# 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
These results underscore the critical importance of adequate POLCA card availability in mitigating deadlock risks, particularly when employing reactive simulation strategies. Notably, in the proactive simulation mode \"opt\", where job start times are strictly set by the CP model, no deadlocks occur. This is because the CP model must produce a valid schedule for all orders that avoids any circular waiting. As a result, the CP model naturally provides a deadlock-free solution, eliminating the need for additional measures like starvation cards. This demonstrates a clear benefit of integrating POLCA with finite capacity scheduling: by properly controlling the timing of POLCA card retrieval, deadlocks can be prevented, thus improving the overall robustness of the system. In addition to the deadlock analysis, a comprehensive set of descriptive statistics was computed for each experimental group to assess system performance. For each order, total tardiness was calculated as the sum of delays beyond the order's due date, and summary statistics were derived including the average tardiness, its standard deviation, as well as the minimum and maximum tardiness observed. The percentage of tardy orders was also computed, indicating the proportion of orders that missed their due dates. Furthermore, the total waiting time per order--aggregating the waiting periods for both machine availability and POLCA card availability--was recorded. The Shop Floor Throughput Time (STT) was measured as the time interval between the start of the first job and the finish of the last job within an order, with both average and maximum STT values reported. Finally, the Total Throughput Time (TTT), defined as the elapsed time from order arrival to the finish of the last job, was determined, again providing average and maximum values. These statistics, as summarized in Table 3, offer a detailed overview of the system's operational performance under the various experimental conditions. \begin{table}
\begin{tabular}{l l l l r r r} \hline \hline
**opt Mode** & **nb Machines** & **PolaCards** & **Simulation Mode** & **Instances** & **with Deadlock** & **pct with Deadlock** \\ \hline early & 6 & 1 & opt & 10 & 0 & 0.00 \\  & & & sim & 10 & 10 & 100.00 \\ \cline{2-6}  & & 2 & opt & 10 & 0 & 0.00 \\  & & & sim & 10 & 0 & 0.00 \\ \cline{2-6}  & & 3 & opt & 10 & 0 & 0.00 \\  & & & sim & 10 & 0 & 0.00 \\ \hline
9 & 1 & opt & 10 & 0 & 0.00 \\  & & & sim & 10 & 10 & 100.00 \\ \cline{2-6}  & & 2 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \cline{2-6}  & & 3 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \hline
12 & 1 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 5 & 50.00 \\ \cline{2-6}  & & 2 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \cline{2-6}  & & 3 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \hline
9 & 1 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 10 & 100.00 \\ \cline{2-6}  & & 2 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \cline{2-6}  & & 3 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \hline
12 & 1 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 5 & 50.00 \\ \cline{2-6}  & & 2 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \cline{2-6}  & & 3 & opt & 10 & 0 & 0.00 \\  & & sim & 10 & 0 & 0.00 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Summary of deadlock occurrences across experimental configurations. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline
Table 3 presents the descriptive statistics for key performance indicators across the different experimental groups. In general, these statistics vary considerably with the number of workcenters, the POLCA card settings, and the simulation mode. For instance, configurations with a higher POLCA card count tend to yield lower average tardiness and reduced waiting times. Moreover, the proactive simulation mode \"opt\", which enforces the start times provided by the CP model, generally exhibits more consistent and lower maximum tardiness and throughput times compared to the reactive simulation mode \"sim\".