Research on the assessment of psycholinguistic teaching effect with triangular fuzzy information

1Introduction

Fuzzy multiple attribute decision making (FMADM) is an important part of modern decision science [1–4]. It assumes that there exists a set of alternatives with multiple attributes which a decision maker (DM) should evaluate and analyze. The aim of FMADM is to find the most desirable alternative or rank the feasible alternatives for supporting decision makings. As an active research area, FMADM problems have been tried to be solved by some classical methods such as the simple additive weighting method (SAW), the analytic hierarchy process (AHP), and the technique for order preference by similarity to ideal solution (TOPSIS) [5–7]. Xu [8] and Fan and Wang [9] developed the fuzzy ordered weighted averaging (FOWA) operator. Xu [10] introduced the fuzzy ordered weighted geometric (FOWG) operator. Xu and Wu [11] proposed the fuzzy induced ordered weighted averaging (FIOWA) operator. Xu and Da [12] developed the fuzzy induced ordered weighted geometric (FIOWG) operator. Xu [13] developed the fuzzy weighted harmonic mean (FWHM) operator, fuzzy ordered weighted harmonic mean (FOWHM) operator, fuzzy hybrid harmonic mean (FHHM) operator. Wei [14] proposed the fuzzy ordered weighted harmonic averaging(FOWHA) operator. Wei [15] developed the fuzzy induced ordered weighted harmonic mean (FIOWHM) operator. Wei [16] proposed the generalized triangular fuzzy correlated averaging operator and applied these operators to multiple attribute decision making. Merigo [17] presented the fuzzy probabilistic ordered weighted averaging (FPOWA) operator. Merigo and Casanovas [18] presented the fuzzy generalized hybrid averaging (FGHA) operator, the fuzzy induced generalized hybrid averaging (FIGHA) operator, the Quasi-FHA operator and the Quasi-FIHA operator. Merigo and Gil-Lafuente [19] proposed the fuzzy induced generalized ordered weighted averaging (FIGOWA) and the fuzzy induced quasi-arithmetic OWA (Quasi-FIOWA) operator. Xu [20] developed some fuzzy ordered distance measures, such as linguistic ordered weighted distance measure, uncertain ordered weighted distance measure, linguistic hybrid weighted distance measure, and uncertain hybrid weighted distance measure, etc. Zhao et al. [21] proposed fuzzy prioritized operators for multiple attribute group decision making. Wei et al. [22] proposed fuzzy power aggregating operators for multiple attribute group decision making. Zhu et al. [23] developed the triangular fuzzy weighted Bonferroni mean (TFWBM) operator for multiple attribute decision making under the triangular fuzzy environments. Guo et al. [24] developed the triangular fuzzy geometric Bonferroni mean (TFGBM) operator and triangular fuzzy weighted geometric Bonferroni mean (TFWGBM) operator, based on which we design two procedure for multiple attribute decision making under the triangular fuzzy environments.

As a powerful method which improves class-teaching effect, the students’ evaluation of the class-teaching effect has become more and more important in recent years in universities of China. It plays an important role in improving instruction, teachers’ job evaluation, promoting rewards and punishments. However because of the congenital deficiency of the students’ evaluation of the class-teaching effect, it restricts seriously the exertion of the function in diagnosing and stimulating. There are some discontented phenomena in universities, such as some universities have to stop the practice on account of all aspects pressure which can’t endure, some universities limit the practice to form while the evaluation’s result is laid aside, some universities only reward little efficacy. There are many factors which effect the exertion of the function of the students’ evaluation of the class-teaching effect. Considering the subject and the object of evaluation, it involves the cognitive aspect of teachers and students, the mental adjustment and so on. Considering the process of evaluation, it involves publicity of the purpose and significance, the design of the index system, the establishment and implementation of evaluation scheme, the data’s disposal and implication. In this paper, we investigate the multiple attribute decision making problems for evaluating the psycholinguistic teaching effect with triangular fuzzy information. Motivated by the traditional idea of geometric Bonferroni mean [25–27] and Einstein operations [28–32], we develop the triangular fuzzy Einstein geometric Bonferroni mean (TFEGBM) operator and triangular fuzzy weighted Einstein geometric Bonferroni mean (TFWEGBM) operator for aggregating the triangular fuzzy information. Then, based on the TFWEGBM operator, we develop the procedure for multiple attribute decision making with the triangular fuzzy information. Finally, a practical example for evaluating the psycholinguistic teaching effect is given to verify the developed approach.

2Preliminaries

In this section, we briefly describe some basic concepts and basic operational laws related to triangular fuzzy numbers.

Definition 1. [33] A triangular fuzzy numbers a˜ can be defined by a triplet (a^L, a^M, a^U). The membership function μa˜(x) is defined as:

Definition 2. [33] Basic operational laws related to triangular fuzzy numbers:

Definition 3. [13] Let b˜=[bL,bM,bU] and a˜=[aL,aM,aU] be two triangular fuzzy numbers, then the degree of possibility of a ≥ b is defined as

Einstein operations [28] includes the Einstein product and Einstein sum, which are examples of t-norms and tconorms, respectively. Einstein product ⊗_ɛ is a t-norm and Einstein sum ⊕_ɛ is a t-conorm, where

In the following, Zhu et al. [25] studied on the geometric Bonferroni mean (GBM).

Definition 4. [25] Let p, q ≥ 0 and a_i (i = 1, 2, ⋯ , n) refers to a collection of non-negative real numbers. Afterwards, the aggregation functions

3Triangular fuzzy Einstein geometric Bonferroni mean operators

The geometric Bonferroni mean [25–27] and Einstein operations [28–32], however, have usually been used in situations where the input arguments are the non-negative real numbers. We shall extend the GBM operators and Einstein operations to accommodate the situations where the input arguments are triangular fuzzy numbers. In this section, we shall investigate the GBM operators and Einstein operations under triangular fuzzy environments. Then, we shall propose the triangular fuzzy Einstein geometric Bonferroni mean (TFEGBM) operator as follows:

Definition 5. Let a˜i=[aiL,aiM,aiU](i=1,2,⋯,n) be a set of triangular fuzzy numbers, and let p, q > 0. If then TFEGBM^p,q is called the triangular fuzzy Einstein geometric Bonferroni mean (TFEGBM) operator.

It can be easily proved that the TFEGBM operator has the following properties.

Theorem 1. (Boundedness) Let a˜i=[aiL,aiM,aiU] (i=1,2,⋯,n) be a set of triangular fuzzy numbers, and let

Then

Theorem 2. (Monotonicity) Let a˜i=[aiL,aiM,aiU](i=1,2,⋯,n) and a˜i′=[ai′L,ai′M,ai′U](i=1,2,⋯,n) be two sets of triangular fuzzy numbers, if a˜j≤a˜j′, for all j, then

Theorem 3. (Idempotency) Let a˜i=[aiL,aiM,aiU](i=1,2,⋯,n) be a set of triangular fuzzy numbers. If all a˜j(a˜j=[ajL,ajM,ajU]) are equal, i.e. a˜j(a˜j=[ajL,ajM,ajU])=a˜(a˜=[aL,aM,aU]) for all j, then

Theorem 4. (Commutativity) Let a˜i=[aiL,aiM,aiU](i=1,2,⋯,n) and a˜i′=[ai′L,ai′M,ai′U](i=1,2,⋯,n) be two sets of triangularfuzzy numbers, where a˜i′=[ai′L,ai′M,ai′U](i=1,2,⋯,n) is any permutation of a˜i=[aiL,aiM,aiU](i=1,2,⋯,n), then

Considering that the input arguments may have different importance, here we define the triangular fuzzy weighted Einstein geometric Bonferroni mean (TFWEGBM) operator.

Definition 6. a˜i=[aiL,aiM,aiU](i=1,2,⋯,n) be a set of triangular fuzzy numbers and p, q > 0, w = (w₁, w₂, ⋯ , w_n) ^T is the weight vector of a˜i=[aiL,aiM,aiU](i=1,2,⋯,n), where w_i indicates the importance degree of a˜i, satisfying w_i > 0 (i = 1, 2, ⋯ , n), and ∑i=1nwi=1. If

4An approach to multiple attribute decision making with triangular fuzzy information

In this section, we shall utilize the triangular fuzzy weighted Einstein geometric Bonferroni mean (TFWEGBM) operator to multiple attribute decision making for evaluating the psycholinguistic teaching effect with triangular fuzzy information. Let A = {A₁, A₂, ⋯ , A_m} be a discrete set of alternatives, G = {G₁, G₂, ⋯ , G_n} be the set of attributes, whose weight vector is ω = (ω₁, ω₂, ⋯ , ω_n), with ω_j ≥ 0, j = 1, 2, ⋯ , n, ∑j=1nωj=1. Suppose that R=(a˜ij)m×n=[rijL,rijM,rijU]m×n is the decision making matrix, where a˜ij is the preference value, which take the form of triangular fuzzy numbers, given by the decision maker, for the alternative A_i ∈ A with respect to the attribute G_j ∈ G.

Then, we utilize the TFWEGBM operator to solve multiple attribute decision making problems for evaluating the psycholinguistic teaching effect with triangular fuzzy information, which can be described as following:

Step 1. Utilize the decision information given in matrix R˜, and the TFWEGBM operator (in general, we can take p = q = 1)

Step 2. To rank these overall preference values r˜i(i=1,2,⋯,m), we first compare each r˜i with all the r˜j(j=1,2,⋯,m) by using Equation (2). For simplicity, we let pij=p(r˜i≥r˜j), then we develop a complementary matrix as P = (p_ij) _m×m, where p_ij ≥ 0, p_ij + p_ji = 1, p_ii = 0.5, i, j = 1, 2, ⋯ , n.

Summing all the elements in each line of matrix P, we have

Step 3. Rank all the alternatives A_i (i = 1, 2, ⋯ , m) and select the best one(s) in accordance with the collective overall preference values p_i (i = 1, 2, ⋯ , m).

5Numerical example

The traditional way of teaching features that teachers impart knowledge to students in classrooms where students may have difficulties in good reception of knowledge due to various factors such as local environment, learning peers and other external interruptions. In fact, every kind of learning must prioritize student’s knowledge acceptance, however, in many classes, the students in the back are more likely to ignore what the teacher have transmitted in classes. In the information era, computer technology has become a great helper in bringing magic to classrooms. In the network setting, web-based learning technology can make learning transcend the constraints of time, place and learning group, facilitate cooperation among learners and bring about the best learning experiences to students. In this process, even students have troubles in understanding the courses, they still can turn to virtual learning environment at their convenience, which can ensure the required knowledge is well received. In this section, we utilize a practical multiple attribute decision making problems for psycholinguistic teaching effect evaluation to illustrate the application of the developed approaches. The expert team selects four attributes to evaluate the five possible college psycholinguistic schools: (1) ➀G₁ is the environment of teaching and studying; ➁G₂ is the management of teaching information; ➂G₃ is the curriculum design and target; ➃G₄ is the empathy and the teaching practice. The five possible five possible college psycholinguistic schools A_i (i = 1, 2, ⋯ , 5) are to be evaluated using the triangular fuzzy numbers by the decision makers under the above four attributes (whose weighting vector is ω = (0.2, 0.1, 0.3, 0.4)), and construct the following matrix A=(a˜ij)5×4 is shown in Table 1.

In the following, in order to select the most desirable college psycholinguistic schools, we utilize the TFWEGBM operator to solve the multiple attribute decision making problems for psycholinguistic teaching effect evaluation with triangular fuzzy information, which can be described as following:

Firstly, aggregate all triangular fuzzy preference value r˜ij(j=1,2,⋯,n) by using the TFWEGBM to derive the overall triangular fuzzy preference values r˜i(i=1,2,3,4,5) of the college psycholinguistic schools A_i.

Then, rank all the college psycholinguistic schools A_i (i = 1, 2, ⋯ , 5) in accordance with the preference degree p_i (i = 1, 2, ⋯ , 5): A₃ ≻ A₄ ≻ A₂ ≻ A₅ ≻ A₁, and thus the most desirable college psycholinguistic school is A₃.

6Conclusion

With the booming of China’s economy in the 21st century, the quality of higher education is facing many challenges, such as the grand scale of development, inadequate investment in higher education, to name a few. New challenges ahead need new moves and policy planning. Among them, how to evaluate the teaching effect becomes a critical point in teaching-learning activities. Moreover, the development of multi-disciplinary science has become a huge opportunity with which we could be enlightened. Colleges and universities must have high-quality education to develop high-quality talent. Teaching supervision plays an important role in evaluating teaching process and teaching quality. Even, it can function to help monitoring school management in colleges. Teaching assessment is a key link of teaching-effect and teaching-management. In this paper, we investigate the multiple attribute decision making problems for evaluating the psycholinguistic teaching effect with triangular fuzzy information. Motivated by the idea of geometric Bonferroni mean and Einstein operations, we develop the triangular fuzzy Einstein geometric Bonferroni mean (TFEGBM) operator and triangular fuzzy weighted Einstein geometric Bonferroni mean (TFWEGBM) operator for aggregating the triangular fuzzy information. Then, based on the TFWEGBM operator, we develop the procedure for multiple attribute decision making with the triangular fuzzy information. Finally, a practical example for evaluating the psycholinguistic teaching effect is given to verify the developed approach. In our future studies, our results may be further extended by using the traditional uncertain and fuzzy decision making theories and techniques [34–49].