An Optimal Lot-Sizing and Offline Inspection Policy in the Case of Nonrigid Demand

A batch production process that is initially in the in-control state can fail with constant failure rate to the out-of-control state. The probability that a unit is conforming if produced while the process is in control is constant and higher than the respective constant conformance probability while the process is out of control. When production ends, the units are inspected in the order they have been produced. The objective is to design a production and inspection policy that guarantees a zero defective delivery in minimum expected total cost. The inspection problem is formulated as a partially observable Markov decision process (POMDP): Given the observations about the quality of the items that have already been inspected, the inspector should determine whether to inspect the next unit or stop inspection and possibly pay shortage costs. We show that the optimal policy is of the control limit threshold (CLT) type: The observations are used to update the probability that the production process was still in control while producing the candidate unit for inspection. The optimal policy is to continue inspection if and only if this probability exceeds a CLT value that may depend on the outstanding demand and the number of uninspected items. Structural properties satisfied by the various CLT values are presented.