# Numeric Evaluation of Game-Theoretic Collaboration Modes in Supplier Development
## 5 Results  
In this section, we are particularly interested in the impact of manufacturers relationships on their revenue on supplier development. To this end, we utilize the MPC method to find the optimal profit of supplier development. The values of the parameters for an academic example from [8] are shown in Table 1. We utilize these variables in (1) to compute the corresponding output values of profit \(J_{T}(u;x_{0})\) for all of the described settings. We first compute the investment schema of \(M_{1}\) and \(M_{2}\) over time by using MPC in a Matlab framework. We apply an interior point gradient-based method to find the local optimum solution. Solving the respective problems, we obtained Figures 5 and 6. We observe that manufacturers in a fully cooperative setting tend to invest for longer periods than the others, and the manufacturers experience faster and greater production cost reduction (Figure 5, right side). In contrast to that, the non-cooperative scenario shows that manufacturers prefer shorter investment periods as the loop \begin{table}
\begin{tabular}{c c c} \hline \hline
**Symbol** & **Description** & **Value** \\ \hline T & Contract period & 60 \\ m & Learning rate & \(-0.1\) \\ a & Prohibitive price & 200 \\ b & Price elasticity & 0.01 \\ \(c_{M_{1}}\) & Variable cost per unit (\(M_{1}\)) & 65 \\ \(c_{M_{2}}\) & Variable cost per unit (\(M_{2}\)) & 70 \\ \(c_{0}\) & Variable cost per unit (S) & 100 \\ \(c_{SD_{1}}\) & Supplier development cost per unit (\(M_{1}\)) & 9000 \\ \(c_{SD_{2}}\) & Supplier development cost per unit (\(M_{2}\)) & 8500 \\ \(u_{1}\) & Number of supplier development projects (\(M_{1}\)) & [0, 5] \\ \(u_{2}\) & Number of supplier development projects (\(M_{2}\)) & [0, 2] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of parameters ([PERSON] et al. 2016 [7]; [PERSON] et al. 2017 [8]). simulation shows their investment is not paying off and they should stop investing in the development of the supplier (Figure 5, left side). The collaboration results (Figure 6) reveal that the bang-bang behavior from [7; 8] is not present in this setting which may be due to manufacturers sharing information and receiving the feedback of the other one's investment, which encourages them to continue investing for longer periods even with the lower amount of investment. The collaboration results show that in the simultaneous scenario, manufacturers invest in supplier for a longer period than in the sequential one. Since MPC assumes a certain amount of profit, knowing that others also invest leads to higher profit expectations during the planning stage in this setting. To make the results clear, we compare the global profit (the sum of two manufacturers' profit) in each setting shown in Figure 7 (left side). Result reveals that in earlier time steps, non-cooperative behavior performs better and reaches faster to higher profits than the rest. However, for longer perspectives, the fully cooperative setting shows the best performance and provides the highest profit for manufacturers. Consequently, for planning longer investments, cooperative behavior leads to the highest profit. However, for short investments, manufacturers should choose the non-cooperative setting to gain more profit from their investment. Figure 5: Supplier development schema for \(M_{1}\) and \(M_{2}\) in the cooperative and non-cooperative settings. Figure 6: Supplier development schema for \(M_{1}\) and \(M_{2}\) in the simultaneous and sequential settings. To investigate the effect of the relationship of manufacturers on the profit that each one of them gains from supplier development, we compare the local profits of all settings for \(M_{1}\) and \(M_{2}\). Figure 7 (right side) depicts the result. According to the results over a longer period, the fully cooperative setting for both manufacturers results in the highest profit.