria are cosidered). The aggregated results of AOWAWA operator with differet decisio strategies are preseted i Tables 4. Table 4. Decisio strategies based o AOWAWA operator At least oe γ 0 Few γ = 0.1 Some γ = 0.5 Half (average) γ = 1 May γ = 2 Most γ = 10 All γ I additio, the rakigs of AOWAWA operator with differet values of β ca be see i Table 5. These values show the effect of the selectio WA ad OWA i the fial evaluatio process. For example, if oly WA is applied, the β = 0, whilst β = 1 implies oly OWA is used. Table 5. Aggregated results of AOWAWA operator based o β values β = 0 β = 0.2 β = 0.4 β = 0.6 β = 0.8 β = Coclusio I this paper, the aalysis o extesios of ME-MCDM model based o the OWA operators has bee coducted. The focus is give o the aggregatio operatio, specifically with respect to the fusios of criteria ad experts' judgmets. The majority cocept based o the IOWA ad liguistic quatifiers to aggregate the experts judgmets is aalyzed, i which cocetrated o the classical ad alterative schemes of group decisio makig model. The, a review o the weightig methods related to the itegratio of WA ad OWA is provided. Correspodigly, the alterative weightig techique is proposed which is called as the AOWAWA operator. The ME-MCDM model based o two-stage aggregatio processes the is developed. A compariso is coducted to see the effect of differet weightig techiques i aggregatig the criteria ad the results of usig differet decisio schemes for the fusio of majority opiio of experts. A 228

19 OWA-based Aggregatio Operatios i Multi-Expert MCDM Model umerical example i the selectio of ivestmets the has bee used for the compariso purpose. I geeral, the coclusio which ca be made from the aalysis ad compariso is that, the selectio of decisio schemes as well as the weightig methods employed i the aggregatio process show differet rakigs for the optios. Moreover, each of the decisio schemes represets the decisio strategy (i.e., with respect to criteria) i a differet way, whether as a idividual expert decisio strategy or as group/majority decisio strategy. Hece, the selectio of both approaches reflects differet results. ACKNOWLEDGEMENTS. Support from Miistry of Higher Educatio Malaysia ad Uiversity of Malaysia Tereggau (UMT) are gratefully ackowledged. REFERENCES [1] Beliakov, G., Pradera, A., Calvo, T. (2007), Aggregatio Fuctios: A Guide for Practitioers. Spriger-Verlag, Berli; [2] Beliakov, G., James, S. (2011), Iduced Ordered Weighted Averagig Operators. I: Yager, R.R., et al., (ed) Recet developmets i the OWA operators; Studies i Fuzziess ad Soft Computig 265. Spriger- Verlag, pp ; [3] Bordoga, G., Sterlacchii, S. (2014), A Multi Criteria Group Decisio Makig Process Based o the Soft Fusio of Coheret Evaluatios of Spatial Alteratives. I: Zadeh, L.A., et al., (ed) Recet developmets ad ew directios i soft computig, Studies i Fuzziess ad Soft Computig 317. Spriger, pp ; [4] Egema, K.J., Filev, D.P., Yager, R.R. (1996), Modellig Decisio Makig Usig Immediate Probabilities. Iteratioal Joural of Geeral Systems, 24, ; [5] Figueira, J., Greco S., Ehrgott, M. (2005), Multiple Criteria Decisio Aalysis: State of the Art Surveys. Spriger, New York; [6] Grabisch, M., Marichal, J.L., Mesiar, R., Pap, E. (2009), Aggregatio Fuctios. Cambridge Uiversity Press, Cambridge; [7] Llamazares, B. (2013), A Aalysis of Some Fuctios that Geeralizes Weighted Meas ad OWA Operators. Iteratioal Joural of Itelliget Systems, 28(4), ; [8] Merigó, J.M. (2012), OWA Operators i the Weighted Average ad their Applicatio i Decisio Makig. Cotrol ad Cyberetics, 41(3), ; [9] Merigó, J.M., Gil-Lafuete, A.M. (2008), Usig OWA Operator i the Mikowski Distace. Iteratioal Joural Social, Humaities ad Sciece, 2, ; [10] Merigó, J.M., Gil-Lafuete, A.M. (2009), The Iduced Geeralized OWA Operator. Iformatio Scieces, 179, ; 229

20 Biyami Yusoff, Jose Maria Merigó, David Ceballos [11] Merigó, J.M., Yager, R.R. (2013), Geeralized Movig Averages, Distace Measures ad OWA Operators. Iteratioal Joural of Ucertaity, Fuzziess ad Kowledge-Based Systems, 21(4), ; [12] Pasi, G., Yager, R.R. (2006), Modellig the Cocept of Majority Opiio i Group Decisio Makig. Iformatio Scieces, 176, ; [13] Taib, C.M.I.C., Yusoff, B., Abdullah, M.L., Wahab, A.F. (2016), Coflictig Bifuzzy Multi-attribute Group Decisio Makig Model with Applicatio to Flood Cotrol Project. Group Decisio ad Negotiatio, 25(1), ; [14] Torra, V. (1997), The Weighted OWA Operator. Iteratioal Joural of Itelliget Systems, 12, ; [15] Xu, Z.S., Da, Q.L. (2003), A Overview of Operators for Aggregatig Iformatio. Iteratioal Joural of Itelliget Systems, 18, ; [16] Xu, Z.S. (2005), A Overview of Methods for Determiig OWA Weights. Iteratioal Joural of Itelliget Systems, 20, ; [17] Yager, R.R. (1988), O Ordered Weighted Averagig Aggregatio Operators i Multi-criteria Decisio Makig. IEEE Trasactios o Systems ad Ma Cyberetics, 18, ; [18] Yager, R.R. (1996), Quatifier Guided Aggregatio Usig OWA Operators. Iteratioal Joural of Itelliget Systems, 11, 49-73; [19] Yager, R.R. (1998), Icludig Importaces i OWA Aggregatios Usig Fuzzy Systems Modellig. IEEE Trasactios o Fuzzy Systems, 6(2), ; [20] Yager, R.R. (2004), Geeralized OWA Aggregatio Operators. Fuzzy Optimizatio ad Decisio Makig, 3, ; [21] Yager, R.R., Egema, K.J., Filev, D.P. (1995), O the Cocept of Immediate Probabilities. Iteratioal Joural of Itelliget Systems, 10, ; [22] Yager, R.R., Filev, D.P. (1999), Iduced Ordered Weighted Averagig Operators. IEEE Trasactios o Systems ad Ma Cyberetics. - Part B, 29, ; [23] Zadeh, L.A. (1983), A Computatioal Approach to Fuzzy Quatifiers i Natural Laguages. Computig ad Mathematics Applicatios, 9,