Hesitant fuzzy linguistic entropy and cross-entropy measures and alternative queuing method for multiple criteria decision making

Abstract

Hesitant fuzzy linguistic term set (HFLTS) is a useful tool for describing people's subjective cognitions in the process of decision making. Multiple criteria decision making (MCDM) involves two important steps: (1) determining the criteria weights; (2) obtaining a suitable ranking of alternatives. In this paper, we propose some hesitant fuzzy linguistic entropy and cross-entropy measures, and then establish a model for determining the criteria weights, which considers both the individual effect of each hesitant fuzzy linguistic element (HFLE) and the interactive effect between any two HFLEs with respect to each criterion. Additionally, we give a hesitant fuzzy linguistic alternative queuing method (HFL-AQM) to deal with the MCDM problems. The directed graph and the precedence relationship matrix make the calculation processes and the final results much more intuitive. Finally, a case study concerning the tertiary hospital management is made to verify the weight-determining method and the HFL-AQM.

Introduction

Hesitant fuzzy linguistic term set (HFLTS) [32] is a more reasonable information expression form to describe people's subjective cognitions than fuzzy set (FS) [49], intuitionistic fuzzy set (IFS) [3], intuitionistic multiplicative set (IMS) [40], hesitant fuzzy set (HFS) [41], Pythagorean fuzzy set (PFS) [47], etc. Based on the continuous linguistic term set (virtual linguistic term set) [42], [45], Liao et al. [19] established the mapping between virtual linguistic terms and their corresponding semantics as shown in Fig. 1. Then, for the purpose of facilitating the calculation process and application, Liao et al. [20] gave the mathematical representation of the HFLTS whose components are the hesitant fuzzy linguistic elements (HFLEs). Up to now, a lot of research work has been done on HFLTSs, such as the hesitant fuzzy linguistic information aggregation operators [12], [17], [51], the hesitant fuzzy linguistic measures [11], [15], [19], [21], the hesitant fuzzy linguistic preference relations [24], [52], [53], and the hesitant fuzzy linguistic decision making methods [[4], [5], [9], [16]–[18], [22], [37], [38]].

Multiple criteria decision making (MCDM) is an effective framework, which has been used to evaluate a finite number of decision alternatives having multiple criteria [6]. Lots of methods have been developed to solve the MCDM problems, such as the TOPSIS method [4], the VIKOR method [22], the TODIM method [38], etc. Furthermore, the Granular Computing techniques [1], [2], [7], [8], [17], [25], [26], [27], [29], [31], [33], [34], [39], [43], [44], [48] can also be used to solve the MCDM problems effectively. In general, MCDM involves two important steps: (1) determining the criteria weights; (2) obtaining a suitable ranking of alternatives. For the first step, when dealing with hesitant fuzzy linguistic MCDM problems, these are few weight-determining methods in the existing literature [11], [30]. Farhadinia [11] defined some entropy measures for HFLTSs, which can be used to deal with the MCDM problems, where the information about criteria weights is incomplete. Peng et al. [30] defined the concept of combination weight, and used it to solve the hesitant fuzzy linguistic MCDM problems and overcome the uncertainty caused by subjective weights. For the second step, many aggregation operators and decision making methods have been proposed to deal with the MCDM problems under hesitant fuzzy linguistic information environment, including the hesitant fuzzy linguistic Bonferroni mean (HFLBM) operator and the weighted hesitant fuzzy linguistic Bonferroni mean (WHFLBM) operator [12], the hesitant fuzzy linguistic TOPSIS methods [4], [10], the hesitant fuzzy linguistic VIKOR method [22], the hesitant fuzzy linguistic TODIM methods [37], [38], the hierarchical hesitant fuzzy linguistic MCDM method [16], and the likelihood-based methods for hesitant fuzzy linguistic MCDM [17], [18].

In the existing weight-determining methods and hesitant fuzzy linguistic MCDM methods, there are the following shortcomings:

• Lots of information will be lost when we only utilize the entropy measure to determine the weights of criteria because we may neglect the interactive effect of the decision information.

• The above aggregation operators and decision making methods are extremely complex or not intuitive.

In order to overcome the above issues, in this paper, we first develop some new hesitant fuzzy linguistic entropy measures and cross-entropy measures. Then, we establish a novel weight-determining model, which considers not only the individual effect of each HFLE, but also the interactive effect between any two HFLEs with respect to each criterion. Furthermore, a hesitant fuzzy linguistic alternative queuing method (HFL-AQM) is proposed to deal with the MCDM problems. This method uses both the graph theory and the precedence relationship matrix skillfully. Especially, the directed graph makes the final ranking results of all alternatives more intuitively to be distinguished.

The rest of the paper is organized as follows: In Section 2, we review some concepts related to HFLTSs. The expectation value and the variance of HFLE are given and a comparison method of HFLEs is established. In Sections 3 and 4, some hesitant fuzzy linguistic entropy measures and cross-entropy measures are proposed, respectively. In Section 5, we establish a weight-determining model based on the hesitant fuzzy linguistic entropy measures and cross-entropy measures, and then propose the HFL-AQM. In Section 6, a case study concerning the tertiary hospital management is made to verify the weight-determining method and the HFL-AQM. Additionally, a comparison analysis is made to show the advantages of the proposed weight- determining method and the HFL-AQM. Finally, we end the paper with some conclusions in Section 7.