## 1 Introduction  
Artillery refers to a type of heavy military ranged weapon that hits remote targets by operating the weapon system as a unit. It differs from small arms or tanks in armored forces that are used by infantry to engage in short-range gunfights against enemies. The process of operating artillery is as follows: First, target information such as a target properties, size, and location are acquired; second, decision-making on firing method is followed to ensure efficient and effective hits against the target; and third, artillery units hit the target. The number of casualties owing to artillery bombardment has been the largest in World War I, World War II, and other wars until the development of modern warfare. Thus, many countries have recognized the power and importance of artillery and have made considerable effort to develop weapon system technologies, artillery operation tactics, and firing methods. Thus, with the development of state-of-the-art technologies, target information has become more accurate owing to information collection assets, and improved and intelligent ammunition performance has increased the destructive power of artillery and the accuracy of hit ratio. In addition, studies on artillery operation tactics such as shoot-and-scoot tactics [1] and firing methods have been increasingly conducted. This study investigates different firing methods to determine artillery firing scheduling. Studies on firing methods are largely divided into weapon-target assignment problems (WTAPs) and fire-scheduling problems (FSPs). Existing studies on WTAPs are as follows: Reference [2] introduced a WTAP for the first time and proposed that this problem could be approximated to a linear model even though it was probabilistic and nonlinear. Reference [3] proved that the WTAP is NP-complete and that the computation time increases exponentially as the problem size increases. References [4; 5] solved the WTAP using a modified evolution algorithm, and Reference [6] solved the WTAP using a hybrid algorithm. Reference [7] modified the WTAP into integer programming and network flow problems and developed an exact algorithm and a heuristic algorithm. The FSP was first introduced by Reference [8]; they proposed a greedy heuristic to minimize the makespan. Reference [9] assumed that the probability of target destruction decreased with time and proposed a scheduling heuristic to minimize the total threat from surviving targets. Reference [10] developed a robust optimization (RO) model that minimized the total threat exposed to friendly forces by considering enemy threat parameters as uncertain parameters since the intangible combat power of enemy targets could not be predicted or estimated accurately. Reference [11] proposed a two-phase method that could evaluate the total threat of an enemy that is inflicted on friendly forces when the enemy threat level is uncertain. Existing studies on WTAPs and FSPs obtain solutions by assuming that the threat times of all targets that should be hit by artillery units are the same. However, the threat times of targets would likely differ in actual wars. Thus, this study proposes an artillery firing scheduling methodology that minimizes the total enemy threat that is inflicted on friendly forces by considering that the targets' threat times are different. The artillery firing scheduling problem considered in this study is the single machine earliness/tardiness problem (SMETP). An artillery unit in this study is regarded as a single machine, and the target that should be hit by the artillery unit is considered a job. Here, the time from attacking the target by an artillery unit to the target's destruction is considered the processing time and the threat of a target inflicted on friendly forces before its destruction is defined as a penalty. If the threat time of each target is considered as the release time, then if an artillery unit does not hit their target as soon as the target threat occurs, a tardiness penalty is inflicted.