An Online Review-Driven Picture Fuzzy Multi-Criteria Group Decision-Making Approach for Evaluating the Online Medical Service Quality of Doctors

Abstract

In order to further investigate the level of online medical services in China and improve the medical experience of patients, this study aims to establish an online review-driven picture fuzzy multi-criteria group decision-making (MCGDM) approach for the online medical service evaluation of doctors. First, based on the Aczel–Alsina t-norm and t-conorm, the normal picture fuzzy Aczel–Alsina operations involving a variable parameter are defined to make the corresponding operations more flexible than other operations. Second, two picture fuzzy Aczel–Alsina aggregation operators are developed, and the corresponding properties are discussed as well. Third, combined with the online review information of China’s medical platform Haodaifu, the online review-driven evaluation attributes and their corresponding weights are obtained, which can make the evaluation model more objective. Fourth, an extended normal picture fuzzy complex proportional assessment (COPRAS) decision-making method for the service quality evaluation of online medical services is proposed. Finally, an empirical example is presented to verify the feasibility and validity of the proposed method. A sensitivity analysis and a comparison analysis are also conducted to demonstrate the effectiveness and flexibility of the proposed approach.

1. Introduction

With the innovation and development of information technology, online medical communications and information for doctors and patients have provided a new way to alleviate the imbalance in the supply and demand of medical resources in China. The establishment of symmetry between doctors and patients needs a comprehensive exchange of information. It is necessary for doctors to gain a clear and accurate understanding of patients’ medical history information. Simultaneously, patients also require evaluation methods that can precisely evaluate doctors’ medical service quality. The equal exchange of information between doctors and patients can enhance the precision and effectiveness of treatment, and improve the satisfaction of patients simultaneously. Thus, it is necessary to evaluate the service quality of online medical treatment to achieve sustainable development. Generally, the service quality of doctors’ online medical treatment is related to multiple factors, such as professional ability, service attitude, follow-up, communication ability, etc. Thus, the evaluation of the online medical service quality of doctors (OMSQD) can be considered a multi-criteria decision-making (MCDM) problem.

Recently, some scholars have studied evaluation methods for assessing medical services. For example, Li et al. [1] proposed an Internet-based diagnosis and treatment system for doctors based on the iceberg model and analyzed the relevant influencing factors. Turk et al. [2] used descriptive statistics and Chi-square tests to examine demographic differences between those doctors from online comments. Xiong et al. [3] used text mining and Affinity Propagation (AP) clustering to construct a table of service feature words and calculated the weight of the service features of each doctor. Gong et al. [4] proposed a Time-constraint Probability Factor Graph model (TPFG) to determine the relationship between doctors and patients in online reviews and ranked nodes according to network characteristics.

1.1. Research Gaps and Motivations

From the existing evaluation methods for OMSQD, some research gaps and motivations can be outlined as follows. Firstly, the existing methods of obtaining preference information cannot accurately describe the ambiguity and uncertainty of decision-makers’ (DMs’) preferences. However, picture fuzzy sets (PFSs) can express DMs’ opinions and uncertainties from different perspectives, including yes, abstain, no, and refusal. Normal picture fuzzy sets (NPFSs) can combine the advantages of PFSs and normal distribution simultaneously, which can enhance the comprehensiveness of uncertainty information. Secondly, most existing evaluation methods for online medical services determine the evaluation attributes and their weights subjectively based on database retrieval and experts [1,2], as well as real healthcare data [4]. Thus, it is necessary to explore the online review-driven weight determination method to improve the objectivity of its evaluation. Finally, the Aczel–Alsina T-norm and T-conorm can offer an additional parameter $\lambda$ [5] to increase the variability and flexibility of aggregation operators, which have excellent data-gathering ability for obtaining DMs’ preference information. Thus, the complex proportional assessment (COPRAS) method based on normal picture fuzzy Aczel–Alsina aggregation operators has the advantage of comprehensively evaluating the quality of online medical services.

1.2. Contributions

Based on these research gaps and motivations, this paper proposed an online review-driven picture fuzzy multi-criteria group decision-making (MCGDM) approach for evaluating OMSQD problems. The main contributions can be summarized as follows:

(1)

The NPFSs mentioned and explained above are used to describe DMs’ preference information.

(2)

Aczel–Alsina operations for normal picture fuzzy numbers (NPFNs) are defined and the corresponding properties are discussed.

(3)

The normal picture fuzzy Aczel–Alsina aggregation operators, including the normal picture fuzzy Aczel–Alsina weighted average operator (NPFAAWA) and normal picture fuzzy Aczel–Alsina weight geometric (NPFAAWG) operator, are developed. The corresponding properties are also explored as well. The two proposed operators involving a variable parameter can make the corresponding operations more flexible compared to other existing operators.

(4)

The online review-driven weight determination method is constructed. Then, the evaluation criteria and their corresponding weights for OMSQD are obtained objectively.

(5)

The extend COPRAS method based on normal picture fuzzy Aczel–Alsina aggregation operators is proposed, which can be more effective to evaluate OMSQD problems.

The remainder of this paper proceeds as follows. Section 2 presents a literature review. Section 3 introduces some definitions related to NPFSs. Section 4 defines novel operation rules for NPFNs and normal picture fuzzy Aczel–Alsina aggregation operators, discussing the corresponding properties. Section 5 shows the online review-driven weight determination method. Section 6 proposes the group decision-making framework based on defined aggregation operators. An empirical example for doctor evaluation is applied in Section 7. Additionally, comparative analyses and a sensitivity analysis are also performed to further prove the effectiveness of the proposed method. Section 8 provides some potential managerial implications. Finally, Section 9 presents our conclusions.

2. Literature Review

2.1. Fuzzy Sets and Extensions

Multi-attribute group decision-making (MCGDM) refers to the preferred decision-making processes of multiple DMs’ in selection, ranking, and evaluating based on multiple unrelated attributes. However, because of the complexity of the decision-making environment and the ambiguity and uncertainty of DMs’ preference information, tradition fuzzy sets (FSs) [6,7] and their extensions, including spherical fuzzy sets [8,9], complemental fuzzy sets [10], Z-numbers [11,12,13], intuitionistic fuzzy sets (IFSs) [14,15,16,17,18,19,20], hesitant fuzzy sets (HFSs) [21,22,23,24,25,26], Pythagorean fuzzy sets [27,28], disc Pythagorean fuzzy sets [29], Fermatean fuzzy sets [30,31,32,33], interval-values fuzzy sets [34,35], and q-rung orthopair fuzzy sets [36,37], have been applied to solve MCGDM problems.

2.2. Picture Fuzzy Sets and Normal Picture Fuzzy Sets

As decision-making environments and decision-making information become more and more complex, traditional FSs and their extensions cannot meet the needs of decision-makers. Based on this, Cuong and Kreinovich [38] defined PFSs, which are an extension of IFSs and contain positive, neutral, and negative membership degrees. PFSs can describe uncertainty information from different aspects, and can be considered as an excellent tool for the expression and processing of DMs’ opinions. In recent years, the research on picture fuzzy MCGDM methods has mainly been divided into two categories. First, many scholars have studied MCGDM methods based on picture fuzzy measures, including similarity measures [39,40,41,42,43,44], distance measures [45], projection measures [46,47], entropy measures [48,49,50,51,52], and dissimilarity measures [53]. Second, some MCGDM methods based on picture fuzzy aggregation operators have also been explored [54,55]. For example, Jana et al. [56] defined the picture fuzzy weighted Dombi average and geometric operators based on an arithmetic operation, and these operators were applied to solve the MCDM problem. Khan et al. [55] proposed a logarithmic picture fuzzy weighted average (WA) operator and weighted geometric (WG) operator. Moreover, some scholars have studied the other extensions of PFSs [57,58]. For instance, the fuzzy concept characterized by a normal membership function exhibited closer alignment with human cognition [59]. Li and Liu [60] pointed out that normal fuzzy numbers (NFNS) have more advantages compared with other existing extensions. Then, Yang et al. [61] developed q-rung picture normal fuzzy environment by using the Heronian mean operator. Nguyen et al. [57] discussed the limitations of picture fuzzy numbers (PFNs) in a metric semi-linear space. Akram et al. [62] used q-rung picture fuzzy information to solve MCDM problems.

2.3. COPRAS Method

The complex proportional assessment (COPRAS) method, initially proposed by Zavadskas et al. [63], was utilized to cope with information more efficiently. COPRAS is an MCDM method which can determine the optimal alternative by comparing the performance of different alternatives under various attributes. Specifically, COPRAS utilizes a comprehensive performance index that combines the weights of different attributes and the scores of each alternative to evaluate the overall performance of each alternative. Recently, some scholars have applied it to solve various problems [64,65,66,67,68]. EK Zavadskas et al. [69] developed a methodology based on COPRAS for multi-attribute assessment for road construction. Ch et al. [70] proposed an MCDM method which combined COPRAS and grey relation analysis to measure the usage of websites. Kumar et al. [71] determined the best parameter combination by using inter-criteria correlation and COPRAS hybrid techniques in a solar air-heating system.

3. Preliminaries

In this section, some related concepts are introduced, including the definitions of PFSs, NPFSs, and the Aczel–Alsina T-norm and T-conorm. The score function and accuracy function of NPFSs are also reviewed.

Definition 1

([38]). The PFS T in universe X is determined as follows:

$$T=\left\{\langle x,{\mathsf{\mu }}_T(x),{\mathsf{\eta }}_T(x),{\mathsf{\nu }}_T(x)\rangle |x\in X\right\},$$

(1)

where ${\mathsf{\mu }}_T(x),{\mathsf{\eta }}_T(x),{\mathsf{\nu }}_T(x)\in [0,1]$ and satisfies $0\le {\mathsf{\mu }}_T(x)+{\mathsf{\eta }}_T(x)+{\mathsf{\nu }}_T(x)\le 1$ for any $x\in X$. The functions ${\mathsf{\mu }}_T(x),{\mathsf{\eta }}_T(x)$, and ${\mathsf{\nu }}_T(x)$ denote the positive membership degree, neutral membership degree, and negative membership degree of $X$ in $T$, respectively. Then, for any $x\in X$, ${\mathsf{\pi }}_T(x)=1-{\mathsf{\mu }}_T(x)-{\mathsf{\eta }}_T(x)-{\mathsf{\nu }}_T(x)$ represents the refusal membership degree of $X$ in $T$.

Based on the definition of q-rung normal picture fuzzy sets [72], the concept of NPFSs is provided in the following. Moreover, if $q=1$, then a q-rung normal picture fuzzy set will be reduced to a normal picture fuzzy set (NPFS).

Definition 2

([72]). Let X be an ordinary fixed non-empty set, and $\left(\mathsf{\alpha },\mathsf{\sigma }\right)\in N$; the NPFS A in X is defined as follows:

$$A\left(x\right)=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_A(x),{\mathsf{\eta }}_A(x),{\mathsf{\nu }}_A(x)\rangle .$$

(2)

Its positive membership function is defined as

$${\mathsf{\zeta }}_A\left(x\right)={\mathsf{\mu }}_A{e}^{-{\left(\frac{x-\mathsf{\alpha }}{\mathsf{\sigma }}\right)}^2},x\in X$$

(3)

Its neutral membership function is defined as

$${\mathsf{\phi }}_A\left(x\right)=1-(1-{\mathsf{\eta }}_A)e^{-{\left(\frac{x-\mathsf{\alpha }}{\mathsf{\sigma }}\right)}^2},x\in X$$

(4)

Its negative membership function is defined as

$${\mathsf{\vartheta }}_A\left(x\right)=1-(1-{\mathsf{\nu }}_A)e^{-{\left(\frac{x-\mathsf{\alpha }}{\mathsf{\sigma }}\right)}^2},x\in X$$

(5)

For any $x$ in the domain $X$, there exists ${\mathsf{\mu }}_T(x),{\mathsf{\eta }}_T(x),{\mathsf{\nu }}_T(x)\in [0,1]$ and $0\le {\mathsf{\mu }}_T(x)+{\mathsf{\eta }}_T(x)$$+{\mathsf{\nu }}_T(x)\le 1$.

Definition 3.

Assume that  $T_1$  and $T_2$ are two NPFNs; then, the comparison operations can be defined as follows:

(1) 

If  $S_1(T_1)>S_1(T_2)$, then $T_1>T_2$;

(2) 

If  $S_1(T_1)=S_1(T_2)$, then

• if  $H_1(T_1)>H_1(T_2)$, then  $T_1>T_2$;
• If  $H_1(T_1)=H_1(T_2)$, then
  • if  $S_2(T_1)<S_2(T_2)$, then  $T_1>T_2$;
  • If  $S_2(T_1)=S_2(T_2)$, then
    • if  $H_2(T_1)<H_2(T_1)$, then  $T_1>T_2$;
    • If $H_2(T_1)<H_2(T_1)$, then $T_1=T_2$.

Where $S_i(T_i)\left(i=1,2\right)$ and $H_i\left(T_i\right)\left(i=1,2\right)$ denote the score function and accuracy function, respectively, and

$$\begin{array}{l}{S}_1(T_i)={\mathsf{\alpha }}_{T_i}\left({\mathsf{\mu }}_{T_i}+1-{\mathsf{\eta }}_{T_1}+1-{\mathsf{\nu }}_{T_1}\right),S_2(T_i)={\mathsf{\sigma }}_{T_i}\left({\mathsf{\mu }}_{T_i}+1-{\mathsf{\eta }}_{T_1}+1-{\mathsf{\nu }}_{T_1}\right),\\ H_1(T_i)={\mathsf{\alpha }}_{T_i}\left({\mathsf{\mu }}_{T_i}+1-{\mathsf{\eta }}_{T_1}+1+{\mathsf{\nu }}_{T_1}\right),H_2(T_i)={\mathsf{\alpha }}_{T_i}\left({\mathsf{\mu }}_{T_i}+1-{\mathsf{\eta }}_{T_1}+1+{\mathsf{\nu }}_{T_1}\right).\end{array}$$

(6)

Definition 4

([5]). Assume that the product of t-norm (TNs) can be written as ${\mathrm{A}}_P$, and ${\mathrm{A}}_M$ denotes the Minimum TN. The addition of t-conorm (TCN) can be written as ${\mathrm{B}}_P$, and the Maximum TCN can be written as ${\mathrm{B}}_M$. Then, the drastic TN ${\mathrm{A}}_D$ and TCN ${\mathrm{B}}_D$ will be denoted, respectively, as

$${\mathrm{A}}_D=\left(a,b\right)=\{\begin{array}{l}a,b=0\\ b,\lambda =\infty \forall a,b\in \left[0,1\right]\\ 1\end{array},$$

(7)

$${\mathrm{B}}_D=\left(a,b\right)=\{\begin{array}{l}a,b=0\\ b,a=0\forall a,b\in \left[0,1\right]\\ 1\end{array},$$

(8)

$${\mathrm{B}}_M=\max \left(a,b\right),{\mathrm{A}}_M=\min (a,b),$$

(9)

$${\mathrm{A}}_P=a·b,{\mathrm{B}}_P=a+b-a·b.$$

(10)

Definition 5

([5]). The Aczel–Alsina t-norm (AATN) and t-conorm (AATCN) can be defined as

$${\mathrm{A}}_A^{\lambda }=\{\begin{array}{l}{\mathrm{A}}_D(a,b),\lambda =0\\ \min (a,b),\lambda =\infty \\ e^{-{({(-\mathrm{In}\left(\mathrm{a}\right))}^{\lambda }+{(-\mathrm{In}\left(\mathrm{b}\right))}^{\lambda })}^{\frac{1}{\lambda }}}\end{array},$$

(11)

$${\mathrm{B}}_A^{\lambda }=\{\begin{array}{l}{\mathrm{B}}_D(a,b),\lambda =0\\ \max (a,b),\lambda =\infty \\ 1-e^{-{({(-\mathrm{In}(1-\mathrm{a}))}^{\lambda }+{(-\mathrm{In}(1-\mathrm{b}))}^{\lambda })}^{\frac{1}{\lambda }}}\end{array}.$$

(12)

where ${\mathrm{A}}_A^0={\mathrm{A}}_D\left(a,b\right),{\mathrm{A}}_A^1={\mathrm{A}}_P,{\mathrm{A}}_A^{\infty }=\min ,{\mathrm{B}}_A^0={\mathrm{B}}_D\left(a,b\right),{\mathrm{B}}_A^1={\mathrm{B}}_P,{\mathrm{B}}_A^{\infty }=\max$ describe the behavior of AATN and AATCN at different values of $\mathsf{\lambda }$. These equations can be adjusted according to a single parameter, allowing for more flexibility to deal with uncertainty and ambiguity in the process of aggregating fuzzy information.

4. Normal Picture Fuzzy Aczel–Alsina Aggregation Operators

In this section, based on the Aczel–Alsina T-norm and T-conorm, the operations for two NPFNs are defined, and the properties are discussed. Then, the corresponding normal picture fuzzy Aczel–Alsina aggregation operators, including the NPFAAWA and NPFAAWG operators, are developed. The corresponding properties are also explored in the following.

4.1. Aczel–Alsina Operations for NPFNs

Definition 6.

Assume that $T_1=\langle \left({\mathsf{\alpha }}_1,{\mathsf{\beta }}_1\right),{\mathsf{\mu }}_1,{\mathsf{\eta }}_1,{\mathsf{\nu }}_1\rangle$ and $T_2=\langle \left({\mathsf{\alpha }}_2,{\mathsf{\beta }}_2\right),{\mathsf{\mu }}_2,{\mathsf{\eta }}_2,{\mathsf{\nu }}_2\rangle$ are two NPFNs; then, the operations for two NPFNs can be defined as

$$\begin{array}{l}(1)\hspace{0.17em}{T}_1\oplus T_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\mathsf{\alpha }}_1+{\mathsf{\alpha }}_2,{\mathsf{\sigma }}_1+{\mathsf{\sigma }}_2\right),e^{-{\left({\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }+{(-\mathrm{In}(1-{\mathsf{\eta }}_{T_2}))}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \end{array};$$

(13)

$$\begin{array}{l}(2)\hspace{0.17em}{T}_1\otimes T_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\mathsf{\alpha }}_1{\mathsf{\alpha }}_2,{\mathsf{\alpha }}_1{\mathsf{\alpha }}_2\sqrt{\frac{{\mathsf{\sigma }}_1^2}{{\mathsf{\alpha }}_1^2}+\frac{{\mathsf{\sigma }}_2^2}{{\mathsf{\alpha }}_2^2}}\right),1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left({\left(-\mathrm{In}\left({\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left({\mathsf{\nu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \end{array};$$

(14)

$$(3)\hspace{0.17em}kT=\langle \left(k\mathsf{\alpha },k\mathsf{\sigma }\right),e^{-{\left(k{\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle ;$$

(15)

$$(4)\hspace{0.17em}{T}^k=\langle \left({\mathsf{\alpha }}^k,k^{\frac{1}{2}}{\mathsf{\alpha }}^{k-1}\mathsf{\sigma }\right),1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k{\left(-\mathrm{In}\left({\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle .$$

(16)

Theorem 1.

Assume that $T_1=\langle \left({\mathsf{\alpha }}_1,{\mathsf{\beta }}_1\right),{\mathsf{\mu }}_1,{\mathsf{\eta }}_1,{\mathsf{\nu }}_1\rangle$ and $T_2=\langle \left({\mathsf{\alpha }}_2,{\mathsf{\beta }}_2\right),{\mathsf{\mu }}_2,{\mathsf{\eta }}_2,{\mathsf{\nu }}_2\rangle$ are two NPFNs, and $k_1,k_2>0$; then, we have the following:

$$\left(1\right)T_1\oplus T_2=T_2\oplus T_1;$$

(17)

$$\left(2\right)T_1\otimes T_2=T_2\otimes T_1;$$

(18)

$$\left(3\right)k_{1}T\oplus k_{2}T=\left(k_1+k_2\right)T;$$

(19)

$$\left(4\right)k(T_1\oplus T_2)=k{T}_1\oplus k{T}_2;$$

(20)

$$\left(5\right)T^{k_1+k_2}=T^{k_1}\otimes T^{k_2}$$

(21)

$$(6)T_1^{k_1}\otimes T_2^{k_1}={\left(T_1\otimes T_2\right)}^{k_1};$$

(22)

Proof. 

Since properties (1) and (2) are obviously held, then properties (3)–(6) are proven in the following.

$$\begin{array}{l}(3)\hspace{0.17em}{k}_{1}T\oplus k_{2}T\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left(k_1\mathsf{\alpha },k_1\mathsf{\sigma }\right),e^{-{\left(k_1{\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \oplus \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\langle \left(k_2\mathsf{\alpha },k_2\mathsf{\sigma }\right),e^{-{\left(k_2{\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_2{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_2{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left(\left(k_1+k_2\right)\mathsf{\alpha },\left(k_1+k_2\right)\mathsf{\sigma }\right),e^{-{\left(k_1{\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }+k_2{\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }+k_2{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }+k_2{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\left(k_1+k_2\right)T.\end{array}$$

$$\begin{array}{l}(4)\hspace{0.17em}k\left(T_1\oplus T_2\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=k\langle \left({\mathsf{\alpha }}_1+{\mathsf{\alpha }}_2\right),\left({\mathsf{\sigma }}_1+{\mathsf{\sigma }}_2\right),e^{-{\left({\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left(k\left({\mathsf{\alpha }}_1+{\mathsf{\alpha }}_2\right),k\left({\mathsf{\sigma }}_1+{\mathsf{\sigma }}_2\right)\right),e^{-{\left(k{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }+k{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }+k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }+k{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left(k{\mathsf{\alpha }}_1,k{\mathsf{\sigma }}_1\right),e^{-{\left(k{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \oplus \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\langle \left(k{\mathsf{\alpha }}_2,k{\mathsf{\sigma }}_2\right),e^{-{\left(k{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle =k{T}_1\oplus k{T}_2.\end{array}$$

$$\begin{array}{l}(5)\hspace{0.17em}{T}^{k_1+k_2}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\mathsf{\alpha }}^{k_1+k_2},{\left(k_1+k_2\right)}^{\frac{1}{2}}{\mathsf{\alpha }}^{\left(k_1+k_2\right)-1}\mathsf{\sigma }\right),1-e^{-{\left(\left(k_1+k_2\right){\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(\left(k_1+k_2\right){(-\mathrm{In}(1-{\mathsf{\eta }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(\left(k_1+k_2\right){(-\mathrm{In}({\mathsf{\nu }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\mathsf{\alpha }}^{k_1}{\mathsf{\alpha }}^{k_2},{\mathsf{\alpha }}^{k_1}{\mathsf{\alpha }}^{k_2}\sqrt{\frac{k_1{\mathsf{\alpha }}^{2k_1-2}{\mathsf{\sigma }}^2}{{\mathsf{\alpha }}^{2k_1}}+\frac{k_2{\mathsf{\alpha }}^{2k_2-2}{\mathsf{\sigma }}^2}{{\mathsf{\alpha }}^{2k_2}}}\right),1-e^{-{\left(\left(k_1+k_2\right){(-\mathrm{In}(1-{\mathsf{\mu }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(\left(k_1+k_2\right){(-\mathrm{In}(1-{\mathsf{\eta }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(\left(k_1+k_2\right){(-\mathrm{In}({\mathsf{\nu }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\mathsf{\alpha }}^{k_1},{k_1}^{\frac{1}{2}}{\mathsf{\alpha }}^{k_1-1}\mathsf{\sigma }\right),1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{(-\mathrm{In}(1-{\mathsf{\eta }}_T))}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_1{\left(-\mathrm{In}\left({\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \otimes \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\langle \left({\mathsf{\alpha }}^{k_2},{k_2}^{\frac{1}{2}}{\mathsf{\alpha }}^{k_2-1}\mathsf{\sigma }\right),1-e^{-{\left(k_2{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_2{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_2{\left(-\mathrm{In}\left({\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle =T^{k_1}\otimes T^{k_2}.\end{array}$$

$$\begin{array}{l}(6)\hspace{0.17em}{T_1}^{k_1}\otimes {T_2}^{k_1}=\langle \left({{\alpha }_1}^{k_1},{k_1}^{\frac{1}{2}}{{\alpha }_1}^{k_1-1}{\sigma }_1\right),1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mu }_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\eta }_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_1{\left(-\mathrm{In}\left({\nu }_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \otimes \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\langle \left({{\alpha }_2}^{k_1},{k_1}^{\frac{1}{2}}{{\alpha }_2}^{k_1-1}{\sigma }_2\right),1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\eta }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_1{\left(-\mathrm{In}\left({\nu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({{\alpha }_1}^{k_1}{{\alpha }_2}^{k_1},{{\alpha }_1}^{k_1}{{\alpha }_2}^{k_1}\sqrt{\frac{k_1{{\alpha }_1}^{2k_1-2}{\sigma }_1^2}{{\alpha }_1^{2k_1}}+\frac{k_1{{\alpha }_2}^{2k_1-2}{\sigma }_2^2}{{\alpha }_2^{2k_1}}}\right),1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\mu }_{T_1}\right)\right)}^{\lambda }+k_1{\left(-\mathrm{In}\left(1-{\mu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1-e^{-{\left(k_1{\left(-\mathrm{In}\left(1-{\eta }_{T_1}\right)\right)}^{\lambda }+k_1{\left(-\mathrm{In}\left(1-{\eta }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_1{\left(-\mathrm{In}\left({\nu }_{T_1}\right)\right)}^{\lambda }+k_1{\left(-\mathrm{In}\left({\nu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\left({\alpha }_1{\alpha }_2\right)}^{k_1},{\left({\alpha }_1{\alpha }_2\right)}^{k_1}{k_1}^{\frac{1}{2}}\sqrt{\frac{{\sigma }_1^2}{{\alpha }_1^2}+\frac{{\sigma }_2^2}{{\alpha }_2^2}}\right),1-e^{-{\left(k_1\left({\left(-\mathrm{In}\left(1-{\mu }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mu }_{T_2}\right)\right)}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}1-e^{-{\left(k_1\left({\left(-\mathrm{In}\left(1-{\eta }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\eta }_{T_2}\right)\right)}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}},e^{-{\left(k_1\left({\left(-\mathrm{In}\left({\nu }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left({\nu }_{T_2}\right)\right)}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}}\rangle \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=\langle \left({\alpha }_1{\alpha }_2,{\alpha }_1{\alpha }_2\sqrt{\frac{{\sigma }_1^2}{{\alpha }_1^2}+\frac{{\sigma }_2^2}{{\alpha }_2^2}}\right),1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mu }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\mu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\eta }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left(1-{\eta }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{e^{-{\left({\left(-\mathrm{In}\left({\nu }_{T_1}\right)\right)}^{\lambda }+{\left(-\mathrm{In}\left({\nu }_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle }^k={\left(T_1\otimes T_2\right)}^k.\end{array}$$

□

4.2. Normal Picture Fuzzy Aczel–Alsina Aggregation Operators

Based on the Aczel–Alsina operations for NPFNs, the NPFAAWA and NPFAAWG operators are defined.

Definition 7.

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs, and NPFAAWA: $NPF{N}^n\to NPFN$; then,

$$NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=\underset{i=1}{\overset{n}{\oplus }}{\mathsf{\omega }}_i{T}_i.$$

(23)

${\mathsf{\omega }}_i={\left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,\dots ,{\mathsf{\omega }}_i\right)}^T$ is the weight vector of $T_i(i=1,2,\dots ,n)$, ${\mathsf{\omega }}_i>0$, and ${\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i}=1$.

Theorem 2.

Let  $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs; then, the result aggregated by the NPFAAWA operator is also an NPFN, and

$$\begin{array}{l}NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\\ =\langle \left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i{\mathsf{\alpha }}_i},{\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}\right),e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle \end{array}.$$

(24)

Proof. 

For $n=2$, by using the operation defined in Definition 6,

$${\mathsf{\omega }}_1{T}_1=\langle \left({\mathsf{\omega }}_1{\mathsf{\alpha }}_1,{\mathsf{\omega }}_1{\mathsf{\sigma }}_1\right),e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle ,$$

$${\mathsf{\omega }}_2{T}_2=\langle \left({\mathsf{\omega }}_2{\mathsf{\alpha }}_2,{\mathsf{\omega }}_2{\mathsf{\sigma }}_2\right),e^{-{\left({\mathsf{\omega }}_2{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_2{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_2{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle .$$

We obtain

$$NPFAAWA\left(T_1,T_2\right)={\mathsf{\omega }}_{1}T\oplus {\mathsf{\omega }}_2{T}_2=\langle \begin{array}{l}\left({\mathsf{\omega }}_1{\mathsf{\alpha }}_1+{\mathsf{\omega }}_2{\mathsf{\alpha }}_2,{\mathsf{\omega }}_1{\mathsf{\sigma }}_1+{\mathsf{\omega }}_2{\mathsf{\sigma }}_2\right),e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_1}\right)\right)}^{\lambda }+{\mathsf{\omega }}_2{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},\\ 1-e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_1}\right)\right)}^{\lambda }+{\mathsf{\omega }}_2{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_1{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_1}\right)\right)}^{\lambda }+{\mathsf{\omega }}_2{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_2}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\end{array}\rangle$$

If $n=k$, we have

$$\begin{array}{l}NPFAAWA\left(T_1,T_2,\dots ,T_m\right)\\ =\langle \left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\alpha }}_i},{\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}\right),e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle \end{array}$$

Then, for $n=k+1$, we have

$$\begin{array}{l}NPFAAWA\left(T_1,T_2,\dots ,T_m,T_{m+1}\right)=NPFAAWA\left(T_1,T_2,\dots ,T_m\right)\oplus {\mathsf{\omega }}_{m+1}{T}_{m+1}\\ =\langle \left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\alpha }}_i},{\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}\right),e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle \\ \oplus \langle \left({\mathsf{\omega }}_{m+1}{\mathsf{\alpha }}_{m+1},{\mathsf{\omega }}_{m+1}{\mathsf{\sigma }}_{m+1}\right),e^{-{\left({\mathsf{\omega }}_{m+1}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{m+1}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_{m+1}{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_{m+1}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\mathsf{\omega }}_{m+1}{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_{m+1}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ =\langle \left({\displaystyle \sum _{i=1}^{m+1}{\mathsf{\omega }}_i{\mathsf{\alpha }}_i},{\displaystyle \sum _{i=1}^{m+1}{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}\right),e^{-{\left({\displaystyle \sum _{i=1}^{m+1}{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{m+1}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{m+1}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle .\end{array}$$

It is true for $n=k+1$. Thus, Theorem 2 is true. □

Further, we will explore the properties of the NPFAAWA operator in the following.

Property 1 (Idempotency).

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs. If $T_i=T=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$, then

$$NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=T$$

(25)

Proof. 

$$\begin{array}{l}NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\\ =\langle \left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i\mathsf{\alpha }},{\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i\mathsf{\sigma }}\right),e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle \\ =\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),e^{-{\left({\left(-\mathrm{In}\left({\mathsf{\mu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_T\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ =\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle =T.\end{array}$$

Then, $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=T$ holds. □

Property 2 (Boundedness).

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs, and assume that $T^{+}=\max (T_1,T_2,\dots ,T_i)$ and $T^{-}=\min (T_1,T_2,\dots ,T_i)$; then, we have

$$T^{-}\le NPFAAWA\left(T_1,T_2,\dots ,T_3\right)\le T^{+}$$

(26)

Proof. 

Since $T^{+}=\max (T_1,T_2,\dots ,T_i)$ and $T^{-}=\min (T_1,T_2,\dots ,T_i)$, then $T^{-}=\langle \left(\min {\mathsf{\alpha }}_{\mathit{ii}=1}^n,\max {\mathsf{\sigma }}_{\mathit{ii}=1}^n\right),\min {\mathsf{\mu }}_{\mathit{ii}=1}^n,\max {\mathsf{\eta }}_{\mathit{ii}=1}^n,\max {\mathsf{\nu }}_{\mathit{ii}=1}^n\rangle$ and $T^{+}=\langle \left(\max {\mathsf{\alpha }}_{\mathit{ii}=1}^n,\min {\mathsf{\sigma }}_{\mathit{ii}=1}^n\right),\max {\mathsf{\mu }}_{\mathit{ii}=1}^n,\min {\mathsf{\eta }}_{\mathit{ii}=1}^n,\min {\mathsf{\nu }}_{\mathit{ii}=1}^n\rangle$.

Because $\min {\mathsf{\alpha }}_{\mathit{ii}=1}^n\le a_i\le \max {\mathsf{\alpha }}_{\mathit{ii}=1}^n,\min {\mathsf{\sigma }}_{\mathit{ii}=1}^n\le {\mathsf{\sigma }}_i\le \max {\mathsf{\sigma }}_{\mathit{ii}=1}^n,$ and $\min {\mathsf{\mu }}_{\mathit{ii}=1}^n\le {\mathsf{\mu }}_i\le \max {\mathsf{\mu }}_{\mathit{ii}=1}^n,$, assume that $a_{\max }=\max {\mathsf{\alpha }}_{\mathit{ii}=1}^n,a_{\min }=\min {\mathsf{\alpha }}_{\mathit{ii}=1}^n$${\mathsf{\alpha }}_{\min }=\min {\mathsf{\alpha }}_{\mathit{ii}=1}^n,{\mathsf{\sigma }}_{\min }=\min {\mathsf{\sigma }}_{\mathit{ii}=1}^n$. Then, we have ${\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\mathsf{\alpha }}_{\min }\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{a}_i\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{a}_{\max }$ and ${\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\mathsf{\sigma }}_{\min }\le$ ${\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\mathsf{\sigma }}_i\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\mathsf{\sigma }}_{\max }$.

(a)

For the positive membership degree of $NPFAAWA\left(T_1,T_2,\dots ,T_3\right)$,

$$\begin{array}{l}\min {\mathsf{\mu }}_{\mathit{ii}=1}^n\le {\mathsf{\mu }}_i\le \max {\mathsf{\mu }}_{\mathit{ii}=1}^n\\ \Rightarrow {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left(\max {\mathsf{\mu }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_i\right)\right)}^{\lambda }\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left(\min {\mathsf{\mu }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }\\ \Rightarrow e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left(\min {\mathsf{\mu }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_i\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left(\max {\mathsf{\mu }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\\ \Rightarrow \min {\mathsf{\mu }}_{\mathit{ii}=1}^n\le e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_i\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\le \max {\mathsf{\mu }}_{\mathit{ii}=1}^n\end{array}$$

(b)

For the neutral membership degree of $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)$,

$$\begin{array}{l}1-\max {\mathsf{\eta }}_{\mathit{ii}=1}^n\le 1-{\mathsf{\eta }}_i\le 1-\min {\mathsf{\eta }}_{\mathit{ii}=1}^n\\ \Rightarrow \mathrm{In}\left(1-\max {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\le \mathrm{In}\left(1-{\mathsf{\eta }}_i\right)\le \mathrm{In}\left(1-\min {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\\ \Rightarrow e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-\max {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_i\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-\min {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\\ \Rightarrow 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-\min {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_i\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-\max {\mathsf{\eta }}_{\mathit{ii}=1}^n\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\\ \Rightarrow \min {\mathsf{\eta }}_{\mathit{ii}=1}^n\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_i\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le \max {\mathsf{\eta }}_{\mathit{ii}=1}^n\end{array}$$

(c)

For the negative membership degree of $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)$, we can also obtain

$$\min {\mathsf{\nu }}_{\mathit{ii}=1}^n\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_i\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le \max {\mathsf{\nu }}_{\mathit{ii}=1}^n$$

Then, if $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=T$, we can obtain

$$T=\langle \left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i\mathsf{\alpha }},{\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i\mathsf{\sigma }}\right),e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle$$

and

$$S_1\left(T\right)=\mathsf{\alpha }\left(e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}+e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}+e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^{\mathrm{n}}{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\right)$$

According to Definition 3, we have

$$\begin{array}{l}{S}_1\left(T\right)\le a_{\max }\left(2+\max {\mathsf{\mu }}_{\mathit{ii}=1}^n-\min {\mathsf{\eta }}_{\mathit{ii}=1}^n-\min {\mathsf{\nu }}_{\mathit{ii}=1}^n\right)=T^{+}\\ T^{-}=a_{\min }\left(2+\min {\mathsf{\mu }}_{\mathit{ii}=1}^n-\max {\mathsf{\eta }}_{\mathit{ii}=1}^n-\max {\mathsf{\nu }}_{\mathit{ii}=1}^n\right)\le S_1\left(T\right)\end{array}$$

So, $T^{-}\le NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\le T^{+}$ is true. □

Property 3 (Commutativity).

Let  $T_i^{\prime }\left(i=1,2,\dots ,n\right)$ be any permutation of $T_i\left(i=1,2,\dots ,n\right)$. Then,

$$NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right).$$

(27)

Proof. 

Since

$$\begin{array}{l}NPFAAWA\left(T_1,T_2,\dots ,T_m\right)\\ =\langle \left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\alpha }}_i},{\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}\right),e^{-{\left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle ,\end{array}$$

and $T_i^{\prime }\left(i=1,2,\dots ,n\right)$ is any permutation of $T_i\left(i=1,2,\dots ,n\right)$,

$$\begin{array}{l}NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)\\ =\langle \left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }{\mathsf{\alpha }}_i^{\prime }},{\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }{\mathsf{\sigma }}_i^{\prime }}\right),e^{-{\left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i^{\prime }}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle ,\end{array}$$

we have ${\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }{\mathsf{\alpha }}_i^{\prime }}$$={\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\alpha }}_i}$,${\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }{\mathsf{\sigma }}_i^{\prime }}={\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i{\mathsf{\sigma }}_i}$. Moreover, $e^{-{\left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}=e^{-{\left({\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i^{\prime }}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i^{\prime }}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}$, $1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}=1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}$, $1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}=1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}$.

Thus, $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)=NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)$ is true. □

Property 4 (Monotonicity).

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle$ and $T_i^{\prime }=\langle \left({\mathsf{\alpha }}_i^{\prime },{\mathsf{\sigma }}_i^{\prime }\right),{\mathsf{\mu }}_{T_i^{\prime }},{\mathsf{\eta }}_{T_i^{\prime }},{\mathsf{\nu }}_{T_i^{\prime }}\rangle$ $(i=1,2,\dots ,n)$ be two groups of PFNs. If ${\mathsf{\alpha }}_i\le {\mathsf{\alpha }}_i^{\prime }$, ${\mathsf{\sigma }}_i^{\prime }\le {\mathsf{\sigma }}_i$ and ${\mathsf{\mu }}_{T_i}\le {\mathsf{\mu }}_{T_i^{\prime }},{\mathsf{\eta }}_{T_i^{\prime }}\le {\mathsf{\eta }}_{T_i},{\mathsf{\nu }}_{T_i^{\prime }}\le {\mathsf{\nu }}_{T_i}$, then

$$NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\le NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)$$

(28)

Proof. 

$${\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i^{\prime }}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i^{\prime }}\right)\right)}^{\lambda }\le {\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i}\right)\right)}^{\lambda }\Rightarrow e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i^{\prime }}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i^{\prime }}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}$$

and

$$e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\Rightarrow 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},$$

$$e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\Rightarrow 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\le 1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},$$

Then,

$$S_1\left(NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\right)=\mathsf{\alpha }\left(2+e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_{\mathrm{i}}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}-\left(1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\right)-\left(1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\right)\right)$$

and

$$S_1\left(NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)\right)=\mathsf{\alpha }\left(2+e^{-{\left({\displaystyle \sum _{i=1}^n{\mathsf{\omega }}_i^{\prime }}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{T_i^{\prime }}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}-\left(1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\right)-\left(1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^n{\mathsf{\omega }}_i^{\prime }{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{T_i^{\prime }}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\right)\right)$$

Thus, $NPFAAWA\left(T_1,T_2,\dots ,T_n\right)\le NPFAAWA\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)$ is true. □

Definition 8.

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs, and $NPFAAWG$: $NPF{N}^n\to NPFN$,

$$NPFAAWG\left(T_1,T_2,\dots ,T_n\right)=\underset{i=1}{\overset{n}{\otimes }}{\mathsf{\omega }}_i{T}_i$$

(29)

where ${\mathsf{\omega }}_i={\left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,\dots ,{\mathsf{\omega }}_i\right)}^T$ is the weight vector of $T_i(i=1,2,\dots ,n)$, ${\mathsf{\omega }}_i>0$, and ${\displaystyle \sum _{i=1}^m{\mathsf{\omega }}_i}=1$.

Theorem 3.

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs. The result of using the NPFAAWG operator is also an NPFN, and

$$\begin{array}{l}NPFAAWG\left(T_1,T_2,\dots ,T_n\right)=\underset{i=1}{\overset{n}{\otimes }}{\mathsf{\omega }}_i{T}_i\\ =\langle \left({\displaystyle \prod _{i=1}^n{\mathsf{\alpha }}_i^{{\mathsf{\omega }}_i}},{\displaystyle \prod _{i=1}^n{\mathsf{\sigma }}_i^{{\mathsf{\omega }}_i}}\sqrt{{\displaystyle \sum _{i=1}^n\frac{{\mathsf{\omega }}_i{\mathsf{\sigma }}_i^2}{{\mathsf{\alpha }}_i^2}}}\right),1-e^{-{\left({\displaystyle \sum _{i=1}^n{w}_i}{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_{T_i}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{i=1}^n{w}_i}{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{T_i}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left({\displaystyle \sum _{i=1}^n{w}_i}{\left(-\mathrm{In}\left({\mathsf{\nu }}_{T_i}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \end{array}$$

(30)

We can use this to prove that this statement is true.

Proof. 

Similar to Theorem 1, Theorem 3 can be proven by using the mathematical induction method. The process is omitted here. □

Further, according to Properties (2)–(5), the desirable properties of the NPFAAWG operator are also investigated in the following.

Property 5 (Idempotency).

Let  $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs, and $T_i=T=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle$; then,

$$NPFAAWG\left(T_1,T_2,\dots ,T_n\right)=T$$

(31)

Property 6 (Boundedness).

Let  $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle (i=1,2,\dots ,n)$ be a set of NPFNs, $T^{+}=\max (T_1,T_2,\dots ,T_i)$, and $T^{-}=\min (T_1,T_2,\dots ,T_i)$; then, we have

$$T^{-}\le NPFAAWG\left(T_1,T_2,\dots ,T_n\right)\le T^{+}$$

(32)

Property 7 (Commutativity).

Let $T_i^{\prime }\left(i=1,2,\dots ,n\right)$ be any permutation of $T_i\left(i=1,2,\dots ,n\right)$; then,

$$NPFAAWG\left(T_1,T_2,\dots ,T_n\right)=NPFAAWG\left(T_1^{\prime },T_2^{\prime },\dots ,T_n^{\prime }\right)$$

(33)

Property 8 (Monotonicity).

Let $T_i=\langle \left(\mathsf{\alpha },\mathsf{\sigma }\right),{\mathsf{\mu }}_T,{\mathsf{\eta }}_T,{\mathsf{\nu }}_T\rangle$ and $T_i^{″}=\langle \left({\mathsf{\alpha }}_i^{″},{\mathsf{\sigma }}_i^{″}\right),{\mathsf{\mu }}_{T_i^{″}},{\mathsf{\eta }}_{T_i^{″}},{\mathsf{\nu }}_{T_i^{″}}\rangle$ $(i=1,2,\dots ,n)$ be two groups of NPFNs. If ${\mathsf{\alpha }}_i\le {\mathsf{\alpha }}_i^{″}$, ${\mathsf{\sigma }}_i^{″}\le {\mathsf{\sigma }}_i$ and ${\mathsf{\mu }}_{T_i}\le {\mathsf{\mu }}_{T_i^{″}},{\mathsf{\eta }}_{T_i^{″}}\le {\mathsf{\eta }}_{T_i},{\mathsf{\nu }}_{T_i^{″}}\le {\mathsf{\nu }}_{T_i}$, $T_i\le T_i^{\prime }$, then

$$NPFAAWG\left(T_1,T_2,\dots ,T_n\right)\le NPFAAWG\left(T_1^{″},T_2^{″},\dots ,T_n^{″}\right)$$

(34)

5. Determining the Evaluation Criteria and Corresponding Weights for OMSQD

In this section, based on the online review information from the Chinese medical platform Haodaifu, the online evaluation attributes and corresponding weights of the online medical service quality of doctors are determined.

5.1. Online Review Collection

In order to construct a medical evaluation system based on the NPFNs, this paper used Python to crawl all the diagnosis and treatment evaluation reviews of the orthopedics department in a certain area from the Chinese online medical platform Haodaifu, including 21,050 online reviews of orthopedic doctors. The corresponding algorithm for crawling online reviews is presented in Algorithm 1.

Algorithm 1:	Crawl website reviews and visualize online comments.
Input: Online URL set $R=\left\{R_1,R_2,\dots ,R_n\right\}$
Output: Online Review set $S$
1	request and parse
2	get_page_num
3	For URL in_set R
4	request and parse
5	get_each_page_link_url_meta
6	return metas
7	for metas in url-metas
8	request and parse
9	if url_meta.class=’description’
10	get url-meta.text
11	get url-meta.id
12	save (id, text)
13	end
14	end
15	end
16	close

5.2. Text Preprocessing

Based on the doctor review text, it was necessary to remove duplicate text, remove stop words, filter out meaningless words, and use a word segmentation algorithm to convert the review text into individual words or phrases. In addition, the reviews that appeared multiple times in the same content needed to be deduplicated to avoid double counting, which may have affected the accuracy of the results. After these preprocessing steps, more than 15,000 pieces of evaluation information were obtained, which provided reliable data support for the subsequent analysis.

5.2.1. Word Segmentation

Word segmentation can be implemented by using the open-source Python package Jieba (https://pypi.org/project/jieba/, accessed on 20 January 2020), which can divide online comments into several words to process the textual information of the comments. Also, unimportant words, such as adverbs and prepositions, can be deactivated by using the HIT stop-word table. According to the DMs’ preferences, we eliminated some words, such as “hello”, “date”, etc. For example, some preprocessed data and the algorithm for word segmentation are presented in

Table 1. An example of the extraction results of word segmentation.

Comments	Word Segmentation Results
腰椎间盘突出，感谢冯医生实施的微创手术治疗，腰椎间盘严重脱出，效果很好 “Lumbar disc herniation, thanks to Dr. Feng for the minimally invasive surgical treatment. The lumbar disc is severely herniated, and the effect is very good”	腰椎间盘 突出 感谢 冯 医生 实施 的 微创 手术 治疗 腰椎间盘 严重 脱出 效果 很 好 Lumbar disc herniation, Thanks to Dr. Feng, the minimally, invasive surgery, the treatment, lumbar disc, severely herniated, is very good
非常感谢滕主任！温州有您这样责任心强，技术高，亲切感十足，乐于助人的医生真是件庆幸的事，金华父老乡亲好羡慕啊！ “Thanks to Dr. Teng! Wenzhou is really fortunate to have such a strong sense of responsibility, high technology, full of cordiality, and helpful doctors. Jinhua parents and villagers are so envious!”	非常感谢 滕 主任 温州 有 您 这样 责任心 强 技术 高 亲切感 十足 乐于助人 医生 真是 件 庆幸 事 金华 父老乡亲 好羡慕 Thanks to Dr. Teng, Wenzhou, such a strong sense, responsibility, high technology, cordiality, helpful, doctors, Jinhua, parents and villagers, are so envious
我觉得滕医生有经验，有知识，有医德，非常感谢他对我老爸的这次治疗。 “I think that Dr. Teng has experience, knowledge and medical ethics. I am very grateful to him for this treatment for my father.”	我 觉得 滕 医生 有 经验 有 知识 有 医德 非常感谢 对 我 老爸 这次 治疗 I think, Dr. Teng, has, experience, knowledge, medical ethics, I, am very grateful, treatment, my father

and Algorithm 2, respectively.

Algorithm 2:		Word Segmentation
Input: Online Review set $S=\left\{S_1,S_2,\dots ,S_n\right\}$, Stopwordslist
Output: Word Segmentation result R
1	For S in review set
2	sentence_depart = jieba.cut (S. strip ())
3	#Introduce a deactivation glossary
4	outstr. append(sentence_depart)
5	For word in Word Segmentation result:
6	stopwords = Stopwordslist ()
7	if word in stopwords then
8	R. remove(word)
9	end if
10	end for each
11	return R close

5.2.2. Visualization of Online Reviews

In order to understand the patients’ focus on the doctors more intuitively, the results of word segmentation needed to be visually analyzed. In this paper, a word cloud map was used to analyze the attributes of patients’ attention based on online reviews, so as to determine the criteria for evaluating the OMSQD. The data collected in this paper were based on patient comments. Therefore, common expressions which provided insufficient and inaccurate descriptions, or evaluations containing only emojis, were excluded from the analysis process to ensure the accuracy of the results. Then, the final word cloud diagram was created, as shown in Figure 1.

From Figure 1, it can be found that words such as “surgery”, “patient”, “director”, “medical skills”, “condition”, and “fracture” account for a large proportion, indicating that these words appear more frequently in the comments. Moreover, the high frequency of words such as “thanks” and “patience” indicates that patients are generally satisfied with their doctors’ treatment.

5.3. Feature Extraction

Based on the word segmentation results, the words with the top 20 frequency proportions were selected for classification. Among these words, some words were very vague and did not have an obvious classification standard, so some words were selected from the words with frequencies in the top 20–30 as supplementary words. Then, the words underwent feature extraction, and the results are shown in Algorithm 2. “Director”, “Surgery”, “Medicine”, “Treatment”, “Meniscus”, “Knee”, and “Lumbar” are all about doctors’ professional skills, and the total number of words that pertained to this attribute was 12,062, with the proportion reaching 0.46; “Description”, “Pathology”, “Case”, and “examine” are all about doctors’ communication skills, and the total number of words that pertained to this attribute was 4662, with the proportion reaching 0.18; “gratefulness”, “Patience”, “Attitude”, “Superb”, and “Thanks” can be summed up as describing doctors’ service attitudes, and the total number of words that pertained to this attribute was 6159, with the proportion reaching 0.23; “Rehabilitation”, “Post operation”, and “Care ” can be generalized as describing doctors’ abilities in terms of the results and follow-up, and the total number of words that pertained to this attribute was 3218, with the proportion reaching 0.13. Then, we obtained the evaluation criteria for evaluating the OMSQD, i.e., professional skills, communication skills, service attitude, and results and follow-up. The corresponding weight vector of the four criteria can be determined as $\mathsf{\omega }={\left(0.46,0.18,0.23,0.13\right)}^T$, as shown in

Table 2. Results for feature extraction.

Category	Keywords (English)	Keywords (中)	Sum	Weight
Professional skills	Director (2667)	主任 (2667)	12,062	0.46
	Surgery (4369)	手术 (4369)
	Medicine (1556)	医术 (1556)
	Treatment (1390)	治疗 (1390)
	Meniscus (703)	半月板 (703)
	Knee (689)	膝关节 (689)
	Lumbar (688)	腰椎间盘 (688)
Communication skills	Description (846)	描述 (846)	4662	0.18
	Pathology (1435)	病情 (1435)
	Case (1146)	情况 (1146)
	Examine (1235)	看病 (1235)
Service attitude	gratefulness (1433)	感谢 (1433)	6159	0.23
	Patience (1345)	耐心 (1345)
	Attitude (1256)	态度 (1256)
	Superb (799)	精湛 (799)
	Thanks (790)	谢谢 (790)
	medical ethics (572)	医德 (572)
Results and follow- up	Rehabilitation (1516)	康复 (1516)	3218	0.13
	Post operation (1087)	术后 (1087)
	Care (615)	照顾 (615)

.

6. An extended Picture Fuzzy MCGDM Method Based on COPRAS for Evaluating OMSQD

The COPRAS method is an MCDM method which can improve the overall efficiency of evaluating alternatives. Moreover, as we discussed earlier, the evaluation of OMSQD can be considered an MCDM problem. Compared to other MCDM methods, the COPRAS method is simple and easy to understand, takes the uncertainty of weight into account, and can be utilized to handle complex evaluation problems. Therefore, it can also be used to solve MCGDM problems. An extended picture fuzzy MCGDM method based on COPRAS is proposed for evaluating the OMSQD in the following.

Assume that there are $m$ alternatives, i.e., $P=\left\{P_1,P_2,\dots P_m\right\}$. The set of evaluation criteria of the alternatives is denoted as $C=\left\{c_1,c_2,\dots ,c_n\right\}$. $\mathsf{\omega }={\left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,\dots ,{\mathsf{\omega }}_n\right)}^T$ is the weight vector of the criteria, satisfying $0\le {\mathsf{\omega }}_j\le 1$ and ${\displaystyle \sum _{j=1}^n{\mathsf{\omega }}_j}=1$. Moreover, the set of DMs is denoted as $I=\left\{I_1,I_2,\dots ,I_k\right\}$, and $\mathsf{\phi }={\left({\mathsf{\phi }}_1,{\mathsf{\phi }}_2,\dots ,{\mathsf{\phi }}_k\right)}^T$ is the weight vector of the DMs, satisfying $0\le {\mathsf{\phi }}_l\le 1$ and ${\displaystyle \sum _{l=1}^k{\mathsf{\phi }}_l}=1$. The individual decision matrix is presented as $A_k={\left(A_{ij}^k\right)}_{m\times n}$. Here, $A_{ij}^k$ represents the evaluation information on the alternative $P_i\left(i=1,2,3,\dots ,m\right)$ associated with the criterion $c_j$ provided by the DM $I_k$ in the form of NPFNs. The methodological framework of an extended picture fuzzy MCGDM method based on COPRAS for evaluating the OMSQD is presented in Figure 2, and details of the procedures are described below.

Step 1.

Determine the evaluation criteria and corresponding weight based on online reviews.

Based on the online review-driven weight determination method presented in Section 5, we can obtain the evaluation criteria set of OMSQD $C=\left\{c_1,\hspace{0.17em}{c}_2,\dots ,c_n\right\}$ and the corresponding weight vector $\mathsf{\omega }={\left({\mathsf{\omega }}_1,{\mathsf{\omega }}_2,\dots ,{\mathsf{\omega }}_n\right)}^T$.

Step 2.

Obtain the normal picture fuzzy evaluation matrix.

From the DMs’ preferences, the normal picture fuzzy evaluation matrix $A_k={\left[A_{ij}^k\right]}_{m\times n}$ of each DM can be obtained as follows:

$$A^k={\left[A_{ij}^k\right]}_{m\times n}=\left[\begin{array}{ccccc}{A}_{11}^k& A_{12}^k& A_{13}^k& \cdots & A_{1n}^k\\ A_{21}^k& A_{22}^k& A_{23}^k& \cdots & A_{2n}^k\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ A_{m1}^k& A_{m2}^k& A_{m3}^k& \cdots & A_{mn}^k\end{array}\right]$$

(35)

Step 3.

Standardize the normal picture fuzzy evaluation matrix.

Since the evaluation problems for OMSQD always involve a benefit criterion and a cost criterion, it is necessary to standardize the normal picture fuzzy evaluation matrix as follows:

$$\widetilde{A_{ij}^k}=\{\begin{array}{l}\langle \left(\frac{{\mathsf{\alpha }}_{ij}^k}{{\max }_i\left({\mathsf{\alpha }}_{ij}^k\right)},\frac{{\mathsf{\sigma }}_{ij}^k}{{\max }_i\left({\mathsf{\sigma }}_{ij}^k\right)}\frac{{\mathsf{\sigma }}_{ij}^k}{{\mathsf{\alpha }}_{ij}^k}\right),{\mathsf{\mu }}_{ij}^k,{\mathsf{\eta }}_{ij}^k,{\mathsf{\nu }}_{ij}^k\rangle ,\hspace{0.17em}\hspace{0.17em}for\hspace{0.17em}benefit\hspace{0.17em}criteria\\ \langle \left(\frac{{\min }_i\left({\mathsf{\alpha }}_{ij}^k\right)}{{\mathsf{\alpha }}_{ij}^k},\frac{{\mathsf{\sigma }}_{ij}^k}{{\max }_i\left({\mathsf{\sigma }}_{ij}^k\right)}\frac{{\mathsf{\sigma }}_{ij}^k}{{\mathsf{\alpha }}_{ij}^k}\right),{\mathsf{\mu }}_{ij}^k,1-{\mathsf{\eta }}_{ij}^k,{\mathsf{\nu }}_{ij}^k\rangle ,\hspace{0.17em}for\hspace{0.17em}cost\hspace{0.17em}criteria\end{array}$$

(36)

Then, the standardized normal picture fuzzy decision evaluation matrix can be determined as follows:

$$\widetilde{A^k}={\left[\widetilde{A_{ij}^k}\right]}_{m\times n}=\left[\begin{array}{ccccc}\widetilde{A_{11}^k}& \widetilde{A_{12}^k}& \widetilde{A_{13}^k}& \cdots & \widetilde{A_{1n}^k}\\ \widetilde{A_{21}^k}& \widetilde{A_{22}^k}& \widetilde{A_{23}^k}& \cdots & \widetilde{A_{2n}^k}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ \widetilde{A_{m1}^k}& \widetilde{A_{m2}^k}& \widetilde{A_{m3}^k}& \cdots & \widetilde{A_{mn}^k}\end{array}\right]$$

(37)

Step 4.

Obtain the general normal picture fuzzy evaluation matrix.

According to the standardized normal picture fuzzy evaluation matrix, i.e., Equation (37), and the NPFAAWA operator, i.e., Equation (24), we can obtain the aggregated normal picture fuzzy evaluation matrix $A$ as follows:

$$A={\left[A_{ij}\right]}_{m\times n}=\left[\begin{array}{ccccc}{A}_{11}& A_{12}& A_{13}& \cdots & A_{1n}\\ A_{21}& A_{22}& A_{23}& \cdots & A_{2n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ A_{m1}& A_{m2}& A_{m3}& \cdots & A_{mn}\end{array}\right].$$

(38)

where

$$A_{ij}=\langle \left({\displaystyle \sum _{i=1}^m{\mathsf{\phi }}_i{\mathsf{\alpha }}_{\widetilde{A_{ij}^k}}},{\displaystyle \sum _{i=1}^m{\mathsf{\phi }}_i{\mathsf{\sigma }}_{\widetilde{A_{ij}^k}}}\right),e^{-{\left({\displaystyle \sum _{i=1}^m{\mathsf{\phi }}_i}{\left(-\mathrm{In}\left({\mathsf{\mu }}_{\widetilde{A_{ij}^k}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\phi }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{\widetilde{A_{ij}^k}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{\mathrm{i}=1}^m{\mathsf{\phi }}_i{\left(-\mathrm{In}\left(1-{\mathsf{\nu }}_{\widetilde{A_{ij}^k}}\right)\right)}^{\lambda }}\right)}^{\frac{1}{\lambda }}}\rangle$$

(39)

Step 5.

Calculate a weighted normalized normal picture fuzzy evaluation matrix.

From Step 1 and Step 3, the weighted normalized normal picture fuzzy evaluation matrix $D={\left[d_{ij}\right]}_{m\times n}$ can be obtained as follows:

$$D={\left[d_{ij}\right]}_{m\times n}=\left[\begin{array}{ccccc}{d}_{11}& d_{12}& d_{13}& \cdots & d_{1n}\\ d_{21}& d_{22}& d_{23}& \cdots & d_{2n}\\ ⋮& ⋮& ⋮& \ddots & ⋮\\ d_{m1}& d_{m2}& d_{m3}& \cdots & d_{mn}\end{array}\right]$$

(40)

where

$$d_{ij}=A_{ij}\times {\mathsf{\omega }}_j,j=1,2,3,4,i=1,2,\dots ,m$$

(41)

Step 6.

Determine the evaluation values of the criteria for benefit and cost.

From the NPFAAWG operator and Equation (40), the evaluation values of the criteria for benefit and cost can be determined, respectively, as follows:

$$\begin{array}{l}{P}_i^{+}=\langle \left({\displaystyle \prod _{j=1}^n{\mathsf{\alpha }}_{d_{ij}^{+}}^{{\mathsf{\omega }}_j}},{\displaystyle \prod _{j=1}^n{\mathsf{\alpha }}_{d_{ij}^{+}}^{{\mathsf{\omega }}_j}}\sqrt{{\displaystyle \sum _{j=1}^n\frac{{\mathsf{\omega }}_j{\mathsf{\sigma }}_{d_{ij}^{+}}^2}{{\mathsf{\alpha }}_{d_{ij}^{+}}^2}}}\right),1-e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_{d_{ij}^{+}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{d_{ij}^{+}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left({\mathsf{\nu }}_{d_{ij}^{+}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \\ P_i^{-}=\langle \left({\displaystyle \prod _{j=1}^n{\mathsf{\alpha }}_{d_{ij}^{-}}^{{\mathsf{\omega }}_j}},{\displaystyle \prod _{j=1}^n{\mathsf{\alpha }}_{d_{ij}^{-}}^{{\mathsf{\omega }}_j}}\sqrt{{\displaystyle \sum _{j=1}^n\frac{{\mathsf{\omega }}_j{\mathsf{\sigma }}_{d_{ij}^{+}}^2}{{\mathsf{\alpha }}_{d_{ij}^{-}}^2}}}\right),1-e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left(1-{\mathsf{\mu }}_{d_{ij}^{-}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},1-e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left(1-{\mathsf{\eta }}_{d_{ij}^{-}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}},e^{-{\left({\displaystyle \sum _{j=1}^n{w}_j}{\left(-\mathrm{In}\left({\mathsf{\nu }}_{d_{ij}^{-}}\right)\right)}^{\lambda }\right)}^{\frac{1}{\lambda }}}\rangle \end{array}.$$

(42)

Step 7.

Determine the scores of each alternative:

$$O_i=S\left(P_i^{+}\right)+\frac{{\displaystyle \sum _{i=1}^{m}S\left(P_i^{-}\right)}}{S\left(P_i^{-}\right){\displaystyle \sum _{i=1}^m\frac{1}{S\left(P_i^{-}\right)}}}$$

(43)

where

$$\begin{array}{l}S(P_i^{+})={\mathsf{\alpha }}_{P_i^{+}}\left({\mathsf{\mu }}_{P_i^{+}}+1-{\mathsf{\eta }}_{P_i^{+}}+1-{\mathsf{\nu }}_{P_i^{+}}\right)\\ S(P_i^{-})={\mathsf{\alpha }}_{P_i^{-}}\left({\mathsf{\mu }}_{P_i^{-}}+1-{\mathsf{\eta }}_{P_i^{-}}+1-{\mathsf{\nu }}_{P_i^{-}}\right)\end{array}$$

(44)

Step 8.

Compute the utility degree.

From Equation (43), the utility degree $G_i$ can be computed as follows:

$$G_i=\frac{O_i}{Max\left\{O_i\right\}}\left(i=1,2,\dots ,m\right)$$

(45)

Step 9.

Rank the alternatives.

According to Equation (45), the ranking of alternatives can be carried out. Generally speaking, the value with the greatest utility degree $G_i$ is the optimal choice.

7. A Case Study

With the development of medical technology and the increasing expectations of patients for medical services and their own health, doctors need to provide higher service standards and adhere to their responsibilities. Thus, it is necessary to establish a reasonable evaluation system to evaluate the service of doctors so that patients can screen doctors with the most accurate feedback and make optimal decisions. However, the existing evaluation systems and evaluation methods for assessing doctors are less applicable to online reviews. As we discussed earlier, the evaluation of doctors is a classic MAGDM problem. Thus, we will use the proposed picture fuzzy MCGDM approach for evaluating the online medical service quality of doctors in the following.

Suppose that three DMs from different departments of a hospital need to evaluate the OMSQD for four doctors. The set of DMs can be denoted as $I_n\left\{n=1,2,3\right\}$, and the corresponding weight of three DMs is ${\mathsf{\phi }}_i=\left(0.3,0.4,0.3\right)$. The set of doctors can be denoted as $P_i\left(i=1,2,3,4\right)$. Then, the DMs should evaluate the four doctors in terms of four attributes, including ① $C_1$ professional skills; ② $C_2$ communication skills; ③ $C_3$ service attitude; and ④ $C_4$ results and follow-up. The evaluation information provided by DMs is in the form of NPFSs. The evaluation process is presented in the following.

7.1. The Evaluation Processes

In this subsection, the procedures to obtain the optimal doctor are described.

Step 1.

Determine the evaluation criteria and corresponding weight based on online reviews.

Based on the online review-driven weight determination method presented in Section 5, we can obtain the four key evaluation criteria of the OMSQD problems: “Professional Skills”, “Communication Skills”, “Service Attitude”, and “Results and Follow-up”. The total word counts for these four criteria are 12,062, 4662, 6159, and 3128, respectively. Thus, the corresponding weight vector of four criteria is determined as $\mathsf{\omega }={\left(0.46,0.18,0.23,0.13\right)}^T$.

Step 2.

Obtain the normal picture fuzzy evaluation matrix.

Generally speaking, DMs should finish some preparation work before providing evaluation information. For example, DMs need to acquire a comprehensive understanding of relevant knowledge about normal distribution and picture fuzzy theory. At the same time, DMs also need to be familiar with the alternative doctors evaluated in online medical services, including their professional skills, communication skills, service attitudes, etc. Then, they can evaluate the four doctors under the known criteria. The evaluation preference matrix information is provided in the form of NPFNs. The average scores of the doctors in each criterion range from 0 to 10, while the standard deviations range from 0 to 1. Then, the corresponding evaluation matrix $A^n={\left[A_{ij}^n\right]}_{m\times n}\left(i=1,2,3,4,j=1,2,3,4,n=3\right)$ of each DM can be obtained, as shown in

Table 3. Normal picture fuzzy evaluation information provided by DM1.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(7.0,0.40\right),\left(0.40,0.20,0.30\right)\rangle$	$\langle \left(7.0,0.60\right),\left(0.40,0.10,0.20\right)\rangle$
$P_2$	$\langle \left(4.0,0.20\right),\left(0.60,0.10,0.20\right)\rangle$	$\langle \left(8.0,0.40\right),\left(0.60,0.10,0.20\right)\rangle$
$P_3$	$\langle \left(3.5,0.30\right),\left(0.30,0.20,0.30\right)\rangle$	$\langle \left(6.0,0.20\right),\left(0.50,0.20,0.30\right)\rangle$
$P_4$	$\langle \left(5.0,0.50\right),\left(0.70,0.10,0.20\right)\rangle$	$\langle \left(7.0,0.50\right),\left(0.60,0.10,0.10\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(6.5,0.40\right),\left(0.24,0.40,0.36\right)\rangle$	$\langle \left(6.0,0.50\right),\left(0.25,0.27,0.40\right)\rangle$
$P_2$	$\langle \left(6,0.70\right),\left(0.30,0.50,0.10\right)\rangle$	$\langle \left(6.0,0.70\right),\left(0.49,0.31,0.20\right)\rangle$
$P_3$	$\langle \left(5.5,0.60\right),\left(0.40,0.20,0.20\right)\rangle$	$\langle \left(6.0,0.45\right),\left(0.43,0.35,0.22\right)\rangle$
$P_4$	$\langle \left(7.0,0.40\right),\left(0.40,0.20,0.30\right)\rangle$	$\langle \left(8.5,0.40\right),\left(0.41,0.19,0.35\right)\rangle$

,

Table 4. Normal picture fuzzy evaluation information provided by DM2.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(6.9,0.42\right),\left(0.30,0.30,0.40\right)\rangle$	$\langle \left(7.5,0.35\right),\left(0.50,0.20,0.20\right)\rangle$
$P_2$	$\langle \left(4.5,0.30\right),\left(0.66,0.19,0.15\right)\rangle$	$\langle \left(8.0,0.25\right),\left(0.60,0.20,0.15\right)\rangle$
$P_3$	$\langle \left(4.5,0.30\right),\left(0.60,0.20,0.10\right)\rangle$	$\langle \left(6.5,0.20\right),\left(0.45,0.25,0.30\right)\rangle$
$P_4$	$\langle \left(6.0,0.40\right),\left(0.45,0.25,0.30\right)\rangle$	$\langle \left(8.5,0.45\right),\left(0.42,0.36,0.22\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(7.5,0.25\right),\left(0.30,0.30,0.40\right)\rangle$	$\langle \left(7.0,0.55\right),\left(0.25,0.35,0.35\right)\rangle$
$P_2$	$\langle \left(7.0,0.25\right),\left(0.15,0.35,0.50\right)\rangle$	$\langle \left(7.5,0.30\right),\left(0.56,0.28,0.16\right)\rangle$
$P_3$	$\langle \left(6.5,0.60\right),\left(0.45,0.20,0.25\right)\rangle$	$\langle \left(7.0,0.45\right),\left(0.45,0.35,0.20\right)\rangle$
$P_4$	$\langle \left(7.0,0.35\right),\left(0.51,0.20,0.29\right)\rangle$	$\langle \left(9.5,0.40\right),\left(0.60,0.26,0.14\right)\rangle$

and

Table 5. Normal picture fuzzy evaluation information provided by DM3.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(6.5,0.45\right),\left(0.48,0.27,0.25\right)\rangle$	$\langle \left(8.5,0.60\right),\left(0.45,0.15,0.20\right)\rangle$
$P_2$	$\langle \left(3.0,0.60\right),\left(0.50,0.30,0.20\right)\rangle$	$\langle \left(7.5,0.45\right),\left(0.50,0.30,0.10\right)\rangle$
$P_3$	$\langle \left(4.5,0.30\right),\left(0.60,0.20,0.10\right)\rangle$	$\langle \left(6.5,0.20\right),\left(0.45,0.25,0.30\right)\rangle$
$P_4$	$\langle \left(5.5,0.40\right),\left(0.75,0.15,0.10\right)\rangle$	$\langle \left(8.0,0.55\right),\left(0.50,0.20,0.10\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(7.0,0.45\right),\left(0.25,0.45,0.30\right)\rangle$	$\langle \left(7.5,0.50\right),\left(0.25,0.25,0.40\right)\rangle$
$P_2$	$\langle \left(6.5,0.30\right),\left(0.20,0.50,0.30\right)\rangle$	$\langle \left(7.0,0.50\right),\left(0.40,0.31,0.10\right)\rangle$
$P_3$	$\langle \left(6.5,0.60\right),\left(0.45,0.20,0.25\right)\rangle$	$\langle \left(7.0,0.45\right),\left(0.45,0.35,0.20\right)\rangle$
$P_4$	$\langle \left(7.5,0.45\right),\left(0.35,0.20,0.25\right)\rangle$	$\langle \left(9.0,0.45\right),\left(0.45,0.20,0.35\right)\rangle$

.

Step 3.

Standardize the normal picture fuzzy evaluation matrix.

Since all the attributes are of the benefits type, we can obtain the standardized decision-making matrix based on Equation (36), as shown in

Table 6. The standardized evaluation matrix provided by DM1.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(1.0,0.03\right),\left(0.40,0.20,0.30\right)\rangle$	$\langle \left(1.0,0.08\right),\left(0.40,0.10,0.20\right)\rangle$
$P_2$	$\langle \left(0.5,0.01\right),\left(0.60,0.10,0.20\right)\rangle$	$\langle \left(1.0,0.02\right),\left(0.60,0.10,0.20\right)\rangle$
$P_3$	$\langle \left(0.58,0.04\right),\left(0.30,0.20,0.30\right)\rangle$	$\langle \left(1.0,0.01\right),\left(0.50,0.20,0.30\right)\rangle$
$P_4$	$\langle \left(0.58,0.10\right),\left(0.70,0.10,0.20\right)\rangle$	$\langle \left(0.82,0.07\right),\left(0.60,0.10,0.10\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(0.92,0.04\right),\left(0.24,0.40,0.36\right)\rangle$	$\langle \left(0.85,0.06\right),\left(0.25,0.27,0.40\right)\rangle$
$P_2$	$\langle \left(0.75,0.11\right),\left(0.30,0.50,0.10\right)\rangle$	$\langle \left(0.75,0.11\right),\left(0.49,0.31,0.20\right)\rangle$
$P_3$	$\langle \left(0.91,0.10\right),\left(0.40,0.20,0.20\right)\rangle$	$\langle \left(1.0,0.05\right),\left(0.43,0.35,0.22\right)\rangle$
$P_4$	$\langle \left(0.82,0.04\right),\left(0.40,0.20,0.30\right)\rangle$	$\langle \left(1.0,0.03\right),\left(0.41,0.19,0.35\right)\rangle$

,

Table 7. The standardized evaluation matrix provided by DM2.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(0.92,0.04\right),\left(0.30,0.30,0.40\right)\rangle$	$\langle \left(1.0,0.03\right),\left(0.50,0.20,0.20\right)\rangle$
$P_2$	$\langle \left(0.56,0.06\right),\left(0.66,0.19,0.15\right)\rangle$	$\langle \left(1.0,0.02\right),\left(0.60,0.20,0.15\right)\rangle$
$P_3$	$\langle \left(0.64,0.03\right),\left(0.60,0.20,0.10\right)\rangle$	$\langle \left(0.92,0.01\right),\left(0.45,0.25,0.30\right)\rangle$
$P_4$	$\langle \left(0.63,0.05\right),\left(0.45,0.25,0.30\right)\rangle$	$\langle \left(0.89,0.05\right),\left(0.42,0.36,0.22\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(1.0,0.01\right),\left(0.30,0.30,0.40\right)\rangle$	$\langle \left(0.93,0.08\right),\left(0.25,0.35,0.35\right)\rangle$
$P_2$	$\langle \left(0.87,0.02\right),\left(0.15,0.35,0.50\right)\rangle$	$\langle \left(0.93,0.04\right),\left(0.56,0.28,0.16\right)\rangle$
$P_3$	$\langle \left(0.92,0.09\right),\left(0.45,0.20,0.25\right)\rangle$	$\langle \left(1.0,0.04\right),\left(0.45,0.35,0.20\right)\rangle$
$P_4$	$\langle \left(0.73,0.03\right),\left(0.51,0.20,0.29\right)\rangle$	$\langle \left(1.0,0.03\right),\left(0.60,0.26,0.14\right)\rangle$

and

Table 8. The standardized evaluation matrix provided by DM3.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(0.76,0.05\right),\left(0.48,0.27,0.25\right)\rangle$	$\langle \left(1.0,0.07\right),\left(0.45,0.15,0.20\right)\rangle$
$P_2$	$\langle \left(0.4,0.19\right),\left(0.50,0.30,0.20\right)\rangle$	$\langle \left(1.0,0.04\right),\left(0.50,0.30,0.10\right)\rangle$
$P_3$	$\langle \left(0.64,0.03\right),\left(0.60,0.20,0.10\right)\rangle$	$\langle \left(0.92,0.01\right),\left(0.45,0.25,0.30\right)\rangle$
$P_4$	$\langle \left(0.61,0.05\right),\left(0.75,0.15,0.10\right)\rangle$	$\langle \left(0.88,0.06\right),\left(0.50,0.20,0.10\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(0.82,0.04\right),\left(0.25,0.45,0.30\right)\rangle$	$\langle \left(0.88,0.05\right),\left(0.25,0.25,0.40\right)\rangle$
$P_2$	$\langle \left(0.86,0.02\right),\left(0.20,0.50,0.30\right)\rangle$	$\langle \left(0.93,0.05\right),\left(0.40,0.31,0.10\right)\rangle$
$P_3$	$\langle \left(0.92,0.09\right),\left(0.45,0.20,0.25\right)\rangle$	$\langle \left(1.0,0.04\right),\left(0.45,0.35,0.20\right)\rangle$
$P_4$	$\langle \left(0.83,0.04\right),\left(0.35,0.20,0.25\right)\rangle$	$\langle \left(1.0,0.04\right),\left(0.45,0.20,0.35\right)\rangle$

.

Step 4.

Obtain the general normal picture fuzzy evaluation matrix.

Since the weight vector of three DMs is ${\mathsf{\phi }}_i={\left(0.3,0.4,0.3\right)}^T$, the general normal picture fuzzy evaluation matrix can be obtained by using the normal aggregation operator, as presented in

Table 9. The collective decision matrix.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(0.89,0.04\right),\left(0.34,0.26,0.33\right)\rangle$	$\langle \left(1.0,0.05\right),\left(0.45,0.16,0.20\right)\rangle$
$P_2$	$\langle \left(0.49,0.08\right),\left(0.58,0.21,0.18\right)\rangle$	$\langle \left(1.0,0.02\right),\left(0.56,0.21,0.15\right)\rangle$
$P_3$	$\langle \left(0.62,0.03\right),\left(0.45,0.20,0.19\right)\rangle$	$\langle \left(0.94,0.01\right),\left(0.46,0.23,0.30\right)\rangle$
$P_4$	$\langle \left(0.60,0.06\right),\left(0.56,0.18,0.23\right)\rangle$	$\langle \left(0.86,0.05\right),\left(0.48,0.26,0.16\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(0.92,0.02\right),\left(0.26,0.38,0.36\right)\rangle$	$\langle \left(0.89,0.06\right),\left(0.25,0.30,0.32\right)\rangle$
$P_2$	$\langle \left(0.83,0.04\right),\left(0.19,0.45,0.38\right)\rangle$	$\langle \left(0.87,0.06\right),\left(0.47,0.29,0.16\right)\rangle$
$P_3$	$\langle \left(0.91,0.09\right),\left(0.43,0.20,0.23\right)\rangle$	$\langle \left(1.0,0.15\right),\left(0.44,0.35,0.20\right)\rangle$
$P_4$	$\langle \left(0.78,0.03\right),\left(0.41,0.20,0.28\right)\rangle$	$\langle \left(1.0,0.03\right),\left(0.48,0.22,0.29\right)\rangle$

.

Step 5.

Calculate a weighted normalized normal picture fuzzy evaluation matrix.

Based on Step 1, the weight vector of the attributes can be determined as $\mathsf{\omega }={\left(0.46,0.18,0.23,0.13\right)}^T$. Then, from Equation (41), the weighted standardized evaluation matrix can be obtained as shown in

Table 10. Weighted standardized evaluation matrix.

	Professional Skills	Communication Skills
$P_1$	$\langle \left(0.41,0.01\right),\left(0.48,0.18,0.24\right)\rangle$	$\langle \left(0.18,0.01\right),\left(0.71,0.07,0.09\right)\rangle$
$P_2$	$\langle \left(0.23,0.04\right),\left(0.69,0.15,0.13\right)\rangle$	$\langle \left(0.18,0.01\right),\left(0.78,0.09,0.06\right)\rangle$
$P_3$	$\langle \left(0.28,0.01\right),\left(0.58,0.14,0.13\right)\rangle$	$\langle \left(0.17,0.01\right),\left(0.72,0.11,0.13\right)\rangle$
$P_4$	$\langle \left(0.28,0.02\right),\left(0.68,0.13,0.16\right)\rangle$	$\langle \left(0.15,0.01\right),\left(0.73,0.12,0.07\right)\rangle$
	Service Attitude	Results and Follow-up
$P_1$	$\langle \left(0.21,0.01\right),\left(0.52,0.20,0.19\right)\rangle$	$\langle \left(0.11,0.01\right),\left(0.60,0.12,0.13\right)\rangle$
$P_2$	$\langle \left(0.19,0.01\right),\left(0.45,0.25,0.20\right)\rangle$	$\langle \left(0.11,0.01\right),\left(0.76,0.12,0.06\right)\rangle$
$P_3$	$\langle \left(0.21,0.02\right),\left(0.66,0.10,0.12\right)\rangle$	$\langle \left(0.13,0.01\right),\left(0.74,0.14,0.08\right)\rangle$
$P_4$	$\langle \left(0.18,0.01\right),\left(0.65,0.10,0.15\right)\rangle$	$\langle \left(0.13,0.01\right),\left(0.77,0.08,0.11\right)\rangle$

.

Step 6.

Determine the evaluation values of the criteria for benefit and cost.

Since all the attributes are of the benefits type, we can calculate the evaluation values of the criteria for benefit as ${P_1}^{+}=\langle \left(0.26,0.01\right),\left(0.57,0.17,0.17\right)\rangle$, ${P_2}^{+}=\langle \left(0.19,0.02\right),\left(0.69,0.17,0.11\right)\rangle$, ${P_3}^{+}=\langle \left(0.22,0.01\right),$ $\left(0.66,0.13,0.12\right)\rangle$, and ${P_4}^{+}=\langle \left(0.21,0.01\right),\left(0.70,0.12,0.13\right)\rangle$.

Step 7.

Determine the scores of each alternative.

From Step 6 and Equation (43), we can determine the scores of each alternative as $O_1=0.57,O_2=0.46,O_3=0.52$, and $O_4=0.50$.

Step 8.

Compute the utility degree.

From Step 7 and Equation (45), we can obtain the utility degree as follows: $G_1=1,G_2=0.80,G_3=0.91$, and $G_4=0.87$.

Step 9.

Ranking all alternatives.

Based on the results presented in Step 8, we can obtain $G_1\ge G_3\ge G_4\ge G_2$. Then, the final ranking is $P_1\succ P_3\succ P_4\succ P_2$, and the optimal alternative is $P_1$.

7.2. Sensitivity Analysis

In this subsection, we will conduct a sensitivity analysis by considering the different values of parameter $\lambda$ in the NPFAAWA and NPFAAWG operators. Different values can reflect the DMs’ preferences. If parameter $\lambda$ varies between the ranges $[0, 1]$ and $[1, 100]$, then the utility degree of the four alternatives can be presented in Figure 3 and Figure 4.

From the results presented in Figure 3 and Figure 4, it can be seen that different $\lambda$ values cannot make difference in the decision-making results. The final ranking is always $P_1\succ P_3\succ P_4\succ P_2$, and the optimal doctor is also $P_1$.

7.3. Comparative Analysis

In order to verify the feasibility and validity of the proposed approach, a comparative analysis is conducted based on the existing picture fuzzy MCDM methods. The comparative analysis methods mainly include two aspects: (1) picture fuzzy MCGDM or MCDM methods based on aggregation operators [54,56,73,74]; (2) picture fuzzy MCGDM or MCDM methods based on measures [46,53,75,76].

(1)

Comparison analysis with picture fuzzy MCGDM or MCDM methods based on aggregation operators.

Since most existing picture fuzzy MCDM methods [54,56,73,74] cannot handle problems where the weight information is completely unknown and the corresponding weights are determined subjectively, the weight vector of the attributes is calculated as $\mathsf{\omega }={\left(0.46,0.18,0.23,0.13\right)}^T$ here. For the picture fuzzy MCGDM methods, the corresponding aggregation operators defined in reference [73] are used to aggregate the DMs’ preference information. For the picture fuzzy MCDM methods [54,56,74], the weighted standardized evaluation matrix can be used directly. Then, the final rankings can be obtained based on the score values, as shown in

Table 11. Comparative analysis results based on aggregation operators.

References	Aggregation Operators	Final Rankings
Wei [54]	Weighted geometric operator	$P_1\succ P_3\succ P_4\succ P_2$
Jana [56]	Picture fuzzy Dombi weighted average operator ($\lambda \in [1, 100]$)	$P_1\succ P_3\succ P_4\succ P_2$
Jana [56]	Picture fuzzy Dombi weighted geometric operator ($\lambda \in [1,10]$)	$P_1\succ P_3\succ P_4\succ P_2$
Jana [56]	Picture fuzzy Dombi weighted geometric operator ($\lambda \in [10, 100]$)	$P_3\succ P_1\succ P_4\succ P_2$
Li [73]	Picture fuzzy weighted interaction averaging operator	$P_1\succ P_3\succ P_4\succ P_2$
Cengiz et al. [74]	Proportional picture fuzzy weighted averaging (PPFWA) operator	$P_1\succ P_3\succ P_4\succ P_2$
The proposed method	NPFAAWA and NPFAAWG operators	$P_1\succ P_3\succ P_4\succ P_2$

.

(2)

Comparison analysis with picture fuzzy MCGDM or MCDM methods based on measures [46,53,75,76].

Since the weight information in most existing picture fuzzy MCDM or MCGDM methods [46,53,75,76] is completely unknown, the weight vector of attributes is still calculated as $\mathsf{\omega }={\left(0.46,0.18,0.23,0.13\right)}^T$ here. For the picture fuzzy MCGDM methods, the aggregation operators defined in references [53,76] are used to aggregate the DMs’ preference information, which will be used in calculating the cross-entropy measures. For the picture fuzzy MCDM methods [75], the weighted standardized evaluation matrix can also be used directly. We use the projection-based VIKOR method from [46], and the result is contrary to other results. Then, the corresponding rankings can be obtained as shown in

Table 12. Comparative analysis results based on measures.

References	Measures	Final Rankings
Wang [46]	Projection-based VIKOR	$P_2\succ P_3\succ P_4\succ P_1$
Tian and Peng [75]	Distance-based TODIM ($\lambda$ = 1 and $\theta$ = 2.25)	$P_1\succ P_3\succ P_4\succ P_2$
Peng et al. [53]	Bidirectional projection and VIKOR	$P_1\succ P_3\succ P_4\succ P_2$
Peng et al. [76]	Cross-entropy measure	$P_1\succ P_3\succ P_4\succ P_2$
The proposed method	NPFAAWA and NPFAAWG operators	$P_1\succ P_3\succ P_4\succ P_2$

.

From Table 11 and Table 12, we can see that the results obtained in references [54,73,74] and using the picture fuzzy Dombi weighted average operator in reference [56] are the same as those obtained using the proposed approach. The final ranking is always $P_1\succ P_3\succ P_4\succ P_2$, and the optimal alternative is $P_1$; while the ranking obtained using the picture fuzzy Dombi weighted geometric operator is $P_3\succ P_1\succ P_4\succ P_2$, and the optimal alternative is $P_3$, the ranking obtained using the integrated picture fuzzy normalized projection-based VIKOR method is $P_2\succ P_3\succ P_4\succ P_1$, and the optimal alternative is $P_3$.

Based on the analysis presented above, the advantages of the proposed method can be summarized. First, the NPFNs, which combine the advantages of a normal distribution and picture fuzzy sets, can better describe the DMs’ uncertain information. Second, the evaluation attributes and corresponding weights can be extracted and determined based on online reviews, which can make the proposed method more objective and avoid the uncertainty from subjective evaluation simultaneously. Third, compared to the existing picture fuzzy operators, the variable parameter in the defined NPFAAWA and NPFAAWG operators can enhance their flexibility in the decision-making process. Finally, the proposed picture fuzzy MCGDM method based on COPRAS can effectively solve the evaluation of OMSQD problems. Therefore, the results obtained by the proposed method are more effective and robust.

8. Discussion and Managerial Implications

Online medical treatment is an important method of modern medical care in China, and its service quality plays a very key role in the sustainable development of medical treatment. Based on the evaluation of OMSQD, medical institutions can not only effectively encourage doctors to improve the quality of medical services, but also obtain real-time patient feedback from patients to enhance the relationships between doctors and patients, so as to further improve patients’ satisfaction with medical services.

First of all, OMSQD can provide a new perspective for medical institutions to encourage doctors to improve the quality of medical services. According to the findings of an existing study [77], the medical service quality score of three-level medical institutions in Jiangsu Province in China was increased by 2.5 points after adopting the evaluation method with Delphi. Apparently, the number of quality control reports was increased from 1.12 to 2.67 per year, indicating that the implementation of this evaluation can effectively promote improvements in both medical service quality and overall medical management. Thus, the proposed evaluation method for OMSQD problems can effectively assess the work performance and service quality of doctors by combining the objective comments and experts’ subjective judgment. It also provides another perspective for current medical management in China.

Secondly, online evaluation plays an important role in enhancing patients’ satisfactions with medical services. Generally, many factors, including the medical environment, service processes, service attitudes, professional skills, patient rights, etc., will all affect patients’ satisfaction [78]. Moreover, the criteria obtained by using the review-driven method exactly integrates these influencing factors. The proposed picture fuzzy MCGDM evaluation method in this paper can significantly contribute to enhancing patients’ satisfaction.

Finally, online medical evaluation can ease relationships between doctors and patients. The online medical platform can facilitate communication, which can improve the effectiveness of patient disease consultations. Moreover, medical institutions can also further improve the quality of online medical services based on the reviews provided by patients and relieve the pressure on offline medical institutions [79]. Thus, the proposed evaluation method can contribute to creating harmonious relationships between doctors and patients.

9. Conclusions

In this study, we determined the evaluation criteria of OMSQD and their corresponding weights based on online reviews on a medical platform in China. Then, we developed normal picture fuzzy Aczel–Alsina operations and their corresponding aggregation operators, namely the NPFAAWA operator and NPFAAWG operator, respectively. The related properties were discussed as well. Furthermore, we constructed an extended normal picture fuzzy COPRAS method based on Aczel–Alsina aggregation operators to evaluate OMSQD problems. Finally, we provided a case study of OMSQD to demonstrate the feasibility and validity of the proposed approach. The sensitivity analysis and comparative analysis results indicated that the proposed approach is effective and efficient. The optimal alternative is always $P_3$. The main limitations of this research include two aspects. Firstly, the correlations among these criteria and the corresponding mathematical proof between the word segmentation results are not discussed. Secondly, the number of evaluation attributes extracted from the online reviews for OMSQD problems is not enough. In future research, we will systematically aggregate and analyze reviews from various online medical platforms in order to enhance the accuracy of feature extraction and weight determination. Moreover, we will continue to explore other decision-making methods, including AHP, TOPSIS, VIKOR, etc., and apply them to solve review-driven MCGDM problems.