Probabilistic Hesitant Fuzzy Decision-Theoretic Rough Set Model and Its Application in Supervision of Shared Parking

Abstract

A sophisticated three-way decision model utilizing a multi-granularity probabilistic hesitant fuzzy rough set is introduced to mitigate the issue of information loss arising from hesitant fuzzy sets when characterizing decision information. Initially, the properties of probabilistic hesitant fuzzy sets are examined, subsequently proposing a distance measure and loss function tailored to these sets. Following this, a multi-attribute group decision-making model incorporating probabilistic hesitant fuzzy information is established, and pertinent decision rules that satisfy minimal risk criteria are presented. Building on this foundation, a series of critical steps for resolving a category of multi-attribute group decision-making problems that involve probabilistic hesitant fuzzy information is proposed. Lastly, the multi-attribute group decision-making model with probabilistic hesitant fuzzy information is applied to the supervision of urban shared parking platforms. The results indicate that the decision-making process based on probabilistic hesitant fuzzy sets is more reliable, and the decision-making outcome aligns with the actual situation, thus providing valuable decision-making references for managers.

1. Introduction

Locating an available parking slot continues to be a formidable challenge for drivers, as a significant scarcity of parking slots remains, particularly in large cities worldwide. In Xi’an, for instance, the number of vehicles reached 3.8 million in 2020, with a deficit of over one million parking slots not addressed by current parking facilities. It is estimated that about 30% of traffic is generated by drivers searching for parking slots. An IBM report reveals that the average time required to find a parking slot in 20 cities worldwide surpasses 12.7 min. Finding a satisfactory parking slot within a limited time remains challenging. On the other hand, the standard number of parking slots per 100 square meters remains constant in developed countries. Similar standards have been established in Chinese cities such as Shanghai, Guangzhou, Nanjing, and Chengdu. This imbalance between parking supply and demand cannot be resolved by merely restricting vehicle flow. The Central Committee of the Communist Party of China and the State Council advocate for a rational allocation of parking facilities to enhance urban planning and construction administration, thus making shared parking a preferable option under existing conditions. Advancements in digital technology have made communication between parking lots and drivers more convenient than ever. The transition from broadcast to network transmission has made smart parking feasible in idle spaces [1]. Due to the shortage of supplies, roadside parking slots or private spaces have become a priority for drivers. However, shared parking may easily lead to disputes between shared users and original residents, because sharing increases polluted emissions and unsafe factors in shared places. In addition, some restrictions have been proposed to protect the interests of the original residents. For example, parking slots in residential areas and government offices in China are not yet allowed to be shared, although there are idle slots left. Therefore, there is an urgent necessity to find a regulatory approach to identify which parking slots can be shared and which cannot, so as to save users’ travel costs and reduce urban pollution.

The regulatory strategies of government (platform) to operators mainly include evolutionary game theory, administrative management [1,2], and three-way decision making. Administrative management and evolutionary game theory provide a suggestion on whether parking lots should be regulated or not in special cases. Subsidies for parking lots are crucial for shared parking, but they may also lead to resource waste through rent-seeking. Therefore, supervision is essential for shared parking at various stages, wherein shared parking systems are created by parking lots and spaces are provided by users; there are platforms, parking lots, and users in the game of shared parking. When the parking lots are not regulated, not only the users’ interests may be hurt, but also the traffic situation of the city may be affected. Administrative management was used in the regulation of shared parking in the past, which brought many practical problems. Compared with the actual effect, game theory and multi-criteria group decision-making are better attempts than administrative management in the supervision of shared parking.

Previous studies on evolutionary game theory have primarily considered platforms, supply, and demand as three interacting game subjects, with their complex relationships aiming to improve operational efficiency. Wu et al. [3] constructed an asymmetric evolutionary game model between charging service operators and private charging pile owners, illustrating that the spillover effect of co-creation is the fundamental reason for the strategic choices of both parties. Yuan et al. [4] analyzed the interaction between government and firms within the context of daily penalties, which aids in controlling pollution and reducing regulatory costs. Zou et al. [5] examined the network effect in the evolutionary game between a shared supply chain platform and manufacturers. Dong et al. [6] developed a dynamic evolutionary game model between government and waste mobile phone recycling companies, proposing effective suggestions such as privacy protection for suppliers. Duo et al. [7] analyzed the evolutionary game of parking supply and demand, suggesting that profit maximization and optimal allocation can be achieved through price adjustment and quantity control, with system cost minimization and profit maximization being consistent under specific threshold conditions. Roberto et al. [8] constructed a dynamic evolutionary game model of carpooling systems, analyzing the Nash equilibrium with both pure and mixed strategies. Mutual trust can also address issues arising from platform supervision in the sharing economy, with regulatory responsibility delegated through peer-to-peer sharing.

Multi-criteria group decision making has been applied to many uncertain evaluations such as safety evaluation [9], hesitant fuzzy ranking [10], and clinical diagnosis [11] in recent years. Chen et al. provided a simple solution to the MCGDM problem by combining the Preference Ranking Organization Method for Enrichment Evaluations II (PROMETHEE II) method with Probabilistic Hesitant Fuzzy Linguistic Sets (PHFLSs) [12]. Harish et al. proposed a probabilistic dual hesitancy fuzzy set model with a Maclaurin symmetric mean aggregation operator to solve the uncertainty in medical diagnosis [11]. Zhang et al. developed multi-granulation decision-theoretic rough sets (MGDTRSs) into a hesitant fuzzy linguistic (HFL) background within the two-universe framework and built a multi-granularity three-way decision model [13]. Khan et al. established probabilistic hesitant fuzzy rough (PHFR) sets over two finite nonempty fixed sets, providing decision-makers with practical and scientific decision-making support [14]. Yahya et al. constructed general algorithms for multi-attribute group decision-making problems based on interval-valued hesitant fuzzy (IVHF) and interval-valued probabilistic hesitant fuzzy (IVPHF) information [15]. Krishankumar presented a systematic procedure for MAGDM with the Muirhead mean (MM) operator and demonstrated its practical use in a renewable energy source selection problem [16]. Wang et al. combined several MCDM techniques with probabilistic hesitant fuzzy linguistic term sets (PHFLTSs) to implement group risk assessment in failure mode and effects analysis [17]. Wu et al. constructed a probabilistic hesitant fuzzy MAGDM method based on the generalized probabilistic hesitant fuzzy Bonferroni mean (GPHFBM) operator to analyze the aggregated information, enabling decision-makers to select the proper parameters in the decision-making process [18]. Li et al. proposed an integrated MAGDM method based on the best-worst method (BWM) and MULTI MOORA method with two-dimensional linguistic intuitionistic fuzzy variables (2DLIFVs) for the selection of unmanned aerial vehicle suppliers [19]. Wang et al. constructed a new linguistic T-spherical fuzzy (Lt-SF) MAGDM model by integrating the Lt-SF Heronian mean (Lt-SFHM) operator, generalized distance measure, and additive ratio assessment (ARAS) method for the selection of sustainable recycling suppliers [20]. However, the probabilistic hesitant fuzzy information given by decision-makers is often encountered in practice, and there are few studies that combine PHFSs with a three-way decision model. Therefore, the decision-theoretic rough set (DTRS) is extended to the probabilistic hesitant fuzzy environment and applied to the regulation of shared parking.

Based on the above arguments and motivations, the main contributions of this paper are presented: (1) By introducing the idea of probabilistic hesitant fuzzy sets into DTRS, the probabilistic hesitant fuzzy DTRS model is established. The corresponding rules are induced and some relative properties are given. (2) By the definition of the distance between two probabilistic hesitant fuzzy elements, the calculation method of loss function is determined. (3) A new group decision-making model based on probabilistic hesitant fuzzy DTRS over two universes is developed.

The remainder of this paper is structured as follows: Section 2 discusses related work; Section 3 introduces basic concepts; Section 4 elaborates on probabilistic hesitant fuzzy DTRS over two universes; Section 5 presents the key steps of group decision-making; Section 6 offers a numerical simulation; Section 7 is a discussion; Section 8 is limitations and recommendations; and the final section provides a conclusion.

2. Related Work

The sharing economy has gained prominence in the context of resource sharing, driven by advancements in digital technology [21]. In most cities in China, parking slots are managed by parking lots, which provide human resources and Parking Guidance and Information Systems (PGIS) for sharing. It is assumed that the platform, parking lot, and users have distinct strategies: the platform’s objective is to create conditions that provide the best service for drivers [7], parking lots aim to rent out idle spaces to enhance operational efficiency and service quality [22], and drivers seek to reduce travel costs in shared parking games [23,24]. The platform, parking lots, and drivers are the primary stakeholders in the shared parking process. These three parties learn from one another and continuously adjust their strategies during the game process, based on bounded rationality.

Three basic conditions are required for shared parking: first, platform support is necessary to ensure the development of shared parking; second, essential facilities such as parking guidance systems and idle parking slots must be prepared in the parking lot; and third, users’ parking demands must be met. For the convenience of analysis, parking time is divided into shared and non-shared segments. Both the space provider and shared users are allowed entry during the shared period, while only the space provider is permitted entry during the non-shared period. Drivers participating in shared parking must leave on time to prevent any disruption to the space provider’s parking before the sharing session ends.

The primary goal of parking lots is to achieve profit maximization while providing advice to drivers [25], such as implementing slot reservation strategies [26], slot auctions [27], and mechanism designs [5]. Slot auction strategies are effective solutions for parking lots, with parking fees periodically updated based on occupancy. Dynamic pricing can reduce the revenue loss caused by drivers’ randomness. Parking lots use slot reservation during emergency situations. Platforms encourage parking lots to create sharing systems and form coalitions or trade associations to share regulatory responsibilities. Users are expected to provide idle slots to parking lots. The platform’s strategy can effectively impact the parking choices of drivers and the creation behavior of parking lots. Social benefits and other factors are considered by the platform during the early stages of shared parking. The platform obtains economic benefits when the sharing system operates regularly. If a parking lot receives a subsidy and “does not create”, the platform punishes the parking lot. However, shared parking is not well regulated in China, except for administrative supervision. Evolutionary game theory and the administration of government can give suggestions on whether parking lots should be regulated or not. There are no recommendations for parking lots that work well, but do not meet the requirement. A new regulatory approach is urgently required in the sharing economy to enable the widespread implementation of shared parking while reducing the platform’s regulatory costs. The platform’s regulation is also an MCGDM problem.

3. Basic Concepts

Definition 1.

Let $U$ be a non-empty finite universe and $R$ be an equivalence relation of $U\times U$. The equivalence relation  $R$ induces a partition of $U$, denoted by $[x]$, and $U/R=\{[x]|x\in U\}$ stands for the equivalence classes of $x$. Then $(U,R)$ is the Pawlak approximation space. For any $x\in U$, its lower and upper approximations are defined as follows:

$$\underset{¯}{R}(x)=\{x\in U|[x]\subseteq X\}=\cup \{[x]|[x]\subseteq U\}$$

$$\overline{R}(x)=\left\{x\in U|[x]\cap X\ne \varnothing \right\}=\cup \left\{[x]|[x]\cap X\ne \varnothing \right\}$$

The positive, boundary, and negative regions of $X$ can be defined as follows:

$$POS(X)=\underset{¯}{R}(X), BND(X)=\overline{R}(X)-\underset{¯}{R}(X), NEG(X)=U-\overline{R}(X)$$

Definition 2.

A hesitant fuzzy set (HFS) has been proposed by Torra, in which one hesitates among several possible values to assess an indicator, alternative variables, etc. Basic elements in HFS are hesitant fuzzy elements (HFSs), which include some possible values for the membership of an element to a set.

Definition 3

([28,29]). Given any non-empty set X, a probabilistic hesitant fuzzy set(PHFS) defined on the finite set X can be expressed as: $H=\{x,h(p)={\gamma }^{\lambda }{p}^{\lambda }|x\in X\}$. ${\gamma }^{\lambda }{p}^{\lambda }$ is the basic tool to describe PHFS, which is usually called a probabilistic hesitant fuzzy element (PHFE). The membership degree ${\gamma }^{\lambda }\in [0,1]$ represents the possibility that the element $x\in X$ belongs to the PHFS. $p^{\lambda }\in [0,1]$ indicates the probability of ${\gamma }^{\lambda }$, $\lambda =1,2,\cdots , l$, ${\displaystyle \sum _{\lambda =1}^l{p}^{\lambda }}=1$. In particular, when $\lambda =\frac{1}{l}$, PHFS degenerates into HFS.

Definition 4.

The score function of PHFE $h(p)={\gamma }^{\lambda }{p}^{\lambda }$ is defined as follows:

$$s(h(p))={\displaystyle \sum _{\lambda =1}^l({\gamma }^{\lambda }}\cdot p^{\lambda })/{\displaystyle \sum _{\lambda =1}^l{p}^{\lambda }}$$

(1)

$$D(h(p))={\displaystyle \sum _{\lambda =1}^l(}{p}^{\lambda }{\gamma }^{\lambda }-p^{\lambda }s(h(p)){)}^2/{\displaystyle \sum _{\lambda =1}^l{p}^{\lambda }}$$

(2)

Definition 5.

Given any two probabilistic hesitant fuzzy elements, the improved probabilistic hesitant fuzzy distance measure can be defined as follows:

$$d(h_1,h_2)=\frac{1}{2}{\displaystyle \sum _{\lambda =1}^l\left(|{\gamma }_1^{\lambda }{p}_1^{\lambda }-{\gamma }_2^{\lambda }{p}_2^{\lambda }|+|{\gamma }_1^{\lambda }-{\gamma }_2^{\lambda }|\cdot p_1^{\lambda }{p}_2^{\lambda }\right)}$$

(3)

Definition 6.

Let$U, V$be two non-empty finite universes. A probabilistic hesitant fuzzy relation $R$ between $U$and $V$is a probabilistic hesitant fuzzy subset of $U\times V$, and $R$ is defined by:

$$R=\left\{⟨(x,y),h_R(x)(y)⟩\left|(x,y)\in U\times V\right.\right\}$$

(4)

where $h_R(x)(y)$ is a set of some different finite values in [0, 1], representing the possible membership degrees and the possibility between x and y.

Definition 7.

Let $U, V$ be two non-empty finite universes.$R\in HF(U\times V)$ is a probabilistic hesitant fuzzy relation between $U$ and $V$. The binary probabilistic hesitant fuzzy relation class is a probabilistic hesitant fuzzy subset of the universe V and expressed by:

$$H_R(x)(y)=\frac{h_R(x)(y_1)}{y_1}+\frac{h_R(x)(y_2)}{y_2}+\cdots +\frac{h_R(x)(y_n)}{y_n}, n=\left|V\right|$$

(5)

Definition 8.

Let $U$ be two non-empty finite universes. Suppose $R\in HF(U\times V)$ is an arbitrary probabilistic hesitant fuzzy relation between $U$ and $V$. Then, for any $x\in U,y\in V$, and $A\in HF(V)$, the conditional probability of A for the object x given the probabilistic hesitant fuzzy description $H_R(x)$, $P(A\left|H_R(x)(y)\right.)$ is given as follows:

$$P(A\left|H_R(x)(y)\right.)=\frac{{\displaystyle {\sum }_{y\in V}(s(h_A(y))\cdot s(h_R(x)(y)))}}{{\displaystyle {\sum }_{y\in V}s(h_R(x)(y))}}$$

(6)

where $0\le (A\left|H_R(x)(y)\right.)\le 1$.

Definition 9.

Given any two probabilistic hesitant fuzzy elements,$h_i(p)=\{{\gamma }_i^{\lambda }(p_i^k)|l=1,2,\cdots ,|h_i(p)|\},i=1,2,\lambda >0$, the basic operation rules are as follows:

$$h_1(p)+h_2(p)=\underset{{\gamma }_{1_l}\in h_1,{\gamma }_{2_k}\in h_2}{\cup }\{[{\gamma }_1^l+{\gamma }_2^k-{\gamma }_1^l{\gamma }_2^k](p_1^l{p}_2^k/{\displaystyle {\sum }_{l=1}^{h_1(p)}{p}_1^l}\cdot {\displaystyle {\sum }_{k=1}^{h_2(p)}{p}_2^k})\}$$

(7)

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

(8)

$$\lambda h_{}(p)=\underset{{\gamma }_{}^l\in h_{}}{\cup }\{[1-{(1-{\gamma }_{}^l)}^{\lambda }]|(p_{}^l)\}$$

(9)

Definition 10.

Given any n probabilistic hesitant fuzzy elements,$h_i(p)=\{{\gamma }_i^{\lambda }(p_i^k)|l=1,2,\cdots ,|h_i(p)|\},i=1,2,\cdots ,n$.

The weight vector of $h_i(p)$ is $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_n)$, and its probabilistic hesitant fuzzy weighted average (PHFWA) operator and probabilistic hesitant fuzzy weighted geometry (PHFWG) operator are defined as:

$$\begin{array}{c}PHFWA(h_1(p),\cdots ,h_n(p))=\underset{i=1}{\overset{n}{\oplus }}{\omega }_i{h}_i(p)=\\ \underset{{\gamma }_{1_l}\in h_1,\cdots ,{\gamma }_{n_l}\in h_n}{\cup }\left\{\left[1-{\displaystyle \prod _{i=1}^n{(1-{\gamma }_{i_l})}^{{\omega }_i}}\right]\left({\displaystyle \prod _{i=1}^n{p}_{i_l}/{\displaystyle \prod _{i=1}^n\left({\displaystyle \sum _{l=1}^{h_i(p)}{p}_{i_l}}\right)}}\right)\right\}\end{array}$$

(10)

$$\begin{array}{c}PHFWG(h_1(p),\cdots ,h_n(p))=\underset{i=1}{\overset{n}{\otimes }}{h}_i{}^{{\omega }_i}(p)=\\ \underset{{\gamma }_{1_l}\in h_1,{\gamma }_{2_l}\in h_2,\cdots ,{\gamma }_{n_l}\in h_n}{\cup }\left\{\left[{\displaystyle \prod _{i=1}^n{\gamma }_{i_l}{}^{{\omega }_i}}\right]\left({\displaystyle \prod _{i=1}^n{p}_{i_l}/{\displaystyle \prod _{i=1}^n\left({\displaystyle \sum _{l=1}^{h_i(p)}{p}_{i_l}}\right)}}\right)\right\}\end{array}$$

(11)

Definition 11.

Let $(U,V,F,P)$ be a multi-granulation probabilistic hesitant fuzzy approximation space over two universes. Two threshold parameters, $\alpha$, $\beta$, and the precision parameter $\delta$ are given, with $0\le \beta <\alpha \le 1$, $0<\delta <1$. For a probabilistic hesitant fuzzy set $A\in HF(V)$, the δ-lower and δ-upper approximations of A with respect to $(U,V,F,P)$, denoted as ${\underset{¯}{\Re }}_{{\displaystyle {\sum }_{k=1}^l}}^{\delta ,\alpha }(A)$ and ${\overline{\Re }}_{{\displaystyle {\sum }_{k=1}^l}}^{\delta ,\alpha }(A)$, are defined as follows:

$${\underset{¯}{\Re }}_{{\displaystyle {\sum }_{k=1}^l}}^{\delta ,\alpha }(A)=\{x|\frac{|\{R_k|p(A|H_{R_k}(x)(y)|\ge \alpha \left\}\right|}{l}\ge \delta ,x\in U,y\in V,k=1,2,\cdots ,l\}$$

(12)

$${\overline{\Re }}_{{\displaystyle {\sum }_{k=1}^l}}^{\delta ,\alpha }(A)=\{x|\frac{|\{R_k|p(A|H_{R_k}(x)(y)|\ge \alpha \}|}{l}\ge 1-\delta ,x\in U,y\in V,k=1,2,\cdots ,l\}$$

(13)

4. Probabilistic Hesitant Fuzzy DTRS over Two Universes

4.1. Parameter Symbols

The parameters and definitions are shown in Table 1 for the convenience of analysis.

Table 1. Parameters and definitions.

Symbols	Definitions
$U,V$	Non-empty finite universes
$A,B$	Probabilistic hesitant fuzzy sets
$C$	State set
$h_R(x)(y)$	Probabilistic hesitant fuzzy subset of $U\times V$
$H_R(x)(y)$	The binary probabilistic hesitant fuzzy relation class
${\lambda }_y^j(x_i)$	Relative loss function of $x_i$ in criterion $y_j$
${\lambda }^C(x_i)$	Aggregated relative loss function
$R$	An arbitrary hesitant fuzzy relation between U and V
$\Re$	A family of binary probabilistic hesitant fuzzy relations between U and V
$\delta$	The hesitant parameter
$\omega$	The criteria weight vector
$u$	The opinion weight vector of the experts

4.2. Probabilistic Hesitant Fuzzy DTRS Model

Pawlak initially introduced the rough set concept and examined its applications in characteristic symptom analysis, automatic classification, and medical diagnosis, among others [30]. In this framework, the universe is divided into three regions based on equivalence relations, resulting in a large boundary region due to strict classification precision. Subsequently, various rough set models were proposed to adjust positive or negative regions, such as Probabilistic Rough Set (PRS) models [31,32], Game-Theoretic Rough Sets (GTRS) [33], and DTRS [34,35].

PRS aims to reduce the boundary region and expand the decision regions using three types of rules. When potential memberships could be fuzzy values instead of crisp values, generalizations of fuzzy set models were proposed, including Intuitionistic Fuzzy Sets [36]. Qian was the first to shift from single granularity to multiple granularity, focusing on group decision-making rather than single-agent decision-making, which extended rough set theory for intelligent decisions involving multiple criteria [37]. Sun then expanded the probabilistic rough set to a fuzzy environment, presenting a decision-theoretic rough fuzzy set model that combined rough set theory with granular computing and established generalized multi-granularity rough set models with uncertain decision backgrounds [38,39].

Previous studies developed a group decision-making model by integrating hesitant fuzzy sets into DTRS [13,40,41]. However, the probability of hesitant fuzzy decision-making problems may not necessarily be the same in reality. Consequently, it is essential to incorporate DTRS into the probabilistic hesitant fuzzy set, leading to the establishment of the probabilistic hesitant fuzzy DTRS model.

Let $d=\{d_1,d_2,d_3\}$ stand for the set of decision actions in classifying an alternative $x\in U$, in which $d_1$, $d_2$, and $d_3$ indicate $x\in POS(A)$, $x\in NEG(A)$, and $x\in BND(A)$, respectively. $\lambda (d_i\left|w_j\right.)$ stands for the loss function regarding the cost incurred for making decision $d_1$, $d_3$, and $d_2$, respectively, when x belongs to A. Similarly, $\lambda (d_1\left|B\right.)$, $\lambda (d_2\left|B\right.)$, and $\lambda (d_3\left|B\right.)$ stand for the loss function regarding the cost incurred for making the same decision actions when x belongs to B. For simplicity, let us denote as follows:

$${\lambda }_{1A}=\lambda (d_1|A), {\lambda }_{2A}=\lambda (d_2|A), {\lambda }_{3A}=\lambda (d_3|A)$$

(14)

$${\lambda }_{1B}=\lambda (d_1|B), {\lambda }_{2B}=\lambda (d_2|B), {\lambda }_{3B}=\lambda (d_3|B)$$

(15)

$${\lambda }_{1C}=\lambda (d_1|C), {\lambda }_{2C}=\lambda (d_2|C), {\lambda }_{3C}=\lambda (d_3|C)$$

(16)

For a given probabilistic hesitant fuzzy subset $A\in F(U)$, the state set $\mathsf{\Omega }=\{w_1,w_2,\cdots ,w_m\}$ is $m$ probabilistic hesitant fuzzy set of the universe, which represents the membership of the alternative’s characteristics. $A=\{a_1,a_2,\cdots ,a_n\}$ is the set of actions and $P(w_j|H_R(x))$ is the conditional probability under the state set $w_j$. $\lambda (d_i\left|A\right.)$ is the loss of action when $x\in A$, and $\lambda (d_i\left|w_j\right.)$ is simply marked ${\lambda }_{ij}$. The expected cost of action $d_1$, $d_3$, and $d_2$ in states $A$ and $A^c$ is determined by the following formula based on the Bayesian minimum risk decision theory:

$$R(d_i\left|H_R(x)\right.)={\displaystyle \sum _{j=1}^m\lambda (d_i\left|w_j\right.)\ast P(w_j\left|H_R(x)\right.)}$$

where $i=1,2,3;w_j=A,A^c$.

On condition that:

$$\left\{\begin{array}{l}R(d_1\left|H_R(x))\right.\le R(d_3\left|H_R(x))\right.\\ R(d_1\left|H_R(x))\right.\le R(d_2\left|H_R(x))\right.\end{array}\right.$$

(17)

The decision-makers will take action.

On condition that:

$$\left\{\begin{array}{l}R(d_3\left|H_R(x))\right.\le R(d_1\left|H_R(x))\right.\\ R(d_3\left|H_R(x))\right.\le R(d_2\left|H_R(x))\right.\end{array}\right.$$

(18)

The decision-makers need further assessment.

On condition that:

$$\left\{\begin{array}{l}R(d_2\left|H_R(x))\right.\le R(d_1\left|H_R(x))\right.\\ R(d_2\left|H_R(x))\right.\le R(d_3\left|H_R(x))\right.\end{array}\right.$$

(19)

The decision-makers will not take a step.

Suppose the state set $\mathsf{\Omega }=\{A,A^c\}$ is the probabilistic hesitant fuzzy set of the universe $U$, $P(A|H_R(x))+P(A^c|H_R(x))=1$, ${\lambda }_{1A}$, ${\lambda }_{3A}$, and ${\lambda }_{2A}$ represent the losses by $d_1$, $d_3$, and $d_2$ when x belongs to $A$, ${\lambda }_{1C}$, ${\lambda }_{3C}$, and ${\lambda }_{2C}$ represents the losses by $d_1$, $d_3$, and $d_2$ when x belongs to $A^c$. The loss function is determined according to the losses under different circumstances.

When the probabilistic hesitant fuzzy set A is concerned, $0<{\lambda }_{1A}\le {\lambda }_{3A}\le {\lambda }_{2A}$, ${\lambda }_{1C}>{\lambda }_{3C}>{\lambda }_{2C}>0$. From equation $P(A|H_R(x))+P(A^c|H_R(x))=1$, the following decision rules can be generated by:

Decide $x\in POS(A)$, if $P(A\left|H_R(x)\right.)\ge \alpha$ and $P(A\left|H_R(x)\right.)\ge \gamma$;

Decide $x\in NEG(A)$, if $P(A\left|H_R(x)\right.)<\beta$ and $P(A\left|H_R(x)\right.)<\gamma$;

Decide $x\in BND(A)$, if $P(A\left|H_R(x)\right.)\ge \beta$ and $P(A\left|H_R(x)\right.)<\gamma$.

Let

$$\alpha =\frac{{\lambda }_{1C}-{\lambda }_{3C}}{({\lambda }_{1C}-{\lambda }_{3C})+({\lambda }_{3A}-{\lambda }_{1A})}$$

(20)

$$\beta =\frac{{\lambda }_{3C}-{\lambda }_{2C}}{({\lambda }_{3C}-{\lambda }_{2C})+({\lambda }_{2A}-{\lambda }_{3A})}$$

(21)

$$\gamma =\frac{{\lambda }_{1C}-{\lambda }_{2C}}{({\lambda }_{1C}-{\lambda }_{2C})+({\lambda }_{2A}-{\lambda }_{1A})}$$

(22)

Thus, the decision rules can be expressed as:

$P(A\left|H_R(x))\right.\ge \alpha$, and then the decision-makers will take action;

$\beta <P(A\left|H_R(x))\right.<\alpha$, the decision-makers need further assessment;

$P(A\left|H_R(x))\right.\le \beta$, and then the decision-makers will not take a step.

To determine whether an alternative x belongs to the class $H_R(x)(y)$ or not, we only need to compare the value of $P(A|H_R(x)(y))$ and threshold $(\alpha ,\beta )$, and then make a decision according to the probabilistic hesitant fuzzy DTRS model. A numerical example is given to illustrate the process of shared parking supervision based on the probabilistic hesitant fuzzy DTRS model.

5. Key Steps of Group Decision

The assumptions of the model are listed below for analyses.

(1)

Experts give probabilistic hesitant fuzzy information when evaluating parking lots, and the hesitancy of decision-making is measured by the hesitant parameters.

(2)

The loss function is measured by the distance measure of PHFSs.

(3)

It is supposed that only four factors including safety, convenience, distance, and parking fee in the parking lots are considered by the platform when the experts evaluate it.

(4)

Experts only identify which parking lots meet the requirements, which ones need to be further assessed, and which ones are not considered.

(5)

Since we assume two cases of workdays and non-workdays (not including Chinese traditional festivals), we express the probabilities of the two cases in the form of a probabilistic hesitant fuzzy element. It is supposed that the security of shared parking is better on non-workdays than on workdays. It is more convenient for the same shared parking lot on non-workdays than on workdays.

(6)

It is supposed that the shared parking lot is close to the transfer station (subway station), and that the distance between the parking lot and the transfer station(subway station) is more attractive on weekdays than on non-weekdays.

(7)

The parking fees of the shared parking lots are more acceptable on workdays than on non-workdays.

The key steps of multi-granularity decision making based on the probabilistic hesitant fuzzy DTRS model are as follows:

Step 1 Experts $R_k$ provide the probabilistic hesitant fuzzy relation, where $R_k\in PHF(U\times V)$, $k=1,2,3$. For example, the relation between parking lot $x_1$ and criterium $y_1$ is {0.6(0.4),0.8(0.5)}, where 0.6 and 0.8 denote the possibility that $x_1$ belongs to PHFSs, and 0.4 and 0.5 denote the probability of the occurrence of possibility, respectively.

Step 2 The loss function is measured by the distance measure of PHFSs, and the hesitant parameter ${\delta }_j$ is used to indicate the degree of hesitation. The loss under $C_j$ is expressed as a distance measure between $h_{ij}$ and the $h_{\min }^j$. The loss under $C_j^c$ is expressed as a distance measure between $h_{ij}$ and the $h_{\max }^j$. The relative loss function ${\lambda }_y^j(x_i)$ of $x_i$ is calculated in Table 2. The aggregated relative loss function ${\lambda }^C(x_i)$ is based on criteria weight vector $\omega =({\omega }_1,{\omega }_2,\cdots ,{\omega }_j)$, which is shown in Table 3.

Table 2. Relative loss function ${\lambda }_y^j(x_i)$ of $x_i$.

${\mathit{x}}_{\mathit{i}}$	${\mathit{C}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{j}}^{\mathit{c}}$
$d_1$	0	$d_h(h_{\max }^j,h_{ij})$
$d_3$	${\delta }_j{d}_h(h_{ij},h_{\min }^j)$	${\delta }_j{d}_h(h_{\max }^j,h_{ij})$
$d_2$	$d_h(h_{ij},h_{\min }^j)$	0

Table 3. Aggregated relative loss function ${\lambda }^C(x_i)$.

${\mathit{x}}_{\mathit{i}}$	${\mathit{C}}_{\mathit{j}}$	${\mathit{C}}_{\mathit{j}}^{\mathit{c}}$
$d_1$	0	$d_h(h_{\max }^j,\underset{j=1}{\overset{n}{\oplus }}{\omega }_j{h}_{ij})$
$d_3$	${\displaystyle {\sum }_{j=1}^n{\omega }_j{\delta }_j{d}_h(h_{ij},h_{\min }^j)}$	${\displaystyle {\sum }_{j=1}^n{\omega }_j{\delta }_j{d}_h(h_{\max }^j,h_{ij})}$
$d_2$	$d_h(\underset{j=1}{\overset{n}{\oplus }}{\omega }_j{h}_{ij},h_{\min }^j)$	0

Step 3 Compute the aggregated loss function ${\lambda }_{}^{k_A}$ by $R_k$ according to the conditional probability $P(A\left|C\right.)$ and the aggregated loss function ${\lambda }_{}^{k_C}$.

Step 4 Compute the threshold parameters ${\alpha }_k$ and ${\beta }_k$ by decision-maker $R_k$ according to (20)–(22).

Step 5 Compute the threshold parameters $\alpha$ and $\beta$ according to Definition 8.

Step 6 Draw a conclusion by comparing the threshold parameters $\alpha$, $\beta$, and the conditional probability of $A$.

6. Numerical Simulation

Let $(U,V,F,P)$ be the probabilistic approximation space, $U=\left\{x_1,x_2\cdots x_n\right\}$ be the set of possible applicants for the parking lots, and $V=\left\{y_1,y_2\cdots y_n\right\}$ be the set of criteria, where $y_1$ stands for safety, $y_2$ stands for convenience, $y_3$ stands for distance, and $y_4$ stands for parking fee. Suppose that three decision-makers are invited by the platform, and the opinion weight vector of the three experts is $u=\{u_1,u_2,u_3\}$ = $(0.3,0.3,0.4)$, and a probabilistic hesitant fuzzy relation $R_k\in HF(U\times V)$, $k=1,2,3$, is provided by each expert by use of Definition 6; the evaluation values represented as probabilistic hesitant fuzzy elements are shown in Table 4, Table 5 and Table 6, respectively. $POS(A)$ corresponds to strict regulation, which means parking lots can receive subsidy, $NEG(A)$ corresponds to non-strict regulation, which means parking lots cannot receive subsidy, and $BND(A)$ corresponds to the delayed decision, which means the parking lots will be further assessed.

Table 4. Relation between parking lots and criteria by $R_1$.

${\mathit{R}}_1$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{0.6(0.5), 0.8(0.2)}	{0.2(0.5), 0.25(0.4)}	{0.2(0.5), 0.3(0.5)}	{0.4(0.5), 0.5(0.4)}
$x_2$	{0.5(0.2), 0.8(0.5)}	{0.2(0.5), 0.4(0.5)}	{0.2(0.5), 0.4(0.5)}	{0.2(0.5), 0.8(0.5)}
$x_3$	{0.4(0.5), 0.6(0.5)}	{0.2(0.5), 0.5(0.1)}	{0.2(0.5), 0.5(0.4)}	{0.3(0.5), 0.5(0.1)}
$x_4$	{0.6(0.5), 0.9(0.3)}	{0.2(0.5), 0.4(0.5)}	{0.5(0.3), 0.6(0.5),}	{0.2(0.5), 0.6(0.5)}

Table 5. Relation between parking lots and criteria by $R_2$.

${\mathit{R}}_2$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{0.1}	{0.1(0.5), 0.3(0.5)}	{0.1(0.5), 0.3(0.5)}	{0.3(0.5), 0.5(0.2)}
$x_2$	{0.2(0.5), 0.5(0.4)}	{0.2(0.5), 0.5(0.2)}	{0.3(0.5), 0.5(0.4)}	{0.4(0.5), 0.5(0.2)}
$x_3$	{0.2(0.5), 0.6(0.5)}	{0.2(0.5), 0.4(0.5)}	{0.1(0.5), 0.5(0.4)}	{0.1(0.5), 0.5(0.3)}
$x_4$	{0.1(0.5), 0.5(0.2)}	{0.2(0.5), 0.5(0.2)}	{0.4(0.5), 0.5(0.2)}	{0.5(0.2), 0.6(0.5)}

Table 6. Relation between parking lots and criteria by $R_3$.

${\mathit{R}}_3$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{0.5(0.4), 0.6(0.5)}	{0.1(0.5), 0.3(0.5)}	{0.1(0.5), 0.5(0.3)}	{0.4(0.5), 0.5(0.4)}
$x_2$	{0.4(0.5), 0.5(0.4)}	{0.4(0.5), 0.6(0.5)}	{0.2(0.5), 0.5(0.4)}	{0.4(0.5), 0.6(0.5)}
$x_3$	{0.1}	{0.2(0.5), 0.5(0.2)}	{0.5(0.2), 0.6(0.5)}	{0.2(0.5), 0.3(0.5)}
$x_4$	{0.2(0.5), 0.5(0.4)}	{0.2(0.5), 0.5(0.2)}	{0.4(0.5), 0.6(0.5)}	{0.2(0.5), 0.5(0.4)}

It is supposed that ${\delta }_1=0.3$, ${\delta }_2=0.5$, ${\delta }_3=0.4$, and ${\delta }_4=0.3$. The relative loss function ${\lambda }_{y_j}^{k_C}(x_i)$ is calculated by expert $R_k$, where $k=1,2,3$, and thus Table 7, Table 8 and Table 9 are obtained, respectively.

Table 7. Relative loss function ${\lambda }_{y_j}^{1_C}(x_i)$ by $R_1$.

${\mathit{R}}_1$		${\mathit{y}}_1$		${\mathit{y}}_2$		${\mathit{y}}_3$		${\mathit{y}}_4$
		${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$	${\mathit{C}}_2^{\mathit{c}}$	${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$	${\mathit{C}}_2^{\mathit{c}}$
$x_1$	$d_1$	0	0.775	0	0.85	0	0.763	0	0.7
	$d_3$	0.0675	0.233	0.075	0.425	0.095	0.305	0.09	0.21
	$d_2$	0.225	0	0.15	0	0.237	0	0.3	0
$x_2$	$d_1$	0	0.700	0	0.775	0	0.775	0	0.625
	$d_3$	0.090	0.210	0.112	0.388	0.090	0.31	0.113	0.187
	$d_2$	0.300	0	0.225	0	0.225	0	0.375	0
$x_3$	$d_1$	0	0.850	0	0.887	0	0.775	0	0.850
	$d_3$	0.045	0.255	0.056	0.444	0.090	0.31	0.045	0.255
	$d_2$	0.150	0	0.113	0	0.225	0	0.150	0
$x_4$	$d_1$	0	0.813	0	0.775	0	0.663	0	0.700
	$d_3$	0.561	0.244	0.112	0.388	0.135	0.265	0.09	0.210
	$d_2$	0.187	0	0.225	0	0.337	0	0.300	0

Table 8. Relative loss function ${\lambda }_{y_j}^{2_C}(x_i)$ by $R_2$.

${\mathit{R}}_2$		${\mathit{y}}_1$		${\mathit{y}}_2$		${\mathit{y}}_3$		${\mathit{y}}_4$
		${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$	${\mathit{C}}_2^{\mathit{c}}$	${\mathit{C}}_3$	${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$
$x_1$	$d_1$	0	0.925	0	0.850	0	0.85	0	0.838
	$d_3$	0.023	0.277	0.075	0.425	0.06	0.34	0.048	0.252
	$d_2$	0.075	0	0.150	0	0.150	0	0.162	0
$x_2$	$d_1$	0	0.775	0	0.85	0	0.738	0	0.775
	$d_3$	0.067	0.233	0.075	0.425	0.105	0.295	0.067	0.233
	$d_2$	0.225	0	0.150	0	0.262	0	0.225	0
$x_3$	$d_1$	0	0.700	0	0.775	0	0.712	0	0.85
	$d_3$	0.090	0.210	0.112	0.388	0.115	0.285	0.045	0.255
	$d_2$	0.300	0	0.225	0	0.288	0	0.15	0
$x_4$	$d_1$	0	0.888	0	0.850	0	0.775	0	0.700
	$d_3$	0.034	0.266	0.075	0.425	0.090	0.310	0.090	0.210
	$d_2$	0.112	0	0.150	0	0.225	0	0.300	0

Table 9. Relative loss function ${\lambda }_{y_j}^{3_C}(x_i)$ by $R_3$.

${\mathit{R}}_3$		${\mathit{y}}_1$		${\mathit{y}}_2$		${\mathit{y}}_3$		${\mathit{y}}_4$
		${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$	${\mathit{C}}_2^{\mathit{c}}$	${\mathit{C}}_3$	${\mathit{C}}_1$	${\mathit{C}}_1^{\mathit{c}}$	${\mathit{C}}_2$
$x_1$	$d_1$	0	0.625	0	0.85	0	0.85	0	0.7
	$d_3$	0.113	0.187	0.075	0.425	0.060	0.340	0.090	0.210
	$d_2$	0.375	0	0.150	0	0.150	0	0.300	0
$x_2$	$d_1$	0	0.700	0	0.625	0	0.775	0	0.725
	$d_3$	0.090	0.210	0.188	0.312	0.09	0.31	0.083	0.217
	$d_2$	0.300	0	0.375	0	0.225	0	0.275	0
$x_3$	$d_1$	0	0.925	0	0.850	0	0.700	0	0.812
	$d_3$	0.023	0.277	0.075	0.425	0.120	0.28	0.056	0.244
	$d_2$	0.075	0	0.15	0	0.3	0	0.188	0
$x_4$	$d_1$	0	0.775	0	0.850	0	0.625	0	0.775
	$d_3$	0.067	0.233	0.075	0.425	0.150	0.250	0.067	0.233
	$d_2$	0.225	0	0.150	0	0.375	0	0.225	0

The criterium weight vector is supposed as $\omega =(0.3,0.3,0.2,0.2)$; the aggregated loss function ${\lambda }^{k_C}(x_i)$ by $R_k$ is represented in Table 10. With the help of the PHFWA operator in Definition 8, the distance measure is calculated according to Definition 9 and Table 3. It is generally considered that $h_{\min }^j=0$, $h_{max}^j=1$; taking $x_1{y}_j$ under expert $R_1$ as an example, when the membership degree is the same, the probability adds up to a total number. The result is as follows:

$$\begin{array}{l}PHFW{A}_{x_1,y}=\underset{j=1}{\overset{n}{\oplus }}{\omega }_j{h}_{1j}\\ =\left\{\begin{array}{l}0.31(0.11),0.32(0.088),0.33(0.198),0.34(0.088),0.35(0.183),0.36(0.15),0.37(0.035),\\ 0.38(0.063),0.39(0.063)\end{array}\right.\end{array}$$

$$d_h(\underset{j=1}{\overset{n}{\oplus }}{\omega }_j{h}_{1j},h_{\min }^1)=0.291$$

Table 10. Aggregated loss function ${\lambda }_{}^{k_C}(x_i)$ by $R_k$, $k=1,2,3$.

		${\mathit{R}}_1$		${\mathit{R}}_2$		${\mathit{R}}_3$
$x_1$	$d_1$	0	0.751	0	0.820	0	0.660
	$d_3$	0.079	0.300	0.051	0.329	0.086	0.294
	$d_2$	0.291	0	0.180	0	0.340	0
$x_2$	$d_1$	0	0.360	0	0.720	0	0.570
	$d_3$	0.101	0.279	0.077	0.303	0.118	0.262
	$d_2$	0.370	0	0.280	0	0.430	0
$x_3$	$d_1$	0	0.795	0	0.700	0	0.780
	$d_3$	0.057	0.323	0.093	0.287	0.065	0.315
	$d_2$	0.205	0	0.300	0	0.220	0
$x_4$	$d_1$	0	0.680	0	0.755	0	0.610
	$d_3$	0.095	0.285	0.069	0.311	0.086	0.294
	$d_2$	0.320	0	0.245	0	0.390	0

So, the aggregated loss function ${\lambda }^{k_C}$ associated with the state $C$ can be determined, and the results are listed in Table 11.

Table 11. Aggregated loss function ${\lambda }_{}^{k_C}$ by $R_k$, $k=1,2,3$.

	${\mathit{\lambda }}_{11}^{{\mathit{k}}_{\mathit{C}}}$	${\mathit{\lambda }}_{31}^{{\mathit{k}}_{\mathit{C}}}$	${\mathit{\lambda }}_{21}^{{\mathit{k}}_{\mathit{C}}}$	${\mathit{\lambda }}_{12}^{{\mathit{k}}_{\mathit{C}}}$	${\mathit{\lambda }}_{32}^{{\mathit{k}}_{\mathit{C}}}$	${\mathit{\lambda }}_{22}^{{\mathit{k}}_{\mathit{C}}}$
$R_1$	0	0.083	0.296	0.646	0.296	0
$R_2$	0	0.073	0.251	0.749	0.308	0
$R_3$	0	0.089	0.345	0.655	0.291	0

The platform conducts a self-evaluation $A\in HF(V)$, which is expressed by:

$$A=\frac{\{0.85(0.4),0.8(0.5)\}}{y_1}+\frac{\{0.9(0.5),0.8(0.5)\}}{y_2}+\frac{\{0.9(0.4),0.8(0.5)\}}{y_3}+\frac{\{0.76(0.5),0.8(0.5)\}}{y_4}$$

Since $C=\frac{\left\{1\right\}}{y_1}+\frac{\left\{1\right\}}{y_2}+\frac{\left\{1\right\}}{y_3}+\frac{\left\{1\right\}}{y_4}$, the conditional probability $P(A\left|C\right.)$ can be computed based on Definition 8, as follows:

$$P(A\left|C\right.)=\frac{{\displaystyle {\sum }_{y\in V}(s(h_A(y))\cdot s(h_C(y)))}}{{\displaystyle {\sum }_{y\in V}s(h_C(y))}}=0.824$$

So, the aggregated loss function ${\lambda }^{k_{\mathrm{A}}}$ can be obtained in Table 12. Conditional probability of $A$ is represented in Table 13.

Table 12. Aggregated loss function ${\lambda }_{}^{k_A}$ by $R_k$, $k=1,2,3$.

	${\mathit{\lambda }}_{11}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{31}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{21}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{12}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{32}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{22}^{{\mathit{k}}_{\mathit{A}}}$
$R_1$	0	0.068	0.244	0.532	0.244	0
$R_2$	0	0.060	0.207	0.617	0.254	0
$R_3$	0	0.073	0.284	0.540	0.240	0

Table 13. Conditional probability of $A$.

${\mathit{R}}_{\mathit{k}}$	${\mathit{R}}_1$	${\mathit{R}}_2$	${\mathit{R}}_3$
$P(A|H_{R_k}(x_1))$	0.817	0.816	0.816
$P(A|H_{R_k}(x_2))$	0.819	0.818	0.822
$P(A|H_{R_k}(x_3))$	0.822	0.825	0.832
$P(A|H_{R_k}(x_4))$	0.823	0.817	0.825

According to (20)–(22), the threshold parameters ${\alpha }_k$ and ${\beta }_k$ are computed by the decision-makers $R_k$. In Table 14, the comprehensive parameters $\alpha$ and $\beta$ are obtained by the opinion weight vector $u=(0.4,0.3,0.3)$ of the three decision-makers.

$$\alpha ={\displaystyle \sum _{k=1}^3{u}_k}{\alpha }_k=0.819, \beta ={\displaystyle \sum _{k=1}^3{u}_k}{\beta }_k=0.581$$

Let $\delta =0.67$

$$\begin{array}{l}Po{s}_{{\displaystyle {\sum }_{k=1}^3}}^{0.819,0.581}(A)={\underset{¯}{\Re }}_{{\displaystyle {\sum }_{k=1}^3}}^{0.67,0.819}(A)=\{x|\frac{\left|\right\{R_k|p(A|H_{R_k}(x)(y)|\ge 0.819\}|}{3}\ge 0.67,\\ x\in U,y\in V,k=1,2,3\}\end{array}$$

$$\begin{array}{l}Ne{g}_{{\displaystyle {\sum }_{k=1}^3}}^{0.819,0.581}(A)=U-{\overline{\Re }}_{{\displaystyle {\sum }_{k=1}^3}}^{0.67,0.819}(A)=\{x|\frac{\left|\right\{R_k|p(A|P{H}_{R_k}(x)(y)|\le 0.581\left\}\right|}{3}>0.33,\\ x\in U,y\in V,k=1,2,3\}\end{array}$$

$$Bn{d}_{{\displaystyle {\sum }_{k=1}^3{R}_k}}^{0.819,0.581}(A)=U-Po{s}_{{\displaystyle {\sum }_{k=1}^3{R}_k}}^{0.819,0.581}(A)-Ne{g}_{{\displaystyle {\sum }_{k=1}^3{R}_k}}^{0.819,0.581}(A)$$

As a result, parking lots $\left\{x_2,x_3,x_4\right\}$ can receive a subsidy of the platform, and $\left\{x_1\right\}$ will be further assessed.

Table 14. Threshold parameters ${\alpha }_k$ and ${\beta }_k$.

${\mathit{R}}_{\mathit{k}}$	${\mathit{R}}_1$	${\mathit{R}}_2$	${\mathit{R}}_3$
${\alpha }_k$	0.804	0.856	0.803
${\beta }_k$	0.580	0.633	0.532

7. Discussion

To show the advantages of the new model with a probabilistic hesitant fuzzy relation, the model in this paper is compared with references [18,40,41]. The evaluation values of the relation between the parking lots and criteria are represented as hesitant fuzzy elements, which are shown in Table 15, Table 16 and Table 17. The hesitant parameter $\delta$, the criteria weight vector $\omega$, and the opinion weight vector $u$ of the experts remain the same.

Table 15. New relation between parking lots and criteria by $R_1$.

${\mathit{R}}_1$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{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 16. New relation between parking lots and criteria by $R_2$.

${\mathit{R}}_2$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{0.1}	{0.1,0.3}	{0.1,0.3}	{0.3,0.5}
$x_2$	{0.2,0.5}	{0.2,0.5}	{0.3,0.5}	{0.4,0.5 }
$x_3$	{0.2,0.6}	{0.2,0.4}	{0.1,0.5}	{0.1,0.5}
$x_4$	{0.1,0.5 }	{0.2,0.5}	{0.4,0.5}	{0.5,0.6}

Table 17. New relation between parking lots and criteria by $R_3$.

${\mathit{R}}_3$	${\mathit{y}}_1$	${\mathit{y}}_2$	${\mathit{y}}_3$	${\mathit{y}}_4$
$x_1$	{0.5,0.6}	{0.1,0.3}	{0.1,0.5}	{0.4,0.5}
$x_2$	{0.4,0.5}	{0.4,0.6}	{0.2,0.5}	{0.4,0.6}
$x_3$	{0.1}	{0.2,0.5}	{0.5,0.6}	{0.2,0.3}
$x_4$	{0.2,0.5}	{0.2,0.5}	{0.4,0.6}	{0.2,0.5}

In accordance with the previous calculation, the new aggregated loss function ${\lambda }_{}^{k_A}$ by $R_k$ and the new conditional probability of $A$ are shown in Table 18 and Table 19.

Table 18. New aggregated loss function ${\lambda }_{}^{k_A}$ by $R_k$, $k=1,2,3$.

	${\mathit{\lambda }}_{11}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{31}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{21}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{12}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{32}^{{\mathit{k}}_{\mathit{A}}}$	${\mathit{\lambda }}_{22}^{{\mathit{k}}_{\mathit{A}}}$
$R_1$	0	0.127	0.365	0.460	0.187	0
$R_2$	0	0.102	0.386	0.554	0.212	0
$R_3$	0	0.118	0.310	0.515	0.196	0

Table 19. New conditional probability of $A$.

${\mathit{R}}_{\mathit{k}}$	${\mathit{R}}_1$	${\mathit{R}}_2$	${\mathit{R}}_3$
$P(A|H_{R_k}(x_1))$	0.833	0.819	0.831
$P(A|H_{R_k}(x_2))$	0.832	0.831	0.832
$P(A|H_{R_k}(x_3))$	0.834	0.836	0.836
$P(A|H_{R_k}(x_4))$	0.838	0.827	0.835

It can be obtained from Formulas (20)–(22) in Section 4 and Table 18 that $\alpha ={\displaystyle \sum _{k=1}^3{u}_k}{\alpha }_k=0.712$ and $\beta ={\displaystyle \sum _{k=1}^3{u}_k}{\beta }_k=0.455$.

Compared with the data in Table 19, the results show that all shared parking lots are eligible for subsidies, which is not realistic under the supervision of the platform. Therefore, the regulation based on the probabilistic hesitant fuzzy set is consistent with reality.

8. Limitations and Recommendations

Decision-making based on evolutionary game theory and administrative management have been studied before, which gives the suggestion that parking lots should be regulated in one case and not in others. There are no recommendations for parking lots that have been operated to conform to shared standards. The original goal of the paper was evaluate the parking lots so that idle parking slots could be shared, which is a better attempt to combine three-way decision-making and PHFS, which means that not only the fuzziness of a dynamic parking lot is taken into account, but also the indecisiveness of experts in some special cases. However, the model of this paper gives the evaluation of a delayed decision for the parking lot, leaving the parking lot to continue to improve in order to meet the requirements. On one hand, there is novelty in the supervision of the sharing economy. On the other hand, parking lots that are not subsidized may refuse to accept drivers who have parking demands, which leads to competition among parking lots and a portion of vehicles cruising on the road.

Two cases of workdays and non-workdays (not including Chinese traditional festivals)are assumed, and we express the probabilities of the two cases in the form of a probabilistic hesitant fuzzy element. Because of the conditions, there are still several limitations in this research. The research of this paper is only aimed at the case of urban shared parking in China, so whether the conclusion is applicable to other areas of the world remains to be verified. Although this method can realize the supervision of shared parking in theory, whether it can promote the sharing behavior continuously in practice remains to be studied. In addition, parking lot rent-seeking for subsidies is not analyzed and the monopolistic behavior of parking lots in shared parking is not considered, which will be considered in future studies.

9. Conclusions

Probabilistic hesitant fuzzy sets offer a more sophisticated means of describing complex situations compared to hesitant fuzzy sets. To address scenarios with incomplete decision information, we propose a three-way decision model based on multi-granularity probabilistic hesitant fuzzy DTRS. The integration of probabilistic hesitant fuzzy sets and DTRS presents an innovative and practical approach for evaluating platform supervision. The primary contributions of this work can be summarized as follows. We propose regulating shared parking by leveraging the advantages of PHFSs to effectively address uncertainty, targeting the growing demand for shared parking spaces. This approach is beneficial in alleviating urban traffic congestion. The proposed PHFWA operators capture the relationships between input parameters. Consequently, we introduce a decision-making algorithm based on mean and geometric operators to accomplish this goal. The model’s strength lies in not only avoiding incorrect decision-making due to information loss, but also in providing guidance for subsequent parking lot construction. This method is applicable to other idle resource-sharing regulations, such as shared cars and shared apartments, as well as various fuzzy environments, including medical diagnosis and supplier selection. Considering the influence of government factors that impact decision-making under uncertain circumstances, future research will focus on refining the scope of subsidies and penalties. Additionally, we plan to integrate prospect theory into PHFSs to address decision-making problems involving risk-averse decision-makers.