Fuzzy AHP analytic hierarchy process
fuzzy ahp – FUZZY RULE BASE
In any diagnostic or prognostic study in meteorology for the application of fuzzy reasoning there are three interdependent steps. A successful execution of these steps leads to the solution of the problem in a fuzzy environment; i.e. the solution procedure digests any type of uncertainty in the basic evolution of the event concerned. A typical fuzzy system is made up of three major components: fuzzifier, fuzzy ahp inference engine (fuzzy ahp rule) and defuzzifier as
described below . …
The fuzzifier transforms input data into linguistic terms using membership functions that represent how much a given numerical value of a particular variable fits the linguistic term being considered.
All meteorological events are considered as having ambiguous characteristics and therefore their domain of change are divided into many fuzzy subsets, which are complete, normal and consistent with each other.
Fuzzy Inference System (FIS) – fuzzy ahp
This step systematically relates all the factors that take place in the solution depending on the purpose of the problem. In fact, this part includes many fuzzy conditional statements to describe a certain situation.
The fuzzy inference engine performs the mapping between the input membership functions and the output membership functions using fuzzy rules that can be obtained from the relationships modeled. Let us consider two events X and Y which are interactive and dependent. This dependency can be verbally expressed as follows.
IF X is A (1) THEN Y is B(1)
IF X is A (2) THEN Y is B (2)
IF X is A (3) THEN Y is B(2)
IF X is A (n) THEN Y is B (n),
where A(1), A(2),: : :, A(n) and B(1), B(2),: : :, B(n) are the linguistic description of X and Y, respectively. These fuzzy subsets cover whole domain of change of X and Y. The fuzzy conditional statements mentioned above can be expressed in the form of a fuzzy relation as R(X,Y)=ALSO (R1; R2; R3; : : :; RN), where Ri denotes the fuzzy relation between X and Y stated by the i th fuzzy conditional statement. The term ALSO combines Ri's into the fuzzy relation R(X,Y). Having the fuzzy relationship R (X,Y) established, the compositional rule inference is applied to infer the fuzzy subset B from a given fuzzy subset A. This inference can be represented by the term B=A○R(X,Y), where o is a compositional operator.
Defuzzification The final result from previous step is in the form of fuzzy statement. A defuzzification
method must then be applied to calculate the deterministic value of a linguistic variable.
Since different output membership functions can be resulted, a defuzzifier combines the output into a single label or numerical value as required. There are many defuzzification methods such as centre of gravity (COG, centroid), bisector of area (BOA), mean of maxima (MOM), leftmost maximum (LM), etc.. In this paper, the centroid method, which is commonly used in related works, has been adopted.
The role of decision makers (DMs) in multi-criteria decision making is undeniable. DMs evaluate alternatives and criteria estimating their relative performances. However, the information available to DMs is often very limited, which makes extremely important to create more effective decision making methods to work in uncertain environments.
In order to account for the uncertainty component we model the pair-wise comparisons on which AHP is based using intuitionistic fuzzy numbers.
Mikhailov and Singh (1999) used fuzzy preference programming to derive priority vectors from a set of interval comparisons. More precisely, given a prioritation problem with elements, the DM is assumed to provide a set of fuzzy comparisons that he uses to form an interval decision matrix:
where and are the lower and upper bounds of the corresponding uncertain judgments.
We extend this approach to include intuitionistic fuzzy numbers. That is, we assume and in Eq. (7) to be the bounds of the following TIFN:
For more details on fuzzy preference programming please refer to Mikhailov (2003) and Liou et al. (2011), among others.
Interval judgments are considered consistent if there exists a priority vector that satisfies the following inequalities:
where stands for “fuzzy less than or equal to”.
The inequalities in Eq. (9) can be represented as a set of fuzzy linear constraints if we define the following membership and non-membership functions where the ratio is linear:
Eq. (11) shows that, unlike triangular fuzzy data, the maximum of membership function of intuitionistic fuzzy data can be a value less than 1. Moreover, it is clear that the membership and non-membership functions linearly increase on and , respectively, while they linearly decrease on and , respectively.
The set of fuzzy linear constraints that can derived from Eq. (9) gives rise to a matrix inequality usually denoted as follows:
where the matrix has dimension , or, equivalently, as the following system of fuzzy linear constraints:
, , (13)
where denotes the -th row of The -th constraint is characterised by one of the linear membership function defined in Eq. (10). Anyway, to simplify the notation, we will use to denote the membership function of the -th constraint
The priority vectors solving the prioritation problem correspond to the nonempty feasible fuzzy area on the simplex
That is, is a solution to the problem if it belongs to the fuzzy set described by the following membership function:
The maximizing solution is a vector for which the maximum of the fuzzy feasible area is obtained, that is:
A general method for finding the maximizing solution to decision making problems with fuzzy ahp objectives and constraints has been proposed by Bellman and Zadeh (1970). This method is based on the max-min operator. By defining the variable as in the following equation:
the max-min fuzzy linear problem changes into a crisp linear problem. Consequently, the objective function in terms of membership functions is defined as follows:
Since we are working in an intuitionistic fuzzy setting, we also need to consider and solve the prioritation problem for the minimum solution with respect to the fuzzy constraints, that is, we need to find the vector for which the minimum of the fuzzy feasible area is obtained, that is:
In order to obtain the minimizing solution, we can use the min-max operator. That is, we can define the variable as follows:
and, consequently, obtain the objective function in terms of the non-membership functions, as follows:
where is the linear membership function characterizing the -th constraint .
We can now combine models (18) and (21) in a final model that accounts for both the membership and non-membership constraints:
To solve the above model, we use a maximizing set with two objective functions. To fuzzify and we calculate their minimum and maximum values solving the following set of optimization problems. + fuzzy ahp +