Approaches to Multiple Attribute Decision-Making with Fuzzy Number Intuitionistic Fuzzy Information and Their Application to English Teaching Quality Evaluation
Many experts and scholars focus on the Maclaurin symmetric mean (MSM) operator, which can reflect the interrelationship among the multi-input arguments. It has been generalized to different fuzzy environments and put into use in various actual decision problems. The fuzzy number intuitionistic fuzzy numbers (FNIFNs) could well depict the uncertainties and fuzziness during the English teaching quality evaluation. And the English teaching quality evaluation is frequently viewed as the multiple attribute decision-making (MADM) issue. We expand the MSM equation with FNIFNs to propose the fuzzy number intuitionistic fuzzy MSM (FNIFMSM) equation and fuzzy number intuitionistic fuzzy weighted MSM (FNIFWMSM) equation in this study. A few MADM tools are developed with FNIFWMSM equation. Finally, taking English teaching quality evaluation as an example, this paper illustrates the depicted approach.
In 1965, Zadeh  established a novel fuzzy set (FS) to deal with decision information in the fuzzy new domain [2–7]. To extend novel FS, the intuitionistic fuzzy sets (IFSs) [8, 9] were developed. Subsequently, FS and its related extension knowledges are exploited into the more and more decision domains [10–17]. Iakovidis and Papageorgiou  defined the cognitive maps for medical decision making under IFSs. Li  built the GOWA operator to MADM using IFSs. Su et al.  proposed the interactive method for dynamic IF-MAGDM. Tan  constructed the Choquet integral-based TOPSIS method for IF-MADM. Wu and Zhang  built the IF-MADM based on weighted entropy. Yu  defined the generalized prioritized geometric operators under IFSs. Yu et al.  defined the derivatives and differentials for multiplicative IFSs. Zhao et al.  defined the interactive intuitionistic fuzzy algorithms for multilevel programming problems. Arya and Yadav  defined the intuitionistic fuzzy super-efficiency slack-based measure. Büyüközkan et al.  selected the transportation schemes with the integrated intuitionistic fuzzy Choquet integral method. De and Sana  defined the (p, q, r, l) method for random demand with Bonferroni mean under IFSs. Garg  proposed the improved cosine similarity measure for IFSs. Joshi et al.  defined the Jensen-alpha-Norm dissimilarity measure for IFSs. Li et al.  defined the time-preference and VIKOR-based dynamic method for IF-MADM. The authors in  built the intuitionistic fuzzy MABAC method based on cumulative prospect theory for MAGDM. Niroomand  defined the multiobjective-based direct solution method for linear programming along with intuitionistic fuzzy parameters. Zhao et al.  perfected TODIM for IF-MAGDM on the strength of cumulative prospect theory. Furthermore, Liu and Yuan  built the fuzzy number IFSs (FNIFSs) to combine the IFSs with the triangular fuzzy sets (TFSs). Li et al.  developed the entropy and similarity measure under FNIFSs. Wang  built the geometric operators under FNIFSs. Verma  defined the GFNIFWBM operator under FNIFSs.
Nevertheless, all the functions and tools proposed by the above scholars do not take into account the relationship between parameters [38–41]. To conquer these shortcomings, the crucial purpose of the article is to connect the FNIFSs with MSM operator [42–47] to build several novel fused formulas under FNIFSs.
Consequently, the rest work would be depicted. Several basic concepts of FNIFSs and MSM formulas would be depicted in the second section. The MSM formulas with FNIFSs would be constructed in the third section. An instance about English teaching quality evaluation is given in the fourth section. The conclusions reached will be depicted the last section.
In this section, we introduced the concept of fuzzy number intuitionistic fuzzy sets (FNIFSs)  and the Maclaurin symmetric mean (MSM) operator .
2.1. Fuzzy Number Intuitionistic Fuzzy Set
Liu and Yuan  gave the definition of FNIFS, and the membership and nonmembership are given in the form of TFNs.
Definition 1 (see ). Supposed is a fixed set, is a FNIFS on and its expression form is given as follows: and are two TFNs between 0 and 1, and , , and , .
Let , , so , is viewed as a FNIFN.
Definition 2. (see [36, 48]). and are two FNIFNs.(1)(2)(3)(4)
Definition 3. (see [36, 48]). Let be a given FNIFN, a score function of a FNIFN can be depicted as follows:
Definition 4. (see [36, 48]). Let be a given FNIFN, an accuracy function of a FNIFN can be defined as follows:Based on the and , next, let us look at the size comparison of the two FNIFNs.
Definition 5. (see [36, 48]). Let and be two FNIFNs, then if , then ; if , then(1)If , then (2)If , then
2.2. MSM Operators
Maclaurin  proposed the MSM formula.
Definition 6. (see ). Let be is a real number greater than 0, and . Ifthen we call the MSM formula, where traverses all the k-tuple combinations of and is the binomial coefficient.
3. FNIFMSM and FNIFWMSM Operators
3.1. The FNIFMSM Operator
Here, we are going to expand MSM to coalesce all FNIFNs and establish the fuzzy number intuitionistic fuzzy MSM (FNIFMSM) operators.
Definition 7. Let , , be a set of given FNIFNs. The FNIFMSM operator could be depicted as follows:
Theorem 1. , be a suite of given FNIFNs. The coalesced data obtained from FNIFMSM equations are still an FNIFN.
Proof. According to Definition 2, we can deriveThus,Thereafter,Therefore,Hence, (6) is kept.
Then, we need to prove that equation (6) is still an FNIFN. We need to prove two following conditions:①②
Proof. ①Since , we get Then, Thus, That is to say, ; similarly, we can get and , so ① is maintained.②For , we can derive ; thus,
Example 1. Let , , and be three FNIFNs and suppose , then according to (6), we haveNext, we explore some properties about the FNIFMSM formula.
Property 1. (idempotency). If are equal, then
Property 2. (monotonicity). Let , and , be two sets of given FNIFNs. If , hold for all , then
Property 3. (boundedness). Let , be a set of given FNIFNs. If and , then
Property 4. (commutativity). Let be a set of given FNIFNs and be any permutation of , then
3.2. The FNIFWMSM Operator
In real-life MADM, it is crucial to fully take attribute weights into account. We shall build the fuzzy number intuitionistic fuzzy weighted MSM (FNIFWMSM) formula.
Definition 8. Let be a set of given FNIFNs with weight vector and , . Ifthen we called the fuzzy number intuitionistic fuzzy weighted MSM (FNIFWMSM) formula.
Theorem 2. Let be a set of given FNIFNs. The coalesced data obtained from the FNIFWMSM formula are still a FNIFN.
Proof. According to Definition 2, we could deriveThus,Thereafter,Furthermore,Therefore,Hence, (21) is kept.
Then, we could prove that equation (21) is an FNIFN. We need to prove two following conditions:①②
Proof. ①Since , we get Then, Thus, That means ; similarly, we can get , and , so ① is maintained.②For , we can derive ; thus,
Example 2. Let , , and be three given FNIFNs and suppose and , then according to (21), we have