# Deterministic and Robust Optimization Approach for Single Artillery Unit Fire Scheduling Problem
## 1 Introduction  
The objective of this study is to determine the artillery firing sequence in military operations when a single artillery unit fires at multiple enemy targets, such that the enemy threat to friendly forces is minimized. 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 determining the firing sequence for the assigned targets can be further separated into the problem of determining the firing sequence of multiple artillery units and that of determining the firing sequence of a single artillery unit. Owing to the lower destructive power and performance of older artillery shells, multiple artillery units were required to fire simultaneously at a single enemy target to achieve surprise effects and sufficient destruction. Thus, the determination of the firing sequence of multiple artillery units was considered to minimize the firing operation completion time [1]. However, modern artillery shells have high destructive power, and the shell launch system is automated. Thus, a single artillery unit can operate independently. Therefore, the firing sequence of a single artillery unit is considered in this study. The objective of an artillery unit is to support friendly forces by targeting remote enemy targets, including infantry, artillery, armored units, intelligence units, and command and communications facilities. Once a war commences, various enemy units may inflict severe damage on friendly forces unless they are destroyed by friendly artillery fire. A single artillery unit cannot fire at multiple targets simultaneously; it must fire at them one at a time. Thus, a single artillery unit must determine the sequence of firing at enemy targets to minimize the total losses to friendly forces. In this study, the degree of threat of an enemy target is defined as the degree of the loss that it causes to friendly forces per unit time. We consider the enemy target threat level, destruction time, and firing preparation time simultaneously. Accordingly, we develop a deterministic model for determining the firing sequence to minimize the total threat of the enemy to friendly units until the end of the firing operation. In general, military operational environments are uncertain, and decision-makers must base their decisions on uncertain battlefield information. In particular, it is impossible to assess the information related to the enemy definitively. Such information is highly uncertain because enemies often attempt to deceive their opponents and minimize their exposure. Thus, the enemy target threat is considered to be an uncertain parameter, and the model for determining the firing sequence is developed based on uncertain enemy information. Such a robust optimization (RO) model enables decision-makers to obtain robust solutions and to anticipate robustly the total threat of the enemy to friendly units. The main contributions of this study are as follows. A deterministic optimization model is developed to determine the firing sequence necessary to minimize the total enemy threat to friendly units by simultaneously considering the threat levels of the targets, destruction time, and firing preparation time. Robust counterpart formulations are developed to determine the firing sequence required to minimize the total enemy threat to friendly units when the threat level of the enemy is uncertain. Our experimental results verify that the robust counterpart formulation with cardinality-constrained uncertainty is equivalent to the deterministic optimization model for the firing sequence when the degree of uncertainty is minimum and that it is equivalent to the robust counterpart formulation with box uncertainty when the degree of uncertainty is at its maximum. The remainder of this paper is organized as follows. Section 2 reviews previous studies related to the problems of artillery firing and RO.