# Quantum-Based Analytical Techniques on the Tackling of Well Placement Optimization
## 4 Results and Discussion
### Performance Evaluation and Statistical Analysis  
To assess the discussed algorithms performance, several benchmark criteria are considered for this problem [6; 60]. Thus, the author's preferred benchmark for the evaluation are discussed in the following subsection:  
Effectiveness is the average value that is in between consecutive tests where the best solution is expressed with respect to the global optima as a percentage is effective and it is simply a measure of performance. Mathematically, effectiveness can be defined as follows:  
\[\overline{f}=\frac{1}{N}\sum\
olimits_{i=1}^{N}\frac{f(\dot{p_{i}})}{f(p*)} \tag{37}\]  
where \(f(p)\) is defined as a solution for \(p\), \(p*\) is denoted for the solution for global optima, \(\dot{p_{i}}\) is referred to as the best solution for \(i\)th trial, and \(N\) is trial number for each algorithm. Efficiency is the parameter that is an indicator of the algorithm's speed at which it reached a specific performance level using a number of distinctive evaluations to obtain a solution that is at least 98 percent of the best value in the experiment. \[\overline{L}=\frac{1}{N}\sum\limits_{i=1}^{N}\frac{L_{i}^{98}}{M}, \tag{38}\]  
where additionally, \(L_{i}^{98}\) is the number of unique function evaluations required to find solution \(q\) such that \(f(q)\_0\cdot 98f(\dot{p})\) for trial \(i\) (for minimization) and \(M\) is the total number of function evaluations per trial. Besides these two criteria, statistical data like standard deviation, average, and min-max are collected in experimental trials and the results are shown in Tables 4 and 5. The mean value and standard deviation indicate the robustness of the algorithm. The results of Case Study 1 show that the QPSO algorithm is optimal on four criteria. However, QBA was better on 2 other criteria. The box plot results are shown in Figure 6(a) indicates that QPSO and QBA have reached almost similar results compared to other algorithms and QPSO has the lowest standard deviation. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **GSA** & **PSO** & **CSA** & **GA** & **DE** & **BA** & **QPSO** & **QBA** \\ \hline Max & \(3.84\times 10^{9}\) & \(5.14\times 10^{9}\) & \(3.72\times 10^{9}\) & \(5.09\times 10^{9}\) & \(5.13\times 10^{9}\) & \(5.30\times 10^{9}\) & \(5.03\times 10^{9}\) & \(5.33\times 10^{9}\) \\ Min & \(2.83\times 10^{9}\) & \(3.43\times 10^{9}\) & \(2.43\times 10^{9}\) & \(3.01\times 10^{9}\) & \(3.38\times 10^{9}\) & \(2.10\times 10^{9}\) & \(4.38\times 10^{9}\) & \(4.29\times 10^{9}\) \\ Average & \(3.33\times 10^{9}\) & \(4.07\times 10^{9}\) & \(3.24\times 10^{9}\) & \(3.67\times 10^{9}\) & \(4.26\times 10^{9}\) & \(3.28\times 10^{9}\) & \(4.77\times 10^{9}\) & \(4.67\times 10^{9}\) \\ Standard deviation & \(2.62\times 10^{8}\) & \(5.72\times 10^{8}\) & \(3.73\times 10^{8}\) & \(5.11\times 10^{8}\) & \(4.59\times 10^{8}\) & \(8.35\times 10^{8}\) & \(1.60\times 10^{8}\) & \(2.74\times 10^{8}\) \\ Effectiveness & \(6.24\times 10^{-1}\) & \(7.63\times 10^{-1}\) & \(6.08\times 10^{-1}\) & \(6.88\times 10^{-1}\) & \(8.00\times 10^{-1}\) & \(6.16\times 10^{-1}\) & \(8.94\times 10^{-1}\) & \(8.76\times 10^{-1}\) \\ Efficiency & \(1.39\times 10^{-1}\) & \(5.53\times 10^{-1}\) & \(5.09\times 10^{-1}\) & \(4.78\times 10^{-1}\) & \(6.46\times 10^{-1}\) & \(8.25\times 10^{-1}\) & \(4.28\times 10^{-1}\) & \(5.38\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Statistical data of applied metaheuristic algorithms on well placement optimization for case study 1.