Elsevier

Information Sciences

Volume 263, 1 April 2014, Pages 1-21
Information Sciences

An ELECTRE-based outranking method for multiple criteria group decision making using interval type-2 fuzzy sets

https://doi.org/10.1016/j.ins.2013.12.012Get rights and content

Abstract

The aim of this paper is to develop an ELECTRE (ELimination Et Choice Translating REality)-based outranking method for multiple criteria group decision-making within the environment of interval type-2 fuzzy sets. Along with considering the context of interval type-2 trapezoidal fuzzy numbers, this paper employs a hybrid averaging approach with signed distances to construct a collective decision matrix and proposes the use of ELECTRE-based outranking methods to analyze the collective interval type-2 fuzzy data. By applying a signed distance approach, this work identifies the concordance and discordance sets to determine the concordance and discordance indices, respectively, for each pair of alternatives. Based on an aggregate outranking matrix, a decision graph is constructed to determine the partial-preference ordering of the alternatives and the ELECTREcally non-outranked solutions. This paper provides additional approaches at the final selection stage to yield a linear ranking order of the alternatives. The feasibility and applicability of the proposed methods are illustrated with an example that addresses supplier selection, and a comparative analysis is performed with other approaches to validate the effectiveness of the proposed methodology.

Introduction

The methodologies in the ELECTRE [7], [55] family are the most widely used outranking methods for multiple criteria decision analysis [35]. Figueira et al. [32] provided a comprehensive survey of the field of ELECTRE family methods for multiple criteria decision analysis. The family of ELECTRE methods includes the ELECTRE I [55], ELECTRE II [56], ELECTRE III [54], ELECTRE IV [57], ELECTRE IS [58], ELECTRE TRI [70], and ELECTREGKMS methods [35]. Additionally, these methods each contain different variants, such as the ELECTRE TRI-C [2] and ELECTRE TRI-NC [3] methods. Many useful and valuable developments have been demonstrated to enrich the ELECTRE decision-making methodologies [1], [4], [5], [9], [31], [50]. Furthermore, the ELECTRE methods have been selected for widespread and extensive use in real-world decision situations [32].

In general, the ELECTRE methods are based on a common rule: with knowledge of the concordance and discordance sets for all ordered pairs of alternatives, one can exploit the outranking relation, which is specific for a particular choice or a ranking problem. However, a limitation of the ELECTRE methods is the need for precise measurements of the performance ratings and criteria weights [32]. In many decision situations, ratings and weights cannot be measured precisely because certain decision makers may express their judgments using linguistic terms [38], especially in multiple criteria group decision-making (MCGDM) problems. Decision-making information provided by a subset of decision makers is imprecise or uncertain because of a lack of data, time pressure, or decision makers’ limited attention and information-processing capabilities [11], [14]. Accordingly, decision makers often make decisions within linguistic environments in real-world problems [30], [40], [52], [53], [59]. The theory of fuzzy sets is ideally suited for addressing the ambiguity involved in multiple criteria decision-making problems [51]. In fact, numerous studies have been conducted on the extended ELECTRE methods in a fuzzy context, such as the application of fuzzy membership functions [6], interval weights and data [62], triangular fuzzy numbers [42], trapezoidal fuzzy numbers [38], [44], duplex linguistic sets [69], triangular interval-valued fuzzy numbers [61], and intuitionistic fuzzy sets [63], [67].

Most extensions of the ELECTRE methods have been discussed within the decision environment of type-1 fuzzy sets (T1FSs). However, T1FSs with crisp membership functions encounter difficulties with respect to the linguistic and numerical uncertainties associated with an unstructured or a complex environment [60]. T1FSs are not suitable for situations in which precise membership functions are difficult or impossible to specify for linguistic concepts [49]. Unlike T1FSs, type-2 fuzzy sets (T2FSs) [71] employ membership functions that are also fuzzy, i.e., the membership is itself a function, called a secondary membership function [47]. The T2FSs are superior to T1FSs because they can model second-order uncertainties [36]. When the secondary membership function is a constant equal to 1, the T2FS is called an interval type-2 fuzzy set (IT2FS) or an interval-valued fuzzy set [25], [46], [47].

The computational complexity of using T2FSs is high, which makes employing them in practical applications challenging [73]. In contrast, the membership values of IT2FSs take the form of crisp intervals. Therefore, IT2FSs are the most widely used type of T2FSs because of their relative simplicity [46]. In addition, the computations associated with IT2FSs are manageable [48], which renders the interval type-2 fuzzy decision-making process considerably practical. Wang and Li [64] defined the interval-valued fuzzy numbers, i.e., the interval type-2 fuzzy numbers (IT2FNs), and provided a starting point for real-world applications. Based on the IT2FN framework, Chen and Chen [22], Wei and Chen [66], Chen [12], [13], [14], [16], and Chen et al. [20] presented a linguistic rating system that contains nine-point linguistic rating scales and the corresponding interval type-2 trapezoidal fuzzy numbers (IT2TrFNs) necessary to measure the importance weights and the alternative ratings. Other examples of the linguistic rating systems for IT2TrFNs include seven-point scales [24], [28], [33], [39], [65], [74], five-point scales [25], [39], [49], four-point scales [25], and three-point scales [25], [37], [39], [72].

The purpose of this paper is to extend the ELECTRE method to the decision environment of IT2TrFNs for solving MCGDM problems. Decision makers commonly employ a linguistic rating system. Because uncertainty necessarily exists in real-world decision situations, modeling the uncertainty of human subjective management becomes increasingly important for addressing MCGDM problems. IT2TrFNs are useful for managing the uncertainty and imprecision arising from mental phenomena in a group decision context. Thus, the linguistic ratings can be appropriately represented by IT2TrFNs to directly account for the uncertainties in complex or ill-defined situations. While considering the IT2TrFN context, this paper also presents a hybrid averaging approach with signed distances to include IT2TrFN information to formulate a collective decision environment. Next, this paper develops an ELECTRE-based outranking method to appropriately interpret the IT2TrFN data for inclusion with MCGDM problems. By employing a signed distance approach, we identify the concordance and discordance sets to determine the concordance and discordance indices, respectively, for each pair of alternatives. Based on the aggregate outranking matrix, we can construct a decision graph to determine the partial-preference ordering of the alternatives. In addition to the proposed method for determining ELECTREcally non-outranked solutions, we provide another method at the selection stage to yield the linear ranking orders of the alternatives. The feasibility and the applicability of the proposed ELECTRE methods are illustrated using the MCGDM example of a supplier selection problem presented by Hatami-Marbini and Tavana [38].

This paper makes several contributions to the existing literature on ELECTRE methodologies and other outranking decision-making models for multiple criteria decision analysis. First, we establish the fundamental structure of the ELECTRE-based outranking methodology within the interval type-2 fuzzy environment to address second-order uncertainties in fuzzy MCGDM problems. Second, we utilize IT2TrFNs to address type-2 fuzzy imprecise or uncertain information in the MCGDM process, particularly with respect to a lack of knowledge or experience, intangible or non-monetary criteria, and a complex and an uncertain environment. Third, we employ the concept of signed distance-based hybrid averages to aggregate group opinions into a collective average that can reflect the relative importance of decision makers and the group consensus of individual opinions. Fourth, depending upon decision makers’ specific requirements, we develop different approaches to determine the priority ranking orders of alternatives, consisting of partial-preference orders, linear ranking orders, and auxiliary final rankings. Finally, the comparative analysis not only validates the effectiveness of the proposed ELECTRE methods but also demonstrates that the obtained results contain more influential decision-aiding outcomes.

This paper is organized as follows. Section 2 briefly reviews the concepts of IT2FSs and IT2TrFNs. Section 3 describes an MCGDM problem within the IT2TrFN environment and introduces a signed distance-based hybrid averaging operation to build a collective decision matrix. Section 4 proposes new measurements for the concordance and discordance indices and develops an ELECTRE-based outranking method to approach an MCGDM problem with IT2TrFNs. Section 5 demonstrates the feasibility and the applicability of the proposed methodology using a supplier selection problem and by conducting a comparative analysis with other methods. Section 6 presents the conclusions.

Section snippets

Preliminaries

Selected relevant definitions and operations of IT2FSs and IT2TrFNs are briefly reviewed in this section. The concept of signed distances in the context of IT2TrFNs is also described.

Collective IT2TrFN environment for the MCGDM

This section first formulates a decision environment based on IT2TrFNs for MCGDM problems. Next, a signed distance-based hybrid averaging operation is described to aggregate the IT2TrFN information and to form a collective decision matrix.

ELECTRE-based outranking method for IT2TrFNs

The classical ELECTRE method uses pairwise comparisons of the alternatives to determine the concordance and discordance sets. The method constructs different types of matrices using the concordance and discordance sets and employs the threshold values to filter the less favorable alternatives and select the better ones [32]. Vahdani et al. [62] proposed using an ELECTRE method within an interval type-2 fuzzy environment through the concepts of interval weights and data. Vahdani and Hadipour [61]

Case illustration and discussion

In this section, we consider the numerical example used by Chen et al. [26] and Hatami-Marbini and Tavana [38] to demonstrate the implementation details of the proposed ELECTRE-based outranking methods. Additionally, a comparative study is subsequently conducted to validate the results of the proposed methods with the results from other approaches.

Conclusions

The subjective opinions and judgments of the decision makers are inherently imprecise and contain many uncertainties, particularly in cases in which hybrid data, vague concepts, and uncertain data are involved in the decision-making process [23]. Within this context, this study represented multiple criteria group decisions in terms of IT2TrFNs and developed the ELECTRE-based outranking method.

To solve MCGDM problems, this paper first presented the signed distance-based hybrid averaging

Acknowledgements

The author is very grateful to the respected editor and the anonymous referees for their insightful and constructive comments, which helped to improve the overall quality of the paper. The author is grateful to the grant funding support of Taiwan National Science Council (NSC 102-2410-H-182-013-MY3) during which the study was completed.

References (74)

  • T.-Y. Chen et al.

    The extended QUALIFLEX method for multiple criteria decision analysis based on interval type-2 fuzzy sets and applications to medical decision making

    Eur. J. Oper. Res.

    (2013)
  • S.-J. Chen et al.

    Fuzzy risk analysis based on measures of similarity between interval-valued fuzzy numbers

    Comput. Math. Appl.

    (2008)
  • S.-M. Chen et al.

    Fuzzy risk analysis based on similarity measures between interval-valued fuzzy numbers and interval-valued fuzzy number arithmetic operators

    Expert Syst. Appl.

    (2009)
  • S.-M. Chen et al.

    Fuzzy multiple attributes group decision-making based on the ranking values and the arithmetic operations of interval type-2 fuzzy sets

    Expert Syst. Appl.

    (2010)
  • C.T. Chen et al.

    A fuzzy approach for supplier evaluation and selection in supply chain management

    Int. J. Prod. Econ.

    (2006)
  • L.-H. Chen et al.

    Fuzzy inventory model for deteriorating items with permissible delay in payment

    Appl. Math. Comput.

    (2006)
  • S.-M. Chen et al.

    Fuzzy multiple attributes group decision-making based on ranking interval type-2 fuzzy sets

    Expert Syst. Appl.

    (2012)
  • J. Chiang

    Fuzzy linear programming based on statistical confidence interval and interval-valued fuzzy set

    Eur. J. Oper. Res.

    (2001)
  • S. Corrente et al.

    Multiple criteria hierarchy process with ELECTRE and PROMETHEE

    Omega

    (2013)
  • S. Greco et al.

    ELECTREGKMS: robust ordinal regression for outranking methods

    Eur. J. Oper. Res.

    (2011)
  • S. Greenfield et al.

    The collapsing method of defuzzification for discretised interval type-2 fuzzy sets

    Inform. Sci.

    (2009)
  • A. Hatami-Marbini et al.

    An extension of the ELECTRE I method for group decision-making under a fuzzy environment

    Omega

    (2011)
  • M. Kabak et al.

    A fuzzy hybrid MCDM approach for professional selection

    Expert Syst. Appl.

    (2012)
  • T. Kaya et al.

    An integrated fuzzy AHP-ELECTRE methodology for environmental impact assessment

    Expert Syst. Appl.

    (2011)
  • D.-F. Li et al.

    Linear programming method for multiattribute group decision making using IF sets

    Inform. Sci.

    (2010)
  • S.-C. Ngan

    A type-2 linguistic set theory and its application to multi-criteria decision making

    Comput. Indus. Eng.

    (2013)
  • Y. Peng et al.

    User preferences based software defect detection algorithms selection using MCDM

    Inform. Sci.

    (2012)
  • B. Pérez-Gladish et al.

    Planning a TV advertising campaign: a crisp multiobjective programming model from fuzzy basic data

    Omega

    (2010)
  • M. Rogers et al.

    A new system for weighting environmental criteria for use within ELECTRE III

    Eur. J. Oper. Res.

    (1998)
  • C.W. Tao et al.

    Simplified type-2 fuzzy sliding controller for wing rock system

    Fuzzy Sets Syst.

    (2012)
  • B. Vahdani et al.

    A new design of the elimination and choice translating reality method for multi-criteria group decision-making in an intuitionistic fuzzy environment

    Appl. Math. Modell.

    (2013)
  • W. Wang et al.

    Multi-attribute group decision making models under interval type-2 fuzzy environment

    Knowl.-Based Syst.

    (2012)
  • S.-H. Wei et al.

    Fuzzy risk analysis based on interval-valued fuzzy numbers

    Expert Syst. Appl.

    (2009)
  • M.-C. Wu et al.

    The ELECTRE multicriteria analysis approach based on Atanassov’s intuitionistic fuzzy sets

    Expert Syst. Appl.

    (2011)
  • W.-E. Yang et al.

    An outranking method for multi-criteria decision making with duplex linguistic information

    Fuzzy Sets Syst.

    (2012)
  • L.A. Zadeh

    The concept of a linguistic variable and its application to approximate reasoning-I

    Inform. Sci.

    (1975)
  • D. Zhai et al.

    Uncertainty measures for general type-2 fuzzy sets

    Inform. Sci.

    (2011)
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