## 4 Numerical Example
Suppose the allocation results are adjusted as follows: coal mining company \(S_{3}\) reduces its supply to power generation company \(d_{2}\) by 5 kilotons and coal mining company \(S_{4}\) reduces its supply to power generation company \(d_{3}\) by 5 kilotons. Now the quantity that coal mining company \(S_{4}\) supplies to power generation company \(d_{2}\) has increased from 0 to 5 kilotons and the supply of coal mining company \(S_{3}\) to power generation company \(d_{3}\) has also increased from 0 to 5 kilotons as shown in Table 4.  
Analysis of the matching preferences indicates that the matching in Table 4 is unstable. Table 1 shows that coal mining company \(S_{4}\) ranks power generation companies \(d_{2}\) and \(d_{3}\) 6 th and 3 rd, respectively, in its order of preferences; it prefers to sell coal to power \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Coal Mining Company \(S_{i}\)**} & \multicolumn{3}{c}{**Power Generation Company \(d_{j}\)**} \\ \cline{2-3}  & \(d_{2}\) & \(d_{3}\) \\ \hline \(S_{3}\) & 15 & 5 \\ \(S_{4}\) & 5 & 0 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Allocation after change. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Coal Mining Company \(S_{i}\)**} & \multicolumn{5}{c}{**Power Generation Company \(d_{j}\)**} & \multirow{2}{*}{**Surplus of Supply Side \(I_{si}\)**} \\ \cline{2-3}  & \(d_{1}\) & \(d_{2}\) & \(d_{3}\) & \(d_{4}\) & \(d_{5}\) & \(d_{6}\) \\ \hline \(S_{1}\) & 0 & 30 & 0 & 0 & 0 & 0 \\ \(S_{2}\) & 5 & 0 & 0 & 0 & 0 & 0 \\ \(S_  
generation company \(d_{3}\) over \(d_{2}\) but ends up signing an agreement with \(d_{2}\) to sell 5 kilotons of coal. Similarly, power generation company \(d_{3}\) ranks coal mining companies \(S_{3}\) and \(S_{4}\) 8 th and 3 rd, respectively, in its order of preference; it prefers to purchase coal from coal mining company \(S_{4}\) instead of \(S_{3}\) but ends up signing an agreement with \(S_{3}\) to purchase 5 kilotons of coal. In this situation, coal mining company \(S_{4}\) and power generation company \(d_{3}\) would find that, if they signed an agreement with each other for the purchase of 5 kilotons of coal and cancelled their respective agreements with power generation company \(d_{2}\) and coal mining company \(S_{3}\), they would be in better trading relationships and both parties would benefit from this change. Therefore, the matching between coal mining companies \(S_{3}\) and \(S_{4}\) and power generation companies \(d_{2}\) and \(d_{3}\) is likely to revert to the original allocation in Table 2.  
In theory, based on matching preferences, regardless of however much more coal coal mining company \(S_{4}\) sells to power generation company \(d_{2}\) and however much more coal coal mining company \(S_{3}\) sells to power generation company \(d_{3}\), the result would be pairwise unstable. This unstable matching result would lead to conflicts between the coal mining and power generation companies. It is even likely that, because the two market participants insist that they should match each other, other parties' interests will be undermined. * Sensitivity of matching results  
We further analyzed the sensitive degree of the pairwise stable matching that is discussed above; namely, the impact of changes in the coal supply or demand on the stable matching relation, which is a different analysis perspective and gives further discussions of stable matching results compared with the studies of [PERSON] et al. (2016) [18], [PERSON] et al. (2020) [21], and [PERSON] et al. (2021) [23]. In the simulation, we only changed the supply or demand of one pair of coal mining and power generation companies without causing changes to the trading relations in order to analyze the sensitivity of the pairwise stable matching to marginal supply and demand changes.