Automated rosterers are usually judged by how close they drive hard-constraint violations to zero. For the 13-person emergency department of Naili DBS Hospital, Padang, Indonesia, we show that zero is unattainable and that the residual has three separable causes. We formulate the rostering problem exactly as implemented, with nine hard constraints and four soft criteria weighted by priorities elicited from the department head through the analytic hierarchy process. A general capacity certificate (a counting argument on worker-hours) proves that two fixed-shift contracts force three violation units on the department’s 30-day planning period that no roster and no hire can remove, and a horizon analysis shows the midwife class within 30 h of an hour shortage. An exact constraint-programming model, cross-validated against the implementation’s scoring code, proves that the minimum is 11 units once the night-pattern rule is respected, attributes every unit to a class and a cause, and shows that one additional midwife removes all midwife coverage gaps yet leaves the instance infeasible (optimum eight) because the residual gaps are nurse positions lost to sequencing. Under an equal budget of 48,100 evaluations and 30 runs per method, a genetic algorithm with heuristic repair and a random-restart hill climber are not separable (Cliff’s |δ| ≤ 0.12), both dominate random sampling (|δ| = 1), and neither reaches the optimum under 50 alternative weight vectors or a fourfold larger budget. The instance, not the metaheuristic, is binding; an autonomous scheduler should certify and attribute its residual alongside the roster.