# A Local Search-Based Generalized Normal Distribution Algorithm for Permutation Flow Shop Scheduling
## 5 Results and Comparisons  
In our experiments, the proposed algorithms are extensively validated on three benchmarks commonly used in the literature: (1) the first dataset is called the Carlier dataset, having eight instances with a number of jobs ranging between 7 and 14, and a number of machines at the interval between 4 and 9 [39]; (2) the second is the Reeves dataset with 21 instances, where the number of machines and the number of jobs ranges between 20 and 75, and 5 and 20, respectively [40]; and (3) finally, the third one is known as the Heller and involves two instances with a number of jobs ranging between 20 and 100, and a number of machines of 10, respectively [41]. Those datasets are taken from [42] with some characteristics about the number of jobs and machines, and the best-known makespan \(z^{*}\) in Table 3. Furthermore, the proposed algorithms are extensively compared with a number of the well-established optimization algorithms: sine cosine algorithm (SCA) [43], shap swarm algorithm (SSA) [44], whale optimization algorithm (WOA) [34], genetic algorithm (GA), equilibrium optimization algorithm (EOA) [45], marine predators optimization algorithm (MPA) [42], and a hybrid tunicate swarm algorithm (HTSA) [46] integrated with the local search strategy to ensure a fair comparison and verify their efficacy in terms of six performance metrics: average relative error (ARE), worst relative error (WRE), best relative error (BRE), an average of makespan (Avg), standard deviation (SD), and computational cost (Time in milliseconds (ms)). BRE indicates how far the best-obtained solution \(Z_{B}\) is close to the best-known solution and is formulated using the following formula:  
\[BRE=\frac{|Z^{*}-Z_{B}|}{Z^{*}} \tag{14}\]  
Meanwhile, WRE calculated using the next equation is a metric used to assess the remoteness between the worst-obtained makespan \(Z_{w}\) and the best known. \[WRE=\frac{|Z^{*}-Z_{w}|}{Z^{*}} \tag{15}\]  
Regarding ARE, it is used to show the relative error with respect to the average makespan values within 30 independent runs and the best-known one. Mathematically, ARE is modeled as follows. \[ARE=\frac{\left|Z^{*}-Z_{Avg}\right|}{Z^{*}} \tag{16}\]  
The algorithms used in our experiments after integrating local search are named a hybrid SCA (HSCA) [43], a hybrid SSA (HSSA) [44], a hybrid WOA (HWOA) [34], a hybrid GA (HGA), a hybrid EOA (HEOA) [45], a hybrid MPA (HMPA) [42], and a hybrid TSA (HTSA) [46]. Regarding the parameters of those algorithms, they were assigned after extensive experiments. The EOA has two parameters: \(a_{1}\) (exploration factor) and \(a_{2}\) (exploitation factor), which are needed to be accurately estimated, and after several experiments for extracting their optimal values, we note that all observed values for \(a_{2}\) were significantly converged; therefore, it is set to 1 as used in the standard algorithm; \(a_{1}\), which is responsible for the exploration operator, is assigned a value of 2 estimated after several experiments, pictured in Figure 3a. The SSA is self-adaptive algorithm since it does not have parameters to be assigned before beginning the optimization process; on the other hand, the HSCA has one parameter called \(a\) responsible for determining where the algorithm will search for the near-optimal solution, and the value to this parameter was set to 3, as shown in Figure 3b. The HMPA has one parameter \(P\) called the scaling factor, and it is set in