Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n − 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process

Incomplete interval fuzzy preference relations for supplier selection in supply chain management

In the analytical hierarchy process (AHP), it needs the decision maker to establish a pairwise comparison matrix requires n(n–1)/2 judgments for a level with n criteria (or alternatives). In some instances, the decision maker may have to deal with the problems in which only partial information and uncertain preference relation is available. Consequently, the decision maker may provide interval fuzzy preference relation with incomplete information. In this paper, we focus our attention on the investigation of incomplete interval fuzzy preference relation. We first extend a characterization to the interval fuzzy preference relation which is based on the additive transitivity property. Using the characterization, we propose a method to construct interval additive consistent fuzzy preference relations from a set of n–1 preference data. The study reveals that the proposed method can not only alleviate the comparisons, but also ensure interval preference relations with the additive consistent property. We also develop a novel procedure to deal with the analytic hierarchy problem for group decision making with incomplete interval fuzzy preference relations. Finally, a numerical example is illustrated and a supplier selection case in supply chain management is investigated using the proposed method. First published online: 05 Feb 201

An eigenvector method for generating normalized interval and fuzzy weights

Crisp comparison matrices produce crisp weight estimates. It is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate. In this paper, an eigenvector method (EM) is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix, which differs from Csutora and Buckley's Lambda-Max method in several aspects. First, the proposed EM produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model, while the Lambda-Max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process. Next, the EM directly solves the principal right eigenvector of an interval or fuzzy comparison matrix, while the Lambda-Max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using a-level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems. Finally, the Lambda-Max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights, while the EM has no such requirement. It is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights. Situations where the EM is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed. Three numerical examples including a hierarchical structure decision-making problem are examined using the EM to demonstrate its applications. (c) 2006 Elsevier Inc. All rights reserved

A Pairwise Comparison Matrix Framework for Large-Scale Decision Making

abstract: A Pairwise Comparison Matrix (PCM) is used to compute for relative priorities of criteria or alternatives and are integral components of widely applied decision making tools: the Analytic Hierarchy Process (AHP) and its generalized form, the Analytic Network Process (ANP). However, a PCM suffers from several issues limiting its application to large-scale decision problems, specifically: (1) to the curse of dimensionality, that is, a large number of pairwise comparisons need to be elicited from a decision maker (DM), (2) inconsistent and (3) imprecise preferences maybe obtained due to the limited cognitive power of DMs. This dissertation proposes a PCM Framework for Large-Scale Decisions to address these limitations in three phases as follows. The first phase proposes a binary integer program (BIP) to intelligently decompose a PCM into several mutually exclusive subsets using interdependence scores. As a result, the number of pairwise comparisons is reduced and the consistency of the PCM is improved. Since the subsets are disjoint, the most independent pivot element is identified to connect all subsets. This is done to derive the global weights of the elements from the original PCM. The proposed BIP is applied to both AHP and ANP methodologies. However, it is noted that the optimal number of subsets is provided subjectively by the DM and hence is subject to biases and judgement errors. The second phase proposes a trade-off PCM decomposition methodology to decompose a PCM into a number of optimally identified subsets. A BIP is proposed to balance the: (1) time savings by reducing pairwise comparisons, the level of PCM inconsistency, and (2) the accuracy of the weights. The proposed methodology is applied to the AHP to demonstrate its advantages and is compared to established methodologies. In the third phase, a beta distribution is proposed to generalize a wide variety of imprecise pairwise comparison distributions via a method of moments methodology. A Non-Linear Programming model is then developed that calculates PCM element weights which maximizes the preferences of the DM as well as minimizes the inconsistency simultaneously. Comparison experiments are conducted using datasets collected from literature to validate the proposed methodology.Dissertation/ThesisPh.D. Industrial Engineering 201