# Numeric Evaluation of Game-Theoretic Collaboration Modes in Supplier Development
## 3 MPC Control Scheme  
The MPC scheme is particularly useful to model and optimize systems, where decision-makers, in this case, manufactures, can utilize different portions of information about the real-world system (about the supplier and their competitors). While the measured state contains all information about all the manufacturers' activities in the last time step, the corresponding system models can be defined to only include knowledge available from the respective kind of cooperation or collaboration. Therefore, the (open loop) simulation is restricted to only use information applicable for the specific setting to make a decision, while the effects of all independent decisions can be integrated using the measurement of the current (closed-loop) real-world system state. As a result, each setting is defined by three different components:  
* **Cost function:** The cost function of each scenario differs in whether the manufacturers pursue a common goal or try to obtain their respective profit.
* **System model:** Depending on the scenario, the system model can differ strongly. For example, in the full cooperation setting, a common system model can be used, which includes all information about both manufacturers. In contrast, the non-cooperation setting uses two fully separate system models as neither manufacturer has any information about the other.
* **Order and sequence of decision-making:** In particular the sequential and simultaneous settings rely on different orders of decision making steps within each time step. For instance, multiple decision-making iterations can be used to represent the sharing of information. Thereby, each manufacturer makes a decision and conveys this decision to the other one. The latter uses the mentioned information to render its own decision and replies back to the first manufacturer. Now, the first manufacturer can again use the newly acquired information to update its decision and so on. As a result, several cycles of decision making can be part of a single time step.  
MPC consists of control parts, which applies short-term control decisions to the real-world system and receives feedback about respective effects, and an open-loop simulation, which acts as the foundation for the long-term optimization. Figure 1 sketches the overall scheme, where the real-world system is shown on top of the figure, which refers to the two manufacturers and a supplier. The real-world system receives control inputs from the closed-loop control at predefined time steps \(t_{i}\), \(i\in 1,\ldots,N\) and provides a measured state of the system \(x_{n}\), \(x_{n}\in x_{0},\ldots,(x_{N})\) and a measured time step \(t_{m}\), \(t_{m}\in t_{0},\ldots,(t_{N})\) to the closed-loop control. In this study, we assume a 5 years supplier development project and compute the optimal profitability by applying the 3-month intervals (\(\Delta t=3\)), which means \(N=20\). The closed-loop control uses this information to initiate an open-loop simulation-based optimization. In contrast to the closed-loop, the open-loop covers an increased time horizon of \(N\) time steps. The effects of control decisions \(u_{i}\) with \(i\in t_{m},\ldots,t_{N}\) are simulated using an underlying _system model_ and the control sequence \(u_{t_{m}},\ldots,u_{t_{N}}\) is optimized using a provided _cost function_. The latter is not shown in the picture but is assumed to be part of the optimization. The obtained optimal control sequence for \(N\) time steps is then returned to the closed-loop control, which forwards the first element of this sequence \(u_{t_{m}}\) to the real-world system as control decision for the current time step. After application, a new measurement is conveyed to the closed-loop control and the overall cycle restarts.