## 6 Discussion and Results
### Evolution of Proposed HMFO with Practical Instances
To make the comparison of proposed algorithm with the benchmark algorithms, several performance measures were reported [53; 54; 55; 56], and these measures were mainly useful for multi-/many-objective optimization problems. To make the process simple and effective, Pareto optimal graphs were plotted for three objectives shown in Figures 16 and 17. Figures 16 and 17 give various Pareto graphs generated by HMFO and NSGA -II, respectively. In Figures 16 and 17, the black color solutions indicate the non-dominated solutions, and blue color solutions indicate the remaining solutions. Considering effective evolution of all of the algorithms' performances gives a better picture. Table 9.0 Energy consumption values of all instances 1–10. Table 9.1 Reliability of different jobs of instance from 6–10. Table 9.2 Pareto optimal graphs showing various solutions for three objectives makespan, energy consumption, and machine utilization for HMFO algorithm. Table 9.3 Pareto optimal graphs showing various solutions for three objectives makespan, energy consumption, and Machine utilization for NSGA-II algorithm. The hypervolume (HV) is most used Performance Indicator (PI) among all for the comparison multi-objective algorithms [56]. HV is the volume occupied by the dominated Pareto front approximation '\(R\)' drawn from a reference point to \(\varepsilon\) O \({}^{\text{L}}\), such that \(Y\,\varepsilon\,R\), \(R<\,\text{o}\). The \(HV\) is given by Equation (14). Here, \(\eta_{L}\) represents \(L\) dimensional lebesgue measure. Table 9.4  
The \(HV\) describes the region of the objective space which is weakly dominated by the approximation set. Until now, there are no particular guidelines for selecting the reference point; however, the worst possible point (i.e., dominated by all points) then the nadir point (1, 1,... 1) is considered in most of the studies [57]. Here, the \(HV\) is calculated by specifying the reference point as (1, 1, 1,... 1). Out of all the available \(HV\) values, the best, median, and least values of hypervolume (\(HV\)) results are indicated with help of box-plot in Table 9.5 The \(HV\) is calculated for various instances of problems of proposed HMFO and NSGA-II algorithms represented in Table 9.6 For instance 1, in the case of HMFO, the best, median, and least values are 0.6997, 0.6297, and 0.5497, and similarly for instance 1, in the case of NSGA II, the best, median, and least \(HV\) values are 0.5653, 0.4853, and 0.3653, respectively. In a similar manner, the box plots were obtained for all 10 instances that are represented in Table 9.7 It is well known from the literature the higher the \(HV\) value, the better the performance of the algorithm. In the figure, after thorough observation, the \(HV\) values indicate the higher values for all the instances (i.e., from instance 1 to instance 10) in case of proposed HMFO when compared to the NSGA -II algorithm. This demonstrates the superiority of the proposed HMFO over the NSGA II algorithm for better approximation. Moreover, \(HV\) results for the first five instances (instances 1-5) fall in a lower range when compared to the other instances (instances 6-10). This may be due to different problem scenarios that were considered in all 10 instances. Apart from HV indicator, the results of various other performance measures for first five instances (1-5) and last five instances (6-10) are shown in Tables 10 and 11, respectively. Out of all the available solutions, the number of Non-Dominated (ND) solutions obtained by a proposed algorithm is denoted as \(\alpha\), and the number of ND solutions that are not identified by the benchmark algorithm is denoted by \(\beta\). For better results, it is suggested to have larger values of \(\alpha\) and \(\beta\) and the ratio of \(\beta/\alpha\) ratio near to one determines the  
Table 9.8 Comparison of HMFO and NSGA-II with Hyper-Volume (HV) results for all the 10 instances of problems. effectiveness of the algorithm's strength.