## 4 Illustrative Example  
### Problem Setup  
This paper employs a synthetic air and missile defense (AMD) example that mimics the U.S. Naval Integrated Fire Control Counter Air (NIFC-CA) system. As future warfare is moving towards more distributed mosaic warfare, smaller-unit systems, unmanned systems, and reconfigurable systems will be gradually added to the combat SoS. The operational architecture can be generated based on historical experiences, anticipated capability requirements, and expert engineering judgements. This case study builds a simplified operational architecture based on the current NIFC-CA operations [37]. As illustrated in Figures 6 and 7, the activities of F_S1, F_S2, and F_S3 refer to long-range detection, mid-range detection, and low-altitude detection respectively. F_D1 refers to the operational command and control (C2) function in the field, such as the C2 function provided by early warning (EW) aircraft. F_D2 represents the command and control function of the entire combat SoS. F_I1 and F_I2 refer to long-range and mid-range strike functions. The dependency parameters SOD and COD are assumed to be (0.5, 50) between different activities and (0.9, 10) between activities and capability. There are a couple of different methods to obtain the dependency parameters in the surrogate model form, including expert judgment based on marginal effect, simulation data based fitting, etc. Since the goal of this case study is to demonstrate the feasibility of the integrated method, we have simplified the selection of dependency parameters. Thus, we simply employ two mid-values to include the impact of both the feeder nodes and receiver nodes. Regarding the utility combination form, the dependency parameters are subjective values based on decision-makers' preferences.  
Figure 6: Operational architecture of a synthetic naval AMD SoS.  
Figure 7: Operational activities mapping to capability.  
This case study considers four types of candidate system, namely destroyer, EW aircraft (e.g., E-2D), stealth fighter (e.g., F-35), and unmanned aerial vehicle (UAV). Each type of system can support several activities in the operational architecture, as listed in Table 2. Each type consists of a series of classes (from class A to class E) that vary in technology advancement, acquisition cost, function capacity, communication bandwidth, and ease of reconfiguration. Specifically, from class A to class E, the technological advancement level increases so that the acquisition cost, function capacity, communication capability, and ease of reconfiguration (i.e., less reconfiguration time) also increase. Some of the input data are provided in Tables 2 and 3. The acquisition budget is given as 2000 million dollars and the upper limit of the reconfiguration time is assumed as two seconds. Although the data are currently synthetic, they are able to demonstrate the concept of the proposed framework.  
If we assume a random attack following uniform distribution occurs to both systems and communications links, when an attack happens to a system, the SE value of this system falls to zero; when an attack occurs to a communication link, the parameter \(\delta_{ik}\) falls to zero. After the random attacks, the SoS reconfigures itself to recover partial capability.