The Production of Service: A Workload View of Complementarity and Substitution

We study optimal coordination in quality-driven rework systems, where each station contributes to an item's quality and the likelihood of rework depends on these contributions through a production function. Our model links quality targets to processing times via Brownian hitting times and aggregates quality into rework probability. By setting quality targets, a planner jointly shapes processing times and rework rates, determining overall workload. We characterize the attainable workload region and use it to minimize effort costs in two-station networks. The optimal policy depends on (i) complementarity versus substitutability of quality contributions, (ii) each station's quality impact, and (iii) relative operational costs. We identify three operating regimes: two corner regimes (one active station) and a network regime (both active). As complementarity increases, effort shifts toward the more cost-effective station, while finite capacity modifies this threshold, pushing effort toward the less constrained station. The analysis highlights how complementarity shapes efficiency trade-offs and capacity requirements. To illustrate the model's practical relevance, we calibrate it to industrial maintenance operations in which equipment undergoes repair followed by preventive service while offline. The results show that the optimal design is robust, efficient, and estimable from operational data.