Fuzzy VIKOR with an application to water resources planning

Abstract

The fuzzy VIKOR method has been developed to solve fuzzy multicriteria problem with conflicting and noncommensurable (different units) criteria. This method solves problem in a fuzzy environment where both criteria and weights could be fuzzy sets. The triangular fuzzy numbers are used to handle imprecise numerical quantities. Fuzzy VIKOR is based on the aggregating fuzzy merit that represents distance of an alternative to the ideal solution. The fuzzy operations and procedures for ranking fuzzy numbers are used in developing the fuzzy VIKOR algorithm. VIKOR (VIsekriterijumska optimizacija i KOmpromisno Resenje) focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and on proposing compromise solution (one or more). It is extended with a trade-offs analysis. A numerical example illustrates an application to water resources planning, utilizing the presented methodology to study the development of a reservoir system for the storage of surface flows of the Mlava River and its tributaries for regional water supply. A comparative analysis of results by fuzzy VIKOR and few different approaches is presented.

Introduction

There are situations when the evaluation of alternatives must handle the imprecision of established criteria, and the development of a fuzzy multicriteria decision model is necessary to deal with either “qualitative” (unquantifiable or linguistic) or incomplete information (Vanegas and Labib, 2001, Zadeh et al., 1987). Imprecision in multicriteria decision making (MCDM) can be modeled using fuzzy set theory to define criteria and the importance of criteria. According to Bellman and Zadeh “much of the decision-making in the real world takes place in an environment in which the goals, the constraints, and consequences of possible actions are not known precisely” (Bellman & Zadeh, 1970). Ribeiro provides an overview of the concepts and theories of decision making in a fuzzy environment (Ribeiro, 1996). Von Altrock explains the elements of fuzzy logic system design, presenting case studies of real-world applications, of which the most visible applications are in the realms of consumer products, intelligent control, and industrial systems (Von Altrock, 1995). Less visible, but of growing importance, are applications relating to decision support systems (Zimmermann, 1991, Zimmermann, 1987). Although fuzzy set theory has been and still remains somewhat controversial, its successes are too clear to be denied. However, Ribeiro warns that “too much fuzzification does not imply better modeling of reality, it can be counterproductive”. Fuzzy ranking methods have been developed that can be used to compare fuzzy numbers (Chen & Hwang, 1992), but this is still an interesting research area.

There are two approaches to MCDM in a fuzzy environment, “conventional” and “fuzzy” (Perny & Roubens, 1998). The conventional approach is based on a nonfuzzy decision model, whereas the fuzziness dissolution (defuzzification) is performed at an early stage (Chen and Hwang, 1992, Wu et al., 2009). The fuzzy approach is based on processing fuzzy data for decision making, then dissolving the fuzziness at a later stage (Opricovic, 2007). In both cases, defuzzification is necessary since MCDM results must provide a crisp conclusion. Defuzzification is selection of a specific crisp element based on the output fuzzy set, and it also includes converting fuzzy numbers into crisp scores. There are several defuzzification methods, although the operation defuzzification cannot be defined uniquely (Chen and Cheng, 2005, Detyniecki and Yager, 2000, Lee and Li, 1988, Opricovic and Tzeng, 2003, Yager and Filev, 1994).

The multicriteria decision making (MCDM) procedure consists of generating alternatives, establishing criteria, evaluation of alternatives, assessment of criteria weights, and application of a ranking method (Vincke, 1992). The alternatives are evaluates according to different criteria depending on the objectives of the problem. The evaluation of alternatives should be performed according to each criterion from the set of established criteria. A comparative analysis of MCDM methods is presented in several publications (Escobar and Moreno-Jimenez, 2002, Opricovic and Tzeng, 2007, Triantaphyllou, 2000).

The VIKOR method has been developed as an MCDM method to solve a discrete multicritea problem with noncommensurable and conflicting criteria (Opricovic, 1998). It focuses on ranking and selecting from a set of alternatives, and determines compromise solutions for a problem with conflicting criteria, which can help the decision makers to reach a final decision. The compromise solution is a feasible solution which is the closest to the ideal (Opricovic & Tzeng, 2004). VIKOR is based on old ideas of compromise programming (Duckstein and Opricovic, 1980, Yu, 1973). An extension of VIKOR to determine fuzzy compromise solution for multicriteria is presented in (Opricovic, 2007).

The fuzzy VIKOR method is developed as a fuzzy MCDM method to solve a discrete fuzzy multicritea problem with noncommensurable and conflicting criteria. It is presented in Section 2. The background for this method, including aggregation, normalization, DM’s preference assessment, and operations on fuzzy numbers are discussed, as a study of rationality that in someway justifies the fuzzy VIKOR method and shows the position of its background in the literature on MCDM. This new method provides a contribution to the practice of MCDM. In Section 3, a numerical example illustrates an application of fuzzy VIKOR to water resources planning, aiming to numerical justification. Comparisons of the results by different methods are presented in Section 4.