A Hesitation-Associated Multi-Attribute Decision-Making Method Based on Generalized Interval-Valued Hesitation Fuzzy Weighted Heronian Averaging Operator

Abstract

In multi-attribute decision making (MADM), complex situations often arise where decision attributes are interval-valued hesitant fuzzy numbers (IVHFNs) and the attributes are interrelated. Traditional decision-making methods may be ineffective in handling such cases, highlighting the practical importance of seeking more effective approaches. Therefore, finding a more effective decision-making approach has important practical significance. By combining the theories of Archimedean S-norms and T-norms, we innovatively propose a multi-attribute decision-making method based on the generalized interval-valued hesitant fuzzy weighted Heronian mean (GIVHFWHM) operator to address the aforementioned issues. Initially, based on the operational laws of IVHFNs and the Heronian mean (HM) operator, we introduce the generalized interval-valued hesitant fuzzy Heronian mean (GIVHFHM) operator and the GIVHFWHM operator. We then examine properties of the GIVHFHM operator, including permutation invariance, idempotency, monotonicity, boundedness, and parameter symmetry. A multi-attribute decision-making model is constructed based on the GIVHFWHM operator. Finally, we validate the proposed model through numerical experiments in MADM. The results demonstrate that the new decision-making method, based on the GIVHFWHM operator, is feasible and effective in handling multi-attribute decision problems involving IVHFNs with interdependent attributes. This approach provides a novel perspective and method for solving MADM problems under interval-valued hesitant fuzzy conditions with interdependent attributes. It enriches the theoretical framework of multi-attribute hesitant decision models and expands the application of the Heronian mean operator within interval-valued hesitant fuzzy environments. This methodology assists decision makers in making more accurate decisions within complex decision-making contexts, enhancing both the scientific rigor and reliability of decision-making processes.

1. Introduction

In the present time of the explosive growth of information, the rapid development of social informatization puts higher demands on the complexity of multi-attribute decision-making (MADM) problems. With the continuous progress of information technology, the amount of data and the type of information involved in the decision-making process are increasing, which means decision makers have to consider more factors and variables when facing MADM problems. At the same time, due to the ambiguity of people’s perception of the MADM problem and the uncertainty of the problem itself, the values of the decision attributes often need to be represented by fuzzy information. In such a context, the complexity of decision makers’ thinking is further exacerbated, and the decision-making information they provide often presents complex correlations.

To solve these problems, the intuitionistic fuzzy set (IFS) theory was developed, which excels in dealing with ambiguities, uncertainties, and correlations among attributes in MADM problems. The core of the IFS theory lies in the fact that it allows the decision maker to express a preference for an attribute by means of the degree of affiliation in the presence of uncertainty. This expression is more in line with the intuitive thinking mode of human beings when faced with complex decisions. Torra and Narukawa further extended IFS to hesitant fuzzy sets (HFSs) and used exact numbers to inscribe the degree of affiliation of HFS, which provides new perspectives and methods for MADM problem solving [1].

However, despite the theoretical progress of HFS, in practical applications, decision-making information is often uncertain or fuzzy due to the complexity of attributes and the limitations of decision makers’ cognitive level. This requires the introduction of a more flexible expression of the degree of affiliation in the model. The hesitant fuzzy set proposed by Torra allows the degree of membership of an object to a fuzzy set to be represented as a set of several possible values, effectively describing fuzzy uncertainty problems [2]. However, it is only suitable for situations where evaluation information is given as one or several precise values, and does not account for cases where the evaluation information is interval-valued. In many instances, experts find it challenging to express evaluation information in precise values, while interval values provide a certain degree of flexibility and rationality in representing decision information. For example, in the selection of pilot sites for environmental projects, the particularity of a candidate site may lead experts to hesitate between interval numbers such as [0.3,0.5] and [0.8,0.9] when evaluating various attributes of that site. In such cases, we can express the evaluation information of decision experts using interval-valued hesitant fuzzy sets in the form of $\left\{[0.3,0.5],[0.8,0.9]\right\}$. This representation effectively captures complex fuzzy uncertainty in decision-making problems and can, to some extent, prevent information loss during the evaluation process. Therefore, the study of interval-valued hesitant fuzzy sets (IVHFSs), which are formed by introducing interval numbers into hesitant fuzzy sets, has significant theoretical and practical importance. Researchers worldwide have conducted further studies on hesitant fuzzy sets. Based on the advantageous properties of hesitant fuzzy sets and interval numbers, interval-valued hesitant fuzzy sets were introduced to effectively characterize fuzzy uncertainty in decision making. Chen, Xu, and Xia combined the strengths of hesitant fuzzy sets and interval numbers to propose the concept of interval-valued hesitant fuzzy sets. Chen et al. [3] used interval numbers on [0,1] to represent the membership degree of hesitant fuzzy numbers, introduced the concept of interval-valued hesitant fuzzy numbers, and explored a multi-attribute decision-making method based on interval-valued hesitant fuzzy preference relations. Both domestic and international scholars have shown high interest in studying aggregation operators related to interval-valued hesitant fuzzy numbers. Reference [3] examines the interval-valued hesitant fuzzy arithmetic weighted average operator, interval-valued hesitant fuzzy geometric weighted average operator, generalized interval-valued hesitant fuzzy arithmetic weighted average operator, generalized interval-valued hesitant fuzzy geometric weighted average operator, interval-valued hesitant fuzzy ordered weighted arithmetic mean operator, interval-valued hesitant fuzzy ordered weighted geometric mean operator, generalized interval-valued hesitant fuzzy ordered weighted arithmetic mean operator, generalized interval-valued hesitant fuzzy ordered weighted geometric mean operator, and their related properties. Wei and others combined hesitant fuzzy sets with interval numbers, researching aggregation operators for hesitant interval-valued fuzzy information, such as hesitant interval-valued fuzzy arithmetic mean operator, hesitant interval-valued fuzzy ordered arithmetic mean operator, hesitant interval-valued fuzzy geometric mean operator, hesitant interval-valued fuzzy ordered geometric mean operator, hesitant interval-valued fuzzy hybrid ordered arithmetic mean operator, and hesitant interval-valued fuzzy hybrid ordered geometric mean operator [4]. They also proved properties of the new operators, including idempotency, monotonicity, boundedness, and permutation invariance. Yu Qian et al. combined interval-valued hesitant fuzzy sets with the traditional ELECTRE method to study a multi-attribute decision-making method based on interval-valued hesitant fuzzy ELECTRE [5].

An important assumption in the study of MADM problems is the mutual independence of attributes. However, in practical decision-making applications, attributes are often interrelated with each other. This correlation may be due to the intrinsic connection between attributes or due to the subjective preference of decision makers in the assessment process. Therefore, the study of integration methods for interrelated relationships among decision-making information not only has an important practical background, but also has far-reaching theoretical significance. Chen et al. proposed the concept of IVHFS and investigated the MADM method based on IVHF preference relationships. They used the number of intervals on [0,1] to represent the affiliation degree of HFS, which provided the basis for the mathematical modeling and algorithm design of IVHFS [3]. Subsequently, many scholars have conducted in-depth studies on IVHFS fusion operators, such as IVHFWA operator, IVHFWG operator, GIVHFWA operator, etc. These operators are able to take the correlation between attributes into account when dealing with IVHFS, which improves the accuracy and reliability of decision making [4]. Among these researches, HM operator (hesitant fuzzy metric) has received a lot of attention due to its advantages in handling the correlation information among attributes. Beliakov et al. proved the properties of HM operator in the field of inequality [6], while Sykora et al. proposed the generalized HM operator and investigated two of its special forms [7]. Liu et al. investigated the Heronian OWA operator and compared it with the Bonferroni mean operator to deal with the inter-attribute association property [8]. Yu Qian and Hou Fujun applied the Heronian average operator to MADM problems in a hesitant fuzzy language environment and proposed the HFLHM operator and HFLGHM operator, which showed good performance in dealing with hesitant fuzzy language information [5]. Xiaonan Wang and Yongfeng Ju et al. combined hesitant fuzzy sets and HM operators based on the Archimedean paradigm, and investigated the HFHM operator and HFWHM operator, which were applied to the selection of traffic flow models [9]. Xiaohui Zhou and Jian Yao combined the HM operator with interval-valued intuitionistic fuzzy numbers and studied a decision-making method based on the IVIFGWHM operator [10]. Xianning Lin combined the hesitant fuzzy language set and HM operator based on the Archimedean paradigm and studied the HLHGM operator and HLHWGM operator [11].

On the basis of these studies, this paper proposes a new approach combining the IVHFS theory, Archimedean S-norm- and T-norm-based theories with HM operators, i.e., the GIVHFHM operator and the GIVHFWHM operator [9,12,13,14,15,16]. These operators take into account not only the correlation between attributes, but also the decision maker’s preference information. By studying the properties of these operators, such as replacement invariance, idempotence, monotonicity, boundedness, and parameter symmetry, this paper establishes a MADM model based on the GIVHFWHM operator. The model effectively captures the interconnections among input variables and allows decision makers to choose different parameters for decision making based on their risk preference attitudes. In order to verify the effectiveness of the proposed operator, this paper applies it to an arithmetic experiment on investment underlying selection. The experimental results show that the GIVHFWHM operator is not only able to effectively deal with the correlation between attributes, but is also able to simplify the decision-making process and make the decision-making results more in line with the actual situation. This research result not only provides a new solution idea for the MADM problem but also provides a useful reference for research in related fields.

In conclusion, the complexity and uncertainty of MADM problems will continue to increase with the deepening of social informatization. Therefore, the research and development of decision support tools that can deal with ambiguity, uncertainty, and correlation among attributes is of great significance to improve the efficiency and quality of decision making. Based on the above background, the GIVHFHM operator and GIVHFWHM operator proposed in this paper provide new perspectives and methods for solving the MADM problem, which has important theoretical and practical significance. In the future, with the deepening of research, these operators are expected to be applied in more fields and provide decision makers with more scientific and rational decision support.

2. Generalized Interval-Valued Hesitant Fuzzy Number Operator

This section introduces some basics, including the concept of the HM operator and the basic theory related to IVHFS.

2.1. IVHFS Concept

Definition 1

([3]). Let $X$ be a non-empty set, call $D=\left\{〈x,g_D(x)〉|x\in X\right\}$ an interval hesitant fuzzy element (IVHFE) over $X$. Where $g_D(x)=[{\gamma }^L,{\gamma }^U],$ $0\le {\gamma }^L\le {\gamma }^U\le 1$, denotes the set consisting of a number of possible degrees of affiliation of the element $x$ belonging to $D$.

2.2. IVHFE Algorithm

Definition 2

([3]). Let $h(x)=[{\gamma }^L,{\gamma }^U]$, $h_1(x)=[{\gamma }_1^L,{\gamma }_1^U]$ and $h_2(x)=[{\gamma }_2^L,{\gamma }_2^U]$ be any three IVHFE, $\lambda >0$, their algorithms are as follows:

$$h^{\lambda }={\displaystyle {\cup }_{\gamma \in h}\{[{({\gamma }^L)}^{\lambda },{({\gamma }^U)}^{\lambda }]\}}$$

(1)

$$\lambda h_1={\displaystyle {\cup }_{{\gamma }_1\in h_1}\{[1-{(1-{\gamma }_1^L)}^{\lambda },1-{(1-{\gamma }_1^U)}^{\lambda }]\}}$$

(2)

$$h_1\otimes h_2={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2}\{[{\gamma }_1^L{\gamma }_2^L,{\gamma }_1^U{\gamma }_2^U]\}}$$

(3)

$$\begin{array}{l}{h}_1\oplus h_2={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2}}\\ \{[{\gamma }_1^L+{\gamma }_2^L-{\gamma }_1^L{\gamma }_2^L,{\gamma }_1^U+{\gamma }_2^U-{\gamma }_1^U{\gamma }_2^U]\}\end{array}$$

(4)

2.3. IVHFE Probability

Definition 3

([17]). Let any two IVHFEs $h_1=[{\gamma }_1^L,{\gamma }_1^U]$, $h_2=[{\gamma }_2^L,{\gamma }_2^U]$, then the degree of likelihood of $h_1\ge h_2$ is

$$p(h_1\ge h_2)=\max \left\{1-\max \left({\displaystyle \frac{{\gamma }_2^U-{\gamma }_1^L}{len(h_1)+len(h_2)}},0\right),0\right\}$$

(5)

where $len(h_1)={\gamma }_1^U-{\gamma }_1^L,len(h_2)={\gamma }_2^U-{\gamma }_2^L$.

Let $p_{ij}=p(h_i\ge h_j)$, can be obtained for all elements of the likelihood matrix $P={\left(p_{ij}\right)}_{n\times n}$, where $p_{ij}\ge 0,p_{ij}+p_{ji}=1$, $p_{ii}=0.5,$$i,j=1,2,\cdots ,n$, order the IVHFE $h_i=[{\gamma }_i^L,{\gamma }_i^U]$ by computing each row $p_i={\displaystyle \sum _{j=1}^n{p}_{ij}}$ of the probability matrix $P$.

2.4. IVHFE Scoring Function

Definition 4

([3,4]). Let any one IVHFE $h$, then the score function of $h$ is

$$S(h)={\displaystyle \frac{1}{\#h}}{\displaystyle {\sum }_{\gamma \in h}\gamma }$$

(6)

where $\#h$ is the number of elements of HTFE $h$ , for any HTFE $h_1$ and $h_2$ , if $S(h_1)\ge S(h_2)$ , then $h_1\ge h_2$.

2.5. HM Calculus

In the process of information integration, there are often certain links between attributes, such as complementarity, redundancy, etc. The HM operator not only captures the intrinsic links between decision attributes but also takes into account each attribute. Since the HM operator not only captures the intrinsic connection between decision attributes but also considers the importance of each attribute, the definition of the HM operator is given below.

Definition 5

([6,7,8,17]). Let the set of non-negative real numbers $\left\{a_1,a_2,\cdots ,a_n\right\}$, constant $p,q\ge 0$, then the HM operator is defined as follows:

$$H{M}^{p,q}(a_1,a_2,\cdots ,a_n)={\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i\le j}^n{a}_i^p{a}_j^q}\right)}^{{\displaystyle \frac{1}{p+q}}}$$

(7)

2.6. Archimedean S-Norms and T-Norms

The concepts related to Archimedean norms are given below.

Definition 6

([14]). If a T-norm $T(x,y)$ is continuous within its domain and satisfies $T(x,x)<x$ for all $x\in (0,1)$, then $T(x,y)$ is called an Archimedean T-norm. If the Archimedean T-norm is a strictly increasing function for all $x,y\in (0,1)$, it is referred to as a strict Archimedean T-norm.

Definition 7

([14]). If an S-norm $S(x,y)$ is continuous within its domain and satisfies $S(x,x)>x$ for all $x\in (0,1)$, then $S(x,y)$ is called an Archimedean S-norm. If the Archimedean S-norm is a strictly increasing function for all $x,y\in (0,1)$, it is referred to as a strict Archimedean S-norm.

According to reference [14], a strict Archimedean T-norm $T(x,x)$ can be represented using an additive generator $g(x)$ as $T(x,y)=g^{-1}(g(x)+g(x))$. Similarly, a strict Archimedean S-norm $S(x,x)$ can be expressed as $S(x,y)=h^{-1}(h(x)+h(x))$, where $h(t)=g(1-t)$. Note that, if the additive generator of a strict Archimedean T-norm is a strictly decreasing function $g:[0,1]\to [0,+\infty ]$ satisfying $g(1)=0$, then $h:[0,1]\to [0,+\infty ]$ is a strictly increasing function with $h(0)=0$.

By assigning specific functions to the additive generator $g$, we can obtain some commonly used T-norms and S-norms:

(1)

Let $g(t)=-\ln (t)$, then $h(t)=-\ln (1-t)$, $g^{-1}(t)=e^{-t}$, and $h^{-1}(t)=1-e^{-t}$. Therefore, we derive the following algebraic S-norm and T-norm:

$$S_1(x,y)=x+y-xy,T_1(x,y)=xy$$

(8)

(2)

Let $g(t)=\ln \left({\displaystyle \frac{2-t}{t}}\right)$, then $h(t)=\ln \left({\displaystyle \frac{2-(1-t)}{1-t}}\right)$,$g^{-1}(t)={\displaystyle \frac{2}{e^t+1}}$, and $h^{-1}(t)=1-{\displaystyle \frac{2}{e^t+1}}$. Therefore, we derive the following algebraic S-norm and T-norm:

$$S_2(x,y)={\displaystyle \frac{x+y}{1+xy}},T_2(x,y)={\displaystyle \frac{xy}{1+(1-x)(1+y)}}$$

(9)

(3)

Let $g(t)=\ln \left({\displaystyle \frac{\lambda +(1-\lambda )t}{t}}\right),\lambda >0$. Then, $h(t)=\ln \left({\displaystyle \frac{\lambda +(1-\lambda )(1-t)}{1-t}}\right)$, $g^{-1}(t)={\displaystyle \frac{\lambda }{e^t+\lambda -1}}$, and $h^{-1}(t)=1-{\displaystyle \frac{\lambda }{e^t+\lambda -1}}$. Thus, we derive the following Hamacher S-norm and T-norm:

$$S_3(x,y)={\displaystyle \frac{x+y-xy-(1-\lambda )xy}{1-(1-\lambda )xy}},\lambda >0$$

(10)

$$T_3(x,y)={\displaystyle \frac{xy}{\lambda +(1-\lambda )(x+y-xy)}},\lambda >0$$

(11)

In particular, when $\lambda =1$, the Hamacher S-norm and T-norm reduce to the algebraic S-norm and T-norm. When $\lambda =-2$, they reduce to the Einstein S-norm and T-norm.

(4)

Let $g(t)=\ln \left({\displaystyle \frac{\lambda -1}{{\lambda }^t-1}}\right),\lambda >1$. Then, $h(t)=\ln \left({\displaystyle \frac{\lambda -1}{{\lambda }^{1-t}-1}}\right)$, $g^{-1}(t)={\displaystyle \frac{1}{\ln \lambda }}\ln \left({\displaystyle \frac{\lambda -1+e^{\lambda }}{e^{\lambda }}}\right)$, and $h^{-1}(t)=1-{\displaystyle \frac{1}{\ln \lambda }}\ln \left({\displaystyle \frac{\lambda -1+e^{\lambda }}{e^{\lambda }}}\right)$. Thus, we derive the following Frank S-norm and T-norm:

$$S_4(x,y)=1-{\log }_{\lambda }\left(1+{\displaystyle \frac{({\lambda }^{1-x}-1)({\lambda }^{1-y}-1)}{\lambda -1}}\right),\lambda >1$$

(12)

$$T_4(x,y)={\log }_{\lambda }\left(1+{\displaystyle \frac{({\lambda }^x-1)({\lambda }^y-1)}{\lambda -1}}\right),\lambda >1$$

(13)

In particular, as $\lambda \to 1$, we have:

$$\underset{\lambda \to 1}{{\displaystyle \lim }}g(t)=\underset{\lambda \to 1}{{\displaystyle \lim }}\ln \left({\displaystyle \frac{\lambda -1}{{\lambda }^t-1}}\right)=\underset{\lambda \to 1}{{\displaystyle \lim }}\ln \left({\displaystyle \frac{\lambda -1}{t{\lambda }^{t-1}-1}}\right)=-\ln t$$

(14)

At this point, the Frank S-norm and T-norm reduce to the algebraic S-norm and T-norm. Hamacher [18] proposed a more general T-norm and S-norm, defined as follows: Hamacher T-norm

$$T_{\xi }^H(x,y)={\displaystyle \frac{xy}{\xi +(1-\xi )(x+y-xy)}},\xi >0$$

(15)

Hamacher S-norm

$$S_{\xi }^H(x,y)={\displaystyle \frac{x+y-xy-(1-\xi )xy}{\xi +(1-\xi )(x+y-xy)}},\xi >0$$

(16)

In particular, when $\xi =1$, $T_{\xi }^H(x,y)$ and $S_{\xi }^H(x,y)$, reduce to the algebraic T-norm and S-norm, respectively. When $\xi =2$, $T_{\xi }^H(x,y)$ and $S_{\xi }^H(x,y)$, reduce to $T_E(x,y)$ and $S_E(x,y)$.

2.7. Novel Decision Modeling

Based on the fact that HM can effectively eliminate the correlations between attributes, combined with Archimedean S-norms and T-norms theories, the GIVHFHM operator and GIVHFWHM definitions are innovatively given in this section for the MADM problem where attribute values are given by IVHFS and there are mutual correlations between attributes.

In decision making in an interval-valued hesitant fuzzy environment, the decision maker gives decision attribute information in the form of interval-valued hesitant fuzzy numbers; however, there are often correlations among the decision attribute information. For this reason, based on the Archimedean S-norms and T-norms, combined with the theory of Heronian average operator and geometric Heronian average operator, the generalized interval-valued hesitation modal Heronian average operator, and the generalized interval-valued hesitation fuzzy weighted Heronian average operator are proposed [19].

3. Generalized Interval-Valued Hesitant Fuzzy Heronian Averaging Operator

3.1. Definition and Theorems for the Generalized Interval-Valued Hesitant Fuzzy Heronian Averaging Mean Operator

Definition 8.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued hesitant fuzzy numbers, and constant $p,q>0$, then the generalized interval-valued hesitant fuzzy Heronian averaging operator is defined as Equation (7)

$$GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^p{\otimes }_A{(h_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}$$

(17)

Theorem 1.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued fuzzy numbers, and constant $p,q>0$, then the result after integration by the generalized interval-valued hesitant modal Heronian averaging operator (Equation (17)) is still an interval-valued hesitant fuzzy number, where

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)}\right)\right)\right)\end{array}\right]\right\}}\end{array}$$

(18)

The proof process is as follows:

Firstly, it is proved that Equation (18) holds, and there is a generalized interval-valued hesitant fuzzy operator (Equation (1)), which shows that for all $i,j$ there are

$${(h_i)}^p={\displaystyle {\cup }_{{\gamma }_i\in h_i}\left\{\left[g^{-1}\left(pg({\gamma }_i^L)\right),g^{-1}\left(pg({\gamma }_i^U)\right)\right]\right\}}$$

(19)

$${(h_j)}^q={\displaystyle {\cup }_{{\gamma }_j\in h_j}\left\{\left[g^{-1}\left(qg({\gamma }_j^L)\right),g^{-1}\left(qg({\gamma }_j^U)\right)\right]\right\}}$$

(20)

that is

$${(h_i)}^p{\otimes }_A{(h_j)}^q={\displaystyle {\cup }_{{\gamma }_i\in h_i,{\gamma }_j\in h_j}\left\{\left[\begin{array}{l}{g}^{-1}\left(g(g^{-1}(pg({\gamma }_i^L)))+g(g^{-1}(qg({\gamma }_j^L)))\right),\\ g^{-1}\left(g(g^{-1}(pg({\gamma }_i^U)))+g(g^{-1}(qg({\gamma }_j^U)))\right)\end{array}\right]\right\}}$$

(21)

From the algorithm (Equation (4)), it can be obtained that

$$\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^p{\otimes }_A{(h_j)}^q\right)={\displaystyle {\cup }_{{\gamma }_i\in h_i,{\gamma }_j\in h_j}\left\{\left[\begin{array}{l}{h}^{-1}\left({\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right),\\ h^{-1}\left({\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\end{array}\right]\right\}}$$

(22)

Thus, from the algorithm (Equation (2)), it can be obtained that

$$\begin{array}{l}{\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^p{\otimes }_A{(h_j)}^q\right)\right)\\ ={\displaystyle {\cup }_{{\gamma }_i\in h_i,{\gamma }_j\in h_j}\left\{\left[\begin{array}{l}{h}^{-1}\left(\frac{2}{n(n+1)}h\left(h^{-1}\left({\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\right),\\ h^{-1}\left(\frac{2}{n(n+1)}h\left(h^{-1}\left({\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\right)\end{array}\right]\right\}}\end{array}$$

(23)

Thus,

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^p{\otimes }_A{(h_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}\\ ={\displaystyle {\cup }_{{\gamma }_i\in h_i,{\gamma }_j\in h_j}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\right)\end{array}\right]\right\}}\end{array}$$

(24)

Thus, Equation (18) is proved.

Second, it is shown that the integration result obtained through the generalized interval-valued hesitant fuzzy Heronian averaging operator is still an interval-valued hesitant fuzzy number.

Proof. 

For $\forall i,j=1,2,\cdots ,n$, there is $0\le {\gamma }_i^L\le {\gamma }_i^U\le 1,0\le {\gamma }_j^L\le {\gamma }_j^U\le 1$, $i,j=1,2,\cdots ,n$. Since the additive operator $g(t)$ is strictly decreasing, it follows that $g^{-1}(t)$ is also strictly monotonically decreasing, and then for $\forall i,j=1,2,\cdots ,n$, there is

$$g(1)\le g({\gamma }_i^L)\le g({\gamma }_i^U)\le g(0),g(1)\le g({\gamma }_j^L)\le g({\gamma }_j^U)\le g(0)$$

(25)

So, there is

$$(p+q)g(1)\le pg({\gamma }_i^L)+qg({\gamma }_j^L)\le (p+q)g(0)$$

(26)

$$(p+q)g(1)\le pg({\gamma }_i^U)+qg({\gamma }_j^U)\le (p+q)g(0)$$

(27)

It can be obtained that

$$g^{-1}((p+q)g(0))\le g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\le g^{-1}((p+q)g(1))$$

(28)

$$g^{-1}((p+q)g(0))\le g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\le g^{-1}((p+q)g(1))$$

(29)

And, since $h(t)$ is a strictly monotonically increasing function, and hence $h^{-1}(t)$ is also strictly monotonically increasing, it can be obtained that for $\forall i,j=1,2,\cdots ,n$, there is

$$\begin{array}{l}n(n+1)\cdot h\left(g^{-1}((p+q)g(0))\right)={\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}((p+q)g(0))\right)}\\ \le 2{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\\ \le 2{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\right)}\\ \le {\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}((p+q)g(1))\right)}\\ =n(n+1)\cdot h\left(g^{-1}((p+q)g(1))\right)\end{array}$$

(30)

thus,

$$\begin{array}{l}h\left(g^{-1}((p+q)g(0))\right)\le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\\ \le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\right)}\le h\left(g^{-1}((p+q)g(1))\right)\end{array}$$

(31)

Both sides of the above inequality act simultaneously on $h^{-1}(t)$, then it can be obtained that

$$\begin{array}{l}{g}^{-1}((p+q)g(0))\le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\\ \le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\right)}\right)\le g^{-1}((p+q)g(1))\end{array}$$

(32)

Since the additive operator $g(t)$ is a strictly decreasing function, thus:

$$\begin{array}{l}(p+q)g(1)\le g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\\ \le g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\right)}\right)\right)\le (p+q)g(0)\end{array}$$

(33)

So, for $\forall i,j=1,2,\cdots ,n$, there is

$$\begin{array}{l}g(1)\le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\\ \le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\right)}\right)\right)\le g(0)\end{array}$$

(34)

Thus,

$$\begin{array}{l}0=g^{-1}(g(0))\le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\\ \le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\le g^{-1}(g(1))=1\end{array}$$

(35)

That is, the integration result obtained by the generalized interval-valued hesitation mode Heronian averaging operator is still the interval-valued hesitation fuzzy number. □

3.2. Properties of the Generalized Interval-Valued Hesitant Fuzzy Heronian Averaging Operator

Some excellent properties of the generalized interval-valued hesitant mode Heronian averaging operator are explored below, including idempotence, substitution invariance, monotonicity, boundedness. and parameter symmetry [20,21].

Property 1. Idempotence.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued hesitant fuzzy numbers, if for any $h_i$ satisfying $h_i=[{\gamma }_i^L,{\gamma }_i^U]=h=[{\gamma }^L,{\gamma }^U]$, then we have

$$GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)=h$$

(36)

Proof. 

Since $h_i=[{\gamma }_i^L,{\gamma }_i^U]=h=[{\gamma }^L,{\gamma }^U]$, there is ${\gamma }_i^L={\gamma }^L,{\gamma }_i^U={\gamma }^U$, then there is

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)=GIVHFH{M}^{p,q}(h,h,\cdots ,h)\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }^L)+qg({\gamma }^L))\right)}\right)\right)\right)\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }^U)+qg({\gamma }^U))\right)}\right)\right)\right)\end{array}\right]\right\}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h,{\gamma }_2\in h,\cdots ,{\gamma }_n\in h}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(h\left(g^{-1}((p+q)g({\gamma }^L))\right)\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(h\left(g^{-1}((p+q)g({\gamma }^U))\right)\right)\right)\right)\end{array}\right]\right\}}\\ {\displaystyle {\cup }_{{\gamma }_1\in h,{\gamma }_2\in h,\cdots ,{\gamma }_n\in h}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(g^{-1}((p+q)g({\gamma }^L))\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(g^{-1}((p+q)g({\gamma }^U))\right)\right)\end{array}\right]\right\}}\\ ={\displaystyle {\cup }_{\gamma \in h}\left\{\left[g^{-1}(g({\gamma }^L)),\right.\left.g^{-1}(g({\gamma }^U))\right]\right\}}={\displaystyle {\cup }_{\gamma \in h}\left\{\left[{\gamma }^L,\right.\left.{\gamma }^U\right]\right\}}=h\end{array}$$

(37)

Thus, Equation (36) is proved. □

Property 2. Replacement invariance.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued fuzzy numbers, and for any permutation ${\overline{h}}_i(i=1,2,\cdots ,n)$ of $h_i$, there is

$$GIVHFH{M}^{p,q}({\overline{h}}_1,{\overline{h}}_2,\cdots ,{\overline{h}}_n)=GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)$$

(38)

Proof. 

According to

$$\begin{array}{l}GIVHFH{M}^{p,q}({\overline{h}}_1,{\overline{h}}_2,\cdots ,{\overline{h}}_n)={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle \oplus }}}\left({({\overline{h}}_i)}^p\otimes {({\overline{h}}_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h,{\gamma }_2\in h,\cdots ,{\gamma }_n\in h}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\overline{\gamma }}^L)+qg({\overline{\gamma }}^L))\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\overline{\gamma }}^U)+qg({\overline{\gamma }}^U))\right)}\right)\right)\right)\end{array}\right]\right\}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h,{\gamma }_2\in h,\cdots ,{\gamma }_n\in h}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }^L)+qg({\gamma }^L))\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }^U)+qg({\gamma }^U))\right)}\right)\right)\right)\end{array}\right]\right\}}\\ ={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle \oplus }}}\left({(h_i)}^p\otimes {(h_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}=GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\end{array}$$

(39)

Therefore, Equation (38) is proved. □

Property 3. Monotonicity.

Let the interval-valued hesitant fuzzy numbers $\left\{h_1,h_2,\cdots ,h_n\right\}$ and $\left\{{\widehat{h}}_1,{\widehat{h}}_2,\cdots ,{\widehat{h}}_n\right\}$ and for any $i=1,2,\cdots ,n$, there is $h_i\ge {\widehat{h}}_i$, i.e., there is ${\gamma }_i^L\ge {\widehat{\gamma }}_i^L,{\gamma }_i^U\ge {\widehat{\gamma }}_i^U$, then there is

$$GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\ge GIVHFH{M}^{p,q}({\widehat{h}}_1,{\widehat{h}}_2,\cdots ,{\widehat{h}}_n)$$

(40)

Proof. 

For $\forall i,(i=1,2,\cdots ,n)$, there exists ${\gamma }_i^L\ge {\widehat{\gamma }}_i^L,{\gamma }_i^U\ge {\widehat{\gamma }}_i^U$. $h(t)$ and $h^{-1}(t)$ are strictly monotonically increasing functions. $g(t)$ and $g^{-1}(t)$ are strictly monotonically decreasing functions, then there are

$$pg({\gamma }_i^L)+qg({\gamma }_j^L)\le pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L)$$

(41)

$$pg({\gamma }_i^U)+qg({\gamma }_j^U)\le pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U)$$

(42)

And, since $g^{-1}(t)$ is a strictly monotonically decreasing function, it follows that

$$g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\ge g^{-1}(pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L))$$

(43)

$$g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\ge g^{-1}(pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U))$$

(44)

Also, since $h(t)$ is a strictly monotonically increasing function, the following inequality holds:

$${\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\ge {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L)\right)\right)}$$

(45)

$${\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\ge {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U)\right)\right)}$$

(46)

Since $h^{-1}(t)$ is a strictly monotonically increasing function, the following inequality holds:

$$\begin{array}{l}{h}^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\\ \ge h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L)\right)\right)}\right)\end{array}$$

(47)

$$\begin{array}{l}{h}^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\\ \ge h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U)\right)\right)}\right)\end{array}$$

(48)

And, since $g(t)$ is a strictly monotonically decreasing function, it follows that

$$\begin{array}{l}{\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\\ \le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L)\right)\right)}\right)\right)\end{array}$$

(49)

$$\begin{array}{l}{\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\\ \le {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U)\right)\right)}\right)\right)\end{array}$$

(50)

And, since $g^{-1}(t)$ is a strictly monotonically decreasing function, it follows that

$$\begin{array}{l}{g}^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\right)\\ \ge g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^L)+qg({\widehat{\gamma }}_j^L)\right)\right)}\right)\right)\right)\end{array}$$

(51)

$$\begin{array}{l}{g}^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\right)\\ \ge g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\widehat{\gamma }}_i^U)+qg({\widehat{\gamma }}_j^U)\right)\right)}\right)\right)\right)\end{array}$$

(52)

Therefore, from Theorem 1, it can be obtained as

$$GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\ge GIVHFH{M}^{p,q}({\widehat{h}}_1,{\widehat{h}}_2,\cdots ,{\widehat{h}}_n)$$

(53)

Thus, Equation (40) is proved. □

Property 4. Boundedness.

Let $\left\{h_1,h_2,\cdots ,h_n\right\}$ be any set of interval-valued fuzzy numbers, and for any $i=1,2,\cdots ,n$, there is

$$h^{-}\le GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\le h^{+}$$

(54)

where

$$\begin{array}{l}{h}^{-}=\min \{h_1,h_2,\cdots ,h_n\}=[\underset{i}{\min }{\gamma }_i^L,\underset{i}{\min }{\gamma }_i^U]\\ h^{+}=\max \{h_1,h_2,\cdots ,h_n\}=[\underset{i}{\max }{\gamma }_i^L,\underset{i}{\max }{\gamma }_i^U]\end{array}$$

(55)

Proof. 

For $\forall i,(i=1,2,\cdots ,n)$, there is $\underset{i}{\min }{\gamma }_i^L\le {\gamma }_i^L\le \underset{i}{\max }{\gamma }_i^L,\underset{i}{\min }{\gamma }_i^U\le {\gamma }_i^U\le \underset{i}{\max }{\gamma }_i^U$. $h(t)$ and $h^{-1}(t)$ are strictly monotonically increasing functions. $g(t)$ and $g^{-1}(t)$ are strictly monotonically decreasing functions, then there are

$$pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{j}{\min }{\gamma }_j^L)\ge pg({\gamma }_i^L)+qg({\gamma }_j^L)\ge pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{j}{\max }{\gamma }_j^L)$$

$$pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{j}{\min }{\gamma }_j^U)\ge pg({\gamma }_i^U)+qg({\gamma }_j^U)\ge pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{j}{\max }{\gamma }_j^U)$$

And, since $g^{-1}(t)$ is a strictly monotonically decreasing function, it follows that

$$g^{-1}(pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{j}{\min }{\gamma }_j^L))\le g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\le g^{-1}(pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{j}{\max }{\gamma }_j^L))$$

$$g^{-1}(pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{j}{\min }{\gamma }_j^U))\le g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U))\le g^{-1}(pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{j}{\max }{\gamma }_j^U))$$

And, since $h(t)$ is a strictly monotonically increasing function, we have that

$$\begin{array}{l}{\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{j}{\min }{\gamma }_j^L)\right)\right)}\\ \le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\\ \le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{j}{\max }{\gamma }_j^L)\right)\right)}\end{array}$$

(56)

$$\begin{array}{l}{\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{j}{\min }{\gamma }_j^U)\right)\right)}\\ \le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\\ \le {\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{j}{\max }{\gamma }_j^U)\right)\right)}\end{array}$$

(57)

Since $h^{-1}(t)$ is a strictly monotonically increasing function, so there is that

$$\begin{array}{l}{h}^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{i}{\min }{\gamma }_j^L)\right)\right)}\right)\\ \le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\\ \le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{i}{\max }{\gamma }_j^L)\right)\right)}\right)\end{array}$$

(58)

$$\begin{array}{l}{h}^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{i}{\min }{\gamma }_j^U)\right)\right)}\right)\\ \le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\\ \le h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{i}{\max }{\gamma }_j^U)\right)\right)}\right)\end{array}$$

(59)

Since $g(t)$ is a strictly monotonically decreasing function, it follows that

$$\begin{array}{l}{\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{j}{\min }{\gamma }_j^L)\right)\right)}\right)\right)\\ \ge {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\\ \ge {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{j}{\max }{\gamma }_j^L)\right)\right)}\right)\right)\end{array}$$

(60)

$$\begin{array}{l}{\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{j}{\min }{\gamma }_j^U)\right)\right)}\right)\right)\\ \ge {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\\ \ge {\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{j}{\max }{\gamma }_j^U)\right)\right)}\right)\right)\end{array}$$

(61)

And, since $g^{-1}(t)$ is a strictly monotonically decreasing function, it follows that

$$\begin{array}{l}{g}^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^L)+qg(\underset{j}{\min }{\gamma }_j^L)\right)\right)}\right)\right)\right)\\ \le g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^L)+qg({\gamma }_j^L)\right)\right)}\right)\right)\right)\\ \le g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^L)+qg(\underset{j}{\max }{\gamma }_j^L)\right)\right)}\right)\right)\right)\end{array}$$

(62)

$$\begin{array}{l}{g}^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\min }{\gamma }_i^U)+qg(\underset{j}{\min }{\gamma }_j^U)\right)\right)}\right)\right)\right)\\ \le g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)\right)}\right)\right)\right)\\ \le g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(pg(\underset{i}{\max }{\gamma }_i^U)+qg(\underset{j}{\max }{\gamma }_j^U)\right)\right)}\right)\right)\right)\end{array}$$

(63)

Therefore, from Theorem 1, it can be obtained as

$$h^{-}\le GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)\le h^{+}$$

Therefore, Equation (54) is proved. □

Property 5. Parameter symmetry.

Let $\left\{h_1,h_2,\cdots ,h_n\right\}$ be any set of interval-valued fuzzy numbers with parameters $p>0,q>0$, for any $i=1,2,\cdots ,n$, there is

$$GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)=GIVHFH{M}^{q,p}(h_1,h_2,\cdots ,h_n)$$

(64)

Proof. 

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^p{\otimes }_A{(h_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^L)+qg({\gamma }_j^L))\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{p+q}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg({\gamma }_i^U)+qg({\gamma }_j^U)\right)}\right)\right)\right)\end{array}\right]\right\}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{q+p}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(qg({\gamma }_i^L)+pg({\gamma }_j^L))\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{q+p}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(qg({\gamma }_i^U)+pg({\gamma }_j^U)\right)}\right)\right)\right)\end{array}\right]\right\}}\\ ={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^q{\otimes }_A{(h_j)}^p\right)\right)\right)}^{{\displaystyle \frac{1}{q+p}}}=GIVHFH{M}^{q,p}(h_1,h_2,\cdots ,h_n)\end{array}$$

(65)

Therefore, Equation (64) is proved. □

3.3. Generalized Interval-Valued Hesitant Mode Weighted Heronian Averaging Operator

Definition 9.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued hesitant fuzzy numbers with constant $p,q>0$, then the generalized interval-valued hesitant mode weighted Heronian average operator.

$$GIVHFWH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{2}{n(n+1)}}\left(\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(w_i{h}_i)}^p{\otimes }_A{(w_j{h}_j)}^q\right)\right)\right)}^{{\displaystyle \frac{1}{p+q}}}$$

(66)

Theorem 2.

Let $h_i(i=1,2,\cdots ,n)$ be a set of interval-valued hesitant fuzzy numbers with constant $p,q>0$, then the result after integration by the generalized interval-valued hesitant mode-weighted Heronian averaging operator (Equation (66)) is still an interval-valued hesitant fuzzy number with

$$\begin{array}{l}GIVHFWH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}}\\ \left\{\left[\begin{array}{l}{g}^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg(h^{-1}(w_{i}h({\gamma }_i^L)))+qg(h^{-1}(w_{j}h({\gamma }_j^L)))\right)}\right)\right)\right),\\ g^{-1}\left({\displaystyle \frac{1}{p+q}}g\left(h^{-1}\left({\displaystyle \frac{2}{n(n+1)}}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}(pg(h^{-1}(w_{i}h({\gamma }_i^U)))+qg(h^{-1}(w_{j}h({\gamma }_j^U)))\right)}\right)\right)\right)\end{array}\right]\right\}\end{array}$$

(67)

The proof of Theorem 2 is similar to that of Theorem 1 and is omitted here.

Special cases by additivity operator.

Case 1. When $g(t)=-\ln (t)$, then $g^{-1}(t)=e^{-t},h(t)=-\ln (1-t),h^{-1}(t)=1-e^{-t}$, then the generalized interval-valued hesitant mode Heronian averaging operator deforms to the following form:

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}}\\ \left\{\left[{\left(1-{\displaystyle \prod _{i=1,j=i}^n{\left(1-{({\gamma }_i^L)}^p{({\gamma }_j^L)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{{\displaystyle \frac{1}{p+q}}},{\left(1-{\displaystyle \prod _{i=1,j=i}^n{\left(1-{({\gamma }_i^U)}^p{({\gamma }_j^U)}^q\right)}^{\frac{2}{n(n+1)}}}\right)}^{{\displaystyle \frac{1}{p+q}}}\right]\right\}\end{array}$$

(68)

Case 2. When $g(t)=\ln \left({\displaystyle \frac{2-t}{t}}\right)$, then $g^{-1}(t)={\displaystyle \frac{2}{e^t+1}}$, $h(t)=\ln \left({\displaystyle \frac{2-(1-t)}{1-t}}\right)$, $h^{-1}(t)=1-{\displaystyle \frac{2}{e^t+1}}$, then the generalized interval-valued hesitant mode Heronian averaging operator transforms into the interval-valued hesitant fuzzy Einstein Heronian averaging operator (IFHFEHM):

$$\begin{array}{l}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}}\\ \left\{\left[\begin{array}{l}{\displaystyle \frac{2{\left(R({\gamma }_i^L,{\gamma }_j^L)-S({\gamma }_i^L,{\gamma }_j^L)\right)}^{\frac{1}{p+q}}}{{\left(R({\gamma }_i^L,{\gamma }_j^L)+3S({\gamma }_i^L,{\gamma }_j^L)\right)}^{\frac{1}{p+q}}-{\left(R({\gamma }_i^L,{\gamma }_j^L)-S({\gamma }_i^L,{\gamma }_j^L)\right)}^{\frac{1}{p+q}}}},\\ {\displaystyle \frac{2{\left(R({\gamma }_i^U,{\gamma }_j^U)-S({\gamma }_i^U,{\gamma }_j^U)\right)}^{\frac{1}{p+q}}}{{\left(R({\gamma }_i^U,{\gamma }_j^U)+3S({\gamma }_i^U,{\gamma }_j^U)\right)}^{\frac{1}{p+q}}-{\left(R({\gamma }_i^U,{\gamma }_j^U)-S({\gamma }_i^U,{\gamma }_j^U)\right)}^{\frac{1}{p+q}}}}\end{array}\right]\right\}\end{array}$$

(69)

where

$$\begin{array}{l}R({\gamma }_i^L,{\gamma }_j^L)={\displaystyle \prod _{i=1,j=i}^n{\left({(2-{\gamma }_i^L)}^p{(2-{\gamma }_j^L)}^q+3{({\gamma }_i^L)}^p{({\gamma }_j^L)}^q\right)}^{\frac{2}{n(n+1)}}},\\ R({\gamma }_i^U,{\gamma }_j^U)={\displaystyle \prod _{i=1,j=i}^n{\left({(2-{\gamma }_i^U)}^p{(2-{\gamma }_j^U)}^q+3{({\gamma }_i^U)}^p{({\gamma }_j^U)}^q\right)}^{\frac{2}{n(n+1)}}},\\ S({\gamma }_i^L,{\gamma }_j^L)={\displaystyle \prod _{i=1,j=i}^n{\left({(2-{\gamma }_i^L)}^p{(2-{\gamma }_j^L)}^q-{({\gamma }_i^L)}^p{({\gamma }_j^L)}^q\right)}^{\frac{2}{n(n+1)}}},\\ S({\gamma }_i^U,{\gamma }_j^U)={\displaystyle \prod _{i=1,j=i}^n{\left({(2-{\gamma }_i^U)}^p{(2-{\gamma }_j^U)}^q-{({\gamma }_i^U)}^p{({\gamma }_j^U)}^q\right)}^{\frac{2}{n(n+1)}}}\end{array}$$

(70)

Some special cases are obtained depending on $p,q$.

Case 1. When $q\to 0$, there is

$$\begin{array}{l}\underset{q\to 0}{{\displaystyle \lim }}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{1}{n}}\underset{i=1}{\overset{n}{{\displaystyle {\oplus }_A}}}{(h_i)}^p\right)}^{{\displaystyle \frac{1}{p}}}={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}}\\ \left\{\left[g^{-1}\left({\displaystyle \frac{1}{p}}g\left(h^{-1}\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}h}\left(g^{-1}(pg({\gamma }_i^L))\right)\right)\right)\right),g^{-1}\left({\displaystyle \frac{1}{p}}g\left(h^{-1}\left({\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^{n}h}\left(g^{-1}(pg({\gamma }_i^U))\right)\right)\right)\right)\right]\right\}\end{array}$$

(71)

Case 2. When $p=1,q\to 0$, there is

$$\begin{array}{l}\underset{p=1,q\to 0}{{\displaystyle \lim }}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\displaystyle \frac{1}{n}}\underset{i=1}{\overset{n}{{\displaystyle {\oplus }_A}}}{h}_i\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[h^{-1}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}h}\left({\gamma }_i^L\right)\right),h^{-1}\left(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}h}\left({\gamma }_i^U\right)\right)\right]\right\}}\end{array}$$

(72)

Case 3. When $p=q={\displaystyle \frac{1}{2}}$, there is

$$\begin{array}{l}\underset{p=q={\displaystyle \frac{1}{2}}}{{\displaystyle \lim }}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\displaystyle \frac{2}{n(n+1)}}\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left({(h_i)}^{{\displaystyle \frac{1}{2}}}{\oplus }_A{(h_j)}^{{\displaystyle \frac{1}{2}}}\right)\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(4g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(\frac{1}{2}g({\gamma }_i^L)+\frac{1}{2}g({\gamma }_j^L)\right)\right)}\right)\right)\right),\\ g^{-1}\left(4g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(\frac{1}{2}g({\gamma }_i^U)+\frac{1}{2}g({\gamma }_j^U)\right)\right)}\right)\right)\right)\end{array}\right]\right\}}\end{array}$$

(73)

Case 4. When $p=q=1$, there is

$$\begin{array}{l}\underset{p=q=1}{{\displaystyle \lim }}GIVHFH{M}^{p,q}(h_1,h_2,\cdots ,h_n)={\left({\displaystyle \frac{2}{n(n+1)}}\underset{i=1,j=i}{\overset{n}{{\displaystyle {\oplus }_A}}}\left(h_i{\oplus }_A{h}_j\right)\right)}^{{\displaystyle \frac{1}{2}}}\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_1,{\gamma }_2\in h_2,\cdots ,{\gamma }_n\in h_n}\left\{\left[\begin{array}{l}{g}^{-1}\left(\frac{1}{2}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(g({\gamma }_i^L)+g({\gamma }_j^L)\right)\right)}\right)\right)\right),\\ g^{-1}\left(\frac{1}{2}g\left(h^{-1}\left(\frac{2}{n(n+1)}{\displaystyle \sum _{i=1,j=i}^{n}h\left(g^{-1}\left(g({\gamma }_i^L)+g({\gamma }_j^L)\right)\right)}\right)\right)\right)\end{array}\right]\right\}}\end{array}$$

(74)

4. A Multi-Attribute Decision-Making Method Based on Generalized Interval-Valued Hesitant Fuzzy Weighted Heronian Averaging Operator

4.1. Introduction to Multi-Attribute Group Decision-Making Problems

For a certain multi-attribute group decision-making problem, where the set of scenarios to be evaluated $\{A_1,A_2,\cdots ,A_m\}$, the set of attributes corresponding to each scenario is $\{G_1,G_2,\cdots ,G_n\}$, and the weights corresponding to each decision attribute are $w_k(k=1,2,\cdots ,n)$, where $0\le w_k\le 1,{\displaystyle \sum _{k=1}^n{w}_k}=1$. The weight information vector of the expert is ${\omega }_k(k=1,2,\cdots ,m)$, where $0\le {\omega }_k\le 1,{\displaystyle \sum _{k=1}^n{\omega }_k}=1$. In the interval-valued hesitant fuzzy environment, the decision maker’s evaluation information for the attribute $G_j(j=1,2,\cdots ,m)$ of $A_i(i=1,2,\cdots ,n)$ is the interval-valued hesitant fuzzy number $h_{ij}=[{\gamma }_{ij}^L,{\gamma }_{ij}^U]$, and the decision information constitutes an interval-valued hesitant fuzzy matrix $H={\left(h_{ij}\right)}_{m\times n}$.

4.2. Decision Model Based on Generalized Interval-Valued Hesitant Fuzzy Weighted Heronian Averaging Operator

Based on the generalized interval-valued hesitant fuzzy weighted Heronian averaging operator investigated in this chapter [22,23], a novel decision-making method for solving multi-attribute group decision-making problems in interval-valued hesitant fuzzy environments is proposed, and the detailed steps are as follows.

Decision Modeling Steps

Step 1. A normalized interval-valued hesitant fuzzy matrix $\tilde{H}={\left({\tilde{h}}_{ij}\right)}_{m\times n}$ is obtained from the interval-valued hesitant fuzzy decision matrix $H={\left(h_{ij}\right)}_{m\times n}$, the attribute weight vector $w$ and the expert weight vector $\omega$, and normalized to $H={\left(h_{ij}\right)}_{m\times n}$ according to the attribute’s properties.

$${\overline{\tilde{h}}}_{ij}=\left\{\begin{array}{l}{\tilde{h}}_{ij}\begin{array}{c}\hspace{1em} \mathrm{For} \mathrm{the} \mathrm{benefit} \mathrm{attribute} a_j\end{array}\\ {\left({\tilde{h}}_{ij}\right)}^c\begin{array}{c} \mathrm{For} \mathrm{the} \mathrm{cost} \mathrm{attributes} a_j\end{array}\end{array}\right.$$

(75)

where ${\left({\tilde{h}}_{ij}\right)}^c$ the complementary operation of the interval-valued hesitant fuzzy number ${\tilde{h}}_{ij}$, i.e., ${\left({\tilde{h}}_{ij}\right)}^c={\displaystyle {\cup }_{\tilde{\gamma }\in {\tilde{h}}_j}\left\{1-\tilde{\gamma }\right\}}$.

Step 2. Firstly, the generalized interval-valued hesitant fuzzy weighted Heronian averaging operator (Equation (66)) is used to integrate the information of scheme $A_i$ to obtain the comprehensive evaluation information ${\tilde{h}}_i$$(i=1,2,\cdots ,m)$. Secondly, the generalized interval-valued hesitant fuzzy weighted geometric Heronian averaging operator (Equation (74)) is utilized to obtain the comprehensive evaluation information ${\widehat{h}}_i$, $(i=1,2,\cdots ,m)$, by information integration for scheme $A_i$.

Step 3. The score function (Equation (6)) of the interval-valued hesitant fuzzy number is used to calculate the score function $s({\tilde{h}}_i)$ of scheme $A_i$ separately.

Step 4. Based on the score function $s({\tilde{h}}_i)$, the likelihood matrix $P={\left(p_{ij}\right)}_{n\times n}$, is constructed from Equation (5), and the magnitude of $p_i={\displaystyle \sum _{j=1}^n{p}_{ij}},i=1,2,\cdots ,n$ is computed to size rank $s({\tilde{h}}_i)$.

Step 5. The solutions are ranked according to the size of the score function $s({\tilde{h}}_i)$ and the best solution $A_i$ is selected.

Step 6. End.

The flowchart of the decision-making model is as follows:

5. Application of Generalized Interval-Based Hesitant Fuzzy Weighted Heronian Averaging Operator in Investment Underlying Selection

5.1. Example Applications

The application case in this section is the same as the case of evaluating environmental protection project alternatives in Chapter III. The generalized interval-valued hesitant fuzzy weighted Heronian averaging operator is mainly used to solve the multi-attribute group decision-making problem where the decision attribute information is interval-valued hesitant fuzzy numbers and the attributes are interrelated [24,25,26,27].

With the rapid development of social modernization, the maintenance of the ecological environment has become an important aspect of progress. In 2016, the Environmental Protection Bureau of Hunan Province focused on cultivating the market for environmental governance and ecological protection, accelerating supply-side structural reforms in the ecological and environmental protection sector. Combining the “13th Five-Year” Environmental Protection Plan of Hunan Province, with goals of significantly improving the province’s ecological environment quality, substantially reducing the total emissions of major pollutants, effectively controlling environmental risks, and basically ensuring ecological safety, preparations are being made to implement pilot ecological and environmental protection projects across the province. The first batch of implementation of the pilot scheme sites are Changsha City $A_1$, Chenzhou City $A_2$, Xiangtan City $A_3$, Yueyang City $A_4$, Changde City $A_5$, and other cities. During the implementation process, comprehensive evaluations are needed for the implementation risk resistance factor $G_1$, air quality factors $G_2$, surface water quality factors $G_3$, and urban functional area noise factors $G_4$. In evaluations, areas with good surface water quality often also have good air quality; simultaneously, noise levels in urban functional areas are significantly lower, thereby reducing the implementation risk coefficient. To accurately reflect the experts’ evaluation information, decision attribute values are given in the form of interval-valued hesitant fuzzy numbers. The weights of each evaluation attribute are $w={(0.25,0.15,0.20,0.40)}^T$. A total of four decision experts participate in the project evaluation, forming the set $D=\left\{d_1,d_2,d_3,d_4\right\}$. The weight vector of each expert is $\omega ={(0.15,0.3,0.35,0.20)}^T$. The interval-valued hesitant fuzzy decision matrix $H={(h_{ij})}_{m\times n}$ is shown in

Table 1. Interval-valued hesitant fuzzy matrix $H_{5\times 4}$ for decision expert $d_1$.

	G1	G2	G3	G4
A1	([0.2,0.3], [0.3,0.4])	([0.2,0.5])	([0.7,0.8], [0.8,0.9])	([0.4,0.5])
A2	([0.4,0.5], [0.5,0.6])	([0.3,0.4], [0.6,0.7])	([0.3,0.4])	([0.5,0.6], [0.8,0.9])
A3	([0.5,0.7])	([0.2,0.3], [0.4,0.5])	([0.8,0.9], [0.9,1.0])	([0.3,0.5])
A4	([0.3,0.4], [0.7,0.8])	([0.1,0.3])	([0.6,0.7], [0.8,0.9])	([0.5,0.7])
A5	([0.2,0.3])	([0.4,0.6])	([0.2,0.3], [0.6,0.7])	([0.6,0.7])

,

Table 2. Interval-valued hesitant fuzzy matrix $H_{5\times 4}$ for decision expert $d_2$.

	G1	G2	G3	G4
A1	([0.2,0.3])	([0.3,0.5])	([0.6,0.7], [0.7,0.9])	([0.3,0.5])
A2	([0.4,0.5], [0.5,0.6])	([0.3,0.4], [0.6,0.7])	([0.3,0.4])	([0.5,0.6], [0.9,1.0])
A3	([0.5,0.7], [0.8,0.9])	([0.2,0.3], [0.4,0.5])	([0.8,0.9])	([0.3,0.5])
A4	([0.4,0.4])	([0.1,0.3], [0.3,0.4])	([0.6,0.7], [0.8,0.9])	([0.5,0.7], [0.7,0.8])
A5	([0.2,0.3])	([0.4,0.6])	([0.1,0.2], [0.3,0.5])	([0.6,0.7])

,

Table 3. Interval-valued hesitant fuzzy matrix $H_{5\times 4}$ for decision expert $d_3$.

	G1	G2	G3	G4
A1	([0.2,0.3], [0.3,0.4])	([0.2,0.5], [0.7,0.8])	([0.7,0.9], [0.8,0.9])	([0.4,0.5])
A2	([0.4,0.5])	([0.3,0.4], [0.5,0.6])	([0.3,0.4])	([0.3,0.5], [0.7,0.9])
A3	([0.3,0.6])	([0.2,0.3], [0.3,0.7])	([0.8,0.9])	([0.5,0.7])
A4	([0.2,0.4], [0.5,0.7])	([0.1,0.3])	([0.6,0.7])	([0.4,0.6])
A5	([0.2,0.3])	([0.4,0.6])	([0.2,0.3], [0.6,0.7])	([0.7,0.8])

and

Table 4. Interval-valued hesitant fuzzy matrix $H_{5\times 4}$ for decision expert $d_4$.

	G1	G2	G3	G4
A1	([0.3,0.4])	([0.4,0.5], [0.6,0.8])	([0.6,0.8], [0.8,0.9])	([0.5,0.8])
A2	([0.4,0.5], [0.5, 0.6])	([0.3,0.4], [0.6,0.7])	([0.3,0.4])	([0.3,0.6], [0.7,0.9])
A3	([0.6,0.8])	([0.2,0.3], [0.4,0.5])	([0.6,0.8])	([0.2,0.3], [0.3,0.5])
A4	([0.3,0.4], [0.7,0.9])	([0.4,0.6], [0.6,0.7])	([0.6,0.7], [0.7,0.9])	([0.4,0.6])
A5	([0.3,0.5])	([0.4,0.6])	([0.4,0.5], [0.7,0.9])	([0.5,0.6])

:

5.2. Decision-Making Process and Results

The specific decision process is as follows:

Parameter setting: $p=1,q=1$, gives the detailed decision process of the decision only in terms of the decision attributes given by a decision $d_1$.

Step 1: because of the four attributes of the implementation risk resistance factor, air quality factor, surface water quality factor, and urban functional area noise factor of the implementation site of the investment target, the air quality factor, surface water quality factor are benefit-type attributes, and the implementation risk resistance factor and the urban functional area noise factor are cost-type attributes; the interval-valued hesitant fuzzy matrix $H$ is standardized by Equation (22).

Step 2: using the generalized interval-valued hesitant fuzzy weighted Heronian averaging operator (Equation (66)) for information integration of the scheme site $A_i$ to obtain the comprehensive evaluation information ${\tilde{h}}_i(i=1,2,3,4,5)$, without loss of generality, take $g(t)=\ln \left({\displaystyle \frac{2-t}{t}}\right)$, i.e.,

$$\begin{array}{l}{\tilde{h}}_1=GIVHFWH{M}^{1,1}(h_{11},h_{12},h_{13},h_{14})=\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_{11},{\gamma }_2\in h_{12},{\gamma }_3\in h_{13},{\gamma }_4\in h_{14}}\left\{\left[0.7388,0.7622\right],\left[0.7994,0.8241\right],\left[0.7635,0,7871\right],\left[0.7513,0.7747\right]\right\}}\end{array}$$

(76)

$$\begin{array}{l}{\tilde{h}}_2=GIVHFWH{M}^{1,1}(h_{11},h_{12},h_{13},h_{14})=\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_{11},{\gamma }_2\in h_{12},{\gamma }_3\in h_{13},{\gamma }_4\in h_{14}}\left\{\left[0.7788,0,8063\right],\left[0.7856,0.8152\right],\left[0.7888,0.8069\right],\left[0.7975,0.8159\right]\right\}}\end{array}$$

(77)

$$\begin{array}{l}{\tilde{h}}_3=GIVHFWH{M}^{1,1}(h_{11},h_{12},h_{13},h_{14})=\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_{11},{\gamma }_2\in h_{12},{\gamma }_3\in h_{13},{\gamma }_4\in h_{14}}\left\{\left[0.7563,0,7839\right],\left[0.7689,0.7965\right],\left[0.7301,0.7584\right],\left[0.7434,0.7713\right]\right\}}\end{array}$$

(78)

$$\begin{array}{l}{\tilde{h}}_4=GIVHFWH{M}^{1,1}(h_{11},h_{12},h_{13},h_{14})=\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_{11},{\gamma }_2\in h_{12},{\gamma }_3\in h_{13},{\gamma }_4\in h_{14}}\left\{\left[0.7113,0.7509\right],\left[0.7359,0,7777\right],\left[0.7644,0.7861\right],\left[0.7909,0.8128\right]\right\}}\end{array}$$

(79)

$$\begin{array}{l}{\tilde{h}}_5=GIVHFWH{M}^{1,1}(h_{11},h_{12},h_{13},h_{14})=\\ ={\displaystyle {\cup }_{{\gamma }_1\in h_{11},{\gamma }_2\in h_{12},{\gamma }_3\in h_{13},{\gamma }_4\in h_{14}}\left\{\left[0,7871,0.8119\right],\left[0.7747,0.7998\right],\left[0.7756,0.82411\right],\left[0.7513,0.7747\right]\right\}}\end{array}$$

(80)

Step 3: the score function of each scheme site $A_i$ is calculated by the score function $s({\tilde{h}}_i)$ of the generalized interval-valued hesitant fuzzy numbers, and the results are obtained by weighting the set of weight vectors of the experts as shown in

Table 5. Score function table for each scheme site.

	Generalized Interval-Valued Hesitant Fuzzy Heronian Mean Operator Results	Generalized Interval-Valued Hesitant Fuzzy Weighted Heronian Mean Operator Results
A1	[0.8304,0.8710]	[0.9773,0.9938]
A2	[0.8423,0.8680]	[0.9903,0.9954]
A3	[0.8537,0.8966]	[0.9919,0.9962]
A4	[0.8434,0.8845]	[0.9804,0.9965]
A5	[0.8308,0.8635]	[0.9892,0.9941]

.

Step 4: based on the score function $s({\tilde{h}}_i)(i=1,2,\cdots ,n)$ in Step 3, the likelihood matrix $P={\left(p_{ij}\right)}_{n\times n}$ and likelihood matrix $\widehat{P}={\left({\widehat{p}}_{ij}\right)}_{n\times n}$ are constructed from Equation (5).

$$P=\left(\begin{array}{ccccc}0.500& 0.418& 0.393& 0.275& 0.482\\ 0.582& 0.500& 0.302& 0.804& 0.518\\ 0.607& 0.782& 0.500& 0.607& 0.788\\ 0.725& 0.196& 0.393& 0.500& 0.698\\ 0.518& 0.482& 0.212& 0.302& 0.500\end{array}\right)$$

$$\widehat{P}=\left(\begin{array}{ccccc}0.500& 0.366& 0.291& 0.135& 0.482\\ 0.634& 0.500& 0.412& 0.804& 0.518\\ 0.709& 0.588& 0.500& 0.607& 0.678\\ 0.875& 0.196& 0.393& 0.500& 0.458\\ 0.518& 0.482& 0.322& 0.542& 0.500\end{array}\right)$$

and calculate the size of $p_i={\displaystyle \sum _{j=1}^n{p}_{ij}},i=1,2,\cdots ,n$ to size order $s({\tilde{h}}_i)$$(i=1,2,\cdots ,n)$; detailed results are shown in

Table 6. Table of ranking results for each score function.

	Sorting of Each Score Function
Generalized Interval-Valued Hesitant Fuzzy Heronian Mean Operator Results	$s({\tilde{h}}_3)>s({\tilde{h}}_4)>s({\tilde{h}}_2)>s({\tilde{h}}_5)>s({\tilde{h}}_1)$
Generalized Interval-Valued Hesitant Fuzzy Heronian Mean Operator Results	$s({\widehat{h}}_3)>s({\widehat{h}}_2)>s({\widehat{h}}_5)>s({\widehat{h}}_4)>s({\widehat{h}}_1)$

.

Step 5: Based on the results in Table 6, we rank the alternatives in terms of their merits. The ranking of each alternative after integration using the generalized interval-valued hesitant fuzzy Heronian mean operator is: A3 > A4 > A2 > A1 > A5.

Step 6: end.

Comparing and analyzing the IVHFWA operator and IVHFWG operator in the literature [3], the HIVFCOA operator and HIVFCOG operator in the literature [4], and the IVHF ELECTRE method in the literature [5], the ranking results of each scheme site are shown in

Table 7. Table of ranking results.

Decision Operators or Methods	Ranking of Pilot Scheme Sites
IVHFWA operator in the literature [3]	A2 > A4 > A3 > A5 > A1
IVHFWG operator in the literature [3]	A2 >A3 > A4 > A5 > A1
IVHF ELECTRE method in the literature [5]	A4 > A5 > A3 > A2 > A1
HIVFCOA operator in the literature [4]	A3 > A4 > A2 > A5 > A1
HIVFCOG operator in the literature [4]	A3 > A2 > A4 > A5 > A1
GIVHFHM in this paper	A3 > A4 > A2 > A5 > A1
GIVHFWHM in this paper	A3 > A2 > A5 > A4 > A1

.

Through comparative analysis, we know that the optimal scheme site obtained by the IVHFWA operator and the IVHFWG operator in the literature [3] is A2, and the optimal scheme site obtained by the IVHF ELECTRE method in the literature [5] is A4, which is different from that of the optimal scheme site A3 in this paper, and the order of the scheme sites is also slightly different. Research [3,5] mainly studied the case of independence between attributes, and did not consider the interconnectivity between input attributes. However, in the evaluation of the pilot scheme of this ecological environmental protection project, where the surface water quality is good, the air quality is often good, and the noise in the urban functional area will be much less, which will reduce the risk coefficient of the implementation, so there will be redundancy and complementarity between these evaluation attributes, and therefore only by taking the correlation between attributes into consideration can we make the decision making more reasonable. Research [4] combines HIVFE and fuzzy measure Choquet integral, and studies the HIVFCOA operator and HIVFCOG operator; these two operators fully consider the correlation between the input information, so the optimal solution is also A3, but the fuzzy measure in the HIVFCOA operator and HIVFCOG operator has a certain degree of subjectivity and the solution process is more complicated. In conclusion, the IVHFWHM algorithm studied in this paper is more suitable for the real situation and has strong operability.

In the following, we further analyze the influence of parameter $p,q$ in the GIVHFWHM operator on the information fusion results, and set parameter $p,q$ to different values for numerical experiments, and the experimental results are shown in Figure 1a,b. When parameter $q$ is fixed, the evaluation scores of the pilot sites show a gradually decreasing tendency as parameter $p$ becomes larger, as shown in Figure 1a. At this time, the advantages and disadvantages of the pilot scheme sites are ranked as follows: A3 > A4 > A2 > A5 > A1, and the optimal pilot scheme site is A3. When parameter $p$ is fixed, the evaluation scores of the pilot scheme sites show a gradual trend of decreasing as parameter $q$ increases, as shown in Figure 1b. When $q\in [1,2]\cup [8,10]$, the order of the advantages and disadvantages of the pilot scheme sites Chenzhou City, Xiangtan City, Yueyang City, and Changde City changes slightly, but the combined value of the attribute scores of the pilot scheme sites, and the advantages and disadvantages of the pilot scheme sites are ranked as follows: A3 > A4 > A2 > A5 > A1, and the optimal pilot scheme site is A3. In conclusion, in the reality of the decision-making process, the evaluation of the attribute information will exist in each other’s redundancy, complementarity, and other correlations, so, only by fully considering the correlations between attributes, can we achieve a more efficient decision making. Therefore, only by fully considering the correlation between attributes can the decision be more reasonable. Therefore, the GIVHFWHM algorithm studied in this paper is more suitable for the real situation and is operable.

6. Conclusions

In today’s society, the acceleration of informatization is presenting unprecedented challenges to multi-attribute decision-making (MADM) problems. With the explosive growth of information, decision experts are often limited by their own experience and knowledge when making decisions. This limitation becomes increasingly evident, especially when evaluating various complex issues.

Taking the comprehensive selection of an ecological environment project pilot scheme as an example, decision attributes often have close interrelated characteristics. Attributes such as the air quality level of the pilot location, surface water quality, and the pollution degree of noise in urban functional areas influence each other. Generally, cities with excellent air and surface water quality tend to have relatively lower noise pollution levels. This correlation makes traditional decision-making methods difficult to accurately reflect the actual situation.

Considering that information fusion operators accounting for interrelated attributes are more aligned with practical decision making, the existing aggregation operators for interval-valued hesitant fuzzy sets (IVHFSs) are only effective when attributes are mutually independent, revealing significant shortcomings. To address this deficiency, this paper innovatively combines the Heronian mean (HM) operator with Archimedean S-norm and T-norm theories to propose a novel decision-making method with important practical significance. We introduce a multi-attribute decision-making method based on the generalized interval-valued hesitant fuzzy weighted Heronian mean (GIVHFWHM) operator to solve MADM problems where decision attributes are interval-valued hesitant fuzzy numbers and attributes are interrelated.

First, based on the operational rules of IVHFNs and the HM operator, we propose the GIVHFHM operator and the GIVHFWHM operator. We study properties of the GIVHFHM operator such as permutation invariance, idempotency, monotonicity, boundedness, and parameter symmetry. We then establish a multi-attribute decision-making model based on the GIVHFWHM operator. Finally, we validate the proposed MADM model through numerical experiments.

By establishing a decision-making model based on the GIVHFWHM operator and applying it to the evaluation and selection of pilot schemes for ecological environment projects, we have achieved significant results. The outcomes strongly demonstrate the correctness of the new decision-making algorithm studied in this paper. This novel algorithm effectively eliminates the influence of attribute interrelations on decision results, making them more authentic and reliable.

Specifically, the new decision-making algorithm has the following innovative features. Firstly, it fully accounts for the correlation among decision attributes, breaking the assumption of mutual independence in traditional algorithms, thus more closely aligning with real-world decision scenarios. Secondly, by combining different operators, it achieves effective information fusion, enhancing the accuracy and reliability of decision making. Finally, the algorithm provides a new pathway for solving MADM problems, offering a powerful tool for decision making in related fields.

In conclusion, the research findings of this paper bring new hope for solving MADM problems. In future studies, we will continue to explore more advanced and effective decision-making algorithms to contribute further to social development and progress.