## 2 Models and Methods  
Artillery units perform missions to destroy remote enemy targets by firing ammunition. Enemy targets that should be hit by artillery units are assigned from a higher level of command as a mission or requested from friendly maneuvering forces. These enemy targets have different properties and sizes, and their threat times to friendly forces differ. Furthermore, when artillery units hit enemy targets, the time taken to attack each target until it is destroyed differs. The overall process of artillery units to hit all enemy targets sequentially is defined as a fire operation, and the process of hitting a single target is defined as a fire task. Then, artillery units determine the priority of fire tasks as artillery units cannot perform two or more fire tasks together in a fire operation. The criteria to determine the priority of fire tasks in a fire operation are minimization of the completion time of fire operations, minimization of the average fire task time, and minimization of the enemy's total threat inflicted on friendly forces.  
This study models the problem of determining the priority of fire tasks, with the aim of minimizing the total threat of enemies inflicted on friendly forces during a fire operation.  
The following assumptions are made in this study:  
1. Artillery units can only hit one target at a time and can only hit the next target after completing the current fire operation.
2. The time taken from attacking an energy target with artillery units to destroying the target is determined by the properties and size of the enemy's target.
3. The threat level of the target per unit time is constant until artillery units fire the target.
4. Once the artillery units start hitting a target, the threat level of the target per unit time decreases linearly; the target is destroyed after passing the required time to destroy the target and, then, the threat level (per unit time) becomes zero.
5. Enemy targets that are not under attack by artillery units may deal damage to friendly forces as much as the threat level per unit time.
6. Artillery units can start firing from the time when the threat of an enemy target arises.  
The notations used in this paper are defined as follows:  
\begin{tabular}{c c} \hline \hline Notation & Definition \\ \hline \hline \multicolumn{2}{c}{Decision variable} \\ \hline \(x_{j}\) & Firing start time of target j \\ \(y_{j\prime}\) & If target j takes precedence over target j\({}^{\prime}\) 1, otherwise 0 \\ \(x_{jk}\) & If target j is fired for kth 1, otherwise 0. \\ \(t_{k}\) & The kth firing time of artillery units \\ Parameter & \\ \hline \(j\) & Target j, j \(\in\) N = {1,..., n} \\ \(f_{j}\) & Nominal threat level of target j \\ \(\frac{f_{j}}{f_{j}}\) & Uncertain threat level of target j \\ \(\frac{f_{j}}{f_{j}}\) & Maximum threat level of target j \\ \(\frac{f_{j}}{f_{j}}\) & Minimum threat level of target j \\ \(\frac{f_{k}}{f_{k}}\) & Threat level of the kth firing target. \\ \(a_{j}\) & Threat time of target j \\ \(a_{j}\) & Uncertain threat time of target j \\ \(\frac{a_{j}}{a_{j}}\) & Max threat time of target j \\ \(a_{j}\) & Min threat time of target j \\ \(\frac{a_{k}}{a_{k}}\) & Threat time of the kth firing target. \\ \(p_{j}\) & Amount of time required to destroy target j \\ \(r_{j}\) & Threat removal rate of target j per unit time \(r_{j}=f_{j}/p_{j}\) \\ M & Big positive value \\ \hline \hline \end{tabular}