# Deterministic and Robust Optimization Approach for Single Artillery Unit Fire Scheduling Problem
## 4 Robust Counterpart Formulation
### Robust Counterpart Formulation with Cardinality-Constrained Uncertainty  
In this section, a robust counterpart formulation with cardinality-constrained uncertainty is proposed. This formulation overcomes the drawback of the excessively conservative solution of the robust counterpart formulation with box uncertainty (RCFB). The RCFB model assumes that all of the target threat levels are maximized in the interval \(\left[f_{j},\ f_{j}+\hat{f}_{j}\right]\); however, the actual probability of such a scenario is quite low. Therefore, by considering the possibility that there may be targets with zero deviation of the uncertain enemy target threat level, a parameter, \(\Gamma\), is included in Equation (9). \(\Gamma\) represents the uncertainty degree and it controls the conservative level of the solution. The value of \(\Gamma\) lies in the interval \([0,\,n-1]\) and is not necessarily an integer. The maximum value of \(\Gamma\) is \(n-1\) because the enemy target that the artillery unit first fires at does not cause damage to friendly forces; only the remaining \(n-1\) enemy targets do so. \(\Gamma\) in Equation (9) allows up to \(\lfloor\Gamma\rfloor\) of \(n\) uncertain enemy target threat levels to attain their worst-case values simultaneously and allows one uncertain enemy target threat level to change by \((\Gamma-\lfloor\Gamma\rfloor)\hat{f}_{t}\). Therefore, the decision-maker adjusts the robustness of the solution by changing the value of \(\Gamma\) in the range \([0,\,n-1]\). The robust counterpart formulation with cardinality-constrained uncertainty (RCFC 1) is given by  
\[\text{minimize }y\qquad(\text{RCFC 1})\]  
subject to  
\[y\geq\sum_{j}^{n}f_{j}x_{j}+\beta(x^{*},\Gamma) \tag{9}\]  
where  
\[\beta(x^{*},\Gamma)=\max_{\{S\cup\{t\}|S\leq\Gamma,\lvert S\rvert=\Gamma f \in\Gamma\}}\left\{\sum_{j\in S}\hat{f}_{j}x_{j}^{*}+(\Gamma-\lfloor\Gamma \rfloor\hat{f}_{t}x_{j}^{*}\right\}\]  
(2), (3), and (4). If \(\Gamma\) is \(0\) in RCFC 1, then RCFC 1 is equivalent to DFS, in which the threat level of all enemy targets is certain. Furthermore, if \(\Gamma\) is \(n-1\) in RCFC 1, then RCFC 1 is equivalent to RCFB with the maximum threat level for all of the targets. RCFC 1 can adjust the conservativeness and robustness of the solution with the uncertainty degree, \(\Gamma\), when the enemy target threat levels are uncertain, but it is a nonlinear optimization problem that is intractable. Therefore, RCFC 1 can be transformed into a linear discrete optimization problem that is tractable by following the method of [PERSON] and [PERSON] [16]. **Proposition 2. **_Equation (9) can be reformulated as a linear optimization problem when the uncertain enemy target threat level \(\widetilde{f}_{j}\) lies within the predefined interval \(\left[f_{j},\ f_{j}+\hat{f}_{j}\right]\)._  
**Proof. ** The objective value of the robustness cost function \(\beta(x^{*},\Gamma)\) (RCF) is equivalent to the objective value of the following linear optimization problem (LP). \[\beta(x^{*},\Gamma)=\text{maximize}\ \sum_{j\in J}\hat{f}_{j}x^{*}z_{j}\qquad \text{RCF}\]  
subject to  
\[\sum_{j\in J}z_{j}\leq\Gamma \tag{10}\]  
\[0\leq z_{j}\leq 1\ \forall j\in J \tag{11}\]  
The dual problem of RCF is given by  
\[\text{minimize}\ \Gamma u+\sum_{j}^{n}\pi_{j}\qquad\text{LP}\]  
subject to  
\[u+\pi_{j}\geq\hat{f}_{j}\ x_{j}\qquad j=1,2,\ \ldots,\ n\]  
\(u\geq 0\), and  
\[\pi_{j}\geq 0\qquad j=1,2,\ \ldots,\ n\]  
where \(u\) is the dual variable corresponding to the inequality given by Equation (10) and \(\pi_{j}\) is the dual variable corresponding to the inequality given by Equation (11). RCF is feasible and it is bounded for \(\Gamma\in[0,\ n-1]\). In addition, LP is feasible and bounded for \(\Gamma\in[0,\ n-1]\) by the strong duality theorem, and the optimal objective values of RCF and LP are the same.