Advances in Fuzzy Systems

Volume 2018, Article ID 1975768, 9 pages

https://doi.org/10.1155/2018/1975768

## Two Approximation Models of Fuzzy Weight Vector from a Comparison Matrix

Graduate School of Applied Informatics, University of Hyogo, Kobe, Hyogo 650-0047, Japan

Correspondence should be addressed to Tomoe Entani; pj.ca.ogoyh-u.ia@inatne

Received 25 August 2017; Revised 26 January 2018; Accepted 25 September 2018; Published 24 October 2018

Academic Editor: Katsuhiro Honda

Copyright © 2018 Tomoe Entani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

In this study, our uncertain judgment on multiple items is denoted as a fuzzy weight vector. Its membership function is estimated from more than one interval weight vector. The interval weight vector is obtained from a crisp/interval comparison matrix by Interval Analytic Hierarchy Process (AHP). We redefine it as a closure of the crisp weight vectors which approximate the comparison matrix. The intuitively given comparison matrix is often imperfect so that there could be various approaches to approximate it. We propose two of them: upper and lower approximation models. The former is based on weight possibility and the weight vector with it includes the comparison matrix. The latter is based on comparison possibility and the comparison matrix with it includes the weight vector.

#### 1. Introduction

AHP (Analytic Hierarchy Process) [1] is one of the well-known multicriteria decision-making methods and applied to various decision situations. The decision problem in AHP is structured hierarchically with the criteria and alternatives. The final decision is a priority weight vector of the alternatives reflecting the importance of the criteria. The priority vector of the alternatives represents the decision-makers preference and is obtained by synthesizing two kinds of weight vectors. They are the importance weight vector of the criteria and the priority weight vector of the alternatives under each criterion. Each weight vector is obtained from a pairwise comparison matrix given by a decision-maker intuitively. The core technique in AHP is to derive a weight vector consisting of multiple items, such as alternatives and criteria, from a pairwise comparison matrix. We focus on this technique and derive a fuzzy weight vector from any given comparison matrix.

The often-used techniques are the eigenvector, geometric mean, and logarithm least square methods, all of which basically derive a crisp weight vector from a crisp comparison matrix. Instead of a crisp weight vector, an interval/fuzzy vector could be obtained from a crisp comparison matrix reflecting the inconsistency among the crisp comparisons. On the other hand, the comparisons could be given as interval/fuzzy since a decision-maker’s thinking on the items is not always accurate and precise. In order to represent our vague judgment, interval/fuzzy numbers may be more preferable and suitable than crisp numbers. Moreover, when linguistic patterns among the items are given, the comparison matrix is constructed based on transitivity so as to be robust for rank reverse [2]. In case of an interval/fuzzy comparison matrix, a crisp or interval/fuzzy weight vector could be obtained. In this way, there are four cases: a crisp or interval/fuzzy weight vector from a crisp or interval/fuzzy comparison matrix. We review some previously proposed ideas to distinguish ours from them.

A crisp weight vector is useful for ranking the items linearly. It is obtained not only from a crisp comparison matrix but also from an interval comparison matrix. Using the upper or lower bound of each interval comparison, multiple crisp weight vectors are derived [3] and the crisp weight vector is determined by taking a Euclidean center of them in [4]. The other method to derive a crisp weight vector from an interval comparison matrix is based on fuzzy programming approach where a deviation parameter of each comparison is introduced to measure the satisfaction [5]. Moreover, some methods to derive a crisp weight vector from a fuzzy comparison matrix have been proposed [6–8] and the applications have been shown and compared in [9]. These obtained crisp weight vectors are understood as a summary of the given comparison matrix, although a part of its information may be curtailed. Then, in order to reflect the uncertainty of each interval comparison, an interval comparison matrix is summarized in an interval weight vector. The interval weight vector is obtained from the viewpoint of the stability of the rank order of items in [10], where well-known eigenvector method is applied. The upper and lower bounds of each interval weight are obtained from the devised interval comparison matrix to be less inconsistent with logarithmic goal programming in [11]. Furthermore, the crisp comparison matrix is processed into an interval comparison matrix based on transitivity and the interval weight vector is obtained from the upper and lower bounds of the interval comparisons [12]. Each interval comparison depends on two crisp comparisons and does not reflect the other comparisons so that all the given comparisons are not considered equally. In case of a fuzzy comparison matrix, the fuzzy weight vector is derived so as to be close to it in various ways, and they are reviewed in [13]. For instance, the degree of consistency of the obtained weight vector is minimized in [14], the deviations based on geometric mean are measured in [15], and the deviations by logarithm are considered in [16]. As an application, Fuzzy AHP was used for a selection problem in [17, 18]. The fuzzy comparison has been processed from a crisp comparison by assuming its uncertainty in a certainty degree [19]. The primal reason why a weight vector is an interval/fuzzy seems to be the fuzziness of the given comparisons so that these interval/fuzzy weight vectors reflect the uncertainty of each comparison.

It is a human nature to make a decision and we often prefer to do it by ourselves. We do not prefer a system to do it. Since his/her final decision does not always equal the derived result which s/he refers to in making a decision, there is no need to summarize it as a rigid value, crisp weight vector. Instead, it is more important to represent a weight vector of the items in a decision-maker’s mind as it is. For a decision-maker, the decision aiding system to derive what s/he cannot easily notice but reflects his/her uncertain judgment would be helpful. The given comparison matrix does not represent the whole decision-maker's thinking, and its imperfectness is embodied in inconsistency among the intuitively given comparisons, as well as fuzziness of each comparison. In Interval AHP [20, 21], the interval weight vector is obtained by reflecting inconsistency among the crisp comparisons. Because of its formulation, the upper and lower bounds of the interval weights are emphasized more than the values within an interval weight. This paper derives a fuzzy weight vector whose membership function is estimated by the interval weight vectors from an interval/crisp comparison matrix. The goal is neither to summarize an interval comparison matrix in a crisp weight vector nor to reflect the fuzziness of each interval comparison into an interval weight vector. We redefine the interval weight vector in Interval AHP based on the relations between the comparison matrix and the weight vector from possibilistic view. The interval weight vector reflects uncertainty in a decision-maker’s thinking by inconsistency among the comparisons and fuzziness of the comparisons.

This paper is organized as follows. Sections 2 and 3 explain a comparison matrix and a fuzzy weight vector, respectively. In Section 4, two approaches to derive interval weight vector from an interval/crisp comparison matrix are proposed based on the idea that the given comparisons are not perfect. Then, in Section 5, two example comparison matrices are given to illustrate the proposed approaches to derive the fuzzy weight vectors. The last section concludes this paper.

#### 2. Pairwise Comparison Matrix and Weight Vector

A decision-maker compares a pair of items and gives the comparison based on his/her intuitive judgment. A crisp comparison is used to denote a decision-maker’s judgment as quantitative data. The interval comparison is more general than the crisp one since an interval becomes crisp when its upper and lower bounds are equal. We start with an interval pairwise comparison matrix of items.whose element is the comparison representing the importance ratio of item to item , and intuitively given by a decision-maker. An interval comparison is sometimes more suitable than a crisp comparison since a decision-maker’s judgment is often vague. It may be expressed as follows: item 1 is “probably very much more important” than item 2. The interval comparison of item 1 over item 2 is denoted such that . In case that item 1 is “very much more important” than item 2, the crisp comparison is such that . These comparisons are identical: and reciprocal in interval sense: . A crisp comparison matrix is the case of . Those who are more familiar with crisp numbers than interval numbers may prefer crisp numbers to represent their judgments. There are well-known methods in conventional AHP, such as eigenvector method, geometric mean method, and the logarithmic least square method, which derive crisp weight vector from crisp comparison matrix .

A decision-maker can focus on the compared items without regarding the other items at the same time. It is easy for him/her to compare a pair of items, although s/he has to repeat comparing for all the pairs. The comparisons are inconsistent with each other since s/he compares an item several times intuitively. The given comparison matrix surely corresponds to a decision-maker’s thinking on the items, and it is reasonable to derive the weight vector of the items from it. However, it is not the best to make the comparison matrix and the weight vector mathematically equal because of the imperfectness of the given comparisons. They do not represent the whole decision-maker’s thinking, and it is unknown how good and how much reliable they are. Therefore, there could be various weight vectors to approximate a comparison matrix, and this study proposes two approaches to obtain interval weight vector from interval/crisp comparison matrix .

#### 3. Membership Function of Fuzzy Weight Vector

The membership function of a triangular/trapezoidal fuzzy number is estimated from two kinds of its -level sets. Let us assume them as interval and interval , whose membership values to fuzzy number are and , respectively. The membership function is denoted as follows.where , , , . It is illustrated in Figure 1.