# A Coordinated Supply Contract for a Two-Echelon Supply Chain Considering Learning Effects
## 5 Numerical Illustrations
The blue-color curve in Figure 2 illustrates the total expected profit of the supply chain with LC effects. This observation emphasizes that considering learning effects during the formulation of supply-contract designs is crucial for achieving overall optimization. The primary goal of our study is for the manufacturer to design the supply contract that achieves channel coordination for the system with LC effects. Two contract parameters are involved in decision processes, \(w\) and \(\phi\). All the (\(w\), \(\phi\)) sets satisfying expression (16) in Section 4.3 lead to channel coordination. For example, a specific parameter (\(w\), \(\phi\)) = (13.681, 0.5) that satisfies expression (16) is one of the possible coordinated solutions. In this case, from the perspective of the retailer, ordering 263.2 units maximizes its expected profit. This decision ultimately yields an expected profit of 2964 for the retailer and 2047 for the manufacturer. (Refer to expressions (12) and (13) for this mathematics in Section 4.3) As a result, the expected total profit becomes 5011, the same as in the centralized system. It is noted that in this scenario, 59.1% of the total supply-chain profit goes to the manufacturer, while the remaining 40.9% goes to the retailer. The profit distributions between the manufacturer and retailer are often determined based on the relative bargaining power of each supply-chain member. One distinct feature of the proposed contract scheme is that any bargaining-power structure can be satisfied with the appropriate manipulation of contract parameters. Table 1 illustrates a set of contract parameters satisfying different bargaining-power structures in the new revenue-sharing contracts in which LC effects are integrated. (Refer to Sections 4.3 and 4.4 for the mathematics.) When all the supply-chain profits are allocated to the manufacturer with \(\alpha=0\), the revenue-sharing ratio for the coordinated supply contract is \(\phi=0.167\), which is calculated from expression (19). Then, the corresponding wholesale price for the coordinated contract is \(w=8.161\). On the other hand, if all the supply-chain profits were to go to the retailer with \(\alpha=1\), the revenue-sharing ratio for the coordinated supply contract would be \(\phi=0.730\), which is obtained from expression (20). The corresponding wholesale price for the coordinated contract is \(w=16.224\). For the range of \(0\leq\alpha\leq 0\), the parameter sets in the coordinated contract, \(w\) and \(\phi\), are obtained from expression (18). It is observed that the value of \(\phi\) has a lower limit, \(\phi=0.617\), and an upper limit, \(\phi=0.730\) in the coordinated contract. The results show that that all the profit split structures with different values of \(\alpha\) can be satisfied with appropriate contract parameters. It is also seen that the revenue-sharing ratio, \(\phi\), and the wholesale price, \(w\), increase as the retailer's profit ratio increases. This finding can be explained by the natural tendency for retailers to retain a significant portion of the profit when the wholesale price is high. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\phi\) & \(w^{*}\) & \(\pi_{Lr}\) & \(\pi_{Lm}\) & \(\pi_{Le}\) & \(\pi\) \\ \hline
0.167 & 8.161 & 0 & 5011 & 5011 & 0.000 \\
0.200 & 10.368 & 296 & 4714 & 5011 & 0.059 \\
0.300 & 11.472 & 1185 & 3825 & 5011 & 0.237 \\
0.400 & 12.576 & 2074 & 2936 & 5011 & 0.414 \\
0.500 & 13.681 & 2964 & 2047 & 5011 & 0.591 \\
0.600 & 14.785 & 3853 & 1158 & 5011 & 0.769 \\
0.700 & 15.889 & 4742 & 269 & 5011 & 0.946 \\
0.730 & 16.224 & 5011 & 0 & 5011 & 1.000 \\ \hline \hline \end{tabular} \(w^{*}\): the optimal wholesale price when a specific \(\phi\) is given.  
\end{table} 
Table 1: Feasible profit distribution given the revenue-sharing with LC effects.