## Materials and Methods
### Capability Gap Based Value Decision
#### 2.1.2 Value Model Construction  
Value is a measurement to denote the importance degree of an object. In the military field, a frequently used measurement for denoting value is the capability gap, meaning the gap between system capabilities and capability requirements. Before calculating the capability gap, it's necessary to obtain system portfolio capabilities, which are the combination of those of component systems. Firstly, let \(CR=\left\{cr_{1},\ldots,cr_{k}\right\}\) be the set of capabilities requirements, proposed by stakeholders, where \(cr_{j}\) denotes the value of the \(j\)th capability requirement. The \(C(s_{i})=(c_{i1},\ldots,c_{ik}),C(s_{i})\in\{0,1\}^{m}\) represents the capability value of the system \(s_{i}\), where \(c_{ij}\) denotes the capability value of \(s_{i}\) on the \(j\)th capability. The \(PC(x_{i})=(pc_{1}(x_{i}),\ldots,pc_{k}(x_{i}))\) denotes the capabilities values of the system portfolio corresponding to scheme \(x_{i}\), where \(pc_{j}(x_{i})\) is the \(j\)th capability value. In the combination process, one case must be considered is that certain systems in a system portfolio may have the same capabilities. It's a key procedure to deal with the combination of this kind of capabilities. Hereon, four rules are introduced as follows to support this combination. Assume there are n systems \(s_{i},i=1,\ldots,n\) in a system portfolio, providing the same capability \(t\), with capability values:\(c_{it},i=1,\ldots,n\). 1. Additive rule: the combined capability value is \(\sum c_{it},i=1,\ldots n\). E.g., Assuming that 3 transportation systems operate at the same time, the freight volume are \(5t\), \(6t\) and \(7t\) respectively, then the portfolio of the 3 systems can provide a freight capability of \(18t\). Figure 1: Division for variable X and Y. 2. Maximal rule: the combined capability value is \(max\{c_{it}\},\ i=1,\ldots,n\). E.g., Assuming that there are three bridges over a river, and each bridge can bear the weight of 100\(t\), 120\(t\) and 130\(t\) respectively, then only object less than 130\(t\) can pass the river, because one object can only pass over one bridge at the same time. 3. Minimal rule: the combined capability value is \(min\{c_{it}\},\ i=1,\ldots,n\). E.g., Assuming that there are 3 tandem oil pipelines with oil flow of 5 _thour_, 6 _thour_ and 3 _thour_, then the max oil flow of the 3 pipeline is 3 _thour_. 4. Average rule: the combined capability value is \(\sum c_{it}/n,\ i=1,\ldots n\). E.g., Assuming there are 3 forecast systems, with correctly predicting probability 50%, 60%, and 70%, then the overall correctly predicting probability is 60%. In addition, the capability can be classified into benefit and cost types. Firstly, the relation between a single capability of a solution and corresponding capability requirement is defined based on three premises: (1) When a capability of a solution is deficient or worse than the inferior value of the corresponding capability requirement interval, which means the solution absolutely cannot meet the capability requirement. Its capability gap is supposed to be 1. (2) When a solution can provide a capability, with its value falling in the interval of corresponding capability requirement, its capability gap should be a number between \([0,1]\). Especially when the solution's capability value exactly falls in the middle of the capability requirement interval, the capability gap is supposed to be 0.5, which means the capability can meet the capability requirement by half. (3) When a capability of a solution equals to or better than the superior value of the corresponding capability requirement interval, it means that the capability can absolutely meet the corresponding capability requirement, and the capability gap is supposed to be 0. A concrete case is given in Figure 2, reflecting the relation between a capability and corresponding capability requirement interval \([l,u]\). Four special points discussed above are marked by circle points.