## 2 Models and Methods
### RO Approach Using Scenario Concept
RFS 1 is reformulated to RFS 2 to search for the robust solution with the idea of RO:  
\[\textit{minimize}\ y\quad(\text{RFS 2}) \tag{15}\]  
_subject to_  
\[\sum_{j=1}^{n}\sum_{k=1}^{n}\left\{\vec{f}_{j}x_{jk}\Big{(}t_{k}-\vec{a}_{j} \Big{)}+\frac{\vec{f}_{j}^{2}}{2r_{j}}x_{jk}\right\}\leq y\quad\vec{f}_{j}\in UF _{j},\qquad\vec{a}_{j}\in UA_{j} \tag{16}\]  
\[t_{1}=\sum_{i}^{n}\vec{a}_{j}x_{j1}\quad\ \ \vec{a}_{j}\in UA_{j} \tag{17}\]  
\[t_{k}=\max\left[\left(t_{k-1}+\sum_{j=1}^{n}x_{jk-1}\frac{f_{j}}{r_{j}}\right) \ \sum_{j}^{n}\vec{a}_{j}x_{jk}\right]\ \ \ \ k\geq 2,\ \ \vec{f}_{j}\in UF_{j},\ \vec{a}_{j}\in UA_{j} \tag{18}\]  
\[\sum_{j=1}^{n}x_{jk}=1\quad\forall\ k \tag{19}\]  
\[\sum_{k=1}^{n}x_{jk}=1\quad\forall\ j \tag{20}\]  
\[x_{jk}=0\ or\ 1\quad\forall\ j,\ k \tag{21}\]  
Equation (17) refers to the first fire task time of artillery units when a firing sequence is provided, which is the time at which the first firing target emerges. Equation (18) is the kth fire task time of artillery units when a firing sequence is given. The possible kth fire task time starts when the \((k-1)\)th fire task is complete and the kth firing target emerges. In RFS 2, solution x refers to a firing sequence, which can be expressed as follows:  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]0  
A set of feasible solutions, X, can be expressed as follows according to Equations (19)-(21):  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]1  
In RFS 2, the objective equation and constraint of \(f^{WORST}(x)\), which searches for the worst scenario (WS) for a solution \(x\), are as follows:  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]2subject to_  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]3  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]4  
WS \(1\) has uncertain parameters in the objective equation and a constraint, and it is still an intractable problem. **Theorem 1**. : _For any solution \(x\in X\), the worst scenario occurs only when \(\tilde{f}_{j}\) is at maximum value in \(\forall j\) and when \(\tilde{a}_{j}\) is at minimum value or maximum value._  
The proof of theorem 1 is given in Appendix A.  
WS 2, which is equivalent to the WS 1 model, can be reformulated as follows based on Theorem 1. \[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]5  
_subject to_  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]6  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]7  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]8  
WS 2 has finite constraints. Now, problem \(\underset{x\in X}{\min}\ \underset{s\in S}{\max}f(x,\ s)\) is divided into master and subproblems, as shown in Figure 1, to find the solution. \"Master problem\" refers to finding the robust optimal solution using GA, and \"subproblem\" refers to finding the worst scenario for any solution \(x\). Model WS 2 helps search for the worst scenario, and the objective value is assigned to the individual fitness function score in the master problem. GA was first invented by [PERSON] [23]. GA is a probabilistic and population-based algorithm, which was inspired by the field of genetics. Generally, GA is applied to problems in which the search space is large for discrete optimization problems [24; 25]. This study employed GA to overcome the complexity of solution combination and realistic situations to determine artillery firing schedules within a brief period of time during wars, as suggested by [PERSON] [17]. In this study, the  
\[f^{WORST}(x)=\max_{s\in S}\ f(x,s) \tag{13}\]9 Robust optimization (RO) approach using scenario concept. encoded to a vector (permutation \(\sigma\)) that contained the target information. For example, permutation (3, 1, 4, 2) means that targets 3, 1, 4, and 2 are determined as the first, second, third, and fourth fire tasks, respectively. The initial solution setup in this model is created by the determination criterion that \(r_{j}\) is decreased [11].