Fractional Orthotriple Fuzzy Dombi Power Partitioned Muirhead Mean Operators and Their Application for Evaluating the Government Information Disclosure on Public Health Emergencies

Abstract

Information disclosure is an important prerequisite and guarantee for the government to answer public health incidents in a timely manner, and is also a basic requirement for the management of emergencies. Evaluating the information disclosure on public health incidents is conducive to improving the quality of emergency information disclosure and comprehensively enhancing the emergency answer and treatment ability of public health incidents. In response to the complex uncertainties in the assessment of information disclosure on public health incidents, this paper proposes a new fuzzy multi-attribute evaluation method. First, a multi-attribute evaluation system for the assessment of information disclosure on public health emergencies is proposed. Then, a novel approach to information disclosure assessment is proposed on the basis of Dombi power divided Muirhead mean operators of fractional orthotriple fuzzy, which can fully consider the relationship between properties and the division of relationships within properties and reduce the distortion in the evaluation process. Meanwhile, it can avoid the impact of singular values on the overall evaluation outcomes of the government. In the end, the effectiveness and flexibility of the approach are validated through an empirical study of a real-life case with comparative analysis and sensitivity analysis.

1. Introduction

Public health emergencies often lead to severe consequences of social disorder and damage to people’s lives, health, and safety. Therefore, we need to carry out the emergency administration of public incidents. In the emergency administration of public health incidents, government information disclosure plays an important role. Timely and adequate government information disclosure helps enhance the degree and efficiency of emergency management of public health emergencies. Scholars are attracted to research on government information disclosure. Fernandes et al. [1] conducted a survey of the public, and learned from the analysis of the survey results that whether government information disclosure is in place will directly affect the public service quality. Conradie et al. [2], based on a study of local government information disclosure services in The Netherlands, pointed out that the main challenge of government information disclosure services is the need for more attention to the impact of internal processes on them. Shields [3] argued that the quality of disclosure information should be improved as much as possible through various channels and methods. Public satisfaction with information disclosure and feelings requires consideration throughout the process. Lee et al. [4] proposed a maturity model for government information disclosure that considers various organizational, technical, and financial challenges. Stark [5] proposed a model that legislatures can use to assess the degree of information disclosure on public health incidents.

However, in reality, the degree of government information disclosure on public health incidents is often influenced by multi-dimensional factors, including the degree of public participation, the quality and speed of government information disclosure, the establishment of government information technology level, and the degree of real-time tracking and reporting of information by the media. Therefore, evaluating government information disclosure on public health emergencies can be considered a multi-property decision-making question. As the decision-making environment becomes increasingly uncertain, fuzzy multi-property decision making has emerged. Fuzzy multi-attribute decision making has received widespread attention from scholars since Zadeh et al. [6] proposed fuzzy sets. Atanassov et al. [7] presented the visualized fuzzy set to compensate for the insufficiency of fuzzy sets by combining membership and non-membership. Although IFS shows excellent potential in MAGDM, membership and non-membership sum to a scope of 0 to 1, and the scope of its application is restricted by its capacity to present fuzzy information. To settle this issue, Yager [8] put forward the Pythagorean fuzzy sets, extending the situation that the squares of membership and non-membership are 0 to 1 in total. Yager [9] defined q-rung orthopair fuzzy sets to make decision making more accurate. Abosuliman et al. [10] pointed out that the factional orthotriple fuzzy set (FOFS) is a novel general device to illustrate uncertainty.

Many scholars have applied aggregation operators to multi-attribute decision making(MADM) methods to aggregate multi-attribute evaluation information. The arithmetic mean and geometric mean operators [11] are the most straightforward aggregation operators. However, both operators need to capture the interrelationships between properties. Several aggregation operators have been proposed that consider the correlation between properties, including the Bonferroni mean (BM) operator [12] and Heronian mean (HM) operators [13]. The BM and HM operators can only respond to the correlation between the two attributes. Consequently, the Muirhead mean (MM) [14] operator was proposed to reflect the relationship between arbitrary attributes. In practical situations, the interrelationships among all attributes may not always be maintained. Therefore, dividing attributes into multiple partitions is necessary when interrelationships among some attributes do not exist. The partitioned BM (PBM) operator based on the above considerations was defined by Dutta et al. [15]. The partitioned HM operator was defined by Liu et al. [16]. Qin et al. [17] proposed the q-rung orthopair fuzzy (weighted) Archimedean power partitioned Muirhead mean (PMM) operator, where all criteria are divided into categories and criteria within the same category are related to each other. In the evaluation process, there may be instances when the decision maker dispenses singular values to the evaluation values, which can disturb the final evaluation results. The Power Average operator can consider the overall balance between properties and decrease the impact of singular values on the evaluation results. He et al. [18] presented the power Bonferroni mean operator. Li et al. [19] put forward the power generalized Heronian mean operator. Punetha et al. [20] put forward the Picture fuzzy power Muirhead mean, Picture fuzzy weighted power Muirhead mean, Picture fuzzy dual power Muirhead mean, and Picture fuzzy weighted dual-power Muirhead mean operator.

The Dombi t-conm and Dombi t-norm (DTT) [21] is a particular type of Archimedean t-norm and t-conm. The Dombi operator [22] has good flexibility under general parameters, prioritizing the variability of the parameter operations. As a powerful tool for information aggregation, scholars have conducted relevant research on it. Jana [23] proposed the Q-rung orthopair fuzzy Dombi weighted average operator, Q-rung orthopair fuzzy Dombi order weighted average operator, and a series of associated weighted average operators. Saha et al. [24] proposed the probable linguistic q-rung orthopair fuzzy weighted Generalized Dombi operator and the probable linguistic q-rung orthopair fuzzy weighted Generalized Dombi Bonferroni mean operator. Khan et al. [25] have given the interval-neutrosophic Dombi power Bonferroni mean operator. Wu et al. [26] put forward interval-valued intuitionistic fuzzy Dombi Heronian mean operators (IVIFDHM). Liu et al. [27] put forward 2-Tuple Linguistic Neutrosophic Dombi Power Heronian mean operators (2-TLNDPHM).

Among the existing studies on aggregation operators, many scholars are concerned with the interrelationship between attributes. However, most research elements do not consider the negative impact of singular values of evaluation results and the division between attributes that partially do not have interrelationships. We take the fractional orthotriple fuzzy set as the analysis object, the Dombi operation as the underlying rule, and the power average (PA) operator and the MM operator as the technical elements, and propose a series of fractional orthotriple fuzzy set Dombi power partitioned MM operators, and apply them to settle the issue of sudden onset. This paper makes the following contributions:

• The ability to take full advantage of the compatibility of Dombi operations, prioritize the variability of parameter operations, and flexibly aggregate fractional orthotriple fuzzy numbers (FOFNs);
• It can genuinely reflect the correlation between attributes and resolve the case where attributes fall into some regions. Properties that belong to the same division are related, but not those belonging to different divisions;
• It can eliminate the negative influence of absurd scores (singular values) of decision makers and play an overall balancing role, making the evaluation results more reasonable and accurate.
• Constructing an index system for information disclosure on public health emergencies and conducting comprehensive examination and evaluation. It is of great significance to improve the quality and effect of information disclosure on public health emergencies, and comprehensively enhance the emergency response and handling capacity of public health emergencies.

The case analysis shows the proposed operators’ feasibility and applicability. The operators are more useful and flexible in information evaluation when facing complex environments and uncertain information.

Based on the above considerations, this research aims to set up a sensible indicator system for government information disclosure on public health incidents and innovate the aggregation operator in the environment of aggregated FOFS to better evaluate government information disclosure on public health incidents. The construction of this paper is as follows. Section 2 introduces fundamental knowledge. Section 3 shows detailed information about the fractional orthotriple fuzzy Dombi weighted power partitioned Muirhead mean (FOFDWPPMM) operator. Section 4 constructs an indicator system for evaluating government information disclosure on public health emergencies. Section 5 presents a new MADM model using the Criteria Importance Through Intercriteria Correlation (CRITIC) assignment method and the FOFDWPPMM operator to assess the disclosure of government information on public health emergencies. Section 6 presents an evaluation case of government information disclosure to validate the possibility and adaptability of the approach, along with sensitivity and comparative analysis to describe the superiority and effectiveness of the approach. Section 7 presents the conclusion of this paper.

2. Preliminaries

2.1. FOFS and Their Operational Rules

This section explores the fundamental concepts of fractional orthotriple fuzzy sets (FOFS).

Definition 1. 

For an arbitrary fixed set of X, a FOFS θ on X is described with the triple of mappings $\mu (\hslash ):X\to [0,1],v(\hslash ):X\to [0,1]$ and $\eta (\hslash ):X\to [0,1]$, where each $\hslash \in X,\mu (\hslash ),v(\hslash )$, and $\eta (\hslash )$ are defined as positive, neutral, and negative levels of $\hslash$, respectively, and $0\le \mu {(\hslash )}^f+v{(\hslash )}^f+\eta {(\hslash )}^f\le 1,\left(f\ge 1\right)$. That is, $\theta =\left\{(\hslash ,\mu (\hslash ),v(\hslash ),\eta (\hslash )):\mu {(\hslash )}^f+v{(\hslash )}^f+\eta {(\hslash )}^f\le 1for each \hslash \in X\right\}$. In fact, $\pi (\hslash )={(1-\mu {(\hslash )}^f-v{(\hslash )}^f-\eta (\hslash ))}^{1/f}$ means the indeterminacy of ℏ. The fractional orthotriple fuzzy number is shown by θ, and FOF(X) for all $\hslash \in X$ means the collection of all FOFNs on X in this paper [10].

Definition 2. 

Suppose ${\theta }_1(\hslash )=({\mu }_1(\hslash ),v_1(\hslash ),{\eta }_1(\hslash ))$ and ${\theta }_2(\hslash )=({\mu }_2(\hslash ),v_2(\hslash ),{\eta }_2(\hslash ))$ are two FOFNs [10]. Their properties are as follows:

• $\begin{array}{c}{\theta }_1(\hslash )\oplus {\theta }_2(\hslash )=\{{({\mu }_1{(\hslash )}^f+{\mu }_2{(\hslash )}^f-{\mu }_1{(\hslash )}^f·{\mu }_2{(\hslash )}^f)}^{\frac{1}{f}},\\ \left.{\nu }_1{(\hslash )}^f·{\nu }_2{(\hslash )}^f,{\eta }_1{(\hslash )}^f·{\eta }_2{(\hslash )}^f\right\}\end{array}$;
• $\begin{array}{c}{\theta }_1(\hslash )\otimes {\theta }_2(\hslash )=\{{\mu }_1{(\hslash )}^f·{\mu }_2{(\hslash )}^f,{\left({\nu }_1{(\hslash )}^f+{\nu }_2{(\hslash )}^f-{\nu }_1{(\hslash )}^f·{\nu }_2{(\hslash )}^f\right)}^{\frac{1}{f}},\\ {\left({\eta }_1{(\hslash )}^f+{\eta }_2{(\hslash )}^f-{\eta }_1{(\hslash )}^f·{\eta }_2{(\hslash )}^f\right)}^{\frac{1}{f}}\} \end{array}$;
• $\mathrm{\Psi }\theta (\hslash )=\{{\left(1-{(1-\mu {(\hslash )}^f)}^{\mathrm{\Psi }}\right)}^{\frac{1}{f}},\nu {(\hslash )}^{\mathrm{\Psi }},\eta {(\hslash )}^{\mathrm{\Psi }}\},\mathrm{\Psi }>0$;
• $\theta {(\hslash )}^{\mathrm{\Psi }}=\{{(\mu (\hslash ))}^{\mathrm{\Psi }},{\left(1-{(1-\nu {(\hslash )}^f)}^{\mathrm{\Psi }}\right)}^{\frac{1}{f}},{\left(1-{(1-\eta {(\hslash )}^f)}^{\mathrm{\Psi }}\right)}^{\frac{1}{f}}\}$.

Definition 3. 

Let ${\theta }_1(\hslash )=({\mu }_1(\hslash ),v_1(\hslash ),{\eta }_1(\hslash ))$ and ${\theta }_2(\hslash )=({\mu }_2(\hslash ),v_2(\hslash ),{\eta }_2(\hslash ))$ be two FOFNs [28]. Then, the Euclidean distance $({\theta }_1(\hslash ),{\theta }_2(\hslash ))$ of ${\theta }_1(\hslash )$ and ${\theta }_2(\hslash )$ is given as

$$d({\theta }_1(\hslash ),{\theta }_2(\hslash ))=\left[|{({\mu }_1(\hslash ))}^f-{({\mu }_2(\hslash ))}^f{|}^2+|{({\nu }_1(\hslash ))}^f-{({\nu }_2(\hslash ))}^f{|}^2+|{({\eta }_1(\hslash ))}^f-{({\eta }_2(\hslash ))}^f{|}^2\right]$$

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The Euclidean distance satisfies the attributes below.

• $d({\theta }_1(\hslash ),{\theta }_2(\hslash ))\ge 0$;
• $d({\theta }_1(\hslash ),{\theta }_2(\hslash ))=d({\theta }_2(\hslash ),{\theta }_1(\hslash ))$;
• $d({\theta }_1(\hslash ),{\theta }_2(\hslash ))=0\iff d({\theta }_1(\hslash ))=d({\theta }_2(\hslash ))$.

Definition 4. 

Let $\theta (\hslash )=(\mu (\hslash ),v(\hslash ),\eta (\hslash ))$ be an FOFN; the mark and precision function of $\theta (\hslash )$ are illustrated as $Sc(\theta (\hslash ))=\left({(\mu (\hslash ))}^f-{(\nu (\hslash ))}^f-{(\eta (\hslash ))}^f\right)$ and $Ac(\theta (\hslash ))=\left({(\mu (\hslash ))}^f+{(\nu (\hslash ))}^f+{(\eta (\hslash ))}^f\right)$[10].

The comparison rules are as follows for any two FOFNs θ1(ℏ) and θ2(ℏ):

• If $Sc({\theta }_1(\hslash ))>Sc({\theta }_2(\hslash ))$, ${\theta }_1(\hslash )>{\theta }_2(\hslash )$;
• If $Sc({\theta }_1(\hslash ))<Sc({\theta }_2(\hslash ))$, ${\theta }_1(\hslash )<{\theta }_2(\hslash )$;
• If $Sc({\theta }_1(\hslash ))=Sc({\theta }_2(\hslash ))$.

If $Ac({\theta }_1(\hslash ))>Ac({\theta }_2(\hslash ))$, ${\theta }_1(\hslash )>{\theta }_2(\hslash )$;

If $Ac({\theta }_1(\hslash ))<Ac({\theta }_2(\hslash ))$, ${\theta }_1(\hslash )<{\theta }_2(\hslash )$;

If $Ac({\theta }_1(\hslash ))=Ac({\theta }_2(\hslash ))$, ${\theta }_1(\hslash )={\theta }_2(\hslash )$.

2.2. The Dombi Operation

The definitions of Dombi sum and Dombi result in executions. The specific forms of t-norms and t-conorms are as follows:

Definition 5. 

Let $(r,f)\in (0,1)\times (0,1)$ and $g\ge 0$ [21]. Dombi t-norm and Dombi t-conrom are defined as follows:

$$D-tn(r,f)=\frac{1}{1+{\left[{\left(\frac{1-r}{r}\right)}^g+{\left(\frac{1-f}{f}\right)}^g\right]}^{\frac{1}{g}}}$$

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$$D-tcn(r,f)=\frac{1}{1+{\left[{\left(\frac{r}{1-r}\right)}^g+{\left(\frac{f}{1-f}\right)}^g\right]}^{\frac{1}{g}}}$$

(3)

2.3. PA Operator

The PA operator can polymerize the parameters in various sub-regions with a similar aggregation operator and gather the polymerization outcomes of various sub-regions [29]. The expressions are given as follows:

Definition 6. 

$a_i(i=1,2,\dots ,n)$ is defined as a set of non-negative real numbers [30]. Then,

$$PA(a_1,a_2,\dots ,a_n)=\frac{{\displaystyle \underset{i=1}{\overset{n}{\Sigma }}}((1+T(a_i))a_i)}{{\displaystyle \underset{i=1}{\overset{n}{\Sigma }}}(1+T(a_i))}$$

(4)

is called PA operator, where

$$T(a_i)={\displaystyle \underset{\begin{array}{l}j=1\\ j\ne i\end{array}}{\overset{n}{\Sigma }}}Sup(a_i,a_j)$$

(5)

where Sup (a, b) supports a and b, and the three conditions have the following form:

• $Sup(a,b)=[0,1]$;
• $Sup(a,b)=Sup(b,a)$;
• $Sup(a,b)\ge Sup(x,y)$, if $|a-b|\le |x-y|$.

2.4. MM Operator

The MM operator [29] has a remarkable property—it captures the interrelationship between several parameters and demonstrates the general shape of various operators. The expression of the MM operator is displayed as follows.

Definition 7. 

If $(r_1,r_2,r_3,\dots ,r_n)$ represents n crisp numbers and $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)$ denotes an array of n real numbers, meeting ${\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\ge 0$ but not concurrently ${\alpha }_1={\alpha }_2={\alpha }_3=\dots ={\alpha }_n=0$, $b(i)$ means any permutation of $(1,2,3,\dots ,n)$, and Bn means all permutations converge of $(1,2,3,\dots ,n)$ [29].

Then, the definition of the MM operator is shown below.

$$M{M}^{\alpha }(r_1,r_2,r_3,\dots ,r_n)={\left(\frac{1}{n!}{\displaystyle \sum _{b\in B_n}}{\displaystyle \underset{i=1}{\overset{n}{\Pi }}}{r}_{b(i)}^{{\alpha }_i}\right)}^{\frac{1}{{\displaystyle {\displaystyle \sum _{i=1}^n}}{\alpha }_i}}$$

(6)

When the parameter vectors are different, the following special forms of MM operators are given:

1.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)({\alpha }_i=\alpha ,i=1,2,3,\dots ,n)$, the MM operator degenerates to the geometric mean (GM) operator:

$$M{M}^{(\alpha ,\alpha ,\alpha ,\cdots ,\alpha )}(r_1,r_2,r_3,\dots ,r_n)={\left({\displaystyle \underset{i=1}{\overset{n}{\Pi }}}{r}_i\right)}^{\frac{1}{n}}$$

(7)

2.

When the parameter vector $A=(1,0,0,\dots ,0)({\alpha }_1=1,{\alpha }_i=0,i=2,3,\dots ,n)$, the MM operator degenerates to the Arithmetic mean (AM) operator:

$$M{M}^{(1,0,0,\cdots ,0)}(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{n}{\displaystyle \sum _{i=1}^n}{r}_i$$

(8)

3.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,0,\dots ,0)({\alpha }_1,{\alpha }_2\ne 0,{\alpha }_i=0,i=3,\dots ,n)$, the MM operator degenerates to the BM operator:

$$M{M}^{({\alpha }_1,{\alpha }_2,0,\cdots ,0)}(r_1,r_2,r_3,\dots ,r_n)={\left(\frac{1}{n(n-1)}{\displaystyle \underset{\begin{array}{c}i,j=1\\ i\ne j\end{array}}{\overset{n}{\Pi }}}{r}_i^{{\alpha }_1}{r}_i^{{\alpha }_2}\right)}^{\frac{1}{{\alpha }_1+{\alpha }_2}}$$

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4.

If $A=\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)({\alpha }_1={\alpha }_2=\dots {\alpha }_k=1,{\alpha }_{k+1}={\alpha }_{k+2}=\dots ={\alpha }_n=0)$, the MM operator is degenerated to the Maclauric sysmmetric mean (MSM) operator:

$$M{M}^{\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)}(r_1,r_2,r_3,\dots ,r_n)={\left(\frac{{\Sigma }_{1\le i_1<\cdots <i_k\le \left|S_h\right|}{\displaystyle \prod _{j=1}^k}{r}_{i_J}}{C_n^k}\right)}^{\frac{1}{k}}$$

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2.5. PMM Operator

This paper uses the partitioned Muirhead mean (PMM) operator to illustrate the actual relation between the standards. The definition is as follows.

Definition 8. 

Let $(r_1,r_2,r_3,\dots ,r_n)$ be any set of non-negative numbers, $S=\left\{r_1,r_2,r_3,\dots ,r_n\right\}$ means the set of $r_1,r_2,r_3,\dots ,r_n$, $S_h=\left\{r_1,r_2,r_3,\dots ,r_{\left|S_h\right|}\right\}$$(h=1,2,3,\dots ,N)$ is N subregions of S, based on the situation $S_1\cup S_2\cup S_3\cup \dots \cup S_N=S$, and $S_1\cap S_2\cap S_3\cap \dots \cap S_N=\varnothing$ [17]. The PMM operator is shown as follows:

$$PM{M}^{\alpha }(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{1}{|S_h|!}{\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\displaystyle \underset{i_h=1}{\overset{S_h}{\Pi }}}\left(r_{b(i_h)}^{{\alpha }_{i_h}}\right)\right)}^{\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{S_h}}}{\alpha }_{i_h}}}$$

(11)

where |S_h| means partition argument $S_h=\left\{r_1,r_2,r_3,\dots ,r_{\left|S_h\right|}\right\}$, $(h=1,2,3,\dots ,N)$ $b(i_h)$ represents any array of $(1,2,3,\dots ,|S_h|)$, and $B_{\left|S_h\right|}$ means all permutations converge of $(1,2,3,\dots ,|S_h|)A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)$ collects n real numbers, meeting ${\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\ge 0$ but not concurrently ${\alpha }_1={\alpha }_2={\alpha }_3=\dots ={\alpha }_n=0$.

Similarly, as the parameter vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)$ takes on different values, the PMM operator can be translated into several unique operators.

1.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)({\alpha }_i=\alpha ,i=1,2,3,\dots ,n)$, the PMM operator degenerates to the partitioned geometric mean (PGM) operator:

$$PM{M}^{(\alpha ,\alpha ,\alpha ,\cdots ,\alpha )}(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left({\displaystyle \underset{i_h=1}{\overset{S_h}{\Pi }}}{r}_{i_h}\right)}^{\frac{1}{\left|S_h\right|}}$$

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2.

When the parameter vector $A=(1,0,0,\dots ,0)({\alpha }_1=1,{\alpha }_i=0,i=2,3,\dots ,n)$, the PMM operator degenerates to the partitioned Arithmetic mean (PAM) operator:

$$PM{M}^{(1,0,0,\cdots ,0)}(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left(\frac{1}{\left|S_h\right|}{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{r}_{i_h}\right)$$

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3.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,0,\dots ,0)({\alpha }_1,{\alpha }_2\ne 0,{\alpha }_i=0,i=3,\dots ,n)$, the PMM operator degenerates to the partitioned Bonferroni mean (PBM) operator:

$$PM{M}^{({\alpha }_1,{\alpha }_2,0,\cdots ,0)}(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left({\left(\frac{1}{\left|S_h\right|(|S_h|-1)}{\displaystyle \underset{\begin{array}{c}{i}_h,j_h=1\\ i_h\ne j_h\end{array}}{\overset{\left|S_h\right|}{\Pi }}}{r}_{i_h}^{{\alpha }_1}{r}_{i_h}^{{\alpha }_2}\right)}^{\frac{1}{{\alpha }_1+{\alpha }_2}}\right)$$

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4.

When $A=\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)({\alpha }_1={\alpha }_2=\dots {\alpha }_k=1,{\alpha }_{k+1}={\alpha }_{k+2}=\dots ={\alpha }_n=0)$, the PMM operator is degenerated to the partitioned Maclauric sysmmetric mean (PMSM) operator:

$$PM{M}^{\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)}(r_1,r_2,r_3,\dots ,r_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{{\Sigma }_{1\le i_1<\cdots <i_k\le \left|S_h\right|}{\displaystyle {\displaystyle \prod _{j=1}^k}}{r}_{i_j}}{C_{\left|S_h\right|}^k}\right)}^{\frac{1}{k}}$$

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3. FOFS Dombi Power Muirhead Mean Operators

3.1. FOFS Dombi Execution

A set of operational rules for FOFNs can be defined on the basis of the DTT as follows:

Definition 9. 

Let $\theta (\hslash )=(\mu (\hslash ),v(\hslash ),\eta (\hslash ))$, ${\theta }_1(\hslash )=({\mu }_1(\hslash ),v_1(\hslash ),{\eta }_1(\hslash ))$, and ${\theta }_2(\hslash )=$$({\mu }_2(\hslash ),v_2(\hslash ),{\eta }_2(\hslash ))$ be three arbitrary FOFNs, and Ψ be a positive real number. The sums, products, multiplies, and powers between FOFNs on TD, $\lambda \left(x,y\right)=f^{-1}\left(f\left(x\right)+f\left(y\right)\right)$ and SD, $\lambda \left(x,y\right)=g^{-1}\left(g\left(x\right)+g\left(y\right)\right)$ are shown below.

$$\begin{array}{ll}{\theta }_1(\hslash )\oplus {\theta }_2(\hslash )=& \left({\left(1-{(1+{({(\frac{{\mu }_1{(\hslash )}^f}{1-{\mu }_1{(\hslash )}^f})}^{\lambda }+{(\frac{{\mu }_2{(\hslash )}^f}{1-{\mu }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left({(1+{({(\frac{1-{\nu }_1{(\hslash )}^f}{{\nu }_1{(\hslash )}^f})}^{\lambda }+{(\frac{1-{\nu }_2{(\hslash )}^f}{{\nu }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left({(1+{({(\frac{1-{\eta }_1{(\hslash )}^f}{{\eta }_1{(\hslash )}^f})}^{\lambda }+{(\frac{1-{\eta }_2{(\hslash )}^f}{{\eta }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

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$$\begin{array}{ll}{\theta }_1(\hslash )\otimes {\theta }_2(\hslash )=& \left({\left({(1+{({(\frac{1-{\mu }_1{(\hslash )}^f}{{\mu }_1{(\hslash )}^f})}^{\lambda }+{(\frac{1-{\mu }_2{(\hslash )}^f}{{\mu }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left(1-{(1+{({(\frac{{\nu }_1{(\hslash )}^f}{1-{\nu }_1{(\hslash )}^f})}^{\lambda }+{(\frac{{\nu }_2{(\hslash )}^f}{1-{\nu }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left(1-{(1+{({(\frac{{\eta }_1{(\hslash )}^f}{1-{\eta }_1{(\hslash )}^f})}^{\lambda }+{(\frac{{\eta }_2{(\hslash )}^f}{1-{\eta }_2{(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(17)

$$\begin{array}{ll}\psi \theta (\hslash )=& \left({\left(1-{(1+{(\psi {(\frac{\mu {(\hslash )}^f}{1-\mu {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left({(1+{(\psi {(\frac{1-\nu {(\hslash )}^f}{\nu {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left({(1+{(\psi {(\frac{1-\eta {(\hslash )}^f}{\eta {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(18)

$$\begin{array}{ll}\theta {(\hslash )}^{\psi }=& \left({\left({(1+{(\psi {(\frac{1-\mu {(\hslash )}^f}{\mu {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left(1-{(1+{(\psi {(\frac{\nu {(\hslash )}^f}{1-\nu {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left(1-{(1+{(\psi {(\frac{\eta {(\hslash )}^f}{1-\eta {(\hslash )}^f})}^{\lambda })}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

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3.2. FOFS Dombi Power Partitioned Muirhead Mean Operators

We can derive the mathematical definition of the FOFS Dombi power divided Muirhead mean operators on the basis of the theoretical analysis above. It completes the effective integration of the Dombi operator with the valuable information of the PMM and PA operators.

More importantly, experts may give attributes inappropriate scores. Adding PA operators to the scoring procedure may mitigate the negative impact of these scores on the aggregated results.

Definition 10. 

Let $\{\theta ({\hslash }_1),\theta ({\hslash }_2),\dots ,\theta ({\hslash }_n)\}$, where $\theta ({\hslash }_i)=(\mu ({\hslash }_i),\nu ({\hslash }_i),\eta \left({\hslash }_i\right))$$\left(i=1,2, \dots ,n\right)$ collect FOFNs, which are divided into N different sorts $S_1,S_2, \dots ,S_N$, $(S_h=\{{\theta }_{1_h},{\theta }_{2_h},\dots ,{\theta }_{i_h}\}\left(h=1,2, \dots ,N\right))$, which satisfies the situation $S_1\cup S_2\cup S_3\cup \dots \cup S_N=S$, $S_1\cap S_2\cap S_3\cap \dots \cap S_N=\varnothing$ and $\left|S_1\right|+\left|S_2\right|+\dots +\left|S_h\right|=n$, $\alpha =\left({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\right)$ collects n real numbers, meeting ${\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\ge 0$ but not concurrently ${\alpha }_1={\alpha }_2={\alpha }_3=\dots ={\alpha }_n=0$, $b\left(i_h\right)$ represents any array of $(1,2,3,\dots ,|S_h|)$ and $B_{\left|S_h\right|}$ collects all permutations of $(1,2,3,\dots ,|S_h|)$. FOFDPMM operator:

$$FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)=\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{1}{|S_h|!}{\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\displaystyle {\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\Pi }}}}\left({(\frac{n(1+T({\theta }_{b(i_h)}))}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}(1+T({\theta }_k))}{\theta }_{b(i_h)})}^{{\alpha }_{i_h}}\right)\right)}^{\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}}$$

(20)

where $T({\theta }_{b(i_h)})={\displaystyle \underset{\begin{array}{l}j=1\\ j\ne i\end{array}}{\overset{n}{\Sigma }}}Sup({\theta }_{b(i)},{\theta }_{b(j)})$, $Sup({\theta }_{b(i)},{\theta }_{b(j)})=1-d({\theta }_{b(i)},{\theta }_{b(j)})$, $d({\theta }_{b(i)},{\theta }_{b(j)})$ is the Euclidean distance between ${\theta }_{b(i)}$ and ${\theta }_{b(j)}$, and $Sup({\theta }_{b(i)},{\theta }_{b(j)})$ represents the support for ${\theta }_{b(i)}$ from ${\theta }_{b(j)}$, which meets three situations:

• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})=\left[0,1\right]$;
• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})=Sup({\theta }_b{}_{\left(j\right)},{\theta }_b{}_{\left(i\right)})$;
• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})\ge Sup({\theta }_b{}_{\left(x\right)},{\theta }_b{}_{\left(y\right)})$, if $d({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})\le ({\theta }_b{}_{\left(x\right)}-{\theta }_b{}_{\left(y\right)})$.

To simplify Equation (20), let ${\omega }_{b(i_h)}^{\prime }=\frac{1+T({\theta }_{b(i_h)})}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}(1+T({\theta }_k))}$.

Then, $\omega \prime \in \left[0,1\right]$ and ${\displaystyle \underset{i=1}{\overset{n}{\Sigma }}}{\omega }_{b(i_h)}^{\prime }=1$.

Theorem 1. 

Suppose $\{\theta ({\hslash }_1),\theta ({\hslash }_2) ,\dots ,\theta ({\hslash }_n)\}$ (where $\theta ({\hslash }_i)=(\mu ({\hslash }_i),\nu ({\hslash }_i),\eta ({\hslash }_i) )$$\left(i=1, 2, \dots ,n\right)$ collects FOFNs; the final result of the aggregation with the FOFDPPMM operator is a FOFN and equates to the full expansion formula:

$$FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)=\left(\Im ,\Re ,\aleph \right)$$

(21)

where

$$\Im ={\left(1-{(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(22)

$$\Re ={\left({(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(23)

$$\aleph ={\left({(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(24)

Theorem 2. 

(Idempotency) Suppose $\{\theta ({\hslash }_1),\theta ({\hslash }_2) ,\dots ,\theta ({\hslash }_n)\}$, where $\theta ({\hslash }_i)=(\mu ({\hslash }_i),\nu ({\hslash }_i),\eta ({\hslash }_i) )\left(i=1, 2, \dots ,n\right)$ is a collection of FOFNs, abbreviated as ${\theta }_i=〈u_{\theta (i)},{\nu }_{\theta (i)},\eta {}_{\theta (i)}〉i=1,2,\dots ,n$, if all FOFNs are equal, that is, ${\theta }_1={\theta }_2=\dots ={\theta }_n=\theta$, then:

$$FOFDPPM{M}^{\alpha }\left({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n\right)=\theta$$

(25)

Proof. 

As ${\theta }_i=\theta =\left(\mu ,\eta ,\nu \right)\forall i$, we have $Sup\left({\theta }_i,{\theta }_j\right)=1$ for $i,j=1,2,\dots ,n$.

Therefore, we can obtain ${\omega }_i^{\prime }=1/n\forall i$. Furthermore,

$$\begin{array}{l}FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)\\ =FOFDPPM{M}^{\alpha }(\theta ,{\theta }_{},\dots ,\theta )\\ =\left({\left(1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}({\alpha }_{i_h}{(n\frac{1}{n}{(\frac{{\mu }^f}{1-{\mu }^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}},\right.\\ {\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}({\alpha }_{i_h}{(n\frac{1}{n}{(\frac{1-{\nu }^f}{{\nu }^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}},\\ \left.{\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle {\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}}({\alpha }_{i_h}{(n\frac{1}{n}{(\frac{1-{\eta }^f}{{\eta }^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\right)\\ =\left({\left(1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}\left({(\frac{{\mu }^f}{1-{\mu }^f})}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}},\right.\\ {\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}\left({(\frac{1-{\nu }^f}{{\nu }^f})}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}},\\ \left.{\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}\left({(\frac{1-{\eta }^f}{{\eta }^f})}^{\lambda }\right)\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\right)\\ =\left(\mu ,\nu ,\eta \right)=\theta \end{array}$$

(26)

Hence, the proof is complete. □

Theorem 3. 

(Monotonicity) Assume ${\theta }_i=〈u_{\theta (i)},{\nu }_{\theta (i)},\eta {}_{\theta (i)}〉i=1,2,\dots ,n$ and ${\varphi }_i=〈u_{\varphi (i)},{\nu }_{\varphi (i)},\eta {}_{\varphi (i)}〉i=1,2,\dots ,n$ are two groups of FOFNs such that $u_{\theta (i)}\le u_{\varphi (i)},{\nu }_{\theta (i)}\ge {\nu }_{\varphi (i)},{\eta }_{\theta (i)}\ge {\eta }_{\varphi (i)}$ for $i=1, 2,\dots ,n$:

$$FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\le FOFDPPM{M}^{\alpha }({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_n)$$

(27)

Proof. 

Because ${\theta }_i\le {\varphi }_i$, then ${\mu }_{{\theta }_i}\le {\mu }_{{\varphi }_i}$, for all i.

$$\begin{array}{l}\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f}\le \frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f}\\ \Rightarrow {(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda }\le {(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda }\\ \Rightarrow {(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}\ge {(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}\\ \Rightarrow {({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}\le {({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}\\ \Rightarrow {(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\ge {(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\\ \Rightarrow {\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\\ \le {\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\\ \Rightarrow {\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\\ \le {\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\\ \Rightarrow 1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\\ \le 1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\\ \\ \Rightarrow {\left(1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\theta }_i}^f}{1-{\mu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\\ \le {\left(1-{\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\mu }_{{\varphi }_i}^f}{1-{\mu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\end{array}$$

(28)

Similarly, ${\nu }_{\theta (i)}\ge {\nu }_{\varphi (i)},{\eta }_{\theta (i)}\ge {\eta }_{\varphi (i)}$, for all i. Then,

$$\begin{array}{l}{\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\nu }_{{\theta }_i}^f}{1-{\nu }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\\ \le {\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\nu }_{{\varphi }_i}^f}{1-{\nu }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\end{array}$$

(29)

and

$$\begin{array}{l}{\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\eta }_{{\theta }_i}^f}{1-{\eta }_{{\theta }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\\ \le {\left({\left(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(n{\omega }_{b(i_h)}^{\prime }{(\frac{{\eta }_{{\varphi }_i}^f}{1-{\eta }_{{\varphi }_i}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }}\right)}^{-1}\right)}^{\frac{1}{f}}\end{array}$$

(30)

Following this way, we have

$$FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\le FOFDPPM{M}^{\alpha }({\varphi }_1,{\varphi }_2,\dots ,{\varphi }_n)$$

(31)

Hence, the proof is complete. □

Theorem 4. 

(Boundedness) Assume ${\theta }_i=〈u_{\theta (i)},{\nu }_{\theta (i)},\eta {}_{\theta (i)}〉i=1,2,\dots ,n$ is a collection of FOFNs, ${\theta }^{\mathrm{m}ax}=\underset{1\le i\le n}{\max }\left\{{\theta }_i\right\},{\theta }^{min}=\underset{1\le i\le n}{\min }\left\{{\theta }_i\right\}$:

$${\theta }^{\min }\le FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\le {\theta }^{\max }$$

(32)

Proof. 

According to Theorem 2.

$$FOFDPPM{M}^{\alpha }({\theta }_1^{\min },{\theta }_2^{\min },{\theta }_3^{\min },\dots ,{\theta }_n^{\min })={\theta }^{\min }$$

(33)

$$FOFDPPM{M}^{\alpha }({\theta }_1^{\max },{\theta }_2^{\max },{\theta }_3^{\max },\dots ,{\theta }_n^{\max })={\theta }^{\max }$$

(34)

Then,

$$FOFDPPM{M}^{\alpha }({\theta }_1^{\min },{\theta }_2^{\min },\dots ,{\theta }_n^{\min })\le FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\le FOFDPPM{M}^{\alpha }({\theta }_1^{\max },{\theta }_2^{\max },\dots ,{\theta }_n^{\max })$$

(35)

Therefore,

$${\theta }^{\min }\le FOFDPPM{M}^{\alpha }({\theta }_1,{\theta }_2,\dots ,{\theta }_n)\le {\theta }^{\max }$$

(36)

Hence, the proof is complete. □

Evidently, if the coefficient vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)$ takes different special values, FFDPMM will degenerate into different operators:

1.

When the coefficient vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)({\alpha }_i=\alpha ,i=1,2,3,\dots ,n)$, the FOFDPPMM operator degenerates to the FOFDPPGM operator:

$$FOFDPPM{M}^{(\alpha ,\alpha ,\alpha ,\cdots ,\alpha )}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left({\displaystyle \underset{i_h=1}{\overset{S_h}{\Pi }}}\omega \prime r_{i_h}\right)}^{\frac{1}{\left|S_h\right|}}\right\}}_{Dom}$$

(37)

2.

When the parameter vector $A=(1,0,0,\dots ,0)({\alpha }_1=1,{\alpha }_i=0,i=2,3,\dots ,n)$, the FOFDPPMM operator degenerates to the FOFDPPAM operator:

$$FOFDPPM{M}^{(1,0,0,\cdots ,0)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left(\frac{1}{\left|S_h\right|}{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}\omega \prime r_{i_h}\right)\right\}}_{Dom}$$

(38)

3.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,0,\dots ,0)({\alpha }_1,{\alpha }_2\ne 0,{\alpha }_i=0,i=3,\dots ,n)$, the FOFDPPMM operator degenerates to the FOFDPPBM operator:

$$\begin{array}{l}FOFDPPM{M}^{({\alpha }_1,{\alpha }_2,0,\cdots ,0)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)\\ ={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left({\left(\frac{1}{\left|S_h\right|(|S_h|-1)}{\displaystyle \underset{\begin{array}{c}{i}_h,j_h=1\\ i_h\ne j_h\end{array}}{\overset{\left|S_h\right|}{\Pi }}}\omega \prime r_{i_h}^{{\alpha }_1}{r}_{i_h}^{{\alpha }_2}\right)}^{\frac{1}{{\alpha }_1+{\alpha }_2}}\right)\right\}}_{Dom}\end{array}$$

(39)

4.

If $A=\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)({\alpha }_1={\alpha }_2=\dots {\alpha }_k=1,{\alpha }_{k+1}={\alpha }_{k+2}=\dots ={\alpha }_n=0)$, the FOFDPPMM operator degenerates to the FOFDPPMSM operator:

$$\begin{array}{l}FOFDPPM{M}^{\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)\\ ={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{{\Sigma }_{1\le i_1<\cdots <i_k\le \left|S_h\right|}{\displaystyle \prod _{j=1}^k}\omega \prime r_{i_J}}{C_{\left|S_h\right|}^k}\right)}^{\frac{1}{k}}\right\}}_{Dom}\end{array}$$

(40)

3.3. FOFS Dombi Weighted Power Partitioned Muirhead Mean Operators

This section puts forward a weighted form of the FOFDPPMM operator, and introduces weights and PA operators to know the relative importance of properties.

Definition 11. 

If $\{\theta ({\hslash }_1),\theta ({\hslash }_2),\dots ,\theta ({\hslash }_n)\}$ (where $\theta ({\hslash }_i)=(\mu ({\hslash }_i),\nu ({\hslash }_i),\eta \left({\hslash }_i\right))$) $\left(i=1,2, \dots ,n\right)$ collects FOFNs that are divided into N distinct sorts $S_1,S_2, \dots ,S_N$, where$S_h=\{{\theta }_{1_h},{\theta }_{2_h},\dots ,{\theta }_{i_h}\}$$\left(h=1,2, \dots ,N\right)$ and $\left|S_1\right|+\left|S_2\right|+\dots +\left|S_h\right|=n$, and $\alpha =\left({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\right)$ collects n real numbers, meeting the situation ${\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n\ge 0$ but not concurrently ${\alpha }_1={\alpha }_2={\alpha }_3=\dots ={\alpha }_n=0$, let ${\omega }_i$ represent the weight of ${\theta }_i$, where ${\omega }_i\in \left[0,1\right]$ and ${\omega }_1+{\omega }_2+\dots +{\omega }_n=1$ . The FOFS Dombi Weighted power partitioned Muirhead mean operator is shown below.

$$\begin{array}{l}FOFDWPPM{M}^{\alpha }({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)\\ =\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{1}{|S_h|!}{\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\Pi }}}\left({(\frac{n{\omega }_{b(i_h)}(1+T({\theta }_{b(i_h)}))}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}{\omega }_k(1+T({\theta }_k))}{\theta }_{b(i_h)})}^{{\alpha }_{i_h}}\right)\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}}\end{array}$$

(41)

where $T({\theta }_{b(i_h)})={\displaystyle \underset{\begin{array}{l}j=1\\ j\ne i\end{array}}{\overset{n}{\Sigma }}}Sup({\theta }_{b(i)},{\theta }_{b(j)})$, $Sup({\theta }_{b(i)},{\theta }_{b(j)})=1-d({\theta }_{b(i)},{\theta }_{b(j)})$, $d({\theta }_{b(i)},{\theta }_{b(j)})$ is the Euclidean distance between ${\theta }_{b(i)}$ and ${\theta }_{b(j)}$, $Sup({\theta }_{b(i)},{\theta }_{b(j)})$ denotes the support for θb(i) from θb(j), which meets the three conditions below.

• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})=\left[0,1\right]$;
• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})=Sup({\theta }_b{}_{\left(j\right)},{\theta }_b{}_{\left(i\right)})$;
• $Sup({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})\ge Sup({\theta }_b{}_{\left(x\right)},{\theta }_b{}_{\left(y\right)})$, if $d({\theta }_b{}_{\left(i\right)},{\theta }_b{}_{\left(j\right)})\le ({\theta }_b{}_{\left(x\right)}-{\theta }_b{}_{\left(y\right)})$.

For simplifying Equation (41), let ${\omega }_{b(i_h)}^{\prime }=\frac{1+T({\theta }_{b(i_h)})}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}(1+T({\theta }_k))}$.

Then, $\omega \prime \in \left[0,1\right]$ and ${\displaystyle \underset{i=1}{\overset{n}{\Sigma }}}{\omega }_{b(i_h)}^{\prime }=1$.

Theorem 5. 

Suppose {θ(ℏ₁), θ (ℏ₂),…, θ(ℏ_n)} (where θ(ℏi) = (μ(ℏi), ν(ℏi), η(ℏi)) (i = 1, 2,..., n) collects FOFNs, the final outcome of the aggregation by using the FOFDPPMM operator is a FOFN and the full expansion formula is equal to

$$FOFDWPPM{M}^{\alpha }({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)=\left(ℝ,\mathbb{Z},\mathbb{N}\right)$$

(42)

where

$$ℝ={\left(1-{(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(\frac{n{\omega }_{b(i_h)}{\omega }_{b(i_h)}^{\prime }}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}{\omega }_k{\omega }_k^{\prime }}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(43)

$$\mathbb{Z}={\left({(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(\frac{n{\omega }_{b(i_h)}{\omega }_{b(i_h)}^{\prime }}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}{\omega }_k{\omega }_k^{\prime }}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(44)

$$\mathbb{N}={\left({(1+{\left(\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\Sigma }}}{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\alpha }_{i_h}}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{|S_{h|}}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}({\alpha }_{i_h}{(\frac{n{\omega }_{b(i_h)}{\omega }_{b(i_h)}^{\prime }}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}{\omega }_k{\omega }_k^{\prime }}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}$$

(45)

This is proven as follows, and let

$$\frac{n{\omega }_{b(i_h)}{\omega }_{b(i_h)}^{\prime }}{{\displaystyle \underset{k=1}{\overset{n}{\Sigma }}}{\omega }_k{\omega }_k^{\prime }}={\tau }_{b(i_h)}$$

(46)

From Definition 9, we obtain

$$\begin{array}{ll}{\tau }_{b(i_h)}{\theta }_{b(i_h)}=& \left({\left(1-{(1+{\left({\tau }_{b(i_h)}{\left(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f}\right)}^{\lambda }\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left({(1+{\left({\tau }_{b(i_h)}{\left(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f}\right)}^{\lambda }\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left({(1+{\left({\tau }_{b(i_h)}{\left(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f}\right)}^{\lambda }\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(47)

Then,

$$\begin{array}{ll}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}=& \left({\left({(1+{\left(a(i_h){\left({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda }\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left(1-{(1+{\left(a(i_h){\left({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda }\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left(1-{(1+{\left(a(i_h){\left({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda }\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(48)

So, it can be obtained

$$\begin{array}{ll}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}=& \left({\left({(1+{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}\left(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left(1-{(1+{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}\left(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left(1-{(1+{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}\left(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(49)

Then,

$$\begin{array}{ll}{\displaystyle \underset{b\in B_{\left|S_h\right|}}{\oplus }}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}=& \left({\left(1-{(1+{\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ & {\left({(1+{\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ & \left.{\left({(1+{\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{\left({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(50)

Then,

$$\begin{array}{l}\frac{1}{\left|S_h\right|!}{\displaystyle \underset{b\in B_{\left|S_h\right|}}{\oplus }}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}=\\ \left({\left(1-{(1+{\left(\frac{1}{\left|S_h\right|!}\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ {\left({(1+{\left(\frac{1}{\left|S_h\right|!}\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ \left.{\left({(1+{\left(\frac{1}{\left|S_h\right|!}\left({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(51)

Then,

$$\begin{array}{l}{\left(\frac{1}{\left|S_h\right|!}{\displaystyle \underset{b\in B_{\left|S_h\right|}}{\oplus }}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}}=\\ \\ \left({\left({(1+{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{\left(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ {\left(1-{(1+{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{\left(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\end{array}$$

(52)

$$\left.{\left(1-{(1+{\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{\left(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1})\right)}^{-1}\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)$$

Then,

$$\begin{array}{l}{\displaystyle \underset{h=1}{\overset{N}{\oplus }}}{\left(\frac{1}{\left|S_h\right|!}{\displaystyle \underset{b\in B_{\left|S_h\right|}}{\oplus }}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}}=\\ \left({\left(1-{(1+{\left({\displaystyle \sum _{h=1}^N}\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\right.\\ {\left({(1+{\left({\displaystyle \sum _{h=1}^N}\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ \left.{\left({(1+{\left({\displaystyle \sum _{h=1}^N}\left(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1}\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(53)

Finally,

$$\begin{array}{l}\frac{1}{N}{\displaystyle \underset{h=1}{\overset{N}{\oplus }}}{\left(\frac{1}{\left|S_h\right|!}{\displaystyle \underset{b\in B_{\left|S_h\right|}}{\oplus }}{\displaystyle \underset{i_h=1}{\overset{\left|S_h\right|}{\otimes }}}{\left({\tau }_{b(i_h)}{\theta }_{b(i_h)}\right)}^{a(i_h)}\right)}^{\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}}=\\ \left({\left(1-(1+{\left(\frac{1}{N}\left({\displaystyle \sum _{h=1}^N}(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{\mu {(\hslash )}_{b(i_h)}^f}{1-\mu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1})\right)\right)}^{\frac{1}{\lambda }}){}_{-1}\right)}^{\frac{1}{f}},\right.\\ {\left({(1+{\left(\frac{1}{N}\left({\displaystyle \sum _{h=1}^N}(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\nu {(\hslash )}_{b(i_h)}^f}{\nu {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1})\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}},\\ \left.{\left({(1+{\left(\frac{1}{N}\left({\displaystyle \sum _{h=1}^N}(\frac{1}{{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}a(i_h)}{(\frac{1}{\left|S_h\right|!}({\displaystyle \sum _{b\in B_{\left|S_h\right|}}}{({\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}(a_{(i_h)}{({\tau }_{b(i_h)}{(\frac{1-\eta {(\hslash )}_{b(i_h)}^f}{\eta {(\hslash )}_{b(i_h)}^f})}^{\lambda })}^{-1}))}^{-1}))}^{-1})\right)\right)}^{\frac{1}{\lambda }})}^{-1}\right)}^{\frac{1}{f}}\right)\end{array}$$

(54)

The FOFDWPPMM operator has three mathematical properties similar to the FOFDPPMM operator: idempotency, monotonicity, and boundedness. For reasons of space, it will not be illustrated here.

Obviously, if $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)$ has different special values, the FOFDWPPMM becomes several operators:

1.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,{\alpha }_3,\dots ,{\alpha }_n)({\alpha }_i=\alpha ,i=1,2,3,\dots ,n)$, the FOFDWPPMM operator degenerates to the FOFDWPPGM operator:

$$FOFDWPPM{M}^{(\alpha ,\alpha ,\alpha ,\cdots ,\alpha )}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left({\displaystyle \underset{i_h=1}{\overset{S_h}{\Pi }}}{\tau }_{b(i_h)}{r}_{i_h}\right)}^{\frac{1}{\left|S_h\right|}}\right\}}_{Dom}$$

(55)

2.

When the parameter vector $A=(1,0,0,\dots ,0)({\alpha }_1=1,{\alpha }_i=0,i=2,3,\dots ,n)$, the FOFDWPPMM operator degenerates to the FOFDWPPAM operator:

$$FOFDWPPM{M}^{(1,0,0,\cdots ,0)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left(\frac{1}{\left|S_h\right|}{\displaystyle \sum _{i_h=1}^{\left|S_h\right|}}{\tau }_{b(i_h)}{r}_{i_h}\right)\right\}}_{Dom}$$

(56)

3.

When the parameter vector $A=({\alpha }_1,{\alpha }_2,0,\dots ,0)({\alpha }_1,{\alpha }_2\ne 0,{\alpha }_i=0,i=3,\dots ,n)$, the FOFDWPPMM operator degenerates to the FOFDWPPBM operator:

$$\begin{array}{l}FOFDWPPM{M}^{({\alpha }_1,{\alpha }_2,0,\cdots ,0)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)\\ ={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}\left({\left(\frac{1}{\left|S_h\right|(|S_h|-1)}{\displaystyle \underset{\begin{array}{c}{i}_h,j_h=1\\ i_h\ne j_h\end{array}}{\overset{\left|S_h\right|}{\Pi }}}{\tau }_{b(i_h)}{r}_{i_h}^{{\alpha }_1}{r}_{i_h}^{{\alpha }_2}\right)}^{\frac{1}{{\alpha }_1+{\alpha }_2}}\right)\right\}}_{Dom}\end{array}$$

(57)

4.

If $A=\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)({\alpha }_1={\alpha }_2=\dots {\alpha }_k=1,{\alpha }_{k+1}={\alpha }_{k+2}=\dots ={\alpha }_n=0)$, the FOFDWPPMSM operator degenerates to the PMSM operator:

$$FOFDWPPM{M}^{\left(\stackrel{k}{\overbrace{1,1,\cdots 1}},\stackrel{n-k}{\overbrace{0,0,\cdots 0}}\right)}({\theta }_1,{\theta }_2,{\theta }_3,\dots ,{\theta }_n)={\left\{\frac{1}{N}{\displaystyle \sum _{h=1}^N}{\left(\frac{{\Sigma }_{1\le i_1<\cdots <i_k\le \left|S_h\right|}{\displaystyle \prod _{j=1}^k}{\tau }_{b(i_h)}{r}_{i_J}}{C_{\left|S_h\right|}^k}\right)}^{\frac{1}{k}}\right\}}_{Dom}$$

(58)

4. Construction of an Evaluation Index System for Government Information Disclosure on Public Health Incidents

Information disclosure on public health emergencies is a very complex process. The information discharger discloses the specific information to the information recipient through certain media, and in this process, every influence factor interpretation or evaluation cannot be a simple conclusion but must be a comprehensive and systematic analysis of the various factors or influences involved in the process. Maletzke [31] proposed a systematic mass communication model, stressing the need for a comprehensive and systematic analysis of the various factors or influences involved in the activity or process of social communication. Therefore, this paper is based on the theoretical framework of Maletzke’s mass communication model and uses the core elements of Maletzke’s model to divide the factors affecting information disclosure into four dimensions: the government dimension, the information dimension, the media dimension, and the public dimension. The government, the media, and the public are the main participants in public health emergencies, playing different roles and assuming other responsibilities in public health incidents. Public health incident information disclosure depends on the conscious and rational attitude and behavior of the government, media, and the public. They should jointly pay attention to public health emergencies, participate in the response to them, and assume social responsibilities. This way, information disclosure can be timely and effective, and public health emergencies can be handled scientifically and reasonably. We integrate the views of related studies and propose the influencing factors of public health emergency information disclosure from four dimensions: government, information, media, and public. Eighteen factors affecting government information disclosure on public health emergencies are summarized. The Decision-Making Trial and Assessment Laboratory (DEMATEL) approach examines the influence of information disclosure on public health incidents. The DEMATEL method is used to screen and extract the specific factors of information disclosure on public health incidents. The preliminary assessment system of information disclosure on public health incidents is constructed, involving 14 attributes in four dimensions (due to space limitations, the process of this method is not repeated).

Table 1. Assessment indicator system for government information disclosure on public health emergencies.

Evaluation index system for government information disclosure on public health emergencies	Information disclosure subject dimension (Government dimension A1)	Development of laws and regulations related to government information disclosure (B1)
		The scope, time, and standards of government information disclosure (B2)
		Government information technology construction level (B3)
		Degree of information disclosure awareness among government officials (B4)
	Information subject dimension (Information dimension A2)	Quality of information (B5)
		Number of messages (B6)
		Timeliness of information (B7)
	Information pathway dimension (Media dimension A3)	Media opinion-monitoring function (B8)
		Media information-processing level (B9)
		Extent of media coverage of information tracking in real time (B10)
		Professional ethical qualities of media personnel (B11)
	Information-receiving subject dimension (Public dimension A4)	Awareness of public information disclosure (B12)
		Public awareness of the right to information, participation, and supervision (B13)
		Public literacy and ability to access information (B14)

shows the details.

The information disclosure subject dimension (government dimension) consists of four aspects: the development of laws and regulations related to government information disclosure [32]; the scope, time, and standards of government information disclosure [33,34]; the government information technology construction level [35]; and the degree of information disclosure awareness among government officials [35,36]. The development of laws and regulations related to government information disclosure refers to the legislative regulation of the government’s actions, such as unclear responsibilities and weak awareness of information disclosure when information about public health incidents is disclosed. The range, time, and standards of government information disclosure refer to the scale of government information disclosure as the real basis of information disclosure on public health emergencies and the implementation standard of government information disclosure. The government information technology construction level refers to several aspects, including electronic management and office electrification. Strengthening the government’s information construction work can effectively promote re-designing government functions, improve government administrative efficiency, and reduce the government’s operating costs. Government officials’ degree of information disclosure awareness refers to the conscious attitude of government information workers to voluntarily disclose government information to the public, legal persons, or other organizations by means of legal forms and procedures.

The subject information dimension (information dimension) consists of three aspects: the quality of information [37,38], the number of messages [39], and the timeliness of information [34,40]. The quality of information includes academic level, credibility, timeliness, and continuity of content, which are the most basic and most important criteria for the evaluation of network information resources and the different levels of network information, and the authenticity and validity of the information have to be examined. The number of messages refers to the total amount of government information actively disclosed in the current year and the total amount of all kinds of periodicals and networks. Timeliness refers to how quickly and efficiently information is received, processed, transmitted, and used. For shorter time intervals, the timelier the use of information, the higher the service level, and the more important the timeliness.

The information pathway dimension (media dimension) includes four aspects: the media opinion-monitoring function [41,42], the media information-processing level [43,44], the extent of media coverage of information tracking in real time, and the professional and ethical qualities of media personnel [37,45,46]. The media opinion-monitoring function means that the media plays their best role in an epidemic to interrupt the spread of rumors, report valid information, and provide correct information. The media information-processing level refers to media taking the initiative to communicate with relevant government departments; release information in the best way and at the best time; ensure the rapid, accurate, and effective dissemination of public health emergency information; and enhance the influence of media. The extent of media coverage of information tracking in real time refers to the timely detection of potential problems through real-time monitoring of relevant information in the media and websites to guarantee the government makes correct decisions. The ethical qualities of media workers refer to the norms and guidelines of journalism that have been developed by journalists in the course of their long-term practical activities in the field of journalism in order to regulate the interrelations between people, and this is the unique requirement of social ethics for journalists.

The information-receiving subject dimension (public dimension) includes three aspects: the awareness of public information disclosure [43]; the public awareness of the right to information, participation, and supervision [34,43]; and the public literacy and ability to access information [47]. The awareness of public information disclosure refers to the extent to which the government and related agencies disclose their information according to the public’s wishes and needs. The most useful information disclosure is when the public can acquire the most information in the shortest time, at the lowest cost, and in the most convenient way. The public awareness of the right to information, participation, and supervision refers to the degree of understanding of the public’s right to information, expression, leadership, and involvement in working with the press and public opinion. Protecting the public’s right to information allows them to fulfill their right to participation and leadership. In contrast, the full realization of citizens’ right to participation and supervision promotes the progress and improvement of the democratic style of government agencies. It enhances the credibility and authority of the government. Public literacy and the ability to access information refers to the public’s ability to obtain, analyze, evaluate, and disseminate various pieces of information as a subject, directly proportional to the effectiveness of information disclosure and dissemination.

5. A New MADM Model on Basis of CRITIC and FOFDWPPMM Operator

In this paper, we use the CRITIC (Criteria Importance Through Intercriteria Correlation) assignment method, an objective weight assignment method proposed by Diakoulaki [48] in 1995. It is a comprehensive measure of indexes on the basis of the comparative strength of the assessment indexes and the conflicting property of the objective weights between the indicators. This method can consider the correlation between indicators while considering the magnitude of variability of indicators and can conduct the scientific evaluation by using the objective properties of the information itself. The honest assessment indicator system of government information disclosure on public health emergencies is relatively complex, so it is not easy to establish a comprehensive index system to evaluate it. On the basis of the above considerations, this paper uses the CRITIC method to calculate the property weights in the problem in the factional orthotriple fuzzy environment, resulting in the FOF-CRITIC process. A MADM problem consists of multiple alternatives, $C=\left\{C_1,C_2,\dots ,C_m\right\}$, and n attributes $a=\left\{a_1,a_2,\dots ,a_n\right\}$. Also, these n attributes are divided into N sub-regions, $S_h=\{a_1,a_2,\dots ,a_{\left|S_h\right|}\} \left(h=1, 2, \dots ,N\right)$ meeting $S_1\cup S_2\cup S_3\cup \dots \cup S_N=a$ and $S_1\cap S_2\cap S_3\cap \dots \cap S_N=\varnothing$. $\left|S_h\right|$ denotes the quantity of parameters in the division $S_h=\{a_1,a_2,\dots ,a_{\left|S_h\right|}\} \left(h=1, 2, \dots ,N\right)$. The major procedures of the FOF-CRITIC approach are shown below.

Step 1: Standardizing the evaluation matrix. Two kinds of criteria may be included in multi-property decision making, namely benefit criterion and cost criterion, which play positive and negative roles in the evaluation outcomes, respectively. To remove the negative effects, the following specification is performed:

$${\mathrm{\Gamma }}_{ij}=\left\{{}_{\left({\chi }_{ij},{\alpha }_{ij},{\beta }_{ij}\right), \mathrm{if} {\Phi }_j \mathrm{is} \mathrm{a} \cos \mathrm{t} \mathrm{criterion}}^{\left({\alpha }_{ij},{\beta }_{ij},{\chi }_{ij}\right), \mathrm{if} {\Phi }_j \mathrm{is} \mathrm{a} \mathrm{benefit} \mathrm{criterion}}\right.$$

(59)

Step 2: The CRITIC assignment approach is adopted to decide the property weights based on the resulting assessment matrix. This method minimizes the perturbing effects of subjective randomness.

1.

$S\left(f\right)$ is the score function of FOFNs, and the inter-attribute correlation coefficient SFCC can be defined as

$$FOFC{C}_{jt}=\frac{{\displaystyle {\displaystyle \sum _{i=1}^m}\left(S\left(f_{ij}\right)-S\left(f_j\right)\right)\cdot \left(S\left(f_{it}\right)-S\left(f_t\right)\right)}}{\sqrt{{\displaystyle {\displaystyle \sum _{i=1}^m}{\left(S\left(f_{ij}\right)-S\left(f_j\right)\right)}^2}}\cdot \sqrt{{\displaystyle {\displaystyle \sum _{i=1}^m}{\left(S\left(f_{it}\right)-S\left(f_t\right)\right)}^2}}}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em},\hspace{0.33em}\hspace{0.33em}j,t=1,2,\dots ,n$$

(60)

where $S\left(f_j\right)=\frac{1}{m}{\displaystyle {\displaystyle \sum _{i=1}^m}S\left(f_{ij}\right)}$ and $S\left(f_t\right)=\frac{1}{m}{\displaystyle {\displaystyle \sum _{i=1}^m}S\left(f_{it}\right)}$.

2.

The standard deviation of the attribute is

$$FOFS{D}_j=\sqrt{\frac{1}{m}{\displaystyle {\displaystyle \sum _{i=1}^m}{\left(S\left(f_{ij}\right)-S\left(f_j\right)\right)}^2}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,2,\dots ,n$$

(61)

where $S\left(f_j\right)=\frac{1}{m}{\displaystyle {\displaystyle \sum _{i=1}^m}S\left(f_{ij}\right)}$.

3.

Then, the attribute weights ${\omega }_j$ are as follows:

$${\omega }_j=\frac{FOFS{D}_j{\displaystyle {\displaystyle \sum _{t=1}^n}\left(1-FOFC{C}_{jt}\right)}}{{\displaystyle {\displaystyle \sum _{j=1}^n}\left(FOFS{D}_j{\displaystyle {\displaystyle \sum _{t=1}^n}\left(1-FOFC{C}_{jt}\right)}\right)}},\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}j=1,2,\dots ,n$$

(62)

where ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle {\displaystyle \sum _{j=1}^n}{\omega }_j}=1$.

Step 3: Calculate the comprehensive assessment value of every alternative $C_i\left(i=1,2,\dots ,m\right)$. Utilizing the property weight vector $\omega =\left\{{\omega }_1,{\omega }_2\dots .{\omega }_n\right\}$, where ${\omega }_j\in \left[0,1\right]$ and ${\displaystyle {\displaystyle \sum _{j=1}^n}{\omega }_j}=1$, and the FOFDWPPMM operator in Definition 11, we calculate the entire assessment value of the alternatives $C_i\left(i=1,2,\dots ,m\right)$.

Step 4: Calculate the score value of every alternative according to Definition 4, and rank the scores of each alternative.

The computational flowchart of the method is shown in Figure 1.

6. Example Analysis

To ensure that the research remains trusted and accurate, ten experts (four scholars, three government department staff, and three media staff) in related fields were invited to give multidimensional scores to 14 attributes in this paper’s assessment indicator system for government information disclosure on public health incidents. The 14 properties, which are interpreted in Section 4, are shown below.

B₁: Development of laws and regulations related to government information disclosure;

B₂: The scope, time, and standards of government information disclosure;

B₃: Government information technology construction level;

B₄: Degree of information disclosure awareness among government officials;

B₅: Quality of information;

B₆: Number of messages;

B₇: Timeliness of information;

B₈: Media opinion-monitoring function;

B₉: Media information-processing level;

B₁₀: Extent of media coverage of information tracking in real time;

B₁₁: Professional ethical qualities of media personnel;

B₁₂: Awareness of public information disclosure;

B₁₃: Public awareness of the right to information, participation, and supervision;

B₁₄: Public literacy and ability to access information.

The 14 attributes were divided into four partitions: A1: information disclosure subject dimension (government dimension), $A_1=\left\{B_1,B_2,B_3,B_4\right\}$; A2: information subject dimension (information dimension) $A_2=\left\{B_5,B_6,B_7\right\}$; A3: information route dimension (media dimension), $A_3=\left\{B_8,B_9,B_{10},B_{11}\right\}$; A4: information-receiving subject dimension (public dimension) $A_4=\left\{B_{12},B_{13},B_{14}\right\}$.

Table 2. Evaluation matrix by FOFS.

Criterion	Attribute	C1	C2	C3	C4
A1	B1	<0.5,0.2,0.4>	<0.3,0.2,0.5>	<0.2,0.2,0.6>	<0.5,0.2,0.3>
	B2	<0.4,0.4,0.3>	<0.2,0.4,0.6>	<0.1,0.3,0.5>	<0.6,0.5,0.5>
	B3	<0.6,0.2,0.2>	<0.4,0.5,0.1>	<0.3,0.5,0.6>	<0.2,0.3,0.4>
	B4	<0.5,0.4,0.1>	<0.3,0.6,0.4>	<0.5,0.4,0.4>	<0.4,0.6,0.6>
A2	B5	<0.8,0.1,0.2>	<0.6,0.3,0.6>	<0.6,0.6,0.2>	<0.3,0.3, 0.4>
	B6	<0.4,0.2,0.3>	<0.2,0.4,0.5>	<0.4,0.2,0.3>	<0.5,0.2,0.6>
	B7	<0.3,0.6,0.4>	<0.6,0.5,0.3>	<0.3,0.7,0.5>	<0.3,0.4,0.2>
A3	B8	<0.5,0.4,0.2>	<0.7,0.2,0.4>	<0.3,0.5,0.6>	<0.4,0.2,0.4>
	B9	<0.4,0.2,0.3>	<0.2,0.5,0.5>	<0.2,0.6,0.4>	<0.6,0.2,0.6>
	B10	<0.2,0.7,0.2>	<0.6,0.2,0.6>	<0.6,0.1,0.2>	<0.5,0.5,0.5>
	B11	<0.8,0.2,0.5>	<0.2,0.2,0.8>	<0.5,0.5,0.1>	<0.2,0.2,0.2>
A4	B12	<0.7,0.4,0.5>	<0.5,0.7,0.4>	<0.2,0.4,0.6>	<0.1,0.7, 0.6>
	B13	<0.5,0.3,0.1>	<0.4,0.5,0.6>	<0.1,0.2,0.2>	<0.3,0.3,0.7>
	B14	<0.6,0.4,0.2>	<0.7,0.4,0.2>	<0.8,0.2,0.3>	<0.4,0.6,0.3>

shows specific information on the evaluation of each attribute by experienced research experts.

6.1. Evaluation Ranking

Step 1: Standardize the evaluation matrix. Because all 14 properties are benefit attributes, no normalization is required.

Step 2: Calculate the attribute weights utilizing the CRITIC assignment approach, as displayed in

Table 3. Property weight calculation.

Criterion	Criterion Weight	Attribute	CRITIC Weight
A1	0.2076	B1	0.0521
		B2	0.0354
		B3	0.0624
		B4	0.0576
A2	0.2004	B5	0.0552
		B6	0.0402
		B7	0.1051
A3	0.3668	B8	0.0951
		B9	0.0545
		B10	0.1315
		B11	0.0857
A4	0.2251	B12	0.0689
		B13	0.0563
		B14	0.0999

.

Step 3: Let λ = 2, the parameter vector $\alpha =\left(\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{3},\frac{1}{3},\frac{1}{3},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{4},\frac{1}{3},\frac{1}{3}\right.,\left.\frac{1}{3}\right)$, and calculate the comprehensive evaluation value of the alternatives $C_i\left(i=1,2,\dots ,m\right)$ according to the FOFDWPPMM operator in Definition 11. The results are

$$\begin{gathered} C_1=\{0.4658,0.3908,0.3239\}, C_2=\{0.3356,0.4337,0.5322\} \\ {C}_3=\{0.3279,0.4049,0.4148\}, C_4=\{0.3119,0.3448,0.5243\} \end{gathered}$$

Step 4: Calculate the score value of every alternative on the basis of Definition 4 to obtain $SC{O}_{C_1}=-0.0408,SC{O}_{C_2}=-0.3587,SC{O}_{C_3}=-0.2285,SC{O}_{C_4}=-0.2965$, and rank the score values to obtain $C_1>C_3>C_4>C_2$. That is, C1 has the highest degree of information disclosure.

6.2. Figures, Tables, and Schemes

The FOFDWPPMM operator herein is significantly related to the parameter λ. The function of parameter λ in the evaluation of information disclosure can be more accurately observed by the perturbation analysis of parameter λ.

Table 4. Role of coefficient λ in ranking outcomes (1–10).

Coefficient Value	Ranking of Score Values	Ranking Outcome
λ = 1	SCOC1 > SCOC2 > SCOC3 > SCOC4	C1 > C2 > C3 > C4
λ = 2	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 3	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 4	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 5	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 6	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 7	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 8	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 9	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2
λ = 10	SCOC1 > SCOC3 > SCOC4 > SCOC2	C1 > C3 > C4 > C2

and Figure 2 show the details. Based on Table 4, there are various ranking orders for various values of parameter λ: when λ = 1, the ranking outcome is $C_1>C_2>C_3>C_4$, and when λ ≥ 2, the ranking outcome is $C_1>C_3>C_4>C_2$. However, the government with the highest degree of information disclosure is always C1 regardless of the change in λ value. The ranking order of the comprehensive score of the programs remains the same starting from λ equal to 2, which is always $C_1>C_3>C_4>C_2$.

At the same time, from Figure 1, we observe that with the increase in λ value, the comprehensive evaluation value of the four schemes also decreases, and their scores generally show a downward trend. However, the comprehensive evaluation value of the C₁ program is significantly higher than the other three programs. From λ = 2, the comprehensive score of the C₁ program decreases slowly and tends to level off in general. In contrast, the score of C₂ decreases significantly faster than C₁ and C₄, so when λ ≥ 2, the comprehensive score of C₂ decreases rapidly and is lower than C₁ and C₄. The score of C₃ decreases more slowly compared to C₄, and with the growth of λ, the change in C₄ comprehensive score tends to be relatively flat. Generally, the magnitude of the coefficient λ in the FOFDWPPMM operator can vary in the relevant order of the four governments.

6.3. Contrast Analysis

The efficacy and flexibility of the aggregation approach herein can be illustrated by the above example of the information disclosure assessment of a public health emergency and the sensitivity analysis of the parameter λ. To further demonstrate the applicability and superiority of our approach, it is compared with the FOFDWPPMM operator, the FOFDWA operator, the FOFDWG operator, the FOFDWPGM operator, the FOFDWPAM operator, and the FOFDWPBM operator for quantitative comparative analysis (

Table 5. Quantitative comparative analysis.

Aggregation Approach	Ranking of Score Values	Ranking Outcome
FOFWGM	SCOC4 > SCOC2 > SCOC3 > SCOC1	C4 > C2 > C3 > C1
FOFWAM	SCOC1 > SCOC3 > SCOC2 > SCOC4	C1 > C3 > C2 > C4
FOFDWPGM	SCOC1> SCOC3> SCOC4> SCOC2	C1 > C3 > C4 > C2
FOFDWPAM	SCOC1> SCOC2> SCOC3> SCOC4	C1 > C2 > C3 > C4
FOFDWPBM	SCOC3> SCOC2> SCOC4> SCOC1	C3 > C2 > C4 > C1
FOFDWPPMM	SCOC1> SCOC3> SCOC4> SCOC2	C1 > C3 > C4 > C2

). The superiority of the operators in the paper is further demonstrated by showing the qualitative comparison of several aggregation methods (

Table 6. Qualitative comparative analysis.

Aggregation Method	Whether to Consider the Interrelationship of Two Attributes	Whether to Consider the Interrelationship of Multiple Attributes	Whether to Consider the Internal Division of Attribute Interrelationships	Whether to Reduce the Negative Impact of Inappropriately High and Inappropriately Low Evaluations	Whether the Parameter Vector Increases the Flexibility of the Method
FOFWGM	NO	NO	NO	NO	NO
FOFWAM	NO	NO	NO	NO	NO
FOFDWPGM	NO	NO	YES	NO	YES
FOFDWPAM	NO	NO	YES	NO	YES
FOFDWPBM	YES	YES	YES	NO	YES
FOFDWPPMM	YES	YES	YES	YES	YES

).

Comparing the ranking outcomes in Table 5, the best alternative acquired by all four aggregation methods except the FOFWGM operator and the FOFDWPBM operator is C₁. Except for the FOFWGM operator, the results of all five aggregation methods consider the C₃ government score higher than the C₄ government score. The final rankings obtained by all four methods except the FOFWGM and FOFDWPAM operators consider the C₃ government to be better than the C₂ government. Although the order of the above five methods is slightly varied, they are consistent in the choice of the optimal solution. The accuracy and validity of the judgment in this paper are further validated by these results. The reasons for the different rankings are summarized as follows from the results of the above comparisons:

• First of all, the reason why the results differ from those of the FOFDWA operator and the FOFDWG operator is that the FOFDWPPMM operator can decrease the negative impact of singular values on the evaluation outcomes but also can fully take into account the relationship between information-related properties and the division of relationships between properties internally. The proposed approach herein can minimize the distortion in the information aggregation process. Therefore, this paper’s proposed method has greater flexibility, greater practicality, and a wider usage scope.
• By comparing with the FOFDWPAM operator, the choice of an optimal solution is consistent with the FOFDWPPMM operator. Still, the sorting order of C₂, C₃, and C₄ is different, because the FOFDWPAM operator only considers the division between different attributes, does not consider the relationship between the two attributes, and cannot remove the negative influence of singular value on the evaluation outcomes. Specifically, the FOFDWPMM operator is reduced to the FFDWPBM operator when $A=({\alpha }_1,{\alpha }_2,0,\dots ,0)({\alpha }_1,{\alpha }_2\ne 0,{\alpha }_i=0,i=3,\dots ,n)$. Therefore, when there is a more complicated relationship between different properties, this paper will be more general and applicable to a broader scope.
• Compared with the FOFDWPBM operator, although the FOFDWPBM operator considers the interrelationships between properties and the division of relationships within properties, the FOFDWPBM operator cannot remove the negative influence of singular values on the assessment outcomes. Therefore, the assessment outcome by the FOFDWPBM operator differs from the aggregation method herein.

After comparative analysis, the FOFDWPPMM operator herein can fully take into account the relationship between properties and the division of relationships within properties to decrease the distortion in the assembly process. It can also avoid the impact of the singular value on the assessment outcome. In summary, the multi-attribute decision-making approach herein has greater flexibility, more robust applicability, and a broader application scope.

6.4. Discussion

Section 6.2 discusses the sensitivity analysis of different parameter perturbations, and Section 6.3 discusses the comparative analysis of the operators proposed in this paper and the existing different operators (see Table 5 and Table 6) which highlight the validity and applicability of the FOFDWPPMM operator proposed in this paper. In the following, we will discuss the superiority of the information disclosure assessment method for public health emergencies based on the FOFDWPPMM operator proposed in this paper from the perspective of the overall process of the assessment method in terms of (1) developing a flexible and practical FOFDWPPMM operator, (2) determining objective and reasonable attribute weights, and (3) constructing an applicable method for assessing the disclosure of public health emergency information.

In this paper, fractional orthotriple fuzzy sets are used to express the evaluation values of each attribute, and the FOFDWPPMM operator is proposed for aggregating multi-attribute evaluation information in uncertain situations. Based on the existing research, fuzzy sets [6] can only express either/or evaluation information; intuitionistic fuzzy sets [7] can only express membership and non-membership, and the sum of the two cannot be greater than 1; Pythagorean fuzzy sets [8] expand the range of values of the intuitionistic fuzzy set, and the sum of squares of the two is not greater than 1; q-rung orthopair fuzzy sets [9] further expand the range of values, and the power of the two is not greater than 1. Based on the above extended fuzzy set, the fractional orthotriple fuzzy sets include positive, neutral, and negative levels, containing more comprehensive uncertainty information, which can more flexibly and completely characterize the evaluation information in the context of uncertainty. Taking fractional orthotriple fuzzy sets as the basis, integrating the compatible qualities of the Dombi operation, the power mean operator to avoid singular value influence, and the Muirhead average operator to deal with attribute correlation, and taking into account the situation of attribute partition, we propose the FOFDWPPMM operator, which includes positive, neutral, and negative levels. It is a more comprehensive information aggregation operator that can function as a better technical tool for solving the fuzzy multi-attribute evaluation problem.

According to the objective assignment method, CRITIC, this paper proposes the FOF-CRITIC assignment method to determine the attribute weights. Compared with the subjective assignment method, AHP [49], which is prone to the subjective consciousness of experts, this method can avoid the influence of the lack of knowledge of experts or decision makers on the results and has better objectivity in application. In addition, the information disclosure on the public health emergency evaluation index system is divided into four dimensions, and there is a correlation between the attributes of each dimension. The objective weighting method, the entropy value method [50], cannot directly obtain attribute weights based on the volatility or correlation of the assessment value, and is not suitable for a public health emergency information assessment. Although the variation coefficient method [51] considers the fluctuation of numerical values, and the principal component analysis method [52] considers the correlation between values, they are still not comprehensive. The CRITIC method measures the amount of information reflected in each evaluation index by introducing a standard deviation. The larger the calculated standard deviation, the greater the difference of the indicator and the more different types of information are reflected. Then, the correlation coefficient is used to reflect the conflict between the indicators; the larger the correlation coefficient, the more repeated information there is reflected between the two indicators, and the lower the weight of the index that should be assigned to it. Then, the two values are summarized and calculated to obtain the objective weight of each evaluation index. The CRITIC method considers both the volatility of the value and the correlation between the values, which can improve the objectivity of the index weight to a certain extent.

Most of the research on information disclosure during public health emergencies focuses on the necessity [1,2], problems [3], and strategies [4], and there are few evaluation studies on public health emergency information disclosure. Based on the mass communication model proposed by Maletsk, this paper constructs the index system of public health emergency information disclosure, which is an expansion of existing research and is conducive to further promoting the disclosure of public health emergency information; improving the quality of information disclosure; strengthening the prediction, prevention, and control of public health emergencies; and improving the emergency handling level and management efficiency during public health emergencies. In addition, there are evaluation models related to emergencies, such as TOPSIS [49], VIKOR [53], ELECTRE [54], PROMETHEE [55], and other methods, which do not consider the different dimensional partitions of attributes, nor have they considered the correlation between attributes in the same dimension. Most of the index systems, in reality, are divided into different dimensions, and there is a strong correlation between attributes in the same dimension, so the multi-attribute evaluation method based on the FOFDWPPMM operator proposed in this paper has high adaptability for solving the problem of public health emergency information disclosure assessment and has a certain reference value for public health emergency information disclosure practice and related research.

7. Conclusions

The core contribution of this paper is proposing the FOFDWPPMM operator and applying it to assess information disclosure methods in public health emergencies. First, FOFS have a wide space in characterizing assessment information, and Dombi operations have compatible qualities. Thus, this paper gives the FOFS Dombi operation rules based on Dombi operation rules. On this basis, the FOFDWPPMM operator is proposed by comprehensively considering the avoidance of singularity influence (power), the attribute partitioning case (partitioned), and the handling of correlations among multiple attributes (Muirhead). Then, considering the objectivity and practicality of the CRITIC method in practical assessment, the FOF-CRITIC objective assignment method is proposed and combined with the FOFDWPPMM operator to give the multi-attribute assessment process. In addition, considering the applicability of the proposed research method, based on the mass communication model, the index system for assessing the information disclosure during public health emergencies is constructed, and the multi-attribute assessment method proposed in this paper is applied to assess the disclosure of information on public health emergencies. Finally, the effectiveness and superiority of the new method are demonstrated through example analysis.

The research in this paper exploits the advantage of FOFS for the expression of uncertain information to make up for the shortcomings of existing information disclosure assessment methods for public health emergencies. The proposed FOFS Dombi arithmetic rule can be used as a computational basis with other operators (e.g., Hamy, Prioritized, Maclaurin). The proposed FOFDWPPMM operator is a novel extension of the fractional orthotriple fuzzy MADM method, which can better deal with the multi-attribute decision-making problems of complex systems. The research results presented in this paper still have limitations in practical applications, especially in the screening of multi-attribute evaluation data. In the future, we will further study the high-ranked connective system of the FOFS to solve the problem of data sources. In addition, in practical fuzzy multi-attribute evaluation, the expert’s evaluation information is often uncertain qualitative linguistic information. We also will extend our proposed operators to linguistic fuzzy sets, such as intuitionistic fuzzy linguistic fuzzy sets, Pythagorean fuzzy linguistic fuzzy sets, q-rung orthopair fuzzy linguistic fuzzy sets, and fractional orthotriple fuzzy linguistic fuzzy sets. We will apply them to other multi-attribute decision-making problems, such as quality assessment, pattern recognition, cluster analysis, medical diagnosis, and so on.