# Deterministic and Robust Optimization Approach for Single Artillery Unit Fire Scheduling Problem
## 5 Experimental Results and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & & \multicolumn{4}{c}{**Objective Values**} & \multicolumn{4}{c}{**CPU Time (s)**} \\ \hline \multirow{2}{*}{\(n\)} & \multirow{2}{*}{\(\varepsilon\)} & \multirow{2}{*}{**DFS**} & \multirow{2}{*}{**RCFB**} & \multicolumn{4}{c}{**RCFC 2**} & \multirow{2}{*}{**DFS**} & \multirow{2}{*}{**RCFB**} & \multirow{2}{*}{**RCFB**} & \multirow{2}{*}{**RCFB**} \\ \cline{5-5} \cline{7-10}  & & & & & \(\Gamma\) = 0 & \(\Gamma\) = \(n-1\) & & & \\ \hline \multirow{5}{*}{6} & 0.2 & & 1924.4 & 1603.7 & 1924.4 & & 0.09 & 0.08 \\  & 0.4 & & 2245.1 & 1603.7 & 2245.1 & & 0.10 & 0.09 \\  & 0.6 & 1603.7 & 2565.8 & 1603.7 & 2565.8 & 0.07 & 0.09 & 0.08 \\  & 0.8 & & 2886.6 & 1603.7 & 2886.6 & & 0.09 & 0.08 \\  & 1.0 & & 3207.3 & 1603.7 & 3207.3 & & 0.10 & 0.08 \\ \hline \multirow{5}{*}{9} & 0.2 & & 4267.9 & 3556.6 & 4267.9 & & 0.61 & 0.50 \\  & 0.4 & & 4979.2 & 3556.6 & 4979.2 & & 0.59 & 0.51 \\ \cline{1-1}  & 0.6 & 3556.6 & 5690.5 & 3556.6 & 5690.5 & 0.45 & 0.62 & 0.49 \\ \cline{1-1}  & 0.8 & & 6401.8 & 3556.6 & 6401.8 & & 0.60 & 0.50 \\ \cline{1-1}  & 1.0 & & 7113.1 & 3556.6 & 7113.1 & & 0.64 & 0.49 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of objective values and CPU times of DFS (deterministic fire scheduling), RCFB (robust counterpart formulation with box uncertainty), and RCFC 2 (robust counterpart formulation with cardinality-constrained uncertainty) with various \(\Gamma\). Next, when number of targets was set to 12, we observed the objective values according to the changes in \(\varepsilon\) and \(\Gamma\) in RCFC 2. In these experiments, the uncertain threat level, \(\widehat{f}_{j}\), belonged to the uncertainty set \(U_{j}=\left\{\widehat{f}_{j}:f_{j}\leq\widehat{f}_{j}\leq f_{j}+\hat{f}_{j} \right\}\), where \(\hat{f}_{j}=\varepsilon f_{j}\) and \(\varepsilon\) was selected from the set {0.2, 0.4, 0.6, 0.8, 1.0}. Table 2 lists the objective values of RCFC 2 for different \(\varepsilon\) and \(\Gamma\). It can be seen that as \(\Gamma\) increases, the rate of increase of the objective values decreases, as shown in Figure 1, regardless of \(\varepsilon\). RCFC 2 provides a solution that does not increase the total threat to friendly forces significantly, even if the decision-makers do not know the uncertainty degree or if the uncertainty degree increases. Regarding the effect of \(\varepsilon\) on the objective value, as \(\Gamma\) increases, the difference of the objective value with respect to \(\varepsilon\) also increases, as shown in Figure 2. For example, the objective value in the case of \(\varepsilon\) = 0.2 lies between 6602.9 and 7923.5, while that in the case of \(\varepsilon\) = 1.0 lies between 6602.9 and 13,205.9. The greater the accuracy of the information about the enemy threat level, the greater is the accuracy with which the total threat of the enemy to friendly forces can be predicted. Therefore, efforts should be devoted toward making information about enemy target threats more accurate. \begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \(\Gamma\) & **0** & **1** & **2** & **3** & **4** & **5** & **6** & **7** & **8** & **9** & **10** & **11** \\ \hline
**0.2** & 6602.9 & 6835.0 & 7033.6 & 7204.3 & 7359.2 & 7499.8 & 7628.1 & 7733.5 & 7813.2 & 7869.2 & 7905.4 & 7923.5 \\
**0.4** & 6602.9 & 7062.7 & 7451.7 & 7793.8 & 8104.8 & 8389.4 & 8646.1 & 8855.7 & 9017.1 & 9132.5 & 9206.9 & 9244.1 \\ \(\varepsilon\) & **0.6** & 6602.9 & 7286.1 & 7895.6 & 8374.8 & 8840.1 & 9269.5 & 9657.7 & 9977.0 & 10219.8 & 10394.3 & 10,507.9 & 10,564.7 \\
**0.8** & 6602.9 & 7505.0 & 8263.6 & 8946.5 & 9568.7 & 10,143.3 & 10,661.8 & 11,095.1 & 11,422.5 & 11,656.0 & 11,808.5 & 11,885.3 \\
**1.0** & 6602.9 & 7791.1 & 8665.0 & 9516.9 & 10,295.0 & 11,014.5 & 11,659.8 & 12,207.0 & 12,624.7 & 12,917.7 & 13,109.0 & 13,205.9 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Objective values for different \(\varepsilon\) and \(\Gamma\) (\(n\) = 12). Figure 1: Objective values for different \(\varepsilon\) (\(n\) = 12).