## 3 Integrated Method for SoS Architecture Selection and Reconfiguration
### Mathematical Formulation for Reconfiguration  
The formulation structure of the SoS architecture design problem with reconfiguration options is illustrated in Figure 4. Given a set of determined but probably incomplete DoDAF based architecture models, the architecture design decision model and reconfiguration decision model can, on one hand, use the provided information, and on the other hand, eventually help develop comprehensive DoDAF models. The focus of Figure 4 is developing and solving the design decision model that requires an appropriate SoS performance evaluator, in addition to the DoDAF models. Plus, as a novelty, the model also considers reconfiguration decisions in advance in the architecture design process. In the reconfiguration decision model, multiple scenarios concerning different forms of disruptions would require different reconfiguration strategies to achieve the desired performance, based upon the reconfigurability of the generated architecture design. The current literature offers numerous methods for defining the resilience metrics, based upon the performance (or utility, effectiveness) change curves as shown in Figure 5. A variety of curves have been explored for different types of system. In a combat SoS facing enemy attacks, we make several assumptions: (1) once a system or a functional module receives attacks, the system or functional module stops functioning immediately; (2) the constituent systems within an SoS are able to reconfigure themselves while providing the original functions continuously; (3) the new configuration starts to work when all the reconfiguration procedures are completed. The three assumptions respectively explain the plummeting effectiveness, the stable effectiveness during reconfiguration, and the perpendicular effectiveness rise in Figure 5.  
Considering resilience as a combination of survivability and recoverability, we develop two objective functions for this problem. The mathematical expressions are written as in (3). One is to maximize the remaining SoS effectiveness \(\mathcal{O}_{n}^{\mathit{disrup}}\) when attacked systems are completely destroyed, implying the essence of survivability. The other is to maximize  
Figure 4: SoS architecture design problem with reconfiguration options. the difference between SoS effectiveness after reconfiguration \(O_{n}^{reconf}\) and after disruptions \(O_{n}^{disrupt}\), indicating the recoverability part of resilience. The time for taking reconfiguration actions also matters, but this study considers it as constraints instead of incorporating it into the resilience metrics. \[\left\{ \begin{aligned} & y_{m,x_{ik}^{min}x_{ik}^{max}}E\Big{[}O_{n}^{disrupt} \Big{]}\\ &\max_{y_{m,x_{ik}^{min}x_{ik}^{max}}}E\Big{[}O_{n}^{reconf}-O_{n}^ {disrupt}\Big{]}\end{aligned} \right. \tag{3}\]  
Although the reconfiguration process concerns both the decisions of system-activity re-allocation and system-system re-connection, the decision variables only include system-activity re-allocation \(x_{ik}^{reconf}\) that indicates the degree that system \(i\in I\) supports activity \(k\in K\). Systems that are allocated to linked activities are assumed to be connected automatically. The other decision variables include the binary variable \(y_{m}\) for initial system selection and architecture allocation \(x_{ik}^{init}\) that can both influence the reconfiguration decision options. All decision variables impact the objective via changing the self-effectiveness values of the activity nodes. The calculation of \(O_{n}^{disrupt}\) and \(O_{n}^{reconf}\) uses (1) repeatedly until reaching the sink node. The initial self-effectiveness values are based on decision variable \(y_{m}\) and \(x_{ik}^{init}\), as well as \(unit\_fc_{ik}\) that refers to the function provided for unit support level if allocating system \(i\) to activity \(k\). The objective functions are subject to the following constraints. The first three depend on reconfigurability while the fourth determines reconfigurability.