On the Methods of Decision Making under Uncertainty with Probability Information

Transcription

1 O the Methods of Decisio Makig uder Ucertaity with Probability Iformatio Xiwag Liu* School of Ecoomics ad Maagemet, Southeast Uiversity, Najig , Chia By cosiderig the decisio maker s attitude of profit ad risk, we propose a alterative selectio method that ca iclude the methods of decisio makig uder igorace ad decisio makig uder risk as special cases. A idex to measure the decisio maker s risk-averse degree is proposed. With a give optimistic level of profit ad risk, the evaluatio results of the alteratives ca be obtaied with a geometric ordered weighted average OWA) operator ad a basic defuzzificatio distributio BADD) eat OWA operator. Some properties of these two kids of OWA operator i the problem of decisio makig uder ucertaity are also proposed Wiley Periodicals, Ic. 1. INTRODUCTION I the last few years, decisio uder ucertaity has become a topic of iterest i AI. 1 5 The problem of decisio makig uder ucertaity ca be described with a decisio matrix A 2,6 u 1 u 2 J u A 1 A 2 a12 J a1 a 21 a 22 J a 2 I I I J I A ma11 a m1 a m2 J a m A i ~i 1,2,...,m! represets the alteratives to the decisio maker, u j ~ j 1,2,...,! represets the ucertaity stats that will happe after the decisio is made, which is described by state variable V, ad a ij represets the payoff the decisio will get whe the decisio maker selects alterative A i ad evet u j happes. * address: xwliu@seu.edu.c. INTERNATIONAL JOURNAL OF INTELLIGENT SYSTEMS, VOL. 19, ) 2004 Wiley Periodicals, Ic. Published olie i Wiley IterSciece DOI /it 转载

2 LIU Geerally, the decisio maker selects oe alterative cosiderig the payoffs i various coditios. Two types of decisio problems have bee ivestigated. 1. Decisio makig uder risk. This assumes that we have some probabilistic iformatio with respect to the state of V. That is, it assumes there exists a probability distributio for u i such that p j Prob~V u j!. I this article, we will assume that j, j 1,2,...,, p j 0. For the selectio of the best alterative i decisio makig uder risk, there are two most commoly used methods i practice: the expected value method ad the most probability method. 1) The expected value method calculates the expected payoff E i for each alterative A i, where Val~A i! E i p j a ij 1) The alterative with the best expected payoff will be selected. 2) The most probability method selects the payoff value that has the biggest probability value as the evaluatio criteria for each alterative, which meas that makes p j max 1r p r. Val~A i! a ij 2) 2. The secod category is decisio makig uder igorace. The decisio maker has o iformatio about the probability iformatio u j except that it is i the state space of V. I this case, the results are determied by the attitude of the decisio maker. A procedure is itroduced to get the appropriate solutio with a give decisio attitude. There are usually three types of decisio attitude i the research literature. 1) The first is called the pessimistic attitude. The decisio maker oly cosiders the worst case for each alterative; the alterative with the best payoff uder the worse case will be selected, that is, the evaluatio value of each alterative is determied by Val~A i! mi$a ij % 3) j 2) The secod is called optimistic. The decisio maker oly cosider the best cases for each alterative; the alterative with the best payoff uder the best case will be selected. The evaluatio value for each alterative is determied by Val~A i! max$a ij % 4) j 3) The third is called eutral, also called the Laplace criterio. The decisio maker cosiders every state equally; the evaluatio value for each alterative is determied by its overall coditio, the arithmetic weighted average of the payoff values. The evaluatio value for each alterative is determied by Val~A i! 1 a ij 5)

3 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1219 A combiatio of the pessimistic ad optimistic is called the Hurwicz method; the evaluatio value for each alterative is determied by Val~A i! ~1 a! mi$a ij % a max$a ij % 6) j j where is the optimistic degree of the decisio maker; whe a 0, it correspods to the pessimistic; whe a 1, it correspods to the optimistic. As a extesio of the Hurwicz method, Yager 7 proposed the ordered weighted average OWA) operator for this multi-attribute decisio-makig problem. The OWA weights ca reflect the optimistic, pessimistic, or eutral attitude of the decisio maker; the method gives set weights correspodig to the payoff value of state for each alterative; the evaluatio process is based o the aggregatio results of each alterative: Val~A i! w j b ij 7) where W ~w 1,w 2,...,w! is the OWA weights ~w j 0, j 1,2,...,, w j 1!, b ij is the jth biggest elemet of a ij for specific i. The pessimistic, optimistic, eutral, ad Hurwicz criteria ca be represeted with W *, W *, W N, ad W H, respectively, 8 where W * ~1,0,...,0!, W * ~0,0,...,1!, W N ~1/, 1/,...,1/!, ad W H ~a,0,...,1 a!. To help classify the decisio maker s attitude associated with a particular weight vector W, Yager 7 ad Yager ad Filev 8 developed a measure associated with ay weighig vector W called the measure of optimismoress). Assume W is a OWA weightig of dimesio ; the the measure of optimism associated with W is defied as Opt~W! 1 ~ j!w j 8) 1 where w j is the jth compoet i W. It ca be easily see that Opt~W! lies i the uit iterval. The larger the optimism value, the more optimistic the aggregatio is. I respect, it ca be show that Opt~W *! 1, Opt~W *! 0, Opt~W N! 1 2 _, ad Opt~W H! a. Geerally, as more of the weights are located ear the top W, the more optimistic the weightig vector becomes, whereas movig weight to the bottom of W makes it more pessimistic. The methods of decisio makig uder risk ad decisio makig uder igorace cosider the objective probability ad the subjective attitude separately, so there is a eed to propose a geeral form to combie these two aspects together, which was poited out by Yager 4 pp. 1 2):

4 LIU What is ofte overlooked is the use of the expected value implicitly implies a particular decisio attitude, oe midway betwee the optimistic ad pessimistic attitude. I the face of havig o iformatio about the decisio maker s attitude, the use of the expected value seems like a reasoable choice. However, as we move more ito the realm of itelliget techologies, where by itelliget techologies we ofte mea techologies that try to iclude as much as possible the subtleties of the huma aget i the automated procedures, we must try to iclude i our systems a more realistic represetatio of the decisio maker s attitude toward ucertaity, we require the ability to iclude the decisio maker s attitude i the decisio fuctio. For the problem of costructig decisio fuctios that allow for the iclusio of decisio attitude ad probabilistic iformatio i ucertaity, Yager 2,4,9 ad Yager ad Filev 8 comprehesively ivestigated this problem i various coditios ad proposed a mechaism for combiig probabilistic iformatio about the state of ature with iformatio about the decisio maker s attitude; the mai idea is to combie the payoff a ij with correspodig probability value p j for each alterative i, the, with the fuzzy modelig techique, get the evaluatio value for alterative i. I the preset article, we propose a alterative method for the problem of decisio makig uder ucertaity with probability iformatio. Obviously, faced with ucertaity, the decisio maker s objective is to get the maximum payoff value ad at the same time to reverse the risk. Whe there is o iformatio o probability, or the probability values of each coditio are equal, the decisio maker oly eeds to select the best alterative accordig to the evaluatio value based o his decisio makig attitude. Whe his attitude is formed as a OWA operator, the weights will chage as the attitude chages. I the latter, we will call this attitude the profit attitude as it has o relatio with the risk of ucertaity probability iformatio). Whe the probability iformatio is provided, there will exist differet risks for differet probability values; the bigger the probability value, the smaller the risk is, so aother aspect of the decisio maker s attitude is the attitude toward risk. Based o this idea, we should cosider the decisio maker s attitude i two aspects: the attitude toward profit that is associated with payoff value, ad the attitude toward risk that is associated with the probability value. As metioed above, the attitude toward profit ca be categorized as optimistic, eutral,orpessimistic, correspodig to the maximum, arithmetic average, ad miimum payoff value of each alterative, respectively. The attitude toward risk is somewhat differet from the attitude toward profit. No decisio maker will select the payoff value with the miimum probability value as the evaluatio criterio for each alterative, but the decisio maker s risk attitude ca also be categorized ito three types: the risk averse, which selects the payoff value with the maximum probability as the evaluatio criterio; the risk eutral, which regards the risk as what it is, that is, the decisio maker cosiders the probability value i the decisio-makig process, but the probability iformatio does ot domiate the decisio result; ad the third type is the risk umb, i which the decisio maker does ot cosider the differet risks that are geerated from the probability values; he/she is idifferet to them, that is, he/she regards the risk as equal whatever the ozero) probability value is.

5 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1221 By cosiderig the decisio maker s attitude toward profit ad risk, we propose a decisio model that combies the decisio maker s attitude ad the probability iformatio together. Sectio 2 gives a explaatio of the OWA operator that describes the decisio maker s profit attitude as a probability distributio, ad combies it with the objective probability iformatio. With a give profit optimistic degree, the evaluatio value for each alterative ca be uiquely obtaied with the geometric OWA operator maximum etropy OWA operator), ad the more optimistic the decisio maker is, the bigger the evaluatio value will be. Sectio 3 gives a method to describe the decisio maker s attitude with a basic defuzzificatio distributio BADD) operator, a kid of eat OWA operator. A measure to represet the decisio maker s risk attitude is proposed. Sectio 4 gives a evaluatio method with a give profit ad risk optimistic degree simultaeously. Sectio 5 summarizes the mai results ad draws coclusios. 2. THE EXPLANATION OF PROFIT ATTITUDE OPERATOR w i From the discussio above we ca see that i the methods of decisio makig uder risk, the expected value does ot reflect the decisio maker s attitude it ca be assumed that the decisio attitude is eutral), ad the methods of decisio makig uder igorace do ot iclude the probability iformatio. I may decisiomakig problems we must cosider these two aspects simultaeously. There eeds to be a method to combie the two paradigms foud i the classical literature, decisio makig uder risk ad decisio makig uder igorace. If we regard the OWA operator as aother kid of probability iformatio that represets the decisio maker s attitude, the problem of decisio makig uder ucertaity with probability iformatio ca formulated as follows: h 1 h 2 J h u 1 u 2 J u A 1 A 2 a12 J a1 a 21 a 22 J a 2 I I I J I A ma11 a m1 a m2 J a m where h i idicates that the decisio maker selects the payoff of state i as the evaluatio criterio of the alteratives. The probability that h i happes depeds o the attitude of the decisio maker. If the decisio maker is pessimistic, he will always selects the lowest payoff as the criterio of each alterative; o the other had, if he is a optimistic, he will always select the highest. Let P~h i! w i, where the pessimistic ad optimistic correspods to W * ad W *, respectively. So a OWA operator ca be see as the probability of the evaluatio criterio for the alteratives that reflect the decisio maker s attitude. Here we have two kids of probabilities: Oe is objective ad determied by the ature of the problem, the other is subjective ad determied by the attitude of the decisio maker. As the OWA

6 LIU Table I. The combied probability distributio. u 1 u 2 J u h 1 p 1 w 1 p 2 w 1 J p w 1 w 1 h 2 p 1 w 2 p 2 w 2 J p w 2 w 2 I I I I I I h p 1 w p 2 w J p w w p 1 p 2 J p operator is oly determied by the order of the payoff value, so the two probabilities ca be see as idepedet, that is P~u i, h j! p i w j The combied probability distributio is show i Table I. Here, we are cosiderig two kids of ucertaity simultaeously: the objective ucertaity, the probability iformatio, ad the subjective ucertaity, the decisio maker s attitude. The evaluatio process is to combie these two kids of ucertaity together. Obviously whe u i ad h j happe, the objective evaluatio value of alterative k is a ik, ad the subjective evaluatio value is a jk. Whe i j, the two evaluatios will be cosistet, that is, whe u i ad h i happe, the payoff value of alterative k is a ik, ad whe u j ad h j happe, the payoff value of alterative k is a jk, but whe i j, the two aspects of the evaluatio results are icosistet. The payoff value of alterative k ca be set as the trade-off of a ik ad a jk, as the evaluatio process attais a ik ad a jk with probability p i w i ad p j w j, respectively. So a reasoable method to set the payoff value is to set it as the weighted average of a ik ad a jk with relative weights p i w i ad p j w j. The geeral expressio of the payoff value of alterative k whe h i ad u i happes is that w i p i a ik w j p j a jk w i p i w j p j The payoff values for alterative k are show i Table II. Table II. The payoff value of alterative k. u 1 u 2 J u h 1 a 1k w 1 p 1 a 1k w 2 p 2 a 2k w 1 p 1 w 2 p 2 J w 1 p 1 a 1k w p a k w 1 p 1 w p h 2 w 1 p 1 a 1k w 2 p 2 a 2k a 2k J w 2 p 2 a 2k w p a k w 1 p 1 w 2 p 2 w 2 p 2 w p I I I I I h w 1 p 1 a 1k w p a k w 1 p 1 w p w 2 p 2 a 2k w p a k J a k w 2 p 2 w p

7 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1223 From the above, the expectatio of alterative k is E k w i p i a ik w j p j a jk w i p j 9) w i p i w j p j Now, we will ivestigate the properties of the proposed approach. Propositio 1. If we set p 1 p 2 {{{ p 1/, the E k 1 w i p i a ik w j p j a jk w i p j w i p i w j p j w i a ik w j a jk w i w i w j w i a ik 10) This become the case of decisio makig uder igorace. Propositio 2. If we set w 1 w 2 {{{ w 1/, similarly, E k p i a ik This become the case of decisio makig uder risk, which correspods to the eutral decisio attitude. Propositio 3. If we set W W *, E k a 1k max a ik 1k This correspods to the optimistic decisio attitude. Propositio 4. If we set W W *, E k a k mi a ik 1k This correspods to the pessimistic decisio attitude. For ease of computatio, istead of 9), we ca also use a simplified form that coicides with the immediate probability method 10 : E k w i p i a ik w j p j 11)

8 LIU It is obvious that the above properties still holds whe w j p j 0. The method is a atural extesio combiig the methods of decisio uder risk ad decisio uder igorace. For elemets to be aggregated x ~x 1, x 2,...,x!, x 1 x 2 {{{ x, ad OWA weights W ~w 1,w 2,...,w!,we deote E W ~x! as E W ~x! w i p i x i w j p j 12) By calculatig the expectatio value of each alterative, we ca select the most suitable alterative. I the latter, we will maily discuss that the attitude of the decisio maker ca be expressed as a OWA operator with mootoic weights, which meas the weights chage icrease or decrease) with the chage of elemets to be aggregated i the same or reverse order directio. If we regard the weights as the importace that represets the decisio maker s attitude, the direct meaig of the mootoic OWA weights is that it reflects the decisio maker s cosistet prefereces about the value of the aggregated elemets: The bigger, the more importat or the smaller, the more importat. We especially ivestigated a class mootoic OWA operator called the geometric OWA operator. Its weights chage gradually with a fixed ratio. Some properties of it are ivestigated, especially the equivalece of the geometric OWA operator ad the maximum etropy OWA operator MEOWA). Here we will give some basic cocepts of the geometric OWA operator. A geometric OWA operator is a class of OWA operator with mootoic weights with which the adjacet weights have a fixed ratio q. A geeral form of it ca be expressed as Cosiderig that 1 w i 1, so w i aq a 0, q 0 13) a 1, q 1 q 1 q 1, q 1 14) w i 1, q 1 ~1 q!q, q 1 1 q 15)

9 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1225 Whe q 1, Opt~W! 1 2 _, ad whe q 0, Opt~W! 1. Hereafter we will oly cosider the coditio that q 0 ad q 1, but all the mai results ca iclude these two special cases: i Opt~W! 1 w i q q 1 ~q 1!~ 1!~q 1! 1 q i i0 ~ 1!~q 1! 2 i0 ~ 1 i!q i 16) 1 ~ 1! q i i0 For a give Opt~W! V ad, q is the root of the solutio of 17) that is trasformed from 16): ~ 1!Vq 1 ~~ 1!V i 1!q i 0 17) i2 Theorem 1. Whe is specified ~ 2!, Opt~W! V f ~,q! as the fuctio of q it is strict mootoe decreasig. Proof. See the Appedix. With a give, Opt~W! ad q are a oe-to-oe mappig, that is, there exists a iverse fuctio of f ~,q! for q, q f 1 ~, V! So formula 17) always has a solutio for V ~0,1# ; whe V 1, q 0, W W * ad whe V 1 2 _, q 1, W W N. With the give optimistic degree V, the geometric OWA weights ca be uiquely determied. The relatioships betwee V ad q for differet are show i Figure 1. We also proved the equivalece of geometric OWA operator ad the maximum etropy OWAMEOWA) operator see the Appedix). The maximum etropy OWA operator was first suggested by O Haga, 11 ad later was discussed by Filev ad Yager 10 amd Fullér ad Majleder. 12 It is a

10 LIU Figure 1. The Optoress) value chages with q. special class of OWA operators havig maximal etropy of the OWA weights for a give optimistic level. The approach is based o the solutio of the followig mathematical programmig problem: max w i l~w i! 1 s.t. ~ i!w i a, 0 a 1 1 w i 1, 0 w i 1, i 1,2,...,m 18) So we ca get the maximum etropy OWA weights with Equatio 17 istead of Equatio 18. Theorem 2. For ordered vector x ~x 1, x 2,...,x!, x 1 x 2 {{{ x ad OWA weights W ~w 1,w 2,...,w!, W ' ~w ' 1,w ' 2,...,w '!, if w i /w w ' ' i /w, i 1,2,..., 1, the E W ~x! E W '~x!.

11 Proof. DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1227 E W ~x! E W '~x! w i p i x i w j p j w i ' p i x i w j ' p j w i p i x i w ' j p j w ' i p i x i w j p j 1 w i p i w ' j p j w i w j ' p i p j 1 w i w j ' p i p j 1 w i w ' j p i p j 1 w i p i x i w ' j p j w i p i w ' j p j ~x i x j! w i w ' j p i p j 1 ji 1j ji 1j w i w ' j p i p j 1 ji 1j ij 1i w i w ' j p i p j ji 1j ji 1j ji 1j w i p i w j ' p j ~x i x j! w i p i w j ' p j ~x i x j! w i p i w j ' p j ~x i x j! w i p i w j ' p j ~x i x j! w i p i w j ' p j ~x i x j! w j p j w i ' p i ~x j x i! w ' j p j x j w i p i p i p j ~x i x j!~w i w j ' w j w i '!

12 LIU As i, j~ j i!, x i x j 0, ad w i /w w ' ' i /w, the w i /w j w ' i /w ' j, that is, w i w ' ' j w j w i 0, so E W ~x! E W '~x!. If we set W ' ~1/, 1/,...,1/!, orw ~1/, 1/,...,1/!, we ca get the followig corollary. Corollary 1. For ordered vector x ~x 1, x 2,...,x!, x 1 x 2 {{{ x, ad weights W ~w 1,w 2,...,w!.Ifw 1 w 2 {{{ w, the E W ~x! p 1 x 1 p 2 x 2 {{{ p x.ifw 1 w 2 {{{ w, the E W ~x! p 1 x 1 p 2 x 2 {{{ p x. Theorem 3. For ay give ad the geometric OWA operator weights W ad W ',ifopt~w! Opt~W '!, the for ay -dimesioal elemet x, E W ~x! E W '~x!. Proof. From Theorem 1 we ca establish whether Opt~W! Opt~W '!. Suppose their correspodig ratios are q ad q ' ; the q q '.Asw i /w 1/q ad w ' ' i /w 1/q ',sow i /w w ' ' i /w. From Theorem 2 we have E W ~x! E W '~x!. For the geometric OWA operator, whe Opt~W! 1 2 _, q 0, W ~1/, 1/,...,1/!, we ca get the followig theorem. Theorem 4. For geometric OWA weights determied by q, whe Opt~W! 1 2 _, the E W ~x! p 1 x 1 p 2 x 2 {{{ p x, especially whe Opt~W! r 0, qr `, E W ~x! r mi 1i $x i %. O the other had, if Opt~W! 1 _ 2, the E W ~x! p 1 x 1 p 2 x 2 {{{ p x, especially whe Opt~W! r 1, qr 0, E W ~x! r max 1i $x i %. Proof. Omitted. From Theorems 3 ad 4, we ca get that for the decisio makig uder risk cosiderig the decisio maker s attitude, with give optimistic level Opt~W! V, we ca always get a uique evaluatio value for each alterative with the geometric OWA operator maximum etropy OWA operator). The process ca be summarized as follow: Algorithm 1. Step 1. Set the optimistic level Opt~W! V; solve Equatio 17 to get q. Step 2. Geerate OWA weights with Equatio 15. Step 3. Calculate E W ~x! for each alterative with Equatio 12. From the process, we ca see that if x i x j for i j, the E W will be uiquely determied by the optimistic level Opt~W! V, ad the more optimistic the decisio maker is, the bigger the evaluatio value will be. Here is a example.

13 Example 1. 4 Cosider the followig decisio matrix: S S 1 S 2 S 3 S 4 A A with probabilities P 1 0.3, P 2 0.2, P 3 0.1, P Set the optimistic degree as 0.4; Equatio 17 becomes 6q 3 q 2 4q 9 0 so q The OWA weights values become w ; w ; w ; w E A1 4 4 x i w i p i w i p i 1 * * * * * * * * * * * * DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1229 Similarly, E A The preferred alterative solutio is A INCLUDING THE DECISION MAKER S RISK ATTITUDE I the proposed method, the OWA operator weights are determied by the order of the payoff value, so the OWA operator oly reflects the decisio maker s attitude toward profit or beefit; it has o relatio to the probability iformatio. I fact, i real decisio makig, the decisio maker ot oly cosiders the profit or beefit but he will also cosider the risk coected with this beefit. As metioed i Sectio 1, the decisio maker s attitude ca be cosidered i two aspect: the attitude toward profit ad the attitude toward risk. The profit is the payoff value ad the risk is the probability value. The method proposed i the previous sectio with Equatio 12 ca be see as the decisio attitude with risk eutral; the objective risk probability value is equal to the subjective probability value. But whe the decisio maker is ot risk eutral, the subjective probability should ot be equal to the objective probability. Ulike the attitude of profit, the decisio maker ca be categorized as pessimistic, eutral, ad optimistic, ad the aggregatio value is betwee the miimum ad maximum payoff values. As o decisio maker will pursue the alterative with maximum risk smallest probability value), the decisio maker s attitude

14 LIU about risk ca be categorized as risk umb, risk eutral, ad risk averse. The risk-averse decisio maker selects the payoff value of the most likely occurrece maximum probability value) as the criterio for each alterative. The riskeutral decisio maker selects the probability value as the oe aspect of the selectig process, that is, the bigger the probability value the smaller the risk), the more he will select it, but he is ot domiated by the probability iformatio. The objective probability values are equal to the subjective values. The risk-umb decisio maker does ot cosider the risk i the decisio process. He is idifferet to it; that is, he regards the risk as equal whatever the probability value is. Obviously, to describe the decisio maker s attitude about risk, we caot use the ordiary OWA operator, as we do ot eed to get aggregatio value ragig from the miimum to the maximum. What we eed to do is to trasform the origial probability iformatio ito the aother probability distributio that icludes the decisio maker s risk attitude. For a OWA operator W associated with the probability value, it will be meaigless for Opt~W! 1 2 _. A suitable formula is the basic defuzzificatio distributio BADD) OWA operator that was proposed by Yager 13 ad Yager ad Filev. 14 The BADD operator is a kid of eat OWA operator, which has the characteristic that the aggregated value is idepedet of the orderig of the elemets to be aggregated. It has bee applied i the liguistic decisio-makig problem. 15,16 For elemets to be aggregated x ~x 1, x 2,...,x!, a BADD OWA operator has the weights associated with x i of the followig form: w i x i a a x j 19) Replacig the x i with probability iformatio p i, we have w i p i a a p j 20) Whe a 0, we get a simple average operator, the weights uder all coditios become the same value, ad the decisio maker is risk umb. Whe a 1, the weights are the origial probability value ad the decisio maker is risk eutral. Whe a 1 the weights have the same order as the probability iformatio; the bigger the probability value is, the bigger the correspodig weight becomes. As a icreases, this tedecy become more acute, which meas the decisio maker is more risk averse. Whe a r `, all the w i s approach zero except the weights of the biggest probability value; this becomes the evaluatio criterio of the maximum probability method. So the decisio-makig method with risk attitude of the decisio maker ca be as follow:

15 E k w i a ik 21) Similar to the Opt~Opt p! ordiary OWA operator, which describes the profit optimistic degree of the decisio maker, the risk attitude of the decisio maker is determied by a, so we have the followig defiitio. Defiitio 1. For a BADD OWA operator determied by Equatio 20, the decisio maker s risk optimistic degree ca be defied as: Opt r 1 22) 1 a Obviously, whe the decisio maker is risk umb, that is, a 0, Opt r 1. Whe the decisio maker is risk eutral, that is, a 1, Opt r 1 2 _. Whe the decisio maker is risk averse, that is, a 1, Opt r 1 2 _. Whe a r `, that is, Opt r r 0, the decisio maker is completely risk averse; he oly cosiders the coditio that has the biggest probability value the miimum risk). The decisio makig uder ucertaity with the risk optimistic degree ca be summarized as follow: Algorithm 2. DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1231 Step 1. Set the risk optimistic level Opt r ; calculate a with Equatio 22 or directly set a). Step 2. Geerate BADD OWA weights with Equatio 20. Step 3. Calculate E W ~x! for each alterative with Equatio 21. I the latter, we will aalyze the properties whe the probability value chages with the payoff value i the same or reverse directio. Whe the probability value chages with the payoff value i the same directio, the bigger payoff value, the bigger the probability value is, that is, the coditio that has the biggest payoff value also has the miimum risk the biggest probability value). O the other had, whe the probability iformatio chages with the payoff value i the reverse directio, the coditio that has the biggest payoff value will have the maximum risk the smallest probability value). Obviously, for a purely risk-averse decisio maker, i the former coditio, he will select the biggest payoff value as the evaluatio value of each alterative, ad i the latter coditio, he will select the smallest payoff value as the evaluatio value of each alterative. Further, we have the followig theorem. Theorem 5. If the the probability chages with the payoff value i the same directio, the the expectatio value E k of alterative k mootoic decreases with the decisio maker s risk-optimistic degree Opt r. O the other had, if the probability chages with the payoff value i the reverse directio, the the expectatio value E k alterative k mootoic icreases with the decisio maker s risk-optimistic degree Opt r.

16 LIU Proof. ]E k ]a ]w i ]a a ik ] p ]a i a p j aa ik p a i a ik l p i a2 p i p a j p a j l p j p a i p a j ~l p i l p j!a ik a2 p i ji 1j ji 1j ji 1j ji 1j p i a p j a ~l p i l p j!a ik a2 p i p i a p j a ~l p i l p j!a ik a2 p i p i a p j a ~l p i l p j!a ik a2 p i ji 1j ij 1i ji 1j p i a p j a ~l p i l p j!~a ik a jk! a2 p i p i a p j a ~l p i l p j!a ik p i a p j a ~l p i l p j!a ik p i a p j a ~l p j l p i!a jk

17 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1233 As Opt r decreases with a, if the probability values p i chage i the same directio with a ik, the ]E k /]a 0. E k decreases with Opt r. O the other had, if the probability values p i chage i the reverse directio with a ik, the ]E k /]a 0, E k icreases with Opt r. From Theorem 5, we ca see that with the risk-optimistic degree Opt r decrease, the evaluatio value of each alterative is closer to the payoff value that has the biggest probability value. Of course, i a real decisio-makig problem, the same or reverse chage directio of p i ad a ik for alterative i may be ot always be satisfied, but it really shows the priciple that risk attitude associated with the probability iformatio iflueces the evaluatio results of the alteratives for the decisio maker. 4. DECISION MAKING UNDER UNCERTAINTY CONSIDERING THE DECISION MAKER S ATTITUDE I Sectios 2 ad 3, we ivestigated the decisio-makig problem uder ucertaity cosiderig the decisio maker s attitude about profit ad risk, respectively. Now we will cosider these two aspects together. The preferece expectatio of the alteratives cosiderig the decisio maker s attitude ca be expressed as E k w pi w ri a ik w pj w rj 23) where w pi ad w ri represet the OWA weights of profit ad risk, respectively. If we set the profit weights with the ordiary OWA operator W p ~w 1,w 2,...,w! ad risk weights with the BADD OWA operator a a p 1 p 2 W r,,..., a a p i p i E k ca be expressed as E k w i p i a a ik w j p j a a p a p i 24) For the evaluatio value E k determied by the geometric ad BADD OWA operators with parameter q ad a, replacig x i with a ik, p i with p i a ad E W ~x! with E k, all the coclusios i Theorem 4 ca be exteded here.

18 LIU If we set Opt r 0, the a 0. The decisio maker is risk umb, ad Equatio 24 becomes the case of decisio makig uder igorace with a OWA operator. If we set Opt p 1 2 _, q 1, w i 1/, the decisio maker is profit eutral, ad Equatio 24 becomes the case of decisio makig uder risk. Especially whe Opt r 1 2 _, a 1, ad the decisio maker is risk eutral. It becomes the traditioal expectatio method, which correspods the decisio maker with eutral profit ad risk attitude. Whe a r `, E W ~x! approximates the evaluatio value with maximum probability criterio. So Equatio 24 is a atural extesio ad combiatio of the decisio-makig model of decisio makig uder risk ad decisio makig uder igorace. By settig the weights of risk ad profit attitude, we ca get the evaluatio results that represet the decisio maker s attitude. For a give attitude of profit ad risk optimistic degree Opt p ~Opt! ad Opt r, the decisio makig problem uder ucertaity ca be solved as follows: Algorithm 3. Step 1. Solve Equatio 17 to get q. Step 2. Geerate OWA weights with Equatio 15 or directly set the OWA weights). Step 3. Calculate a with Equatio 22 or directly set a). Step 4. Calculate E W ~x! for each alterative with Equatio 24. Example 2. Cosider the followig example: S S 1 S 2 S 3 A A A with the probabilities P 1 0.3, P 2 0.4, P ) Suppose that the decisio maker is a bit pessimistic; the optimistic levels are set as Opt p 1 3 _ ad Opt r 1 3 _. From Equatio 17 we ca get 3q 2 2q 7 0, q 1 4 _ ~1 M33!, so w , w , w From Equatio 22, a 2. E E * * * * * * * * * E * * * * * * * * * The most preferable alterative is A 1.

19 2) If we set Opt p 2 3 _, we ca get that E E * * * * * * * * * E * * * * * * * * * DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1235 The most preferable alterative is A CONCLUSIONS I the preset article, by extedig the decisio-makig methods uder igorace ad uder risk, we proposed a method that ca combie these two types of decisio models together, which ca iclude both the decisio maker s attitude ad the probability iformatio. By cosiderig the profit attitude ad risk attitude, the decisio maker s attitude ca be preseted with the geometric OWA operator ad BADD OWA operator. With a give risk ad profit optimistic degree, the evaluatio results that represet the decisio maker s attitude ca be obtaied, ad the evaluatio value is always cosistet with the optimistic attitude idex of the decisio maker. Ackowledgmet The work is supported by the Natioal Natural Sciece Foudatio of Chia NSFC) uder project Refereces 1. Dubois D, Grabisch M, Modave F, Prade H. Relatig decisio uder ucertaity ad multicriteria decisio makig models. It J Itell Syst 2000;15: Yager RR. O the valuatio of alteratives for decisio-makig uder ucertaity. It J Itell Syst 2002;17: Yager RR, Kreiovich V. Decisio makig uder iterval probabilities. It J Approx Reaso 1999;22: Yager RR. Icludig decisio attitude i probabilistic decisio makig. It J Approx Reaso 1999;21: Yager RR. Decisio makig with fuzzy probability assessmets. IEEE Tras Fuzzy Syst 1999;7: Chakog V, Haimes YY. Multiobjective decisio makig: Theory ad methodology. New York: Elsevier Sciece B.V.; Yager RR. O ordered weighted averagig aggregatio operators i multicriteria decisiomakig. IEEE Tras Syst Ma Cyber 1988;18: Yager RR, Filev DP. Iduced ordered weighted averagig operators. IEEE Tras Syst Ma Cyber 1999;29:

20 LIU 9. Yager RR. Fuzzy modelig for itelliget decisio makig uder ucertaity. IEEE Tras Syst Ma Cyber 2000;30: Filev D, Yager RR. Aalytic properties of maximum etropy OWA operators. Iform Sci 1995;85: O Haga M. Aggregatig template or rule atecedets i real-time expert systems with fuzzy set. I: Grove P, editor. Proc 22d Au. IEEE Asilomar Cof o Sigals, Systems, Computers, CA, pp Fullér R, Majleder P. A aalytic approach for obtaiig maximal etropy OWA operator weights. Fuzzy Set Syst 2001;124: Yager RR. Families of OWA operators. Fuzzy Set Syst 1993;59: Yager RR, Filev DP. Parameterized ad-like ad or-like OWA operators. It J Ge Syst 1994;22: Marimi M, Umao M, Hatoo I, Tamura H. Hierarchical semi-umeric method for pairwise fuzzy group decisio makig. IEEE Tras Syst Ma Cyber 2002;325): Marimi M, Umao M, Hatoo I, Tamura H. Liguistic labels for expressig fuzzy preferece relatios i fuzzy group decisio makig. IEEE Tras Syst Ma Cyber 1998;282): APPENDIX Lemma 1. For 2, ad q 0, q 1, we have 2 q 1 ~1 q! 2 ~1 q! 2 0. Proof. Whe 2, 2 q 1 ~1 q! 2 ~1 q! 2 ~1 q! 4 0. I the latter we will assume 2. 2 q 1 ~1 q! 2 ~1 q! q 1 ~1 q! 2 ~1 q! i2 q i0 2 1 q i0 j0 1 1 i0 j0 1 1 i0 j0 k0 22 i0 1 q 1 i0 1 1 q 1 i0 j0 q ~q 1 q ij! ijk 0i, j1 q i2 ~1 q!2 1 q i q j ~1 q!2 j0 ij ~1 q!2 ~1 q!2 ~q 1 q k! ~1 q!2

21 k0 DECISION MAKING UNDER UNCERTAINTY WITH PROBABILITY 1237! 2 22 ~q 1 q k! ~q 1 q k ~1 q!2 k 2 ijk 0i, j1 k0 ijk 0i, j1 k0 2 2 ijk 0i, j1 k0 ijk 0i, j1 k0 2 2 k0 2 k0 2 r0 ijk 0i, j1 22 ~q 1 q k! k 2 ~q 1 q k! r0 2 ~q k1 1!q k r0 ijk 0i, j1 ijk 0i, j1 ijr 0i, j1 ijr2 0i, j1! ~q 1 q k ~1 q!2 ~q 1 q r! ~1 q!2 ~1 q r1!q ~1 q!2 1 2 ~q k1 1!q k ~1 q k1!q ~1 q!2 k0 ijk2 1 0i, j1 a 2 ~k 1!~q k1 1!q k ~ k 1!~1 q k1!q 1 ~1 q!2 k0 2 ~k 1!~q k1 1!q k ~ k 1!~1 q k1!q 1 ~1 q!2 k0 1 2 ~ r 1!~q r1 1!q r2 ~ k 1!~1 q k1!q ~1 q! 2 2 k0 k0 1 2 ~ k 1!~q k1 1!q k2 ~ k 1!~1 q k1!q ~1 q! 2 2 k0 ~ k 1!~1 q k1!~q 1 q k2!~1 q! 2 k0 2 ~ k 1!~1 q k1! 2 q k2 ~1 q! 2 0 k0

22 LIU Proof of Theorem 1. ] Opt~W! ] ]q ]q i0 2 ~ 1 i!q i 1 q i ~ 1! i0 q 2 2q 2 1 q 1 2 2q 2 q 1 2 ~q 1! 2 ~q 1! 2 ~ 1! 2 q 1 ~q 1! 2 ~q 1! 2 ~q 1! 2 ~q 1! 2 ~ 1! From the results of Lemma 1, it is obvious that ] Opt~W!/]q 0 for q ~0,`!, q 1. Theorem 6. The geometric OWA operator is the uique maximum etropy OWA operator for a give Optoress) value, that is, the MEOWA operator ad the geometric OWA operator are equivalet. Proof. Usig the method of Lagrage multipliers, Equatio 18 ca be trasformed to a formula as follows: L~W, a, b! 0 i w i l~w i! l 1 1 w i a l 2 w i 1 25) The ecessary coditios of the solutio are ]L i l w i 1 l 1 ]w i 1 l 2 0 ]L i ]l 1 1 w i a 0 26) ]L ]l 2 w i 1 0 We ca get that w i e l 2 1 w i e l 1 /~1! 27) w So the MEOWA operator W ~w 1,w 2,...,w! is a geometric OWA operator. From Theorem 1, we ca see that for a geometric OWA operator, the ratio q ad the weights are uiquely determied by its optimistic degree, so the geometric OWA operator is the uique maximum etropy OWA operator for a give Optoress) value, that is, the MEOWA operator ad the geometric OWA operator are equivalet.