The detection of an upward shift in the fraction defective of a repetitive process is considered using the geometric CUSUM. This CUSUM makes use of the information provided by the run-lengths of non-defective items between successive defective items, and was initially developed for the case of 100% inspection. This paper considers the geometric CUSUM under sampling inspection, and emphasizes that the pattern of sampling inspection can be quite haphazard without causing any difficulty for the operation of the CUSUM. Two separate mechanisms for the occurrence of a shift are considered. Methods for evaluating zero-state and steady-state ARL are presented for both 100% inspection and sampling inspection. Parameter choice is also considered, and recommendations made. Comparisons with some np -charts are provided.
The Binomial CUSUM is used to monitor the fraction defective p of a repetitive process, particularly for detecting small to moderate shifts. The number of defectives from each sample is used to update the monitoring CUSUM. When 100% inspection is in progress, the question arises as to how many sequential observations should be grouped together in forming successive samples. The tabular form of the CUSUM has three parameters: the sample size n, the reference value k, and the decision interval h, and these parameters are usually chosen using statistical or economic-statistical criteria, which are based on Average Run Length (ARL). Unlike earlier studies, this investigation uses steady-state ARL rather than zero-state ARL, and the occurrence of the shift can be anywhere within a sample. The principal ®nding is that there is a signi®cant gain in the performance of the CUSUM when the sample size n is set at one, and this CUSUM might be termed the Bernoulli CUSUM. The advantage of using n 1 is greater for larger shifts and for smaller values of in-control ARL.
A Continuous Sampling Plan, CSP-CUSUM, is proposed based on the use of Cumulative Sums (CUSUMs) for deciding when to switch between the phases of sampling inspection and 100% inspection. The Geometric CUSUM, also termed the Run-length CUSUM, is chosen for this purpose, and two separate CUSUMs are to be operated, one for each inspection phase. The conventional measures of performance for CSPs such as average outgoing quality, average fraction inspected, and average proportion passed under sampling inspection are evaluated for CSP-CUSUM, and comparisons with some standard CSPs are presented. An additional performance-measure, Average Cycle Length, is proposed. A table is provided to aid the choice of parameters for the operation of CSP-CUSUM. It is recommended that a Geometric CUSUM control chart be maintained in parallel with CSP-CUSUM to detect significant upward shifts in the incoming fraction defective.
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