# A Hybrid Discrete Memetic Algorithm for Solving Flow-Shop Scheduling Problems
## 8 Parameter Sensitivity Analysis  
Since the proposed HDBMEA algorithm contains three distinct search algorithms, the number of parameters adds up to eleven. The determination of each parameter must be backed by an analysis. A subset of the Taillard benchmark set was used, where the number of machines and jobs equals twenty, to visualize the impact of each parameter on the overall scheduling performance and runtime. Each of the ten instances was run ten times for each parameter set (100 runs total). We considered the mean and standard deviation. The formula for obtaining the quality of each solution is detailed in Equation (11). Each plot shows the overall set of values obtained (light blue ribbon), the standard deviation (mid-blue ribbon), and the mean of all runs (dark blue line). Plots showing the change in standard deviation are also included. These plots confirm our chosen set of parameters for the final benchmark results. The iteration count of the simulated annealing algorithm is the maximum number of iterations since the last improvement in the makespan. This parameter dramatically \begin{table}
\begin{tabular}{l c} \hline \hline
**Parameter** & **Value** \\ \hline Maximum iteration count (SA) & 100 \\ \hline Initial temperature (SA) & 300 \\ \hline \(\alpha\) (SA) & 0.1 \\ \hline Iteration count (MCTS) & 10,000 \\ \hline \(N_{ind}\) & 8 \\ \hline Maximum iteration count (DBMEA) & 2 \\ \hline \(N_{clones}\) & 8 \\ \hline \(N_{inf}\) & 40 \\ \hline \(l_{seg}\) & 4 \\ \hline \(l_{trans}\) & 5 \\ \hline \(N_{mort}\) & 0.05 \\ \hline \end{tabular}
\end{table} 
Table 1: Parameters used for the HDBMEA algorithm. impacts the results and runtime and their standard deviation (See Figure 11). The omega and standard deviation decrease dramatically until our chosen iteration count of 100 (see Figures 10(b) and 11(d)), after which the improvements are negligible while the runtime keeps increasing linearly (See Figure 10(a)). The starting temperature for the simulated annealing algorithm has little impact on the results and runtime (Figure 12); however, there is a slight dip in \(\omega\) at the 300 mark (See Figure 11(b)), which is our chosen final value. The \(\alpha\) parameter of the simulated annealing algorithm is the temperature decrease in each iteration. This method creates an exponential multiplicative cooling strategy, which was proposed by [PERSON] et al. [68], and the effects of which are considered in [69]. The parameter did not significantly affect the overall performance of the algorithm, however (Figure 13). The iteration count of the MCTS algorithm is fixed; it does not take the number of iterations elapsed since the last improvement into account. This parameter improves the search results drastically (See Figure 11(b)), while the runtime increases only linearly (See Figure 11(a)). A high value of 10,000 was maintained throughout our testing since this proved to be a good middle-ground between scheduling performance (\(\omega\) ) and runtimes. The benefit of a high iteration count is the decrease in the standard deviation of \(\omega\) (See Figure 11(d)), meaning that the scheduling performance is increasingly less dependent on the processing times in the benchmark set. One downside of a high iteration count is the increase in the standard deviation of runtimes (See Figure 11(c)) since the SA algorithm is called every iteration, which has no fixed iteration count. Figure 11: Analysis of the maximum iteration count (SA) parameter. Figure 12: Analysis of the starting temperature (SA) parameter. Figure 13: Analysis of the \(\alpha\) parameter. The number of individuals (\(N_{ind}\)) in an iteration of the Discrete Bacterial Memetic Evolutionary Algorithm (the population size) dramatically affects both the scheduling performance and the runtime (See Figure 15). A larger number of individuals means a broader range of solutions that MCTS and SA can further improve. However, a lower population count was maintained with a high iteration count in both MCTS and SA to reduce runtimes while keeping omega values low.