3. Fuzzy Multiple Criteria Decision Analysis for Evaluation
3.2 Fuzzy Analytic Hierarchy Process
In real MCDM problems, it is necessary to divide the process into distinct stages. Firstly, based on a general problem statement, the various stakeholders are defined, typically including the decision-makers, various interest groups affected by the decision, experts in the appropriate fields, as well as planners and analysts responsible for the preparations and managing the process. The overall objective will be set up in this stage. Secondly, based on various points of view from stakeholders, the problems can be categorized into distinct aspects. Thirdly, defining alternatives/strategies and criteria, a discrete MCDM problem consisting of a finite set of alternatives/strategies can be evaluated in terms of multicriteria.
Finally, choosing a suitable method to measure the criteria can help the evaluators and analysts to process the evaluating cases.
3.2.1 Building a hierarchical system for evaluation
Analytic Hierarchy Process (AHP) is a popular technique often used to model subjective decision-making processes based on multiple attributes (Saaty 1977; 1980). From that moment on, it is being widely used in corporate planning, portfolio selection, and benefit/cost analysis by government agencies for resource allocation purposes. And it is being used more widely on an international scale for planning infrastructure in developing countries and for evaluating natural resources for investment.
When all the aspects for consideration have been set up, the final set of criteria should meet the following requirements (1)Completeness; (2)Operationality; (3)Nonredundancy;
(4)Minimality (Keeney and Raiffa, 1976 ).
In this study, we firstly establish a hierarchy system for analysis and evaluation through scenario writing and brainstorming. Phase 1 includes our overall objectives. Secondly, we consider related aspects for achieving goals in Phase 2. Thirdly, list considered in Phase 3. All considered criteria measured by evaluators, consisting of individuals with different viewpoints.
Finally, the alternatives/strategies will listed in Phase 4 (Figure 3.3).
3.2.2 Determining the evaluated criteria weights
Because the evaluation of criteria entails diverse and meanings, we cannot assume that each evaluation criterion is of equal importance. There are many methods that can be employed to determine weights (Hwang and Yoon, 1981), such as the eigenvector method, weighted least square method, entropy method, AHP, DEMATEL (Gabus & Fontela, 1972, 1973; Tamura et al, 2002), as well as linear programming techniques for multidimension of
analysis preference (LINMAP). The selection of method depends on the nature of the problems to express the preference relation of perception from evaluators. In this section, we introduce a revised AHP method to assess the weights of criteria for our study.
Goal Overall objective
Aspects Dimension 1 … Dimension j … Dimension k
Criteria
C1,1 Cj,1
1,n1
C Cj,nj C k,1 Ck,nk
… … …
Alternatives A1 … Ai … An
Figure 3.3 Analytic Hierarchy System for Evaluation
Saaty (1980) originally introduced the Analytic Hierarchy Process to systematically cope with complex problems in social system. He used the principal eigenvector of the comparison matrix to find the comparative weight among the criteria of the hierarchy systems. If we wish to compare a set of n criteria pairwise according to their relative importance (weights), then denote the criteria by and their weights by . If
is given, the pairwise comparisons may be represented by matrix A of the following formulation:
Cn
C C , ,...,
2
1 w ,w ,...,wn
2
1 w= T
wn
w
w, ,..., ) ( 1 2
(A−λmaxI w) =0 (3.1) Eqs.(3.1) denotes that A is the positive reciprocal matrix of pairwise comparison values derived by intuitive judgment for ranking order. In order to derive the priority eigenvector, we must find the eigenvector w with respective
λmaxwhich satisfiesAw w λmax
= . Saaty (1980) suggested the consistency index ( C.I.=
(
λmax −n) (
n−1 ))
to test the consistency of the intuitive judgment. In general, a value of C.I. is less than 0.1 is satisfactory (i.e. C.I. 0.1)≤ .The procedure for AHP can be summarized in four steps, as follows:
Step 1. Set up the decision system by decomposing the problem into a hierarchy of
interrelated elements.
Step 2. Generate input data consisting of pairwise comparative judge of decision elements.
Step 3. Synthesize the judgment and estimate the relative weight.
Step 4. Determine the aggregating weights of the decision elements to arrive at a set of ratings for the alternatives/strategies.
Besides Saaty’s method to aggregate the relative weights by participating evaluators, Buckley (1985b) proposed geometric mean method to calculate the final fuzzy weights for each fuzzy matrix. Given a m×m positive reciprocal matrix is derived by
pairwise comparison from m participating evaluators, then represents the geometric mean of each raw. According to Saaty’s definition,
[ ]aij
λmaxbe the largest eigenvalue of A and the weights, as the components of the normalized eigenvector corresponding to
Buckley (1985b) further considered a fuzzy positive reciprocal matrix , extending the geometric mean method to the fuzzy geometric mean method and exploited which to find the final fuzzy weights of each criterion as follows
[ ]aij where and called additive and multiplicative operators of two fuzzy number, respectively. These arithemetic operations will describe in next section.
⊕ ⊗
3.2.3 Driving the fuzzy performance score and fuzzy synthetic value
The evaluators choose a performance score for each participating company based on their subjective judgments. This way of estimating the achievement level of each criterion on each strategy can use the methods of fuzzy theory for treating the fuzzy environment.
In evaluating process, after well define the criteria and their relationship, we have to determine the weights or measure of these criteria, and obtain then performance score of each alternative with respect to evaluated criteria. Furthermore, choose an suitable aggregateing operator to derive the synthetic value of these alternatives respectively, the final step is to assign the preferred order for all alternatives based on their synthetic values.
In general case, we can employ triangular fuzzy number to express the aggregated fuzzy weights of j-th criterion as follows:
(
i i i i
w = l ,m ,u
)
(3.3)where wi is derived by Eqs.(3.2).
It can be assumed that evaluation expert k has his fuzzy performance score of for the criteria j under alternative i, and all the items to be evaluated is defined in feasible set S.
k Each expert may has his different academic and business careers, so as his objective understanding on the linguistic variables. This study utilize the average number to integrate the fuzzy judgment values given by m experts. That is, express the average fuzzy judgment given by the participated evaluators. Its triangular fuzzy number is shown below:
Eij
Specifically, Eij can be calculated by Buckley (1985a):
( ) ( ) ( )
Moreover, the fuzzy synthetic matrix R can then be developed from both fuzzy weighting vector and fuzzy performance matrix as following:
= ⇔ “⇔” in Eqs.(3.6) indicates the aggregating operator of fuzzy weighting vector and fuzzy performance matrix.
How to assess the measure of evaluated criteria is the critical process. In traditional evaluation methods such as Analytic Hierarchy Process, ELECTRE, PROMETHEE, TOPSIS, VIKOR and Grey Relation Analysis, assume definitely mutually independent between each pair criteria, and the simple additive weighted (SAW) method is appropriately to aggregate the synthetic value from criteria weights with performance scores. However, in most of MCDM problems, dependence or feedback may exist in evaluating structure. This independent relationship can not satisfy the nature of real situations. we can not employ SAW to derive the synthetic values if the relationship among these criteria are not independent, then the other aggregating tools is more suitable. For instance, fuzzy integral will provide
appropriate estimate of synthetic values while in dependent situation; Analytic Network Process (ANP) can be applied to estimate the synthetic values while in the situation of feedback exists in considered dimension with its lower level of hierarch system, i.e. criteria.
Finally, the final fuzzy synthetic judgment of individual alternative for j evaluated criteria can be illustrated as follows:
(
, ,)
3.3 Linguistic Variables in Fuzzy Decision Making EnvironmentAccording to Dubois and Prade (1978), a fuzzy number A~
is a fuzzy subset of a real number, and its membership function is ~(x):
µA ] , where x represents the criteria, and is described by enshrined with the following characteristics:
1
It cane be called fuzzy number if all the conditions above are satisfied. The triangular fuzzy number µA( )x =(L M U, , ) can be defined as Eqs.(3.8) and Figure 3.4:
Figure 3.4 Membership Function of Triangular Fuzzy Number
According to the extension principle of triangular fuzzy numbers put forward by Zadeh (1975), the arithmetic operations of two triangular fuzzy numbers and
can be expressed as follows:
(1) Addition of two fuzzy numbers ⊕
) (2) Subtraction of two fuzzy numbers Θ
1 2 3 1 2 3 1 3 2 2 3 1
( ,a a a ) ( , , )Θ b b b =(a −b a, −b a, −b ) (3.10) (3) Multiplication of two fuzzy numbers ⊗
1 2 3 1 2 3 1 1 2 2 3 3
( ,a a ,a )⊗ ( ,b b b, ) ≅ (a b a b, ,a b ) (3.11) (4) Multiplication of any real number k and a fuzzy number
1 2 3 1 2 3 On the other hand, the concept of linguistic variables is fundamental within fuzzy set theory. In formally, a linguistic variable is a variable whose values are words or sentences rather than numbers. For instance, when we refer to environmental conditions, we may express our observations by statement like warm place or, clean and green place or, very wild and quite cute place, and so on. The state of being warm could be translated by the variable temperature, with values in a set such as the interval . Alternatively, temperature could be quantified using labels such as cold, warm, hot. Clearly, a precise numerical value such as seems simpler than the ill-defined term warm. But the linguistic label warm is a choice of one out of three possible values, whereas is a choice of one out of many, perhaps, in the entire range. Linguistic characterizations are, in general, less specific than numerical, but it would certainly be much safer, unless one actually knew the exact temperature, to state that an environment temperature is warm than that is . The statement could be strengthened if the underlying meaning of warm is conceived as around . In this setting, whereas the numerical value 25 can be visualized as a point in a set, the linguistic value warm can be viewed as a collection of objects (temperatures) within a bounded region whose center is at 25. the situation with the state of being clean and green or very wild and quite cute is more complex, because the scale involved in their quantification is quite subjective, and is not natural to translate them into numerical values. But they do
0 50 C− o
convey useful information (Pedrycz and Gomide, 1998).
Briefly speaking, the concept of linguistic variable plays a major role in many applications of fuzzy set theory. Specifically, it is very difficult for conventional quantification to express reasonably those situations that are overtly complex or hard to define in the evaluating process for real MCDM problems; thus the notion of a linguistic variable is necessary in such situations. For example, we can use this kind of expression to compare two evaluated criteria by linguistic variables in a fuzzy environment for AHP weighting assessment as “absolutely important”, “very strongly important”, “essentially important”,
“weakly important”, and “equally important”. We also can employ linguistic variables as a way to measure the performance score of considered alternatives/strategies for each criterion as “very low”, “low”, “fair”, “high”, and “very high”. In this paper we employ the triangular fuzzy numbers to express the fuzzy scale as above. In order to accomplish the data analysis, we can further define these linguistic variables using a fuzzy five level scale, here we give a typical example as Table 3.1 and Figure 3.5.
Table 3.1 Linguistic Variable Expression in Fuzzy Five Level Scale Intensity of fuzzy scale Definition of linguistic variables
2, 4,6,8 Intermediate values between two adjacent judgments
1 3 5 7 9
Figure 3.5 Membership Function for the Five-level Linguistic Variables
3.4 Fuzzy Measure and Fuzzy Integral for Synthetic Judgment
Once the mutual dependence exist among the evaluated criteria, it will overestimate or underestimate the synthetic value if we apply traditional simple additive weighting method in this situation. Sugeno (1974) extended fuzzy measure to proposed fuzzy integral for copeing with multiplicative utility function. In this section, we describe the detail procedure about how to use fuzzy integral technique to aggregate synthetic judgment in the evaluation process.
If the evaluated criteria are the situation of mutually independence, we can use SAW to aggregate the relative weights and performance scores for each possible alternative. Actually, this situation seldom holds in real FMADM problems, in this section we refer to multi-attribute utility theory (Keeney and Raiffa, 1976) to demonstrate non-additive aggregating method called fuzzy integral technique to overcome the criteria are non-independent cases.
In 1974, Sugeno introduced the concept of fuzzy measure and fuzzy integral, generalizing the usual definition of a measure by replacing the usual additive property with a weaker requirement, i.e. the monotonicity property with respect to set inclusion. In this section, we give a short introduction to some notions from the theory of fuzzy measure and fuzzy integral. For a more detailed account, please refer to Dubois and Prade (1980), Grabisch (1995), Hougaard and Keiding (1996), etc.
Definition 3.4.1 Let X be a measurable set that is endowed with properties of σ-algebra, where ℵ is all subsets of X. A fuzzy measure g defined on the measurable space (X,ℵ) is a set function g:ℵ→[0,1] , which satisfies the following properties:
(1)g(φ)=0,g(X)=1;
(2) for all A,B∈ℵ,if A⊆B then g(A)≤g(B) (monotonicity).
As in the above definition, (X,ℵ,g is said to be a fuzzy measure space. Furthermore, ) as a consequence of the monotonicity condition, we can obtain:
)}
In the case where , the set function g is called a possibility measure (Zadeh 1978), and if
)}
Definition 3.4.2 Let be a simple function, where is the characteristic function of the set
measure of . Then the Lebesque integral of h is
where is a linear combination of a characteristic function such that ,and .
From the beginning of the application of fuzzy measures and fuzzy integrals to FMCDA problems, it seems to have been felt that there was dependent relation between criteria.
Keeney and Raiffa (1976) advocated the multi-attributes multiplicative utility function, called non-additive multi-criteria evaluation technique, to refine the situations do not conform to the assumption of independence between criteria (Ralescu and Adams, 1980; Chen and Tzeng, 2001; Chen et al., 2000).
Let g be a fuzzy measure which is defined on a power set P(x) and satisfies the definition 3.4.1 as above. For any two disjoint sets A and B, A B∩ = ∅ , the value of the fuzzy measure it takes upon its union, gλ
(
A∪B)
, is computed as( ) ( ) ( ) ( ) ( ) for 1
gλ A∪B =gλ A +g Bλ +λgλ A gλ B − ≤ < ∞ (3.17) λ where λ≥ −1. The parameter of the fuzzy measure λ is used to describe an “interaction”
between the components that are combined (Pedrycz and Gomide, 1998).
If λ=0, then the above expression reduces to the additive measure,
( ) ( ) ( )
gλ A∪B =gλ A +g Bλ (3.18) If λ >0, we obtain
( ) ( ) ( )
gλ A∪B >gλ A +g Bλ (3.19) This so-called super-additivity relationship quantifies a synergy effect, meaning that an
evidence associated with the union of A and B is greater than the sum of the evidences arising from these two sources support each other.
On the other hand, if λ<0, leading to the sub-additivity effect, these two sources of evidence are in competition (or redundancy), and their effect translates into the form
) The value of the parameter of the λ -fuzzy measure is obtained from the normalization condition gλ
( )
X =1 . Generally speaking, setting X ={ , , , }x x1 2 ⋅⋅⋅ xn , the density of fuzzyFigure 3.6 Conceptual Diagram of Fuzzy Integral
In order to clarify the operation of the fuzzy integral technique, we give some numerical examples in Appendix B. In practical application of real FMCDA problems,
1 2
{ , , , }n
X = x x ⋅⋅⋅ x , probably represents the set of criteria, in case the criteria are not necessary mutually independent, in order to drive the synthetic utility values, we first exploit the factor analysis technique to extract the criteria in some common factors. The criteria within the same factor are not independent, a non-additive measurement case, utilizing non-additive fuzzy integral technique as Eqs.(3.22) to find the synthetic utilities of each alternative within the same factor. On the other hand, there is mutually independent between factors, and the measurement is an additive case, so we can utilize the additive aggregate method to conduct the synthetic utility values for all alternatives. In order to clarify the concept of fuzzy measure in evidence theory, we also summarize the generalized fuzzy measure in Appendix C.
On the other hand, the basic assumption of the Analytic Hierarchy Process (AHP) is that it can be used in the circumstances where a problem can be decomposed as a linear top-to-bottom form of a hierarchy, which the upper level is functionally independent from all its lower levels and the elements in each level are also independent. However, many decision problems cannot be structured hierarchically because of the interactions and dependencies of inter-level or intra-level elements. Saaty (1994) further introduced a revised model so called Analytic Network Process (ANP) for dealing with evaluating process while exists dependent or feedback relationship among considered criteria. The ANP was proposed (Saaty, 1996;
Saaty and Vargas, 1998) to overcome the problem of interdependence and feedback between criteria or alternatives. The ANP is the general form of the analytic hierarchy process (AHP) (Saaty, 1980) which has been used in multicriteria decision making (MCDM) to release the restriction of hierarchical structure, and has been applied to project selection (Meade and Presley, 2002; Lee and Kim, 2000), product planning, strategic decision (Sarkis, 2003; Karsak et al., 2002), optimal scheduling (Momoh and Zhu, 2003) and so on.
3.5 Defuzzification for Determining Preferred Order
In many applications, we employ some fuzzy-based technique and obtain a fuzzy result.
For instance, while the fuzzy synthetic value of each alternative is drived, the next step is how to determine the preferred order for these alternatives. However, the derived fuzzy synthetic value is not a crisp value, it couldn’t be used for comparing from each other. Therefore, the defuzzification method for fuzzy numbers will be utilized to obtain comparable crisp value.
Actually, in virtually all real world systems it is a crisp (non-fuzzy) result that should be
implemented. The most commonly used defuzzification procedure in fuzzy control is certainly the center of area (Centroid), also called center of gravity method. Setting R i represents the fuzzy result in synthetic decision of i-th alternative, then its defuzzified value is illustrated as follows (Kacprzyk, 1997)
1 Here we transform this formula and rewrite the centroid defuzzified value as below:
( )
As mentioned above, defuzzification is selection of a specific crisp element based on the output fuzzy set, and it also includes converting fuzzy numbers into crisp scores. The commonly used defuzzification method, Center of Area (Centroid), together with several other procedures, are presented by Yager and Filev (1993). The other two widely used defuzzified methods include mean of maximal, and α-cut method (Zhao and Govind, 1991;Tsaur et al., 1997; Teng and Tzeng, 1994), however the operation defuzzification can not be defined uniquely (Opricovic and Tzeng, 2003).
3.6 Fuzzy Classification for Solving Optimal Strategy Combination
Clustering is one of the most fundamental issues in pattern recognition. It plays an important role in many engineering fields such as pattern recognition, system modeling, image analysis, communication, data mining, and so on. In briefly, given a finite set of data,
{
1, , ,2 n}
X = x x x , the problem of clustering in X is to find several cluster centers that can properly characterize relevant classes of X.
The objective of most clustering method is to provide useful information by grouping (unlabelled) data in clusters; within each cluster the data exhibits similarity. Similarity is defined by a distance measure, and global objective functional or regional graphic-theoretic criteria are optimized to find the optimal partitions od data.
Numerous tools investigated for hard and fuzzy clustering have been developed, the most widely used algorithms are the Hard c-Means Algorithm (HCMA) and the Fuzzy c-Means Algorithm (FCMA) (Dunn, 1974; Bezdek, 1980). However, the user of these algorithms is usually required to specify the number c of clusters and some other parameters. Different
choices of these parameters may lead to different c-partitions, and consequently to different clustering results. Thus, the difficult problem encountered is the evaluation of the quality of the c-partitions resulting from the algorithms with different parameters setting. Many functions, called cluster-validity or validity criteria, have been proposed in the literature, which are used to measure the effectiveness of the clustering algorithms.
choices of these parameters may lead to different c-partitions, and consequently to different clustering results. Thus, the difficult problem encountered is the evaluation of the quality of the c-partitions resulting from the algorithms with different parameters setting. Many functions, called cluster-validity or validity criteria, have been proposed in the literature, which are used to measure the effectiveness of the clustering algorithms.