Theory and Methodology
A discussion on Extent Analysis Method and applications of fuzzy AHP

https://doi.org/10.1016/S0377-2217(98)00331-2Get rights and content

Abstract

This paper proves the basic theory of the triangular fuzzy number and improves the formulation of comparing the triangular fuzzy number's size. On this basis, a practical example on petroleum prospecting is introduced.

Introduction

Van Laarhoven (1983) [1]proposed a method of fuzzy judgement by comparison of the triangular fuzzy number. He applied the operation of the triangular fuzzy number and the logarithmic least squares method to obtain element sequencing. Chang (1995) proposed the principle for comparison between the elements of the fuzzy number. Meanwhile he wrote an article entitled Applications of the Extent Analysis Method on Fuzzy AHP [2], which has been published in the European Journal of Operational Research in December, 1996.

However, when the model of Chang (1996) is calculated by using a computer, it continually outputs such information as “zero is used as divisor”, “data is out of range”. Therefore, the following aspects of work are done to improve the model.

  • 1.

    Document of Theorem 1 to provide the theoretical basis necessary.

  • 2.

    Document of a particular case of the comparison of two triangular fuzzy numbers.

  • 3.

    The output mistake of the computer lies in the imperfect expression about μ(d), that is, the term μ(d) is unreasonable when two triangular fuzzy numbers do not intersect. μ(d) should be taken to be zero.

  • 4.

    Sugguestion that only 12⩽δ⩽1 is a proper term after massive analogue calculation.

Section snippets

The basic theory of the triangular fuzzy number

Definition 1. Let F(R) is the total fuzzy set of R, and suppose M∈F(R). If ∃x0R, which makes μm(x0)=1; ∀λ∈(0,1), Mλ=[x μM(x)⩾λ] is a convex set; μM is the subordinate degree function of M:R→[0,1] which is denoted asμM(x)=1m−lx−1m−l,x∈[l,m],1m−ux−um−u,x∈[m,u],0otherwise,then we call M the triangular fuzzy number. In this formula u and l are the upper bound and the lower bound of M, resepectively, where m is the mid-value of M. Generally speaking, the triangular fuzzy number is denoted as (l, m, u

Constructing a fuzzy judgement matrix

For some factors (criteria) of the (k  1)th layer, there are altogether nk related factors of the kth layer. When these nk factors are made for pair-by-pair comparison, we denote comparison quantitatively with the triangular fuzzy number, then we get the fuzzy judgement matrix.

aij=[lij,mij,uij], one of the elements of A=(aij)nk×nk, is a closed interval whose mid-value is mij. Then, mij is just one of the integers from one to nine which are used in the method of the AHP.

Let: mijlij=uijmij=δ, δ

Example

After regional prospecting in some oil field in China, we found a batch of blocks. According to the following nine indexes, we evaluated the potential for synthetically prospecting to locate possible drilling sites for oil.

Fig. 3 illustrates the order of the layers follow.

The first step: According to the total goal's requirement, the decision makers (There are two experts.) took part in the evaluation, marked the result of pair-by-pair comparison and then got the fuzzy judgement matrix A (see

References (3)

  • P.J.M. Van Laarhoven, W. Pedrycz, A fuzzy extention of Saaty's priority theory, Fuzzy Sets and Systems 11 (3) (1983)...
There are more references available in the full text version of this article.

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