## 5 Numerical Experiments and Discussion
### Case Study-3  
Case Study-3 is similar to the setup of Case Study-2. The key difference is in the data. The interval widths in Case Study-2 are constant, and the intervals are non-overlapping. In this case study, the interval widths are not constant, and the intervals are overlapping for a given row or column. The data for Case Study-3 are presented as follows: The survival time payoff matrix \(\mathbf{H}\) containing the interval-based survival times are given in Table 6, and Table 7 presents the energy \(\mathbf{p}\) and the latency \(\mathbf{q}\) values. All the models are solved using the open source GLPK solver. The Pareto surface obtained after solving Formulation (12a-f) is depicted in Figure 1(a), and the Pareto surfaces obtained after solving Formulations (13a-f) and (14a-f) are illustrated in blue and green colored surfaces in Figure 1(b), respectively. Table 8 displays the results of the case study at \(\varepsilon_{1}=\varepsilon_{2}=0\), which indicates the minimax game values. From Table 8, it can be concluded that if the defender ignores the latency- and the energy-related objectives, then, under optimistic circumstances, playing \(\mathbf{R3}\), \(\mathbf{R5}\), \(\mathbf{R6}\), \(\mathbf{R7}\) or \(\mathbf{R8}\) with the given probabilities in Table 8 is the best for LPP. In practice, this could mean that \(\mathbf{R7}\) is deployed 36.9% of the time, \(\mathbf{R3}\) is deployed 26.3% of the time, \(\mathbf{R8}\) is deployed 14.4% of the time, and so on. Similarly, under pessimistic circumstances, playing \(\mathbf{R1}\), \(\mathbf{R2}\), \(\mathbf{R4}\), \(\mathbf{R5}\), \(\mathbf{R6}\) or \(\mathbf{R7}\) with the given probabilities in Table 8 is the best for LPP.  
Figure 1: Case Study-2, Pareto surfaces. (**a**) Formulation (12a–f) Pareto. (**b**) Formulations (12a–f)–(14a–f) Pareto surfaces.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Optimistic** & **Average** & **Pessimistic** \\ \hline
**R1** & 0.333 & 0.333 & 0.333 \\ \hline
**R2** & 0.333 & 0.333 & 0.333 \\ \hline
**R3** & 0 & 0 & 0 \\ \hline
**R4** & 0 & 0 & 0 \\ \hline
**R5** & 0 & 0 & 0 \\ \hline
**R6** & 0.333 & 0.333 & 0.333 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Minimax game values (at \(\varepsilon_{1}=\varepsilon_{2}=0\)) of Case Study-2.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **A1** & **A2** & **A3** & **A4** & **A5** \\ \hline
**R1** & [0.38, 0.83] & [0.23, 0.63] & [0.33, 0.59] & [0.09, 0.74] & [0.39, 0.99] \\ \hline
**R2** & [0.36, 0.81] & [0.24, 0.79] & [0.0, 0.78] & [0.39, 0.74] & [0.25, 0.5] \\ \hline
**R3** & [0.16, 0.91] & [0.28, 0.73] & [0.13, 0.73] & [0.0, 0.85] & [0.16, 0.56] \\ \hline
**R4** & [0.0, 0.61] & [0.2, 0.85] & [0.36, 0.76] & [0.33, 0.59] & [0.0, 0.59] \\ \hline
**R5** & [0.09, 0.95] & [0.2, 0.7] & [0.21, 0.51] & [0.30, 0.8] & [0.4, 0.8] \\ \hline
**R6** & [0.09, 0.55] & [0.01, 0.91] & [0.25, 0.5] & [0.42, 0.52] & [0.10, 0.7] \\ \hline
**R7** & [0.11, 0.96] & [0.02, 0.62] & [0.16, 0.76] & [0.09, 0.75] & [0.38, 0.88] \\ \hline
**R8** & [0.0, 0.59] & [0.0, 0.89] & [0.47, 0.97] & [0.04, 0.5] & [0.33, 0.58] \\ \hline \hline \end{tabular}
\end{table}
Table 6: Survival time payoffs for the defender in Case Study-3.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **R1** & **R2** & **R3** & **R4** & **R5** \\ \hline
**P** & 0.8 & 0.3 & 0.5 & 0.5 & 0.4 \\ \hline
**q** & 0.7 & 0.5 & 0.4 & 0.2 & 1 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Energy (**p**) and Latency (**q**) for each routing mechanism in Case Study-3.  
Figure 2: Case Study-3, Pareto surfaces. (**a**) Formulation (12a–f) Pareto. (**b**) Formulations (12a–f)–(14a–f) Pareto surfaces.