# Beyond Kanban: POLCA-Constrained Scheduling for Job Shops
## 3 POLCA and the Enhanced Contraint Programming: A Basic Case
### Computational Experiments
#### 3.4.3 Computational Results
To evaluate the influence of experimental parameters and the effectiveness of the scheduling approach, we first performed a series of independent two-sample \(t\)-tests. The outcome of the statistical tests is visually reflected in the boxplots (Figures 2-4), where the distribution for the optimization-based approach (\"opt\") typically appears lower and less dispersed than for the simulation-driven approach (\"sim\") across all key metrics and most configurations. The position of the medians and the clear separation between the boxplots directly mirror the negative values of the \(t\)-statistic and the highly significant \(p\)-values observed. For every experimental configuration, defined by the combination of optimization mode, number of workcenters, and number of POLCA cards, we compared the proactive optimization approach (\"opt\") with the reactive simulation-driven approach (\"sim\"). The comparison was conducted on three key performance metrics: average Tardiness (avg_tardiness), average Shop Throughput Time (avg_stt), and maximum Shop Throughput Time (max_stt). For each solved instance, the average tardiness is calculated as the mean tardiness across all orders, providing a direct measure of the system's ability to deliver jobs on time and to minimize late deliveries. STT represents the lead time of an order, that is, the total time from the order's release to its completion. Considering both the average and the maximum STT allows us to assess not only the overall efficiency of the system but also its robustness against extreme delays, which is particularly relevant in high-mix, low-volume production environments. The average STT reflects how quickly, on average, orders are completed, while the maximum STT captures the worst case scenario within each configuration, highlighting situations where a single order could experience excessive delay. For each configuration and metric, the null hypothesis was that the distributions of the two methods are equal. Before each \(t\) test, [PERSON]'s test was used to verify the equality of variances; whenever the assumption was rejected (\(p<0.05\)), [PERSON]'s correction was applied to account for heteroskedasticity. For each comparison, the mean and standard deviation for both approaches, the \(t\) statistic, the \(p\)-value, [PERSON]'s \(p\)-value, and the equal variance assumption are reported. The detailed results are presented in Tables 4-6. The results of the two-sample \(t\)-tests demonstrate a clear and consistent advantage of the optimization-based approach (\"opt\") over the simulation-driven approach (\"sim\") across almost all experimental configurations and performance metrics. For both avg_tardiness and avg_stt, the mean values obtained under the \"opt\" mode are systematically and substantially lower than those observed with \"sim\", often by several hundreds of units, indicating that the proactive scheduling method is markedly more effective in reducing both average tardiness and mean lead times. This difference is strongly statistically significant, as shown by \(p\)-values well below the conventional 0.05 threshold in the vast majority of cases. Notably, the effect is especially pronounced in configurations with higher numbers of workcenters, where the gap between the two approaches becomes even more evident. This trend is particularly visible in the boxplots of avg_stt and max_stt, where not only the medians but also the entire range of the \"opt\" distribution remain consistently below those of \"sim\", especially as the number of workcenters increases. Furthermore, the extreme values (upper whiskers and outliers)for \"sim\" are much more pronounced, highlighting the lower robustness of the simulation-driven approach in the worst cases.