Proceedigs of the World Cogress o Egieerig 200 Vol I, Jue 30 - July 2, 200, Lodo, U.K. A Uified Model betwee the OWA Operator ad the Weighted Average i Decisio Makig with Dempster-Shafer Theory José M. Merigó, Member, IAENG, Kurt J. Egema Abstract We preset a ew decisio makig model by usig the Dempster-Shafer belief structure that uses probabilities, weighted averages ad the ordered weighted averagig (OWA) operator. Thus, we are able to represet the decisio makig problem cosiderig objective ad subjective iformatio ad the attitudial character of the decisio maker. For doig so, we use the ordered weighted averagig weighted average (OWAWA) operator. It is a aggregatio operator that uifies the weighted average ad the OWA i the same formulatio. As a result, we form the belief structure OWAWA (BS-OWAWA) aggregatio. We study some of its mai properties ad particular cases. We also preset a applicatio of the ew approach i a decisio makig problem cocerig political maagemet. Idex Terms Dempster-Shafer belief structure; Decisio makig; OWA operator; Weighted average; Aggregatio operators. I. INTRODUCTION The Dempster-Shafer (D-S) theory of evidece was itroduced by Dempster [3] ad by Shafer [0]. Sice its itroductio, this theory has bee studied ad applied i a lot of situatios such as [4,8-,4,7]. It provides a uifyig framework for represetig ucertaity because it icludes as special cases the situatios of risk (probabilistic ucertaity) ad igorace (imprecisio). Oe of the key applicatio areas of the D-S theory is i decisio makig because it allows to use risk ad ucertai eviromets i the same framework. This framework ca be carried out with a lot of aggregatio operators [-2,5-7,2-6]. Some authors [4,8,4] have cosidered the possibility of usig the ordered weighted averagig (OWA) operator. The OWA operator [3] is a aggregatio operator that provides a parameterized family of aggregatio operators betwee the maximum ad the miimum. Sice its itroductio, it has bee applied i a wide rage of situatios [-2,5-8,2-6]. Recetly [6-7], Merigó has itroduced the ordered weighted averagig weighted average (OWAWA) operator. It is a aggregatio operator that uifies the weighted average Mauscript received March 22, 200. J.M. Merigó is with the Departmet of Busiess Admiistratio, Uiversity of Barceloa, Av. Diagoal 690, 08034 Barceloa, Spai (correspodig author: +34-93-402962; fax: +34-93-4039882; e-mail: jmerigo@ ub.edu). K.J. Egema is with the Departmet of Iformatio Systems, Haga School of Busiess, IONA College, 080 New Rochelle, NY, USA (email: kegema@ioa.edu). (WA) ad the OWA operator i the same formulatio cosiderig the degree of importace that each cocept has i the aggregatio. The aim of this paper is to preset a ew decisio makig model with D-S theory by usig the OWAWA operator. The mai advatage of usig this framework is that we are able to cosider probabilistic iformatio with WAs ad OWAs. Thus, we are able to cosider a decisio makig problem with objective ad subjective iformatio ad cosiderig the attitudial character of the decisio maker. For doig so, we preset a ew aggregatio operator, the belief structure OWAWA (BS-OWAWA) operator. It is a ew aggregatio operator that aggregates the belief structures with the OWAWA operator. We study some of its mai properties ad particular cases. We also develop a illustrative example of the ew approach i a decisio makig problem cocerig the selectio of policies. We study a problem where a govermet is plaig the moetary policy for the ext year. The mai advatage of usig this approach is that we are able to cosider a wide rage of scearios ad select the oe closest with our iterests. This paper is orgaized as follows. I Sectio 2, we briefly review some basic cocepts about the D-S theory, the WA, the OWA ad the OWAWA operator. I Sectio 3 we preset the ew decisio makig approach. Sectio 4 itroduces the BS-OWAWA operator ad i Sectio 5 we develop a illustrative example. Sectio 6 summarizes the mai coclusios of the paper. II. PRELIMINARIES A. Dempster-Shafer Belief Structure The D-S theory [3,0] provides a uifyig framework for represetig ucertaity as it ca iclude the situatios of risk ad igorace as special cases. Note that the case of certaity is also icluded as it ca be see as a particular case of risk ad igorace. Defiitio. A D-S belief structure defied o a space X cosists of a collectio of oull subsets of X, B j for j =,,, called focal elemets ad a mappig m, called the basic probability assigmet, defied as, m: 2 X [0, ] such that: ) m(b j ) [0, ]. 2) j= m ( B j ) =. 3) m(a) = 0, A B j.. ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
Proceedigs of the World Cogress o Egieerig 200 Vol I, Jue 30 - July 2, 200, Lodo, U.K. As said before, the cases of risk ad igorace are icluded as special cases of belief structure i the D-S framework. For the case of risk, a belief structure is called Bayesia belief structure if it cosists of focal elemets such that B j = {x j }, where each focal elemet is a sigleto. The, we ca see that we are i a situatio of decisio makig uder risk eviromet as m(b j ) = P j = Prob {x j }. The case of igorace is foud whe the belief structure cosists i oly oe focal elemet B, where m(b) essetially is the decisio makig uder igorace eviromet as this focal elemet comprises all the states of ature. Thus, m(b) =. Other special cases of belief structures such as the cosoat belief structure or the simple support fuctio are studied i [0]. Note that two importat evidetial fuctios associated with these belief structures are the measures of plausibility ad belief [0]. B. The OWA Operator The OWA operator [3] 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 2. A OWA operator of dimesio is a mappig OWA: R R that has a associated weightig vector W of dimesio with w j [0, ] ad j= w j =, such that: OWA (a,, a ) = w j b j i= where b j is the jth largest of the a i. Note that differet properties could be studied such as the distictio betwee descedig ad ascedig orders, differet measures for characterizig the weightig vector ad differet families of OWA operators [-2,6-8,2,5-6]. C. The Weighted Average The weighted average (WA) is oe of the most commo aggregatio operators foud i the literature. It has bee used i a wide rage of applicatios. It ca be defied as follows. Defiitio 3. A WA operator of dimesio is a mappig WA: R R that has a associated weightig vector V, with v j [0, ] ad i = v i =, such that: WA (a,, a ) = v i a i j= 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 [-2,6]. () (2) D. The OWAWA Operator The ordered weighted averagig weighted average (OWAWA) operator [6-7] 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. It ca be defied as follows. Defiitio 4. 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) j= where b j is the jth largest of the a i, each argumet a i has a associated weight (WA) v i with i = v i = 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. As we ca see, if β =, we get the OWA operator ad if β = 0, the WA. The OWAWA operator accomplishes similar properties tha the usual aggregatio operators. Note that we ca distiguish betwee descedig ad ascedig orders, exted it by usig mixture operators, ad so o. III. DECISION MAKING WITH D-S THEORY USING THE OWAWA OPERATOR A ew method for decisio makig with D-S theory is possible by usig the OWAWA operator. The mai advatage of this approach is that we ca use probabilities, WAs ad OWAs i the same formulatio. Thus, we are able to represet the decisio problem i a more complete way because we ca use objective ad subjective iformatio ad the attitudial character (degree of optimism) of the decisio maker. The decisio process ca be summarized as follows. Assume we have a decisio problem i which we have a collectio of alteratives {A,, A q } with states of ature {S,, S }. a ih is the payoff if the decisio maker selects alterative A i ad the state of ature is S h. The kowledge of the state of ature is captured i terms of a belief structure m with focal elemets B,, B r ad associated with each of these focal elemets is a weight m(b k ). The objective of the problem is to select the alterative which gives the best result to the decisio maker. I order to do so, we should follow the followig steps: Step : Calculate the results of the payoff matrix. Step 2: Calculate the belief fuctio m about the states of ature. Step 3: Calculate the attitudial character (or degree of oress) of the decisio maker α(w) [6-7,3]. Step 4: Calculate the collectio of weights, w, to be used i the OWAWA aggregatio for each differet cardiality of focal elemets. Note that it is possible to use differet methods depedig o the iterests of the decisio maker [6,2,5]. Note that for the WA aggregatio we have to ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
Proceedigs of the World Cogress o Egieerig 200 Vol I, Jue 30 - July 2, 200, Lodo, U.K. calculate the weights accordig to a degree of importace (or subjective probability) of each state of ature. This ca be carried out by usig the opiio of a group of experts that has some iformatio about the possibility that each state of ature will occur. Step 5: Determie the results of the collectio, M ik, if we select alterative A i ad the focal elemet B k occurs, for all the values of i ad k. Hece M ik = {a ih S h B k }. Step 6: Calculate the aggregated results, V ik = OWAWA(M ik ), usig Eq. (4), for all the values of i ad k. Step 7: For each alterative, calculate the geeralized expected value, C i, where: C = r i Vik r= m ( Bk ) (4) Step 8: Select the alterative with the largest C i as the optimal. Note that i a miimizatio problem, the optimal choice is the lowest result. From a geeralized perspective of the reorderig step, it is possible to distiguish betwee ascedig ad descedig orders i the OWAWA aggregatio. IV. THE BS-OWAWA OPERATOR Aalyzig the aggregatio i Steps 6 ad 7 of the previous subsectio, it is possible to formulate i oe equatio the whole aggregatio process. We will call this process the belief structure OWAWA (BS-OWAWA) aggregatio. It ca be defied as follows. Defiitio 5. A BS-OWAWA operator is defied by where = r qk i k= = C m ( B k v j b k j (5) k ) ˆ vˆ j k is the weightig vector of the kth focal elemet such that j= v ˆ = ad vˆ j k [0, ], b is the j k th largest of the a ik, each argumet a ik has a associated weight (WA) v i k with i = v i = ad v k i k [0, ], ad a weight (OWA) w with j = w j k = ad w [0, ], v ˆ = βw + ( β ) v 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 ik, ad m(b k ) is the basic probability assigmet. Note that q k refers to the cardiality of each focal elemet ad r is the total umber of focal elemets. The BS-OWAWA operator is mootoic, bouded ad idempotet. By choosig a differet maifestatio i the weightig vector of the OWAWA operator, we are able to develop differet families of BS-OWAWA operators [6]. As it ca be see i Defiitio 5, each focal elemet uses a differet weightig vector i the aggregatio step with the OWAWA operator. Therefore, the aalysis eeds to be doe idividually. Remark. For example, it is possible to obtai the followig cases: The maximum-wa is formed if w = ad w j = 0, for all j. The miimum-wa is obtaied if w = ad w j = 0, for all j. The average is foud whe w j = / ad v i = /, for all a i. The step-owawa operator is foud whe w k = ad w j = 0, for all j k. The arithmetic-wa is obtaied whe w j = / for all j. Note that if v i = /, for all i, we get the arithmetic-owa (A-OWA). 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. V. NUMERICAL EXAMPLE I the followig, we are goig to develop a umerical example about the use of the OWAWA i a decisio makig problem with D-S theory. We focus o the selectio of moetary policies. Assume a govermet that it is plaig his moetary policy for the ext year ad cosiders five possible alteratives. A = Develop a strog expasive moetary policy. A 2 = Develop a expasive moetary policy. A 3 = Do ot make ay chage. A 4 = Develop a cotractive moetary policy. A 5 = Develop a strog cotractive moetary policy. I order to evaluate these moetary policies, the group of experts of the govermet cosiders that the key factor is the ecoomic situatio of the world for the ext year. After careful aalysis, the experts have cosidered five possible situatios 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. Depedig o the situatio that could happe i the future, the experts establish the payoff matrix. The results are show i Table. Table : Payoff matrix. S S 2 S 3 S 4 S 5 A 70 60 80 40 50 A 2 30 60 80 50 70 A 3 50 40 50 70 80 A 4 40 60 90 70 40 A 5 50 50 40 70 70 ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
Proceedigs of the World Cogress o Egieerig 200 Vol I, Jue 30 - July 2, 200, Lodo, U.K. After careful aalysis of the iformatio, the experts have obtaied some probabilistic iformatio about which state of ature will happe i the future. This iformatio is represeted by the followig belief structure about the states of ature. Focal elemet B = {S, S 2, S 3 } = 0.3 B 2 = {S, S 3, S 5 } = 0.3 B 3 = {S 3, S 4, S 5 } = 0.4 The attitudial character of the eterprise is very complex because it ivolves the opiio of differet members of the board of directors. After careful evaluatio, the experts establish the followig weightig vectors for both the WA ad the OWA operator: W = (0.2, 0.4, 0.4) ad V = (0.3, 0.3, 0.4). Note that they assume that the OWA has a degree of importace of 30% ad the WA a degree of 70%. With this iformatio, we ca obtai the aggregated results. They are show i Table 2. Table 2: Aggregated results. AM WA OWA OWAWA V 70 7 68 70. V 2 66.6 65 64 64.7 V 3 56.6 56 52 54.8 V 2 56.6 59 52 56.9 V 22 60 6 56 59.5 V 23 66.6 67 64 66. V 3 46.6 47 46 46.7 V 32 60 62 56 60.2 V 33 66.6 68 64 66.8 V 4 63.3 66 58 63.6 V 42 56.6 55 50 53.5 V 43 66.6 64 62 63.4 V 5 46.6 46 46 46 V 52 53.3 55 50 53.5 V 53 60 6 58 60. Oce we have the aggregated results, we have to calculate the geeralized expected value. The results are show i Table 3. Table 3: Geeralized expected value. AM WA OWA OWAWA A 63.62 63.2 60.4 62.36 A 2 6.62 62.8 58 6.36 A 3 58.62 60 56.2 58.79 A 4 62.6 6.9 57.2 60.49 A 5 54 54.7 52 53.89 As we ca see, depedig o the aggregatio operator used, the results ad decisios may be differet. Note that i this case, our optimal choice is the same for all the aggregatio operators but i other situatios we may fid differet decisios betwee each aggregatio operator. A further iterestig issue is to establish a orderig of the policies. Note that this is very useful whe the decisio maker wats to cosider more tha oe alterative. The results are show i Table 4. Table 4: Orderig of the policies. Orderig Orderig AM A A 4 A 2 A 3 A 5 OWA A A 2 A 4 A 3 A 5 WA A A 2 A 4 A 3 A 5 OWAWA A A 2 A 4 A 3 A 5 As we ca see, depedig o the aggregatio operator used, the orderig of the moetary policies may be differet. Note that i this example the optimal choice is clearly A. VI. CONCLUSION We have preseted a ew decisio makig approach with D-S belief structure by usig the OWAWA operator. The mai advatage of this approach is that it deals with probabilities, WAs ad OWAs i the same framework. Therefore, we are able to cosider subjective ad objective iformatio ad the attitudial character of the decisio maker. For doig so, we have developed the BS-OWAWA operator. It is a ew aggregatio operator that uses belief structures with the OWAWA operator. We have studied some families of BS-OWAWA operators ad we have see that it cotais the OWA ad the WA aggregatio as particular cases. Moreover, by usig the OWAWA we ca cosider a wide rage of iter medium results givig differet degrees of importace to the WA ad the OWA. We have also developed a umerical example of the ew approach. We have focused o a decisio makig problem about selectio of moetary policies. The mai advatage of this approach is that it provides a more complete represetatio of the decisio process because the decisio maker ca cosider may differet scearios depedig o his iterests. I future research, we expect to develop further extesios of this approach by cosiderig more complex aggregatio operators such as those oes that use ucertai iformatio or order-iducig variables. ACKNOWLEDGEMENTS Support from the Spaish Miistry of Sciece ad Iovatio uder project JC2009-0089 is gratefully ackowledged. REFERENCES [] G. Beliakov, A. Pradera ad T. Calvo, Aggregatio Fuctios: A Guide for Practitioers. Berli: Spriger-Verlag, 2007. [2] T. Calvo, G. Mayor ad R. Mesiar, Aggregatio Operators: New Treds ad Applicatios. New York: Physica-Verlag, 2002. [3] A.P. Dempster, Upper ad lower probabilities iduced by a multi-valued mappig, Aals of Mathematical Statistics, 38:325-339, 967. [4] K.J. Egema, H.E. Miller ad R.R. Yager, Decisio makig with belief structures: A applicatio i risk maagemet, It. J. Ucertaity, Fuzziess ad Kowledge-Based Systems, 4:-26, 996. [5] J. Fodor, J.L. Marichal ad M. Roubes, Characterizatio of the ordered weighted averagig operators, IEEE Tras. Fuzzy Systems, 3.236-240, 995. [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, 2008. [7] J.M. Merigó, O the use of the OWA operator i the weighted average ad its applicatio i decisio makig, i: Proceedigs of the WCE 2009, Lodo, UK, pp. 82-87, 2009. ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)
Proceedigs of the World Cogress o Egieerig 200 Vol I, Jue 30 - July 2, 200, Lodo, U.K. [8] J.M. Merigó ad M. Casaovas, Iduced aggregatio operators i decisio makig with Dempster-Shafer belief structure, It. J. Itelliget Systems, 24:934-954, 2009. [9] M. Reformat, R.R. Yager, Buildig esemble classifiers usig belief fuctios ad OWA operators, Soft Computig, 2: 543-558, 2008. [0] G.A. Shafer, Mathematical Theory of Evidece. Priceto, NJ: Priceto Uiversity Press, 976. [] R.P. Srivastava ad T. Mock, Belief Fuctios i Busiess Decisios. Heidelberg: Physica-Verlag, 2002. [2] Z.S. Xu, A overview of methods for determiig OWA weights, It. J. Itelliget Systems, 20:843-865, 2005 [3] R.R. Yager, O ordered weighted averagig aggregatio operators i multi-criteria decisio makig, IEEE Tras. Systems, Ma ad Cyberetics B, 8:83-90, 988. [4] R.R. Yager, Decisio makig uder Dempster-Shafer ucertaities, It. J. Geeral Systems, 20:233-245, 992. [5] R.R. Yager, Families of OWA operators, Fuzzy Sets ad Systems, 59:25-48, 993. [6] R.R. Yager ad J. Kacprzyk, The Ordered Weighted Averagig Operators: Theory ad Applicatios. Norwell: Kluwer Academic Publishers, 997. [7] R.R. Yager ad L. Liu, Classic Works of the Dempster-Shafer Theory of Belief Fuctios. Berli: Spriger-Verlag, 2008. ISSN: 2078-0958 (Prit); ISSN: 2078-0966 (Olie)