Hesitant Fuzzy Multi-Granulation Rough Set Model Based on Similarity Assessment

Abstract

A novel multiple attribute group decision-making (MAGDM) model is introduced in this study, utilizing a diversified hesitant fuzzy multi-granulation information system to address challenges in incomplete information settings. The analysis commences with an exploration of hesitant fuzzy sets and multi-granulation approximation. Subsequently, the integration of cumulative prospect theory into Technique for Order of Preference by Similarity to Ideal Solution (TOPSIS) within a hesitant fuzzy framework is discussed, emphasizing the incorporation of a risk preference coefficient in GDM to enhance individual assessments. The model proposes a systematic approach to address various MAGDM scenarios under hesitant fuzzy conditions. An illustrative case study on resource-sharing is provided to demonstrate the efficacy of the diversified MAGDM model, with evaluation outcomes expressed using hesitant fuzzy elements, offering valuable insights into group decision-making in hesitant fuzzy environments.

1. Introduction

1.1. Literature Review

Individuals often weigh various conditional attributes and costs when making decisions in their daily lives. Due to inherent uncertainty, individual decision-making may be ineffective in practice. In contrast, group decision-making (GDM) merges individual opinions to reach a consensus that satisfies all experts. In three-way decision-making, costs and losses are factored in for comparison, leading to the ranking of alternatives based on scores [1]. The aim is to arrive at a decision with minimal cost [2]. Since the rough set and fuzzy set have been introduced into decision-making, interval-valued fuzzy rough sets [3], interval-valued hesitant fuzzy sets [4], intuitionistic hesitant fuzzy connection sets [5], double-quantitative multi-granularity kernel fuzzy rough sets [6], probabilistic hesitant fuzzy sets (PHFS) [7,8], and type-2 fuzzy sets have been proposed [9], which are applied to green supplier selection [10,11], risk assessment decision [12,13], and medical diagnosis [14]. Decision-making processes are influenced by multiple uncertain factors, for example, the availability of shared parking spaces can be impacted by variables like surrounding traffic and holiday events. Traditional fuzzy sets, hesitant fuzzy sets (HFS), and multi-granulation rough sets (MGRS) may not adequately address such scenarios. By leveraging hesitant fuzzy sets and rough sets to manage uncertainties, the hesitant fuzzy multi-granulation rough-set model can effectively handle intricate uncertain information [15]. Multiple important issues arise in multiple attribute group decision-making (MAGDM) problems, including in the consensus reaching process [16], optimal alternative selection [17], and evaluation within incomplete information systems [18]. Three key research directions include aggregation operators [19], information fusion through granular computing, and distance measures for alternatives using fuzzy relations such as probabilistic hesitant fuzzy preference relation, interval-valued fuzzy preference relation, and similarity relation [20]. Additionally, various MAGDM models have been developed for uncertain environments where decision-makers struggle to effectively express preferences across all attributes [21]. Recent advancements in MAGDM encompass the integration of soft expert sets into dual hesitant fuzzy sets [22,23] and the application of formal concept analysis in fuzzy sets [24,25]. Decision-makers exhibit diverse preferences, leading to the emergence of the diversified MAGDM problem when considering varied attribute sets [26], such as the three-way MAGDM method with fuzzy sets, variable precision fuzzy rough sets [13], multi-granulation fuzzy rough sets [27], and multi-granulation soft fuzzy rough sets [28].

1.2. Research Gap and Motivation

There are several challenges in the process of multi-granularity hesitant fuzzy decision-making. The hesitance degree and the psychological behavior preferences of decision-makers are rarely considered in GDM. The hesitancy degree measure and variance measure are rarely considered in distance measurements between hesitant fuzzy elements. The risk coefficient of experts may have an impact on decision-making outcomes. However, there are relatively few studies on the combination of experts’ risk attitudes and psychological behaviors. Therefore, it is necessary to take into account the psychological behavior of experts in MAGDM.

Experts may provide overestimations or underestimations when dealing with hesitant fuzzy values, the fairness of experts is rarely taken into account in the decision-making process.

The evaluation of all attributes by experts may not capture the distinctiveness of individual experts in their respective professional domains, potentially leading to inappropriate judgments and overlooking fairness considerations in evaluations. In the context of MAGDM based on a diversified hesitant fuzzy multi-granulation rough set (DHFMGRS), the issue is delineated from two perspectives: firstly, experts can only offer comments on specified attributes, and secondly, the hesitance and risk appetite of experts need to be taken into account, especially in extreme cases where various factors are weighed in the applicant-ranking process.

1.3. Contribution and Paper Organization

The contributions of the proposed work are as follows:

A multi-granularity hesitant fuzzy set is combined with cumulative prospect theory to consider the influence of hesitance degree in the hesitant fuzzy decision-making environment. The uncertainty, hesitance, and psychological behavior preferences of decision-makers in the decision-making process are analyzed. DHFMGRS adeptly manages uncertainties stemming from multiple sources (granularities) and inherent hesitant fuzziness within each source. DHFMGRS facilitates the resolution of intricate decision-making problems characterized by fragmented information and uncertainty; the combination of DHFMGRS and TOPSIS can better handle group decision-making problems in hesitant and fuzzy environments. The influence of sensitivity parameters and expert risk coefficients on decision-making results are analyzed. When decision-makers hold a neutral attitude and the sensitivity coefficient is greater than 0.88, the ranking tends to stabilize.

Decision-makers only select certain attributes that they are familiar with to articulate their opinions, using an inconsistency evaluation index based on diverse attributes. Thus, decision-makers only evaluate professional fields rather than all attributes; the issue of fairness is taken into account in the decision-making process. To mitigate experts’ tendencies towards the overestimation or underestimation of applicants, the deviation between individual and collective opinions is assessed, along with each expert’s evaluation variance for a given alternative. The subsequent sections are structured as follows: Section 2 reviews the mathematical foundations. Section 3 introduces the MAGDM based on the DHFMGRS over two universes by integrating hesitant fuzzy sets with multi-granularity rough sets. Section 4 and Section 5 present a novel methodology and an illustrative example for addressing the MAGDM problem. Section 6 comprises our discussions and recommendations, and Section 7 offers our conclusions.

2. Mathematical Foundation

This section introduces the mathematical foundation of decision-making based on hesitant fuzzy sets, along with its fundamental operations. DHFMGRS is designed to handle scenarios involving conflicting or ambiguous information provided by experts. Various DHFMGRS models can be developed using variable-precision approaches, enabling decision-makers to choose the most appropriate model for particular applications and risk-tolerance levels. DHFMGRS proves to be especially beneficial for MAGDM issues, where distance and similarity measures are introduced into MAGDM [29] and diverse hesitant fuzzy evaluations are offered by multiple experts. A new model incorporating a risk coefficient is devised to counteract individual biases and ensure a balanced evaluation process among experts [30]. The DHFMGRS framework adeptly synthesizes these evaluations to facilitate consensus decision-making.

Definition 1. 

A hesitant fuzzy set (HFS) defined on the finite set $U$ can be expressed as $E=\{<x,h_E(x)>|x\in U\}$. The membership degree $h_E(x)\in [0,1]$ is the hesitant fuzzy element (HFE) of $E$, which represents the possibility that the element $x\in U$ belongs to the set $E$. $H=\cup h_E(x)$ is the set of all HFEs of $E$.

The definitions of typical score of HFE are presented below.

Definition 2

([31]). Let $h={\cup }_{\gamma \in h}\{\gamma \}={\left\{{\gamma }_j\right\}}_{j=1}^{l(h)}$ be a HFE, where $l(h)$ is the length of $h$; the score and the variance function of $h$ is presented below.

$$s(h)={\displaystyle \frac{1}{l(h)}}{\displaystyle \sum _{\gamma \in h}\gamma }$$

(1)

$$v(h)=\sqrt{{\displaystyle \frac{1}{l(h)}}{(\gamma -s(h))}^2}$$

(2)

Farhadinia [32] defines the score function as below:

$$s(h)={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^{l(h)}\delta (j){\gamma }_j}}{{\displaystyle {\sum }_{j=1}^{l(h)}\delta (j)}}}$$

(3)

Let $h\subseteq {\mathrm{{\rm I}}}^U([0,1])$, ${\mathrm{{\rm I}}}^{\cup }([0,1])$ be the subset consisting of all finite unions in the interval $[0,1]$, when $a_1,\dots ,a_n\in [0,1]$, $b_1,\dots ,b_n\in [0,1]$, and $b_i\le a_{i+1}$; thus, $h$ is the union of $l(h)$ intervals with extremes $a_i$ and $b_i$. Alcantud [33] extends the typical scores as below:

$$s_{XX}(h)={\displaystyle \frac{1}{2l(h)}}{\displaystyle {\sum }_{j=1}^{l(h)}(a_j}+b_j)$$

(4)

$$S_F(h)={\displaystyle \frac{{\displaystyle {\sum }_{j=1}^{l(h)}\delta (j)(a_j+b_j)/2}}{{\displaystyle {\sum }_{j=1}^{l(h)}\delta (j)}}}$$

(5)

Definition 3

([34]). Let $h$ be an HFS on $U=\{x_1,x_2,\dots x_m\}$, and for any $x_i\in U$, $l(h(x_i))$ be the length of $h(x_i)$. The hesitance degree of $h(x_i)$ and $h$ are expressed as follows, respectively.

$$u(h(x_i))=1-{\displaystyle \frac{1}{l(h(x_i))}}$$

(6)

$$u(h)={\displaystyle \frac{1}{m}}{\displaystyle \sum _{i=1}^{m}u(h(x_i)})$$

(7)

Definition 4

([35]). Given any two HFSs $h_1,h_2$ on $U=\{x_1,x_2,\dots x_m\}$, the normalized generalized distance measures distance between $h_1(x)$ and $h_2(x)$ is below:

$${\zeta }_h(h_1,h_2)={\left[{\displaystyle \frac{1}{2m}}{\displaystyle \sum _{i=1}^m\left(|u(h_1(x_i)-u(h_2(x_i)){|}^{\sigma }+\frac{1}{l_{x_i}}{\displaystyle \sum _{\gamma =1}^{l_{x_i}}|h_1^{\gamma }(x_i)-h_2^{\gamma }(x_i){|}^{\sigma }}\right)}\right]}^{1/\sigma }$$

(8)

where $\gamma >0$, $l_{x_i}=\max \{l{h}_1(x_i),l{h}_2(x_i)\}$. $h_1^{\gamma }(x_i)$ is the $\gamma \hspace{0.17em}th$ value in $h_1(x_i)$ and $h_2^{\gamma }(x_i)$ is the $\gamma \hspace{0.33em}th$ value in $h_2(x_i)$. $\left|u\right(h_1(x_i)-u(h_2(x_i))|$ is the hesitance degree measures.

Definition 5.

Based on Reference [36], the new normalized generalized distance measures between hesitant fuzzy elements $h_1(x)$ and $h_2(x)$, which is listed below:

$${\zeta }_n(h_1,h_2)={\{{\displaystyle \frac{1}{3}}[|u(h_1(x_i))-u(h_2(x_i)){|}^{\sigma }+|s(h_1(x_i))-s(h_2(x_i)){|}^{\sigma }+|v(h_1(x_i))-v(h_2(x_i)){|}^{\sigma }]\}}^{1/\sigma }$$

(9)

where $\left|u\right(h_1(x_i))-u(h_2(x_i))|$, $\left|s\right(h_1(x_i))-s(h_2(x_i))|$ and $\left|v\right(h_1(x_i))-v(h_2(x_i))|$ are hesitance degree measures, score measures, and variance measures, respectively.

In Formula (9), the weighted normalized generalized distance measures between $h_1(x)$ and $h_2(x)$ are calculated:

$${\zeta }_w(h_1,h_2)={\displaystyle {\sum }_{i=1}^m{\omega }_i}{\zeta }_n(h_1,h_2)$$

(10)

Definition 6.

The weighted normalized generalized distance-based similarity degree of $h_1(x)$ and $h_2(x)$ is defined below:

$$si{m}_w(h_1,h_2)=\left\{\begin{array}{l}0,\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}{h}_1(x)=h_2(x)\\ 1-\left|{\zeta }_w\right(h_1,h_2)|,h_1(x)\ne h_2(x)\end{array}\right.$$

(11)

Definition 7.

Given any two HFEs, $h_i(x)=\{h_i^{\gamma }|\gamma =1,2,\dots ,l\},i=1,2,\gamma >0$, the basic operation rules are listed as follows:

$$h_1\oplus h_2=\underset{{\gamma }_1^l\in h_1,{\gamma }_2^k\in h_2}{\cup }\{{\gamma }_1^l+{\gamma }_2^k-{\gamma }_1^l{\gamma }_2^k\}$$

$$h_1\otimes h_2=\underset{{\gamma }_1^l\in h_1,{\gamma }_2^k\in h_2}{\cup }\{{\gamma }_1^l{\gamma }_2^k\}$$

$$\lambda h=\underset{{\gamma }^l\in h}{\cup }\{[1-{(1-{\gamma }^l)}^{\lambda }]\}$$

$$h^{\lambda }=\underset{{\gamma }^l\in h}{\cup }{\{{\gamma }^l\}}^{\lambda }$$

$$h^c=\underset{{\gamma }^l\in h}{\cup }\{1-{\gamma }^l\}$$

$$h_1\cup h_2=\underset{{\gamma }_1^l\in h_1,{\gamma }_2^k\in h_2}{\cup }\max \{{\gamma }_1^l,{\gamma }_2^k\}$$

$$h_1\cap h_2=\underset{{\gamma }_1^l\in h_1,{\gamma }_2^k\in h_2}{\cap }\min \{{\gamma }_1^l,{\gamma }_2^k\}$$

where $h^c$ is the complement of $h$.

Definition 8. 

A HF information system is denoted as $I=\{U,C\cup d,V,f\}$, where $U$ is a non-empty finite universe; $C$ is a condition attribute set; $d$ is a decision attribute set, $d\in D$; and $V=V_C\cup V_d$ is the domain of all attributes. For each $x\in U$ and $c\in C$, $f(x,c)$ is an HFE, representing possible values between $x$ and $c$.

3. The MAGDM Based on a Diversified Multiple-Attribute Hesitant Fuzzy Information System

Parameters and definitions are presented in

Table 1. Parameters and definitions.

Parameters	Definitions
$U$	Non-empty finite universes
$C$	A condition attribute set
$D$	A decision attribute set
$A$	Decision-making object
$k$	The serial number of the decision-maker
$\omega$	The weight of attributes
$\zeta$	Distance measure
$R_k$	The binary hesitant fuzzy relation between $U$ and $C$
$\alpha ,\beta$	Sensitivity coefficient, $\alpha =\beta =0.88$
$\lambda$	Loss aversion coefficient, $\lambda =2.25$
$\chi$	Risk preference coefficient, $\chi =0.5$
$\sigma$	Priority order of alternative
$u$	The weight of the experts
$\epsilon$	Consensus threshold, $\epsilon =0.85$
$\iota$	The threshold for assessment, $\iota =0.15$

.

The fuzzy rough set is the generalization of the Pawlak rough set in which an equivalence relation is replaced by a fuzzy similarity relation. The fuzzy rough set focuses on the rough approximation of fuzzy decision-making objects over the universe of discourse. The multi-granulation rough set was proposed and established by two equivalence relations of the universe, which is the generalization of the rough set and extends the single binary relation to multiple binary relations over the universe of discourse. The multi-granulation hesitant fuzzy rough sets are problems from rough approximations of a hesitant fuzzy set based on multiple classical equivalence relations, and are applied to deal with MAGDM problems in hesitant fuzzy environments. Experts may hesitate between positive and negative when they are undergoing an important decision. Consequently, it is essential to combine the MAGDM and hesitant fuzzy set; thus, we establish the diversified multiple-attribute hesitant fuzzy decision-making information system in Section 3.1.

3.1. Diversified Multiple Attribute Hesitant Fuzzy Information System

Let six-tuple $(U,C,D,F,\psi ,\mu )$ be a diversified attribute hesitant fuzzy decision-making information systems, where $U=\{x_1,x_2,\dots ,x_m\}$ is a decision-making object, $C$ is an attribute set, $D$ is a decision-maker’s set, and $\mu$ is a decision-makers weight vector, where $C=(c_1,c_2,\dots ,c_n)$, $D=\{d_1,d_2,\dots ,d_l\}$, and $\mu =\{{\mu }_1,{\mu }_2,\dots ,{\mu }_l\}$ are satisfied with $0\le {\mu }_k\le 1$ and ${\displaystyle {\sum }_{k=1}^l{\mu }_k}=1$. $c_{{\Psi }_k}$ is the selected attribute set of the decision-maker $k$$(k=1,2,\dots ,l)$, i.e., $c^k=\{c_1^k,\dots ,c_{{\psi }_k}^k\}$ for any decision-maker $k$. $F$ is the set of mappings from $U$ to $c^k(k=1,2,\dots ,l)$, $F=\{f_k(x_i,c^k)|i=1,2,\dots ,m;c^k=(c_1^k,\dots ,c_{{\psi }_k}^k);k=1,2,\dots ,l\}$, where $f_k:U\times c^k\to [0,1]\hspace{0.33em}(c^k\subseteq C,k=1,2,\dots ,l)$. $f_k$ is the hesitant fuzzy evaluation results given by the $kth$$(k=1,2,\dots ,l)$ decision-maker corresponding to the attribute subset $c^k$.

Definition 9.

Let $(U,C,D,F,\psi ,\mu )$ be the diversified-attribute hesitant fuzzy decision-making information systems. $x_i,x_{i’}\in U$, $(x_i,x_{i’}=1,2,\dots m)$, $c_h^k\in c^k$, $d_k\subseteq D$$(k=1,2,\dots ,l)$. By Formula (10), ${\zeta }_h^k(x_i,x_{i’})$ be the distance measure between alternatives $x_i$ and $x_{i’}$ on attribute $c_h^k\in c^k$, ${\omega }_h^k={\displaystyle \frac{{\displaystyle {\sum }_{i’}{\displaystyle {\sum }_i{\zeta }_h^k(x_i,x_{i’})}}}{{\displaystyle {\sum }_{h\in {\psi }_k}{\displaystyle {\sum }_{i’}{\displaystyle {\sum }_i{\zeta }_h^k(x_i,x_{i’})}}}}}$ be the weight of the attribute $c_h^k$ over the attribute set $c^k$ with the hesitant fuzzy assessment by the expert $k$. The weighted distance measures between alternatives $x_i$ and $x_{i’}$ on the basis of Definition 5 are presented below.

$${\zeta }_{c^h}^k(x_i,x_{i’})={\displaystyle \sum _{h={\psi }_1}^{{\psi }_k}{\omega }_h^k|f_k(c_h^k(x_i))-f_k(c_h^k(x_{i’}))|}$$

(12)

In the MAGDM problem, let $U=\{x_1,x_2,\dots ,x_m\}$, $D=\{d_1,d_2,\dots ,d_l\}$, and $c^k=\{c_1^k,\dots ,c_{{\psi }_k}^k\}$; the diversified-attribute hesitant fuzzy evaluation from different experts is shown in

Table 2. Diversified-attribute hesitant fuzzy evaluation from different experts.

	${\mathit{c}}_1^1$	$\dots$	${\mathit{c}}_{{\mathit{\psi }}_1}^1$	$\dots$	${\mathit{c}}_1^{\mathit{k}}$	$\dots$	${\mathit{c}}_{{\mathit{\psi }}_{\mathit{k}}}^{\mathit{k}}$
$x_1$	(0.2, 0.3)	$\dots$	(0.1, 0.2)	$\dots$	(0.6, 0.8)	$\dots$	(0.4, 0.5)
$x_2$	(0.1, 0.3)	$\dots$	(0.1, 0.2)	$\dots$	(0.4, 0.5)	$\dots$	(0.6, 0.8)
$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$	$\dots$
$x_n$	(0.3, 0.6)	$\dots$	(0.3, 0.5)	$\dots$	(0.6, 0.8)	$\dots$	(0.5, 0.7)

.

Definition 10.

The value functions of benefits and losses of alternative $i$ in the cumulative prospect theory are defined as follows.

$$v_{ij}(\zeta )=\left\{\begin{array}{l}\zeta {(h_{ij},h_j^{-})}^{\alpha }\hspace{1em}\hspace{1em}\hspace{1em}\zeta (h_{ij},h_j^{-})\ge 0\\ -\lambda \zeta {(h_{ij},h_j^{+})}^{\beta }\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\zeta (h_{ij},h_j^{+})<0\end{array}\right.$$

(13)

where $\alpha$ and $\beta$ are sensitivity coefficients and $\lambda$ is loss aversion coefficient. $\alpha =\beta =0.88$, $\lambda =2.25$.

The comprehensive cumulative benefit and loss of alternative $i$ with respect to expert $k$ is below.

$$V_{i,k}^{+}(\zeta )={\displaystyle {\sum }_{c_h^k\in c^k}{v}_{i{c}_h^k}^{+}(\zeta ){\Pi }^{+}({\omega }_h^k)}$$

(14)

$$V_{i,k}^{-}(\zeta )={\displaystyle {\sum }_{c_h^k\in c^k}{v}_{i{c}_h^k}^{-}(\zeta ){\Pi }^{-}({\omega }_h^k)}$$

(15)

where

$${\Pi }^{+}({\omega }_h^k)={\displaystyle \frac{{({\omega }_h^k)}^{0.61}}{{({({\omega }_h^k)}^{0.61}+{(1-{\omega }_h^k)}^{0.61})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$0.61$}\right.}}}\hspace{1em}\hspace{1em}\hspace{1em},\hspace{1em}{\Pi }^{-}({\omega }_h^k)={\displaystyle \frac{{({\omega }_h^k)}^{0.69}}{{({({\omega }_h^k)}^{0.69}+{(1-{\omega }_h^k)}^{0.69})}^{\raisebox{1ex}{$1$}\!\left/ \!\raisebox{-1ex}{$0.69$}\right.}}}\hspace{1em}\hspace{1em}\hspace{1em}$$

Definition 11.

The total satisfaction based on expert k is presented below.

$$s_i^k={\displaystyle \frac{(1-\chi )V_{i,k}^{+}}{(1-\chi )V_{i,k}^{+}+\chi V_{i,k}^{-}}}$$

(16)

where $\chi$ is the risk preference coefficient, $\chi <0.5$ means that the decision-maker is risk-averse, and $\chi >0.5$ means that the decision-maker is risk-seeking.

The total satisfaction based on experts’ preferences is presented below.

$$S_i=u_k{s}_i^k$$

(17)

The best alternative $x_i$ is obtained by the highest satisfaction of decision-makers.

3.2. The Ranking of MAGDM in Hesitant Fuzzy Environments

Let $R_k(x_i,c^k)$ be the binary hesitant fuzzy relation between universe $U$ and $c^k$, Then, six-tuple $(U,C,D,F,\psi ,\mu )$ be a diversified-attribute hesitant fuzzy decision-making information system, where $U=\{x_1,x_2,\dots ,x_m\}$ is a decision-making object and $C=(c_1,c_2,\dots ,c_n)$ is an attribute set; the attribute set $c^k\subseteq C$ is selected by the decision-maker $k$ $(k=1,2,\dots ,l)$, $D=\{d_1,d_2,\dots ,d_l\}$ is a decision attribute set, and $F$ is the set of mapping from $U$ to $c^k$. ${\mu }_k$ is the weight of the decision-maker $k$, which satisfies $0\le {\mu }_k\le 1$ and ${\displaystyle {\sum }_{k=1}^l{\mu }_k}=1$. Given $A\in F(U)$, the binary hesitant fuzzy relation between universe $U$ and $c^k$ is expressed below:

$$R_k(x_i,c^k)={\displaystyle \frac{r_k(x_i,c_1^k)}{c_1^k}}+{\displaystyle \frac{r_k(x_i,c_2^k)}{c_2^k}}+\dots +{\displaystyle \frac{r_k(x_i,c_{{\psi }_k}^k)}{c_{{\psi }_k}^k}},(i=1,2,\dots ,m,c^k=\{c_1^k,\dots ,c_{{\psi }_k}^k\})$$

(18)

where $r_k(x_i)(c_{{\psi }_k}^k)$ is a hesitant fuzzy set, representing the possible membership degrees between $x_i$ and $c_{{\psi }_k}^k$. When $c_1^k=c_2^k=\dots =c_{{\psi }_k}^k$, the binary hesitant fuzzy classes $R_k(x_i,c^k)$ of object $x_i$ will degenerate into hesitant fuzzy relation $R$ between $U$ and $c^k$. Given $A\in F(U)$, the lower and upper approximations of the hesitant fuzzy decision-making object $A$ with respect to expert k can be expressed in another way:

$$\underset{¯}{R_k}(A)(x_i)=\underset{c_j\in C}{\inf }S(N(R_k(x_i)(c_{{\psi }_k}^k)),A(c_j))$$

(19)

$$\overline{R_k}(A)(x_i)=\underset{c_j\in C}{\sup }T(R_k(x_i)(c_{{\psi }_k}^k),A(c_j))$$

(20)

Hesitant fuzzy relation $R$ degenerates into fuzzy relation, which is the diversified multiple fuzzy decision-making information system over two universes in Reference [27].

When ${\mu }_1={\mu }_2=\dots ={\mu }_l$, we obtain the following equations, which are the multiple attribute group decision-making information system with respect to $(U,C,D,F,\psi ,\mu )$ proposed in Reference [28].

$$\underset{¯}{{\displaystyle {\sum }_{k=1}^l{R}_k}}(A)(x_i)={\displaystyle \sum _{k=1}^l\underset{c_j\in C}{\inf }\left[S(N(R_k(x_i)(c_{{\psi }_k}^k)),A(c_j))\right]}$$

$$\overline{{\displaystyle {\sum }_{k=1}^l{R}_k}}(A)(x_i)={\displaystyle \sum _{k=1}^l\underset{c_j\in C}{\sup }\left[T(R_k(x_i)(c_{{\psi }_k}^k),A(c_j))\right]}$$

So, specifically, the fuzzy multi-granulation rough set is a special case of the hesitant fuzzy multi-granulation rough set over two universes. The ranking based on expert k is presented below.

$${\sigma }_{0.5}^k(x_i)=0.5\underset{¯}{R_k}(A)(x_i)+0.5\overline{R_k}(A)(x_i)$$

(21)

The total ranking based on all experts is presented below.

$${\sigma }_{0.5}(x_i)={\displaystyle \sum _{k=1}^l{\mu }_k{\delta }_{0.5}^k}(x_i)$$

(22)

4. Key Steps for the Solution of the Diversified MAGDM Problem

A triangular norm satisfies the boundary condition $S(x,1)=x$ for any $x\in [0,1]$. Meanwhile, a triangular co-norm satisfies the boundary condition $S(x,0)=x$ for any $x\in [0,1]$. The most common continuous triangular norms and triangular co-norms are presented below:

$$T_M(x^{(1)},x^{(2)})=\min (x^{(1)},x^{(2)});$$

$$S_M(x^{(1)},x^{(2)})=\max (x^{(1)},x^{(2)});$$

$$T_P(x^{(1)},x^{(2)})=x^{(1)}\ast x^{(2)};$$

$$S_P(x^{(1)},x^{(2)})=x^{(1)}+x^{(2)}-x^{(1)}\ast x^{(2)};$$

$$T_L(x^{(1)},x^{(2)})=\max \{0,x^{(1)}+x^{(2)}-1\};$$

$$S_L(x^{(1)},x^{(2)})=\min \{1,x^{(1)}+x^{(2)}\}.$$

$$T_{\epsilon }(x^{(1)},x^{(2)})={\displaystyle \frac{x^{(1)}\times x^{(2)}}{1+(1-x^{(1)})(1-x^{(2)})}}$$

$$S_{\epsilon }(x^{(1)},x^{(2)})={\displaystyle \frac{x^{(1)}+x^{(2)}}{1+x^{(1)}\times x^{(2)}}}$$

The key steps of hesitant fuzzy multi-granularity decision-making based on the diversified preference model are below:

Step 1. All alternatives are evaluated by decision-makers when decision-makers select parts of the attributes.

Step 2. The weight of attribute $c_h^k$ with respect to the attribute set $c^k$ is calculated by Definition 9.

Step 3. The positive ideal solution and negative ideal solution are denoted by $A^{+}$ and $A^{-}$; the cumulative prospect loss value $v_i^{-}(\zeta )$ and benefit value $v_i^{+}(\zeta )$ of each alternative with reference points are calculated by Formula (13).

Step 4. Calculate the comprehensive cumulative benefit $V_{i,k}^{+}$ and loss $V_{i,k}^{-}$ of each alternative $i$ with respect to expert $k$ by Formulas (14) and (15), respectively.

Step 5. The comprehensive satisfaction based on decision-makers’ preferences is calculated by Formula (17).

The flowchart of the proposed model is presented in Figure 1.

In Section 5, an example is given to illustrate the process of evaluation of shared parking lots based on the multiple-attribute hesitant fuzzy information system.

5. An Illustrative Example

This section presents an assessment case of shared parking lots, which is a hot issue in China. A parking lot is usually evaluated in order to carry out subsidies, penalties, and other measures in the early stage of shared parking. Therefore, three experts are invited to evaluate the safety, feasibility, convenience, profitability, environmental protection, and sustainability. Due to the uncertainty of government policies, the experts could not give an accurate assessment, but only give hesitant fuzzy information. Each expert only assesses a portion of the identified attributes and not others; the diversified MAGDM model is utilized to solve this problem.

Let $U=\{x_1,x_2,x_3,x_4\}$ be the set of possible constructors of applicants for the parking lots and $C=\{c_1,c_2\dots c_6\}$ be the set of attributes, where $c_1$ represents safety, $c_2$ represents feasibility, $c_3$ represents convenience, $c_4$ represents profitability, $c_5$ represents environmental protection, and $c_6$ represents sustainability. The first expert is a transportation planning specialist who only focuses on safety, feasibility, convenience, and profitability. The second expert is an engineering specialist who focuses solely on convenience, profitability, environmental protection, and sustainability. The third expert is a management specialist who focuses only on safety, feasibility, environmental protection, and sustainability. The experts assessed to what extent the four parking lots met the requirements of external sharing. The opinion weights of the experts are $u=\{u_1,u_2,u_3\}$ = $(0.3,0.3,0.4)$.

Experts give hesitant fuzzy information when they assess whether the shared parking lots meet the requirements of sharing or not [37,38]. It is assumed that only six factors, including safety, feasibility, convenience, profitability, environmental protection, and sustainability, are considered by the experts. Experts were required to make judgments on both working days and non-working days. A well-known direction of the MAGDM problem is to find an alternative who best meets the requirements of all of the experts; the key steps to solve the problem are presented below.

Step 1. The hesitant fuzzy relation between parking lots and attributes are represented in

Table 3. Hesitant fuzzy relation $R_1$ between shared parking lots and attributes.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_3$	${\mathit{c}}_4$
$x_1$	{0.6, 0.8}	{0.2, 0.25}	{0.2, 0.3}	{0.4, 0.5}
$x_2$	{0.5, 0.8}	{0.2, 0.4}	{0.2, 0.4}	{0.2, 0.8}
$x_3$	{0.4, 0.6}	{0.2, 0.5}	{0.2, 0.5}	{0.3, 0.5}
$x_4$	{0.6, 0.9}	{0.2, 0.4}	{0.5, 0.6}	{0.2, 0.6}

,

Table 4. Hesitant fuzzy relation $R_2$ between shared parking lots and attributes.

	${\mathit{c}}_3$	${\mathit{c}}_4$	${\mathit{c}}_5$	${\mathit{c}}_6$
$x_1$	{0.5}	{0.6, 0.8}	{0.1, 0.3}	{0.3, 0.5}
$x_2$	{0.2, 0.5}	{0.7, 0.9}	{0.3, 0.5}	{0.4, 0.5}
$x_3$	{0.2, 0.6}	{0.8}	{0.1, 0.5}	{0.1, 0.5}
$x_4$	{0.5}	{0.2, 0.5}	{0.4, 0.5}	{0.5, 0.6}

, and

Table 5. Hesitant fuzzy relation $R_3$ between shared parking lots and attributes.

	${\mathit{c}}_1$	${\mathit{c}}_2$	${\mathit{c}}_5$	${\mathit{c}}_6$
$x_1$	{0.5, 0.6}	{0.4, 0.5}	{0.5}	{0.4, 0.5}
$x_2$	{0.4, 0.5}	{0.4, 0.6}	{0.2, 0.5}	{0.4, 0.6}
$x_3$	{0.3, 0.5}	{0.5}	{0.5, 0.6}	{0.2, 0.3}
$x_4$	{0.5}	{0.4, 0.5}	{0.4, 0.6}	{0.2, 0.5}

, respectively. From Table 3, Table 4 and Table 5, it can be seen that the evaluations of experts on parts of attributes are similar, but asymmetrical.

Step 2. By Definition 9, the weight of attribute $c_h^k$ with respect to the attribute set $c^k$ is presented below.

$${\omega }^1=({\omega }_1^1,{\omega }_2^1,{\omega }_3^1,{\omega }_4^1)=(0.2384,0.1682,0.3031,0.2902)$$

$${\omega }^2=({\omega }_3^2,{\omega }_4^2,{\omega }_5^2,{\omega }_6^2)=(0.2890,0.3756,0.1713,0.1641)$$

$${\omega }^3=({\omega }_1^3,{\omega }_2^3,{\omega }_5^3,{\omega }_6^3)=(0.2573,0.2032,0.3041,0.2354)$$

Step 3. The positive and negative ideal solution are set; let $A^{+}=\{1,1,\dots ,1\}$, $A^{-}=\{0,0,\dots 0\}$. The value functions of benefit and loss of alternative $i$ with reference point are calculated, respectively. Take $v_{1,1}^{+}$ corresponding to expert 1 as an example.

$$v_{1,1}^{+}=\xi {(h_{11},h_1^{-})}^{0.88}=|0.733+0.1053+0.5{|}^{0.88}={1.338}^{0.88}=1.292$$

Step 4. The comprehensive cumulative prospect benefit and loss of each alternative corresponding to expert $k$ are calculated, respectively. Take $V_{1,1}^{+}$ corresponding to expert 1 as a example.

$$\begin{array}{l}{V}_{1,1}^{+}=\xi (h_{1j},h_j^{-})={\displaystyle \sum _{j=1}^4\xi {(h_{1j},h_j^{-})}^{0.88}{\Pi }^{+}({\omega }_j^1)}\\ ={1.338}^{0.88}\times 0.284+{0.765}^{0.88}\times 0.24+{0.822}^{0.88}\times 0.320+{1.022}^{0.88}\times 0.313=1.114\end{array}$$

Each alternative corresponding to expert $k$ are listed in

Table 6. Comprehensive gain value and loss value of each alternative with respect to expert 1.

	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$V_{i,1}^{+}$	1.114	1.317	1.238	1.307
$V_{i,1}^{-}$	2.936	3.038	3.026	2.815

,

Table 7. Comprehensive gain value and loss value of each alternative with respect to expert 2.

	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$V_{i,2}^{+}$	1.083	1.298	1.166	1.038
$V_{i,2}^{-}$	2.421	2.682	2.421	2.466

, and

Table 8. Comprehensive gain value and loss value of each alternative with respect to expert 3.

	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$V_{i,3}^{+}$	1.057	1.237	1.072	1.037
$V_{i,3}^{-}$	2.376	3.391	2.682	2.572

, respectively.

Step 5. Let $\chi =0.5$, $u=\{u_1,u_2,u_3\}=\{0.3,0.3,0.4\}$; the comprehensive satisfaction $S_i$ based on experts’ preferences is presented in

Table 9. Comprehensive satisfaction $S_i$ based on experts’ preferences.

	$\mathit{i}=1$	$\mathit{i}=2$	$\mathit{i}=3$	$\mathit{i}=4$
$s_i$	0.300	0.310	0.295	0.304

.

It is easy to find that the priority order of the four alternatives are arranged as $\sigma (x_2)\succ \sigma (x_4)\succ \sigma (x_1)\succ \sigma (x_3)$. When the risk coefficient changes, the obtained comprehensive satisfaction is different. The similarity of comprehensive satisfaction with the risk preference coefficient is shown in Figure 2.

When the sensitivity coefficient changes, the similarity of comprehensive satisfaction with the sensitivity coefficient is shown in Figure 3.

From Figure 3, when $\alpha =\beta \ge 0.88,$ the ranking based on experts’ preferences tend to be consistent, which are similar, but asymmetrical.

6. Discussion and Recommendations

6.1. Comparison with Other Methods

Experts were selected from the expert database; the mutual trust between experts is uncertain in most cases. However, experts are not interested in all attributes; each expert has their own preferences for the different attributes. For instance, $c_5$ and $c_6$ are missing in the decision-making of expert 1 in Table 3. Each decision-maker selects the attributes that they regard as the most valuable. When the opinion of the $kth$ decision-maker is missing on $c_j$, that is, the $kth$ decision-maker dismisses the individual preference opinion of an alternative $x_i\in U$ relative to $c_i\in C$, it is supposed that $A_k(c_j)=0.5$. In the same way,$A_1(c_5)=A_1(c_6)=A_2(c_1)=A_2(c_2)=A_3(c_3)=A_3(c_4)=0.5$. To verify the validity of the GDM approach based on the MGHFRS, the GDM object $A$ is presented below, which means that the alternative’s general performance is good.

$$A={\displaystyle \frac{(0.3,0.4)}{c_1}}+{\displaystyle \frac{(0.3,0.5)}{c_2}}+{\displaystyle \frac{(0.15,0.4)}{c_3}}+{\displaystyle \frac{(0.2,0.4)}{c_4}}+{\displaystyle \frac{(0.4,0.6)}{c_5}}+{\displaystyle \frac{(0.2,0.4)}{c_6}}$$

According to the evaluation of each expert, the lower and upper approximations of A with respect to $R_1$, $R_2$, and $R_3$ are presented below.

$$\mathrm{A}\underset{¯}{R_1}(A)={\displaystyle \frac{(0.3,04)}{x_1}}+{\displaystyle \frac{(0.2,0.5)}{x_2}}+{\displaystyle \frac{0.5}{x_3}}+{\displaystyle \frac{(0.3,0.4)}{x_4}},$$

$$\overline{R_1}(A)={\displaystyle \frac{(0.4,0.5)}{x_1}}+{\displaystyle \frac{(0.4,0.5)}{x_2}}+{\displaystyle \frac{(0.4,0.5)}{x_3}}+{\displaystyle \frac{(0.4,0.5)}{x_4}},$$

$$\underset{¯}{R_2}(A)={\displaystyle \frac{(0.2,0.4)}{x_1}}+{\displaystyle \frac{(0.2,0.4)}{x_2}}+{\displaystyle \frac{(0.2,0.4)}{x_3}}+{\displaystyle \frac{0.5}{x_4}},$$

$$\overline{R_2}(A)={\displaystyle \frac{(0.4,0.5)}{x_1}}+{\displaystyle \frac{(0.3,0.5)}{x_2}}+{\displaystyle \frac{(0.3,0.5)}{x_3}}+{\displaystyle \frac{(0.4,0.5)}{x_4}},$$

$$\underset{¯}{{\mathrm{R}}_3}(A)={\displaystyle \frac{(0.4,0.5)}{x_1}}+{\displaystyle \frac{0.5}{x_2}}+{\displaystyle \frac{0.5}{x_3}}+{\displaystyle \frac{0.5}{x_4}},$$

$$\overline{R_3}(A)={\displaystyle \frac{(0.4,0.5)}{x_1}}+{\displaystyle \frac{(0.4,0.5)}{x_2}}+{\displaystyle \frac{(0.4,0.6)}{x_3}}+{\displaystyle \frac{(0.4,0.6)}{x_4}}.$$

If the experts adopt a neutral attitude, the evaluation result ${\sigma }_{0.5}^1(x_i)$ given by expert 1 is presented in Formula (21).

$${\sigma }_{0.5}^1(x_1)=(0.3519,0.4469), {\sigma }_{0.5}^1(x_2)=(0.3070,0.4998),$$

$${\sigma }_{0.5}^1(x_3)=(0.4522,0.4998), {\sigma }_{0.5}^1(x_4)=(0.3519,0.4469).$$

The evaluation result ${\sigma }_{0.5}^2(x_i)$ given by expert 2 is below.

$${\sigma }_{0.5}^2(x_1)=(0.3070,0.4469), {\sigma }_{0.5}^2(x_2)=(0.2515,0.4469),$$

$${\sigma }_{0.5}^2(x_3)=(0.3070,0.4082), {\sigma }_2(x_4)=(0.4469,0.4998).$$

The evaluation result ${\sigma }_{0.5}^3(x_i)$ given by expert 3 is below.

$${\sigma }_{0.5}^3(x_1)=(0.4019,0.4998), {\sigma }_{0.5}^3(x_2)=(0.2928,0.4469),$$

$${\sigma }_{0.5}^3(x_3)=(0.4469,0.5226), {\sigma }_{0.5}^3(x_4)=(0.4469,0.5226).$$

Based on the weights of the experts in Section 5, the total evaluation result ${\delta }_{0.5}(x_i)$ given by all experts is calculated by Formula (22) below.

$${\sigma }_{0.5}(x_1)=(0.3592,0.4693), {\sigma }_{0.5}(x_2)=(0.2841,0.4632),$$

$${\sigma }_{0.5}(x_3)=(0.4094,0.4843), {\sigma }_{0.5}^3(x_4)=(0.4197,0.4941).$$

According to the method in Reference [17], we get $\sigma (x_3)\succ \sigma (x_2)\succ \sigma (x_4)\succ \sigma (x_1)$. The selection of aggregation operators cannot effectively handle the evaluations of different experts in a hesitant fuzzy environment, which will lead to errors in the aggregation results. The ranking function for alternatives based on aggregation operators is not applicable to the diversified MAGDM problem in a hesitant fuzzy environment.

According to the method in Reference [27], the ranking function for alternatives may be selected based on fuzzy logical operators in the diversified MAGDM problem, the fuzzy logic operator generates significant errors during the calculation process, the result is $\sigma (x_4)\succ \sigma (x_1)\succ \sigma (x_3)\succ \sigma (x_2)$. The assumption of assigning a value of 0.5 to the missing evaluations is a relatively risky assumption in Section 6.1 and Reference [27]. The assumption of neutrality might not accurately represent a real expert’s perspective; the inappropriate assignment could therefore lead to inaccurate results.

Form the analysis above, the results compared with other existing methods are shown in

Table 10. The results compared with other existing methods.

Methods	The Rankings
Method in Reference [17]	$\sigma (x_3)\succ \sigma (x_2)\succ \sigma (x_4)\succ \sigma (x_1)$
Method in Reference [27]	$\sigma (x_4)\succ \sigma (x_1)\succ \sigma (x_3)\succ \sigma (x_2)$
The proposed method	$\sigma (x_2)\succ \sigma (x_4)\succ \sigma (x_1)\succ \sigma (x_3)$

.

The proposed method overcomes the adverse effects brought by the aggregation operator, which is combined with the cumulative prospect theory to consider the influence of hesitance degree and psychological behavior preferences of decision-makers in the decision-making process. The proposed method does not take into account the assumption of expert neutrality, as decision-makers only select certain attributes they are familiar with to articulate their opinions. Therefore, it is reasonable for minor differences in the results when different algorithms are adopted. There are two evaluations corresponding to x3, with a possibility of 0.1 when attributes c5 and c6 were evaluated by expert 2. Thus, the evaluation method we proposed is more reasonable. According to the methods in References [17,27], ideal conclusions can be obtained in a fuzzy environment, but not in hesitant fuzzy environments; the combination of DHFMGRS and TOPSIS can reasonably handle hesitant fuzzy problems.

6.2. The Fairness of Evaluation

In this section, the main analysis is whether the assessment of experts is fair or not. Liang [30] proposed amendments to the three characteristics of identifying manipulative behavior, the two criteria by which we examine the fairness of experts are as follows:

• There should be no significant difference in expert evaluation for each applicant.
• There should be no significant difference in the evaluation of the same applicant by various experts.

Let ${\mathrm{\Xi }}_i^k$ be the deviation between the evaluation of expert $k$ on an applicant $x_i$ and other applicants.

$${\mathrm{\Xi }}_i^k=|{\displaystyle \sum _{j=1}^n{\omega }_j{f}_{ij}^k}-{\displaystyle \frac{{\displaystyle {\sum }_{i’=1,i’\ne i}^m{\displaystyle {\sum }_{j=1}^n{\omega }_j{f}_{i’j}^k}}}{m-1}}|,i=1,2,\dots ,m,k=1,2,\dots l.$$

where ${\omega }_j$ is the weight of the attribute $c_j$, $f_{ij}^k$ is evaluation of expert $k$ on an applicant $x_i$ with respect to attributes $c_j$. The consensus threshold is $\epsilon$; when ${\mathrm{\Xi }}_i^k>\epsilon$, the evaluation of expert $k$ on applicant $x_i$ is significantly different from other schemes.

Let ${\mathrm{H}}_i^k$ be the deviation between evaluation of expert $k$ on an applicant $x_i$ and that of other experts. When ${\mathrm{H}}_i^k>\epsilon$, the evaluation of expert $k$ on an applicant $x_i$ is significantly different from that of other experts.

$${\mathrm{H}}_i^k={\displaystyle \sum _{j=1}^n{\omega }_j|f_{ij}^k}-{\overline{f}}_{ij}^{g/k}|,i=1,2,\dots ,m,k=1,2,\dots l.$$

where ${\overline{f}}_{ij}^{g/k}={\displaystyle \frac{{\displaystyle {\sum }_{h\ne k}{u}_h^g{f}_{ij}^h}}{1-u_k^g}}$ is the collective evaluation of all experts, except expert $k$.

The consensus threshold $\epsilon$ is set to 0.85 and the threshold $\iota$ for assessment is set to $\iota =1-\epsilon =0.15$. The deviation from evaluation on an applicant $x_i$ and that of other applicants are presented in

Table 11. The deviation from evaluation on an applicant $x_i$ and that of other applicants.

	$\mathit{k}=1$	$\mathit{k}=2$	$\mathit{k}=3$
$x_1$	0.026	0.014	0.030
$x_2$	0.015	0.030	0.003
$x_3$	0.032	0.014	0.025
$x_4$	0.057	0.003	0.003

, and the deviation between evaluation of expert $k$ and that of other experts on an applicant $x_i$ are presented in

Table 12. The deviation between evaluation of expert $k$ and that of other experts on an applicant $x_i$.

	k = 1	k = 2	k = 3
$x_1$	0.041	0.146	0.111
$x_2$	0.132	0.091	0.112
$x_3$	0.143	0.130	0.116
$x_4$	0.098	0.089	0.087

.

It is easy to find that all applicants satisfy the threshold above. That is, the expert’s assessment is fair and unbiased. The proposed model is compared with the three-way decision model, which gives the evaluation of a delayed decision for the alternatives [37]. On the whole, the evaluation results of the proposed model are consistent with the results of the existing models.

The proposed model provides a reference for a diversified multiple-attribute hesitant fuzzy information system, and the model avoids the extremely high or extremely low evaluation of individual experts to a certain extent. However, the hesitant fuzzy relationship evaluated by experts may not necessarily conform to the actual situation. There is a lack of a unified standard for how to select the appropriate hesitant fuzzy relationship. As the number of granularities increases, the computational complexity will rise, and a more concise algorithm may be required. We state that the expert is neutral to the attributes that are not evaluated by the expert, which is not necessarily accurate; the precise parameters given by experts do not necessarily match the actual situation in some cases.

7. Conclusions

A diversified-attribute hesitant fuzzy multi-granulation rough-set model combined with cumulative prospect theory is proposed to overcome the adverse effects brought by the aggregation operator and assumption of expert neutrality. The key steps are proposed to solve a series of diversified MAGDM problems under hesitant fuzzy circumstances. The manipulative behavior among the experts was examined to guarantee fairness in the assessment. Finally, the proposed model is applied to the evaluation of shared parking. The advantage of the proposed model lies in ensuring that the influence of hesitance degree and the psychological behavior preferences of decision-makers are considered in the GDM. Decision-makers only select the certain attributes that they are familiar with to articulate their opinions. It also offers a reference for the assessment of diversified multi-attribute group decision-making. Additionally, in the subsequent research, we will employ a formal concept analysis to analyze the issue of hesitant fuzzy multi-granulation rough sets in incomplete circumstances.