## 2 Methodology
### Network Optimization  
The numerical solutions of the second kind of LE-NSM are acquired to optimize the GNN by applying the hybrid computing scheme, i.e., GAASM. GA is one of the intelligent evolutionary computing schemes that is based on natural development. In the 7 th decade of the 19 th century, GA is discovered with the innovator's work of [PERSON] [39], and later it is employed as a key leading derivative to optimize the models based on constrained/un-constrained arrangements. GA works through the optimal process of mutation, selection, heuristic, and crossover. GA is widely applied in many areas such as robotics, astrophysics, optics, digital communication, bioinformatics, signal processing, financial mathematics, nuclear power system, economics, chemical industry, and materials. Some recent submissions of GAs that works as an optimization model arewind power model [40], heart disease prediction [41], intrusion detection performance model [42], energy managing systems [43], metal-organic constructions [44], heterogeneous celebration [45], heartbeat systems [46], a study of military systems [47], aquatic weed model [48]. These presented applications inspired the authors to apply the GA as an optimization process using the GNN to find the approximate outputs of the second kind of LE-NSM. GA hybridizes with local search ASM, i.e., local search approach to use the quick convergence by assigning the best GA values as a start/initial point. Hence, ASM is suitable to regulate the parameters. ASM has been implemented in many recent submissions, such as trade and industry load dispatch models [49], short-term hydrothermal supervision [50], bipedal walking robot dynamics [51], economic multiproduct manufacture [52], LNG process [53], model of heating in the thermal blow frame cycling [54], aircraft transportation [55], wind turbine support structures [56] and quadratic convex bilevel programming models [57]. In this study, the combination of GAASM is implemented to solve the second kind of LE-NSM, and the optimization process of GAASM is provided in Table 1. The variety of the procedure is introduced for selection, i.e., stochastic uniform, remainder, roulette, and tournament, for mutation, i.e., Gaussian, uniform and adaptive feasible, as well as for crossover, scattered, single point, two points, intermediate, arithmetic and heuristic, however, we set stochastic uniform for the selection, heuristics for the crossover and adaptive feasible for mutation. These parameter settings are adopted after a lot of experiments, knowledge, experience, and performance advantages on different applications in the presented study. **GA procedure**  
**Inputs: Indicate the chromosomes with equal number of model entries as: \(W=[\mathbf{q},\mathbf{w},\mathbf{p}]\)**  
**Population:** The chromosomes set is signified as: \(\mathbf{q}=[q_{1},q_{2},q_{3},\ldots,q_{m}]\), \(\mathbf{w}=[w_{1},w_{2},w_{3},\ldots,w_{m}]\) and \(\mathbf{p}=[p_{1},p_{2},p_{3},\ldots,p_{m}]\). **Output:** GA best global weights are symbolized as \(\mathbf{W_{\text{GA-Best}}}\)  
**Initialization:** Create a \(\mathbf{W}\) called a weight vector containing real entries to select a chromosome. '\(\mathbf{W}\)' is applied to design an initial population with [Population Size = 270]. Regulate the values of generation as well as assertion for the ga optimset. **Fitness valuation:** Accomplished the fitness (\(E\)) in the population for all the weight vectors using the Equations (5)-(7). **Stopping criteria:** Terminate, when any of the value is achieved  
\(\bullet\) [Fit = \(10^{-19}\)], [StallLimit = 100], [Generations = 75], [TolCon = TolFun = \(10^{-22}\)],  
Other: default  
Move to **storage**, when meets the stopping standards  
**Ranking:** Rank the weight vector of Population for the brilliance of Fit  
**Reproduction:**  
\(\bullet\) [Selection = @ uniform],  
\(\bullet\) [Crossover:@ heuristic],  
\(\bullet\) [Mutations:@adapt feasible]. **Store:** Save \(W_{\text{GA-Best}}\), \(E\), Generations, function counts and time for the existing GAs runs.