## 5 Numerical Experiments
Reliability is represented by how far the algorithm predictions of the minimum are from the known global minimum, while efficiency is represented by the number of function evaluations needed for the calculation of this best value. Figure 4: _Cont._  
To complete the evaluation of the MAKHA in comparison with the original MA and KHA algorithms, we have employed the performance profile (_PP_) reported by [PERSON] _et al_. [55], who introduced PP as a tool for evaluating and comparing the performance of optimization software. In particular, PP has been proposed to represent compactly and comprehensively the data collected from a set of solvers for a specified performance metric. For instance, the number of function evaluations or computing time can be considered performance metrics for solver comparison. The PP plot allows visualization of the expected performance differences among several solvers and to compare the quality of their solutions by eliminating the bias of failures obtained in a small number of problems. To introduce _PP_, consider \(n_{s}\) solvers (_i.e._, optimization methods) to be tested over a set of \(n_{\text{p}}\) problems. For each problem \(p\) and solver \(s\), the performance metric \(t_{ps}\) must be defined. In our study, reliability of the stochastic method in accurately finding the global minimum of the objective function is considered as the principal goal, and hence the performance metric is defined as  
\[t_{ps}=f_{calc}-f^{*} \tag{24}\]  
Figure 4: Surface plots of the two-variable benchmark functions used in this study: (**a**) [PERSON], (**b**) [PERSON], (**c**) Booth, (**d**) Carrom table, (**e**) Cross-leg table, (**f**) [PERSON], (**g**) Levy 13, and (**h**) Schaffer. where \(f^{\ast}\) is the known global optimum of the objective function and \(f_{calc}\) is the mean value of that objective function calculated by the stochastic method over several runs. In our study\(,f_{calc}\) is calculated from 30 runs to solve each test problem by each solver; note that each run is different because of the random number seed used and the stochastic nature of the method. So, the focus is on the average performance of stochastic methods, which is desirable for comparison purposes. For the performance metric of interest, the performance ratio, \(r_{ps}\), is used to compare the performance on problem, \(p\), by solver, \(s\), with the best performance by any solver on this problem. This performance ratio is given by  
\[r_{m}=\frac{t_{m}}{\min\left\{t_{ps}:1\leq s\leq n_{s}\right\}} \tag{25}\]  
The value of \(r_{ps}\) is 1 for the solver that performs the best on a specific problem \(p\). To obtain an overall assessment of the performance of solvers on \(n_{p}\) problems, the following cumulative function for \(r_{ps}\) is used:  
\[\rho_{s}\left(\varsigma\right)=\frac{1}{n_{p}}size\left\{p:r_{ps}\leq\varsigma\right\} \tag{26}\]  
where \(\rho(\varsigma)\) is the fraction of the total number of problems, for which solver \(s\) has a performance ratio \(r_{ps}\) within a factor of \(\varsigma\) of the best possible ratio. The PP of a solver is a plot of \(\rho_{s}(\varsigma)\)_versus_\(\varsigma\); it is a non-decreasing, piece-wise constant function, continuous from the right at each of the breakpoints. To identify the best solver, it is only necessary to compare the values of \(\rho_{s}(\varsigma)\) for all solvers and to select the highest one, which is the probability that a specific solver will \"win\" over the rest of solvers used. In our case, the PP plot compares how accurately the stochastic methods can find the global optimum value relative to one another, and so the term \"win\" refers to the stochastic method that provides the most accurate value of the global minimum in the benchmark problems used.