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i=1, 2, …, n. The experiment is straightforward. When prudently arranged and carried out, it can deliver reliable and significant estimates of membership grades. Moreover, the results (estimated membership values) can be evaluated with respect to their statistical relevance. To accomplish that we associate with each membership value its standard deviation coming from the results (response) of the experiments. Assuming that the results comply with a certain binomial distribution, the standard deviation of the estimate A(xi) is computed as

Associating these results with the previous membership estimates we derive the bounds defined as

3.5.2. Vertical method of membership estimation

This method takes advantage of the identity principle and “reconstructs” a fuzzy set via identifying its α-cuts. Selecting several levels of α, the testees are requested to identify the corresponding subset of X whose elements belong to A to a degree not less than α. The fuzzy set is built by piling up the successive a-cuts.

In comparison to the horizontal approach, the factor of uncertainty emerging in this class of experiment is distributed along the membership axis. Observe that in the latter method the uncertainty component resides along the universe of discourse (α-levels). Thus the methods could be regarded as highly orthogonal. To some degree they are also complementary since the estimation results could be put together by selecting several xi’s and α -levels. In this fashion one can achieve some capabilities of crossvalidation of the derived results.

The main advantage of these two methods lies in their conceptual clarity. On the other hand, the most prominent shortcoming stems from the fact of a “local” (as opposed to “global”) character of’ the experiments set up in each of these environmentsþthe expert becomes exposed to a single element either in the universe of discourse or the membership scale. This makes the individual experiments quite isolated - that in fact goes against the fundamental concept of membership continuity being so intensively advocated in fuzzy sets. As a straightforward consequence of this unrelated treatment of the elements, the outcomes of the experiment could be scattered and inconsistent.

3.5.3. Pairwise comparison method of membership function estimation

The method proposed by Saaty (1980) mitigates one of the deficiencies of the horizontal and vertical approaches by determining the membership function through a sequence of pairwise comparisons of the individual objects of the universe of discourse. This method applies to the membership functions defined in a finite universe of discourse. To explain the rationale behind it, let us assume for the time being that the membership function is given and its values at x1, x2, … xn are known and equal to A (x1) A (x2) … A(xn), respectively. Consider the ratios A(xi)/Α(xj) , i, j=1, 2,…, n and arrange them in the form of a square matrix

namely

A is usually referred to as a reciprocal matrix. Note that:

1  - all diagonal elements of A are equal to unity,

2  - A is reciprocal meaning that

i, j=1,2,…,n,

3  - A is transitive in the sense that aik akj = aij. This property can be easily verified by observing that

Now multiplying A by the vector

one obtains,

namely,

From the basic matrix algebra (A is a positive definite matrix; I is the identity matrix) one concludes that n denotes the largest eigenvalue of A. Moreover the corresponding eigenvector associated with n is just equal to a.

Now let us reverse the problem (as this is a scenario we have to deal with in reality) and consider that the entries of A are not known and need to be estimated through a series of pairwise comparisons. In designing the experiment we can easily preserve the property of reciprocality however maintaining the aspect of transitivity of the entries of A becomes next to impossible.

As the elements of A are collected in an experimental way, one proceeds with selecting a suitable ratio scale (that usually involves 7 ± 2 quantization levels) with the aid of which the objects are to be pairwise compared in the context of A. The level of preference of xi over xj is quantified numerically: the more xi is preferred over xj, the higher the numerical level associated with this pair. The level of preference of xi over xi is always equal to one as shown by the diagonal elements of A. If xi is not preferred over xj then one considers the level of preference attached to the swapped pair of the elements. The number of necessary comparisons is therefore n(n-1)/2. As the transitivity property cannot always be strictly enforced, the maximal eigenvalue is no longer equal to n. As shown in Saaty (1980), it exceeds n. Interestingly enough, the higher the value of the maximal eigenvalue, the more significant the transitivity inconsistencies within the collected data. This furnishes us with a useful tool to monitor the quality of estimates; if too low, the experiments may need to be repeated.

3.5.4. Problem specification-based membership determination

This way of determining membership function relies directly on computations that take into account a certain numerical objective function emerging in the problem. For illustrative purposes let us study a problem in which a nonlinear function y=f(x) has to be linearly approximated around a certain point of the universe of discourse, say x*. It is well known that the linear approximation y= a(×-×*) holds only in a small neighborhood of × = ×*. How small, this can be quantified by introducing a fuzzy set describing an acceptable error of approximation (or briefly, the quality of linear approximation). The approximation error can be quantified as F(x). In particular, one considers


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