In previous research on two-sided matching (TSM) decision, agents’ preferences were often given in the form of exact values of ordinal numbers and linguistic phrase term sets. Nowdays, the matching agent cannot perform the exact evaluation in the TSM situations due to the great fuzziness of human thought and the complexity of reality. Probability hesitant fuzzy sets, however, have grown in popularity due to their advantages in communicating complex information. Therefore, this paper develops a TSM decision-making approach with multi-attribute probability hesitant fuzzy sets and unknown attribute weight information. The agent attribute weight vector should be obtained by using the maximum deviation method and Hamming distance. The probabilistic hesitancy fuzzy information matrix of each agent is then arranged to determine the comprehensive evaluation of two matching agent sets. The agent satisfaction degree is calculated using the technique for order preference by similarity to ideal solution (TOPSIS). Additionally, the multi-object programming technique is used to establish a TSM method with the objective of maximizing the agent satisfaction of two-sided agents, and the matching schemes are then established by solving the built model. The study concludes by providing a real-world supply-demand scenario to illustrate the effectiveness of the proposed method. The proposed method is more flexible than prior research since it expresses evaluation information using probability hesitating fuzzy sets and can be used in scenarios when attribute weight information is unclear.

Two-sided matching decision-making (TSMDM) consists of two sides of matching agents and belongs to the decision-making problems [

TSMDM is now used in a wide range of fields, such as marriage matching problems [

Meantime, a significant number of scholars have studied the multi-attribute TSMDM problem. For instance, Jiang et al. [

Some researchers studied the TSMDM problem with complete or partial preference ordering in the previous studies. For instance, Pu et al. [

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, decision-makers 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 [

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 [

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.

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 matching agent sets,

Then,

HFS was first proposed by Torra [

According to the score function, the deviation function is defined by:

According to the score functions, we can compare PHFEs

if

if

if

Zhang et al. [

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:

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 [

The closeness coefficients

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

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:

For the matching object

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 [

In relation to the

The total deviation among

We can evaluate the total variance between

Then, a constrained condition

Therefore, a model for computing the attribute weight for

By using the Lagrange multiplier method and normalizing the weight vector, we have

Similar to that, the attribute weight vector for

Furthermore, the matrices

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

Additionally, a TSMDM model could be developed to obtain the best matching. Suppose

In model

We apply the linear weighted method to solve the multi-objective optimization model, and thus model

In conclusion, the steps of the proposed TSMDM approach can be summed up as follows (

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

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

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

Matching is performed by the municipal education bureau/commission using probabilistic hesitant fuzzy evaluation matrices. The information regarding how much each attribute weighs in the evaluation is not provided by two-sided agents. In accordance with the above given steps, the weight information of each attribute can be calculated by

On this basis, the comprehensive assessment

Therefore, the comprehensive assessment

The comprehensive PHFS evaluation information of the primary schools can be obtained, as shown in matrix

Owing to the number of each PHFE is not equal for

According to the satisfaction degree formula, the satisfaction degree of the matching agent

Based on the satisfaction matrices

The optimization model can be solved by utilizing the Lingo 11 software, so that we can get _{3}, _{3}), which shows that primary school

To better illustrate the impact of the weights

Case | Weight values | Matching pairs |
---|---|---|

1 | ||

2 | ||

3 | ||

4 | ||

5 | ||

6 | ||

7 | ||

8 | ||

9 | ||

10 | ||

11 |

The matching objects can express their evaluations more flexibly in PHFS since the decision-maker 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 [

Furthermore, the tendency of the object function value

From

The tendency of matching scheme

From

The tendency of the relationship between object function value

As shown in

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 [

Method | Matching result |
---|---|

Method given in this paper | |

Method given in [ |

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.

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.

The authors would like to thank the editor and anonymous reviewers for their valuable comments and suggestions, thus improving the version of this article.

This work is supported by the National Natural Science Foundation in China (Yue Qi, Project No. 71861015).

The authors declare that they have no conflicts of interest regarding the present study.