Potential Cases, Methodologies, and Strategies of Synthesis of. Solutions in Distributed Expert Systems. Minjie Zhang

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1 Potential Cases, Methodologies, and Strategies of Synthesis of Solutions in Distributed Expert Systems Minjie Zhang School of Computer, and Information Sciences Edith Cowan University, WA 6027, Australia Chengqi Zhang School of Mathematical, and Computing Sciences University of New England, NSW 2351, Australia Abstract In this paper, rstly, potential synthesis cases in distributed expert systems (DESs) and types of DESs are identied. Based on these results, necessary conditions of synthesis strategies in dierent synthesis cases are recognized. Secondly, two methodologies for designing synthesis strategies in distributed expert systems are investigated. They are analysis methods, and inductive methods. Thirdly, two methodologies are discussed based on the points of performance, complexity, and requirements. Key words: distributed expert systems, methodologies, synthesis of solutions, synthesis strategies, analysis methods, inductive methods 1 Introduction A distributed expert system (DES) is one of the special congurations of distributed problem solving. It consists of a number of dierent expert systems (ESs) connected by computer networks. In a DES, each expert system (ES) can either work individually to solve some specic problems, or cooperate with the others to deal with complex problems [7, 3, 4]. If more than one ES solves the same task, each ES could obtain a solution. It is thus important to synthesize such multiple solutions to the same task (called inputs) from dierent ESs in order to obtain the desired nal solution (called outputs) to the task. Let us look at an example. Two ESs predict an earthquake in a particular area. ES1 believes that the possibility of the potential earthquake being class 5 is x1 = 0:8, while ES2 believes that the possibility of the potential earthquake being class 5 is x2 = 0:5. Consider the following two cases based on the the example. Case (1): Two ESs obtain the uncertainties of the solution (a class 5 earthquake in a particular area) x1 = 0:8 and x2 = 0:5, respectively, based on the same geochemical results. This case demonstrates a belief conict between ES1 and ES2 because they obtained the same solution with dierent uncertainties given the same evidence. The nal uncertainty S(x1; x2) should be between x1 and x2 (i.e., minfx1; x2g S(x1; x2) maxfx1; x2g). Case (2): ES1 predicts a class 5 earthquake in an area with an uncertainty of y1 = 0:8 based on the evidence from a geophysical experiment and ES2 obtains the same solution with an uncertainty of y2 = 0:5 based on the geological structure of this area. The nal uncertainty for the solution of a class 5 earthquake in this case should be bigger than any one of y1 and y2 (i.e., S(y1; y2) maxfy1; y2g) because two ESs obtain the same solution from dierent evidence, so this solution is more reliable than the solution which comes from the same evidence. The conclusion is, that if two ESs obtain the same solution with two identical uncertainties, such as x1 = y1, x2 = y2 (using the above example), the synthesis of x1 and x2 represented by S(x1; x2) may be dierent from S(y1; y2) if these solutions originate from dierent evidence. 1

2 The above two cases indicate that a right synthesis strategy is not only based on the uncertainties of solutions, but also based on the relationship between evidence of solutions. If an improper synthesis strategy is chosen in a situation, a wrong solution may result. Therefore, how to classify, design and choose synthesis strategies is one of the critical research issues in the DES eld. This paper concentrates on the classication of the potential synthesis cases, and investigations of methodologies for designing synthesis strategies. In Section 2, the synthesis problem is formally described. In Section 3, the potential synthesis cases in DESs are identied, and necessary conditions for developing synthesis strategies are proposed. In Section 4, the principle of methodologies for designing synthesis strategies are discussed and two methodologies are compared. Finally, in Section 5, this paper is concluded and further proposed work is outlined. 2 Problem description Suppose there are n ESs in a DES to evaluate the values of an attribute of an object (e.g. in a medical DES, the identity of an organism infecting a specic patient). The solution for an ES i can be represented as (< object >< attribute > (V1 CF i1 A i ) (V2 CF i2 A i )... (V m CF im A i )) 2.1 where V j (1 j m) represents jth possible value, CF ij (1 i n; 1 j m) represents the uncertainty for jth value from ES i, A i (1 i n) represents the authority of ES i, and m indicates that there are m possible values for this attribute of the object. For example, there are 6 possible values for the face-up of a die. The authority A i is the condence level for the solution from ES i. The value range of the authority is [0, 1]. The higher the authority, the more reliable the solution. It can be assigned for each ES by human experts or generated based on the historical performance of ESs. >From the synthesis point of view, all ESs are concerned with the same attribute of an object. So we will only keep the attribute values, uncertainties, and authorities in the representation. Here is the representation of m possible values with uncertainties from n ESs: 2 64 CF11 CF12::: CF1m A1 CF21 CF22::: CF2m A2 :::::::::::::::::::::::::::::::::::::: CF n1 CF n2::: CF nm A n :2 The synthesis strategy is responsible for obtaining nal uncertainties (CF 1 CF 2... CF m ) based on Matrix 2.2, where * indicates the synthesis result from corresponding values with the subscripts 1; 2; :::; n in the same place. 3 Synthesis of solutions 3.1 Potential synthesis cases in DESs In this subsection, we will analyze the synthesis cases in DESs based on the relationship between evidence sets of a solution from dierent ESs. Informally, we know that there are four relationships between evidence sets of a solution. That is, the evidence sets of a solution from dierent ESs are (a) identical, (b) inclusion, (c) overlap, and (d) disjoint. 2

3 Before we formally dene synthesis cases, some preparation work should be done. We use propositions to represent evidence and use conclusions to represent solutions. In order to simplify the explanation, we will rst work on the assumption that there are two ESs in a DES; then we will extend to any number of ESs in DESs. Let P be a set of propositions, R be a set of rules, and CF represent the uncertainty of a proposition in an ES. Denition 1: An inference network G in an ES is dened as a directed acyclic graph in which the nodes are propositions in P, and the arcs are activated rules in R. (Suppose the rule format is A?! B in this denition). The root of such a network is a proposition in P which is not the premise of any rule in R. In contrast, a leaf is a proposition in P which is not the conclusion of any rule in R. Denition 2: A rule chain is dened as any chain from one node A to another node B in an inference network G (1) if there exists a rule in which A is a premise of the rule and B is a conclusion of the rule ; or (2) if there exist a sequence of rules in which A is a premise of the rule and there exists a rule chain from a node in the conclusion of the rule to node B. Denition 3: A general rule chain is dened as a rule chain from a leaf to the root of an inference network G. Denition 4: The original evidence set of a proposition B is represented by E(B), where E(B) is a unique set of leaf propositions which satisfy the condition that there is a rule chain to connect such a leaf to the proposition B. For example, if there is an inference network G i in ES i, where G i is: D?! C?! A?! B, F?! E?! B, T?! A?! C?! B, then E i (B) = fd; F; T g. The next four denitions are our formal denitions of dierent synthesis cases. In the following denitions, only the evidences in original evidence sets are considered because they are objective. Denition 5: A conict synthesis case [11] occurs when the original evidence sets of a proposition from dierent ESs are equivalent, but dierent ESs produce the same solution with dierent uncertainties. That is, for a proposition B, if there exist E i (B) = E j (B), where E i (B) is in ES i, E j (B) is in ES j, and CF i 6= CF j, where CF i is the uncertainty of the proposition B from ES i and CF j is the uncertainty of the proposition B from ES j. For example, if there are inference networks G i in ES i and G j in ES j, where G i : D?! C?! A?! B, F?! H?! B and G j : D?! H?! B, F?! J?! B, then E i (B) = fd; F g and E j (B) = fd; F g are equivalent. Denition 6: An inclusion synthesis case occurs when the original evidence set of a proposition from one ES is a subset of an original evidence set of another ES. Formally, for a proposition B, E i (B) E j (B), or vice versa, where E i (B) is in ES i and E j (B) is in ES j. For example, for a proposition B, there are inference networks G i in ES i and G j in ES j, where G i : A?! C?! B; D?! B; and G j : A?! K?! B; D?! G?! B; C?! B; E?! B. In this example, E i (B) = fa; Dg, E j (B) = fa; D; C; Eg, so E i (B) E j (B). Note: for ES j, C is an original evidence while for ES i, C is a derived evidence, so that C is not in the E i (B) but in E j (B). Denition 7: An overlap synthesis case occurs when the original evidence sets of a proposition from dierent ESs are not equivalent, but the intersection of original evidence sets is not empty. Formally, for a 3

4 proposition B, E i (B) \ E j (B) 6=, E i (B) \ E j (B) 6= E i (B), and E i (B) \ E j (B) 6= E j (B) where E i (B) is in ES i and E j (B) is in ES j. For example, for a proposition B, there are inference networks G i in ES i and G j in ES j, where G i : A?! C?! B, D?! B and G j : A?! E?! F?! B, H?! G?! B. In this example, E i (B) = fa; Dg, E j (B) = fa; Hg, so E i (B) \ E j (B) = fag 6=, E i (B) \ E j (B) 6= E i (B), and E i (B) \ E j (B) 6= E j (B). Denition 8: A disjoint synthesis case occurs when the intersection of original evidence sets of a proposition from dierent ESs is empty. Formally, for a proposition B, E i (B) \ E j (B) =, where E i (B) is in ES i, and E j (B) is in ES j. For instance, for a proposition B, there are inference networks G i in ES i and G j in ES j where G i : A?! C?! B, X?! H?! B and G j : F?! D?! B. In this example, E i (B) = fa; Xg, E j (B) = ff g, so E i (B) \ E j (B) =. The analysis of the above four synthesis cases is based on only two ESs. Now we extend above denitions to n ESs. Denition 9: A conict synthesis case occurs among n ESs when the original evidence sets of a proposition from n ESs are equivalent, but at least two dierent ESs produce the same solution with dierent uncertainties. Formally, for a proposition B, if there exist E1(B) = E2(B) = ::: = E n (B), where E1(B) is in ES1, E2(B) is in ES2,..., and E n (B) is in ES n, and 9i; j, (1 i; j n; i 6= j), CF i 6= CF j, where CF i is the uncertainty of the proposition B from ES i and CF j is the uncertainty of the proposition B from ES j. Denition 10: An inclusion synthesis case among n ESs occurs if these exists an original evidence set of a proposition from ES i (E i (B)) in which E i (B) strictly includes all other E j (B). Formally, for a proposition B, 8j (1 j n; j 6= i), E i (B) E j (B) where E1(B) is in ES1, E2(B) is in ES2,..., and E n (B) is in ES n. Denition 11: A disjoint synthesis case occurs among n ESs when the intersection of any two original evidence sets of a proposition from n ESs is empty. Formally, for a proposition B, 8i; j, (1 i; j n; i 6= j), E i (B) \ E j (B) =, where E1(B) is in ES1, E2(B) is in ES2,..., and E n (B) is in ES n. If the number of ESs in a DES is more than 2, rst we identify whether the synthesis case belongs to a conict, inclusion, or disjoint case. If the synthesis case does not satisfy the conditions of the above three cases, then it is an overlap synthesis case. 3.2 Classication of Types of DESs In this subsection, we will discuss the classication of types of DESs based on the relationship between the sets of knowledge and the sets of available data of ESs. Denition 12: An available data set of an ES is the data set which the ES can access. It is represented by DAT A. >From the above denition, we know that an available data set is a superset of an original evidence set (refer to denition 4). First, every element in an original evidence set must come from an available data set. Second, based on the knowledge base of an ES, some elements in an available data set may not be used either because they are not in the premise part of any rule or because the rule is not activated. For example, suppose there are two rules in an ES, A1?! B and A2?! B. It is possible that the ES obtains the solution B from A1 only. It might not need to use A2 even if A2 may be available. In this case, the original evidence set E(B) = fa1g while the available data set DAT A = fa1; A2g. 4

5 Denition 13: ES i = ES j if (1) both ES i and ES j have the same set of propositions, P i = P j ; and (2) both ES i and ES j have the same set of rules, R i = R j ; supposing both ES i and ES j use the same inference theory. Otherwise ES i 6= ES j. The relationships among the original evidence set (E(B)), the available data set (DAT A) and the proposition set (P ) is E(B) DAT A P. The proposition set (P ) includes the available data set DAT A and other derived propositions. According to the relationships between knowledge and available data of ESs, DESs can be classied into four types: Denition 14: A homogeneous DES is a DES in which all of the ESs have the same knowledge and can access the same available data set. This type of DESs can be dened as: 8i; j, ES i = ES j, 1 i, j n 8i; j, DAT A i = DAT A j, 1 i, j n In this type of DESs, all of the ESs can perform the same type of job in parallel. For example, in a VLSI system, \input data and the design constraints are available to all nodes" and \knowledge (or rules of ESs) could be distributed among multiple nodes which would operate in parallel " [8]. Denition 15: A partially homogeneous DES is a DES in which all of the ESs have the same knowledge but at least one ES accesses a dierent data set. This type of DESs can be dened as: 8i; j; ES i = ES j ; 1 i; j n 9i; j; DAT A i 6= DAT A j ; 1 i; j n For example, there are n ESs cooperate to determine the dangerous area of earthquakes for a specic area. All of ESs are identical knowledge based systems. Some of them use data from geochemical experiments while others use the information of geophysics test. Denition 16: A partially heterogeneous DES is a DES in which at least one ES has a dierent knowledge from other ESs, but all access the same available data set. This type of DESs can be dened as: 9i; j; ES i 6= ES j ; 1 i; j n 8i; j, DAT A i = DAT A j, 1 i, j n For instance, consider the situation when several medical ESs in one eld diagnose a patient. Each ES uses its own domain knowledge to make the decision for the patient, based on the same available evidences. Denition 17: A heterogeneous DES is a DES in which at least one ES in a DES has dierent knowledge from other ESs, and at least one ES accesses the dierent available data set from other ESs. It can be dened as: 9i; j; ES i 6= ES j ; 1 i; j n 9i; j; DAT A i 6= DAT A j ; 1 i; j n Suppose that two ESs cooperate to predict earthquakes in the same area. One ES is a geologist and the other is a geochemist. When they cooperate, they may use dierent knowledge and evidences to solve the problem. 3.3 Relationships between synthesis cases and DES types We have identied synthesis cases and classied DES types in the above two subsections. Now we can briey summarize the relationships between synthesis cases and DES types as follows. 5

6 Theorem 1: A conict synthesis case (refer to Denition 5 in Subsection 3.1) does not exist in a homogeneous DES, nor in a partially homogeneous DES, but may exist in a partially heterogeneous DES and a heterogeneous DES. Proof: (1) In both a homogeneous DES and a partially homogeneous DES, if two ESs have the same original evidence set, the solution is the same (no conict) because ES i = ES j. (2) However, in both a partially heterogeneous DES and a heterogeneous DES, the knowledge bases of dierent ESs can be dierent. In this situation, for the same original evidence set, dierent ESs may use dierent evidences to obtain the same solution with dierent uncertainties. Theorem 2: An inclusion synthesis case (refer to Denition 6 in Subsection 3.1) does not exist in a homogeneous DES, but may exist in a partially homogeneous DES, a partially heterogeneous DES and a heterogeneous DES. Proof: (1) In a homogeneous DES, all ESs have the same knowledge and choose the same original data sets, so there is no inclusion synthesis case. (2) In a partially homogeneous DES and a heterogeneous DES, the available data sets can be dierent. In these cases, for the same solution, the original evidence set from one ES can be a subset of an original evidence set of another ES. For instance, in a partially homogeneous DES, the rules A?! B; C?! B, D?! B, E?! B exist in both ES i and ES j, and suppose A, C, D, E 2 DAT A i, C, D 2 DAT A j. An inclusion synthesis case may also exist in a partially heterogeneous DES because in a partially heterogeneous DES, ESs can share the same available data set. For example, ES i have rules A?! B, C?! B, D?! B, ES j has rules A?! B, D?! B. E i (B) = fa; C; Dg, E j (B) = fa; Dg, and E j (B) E i (B). Theorem 3: An overlap synthesis case (refer to Denition 7 in Subsection 3.1) does not exist in a homogeneous DES but may exist in a partially homogeneous DES, a partially heterogeneous DES, and a heterogeneous DES. Proof: (1) In a homogeneous DES, it is impossible to have an overlap synthesis case because all ESs have the same knowledge and choose the same original data sets. (2) However, in a partially homogeneous DES and a heterogeneous DES, the available data sets of dierent DESs can be dierent. In this situation, for the same solution, dierent ESs can use dierent original evidence sets. For example, in a partially homogeneous DES, the rules A?! B; C?! B; D?! B exist in both ES i and ES j, and suppose A; C 2 DAT A i, C; D 2 DAT A j. This kind of situation may also exist in a heterogeneous DES. In a partially heterogeneous DES, ESs can use dierent data to obtain the same solution by using dierent knowledge, even if the available data sets are the same. For example, ES i has rules A?! B; C?! B, ES j has rules A?! B; D?! B. The available data set is fa; C; Dg. From this example, we know that a partial overlap synthesis case exists in a partially heterogeneous DES. Theorem 4: A disjoint synthesis case (refer to Denition 8 in Subsection 3.1) does not exist in a homogeneous DES but might appear in a partially homogeneous DES, a partially heterogeneous DES and a heterogeneous DES. Proof: This proof is quite similar to the proof of Theorem 3. The only dierence is that the original evidence sets of dierent ESs have no overlap. For example, in a partially homogeneous DES and a heterogeneous DES, both rule A i?! B and rule A j?! B exist in both ES i and ES j, and suppose A i 2 DAT A i and A j 2 DAT A j (and vice versa ). In a partially heterogeneous DES, ES i has rule sets A?! B, C?! B; 6

7 ES j has rule sets D?! B; E?! B. The available data set is fa; C; D; Eg. This is the disjoint synthesis case. 3.4 Necessary conditions of synthesis strategies in DESs In this subsection, we will describe both general necessary conditions for all synthesis strategies and specic necessary conditions of synthesis strategies for each synthesis case. The general conditions are on a more abstract level, while specic conditions are on a more concrete level General necessary conditions for all synthesis strategies Let S represent the synthesis function of CF i and CF j (S(CF i ; CF j )) where CF i and CF j represent the uncertainties of a proposition B from ES i and ES j, respectively. S could be any of the synthesis functions appropriate to the dierent situations of conict, inclusion, overlap, or disjoint. The following properties are the fundamental consistency conditions which an acceptable synthesis strategy in DESs must satisfy. (a) Suppose that X is the set of uncertainties of propositions in an inexact reasoning model. If 8i; j; CF i 2 X & CF j 2 X, then S(CF i ; CF j ) 2 X. The reason for this property is that the value of uncertainty after synthesis should be still in the same uncertainty range, otherwise the synthesis result will be meaningless. (b) The synthesis function S on X must satisfy the associative law. The reason for this property is that in the real world, the nal solution of the problem is only based on the evidence which is used to obtain the solution, not on the order of evidence. That is, S(S(CF i ; CF j ); CF k ) = S(CF i ; S(CF j ; CF k )). (c) The synthesis function S on X must satisfy the commutative law. The reason for this property is the same as (b). The general necessary conditions are valid only when the synthesis strategies synthesize dierent uncertainties in accumulative manners (i.e., synthesize two uncertainties at once) Specic necessary conditions of synthesis strategies for each synthesis case In the conict synthesis case, the necessary condition for the synthesis function should be minfcf i ; CF j g S conflict (CF i ; CF j ) maxfcf i ; CF j g, where S conflict is a synthesis function for a conict synthesis case, because both uncertainties of CF i and CF j come from the same original evidence set (E i (B) = E j (B)). This condition is nothing related to any inexact reasoning model even we use min and max. In other words, the dierence between CF i and CF j only comes from the dierent subjective interpretation of dierent ESs on the same objective evidences. Since there is no additional evidence for each of the ESs, the opinion from all of them should be considered, and they constrain each other. In the disjoint synthesis case, there is no overlap between E i (B) and E j (B). Either E i (B) or E j (B) can contribute positively or negatively to the uncertainty of proposition B being true independently. Therefore, if both ESs favor the proposition B being true (CF i > 0 and CF j > 0 under the EMYCIN [5] inexact reasoning model), the necessary condition for the synthesis function should be S disjoint (CF i ; CF j ) > maxfcf i ; CF j g; where S disjoint represents a synthesis function for the disjoint synthesis case. If both ESs are against the proposition B being true (CF i < 0 and CF j < 0), then S disjoint (CF i ; CF j ) < minfcf i ; CF j g. In all other cases, it should be minfcf i ; CF j g S disjoint (CF i; CF j ) maxfcf i ; CF j g. 7

8 In the inclusion synthesis case, if E i (B) is the subset of E j (B), the necessary condition of inclusion synthesis case should be S inclusion (CF i ; CF j ) = CF j where S inclusion is the synthesis function for the inclusion case. The idea behind this is that ES i gets the solution based on less evidences than ES j. Evidences used by ES i are already used by ES j, so ES i makes no more contribution to the nal solution. In the overlap synthesis case, there are some additional evidences between E i (B) and E j (B). The necessary condition for this kind of case should be S overlap (CF i ; CF j ) S conflict (CF i ; CF j ) ( if CF i 0; CF j 0); or S overlap (CF i ; CF j ) < S conflict (CF i ; CF j ) (if CF i < 0; CF i < 0) where S overlap is the synthesis function for the overlap synthesis case. That means, S overlap is stronger than S conflict. 4 Methodologies for designing synthesis strategies 4.1 Measurements for synthesis strategies Firstly, we would like to formally dene synthesis strategies as follows: Denition 18: A perfect synthesis strategy f can be dened as: 8X i 2 X; f(x i ) = Y i where X i represents a matrix (n (m + 1)) (refer to Matrix 2.2) of multiple solutions to a problem from dierent ESs (an input), Y i represents a vector (m) of the desired nal solution after synthesis from X i (an output), and X is the set of all X i. As described above, we always know the set X (all input matrices). Y i is our desired nal solution after synthesis from X i. Questions here are: (1) How do we know whether Y i is our desired nal solution for any X i? and, (2) For how many X i do we know the corresponding Y i? For the rst question, it is reasonable to dene Y i as a synthesis of solution for any X i from human experts. For the second question, if, for any X i, we know Y i, we have nothing to do about synthesis of solutions. In fact, for only a limited number of X i, we may know corresponding Y i, and f is dened as a psuedo function to map the limited X i to corresponding Y i perfectly. The goal of designing synthesis strategies is to nd a mapping function (strategy) f 0 in which, for any X i, f 0 (X i ) = Y 0 i should be very close to Y i. Denition 19: A better synthesis strategy is dened as the strategy which can map every X i to the corresponding Y i, with less errors. Denition 20: The specic error f 0(X i ) for a synthesis strategy f 0 for X i is dened as f 0(X i ) = jy i? f 0 (X i )j = jy i? Y 0 i j = fj 1j; j2j; :::; j m jg where m is the number of possible values for an attribute. Denition 21: The specic mean error f 0 (X i ) for a synthesis strategy f 0 from X i is dened as f 0 (X i ) = j 1j + j2j + ::: + j m j m as Denition 22: The specic maximum error max (X i ) for the synthesis strategy f 0 max (X i ) = maxfj1j; 2j; :::; j m jg from X i is dened 8

9 . Denition 23: The general mean error f 0 for the synthesis strategy f 0 from all X i is dened as Supposing the number of elements in the set X is N. 8X i 2 X; f 0 = P N i=1 f 0(X i ) N Denition 24: The general maximum error for the synthesis strategy f 0 from all X i is dened as 8X i 2 X; max = maxf max (X1); max (X2); :::; max (X N )g Thus, the strategy f 0 is said to be better than g 0 if f 0 < g 0. The best synthesis strategy f is dened as f = 0. Such a measurement for a synthesis strategy is known as `quantitatively measured '. In practice, it is nearly impossible to dene the best f. The above denition is valid only when you know a certain amount of Y i from X i. There are two main steps to judge a synthesis strategy. The rst step: if some answers are known for some examples, these examples can be the benchmark to test the methods. Otherwise, the necessary condition can be used to judge the valid of the methods (refer to Subsection 3.4). 4.2 Principles of two methodologies In the literature, there are two methodologies to dene f 0. One is the method used for the analysis of characteristics of the input X i thoroughly to dene f 0 (analysis methods) and the other is from the number of X i and the corresponding Y i to dene f 0 (inductive methods). We analyze these methodologies below Analysis methods An analysis method is a methodology which can be used to dene a synthesis strategy from X to Y (refer to Denition 18) by analyzing the characteristics of X. These characteristics may include relationships among original evidence sets from ESs which derive the input matrix, the factors which aect the desired nal solution, and the weights for all factors. For example, this method can be used to synthesize committee members' opinions to select the best movie in a movie festival; to synthesize the individual comments from assessors in order to decide whether a project should be funded; and to synthesize multiple opinions from dierent experts to decide whether a new product should be produced. This method is useful for areas in which the individual solution from each expert can be described by uncertainties, or numbers, and relationships between X i and Y i are not so complicated. In particular, this method is useful for areas in which patterns are dicult to obtain. Normally, analysis methods require some preconditions. If synthesis cases satisfy these preconditions, this kind of strategies can work well [11]. The typical examples of analysis methods are: (1) uncertainty management in which the outputs are based on not only the mean value of corresponding inputs but also the uniformity about corresponding inputs [2]; (2) a synthesis strategy for heterogeneous DESs which was developed based on both transformation functions among dierent inexact reasoning models among heterogeneous ESs and mean values of inputs from ESs [9]; and (3) a synthesis strategy based on the factors of authorities from ESs, mean values of inputs, inuence among ESs for decision making, and uniformity of inputs from ESs. All of these strategies implemented analysis methods by some mathematical theories. 9

10 Popular mathematical theories used to implement analysis methods could be `decision theory', `evidential theory' [6], `probability theory' [1], and so on. Generally, there are ve levels to be explored in this kind of method: (1) analysis of the characteristics of the evidence to derive an input X i (such as conict cases and non-conict cases. (2) analysis of input X i itself (such as all experts give consistently positive solutions, or negative solutions, or some experts give positive solutions while other experts give negative solutions); (3) analysis of the factors which aect Y i (such as, average of individual solutions, the consistency among individual solutions, condence of each expert, and so on); (4) how to weight each factor; and (5) how to combine dierent factors. We have developed a synthesis strategy [10] to demonstrate how an analysis method was used. For this method, patterns are not absolutely necessary. If there are some patterns, they can be used to test strategies, and then give some clues to generate better strategies. If the patterns are dicult to obtain, or too few patterns are available, we will test the strategies to see whether they satisfy the necessary conditions. For example, one typically necessary condition is that the nal solution after synthesis should not be negative if all individual solutions are positive Inductive methods The idea of inductive methods diers from the idea of analysis methods. An inductive method is a methodology which can be used to nd the general relationship between X and Y based on sucient patterns. The principle of inductive methods is from specic to general. Suppose 8i(1 i k)f(x i ) = Y i, in which f is a synthesis strategy, X i is the input matrix of multiple solutions, and Y i is the known synthesis solution from X i, we believe that f also works well for any X j (j > k) although we cannot guarantee that it is always true. It is obvious that the rst condition of using this method for dening a strategy is that enough patterns must be known (this requirement diers from the analysis methods above). The more patterns, the better. The second condition is that the patterns are distributed fairly randomly. The third condition is that the of the strategy is closed to 0, or the strategy `converges' in neural network terms. For some synthesis problems, (if the relationship between inputs and outputs is linear,) we can use mathematical models such as \The Least Square Method" to nd this relationship. However, it does not work well for complicated problems. Another tool used to implement inductive methods is the neural network. We have developed two synthesis strategies using the neural network technique [12]. The characteristics of these strategies are based on patterns of both inputs and corresponding outputs in order to nd the best mapping functions from inputs to corresponding outputs. A neural network is a good mechanism to simulate certain complicated relationships [12]. The general structure of a neural network includes an input layer, a hidden layer, and an output layer. The three layers are connected by a certain method based on requirements. A neural network can be generated by a number of learning patterns. A pattern includes an input pattern and an output pattern. Generally, neural networks can be trained by supervised training. That is, the input and output patterns are known during training. 10

11 The output patterns oer hints as to the correct output for a particular input pattern. During training, the hidden layer continuously adjusts the weight to reduce the dierence between desired outputs and actual outputs. Once the relationship is found, actual outputs Y 0 come from the mapping function. In Section 6, a neural network strategy will be briey discussed. 5 Conclusion In this paper, we have identied the potential cases of synthesis in DESs and classied the types of DESs. Based on these results, necessary conditions of synthesis strategies in dierent synthesis cases are recognized. Two methodologies (analysis and inductive methods) have been proposed to design synthesis strategies in DESs. Two methodologies compensate each other in the following ways: (1) Performance Analysis methods can work well for some simple problems. In particular, this method is perfect when the desired nal solutions are derived from some formulas based on inputs. Inductive methods can be used to solve complicated problems. For complicated cases, inductive methods are better than analysis methods because they can simulate complicated relationships quite closely if inductive functions exist. (2) Complexity The complexity of analysis methods is less than that of inductive methods. In this kind of method, the relationship between inputs and outputs is concise, and the calculation is inexpensive. An inductive method is the complex method. The selection of suitable samples is a big job. To nd a nal mapping function is time consuming such as training of a neural network. (3) Requirements Analysis methods virtually need no patterns. In analysis methods, the only requirement is that the relationship between inputs and outputs can be summarized by an analysis method. Inductive methods need a lot of patterns. Some of the additional requirements are: (1) samples should be distributed randomly and cover most cases, and (2) a mapping function should exist (neural network should be converged for the problems). After comparison, our conclusion is that they compensate each other. Based on dierent kinds of synthesis problems and available information, dierent methodologies should be used to design synthesis strategies. This paper gives a clear idea of two potential ways for doing research in synthesis of solutions and also oers a guideline for developing and choosing synthesis strategies in DESs by using two dierent methodologies. Further work intends to (1) investigate the analogical methodology and develop new synthesis strategies by case-based reasoning based on an analogical method, and (2) investigate how to combine dierent methodologies together. 11

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