Utility Analysis of Belief in Evidence Theory

Systems Engineering — Theory & Practice Volume 28, Issue 3, March 2008 Online English edition of the Chinese language journal Cite this article as: SETP, 2008, 28(3): 103–110

Utility Analysis of Belief in Evidence Theory LIU Ye-zheng∗ , JIANG Yuan-chun, ZHANG Jie-kui School of Management, Hefei University of Technology, Hefei 230009, China Key Laboratory of Process Optimization and Intelligent Decision-making, Ministry of Education, Hefei 230009, China

Abstract: This article proposes, for the first time, a new issue called utility paradox problem, which can not be solved effectively by existing methods in evidence theory. A utility analysis method considering the subjective preference of decision maker is presented to solve the utility paradox problem. First, with a utility-belief function, the belief degrees given by experts are transformed into the utility-belief degrees, which determine decision makers’ actions. Second, the rule of combination in evidence theory is used to combine the utility-belief degrees. This article presents properties the utility-belief function should have, and a specific function is given with its validation proved. Experiment shows the method is effective to solve the utility paradox problem. Key Words: evidence theory; utility paradox; utility analysis; utility-belief function; utility belief

1 Introduction In our daily decision making process, a lot of uncertain information including incompletion, imprecision, and inaccuracy should be handled. In order to use the uncertain information to support our decision, some uncertainty theories, such as Bayesian decision theory, fuzzy set theory[1] , rough set theory[2] , and evidence theory[3] are proposed. Specifically, evidence theory gives a general framework, in which information can be described and combined effectively. In the last few years, evidence theory has been used successfully in many areas, such as decision science, expert system, and information fusion. However, some limitations still exist in the application of evidence theory. To solve these limitations, much work has been done on the acquisition of independent evidence bodies[4] , the rule of combination[5] , and the resolution of conflicting evidence bodies[6,7] . In the framework of evidence theory, a crucial role is played by Dempster’s rule of combination, which has several interesting mathematical properties. However, illogical results might be obtained by Dempster’s rule of combination in many paradox problems. Solution for the paradox problems is one of the main research contents in evidence theory. For example, to solve the Zadeh paradox problem [8] , Yager [9] , Yang[10] , Liang[11] and Sun[12] proposed their methods, respectively. In addition, Liang[11] gave a useful method based on the weight degrees of evidence bodies to solve the three paradox problems proposed by Zhang[13] . Although existing methods can get better results for the above paradox problems, they can not deal with the following problem. Example 1. One person is planning to apply for the service of a credit card but afraid of the security issues. In his opinion, he will apply for a credit card only if the probability of credit fraud is less than 0.01. Otherwise, he will not lodge the application. To determine the probability of credit fraud, he consults three experts who are familiar with the card sys-

tem. The opinions given by them are presented in Table 1. As can be seen from Table 1, the belief degrees given by the three experts for the credit security are all smaller than 0.9. That is to say, all the expert opinions do not support the person to use the credit card. However, if we use the Dempster’s rule of combination to combine the three evidence bodies, the collective belief degree for security is larger than 0.995, which supports the person to use the credit card. Obviously, the collective belief degree might mislead the person to make a wrong decision. The main reason for the above mistake is that the Dempster’s rule of combination does not take the objective preference of decision makers into account. In real world, decisions are usually made by decision makers while the main role of experts is to give advices. For a specific decision problem, the objective preference of the decision maker is normally decisive. However, the existing methods often combine the expert opinions directly and few of them consider the preference of decision makers. In real decision process, different decision makers might have distinct objective preference because of the diversity of their knowledge, experience, or expertise related to the problem. This article uses the concept of “utility” in the decision science to describe the objective preference of decision makers, and name the belief degree, which accounts for the preference of the utility-belief degree. The above paradox problem is called the “utility paradox” problem in

Table 1. Expert opinions for the credit card system

Expert E1 E2 E3

Belief degree of security 85% 87% 85%

Received date: November 14, 2006 ∗ Corresponding author: Tel: +86-551-2904991; E-mail: [email protected] Foundation item: Supported by the National Natural Science Foundation (No.70672097, No.70631003) c 2008, Systems Engineering Society of China. Published by Elsevier BV. All rights reserved. Copyright 

Belief degree of fraud 15% 13% 15%

LIU Ye-zheng, et al./Systems Engineering — Theory & Practice, 2008, 28(3): 103–110

evidence theory. That is, the belief degrees given by experts for a hypothesis are not consistent with the decision maker’s preference, i.e., the utility of the decision maker, but the collective belief degree derived from the experts using the combination rules is consistent with the decision maker’s utility. This article proposes a utility analysis method to solve the problem. The main contribution of this article includes: first, we propose the utility paradox problem in evidence theory for the first time. Second, the utility analysis method is proposed based on the utility-belief function and the properties of the belief-utility function are discussed as well. Third, a specific equation of the utility-belief function is proposed, and some proofs are also given to verify the validity of the equation. The remainder of this article is organized as follows. Section 2 describes the utility paradox problem and our proposed method in detail. Section 3 presents a numerical example to highlight the procedure of our methods. Section 4 gives some discussions of the proposed methods. Summary and further research is concluded in section 5.

2 Problem description and analysis 2.1 Problem description Suppose the state space of a discrete stochastic event X is as follows: x1 · · · xi · · · xn p1 · · · p i · · · p n , where xi is the state of X and pi is its corresponding occurrence probability, i = 1, 2, · · · , n. The occurrence probability for each state is usually supposed to be equal without any prior information about X, that is, pi = 1/n, i = 1, 2, · · · , n. We denote this probability as δ, δ ∈ (0, 1]. Generally, decision makers make decisions according to δ if there is no any priori information about X. That is, if the occurrence probability of state xi is larger than δ, they will prefer to take the action corresponding to the state. Otherwise, they will not prefer to take the action. The probability of the action is consistent with the occurrence probability of the state. However, in many decision problems, decision makers usually have some knowledge and experience about the problem, and hence, they also have different utilities (objective preferences) for the problem. On this condition, they will make decision based on their utilities but not δ. In Example 1, the probability of the credit security and fraud is both 0.5, that is, δ = 0.5 without any prior information. However, it is impossible that the utility of the customer is merely 0.5. In the credit card application problems, even if the probability of credit fraud is 0.05, some customers might still think that it is too dangerous and will not apply for the service. The utility of decision makers has an important role in the decision making process and significantly affects the decision results. Suppose ai is the corresponding action of state xi , (a1 , a2 , · · · , an ) is the set of actions with respect to the discrete stochastic event X, and εi is the utility of the decision maker for state xi , . In general, if the decision maker estimates pi , the occurrence probability state xi , is larger than his utility εi , and he will prefer to take the corresponding action, that is p(ai ) > δ. Otherwise, he will not prefer to take

the corresponding action, that is p(ai ) < δ. In addition, if pi is equal to εi , the decision maker will make a stochastic decision, that is p(ai ) = δ. Obviously, there are some relationships between pi and p(ai ), which is determined by the utility of the decision maker. In evidence theory, the above problem is not described by probability but by the basic belief assignment (BBA). Suppose the framework of discernment of event X is Θ = {x1 , · · · , xi , · · · , xn }, xi is a state of event X, i = 1, 2, · · · , n. Then, the BBA given by experts can be represented using the following form: ··· Xl ··· Xt X1 m(X1 ) · · · m(Xl ) · · · m(Xt ) where ∀(Xl ) ∈ 2Θ , m(Xl ) is the belief degree of state set Xl give by experts. For convenience, we can give some restriction to the framework of discernment like [10] to get a simple BBA: ··· xi ··· xn xu x1 m(x1 ) · · · m(xi ) · · · m(xn ) m(xu ) where m(xi ) is the belief degree of state xi , xu is the uncertain state and m(xu ) is its corresponding belief degree. On this condition, the action the decision maker will take depends on his utility of the event and the BBAs given by experts. And the utility of the decision maker is decisive in the decision process once the BBAs are established by experts. 2.2

Utility-belief function

As shown above, decision makers usually have their utilities for each state in the framework of discernment. This utility is determined by not only their knowledge and experience but also their preference degree for the uncertainty and risk. This article presents a new method, utility analysis method, to take the knowledge, experience, and preference of decision makers into account to solve the utility paradox problem. The following is the definition of the utility-belief function and its properties. Definition 2.1 (Utility-belief function). Suppose discrete stochastic event X has n independent states, δ = 1/n. x is one state of X and the utility of the decision maker for state x is ε. If the belief degree for state x given by experts is m, then the corresponding utility-belief function is defined as U (m, ε, δ), the utility-belief degree derived from U (m, ε, δ) is denoted by u(x). The properties of U (m, ε, δ) are as follows: (1) If m = 0, then u(x) = 0; (2) If m = 1, then u(x) = 1; (3) If m = ε, then u(x) = δ; (4) If ε = δ, then u(x) = m. Furthermore, u(x) is the strictly convex function and strictly concave function when ε < δ and ε > δ, respectively. (5) U (m, ε, δ) is a monotonously increasing function. Using the utility-belief function defined above, the belief degrees give by experts can be modified based on the utility of the decision maker. The modified belief degrees are called utility-belief degrees, which are the integration of experts’ advices, preference of decision makers, and the states

LIU Ye-zheng, et al./Systems Engineering — Theory & Practice, 2008, 28(3): 103–110

of event X. As can be seen in section 3, the utility-belief degrees can modify the belief degrees given by experts properly and get better results for the utility paradox problem. The elaboration of the five properties is give as follows. Properties (1) and (2) indicate that if state x will occur or will not occur with a probability of 100%, the decision maker will make sure to take or refuse to take the corresponding action, respectively. Property (3) indicates that the decision making process is similar with the stochastic decision without any prior information if the occurrence probability of state x is equal to the utility of the decision maker. So the utility-belief degree to take action is equal to δ. For a specific state, different decision makers usually have distinct utilities. Therefore, even though the occurrence probability of the state is given by the experts, the probabilities for the decision makers to take the corresponding actions might still be different. For the decision makers with lower utilities, the probabilities to take action are larger than that of the decision makers with higher utilities. As shown in property (4), the utility-belief degrees are equal to the belief degrees given by experts when the utility of the decision maker is equal to δ. If the utility of the decision maker is smaller than δ, the utility-belief degree is larger than the belief degree and the smaller the utility is, the larger the probability is for the decision maker to take action. On the contrary, the utility-belief degree is larger than the belief degree if the utility is smaller than δ and the larger the utility is, the smaller the probability is. Property (5) indicates that the decision maker is a rational person. The larger the occurrence probability of a state is, the larger the probability is for the decision maker to take action, and the larger the utility-belief degree is. In addition, the sum of the utility-belief degrees for the certain states should satisfy Eq.(1) because of the existence of uncertainty, and the utility-belief degree of the uncertainty can be computed by Eq.(2): n 

Figure 1. Utility-belief function (δ = 0.25)

Proof. (1) Obviously, U (0, ε) = 0, U (1, ε) = 1, U (ε, ε) = δ. So Eq. (3) is true of Property (1), (2) and (3). ε 1 1 − (1 − δε )( m dU ε ) − m(1 − δ )( ε ) = ε m dm [1 − (1 − δ )( ε )]2

(2)

= = =

(3)

(2)

=

In this way, the basic belief assignment given by experts can be transformed to be the utility belief assignment as follows: ··· xi ··· xn xu x1 u(x1 ) · · · u(xi ) · · · u(xn ) u(xu )

=

u(xu ) = 1 −

n 

(u(xi ))

[1 −

>0

d2 U dm2 =

i=1

[1 −

δε−δ+ε δε 2 (1 − δε )( m ε )] ε(1−δ) (1−ε)δ 2 (1 − δε )( m ε )]

Therefore, Eq. (3) is a monotonously increasing function and is true of Property (5).

(1)

(u(xi )) ≤ 1

1 − (1 − δε )( m+m ε ) ε m 2 [1 − (1 − δ )( ε )]

i=1

=

ε(1−δ) (1−ε)δ −2 × 3 [1 − (1 − δε )( m ε )] ε(1−δ) (1−ε)δ −2 × 3 [1 − (1 − δε )( m ε )] 2 × ( δε − 1)( 1ε ) dU × dm [1 − (1 − δε )( m ε )]

ε m × [1 − (1 − )( )] δ ε ε 1 × (1 − )( ) δ ε

U ε−δ 1 dU × × 2( )( ) dm m ε ε 2

(3)

As can be seen from the above equation, ddmU2 is smaller 2 than 0 if ε is smaller than δ. Otherwise, ddmU2 is larger than 0. Therefore, Eq. (3) is a strictly convex function if ε is smaller than δ and a strictly concave function if ε is larger than δ. Figure 1 shows the relationship between the belief degrees given by experts and the utility-belief degrees. δ is set to be 0.25.

Theorem 2.1 Eq. (3) satisfies all the five properties of the utility-belief function.

Theorem 2.2 If the belief degrees for state xi is as follows: ¬xi xu xi m(xi ) ¬m(xi ) m(xu )

2.3 Specific utility-belief function According to the definition of utility-belief function, this article gives the following equation to solve the utility paradox problems when n = 2. U (m, ε) =

m 1 − (1 − δε )( m ε )

where x = 1 − x(x = m, ε).

LIU Ye-zheng, et al./Systems Engineering — Theory & Practice, 2008, 28(3): 103–110

Then, the utility-belief degrees for state xi calculated by Eq. (3) satisfy:

Table 2. BBAs for the credit card system

ui + ¬ui ≤ 1.

E1(m1 ) E2(m2 ) E3(m3 )

And ui + ¬ui = 1 if mu = 0. Proof. (1) If mu = 0, then ¬mi = m , so we can calculate the sum of ui and ¬ui as follows: mi mi + ε mi 1 − (1 − δ )( ε ) 1 − (1 − ε )( mεi ) δ

(1 − ε)δmi εmi δ = + δ(1 − ε) − (δ − ε)(1 − mi ) δε + (δ − ε)mi =1 (2) If mu > 0, suppose mu1 + mu2 = mu , mu1 ≥ 0, mu2 ≥ 0, then mi + ¬mi + mu = (mi + mu1 ) + (¬mi + mu2 ) = mi + mi = 1 and 1 = U (mi , ε) + U (mi , 1 − ε) = U (mi + mu1 , ε) + U (mi + mu2 , 1 − ε) Because U is the monotonously increasing function for m and mi + mu1 > mi , mi + mu2 > mi , then 1 = U (mi + mu1 , ε) + U (mi + mu2 , 1 − ε) ≥ U (mi , ε) + U (mi , 1 − ε). In this way, the corresponding utility-belief degree of mu can be calculated by 1 − ui − ¬ui . As mentioned above, Eq.(3) can deal with the utility paradox problems with only two states. For the problems with more than 2 states, we adopt the following strategy. Suppose the states of event X are independent with each other, for any state xi whose corresponding action is ai , we can transform the evidence source as follows: ¬xi xu xi mi ¬mi mu where mi is the belief degree the state xi given by experts, ¬xi is the set of states that are independent with xi , ¬mi is the sum of the belief degrees of the states in ¬xi , and mu = 1 − mi − ¬mi is the belief degree assigned to the uncertainty. Based on the above transformation, Eq.(3) can be used to transform mi , ¬mi , and mu to be the utilitybelief degrees. Then, the combination rules can be used to combine the relative evidence bodies and a collective BBA can be obtained for state xi . For other states of event X, the same strategy can be used to determine the corresponding actions.

Fraud (x2 ) 5% 5% 2%

Uncertainty (xu = x1 , x2 ) 5% 3% 3%

Table 3. Collective BBAs of traditional methods

Dempster’s rule of combination Yager’s rule of combination

ui + ¬ui =

Security (x1 ) 90% 92% 95%

3 3.1

m(x1 ) 99.95% 88.44%

m(x2 ) 0.04% 0.04%

m(xu ) 0.01% 11.53%

Illustration Credit card application

Suppose two persons A and C are both planning to apply for the service of a credit card. Before applying for the service, they consult three specialists on the security of the credit card system. Based on their knowledge and experience, the three specialists give three basic belief assignments of the security, fraud, and uncertainty of the credit card. The BBAs are presented in Table 2. Suppose A is a defender, whose utility of the credit security is 95%. That is, if the probability of credit security exceeds 0.95, he will prefer to use the credit card. On the contrary, C is a prospector whose utility of the credit security is 90%. Intuitively, the probability of A to use the credit card is smaller than that of C. As can be seen in Table 2, A might prefer to use the card and C prefers to refuse the card if the three experts’ advices are given to them, respectively. Now, we will give the contrastive analysis whether A and C should use the credit card using the traditional methods and the utility analysis method. (1) Results of the traditional methods We first use the traditional methods, such as Dempster’s and Yager’s rule of combination to combine the three BBAs in Table 2. The results are shown in Table 3. As can be seen in Table 3, if the BBA obtained by Dempster’s rule of combination is given to A and C, both of them will believe that the credit card is secure because the probability of credit security exceeds 0.999 on the basis of the collective BBA. But it is not the fact. Intuitively, the belief degrees given by the three experts for credit security are all less than the utility of A. So the collective belief degrees might mislead A to make a wrong decision. Similarly, if the collective belief degrees obtained by Yager’s rule of combination are provided to the two persons, both of them will think that this credit card system is not secure because the collective belief degree for the credit security is less than 0.9. This will mislead C to make a wrong decision because the belief degrees given by the three experts are all support C to use the credit card. (2) Results of the utility analysis method The main limitation of the traditional methods to solve the utility paradox problem is that they do not take the utilities of A and C into account. As a result, the traditional combination rules always magnify or minify the real belief degrees given by experts and get illegal combination results. In addition, they provide the same collective BBA for the

LIU Ye-zheng, et al./Systems Engineering — Theory & Practice, 2008, 28(3): 103–110

Table 4. Utility-belief degrees of A and C

Customer

Expert

A (ε1 = 0.95) (ε1 = 0.05) C (ε1 = 0.90) (ε1 = 0.10)

E1(u1 ) E2(u2 ) E3(u3 ) E1(u1 ) E2(u2 ) E3(u3 )

Security (x1 ) 0.3214 0.377 0.5 0.5 0.561 0.6786

Fraud (x2 ) 0.5 0.5 0.2794 0.3214 0.3214 0.1552

Table 6. Transformed BBAs of stock B

Uncertainty (xu ) 0.1786 0.123 0.2206 0.1786 0.1176 0.1663

E1 E2

Advancing (x1 ) 37% 41%

Declining (x2 ) 23% 25%

Unchanged (x3 ) 28% 21%

Advancing 37% 41%

Investor A

Uncertain 12% 13%

decision makers with different utility. Obviously, it is unreasonable and might mislead the decision makers to make wrong decisions. To get reasonable results and support A and C to make correct decisions, the utility analysis method is used to analyze the above problem. In this example, δ is equal to 0.5 because there are only two states in the problem. Based on the utilities of A and C, Eq.(3) is used to transform the belief degrees in Table 2 to be the utility-belief degrees presented in Table 4. For the utility-belief degrees in Table 4, if the Dempster’s rule of combination is used, the collective results for A and C are as follows: uA : uA (x11 ) = 0.4533, uA (x12 ) = 0.5343, uA (xu ) = 0.0124 uC : uC (x11 ) = 0.8452, uC (x12 ) = 0.1471, uC (xu ) = 0.0077 Similarly, if the Yager’s rule of combination is used, the collective results for A and C are as follows: uA : uA (x11 ) = 0.1753, uA (x12 ) = 0.2066, uA (xu ) = 0.6181 uC : uC (x11 ) = 0.3855, uC (x12 ) = 0.0671, uC (xu ) = 0.5474 As can be seen from the above results, the utility analysis method can get distinct results for different decision makers based on the same BBAs. The collective results suggest that the probability of A to apply for the credit card is less than that of C. Furthermore, both of the two results obtained by the Dempster’s and Yager’s rule of combination support C to apply for the card (uC (x12 ) < uC (x11 )) and do not support A to apply for the card (uA (x12 ) < uA (x11 )). The result is consistent with our intuition. 3.2 Stock investment Suppose A and C are planning to invest in stock B and a consultation is held with two senior stock experts before investment. The belief degrees given by the experts are presented in Table 5. In this example, there are three certain states for the stock, including advancing, declining, and unchanged state (n = 3, δ = 1/3). A is a conservative investor, his utility for the advancing state is 0.45 and declining

Declining 23% 25%

Unchanged and uncertain 40% 34%

Table 7. Utility-belief assignment of stock B

C

Table 5. Belief degrees of stock B

Expert

Expert E1 E2

Expert E1 E2 E1 E2

Buy 0.2641 0.2981 0.3529 0.3922

Sell 0.374 0.4 0.2584 0.28

Wait and see 0.3619 0.3019 0.3887 0.3278

Table 8. Utility-belief degrees of investor A and C

Traditional methods Utility methods

Rule of combination Dempster Yager Dempster Yager

A/C A/C A C A C

Buy

Sell

0.5429 0.4415 0.3401 0.5083 0.2663 0.4066

0.2898 0.2357 0.5203 0.3322 0.4073 0.2658

Wait and see 0.1672 0.3228 0.1396 0.1595 0.3264 0.3276

state is 0.20. That is, A will buy stock B on condition that p(x1 ) > 0.45, and sell if p(x2 ) > 0.2 . On the contrary, C is a positive investor, and he will buy the stock if p(x1 ) > 0.35 and sell p(x2 ) > 0.3 . Based on the belief degrees presented in Table 5, what is the possibility for investors A and C to buy stock B? In this example, the corresponding set of actions is buying, selling, waiting-and-seeing. Generally speaking, the decision of buying is made according to the advancing state and selling is based on the declining state. Waiting-andseeing is one kind of the decisions which is made because the tendency of the stock is uncertain. In this example, the decision of Waiting-and-seeing relates to two states, including the unchanged and uncertain states. To get states as independent as possible, we can transform Table 5 to Table 6 according to the above analysis. Using the utility analysis method, we can get the utilitybelief degrees of the stock B for each investor based on their utilities for each state. The results are presented in Table 7. Using Dempster’s and Yager’s rule of combination, we get the collective results in Table 8. From Table 8, we can see that both A and C are likely to buy the stock if the results obtained by the traditional methods are presented to them. But we can get a more reasonable conclusion according to the results of the utility analysis. That is, the possibility of A to buy the stock is larger than that of C. In this example, the states of the stock are dependent with each other. As a result, the corresponding actions are not independent and can not be analyzed by Eq.(3) directly. The aim of this example is to illustrate how to deal with the problems with dependent states. That is, transferring the dependent states to be independent states. Another way to deal with the dependent problem is to redistribute the belief degrees of the dependent states to the

LIU Ye-zheng, et al./Systems Engineering — Theory & Practice, 2008, 28(3): 103–110

independent states. Limited to the length, the elaboration is omitted in this article.

4 Discussion (1) The relationship between the utility analysis and traditional method As can be seen in the properties of the utility-belief function, the following equation is true when ε is equal to δ: U (m, ε, δ) = m That is to say, the suggested utility analysis method is the same with the traditional methods if the utility of the decision makers are equal to δ. (2) Profit and loss, belief degree, and utility-belief degree. In the real decision making process, decision makers usually make decisions based on their utilities of the problem. In utility theory, the main goal is to analyze the utilities of decision makers when the probability of each states and the corresponding profit and loss are known. The utilities determine the actions decision makers will take. In this article, we suppose that the utilities of decision makers are known. With the known utilities, we analyze the decision makers’ preference for the occurrence probabilities of the event and confirm whether the decision makers should take the corresponding actions. As can be seen from the properties of the utility-belief function, the utility-belief degrees are obtained based on the belief degrees given by experts and the preference of the decision makers. The belief degrees given by experts are the measure of the occurrence probability of the states. The preference of the decision makers shows not only the decision maker’s preference on risk but also the decision maker’s comprehensive consideration to the profit and loss. Therefore, the utility analysis method combines the experts’ advices, the profit and loss of states, and the preference of decision makers to support our decision and the results obtained by the utility analysis method are more effective and reliable. (3) The utility analysis method proposed in this article is based on the hypothesis that the states of an event are independent with each other. If the states are not independent, the corresponding decision is usually affected by more than one of them. There are two ways to solve the problem. The first way is using the Eq.(3) as shown in section 3.2, but it needs transformation to get independent states. Another way is to design a more general utility-belief function that can deal with the problems with more than two dependent states. Furthermore, how to make decisions based on the dependent states and the conflicting conditions are the main directions in the utility paradox research.

5 Conclusion In this article, the utility paradox problem in evidence theory was investigated for the first time. The outcome is

a utility analysis method that utilizes the utility of decision makers to modify the belief degrees given by experts. The belief degrees given by experts are transformed into utilityconstraint belief degrees first. Then, the combination rules in evidence theory are used to combine the utility-constraint belief degrees to get reasonable results. The experiments demonstrate that utility analysis method can achieve better results than the traditional methods for the paradox problem. In addition, Eq.(3) is just one of the utility-belief functions and can only deal with the problems with two states. Therefore, our research will focus on the design of more general utility-belief function in the future to reflect the relationship between the advices of experts and the decisions made by decision makers.

Acknowledgments The authors would like to acknowledge the support from the National Natural Science Foundation under contract No.70672097 and No.70631003.

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