## 3 Integrated Method for SoS Architecture Selection and Reconfiguration
### SoS Effectiveness Evaluator  
SoS resilience evaluation requires proper resilience metrics as well as an effective method to assess SoS operational effectiveness. We propose a three-layer SoS effectiveness evaluator based on the functional dependency network analysis (FNDA) method [35; 36], as illustrated in Figure 3. SoS effectiveness herein is a representative utility index for evaluating the overall functional capability of a given SoS under a specific condition from a decision-maker's perspective. It is a prerequisite for an SoS resilience evaluator. As shown on the left of Figure 3, the bottom layer is the network of equipment systems (reflecting the physical architecture in Figure 2), which yields combat performance through operational activities (e.g., detect, classify, command, control, strike) in the mid-layer (reflecting operational architecture in Figure 2). The connection between the bottom layer and the mid-layer are the functions provided by the equipment systems. The output of the mid-layer includes various SoS performances, such as task completion time and ratio of neutralized targets. While the performances vary in units and scales, the upper level aims to aggregate the values of different performance indicators into one representative indicator based on decision-makers' preferences. Figure 3: Structure of SoS effectiveness evaluator. The FDNA and its extensions are typically used as either surrogate models or utility combination models. While the original FDNA method focuses on the utility perspective, the extended methods regarding SoS capability (i.e., operability, effectiveness), often consider themselves as surrogate models that can partially replace high-resolution simulation models in the early design phase. However, the current research does not clearly distinguish between these two perspectives, leading to concept confusions and method misuses. In fact, when using a utility combination model, the utility or effectiveness of each node should be computed based upon the operational output information (e.g., ratio of detected targets) instead of the operational process information (e.g., location and speed of a target). In this paper, we adopt the FDNA method as both the utility combination model and surrogate model. We apply the surrogate FDNA model to the mid-layer and bottom layer, as shown on the right bottom of Figure 3, in which the ellipses represent operational activities in the mid-layer that are supported by the physical systems in the bottom layer. At the upper level, we use the FDNA method as the utility combination model, in which the values of the leaf nodes are generated by the mid-layer and the non-leaf nodes are virtual effectiveness nodes (e.g., detection capability). The mathematical formulation of the regular FDNA method [35] with an extension regarding the self-effectiveness (SE) [36] is given as in (1). The right bottom network in Figure 3 is an example. The effectiveness \(O_{k}\) and SE value of each node are both normalized between 0 and 100. The SE values of the virtual nodes in the utility combination model should all be set to 100. The effectiveness \(O_{i}\) of the leaf nodes are equal to their self-effectiveness values \(SE_{i}\). \[O_{k}=\min\biggl{(}\sum_{i=1}^{m}\frac{O_{i}\alpha_{ik}\delta_{ik}+SE_{i}(1- \alpha_{ik})}{m},\min(O_{1}+\beta_{11},\ldots,O_{i}+\beta_{ik},\ldots,O_{m}+ \beta_{mk})\biggr{)}, \tag{1}\]  
\[0\leq\alpha_{ik}\leq 1,\ \ 0\leq\beta_{ik}\leq 100,\ 0\leq\delta_{ik}\leq 1, \ 0\leq SE_{i},O_{i},O_{k}\leq 100, \tag{2}\]  
In a two-node pair (i.e., \(m=1\)), the FDNA method is a piece-wise linear parametric model using two dependency defining parameters--strength of dependency (SOD) \(\alpha_{ik}\) and criticality of dependency (COD) \(\beta_{ik}\). The SOD defines the fraction of the effectiveness of node \(k\) that depends on the effectiveness of node \(i\). The COD defines the remaining effectiveness of node \(k\) when the effectiveness of node \(i\) is equal to zero.