# Selecting Personnel with the Weighted Cross-Entropy TOPSIS of Hesitant Picture Fuzzy Linguistic Sets

Authors: Xiao-Hui Wu; Lin Yang; Jie Qian
Journal: Journal of Mathematics
 (2021)

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## Abstract

Personnel selection is a key important role for the human resource department of organization, and hesitant picture fuzzy linguistic sets (HPFLSs) elaborated the advantages of both hesitant linguistic set and picture fuzzy set, which is more flexible and effective to solve the decision-making problems of personnel selection than other extension of fuzzy linguistic sets (FLSs). Cross-entropy, as effective measurement tools, is wildly used under fuzzy multicriteria decision-making (FMCDM) environment; thus, in order to elaborate the advantages of both cross-entropy and HPFLSs under FMCDM environment, the cross-entropy definition of HPFLSs is firstly given in this paper. Meanwhile, several novel cross-entropy measures between two HPFLSs are introduced, and their related properties are proved. Then, an approach based on the weighted cross-entropy measures and TOPSIS under hesitant picture fuzzy linguistic environment is proposed. Finally, the proposed method is applied to the real personnel’s selection, and the ranking results show that the proposed methods are practical and effective.


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## Body

## 1. Introduction

Selecting personnel is the key role faced by the human resource (HR) department in organization, and multicriteria decision-making (MCDM) methods, as one of the effective tools, can be used to solve these problems. Due to the existing vague and uncertainty evaluation information in the process of personnel selection, fuzzy set-based MCDM approaches [1–7] were proposed by many researchers. Sang et al. [2] proposed the fuzzy TOPSIS-based method to solve personnel selection for knowledge-intensive enterprise. Baležentis et al. [4] conducted personnel selection approach based on computing with words and fuzzy MULTIMOORA, and Lin [5] proposed the personnel selection method based on the analytic network process and fuzzy data envelopment analysis. However, FSs-based approaches can only express evaluation information with membership degree and cannot express evaluation information with nonmembership degree. Thus, intuitionistic fuzzy set (IFS) based MCDM approaches [8–10] were put forward to solve personnel selection problems. Zhang and Liu [8] proposed GRA-based intuitionistic fuzzy (IF) multicriteria group decision-making method for personnel selection. Boran et al. [9] presented the IFS-based method for personnel selection. Kilic et al. [10] proposed an integrated decision analysis methodology based on IF-DEMATEL and IF-ELECTRE for personnel selection. Considering the real personnel selection situation, human resource managers can be hesitant to express preference while evaluating personnel, so Yu et al. [11] conducted hesitant fuzzy sets (HFSs) based MCDM approaches to aid managers of the human resource (HR) department for selecting excellent personnel.As so far, many extensive FS-based MCDM approaches [2–11] have been conducted and helped managers of HR department to well solve decision-making problems of personnel selection. In some real circumstance, personnel evaluation may involve indeterminate and inconsistent information; thus, several neutrosophic-based MCDM methods [12–15] for personnel selection have been proposed by many researchers, in order to elaborate the advantages of TODIM and multivalued neutrosophic set. Pu et al. [12] conducted a projection-based TODIM method with a multivalued neutrosophic set for personnel selection. Nabeeh et al. [13] proposed an integrated neutrosophic-TOPSIS approach for personnel selection, considering cross-entropy as the major measurement for MCDM approaches. Wu et al. [14, 15] proposed two methods for middle-level manager selections with cross-entropy of probability hesitant interval neutrosophic set and multivalued neutrosophic sets.Although these above proposed MCDM approaches for personnel selection can demonstrate the effectiveness under practical environment, these approaches cannot be effectively utilized to solve personnel selection decision-making problems with different attitude information from decision-makers or experts. For such kind of personnel selection situations, it is required to present new approaches.Picture fuzzy set (PFS) [16] was firstly proposed by Cường and used to describe different attitude information by many researchers. Until now, many MCDM methods based on the picture fuzzy set [17–22] were proposed. Tian et al. [17, 20, 21] proposed PFS-based MCDM methods, which are used for tourism attraction recommendation and tourism environmental impact assessment. Wang et al. [19] proposed a bounded rationality behavioral decision support model with picture fuzzy information, which is applied to select the different hotel among various types of travelers. However, in the real personnel selection decision-making environment, the interviewers might be hesitant to give the evaluation among several linguistic terms, such as “outstanding excellent,” “excellent,” and “merit” linguistic terms. Meanwhile, different interviewer has different attitude for the evaluation value. For example, one listed company carried out personnel selection interview; there are several job applicants for the interview by company interviewers; in the interview process, ten interviewers are required to give the evaluation score for the job applicants. Three interviewers are unsatisfied with all job applicants and refused to give any evaluation value; seven interviewers opposed to give the applicant with the score “outstanding excellent,” but they support to give the evaluation value with the score “excellent.” Due to such kinds of decision-making issues, PFS cannot be used to well describe the evaluation information. In order to avoid the information loss, the proposed new extensive PFS-based methods are required.HPFLSs [23], as the extensive set of PFS, elaborated the advantages of both hesitant linguistic set and picture fuzzy set, which are more suitable for the practical hesitant decision-making problem together with different attitude information. As mentioned, the above example of job applicant interview, if the evaluation linguistic terms set is s5=outstading excellent;s4=excellent;s3=merit;s2=average;s1=pass;s0=failure, then the evaluation value with different attitude information provided by the interviewers can be collected as a set of s4,0.7,0,0,s5,0,0,0.7, and it is called as HPFLSs.In the following, the contributions of this paper are presented as follows:(1)
The cross-entropy measure definition of hesitant picture fuzzy linguistic number is given, and its properties needed to be satisfied are listed. Meanwhile, the comparison rules of hesitant fuzzy linguistic set are defined.(2)
Based on the cross-entropy definition of hesitant picture fuzzy linguistic number, several formulas of cross-entropy of hesitant fuzzy linguistic number are constructed, and the relative properties are proven as well.(3)
Considering the real-life application, the weight of criteria cannot be provided in time due to some pressures or reasons. The method is presented to obtain the unknown weight of criteria.(4)
In accordance with the proposed cross-entropy formula, the MCDM approaches based on cross-entropy of hesitant picture fuzzy linguistic numbers and TOPSIS are constructed.The rest of the paper is organized as follows: some definitions related to picture fuzzy linguistic set and HPFLSs are introduced in Section2. In Section 3, several novel formulas of cross-entropy measure under HPFLSs environment are proposed, and the related mathematics-induced proofs are shown. By utilizing the proposed cross-entropy measure, in Section 4, an approach of the detailed decision-making steps for solving the MCDM problem under HPFLSs environment with completely unknown criteria weight is presented. Furthermore, a practical application on the MCDM problem of personnel’s selection is conducted to demonstrate the effectiveness of the proposed method in Section 5. Finally, conclusion and further direction is drawn in Section 6.


## 2. Preliminaries

In this section, the definitions of cross-entropy measurement, the PFS, 2TFLSs, 2TLPFSs, HPFLSs, and cross-entropy of FS and PFS are presented to lay the groundwork for later analysis.


### 2.1. Definitions of PFS, 2TFLSs, 2TLPFSs, and HPFLSs

Definition 1 (see [16]).
LetX be a universe space. A PFS is defined as(1)A=x,μAx,ηAx,νAx|x∈X,where μAx∈0,1 is called the degree of positive membership of x in A, ηAx∈0,1 is called the degree of neutral membership of x in A, νAx∈0,1 is called the degree of negative membership of x in A, and μAx, ηAx, νAx satisfy the condition 0≤μAx+ηAx+νAx≤1. Moreover, πAx=1−μAx+ηAx+νAx can be called the degree of refusal.
LetS=sj|j=0,1,2,…,g be a linguistic term set. Symbolic method aggregation linguistic information obtains a value β∈0,g, and if β∉0,g, then an approximation function app2⋅ is used to express the index result of S.Definition 2 (see [24]).
Letβ be the aggregation result of the indices of a set of labels assessed in a linguistic term set S, i.e., the results of symbolic aggregation operation, β∈1,t be the t cardinality of S. Let i=roundβ and α=β−i be two values, such that i∈1,t and α∈−0.5,0.5; then, α is called the symbolic translation.
For convenience to collect the data in the selection of the proxy advisor firm problems, Nie et al. [25] defined the concept of 2-tuple linguistic picture fuzzy sets (2TLPFSs).Definition 3.
LetS=sj|j=0,1,2,…,g be a linguistic set and S¯=sq,αq|sq∈S,αq∈0.5,0.5 be a linguistic 2-tuple fuzzy set. Suppose that su,αu, sη,αη, and sν,αν∈S¯; if 0≤μ+η+ν≤g, then r¯=su,αu,sη,αη,sν,αν is called 2TLPFSs, and su,αu, sη,αη, and sν,αν could be called the positive membership degree, neutral membership degree, and negative membership degree of r¯, respectively.
Considering the real-life decision-making environment, decision-makers might be hesitant to provide the evaluation information with different attitude; thus, Yang et al. [23] defined hesitant picture fuzzy linguistic sets (HPFLSs).Definition 4.
LetS=sJ|J=0,1,2,…,m be an LTS. An HPFLSs is defined as(2)Hs=sik,μik,ηik,νik|i∈0,1,2,…,m;k=1,2,…,α,where μik∈0,1, ηik∈0,1, νik∈0,1, and 0≤μik+ηik+νik≤1. μik, ηik, and νik represent the positive membership, indeterminacy membership, and negative membership of the linguistic term sik, respectively; πik=1−μik+ηik+νik is the refusal membership of the linguistic term sik. And, its complementary set is Hsc=sik,νik,ηik,μik|i∈0,1,2,…,m;k=1,2,…,α.


### 2.2. Cross-Entropy Measure of FS and PFS

The cross-entropy measure was introduced by Kullback [26], and its definition is shown as follows.Definition 5 (see [26]).
LetP=p1,p2,…,pn and Q=q1,q2,…,qn be two given probability distributions, where pi≥0∑i=1npi=1 and qi≥0∑i=1nqi=1 for i=1,2,…,n. The cross-entropy measure of P to Q is defined as(3)HP,Q=∑i=1nPi∗lnpiqi.
Based on Kullback’s entropy definition, Shang and Jiang [27] proposed the cross-entropy measure between two fuzzy sets.Definition 6.
Assume thatA=Ax1,Ax2,…,Axn and B=Bx1,Bx2,…,Bxn are two fuzzy sets in the universe of discourse X=x1,x2,…,xn. The fuzzy cross-entropy of A from B is defined as follows:(4)HA,B=∑i=1nAxilog2Axi/1/2Axi+Bxi+1−Axilog21−Axi/1−1/2Axi+Bxi,which indicates the degree of discrimination of A from B.
However,HA,B is not symmetric with respect to its arguments. Shang and Jiang [27] proposed a symmetric discrimination information measure: IA,B=HA,B+HB,A.
Wei [28] extended cross-entropy of fuzzy sets to picture fuzzy sets and conducted the cross-entropy of the picture fuzzy set.Definition 7.
LetA and B be two picture fuzzy sets, then the cross entropy measure of A and B are obtained as follows:(5)CA,B=∑i=1nμAxilnμAxi1/2μAxi+μBxi+1−μAxiln1−μAxi1−1/2μAxi+μBxi+∑i=1nηAxilnηAxi1/2ηAxi+ηBxi+1−ηAxiln1−ηAxi1−1/2ηAxi+ηBxi+∑i=1nνAxilnνAxi1/2νAxi+νBxi+1−νAxiln1−νAxi1−1/2νAxi+νBxi.
Due toCA,B being nonsymmetric, a symmetric discrimination cross-entropy measure is presented as IA,B=CA,B+CB,A.


## 2.1. Definitions of PFS, 2TFLSs, 2TLPFSs, and HPFLSs

Definition 1 (see [16]).
LetX be a universe space. A PFS is defined as(1)A=x,μAx,ηAx,νAx|x∈X,where μAx∈0,1 is called the degree of positive membership of x in A, ηAx∈0,1 is called the degree of neutral membership of x in A, νAx∈0,1 is called the degree of negative membership of x in A, and μAx, ηAx, νAx satisfy the condition 0≤μAx+ηAx+νAx≤1. Moreover, πAx=1−μAx+ηAx+νAx can be called the degree of refusal.
LetS=sj|j=0,1,2,…,g be a linguistic term set. Symbolic method aggregation linguistic information obtains a value β∈0,g, and if β∉0,g, then an approximation function app2⋅ is used to express the index result of S.Definition 2 (see [24]).
Letβ be the aggregation result of the indices of a set of labels assessed in a linguistic term set S, i.e., the results of symbolic aggregation operation, β∈1,t be the t cardinality of S. Let i=roundβ and α=β−i be two values, such that i∈1,t and α∈−0.5,0.5; then, α is called the symbolic translation.
For convenience to collect the data in the selection of the proxy advisor firm problems, Nie et al. [25] defined the concept of 2-tuple linguistic picture fuzzy sets (2TLPFSs).Definition 3.
LetS=sj|j=0,1,2,…,g be a linguistic set and S¯=sq,αq|sq∈S,αq∈0.5,0.5 be a linguistic 2-tuple fuzzy set. Suppose that su,αu, sη,αη, and sν,αν∈S¯; if 0≤μ+η+ν≤g, then r¯=su,αu,sη,αη,sν,αν is called 2TLPFSs, and su,αu, sη,αη, and sν,αν could be called the positive membership degree, neutral membership degree, and negative membership degree of r¯, respectively.
Considering the real-life decision-making environment, decision-makers might be hesitant to provide the evaluation information with different attitude; thus, Yang et al. [23] defined hesitant picture fuzzy linguistic sets (HPFLSs).Definition 4.
LetS=sJ|J=0,1,2,…,m be an LTS. An HPFLSs is defined as(2)Hs=sik,μik,ηik,νik|i∈0,1,2,…,m;k=1,2,…,α,where μik∈0,1, ηik∈0,1, νik∈0,1, and 0≤μik+ηik+νik≤1. μik, ηik, and νik represent the positive membership, indeterminacy membership, and negative membership of the linguistic term sik, respectively; πik=1−μik+ηik+νik is the refusal membership of the linguistic term sik. And, its complementary set is Hsc=sik,νik,ηik,μik|i∈0,1,2,…,m;k=1,2,…,α.


## 2.2. Cross-Entropy Measure of FS and PFS

The cross-entropy measure was introduced by Kullback [26], and its definition is shown as follows.Definition 5 (see [26]).
LetP=p1,p2,…,pn and Q=q1,q2,…,qn be two given probability distributions, where pi≥0∑i=1npi=1 and qi≥0∑i=1nqi=1 for i=1,2,…,n. The cross-entropy measure of P to Q is defined as(3)HP,Q=∑i=1nPi∗lnpiqi.
Based on Kullback’s entropy definition, Shang and Jiang [27] proposed the cross-entropy measure between two fuzzy sets.Definition 6.
Assume thatA=Ax1,Ax2,…,Axn and B=Bx1,Bx2,…,Bxn are two fuzzy sets in the universe of discourse X=x1,x2,…,xn. The fuzzy cross-entropy of A from B is defined as follows:(4)HA,B=∑i=1nAxilog2Axi/1/2Axi+Bxi+1−Axilog21−Axi/1−1/2Axi+Bxi,which indicates the degree of discrimination of A from B.
However,HA,B is not symmetric with respect to its arguments. Shang and Jiang [27] proposed a symmetric discrimination information measure: IA,B=HA,B+HB,A.
Wei [28] extended cross-entropy of fuzzy sets to picture fuzzy sets and conducted the cross-entropy of the picture fuzzy set.Definition 7.
LetA and B be two picture fuzzy sets, then the cross entropy measure of A and B are obtained as follows:(5)CA,B=∑i=1nμAxilnμAxi1/2μAxi+μBxi+1−μAxiln1−μAxi1−1/2μAxi+μBxi+∑i=1nηAxilnηAxi1/2ηAxi+ηBxi+1−ηAxiln1−ηAxi1−1/2ηAxi+ηBxi+∑i=1nνAxilnνAxi1/2νAxi+νBxi+1−νAxiln1−νAxi1−1/2νAxi+νBxi.
Due toCA,B being nonsymmetric, a symmetric discrimination cross-entropy measure is presented as IA,B=CA,B+CB,A.


## 3. The Definition and Formula of HPFLS Cross-Entropy Measure

In this section, the cross-entropy definition of HPFLSs is given. Considering that the previous cross-entropy formula [28] ignored the influence of refusal membership and different decision maker (DM) might have different support preference for different memberships, several formulas of HPFLSs, which overcome the limitations of previous cross-entropy, are constructed.Definition 8.
LetHs1=si1k1,μi1k1,ηi1k1,νi1k1|i1∈0,1,2,…,m;k1=1,2,…,α1 and Hs2=si2k2,μi2k2,ηi2k2,νi2k2|i2∈0,1,2,…,m;k2=1,2,…,α2 be two HPFLSs. Hs1 is greater than or equal to Hs2, denoted by Hs1≥Hs2, if and only if μ1≥μ2, η1≤η2, ν1≤ν2, where(6)μ1=∑k1=1α1μk1×ik1α1α1α1,μ2=∑k2=1α2μk2×ik2α2α2,α2η1=∑k1=1α1ηk1×ik1α1α1α1,η2=∑k2=1α2ηk2×ik2α2α2α2,ν1=∑k1=1α1νk1×ik1α1α1α1,ν2=∑k2=1α2νk2×ik2α1α2α2,ikj=1+expij−m/2×θ−nj=1,2 and ij are the subscript values of the linguistic assessment term sijkj and n>0, θ>0, μkj=μijkj+1−ηijkj−νijkj/2,ηkj=ηijkj+1−μijkj−νijkj/2, and νkj=νijkj+1−ηijkj−μijkj/2,j=1,2.Definition 9.
Assume thatHs1=si1k1,μi1k1,ηi1k1,νi1k1|i1∈0,1,2,…,m;k1=1,2,…,α1 and Hs2=si2k2,μi2k2,ηi2k2,νi2k2|i2∈0,1,2,…,m;k2=1,2,…,α2 are two HPFLS, CE∗:HPFLS×HPFLS⟶R+. The cross-entropy CE∗Hs1,Hs2 of Hs1 and Hs2 should satisfy the following conditions:(1)
CE∗Hs1,Hs2=CE∗Hs2,Hs1,∀Hs1,Hs2∈HPFLS(2)
CE∗Hs1c,Hs2c=CE∗Hs1,Hs2,∀Hs1,Hs2∈HPFLS(3)
CE∗Hs1,Hs2≥0,∀Hs1,Hs2∈HPFLS, ifHs1=Hs2,thenCE∗Hs1,Hs2=0(4)
CE∗Hs1,Hs3≥CE∗Hs1,Hs2andCE∗Hs1,Hs3≥CE∗Hs2,Hs3,∀Hs1,Hs2,Hs3∈HPFLS and ifHs1≥Hs2≥Hs3
Based on the above definitions, the cross-entropy formulas of HPFLSs are obtained as follows:(7)CE1Hs1,Hs2=ρ×sin∑k1=1α1μk1∗ik1α1α1α1×sin∑k1=1α1μk1∗ik1α1α1α1−∑k2=1α2μk2∗ik2α2α2α2+φ×sin∑k1=1α1ηk1∗ik1α1α1α1×sin∑k1=1α1ηh1∗ik1α1α1α1−∑k2=1α2ηk2∗ik2α2α2α2+δ×sin∑k1=1α1νk1∗ik1α1α1α1×sin∑k1=1α1νk1∗ik1α1α1α1−∑k2=1α2νk2∗ik2α2α2α2.
Similarly, other three cross-entropy formulas of HPFLSs are obtained as follows:(8)CE2Hs1,Hs2=ρ×tan∑k1=1α1μk1×ik1α1α1α1×tan∑k1=1α1μk1×ik1α1α1α1−∑k2=1α2μk2×ik2α2α2α2+φ×tan∑k1=1α1ηk1×ik1α1α1α1×tan∑k1=1α1ηk1×ik1α1α1α1−∑k2=1α2ηk2×ik2α2α2α2+δ×tan∑k1=1α1νk1×ik1α1α1α1×tan∑k1=1α1νk1×ik1α1α1α1−∑k2=1α2νk2×ik2α2α2α2,(9)CE3Hs1,Hs2=ρ×∑k1=1α1μk1×ik1α1α1α1+1×ln2×∑k1=1α1μk1×ik1α1/α1α1+12+∑k1=1α1μk1×ik1α1/α1α1+∑k2=1α2μk2×ik2α2/α2α2+φ×∑k1=1α1ηk1×ik1α1α1α1+1×ln2×∑k1=1α1ηk1×ik1α1/α1α1+12+∑k1=1α1ηk1×ik1α1/α1α1+∑k2=1α2ηk2×ik2α2/α2α2+δ×∑k1=1α1νk1×ik1α1α1α1+1×ln2×∑k1=1α1νk1×ik1α1/α1α1+12+∑k1=1α1νk1×ik1α1/α1α1+∑k2=1α2νk2×ik2α2/α2α2,(10)CE4Hs1,Hs2=ρ∑k1=1α1μk1∗ik1α1α1α1×exp∑k1=1α1μk1×ik1α1α1α1−1−exp∑k2=1α2μk2×ik2α2α2α2−1+φ×∑k1=1α1ηk1×ik1α1α1α1×exp∑k1=1α1ηk1×ik1α1α1α1−1−exp∑k2=1α2ηk2×ik2α2α2α2−1+δ×∑k1=1α1νk1×ik1α1α1α1×exp∑k1=1α1νk1×ik1α1α1α1−1−exp∑k2=1α2νk2×ik2α2α2α2−1,where ikj=1+expij−m/2×θ−nj=1,2 and ij are the subscript values of the linguistic assessment term sijkj and n>0, θ>0, μkj=μijkj+1−ηijkj−νijkj/2,ηkj=ηijkj+1−μijkj−νijkj/2,νkj=νijkj+1−ηijkj−μijkj/2,j=1,2,ρ+φ+δ=1,and normallyρ=φ=δ=1/3.
However, considering that the formulasCEkHs1,Hs2k=1,2,3 are asymmetric, the above formulas are modified as follows:(11)CEk∗Hs1,Hs2=CEkHs1,Hs2+CEkHs2,Hs1,k=1,2,3.Property 1.
LetHs1=si1k1,μi1k1,ηi1k1,νi1k1|i1∈0,1,2,…,m;k1=1,2,…,α1 and Hs2=si2k2,μi2k2,ηi2k2,νi2k2|i2∈0,1,2,…,m;k2=1,2,…,α2 be two HPFLSs. The cross-entropy measure CEk∗Hs1,Hs2 is defined as the discrimination degree from Hs1 to Hs2, and then CEk∗Hs1,Hs2 satisfies conditions (1)–(4) in Definitions 1 and 1.Proof.
Since it is easily proven that conditions (7) and (8) hold, their proofs are omitted here. For condition (9), the detailed proof is shown as follows.
Given thatu1=∑k1=1α1μk1×ik1α1/α1α1 and u2=∑k2=1α2μk2×ik2α2/α2α2, it is obvious to prove that the following constructed functions hold:(12)f1u1,u2=sinu1×sinu1−u2+sinu2×sinu2−u1=sinu1×sinu1−u2−sinu2×sinu1−u2=sinu1−sinu2×sinu1−u2≥0,f2u1,u2=tanu1×tanu1−u2+tanu2×tanu2−u1=tanu1×tanu1−u2−tanu2×tanu1−u2=tanu1−tanu2×tanu1−u2≥0,f3u1,u2=μ1+1×ln2×μ1+12+μ1+μ2+μ2+1×ln2×μ2+12+μ1+μ2=μ1+1×lnμ1+1μ1+μ2+2/2+μ2+1×lnμ2+12+μ1+μ2/2=μ1+1×lnμ1+1+μ2+1×lnμ2+1−2+μ1+μ2×ln2+μ1+μ2/2,where μ∈1,2, fμ=μlnμ is a concave function [29]; thus, it satisfies fμ1+1+fμ2+1/2≥f2+μ1+μ2/2, so μ1,μ2∈0,1 and f3u1,u2≥0 hold.(13)f4u1,u2=u1×expu1−1−expu2−1+u2×expu2−1−expu1−1=u1−u2×expu1−1−expu2−1≥0.
Similarly, given thatη1=∑k1=1α1ηk1×ik1α1/α1α1, η2=∑k2=1α2ηk2×ik2α2/α2α2, ν1=∑k1=1α1νk1×ik1α1/α1α1 and ν2=∑k2=1α2νk2×ik2α2/α2α2 , then it can be proven that the following inequalities fiη1,η2≥0, fiν1,ν2≥0, i=1,2,3,4 hold.
According to Definitions8 and 9, the following function can be obtained.(14)CE1∗Hs1,Hs2=CE1Hs1,Hs2+CE1Hs2,Hs1=f1u1,u2+f1η1,η2+f1ν1,ν2≥0.
Thus, the inequalitiesCE2∗Hs1,Hs2≥0, CE3∗Hs1,Hs2≥0, CE4∗Hs1,Hs2≥0 can hold; next, the detailed proof of condition (10) is shown as follows:
Assume thatHs1≥Hs2≥Hs3; in accordance with Definition 8, the inequalities of μ1≥μ2≥μ3, η1≤η2≤η3, and ν1≤ν2≤ν3 can be obtained. Then,(15)f1u1,u2=sinu1−sinu2×sinu1−u2≤sinu1−sinu3×sinu1−u3≤f1u1,u3,f2u1,u2=tanu1−tanu2×tanu1−u2≤tanu1−tanu3×tanu1−u3≤f2u1,u3,f3u1,u2−f3u1,u3=μ2+1×lnμ2+1−μ3+1×lnμ3+1+2+μ1+μ3×ln2+μ1+μ3/2−2+μ1+μ2×ln2+μ1+μ2/2≤μ2+1×lnμ2+1−μ3+1×lnμ3+1+μ1+μ2+2×lnμ1+μ2+2/2−μ1+μ2+2×lnμ1+μ2+2/2≤0,f4u1,u2−f4u1,u3=u1×expu1−1−expu2−1+u2×expu2−1−expu1−1−u1×expu1−1−expu3−1−u3×expu3−1−expu1−1≤0.
Similarly,fiη1,η2≤fiη1,η3, fiν1,ν2≤fiν1,ν3i=1,2,3,4 can be proven. Thus, the proof of condition (10) is completely done.


## 4. The MCDM Approach with Weighted Cross-Entropy and TOPSIS of HPFLSs

In this section, the method based on Hamming distance and Lagrangian function is proposed to solve the MCDM problems under HPFLSs environment with unknown criteria weight, and the detailed decision-making steps under HPFLSs environment are summarized. For this sake, the hierarchical frame of the proposed method is presented as shown in Figure1.Figure 1
The hierarchical frame of the proposed cross-entropy-based HPFLSs MCGDM approach.


### 4.1. The Method of Unknown Criteria Weight Determination

In the practical decision-making environment, due to some emergency situation, the weight of the criterion cannot be provided in time. Therefore, the method based on the maximum-minimum distance is proposed to calculate the unknown criteria weight. In general, the criterion with the bigger distance value among the alternative should be assigned with bigger weight and vice versa [30, 31]. In this section, the Hamming distance is used to calculate the distance. For a specific criterion a4, the distance s5,0.5,0.3,0.1,s4,0.5,0,0 between one alternative and other alternatives can be obtained by the following calculation formula:(16)Dcq=1m−1∑t=1m∑e=1,e≠tmHstq−Hseq=1m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq.The larger the value ofc3 is, the more important the corresponding criterion is. Next, the following model is constructed to determine the weight value.(17)MmaxDcw=∑q=1nwqDcqs.t.∑q=1nwq2=1,wq≥0,q=1,2,…,n.Then, the above model (M) is solved by constructing the Lagrangian function.(18)Lw,ε=∑q=1nwqDcq+ε2∑q=1nwq2−1.In order to compute the partial derivatives ofs5,0.6,0,0,s4,0.2,0,0 and s5,0.8,0.1,0,s4,0.2,0.2,0, the following partial derivative equations are obtained.(19)∂wq,εwq=Dcq+εwq=0,∂wq,εε=∑q=1nwq2−1=0.By solving formula (19), the formula for determining the weight vector is obtained as follows:(20)wq=1/m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq∑q=1n1/m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq.


### 4.2. The Method with Weighted Cross-Entropy TOPSIS of HPFLSs

According to the above formula (11), weighted cross-entropy values from the alternative to positive ideal solution Hs+=smax,1,0,0,…,smax,1,0,0︷n and the negative ideal solution Hs−=smin,0,0,1,…,smin,0,0,1︷n can be obtained. The detailed process is shown as follows:(21)CE1∗Hs,Hs+=CE1Hs,Hs++CE1Hs+,Hs=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj×sin∑kj=1αjμkj×ikjαjαjαj−ik++φ×sin∑kj=1αjηkj×ikjαjαjαj×sin∑kj=1αjηkj×ikjαjαjαj+δ×sin∑kj=1αjνkj×ikjαjαjαj×sin∑kj=1αjνkj×ikjαjαjαj+ρ×sinik+×sinik2−∑kj=1αjμkj×ikjαjαjαj+0+0=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj−sinik+×sin∑kj=1αjμkj×ikjαjαjαj−ik++φ×sin∑kj=1αjηkj×ikjαjαjαj2+δ×sin∑kj=1αjνkj×ikjαjαjαj2,CE1∗Hs,Hs−=CE1Hs,Hs−+CE1Hs−,Hs=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj×sin∑kj=1αjμkj×ikjαjαjαj−ik−+φ×sin∑kj=1αjηkj×ikjαjαjαj×sin∑kj=1αjηkj×ikjαjαjαj+δ×sin∑kj=1αjνkj×ikjαjαjαj×sin∑kj=1αjνkj×ikjαjαjαj+ρ×sinik−×sinik2−∑kj=1αjμkj×ikjαjαjαj+0+0=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj−sinik+×sin∑kj=1αjμkj×ikjαjαjαj−ik−+φ×sin∑kj=1αjηkj×ikjαjαjαj2+δ×sin∑kj=1αjνkj×ikjαjαjαj2.In a similar way, we can also obtain that(22)CE2∗Hs,Hs+=CE2Hs,Hs++CE2Hs+,Hs=∑j=1nwj×ρ×tan∑kj=1αjμkj×ikjαjαjαj−tanik+×tan∑kj=1αjμkj×ikjαjαjαj−ik++φ×tan∑kj=1αjηkj×ikjαjαjαj2+δ×tan∑kj=1αjνkj×ikjαjαjαj2,CE2∗Hs,Hs−=CE2Hs,Hs−+CE2Hs−,Hs=∑j=1nwj×ρ×tan∑kj=1αjμkj×ikjαjαjαj−tanik+×tan∑kj=1αjμkj×ikjαjαjαj−ik++φ×tan∑kj=1αjηkj×ikjαjαjαj2+δ×tan∑kj=1αjνkj×ikjαjαjαj2,CE3∗Hs,Hs+=CE3Hs,Hs++CE3Hs+,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj+1×ln2×∑kj=1αjμkj×ikjαj/αjαj+12+∑kj=1αjμkj×ikjαj/αjαj+ik++ik++1ln2ik++12+ik++∑kj=1αjμkj×ikjαj/αjαjφ×∑kj=1αjηkj×ikjαj/αjαj+1×ln2×∑kj=1αjηkj×ikjαj/αjαj+12+∑kj=1αjηkj×ikjαj/αjαj+ln22+ηkj×ikjαj/αjαjδ×∑kj=1αjνkj×ikjαj/αjαj+1×ln2×∑kj=1αjνkj×ikjαj/αjαj+12+∑kj=1αjνkj×ikjαj/αjαj+ln22+∑kj=1αjνkj×ikjαj/αjαj,CE3∗Hs,Hs−=CE3Hs,Hs−+CE3Hs−,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj+1×ln2×∑kj=1αjμkj×ikjαj/αjαj+12+∑kj=1αjμkj×ikjαj/αjαj+ln22+∑kj=1αjμkj×ikjαj/αjαjφ×∑kj=1αjηkj×ikjαjαjαj+1×ln2×∑kj=1αjηkj×ikjαj/αjαj+12+∑kj=1αjηkj×ikjαj/αjαj+ln22+∑kj=1αjηkj×ikjαj/αjαjδ×∑kj=1αjνkj×ikjαjαjαj+1×ln2×∑kj=1αjνkj×ikjαj/αjαj+12+∑kj=1αjνkj×ikjαj/αj+αjik−+ik−+1ln2ik−+12+ik−+∑kj=1αjνkj×ikjαj/αjαj,CE4∗Hs,Hs+=CE4Hs,Hs++CE4Hs+,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj−ik+×exp∑kj=1αjμkj×ikjαjαjαj−1−expik+−1φ×∑kj=1αjηkj×ikjαjαjαj×exp∑kj=1αjηkj×ikjαjαjαj−1−exp−1δ×∑kj=1αjνkj×ikjαjαjαj×exp∑kj=1αjνkj×ikjαjαjαj−1−exp−1,CE4∗Hs,Hs−=CE4Hs,Hs−+CE4Hs−,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj−ik−×exp∑kj=1αjμkj×ikjαjαjαj−1−expik+−1φ×∑kj=1αjηkj×ikjαjαjαj×exp∑kj=1αjηkj×ikjαjαjαj−1−exp−1δ×∑kj=1αjνkj×ikjαjαjαj×exp∑kj=1αjνkj×ikjαjαjαj−1−exp−1.


### 4.3. The MCDM Method of HPFLS-Based Cross-Entropy and TOPSIS

In this section, an MCDM approach of HPFLS-based cross-entropy and TOPSIS is described in detail, and the detailed decision-making steps are as follows:LetA=a1,a2,…,am be the collection of m alternatives, E=e1,e2,…,ek be the collection of k decision-maker, C=c1,c2,…,cn be a set of n criteria, and w=w1,w2,…,wn be the weight collection of criteria. Next, according to the criteria, the evaluation score for alternatives Ai on criterion Cj, which is provided by decision makers (DMs), can be expressed as decision-making matrix M=β˜ijm×n with HPFLSs.The detailed decision-making steps are as follows.Step 1.
Normalize the evaluation value of the decision matrix
All criteria in the decision-making matrix must be distinguished as benefit-type or cost-type criteria. In order to normalize the criteria values, normally, the evaluation values of the benefit criteria do not need to be changed, and the evaluation values of the cost criteria need to be replaced with their complementary sets.
The following formula is used to normalize the decision-making matrix:(23)β˜⌢ij=β˜ij,Cj∈BS,β˜ijc,Cj∈CS,where BS is the set of benefit criteria, CS is the set of cost criteria, and β˜ijc is the complementary set of β˜ij. The normalized decision-making matrix is denoted by M⌢=β˜⌢ijm×n.Step 2.
Calculate the vector of criteria weight
Due to the unknown weightwj of criteria cjj=1,2,…,n, the value of the criteria weight needs to be firstly determined. In accordance with information theory, if the criterion has similar or the same value for all schemes, then the criterion is assigned with smaller weight because these criteria are less helpful to distinguish the advantage and disadvantage of alternatives. Then, according to formula (19), the weight value wj of each criterion Cj can be obtained.Step 3.
Calculate the score of each alternative
According to the formulas in Section4.2, the weighted cross-entropy values from each alternative to the positive ideal solution Hs+=smax,1,0,0,…,smax,1,0,0︷n and the negative ideal solution Hs−=smin,0,0,1,…,smin,0,0,1︷n under hesitant picture fuzzy linguistic environment are gained. Based on the value of weighted cross-entropy, the score value of each alternative can be obtained by the following formula:(24)Ri=CEj∗Hsi,Hs+CEj∗Hsi,Hs++CEj∗Hsi,Hs−,j=1,2,3,4.Step 4.
Rank the alternative with ordered scores
The obtained scoreR is ordered from small to large. The smaller the score is, the better the alternative is, and vice versa.


## 4.1. The Method of Unknown Criteria Weight Determination

In the practical decision-making environment, due to some emergency situation, the weight of the criterion cannot be provided in time. Therefore, the method based on the maximum-minimum distance is proposed to calculate the unknown criteria weight. In general, the criterion with the bigger distance value among the alternative should be assigned with bigger weight and vice versa [30, 31]. In this section, the Hamming distance is used to calculate the distance. For a specific criterion a4, the distance s5,0.5,0.3,0.1,s4,0.5,0,0 between one alternative and other alternatives can be obtained by the following calculation formula:(16)Dcq=1m−1∑t=1m∑e=1,e≠tmHstq−Hseq=1m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq.The larger the value ofc3 is, the more important the corresponding criterion is. Next, the following model is constructed to determine the weight value.(17)MmaxDcw=∑q=1nwqDcqs.t.∑q=1nwq2=1,wq≥0,q=1,2,…,n.Then, the above model (M) is solved by constructing the Lagrangian function.(18)Lw,ε=∑q=1nwqDcq+ε2∑q=1nwq2−1.In order to compute the partial derivatives ofs5,0.6,0,0,s4,0.2,0,0 and s5,0.8,0.1,0,s4,0.2,0.2,0, the following partial derivative equations are obtained.(19)∂wq,εwq=Dcq+εwq=0,∂wq,εε=∑q=1nwq2−1=0.By solving formula (19), the formula for determining the weight vector is obtained as follows:(20)wq=1/m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq∑q=1n1/m−1∑t=1m∑e=1,e≠tm∑ktq=1αtqitq−∑keq=1αeqieq+∑ktq=1αtqμitqktq−∑keq=1αeqμieqkeq+∑ktq=1αtqηitqktq−∑keq=1αeqηieqkeq+∑ktq=1αtqνitqktq−∑keq=1αeqνieqkeq.


## 4.2. The Method with Weighted Cross-Entropy TOPSIS of HPFLSs

According to the above formula (11), weighted cross-entropy values from the alternative to positive ideal solution Hs+=smax,1,0,0,…,smax,1,0,0︷n and the negative ideal solution Hs−=smin,0,0,1,…,smin,0,0,1︷n can be obtained. The detailed process is shown as follows:(21)CE1∗Hs,Hs+=CE1Hs,Hs++CE1Hs+,Hs=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj×sin∑kj=1αjμkj×ikjαjαjαj−ik++φ×sin∑kj=1αjηkj×ikjαjαjαj×sin∑kj=1αjηkj×ikjαjαjαj+δ×sin∑kj=1αjνkj×ikjαjαjαj×sin∑kj=1αjνkj×ikjαjαjαj+ρ×sinik+×sinik2−∑kj=1αjμkj×ikjαjαjαj+0+0=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj−sinik+×sin∑kj=1αjμkj×ikjαjαjαj−ik++φ×sin∑kj=1αjηkj×ikjαjαjαj2+δ×sin∑kj=1αjνkj×ikjαjαjαj2,CE1∗Hs,Hs−=CE1Hs,Hs−+CE1Hs−,Hs=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj×sin∑kj=1αjμkj×ikjαjαjαj−ik−+φ×sin∑kj=1αjηkj×ikjαjαjαj×sin∑kj=1αjηkj×ikjαjαjαj+δ×sin∑kj=1αjνkj×ikjαjαjαj×sin∑kj=1αjνkj×ikjαjαjαj+ρ×sinik−×sinik2−∑kj=1αjμkj×ikjαjαjαj+0+0=∑j=1nwj×ρ×sin∑kj=1αjμkj×ikjαjαjαj−sinik+×sin∑kj=1αjμkj×ikjαjαjαj−ik−+φ×sin∑kj=1αjηkj×ikjαjαjαj2+δ×sin∑kj=1αjνkj×ikjαjαjαj2.In a similar way, we can also obtain that(22)CE2∗Hs,Hs+=CE2Hs,Hs++CE2Hs+,Hs=∑j=1nwj×ρ×tan∑kj=1αjμkj×ikjαjαjαj−tanik+×tan∑kj=1αjμkj×ikjαjαjαj−ik++φ×tan∑kj=1αjηkj×ikjαjαjαj2+δ×tan∑kj=1αjνkj×ikjαjαjαj2,CE2∗Hs,Hs−=CE2Hs,Hs−+CE2Hs−,Hs=∑j=1nwj×ρ×tan∑kj=1αjμkj×ikjαjαjαj−tanik+×tan∑kj=1αjμkj×ikjαjαjαj−ik++φ×tan∑kj=1αjηkj×ikjαjαjαj2+δ×tan∑kj=1αjνkj×ikjαjαjαj2,CE3∗Hs,Hs+=CE3Hs,Hs++CE3Hs+,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj+1×ln2×∑kj=1αjμkj×ikjαj/αjαj+12+∑kj=1αjμkj×ikjαj/αjαj+ik++ik++1ln2ik++12+ik++∑kj=1αjμkj×ikjαj/αjαjφ×∑kj=1αjηkj×ikjαj/αjαj+1×ln2×∑kj=1αjηkj×ikjαj/αjαj+12+∑kj=1αjηkj×ikjαj/αjαj+ln22+ηkj×ikjαj/αjαjδ×∑kj=1αjνkj×ikjαj/αjαj+1×ln2×∑kj=1αjνkj×ikjαj/αjαj+12+∑kj=1αjνkj×ikjαj/αjαj+ln22+∑kj=1αjνkj×ikjαj/αjαj,CE3∗Hs,Hs−=CE3Hs,Hs−+CE3Hs−,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj+1×ln2×∑kj=1αjμkj×ikjαj/αjαj+12+∑kj=1αjμkj×ikjαj/αjαj+ln22+∑kj=1αjμkj×ikjαj/αjαjφ×∑kj=1αjηkj×ikjαjαjαj+1×ln2×∑kj=1αjηkj×ikjαj/αjαj+12+∑kj=1αjηkj×ikjαj/αjαj+ln22+∑kj=1αjηkj×ikjαj/αjαjδ×∑kj=1αjνkj×ikjαjαjαj+1×ln2×∑kj=1αjνkj×ikjαj/αjαj+12+∑kj=1αjνkj×ikjαj/αj+αjik−+ik−+1ln2ik−+12+ik−+∑kj=1αjνkj×ikjαj/αjαj,CE4∗Hs,Hs+=CE4Hs,Hs++CE4Hs+,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj−ik+×exp∑kj=1αjμkj×ikjαjαjαj−1−expik+−1φ×∑kj=1αjηkj×ikjαjαjαj×exp∑kj=1αjηkj×ikjαjαjαj−1−exp−1δ×∑kj=1αjνkj×ikjαjαjαj×exp∑kj=1αjνkj×ikjαjαjαj−1−exp−1,CE4∗Hs,Hs−=CE4Hs,Hs−+CE4Hs−,Hs=∑j=1nwj×ρ×∑kj=1αjμkj×ikjαjαjαj−ik−×exp∑kj=1αjμkj×ikjαjαjαj−1−expik+−1φ×∑kj=1αjηkj×ikjαjαjαj×exp∑kj=1αjηkj×ikjαjαjαj−1−exp−1δ×∑kj=1αjνkj×ikjαjαjαj×exp∑kj=1αjνkj×ikjαjαjαj−1−exp−1.


## 4.3. The MCDM Method of HPFLS-Based Cross-Entropy and TOPSIS

In this section, an MCDM approach of HPFLS-based cross-entropy and TOPSIS is described in detail, and the detailed decision-making steps are as follows:LetA=a1,a2,…,am be the collection of m alternatives, E=e1,e2,…,ek be the collection of k decision-maker, C=c1,c2,…,cn be a set of n criteria, and w=w1,w2,…,wn be the weight collection of criteria. Next, according to the criteria, the evaluation score for alternatives Ai on criterion Cj, which is provided by decision makers (DMs), can be expressed as decision-making matrix M=β˜ijm×n with HPFLSs.The detailed decision-making steps are as follows.Step 1.
Normalize the evaluation value of the decision matrix
All criteria in the decision-making matrix must be distinguished as benefit-type or cost-type criteria. In order to normalize the criteria values, normally, the evaluation values of the benefit criteria do not need to be changed, and the evaluation values of the cost criteria need to be replaced with their complementary sets.
The following formula is used to normalize the decision-making matrix:(23)β˜⌢ij=β˜ij,Cj∈BS,β˜ijc,Cj∈CS,where BS is the set of benefit criteria, CS is the set of cost criteria, and β˜ijc is the complementary set of β˜ij. The normalized decision-making matrix is denoted by M⌢=β˜⌢ijm×n.Step 2.
Calculate the vector of criteria weight
Due to the unknown weightwj of criteria cjj=1,2,…,n, the value of the criteria weight needs to be firstly determined. In accordance with information theory, if the criterion has similar or the same value for all schemes, then the criterion is assigned with smaller weight because these criteria are less helpful to distinguish the advantage and disadvantage of alternatives. Then, according to formula (19), the weight value wj of each criterion Cj can be obtained.Step 3.
Calculate the score of each alternative
According to the formulas in Section4.2, the weighted cross-entropy values from each alternative to the positive ideal solution Hs+=smax,1,0,0,…,smax,1,0,0︷n and the negative ideal solution Hs−=smin,0,0,1,…,smin,0,0,1︷n under hesitant picture fuzzy linguistic environment are gained. Based on the value of weighted cross-entropy, the score value of each alternative can be obtained by the following formula:(24)Ri=CEj∗Hsi,Hs+CEj∗Hsi,Hs++CEj∗Hsi,Hs−,j=1,2,3,4.Step 4.
Rank the alternative with ordered scores
The obtained scoreR is ordered from small to large. The smaller the score is, the better the alternative is, and vice versa.


## 5. Practical Application

In this section, the middle-level managers’ selection of Dongfeng Commercial Vehicle Co., Ltd., with the approach of weighted cross-entropy of HPFLSs is introduced, and the further analysis is presented.


### 5.1. Illustration of Practical Application

Practical application of the recruitment of middle-level management personnel in Dongfeng Commercial Vehicle Co., Ltd., is adopted as the illustration of the proposed method. Dongfeng Commercial Vehicle Co., Ltd., is the largest commercial vehicle manufacturer in China, which is mainly engaged in the production of heavy trucks, engines, gearboxes, and so forth. Due to the demands of business development, the company needed to recruit professional personnel as the company’s middle-level managers with automotive engineering background and high-level automotive marketing capabilities. After several rounds of interviews, four candidates were selected for the final interview, and the evaluation of each candidate was scored from four criteria, which are negotiation and communication skillsc1, business development capability c2, sales and marketing capability c3, and technical background c4. Weight for each criterion was not given in advance. Ten higher-level managers from the company are chosen as the interviewers; they are from HR, technical, sales, and financial department; all interviewers discussed and gave the evaluation value, which are collected as shown in Table 1.Step 1: normalize the evaluation value of decision-making matrixAll criteria are identified as benefit type, so the decision-making matrix does not need to be normalizedStep 2: calculate the weight vector for criteriaSince the weights of the criteria are completely unknown, formula (20) is used to obtain the weight vector as w=0.2875, 0.4938, 0.1250, 0.0938Step 3: calculate the score of each alternativeAccording to the formula in Section 1.3.2, the weighted cross-entropy value of HPFLSs from each alternative to PISHs+=smax,1,0,0,…,smax,1,0,0︷n and NIS Hs−=smin,0,0,1,…,smin,0,0,1︷n can be obtained. For the convenience of demonstration, the parameter is assigned with θ=4, n=1, thus, the results of each alternative with four cross-entropy formulas are obtained as follows:(25)R1=CE1∗Hs1,Hs+CE1∗Hs1,Hs++CE1∗Hs1,Hs−=0.17120.1712+0.8645=0.1653,R1=CE2∗Hs1,Hs+CE2∗Hs1,Hs++CE2∗Hs1,Hs−=0.46490.4649+2.1757=0.1761,R1=CE3∗Hs1,Hs+CE3∗Hs1,Hs++CE3∗Hs1,Hs−=0.03650.0365+0.1941=0.1583,R1=CE4∗Hs1,Hs+CE4∗Hs1,Hs++CE4∗Hs1,Hs−=0.16780.1678+0.6810=0.1977.In similar way, the value ofR2, R3, and R4 can be obtained. The detailed results are shown in Table 2.Step 4: rank the alternative with ordered scoresTable 1
Evaluation value decision-making matrixM.
c1c2c3c4a1s5,0.6,0.1,0.3,s4,0.4,0,0s5,0.5,0.3,0.2,s4,0.7,0,0s5,0.8,0,0,s4,0.2,0,0s5,0.8,0.1,0,s4,0.2,0,0a2s5,0.5,0.2,0.1,s3,0.6,0,0s4,0.6,0.1,0.1,s3,0.4,0,0s5,0.5,0.2,0.1,s4,0.5,0,0.3s5,0.5,0.2,0.1,s4,0.5,0,0.1a3s5,0.5,0.3,0.1,s4,0.5,0,0s5,0.7,0.2,0,s4,0.3,0,0s5,0.7,0.1,0,s4,0.3,0,0s5,0.7,0,0.1,s4,0.3,0,0a4s5,0.5,0.3,0.1,s4,0.5,0,0s5,0.6,0.2,0.1,s4,0.4,0,0.1s5,0.5,0.2,0.1,s4,0.5,0,0s5,0.6,0.1,0.1,s4,0.5,0.1,0Table 2
Score of each alternative.
R1R2R3R4CE1∗0.16530.19150.16140.1884CE2∗0.17610.20770.16780.2014CE3∗0.15830.17970.14870.1775CE4∗0.19770.23660.20200.2297According to the obtained scoresRii=1,2,3,4, the scores are ordered from high to low. The lower the score is, the better the alternative is. It is obviously to see that if cross-entropy CE4∗ is used, the alternative order is a1≻a3≻a4≻a2; if the cross-entropy CE1∗, CE2∗, or CE3∗ is used, the alternative ranking is a3≻a1≻a4≻a2.


### 5.2. Further Analysis

In order to further analyze the influence of each alternative valueRii=1,2,3,4 and its ranking with four different cross-entropy formulas under different parameter values environment, the alternative ranking of the overall trend with different parameter combinations are conducted. The detailed alternative score is shown as follows:According to the results of each subgraph in Figures2–5, the over trend of alternative scores is similar for different cross-entropies with parameter values 1≤θ≤6 and −2≤n≤10. If parameter θ approaches to 1 and n approaches to −2, the score of each scheme is minimum; when parameter θ approaches to 1 and n approaches to 10, the score of each alternative is maximum. The difference is that if the parameters θ and n are assigned with the same value, the increasing or decreasing value of each alternative score is different, which is mainly caused by the usage of different functions for each cross-entropy.Figure 2
Score of alternativea1 with parameter values 1≤θ≤6, −2≤n≤10.Figure 3
Score of alternativea2 based on parameter values 1≤θ≤6, −2≤n≤10.Figure 4
Score of alternativea3 based on parameter values 1≤θ≤6, −2≤n≤10.Figure 5
Score of alternativea4 based on parameter values 1≤θ≤6, −2≤n≤10.In order to easily understand the influence of parametersθ and n on the ranking score of alternative, the fixed values are assigned for parameter θ, that is, θ=1,2,5,10, and the parameter n value range is assigned with −2≤n≤10. The ranking score trends of each alternative with four different cross-entropy formulas are shown in Figures 6–9 as follows.Figure 6
Ranking score of each alternative based on parametersθ=1, −2≤n≤10.Figure 7
Ranking score of each alternative based on parametersθ=2, −2≤n≤10.Figure 8
Ranking score of each alternative based on parametersθ=5, −2≤n≤10.Figure 9
Ranking score of each alternative based on parametersθ=10, −2≤n≤10.As shown in Figures6–9, the smaller the value of parameter θ is, the value change of the parameter n has a significant influence on the score and ranking results of each alternative. However, as the value of parameter θ is increasing, the value changes of parameter n are less and less able to influence the scores and ranking results of each alternative.Next, the ranking score results of each alternative influenced by the parameter value range1≤θ≤8 are analyzed under the condition in which the value of parameter n is fixed, that is, n=−2,0,3,8,18. The detailed results are shown in Figures 10–14 as follows.Figure 10
Ranking score of each alternative based on parametersn=−2, 1≤θ≤8.Figure 11
Ranking score of each alternative based on parametersn=0, 1≤θ≤8.Figure 12
Ranking score of each alternative based on parametersn=3, 1≤θ≤8.Figure 13
Ranking score of each alternative based on parametersn=8, 1≤θ≤8.Figure 14
Ranking score of each alternative based on parametersn=18, 1≤θ≤8.From the ranking and score results presented in Figures10–14, it is obvious to see that if n<0, with the increasing of parameter value θ, the score of each alternative is increasing and the increasing amplitude is different. Meanwhile, the ranking order of each alternative also changes. If n=0, the ranking value of each alternative will not change with the increasing value of parameter θ, and the ranking order of each alternative is relatively fixed. If n>0, the score of each alternative is decreasing, and the final ranking results change slightly as the increasing value of parameter θ.Ifn<0 and θ<5, the best alternative is a4, the worst alternative is a2; if θ>5, the best alternative is still a1, but the worst alternative becomes a3 or a1.Ifn<0, the best alternative is a4, and the worst alternative is a3 or a1.Ifak, with the increasing of the value of parameter ϑai,ak=0−0.0691−0.0457−0.0537−0.01340−0.0061−0.0074−0.0220−0.04480−0.0210−0.0069−0.0400−0.00850, the best alternative is a2, but the worst alternative is a3 or a1.From the above analysis of Figures9–13, the following conclusions can be obtained:(i)
If the value of parameterθ is fixed, while the value of parameter n increases, the score of the alternatives become bigger, which indicates that the criteria with highest evaluation value can bring biggest influence for the ranking order of alternatives, and vice versa.(ii)
If parametern<0 and its values are fixed, with the continuous increasing of parameter θ, the score of the alternative grows bigger slightly, which indicates that the integration value of each alternative is more influenced by the criterion with high evaluation value, and vice versa.(iii)
If parametern>0 and its values are fixed, with the continuous increasing of parameter θ, the score of the alternative become smaller, which indicates that the integration value of each alternative is more influenced by the criteria with the low evaluation value.From the above analysis results, considering practical applications, decision-makers pay more attention to the final ranking results affected by the criteria with high evaluation values. The parameter valuen<0 is selected, and the value of parameter θ should be as large as possible. If decision-makers pay more attention to the criteria with low evaluation values which affect the final ranking results, the parameter value n>0 is selected, and the value of parameter θ should also be as large as possible.


### 5.3. Comparative Analysis

In order to verify the effectiveness of the constructed approach, the comparative analysis is carried out with the representative method of PFS-based weighted cross-entropy, which is proposed by Wei [28]. For convenient comparison, the HPFLSs have been transformed to PFS. In accordance with the further analysis presented in Subsection 5.2, four pairs of representative values for parameter θ and n are selected. Then, the comparative results for the different methods are shown in Table 3.Table 3
Comparative results obtained by different methods.
MethodsCross-entropyParameterRank resultsWei [28]WeightedNonea4≻a2≻a3≻a1Our methodsWeightedθ=5,n=−2a4≻a3≻a1≻a2θ=8,n=−2a4≻a2≻a3≻a1θ=1,n=2a2≻a4≻a1≻a3θ=10,n=2a4≻a3≻a1≻a2Table3 shows that the ranking result obtained by Wei [28] has the difference with obtained by our methods. The reasons causing the difference mainly are the parameters, that is, there is no parameter in Wei’s study [28]. However, there are two parameters in our proposed methods. For our methods, if different pairs of values for parameter θ and n are chosen, the final ranking of alternative is slightly different. In some cases, alternative a4 is superior to alternative a2; in some cases, alternative a2 is superior to alternative a4. The reasons causing different ranking results are discussed in Section 5.2.In accordance with the comparative analysis and further discussion in Section 5.2, it can be concluded that our proposed method owns the advantages as follows:(1)
The proposed method elaborates the advantages of hesitant fuzzy set and picture fuzzy linguistic set and is better to describe the hesitant degree and risk attitude information of DM(2)
In accordance with the information theory, the criteria weights in the proposed method are obtained by the maximum-minimum distance algorithm; thus, the results are more objective and precise(3)
Under the practical decision-making environments, some DMs might pay higher attention to membership, such as membership degree of “vote for” or “vote against,” and the constructed approach is more flexible and reasonable to provide the different ranking results by adjusting the value of the parameterθ and n; thus, the more acceptable results are provided for DMS.


## 5.1. Illustration of Practical Application

Practical application of the recruitment of middle-level management personnel in Dongfeng Commercial Vehicle Co., Ltd., is adopted as the illustration of the proposed method. Dongfeng Commercial Vehicle Co., Ltd., is the largest commercial vehicle manufacturer in China, which is mainly engaged in the production of heavy trucks, engines, gearboxes, and so forth. Due to the demands of business development, the company needed to recruit professional personnel as the company’s middle-level managers with automotive engineering background and high-level automotive marketing capabilities. After several rounds of interviews, four candidates were selected for the final interview, and the evaluation of each candidate was scored from four criteria, which are negotiation and communication skillsc1, business development capability c2, sales and marketing capability c3, and technical background c4. Weight for each criterion was not given in advance. Ten higher-level managers from the company are chosen as the interviewers; they are from HR, technical, sales, and financial department; all interviewers discussed and gave the evaluation value, which are collected as shown in Table 1.Step 1: normalize the evaluation value of decision-making matrixAll criteria are identified as benefit type, so the decision-making matrix does not need to be normalizedStep 2: calculate the weight vector for criteriaSince the weights of the criteria are completely unknown, formula (20) is used to obtain the weight vector as w=0.2875, 0.4938, 0.1250, 0.0938Step 3: calculate the score of each alternativeAccording to the formula in Section 1.3.2, the weighted cross-entropy value of HPFLSs from each alternative to PISHs+=smax,1,0,0,…,smax,1,0,0︷n and NIS Hs−=smin,0,0,1,…,smin,0,0,1︷n can be obtained. For the convenience of demonstration, the parameter is assigned with θ=4, n=1, thus, the results of each alternative with four cross-entropy formulas are obtained as follows:(25)R1=CE1∗Hs1,Hs+CE1∗Hs1,Hs++CE1∗Hs1,Hs−=0.17120.1712+0.8645=0.1653,R1=CE2∗Hs1,Hs+CE2∗Hs1,Hs++CE2∗Hs1,Hs−=0.46490.4649+2.1757=0.1761,R1=CE3∗Hs1,Hs+CE3∗Hs1,Hs++CE3∗Hs1,Hs−=0.03650.0365+0.1941=0.1583,R1=CE4∗Hs1,Hs+CE4∗Hs1,Hs++CE4∗Hs1,Hs−=0.16780.1678+0.6810=0.1977.In similar way, the value ofR2, R3, and R4 can be obtained. The detailed results are shown in Table 2.Step 4: rank the alternative with ordered scoresTable 1
Evaluation value decision-making matrixM.
c1c2c3c4a1s5,0.6,0.1,0.3,s4,0.4,0,0s5,0.5,0.3,0.2,s4,0.7,0,0s5,0.8,0,0,s4,0.2,0,0s5,0.8,0.1,0,s4,0.2,0,0a2s5,0.5,0.2,0.1,s3,0.6,0,0s4,0.6,0.1,0.1,s3,0.4,0,0s5,0.5,0.2,0.1,s4,0.5,0,0.3s5,0.5,0.2,0.1,s4,0.5,0,0.1a3s5,0.5,0.3,0.1,s4,0.5,0,0s5,0.7,0.2,0,s4,0.3,0,0s5,0.7,0.1,0,s4,0.3,0,0s5,0.7,0,0.1,s4,0.3,0,0a4s5,0.5,0.3,0.1,s4,0.5,0,0s5,0.6,0.2,0.1,s4,0.4,0,0.1s5,0.5,0.2,0.1,s4,0.5,0,0s5,0.6,0.1,0.1,s4,0.5,0.1,0Table 2
Score of each alternative.
R1R2R3R4CE1∗0.16530.19150.16140.1884CE2∗0.17610.20770.16780.2014CE3∗0.15830.17970.14870.1775CE4∗0.19770.23660.20200.2297According to the obtained scoresRii=1,2,3,4, the scores are ordered from high to low. The lower the score is, the better the alternative is. It is obviously to see that if cross-entropy CE4∗ is used, the alternative order is a1≻a3≻a4≻a2; if the cross-entropy CE1∗, CE2∗, or CE3∗ is used, the alternative ranking is a3≻a1≻a4≻a2.


## 5.2. Further Analysis

In order to further analyze the influence of each alternative valueRii=1,2,3,4 and its ranking with four different cross-entropy formulas under different parameter values environment, the alternative ranking of the overall trend with different parameter combinations are conducted. The detailed alternative score is shown as follows:According to the results of each subgraph in Figures2–5, the over trend of alternative scores is similar for different cross-entropies with parameter values 1≤θ≤6 and −2≤n≤10. If parameter θ approaches to 1 and n approaches to −2, the score of each scheme is minimum; when parameter θ approaches to 1 and n approaches to 10, the score of each alternative is maximum. The difference is that if the parameters θ and n are assigned with the same value, the increasing or decreasing value of each alternative score is different, which is mainly caused by the usage of different functions for each cross-entropy.Figure 2
Score of alternativea1 with parameter values 1≤θ≤6, −2≤n≤10.Figure 3
Score of alternativea2 based on parameter values 1≤θ≤6, −2≤n≤10.Figure 4
Score of alternativea3 based on parameter values 1≤θ≤6, −2≤n≤10.Figure 5
Score of alternativea4 based on parameter values 1≤θ≤6, −2≤n≤10.In order to easily understand the influence of parametersθ and n on the ranking score of alternative, the fixed values are assigned for parameter θ, that is, θ=1,2,5,10, and the parameter n value range is assigned with −2≤n≤10. The ranking score trends of each alternative with four different cross-entropy formulas are shown in Figures 6–9 as follows.Figure 6
Ranking score of each alternative based on parametersθ=1, −2≤n≤10.Figure 7
Ranking score of each alternative based on parametersθ=2, −2≤n≤10.Figure 8
Ranking score of each alternative based on parametersθ=5, −2≤n≤10.Figure 9
Ranking score of each alternative based on parametersθ=10, −2≤n≤10.As shown in Figures6–9, the smaller the value of parameter θ is, the value change of the parameter n has a significant influence on the score and ranking results of each alternative. However, as the value of parameter θ is increasing, the value changes of parameter n are less and less able to influence the scores and ranking results of each alternative.Next, the ranking score results of each alternative influenced by the parameter value range1≤θ≤8 are analyzed under the condition in which the value of parameter n is fixed, that is, n=−2,0,3,8,18. The detailed results are shown in Figures 10–14 as follows.Figure 10
Ranking score of each alternative based on parametersn=−2, 1≤θ≤8.Figure 11
Ranking score of each alternative based on parametersn=0, 1≤θ≤8.Figure 12
Ranking score of each alternative based on parametersn=3, 1≤θ≤8.Figure 13
Ranking score of each alternative based on parametersn=8, 1≤θ≤8.Figure 14
Ranking score of each alternative based on parametersn=18, 1≤θ≤8.From the ranking and score results presented in Figures10–14, it is obvious to see that if n<0, with the increasing of parameter value θ, the score of each alternative is increasing and the increasing amplitude is different. Meanwhile, the ranking order of each alternative also changes. If n=0, the ranking value of each alternative will not change with the increasing value of parameter θ, and the ranking order of each alternative is relatively fixed. If n>0, the score of each alternative is decreasing, and the final ranking results change slightly as the increasing value of parameter θ.Ifn<0 and θ<5, the best alternative is a4, the worst alternative is a2; if θ>5, the best alternative is still a1, but the worst alternative becomes a3 or a1.Ifn<0, the best alternative is a4, and the worst alternative is a3 or a1.Ifak, with the increasing of the value of parameter ϑai,ak=0−0.0691−0.0457−0.0537−0.01340−0.0061−0.0074−0.0220−0.04480−0.0210−0.0069−0.0400−0.00850, the best alternative is a2, but the worst alternative is a3 or a1.From the above analysis of Figures9–13, the following conclusions can be obtained:(i)
If the value of parameterθ is fixed, while the value of parameter n increases, the score of the alternatives become bigger, which indicates that the criteria with highest evaluation value can bring biggest influence for the ranking order of alternatives, and vice versa.(ii)
If parametern<0 and its values are fixed, with the continuous increasing of parameter θ, the score of the alternative grows bigger slightly, which indicates that the integration value of each alternative is more influenced by the criterion with high evaluation value, and vice versa.(iii)
If parametern>0 and its values are fixed, with the continuous increasing of parameter θ, the score of the alternative become smaller, which indicates that the integration value of each alternative is more influenced by the criteria with the low evaluation value.From the above analysis results, considering practical applications, decision-makers pay more attention to the final ranking results affected by the criteria with high evaluation values. The parameter valuen<0 is selected, and the value of parameter θ should be as large as possible. If decision-makers pay more attention to the criteria with low evaluation values which affect the final ranking results, the parameter value n>0 is selected, and the value of parameter θ should also be as large as possible.


## 5.3. Comparative Analysis

In order to verify the effectiveness of the constructed approach, the comparative analysis is carried out with the representative method of PFS-based weighted cross-entropy, which is proposed by Wei [28]. For convenient comparison, the HPFLSs have been transformed to PFS. In accordance with the further analysis presented in Subsection 5.2, four pairs of representative values for parameter θ and n are selected. Then, the comparative results for the different methods are shown in Table 3.Table 3
Comparative results obtained by different methods.
MethodsCross-entropyParameterRank resultsWei [28]WeightedNonea4≻a2≻a3≻a1Our methodsWeightedθ=5,n=−2a4≻a3≻a1≻a2θ=8,n=−2a4≻a2≻a3≻a1θ=1,n=2a2≻a4≻a1≻a3θ=10,n=2a4≻a3≻a1≻a2Table3 shows that the ranking result obtained by Wei [28] has the difference with obtained by our methods. The reasons causing the difference mainly are the parameters, that is, there is no parameter in Wei’s study [28]. However, there are two parameters in our proposed methods. For our methods, if different pairs of values for parameter θ and n are chosen, the final ranking of alternative is slightly different. In some cases, alternative a4 is superior to alternative a2; in some cases, alternative a2 is superior to alternative a4. The reasons causing different ranking results are discussed in Section 5.2.In accordance with the comparative analysis and further discussion in Section 5.2, it can be concluded that our proposed method owns the advantages as follows:(1)
The proposed method elaborates the advantages of hesitant fuzzy set and picture fuzzy linguistic set and is better to describe the hesitant degree and risk attitude information of DM(2)
In accordance with the information theory, the criteria weights in the proposed method are obtained by the maximum-minimum distance algorithm; thus, the results are more objective and precise(3)
Under the practical decision-making environments, some DMs might pay higher attention to membership, such as membership degree of “vote for” or “vote against,” and the constructed approach is more flexible and reasonable to provide the different ranking results by adjusting the value of the parameterθ and n; thus, the more acceptable results are provided for DMS.


## 6. Conclusion and Further Direction

Considering the actual personnel selection decision-making problem, different DMs could have different attitudes (e.g., attitude for support, neutral, oppose, and refusal) for the evaluation value of criteria or alternative and might pay high attention to the influence of job candidate ranking by the criteria with high or low evaluation value; meanwhile, the criteria weight objectively obtained by criteria evaluation value is more reasonable than subjectively given by experts; thus, an approach to elaborate the advantages of both cross-entropy and HPFLSs is proposed.The main contribution of the paper is defining the cross-entropy of HPFLN and listing its properties needed to be satisfied. Based on the definition of cross-entropy of HPFLN, several formulas of cross-entropy of HPFLN are constructed, and the relative properties are proven. In accordance with the proposed cross-entropy formula, an MCDM approach based on cross-entropy and TOPSIS under the HPFLSs environment are constructed. Finally, the real personnel selection decision-making problem from Dongfeng Commercial Vehicle Co., Ltd., is used to verify the proposed method and demonstrate effectiveness.The limitation of our proposed method is in most real-life MCGDM environment; DMs more prefer to provide the evaluation value with simple crisp number or linguistic term; thus, how to transfer the simple crisp number and linguistic term to HPFLSs is necessary to study in the future. Additionally, according to the requirements of real-life decision-making, the proposed method will be extended to other applications [32–42], or constructing the novel MCDM method with entropy [43–45], evidential reasoning [46], cloud model [47, 48], interactive operators [49, 50], power operators [51], TODIM [52], and so forth.



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*Source: 7104045-2021-11-20.xml*