Teaching Performance Evaluation Based on the Proportional Hesitant Fuzzy Linguistic Prioritized Choquet Aggregation Operator

Lei Wang; Lili Rong; Fei Teng; Peide Liu

doi:10.2991/ijcis.d.210112.001

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Volume 14, Issue 1, 2021, Pages 635 - 650

Teaching Performance Evaluation Based on the Proportional Hesitant Fuzzy Linguistic Prioritized Choquet Aggregation Operator

Authors

Lei Wang¹, Lili Rong²^{, 3}^{, *}^,, Fei Teng², Peide Liu²^,

¹School of International Education, Shandong University of Finance and Economics, Jinan Shandong, 250014, China

²School of Management Science and Engineering, Shandong University of Finance and Economics, Jinan Shandong, 250014, China

³School of Science and Technology, Shandong TV University, Jinan Shandong, 250014, China

^*Corresponding author. Email: ronglili@126.com

Corresponding Author

Lili Rong

Received 11 August 2020, Accepted 29 December 2020, Available Online 19 January 2021.

DOI: 10.2991/ijcis.d.210112.001 How to use a DOI?
Keywords: Choquet integral; Proportional hesitant fuzzy linguistic term set; Multi-attribute group decision-making; Prioritized aggregation operators; Teaching performance evaluation
Abstract: The quality of teaching can be improved by teaching performance evaluation from multiple experts, which is a multiple attribute group decision-making (MAGDM) problem. In this paper, a group decision-making method under proportional hesitant fuzzy linguistic environment is proposed to evaluate teaching performance. Firstly, proportional hesitant fuzzy linguistic term set (PHFLTS) is applied to express the decision makers' (DMs) preferences for teaching performance index. Secondly, the PHFLP_rCA operator is developed and its properties are discussed. Then based on the PHFLP_rCA operator, a MAGDM method is formulated. Thirdly, the method is applied in teaching performance evaluation of Chinese-foreign cooperative education project. Finally, this method is proved more scientific, objective and accurate by compared with other two methods.
Copyright: © 2021 The Authors. Published by Atlantis Press B.V.
Open Access: This is an open access article distributed under the CC BY-NC 4.0 license (http://creativecommons.org/licenses/by-nc/4.0/).

1. INTRODUCTION

Chinese-foreign cooperative education project is the main form of transnational education in China. With the continuous expansion of its scale and the gradual upgrade of project-running, its sustainable development has become an inevitable requirement, which depends on the improvement of teaching quality. In the process of quality improving, teachers play an important role, such as curriculum designers and implementers. Teachers' qualifications, teaching methods, and teaching ability have become the factors that affect the teaching quality. In order to improve the teaching quality, it is necessary to conduct the teaching performance evaluation of Chinese-foreign cooperative education project.

The teaching performance evaluation of Chinese-foreign cooperative education project can be performed by a group of decision makers (DMs) based on multiple attributes, which may be inaccuracy, ambiguity, and uncertainty. Therefore, the teaching performance evaluation of Chinese-foreign cooperative education project can be considered as a fuzzy multiple attribute group decision-making (MAGDM) problem. It plays a vital role in determining the quality of teaching performance evaluation to apply scientific methods comprehensively and effectively. In the aspect of teaching performance evaluation, some researches on this topic have been done by using MAGDM methods [1–6]. However, there are two drawbacks in the current researches. Firstly, most of the relevant studies are hard to deal with group information. Secondly, these studies assumed that the indicators were independent of each other, without considering the priority or correlation between the indicators. However, in the process of practical evaluation, there will be mutual relationship between the indicators.

In order to overcome these two drawbacks of current researches, a better method should be proposed to solve them. In the first place, the appropriate linguistic expression is chosen for teaching performance evaluation. On the one hand, appropriate linguistic expression should enable DMs to express their preferences as clearly as possible, and reduce the subjectivity and uncertainty in the decision-making process; on the other hand, the rational expression of linguistic information is the premise and basis for solving MAGDM problems effectively [7–11]. Based on the above two aspects, this paper adopts the proportional hesitant fuzzy linguistic term sets (PHFLTSs) [12–15]. PHFLTSs are developed from traditional linguistic information, which are used to reasonably describe the subjective preference information given by DMs. Experts put forward several linguistic information representation models based on different situations, such as linguistic variables [16–18], hesitant fuzzy linguistic variables [19,20], extended hesitant fuzzy linguistic variables [21,22], linguistic distribution assessment variables [23,24], etc. By reducing the implicit constraints of the above linguistic information representation models, a more general linguistic information representation model, i.e., proportional hesitant fuzzy linguistic elements (PHFLEs), can be proposed. PHFLEs are general forms of the above linguistic variables, which are more conducive for DMs freely expressing their subjective preferences, and more suitable for teaching performance evaluation of Chinese-foreign cooperative education project.

In the second place, after choosing the appropriate linguistic expression, another key problem is to select the information aggregation operator, that is, to find the appropriate tool for effective integration of teaching performance evaluation values. Choosing the suitable information aggregation tool is a crucial step. At present, information aggregation operators have drawn extensive attention and achieved fruitful results, especially several kinds of aggregation operators, such as Choquet integration operator [25], and prioritized integration operator [26,27]. However, the research on information aggregation operators based on PHFLEs is rare, and fails to be applied directly in teaching performance evaluation with priority and correlation.

Since Yager [26] proposed the prioritized aggregation (PrA) operator, it has been extensively improved [28–30], but the existing PrA operators cannot deal with the PHFLEs. Therefore, this paper enriches the PrA operators and proposes the proportional hesitant linguistic priority weighted average (PHFLP_rWA_m) operator.

Besides prioritized relation among elements, there are interdependence or interrelated features among elements, thus it is unreasonable to aggregate elements by additive measures. Sugeno [31] put forward the concept of non-additive measure (fuzzy measure), which just has monotonicity but not additive property. Choquet integral (CI) based on fuzzy measure can capture the interaction between different elements, but it hasn't been applied into the proportional hesitant linguistic information. Therefore, it is important to propose an innovative CI operator to integrate the PHFLEs and deal with the MAGDM problems in the proportional hesitant linguistic environment.

Three innovative points in this paper are:

The first one is to enable DMs to express their preferences more freely and accurately. The proportional hesitant linguistic information can show the advantage of a group of DMs who have their own preferences when they make decisions.
The second one is to make the PHFLEs be applied in the situation with priority and correlation. The phenomenon of attributes with priority and correlation in MAGDM problems is common in real life, but it has never been extended to the PHFLEs.
The third one is to provide a new solution to the teaching performance evaluation of Chinese-foreign cooperative education project.

The remaining sections of this paper are outlined as follows: Section 2 introduces PHFLEs and its related concepts; Section 3 proposes proportional hesitant fuzzy linguistic Choquet aggregation PHFLCAμ operator, PHFLP_rWA_m operator, and PHFLP_rCA operator; Section 4 establishes a model based on PHFLP_rCA operator method; Section 5 demonstrates a practical example about teaching performance evaluation of Chinese-foreign cooperative education project, then shows the advantages of the proposed method by comparing with other two methods. Finally, Section 6 comes to the overall conclusion.

2. PRELIMINARIES

This section introduces the concept of PHFLEs, which can be transformed into different linguistic information representation models according to the decrease of proportional constraints, and some related theories such as expectation function, deviation function, and their corresponding ranking methods are applied. In addition, the CI and the prioritized integration operators are reviewed.

2.1. PHFLEs and its Related Concepts

Definition 1.

[32] Let L=li|i=1,2,…,t be a fully ordered finite discrete set, which li represents a linguistic term. If L meets the following conditions, L is called a linguistic term set (LTS): (1) The set L is ordered: if i≥κ, then li≥lκ; (2) there is a negative operator: Negli=t−i.

For example, when t=5, a linguistic glossary is presented as follows:

L=l1:very low,l2:low, l3:moderate l4:high,l5:very high.

Definition 2.

[33] Suppose X is a given domain ∀xj∈X, and L=li|i=1,2,…,t be a LTS, then HL=〈xj,hlxj〉|xj∈X is called hesitant fuzzy LTS (HFLTS), hlxj=lφτxj|lφτxj∈L,τ=1,2,…,# is a set of elements of L, # is the number of linguistic terms in hlxj. For convenience, hlxj is called hesitant fuzzy linguistic element (HFLE) and HL is a set of HFLEs.

Definition 3.

[34] Let X be a given domain and ∀xj∈X, L be a given LTS and ∀li∈L, then HLp=〈xj,hLjp〉|xj∈X of X is called probabilistic LTS (PLTS), where hLjp=ljτpτ|ljτ∈L,pτ≥0,τ=1,2,…,#,∑τ=1#pτ≤1, ljτpτ is the τth linguistic term ljτ and its probability distribution pτ, # is the number of different linguistic terms in hLjp. Linguistic terms ljττ=1,2,…,# are arranged in ascending order. For convenience, hLjp is called the probabilistic linguistic element (PLE).

Definition 4.

[24] Let X be a given domain and ∀xj∈X, L be a given LTS and ∀li≥L, then H¨Lp=〈xj,h¨Ljp〉|xj∈X of X is called a distributed LTS, where h¨Ljp=lipij|li∈L,pij≥0,i=1,2,…,t,∑i=1tpij=1 is called a distributed linguistic element (DLE).

Definition 5.

[12] Let X be a given domain and ∀xj∈X, L is a given LTS and ∀li∈L, then H˜˙PHp˙=〈xj,h˜˙PHjp˙〉|xj∈X of X is called PHFLTS, where h˜˙PHjp˙=li,p˙ij|li∈L,0≤p˙ij≤1,i=1,2,…,t, ∑i=1tp˙ij=1, li,p˙ij is LTS li and p˙ij denotes the degree of possibility that the alternative carries an assessment value li provided by a group of DM. For convenience, h˜˙PHjp˙ is called PHFLE.

Definition 6.

Let h˜˙PHjp˙=li,p˙ij|li∈L,i=1,2,…,t be a PHFLE, then the expectation function of h˜˙PHjp˙ is

Eh˜˙PHjp˙=∑i=1ti⋅p˙ij(1)

Definition 7.

Let h˜˙PHjp˙=li,p˙ij|li∈L,i=1,2,…,t be a PHFLE, Eh˜˙PHjp˙=∑i=1ti⋅p˙ij, then the deviation function of h˜˙PHjp˙ is

σh˜˙PHjp˙=∑i=1tp˙iji−Eh˜˙PHjp˙21/2(2)

Definition 8.

Let h˜˙PH1p˙ and h˜˙PH2p˙ be any two PHFLEs, then there are

if Eh˜˙PH1p˙>Eh˜˙PH2p˙, then h˜˙PH1p˙>h˜˙PH2p˙;
if Eh˜˙PH1p˙<Eh˜˙PH2p˙, then h˜˙PH1p˙<h˜˙PH2p˙;
if Eh˜˙PH1p˙=Eh˜˙PH2p˙,
1. if σh˜˙PH1p˙<σh˜˙PH2p˙, then h˜˙PH1p˙>h˜˙PH2p˙;
2. if σh˜˙PH1p˙>σh˜˙PH2p˙, then h˜˙PH1p˙<h˜˙PH2p˙;
3. if σh˜˙PH1p˙=σh˜˙PH2p˙, then h˜˙L1p˙=h˜˙L2p˙.

2.2. CI Operator and PrA Operator

2.2.1. Fuzzy measure and CI

Definition 9.

[31] Let PX be the power set of X=x1,x2,…,xn, and the fuzzy measure μ:PX→0,1 of X satisfies the following conditions:

μ∅=0,μX=1;
if ∀A,B∈PX and A⊆B, then μA≤μB.

Fuzzy measure can be regarded as monotone set function, the fuzzy measure on X has the following characteristics:

Additivity: μA∪B=μA+μB.
Sub-additivity: ∀A,B∈PX, μA∪B≤μA+μB.
Super-additivity: ∀A,B∈PX, μA∪B≥μA+μB.

In MAGDM, μ(A) can be regarded as the importance of attribute subset A∈P(X). The monotony of fuzzy measure means that when new attributes are added to attribute subset, the importance of attribute subset will not decrease [35]. Nonadditivity is the main feature of fuzzy measure, which can more flexibly express the relationships between decision attributes from redundancy (negative interaction) to complementarity (positive interaction) [25,36].

Since the fuzzy measure is a function defined on the power set, it is necessary to determine 2n−2 parameters for calculating the fuzzy measure of n attribute, and the calculation amount for solving the fuzzy measure is large. Facing the large-scale calculation, Sugeno [31] proposed λ fuzzy measure to replace the general fuzzy measures, which simplified the computational complexity of fuzzy measures.

Definition 10.

[31] Let P(X) be the power set of X=x1,x2,…,xn, ∀A,B∈P(X),A∩B=∅, if the fuzzy measure g on X satisfies the following conditions:

gA∪B=g(A)+g(B)+λg(A)g(B), λ∈−1,∞

then g is called λ fuzzy measure.

For ∀A,B∈P(X), A∩B=ϕ: if λ=0, g(A∪B)=g(A)+g(B), attribute subsets A and B are independent; if −1<λ<0, g(A∪B)<g(A)+g(B), attribute subset A and B are redundant; if 0<λ<1, g(A∪B)>g(A)+g(B), attribute subsets A and B are complementary. In attribute-related multi-attribute decision-making, the role of attribute subset D∈P(X) in decision-making process is determined not only by g(D) itself, but also by other attribute subsets. If g(D)=0, then attribute subset D is irrelevant. For attribute subset H∈P(X), g(H∪D)−g(H)>0 indicates that attribute subset H is important.

According to the definition of λ fuzzy measure g, ∀xj∈X, j,k=1,2,…,n, j≠k, xj∩xk=∅, ∪j=1nxj=X, then λ fuzzy measure is shown as follows:

gX=g∪j=1nxj=1λ∏j=1n1+λgxi−1,λ≠0∑j=1ngxi,λ=0(3)

Since gX=1, when λ≠0, the value of λ is determined according to the following formula:

λ+1=∏j=1n1+λgxj(4)

Definition 11.

[25] Let X=x1,x2,…,xn be a nonempty set, f be a nonnegative real value function defined on X, g be a fuzzy measure defined on X, and the CI of function f on g is defined as follows:

∫fdg=∑j=1nfxj−fxj−1⋅gXj(5)

It can also be expressed as

∫fdg=∑j=1ngXj−gXj+1⋅fxj(6)

where fx1≤fx2≤…≤fxn, fx0=0, Xj=xj,xj+1,…,xn and xn+1=∅.

Aggregation characteristics of CI are idempotency, compensation, monotone additivity, etc. In addition, CI is an extension of weighted average and orderly weighted average. As long as the fuzzy measure is additive, the CI degenerates into a weighted average or an ordered weighted average.

2.2.2. PrA operator

Definition 12.

[26] Let C=C1,C2,…,Cn be a set of attributes, then there exists a linear ordered prioritized relationship between attributes, which can be expressed as C1≻C2≻⋯≻Cn, that is Cj priority ranks are higher than Ck, ∀j<k. Cjx is the evaluation value of alternative x under attribute Cj, which satisfies Cjx∈0,1. If

PrACjx=∑j=1nwjCjx(7)

where wj=Tj/∑j=1nTj, Tj=∏k=1j−1Ckx j=2,3,…,n, T1=1 then PrA is called the PrA operator.

The PrA operator [26] based on priority measure cannot solve all of priority decision problem. To be specific, for two alternatives, this problem cannot be solved when the satisfaction of their highest priority attribute is the same and the satisfaction of these attributes is the smallest of all priority attributes. The fundamental reason is that the measurement of priority is too strict to compensate for any priority attributes. Based on this, Chen et al. [37] proposed a generalized prioritized operator to optimize the prioritized operator [9].

Definition 13.

[37] Let C=C1,C2,…,Cn be a set of attributes, then there exists a linear ordered priority relationship between attributes, which can be expressed as C1≻C2≻…≻Cn. Let A=Ct1,Ct2,…,Ctl be a subset of attribute set C, by reordering the elements in subset A, A=Cσ1,Cσ2,…,Cσl can be obtained, of which σ1,σ2,⋯,σl is an arbitrary sequence of t1,t2,…,tl, satisfying σ1<σ2<⋯<σl. cix∈0,1 (i=1,2,…,n) is the satisfaction of alternative x under attribute Ci. A numerical value of bi is given for each priority attribute Ci in attribute set C, which satisfies bi∈0,1 and ∑i=1nbi=1, b1≥b2≥…≥bn. The generalized prioritized measure m:PC→0,1 is defined as follows:

mA=∑j=1lbσjfjσj and m∅=0(8)

where, P(C) is the power set of attribute set C, f(⋅) is a monotone decreasing function which satisfies fj(j)=1, fj(n)=0 (j=1,2,…,n).

If fjσj=n+1−σjn+1−j (j=1,2,…,n), then the new prioritized measures are expressed as follows:

mA=∑j=1lbσjn+1−σjn+1−j(9)

Based on this, Chen et al. [37] proposed a generalized PrA operator, i.e.

cx=Chmc1x,c2x,…,cnx=∑i=1ncixmhi−mhi−1

where cix is the ith largest value in c1x,c2x,…,cnx, hi=C1,C2,…,Ci and h0=∅, mhi i=0,1,…,n is generalized prioritized measures based on hi.

In this section, the basic theory of MAGDM based on PHFLTS is studied. Firstly, the related linguistic information representation models are briefly reviewed. In order to overcome the shortcoming of several kinds of linguistic information representation models, the PHFLTS is proposed, and related theories, such as expectation function, deviation function, and ranking method are put forward. Secondly, two kinds of typical functional information integration operators, namely CI integration operator and PrA operator, are reviewed.

3. PROPORTIONAL HESITANT FUZZY LINGUISTIC PRIORITIZED CHOQUET AGGREGATION OPERATOR

This section focuses on the aggregation of PHFLEs, and proposes the PHFLP_rCA operator to deal with the MAGDM problems with both priority and correlation.

3.1. PHFLCA_µ Operator Based on Fuzzy Measure

Definition 14.

Let H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be the PHFLTS, where h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, and μ is fuzzy measure based on H˜˙PHp˙. Then the PHFLCAμ operator is defined as follows:

PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=⊕j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙h˜˙PHjp˙(10)

where h˜˙PH1p˙≤h˜˙PH2p˙≤…≤h˜˙PHnp˙ and H˜˙PHjp˙=h˜˙PHjp˙,h˜˙PHj+1p˙,…,h˜˙PHnp˙, j=1,2,…,n, H˜˙PHn+1p˙=∅.

Theorem 1.

Let H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be the PHFLTS, where h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, μ be fuzzy measure based on H˜˙PHp˙, then the result of PHFLCAμ operator is as follows:

PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=li,∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙ij|li∈L,i=1,2,…,t(11)

Next, the properties of PHFLCAμ operator are idempotency, permutation invariance, monotonicity, and boundedness.

Property 1. (Idempotency)

Let H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be the PHFLTS, where h˜˙PHjp˙=h˜˙PHp˙=li,p˙i|li∈L,∑i=1tp˙i=1,i=1,2,…,t, j=1,2,…,n, then

PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=h˜˙PHp

Proof.

Because of h˜˙PHjp˙=h˜˙PHp˙, μH˜˙PHjp˙−μH˜˙PHj+1p˙≥0 and ∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙=1, it is available that

PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=li,∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙i|li∈L=h˜˙PHp˙

Property 2. (Commutativity)

Let h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be any PHFLTS, and h˜˙PH1p˙',h˜˙PH2p˙',…,h˜˙PHnp˙' is any sequence of h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙, then

PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=PHFLCAμh˜˙PH1p˙′,h˜˙PH2p˙′,…,h˜˙PHnp˙′

Proof:

Since h˜˙PH1p˙′,h˜˙PH2p˙′,…,h˜˙PHnp˙′ is an arbitrary sequence of h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙, CI operator can be regarded as an extended ordered weighted average operator. Based on the above formula, we can find that PHLCAμ operators have permutation invariance.

Property 3. (Monotonicity)

Let h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ and h˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙ be any two PHFLTSs, of which h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PHjq˙=li,q˙ij|li∈L,∑i=1tq˙ij=1,i=1,2,…,t and Eh˜˙PH1p˙≤Eh˜˙PH2p˙≤…≤Eh˜˙PHnp˙, Eh˜˙PH1q˙≤Eh˜˙PH2q˙≤…≤Eh˜˙PHnq˙, when Eh˜˙PHjp˙≥Eh˜˙PHjq˙ for all j, then

EPHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥EPHFLCAμh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙

Proof:

Since H˜˙PHj+1p˙⊆H˜˙PHjp˙, then μH˜˙PHjp˙−μH˜˙PHj+1p˙≥0. Since ∀j, Eh˜˙PHjp˙≥Eh˜˙PHjq˙, then

∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙ij≥∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙q˙ij.

According to Definition 6,

EPHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥EPHFLCAμh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙

then, PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥PHFLCAμh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙.

Property 4. (Boundedness)

Let h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be a PHFLTS, where h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PH+p˙=l1,0,l2,0,…,lt,1 and h˜˙PH−p˙=l1,1,l2,0,…,lt,0, then

h˜˙PH−p˙≤PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≤h˜˙PH+p˙

Proof:

According to Definition 6, we get Eh˜˙PH−p˙=1, Eh˜˙PH+p˙=t.

EPHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙i≤t∑i=1t∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙i=t

The same, we get

EPHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙i≥∑i=1t∑j=1nμH˜˙PHjp˙−μH˜˙PHj+1p˙p˙i=1

so, Eh˜˙PH−p˙≤EPHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≤Eh˜˙PH+p˙ i.e., h˜˙PH−p˙≤PHFLCAμh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≤h˜˙PH+p˙.

Noted: When the PHFLEs h˜˙PHjp˙ j=1,2,…,n are independent of each other, the fuzzy measure μ degenerates into an additive measure, i.e., μH˜˙PHjp˙=∑h˜˙PHjp˙∈H˜˙PHjp˙μh˜˙PHjp˙, ∀H˜˙PHjp˙∈H˜˙PHp˙. The PHFLCAμ operator degenerates to the proportional hesitant linguistic orderly weighted average operator.

3.2. PHFLP_rWA_m Operator Based on Generalized Prioritized Measure

The generalized prioritized measure proposed by Chen et al. [37], can solve the following problems when the satisfaction degree of the highest priority attribute is same and the satisfaction degree is the lowest among all priority attributes. This is because the prioritized measure is too strict to compensate for any priority attributes. Therefore, this section proposes PHFLPrWAm operator based on generalized prioritized measure.

Definition 15.

Let C=C1,C2,…,Cn be a set of attributes, then there exists a linear ordered priority relationship between attributes, which can be expressed as C1≻C2≻…≻Cn, i.e., Cj priority ranks are higher than Ck, ∀j<k. h˜˙PHjp˙ is the evaluation result of alternative x under attribute Cj, i.e., h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PHjp˙∈H˜˙PHp˙, H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ is the PHFLTS. The PHFLPrWAm operator is defined as follows:

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑j=1nh˜˙PHjp˙mHj−mHj−1(12)

where h˜˙PHjp˙ is the jth largest element in h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙, Hτ=Cσ1,Cσ2,…,Cστ, σ1,σ2,…,στ is the sequence of 1,2,…,τ, satisfying if ε<k, Cσε≻Cσk and H0=∅, mHj j=1,2,…,n is generalized prioritized measure.

mHτ=∑ε=1τbσεn+1−σεn+1−ε(13)

Theorem 2.

Let C=C1,C2,…,Cn be a set of attributes, then there exists a linear ordered priority relationship between attributes, which can be expressed as C1≻C2≻…≻Cn, i.e., Cj priority ranks are higher than Ck, ∀j<k. h˜˙PHjp˙ is the evaluation result of alternative x under attribute Cj, i.e., h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PHjp˙∈H˜˙PHp˙, H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ is the PHFLTS, then the integration result of PHFLPrWAm operator is as follows:

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=li,∑j=1np˙ijmHj−mHj−1|i=1,2,…,t(14)

Through the above theorem, we can easily find that PHFLPrWAm operator has the following properties:

Property 5. (Idempotency)

Let H˜˙PHp˙=h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be the PHFLTS, of which h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t. If h˜˙PHjp˙=h˜˙PHp˙=li,p˙i|i=1,2,…,t, j=1,2,…,n, then

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=h˜˙PHp˙

Proof:

Because of h˜˙PHjp˙=h˜˙PHp˙, mHj−mHj−1≥0 and ∑j=1nmHj−mHj−1=1, it is available that

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=li,∑j=1np˙ijmHj−mHj−1|li∈L=h˜˙PHp˙

Property 6. (Monotonicity)

Let h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ and h˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙ be any two PHFLTSs, of which h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PHjq˙=li,q˙ij|li∈L,∑i=1tq˙ij=1,i=1,2,…,t and h˜˙PHjp˙≥h˜˙PHjq˙, i.e., the proportional distribution of corresponding linguistic term is p˙ij≥q˙ij i=1,2,…,t, then

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥PHFLPrWAmh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙

Proof:

Based on Theorem 2, we get

PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=li,∑j=1np˙ijmHj−mHj−1|li∈L,

PHFLPrWAmh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙=li,∑j=1nq˙ijmHj−mHj−1|li∈L.

According to the Definition 8, the expectation values are

EPHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1np˙ijmHj−mHj−1,

EPHFLPrWAmh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙=∑i=1ti⋅∑j=1nq˙ijmHj−mHj−1.

Because p˙ij≥q˙ij, we get p˙ijmHj−mHj−1≥q˙ijmHj−mHj−1.

Then ∑j=1np˙ijmHj−mHj−1≥∑j=1nq˙ijmHj−mHj−1, we can get

EPHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥EPHFLPrWAmh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙.

Therefore, PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≥PHFLPrWAmh˜˙PH1q˙,h˜˙PH2q˙,…,h˜˙PHnq˙.

Property 7. (Boundedness)

Let h˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙ be a PHFLTS, where h˜˙PHjp˙=li,p˙ij|li∈L,∑i=1tp˙ij=1,i=1,2,…,t, h˜˙PH+p˙=l1,0,l2,0,…,lt,1 and h˜˙PH−p˙=l1,1,l2,0,…,lt,0, then

h˜˙PH−p˙≤PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≤h˜˙PH+p˙

Proof:

Based on the Definition 8, we get that

Eh˜˙PH+p˙=t, Eh˜˙PH−p˙=1,

EPHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1np˙ijmHj−mHj−1.

Because

EPHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1np˙imHj−mHj−1≤t∑i=1t∑j=1np˙imHj−mHj+1=t.

Similarly,

EPHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙=∑i=1ti⋅∑j=1np˙imHj−mHj−1≥∑i=1t∑j=1np˙imHj−mHj+1=1.

So, we get h˜˙PH−p˙≤PHFLPrWAmh˜˙PH1p˙,h˜˙PH2p˙,…,h˜˙PHnp˙≤h˜˙PH+p˙.

3.3. PHFLP_rCA Operator

Yager [26] pointed out that prioritized decision problems can be categorized in two forms: (1) one form is strictly ordered priority, i.e., each priority corresponds to one attribute; (2) the other form is weakly ordered priority, i.e., each priority corresponds to one or more attributes. PHFLPrWAm operator can only deal with the first kind of priority problems, but it cannot deal with the second kind of MAGDM problems. However, the second kind of MAGDM problems often occur in real life, thus a new integration operator needs proposing to handle the problems of weakly ordered priority, i.e., PHFLPrCA operator. The operation of PHFLPrCA operator can be described in two steps. Firstly, the PHFLCAg operator is based on λ fuzzy measure to get the satisfaction, which is displayed in each priority level. Secondly, the PHFLPrWAm operator based on prioritized measure is to get the overall satisfaction, which is displayed in each alternative. Thus the PHFLPrCA operator proposed in this section considers both the priority relationship among priority levels and the correlation among attributes.

In the MAGDM problems with weakly ordered priority, attribute set C=C1,C2,…,Cn is divided into q independent priority levels H=H1,H2,…,Hq, Hτ=C1τ,C2τ,…,Cnττ, of which nτ is the number of attributes contained in priority level Hτ. Assuming that there is a priority relation H1≻H2≻…≻Hq in the independent priority level H1,H2,…,Hq, when the attribute priority of k>ε in Hk is higher than that in Hε. Attribute set C=∪τ=1qHτ, numbers of attributes n=∑τ=1qnτ, evaluation value of alternative x under attribute Ckττ∈Hτ is PHFLE, expressed as hτkτp.

PHFLCAg operator is based on λ fuzzy measure to calculate the satisfaction hτp, which is displayed in each priority level:

PHFLCAgh˜˙τ1p˙,h˜˙τ2p˙,…,h˜˙τnτp˙=⊕j=1nτh˜˙τjp˙gHτj−gHτj−1(15)

where h˜˙τjp˙ is the jth largest value in h˜˙τ1p˙,h˜˙τ2p˙,…,h˜˙τnτp˙, and Hτj=Cτ1,Cτ2,…,Cτj, Hτ0=∅, gHτj is λ fuzzy measure of Hτj.

According to the above formulas, the ensemble results of PHFLCA_g operators based on λ fuzzy measure is as follows:

h˜˙τp˙=PHFLCAgh˜˙τ1p˙,h˜˙τ2p˙,…,h˜˙τnτp˙=li,∑j=1nτp˙τjgHτj−gHτj−1|i=1,2,…,t(16)

The PHFLPrAm operator is based on prioritized measure to calculate the overall satisfaction h˜˙p˙, which is displayed in each alternative,

PHFLPrAmh˜˙1p˙,h˜˙2p˙,…,h˜˙qp˙=⊕τ=1qh˜˙τp˙mhτ−mhτ−1(17)

where h˜˙τp˙ is τth largest value in h˜˙1p˙,h˜˙2p˙,…,h˜˙qp˙, hτ=C1,C2,…,Cτ, h0=∅, mhτ is generalized prioritized measure.

According to the above formula, the integration result of PHFLPrAm operator based on prioritized measure is as follows:

PHFLPrAmh˜˙1p˙,h˜˙2p˙,…,h˜˙qp˙=li,∑τ=1qp˙τmhτ−mhτ−1(18)

Through the above formulas, it is easy to prove that the operators proposed in this section are idempotent, monotonic and bounded.

4. MAGDM METHOD BASED ON PHFLPrCA Operator

4.1. MAGDM Problems with Priority Relations

In a MAGDM problem with both priority and correlation, it is assumed that there are m alternatives, i.e., A1,A2,…,Am. |G| DMs evaluate the alternative according to n attributes, i.e., C1,C2,…,Cn. n attributes are divided into q independent priority levels, i.e., H=H1,H2,…,Hq. Priority level Hτ contains nτ attributes, i.e., Hτ=C1τ,C2τ,…,Cnττ. Assuming that the attributes in the same priority level are interactive, the priority relationship among the q independent priority levels is H1≻H2≻…≻Hq. The attributes in all priority levels constitute the whole set of attributes, C=∪τ=1qHτ. |G| DMs evaluated alternative Aς under attribute Ckττ. The evaluation information was hesitant fuzzy linguistic hςkτeτ τ=1,2,…,q;kτ=1,2,…,nτ;ς=1,2,…,m;e=1,2,…,|G|. The evaluation information of |G| DMs could be transformed into PHFLEs h˜˙ςkττp˙.

4.2. Model of Priority Level in Generalized Prioritized Measure

In order to determine the priority weight, it is necessary to determine the value bττ=1,2,…,q in the generalized prioritized measure. Based on O'Hagan's maximum entropy method, a mathematical programming model is established, with predefined priority attitudes as constraints, and with entropy as objective function to determine bττ=1,2,…,q. Attitude eigenvalue is called Orness measure, which reflects the optimism of DMs. The greater the Orness measure is, the more optimistic the DM will be. Entropy is measured by the degree of discreteness. The greater the degree of discreteness is, the more information will be involved in the process of information integration. Specific models are as follows:

max−∑τ=1qbτlnbτs.t.∑τ=1qq−τq−1bτ=Ω,0.5≤Ω≤1∑τ=1qbτ=1,bτ∈0,1(19)

where Ω represents the eigenvalue of preferential attitude. Generally speaking, attributes with higher priority are more important, so the value of Ω is between 0.5 and 1. −∑τ=1qbτlnbτ is about the dispersion of bτ(τ=1,2,…,q).

4.3. The Steps of MAGDM Method

Priority relation among attributes is a common phenomenon in decision-making problems, as well as, interaction between attributes, i.e., correlation, is another common phenomenon in decision-making process. Therefore, this section proposes a PHFLP_rCA operator-based decision-making method to solve the multi-attribute decision-making problem with weakly ordered priority and association. The framework is shown in Figure 1. The specific steps are as follows:

Step 1: DMs evaluate the alternatives under each attribute to give HFLEs hςkτeτ, convert HFLEs hςkτeτ into PHFLEs h˜˙ςkττp˙, thus forming the PHFLEs decision matrix Y=h˜˙ςkττp˙.

Step 2: Use formula (4) to determine λτ (τ=1,2,…,q) in each priority level Hτ.

Step 3: Use formula (3) to determine the optimal fuzzy measure gλτ(Sτ) of attribute set Sτ⊂C1τ,C2τ,…,Ckττ in each priority level Hτ.

Step 4: Use formula (16) to calculate the satisfaction h˜˙ςτp˙ of alternative Aς (ς=1,2,…,m) under each priority level Hτ (τ=1,2,…,q).

Step 5: Use formula (19) to determine bτ (τ=1,2,…,q) of each priority level in the generalized prioritized measure.

Step 6: Use formula (18) to calculate the overall satisfaction h˜˙ςp˙ of alternative Aς (ς=1,2,…,m).

Step 7: Use formula (1) and formula (2) to calculate the expectation value Eh˜˙ςp˙ and deviation σh˜˙ςp˙ of the overall satisfaction of each alternative.

Step 8: Rank the alternatives according to the expectations and deviations.

5. CASE STUDY: TEACHING PERFORMANCE EVALUATION OF CHINESE-FOREIGN COOPERATIVE EDUCATION PROJECT OF S COLLEGE

Chinese-foreign cooperative education project is an educational model with the background of globalization. It usually refers to the educational projects that foreign educational institutions and domestic educational institutions have set up in China. Under such background S college carries out many projects with several foreign universities.

However, with the expansion of the projects' scale and the increasing number of students enrolled, the problems of resource integration, personnel training, and scientific research within the S college has gradually emerged. In order to solve the hidden problems within the college, improve the efficiency of teachers' performance and promote the realization of project objectives, S college has the demand that carries out a reasonable evaluation index system. According to the requirement of trinity (teaching ability, scientific research ability, and other ability), the college evaluation committee has established an evaluation index system to combine the characteristics and development goals of Chinese-foreign cooperative education project in S college, as shown in Table 1.

Serial Number	Elements	Specific Indicators
1	Teaching ability H1	Bilingual teaching ability C11
		Developing international course ability C21
		Cooperative teaching ability C31
2	Scientific research ability H2	The level of scientific research projects C12
		The level of academic papers C22
		International research cooperation ability C32
3	Other abilites H3	Project management ability C13
3	Other abilites H3	Cross-cultural ability C23

Table 1

Teaching performance evaluation of Chinese-foreign cooperative education project of S college.

The focus of the evaluation is teachers' teaching ability, so the priority of three elements is Teaching abilityH1≻Scientific research abilityH2≻Other abilitesH3. It shows that the lack of teaching ability cannot be compensated by the increase of the corresponding scientific research ability. In addition, some correlations are displayed between the specific indicators contained in each element, so the sum of the importance of elements may not be 1. In the process of formulating the evaluation index system, the importance of each indicator is obtained from expert interviews, i.e., in the teaching ability H1, the importance of three specific indicators are gλ1C11=gλ1C31=2/3, gλ1C21=1/3, respectively; in the scientific research ability H2, the importance of three specific indicators are gλ2C12=gλ2C22=1/3, gλ2C32=2/3, respectively; in the other abilities H3, the importance of two specific indicators are gλ3C13=gλ3C23=2/5, respectively.

Evaluation team is composed of 11 experts. Experts express evaluation values in the form of linguistic words, because the evaluation values are highly uncertain and difficult to express in precise numbers. Experts may be more accustomed to using qualitative words such as “very low,” “low,” “medium,” “high,” and “very high.” Experts can choose one or more linguistic words from S=s1:very low,s2:low,s3:moderate,s4:high,s5:very high to evaluate according to their own understanding. When they know nothing about it, they can also give no evaluation value. Therefore, the evaluation results of 11 experts are summarized in Appendix (Table A1).

5.1. Decision-Making Steps

According to the MAGDM method described in the previous section, the teaching performance in S college is evaluated. The specific steps are as follows:

Step 1: The evaluation information given by experts is transformed into PHFLEs, and the PHFLE matrix is constructed, as shown in Table 2.

	A1	A2	A3	A4
C11	{l3,0.230,l4,0.385,l5,0.385}	l4,0.462,l5,0.538	{l2,0.385,l3,0.538,l4,0.077}	{l2,0.250,l3,0.583,l4,0.417}
C21	{l3,0.357,l4,0.429,l5,0.214}	l4,0.571,l5,0.429	{l2,0.308,l3,0.538,l4,0.154}	{l2,0.214,l3,0.571,l4,0.214}
C31	{l3,0.286,l4,0.500,l5,0.214}	l4,0.538,l5,0.462	{l2,0.357,l3,0.571,l4,0.072}	{l2,0.333,l3,0.500,l4,0.167}
C12	{l3,0.357,l4,0.500,l5,0.143}	l4,0.500,l5,0.500	{l2,0.231,l3,0.538,l4,0.231}	{l2,0.308,l3,0.385,l4,0.308}
C22	{l3,0.072,l4,0.571,l5,0.357}	l4,0.538,l5,0.462	{l2,0.214,l3,0.572,l4,0.214}	{l2,0.333,l3,0.250,l4,0.417}
C32	{l3,0.214,l4,0.429,l5,0.357}	{l3,0.076,l4,0.462,l5,0.462}	{l2,0.231,l3,0.461,l4,0.308}	{l2,0.417,l3,0.250,l4,0.333}
C13	{l3,0.076,l4,0.462,l5,0.462}	{l3,0.071,l4,0.571,l5,0.357}	{l2,0.154,l3,0.538,l4,0.308}	{l2,0.417,l3,0.417,l4,0.167}
C23	{l3,0.286,l4,0.428,l5,0.286}	l4,0.429,l5,0.571	{l2,0.214,l3,0.572,l4,0.214}	{l2,0.333,l3,0.417,l4,0.250}

Table 2

Proportional hesitant fuzzy linguistic elements (PHFLEs) decision matrix.

Step 3: Determine the λ fuzzy measure of each attribute subset under each priority level λτ τ=1,2,3 according to formula (3) and formula (4).

For priority level H1, gλ1C11=gλ1C31=2/3, gλ1C21=1/3, according to formula (4), λ1=−0.879 can be obtained.

According to formula (3), gλ1C11,C21=0.805, gλ1C11,C31=0.943, gλ1C21,C31=0.805, and gλ1C11,C21,C31=1 can be obtained.
For priority level H2, gλ2C12=gλ2C22=1/3, gλ2C32=2/3, according to formula (4), λ2=−0.658 can be obtained.

According to formula (3), gλ2C12,C22=0.594, gλ2C12,C32=0.854, gλ2C22,C32=0.854, and gλ2C12,C22,C32=1 can be obtained.
For priority level H3, gλ3C13=gλ3C23=2/5, according to formula (4), λ3=1.25 can be obtained.

According to formula (3), gλ3C13,C23=1 can be obtained.

Step 4: Use formula (16) to calculate the satisfaction h˜˙ςτp˙ of Aςς=1,2,3,4 under priority Cττ=1,2,3.

Computing the satisfaction of A1 at a priority level:

h˜˙11p˙=gλ1C31−gλ1∅h˜˙131p˙⊕gλ1C21,C31−gλ1C31h˜˙121p˙⊕gλ1C11,C21,C31−gλ1C21,C31h˜˙111p˙=l1,0,l2,0,l3,0.263,l4,0.409,l5,0.328

h˜˙12p˙=gλ2C22−gλ2∅h˜˙122p˙⊕gλ2C22,C32−gλ2C22h˜˙132p˙⊕gλ2C12,C22,C32−gλ2C22,C32h˜˙112p˙=l1,0,l2,0,l3,0.147,l4,0.538,l50.315

h˜˙13p˙=gλ3C13−gλ3∅h˜˙113p˙⊕gλ3C13,C23−gλ3C13h˜˙123p˙=l1,0,l2,0,l3,0.130,l4,0.426,l5,0.387

In the same way, the satisfaction of A2,A3,A4 at each priority level can be obtained as follows:

h˜˙21p˙=l1,0,l2,0,l3,0,l4,0.539,l5,0.461;

h˜˙22p˙=l1,0,l2,0,l3,0.004,l4,0.508,l5,0.487;

h˜˙23p˙=l1,0,l2,0,l3,0.048,l4,0.499,l5,0.396;

h˜˙31p˙=l1,0,l2,0.329,l3,0.543,l4,0.127,l5,0;

h˜˙32p˙=l1,0,l2,0.228,l3,0.519,l4,0.253,l5,0;

h˜˙33p˙=l1,0,l2,0.162,l3,0.517,l4,0.264,l5,0;

h˜˙41p˙=l1,0,l2,296,l3,0.524,l4,0.180,l5,0;

h˜˙42p˙=l1,0,l2,0.387,l3,0.258,l4,0.355,l5,0;

h˜˙43p˙=l1,0,l2,0.370,l3,0.393,l4,0.180,l5,0.

Step 5: Use formula (19) to calculate the values bττ=1,2,3 in the generalized prioritized measure.

max−b1lnb1−b2lnb2−b3lnb3s.t.b1+12b2+0×b3=0.75b1+b2+b3=1b1,b2,b3≥0

b1=0.612,b2=0.276,b3=0.112.

Step 6: Use Formula (18) to calculate the overall satisfaction of each alternative at all priority levels.

h˜˙1p˙=l1,0,l2,0,l3,0.227,l4,0.435,l5,0.332;

h˜˙2p˙=l1,0,l2,0,l3,0.008,l4,0.513,l5,0.472;

h˜˙3p˙=l1,0,l2,0.301,l3,0.537,l4,0.158,l5,0;

h˜˙4p˙=l1,0,l2,0.321,l3,0.461,l4,0.212,l5,0.

Step 7: Calculating expectation Eh˜˙ςp˙ ς=1,2,3,4

Eh˜˙1p˙=4.079, Eh˜˙2p˙=4.437, Eh˜˙3p˙=2.847, Eh˜˙4p˙=2.871.

Step 8: Rank the alternatives according to the expectation value.

Eh˜˙2p˙>Eh˜˙1p˙>Eh˜˙4p˙>Eh˜˙3p˙, so A2≻A1≻A4≻A3. That is the performance of the four teachers in S College, A2 is the best and A3 is the worst.

5.2. The Impact of Prioritized Attitudinal Character on Decision-Making Results

The influence of prioritized attitudinal character Ω on priority decision results is further analyzed. Firstly, O'Hagan's maximum entropy method is used to calculate the corresponding bττ=1,2,3 based on different Ω values. As can be seen from Table 3, Ω can be used to adjust the priority of attributes. As far as Ω∈0.6,0.7,0.75,0.8,0.9 is concerned, it can be found from Table 3 and Figure 2 that the bττ=1,2,3 value forms a nonsubtractive sequence, i.e., b1≥b2≥b3. Then, the PHFLPrCA operator based on the corresponding bττ=1,2,3 value is used to rank alternatives. From the ranking results listed in Table 3 and Figure 3, no matter how Ω changes, A2 is always the best solution. The prioritized attitudinal character Ω can be used to describe the DM's psychology. The greater the value of Ω is, the more optimistic the DM is. On the contrary, the smaller the Ω is, the more pessimistic the DM is. Therefore, DMs can choose an appropriate Ω values according to their own preference and practical problems.

Ω	bττ=1,2,3	Eh˜˙ςp˙	Ranking
0.6	b1=0.440,b2=0.320,b3=0.240	Eh˜˙1p˙=4.077,Eh˜˙2p˙=4.390,Eh˜˙3p˙=2.862,Eh˜˙4p˙=2.842.	A2≻A1≻A3≻A4
0.7	b1=0.550,b2=0.300,b3=0.150	Eh˜˙1p˙=4.080,Eh˜˙2p˙=4.425,Eh˜˙3p˙=2.854,Eh˜˙4p˙=2.865.	A2≻A1≻A4≻A3
0.75	b1=0.612,b2=0.276,b3=0.112	Eh˜˙1p˙=4.079,Eh˜˙2p˙=4.437,Eh˜˙3p˙=2.847,Eh˜˙4p˙=2.871.	A2≻A1≻A4≻A3
0.8	b1=0.684,b2=0.232,b3=0.084	Eh˜˙1p˙=4.077,Eh˜˙2p˙=4.448,Eh˜˙3p˙=2.838,Eh˜˙4p˙=2.876.	A2≻A1≻A4≻A3
0.9	b1=0.846,b2=0.108,b3=0.046	Eh˜˙1p˙=4.070,Eh˜˙2p˙=4.464,Eh˜˙3p˙=2.817,Eh˜˙4p˙=2.878.	A2≻A1≻A4≻A3

Table 3

Decision-making results corresponding to prioritized attitudinal character.

5.3. Validity Test of the Method

This subsection focuses on the reliability and validity of the created method by the test criteria [38]. This is because different MAGDM method may lead to different ranking results for the same decision-making problem.

Test criterion 1. Under the condition that attributes' weights keep unchanged, the optimal alternative still maintain its first place when any nonoptimal alternative are substituted by another non-optimal alternative. Thus, if the result meets the above conditions, this MAGDM method is effective.

Among four alternatives, A3 is a nonoptimal alternative. C13 can C23 are taken place by C13=l2,0.385,l3,0.538,l4,0.077, C23=l2,0.357,l3,0.643, by using test criterion 1. The overall satisfaction values of each alternative are obtained:

h˜˙1p˙=l1,0,l2,0,l3,0.227,l4,0.435,l5,0.332;

h˜˙2p˙=l1,0,l2,0,l3,0.008,l4,0.513,l5,0.472;

h˜˙3p˙=l1,0,l2,0.329,l3,0.534,l4,0.097,l5,0;

h˜˙4p˙=l1,0,l2,0.321,l3,0.461,l4,0.212,l5,0.

The expectations of the alternatives are

Eh˜˙1p˙=4.079, Eh˜˙2p˙=4.437, Eh˜˙3p˙=2.647, Eh˜˙4p˙=2.871.

Thus, according to the ranking results A2≻A1≻A4≻A3, A2 is still the best techer in S College. Therefore, the created MAGDM method is verified effectively by test criterion 1.

Test criterion 2. An effective MAGDM method should be proved transitive property, which will be demonstrated in test criterion 3.

Test criterion 3. If the same method is used to solve sub-problems, which are divided from the original MAGDM problem, the ranking result should be the same as the original MAGDM problem.

According to criteria 2 and 3 the initial MAGDM problem is divided to two sub-problems, A1,A2,A4 and A2,A3,A4. These two sub-problems are solved by the created MAGDM method, so the ranking results of them are A2≻A1≻A4 and A2≻A4≻A3. The integration of the above ranking results is A2≻A1≻A4≻A3, that is the same ranking result of original MAGDM problem, which proved transitive property. Therefore, the created MAGDM method is verified effectively by test criterion 2 and 3.

5.4. Advantages of the Method

To further illustrate the advantages of the proposed method, the PHFLP_rCA operator-based MAGDM method proposed in this paper is compared with other two decision-making methods [12], which are PHFLWA operator and PHFLOWA operator. The two methods mentioned above are used to rank the alternatives of the cases in this paper. The ranking results are shown in the following table:

Methods		Ranking Values (R)	Ranking Results
Method 1 (PHFLWA)		RA1=2.568,RA2=2.944,RA3=1.084,RA4=1.404.	A2≻A1≻A4≻A3
Method 2 (PHFLOWA)		RA1=2.751,RA2=3.234,RA3=1.006,RA4=1.009.	A2≻A1≻A4≻A3
Our method	Ω=0.6	RA1=4.077,RA2=4.390,RA3=2.862,RA4=2.842.	A2≻A1≻A3≻A4
Our method	Ω=0.8	RA1=4.077,RA2=4.448,RA3=2.838,RA4=2.878.	A2≻A1≻A4≻A3

Table 4

Ranking results of different methods.

Compared with the other two methods, this method has the following advantages:

The created method is flexible. By changing the Ω value, the result of decision-making problem can be solved more flexibly. When Ω=0.6, the ranking results of the proposed method are slight differences with those of other two methods (shown in Table 4); when Ω=0.8, the ranking results of the proposed method are consistent with them. This shows that the proposed method is more flexible than the other two methods in dealing with decision-making problems.
The created method can solve the decision-making problems with both priority and correlation. The existing two methods in this section have the same shortcoming, i.e., they do not consider the relationship between attributes, and cannot deal with the priority relationship between different levels. However, according to the actual situation, this paper can solve the decision-making problems of priority and correlation (complementary relationship, redundant relationship, independent relationship) among attributes, whereas the above two methods are not suitable for dealing with such problems.
The created method owns a simple calculation process. Both methods 1 and 2 aggregate indicators by means of two times. If there are more than two indicators involved, they have to aggregate them one by one, which results in a complicated calculation process. On the contrary, the created method can aggregate indicators by means of only one time, no matter how many indicators are. Fewer steps make the created method simpler than methods 1 and 2.

In this section, seven steps are applied in teaching performance evaluation to obtain the best alternative. And the best alternative won't change by adjusting Ω, thus DMs can choose an appropriate Ω values according to their own preference and practical problems. Moreover, this method passed validity test and is proved to own many advantages by compared with other two methods.

6. CONCLUSION

In order to solve teaching performance evaluation of Chinese-foreign cooperative education project, this paper mainly studies the PHFLE information aggregation operator with priority and correlation function and its application. Firstly, the PHFLCA_g operator is proposed for MAGDM problems with correlations among attributes (redundancy, complementarity, and independence). Secondly, the PHFLP_rCA operator is proposed for the case that attributes have both priority and correlation. In this paper, the properties of correlation operators are briefly introduced while corresponding operators are proposed. Thirdly, based on the PHFLP_rCA operator, a new MAGDM method is proposed. In order to verify the effectiveness of the proposed method, this paper chooses the typical MAGDM problem on teaching performance evaluation of Chinese-foreign cooperative education project in S College, which has priority and correlation among its attributes. Then this paper illustrates the role of the proposed method in solving the problem. Finally, the advantages of this method are illustrated by comparing with other two methods.

The limitations of this paper are shown in following two aspects, which will be optimized in our future research. The first limitation is the LTS supposed in balanced and symmetric environment, however, in real life the LTS is unbalanced and asymmetric sometimes. The second limitation is that the DMs adopt the same LTS, however, DMs often choose different LTSs by their experience and preference. In the future PHFLE will be applied into unbalanced and multi-granular environment.

CONFLICTS OF INTEREST

The authors declare no conflicts of interest.

AUTHORS' CONTRIBUTIONS

Conceptualization, Lei Wang and Peide Liu; methodology, Lili Rong and Fei Teng; formal analysis, Lei Wang and Lili Rong; writing-original draft preparation, Lei Wang and Fei Teng; writing-review and editing, Lei Wang and Peide Liu; supervision, Peide Liu; funding acquisition, Lei Wang, Lili Rong, Fei Teng and Peide Liu.

ACKNOWLEDGMENTS

This work is supported by the National Natural Science Foundation of China (Nos. 71471172 and 71801142), the Special Funds of Taishan Scholars Project of Shandong Province (No. ts201511045), the Natural Science Foundation of Shandong Province (No. ZR2020QG002), the Social science planning project of Shandong Province (No. 20CSDJ23), the Scientific research project of Shandong TV University (Nos. 2019KD002 and 2020JXYJ001Z), Key projects of Art Science in Shandong Province (Nos. ZD202008366 and ZD202008492).

APPENDIX

	A1	A2	A3	A4
C11	l4, l5, l3, l3, l4, l5, l5, l5, l4, l4,l5, l3,l4	l4, l4, l5, l5, l5, l4, l5, l4, l5, l4,l5, l4,l5	l2, l3, l3, l4, l2, l3, l3, l2, l3, l2,l3, l2,l3	l3, l3, l2, l4, l3, l3, l2, l3, l4, l3, l2,l3
C21	l4, l3, l3, l4, l4, l5, l3, l3, l4,l5, l3,l4, l4,l5	l4, l5, l4, l4, l4, l4, l5, l5, l4,l5, l4,l5, l4,l5	l2, l2, l2, l3, l3, l4, l3, l3, l3, l2,l3, l3,l4	l3, l3, l2, l3, l3, l3, l2, l4, l2,l3, l3,l4, l3,l4
C31	l3, l4, l3, l4, l4, l4, l5, l3, l4,l5, l4,l5, l3,l4	l5, l4, l4, l4, l5, l4, l4, l5, l5, l4,l5, l4,l5	l3, l3, l2, l3, l2, l3, l3, l4, l2,l3, l2,l3, l2,l3	l3, l2, l3, l4, l2, l3, l3, l2, l2, l3, l3,l4
C12	l3, l3, l3, l4, l5, l4, l4, l4, l3,l4, l3,l4, l4,l5	l5, l4, l4, l5, l5, l4, l5, l4, l4, l5, l4,l5	l2, l3, l4, l2, l3, l4, l3, l3, l3, l3,l4, l2,l3	l3, l2, l3, l4, l2, l4, l2, l3, l4, l2,l3, l3,l4
C22	l4, l4, l3, l4, l5, l4, l4, l5, l4,l5, l4,l5, l4,l5	l4, l4, l4, l4, l5, l5, l5, l4, l5, l4,l5, l4,l5	l3, l2, l3, l3, l4, l3, l2, l3, l2,l3, l3,l4, l3,l4	l2, l4, l2, l3, l2, l3, l4, l2, l4, l4, l3,l4
C32	l3, l4, l5, l3, l4, l5, l4, l5, l4,l5, l3,l4, l4,l5	l5, l4, l5, l4, l5, l5, l5, l4, l4, l3,l4, l4,l5	l2, l2, l3, l3, l4, l3, l3, l4, l4, l2,l3, l3,l4	l3, l2, l2, l3, l4, l2, l4, l2, l4, l2, l3,l4
C13	l4, l4, l4, l5, l5, l4, l5, l3, l5, l4,l5, l4,l5	l4, l5, l5, l4, l4, l4, l5, l4, l3,l4, l4,l5, l4,l5	l3, l2, l3, l4, l2, l3, l3, l4, l3, l3,l4, l3,l4	l2, l3, l3, l2, l3, l4, l2, l3, l2, l2, l3,l4
C23	l3, l4, l3, l4, l5, l4, l5, l3, l4,l5, l4,l5, l3,l4	l5, l5, l4, l5, l4, l5, l4, l5, l4,l5, l4,l5, l4,l5	l3, l3, l4, l3, l2, l3, l2, l4, l2,l3, l2,l3, l3,l4	l2, l3, l3, l4, l3, l3, l2, l2, l4, l2, l3,l4

Table A1

Information on performance evaluation of four teachers in S college.

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Journal: International Journal of Computational Intelligence Systems
Volume-Issue: 14 - 1
Pages: 635 - 650
Publication Date: 2021/01/19
ISSN (Online): 1875-6883
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TY  - JOUR
AU  - Lei Wang
AU  - Lili Rong
AU  - Fei Teng
AU  - Peide Liu
PY  - 2021
DA  - 2021/01/19
TI  - Teaching Performance Evaluation Based on the Proportional Hesitant Fuzzy Linguistic Prioritized Choquet Aggregation Operator
JO  - International Journal of Computational Intelligence Systems
SP  - 635
EP  - 650
VL  - 14
IS  - 1
SN  - 1875-6883
UR  - https://doi.org/10.2991/ijcis.d.210112.001
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International Journal of Computational Intelligence Systems

Teaching Performance Evaluation Based on the Proportional Hesitant Fuzzy Linguistic Prioritized Choquet Aggregation Operator

1. INTRODUCTION

2. PRELIMINARIES

2.1. PHFLEs and its Related Concepts

Definition 1.

Definition 2.

Definition 3.

Definition 4.

Definition 5.

Definition 6.

Definition 7.

Definition 8.

2.2. CI Operator and PrA Operator

2.2.1. Fuzzy measure and CI

Definition 9.

Definition 10.

Definition 11.

2.2.2. PrA operator

Definition 12.

Definition 13.

3. PROPORTIONAL HESITANT FUZZY LINGUISTIC PRIORITIZED CHOQUET AGGREGATION OPERATOR

3.1. PHFLCAµ Operator Based on Fuzzy Measure

Definition 14.

Theorem 1.

Property 1. (Idempotency)

Proof.

Property 2. (Commutativity)

Proof:

Property 3. (Monotonicity)

Proof:

Property 4. (Boundedness)

Proof:

3.2. PHFLPrWAm Operator Based on Generalized Prioritized Measure

Definition 15.

Theorem 2.

Property 5. (Idempotency)

Proof:

Property 6. (Monotonicity)

Proof:

Property 7. (Boundedness)

Proof:

3.3. PHFLPrCA Operator

4. MAGDM METHOD BASED ON PHFLPrCA Operator

4.1. MAGDM Problems with Priority Relations

4.2. Model of Priority Level in Generalized Prioritized Measure

4.3. The Steps of MAGDM Method

5. CASE STUDY: TEACHING PERFORMANCE EVALUATION OF CHINESE-FOREIGN COOPERATIVE EDUCATION PROJECT OF S COLLEGE

5.1. Decision-Making Steps

5.2. The Impact of Prioritized Attitudinal Character on Decision-Making Results

5.3. Validity Test of the Method

5.4. Advantages of the Method

6. CONCLUSION

CONFLICTS OF INTEREST

AUTHORS' CONTRIBUTIONS

ACKNOWLEDGMENTS

APPENDIX

REFERENCES

Cite this article

3.1. PHFLCA_µ Operator Based on Fuzzy Measure

3.2. PHFLP_rWA_m Operator Based on Generalized Prioritized Measure

3.3. PHFLP_rCA Operator