# Deterministic and Robust Optimization Approach for Single Artillery Unit Fire Scheduling Problem
## 5 Experimental Results and Discussion  
We generated and tested a set of problems using the proposed models. First, we verified that the robust counterpart formulation with cardinality-constrained uncertainty (RCFC 2) is equivalent to the deterministic optimization model (DFS) for fire scheduling when the uncertainty degree is zero and that it is equivalent to the robust counterpart formulation with box uncertainty when the uncertainty degree is \(n-1\). Next, we observed the objective values according to the changes in \(\Gamma\) and \(\varepsilon\) in the robust counterpart formulation with cardinality-constrained uncertainty, where \(\Gamma\) is the uncertainty degree and \(\varepsilon\) is the uncertainty level. The number of targets was set to 5, 10, 15, and 20. Threat levels, \(f_{j}\), were randomly generated in [25,100] as specified by [PERSON] et al. [7]. \(p_{j}\) and \(s_{j\!j\prime}\) were generated based on field artillery tactics [21]. When the single artillery unit fires at one target, \(p_{j}\) is the amount of time required to destroy target \(j\) and is determined based on the type and size of the enemy target. This value can range from 1 min to 3 min. \(s_{j\!j\prime}\), is the preparation time required to fire at target \(j^{\prime}\) after firing at target \(j\) and is determined based on the location of target \(j\) and the type of shell. This value can range from 0 min to 2 min. Accordingly, \(p_{j}\) and \(s_{j\!j\prime}\) are randomly generated in [1,3] and [0,2], respectively. The numerical experiments were implemented using the IBM ILOG CPLEX 12.6.1 (IBM software, New York, NY, USA). The computer environment was an Intel Core i5-6600 CPU (3.3 GHz) with 8 GB RAM (random-access memory). First, we solved the test problems using the proposed models and verified the robust counterpart formulation with cardinality-constrained uncertainty (RCFC 2). In these experiments, the uncertain threat level, \(\widetilde{f}_{j}\), belonged to the uncertainty set \(U_{j}=\left\{\widetilde{f}_{j}:f_{j}\leq\widetilde{f}_{j}\leq f_{j}+\hat{f}_ {j}\right\}\), where \(\hat{f}_{j}=\varepsilon f_{j}\) and \(\varepsilon\) was selected from the set {0.2, 0.4, 0.6, 0.8, 1.0}. The numerical experiments were repeated 100 times for each case. Table 1 lists the objective values and CPU times of DFS, RCFB, and RCFC 2 with various \(\Gamma\). The objective value of RCFC 2 is the same as that of DFS when the uncertainty degree is zero, regardless of the number of enemy targets. Moreover, the objective value of RCFC 2 is the same as that of RCFB when the uncertainty degree is maximized, regardless of the number of enemy targets. Therefore, RCFC 2 with \(\Gamma\) = 0 is equivalent to DFS and RCFC 2, with \(\Gamma\) = \(n-1\) is equivalent to RCFB, so the DFS and RCFB problems can be solved by changing only \(\Gamma\) in the RCFC 2 model. No significant differences were observed among the CPU times of the proposed models. However, as the number of targets increased, the CPU time increased exponentially. The computational complexity of a large-scale fire scheduling problem is extremely high. Therefore, heuristics are required to solve such large-scale problems.