# Deterministic and Robust Optimization Approach for Single Artillery Unit Fire Scheduling Problem
## 2 Literature Review  
The artillery firing problem can be separated into the problem of assigning targets to multiple artillery units and that of determining the firing sequence for the assigned targets. The problem of assigning multiple enemy targets to multiple artillery units is a typical assignment problem, i.e., an NP-complete problem [2]. The problem of weapon-target assignment was first studied by [PERSON] [3]. Since then, various methods have been investigated and developed by many researchers. [PERSON] and [PERSON] [4] proposed a multi-stage version of the weapon-target assignment problem and developed a method to obtain the solution by transforming the problem into a deterministic one even when the number of targets is considerably large. [PERSON] and [PERSON] [5] developed an evolutionary approach to the weapon-target assignment problem, whereas [PERSON] and [PERSON] [6] solved the problem using a hybrid algorithm that combines the ant colony optimization algorithm and the genetic algorithm. [PERSON] et al. [7] proposed exact and heuristic algorithms to find optimal and approximate solutions by transforming the weapon-target assignment problem into integer and network flow problems, respectively. In contrast to the weapon-target assignment problem, the problem of determining the firing sequence has not been studied extensively. The firing sequence problem was first studied by [PERSON] et al. [1], who proposed a greedy heuristic to minimize the firing completion time of multiple artillery units. In their work, multiple artillery units were restricted to fire at a single target simultaneously to achieve surprise effects and sufficient destruction. Modern artillery shells can achieve sufficient surprise effects and destruction even if there is only a single firing unit. Hence, the restriction described above is not required. Instead, a method for determining the firing sequence of a single unit should be developed. Therefore, the firing sequence of a single artillery unit is considered in this study. [PERSON] and [PERSON] [8] assumed that the probability of target destruction by artillery fire decreases with time and developed a heuristic that minimizes the total threat of the surviving targets. In addition, [PERSON] and [PERSON] [8] assumed that the threat posed by an enemy target is independent of time. However, it is reasonable to assume that such a parameter is related to time, because while enemy targets exist, they can continue to cause damage to friendly units. We consider the threat level of an enemy target to increase with its survival time and develop a scheduling method to minimize the total threat of the enemy to friendly units until the end of the firing operation. Most existing studies define the factors underlying the solution of the artillery firing problem as deterministic parameters. However, there are many uncertainties among these parameters that must be considered for operational decision-making. In particular, it is impossible to evaluate information related to the enemy definitively. Nevertheless, [PERSON] and [PERSON] [5] and [PERSON] et al. [7] solved the weapon-target assignment problem by considering the enemy target threats to be certain parameters. Similarly, [PERSON] and [PERSON] [8] solved the fire scheduling problem by considering the enemy target threat to be a certain parameter. However, in the present study, the enemy target threat is considered to be an uncertain parameter, and an RO formulation is proposed to determine the firing sequence. The RO approach was first studied by [PERSON] [9], and it was extended by [PERSON] and [PERSON] [10] and [PERSON] and [PERSON] [11]. RO has been applied to various problems, such as the vehicle routing problem (VRP) [12], supply chain management [13], location and allocation [14], and portfolio optimization [15]. However, to the best of our knowledge, no study has applied RO to the artillery fire scheduling problem thus far.