# Beyond Kanban: POLCA-Constrained Scheduling for Job Shops
## 4 Industrial Case: Aerospace Manufacturing Firm
### The Proposed Methodology
#### 4.2.2 Constraint Programming Model
Sequencing of jobs is achieved via the precedence constraint in Equation (19), which mandates that the overall processing interval of job \((i,j-1)\) must end before that of job \((i,j)\) can start, with the delay between them captured by the integer variable \(\delta_{i,j}\). The allowable delay is then bounded by \(\Delta_{i,j}^{MIN}\) and \(\Delta_{i,j}^{MAX}\) in Equation (20), parameters that are detailed in \[\text{SizeOf}(\text{jobItv}_{i,j}^{WK})=p_{i,j}^{WK}+p_{i,j}^{WC },\forall i\in\mathcal{N},j\in\mathcal{J}_{i}\]0 Additionally, Equation (21) employs an Alternative constraint to assign each job to a qualified worker chosen from the set \(\mathcal{K}_{i,j}\). Temporal feasibility for orders is maintained by Equations (22) and (23), which ensure, respectively, that the order interval \(\mathrm{ordItv}_{i}\) does not begin before the order's arrival time \(a_{i}\) and does not extend beyond its due date \(d_{i}\) plus tardiness \(\tau_{i}\). Resource capacity is further safeguarded by the NoOverlap constraints. In Equation (24), the workcenter intervals for jobs assigned to a workcenter \(m\) (where \(\pi_{m}^{WC}=0\), as indicated in Table 7) are prevented from overlapping, and a similar restriction is imposed on worker intervals via Equation (25) when \(\pi_{w}^{WK}=0\). The model also integrates external availability restrictions through the ForbidExtent constraints. Equation (26) prohibits the execution of workcenter intervals during time slots when a workcenter is unavailable, as defined by \(\mathcal{S}_{m}^{WC}\), while Equation (27) applies the same logic to worker intervals using the set \(\mathcal{S}_{w}^{WK}\). These constraints utilize step functions to block out periods when the resources cannot be utilized. A novel feature of the model is its incorporation of POLCA card-based workload control. Equations (28) and (29) define the POLCA interval variable \(\mathrm{loopItv}_{i,j_{1},j_{2}}\) for transitions between production cells. Specifically, Equation (28) ensures that the start of the POLCA interval coincides with the start of the overall processing interval \(\mathrm{jobItv}_{i,j_{1}}^{PN}\) of the first job in the transition, while Equation (29) forces the end of the POLCA interval to match the end of the overall processing interval \(\mathrm{jobItv}_{i,j_{2}}^{PN}\) of the last job. Finally, Equation (30) employs the AlwaysIn constraint, together with a Pulse function weighted by the quantum \(q_{i,j_{1},j_{2}}\), to limit the cumulative number of concurrently active POLCA cards to \(C_{S_{1},S_{2}}\) for transitions between cells \(g_{1}\) and \(g_{2}\), as defined in \[\text{SizeOf}(\text{jobItv}_{i,j}^{WK})=p_{i,j}^{WK}+p_{i,j}^{WC },\forall i\in\mathcal{N},j\in\mathcal{J}_{i}\]1 Overall, by carefully integrating processing time definitions, synchronization and precedence constraints, resource capacity restrictions, and advanced POLCA card-based workload control, this constraint programming model provides a comprehensive framework for optimizing job scheduling in high-mix, low-volume manufacturing environments. The formulation not only ensures operational feasibility through its detailed temporal and resource constraints but also enhances flexibility and responsiveness in dynamic industrial settings.