An Improved Interval-Valued Hesitant Fuzzy Weighted Geometric Operator for Multi-Criterion Decision-Making

Abstract

In this paper, an improved interval-valued hesitant fuzzy weighted geometric (IIVHFWG) operator for multi-criterion decision-making is proposed. This operator is free of the limitations of the existing interval-valued hesitant fuzzy weighted average operator, interval-valued hesitant fuzzy weighted geometric operator, generalized interval-valued hesitant fuzzy weighted geometric operator, interval-valued hesitant fuzzy Hammer weighted average operator, and interval-valued hesitant fuzzy Hammer weighted geometric operator, which are prone to being influenced by extreme values. Based on the proposed IIVHFWG operator, a new method to solve the multi-criterion decisionmaking problems with interval-valued hesitant fuzzy elements is presented. Several numerical examples together with comparisons are introduced to demonstrate the effectiveness and advantages of this method.

1. Introduction

Multi-criterion decision-making (MCDM) involves considering the impacts and interactions of various factors while evaluating and comparing alternatives through a variety of methodologies and techniques. The ultimate goal of MCDM is to identify the optimal solution or decision that achieves the predetermined objectives and adheres to the defined constraints. Researchers have proposed numerous techniques to address this problem [1,2,3]. Zadeh [4] introduced the concept of fuzzy sets (FSs), which has achieved widespread applications in various domains. However, FSs fall short in capturing the nuanced degree of dissatisfaction and hesitation associated with complex and ambiguous information. Atanassov [5] extended the theory of FSs to intuitionistic fuzzy sets (IFSs), and subsequently developed several MCDM techniques based on the intuitionistic fuzzy sets. IFSs possess membership and non-membership degrees, which can respectively represent the degrees of satisfaction and dissatisfaction. Later, the concept of interval-valued intuitionistic fuzzy sets was proposed by Atanassov [6] and several MCDM techniques were developed accordingly. The concept of linguistic interval-valued intuitionistic fuzzy sets was presented by Liu and Qin [7] and Garg and Kumar [8]. Linguistic interval-valued intuitionistic fuzzy sets make decision results easier to understand and explain than interval-valued intuitionistic fuzzy sets.In some real-world scenarios, elements exhibit diverse membership and non-membership situations, between which there exists a hesitation relationship. To this end, Chen et al. [9] introduced the concept of interval-valued hesitant fuzzy sets, which provides new methods and techniques for effectively addressing problems featuring the hesitation relationship. For instance, decision-making methods grounded on intervalvalued hesitant fuzzy sets can be applied to interval-valued hesitant fuzzy decision-making, whereby the best solution or decision is determined by weighing the uncertainty ranges and memberships of conflicting factors.

Among the universal fuzzy sets, many are applied to solve MCDM problems [10]. Ma et al. [11] introduced the determinacy and consistency of linguistic terms to MCDM problems and presented a fuzzy-set-based model to address linguistic information. Liu et al. [12] define a series of new scoring functions for MCDM problems. Chen et al. [13] present a method for linking optimism and pessimism with multicriteria decision analysis. Augustine et al. [14] proposed the correlation coefficient of IFSs and applied it to MCDM. Qin et al. [15] introduced power Muirhead mean operators, and Zhong et al. [16] presented hesitant fuzzy Archimedean Muirhead mean operators applied to MCDM. Gal et al. [17] establish a fuzzy MCDM framework based on intuitionistic fuzzy sets. Interval-valued intuitionistic fuzzy sets can provide a more precise description of uncertainty by using intervals to express the membership and non-membership values of elements. Then, a new method for handling multi-criteria fuzzy decision-making problems based on interval-valued intuitionistic fuzzy sets is presented [18,19]. Wu et al. [20] investigate an approach for MCDM problems with interval-valued intuitionistic fuzzy preference relations. Nayagam et al. [21] introduce fuzzy numbers and intuitionistic fuzzy numbers to model problems involving incomplete and imprecise numerical quantities. Nayagam et al. [22] introduce and study a new method for sorting interval-valued intuitionistic fuzzy sets. Kokocc et al. [23] propose new ranking functions. Fares et al. [24], Joshi et al. [25], and Beg et al. [26] apply distance measures to multi-criteria decision-making. Zhong et al. [27] propose power Muirhead mean operators and apply them to multi-criteria decision-making. Qin et al. [28] and Rashid et al. [29] developed a novel method for multi-criteria decision-making. Gong et al. [30] and Wang et al. [31] presented several real-world examples relevant to MCDM problems. These operators or methods have their own characteristics and application conditions. Traditional VIKOR-based methods [32,33] can also be applied to MCDM problems, but they demonstrate inferior ranking results in terms of credibility and accuracy compared to MCDM methods based on aggregation operators [34].

Recently, interval-valued hesitant fuzzy sets have been widely used in MCDM problems. In [35], Zhang presented two interval-valued hesitant fuzzy QUA-LIFLEX ranking techniques to handle MCDM problems. In [36], Peng et al. developed an improved relative ratio approach to solve interval-valued hesitant fuzzy MCDM problems. In [37], Bai introduced the Hamming distance, Euclidean distance, Hausdorff distance, and generalized distance between interval-valued hesitant fuzzy sets and applied a generalized normalized distance similarity measure and a generalized Hausdorff normalized distance similarity measure to MCDM. In [38], Zhang and Wu developed an interval-valued hesitant fuzzy Hammer weighted average operator and an interval-valued hesitant fuzzy Hammer weighted geometric operator to tackle MCDM problems. In [9], Chen et al. proposed an interval-valued hesitant fuzzy weighted average operator, an interval-valued hesitant fuzzy weighted geometric operator, a generalized interval-valued hesitant fuzzy weighted average operator, and a generalized interval-valued hesitant fuzzy weighted geometric operator. In [39], Li and Peng applied the interval-valued hesitant fuzzy sets to an MCDM problem of determining shale gas regions. These operators can be applied to multi-criteria decision-making problems, but some operators may lead to uncertain aggregation values and cannot handle extreme cases. Based on this, the motivations of this paper are summarized as follows:

(1)

To develop an improved interval-valued hesitant fuzzy weighted geometric operators (A-IIVHFWG) based on Archimedean t-conorm and t-norm capable of overcoming the influence of extreme values, a parameter called $\lambda$ is introduced. The parameter $\lambda$ is set to a value close to 1, which helps prevent the operator from being affected by extreme values.

(2)

For a better aggregation of the decision-maker’s representations, the A-IIVHFWG operator is instantiated as the IIVHFWG operator. The IIVHFWG operator represents a specific manifestation of the A-IIVHFWG operator.

Related Work

In this work, we first illustrate that the four operators in [9] and the two operators in [38] could lead to uncertain aggregation values. Specifically, the same lower limit of interval-valued hesitant fuzzy elements and different upper limits or different lower limits and the same upper limit of interval-valued hesitant fuzzy numbers result in the same aggregation value in some cases. Then, we propose an improved interval-valued hesitant fuzzy weighted geometric (IIVHFWG) operator that is free of these limitations. Furthermore, we illustrate a shortcoming of an MCDM method presented by Zhang and Wu [38] and develop a new MCDM method that can overcome this shortcoming. We also introduce several numerical examples together with comparisons to demonstrate the developed method. The major contributions of this paper are as follows: (1) an improved interval-valued hesitant fuzzy weighted geometric operator is proposed. (2) an MCDM method based on the presented operators is developed.

The rest of this paper is structured as follows. Section 2 outlines some prerequisites for this study and analyzes the limitations of the six operators. Section 3 explains the details of the proposed operator. Section 4 illustrates a shortcoming of Zhang and Wu’s MCDM approach [38]. Section 5 describes the details of the new MCDM approach. Section 6 validates the new MCDM approach. Section 7 draws a conclusion.

2. Preliminaries

In this paper, we denote the discourse set by $X=\{x_1,x_2,\dots ,x_n\}$. We also use HFE and HFS to denote hesitant fuzzy element and hesitant fuzzy set, respectively. Similarly, IVHFE and IVHFS represent interval-valued hesitant fuzzy element and interval-valued hesitant fuzzy set, respectively. We use $\tilde{A}$ and $\tilde{h}$ to refer to an IVHFS and an IVHFE, respectively.

Definition 1 

([40]). Let X be a reference set, a hesitant fuzzy set A on X is defined in terms of a function $h_A\left(x\right)$ that returns a subset of [0, 1] when it is applied to X

$$\begin{array}{c}\hfill A=\left\{⟨X,h_A\left(x\right)⟩\right|x\in X\}.\end{array}$$

(1)

where $h_A\left(x\right)$ is a set of some different values in [0, 1], representing the possible membership degrees of the element $x\in X$ to A. $h_A\left(x\right)$ is called a HFE, which is a basic unit of HFS.

Definition 2 

([9]). Let X be a reference set, and $D[0,1]$ be the set of all closed subintervals of [0, 1]. An IVHFS on X is

$$\begin{array}{c}\hfill \tilde{A}=\left\{⟨x,h_{\tilde{A}}\left(x\right)⟩\right|x\in X\}.\end{array}$$

(2)

where ${\tilde{h}}_{\tilde{A}}\left(x\right)$ denotes all possible interval-valued membership degrees of the element $x\in X$ to the set $\tilde{A}$. ${\tilde{h}}_{\tilde{A}}\left(x\right)=\tilde{h}\left(x\right)=\left\{\tilde{\gamma }\right|\tilde{\gamma }\in h_{\tilde{A}}\left(x\right)\}$ is called an IVHFE. Here, $\tilde{\gamma }=[{\tilde{\gamma }}^L,{\tilde{\gamma }}^U]$ is an interval number. ${\tilde{\gamma }}^L=$ inf $\tilde{\gamma }$ and ${\tilde{\gamma }}^U=$ sup $\tilde{\gamma }$ represent the lower and upper limits of $\tilde{\gamma }$, respectively.

Definition 3 

([41]). Let $\tilde{a}=[{\tilde{a}}^L,{\tilde{a}}^U]$ and $\tilde{b}=[{\tilde{b}}^L,{\tilde{b}}^U]$ be two interval numbers, then

(1) 

$\tilde{a}=\tilde{b}⟺{\tilde{a}}^L={\tilde{b}}^L$ and ${\tilde{a}}^U={\tilde{b}}^U$;

(2) 

$\tilde{a}+\tilde{b}=[{\tilde{a}}^L+{\tilde{b}}^L,{\tilde{a}}^U+{\tilde{b}}^U]$.

Definition 4 

([38]). For a given IVHFE $\tilde{h}=\left\{\tilde{\gamma }|\tilde{\gamma }\in \tilde{h}\right\}$ = $\left\{\left[{\tilde{\gamma }}^L,{\tilde{\gamma }}^U\right]|\tilde{\gamma }\in \tilde{h}\right\}$, $s\left(\tilde{h}\right)=$$\frac{{\sum }_{\gamma \in \tilde{h}}\left({\tilde{\gamma }}^L+{\tilde{\gamma }}^U\right)}{2l_{\tilde{h}}}$ is called a score function of $\tilde{h}$, where $l_{\tilde{h}}$ is the number of intervals in $\tilde{h}$.

Definition 5 

([38]). For an IVHFE $\tilde{h}=\left\{\tilde{\gamma }\mid \tilde{\gamma }\in \tilde{h}\right\}$ = $\left\{\left[{\tilde{\gamma }}^L,{\tilde{\gamma }}^U\right]|\tilde{\gamma }\in \tilde{h}\right\}$, $\nu \left(\tilde{h}\right)=\frac{{\sum }_{\gamma \in \tilde{h}}\left(|{\tilde{\gamma }}^L-s\left(\tilde{h}\right)\left| + \right|{\tilde{\gamma }}^U-s\left(\tilde{h}\right)|\right)}{2l_{\tilde{h}}}$ is referred to as a variance function of $\tilde{h}$, where $s\left(\tilde{h}\right)$ is the score function of $\tilde{h}$.

Definition 6 

([38]). Let ${\tilde{h}}_1$ and ${\tilde{h}}_2$ be any two IVHFEs, and let $s\left({\tilde{h}}_i\right)$ and $\nu \left({\tilde{h}}_i\right)(i=1,2)$ be the score and variance functions of ${\tilde{h}}_i(i=1,2)$, respectively. Then, the following conditions hold:

(1) 

If $s\left({\tilde{h}}_1\right)>s\left({\tilde{h}}_2\right)$, then ${\tilde{h}}_1>{\tilde{h}}_2$.

(2) 

If $s\left({\tilde{h}}_1\right)=s\left({\tilde{h}}_2\right)$, then

(i) 

if $\nu \left({\tilde{h}}_1\right)<\nu \left({\tilde{h}}_2\right)$, then ${\tilde{h}}_1>{\tilde{h}}_2$.

(ii) 

if $\nu \left({\tilde{h}}_1\right)=\nu \left({\tilde{h}}_2\right)$, then ${\tilde{h}}_1={\tilde{h}}_2$.

We analyze the limitations of the interval-valued hesitant fuzzy weighted average (IVHFWA) operator, interval-valued hesitant fuzzy weighted geometric (IVHFWG) operator, generalized interval-valued hesitant fuzzy weighted average (GIVHFWA${}_{\lambda }$) operator, generalized interval-valued hesitant fuzzy weighted geometric (GIVHFWG${}_{\lambda }$) operator, interval-valued hesitant fuzzy Hammer weighted average (IVHFHWA) operator, and interval-valued hesitant fuzzy Hammer weighted geometric (IVHFHWG) operator:

(1)

IVHFWA operator [9]:

$$\begin{array}{c}IVHFWA\left({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n\right)={\displaystyle \underset{j=1}{\overset{n}{⨁}}}\left(w_j{\tilde{h}}_j\right)=\{\left[1-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^L\right)}^{w_j},1-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^U\right)}^{w_j}\right]\hfill \\ \mid {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\}\hfill \end{array}$$

(3)

where ${\omega }_j$ denotes the weight of the IVHFE ${\tilde{h}}_j=\left\{\left[{\tilde{\gamma }}_j^L,{\tilde{\gamma }}_j^U\right]\right\}$, ${\omega }_j\in [0,1],j=1,2,\dots ,n$, and ${\sum }_{j=1}^n{\omega }_j=1.$

(2)

IVHFWG operator [9]:

$$\begin{array}{l}IVHFWG\left({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n\right)={\displaystyle \underset{j=1}{\overset{n}{⨂}}}{\tilde{h}}_j^{w_j}=\left\{\left[{\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^L\right)}^{w_j},{\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^U\right)}^{w_j}\right]\right.\\ \mid {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\}\end{array}$$

(4)

(3)

GIVHFWA${}_{\lambda }$ operator [9]:

$$\begin{array}{c}{GIVHFWA}_{\lambda }({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)={\left({\displaystyle \underset{j=1}{\overset{n}{⨁}}}\left(w_j{\tilde{h}}_j^{\lambda }\right)\right)}^{1/\lambda }\hfill \\ =\left\{\left[{\left(1-{\displaystyle \prod _{j=1}^n}{\left(1-{\left({\gamma }_j^L\right)}^{\lambda }\right)}^{w_j}\right)}^{1/\lambda }\right.\right.,\left.{\left(1-{\displaystyle \prod _{j=1}^n}{\left(1-{\left({\gamma }_j^U\right)}^{\lambda }\right)}^{w_j}\right)}^{1/\lambda }\right]\hfill \\ \mid {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\},\hfill \end{array}$$

(5)

where $\lambda \ge 0$.

(4)

GIVHFWG${}_{\lambda }$ operator [9]:

$$\begin{array}{c}{GIVHFWG}_{\lambda }\left({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n\right)=\frac{1}{\lambda }\left({\displaystyle \underset{j=1}{\overset{n}{⨂}}}{\left(\lambda {\tilde{h}}_j\right)}^{w_j}\right)\\ =\left\{\left[1-{\left(1-{\displaystyle \prod _{j=1}^n}{\left(1-{\left(1-{\gamma }_j^L\right)}^{\lambda }\right)}^{w_j}\right)}^{\frac{1}{\lambda }},\right.\right.\left.1-{\left(1-{\displaystyle \prod _{j=1}^n}{\left(1-{\left(1-{\gamma }_j^U\right)}^{\lambda }\right)}^{w_j}\right)}^{\frac{1}{\lambda }}\right]\\ \mid {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\}\end{array}$$

(6)

where $\lambda \ge 0$.

(5)

IVHFHWA operator [38]:

$$\begin{array}{c}IVHFHWA({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)\\ =\left\{\left[\frac{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^L\right)}^{{\omega }_j}-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^L\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^L\right)}^{{\omega }_j}+(\theta -1){\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^L\right)}^{{\omega }_j}},\right.\right.\left.\frac{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^U\right)}^{{\omega }_j}-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^U\right)}^{{\omega }_j}+(\theta -1){\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}\right]\\ {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\}\end{array}$$

(7)

where $\theta >0$.

(6)

IVHFHWG operator [38]:

$$\begin{array}{c}IVHFHWG({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)\\ =\left\{\left[\frac{\theta {\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^L\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right)\left(1-{\tilde{\gamma }}_j^L\right)\right)}^{{\omega }_j}+\left(\theta -1\right){\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^L\right)}^{{\omega }_j}},\right.\right.\left.\frac{\theta {\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right)\left(1-{\tilde{\gamma }}_j^U\right)\right)}^{{\omega }_j}+\left(\theta -1\right){\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^U\right)}^{{\omega }_j}},\right]\\ {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\}\end{array}$$

(8)

In the following, we utilize some numerical examples to illustrate the shortcomings of the six operators above, respectively.

Example 1. 

Let $h_1\left(x\right)=\left[1,1\right],h_2\left(x\right)=\left[0.2,0.3\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, the aggregation result of the IVHFWA operator in Equation (3) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the IVHFWA operator may not be able to address the related multi-criteria decision-making problems.

Example 2. 

Let $h_1\left(x\right)=[0,0]$, $h_2\left(x\right)=[0.5,0.8]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, the aggregation result of the IVHFWG operator in Equation (4) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the IVHFWG operator may not be able to address the related multi-criteria decision-making problems.

Example 3. 

Let $h_1\left(x\right)=\left[1,1\right],h_2\left(x\right)=\left[0.3,0.4\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, the aggregation result of the GIVHFWA${}_{\lambda }$ operator in Equation (5) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the GIVHFWA${}_{\lambda }$ operator may not be able to address the related multi-criteria decision-making problems.

Example 4. 

Let $h_1\left(x\right)=\left[0,0\right],h_2\left(x\right)=\left[0.4,0.6\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, the aggregation result of the GIVHFWG${}_{\lambda }$ operator in Equation (6) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the GIVHFWG${}_{\lambda }$ operator may not be able to address the related multi-criteria decision-making problems.

Example 5. 

Let $h_1\left(x\right)=\left[0.3,0.8\right],h_2\left(x\right)=\left[1,1\right]$ be two IVHFEs with the weights ${\omega }_1=1$ and ${\omega }_2=0$, respectively. Then, the aggregation result of the IVHFHWA operator in Equation (7) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the IVHFHWA operator may not be able to address the related multi-criteria decision-making problems.

Example 6. 

Let $h_1\left(x\right)=\left[0,0\right],h_2\left(x\right)=\left[0.4,0.6\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, the aggregation result of the IVHFHWG operator in Equation (8) is $0^0$, which is an indeterminate value. When indeterminate values are present in the data, the IVHFHWG operator may not be able to address the related multi-criteria decision-making problems.

3. The Proposed IIVHFWG Operator

In this section, we propose an IIVHFWG operator that can overcome the limitations illustrated in the previous section.

Definition 7 

([38]). Let ${\tilde{h}}_j(j=1,2,\dots ,n)$ be a collection of IVHFEs, and $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{h}}_j(j=1,2,\dots ,n)$, with $\omega \in [0,1]$ and ${\sum }_{j=1}^n{\omega }_j=1$. Then, an Archimedean t-conorm- and t-norm-based interval-valued hesitant fuzzy weighted geometric (A-IVHFWG) operator is an mapping ${\tilde{H}}^n\to \tilde{H}$, where

$$A-IVHFWG({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)={\otimes }_{j=1}^n\left({\tilde{h}}_j^{{\omega }_j}\right)$$

(9)

Definition 8 

([38]). Let ${\tilde{h}}_j(j=1,2,\dots ,n)$ be a collection of IVHFEs, and $\omega ={({\omega }_1,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{h}}_j(j=1,2,\dots ,n),$ where ${\omega }_i$ indicates the degree of importance of ${\tilde{h}}_i$, statisfying ${\omega }_j\in [0,1]$ and ${\sum }_{j=1}^n{\omega }_j=1$. Then, the aggregated value by using the A-IVHFWG operator is an IVHFE, and

$$\begin{array}{cc}\hfill & A-IVHFWG({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)=\hfill \\ \hfill & \left\{\left[g^{-1}\left(\sum _{j=1}^n{\omega }_{j}g\left({\tilde{\gamma }}_j^L\right)\right),g^{-1}\left(\sum _{j=1}^n{\omega }_{j}g\left({\tilde{\gamma }}_j^U\right)\right)\right]{\tilde{\gamma }}_1,{\tilde{\gamma }}_2,\dots ,{\tilde{\gamma }}_n\in {\tilde{h}}_n\right\}\hfill \end{array}$$

(10)

The A-IVHFWG operator is inherently vulnerable to extreme values. If the function $g\left(t\right)$ is expressed as $-log\left(t\right)$, the A-IVHFWG operator reduces to the interval-valued hesitant fuzzy weighted geometric (IVHFWG) operator, as illustrated in [9]. If $g\left(t\right)$ is instead expressed as $log\left(\right(\theta +(1-\theta )t)/t)$ where $\theta$ > 0, the A-IVHFWG operator reduces to the interval-valued hesitant fuzzy Hammer weighted geometric (IVHFHWG) operator, as elaborated in [38]. The shortcomings of these operators are outlined in section three. Consequently, we propose an improved interval-valued hesitant fuzzy weighted geometric operator (A-IIVHFWG) based on Archimedean t-conorm and t-norm, wherein a specific IIVHFWG operator is given when $g\left(t\right)=-log\left(t\right)$.

Definition 9. 

Let ${\tilde{h}}_j(j=1,2,\dots ,n)$ be a collection of IVHFEs, and $\omega ={({\omega }_j,{\omega }_2,\dots ,{\omega }_n)}^T$ be the weight vector of ${\tilde{h}}_j(j=1,2,\dots ,n),$ where ${\omega }_j$ indicates the degree of importance of ${\tilde{h}}_j$, statisfying ${\omega }_j\in [0,1]$ and ${\sum }_{j=1}^n{\omega }_j=1$. Then, the aggregated value by using the A-IIVHFWG operator is an IVHFE, and

$$\begin{array}{cc}\hfill & \mathrm{A-IIVHFWG}({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n)=\hfill \\ \hfill & \left\{\left[1-\frac{1}{\lambda }\left(1-g^{-1}\left(\sum _{j=1}^{n}g\left(1-\lambda \left(1-{\tilde{\gamma }}_j^L\right)\right)\right)\right),1-\frac{1}{\lambda }\left(1-g^{-1}\left(\sum _{j=1}^{n}g\left(1-\lambda \left(1-{\tilde{\gamma }}_j^U\right)\right)\right)\right)\right]\right\}\hfill \end{array}$$

(11)

When g (t) = −log (t) :

$$\begin{array}{cc}\hfill & \mathrm{IIVHFWG}\left({\tilde{h}}_1,{\tilde{h}}_2,\dots ,{\tilde{h}}_n\right)\hfill \\ \hfill & =\left\{\left[1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\tilde{\gamma }}_j^L\right)\right)}^{w_j}\right),1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\tilde{\gamma }}_j^U\right)\right)}^{w_j}\right)\right.\right.\hfill \\ \hfill & {\tilde{\gamma }}_1\in {\tilde{h}}_1,{\tilde{\gamma }}_2\in {\tilde{h}}_2,{\tilde{\gamma }}_n\in {\tilde{h}}_n\left]\right\}\hfill \end{array}$$

(12)

where $w_j$ is the weight of ${\tilde{h}}_j,0\le w_j\le 1,{\sum }_{j=1}^n{w}_j=1,j=1,2,\cdots ,n$ and $0<\lambda <1$. In this paper, we let $\lambda =0.99$. Particularly, if $\lambda =1,$ then the IIVHFWG operator will reduce to the IVHFWG operator in Equation (4). The proposed IIVHFWG operator has the following properties:

Property 1 

(Idempotency). Let $h_1,h_2,\dots ,$$h_n$ be n IVHFEs and ${\omega }_1,{\omega }_2,\dots ,$${\omega }_n,$ be their weights such that ${\omega }_j\in [0,1],{\sum }_{j=1}^n{\omega }_j=1$ and $j=1,2,\dots ,n$. If $h_1=h_2=\cdots =h_n=h,$ then IIVHFWG$(h_1,h_2,\dots ,h_n)=h$.

Proof. 

If $h_1=h_2=\cdots =h_n=h$, then according to the proposed IIVHFWG operator in Equation (12), we obtain

$$\begin{array}{cc}\hfill & \mathrm{IIVHFWG}\left(h_1,h_2,\cdots ,h_n\right)\hfill \\ \hfill & ={\omega }_1{h}_1\oplus {\omega }_2{h}_2\oplus \cdots \oplus {\omega }_n{h}_n\hfill \\ \hfill & ={\omega }_{1}h\oplus {\omega }_{2}h\oplus \cdots \oplus {\omega }_{n}h\hfill \\ \hfill & =\left\{\left[1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^L\right)\right)}^{w_j}\right),\right.\right.\hfill \\ \hfill & \left.\left.1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^U\right)\right)}^{w_j}\right)\right]\right\}\hfill \\ \hfill & =\left\{\left[1-\frac{1}{\lambda }\left(1-{\left(1-\lambda \left(1-{\gamma }_j^L\right)\right)}^{w_1+w_2+\dots +w_n}\right),\right.\right.\hfill \\ \hfill & \left.\left.1-\frac{1}{\lambda }\left(1-{\left(1-\lambda \left(1-{\gamma }_j^U\right)\right)}^{w_1+w_2+\dots +w_n}\right)\right]\right\}\hfill \\ \hfill & =h({\omega }_1+{\omega }_2+\cdots +{\omega }_n)\hfill \\ \hfill & =h\hfill \end{array}$$

□

Property 2 

(Boundedness). Let $h_1,h_2,\dots ,$$h_n$ be n IVHFEs, $h^{-}$ = min$\{h_1,h_2,\dots ,h_n\}$ and $h^{+}$ = max$\{h_1,h_2,\dots ,h_n\}.$ Then, $h^{-}\le \mathrm{IIVHFWG}(h_1,$ $h_2,\dots ,h_n)\le h^{+}$.

Proof. 

According to the proposed IIVHFWG operator in Equation (12), we obtain

$$\begin{array}{cc}\hfill & \mathrm{IIVHFWG}\left(h_1,h_2,\dots ,h_n\right)={\oplus }_{j=1}^n{\omega }_j{h}_j\hfill \\ \hfill & \le {\oplus }_{j=1}^n{\omega }_j{h}^{+}=h^{+}\sum _{j=1}^n{\omega }_j,\hfill \\ \hfill & \mathrm{IIVHFWG}\left(h_1,h_2,\dots ,h_n\right)={\oplus }_{j=1}^n{\omega }_j{h}_j\hfill \\ \hfill & \ge {\oplus }_{j=1}^n{\omega }_j{h}^{-}=h^{-}\sum _{j=1}^n{\omega }_j,\hfill \end{array}$$

Due to the fact that ${\sum }_{j=1}^n{\omega }_j=1$, we obtain $h^{-}\le \mathrm{IIVHFWG}(h_1,$ $h_2,\dots ,h_n) \le h^{+}$.   □

In the following, we use the six examples in the previous section again to show that the proposed IIVHFWG operator is free of the limitations of the six existing operators.

Example 7. 

Let $h_1\left(x\right)=\left[1,1\right],h_2\left(x\right)=\left[0.2,0.3\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.2,0.3]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the IVHFWA operator in Example 1.

Example 8. 

Let $h_1\left(x\right)=[0,0]$, $h_2\left(x\right)=[0.5,0.8]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.5,0.8]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the IVHFWG operator in Example 2.

Example 9. 

Let $h_1\left(x\right)=\left[1,1\right],h_2\left(x\right)=\left[0.3,0.4\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.3,0.4]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the GIVHFWA${}_{\lambda }$ operator in Example 3.

Example 10. 

Let $h_1\left(x\right)=\left[0,0\right],h_2\left(x\right)=\left[0.4,0.6\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.4,0.6]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the GIVHFWG${}_{\lambda }$ operator in Example 4.

Example 11. 

Let $h_1\left(x\right)=\left[0.3,0.8\right],h_2\left(x\right)=\left[1,1\right]$ be two IVHFEs with the weights ${\omega }_1=1$ and ${\omega }_2=0$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.3,0.8]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the IVHFHWA operator in Example 5.

Example 12. 

Let $h_1\left(x\right)=\left[0,0\right],h_2\left(x\right)=\left[0.4,0.6\right]$ be two IVHFEs with the weights ${\omega }_1=0$ and ${\omega }_2=1$, respectively. Then, according to the proposed IIVHFWG operator, we obtain IIVHFWG$(h_1,h_2)$ = $[0.4,0.6]$. The IIVHFWG operator is not affected by indeterminate values during the aggregation process and does not have the drawbacks of the IVHFHWG operator in Example 6.

4. Limitations of Zhang and Wu’s MCDM Method

In this section, we analyze the limitations of Zhang and Wu’s MCDM method [38]. Let Y = $Y_1,Y_2,\dots ,Y_m$ be a discrete set of alternatives, $G=\{G_1,G_2,\dots ,G_n\}$ be a set of criteria, and $\omega$ = $[{\omega }_1,{\omega }_2,\dots ,$ ${\omega }_n{]}^T$ be the weight vector of G, where ${\sum }_{i=1}^n{\omega }_j=1$ and ${\omega }_j\ge 0,j=1,2,\dots ,n$.

Step 1: 

Transform the interval-valued hesitant fuzzy decision matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}$ into a normalized interval-valued hesitant fuzzy decision matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$:

$$\tilde{A}=\left({\tilde{a}}_{ij}\right)=\left\{\begin{array}{c}{\tilde{r}}_{ij},\mathrm{for}\mathrm{benefit}\mathrm{attribute}{\mathrm{G}}_{\mathrm{j}}\hfill \\ {\tilde{r}}_{ij}^c,\mathrm{for}\mathrm{cost}\mathrm{attribute}{\mathrm{G}}_{\mathrm{j}}\hfill \end{array}\right.$$

(13)

where ${\tilde{r}}_{ij}^c$ is the complement of ${\tilde{r}}_{ij}$ such that ${\tilde{r}}_{ij}^c$$=\left.\left\{\left[1-{\tilde{\gamma }}_{ij}^U,1-{\tilde{\gamma }}_{ij}^L\right]\right|{\tilde{\gamma }}_{ij}\in {\tilde{r}}_{ij}\right\}$, $i=1,2,\dots ,m$, $j=1,2,\dots ,n$.

Step 2: 

Aggregate the IVHFEs ${\tilde{a}}_{i1},{\tilde{a}}_{i2},\dots ,$ ${\tilde{a}}_{ij}$ in the decision matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$ using the IVHFHWA operator, shown as follows:

$$\begin{array}{c}\mathrm{IVHFHWA}({\tilde{a}}_{i1},{\tilde{a}}_{i2},\dots ,{\tilde{a}}_{ij})\\ =\left\{\left[\frac{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^L\right)}^{{\omega }_j}-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^L\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^L\right)}^{{\omega }_j}+(\theta -1){\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^L\right)}^{{\omega }_j}},\right.\right.\left.\left.\frac{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^U\right)}^{{\omega }_j}-{\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right){\tilde{\gamma }}_j^U\right)}^{{\omega }_j}+(\theta -1){\displaystyle \prod _{j=1}^n}{\left(1-{\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}\right]\right\}\end{array}$$

or the IVHFHWG operator, shown as follows:

$$\begin{array}{c}\mathrm{IVHFHWG}({\tilde{a}}_{i1},{\tilde{a}}_{i2},\cdots ,{\tilde{a}}_{ij})\\ =\left\{\left[\frac{\theta {\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^L\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right)\left(1-{\tilde{\gamma }}_j^L\right)\right)}^{{\omega }_j}+\left(\theta -1\right){\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^L\right)}^{{\omega }_j}},\right.\right.\left.\left.\frac{\theta {\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^U\right)}^{{\omega }_j}}{{\displaystyle \prod _{j=1}^n}{\left(1+\left(\theta -1\right)\left(1-{\tilde{\gamma }}_j^U\right)\right)}^{{\omega }_j}+\left(\theta -1\right){\displaystyle \prod _{j=1}^n}{\left({\tilde{\gamma }}_j^U\right)}^{{\omega }_j}},\right]\right\}\end{array}$$

In this paper, we let $\theta =3.$

Step 3: 

Calculate the score values and variance values of ${\tilde{a}}_i(i=1,2,\dots ,m)$ according to Definitions 4 and 5.

Step 4: 

Generate a ranking of all alternatives according to Definition 6.

Example 13. 

Let $Y_1,Y_2,Y_3$ and $Y_4$ be four alternatives and $G_1,G_2,G_3$ be three criteria. Assume the weights of criteria are ${\omega }_1=0.36,{\omega }_2=0.33$ and ${\omega }_3=0.31$ and the decision matrix is $\widetilde{R_1}$.

$$\widetilde{R_1}=\begin{array}{c}\\ Y_1\\ Y_2\\ Y_3\\ Y_4\end{array}\begin{array}{ccccc}& G_1& G_2& G_3& \\ (& \begin{array}{c}\left\{\right[1,1\left]\right\}\\ \left\{\right[0.5,0.6],[0.7,0.9\left]\right\}\\ \left\{\right[1,1\left]\right\}\\ \left\{\right[0.2,0.4\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.2,0.8],[0.5,0.9\left]\right\}\\ \left\{\right[0.7,0.8],[0.6,0.9\left]\right\}\\ \left\{\right[0.2,0.6],[0.2,0.8\left]\right\}\\ \left\{\right[0.2,0.3\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.3,0.5],[0.2,0.5\left]\right\}\\ \left\{\right[0.2,0.6],[0.5,0.7\left]\right\}& \left\{\right[0.3,0.4],[0.2,0.6\left]\right\}\\ \left\{\right[0.5,0.7\left]\right\}\end{array}& )\end{array}$$

Using the MCDM method above (based on the IVHFHWA operator), a ranking of the four alternatives is generated as $Y_2\succ Y_1=Y_3\succ Y_4$. It can be seen that Zhang and Wu’s MCDM method cannot distinguish the order of the alternatives $Y_1$ and $Y_3$ in this case.

Example 14. 

Let $Y_1,Y_2$ and $Y_3$ be three alternatives and $G_1,G_2,G_3$ be three criteria. Assume the weights of criteria are ${\omega }_1=0.25,{\omega }_2=0.36$ and ${\omega }_3=0.39$ and the decision matrix is $\widetilde{R_2}$.

$$\widetilde{R_2}=\begin{array}{c}\\ Y_1\\ Y_2\\ Y_3\end{array}\begin{array}{ccccc}& G_1& G_2& G_3& \\ (& \begin{array}{c}\left\{\right[0,0\left]\right\}\\ \left\{\right[0,0\left]\right\}\\ \left\{\right[0.3,0.4],[0.5,0.8\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.6,0.8],[0.2,0.7\left]\right\}\\ \left\{\right[0.3,0.8],[0.5,0.7\left]\right\}\\ \left\{\right[0.6,0.8],[0.2,0.3\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.2,0.7],[0.3,0.9\left]\right\}\\ \left\{\right[0.5,0.7],[0.3,0.9\left]\right\}\\ \left\{\right[0.2,0.6],0.4,0.9\}\end{array}& )\end{array}$$

Using the MCDM method above (based on the IVHFHWG operator), a ranking of the three alternatives is generated as $Y_3\succ Y_1=Y_2$. It can be seen that Zhang and Wu’s MCDM method cannot distinguish the order of the alternatives $Y_1$ and $Y_2$ in this case.

Example 15. 

Let $Y_1,Y_2$, and $Y_3$ be three alternatives and $G_1,G_2,G_3$ be three criteria. Assume the weights of criteria are ${\omega }_1=0.36,{\omega }_2=0.28$, and ${\omega }_3=0.36$ and the decision matrix is $\widetilde{R_3}$.

$$\widetilde{R_3}=\begin{array}{c}\\ Y_1\\ Y_2\\ Y_3\end{array}\begin{array}{ccccc}& G_1& G_2& G_3& \\ (& \begin{array}{c}\left\{\right[0,0\left]\right\}\\ \left\{\right[0.3,0.7],[0.5,0.6\left]\right\}\\ \left\{\right[0,0\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.3,0.8],[0.3,0.5\left]\right\}\\ \left\{\right[0.4,0.8],[0.4,0.7\left]\right\}\\ \left\{\right[0.2,0.8],[0.4,0.6\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.2,0.7],[0.2,0.4\left]\right\}\\ \left\{\right[0.5,0.6],[0.7,0.9\left]\right\}\\ \left\{\right[0.4,0.7],[0.6,0.8\left]\right\}\end{array}& )\end{array}$$

Using the MCDM method above (based on the IVHFHWG operator), a ranking of the three alternatives is generated as $Y_2\succ Y_1=Y_3$. It can be seen that Zhang and Wu’s MCDM method cannot distinguish the order of the alternatives $Y_1$ and $Y_3$ in this case.

5. New MCDM Method

In this section, we develop a new MCDM method based on the proposed IIVHFWG operator. Let Y = $Y_1,Y_2,\dots ,Y_m$ be a discrete set of alternatives, $G=\{G_1,G_2,\dots ,G_n\}$ be a set of criteria, $\omega$ = $[{\omega }_1,{\omega }_2,\dots ,$ ${\omega }_n{]}^T$ be the weight vector of G, where ${\sum }_{i=1}^n{\omega }_j=1$ and ${\omega }_j\ge 0,j=1,2,\dots ,n$, and the decision matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}$ is

$$\tilde{R}=\begin{array}{c}\\ Y_1\\ Y_2\\ ⋮\\ Y_m\end{array}\begin{array}{cccccc}& G_1& G_2& \cdots & G_n& \\ (& \begin{array}{c}{\tilde{r}}_{11}\\ {\tilde{r}}_{21}\\ ⋮\\ {\tilde{r}}_{m1}\end{array}& \begin{array}{c}{\tilde{r}}_{12}\\ {\tilde{r}}_{22}\\ ⋮\\ {\tilde{r}}_{m2}\end{array}& \begin{array}{c}\cdots \\ \cdots & \ddots \\ \cdots \end{array}& \begin{array}{c}{\tilde{r}}_{1n}\\ {\tilde{r}}_{2n}\\ ⋮\\ {\tilde{r}}_{mn}\end{array}& )\end{array},$$

The proposed MCDM method mainly consists of the following steps:

Step 1: 

Transform the interval-valued hesitant fuzzy decision matrix $\tilde{R}={\left({\tilde{r}}_{ij}\right)}_{m\times n}$ into a normalized interval-valued hesitant fuzzy decision matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$:

$$\tilde{A}=\left({\tilde{a}}_{ij}\right)=\left\{\begin{array}{cc}{\tilde{r}}_{ij},\hfill & \mathrm{for}\mathrm{benefit}\mathrm{attribute}{G}_j\hfill \\ {\tilde{r}}_{ij}^c,\hfill & \mathrm{for}\mathrm{cost}\mathrm{attribute}{G}_j\hfill \end{array}\right.$$

where ${\tilde{r}}_{ij}^c$ is the complement of ${\tilde{r}}_{ij}$ such that ${\tilde{r}}_{ij}^c$$=\left.\left\{\left[1-{\tilde{\gamma }}_{ij}^U,1-{\tilde{\gamma }}_{ij}^L\right]\right|{\tilde{\gamma }}_{ij}\in {\tilde{r}}_{ij}\right\}$, $i=1,2,\dots ,m$, $j=1,2,\dots ,n$.

Step 2: 

Aggregate the IVHFEs ${\tilde{a}}_{i1},{\tilde{a}}_{i2},\cdots ,$ ${\tilde{a}}_{ij}$ in the decision matrix $\tilde{A}={\left({\tilde{a}}_{ij}\right)}_{m\times n}$, where $i=1,2,\dots ,m$ and $j=1,2,\dots ,n$, using the proposed IIVHFWG operator, shown as follows:

$$\begin{array}{cc}\hfill & \mathrm{IIVHFWG}\left(h_1,h_2,\dots ,h_n\right)\hfill \\ \hfill & =\left\{\left[1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^L\right)\right)}^{w_j}\right),\right.\right.\left.\left.1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^U\right)\right)}^{w_j}\right)\right]\right\}\hfill \end{array}$$

Step 3: 

Calculate the score and variance values of ${\tilde{a}}_i(i=1,2,\cdots ,m)$ according to Definitions 4 and 5.

Step 4: 

Generate a ranking of all alternatives according to Definition 6.

The time complexity of the aforementioned MCDM methods mainly concentrates on the aggregation of the operator IIVHFWG in Step 2. Taking the matrix $\tilde{R}$ as an example, the time complexity of the operator can be determined as $O(l_{m1}^1{l}_{m2}^2\dots l_{mn}^n)$. The time complexity of Step 3 depends on the size of the data after operator aggregation and is linear. Step 4 involves data sorting, and its time complexity depends on the choice of sorting algorithm and the size of the data. The overall time complexity of MCDM is concentrated in Step 2, which is $O(l_{m1}^1{l}_{m2}^2\dots l_{mn}^n)$.

The operators IVHFHWA, IVHFWG, and IVHFHWG have been observed to possess inherent structural defects. The incidence of either 0 or 1 within the data may generate identical aggregation values. This phenomenon can occur specifically when the lower limit of interval-valued hesitant fuzzy elements is constant while the upper limits fluctuate, or when the lower limits vary whilst the upper limit remains the same. In such situations, identical aggregation values may emerge for interval-valued hesitant fuzzy numbers.

The IVHFHWG and IVHFWG operators display common limitations. When $\theta$ equals 1, the IVHFHWG operator is transformable into the IVHFWG operator. In the forthcoming section, we will put forth theoretical justifications for the proposition that these limitations can be mitigated by the IIVHFWG operator.

Let

$$\begin{array}{cc}\hfill & IVHFWG=\left\{\left[\prod _{j=1}^n{\left({\tilde{\gamma }}_j^L\right)}^{w_j},\prod _{j=1}^n{\left({\tilde{\gamma }}_j^U\right)}^{w_j}\right]\right\}\hfill \end{array}$$

If ∃ j $\in \{1,2,\dots ,n\}$, let ${\tilde{\gamma }}_j^L$ = 0, then IVHFWG ≡ 0.

Since ${\tilde{\gamma }}_j^L$=0, this will cause other ${\tilde{\gamma }}_j^L$ to have no effect in the aggregation process.

Therefore, If both IVHFWG${}_1$ and IVHFWG${}_2$ contain ${\tilde{\gamma }}_j^L=0$, it may result in IVHFWG${}_1$ = IVHFWG${}_2$, the aggregation value of the two is the same, and the ranking cannot be accurately distinguished.

Let IIVHFWG

$$\begin{array}{cc}\hfill & =\left\{\left[1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^L\right)\right)}^{w_j}\right),\right.\right.\left.\left.1-\frac{1}{\lambda }\left(1-\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^U\right)\right)}^{w_j}\right)\right]\right\}\hfill \\ \hfill & =\left\{\left[1-\frac{1}{\lambda }+\frac{1}{\lambda }\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^L\right)\right)}^{w_j},\right.\right.\left.\left.1-\frac{1}{\lambda }+\frac{1}{\lambda }\prod _{j=1}^n{\left(1-\lambda \left(1-{\gamma }_j^U\right)\right)}^{w_j}\right]\right\}\hfill \end{array}$$

If there exists a $j\in \{1,2,\cdots ,n\}$ such that $1-\lambda (1-{\gamma }_j^L)=0$, or equivalently, ${\gamma }_j^L=1-\frac{1}{\lambda }$, then IIVHFWG $\equiv 1-\frac{1}{\lambda }$, rendering other ${\gamma }_j^L$ values ineffective. In such a case, the value of IIVHFWG is fixed. If both IIVHFWG${}_1$ and IIVHFWG${}_2$ contain ${\tilde{\gamma }}_j^L=1-\frac{1}{\lambda }$, then IIVHFWG${}_1$ = IIVHFWG${}_2$ and their rank cannot be distinguished accurately.

If ${\gamma }^L=1-\frac{1}{\lambda }$, then the IIVHFWG operator and the IVHFWG operator are equivalent, and $1-\frac{1}{\lambda }\in [0,1]$, indicating that $\lambda \in [1,\infty )$. When $\lambda <1$, then $1-\lambda (1-{\gamma }_j^L)\ne 0$, and consequently, IIVHFWG $\ne 1-\frac{1}{\lambda }$. Therefore, this paper considers the interval of values for $\lambda$ to be (0, 1), allowing the IIVHFWG operator to avoid the disadvantages of the above operators. The proofs for other operators are similar.

Using the developed MCDM method, a ranking for Example 13, a ranking for Example 14, and a ranking for Example 15 are generated as $Y_2\succ Y_1\succ Y_3\succ Y_4$, $Y_3\succ Y_1\succ Y_2$, and $Y_2\succ Y_3\succ Y_1$, respectively. It can be seen that the developed MCDM method does not have the limitations illustrated in the previous section.

6. Validation of the Developed Method

In this section, we validate the developed method via quantitative comparisons based on two application examples.

Example 16. 

An investment institution is considering investing in one out of four wind power projects but being unsure about how to evaluate the differences between them. To this end, four criteria are used to evaluate the different wind power projects: power generation ($G_1$), return on investment ($G_2$), environmental impact ($G_3$), and technical feasibility ($G_4$). A decision matrix $\tilde{R}={\left(\tilde{r}ij\right)}_{4\times 4}$ is constructed using IVHFEs and the weights of the criteria are assigned as ${\omega }_1=0.25$, ${\omega }_2=0.26$, ${\omega }_3=0.23$, and ${\omega }_4=0.26$. The decision matrix is $\widetilde{R_4}$.

$$\widetilde{R_4}=\begin{array}{c}\\ Y_1\\ Y_2\\ Y_3\\ Y_4\end{array}\begin{array}{cccccc}& G_1& G_2& G_3& G_4& \\ (& \begin{array}{c}\left\{\right[0.7,0.8],[0.4,0.5\left]\right\}\\ \left\{\right[0.2,0.4],[0.6,0.8\left]\right\}\\ \left\{\right[0.4,0.6],[0.2,0.5\left]\right\}\\ \left\{\right[0.1,0.4],[0.7,0.8\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.3,0.5],[0.6,0.7\left]\right\}\\ \left\{\right[0,0.4],[0.3,0.6\left]\right\}\\ \left\{\right[0.6,0.8],[0.2,0.4\left]\right\}\\ \left\{\right[0,0.4\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.5,0.8],[0.7,0.8\left]\right\}\\ \left\{\right[0.3,0.6],[0.5,0.6\left]\right\}\\ \left\{\right[0.3,0.7],[0.6,0.7\left]\right\}\\ \left\{\right[0.1,0.6],[0.5,0.6\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.7,0.8],[0.2,0.9\left]\right\}\\ \left\{\right[0.5,0.7],[0.3,0.5\left]\right\}\\ \left\{\right[0.8,0.8\left]\right\}\\ \left\{\right[0.4,0.7],[0.3,0.5\left]\right\}\end{array}& )\end{array}$$

We apply the developed method to address the following example:

Step 1: 

Considering that all the criteria $G_j(j=1,2,3,4)$ are of the benefit type, the performance values of the alternatives $Y_i(i=1,2,3,4)$ do not require normalization.

Step 2: 

By applying the IIVHFWG operator, we obtain ${\tilde{a}}_1,{\tilde{a}}_2,{\tilde{a}}_3,{\tilde{a}}_4$. Considering the page size, the aggregated result value is no longer displayed.

Step 3: 

According to Definitions 4 and 5, we obtain $s\left({\tilde{a}}_1\right)=0.5947,s\left({\tilde{a}}_2\right)=0.3369,s\left({\tilde{a}}_3\right)=0.5428$, $s\left({\tilde{a}}_4\right)=0.3245$, and $\nu \left({\tilde{a}}_1\right)=0.1223,\nu \left({\tilde{a}}_2\right)=0.196,\nu \left({\tilde{a}}_3\right)=0.1104,\nu \left({\tilde{a}}_4\right)=0.2084$.

Step 4: 

Based on Definition 6, we obtain a ranking $Y_1\succ Y_3\succ Y_2\succ Y_4$.

Using other three MCDM methods in Table 1, the example can also be addressed and the results are listed in Table 1. It can be seen that the best alternative of the developed method is the same as that of other methods (except for Zhang and Wu’s method using the IVHFHWG operator, Li and Peng’s MCDM method). This demonstrates the effectiveness of the developed method.

Table 1. Rankings of the alternatives generated by different MCDM methods for Example 16.

MCDM Method	Generated Preference
Zhang and Wu’s method [38] using the IVHFHWG operator	$Y_1\succ Y_3\succ Y_2=Y_4$
Zhang and Wu’s method [38] using the IVHFHWA operator	$Y_1\succ Y_3\succ Y_2\succ Y_4$
Kahraman et al.’s MCDM method [1]	$Y_1\succ Y_3\succ Y_2\succ Y_4$
Li and Peng’s MCDM method [39] using the IVHFWG${}_{Hamacher}$ ($\eta =1$) operator	$Y_1\succ Y_3\succ Y_2=Y_4$
The proposed MCDM method	$Y_1\succ Y_3\succ Y_2\succ Y_4$

Example 17. 

There is an energy company that needs to choose a new electricity generation technology to meet future energy demands. There are three candidate power generation technologies: solar photovoltaic ($Y_1$), wind power ($Y_2$), and coal-fired power ($Y_3$), and there are four key attributes to assess their merits. These four attributes are generation efficiency ($G_1$), reliability ($G_2$), environmental impact ($G_3$), and cost ($G_4$). Assume the corresponding weights are ${\omega }_1=0.29$, ${\omega }_2=0.38$, ${\omega }_3=0.33$ and the decision matrix is $\widetilde{R_5}$.

$$\widetilde{R_5}=\begin{array}{c}\\ Y_1\\ Y_2\\ Y_3\end{array}\begin{array}{cccccc}& G_1& G_2& G_3& G_4& \\ (& \begin{array}{c}\left\{\right[0.3,1],[0,0.4\left]\right\}\\ \left\{\right[0,0.5],[0.1,0.3\left]\right\}\\ \left\{\right[0.5,1],[0,0.6\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.6,0.7],[0.5,0.6\left]\right\}\\ \left\{\right[0.3,0.5],[0.2,0.7\left]\right\}\\ \left\{\right[0.3,0.5],[0.5,0.8\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.7,1],[0.5,0.7\left]\right\}\\ \left\{\right[0.5,0.7],[0.4,0.9\left]\right\}\\ \left\{\right[0.5,1],[0.5,0.9\left]\right\}\end{array}& \begin{array}{c}\left\{\right[0.5,0.8],[0.7,0.8\left]\right\}\\ \left\{\right[0.7,0.8],[0.3,0.8\left]\right\}\\ \left\{\right[0.4,0.6],[0.3,0.8\left]\right\}\end{array}& )\end{array}$$

We apply the proposed MCDM method to address the following example:

Step 1: 

Considering that all the criteria $G_j(j=1,2,3,4)$ are of the benefit type, the performance values of the alternatives $Y_i(i=1,2,3)$ do not require normalization.

Step 2: 

By applying the IIVHFWG operator, we obtain ${\tilde{a}}_1,$ ${\tilde{a}}_2$ and ${\tilde{a}}_3$. Considering the page size, the aggregated result value is no longer displayed.

Step 3: 

According to Definitions 4 and 5, we obtain $s\left({\tilde{a}}_1\right)=0.5147,s\left({\tilde{a}}_2\right)=0.3764,s\left({\tilde{a}}_3\right)=0.535$ and $\nu \left({\tilde{a}}_1\right)=0.1938,\nu \left({\tilde{a}}_2\right)=0.2039,$ $\nu \left({\tilde{a}}_3\right)=0.2375$.

Step 4: 

Based on Definition 6, we obtain a ranking $Y_3\succ Y_1\succ Y_2$.

Using other three MCDM methods in Table 2, the example can also be addressed and the results are listed in Table 2. It can be seen that the best alternative of the proposed method is the same as that of other methods (except for Zhang and Wu’s method using the IVHFHWA operator, Li and Peng’s MCDM method). This also demonstrates the effectiveness of the developed method.

Table 2. Rankings of the alternatives generated by different MCDM methods for Example 17.

MCDM Method	Generated Ranking
Zhang and Wu’s method [38] using the IVHFHWG operator	$Y_3\succ Y_1\succ Y_2$
Zhang and Wu’s method [38] using the IVHFHWA operator	$Y_1\succ Y_3\succ Y_2$
Li and Peng’s MCDM method [39] using the IVHFWA${}_{Hamacher}$ ($\eta =1$) operator	$Y_1\succ Y_3\succ Y_2$
Kahraman et al.’s MCDM method [1]	$Y_3\succ Y_1\succ Y_2$
The proposed MCDM method	$Y_3\succ Y_1\succ Y_2$

7. Conclusions

In this paper, an improved interval-valued hesitant fuzzy weighted geometric operator for multi-criterion decision-making is presented. First, the limitations of six existing interval-valued hesitant fuzzy aggregation operators are illustrated via numerical examples. Aiming at these limitations, an improved interval-valued hesitant fuzzy weighted geometric operator is then constructed using two new operation laws. The constructed operator is shown to be free of the limitations. After that, the shortcomings of an existing method are discussed via numerical examples and a new method based on the constructed operator is developed. As illustrated via the same numerical examples, the established method can overcome the shortcomings. Finally, two application examples are presented and quantitatively compared to showcase the efficacy of the implemented approach. The illustration and demonstration results suggest that the proposed approach is as effective as some existing methods while being free of their limitations. The A-IVHFWG operator was proposed [38] and the cases wherein $g\left(t\right)$ equals $-log\left(t\right)$, $log\left(\frac{2-t}{t}\right)$, and $log\left(\frac{\theta +(1-\theta )t}{t}\right)$ were discussed. However, this paper only presents the specific form of the operator for the case of $g\left(t\right)$ = $-log\left(t\right)$ and does not discuss the cases of $g\left(t\right)=log\left(\frac{2-t}{t}\right)$ and $g\left(t\right)=log\left(\frac{\theta +(1-\theta )t}{t}\right)$. Future work will focus on further refining the A-IIVHFWG operator and providing specific forms of the operator for $g\left(t\right)=log\left(\frac{2-t}{t}\right)$ and $g\left(t\right)=log\left(\frac{\theta +(1-\theta )t}{t}\right)$. These operators will be applied to multi-criterion decision-making problems.