Optimal Inspection Policies for Imperfect Production Systems with Learning Effects and Bayesian Demand Updating

This study develops a mathematical optimization framework to determine optimal inspection policies for imperfect production systems subject to stochastic deterioration. System degradation is modeled using a Weibull power law process, which captures the increasing likelihood of transitions from in-control to out-of-control states over time. When deterioration occurs, a reverse-order inspection strategy based on negative binomial sampling is employed, wherein an inspection continues until a predefined number of conforming items is obtained. The proposed model integrates inspection decisions with production learning effects and Bayesian demand updating. Learning-by-doing is incorporated through an experience-dependent production cost function, while demand uncertainty is addressed using Bayesian posterior estimation. A comprehensive expected total cost function is formulated, including production, inspection, inventory holding, warranty, and rework costs. The analytical properties of the model are examined, demonstrating that the expected total cost function is strictly convex with respect to the inspection decision variable. This convexity guarantees the existence and uniqueness of the optimal solution. Numerical experiments and sensitivity analyses illustrate the effects of defect rates, learning parameters, warranty periods, and demand uncertainty on the optimal inspection policy. The results show that jointly optimizing inspection intensity, learning effects, and demand information leads to significant cost reductions and robust decision-making in deteriorating production systems.