# Robots in Partially Known and Unknown Environments:
## 5 Experiments and Results
### OfflineRMP Analysis
#### 5.2.1 Parameter Tuning  
A tuning procedure consists of determining the best parameter values for a given metaheuristic algorithm considering a set of problem instances and its stochastic nature. A couple of automated tuning methods for metaheuristic algorithms have been proposed in the literature [37]. We used ParamILS [38], a well-known tuning approach, to set the cooling factor \(\alpha\), the initial temperature \(t_{0}\), and the maximum number of iterations. ParamILS is an iterated local search heuristic that searches for the best parameter configuration. Starting from an initial parameter configuration, at each step, the value of one parameter is changed, and the quality of the new configuration is tested. A new configuration is considered to be better only if its average performance is better, as measured by spending a higher number of executions.  
The parameters tuned, their possible values, and their initial values are listed in Table 4. The possible values are listed between curly brackets, and the corresponding initial values are shown between square brackets. Moreover, we used the set of instances listed in Table 3 to perform the process. The distance to the best-known solution is used to compare the quality of different parameter configurations. A maximum of 1000 iterations was considered to be the default budget that was not increased during the tuning process. A total budget of 25,000 executions of the OfflineRMP algorithm was considered to be the stopping criterion of the tuning process.  
Table 5 summarizes the performance of the proposed approach considering the parameter set obtained from the tuning process: \(\alpha=0.95\), \(t_{0}=500\), and \(max\_it=1000\). It shows, for each problem instance, the average length and standard deviation of the initial solution, of the solution obtained once the simulated annealing algorithm finishes, and the average length and standard deviation of the solution obtained after the post-processing phase. Moreover, it also shows the average smoothness and standard deviation of the final solution, and the average iteration of the best solution found. Smoothness was measured using Equation (6).  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Description** & **Values** \\ \hline \(\alpha\) & Cooling factor of simulated annealing & \{0.8, 0.85, 0.9, 0.95, 0.99\} [0.95] \\ \(t_{0}\) & Initial temperature & \{100, 250, 500, 750, 1000\} [500] \\ max\_it & Maximum number of iterations & \{100, 500, 1000, 2000, 5000\} [1000] \\ \hline \hline \end{tabular}
\end{table}
Table 4: Tuning process.  
Figure 14: Environments’ variety.  
\[\text{Smoothness}=\frac{1}{n-2}\sum_{p=2}^{n-1}cos^{-1}\frac{v^{f}(p)\cdot v^{l}(p )}{\|v^{f}(p)\|\cdot\|v^{l}(p)\|} \tag{6}\]  
where \(n\) is the number of vertices, and \(v^{f}\) and \(v^{l}\) correspond to the vectors from the last vertex to the current and from the current to the following. The smoothness computes the average between the angles formed at each vertex measured in degrees from 0 to 180; a higher number shows less movement between vertices, hence a smoother path.