Sequential Probability Ratio Test Chart Research Articles

Economic modelling and optimisation of SPRT-based quality control schemes for individual observations

In many industrial processes, it may not be practical to obtain sample sizes larger than one due to various reasons. As a result, practitioners must rely on control charts based on individual observations for statistical process monitoring and control. The sequential probability ratio test (SPRT) chart is an appropriate solution for such situations. Existing literature predominantly delves into the statistical aspects of designing or optimising an SPRT chart, neglecting the economic and associated design aspects. However, past experience suggests that statistical design may not always be optimal from an economic perspective. This paper aims to bridge this research gap by proposing a two-stage Markov chain approach to develop an economic model for the optimal design of the SPRT chart with individual observations. Following the principles of economic-statistical designs, optimal parameters are determined by solving a mixed-integer non-linear programming problem that minimises the long-run cost per item while ensuring compliance with practical constraints. Through an extensive study, we find that the economic performance of the SPRT chart surpasses that of competing individual control charts in many cases. This indicates that the economic-statistical design of the SPRT chart can yield significant economic benefits while maintaining desirable statistical properties. Moreover, enhancements have been introduced to refine the optimal design of the SPRT chart, retaining excellent overall performance across a wide range of shifts. Finally, we provide two industrial examples to illustrate the effectiveness of the proposed SPRT chart.

An integrated optimal-GICP design for the SPRT control chart with estimated process parameters based on the average number of observations to signal

ABSTRACT As a variable-sample-size control scheme, the sequential probability ratio test (SPRT) chart is favoured due to its sensitivity and high sampling efficiency. It is frequently documented that the SPRT chart inspects only a small number of observations at each sampling stage, making the said chart extremely appealing to quality engineers. In the current literature, most developments of the SPRT chart are established based on the average run length and average time to signal metrics. However, these metrics do not consider sampling efficiency. In this paper, we develop two optimal designs of the SPRT chart, based on (i) the average value of the average number of observations to signal (AANOS) and (ii) the expected value of the AANOS, with consideration of Phase-I process parameter estimation. We develop a model that integrates the concept of optimization and the guaranteed in-control performance (GICP) framework to neutralize the adverse effects of parameter estimation. Results show that the proposed optimal-GICP design preserves satisfactory out-of-control performances for moderate and large mean shifts, while keeping the false alarm rates at reasonably low levels. Finally, we illustrate an example of the SPRT chart with estimated process parameters for monitoring loop height measurements from a wire bonding dataset.

Optimal designs of the omnibus SPRT control chart for joint monitoring of process mean and dispersion

The vast majority of control schemes related to the sequential probability ratio test (SPRT) are designed for the purpose of monitoring only the process mean. Nonetheless, most manufacturing processes are vulnerable to external factors that cause the process mean and variability to change simultaneously. It is, therefore, crucial to consider a joint scheme for monitoring both the location and scale parameters of a production process. In this article, we develop a scheme that combines both mean and variance information in a single SPRT, known as the omnibus SPRT (OSPRT) chart. Expressions for the run-length properties of the OSPRT chart are derived by means of the Markov chain approach. We also propose optimal designs for the OSPRT chart based on two different metrics, i.e. by minimising the average time to signal and the average extra quadratic loss. Through a comprehensive analysis, this article reveals that the optimal OSPRT chart outperforms the classical -S, weighted-loss cumulative sum, absolute-value SPRT, and two maximum weighted-moving-average-type charts. The optimal OSPRT chart also has the advantage of collecting a small number of samples on average before producing a decision. Finally, the implementation of the OSPRT chart is presented with a wire bonding industrial dataset.

On designing an optimal SPRT control chart with estimated process parameters under guaranteed in-control performance

Being one of the most sophisticated control charts, the sequential probability ratio test (SPRT) chart possesses fast detection ability across a broad range of process shifts. The SPRT chart has the advantage of sampling only a modest number of observations. Thus far, the SPRT chart has been developed under the assumption that the process parameters are known. As process parameters are often unknown in practice, this article reveals that parameter estimation from a limited amount of Phase-I data, leads to excessive false alarms and unstable chart’s performances. To counter these problems, this article advocates the use of adjusted control limits to ensure that a sufficiently high proportion of the in-control conditional average time to signal value, is greater than a pre-specified level. This increasingly prevalent design of control charts is known as Guaranteed In-Control Performance (GICP). In this article, we propose an optimization design for the SPRT chart with estimated process parameters, by minimizing the expected value of the average extra quadratic loss under the GICP framework. Theoretical derivations by means of the Markov chain approach are developed in this article to evaluate the run-length properties of the SPRT chart with estimated process parameters. Results show that the overall performance of the proposed optimal SPRT chart is almost twice as good as the optimal CUSUM chart over a given range of process mean shifts. Finally, an implementation of the proposed optimal SPRT chart is demonstrated with real industrial data obtained from an epitaxial process.

A two-sided SPRT control chart for process dispersion

This paper proposes a two-sided sequential probability ratio test chart for monitoring dispersion of a normally distributed process. Accurate approximate expressions that relate the statistical performance measures of this chart to that of its corresponding one-sided charts are obtained through simulations. An approach to design the chart having the specified in-control performance and the optimal out-of-control performances for detecting the particular upper and lower shifts that are desired to detect as efficiently as possible is provided. It is found through numerical comparisons that the overall statistical performance of the proposed chart is superior to that of the competitor charts in the existing literature. The application of the chart is illustrated through real-data on the thickness of dielectric polyimide coating layer that is applied on a wafer used in an integrated circuit.

The SPRT sign chart for process location

This paper develops a nonparametric sequential probability ratio test (SPRT) control chart for monitoring the process location. The chart is based on a sequential sign test. Young’s algorithm to determine the properties of the SPRT for a parameter of discrete Koopman-Darmois density is used to obtain the exact statistical performance measures of the proposed chart. The proposed chart substantially outperforms the Shewhart sign, synthetic Shewhart sign, CUSUM sign, and EWMA sign charts in detecting a location shift of any size. It is a preferable alternative to the parametric SPRT chart for mean when the underlying process distribution is not precisely known. The implementation of the chart is illustrated through an example.

Repeated SPRT charts for monitoring INAR(1) processes

AbstractPoisson integer valued autoregressive (INAR) models have been proposed for modeling correlated count data. Poisson lognormal (PLN) INAR models extend their use to overdispersed contexts. In this paper, we will propose the use of a repeated Sequential Probability Ratio Test (SPRT) procedure to detect change in first‐order INAR and PLN INAR models. We consider change in the mean, the autocorrelation parameter, and the overdispersion parameter. Simulation results show the repeated SPRT procedure performs favorably relative to previously proposed CUSUM procedures that are based on either the observations themselves or residuals of the observations from predicted values. A dataset on invasive insect species is used to illustrate the repeated SPRT procedure.

An economically designed Sequential Probability Ratio Test control chart for short-run production

This paper proposes an economic model for the design of an SPRT (Sequential Probability Ratio Test) chart for monitoring the process mean in short-run production. The model expresses the short-run cost per unit of operating the SPRT chart as a function of the cost parameters associated with the operation. A simple algorithm capable of optimizing the charting parameters is also proposed. The model can be used to quantify cost reductions achievable by substituting a traditional control policy by SPRT control. Numerical examples illustrate the effectiveness of the proposed procedure. It is shown that the resulting cost reduction can range from modest to substantial as the out-of-control probability of the process increases.

An efficient multivariate control charting mechanism based on SPRT

This article devises a feasible multivariate Sequential Probability Ratio Test chart (MSPRT chart) to monitor the mean of the multivariate normal distribution. To reduce the time required by the MSPRT chart to detect shifts of a wide range, the charting parameters are optimised to minimise the Average Extra Quadratic Loss (AEQL). The comparative study reveals that the MSPRT chart can improve the performance of the typical Hotelling’s T2 chart and the multivariate synthetic (Msynthetic) chart. It also shows that the MSPRT chart outperforms the multivariate Exponentially Weighted Moving Average (MEWMA) chart for moderate and large shifts but the latter prevails in detecting small shifts. For ease of use, we also provide two design tables for the quality engineers to adopt the MSPRT and Hotelling’s T2 charts more conveniently. Finally, we used an example concerning the white wine production process to demonstrate the application of the proposed scheme.

A rational sequential probability ratio test control chart for monitoring process shifts in mean and variance

The sequential probability ratio test (SPRT) chart is a very effective tool for monitoring manufacturing processes. This paper proposes a rational SPRT chart to monitor both process mean and variance. This SPRT chart determines the sampling interval d based on the rational subgroup concept according to the process conditions and administrative considerations. Since the rational subgrouping is widely adopted in the design and implementation of control charts, the studies of the rational SPRT have a practical significance. The rational SPRT chart is designed optimally in order to minimize the index average extra quadratic loss for the best overall performance. A systematic performance study has also been conducted. From an overall viewpoint, the rational SPRT chart is more effective than the cumulative sum chart by more than 63%. Furthermore, this article provides a design table, which contains the optimal values of the parameters of the rational SPRT charts for different specifications. This will greatly facilitate the potential users to select an appropriate SPRT chart for their applications. The users can also justify the application of the rational SPRT chart according to the achievable enhancement in detection effectiveness.

Optimal average sample number of the SPRT chart for monitoring fraction nonconforming

The Sequential Probability Ratio Test (SPRT) control chart is a powerful tool for monitoring manufacturing processes. It is highly suitable for the applications where testing is destructive or very expensive, such as the automobile airbags test. This article studies the effect of the Average Sample Number (ASN) (i.e., the average sample size) on the chart’s performance. A design algorithm is proposed to develop the optimal SPRT chart for monitoring the fraction nonconforming p of Bernoulli processes. By optimizing the ASN and other charting parameters, the average detection speed of the SPRT chart is almost doubled. It is also found that the optimal SPRT chart significantly outperforms the optimal np and binomial CUSUM charts, in terms of Average Number of Defectives (AND), under different combinations of the design specifications. It is observed that the SPRT chart using a relatively smaller ASN and a shorter sampling interval (h) has a higher overall detection effectiveness.

An optimal design algorithm of the SPRT chart for minimizing weighted ATS

Abstract The Sequential Probability Ratio Test (SPRT) chart is a very effective control chart. It is much faster to detect out-of-control cases compared with the X ¯ and CUSUM charts, even their VSSI (Variable Sample Sizes and Sampling Intervals) versions. The new contribution of this paper is the development of an algorithm for the optimal design of the SPRT chart based on the Average Extra Quadratic Loss (AEQL). More importantly, the results of the systematic comparative studies show that this design algorithm is indeed effective. Using the design algorithm proposed here yields charts that significantly outperform other designs in terms of the proposed AEQL performance criteria for detecting process mean shifts. Moreover, this paper conducts a designed experiment to study the effect of the Average Sample Number (ASN) (i.e., the average sample size) on the chart's performance. It is found that, in order to achieve higher overall detection effectiveness, the ASN of the SPRT chart should be set substantially smaller compared with that recommended by the existing guidelines.

A comparison study of effectiveness and robustness of control charts for monitoring process mean

This article compares the effectiveness and robustness of nine typical control charts for monitoring the mean of a variable, including the most effective optimal and adaptive Sequential Probability Ratio Test (SPRT) charts. The nine charts are categorized into three types (the X¯ type, CUSUM type and SPRT type) and three versions (the basic version, optimal version and fully adaptive (FA) version). While the charting parameters of the basic charts are determined by common wisdoms, the parameters of the optimal and fully adaptive charts are designed optimally in order to minimize an index, Average Extra Quadratic Loss ( AEQL ), for the best overall performance. A Performance Comparison Index, PCI , is also proposed as the measure of the relative overall performance of the charts. This comparison study does not only compare the detection effectiveness of the charts, but also investigate their robustness in performance. Moreover, the probability distribution of the mean shift δ is studied explicitly as an influential factor in a factorial experiment. Apart from many other findings, the results of this study reveal that the SPRT chart is more effective than the CUSUM chart and X ¯ chart by 58% and 126%, respectively, from an overall viewpoint. Moreover, it is found that the optimization design of charting parameters can increase the detection effectiveness by 29% on average, and the adaptive features can further enhance the detection power by 35%. Finally, a set of design tables are provided to facilitate the users to select a chart for their applications. ► Compare 9 control charts including X-bar, CUSUM and SPRT types and basic, optimal and adaptive versions. ► Optimization design minimizes Average Extra Quadratic Loss for the best overall performance. ► Compare not only the detection effectiveness, but also the robustness in performance. ► We found that SPRT chart is more effective than CUSUM and X-bar charts by 58% and 126%, respectively. ► Optimization design and adaptive feature increase detection effectiveness by 29% and 35%, respectively.

A comparison study on effectiveness and robustness of control charts for monitoring process mean and variance

AbstractThis article compares the effectiveness and robustness of nine typical control charts for monitoring both process mean and variance, including the most effective optimal and adaptive sequential probability ratio test (SPRT) charts. The nine charts are categorized into three types (the type, CUSUM type and SPRT type) and three versions (the basic version, optimal version and adaptive version). While the charting parameters of the basic charts are determined by common wisdoms, the parameters of the optimal and adaptive charts are designed optimally in order to minimize an index average extra quadratic loss for the best overall performance. Moreover, the probability distributions of the mean shift δµ and standard deviation shift δσ are studied explicitly as the influential factors in a factorial experiment. The main findings obtained in this study include: (1) From an overall viewpoint, the SPRT‐type chart is more effective than the CUSUM‐type chart and type chart by 15 and 73%, respectively; (2) in general, the adaptive chart outperforms the optimal chart and basic chart by 16 and 97%, respectively; (3) the optimal CUSUM chart is the most effective fixed sample size and sampling interval chart and the optimal SPRT chart is the best choice among the adaptive charts; and (4) the optimal sample sizes of both the charts and the CUSUM charts are always equal to one. Furthermore, this article provides several design tables which contain the optimal parameter values and performance indices of 54 charts under different specifications. Copyright © 2011 John Wiley & Sons, Ltd.

A new SPRT chart for monitoring process mean and variance

This article proposes a new control chart (called the ABS SPRT chart) for Statistical Process Control (SPC) based on Sequential Probability Ratio Test (SPRT). This chart is able to monitor the mean and variance of a variable x simultaneously by inspecting the absolute sample shift | x− μ 0| (where μ 0 is the in-control mean or target value of x). The ABS SPRT chart is designed by an optimization algorithm, aiming at minimizing the Average Extra Quadratic Loss ( AEQL) over the process shift domain. The results of intensive performance studies show that the ABS SPRT chart not only uniformly outperforms the CUSUM chart with a Variable Sample Size (VSS) feature, but is also more effective than a 2-SPRT scheme which incorporates a lower-sided SPRT chart and an upper-sided one. From a holistic viewpoint, the ABS SPRT chart detects process shifts in mean and variance faster than the VSS CUSUM chart and 2-SPRT scheme by more than 30% and 13%, respectively. Noteworthily, the design of an ABS SPRT chart is relatively easier than that of a VSS CUSUM chart, and much simpler than the design of a 2-SPRT scheme.

An SPRT control chart with variable sampling intervals

The sequential probability ratio test (SPRT) chart is a very effective control chart. It is much faster to detect mean shifts compared with the \( \bar{X} \) and CUSUM charts, even their VSSI (variable sample sizes and sampling intervals) versions. This article proposes an SPRT scheme, called the VSI SPRT chart, which uses the variable sampling intervals (VSI) scheme to further increase the effectiveness of the SPRT chart for detecting process mean shifts. This VSI SPRT chart employs a long sampling interval d1 when the process is likely to be in control and adopts a short sampling interval d2 when the process seems close to an out-of-control condition. The results of the case studies show that the adaptive feature has the potential to further increase the overall detection effectiveness of the SPRT chart compared with the fixed sampling interval SPRT chart and, therefore, to reduce the manufacturing costs. This new chart is particularly effective when mean shift is moderate.

The use of statistical process control (risk‐adjusted CUSUM, risk‐adjusted RSPRT and CRAM with prediction limits) for monitoring the outcomes of out‐of‐hospital cardiac arrest patients rescued by the EMS system

Based on previous experience from surgical surveillance, risk-adjusted cumulative sum (CUSUM)-type charts were applied to monitor out-of-hospital cardiac arrest (OHCA) patient mortality. Data from 2356 OHCA patients were collected by the Taipei County Fire Bureau from June 2006 to November 2007. Logistic regression analysis was applied to create a risk-adjusted model. Next, a risk-adjusted CUSUM chart, a risk-adjusted resetting sequential probability ratio test chart and a cumulative risk-adjusted mortality with prediction limits chart were used to detect excess deaths of the OHCA patients rescued by the emergency medical service (EMS) system. The overall mortality rate, defined as having no return of spontaneous circulation, was 79.3%. These three charts signalled an increase in the death rate at similar sites, and also suggested a small process shift. A visual approach to EMS systems monitoring that combines the risk-adjusted cumulative sum, Risk-adjusted resetting sequential probability ratio test and cumulative risk-adjusted mortality with prediction limits charts was established. It was found that this approach can be effectively used by the EMS community to monitor OHCA outcomes in real time.

An improved SPRT control chart for monitoring process mean

The sequential probability ratio test (SPRT) chart is a desirable tool for monitoring manufacturing processes due to its high effectiveness. It is especially suitable for applications where testing is very expensive or destructive, such as the automobile airbags test and unaxial tensile test. This article proposes a design algorithm for the SPRT chart in which the reference value (γ) of the SPRT chart is optimized. The design algorithm increases the overall effectiveness of the SPRT chart by more than 10% on average. Moreover, the improvement in detection effectiveness is achieved without any additional difficulty in implementation. A design table is also provided to facilitate quality engineers to design the SPRT chart.

Continuous monitoring tools for pediatric surgical outcomes: an example using biliary atresia

Purpose Continuous monitoring tools can be used to monitor surgical outcomes over time. We illustrate the use of CUmulative SUM (CUSUM) charts in monitoring outcomes of Kasai portoenterostomy for treatment of biliary atresia at a supraregional unit. Methods Data on 57 consecutive infants who underwent a Kasai portoenterostomy performed by a single surgeon between June 1994 and June 2006 were collected. A procedure was defined as successful if clearance of jaundice (plasma bilirubin level <20 μmol/l) was achieved within 6 months of surgery. We applied cumulative observed-minus-expected, sequential probability ratio test (SPRT), and zero-resetting SPRT CUSUM charts and compared the results with those of standard aggregate data analyses. An expected failure rate of 43.0%, based on the national average failure rate, was used. Results The failure rate observed after 57 operations was 29.8%. The zero-resetting SPRT chart indicated a lower-than-expected failure rate earlier than did the aggregate data analyses and any of the other continuous monitoring techniques. Conclusions The CUSUM chart method provides ongoing feedback that can be used for continuous monitoring of the outcome of a procedure to ensure that standards of care are maintained. Its use as a routine monitoring tool in pediatric surgery deserves wider recognition.

The SPRT Control Chart for Standard Deviation Based on Individual Observations

When the production rate is low or only one observation is available at each time that a sample is to be taken, the traditional control chart for monitoring the process variation is the moving range chart. In this paper, we propose a chart based on the sequential probability ratio test (SPRT) for detecting increases in process variability, which can have some impact on process capability and product quality. Critical values for the control chart are presented. To demonstrate an application of the SPRT chart, an example using real manufacturing data is presented.