# A Coordinated Supply Contract for a Two-Echelon Supply Chain Considering Learning Effects
## 5 Numerical Illustrations  
In this section, we provide numerical illustrations to demonstrate the performance of our proposed model compared to the existing benchmark models. Additionally, we demonstrate how the contract parameters affect the bargaining structure of each member in the supply chain. We use specific parameter values, \(p=60\), \(s=10\), \(w=40\) (under wholesale contract), \(c_{1}=170.3\), and \(r=0.8\) (under learning effect case). We assume that the demand variable \(X\) follows a uniform distribution, \(X\sim\)_Uniform_ (\(100,300\)). To visually represent and analyze these results, we leverage _an open-source'matplotlib.pyplot'_ library ([[https://matplotlib.org/](https://matplotlib.org/)]([https://matplotlib.org/](https://matplotlib.org/))) in _Python version 3.10_. Figure 1 shows the production cost over production quantity under learning-curve effects. First, we consider a supply-chain system where LC effects are not considered. Here, we assume that the per-unit production cost remains constant at \(30.0\) (i.e., \(c=30.0\)). (This corresponds to the average production cost when \(220\) items are produced in the system under learning-curve effects, i.e., \(c_{20}=170.3\times 220^{\log_{2}0.8}=30.0\)). In the centralized system without learning effects, we can derive the optimal production quantity (\(Q_{Nc}^{*}\)) using expression (6): \(Q_{Nc}^{*}=F^{-1}\left(\frac{p-c}{p-s}\right)=F^{-1}(0.6)=220\). The total expected supply-chain profit, calculated from Equation (3), is \(4800\). In a decentralized system with independent supply-chain members considering a wholesale contract (\(w=40\)), the retailer's optimal order quantity (\(Q_{Nbn}^{*}\)) is derived as follows: \(Q_{Nbn}^{*}=F^{-1}\left(\frac{p-w}{p-s}\right)=F^{-1}(0.4)=180\), resulting in the total expected supply-chain profit of \(4600\). The finding shows that the wholesale contract yields a supply-chain profit that falls short by 200 compared to the centralized system due to the double marginalization problem. To address the double marginalization problem in the decentralized system, a revenue-sharing contract can be adopted with appropriate parameters, wholesale price (\(w\)), and revenue-sharing ratio (\(\phi\)). For example, if we set \(\phi\) = 0.5 in the revenue-sharing contract, we can obtain \(w\) = 17.2 by the channel coordination relationship \(\frac{\varphi p-w}{\varphi p-s}=\frac{p-c}{p-s}\). With the revenue-sharing contract with parameters (\(w\), \(\phi\)) = (17.2, 0.5), it is possible to achieve the same overall optimal profit of 4800 as would be attained in a centralized decision-making scenario. In Figure 2, the orange-color curve shows supply-chain performance over different order quantities without the consideration of learning-curve effects. It is seen that the maximum supply-chain profit is realized when 220 products are ordered for the system under static production costs. Now, let us consider the proposed contract for a supply-chain system under learning-curve effects with \(r\) = 0.8. The first step in the proposed contract is to find the optimal production (order) quantity, considering the learning effects. The proposed iteration  
Figure 1: The per-unit production cost under the production quantity with learning curve. Figure 2: Supply-chain performance over varying production quantities with and without LC effects. method (the Newton-Raphson method) in Section 4.2 provides the optimal order quantity of 263.2. (In this scenario, only five iterations are enough to find the optimal quantity.) This yields the corresponding production cost, \(c_{263}=c_{1}\Omega^{\log_{2}r}=170.3\times 263.2^{\log_{2}0.8}=28.32\), and \(z^{*}=F(263.2)=0.816\). (Refer to expressions (2) and (6) for the calculations.) According to Equation (7), the resulting total expected supply-chain profit is 5011. It should be noted that the resulting profit is increased by 211, compared with the profit from the optimal decision without the consideration of LC effects.