Two-Sided Matching Decision Making with Multi-Attribute Probabilistic Hesitant Fuzzy Sets

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Introduction
Two-sided matching decision-making (TSMDM) consists of two sides of matching agents and belongs to the decision-making problems [1], which was first proposed and applied by Gale and Shapley in a study of matchmaking and college admission [2].The agent, also called as the intermediary, is what determines the result of the matching process but does not actually participate in the matching process directly.The matching agents could play various roles in various scenarios requiring decisionmaking [3].If you search or buy a product online today, you will find more of the same products on the website tomorrow; this indicates that the matching system has automatically matched you as a potential customer.Both sides of matching agents participate in the matching process, and using a given matching technique, all matching agents are evaluated by those on the other side.The formation of a perfect matching scheme which is satisfied by all matching agents must be facilitated by the intermediary based on the preferences or evaluations provided by the matching agent.
TSMDM is now used in a wide range of fields, such as marriage matching problems [4,5], colleges admissions [6], volunteer team dispatching [7], human resource management [8], economic management matching [9], public-private partnership [10], and knowledge service matching [11].The two-sided matching theory has been used in a variety of areas recently, along with the development of the market economy and the level of science and technology.For instance, Wu et al. [12] studied the supply-demand issue in transport services and developed the corresponding TSMDM method.Wang et al. [13] has developed a novel matching method for solving the freight source matching problem.Liu et al. proposed a unique TSMDM method with the objective of achieving satisfying matching between surgeons and patients [14].For the purpose of resolving the existing house sellerbuyer matching problem, Jiang et al. [15] proposed a novel TSMDM.
Meantime, a significant number of scholars have studied the multi-attribute TSMDM problem.For instance, Jiang et al. [16] proposed a multi-objective non-linear model to address one-shot multi-attribute exchange problems.Based on the Choquet integral aggregation operator, a novel TSMDM method was developed by Liang et al. [17], which was used to solve the multi-attribute TSMDM problems considering q-Rung Orthopair fuzzy information.Liang et al. [18] developed a novel TSMDM model based on the strict order number information.In light of multi-granularity probabilistic linguistic MARCOS, Lu et al. [19] proposed a multi-attribute two-sided matching method.Some researchers studied the TSMDM problem with complete or partial preference ordering in the previous studies.For instance, Pu et al. [20] proposed a novel TSMDM method under the goal of maximizing intermediary revenue and overall agent satisfaction.Fan et al. [21] studied the TSMDM problem with uncertain preference ordinal and proposed the corresponding TSMDM model.A novel TSMDM method was developed by Chen et al. [22] after studying the TSMDM problem in perspective of psychological behaviors.Qin et al. [23] proposed an optimization method for resolving the TSMDM problem with incomplete weak preference ordering.To express the evaluation information in the practical TSMDM problem, linguistic evaluation was also applied.For instance, Lin et al. [24] studied the multi-attribute TSMDM problems with 2-tuple linguistic information and proposed a new TSMDM method.Considering the multi-granular HFLTSs environment, Zhang et al. [25] developed a comprehensive TSMDM method considering the stable one-to-one matching constraints.
Researches conducted by many scholars have provided sufficient solutions for the two-sided matching problem under different scenarios.However, decision-makers find it difficult to deliver precise and concise information given the complexity of the real world.In certain situations, decisionmakers are unable to convey the whole significance of the evaluation information.To describe the evaluation information, several fuzzy information express tools would be employed, such as interval numbers, fuzzy numbers, hesitant fuzzy sets, intuitionistic fuzzy sets [26] and others.PHFS, which is an important extended form of the hesitant fuzzy set and a close and effective combination of membership degree and probability distribution information, was first proposed by Zhu [27].Probabilistic hesitating fuzzy sets, as compared to HFS, are more capable of dealing with and explain the information of preference evaluation in complex decision-making situations.In real world, significant and vital decisions are usually made by the group which consists of multiple members.Due to the inaccurate estimations and lack of expertise, the evaluation information provided by experts may have different assessment values.Therefore, the information of the probabilistic hesitant fuzzy element (PHFE) may be used to express the overall opinion formed by these different evaluation results.For instance, suppose a company invites ten experts to evaluate a scheme (score between [0, 1]), and let's say that six of the ten experts give the scheme a score of 0.5 and four experts give it a score of 0.7.The evaluation information can be provided by the PHFE {0.5 |0.6, 0.7| 0.4}.The PHFS theory can be applied to handle imperfect, vague, and imprecise information.Thus PHFS has been widely applied in economics [28], emergency management [29,30], and decision-making [31].Meanwhile, some theoretical foundations have been established, such as aggregation operations [32], distance measures [33], and entropy measures [34].Because of the major difference in individual cognition and the lack of knowledge structure, the two-sided agents are used to express the evaluation information by PHFS.Based on this overview of previous literature, the probability hesitant fuzzy environment in multi-attribute TSMDM problems has not been given much attention.Therefore, a TSMDM method, considering the PHFS with incomplete attribute weight information, will be proposed in this paper.By using the TOPSIS method, the solution strategy is developed and the matching result takes into account the maximum degree of satisfaction on both sides of the matching agents.Additionally, a supply-demand example for catering services will be provided to highlight the usefulness and viability of the proposed model.The new aspects of this paper are: (1) the development of a novel two-sided matching decision-making method in a probability hesitant fuzzy environment; and (2) the application of the proposed method to the two-sided matching problem with incomplete attribute weight.
The rest of this paper is organized as follows.Some preliminaries about TSMDM and PHFS will be presented in Section 2. The proposed problem is explained in Section 3, and builds the framework for the proposed method.In Section 4, a case is provided to demonstrate the effectiveness of the proposed method.Additionally, Section 4 explains the characteristics of the proposed TSMDM method.Finally, the main contributions of the proposed method are presented in Section 5.

Preliminaries
The concept of two-sided matching is given in this section, and then some related theories of PHFSs are introduced, including the concept, operation rules, score function and aggregation operation.Finally, it discusses the distance measure of PHFSs and TOPSIS method.

Two-Sided Matching
Two matching agent sets, A and B, are given in this subsection.
be the set of n matching objects, where B j denotes the jth agent in B, and thus satisfies n Definition 1 [35].Suppose μ: Then, μ refers to the matching scheme between set A and set B, where μ(A i ) = B j denotes that A i is matched with B j in μ and the (A i , B j ) is defined as a matching pair, μ(A i ) = B k denotes that A i is not matched with B k in μ.Specially, when μ(B j ) = B j , then B j is unmatched in μ.As n ≥ m ≥ 2, each matching object in A must be matched with one matching object in B and n − m matching objects in B are unmatched.

Probabilistic Hesitant Fuzzy Sets
HFS was first proposed by Torra [26] and further expressed in mathematical symbol by Xia et al. [36].Unlike other extensions of the fuzzy set, HFS allows the membership degree of an element to a set presented by several possible values, which can express the uncertain information more comprehensively.However, the probability of each membership degree is the same.To better represent the preference information, by incorporating probability to describe the preference information in HFS, Zhu [27] proposed the concept of PHFS.Compared with HFSs, the following PHFSs can reserve more original information.
Definition 2 [27,32].Let X be a fixed set.Then, a PHFS on X is ) is a set of elements and γ i (x) |p i (x) is a subset of [0, 1] denoting the hesitant fuzzy information with probabilities to the set H. For convenience, h(γ i (x) |p i (x)) is called a probabilistic hesitant fuzzy element (PHFE), and PHFE h(γ i (x) where p i is the probability of the membership degree γ i and #h i=1 p i = 1,γ i |p i is considered as a term of the PHFE, #h is the total number of different membership degrees.In this paper, we arrange the terms γ i |p i in increasing order based on the value of γ i .

Remark 1: In Definition 2, #h
i=1 p i = 1 states that the PHFE includes the complete probabilistic information.
#h i=1 p i < 1 states that the probability information is incomplete.Therefore, the following formula can be used to transform the PHFE with incomplete probability information into the PHFE with complete probability information: may be revised according to the multiplier of the membership degree γ i and the corresponding probability p i , and then re-order the terms γ i p i in increasing order.For instance (0.5 |0.4,0.6 |0.2, 0.7 |0.4 ) can be revised as (0.12, 0.2, 0.28).
Definition 3 [27,32]: Let h(p), h 1 (p 1 ), h 2 (p 2 ) be three PHFEs and α > 0, then some operation rules among them are defined as: Definition 4 [27,32] According to the score function, the deviation function is defined by: According to the score functions, we can compare PHFEs h 1 (p) and h 2 (p) based on the following rules:

Aggregation Operators and Distance Measures for PHFEs
Zhang et al. [32] proposed the probabilistic hesitant fuzzy weighted averaging (PHFWA) operator and the probabilistic hesitant fuzzy weighted geometric (PHFWG) operator, both of which are very useful in decision-making using probabilistic hesitant fuzzy information.
Definition 5 [32].Let h 1 (p 1 ), h 2 (p 2 ) • • • h n (p n ) be PHFEs, and w j n j=1 w j = 1 be the weight of PHFEs h j (p j ), then a probabilistic hesitant fuzzy weighted averaging (PHFWA) operator and a probabilistic hesitant fuzzy weighted geometric (PHFWG) operator are stated as follows: These two operators can transform the fuzzy decision information of the decision makers into aggregated decision information to identify the best option out of all feasible choices.
The definitions of the Hamming distance and the Euclidean distance of PHFEs are given as follows: Definition 6 [29].Let h 1 (p 1 ) and h 2 (p 2 ) be two PHFEs.If #h = #h 2 = #h 1 , the number of elements in h 1 (p 1 ) and h 2 (p 2 ) are the same, then the normalized probabilistic hesitant fuzzy Hamming distance measure between h 1 (p 1 ) and h 2 (p 2 ) is defined as d H (h 1 , h 2 ), and the normalized probabilistic hesitant fuzzy Euclidean distance measure between h 1 (p 1 ) and h 2 (p 2 ) is defined as where γ 1σ (i) and γ 2σ (i) are the ith biggest values in h 1 (p 1 ) and h 2 (p 2 ).
Remark3: In most cases, #h 2 = #h 1 .For the sake of calculation convenience, we need to treat the small-length probabilistic hesitancy fuzzy elements with equal length.The short-length probability hesitancy fuzzy element is supplemented by a smaller or larger membership degree element depending on the decision-makers' various risk preferences.If the decision maker is risk-averse, add the element with the lowest value; if the decision maker likes taking risk, add the element with the highest value; if the decision maker is risk-neutral, add the average of the maximum and minimum values in the set.Let the probability of adding elements be 0, so that the correlation calculation remains unchanged even after adding elements.In this paper, the decision makers are assumed to be pessimists who expect unfavorable outcomes and may add the minimum value, and we define that the probability of each added value is 0. For example, h 1 (p 1 ) and h 2 (p 2 ) are two PHFEs, let h 1 (p 1 ) = (0.5 |0.5, 0.7 |0.5 ), h 2 (p 2 ) = (0.5 |0.3, 0.6 |0.3, 0.7 |0.4 ).Apparently, #h 2 > #h 1 ; then, we extend h 1 (p 1 ) until it has the same length as h 2 (p 2 ), that is h 1 (p 1 ) = (0.5 |0, 0.5 |0.5, 0.7 |0.5 ).According to the Hamming distance measure, d H (h 1 , h 2 ) can be calculated as: Definition 7 [29].The probabilistic hesitant fuzzy Positive Ideal Solution (PIS) is given as: , and the probabilistic hesitant fuzzy Negative Ideal Solution (NIS) is

The TOPSIS Method for PHFS
The TOPSIS method, which was first proposed by Hwang and Yoon, is essential for processing information from multi-attribute evaluations and is a topic of much scholarly discussion [37].According to the PHFSs Hamming distance measure and the basic idea of traditional TOPSIS method, let D − be the distance measure from the NIS and D + be the distance measure from the PIS.Obviously, the larger the distance D + the better the alternative, while the smaller the distance D − the better the alternative [33].The alternative is defined as follows in order to represent the alternative closeness coefficient: The closeness coefficients C for each alternative are calculated for both, the distance D + and the distanceD − simultaneously.The optimal solution is the one that comes closest to the positive ideal solution and is farthest from the negative ideal solution.From the above analysis, we have come to know that the higher the closeness coefficient, the better the alternative.

The Approach of TSMDM with PHFSs Information
In this section, the TSMDM problem with PHFSs is introduced and a novel TSMDM method is proposed.

Problems Description
Two agent sets are given in the two-sided matching problem which is addressed in this paper.Considering the fuzziness of the thinking of the matching subject and the disparity in knowledge organization, the evaluation information given by the two-sided agent is presented in the form of PHFS, and the attribute weight information given is also incomplete.The problem studied in this paper is to promote the formation of equitable bilateral matching results based on the information of probabilistic hesitancy fuzzy sets given by two-sided agents.The main notations used in this section are presented as follows: The set of matching objects on the side B, B j denotes the j th matching object in B.
The set of attributes used by matching objects A when providing PHFE assessments over matching objects B, where is the kth attribute that the matching objects A pay attention to.
The set of attributes used by matching objects B when providing PHFE assessments over matching objects A, where is the lth attribute that the matching objects B pay attention to.
The weight vector of the attribute used by A i , where w The weight vector of the attributes used is the weight of the lth attribute in

The Maximizing Deviation Method to Obtain the Weights of Attribute
Weight plays an essential role in reaching a reasonable decision result.The matching subject is unable to provide a comprehensive attribute weight vector due to the diversity and fuzziness of the information in the real decision-making environment.At the same time, Wang [38] puts forward a maximum deviation method to determine the attribute weight, and the core idea is to give the attribute weight based on the evaluation information given by the decision-maker.Some models will be proposed to determine the attribute weight vector for a two-sided agent based on the maximum deviation method.
In relation to the kth attribute C A k in C A , if matching object A i provides assessment to the matching object B j and B f , denoted by h A ijk and h A ifk , then Definition 6 states that: The total deviation among A i 's PHFE assessments over B can be calculated by: We can evaluate the total variance between A i 's PHFE assessments over B with regard to all attributes by considering the weights of each attribute.
Then, a constrained condition s k=1 (w } is introduced based on Wang's analysis.On this basis, the optimal weight vector is determined.
Therefore, a model for computing the attribute weight for A i can be presented as: By using the Lagrange multiplier method and normalizing the weight vector, we have Similar to that, the attribute weight vector for B j , i.e., w B j = {w t } can be calculated as: Furthermore, the matrices ) m×n can be calculated by aggregating the evaluation values based on the PHFWA operator.

The Satisfaction Degrees
First, the Hamming distance measure between the comprehensive evaluation of each matching agent and their corresponding positive and negative ideal evaluation is calculated, that is: Then, the closeness coefficients of each matching agent could be computed by: The degree of divergence within the matching agent sets could clearly be observed in the closeness coefficients of each matching.For the matching agent A i to the matching agent B j , the small the distance d + aij and the larger the distance d − aij , the higher the closeness coefficient ϕ aij .For the matching agent B j to the matching agent A i , the small the distance d + bij and the larger the distance d − bij , the higher the closeness coefficient ϕ bij .The higher the closeness coefficient ϕ aij meaning the higher the satisfaction of the matching agent A i to the matching agent B j ; the higher the closeness coefficient ϕ bij meaning the higher the satisfaction of the matching agent B j to the matching agent A i .From the above analysis, we can find that the closeness coefficient information can be utilized to measure the satisfaction degree.Therefore, we can establish the satisfaction degree matrices S A = (ϕ aij ) mn and S B = (ϕ bij ) mn to represent the satisfaction degree of the matching agent A i to the matching agent B j , and the satisfaction degree of the matching agent B j to the matching agent A i , respectively.

The TSMDM Model for PHFS
Additionally, a TSMDM model could be developed to obtain the best matching.Suppose x ij represents a binary variable, when x ij = 0; it implies that A i is not matched with B j ; x ij = 1 implies that A i is matched with B j and thus we have a matching pair (A i , B j ).Consequently, a multiobjective programming model can be constructed as follows based on thorough consideration of agent satisfaction maximization: In model (16), formula (a) and formula (b) aim to maximize the overall satisfaction.And formula (c) and formula (d) refer to the one-to-one matching constrains.

Model Solution
We apply the linear weighted method to solve the multi-objective optimization model, and thus model ( 17) could be constructed as follows: where ϕ ij = w A ϕ aij + w B ϕ bij ; w A and w B refer to the different importance of two-sided agent sets, which satisfies w A , w B ∈ (0, 1), w A + w B = 1.For both sides in the actual TSMDM, the goal Z A and Z B have the same status, we can allow w A = w B , if the goal Z A is more important than Z B .We can also let w A > w B .Finally, we can use Matlab or Lingo 11 to solve model (17).

The Steps
In conclusion, the steps of the proposed TSMDM approach can be summed up as follows (see Fig. 1 below): Algorithm: 1 Input: evaluation matrices H A i = (h A ijk ) m×s and H B j = (h B ijl ) n×t Output: matching scheme Begin For PHFE in evaluation matrices ) n×t Compute the attribute weight vector according to the Eqs.( 8)-( 13) For attribute weight in attribute weight vector w A i For attribute weight in attribute weight vector w B j Do Determine the comprehensive evaluation matrices mn Do Construct the satisfaction degree matrices S A = (ϕ aij ) mn , S B = (ϕ bij ) mn For satisfaction degree in satisfaction degree matrices S A = (ϕ aij ) mn , S B = (ϕ bij ) mn Do Construct multiple objective model ( 16) For multiple objective model ( 16) Do Transform the multiple objective model ( 16) into a single objective model (17) Then determine the matching scheme Step 1: Construct evaluation matrices the PHFEs information given by two-sided agents.
Step 2: Compute the attribute weight vector w A i = {w t } according to Eqs. ( 12) and (13).

on the attribute weight vector w A i = {w
Step 4: According to Eq. ( 13), determine the distance matrix mn by calculating the distance between the comprehensive evaluation of the matching agent and the positive and negative ideal evaluation value d Step 5: Construct the satisfaction degree matrices S A = (ϕ aij ) mn , S B = (ϕ bij ) mn by using Eq. ( 14).
Step 6: Construct model ( 16) based on the satisfaction matrix S A = (ϕ aij ) mn and S B = (ϕ bij ) mn , and then transform it into model (17) using the linear weighted method.

Illustrative Example and Discussion
To demonstrate the proposed approach, an example of lunch supply and demand matching is provided in this section.

Illustrative Example
Because of operational constraints and limited space, primary schools are unable to provide lunch for both students and teachers.Therefore, their lunch supply needs to be outsourced and completed by external catering companies.The local education authority of a city intends to provide lunch to students in three primary schools in order to improve public service and reduce chances for students to leave school.The local education authority conducts a comprehensive survey of the situation and information summary of the actual activity, and invites five catering companies to bid for three primary schools.An enterprise can only bid effectively in one school for a prolonged period of time due to the financial strain and service quality.To establish a compatible match and maintain a stable, strong partnership that benefits both sides, the local education authority makes the three primary schools A i (i = 1, 2, 3) evaluate five catering companies B j (j = 1, 2, 3, 4, 5) at four aspects of the management experience, safety, health and the professional qualifications of the company, which are regarded as the four attributes C A k (k = 1, 2, 3, 4).Moreover, let the five companies B j (j = 1, 2, 3, 4, 5) evaluate the three primary schools A i (i = 1, 2, 3) at these four given attributes C B k (k = 1, 2, 3, 4), which are the return of investment, the distance between school and company, the student amount, and the credibility of the school.To be able to obtain a reasonable evaluation, 3 schools and 5 companies invite experts to give the evaluation value anonymously based on the given attributes.By summarizing the expert assessments, they provide the assessment information to the education bureau/commission as a probability hesitation fuzzy set.We can take an example of the assessment of A 1 to B 1 with respect to the attribute C A 1 .If three experts support 0.5, one expert suggests 0.4, and one expert insists on 0.3, then the final assessment is determined by PHFE {0.5 |0.6, 0.4 |0.2, 0.3 |0.2 }.Certainly, the PHFS assessments will be more reasonable and include more information.Similarly, all the PHFS evaluation matrices of candidates on the other side are obtained and shown in Tables 1-8.
Step 1: Construct evaluation matrices.At the same time, there are differences in the number of membership elements among different PHFEs.Decision-makers are therefore considered to be risk-averse.Based on this, the PHFE, which has a smaller number of membership degrees, is added with the minimal value, and the probability of each additive value is 0. In accordance with the revised method of Definition 1, the PHFE of PHFS evaluation matrix can be revised, and then the PHFS evaluation matrices can be modified and shown in Tables 9-16.
According to the satisfaction degree formula, the satisfaction degree of the matching agent A i to the matching agent B j , and the satisfaction degree of the agent B j to the agent A i can be calculated and stated as follows: The optimization model can be solved by utilizing the Lingo 11 software, so that we can get x 12 = x 25 = x 33 = 1, and the others x ij = 0. Therefore, the optimal matching pairs are: (A 1 , B 2 ), (A 2 , B 5 ), (A 3 , B 3 ), which shows that primary school A 1 matches the catering company B 2 , primary school A 2 matches the catering company B 5 , and primary school A 3 matches the catering company B 3 .

Sensitivity Analysis and Discussion
To better illustrate the impact of the weights w A and w B in solving model ( 16), the paper conducts sensitivity analysis of the model.For simplicity, different values of weights S A = (ϕ aij ) mn and S B = (ϕ bij ) mn are discussed, and the results are shown in Table 17.

Case Weight values
Matching pairs The matching objects can express their evaluations more flexibly in PHFS since the decisionmaker may hesitate while describing the preference evaluation information.As can be observed, in comparison to the matching objects, information on their assessments is provided in triangular intuitionistic fuzzy sets [39], hesitant fuzzy numbers [40] and intuitionistic fuzzy numbers [8], the method proposed in this paper has more advantages.Furthermore, the tendency of the object function value Z from cases 1-3 is displayed in Fig. 2. The tendency of the matching scheme x ij from cases 1-11 is displayed in Fig. 3. Fig. 4 reveals the tendency of the object function value Z from cases 8-11.Based on Figs.2-4 the comprehensive tendency of the object function value Z from cases 1-11 could be acquired, as displayed in Fig. 4. From Figs. 2 to 4, the paper analyzes the role of two-sided agent weight values on the objective function value, as displayed in Fig. 5.As the weight of the two-sided agent changes from cases 1-2, the object function drops significantly.As the weight changes from cases 2 to 9, the function value gradually decreases until it reaches its lowest point in case 9.However, from cases 9-11, results show that the value of object functioninitially rises and then falls.The tendency of matching scheme x ij from cases 1-3 is displayed in Fig. 6.The tendency of matching scheme x ij from cases 1-11 is displayed in Fig. 7.The tendency of matching scheme x ij from cases 8-11 is revealed in Fig. 8. Based on Figs. 6 to 8, the comprehensive tendency of matching scheme x ij from cases 1 to 11 could be acquired, and is displayed in Fig. 9. From Figs. 6 to 8, the function of two-sided agent weight values on the matching scheme is examined in the paper.Fig. 9 shows an overall comparison.It can be concluded that the matching results slightly changes with the change in the two-sided agent's weight.Therefore, it is clear that the weight of the two-sided agent plays no significant role in the change of the matching scheme.
The tendency of the relationship between object function value Z and matching scheme x ij from cases 1-3 is displayed in Fig. 10.The tendency of the relationship between object function value Z and matching scheme x ij from cases 1-11 is displayed in Fig. 11.The tendency of the relationship between object function value Z and matching scheme x ij from cases8-11 is revealed in Fig. 12.The comprehensive tendency of the relationship between object function value Z and matching scheme x ij from cases 1-11 could be acquired based on Figs. 10 to 12, as displayed in Fig. 13.
As shown in Figs. 10 to 12, the paper analyzes the role of two-sided agent weight values on the objection function value and matching scheme.Fig. 13 displays the overall comparison.This leads to the conclusion that a key factor in the change in the value of the objection function is the weight of the two-sided agent.However, the result reveals that the weight of the two-sided agent has no vital role in the change of matching scheme.Firstly, to better illustrate the advantages of the proposed method for handling fuzzy preference information, the paper compares the proposed method with the one given in [35].In Reference [35], the score function is applied to the preference information provided by two-sided agents.Table 18 shows the matching results resulting from two methods.As can be seen in Table 18, there are many differences in the bilateral matching scheme obtained by the above two separate methods.Compared with the method used in reference [35] to process fuzzy preference information, the distance metric applied in this paper accurately processes the preference evaluation information given by matching agents, thus avoiding the loss and misunderstanding of preference information.

Method
Matching result Method given in this paper (A 1 , B 2 ), (A 2 , B 5 ), (A 3 , B 3 ) Method given in [35] (A 1 , B 5 ), (A 2 , B 2 ), (A 3 , B 1 ) Second, in contrast to the traditional TSM research, the methodology used in this paper retains complete information while taking into account hesitant fuzziness and the corresponding probability when evaluating matching objects, making the evaluation more scientific and reasonable, and the results under the probability hesitation fuzzy environment correspond to reality.Third, multiple attributes are considered, and their weights are calculated by using the maximum deviation method, which can ensure a scientific matching result.Simultaneously, the satisfaction of TSM is acquired by the closeness coefficients of each matching subject, making the entire process more interpretable.

Conclusions
In this paper, the TSMDM approach, by considering the attribute of matching objects under probability hesitant fuzzy environment, is developed.The main contributions of this paper are as follows: First, the PHFS is applied to describe the evaluation information provided by matching agents, and the weight vector is incomplete.Second, attribute weight vectors for matching objects are determined using the Hamming distance measures of PHFSs and the extended maximum deviation method based on the unknown weight and probability hesitant fuzzy environment of each matching object.Third, agent satisfaction degrees are gained in accordance with the multi-attribute probability hesitant fuzzy sets comprehensive assessments by using PHFOWA operator.Additionally, the relative closeness coefficient is used to calculate the satisfaction degrees of matched agents.Finally, we build a TSMDM model with the intention of maximizing the overall satisfaction of matching agents, and then the best matching result is determined by solving the constructed TSMDM model with linear weighted method.Overall, the proposed TSMDM approach not only has great potential for application in some real-world TSMDM scenarios, but it could also enhance the theoretical framework for the study of TSMDM problems.The psychology and expectations of the matching subjects when the evaluation information is provided, which is also a future research direction, are not taken into account in this paper.In terms of future research, the future research direction could be presented as follows: (1) For some TSMDM problems, matching objects may provide more evaluation value in hybrid formats, such as the PHFS, probability interval-valued hesitant fuzzy set (PIVHFS) and PHFLTS.In the meanwhile, the matching agents might provide insufficient attribute information.Hence, it could be useful to carry on the discussion of the TSMDM issues with multiple fuzzy information and insufficient attribute values.(2) In the TSMDM process, each matching individual should determine the automatic matching of recommended matching alternatives in the evolutionary environment as a result of the development of artificial intelligence computing systems.In light of these conditions, it could be necessary to develop a novel TSMDM method.

Figure 1 :
Figure 1: The procedure of two-sided matching

Step 4 :
To calculate the distance.

Figure 4 :
Figure 4: The object function value Z from cases 8-11

Figure 5 :
Figure 5: The object function value Z from cases 1-11

Figure 10 :Figure 11 :Figure 12 :Figure 13 :
Figure 10: The relationship between object function value Z and matching scheme x ij from cases 1-3 ijk ) n×s : The matrix about matching objects B provided by A i , where h A ijk is A i 's PHFE assessment over B j with respect to C A k .H B j = (h B ijl ) m×t : The evaluation matrix from objects A to B j , where h B ijl is B j 's PHFE assessment over A i with respect to C B l .For the matching object A i , let h A ijk denote his/her assessment for the matching object B j in accordance with the attribute C A k .The kth attribute's weight in C A is due to w k , all these HPFEs are contained in the probabilistic hesitant fuzzy matrixesH A i = (h A ijk ) n×s (i = 1, • • • m).For the matching object B j , let h B ijl denote his/her assessment for the matching object A i with respect to the attribute C B l .Owing to w l , all these PHFEs are contained in the probabilistic hesitant fuzzy matrixes

Table 10 :
Continued A

:
The object function value Z from cases 1-3