Fuzzy MADM: An outranking method

doi:10.1016/S0377-2217(02)00537-4

European Journal of Operational Research

Volume 145, Issue 2, 1 March 2003, Pages 317-328

https://doi.org/10.1016/S0377-2217(02)00537-4 Get rights and content

Abstract

Multi-attribute decision making forms an important part of the decision process for both the small (an individual) and the large (an organization) problems. When available information is precise, many methods exist to solve this problem. But the uncertainty and fuzziness inherent in the structure of information make rigorous mathematical models inappropriate for solving this type of problems. This paper incorporates the fuzzy set theory and the basic nature of subjectivity due to the ambiguity to achieve a flexible decision approach suitable for uncertain and fuzzy environment. The proposed method can take both crisp and fuzzy inputs. An outranking intensity is introduced to determine the degree of overall outranking between competing alternatives, which are represented by fuzzy numbers. The comparison of these degrees is made through the concept of overall existence ranking index. A numerical example is given to illustrate the approach.

Introduction

Multiple attribute decision-making (MADM) problems are encountered under various situations where a number of alternatives and actions or candidates need to be chosen based on a set of criteria or attributes. Comparing the alternatives is the key of making the decision in such cases. However, in case of conflicting alternatives, a decision maker (DM) must also consider imprecise or ambiguous data, which is norm in this type of decision problems. The theory of fuzzy set is ideally suited for handling this ambiguity encountered in solving MADM problems.

Fuzzy set theory was first used to solve decision-making problems by Bellman and Zadeh [2]. Since then, many investigators such as Zimmerman [15], [16], Baas and Kwarnaak [1], Yager [10], [11], among many others, have proposed approaches to handle fuzzy optimization problems. A fuzzy MADM problem [4], [5] is consisted of two phases. The first phase requires the finding of the fuzzy utility function (fuzzy ratings) for each alternative. The second phase requires the application of fuzzy ranking methods. Most of these methods are based on the existence of a rigid frontier between the admissible and inadmissible actions, which are often fuzzy. The preferences between the actions from the standpoint of the DM are seldom examined.

Many fuzzy relations have been introduced to model individual preferences. Zadeh [12], [13] first introduced the concept of fuzzy relation. The other types of relations include fuzzy preference relation [6], fuzzy domination relation [6], [9], and fuzzy outranking relation [7], [8]. Roy [7] and Siskos et al. [8] used outranking relations effectively by introducing fuzzy concordance relations and fuzzy discordance relations. A fuzzy concordance relation is an aggregation of fuzzy partial relations, each is considered as a model for a unique criterion. The fuzzy discordance relation takes into the consideration the importance of the differences between the performances of alternatives for each criterion. Both Roy [7] and Siskos et al. [8] used crisp data as attributes.

This paper introduces a new methodology that combines both the concept of fuzzy outranking and fuzzy attributes to provide a more flexible way for comparing alternatives. Specifically, attributes can be crisp or fuzzy and the concept of overall outranking intensity is introduced, and Chang and Lee’s [3] overall existence ranking index (OERI) is adopted to compare the concordance and discordance degrees.

Section snippets

The outranking relation

For any given two alternatives a and b, the outranking relation principle is based on the fact that even though a and b do not dominate each other, a DM accepts the risk of regarding a as at least as good as b, given the available information. Deriving an outranking relation is the key in classical outranking methods. The problem of uncertainty and fuzziness was later considered and led researchers to derive a fuzzy outranking structure which made the comparison more realistic and accurate.

The proposed approach

In the proposed method, we define a fuzzy outranking function in A×A as a function f:A×A→R in which the different values f(a,b) indicate the degree of outranking associated with the pair of alternatives (a,b). This function reflects the credibility of a preference existing of a over b. This function still satisfies Roy’s [7] first property:

f(a,b) increases with the reliability of the outranking of a by b; thus for all the attributes $j, f(a,b)$ is a non-decreasing function of g_j(a) and a

Sensitivity analysis

We now provide another example for a sensitivity study that compares the results of Siskos et al.’s and the proposed method. This study focuses on the effects when the emphasis on discordance is changed. In this example, 200 alternatives with 10 crisp attributes are considered. We randomly generate the performance scores of the alternatives for each attribute between 0 and 1 as if the scores have been normalized. A portion of the table is presented in Table 6. Three alternatives which our

Conclusions

It is important for MADM methods to provide sound advises and insights for seemingly conflicting problems under various situations. As demonstrated by the very simple examples in this paper, MADM problems are generally very difficult to define and handle due to ambiguities in attributes and initial parameters, such as the veto thresholds. In this paper, we propose a fuzzy outranking method based on Siskos et al.’s structure. The proposed method introduces various fuzzy numbers for determining

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Fuzzy MADM: An outranking method

Abstract

Introduction

Section snippets

The outranking relation

The proposed approach

Sensitivity analysis

Conclusions

Automatica

Computers and Mathematics with Applications

Fuzzy Sets and Systems

International Journal of Man–Machine Studies

International Journal of Man–Machine Studies

Information and Control

Information Sciences

Decision making in a fuzzy environment

Management Science