Response Fusion in Multi-Agent Environment

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INTRODUCTION
One of the main purpose in agents communication is to achieve the goals better [10] . Suppose that in response to a query, each agent can produce a response set which has a level of uncertainty. Uncertainty in agent response may be attributed to two main reasons, namely, deficiency in agent capability and the ambiguity of application information. The first aspect is generally due to agent design/implementation deficiency. The second aspect is due to information overload. For example, in an environment such as the Internet, properties like inaccessibility, nondeterministic and dynamic nature of the information space are sources of agent imprecise decisions. Our goal is the utilization of the other agents in order to reduce the uncertainty and consequently to obtain response with higher quality. For this purpose, the query should propagate in multi-agent environment and agents' response to be aggregate.
For aggregation, these issues should be considered. Agent responses have different degree of accuracy, so final response should be a function of each agent response with considering the degree of credibility. In other words, the opinion of agent with higher credibility has more effect on final answer and vice versa.
For each agent response, the degree of its credibility should be determined. In order to fuse agent responses, we need an operator with considering the problem of credibility degree of resources.
Credibility assignment can be user generated or sanctioned knowledge base [2] . In sanction based system a central agent decides about the credibility of the other agents. As there is no centralized element in multiagent environment, sanctioned knowledge base methods might not be used. In our suggested method, the opinion of agents' community is used to assign the credibility to each agent. There are currently some works based on weighting and probability theory for credibility assignment [4,12,13,14,15] . Also bayesian [5,6] and fuzzy approach [7,8] are used for credibility assessment. Also in our work the possibility of credibility is assigned to each agent, not the probability of credibility. As the credibility, value assigned by each agent to the other agents is an approximate value and with some uncertainty, assigning credibility possibility is more appropriate than credibility probability.
In the next step, we assume that when an agent receives a query, it propagates the query to the other agents to attain their abilities and opinions. After receiving a query, each agent generates a fuzzy answer set. We need a fusion operator, which considers agent credibility to fuse these answers sets with considering agent credibility. For information fusion by considering source credibility, an operator is suggested by Yager [13] and Prade [9] . A problem with this operator applying to fuse agent answers is that the operator on the final decisions does not reflect the low credibility of all agents within the multi-agent environment. Therefore, an environment with low creditable decision makers cannot be distinguished from the ones with highly creditable agents. Consequently, a comparison of agent community response is not possible. To resolve this problem, with considering the credibility of the agents, an improved version of this decision fusion operator based upon the assumption that each agent generates a fuzzy decision set is presented.

MATERIALS AND METHODS
Multi-agent environment is an infrastructure that enables collaborative decision-making. Decision makers may have different degrees of credibility. I A = {A 1 , A 2 , A 3 ,.., A N }. Each of these agents collects information from its accessible knowledge resources and has special capability of decision-making. In this regard, each agent, such as A i, defines an assigned credibility possibility distribution for each subject. DP N is the Nth decision problem, D k is the decision which is made for DP N and R N is the membership degree of D k To decision set which is made against DP N . The crePosss ij values for each subject are kept in a matrix called the credibility matrix for each subject, as shown in Fig. 1.

CREDIBILITY ASSIGNMENT TO AGENTS
After the matrix CrePoss is established, the credibility possibility of each agent crePosij, should be influenced by the opinions of the other agents.
Credibility assignment to each agent is defined by a fuzzy relation implemented as a max-min composition. At each instance of time, t+1, the maxmin composition influences the opinion of each agent, Our goal is to gain a possibility distribution for the credibility of the agents by a distributed model. In our model, crePosss ij , the credibility of the agent Ai is dependent on, the amount of the credibility that all agents DM k for k = 1 to N, have assigned to the ith agent. The credibility of an agent, crePosss ij , is a function of crePosss ij k = 1,…, N. Considering theorem 2, to work out the value of the credibility vector, crePoss, instead of using the relation crePossoP n = crePoss, the relation crePoss o AssCreM`crePoss = can be used to obtain a distribution, crePoss. Such a distribution is called stationary distribution.

Example 1:
Suppose there are three agents A = {A1, A2, A3}, a max function, f and a min function, g. It is desirable to calculate the credibility possibility of the second agent such that the condition of the function F, defined in relation (2), is satisfied.
Using theorem 2 and Eq. 2, the possibility measure of the second agent credibility computes as follows: where, 1 2 crePoss ,crePoss and 3 crePoss represent the credibility of the first, second and third agents respectively and crePoss is the credibility matrix.

DECISION FUSION
As described above, for a collection of N agents, indexed by the set A = {A1, A2,…, AN} , the credibility possibility could be estimated for each of the agents, based upon the other agents opinions. The distribution of the estimated credibility possibility values is represented as a vector 1 2 N crePoss crePoss ,crePoss ,......,crePoss = , where crePoss i is the credibility of Ai. In this research, the vector crePoss is applied to fuse decisions made by the members of agent community. In response to any decision making problem such as q, Ai generates a fuzzy decision set i R (DP) = {D1, D2,..., Dm}, with the perception of its accessible knowledge environment and its belief, such that each fuzzy member k We choose function F such that it has the following eight properties which are required for any information fusion operator [14] . It implies that if all the members of agent community, DM, make two decisions D k and Dl then based on the assumption that all agents agreement on Dk being either identical to or more acceptable than Dl, it can be concluded that Dk has greater membership degree to final decision set than Dl has. . In this regard, the membership degree D l to final decision set, F l R (DP, D ) ′ , will be more than or equal to the membership degree of D k to final Decision set, F k R (DP,D ) .This property is applicable when two or more knowledge multi-agent are compared.

Property 5:
If credibility possibility of an agent is zero, then its opinion has no effect on the result of function f.

Property 6:
If the credibility possibility of an agent is one, then its decision impact should not be reduced and its constraints on the answer should be imposed completely.

SOLUTION OF THE EQUATION SET
In order to solve the equation set crePoss o crePossM crePoss = , the fuzzy markov chain model [1] is used. Since crePossM is a fuzzy transitive matrix, crePoss should be an eigen fuzzy set. Also, since crePoss is a possibility distribution, it is appropriate to obtain the greatest eigen fuzzy set satisfying the equation set crePoss o crePossM crePoss = .
Example 2: Suppose that the credibility matrix crePossM is as shown Fig. 2. It is desired to obtain he credibility vector, crePoss . For obtaining the greatest eigen fuzzy set we use the algorithm which presented in [15] .
If we have xo P = P equation the stages of algorithm are as follows: • Determine first x1 with the elements corresponding to the greatest elemen tcolumn of P • Compute P2 = Po P and determine the greatest elements in each column of P2 they give x2 • Compare x2 with x1: if they are different, compute P3 = P o P2 to get x3 • Compare x3 with x 1: if they are different, compute P4 = Po P3 to get x4, stop It is found m such that xm+1 = xm, that is xm = xmo P. Applying this algorithm, the greatest eigen fuzzy set, or in the other words the possibility distribution vector, crePoss and the possibility of crePoss will be as follows:

SUGGESTED OPERATOR FOR DECISION FUSION
In this research, to evaluate RF function, an information fusion operator considering the properties discussed in decision fusion section is presented. Some works on information fusion operator based on source credibilty have been done by Dubios and Prade [3] .
According to relation 2, This operator modifies rank of answers and then selects the minimum of these modified values. The drawback of this operator is that it has not monotonic properties with respect to credibility possibility. Therefore, it could not be used for comparing different multi-agent environment or aggregating of decision sets of more than one multiagent environment. In order to use the suggested operator, considering the degree of credibility possibility of each agent, i Then we use of i k R (DP, D ) ′ to gain R F (DP,D K ) in the following way:     Table 1 as decision sets for agents' community then the modified memberships are the values presented in Table 2. The membership degree of D1 is obtained in the following way.

PROOF THE CORRECTNESS OF THE OPERATOR
In this research, it is proved that the new decision fusion operator, R f , introduced in previous section, satisfies properties number 1 to 8 discussed in decision fusion section. In addition, it is obvious that: . Considering the first parameter, it is observed that only the highest credible agent DMi determines the value of the first parameter.
Considering the second parameter which is a conjunction of (1-crePoss i ) and R i (DP,D 1 ), it is observed that if: crePossj ≥ crePossk and 1-crePossj ≤ 1-crePossk then according to the T-conorm property the following relations are always true: , so it has no effect in determination of RF. So the introduced operator satisfies the forth property.

Property 5:
If the credibility possibility of the ith agent is zero, then according to the relation (2), the ith agent has no effect on the value of RF , because:

RESULTS AND DISCUSSION
Our goal is to fuse agent responses to a query in order to improve the accuracy of answer and reduce the uncertainty. As agents have different degree of credibility in response to a query, agent responses should be fused with considering the answers credibility. In this research, we first suggest a method for assigning credibility to each agent response. This approach is based on possibility theory. According to this approach, we reach a fuzzy equation set which is solved by use of fuzzy markov chain. The assigned credibility value is of the type of possibility and have a fuzzy nature. In the next stage, we assume that in response to any query each agent produce a fuzzy answer set. Afterwards we suggest a new operator for fusing these fuzzy answer sets with considering response credibility. This operator has a set of properties which is necessary for information fusion. We discussed eight properties in this regard and then we proof that this operator has these properties. The proposed operator modifies the rank of agents' answers considering the credibility of agent which gives it and then obtains a final answer with higher degree of credibility.