# Quantum-Based Analytical Techniques on the Tackling of Well Placement Optimization
## 4 Results and Discussion
### Exploration and Exploitation Analysis  
Dimension wise analysis sheds light on the internal behavior of the algorithms [63]. In this context, it is important to have a strong diversification to better exploit the search space. A small degree of diversification implies local convergence. To measure the diversity in each dimension, the authors have considered the equation proposed in Reference [64]:  
\[Div_{j}=\frac{1}{n}\sum_{i=1}^{n}|median(x_{j})-x_{ij}|, \tag{39}\]  
where \(x_{ij}\) is the dimension \(j\) of the \(i\)th swarm, \(median(x_{j})\) refers to the median of dimension \(j\) in the whole population, and \(n\) is the total number of the population. So, the average diversity can be calculated using Equation (40):  
\[Div=\frac{1}{D}\sum\
olimits_{i=1}^{n}Div_{j}, \tag{40}\]  
where \(Div\) is the diversity measurement of the whole population in an iteration. Hence, Exploration and Exploitation can be measured using the following equation:  
\[Exploration=\frac{Div}{Div_{max}}\times 100, \tag{41}\]  
\[Exploitation=(1-\frac{Div}{Div_{max}})\times 100, \tag{42}\]  
where, \(Div_{max}\) presents the maximum diversity of whole populations in one run. Without revealing the behavior of swarms in the iterative process, it is difficult to understand the end result and the value of the objective function. Therefore, to understand the end results of these techniques or convergence curve in Figure 6, this study initiates a graphical illustration to show the algorithms' exploration and exploitation behavior over the whole iteration. From Figure 8, it can be inferred that a higher NPV can be achieved with an adequate exploration and development ratio. On the other hand, these figures clearly indicate a gradual change in intensity during exploration and exploitation during the iterative process. Figure 8: _Cont._Figure 8: _Cont_. The graphs shown in Figure 6 shows that while QPSO and QBA has a high average exploration rate. On the other hand, CSA, GSA, and GA had a relatively low exploration rate. In case Study 1, the overall exploration rate of QBA and QPSO is 74% and 60%, respectively, and a quantum behaved algorithm offered a better optimal solution. Again, among all algorithms, the least average exploration group contains BA, GA, CSA, and GSA and their exploration rate was 6.6%, 8.2%, 17.4%, and 6.4%, respectively. This shows that the average diversity obtained by the other algorithms is much less than the quantum-behaved algorithms. Furthermore, it can be noted that BA, GA, CSA, and GSA obtained the least average NPV value and quantum-behaved algorithms offer a better optimal solution. Again, PSO and DE obtained 27.4% and 54.6% exploration rate, respectively. Figure 8: Exploration and exploitation. On the other hand, in case study 2, the average diversity of BA, CSA, and DE was 85.1%, 83.3%, and 85.2%, respectively. In this case, BA provides the highest exploration rate. However, BA and CSA were unable to achieve better optimization results. Hence, it showcases that the higher exploration rates are not necessary to optimize the well location. Again, PSO, GSA, and GA achieved 39%, 21.93%, 14.2% exploration rates. Moreover, PSO was able to achieve 3 rd best average NPV. Similarly, 1 st and 2 nd best average NPV provider QBA and QPSO achieved 66.4%, 71% exploration, respectively. Therefore, the correct exploration-exploitation ratio should be carefully calculated according to the search field. In general, a roughly balanced response indicates that the algorithm is more efficient. This seems to be one of the explanations for the successful results of The QPSO and QBA algorithm. Another important finding in this study is the successful search is produced through exploration mechanisms with the effective exploration of mechanisms and appropriate balanced responses. In some cases, the two metaheuristic algorithms offer very different performances to the quality of the solution, but the balanced responses are similar (i.e., in case study 2, DE and BA's Exploration percentage = 85, exploitation percentage = 15).