On the Use of the OWA Operator in the Weighted Average and its Application in Decision Making

Transcription

1 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. O the Use of the OWA Operator i the Weighted Average ad its Applicatio i Decisio Makig José M. Merigó, Member, IAENG Abstract We itroduce a ew aggregatio operator that uifies the weighted average ad the ordered weighted averagig (OWA) operator i the same formulatio. We call it the ordered weighted averagig weighted averagig (OWAWA) operator. This aggregatio operator provides a more complete represetatio of the weighted average ad the OWA because it icludes them as particular cases of a more geeral cotext. We study differet properties ad families of the OWAWA operator. We also develop a illustrative example of the ew approach i a decisio makig problem about selectio of strategies. Idex Terms OWA operator; Weighted average; Decisio makig; Selectio of strategies. I. INTRODUCTION The weighted average (WA) is oe of the most commo aggregatio operators foud i the literature. It ca be used i a wide rage of differet problems icludig statistics, ecoomics, egieerig, etc. Aother iterestig aggregatio operator that has ot bee used so much i the literature, especially because it appeared i 988, is the ordered weighted averagig (OWA) operator [2]. The OWA operator provides a parameterized family of aggregatio operators that rage from the maximum to the miimum. For further readig o the OWA operator ad some of its applicatios, refer to [-4,6,8-9]. Recetly, some authors [9-] have tried to uify both cocepts i the same formulatio. It is worth otig the work developed by Torra [9] with the itroductio of the weighted OWA (WOWA) operator ad the work of Xu [] about the hybrid averagig (HA) operator. Both models arrived to a uificatio betwee the OWA ad the WA because both cocepts were icluded i the formulatio as particular cases. However, as it has bee studied i [6], these models seem to be a partial uificatio but ot a real oe because they ca uify them but they caot cosider how relevat these cocepts are i the specific problem cosidered. For example, i some problems we may prefer to give more importace to the OWA operator because we believe that it is more relevat ad vice versa. I this paper, we preset a ew approach to uify the OWA operator with the WA. We call it the ordered weighted averagig weighted averagig (OWAWA) operator. We could also refer to it as the WOWA operator but we have ot Mauscript received March 23, J.M. Merigó is with the Departmet of Busiess Admiistratio, Uiversity of Barceloa, Av. Diagoal 690, Barceloa, Spai (correspodig author: ; fax: ; jmerigo@ ub.edu). doe so because i the literature there is aother approach that already uses this ame [9]. The mai advatage of this approach is that it uifies the OWA ad the WA takig ito accout the degree of importace of each case i the formulatio. Thus, we are able to cosider situatios where we give more or less importace to the OWA ad the WA depedig o our iterests ad the problem aalysed. We study differet properties of the OWAWA operator ad differet particular cases. We see that the OWA ad the WA are particular cases of this geeral formulatio. Moreover, we are also able to uify the arithmetic mea (or simple average) with the OWA operator whe the weights of the WA are equal. We study other families such as the step-owawa, the media-owawa, the olympic-owawa, the S-OWAWA, the cetered-owawa, etc. We also aalyze the applicability of the ew approach ad we see that it is possible to develop a astoishigly wide rage of applicatios. The reaso is that the OWAWA icludes the WA ad the OWA as special cases. Therefore, all the studies that use the WA or the OWA ca be revised by usig this ew approach i order to obtai a more complete aalysis of the problem aalysed. For example, we ca apply it i a lot of problems about statistics, ecoomics, egieerig, decisio theory, etc. I this paper we focus o a decisio makig problem about selectio of strategies. The mai advatage of the OWAWA i these problems is that it is possible to cosider the subjective probability (or the degree of importace) ad the attitudial character of the decisio maker. This paper is orgaized as follows. I Sectio 2 we briefly revise the WA ad the OWA operator. I Sectio 3 we preset the ew approach. Sectio 4 aalyzes differet families of OWAWA operators. I Sectio 5 we study the applicability of the ew approach i a decisio makig problem. Sectio 6 presets a umerical example ad i Sectio 7 we summarize the mai coclusios of the paper. A. The OWA Operator II. PRELIMINARIES The OWA operator [2] is a aggregatio operator that provides a parameterized family of aggregatio operators betwee the miimum ad the maximum. It ca be defied as follows. Defiitio. A OWA operator of dimesio is a mappig OWA: R R that has a associated weightig vector W of dimesio such that w j [0, ] ad j = w j =, the:

2 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. OWA(a,, a ) = w j b j where b j is the jth largest of the a i. Note that differet properties ca be studied such as the distictio betwee descedig ad ascedig orders, differet measures for characterizig the weightig vector ad differet families of OWA operators. Note that it is commutative, mootoic, bouded ad idempotet. For further readig, refer, e.g., to [-4,6,8-9]. B. The Weighted Average The weighted average (WA) is oe of the most commo aggregatio operators i the literature. It has bee used i a icredible wide rage of applicatios. It ca be defied as follows. Defiitio 2. A WA operator of dimesio is a mappig WA: R R that has a associated weightig vector W, with w j [0, ] ad w j =, such that WA(a,, a ) = w j a i where a i represets the argumet variable. The WA operator accomplishes the usual properties of the aggregatio operators. For further readig o differet extesios ad geeralizatios of the WA, see for example [-3,5-7,0-]. III. THE ORDERED WEIGHTED AVERAGING - WEIGHTED AVERAGING OPERATOR The ordered weighted averagig weighted averagig (OWAWA) operator is a ew model that uifies the OWA operator ad the weighted average i the same formulatio. Therefore, both cocepts ca be see as a particular case of a more geeral oe. This approach seems to be complete, at least as a iitial real uificatio betwee OWA operators ad WAs. It ca also be see as a uificatio betwee decisio makig problems uder ucertaity (with OWA operators) ad uder risk (with probabilities). Note that some previous models already cosidered the possibility of usig OWA operators ad WAs i the same formulatio. The mai models are the weighted OWA (WOWA) operator [9-0] ad the hybrid averagig (HA) operator []. Although they seem to be a good approach, they are ot so complete tha the OWAWA because it ca uify OWAs ad WAs i the same model but they ca ot take i cosideratio the degree of importace of each case i the aggregatio process. Moreover, i some particular cases we also fid icosistecies [6]. Other methods that could be cosidered are the cocept of immediate probability [4,6,5,9]. This method is focused o the probability but it is easy to exted it to the use of WAs because sometimes the WA is used as a subjective probability. As said before, these a other approaches are useful for some particular situatios but they does ot seem to be so complete tha the OWAWA because they ca uify OWAs with WAs (or with probabilities) but they ca ot uify them givig differet () (2) degrees of importace to each case. Note that i future research we will also prove that these models ca be see as a special case of a geeral OWAWA operator (or its respective model with probabilities) that uses quasi-arithmetic meas. Obviously, it is possible to develop more complex models of the WOWA, the HA ad the IP-OWA that takes ito accout the degree of importace of the OWAs ad the WAs (or probabilities) i the model but they seem to be artificial ad ot a atural uificatio as it will be show below. I the followig, we are goig to aalyze the OWAWA operator. It ca be defied as follows. Defiitio 3. A OWAWA operator of dimesio is a mappig OWAWA: R R that has a associated weightig vector W of dimesio such that w j [0, ] ad j = w j =, accordig to the followig formula: OWAWA (a,, a ) = v ˆ j b j (3) where b j is the jth largest of the a i, each argumet a i has a associated weight (WA) v i with i = vi = ad v i [0, ], v ˆ j = βw j + ( β ) v j with β [0, ] ad v j is the weight (WA) v i ordered accordig to b j, that is, accordig to the jth largest of the a i. Note that it is also possible to formulate the OWAWA operator separatig the part that strictly affects the OWA operator ad the part that affects the WAs. This represetatio is useful to see both models i the same formulatio but it does ot seem to be as a uique equatio that uifies both models. Defiitio 4. A OWAWA operator is a mappig OWAWA: R R of dimesio, if it has a associated weightig vector W, with j = w j = ad w j [0, ] ad a weightig vector V that affects the WA, with i = vi = ad v i [0, ], such that: OWAWA (a,, a ) = β w jb j + ( β) viai (4) i= where b j is the jth largest of the argumets a i ad β [0, ]. I the followig, we are goig to give a simple example of how to aggregate with the OWAWA operator. We cosider the aggregatio with both defiitios. Example. Assume the followig argumets i a aggregatio process: (30, 50, 20, 60). Assume the followig weightig vector W = (0.2, 0.2, 0.3, 0.3) ad the followig probabilistic weightig vector V = (0.3, 0.2, 0.4, 0.). Note that the WA has a degree of importace of 70% while the weightig vector W of the OWA a degree of 30%. If we wat to aggregate this iformatio by usig the OWAWA operator, we will get the followig. The aggregatio ca be solved either with (3) or (4). With (3) we calculate the ew weightig vector as:

3 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. vˆ = = 0.3 vˆ2 = = 0.2 vˆ3 = = 0.3 v ˆ4 = = 0.37 Ad the, we calculate the aggregatio process as follows: OWAWA = = With (4), we aggregate as follows: OWAWA = 0.3 ( ) ( ) = Obviously, we get the same results with both methods. From a geeralized perspective of the reorderig step, it is possible to distiguish betwee the descedig OWAWA (DOWAWA) ad the ascedig OWAWA (AOWAWA) operator by usig w j = w* j+, where w j is the jth weight of the DOWAWA ad w* j+ the jth weight of the AOWAWA operator. If B is a vector correspodig to the ordered argumets b j, we shall call this the ordered argumet vector ad W T is the traspose of the weightig vector, the, the OWAWA operator ca be expressed as: OWAWA (a,, a ) = W T B (5) Note that if the weightig vector is ot ormalized, i.e., W = j = w j, the, the OWAWA operator ca be expressed as: OWAWA (a,, a ) = v j b j W ˆ The OWAWA is mootoic, commutative, bouded ad idempotet. It is mootoic because if a i u i, for all a i, the, OWAWA(a, a 2,, a ) OWAWA(u, u 2, u ). It is commutative because ay permutatio of the argumets has the same evaluatio. That is, OWAWA(a, a 2,, a ) = OWAWA(u, u 2,, u ), where (u, u 2,, u ) is ay permutatio of the argumets (a, a 2,, a ). It is bouded because the OWAWA aggregatio is delimitated by the miimum ad the maximum. That is, Mi{a i } OWAWA(a, a 2,, a ) Max{a i }. It is idempotet because if a i = a, for all a i, the, OWAWA(a, a 2,, a ) = a. Aother iterestig issue to aalyze are the measures for characterizig the weightig vector W. Followig a similar methodology as it has bee developed for the OWA operator [6,2] we ca formulate the attitudial character, the etropy of dispersio, the divergece of W ad the balace operator. Note that these measures affect the weightig vector W but ot the WAs because they are give as some kid of objective iformatio. IV. FAMILIES OF OWAWA OPERATORS First of all we are goig to cosider the two mai cases of the OWAWA operator that are foud by aalyzig the coefficiet (6) β. Basically, if β = 0, the, we get the WA ad if β =, the OWA operator. Note that if v i = /, for all i, the, we get the uificatio betwee the arithmetic mea (or simple average) ad the OWA operator. By choosig a differet maifestatio of the weightig vector i the OWAWA operator, we are able to obtai differet types of aggregatio operators. For example, we ca obtai the partial maximum, the partial miimum, the partial average ad the partial weighted average. Remark. The partial maximum is foud whe w = ad w j = 0 for all j. The partial miimum is formed whe w = ad w j = 0 for all j. More geerally, the step-owawa is formed whe w k = ad w j = 0 for all j k. Note that if k =, the step-owawa is trasformed to the partial maximum, ad if k =, the step-owawa becomes the partial miimum operator. Remark 2. The partial average is obtaied whe w j = / for all j, ad the partial weighted average is obtaied whe the ordered positio of i is the same as the ordered positio of j. Remark 3. Aother iterestig family is the S-OWAWA operator. It ca be subdivided ito three classes: the or-like, the ad-like ad the geeralized S-OWAWA operators. The geeralized S-OWAWA operator is obtaied if w = (/)( (α + β)) + α, w = (/)( (α + β)) + β, ad w j = (/)( (α + β)) for j = 2 to, where α, β [0, ] ad α + β. Note that if α = 0, the geeralized S-OWAWA operator becomes the ad-like S-OWAWA operator, ad if β = 0, it becomes the or-like S-OWAWA operator. Remark 4. Aother family of aggregatio operator that could be used is the cetered-owawa operator. We ca defie a OWAWA operator as a cetered aggregatio operator if it is symmetric, strogly decayig ad iclusive. Note that these properties have to be accomplished for the weightig vector W of the OWAWA operator but ot ecessarily for the weightig vector V of the WA. It is symmetric if w j = w j+. It is strogly decayig whe i < j ( + )/2 the w i < w j ad whe i > j ( + )/2 the w i < w j. It is iclusive if w j > 0. Note that it is possible to cosider a softeig of the secod coditio by usig w i w j istead of w i < w j, ad it is also possible to remove the third coditio. We shall refer to it as a o-iclusive cetered-owawa operator. Remark 5. For the media-owawa, if is odd we assig w ( + )/2 = ad w j* = 0 for all others. If is eve we assig for example, w /2 = w (/2) + = 0.5 ad w j* = 0 for all others. For the weighted media-owawa, we select the argumet b k that has the kth largest argumet such that the sum of the weights from to k is equal or higher tha 0.5 ad the sum of the weights from to k is less tha 0.5. Remark 6. Aother type of aggregatio that could be used is the E-Z OWAWA weights. I this case, we should distiguish betwee two classes. I the first class, we assig w j* = (/q) for j* = to q ad w j* = 0 for j* > q, ad i the secod class, we assig w j* = 0 for j* = to q ad w j* = (/q) for j* = q + to.

4 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. Remark 7. The olympic-owawa is geerated whe w = w = 0, ad for all others w j* = /( 2). Note that it is possible to develop a geeral form of the olympic-owawa by cosiderig that w j = 0 for j =, 2,, k,,,, k +, ad for all others w j* = /( 2k), where k < /2. Note that if k =, the this geeral form becomes the usual olympic-owawa. If k = ( )/2, the this geeral form becomes the media-owawa aggregatio. That is, if is odd, we assig w ( + ) / 2 =, ad w j* = 0 for all other values. If is eve, we assig, for example, w /2 = w ( / 2) + = 0.5 ad w j* = 0 for all other values. Remark 8. Note that it is also possible to develop the cotrary case, that is, the geeral olympic-owawa operator. I this case, w j = (/2k) for j =, 2,, k,,,, k +, ad w j = 0, for all other values, where k < /2. Note that if k =, the we obtai the cotrary case for the media-owawa. Remark 9. A further iterestig type is the o-mootoic-owawa operator. It is obtaied whe at least oe of the weights w j is lower tha 0 ad j = w j =. Note that a key aspect of this operator is that it does ot always achieve mootoicity. Therefore, strictly speakig, this particular case is ot a OWAWA operator. However, we ca see it as a particular family of operators that is ot mootoic but evertheless resembles a OWAWA operator. Remark 0. Note that other families of OWAWA operators could be used followig the recet literature about differet methods for obtaiig the OWAWA weights such as [-3,6,8,2,4,6]. V. SELECTION OF STRATEGIES WITH THE OWAWA OPERATOR The OWAWA operator is applicable i a wide rage of situatios where it is possible to use the WA ad the OWA operator. Therefore, we see that the applicability is icredibly broad because all the previous models, theories, etc., that uses the WA ca be exteded by usig the OWAWA operator. The reaso is that most of the problems with WAs deal with ucertaity. Usually, i most of the problems it is assumed a eutral attitudial character agaist the WA but we are still uder ucertaity. Thus, sometimes we may prefer to be more or less optimistic agaist this iformatio. Moreover, by usig the OWA i the WA, we ca uder or overestimate the results of a specific problem. Note also that the WA ca be see as a subjective probability. Summarizig some of the mai fields where it is possible to develop a lot of applicatios with the OWAWA operator, we ca metio: Statistics. Mathematics Ecoomics Decisio theory Egieerig Physics Etc. Note that we ca use the OWAWA operator i practically all the previous studies that have used the WA or the OWA i the aalysis. I this paper, we will cosider a decisio makig applicatio i the selectio of strategies. The use of the OWAWA operator ca be useful i a lot of situatios, but the mai reaso for use it is whe we wat to cosider the subjective probability (or degree of importace) of each state of ature (or characteristic) ad the attitudial character of the decisio maker i the same problem. The process to follow i the selectio of strategies with the OWAWA operator is similar to the process developed i [5-6], with the differece that ow we are cosiderig a strategic maagemet problem. The 5 steps of the decisio process ca be summarized as follows: Step : Aalysis ad determiatio of the sigificat characteristics of the available strategies for the compay. Theoretically, it is represeted as: C = {C, C 2,, C i,, C }, where C i is the ith characteristic of the strategy ad we suppose a limited umber of characteristics. Step 2: Fixatio of the ideal levels of each characteristic i order to form the ideal strategy. Table : Ideal strategy C C 2 C i C P = µ µ 2 µ i µ where P is the ideal strategy expressed by a fuzzy subset, C i is the ith characteristic to cosider ad µ i [0, ]; i =, 2,,, is a umber betwee 0 ad for the ith characteristic. Step 3: Fixatio of the real level of each characteristic for all the strategies cosidered. Table 2: Available alteratives C C 2 C i C P k = µ µ 2 µ i µ with k =, 2,, m; where P k is the kth strategy expressed by a fuzzy subset, C i is the ith characteristic to cosider ad µ i [0, ]; i =,,, is a umber betwee 0 ad for the ith characteristic of the kth strategy. Step 4: Compariso betwee the ideal strategy ad the differet alteratives cosidered usig the OWAWA operator. I this step, the objective is to express umerically the removal betwee the ideal strategy ad the differet alteratives cosidered. Note that it is possible to cosider a wide rage of OWAWA operators such as those described i Sectio 3 ad 4. Step 5: Adoptio of decisios accordig to the results foud i the previous steps. Fially, we should take the decisio about which strategy select. Obviously, our decisio is to select the strategy with the best results accordig to the type of OWAWA operator used i the aalysis. VI. NUMERICAL EXAMPLE I the followig, we preset a umerical example of the ew approach i a decisio makig problem about selectio of strategies. We aalyze a ecoomic problem about the moetary policy of a coutry. Assume the govermet of a coutry has to decide o the type of moetary policy to use the ext year. They cosider five alteratives:

5 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. A = Develop a strog expasive moetary policy. A 2 = Develop a expasive moetary policy. A 3 = Do ot develop ay chage i the moetary policy. A 4 = Develop a cotractive moetary policy. A 5 = Develop a strog cotractive moetary policy. I order to evaluate these strategies, the govermet has brought together a group of experts. This group cosiders that the key factor is the ecoomic situatio of the world ecoomy for the ext period. They cosider 5 possible states of ature that could happe i the future: S = Very bad ecoomic situatio. S 2 = Bad ecoomic situatio. S 3 = Regular ecoomic situatio. S 4 = Good ecoomic situatio. S 5 = Very good ecoomic situatio. The results of the available strategies, depedig o the state of ature S i ad the alterative A k that the decisio maker chooses, are show i Table. Table : Available alteratives S S 2 S 3 S 4 S 5 A A A A A I this problem, the experts assume the followig weightig vector: W = (0.3, 0.2, 0.2, 0.2, 0.). They assume that the WA that each state of ature will have is: V = (0., 0.2, 0.3, 0.3, 0.). Note that the OWA operator has a importace of 40% ad the probabilistic iformatio a importace of 60%. For doig so, we will use Eq. (3) to calculate the OWAWA aggregatio. The results are show i Table 2. Table 2: OWAWA weights ˆv ˆv 2 ˆv 3 ˆv 4 ˆv 5 V* With this iformatio, we ca aggregate the expected results for each state of ature i order to make a decisio. I Table 3, we preset differet results obtaied by usig differet types of OWAWA operators. Table 3: Aggregated results WA OWA WAM OWAWA A A A A A Note that we ca also obtai these results by usig Eq. (4). The, we will calculate separately the OWA ad the probabilistic approach as show i Table 4. Table 4: First aggregatio process WA AM OWA A A A A A After that, we will aggregate both models i the same process cosiderig that the OWA model has a degree of importace of 40% ad the probabilistic iformatio 60% as show i Table 5. Table 5: Fial aggregated results WA OWA WAM OWAWA A A A A A Obviously, we get the same results with both methods. If we establish a orderig of the alteratives, a typical situatio if we wat to cosider more tha oe alterative, the, we get the results show i Table 6. Note that the first alterative i each orderig is the optimal choice. Table 6: Orderig of the strategies Orderig WA A A 4 =A 5 A 2 A 3 OWA A 2 A A 5 A 3 =A 4 WAM A A 2 =A 5 A 4 A 3 OWAWA A A 2 A 5 A 4 A 3 As we ca see, depedig o the aggregatio operator used, the orderig of the strategies may be differet. Therefore, the decisio about which strategy select may be also differet. VII. CONCLUSION We have developed a ew aggregatio operator that uifies the WA with the OWA operator. We have called it the OWAWA operator. The mai advatage is that it provides a uified framework betwee the WA ad the OWA that allows us to use both of them i the same formulatio ad cosiderig how relevat they are i the specific problem cosidered. We have studied some of its mai properties ad we have see that it is possible to use a wide rage of particular cases i the OWAWA operator. We have also studied the applicability of the OWAWA operator ad we have see that there are a lot of potetial applicatios that ca be developed because i almost all the studies where it appears the WA or the OWA, it is possible to exted the aalysis by usig the OWAWA operator. The reaso is that the OWAWA geeralizes the WA ad the OWA, so for the extreme case that we oly wat to cosider oe of them, we simply have to use the particular case of the OWAWA whe we oly cosider oe of these two cocepts. We have focused o a decisio makig problem about selectio of strategies. We have see the usefuless of usig

6 Proceedigs of the World Cogress o Egieerig 2009 Vol I WCE 2009, July - 3, 2009, Lodo, U.K. the OWAWA operator because we are able to cosider WAs ad OWAs at the same time. I future research, we expect to develop further extesios to this approach by addig ew characteristics i the problem such as the use of order iducig variables, ucertai iformatio (iterval umbers, fuzzy umbers, liguistic variables, etc.), geeralized ad quasi-arithmetic meas ad distace measures. We will also exted this approach to situatios where we use the probability istead of the WA ad further developmets that have bee iitially developed i [6]. We will also cosider differet applicatios givig special attetio to busiess decisio makig problems such as ivestmet ad product maagemet. REFERENCES [] G. Beliakov, A. Pradera ad T. Calvo, Aggregatio Fuctios: A Guide for Practitioers. Berli: Spriger-Verlag, [2] H. Bustice, F. Herrera ad J. Motero, Fuzzy Sets ad Their Extesios: Represetatio, Aggregatio ad Models. Berli: Spriger-Verlag, [3] T. Calvo, G. Mayor ad R. Mesiar, Aggregatio Operators: New Treds ad Applicatios. New York: Physica-Verlag, [4] K.J. Egema, D.P. Filev ad R.R. Yager, Modellig decisio makig usig immediate probabilities. Iteratioal Joural of Geeral Systems, 24:28-294, 996. [5] J. Gil-Aluja, The iteractive maagemet of huma resources i ucertaity. Dordrecht: Kluwer Academic Publishers, 998. [6] J.M. Merigó, New extesios to the OWA operators ad its applicatio i decisio makig (I Spaish). PhD Thesis, Departmet of Busiess Admiistratio, Uiversity of Barceloa, [7] J.M. Merigó ad A.M. Gil-Lafuete, Uificatio poit i methods for the selectio of fiacial products. Fuzzy Ecoomic Review, 2:35-50, [8] J.M. Merigó ad A.M. Gil-Lafuete, The iduced geeralized OWA operator. Iformatio Scieces, 79:729-74, [9] V. Torra, The weighted OWA operator. Iteratioal Joural of Itelliget Systems, 2:53-66, 997. [0] V. Torra ad Y. Narukawa, Modelig Decisios: Iformatio Fusio ad Aggregatio Operators. Berli: Spriger-Verlag, [] Z.S. Xu ad Q.L. Da, A overview of operators for aggregatig iformatio. Iteratioal Joural of Itelliget Systems, 8: , 2003 [2] R.R. Yager, O ordered weighted averagig aggregatio operators i multi-criteria decisio makig. IEEE Trasactios o Systems, Ma ad Cyberetics B, 8:83-90, 988. [3] R.R. Yager, Decisio makig uder Dempster-Shafer ucertaities. Iteratioal Joural of Geeral Systems, 20: , 992. [4] R.R. Yager, Families of OWA operators. Fuzzy Sets ad Systems, 59:25-48, 993. [5] R.R. Yager, Icludig decisio attitude i probabilistic decisio makig. Iteratioal Joural of Approximate Reasoig, 2:-2, 999. [6] R.R. Yager, Cetered OWA operators, Soft Computig, :63-639, [7] R.R. Yager ad D.P. Filev, Parameterized adlike ad orlike OWA operators. Iteratioal Joural of Geeral Systems, 22:297-36, 994. [8] R.R. Yager ad D.P. Filev, Iduced ordered weighted averagig operators. IEEE Trasactios o Systems, Ma ad Cyberetics B, 29:4-50, 999. [9] R.R. Yager ad J. Kacprzyk, The Ordered Weighted Averagig Operators: Theory ad Applicatios. Norwell: Kluwer Academic Publishers, 997.