A continuous sampling plan using CUSUMs

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.     

Effect of Inspection Error on Out-of-Control Average Run Length of CUSUM Charts

The cumulative sum (CUSUM) chart is commonly used for detecting small or moderate shifts in the fraction of defective manufactured items. However, its construction relies on the error-free inspection assumption, which can seldom be met in practice. In this article, we discuss the construction of an upward CUSUM chart in the presence of inspection error, study the effects of inspection error on the out-of-control ARL of the CUSUM chart, and present a formula for determining the sampling size that compensates for the effect of inspection error on the out-of-control ARL.     

A bayesian analysis of imperfect inspection models

An examination of sampling inspection models is provided for the case where inspection is not perfect and classification errors can be made. The conjugate family of distributions is obtained under t h e assumption that defective items are generated according to a Bernoulli process. To simplify analysis, the use of a single beta prior distribution is considered. Relevant predictive distributions are obtained.     

Repeated screening with inspection error and no false positive results with application to pharmaceutical pill production

Summary.  Repeated screening is a 100% sampling inspection of a batch of items followed by removal of the defective items and further iterations of inspection and removal. The reason for repeating the inspection is that the detection of a defective item happens with probability p <1. A missed defective item is a false negative result. The no false positive result is contemplated in this paper, which is motivated by a problem coming from the production of pharmaceutical pills. Bayesian posterior distributions for the quality of the lot are obtained for the case of both p known and p unknown. Batch rejection and batch acceptance control limits for the number of defective items at subsequent iterations can then be calculated. Theoretical connections to the problem of estimating the number-of-trials parameter of a binomial distribution are drawn.     

A new policy for designing acceptance sampling plan based on Bayesian inference in the presence of inspection errors

The purpose of this article is to present a new policy for designing an acceptance sampling plan based on the minimum proportion of the lot that should be inspected in the presence of inspection errors. It is assumed that inspection is not perfect and every defective item cannot be detected with complete certainty. The Bayesian method is used for obtaining the probability distribution function of the number of defective items in the lot. To design this model, two constraints of producer risk and consumer risk are considered during the inspection process by using two specified points on operating characteristic curve. In order to illustrate the application of the proposed model, an example is presented. In addition, a sensitivity analysis is performed to analyze the model performance under different scenarios of process parameters and the results are elaborated. Finally, the efficiency of the proposed model is compared with the sampling method of Spencer and Kevan de Lopez (2017) at the same conditions.     

The RL2 chart versus the np chart for detecting upward shifts in fraction defective

The use of the np chart for monitoring fraction-defective is well-established, but there are a number of relatively simple alternatives based on run-lengths of conforming items. Here, the RL₂ chart, based on the moving sum of two successive conforming run-lengths, is investigated in order to provide SPC practitioners with clear-cut guidance on the comparative performance of these competing charts. Both sampling inspection and 100% inspection are considered here, and it is shown that the RL₂ chart can often be considerably more efficient than the np chart, but the comparative performance depends on the false-alarm rate used for the comparison. Graphs to aid parameter-choice for the RL₂ chart are also provided.     

A likelihood ratio test for the equality of proportions of two normal populations

In sampling inspection by variables, an item is considered defective if its quality characteristic Y falls below some specification limit L₀. We consider switching to a new supplier if we can be sure that the proportion of defective items for the new supplier is smaller than the proportion defective for the present supplier.

Assume that Y has a normal distribution. A test for comparing these proportions is developed. A simulation study of the performance of the test is presented.     

Dodge-Romig AOQL single sampling plans for inspection by variables

Acceptance sampling plans for inspection by variables, which minimize the mean inspection cost per lot of process average quality (assuming that the remainder of rejected lots is inspected), are derived under the condition that the maximum value of the mean fraction defective after sampling inspection, replacing all defective items found by good ones, shall be equal top _L . These plans are tabulated for chosen values of the given parameters, and compared with the Dodge-Romig AOQL attribute sampling plans. From the comparison it follows that under the same protection of consumer the AOQL plans for inspection by variables are in some situations more economical than the corresponding Dodge-Romig plans.     

Modified tightened two-level continuous sampling plans

In this paper, a modification is proposed on the tightened two-level continuous sampling plan. The tightened two-level plan is one of the three tightened multi-level continuous sampling plans of Derman et al. (1957) with two sampling levels. A modified tightened two-level continuous sampling plan is considered, for which the rules concerning partial inspection depend, in part, on the length of time it takes to decide that the process quality is good enough that 100% inspection may be suspended (e.g. the time required to find i consecutive items free of defects). Using a Markov chain model, expressions for the performance measures of the modified MLP-T-2 plan are derived. The modified MLP-T-2 plan is shown to be identical to the MLP-T-2 plan. Tables are also presented for the selection of the modified MLP-T-2 plan when the AQL or LQL and AOQL are specified.     

Optimal acceptance sampling policy considering Bayesian risks

In this paper, we propose a sampling policy considering Bayesian risks. Various definitions of producer's risk and consumer's risk have been made. Bayesian risks for both producer and consumer are proven to give better information to decision-makers than classical definitions of the risks. So considering the Bayesian risk constraints, we seek to find optimal acceptance sampling policy by minimizing total cost, including the cost of rejecting the batch, the cost of inspection, and the cost of defective items detected during the operation. Proper distributions to construct the objective function of the model are specified. In order to demonstrate the application of the proposed model, we illustrate a numerical example. Furthermore, the results of the sensitivity analysis show that lot size, the cost of inspection, and the cost of one defective item are key factors in sampling policies. The acceptable quality level, the lot tolerance proportion defective, and Bayesian risks also affect the sampling policy, but variations of acceptable quality level and producer Bayesian risks, for values more than a specified value, cause no changes in sampling policy.     

Enhanced adaptive CUSUM charts for process mean

A statistical quality control chart is an important tool of the statistical process control, which is widely used to control and monitor a production process. The CUSUM chart is designed to detect a specific shift, provided that the shift size is known in advance. In practice, however, shift sizes are rarely known. It is then customary to use an adaptive CUSUM chart, which can effectively detect a range of shift sizes. In this paper, we enhance the sensitivities of the improved adaptive CUSUM mean charts using an auxiliary-information-based (AIB) mean estimator. The run length performances of the proposed charts are compared with those of the AIB adaptive and non-adaptive CUSUM charts in terms of the average run length (ARL), extra quadratic loss, and integral relative ARL. These run length comparisons reveal that the proposed charts are more sensitive than the existing charts when detecting different kinds of shift in the process mean. An example is given to demonstrate the implementation of existing and proposed charts.     

An economic inspection interval for control of defective items in a hot rolling mill

The article addresses a real-life problem on determining the optimum sampling interval for control of defective items in a hot rolling mill. Having observed that the pattern of appearance of mill defects indicates a geometric process failure mechanism, an economic model is developed in line with the method suggested by Taguchi and critically examined by Nayebpour & Woodall. An expression for the expected loss per product as a function of the sampling interval is derived and the optimum interval is obtained by minimizing this loss function. The practical issues involved in this exercise, such as estimation of various cost components, are also discussed and the effect of erroneous estimation of cost components is studied through a sensitivity analysis.     

Design and Construction of a Variables Repetitive Group Sampling Plan for Unilateral Specification Limit

This paper attempts to develop a repetitive group sampling (RGS) plan by variables inspection for controlling the process fraction defective or the number of nonconformities when the quality characteristic follows a normal distribution and has only the lower or upper specification limit. The proposed sampling plan is derived by the exact sampling distribution rather than the approximation approach. The plan parameters are solved by a nonlinear optimization model which minimizes the average sample number required for inspection and fulfills the classical two-point conditions on the operating characteristic (OC) curve. The efficiency of the proposed variables RGS is examined and also compared with the existing variables single sampling plan in terms of the sample size required for inspection. The results indicate that the proposed variables RGS plan could significantly reduce samples required for inspection compared to the traditional variables single sampling plan.     

An optimal design of CUSUM control charts for binomial counts

The Shewhart p-chart or np-chart is commonly used for monitoring the counts of non-conforming items which are usually well modelled by a binomial distribution with parameters n and p where n is the number of items inspected each time and p is the process fraction of non-conforming items produced. It is well known that the Shewhart chart is not sensitive to small shifts in p. The cumulative sum (CUSUM) chart is a far more powerful charting procedure for detecting small shifts in p and only marginally less powerful in detecting large shifts in p. The choice of chart parameters of a Shewhart chart is well documented in the quality control literature. On the other hand, very little has been done for the more powerful CUSUM chart, possibly due to the fact that the run length distribution of a CUSUM chart is much harder to compute. An optimal design strategy is given here which allows the chart parameters of an optimal CUSUM chart to be determined easily. Optimal choice of n and the relationship between the CUSUM chart and the sequential probability ratio test are also investigated.     

New cumulative sum control charts for monitoring process variability

In this article, we propose new cumulative sum (CUSUM) control charts using the ordered ranked set sampling (RSS) and ordered double RSS schemes, with the perfect and imperfect rankings, for monitoring the variability of a normally distributed process. The run length characteristics of the proposed CUSUM charts are computed using the Monte Carlo simulations. The proposed CUSUM charts are compared in terms of the average and standard deviation of run lengths with their existing competitor CUSUM charts based on simple random sampling. It turns out that the proposed CUSUM charts with the perfect and imperfect rankings are more sensitive than the existing CUSUM charts based on the sample range and standard deviation. A similar trend is present when these CUSUM charts are compared with the fast initial response features. An example is also used to demonstrate the implementation and working of the proposed CUSUM charts.     

An empirical bayes approach to a variables acceptance sampling plan problem

An empirical Bayes approach to a variables acceptance sampling plan problem is presented and an empirical Bayes rule is developed which is shown to be asymptotically optimal under general conditions. The problem considered is one in which the ratio of the costs of accepting defective items and rejecting non-defective items is specified. Sampling costs are not considered and the size of the sample taken from each lot is fixed and constant. The empirical Bayes estimation of the Bayes rule is shown to require the estimation of a conditional probability. An estimator for conditional probabilities of the form needed is derived and shown to have good asymptotic properties.     

Economic design of quick switching sampling system for resubmitted lots

This paper proposes a quick switching sampling system for the inspection of attributes quality characteristics for resubmitted lots. The optimal parameters for both fraction non conforming items and fraction non conformities of the proposed sampling system are determined using an optimization procedure so that producer’s risk and consumer’s risk are simultaneously satisfied. Tables are also constructed for the selection of parameters for specified AQL and LQL. The advantage of the proposed plan over the existing plan is discussed and illustrate. An economic design of the proposed sampling system is also discussed and shown that the proposed sampling system has minimum total cost and average total inspection compared to other existing sampling plans.     

Acceptance sampling by variables when the remainder of rejected lots is inspected

Acceptance sampling plans for inspection by variables, which minimize the mean inspection cost per lot of process average quality (assuming that the remainder of rejected lots is, inspected), were derived under the condition that the probability of accepting a submitted lot of tolerance quality, shall be 0.1. These plans were tabulated for chosen values of the given parameters, and compared with the Dodge-Romig LTPD attribute sampling plans. From the comparison it follows that under the same protection of consumer the LTPD plans for inspection by variables are in many situations more economical than the corresponding Dodge-Romig plans. This result is valid especially for the large lots and for the small values of the tolerance fraction defective.     

Dynamic sampling plans in on-line control charts

Three simple dynamic sampling plans for detecting the change point are investigated in the discrete-time case. The first is a two-rate sampling CUSUM procedure. The second is a two-rate sampling Shiryayev-Roberts procedure. The third is a periodic sequential testing procedure. Two problems are discussed. First, simple design methods are provided for practical use. Second, a comparison between the three plans is made in the continuous-time case, which shows that by properly choosing the design parameters, the three plans can be made equally efficient in certain senses.     

Selection of csp-t plans

Various continuous sampling plans have been proposed for monitoring the quality of continuous production processes. The multi-level continuous sampling plan of MIL-STD-1235C (1988) is designated as CSP-T Plan. CSP-T plan is a three-level tightened continuous sampling plan. It requires a switch to 100% inspection, at any level, whenever a nonconforming unit is found. This provides quick rectification in the event of a shift in quality. In this paper certain performance characteristics of CSP-T plan are derived using the approach of Stephens (1979) under the assumption that the production process is in statistical control. For the selection of CSP-T plans, two tables are given. These tables can be used to obtain parameters i (clearance interval) and f (sampling fraction) of the CSP-T plan for given acceptable quality level (AQL) with producer’ risk α=0.05 or the limiting quality level(LQL) with consumer’ risk β=0.10 and the outgoing quality limit(AOQL). Two examples are also given to illustrate the selection of plans from these tables