An outranking approach for multi-criteria decision-making with hesitant fuzzy linguistic term sets
Introduction
In practice, multi-criteria decision-making (MCDM) methods are widely used to rank alternatives or select the best one with respect to several concerned criteria. In classical MCDM methods, the assessments of alternatives are precisely known [14], [45]. Due to the fuzziness and uncertainty of decision-making problems and the inherent vagueness of human preferences, however, the best expression of decision-makers comes in natural language. That is why using linguistic assessments is much more realistic than using numerical values. Therefore, linguistic variables [68] whose values are words or sentences from natural or artificial languages are widely used in assessing alternatives with respect to criteria.
In recent years, MCDM problems with criteria values derived from linguistic terms have attracted much attention of the researchers [9], [17], [18], [21], [27], [35], [46], [52], [53], [54], [60] in which the semantics of linguistic terms are usually given priority on a qualitative scale. This type of priority is more expressive for decision-makers than a simple order because of the presence of absolute landmarks [13]. One of the four main schemes in aggregating such linguistic information uses operations that are performed with associated membership functions [8], [25] and a variety of aggregation operators for different fuzzy sets [4], [71]. This scheme transforms linguistic information into fuzzy sets such as triangular fuzzy numbers, trapezoidal fuzzy numbers or type-2 fuzzy sets [30], [31] and several operators such as type-1 OWA and type-2 OWA operators have been recently introduced using this scheme [70], [72]. The second scheme involves the direct use of linguistic labels and requires a demand for computing with words (CWW) and several aggregation operators have been developed to meet the demand [5], [10], [20], [65]. The third scheme deals with linguistic information based on a model with 2-tuple fuzzy linguistic representation [21], with some researchers proposing extended 2-tuple fuzzy linguistic models [11], [12], [19], [47] and developing a variety of 2-tuple linguistic aggregation operators [50], [56], [57], [58], [61], [67]. The fourth scheme explores the application of the related theories in transforming linguistic information. The cloud model, a representative of these theories, can correctly depict the uncertainty of a qualitative concept. This model has been successfully utilized for handling uncertain linguistic variables [48] and Atanassov’s interval-valued intuitionistic linguistic numbers [51].
The aforementioned approaches have a common characteristic in which each linguistic variable is expressed by a single linguistic term. However, because of uncertainty a single term is not suitable or sufficient for decision-makers to express their preferences or opinions. Therefore, some investigations have focused on decision-making in uncertain linguistic environments, and use uncertain linguistic variables that are actually interval variables to handle uncertainty while others have used uncertain linguistic aggregation operators and methods for multi-criteria group decision-making (MCGDM) problems [62]. Additionally, more aggregation operators for uncertain linguistic variables and their application in decision-making can be found in other investigations [34], [36], [49], [63], [64], [66]. An uncertain linguistic variable is continuous in a linguistic interval, which means that the probabilities of all values in this interval are equal, or follow a specified distribution. However, decision-makers may deem that linguistic intervals cannot reflect their preferences in fact so they prefer to use multiple terms for better expression.
Hesitant fuzzy linguistic term sets (HFLTSs) [37], [38], [40] were proposed and used to deal with the situations where decision-makers think of several possible linguistic values or richer expressions than a single linguistic term for an indicator, alternative, variable, etc. Compared to other fuzzy linguistic approaches, HFLTSs are more convenient and flexible to reflect decision-makers’ preferences. Recently, the studies on MCDM problems with HFLTSs are growing. For example, Lee and Chen [24] proposed a new fuzzy decision-making method based on likelihood-based comparison relations of HFLTSs, while Rodríguez et al. [39] proposed a new linguistic group decision model that facilitates the elicitation of flexible and rich linguistic expressions. Zhu and Xu [73] proposed a hesitant fuzzy linguistic preference relation and its consistency measures based on HFLTSs while Beg and Rashid [3] proposed a fuzzy TOPSIS method to aggregate the opinion of experts or decision-makers in MCGDM problems, where the opinions of experts are represented by HFLTSs. Wei et al. [55] defined the operations and two aggregation operators for HFLTSs, and then proposed the decision-making methods to deal with MCDM problems. Liao et al. [26] gave the distance and similarity measures for HFLTSs and applied them in MCDM problems while the fuzzy envelope of an HFLTS [28] was constructed using a fuzzy membership function and then combined with the fuzzy TOPSIS model to solve MCDM problems. Furthermore, Zhang and Wu [69] extended HFLTSs and introduced hesitant fuzzy linguistic sets (HFLSs) that can contain inconsecutive linguistic terms. Also, hesitant fuzzy linguistic aggregation operators were developed to aggregate the assessment values in the form of HFLTSs. Finally, Meng et al. [32] defined linguistic hesitant fuzzy sets (LHFSs) that considered the possible membership degrees of each linguistic term. The same group developed aggregation operators that could be applied in MCDM problems under a linguistic hesitant fuzzy environment.
Although function models have been adopted by most of the existing methods based on HFLTSs, using aggregation operators may bring some drawbacks in operations. (1) If the aggregation operators of HFLTSs were used, the defined discrete linguistic values would be necessarily extended to the continuous ones as in a simple operation using the subscripts of linguistic terms: 0.5 ⊗ {s1} ⊕ 0.5 ⊗ {s2} = {s1.5}. In such cases, there is an awareness that s1.5 does not have any syntax or semantics assigned, because such a virtual linguistic term makes sense only in comparison and in operation. Nevertheless, the existence of s1.5 may affect the effectiveness of HFLTSs to some extent because HFLTSs would be the same as uncertain linguistic variables in this situation. By contrast, the use of the relation models can avoid the appearance of virtual linguistic terms which are unavoidable in the case of using the aggregation operators. (2) Furthermore, the union of two HFLTSs may result in a set containing inconsecutive linguistic terms, e.g., {s1} ⋃ {s5} = {s1,s5}, which contradicts the definition of HFLTSs and (3) the results using aggregation operators are possibly not consistent with common sense. For example, Wei et al. [55] discussed that in Rodríguez’s comparison results [38] it seems to be unreasonable to declare one HFLTS absolutely superior to another if two HFLTSs have common elements. Therefore, an outranking method ought to be recommended to handle MCDM problems with HFLTSs.
All outranking methods permit comparability and intransitivity of preferences [42] and enable the utilization of incomplete information such as judgments on ordinal measurement scales and partial prioritization. Furthermore, they are able to take ordinal scales into account without converting them into abstract ones with an arbitrary imposed range [43] while maintaining the original verbal meanings [16]. Meanwhile, the thresholds can be considered when modeling imperfect knowledge [29]. Outranking methods, including ELECTRE I, ELECTRE II, ELECTRE III, etc., are important methods in MCDM problems, and have been widely used [74]. In these methods, all couples of alternatives are respectively compared under each criterion before determining the preferred one. Among all outranking methods, ELECTRE I [41] and its derivatives [15] are most prominent and have been successfully used in many fields [1], [2], [6], [7], [22], [23], [33], [44], [67].
As mentioned previously, the outranking method is one of the most appropriate methods to solve MCDM problems because of its simple logic. Nevertheless, the literature shows that none of current studies use an outranking method to solve MCDM problems with HFLTSs. Furthermore, many previous studies on outranking methods have focused on certain data while our research is based on uncertain data as an extension of outranking methods.
This paper focuses on how to solve an MCDM problem using HFLTSs, and the proposed approach is an integration of both HFLTSs and ELECTRE I. The assessments of alternatives with respect to criteria are expressed in the form of HFLTSs, and the outranking relations between alternatives are determined by systematically comparing the assessment values for each criterion. The rest of the paper is organized as follows. In Section 2, the concepts and operations of HFLTSs are briefly reviewed together with a distance for HFLTSs. In Section 3, the dominance relations between HFLTSs and some useful properties are defined. Subsequently, the concordance and discordance indices are put forward and an outranking approach for MCDM problems using HFLTSs is developed in Section 4. In Section 5, the proposed approach is demonstrated by using an illustrative example and the results are compared to those using other existing methods. Finally, conclusions are drawn in Section 6.
Section snippets
Preliminaries
In this section, some concepts and operations of HFLTSs are briefly reviewed, and then a distance for HFLTSs is introduced.
Let S = {si∣i = 0, … , g,g ∈ N} be a linguistic term set, where si represents a possible value for a linguistic variable, and the following characteristics should be satisfied:
- (1)
The set is ordered: if α > β, then sα > sβ.
- (2)
There exists a negation operator: neg(sα) = sg−α.
- (3)
If sα > sβ, then max{sα, sβ} = sα and min{sα,sβ} = sβ.
The outranking method based on HFLTSs
In this section, the dominance relations between HFLTSs are put forward and some desirable properties are explored to support these definitions. Then, in order to quickly determine the dominance relations between HFLTSs, some propositions are proposed. Definition 6 Let S = {s0, s1, … , sg} be a linguistic term set and si and sj be two arbitrary linguistic terms on S. A binary relation P can be then defined as follows: Definition 7 Let S = {s0, s1, … , sg} be a linguistic term set, and be two arbitrary
An MCDM outranking approach based on HFLTSs
This section describes a new MCDM approach for decision-making by integrating HFLTSs and the outranking method with evaluation information.
In a multi-criteria linguistic ranking or selection problem, there are n alternatives, denoted by A = {a1, a2, … , an}. Each alternative is assessed by means of m criteria, denoted by C = {c1, c2, … , cm}, and the evaluations are undertaken using linguistic expressions. The weight of the criterion cj is wj, where wj ⩾ 0 (j = 1, 2, … , m), and .
In this decision-making
Illustrative example
As is known, one of the important capabilities in information warfare is the offensive warfare capability – the ability of interfering and damaging the hostile information system. The offensive warfare capability may be decomposed into four evaluation criteria – the capability of information suppression C1, the capability of hard destruction C2, the capability of network resistance C3 and the capability of psychological resistance C4. Assuming the weight vector of the four criteria is W = (0.3,
Conclusions
A HFLTS, defined as an ordered finite subset of consecutive linguistic terms, is more suitable than traditional fuzzy linguistic approaches when experts hesitate among several values to assess a linguistic variable. However, while outranking methods are well-established MCDM methods and their application has shown considerable success in practice, little attention has been paid to the combination of outranking methods with HFLTSs. The main purpose of this paper is to call attention to the
Acknowledgements
The authors thank the editors and anonymous reviewers for their helpful comments and suggestions. This work was supported by the National Natural Science Foundation of China (Nos. 71221061 and 71271218) and the Hunan Provincial Natural Science Foundation of China (No. 14JJ2009).
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