An extended outranking approach for multi-criteria decision-making problems with linguistic intuitionistic fuzzy numbers

Highlights

• Some outranking relations for LIFNs are outlined.

• The associated alternatives under certain criteria can be ranked based on the defined outranking relations.

• An extended outranking method with LIFNs is developed based on the elicitation of the classic and popular relation model.

Abstract

Linguistic intuitionistic fuzzy numbers (LIFNs), characterized by a linguistic membership degree, linguistic non-membership degree, and linguistic indeterminacy degree, represent a helpful tool for depicting uncertain information under complex environments. This paper focuses on developing an innovative method to address multi-criteria decision-making (MCDM) problems with LIFNs in which the weight information is completely unknown. First, the distance of LIFNs is defined with the aid of linguistic scale functions (LSFs). Second, some extended outranking relationships between each pair of LIFNs are proposed based on the elicitation of the classic relation models. Moreover, a ranking method is constructed to deal with MCDM problems according to the proposed outranking relationships of LIFNs. Finally, an illustrative example concerning coal mine safety evaluation is provided to demonstrate the proposed method, and its feasibility and validity are further verified by a sensitivity analysis and comparison with other existing methods.

Introduction

Multi-criteria decision-making (MCDM) refers to making a decision based on many irreplaceable criteria in a complicated environment. As society has evolved, decision-making problems have become more complex, and there is high demand for new theoretical tools to help decision-makers specifically express their preference in uncertain environments. To deal with fuzzy information, Zadeh [1] proposed fuzzy sets (FSs), which are now considered to be useful tools in the context of decision-making problems [2], [3], [4]. However, in some cases, the membership degree alone cannot precisely describe the information in practical problems. In order to address this issue, Atanassov [5] introduced intuitionistic fuzzy sets (IFSs), which measure both membership degree and non-membership degree. Since their introduction, IFSs have been researched in great detail, and some extensions of IFSs have been developed and applied in various fields [6], [7], [8]. Torra and Narukawa [9] further introduced the hesitant fuzzy sets (HFSs), an extension of traditional fuzzy sets that permits the membership degree of an element to be a set of several possible values in [0,1], and whose main purpose is to model the uncertainty produced by human doubt when eliciting information.

Although fuzzy set theory has been developed and generalized, the need to define the membership or non-membership degree of an element as a specific value ranging from zero to one greatly confines their applications in practice. Therefore, they are limited to use in quantitative environments and are incapable of handling qualitative information. However, the best expression of decision-makers’ preferences or opinions in most decision-making environments takes a natural linguistic form because of the complexity of problems and the inherent vagueness of human cognition. Therefore, it is preferable for a decision-maker to employ linguistic variables [10] rather than real numbers to conduct an assessment of practical alternatives. The linguistic variable is a valid tool because the use of linguistic information reinforces the flexibility and reliability of classical decision models [11].

In recent years, linguistic variables that take values of words or sentences from natural or artificial languages have been studied deeply by many scholars. Herrera and Herrera-Viedma [12] investigated the characteristics and semantics of linguistic term set and introduced a linguistic decision-making method based on aggregation operators. Subsequently, Xu [13] proposed uncertain linguistic variables (ULVs), which employ a linguistic interval rather than a single linguistic value to depict fuzzy information; ULVs also suggest that the probabilities of all linguistic values in an interval are equal or obey a specific distribution [14]. Along with the promotion of preliminary linguistic models, some extended linguistic concepts have been developed. Linguistic sets consisting of several discrete linguistic terms have been proposed based on linguistic variables and HFSs, such as hesitant fuzzy linguistic term sets (HFLTSs) [15], hesitant fuzzy linguistic sets (HFLSs) [16], and linguistic hesitant fuzzy sets (LHFSs) [17]. To improve the applicability of IFSs and accommodate them to complex and qualitative environments, Chen et al. [18] proposed linguistic intuitionistic fuzzy numbers (LIFNs) by integrating linguistic models and IFSs. LIFNs simultaneously consider the linguistic membership degree and linguistic non-membership degree, and they can effectively handle ambiguous and uncertain information. Subsequently, a series of aggregation operators for LIFNs were proposed, including the linguistic intuitionistic fuzzy weighted averaging (LIFWA) operator, linguistic intuitionistic fuzzy ordered weighted averaging (LIFOWA) operator, linguistic intuitionistic fuzzy hybrid averaging (LIFHA) operator, and corresponding geometric operators.

Addressing decision-making problems with linguistic information implies the need for computing with words (CWW) [19]. Several computational models have been developed to deal with linguistic information, and the primary ones are briefly described as follows. (1) One method involves the direct use of linguistic labels [12], [13], [16], [18], [20]; this method is easy to employ, but it cannot make the maximum use of the original information. (2) Another method represents linguistic information using fuzzy membership functions, which converts linguistic information into fuzzy numbers or sets, such as triangular fuzzy numbers, trapezoidal fuzzy numbers, and type-2 fuzzy sets [21], [22]. However, this method inevitably leads to the loss and distortion of original information in the transformation process. (3) Another method applies the 2-tuple linguistic representation model to reinforce the accuracy of linguistic computations while improving the fuzzy linguistic method by utilizing a symbolic translation parameter [23], [24]. (4) One method was developed based on a novel cloud model, which can precisely depict the fuzziness and randomness of qualitative concepts, and its application in decision-making problems can be found in lots of studies [25], [26]. (5) Finally, one method utilizes linguistic scale functions (LSFs) to accommodate different semantic circumstances [27]. LSFs are greatly flexible in handling linguistic information, and they can effectively prevent the loss and distortion of original information; therefore, they have successfully been applied in all kinds of linguistic problems [39]. Existing studies using LIFNs merely depend on the direct use of linguistic labels, which cannot effectively handle linguistic information. To address this issue, this paper employs LSFs to deal with linguistic intuitionistic fuzzy information.

Although linguistic intuitionistic aggregation operators can help solve decision-making problems to some extent, they have some prominent drawbacks that cannot be ignored. First, the operations of LIFNs rely on the simple handling of subscripts of linguistic terms, which can fail to process original information. Second, virtual linguistic terms (VLTs) appear in the proposed aggregation operations, as shown in the aggregation result in Example 3 in Chen et al. [18], (s_3.349, s_3.959). Although the syntax and semantics of VLTs can be determined based on the corresponding rules [28], it is difficult to identify VLTs by words or sentences explicitly. For example, s₃ can be called “medium” from the predefined linguistic term set in Chen et al. [18], however, s_3.349 or s_3.959 cannot be assigned to a linguistic name. Moreover, the aggregation operators may be limited to a complete compensability hypothesis, which leads to information loss and fails to meet the actual demands of decision-makers. For example, even though alternative a_i is better than alternative a_j with respect to one criterion, this advantage may be balanced by other criteria such that the final ranking could be a_i ≺ a_j. On the contrary, there is a method that can counter the aforementioned drawbacks in the case of using aggregation operators, that is called the relation models. Relation models employ outranking relationships or priority functions in order to rank alternatives according to their priority under given criteria. Therefore, an outranking approach can be adopted to deal with MCDM problems with LIFNs.

Among outranking methods, the elimination and choice translating reality (ELECTRE) methods, originally proposed by Roy [29], are representative in this field. ELECTRE methods permit intransitivity and comparability of preferences, and they properly make use of incomplete information, such as judgments on partial prioritization and ordinal measurement scales. Due to their simple logic, ELECTRE methods and their extensions have been studied widely and applied in various fields [30], [31], [32], [33]. In most outranking methods, outranking relationships can be classified as strong dominance and weak dominance. In some cases, however, these two dominance relationships are too broad to accurately characterize the dominance degree and definitely illustrate the dominance relationship between different objects. In terms of LIFNs, the linguistic membership and non-membership degrees can be used to present strong and weak dominance relationships. With an aim to more precisely reflect the dominance degree between the pairwise LIFNs, this paper develops the use of three parameters in LIFNs to compare the outranking relationships for each pair of LIFNs. Obviously, a higher linguistic membership degree, a lower linguistic non-membership degree, and a lower linguistic indeterminacy degree indicate a larger LIFN. Moreover, the comparison of different levels of the three parameters in LIFNs can be embodied in three cases, which reflect three distinct dominance degrees for two different LIFNs. Therefore, the outranking relationships for LIFNs can be extended to three cases: strong dominance, moderate dominance, and weak dominance.

As discussed above, the extended outranking method is highly suitable for addressing MCDM problems with LIFNs, and can remove the existing drawbacks of linguistic intuitionistic decision-making method. However, no research has yet investigated integrating the outranking model and LIFNs. This paper focuses on handling MCDM problems utilizing LIFNs, and the proposed method is an integration of LIFNs and extended outranking relationships. The evaluation values of alternatives with respect to criteria are presented in the form of LIFNs, and the outranking relationships between different alternatives are determined by systematically comparing the evaluation values.

The rest part of this paper is organized as follows. In Section 2, some concepts, such as linguistic term sets, LSFs, IFSs and LIFNs are reviewed briefly; and the comparison method and distance of LIFNs are proposed. In Section 3, the outranking method of LIFNs is provided, and the corresponding properties are discussed. In Section 4, the concordance index and discordance index are presented, and the MCDM approach with LIFNs is developed. In Section 5, an illustrative example is used to verify the validity of the proposed approach, and a sensitivity analysis and comparison analysis are conducted. Finally, the conclusion is drawn in Section 6.