# Bilateral Matching Decision Model and Calculation of High-Speed Railway Train Crew Members
## 2 Problem Description and Modeling
### Construction of the Train Crew Matching Model  
The one-to-many bilateral matching decision model in the train crew matching problem is illustrated in Figure 1. In this model, \(A_{1}\) to \(A_{m}\) denote the set of stewards, and \(B_{1}\) to \(B_{n}\) denote the set of chief stewards. The thick solid lines represent the final constructed matching pairs, while the dashed lines represent the mutual preference information between the matching subjects. The evaluation of the matching scheme is based on the satisfaction of the bilateral matching results between the chief stewards and stewards.  
\begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Meaning** \\ \hline \(M\) & Set of stewards, \(M=\{1,2,\ldots,m\}\) \\ \hline \(N\) & Set of chief stewards, \(N=\{1,2,\ldots,n\}\) \\ \hline \(A\) & Set of stewards \(A=\{A_{1},A_{2},\ldots,A_{m}\}\), where \(A_{i}\) represents the steward number \(i\), \(i\in M\) \\ \hline \(B\) & Set of chief stewards \(B=\{B_{1},B_{2},\ldots,B_{n}\}\), where \(B_{j}\) represents the steward number \(j\), \(j\in N\) \\ \hline \(c_{j}\) & Total number of stewards matched by chief steward number \(j\), \(\sum\limits_{j=1}^{n}c_{j}=m\) (total number of stewards matches the total number of places available) \\ \hline \(r_{ij}\) & The complete preference order vector of steward \(A_{i}\) prefers chief steward \(B_{j}\) over chief steward, \(r_{ij}\) indicates that steward \(A_{i}\) ranks chief steward \(B_{j}\) in position \(r_{ij}\). \\ \hline \(s_{ij}\) & The complete preference order vector of steward \(B_{j}\) prefers chief steward \(A_{i}\) over chief steward, \(s_{ij}\) indicates that steward \(B_{j}\) ranks chief steward \(A_{i}\) in position \(s_{ij}\). \\ \hline \(R\) & Steward’s complete preference order matrix for chief stewards, \(R=\left[r_{ij}\right]\) \\ \hline \(S\) & Chief steward’s complete preference order matrix for stewards, \(S=\left[s_{ij}\right]\) \\ \hline \(\mu\) & Mapping function \(\mu:A\cup B\to A\cup B\), a multi-valued that assigns stewards from set A to chief stewards in set B. \\ \hline \(\widetilde{\mu}\) & Modified mapping function \(\widetilde{\mu}:A\cup\widetilde{B}\to A\cup\widetilde{B}\), a single-valued mapping that assigns stewards from set A to chief stewards in set B. \\ \hline \(\widetilde{B}\) & Set of Virtual Chief stewards \(\widetilde{B}=\left\{\widetilde{B}_{1}^{1},\cdots,\widetilde{B}_{1}^{c_{1}}, \widetilde{B}_{j}^{d},\widetilde{B}_{n}^{1},\cdots,\widetilde{B}_{n}^{c_{n}} \right\}\), where \(q\in\left\{1,2,\ldots c_{j}\right\}\) \\ \hline \(\widetilde{\tau}_{ij}^{d}\) & \(\widetilde{\tau}_{ij}^{d}\) represents the complete preference order vector of steward \(A_{i}\) for the q-th virtual representation of chief steward \(B_{j}\), where \(q\in\left\{1,2,\ldots c_{j}\right\}\). \\ \hline \(\widetilde{s}_{ij}^{d}\) & \(\widetilde{s}_{ij}^{d}\) represents the complete preference order vector of steward \(B_{j}\) for the q-th virtual representation of chief steward \(A_{i}\), where \(q\in\left\{1,2,\ldots c_{j}\right\}\). \\ \hline \(\widetilde{R}\) & The complete preference order matrix of stewards for virtual chief stewards, \(\widetilde{R}=\left[\widetilde{\tau}_{ij}^{d}\right]\) \\ \hline \(\widetilde{S}\) & The complete preference order matrix of stewards for virtual chief stewards, \(\widetilde{S}=\left[\widetilde{s}_{ij}^{d}\right]\) \\ \hline \end{tabular}
\end{table}
Table 1: Symbol representation and meaning.