Optimal inspection policies for a scale

In the final stage of manufacturing some specific products such as chemical ones, they weigh each product using a scale and mark the weighing result on each product. However, the scale will occasionally undergo malfunction or a failure during the weighing process. The products weighed by such a scale will be shipped with different marks from their actual weights. In the case of chemical products, those products with wrong marks can be regarded as defective products. This study proposes two types of periodical inspection policies for a scale to detect its malfunction or a failure followed by adjustment. The inspection in this study involves adjustment operations by which the malfunction or failures of the scale can be detected and the scale will recover from its malfunction or failure. Under Policy I, the scale is inspected at time iT (i=1,2,…). Under Policy II, we consider a situation where the scale is inspected every morning before we start daily work of weighing products, which can be observed in the actual circumstances. For such a case, we can carry out an inspection at iτ/N(i=0,1,2,…, N – 1), whereτ signifies the working hours per day and an integer N denotes the frequency of inspections to be conducted per day. Two types of objective functions are considered: (1) the fraction defective, and (2) the long-run average cost under each policy. Under Policy I (Policy II), we examine the existence of an inspection interval Tα (inspection frequency Nα) which guarantees that the fraction defective does not exceed a prespecified value α (0 < α < 1). An economical inspection interval T* (inspection frequency N*) minimizing the long-run average cost is also discussed. Numerical examples are presented to illustrate the theoretical underpinnings of the proposed inspection policy formulations.