In this paper, a stable twosided matching (TSM) method considering the matching intention of agents under a hesitant fuzzy environment is proposed. The method uses a hesitant fuzzy element (HFE) as its basis. First, the HFE preference matrix is transformed into the normalized HFE preference matrix. On this basis, the distance and the projection of the normalized HFEs on positive and negative ideal solutions are calculated. Then, the normalized HFEs are transformed into agent satisfactions. Considering the stable matching constraints, a multiobjective programming model with the objective of maximizing the satisfactions of twosided agents is constructed. Based on the agent satisfaction matrix, the matching intention matrix of twosided agents is built. According to the agent satisfaction matrix and matching intention matrix, the comprehensive satisfaction matrix is set up. Furthermore, the multiobjective programming model based on satisfactions is transformed into a multiobjective programming model based on comprehensive satisfactions. Using the GS algorithm, the multiobjective programming model based on comprehensive satisfactions is solved, and then the best TSM scheme is obtained. Finally, a terminal distribution example is used to verify the feasibility and effectiveness of the proposed method.
In the process of twosided matching decisionmaking (TSMDM), twosided agents yield the preference information according to their own needs and produce a reasonable matching scheme. The TSMDM problem widely exists in various fields, such as the freight source matching problem [
Currently, the preference information provided by people is often vague or uncertain due to the subjectivity and fuzziness of human thinking [
Relevant studies on HFS have emerged in different fields. First, the basic theory regarding HFS has been generalized. For instance, Xu et al. [
In addition, according to the aforementioned literature, there have been many studies on the application of HFS in the TSMDM problem. For instance, Zhang et al. [
Although a considerable number of scholars have explored the TSMDM problem under HFS preferences from a variety of perspectives and have achieved some promising results, there are some deficiencies in the existing research.
First, in the TSMDM with HFS preference information, it is vital for decisionmakers to find a reasonable tool to transform HFS preferences into exact values. In most of the existing studies, the TOPSIS method based on the distance measure or biprojection technology has been widely used in the TSMDM problem with HFS preference information. However, the above methods may still result in the loss of decisionmaking information in some cases. Hence, it would be meaningful to create an effective method to compute agent satisfaction.
Second, the matching intention coefficient, which reflects the matching intention of twosided agents, plays an important role in the TSMDM problem. For some TSMDM problems, the matching intention coefficient may affect the final TSM result. However, it seems that the matching intention coefficient has been neglected in the aforementioned TSMDM model. Therefore, it would be practicable to develop a method to obtain the matching intention coefficient of twosided agents in the TSMDM problem with HFS preference information.
Third, stability is of great significance for twosided agents to achieve a reasonable and satisfactory matching result. However, this topic has not been fully discussed by many existing studies on the TSMDM problem in HFS preference environments. Thus, it is necessary to consider the stability of the TSMDM problem based on HFS preference information.
To overcome these deficiencies, this paper aims to develop a reasonable and effective method to cope with the TSMDM problem considering HFS preferences. Moreover, a TSMDM model with HFS preferences considering the matching intention coefficient is constructed. The key contributions of this work are as follows. First, we create an effective method to compute the agent satisfaction, which uses the distance measure and biprojection technology. Second, we develop an effective method for calculating the matching intention coefficient of twosided agents. Finally, a TSMDM model with HFS preference information considering stable matching constraints and matching intention is constructed.
The remaining structure of this paper is as follows: In
In this section, we first introduce the basic concepts of the TSM, HFS and HFE. Then, the stable matching based on satisfaction is briefly described.
There are two agent sets in the TSM problem, namely,
In the real TSMDM scenario, we assume that
then
In this section, we introduce the TSMDM problem based on HFE preference information considering the matching intention.
In the TSMDM problem, twosided agents usually hesitate to assign their own preferences among multiple membership degrees. Therefore, to show the hesitancy and fuzziness of the evaluation information in TSMDM activities, we assume that twosided agents provide hesitant fuzzy preference information. Furthermore, assume
The problem studied in this paper is as follows: how to construct a TSM model considering the matching intention and stable matching conditions according to the HFE preference matrices
In this section, the TSMDM method based on HFE considering matching intention is presented. First, the HFE matrices are transformed into standardized HFE matrices. Then, the agent satisfaction matrices are calculated by using the TOPSIS and biprojection technologybased methods. Third, the TSM model considering the matching intention and stability is constructed by using the multiobject planning method. Finally, the optimal TSM scheme is obtained through the GS algorithm.
According to Definition 2, we know that
The normalized HFE matrices
Therefore, the length
The normalized HFE matrix is processed according to TOPSIS technology, and the following definitions are given with reference to the reference [
According to Definition 6 and Definition 7 and the calculation theory of closeness in reference [
In
According to Definition 9, Definition 10, and the calculation theory of closeness in reference [
In
Based on the satisfaction matrices
Referring to the references [
To solve Model (M2), the Lagrange function is introduced as follows:
Then, let
Hence, the optimal solution can be obtained on the basis of
Furthermore, the standardized matching intention coefficient can be computed by normalizing
To concretely illustrate the calculation method of the matching intention coefficient given above, we take an example with three agents of each side.
Example 1. Let
According to
Therefore, Model (M1) is transformed into Model (M3) based on the comprehensive satisfactions
The framework of the solution process for TSM Model (M3) based on the GS algorithm is shown in Algorithm 1; then, the detailed procedures are presented as follows:
In summary, the steps of stable HFE TSMDM considering matching intention are given below:
In the business activities of enterprises, the terminal distribution problem is the last link in the enterprise supply chain. Currently, the quality of terminal distribution will affect the overall benefit of enterprises, so the terminal distribution plays an important role in the supply chain. At the same time, there are two decisionmakers in a terminal distribution problem, namely, freight cars and terminal distribution centers. The decisionmaking process may be affected by many factors; therefore, we will consider a distribution problem between freight cars and terminal delivery points as an example to verify the validity and feasibility of the TSM model proposed in this paper. The framework of the proposed case is given in
GH is a thirdparty logistics enterprise specializing in LCL transportation. In a certain terminal distribution task, GH Company needs to arrange four trucks
Van  Distribution center  

{0.2, 0.3, 0.4}  {0.2, 0.5, 0.6, 0.7}  {0.1, 0.2, 0.9}  {0.2, 0.6, 0.8}  {0.1, 0.6, 0.7}  
{0.4, 0.6, 0.8}  {0.2, 0.3, 0.5, 0.9}  {0.1, 0.4, 0.6}  {0.1, 0.4, 0.7}  {0.3, 0.6, 0.7}  
{0.1, 0.4, 0.5, 0.6}  {0.2, 0.7, 0.8}  {0.3, 0.5, 0.9}  {0.3, 0.6}  {0.3, 0.7, 0.8}  
{0.2, 0.7, 0.9}  {0.3, 0.5, 0.6}  {0.4, 0.5}  {0.2, 0.4, 0.5}  {0.1, 0.3, 0.5, 0.9} 
Van  Distribution center  

{0.5, 0.6, 0.8}  {0.2, 0.4, 0.7}  {0.1, 0.3, 0.8, 0.9}  {0.2, 0.3, 0.5}  {0.4, 0.5}  
{0.3, 0.5, 0.6}  {0.1, 0.3, 0.6, 0.8}  {0.2, 0.4, 0.5}  {0.3, 0.7}  {0.1, 0.2, 0.7}  
{0.1, 0.3, 0.6}  {0.2, 0.4, 0.9}  {0.4, 0.6}  {0.5, 0.7, 0.8}  {0.3, 0.5, 0.6}  
{0.1, 0.3, 0.7}  {0.2, 0.4, 0.5, 0.6}  {0.2, 0.4, 0.7}  {0, 2, 0.6, 0.9}  {0.3, 0.4, 0.6} 
The proposed method is adopted to solve the problem, and the solution steps are displayed as follows.
Van  Distribution center  

{0.2, 0.3, 0.4, 0.4}  {0.2, 0.5, 0.6, 0.7}  {0.1, 0.2, 0.9}  {0.2, 0.6, 0.8}  {0.1, 0.6, 0.7, 0.7}  
{0.4, 0.6, 0.8, 0.8}  {0.2, 0.3, 0.5, 0.9}  {0.1, 0.4, 0.6}  {0.1, 0.4, 0.7}  {0.3, 0.6, 0.7, 0.7}  
{0.1, 0.4, 0.5, 0.9}  {0.2, 0.7, 0.8, 0.8}  {0.3, 0.5, 0.9}  {0.3, 0.6, 0.6}  {0.3, 0.7, 0.8, 0.8}  
{0.2, 0.7, 0.9, 0.9}  {0.3, 0.5, 0.6, 0.6}  {0.4, 0.5, 0.7}  {0.2, 0.4, 0.5}  {0.1, 0.3, 0.5, 0.9} 
Van  Distribution center  

{0.5, 0.6, 0.8, 0.8}  {0.2, 0.4, 0.7, 0.7}  {0.1, 0.3, 0.8, 0.9}  {0.2, 0.3, 0.5, 0.5}  {0.4, 0.5, 0.5, 0.5}  
{0.3, 0.5, 0.6, 0.6}  {0.1, 0.3, 0.6, 0.8}  {0.2, 0.4, 0.5, 0.5}  {0.3, 0.7, 0.7, 0.7}  {0.1, 0.2, 0.7, 0.7}  
{0.1, 0.3, 0.6}  {0.2, 0.4, 0.9}  {0.4, 0.6, 0.6}  {0.5, 0.7, 0.8}  {0.3, 0.5, 0.6}  
{0.1, 0.3, 0.7, 0.7}  {0.2, 0.4, 0.5, 0.6}  {0.2, 0.4, 0.7, 0.7}  {0, 2, 0.6, 0.9, 0.9}  {0.3, 0.4, 0.6, 0.6} 
Construct the agent satisfaction matrices
Van  Distribution center  

0.177  0.321  0.311  0.632  0.435  
0.637  0.256  0.217  0.245  0.551  
0.387  0.734  0.753  0.516  0.811  
0.804  0.310  0.669  0.115  0.147 
Van  Distribution center  

0.908  0.367  0.456  0.082  0.324  
0.481  0.360  0.240  0.780  0.269  
0.000  0.390  0.484  0.790  0.326  
0.246  0.142  0.359  0.878  0.280 
Van  Distribution center  

0.048  0.050  0.051  0.035  0.052  
0.074  0.043  0.034  0.051  0.049  
0.021  0.064  0.075  0.079  0.059  
0.052  0.030  0.060  0.043  0.029 
Van  Distribution center  

0.0085  0.0161  0.0159  0.0221  0.0226  
0.0471  0.0110  0.0074  0.0125  0.0270  
0.0081  0.0470  0.0565  0.0408  0.0478  
0.0418  0.0093  0.0401  0.0049  0.0043 
Van  Distribution center  

0.0445  0.0184  0.0233  0.0029  0.0168  
0.0356  0.0155  0.0082  0.0398  0.0132  
0.0000  0.0250  0.0363  0.0624  0.0192  
0.0128  0.0043  0.0215  0.0378  0.0081 
Van  Distribution center  

0  0  0  0  1  
1  0  0  0  0  
0  0  1  0  0  
0  1  0  0  0 
Therefore, the best TSM is
Furthermore, we reconstruct Model (M4) without the stable constraint:
Then, the matching scheme can be obtained by solving TSM Model (M4), as shown in
Van  Distribution center  

0  0  0  0  1  
1  0  0  0  0  
0  0  0  1  0  
0  0  1  0  0 
Hence, the best TSM scheme is determined from
TSM method considering stable constraints (Proposed Method I)  
TSM method without stable constraints (Method II) 
As seen from
First, there are several kinds of methods for dealing with fuzzy preference information, such as the TOPSIS method based on distance measures or biprojection technology. This paper adopted a brandnew method to deal with fuzzy preference information, which contains distance measures and biprojection technology. To further state the effectiveness of the method proposed in this paper, we compare the proposed method for handling hesitant fuzzy information with two methods given in references [
Method proposed in this paper  
Method proposed in reference [ 

Method proposed in reference [ 
Second, many of the existing TSMDM studies ignored the matching intention of twosided agents. In this paper, the proposed TSMDM method considered the matching intention of matching agents, which can be applied to minimize the diversity of the TSM scheme.
Third, stability plays a significant role in the TSMDM process because the unstable matching pair may hinder the formation of matching results even if the satisfaction of twosided agents is high. From the aforementioned research, we find that stability has not been fully considered in TSMDM problems in hesitant fuzzy preference environments. Hence, the onetoone stable matching constraints are integrated into the TSMDM model developed in this paper, which can not only maximize the satisfaction of twosided agents but also ensure a stable TSM scheme.
In this paper, a TSM decisionmaking model considering the matching intention and stability of twosided agents under hesitant fuzzy preference is proposed. In this method, the normalized HFE is transformed into the agent satisfaction after normalizing the HFE preference information. A multiobjective programming model considering the agent satisfactions and the stable matching constraint is constructed. Based on the agent satisfaction matrices, the matching intention coefficient matrix is calculated, and then the comprehensive satisfaction matrices are calculated. Furthermore, the multiobjective programming model based on the satisfactions is transformed into the multiobjective programming model based on the comprehensive satisfactions. The best TSM scheme can be obtained by solving the developed model.
The main innovation of this paper can be denoted as follows: First, the TSM scheme obtained by the method proposed in this paper can reflect the matching intention of twosided agents; second, the proposed method ensures the stability of the matching scheme. The TSM method proposed in this paper provides a new perspective for studying the TSM problem under the condition of HFS.
Further research work can be conducted as follows:
In the proposed TSMDM method, we present a novel method including distance measures and biprojection technology for dealing with hesitant fuzzy preference information. However, due to the complex decisionmaking environment, the proposed distance measures may not fully reveal the comments of the decisionmakers who have various preferences. Therefore, it could be practicable to develop a brandnew distance measure for HFS and the corresponding TOPSIS method.
For some TSMDM problems, twosided agents may provide multiple hesitant fuzzy preference information, such as probability hesitant fuzzy sets, interval hesitant fuzzy sets, and probability interval hesitant fuzzy sets. These would be interesting topics for developing a TSMDM model with multiple hesitant fuzzy preference information.
It is meaningful to conduct research on the TSMDM problem considering bipolar hesitant fuzzy sets and Tspherical fuzzy sets [