An interval type-2 fuzzy PROMETHEE method using a likelihood-based outranking comparison approach

Highlights

• We develop an interval type-2 fuzzy PROMETHEE method to address MCDA problems.

• The proposed method uses the approach of likelihood-based outranking comparisons.

• We present novel likelihood-based preference functions based on outranking indices.

• We develop two algorithmic procedures to acquire partial and complete rankings.

• Comparative analysis validates the effectiveness of the proposed method.

Abstract

Based on the preference ranking organization method for enrichment evaluations (PROMETHEE), the purpose of this paper is to develop a new multiple criteria decision-making method that uses the approach of likelihood-based outranking comparisons within the environment of interval type-2 fuzzy sets. Uncertain and imprecise assessment of information often occurs in multiple criteria decision analysis (MCDA). The theory of interval type-2 fuzzy sets is useful and convenient for modeling impressions and quantifying the ambiguous nature of subjective judgments. Using the approach of likelihood-based outranking comparisons, this paper presents an interval type-2 fuzzy PROMETHEE method designed to address MCDA problems based on interval type-2 trapezoidal fuzzy (IT2TrF) numbers. This paper introduces the concepts of lower and upper likelihoods for acquiring the likelihood of an IT2TrF binary relationship and defines a likelihood-based outranking index to develop certain likelihood-based preference functions that correspond to several generalized criteria. The concept of comprehensive preference measures is proposed to determine IT2TrF exiting, entering, and net flows in the valued outranking relationships. In addition, this work establishes the concepts of a comprehensive outranking index, a comprehensive outranked index, and a comprehensive dominance index to induce partial and total preorders for the purpose of acquiring partial ranking and complete ranking, respectively, of the alternative actions. The feasibility and applicability of the proposed method are illustrated with two practical applications to the problem of landfill site selection and a car evaluation problem. Finally, a comparison with other relevant methods is conducted to validate the effectiveness of the proposed method.

Introduction

Uncertain and imprecise assessment information is usually present in practical multiple criteria decision analysis (MCDA) problems [1] because decision makers are not always certain of their given decision or preference information and often use a certain degree of uncertainty to express their subjective judgments [2]. Type-2 fuzzy sets, initially introduced by Zadeh [3], are valuable for modeling impressions and for quantifying the ambiguous nature of subjective evaluations and judgments [4]. The concept of type-2 fuzzy sets is an extension of an ordinary fuzzy set (i.e., a type-1 fuzzy set) in which the membership function falls into a fuzzy set on the interval [0, 1] [5], [6]. However, the computational complexity of using type-2 fuzzy sets is quite high, which makes it difficult to employ these sets in practical applications [7]. Previous research has suggested that as a special case of type-2 fuzzy sets, interval type-2 fuzzy sets offer an alternative that is able to address vagueness and uncertainty [8]. Therefore, considerable concern has arisen over interval type-2 fuzzy sets in practical fields [5], [9]. Interval type-2 fuzzy sets have been applied productively in the decision-making field, and numerous useful methods have been developed to address MCDA problems with interval type-2 trapezoidal fuzzy (IT2TrF) numbers [6], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19].

Decision makers commonly use linguistic variables to evaluate the importance weights of criteria and the ratings of alternatives with respect to various criteria [20]. In particular, the concept of linguistic variables is useful in the case of complex or ill-defined situations. The linguistic values generally can be represented with ordinary fuzzy numbers. Nevertheless, interval type-2 fuzzy sets have a better ability to address linguistic uncertainties by modeling the vagueness and unreliability of information [21], [22], [23]. In the interval type-2 fuzzy context, several useful linguistic rating systems have been presented to transform the linguistic values into appropriate IT2TrF numbers, i.e., three-point scales [24], [25], [26], four-point scales [24], five-point scales [24], [27], seven-point scales [18], [28], [29], [30], [31], and nine-point scales [4], [11], [16], [32], [33], [34]. Using these linguistic rating systems, decision makers or analysts can conveniently convert the linguistic responses into IT2TrF numbers. Consequently, the current paper primarily focuses on the development of a new interval type-2 fuzzy MCDA method within the IT2TrF environment.

The preference ranking organization method for enrichment evaluations (PROMETHEE), concepts developed by Brans [35], Brans and Vincke [36], and Brans et al. [37] is a well-known outranking method for handling MCDA problems [15], [38], [39]. The PROMETHEE (and other methods in the PROMETHEE family) can address MCDA problems in criteria-based ranking and selection among a finite set of alternative actions that are often incomparable [39]. The PROMETHEE methods compute positive and negative preference flows for each alternative and facilitate the selection of a final alternative by the decision maker [40]. Although many MCDA methods are available, the PROMETHEE methods are widely used because they are easy to apply compared with other outranking methods [38]. For example, the PROMETHEE methods require fewer parameters from decision makers than the method of elimination and choice expressing reality (ELECTRE) [38]. Moreover, the ranking method of PROMETHEE is less computationally expensive than ELECTRE, and it offers a more useful preference function for selection [41]. Furthermore, the PROMETHEE methods are considered quite straightforward and user-friendly [36], [39]. Consequently, the proposed decision-analysis framework in this paper is based on the PROMETHEE methods.

The PROMETHEE methodology has been extended to the fuzzy context, i.e., the applications of fuzzy membership functions [42], triangular fuzzy numbers [43], [44], [45], trapezoidal fuzzy numbers [46], [47], [48], generalized fuzzy numbers [49], and interval type-2 fuzzy sets [15]. Most extensions of the fuzzy PROMETHEE methods have been developed within the decision environment of ordinary fuzzy sets. However, ordinary fuzzy sets cannot fully address all of the uncertainty present in real-world problems [50], [51]. Human judgment is often vague in many practical decision-making problems, especially with respect to time pressures, lack of knowledge or data, intangible or non-monetary criteria, limited attention and information-processing capabilities of the decision makers, and complex and uncertain environments [6], [34]. Under these circumstances, the available information is often insufficient to determine an exact definition of the degree of membership for certain elements [15]. Ordinary fuzzy sets represent uncertainty using numbers in the range [0, 1]. If an entity is uncertain, it is difficult to specify an exact value of the membership function [23]. Accordingly, it is not reasonable to use an accurate membership function for an uncertain quantity [22], [23].

In contrast, interval type-2 fuzzy sets with interval-type membership grades are appropriate for addressing situations of high-order uncertainties, and interval type-2 fuzzy sets involve greater uncertainties than ordinary fuzzy sets [6], [15], [18]. The theory of interval type-2 fuzzy sets is generally able to encompass higher degrees of uncertainty [52] because it can model and reduce the uncertainties to the minimum of their effects [51], [53]. Based on the above considerations, this paper attempts to develop a new PROMETHEE method to address the MCDA problems within the interval type-2 fuzzy environment. Currently, only Chen [15] has constructed a PROMETHEE-based outranking method for solving MCDA problems in an interval type-2 fuzzy context and used a signed distance-based procedure to identify the ordering of IT2TrF numbers. More specifically, Chen [15] applied the concept of signed distances, which are also referred to as oriented distances or directed distances, to establish certain preference functions. Next, several signed distance-based generalized criteria were employed to determine the corresponding signed-distance-based comprehensive preference indices. Based on these developed indices, the extended definitions of leaving flows, entering flows, and net flows were subsequently proposed to construct relevant measures of outranking and outranked relationships. Finally, novel PROMETHEE I and II methods were developed to acquire partial ranking and complete ranking, respectively, of the alternatives. Concisely, the core concept in the PROMETHEE-based outranking method proposed by Chen [15] is a comparison approach using signed distances. In contrast to the work of Chen [15], this paper applies the concept of likelihood-based outranking indices instead of signed distances to build up a different interval type-2 fuzzy PROMETHEE method.

The purpose of this paper is to develop a new interval type-2 fuzzy PROMETHEE method using the approach of likelihood-based outranking comparisons based on IT2TrF numbers. In contrast to the existing PROMETHEE methodology, this paper presents a novel PROMETHEE procedure that is able to address IT2TrF data within the interval type-2 fuzzy environment. Unlike the signed distances used in Chen [15], this paper uses the concept of likelihood-based outranking indices based on the likelihoods of IT2TrF binary relationships to construct the interval type-2 fuzzy PROMETHEE method. The approach using likelihood-based comparisons has been successfully applied to advance MCDA models and methods [24], [28], [54], [55], [56], [57], [58], [59], [60]. This paper extends the concept of fuzzy preference relationships [24] and likelihoods between trapezoidal fuzzy numbers [28], [54], [55] to present the concepts of lower and upper likelihoods for determination of the likelihood of IT2TrF binary relationships. Using the obtained likelihoods, this paper defines the likelihood-based outranking index and presents an approach for likelihood-based outranking comparisons to establish the interval type-2 fuzzy PROMETHEE method.

The proposed method is not a simple extension of the classical PROMETHEE method to the interval type-2 fuzzy environment. In fact, several new concepts (e.g., likelihood-based outranking indices, likelihood-based preference functions, comprehensive preference measures, comprehensive outranking/outranked indices, and comprehensive dominance indices) are developed to adapt the main structure of PROMETHEE to address the MCDA problems in the IT2TrF context and to avoid the computational complexity attached to type-2 fuzzy sets. Specifically, to take a pragmatic approach to likelihood-based outranking comparisons in the context of interval type-2 fuzzy sets, the novel concepts of lower and upper likelihoods serve to quantify the likelihood of an IT2TrF binary relationship. Subsequently, based on the likelihood-based outranking indices, this paper develops new likelihood-based preference functions corresponding to several generalized criteria to better define the indifference area between IT2TrF evaluative ratings. By incorporating the IT2TrF importance weights, the concept of comprehensive preference measures are subsequently proposed to determine the IT2TrF flows in the valued outranking relationships, i.e., IT2TrF exiting flows, IT2TrF entering flows, and IT2TrF net flows, for each alternative. These novel IT2TrF flows provide a sound basis for construction of a technical tool that will produce partial and total preorders for the purpose of acquiring partial ranking and complete ranking of the alternatives. This paper introduces the concepts of a comprehensive outranking index, a comprehensive outranked index, and a comprehensive dominance index corresponding to the three IT2TrF flows to establish two ranking procedures of partial preorders and total preorders. Two algorithmic procedures are developed to determine partial ranking and complete ranking of the alternatives. Finally, to demonstrate the feasibility and the applicability of the proposed method, this paper illustrates the detailed procedures via two practical applications related to the landfill site selection problem and the car evaluation problem. This paper also conducts a comparison with other relevant methods and validates the effective use of the proposed interval type-2 fuzzy PROMETHEE method.

The remainder of this paper is organized as follows. Section 2 briefly reviews the concepts of interval type-2 fuzzy sets and IT2TrF numbers. Section 3 describes an interval type-2 fuzzy MCDA problem in an IT2TrF setting and introduces the concept of likelihoods of IT2TrF binary relationships. Based on pairwise comparisons of likelihood-based outranking indices and the determination of likelihood-based preference functions, Section 4 develops an interval type-2 fuzzy PROMETHEE method for addressing MCDA problems involving IT2TrF data. Section 5 demonstrates the feasibility and the applicability of the proposed methodology using two practical applications, i.e., the landfill site selection problem and the car evaluation problem. This section also compares the solution results yielded by other MCDA methods with those of the proposed method. Finally, Section 6 presents the conclusions.