# A Robust Heuristics for the Online Job Shop Scheduling Problem
## 5 Experimental Results  
The goal of exhaustive experiments was to investigate the effect of parameters \(\lambda\) and \(T\) on the final result. By conducting a large number of scenarios, we aimed to determine the optimal values for these parameters for each problem. The results presented in Table 1 highlight the best objective function values obtained for each \(\lambda\) and \(T\) combination, with the value of \(T\) at which the best result was first found shown in brackets. Bolded results indicate that the solution obtained is the same as the best known solution (BKS), while results in bold and italic indicate our best result, which is not the same as the BKS. These experiments provide valuable insights into the performance of the algorithm under different parameter settings, which can be used to inform future research and development efforts. In this study, we present the results of experiments conducted on 17 well-known JSSP problems. For each problem, we conducted a number of scenarios in which two parameters were varied. The parameter \(\lambda\) runs over six values, i.e., 0%, 1%, 2%, 3.5%, 5%, and 10%, and the parameter \(T\) over 11 different values, i.e., 0.10, 0.20, 0.40, 0.60, 0.80, 1.00, 2.00, 3.00,5.00, 7.00, and 10.00. For each combination of \(\lambda\) and \(T\), we conducted 20,000 experiments. Table 1 shows the best objective function values obtained for each \(\lambda\) and \(T\) combination. Note that Table 1 only displays the best results obtained for each value of \(\lambda\) and the corresponding value of \(T\). The full set of results for each scenario is available upon request. Figure 2 presents the Gantt chart of the best-known solution for the LA01 instance. The solution achieves an optimum time of 666. The results summarized in Table 1 indicate that very different values of parameter \(T\) were used in the best runs; note that only one, the first \(T\), is given. Regarding parameter \(\lambda\), there are several instances for which the best results were obtained with \(\lambda=0\), while there are some other instances where nondelay schedules were inferior. This is expected, as in some instances, it is known that the optimal or best-known solutions must have delays. It may be interesting to note that for some instances (FT10, FT20, ABZ5, ABZ6, LA03), the best solutions were obtained with only one (small nonzero) \(\lambda\). It is not clear whether this was pure luck or whether there is some reason for such behavior. As the approach is based on the priority rule, which only uses the information on the jobs currently in the local queue, we can write our first observation. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline  & & & & & **Our Best with \(\lambda\) and (\(T\))** & \\ \hline
**Instance** & **Size (n \(\times\) m)** & **BKS** & **0\%** & **1\%** & **2\%** & **3.5\%** & **5\%** & **10\%** \\ \hline  
\ \hline \hline \end{tabular}
\end{table} 
Table 1: Results of the algorithm for different \(\lambda\) values. The value of \(T\) at which the best result was first found is in parentheses. Results in bold indicate the best value obtained by our heuristics. Results in bold are BKS; results in in bold and italic are the best results that are not BKS. Figure 2: Gantt chart of the best solution for LA01 instance with a makespan of 666. **Observation 1**. : _The heuristics running online (without full information on the instance available a priori) is able to find near-optimal solutions (in some cases even optimal)._  
In addition to the main experiments, we also conducted advanced experiments to analyze the quality of the obtained solutions. Specifically, we evaluated the distribution of the obtained results relative to the best known solution we obtained. In the first experiment, we have performed multiple runs of the randomized simulations to see how good solutions may be found, and we have seen that the best solutions are close to the performance of the best offline algorithms.