# The evidential reasoning approach for MADA under both probabilistic and fuzzy uncertainties

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4 312 J.B. Yang et al. / European Journal of Operational Research 171 (2006) probabilistic uncertainty, as briefly introduced in this section. Before the introduction, some basic concepts of the Dempster Shafer theory of evidence are discussed Basics of the evidence theory The evidence theory was first developed by Dempster (1967) in the 1960s and later extended and refined by Shafer (1976) in the 1970s. The evidence theory is related to Bayesian probability theory in the sense that they both can update subjective beliefs given new evidence. The major difference between the two theories is that the evidence theory is capable of combining evidence and dealing with ignorance in the evidence combination process. The basic concepts and definitions of the evidence theory relevant to this paper are briefly described as follows. Let H ={H 1,...,H N } be a collectively exhaustive and mutually exclusive set of hypotheses, called the frame of discernment. A basic probability assignment (bpa) is a function m:2 H! [0,1], called a mass function and satisfying X mðuþ ¼0 and mðaþ ¼1; AH where U is an empty set, A is any subset of H, and 2 H is the power set of H, which consists of all the subsets of H, i.e. 2 H ={U,{H 1 },...,{H N },{H 1 [ H 2 },...,{H 1 [ H N },...,H}. The assigned probability (also called probability mass) m(a) measures the belief exactly assigned to A and represents how strongly the evidence supports A. All assigned probabilities sum to unity and there is no belief in the empty set U. The probability assigned to H, i.e. m(h), is called the degree of ignorance. Each subset A H such that m(a) > 0 is called a focal element of m. All the related focal elements are collectively called the body of evidence. Associated with each bpa are a belief measure (Bel) and a plausibility measure (Pl) which are both functions: 2 H! [0, 1], defined by the following equations, respectively: BelðAÞ ¼ X mðbþ and PlðAÞ ¼ X mðbþ; BA A\B6¼U where A and B are subsets of H. Bel(A) represents the exact support to A, i.e. the belief of the hypothesis A being true; Pl(A) represents the possible support to A, i.e. the total amount of belief that could be potentially placed in A. [Bel(A),Pl(A)] constitutes the interval of support to A and can be seen as the lower and upper bounds of the probability to which A is supported. The two functions can be connected by the equation PlðAÞ ¼1 BelðAÞ; where A denotes the complement of A. The difference between the belief and the plausibility of a set A describes the ignorance of the assessment for the set A (Shafer, 1976). Since m(a), Bel(A) and Pl(A) are in one-to-one correspondence, they can be seen as three facets of the same piece of information. There are several other functions such as commonality function, doubt function, and so on, which can also be used to represent evidence. They all represent the same information and provide flexibility in a variety of reasoning applications. The kernel of the evidence theory is the DempsterÕs rule of combination by which the evidence from different sources is combined. The rule assumes that the information sources are independent and use the orthogonal sum to combine multiple belief structures m = m 1 m 2 m K, where represents the operator of combination. With two belief structures m 1 and m 2, the DempsterÕs rule of combination is defined as follows:

6 314 J.B. Yang et al. / European Journal of Operational Research 171 (2006) and to 30% excellent, then the assessment will be incomplete with a degree of incompleteness of 20%. The individual assessments of all alternatives on each basic attribute are represented by a belief decision matrix, defined as follows: D g ¼ðSðe i ða l ÞÞÞ LM : ð2þ 2.3. The recursive ER algorithm One simple approach for attribute aggregation is to transform a belief structure into a single score and then aggregate attributes on the basis of the scores using traditional methods such as the additive utility function approach. However, such transformation would hide performance diversity shown in a distribution assessment, leading to the possible failure of identifying strengths and weaknesses of an alternative on higher level attributes. In the ER approach, attribute aggregation is based on evidential reasoning rather than directly manipulating (e.g. adding) scores. In other words, the given assessments of alternatives on the individual basic attributes are treated as evidence and which evaluation grades the general attribute should be assessed to is treated as hypotheses (Yang and Singh, 1994; Yang and Sen, 1994a,b). DempsterÕs evidence combination rule is then employed and revised to create a novel process for such attribute aggregation (Yang, 2001; Yang and Xu, 2002a). The revision of the rule is necessary due to the need to handle conflicting evidence and follow common sense rules for attribute aggregation in MADA. The detailed analysis and the rationale on the development of the attribute aggregation process can be found in the references (Yang, 2001; Yang and Xu, 2002a). The process is briefly described as the following steps. First of all, a degree of belief given to an assessment grade H n for an alternative a l on an attribute e i is transformed into a basic probability mass by multiplying the given degree of belief by the relative weight of the attribute using the following equations: m n;i ¼ m i ðh n Þ¼w i b n;i ða l Þ; n ¼ 1;...; N; i ¼ 1;...; L; ð3þ m H;i ¼ m i ðhþ ¼1 XN n¼1 m n;i ¼ 1 w i X N n¼1 b n;i ða l Þ; i ¼ 1;...; L; ð4þ m H;i ¼ m i ðhþ ¼1 w i ; i ¼ 1;...; L; ð5þ ~m H;i ¼ ~m i ðhþ ¼w i ð1 XN n¼1 b n;i ða l ÞÞ; i ¼ 1;...; L; ð6þ with m H;i ¼ m H;i ~m H;i and P L w i ¼ 1. Note that the probability mass assigned to the whole set H,m H,i which is currently unassigned to any individual grades, is split into two parts: m H;i and ~m H;i, where m H;i is caused by the relative importance of the attribute e i and ~m H;i by the incompleteness of the assessment on e i for a l. The second step is to aggregate the attributes by combining the basic probability masses generated above, or reasoning based on the given evidence (Yang and Singh, 1994). Due to the assumptions that the evaluation grades are mutually exclusive and collectively exhaustive and that assessments on a basic attribute are independent of assessments on other attributes, or utility independence among attributes (Keeney and Raiffa, 1993), the DempsterÕs combination rule can be directly applied to combine the basic probability masses in a recursive fashion. In the belief decision matrix framework, the combination process can be developed into the following recursive ER algorithm (Yang, 2001; Yang and Xu, 2002a):

7 fh n g : m n;iði1þ ¼ K Iði1Þ ½m n;iðiþ m n;i1 m n;iðiþ m H;i1 m H;IðiÞ m n;i1 Š; ð7þ m H;IðiÞ ¼ m H;IðiÞ ~m H;IðiÞ ; n ¼ 1;...; N; fhg : ~m H;Iði1Þ ¼ K Iði1Þ ½~m H;IðiÞ ~m H;i1 ~m H;IðiÞ m H;i1 m H;IðiÞ ~m H;i1 Š; ð8þ fhg : m H;Iði1Þ ¼ K Iði1Þ ½m H;IðiÞ m H;i1 Š; ð9þ K Iði1Þ ¼ 1 XN X N 6 4 m n;iðiþ m t;i1 7 5 ; i ¼ 1;...; L 1; ð10þ fh n g : b n ¼ fhg : b H ¼ J.B. Yang et al. / European Journal of Operational Research 171 (2006) n¼1 t¼1 t6¼n m n;iðlþ 1 m H;IðLÞ ; n ¼ 1;...; N; ð11þ ~m H;IðLÞ 1 m H;IðLÞ : ð12þ In the above equations, m n,i(i) denotes the combined probability mass generated by aggregating i attributes; m n,i(i) m n,i+1 measures the relative support to the hypothesis that the general attribute should be assessed to the grade H n by both the first i attributes and the (i + 1)th attribute; m n,i(i) m H,i+1 measures the relative support to the hypothesis by the first i attributes only; m H,I(i) m n,i+1 measures the relative support to the hypothesis by the (i + 1)th attribute only. It is assumed in the above equations that m n,i(1) = m n,1 (n =1,...,N), m H,I(1) = m H,1, m H;Ið1Þ ¼ m H;1 and ~m H;Ið1Þ ¼ ~m H;1. Note that the aggregation process does not depend on the order in which attributes are combined. b n and b H represent the belief degrees of the aggregated assessment, to which the general attribute is assessed to the grades H n and H, respectively. The combined assessment can be denoted by S(y(a l )) = {(H n,bn(a l )), n =1,...,N}. It has been proved that P N n¼1 b n b H ¼ 1(Yang and Xu, 2002a). Yang and Xu also put forward four axioms and proved the rationality and validity of the above recursive ER algorithm. The nonlinear features of the above aggregation process were also investigated in detail (Yang and Xu, 2002b). In the above ER algorithm, Eqs. (7) (10) are the direct implementation of the DempsterÕs evidence combination rule within the belief decision matrix; the weigh normalisation process shown in Eq. (1), the assignment of the basic probability masses shown in Eqs. (3) (6) and the normalisation of the combined probability masses shown in Eqs. (11) and (12) are developed to ensure that the ER algorithm can process conflicting evidence rationally and satisfy common sense rules for attribute aggregation in MADA (Yang and Xu, 2002a) The utility interval based ER ranking method The above ER algorithm allows each attribute to have its own set of evaluation grades. Before the aggregation, however, all different sets of evaluation grades have to be transformed into a unified set of assessment grades using either the rule or utility-based equivalence transformation techniques. The interested reader may refer to the paper by Yang (2001). In order to compare M alternatives at the presence of incomplete assessments, maximum, minimum and average utilities are introduced and used to rank them. Suppose the utility of an evaluation grade H n is u(h n ), then the expected utility of the aggregated assessment S(y(a l )) is defined as follows: uðsðyða l ÞÞÞ ¼ XN n¼1 b n ða l ÞuðH n Þ: ð13þ

9 3.2. The new ER algorithm for MADA under both probabilistic and fuzzy uncertainties In the derivation of Eqs. (7) (10), it was assumed that the evaluation grades are independent of each other. Due to the dependency of the adjacent fuzzy assessment grades on each other as shown in Fig. 1, Eqs. (7) (10) can no longer be employed without modification to aggregate attributes assessed using such fuzzy grades. However, the evidence theory provides scope to deal such fuzzy assessments. The ideas similar to those used to develop the non-fuzzy evidential reasoning algorithm (Yang and Singh, 1994; Yang and Sen, 1994a,b) are used to deduce the following fuzzy evidential reasoning algorithm. The new challenge is that the intersection of two adjacent evaluation grades H n and H n +1 is H n,n+1, which is not empty as shown in Fig. 1. Another difference is that the normalisation has to be conducted after all pieces of evidence have been combined in order to preserve the property that the generated belief and plausibility functions still represent the lower and upper bounds of the combined degrees of belief (Yen, 1990). In this subsection, a fuzzy ER algorithm will be developed using the similar technique used in Yang and Singh (1994). Following the assumptions on the fuzzy assessment grades made in the previous subsection, based on the belief decision matrix as shown in Eq. (2) and the basic probability masses generated using Eqs. (3) (6), it is proven in Appendix A that the following analytical (non-recursive) fuzzy ER algorithm can be used to aggregate the L basic attributes for alternative a l (l =1,...,M): ( ) fh n g : mðh n Þ¼k YL ½m i ðh n Þm i ðhþš YL m i ðhþ ; n ¼ 1;...; N; ð17þ fh n;n1 g : mðh n;n1 Þ¼kl max H n;n1 YL ( Y L ½m i ðh n Þm i ðh n1 Þm i ðhþš YL ½m i ðh n1 Þm i ðhþš YL ½m i ðh n Þm i ðhþš ) m i ðhþ ; n ¼ 1;...; N 1; ð18þ ( ) fhg : ~mðhþ ¼k YL m i ðhþ YL m i ðhþ ; ð19þ fhg : " # mðhþ ¼k YL m i ðhþ ; k ¼ J.B. Yang et al. / European Journal of Operational Research 171 (2006) ( XN 1 YL n¼1 XN 1 n¼1 Y L 1 l max H n;n1 l max H n;n1 Y L ½m i ðh n Þm i ðhþš YL ½m i ðh n Þm i ðh n1 Þm i ðhþš YL m i ðhþ! ½m i ðh n1 Þm i ðhþš ½m i ðh N Þm i ðhþš) 1 ; ð20þ fh n g : b n ¼ mðh nþ ; n ¼ 1;...; N; ð21þ 1 mðhþ fh n;n1 g : b n;n1 ¼ mðh n;n1þ ; n ¼ 1;...; N 1; ð22þ 1 mðhþ fhg : b H ¼ ~mðhþ 1 mðhþ ; ð23þ!

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