Two Approximation Models of Fuzzy Weight Vector from a Comparison Matrix

Tomoe Entani

doi:10.1155/2018/1975768

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.

Highlights

AHP (Analytic Hierarchy Process) [1] is one of the wellknown multicriteria decision-making methods and applied to various decision situations
This paper derives a fuzzy weight vector whose membership function is estimated by the interval weight vectors from an interval/crisp comparison matrix
The membership value of the interval weight vector is the sum of the lower bounds of the interval weights, which represents its sure degree

Summary

Introduction

AHP (Analytic Hierarchy Process) [1] is one of the wellknown multicriteria decision-making methods and applied to various decision situations. 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. 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. 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.

Pairwise Comparison Matrix and Weight Vector

Interval Weight Vectors from Interval Pairwise Comparison Matrix

Membership Function of Fuzzy Weight Vector

Numerical Example

Conclusion