Cédric Heuchenne

Guaranteed Conditional Performance of the S2 Control Chart with Estimated Parameters

We evaluate the in-control performance of the S2 control chart with estimated parameters conditional on the Phase I sample. Simulation results indicate no realistic amount of Phase I data is enough to have confidence that the in-control average run length (ARL) obtained will be near the desired value. To overcome this problem, we adjust the S2 chart’s control limits such that the in-control ARL is guaranteed to be above a specified value with a certain specified probability. The required adjustment does not have too much of an adverse effect on the out-of-control performance of the chart.

Double Objective Economic Statistical Design of the VP T^{2} Control Chart: Wald’s identity approach

Research has shown that applying the T2 control chart by using a variable parameters (VP) scheme yields rapid detection of out-of-control states. In this paper, the problem of economic statistical design of the VP T2control chart is considered as a double-objective minimization problem with the statistical objective being the adjusted average time to signal and the economic objective being expected cost per hour. We then find the Pareto-optimal designs in which the two objectives are met simultaneously by using a multi-objective genetic algorithm. Through an illustrative example, we show that relatively large benefits can be achieved by applying the VP scheme when compared with usual schemes, and in addition, the multi-objective approach provides the user with designs that are flexible and adaptive.

Nonlinear Regression With Censored Data

Suppose that the random vector (X, Y) satisfies the regression model Y = m(X) + σ(X)ϵ, where m (·) = E(Y||·) belongs to some parametric class {mθ(·):θ∈} of regression functions, σ2(·) = var(Y||·) is unknown, and ϵ is independent of X. The response Y is subject to random right censoring, and the covariate X is completely observed. A new estimation procedure for the true, unknown parameter vector θ0 is proposed that extends the classical least squares procedure for nonlinear regression to the case where the response is subject to censoring. The consistency and asymptotic normality of the proposed estimator are established. The estimator is compared through simulations with an estimator proposed by Stute in 1999, and both methods are also applied to a fatigue life dataset of strain-controlled materials.

Optimal T2 Control Chart with a Double Sampling Scheme – An Alternative to the MEWMA Chart

Recent studies have shown that a double sampling (DS) scheme yields improvements in detection times of process shifts over variable ratio sampling (VRS) methods that have been extensively studied in the literature. Additionally, a DS scheme is more practical than some of the VRS methods since the sampling interval is fixed. In this paper, we investigate the effect of double sampling on cost, a criterion as important as detection rate. We study economic statistical design of the DS T2 chart (ESD DS T2) so that designs are found that are economically optimal but yet meet desired statistical properties such as having low probabilities of false searches and high probabilities of rapid detection of process shifts. Through an illustrative example, we show that relatively large benefits can be achieved in a comparison with the classical T2 chart and the statistical DS T2 charts with our ESD DS T2 approach. Furthermore, the economic performance of the ESD DS T2 charts is favorably compared to the MEWMA and other VRS T2 control charts in the literature. Copyright

A modified economic-statistical design of the T2 control chart with variable sample sizes and control limits

Recent studies have shown that using variable sampling size and control limits (VSSC) schemes result in charts with more statistical power than variable sampling size (VSS) when detecting small to moderate shifts in the process mean vector. This paper presents an economic-statistical design (ESD) of the VSSC T2 control chart using the general model of Lorenzen and Vance [22]. The genetic algorithm approach is then employed to search for the optimal values of the six test parameters of the chart. We then compare the expected cost per unit of time of the optimally designed VSSC chart with optimally designed VSS and FRS (fixed ratio sampling) T2 charts as well as MEWMA charts.

Monitoring delivery chains using multivariate control charts

Delivery chains are concerned with the delivery of goods and services to customers within a specific time interval; this time constraint is added to the usual consumer demand for product or service quality. In this context, we address the idea of using process control tools to monitor this key variable of delivery time. In applications, there are usually several production and delivery sites and a variety of different ways to transport, treat and provide goods and services; that makes the problem multivariate in nature. We therefore propose to control the process using multivariate T2 control charts economically designed with the addition of statistical constraints, a design method called economic-statistical design. We illustrate the application in general through an illustrative example.

Shewhart Control Charts for Monitoring Reliability with Weibull Lifetimes

In this paper, we present Shewhart-type Z and S2 control charts for monitoring individual or joint shifts in the scale and shape parameters of a Weibull distributed process. The advantage of this method is its ease of use and flexibility for the case where the process distribution is Weibull, although the method can be applied to any distribution. We illustrate the performance of our method through simulation and the application through the use of an actual data set. Our results indicate that Z and S2 control charts perform well in detecting shifts in the scale and shape parameters. We also provide a guide that would enable a user to interpret out-of-control signals.

A statistically adaptive sampling policy to the Hotelling's T2 control chart: Markov chain approach

ABSTRACT We present an alternative sampling scheme for the Hotellings T2 control chart with variable parameters (VP T2) which allows the sampling interval h, the sample size n, and control limit k to vary between minimum and maximum values while keeping the warning line fixed over time. Our method uses only one measurement scale to overcome the difficulties of using two scales in practice. Later, we demonstrate the merits of the method in terms of its performance in detecting small-to-moderate shifts and its ease of application.

The Robust Economic Statistical Design of the Hotelling’s T^2 Chart

ABSTRACT Economic statistical designs aim at minimizing the cost of process monitoring when a specific scenario or a set of estimated process and cost parameters is given. But, in practice the process may be affected by more than one scenario which may lead to severe cost penalties if the wrong design is used. Here, we investigate the robust economic statistical design (RESD) of the T2  chart in an attempt to reduce these cost penalties when there are multiple scenarios. Our method is to employ the genetic algorithm (GA) optimization method to minimize the total expected monitoring cost across all distinct scenarios. We illustrate the effectiveness of the method using two numerical examples. Simulation studies indicate that robust economic statistical designs should be encouraged in practice.

The np Chart with Guaranteed In-control Average Run Lengths

We evaluate the in-control performance of the np-control chart with estimated parameter conditional on the Phase I sample. We then apply the bootstrap method to adjust the control chart limits to guarantee the desired in-control average run length (ARL0) value in the monitoring stage. The adjusted limits ensure that the ARL0 would take a value greater than the desired value (say, B) with a certain specified probability, that is, Pr(ARL0 > B) = 1 − ρ. The results indicate that adjusting control limits is not always necessary. We present a method to design control charts such that in control and out of control run lengths are guaranteed with pre specified probabilities. This method is an improvement of the classical statistical design approach employing constraints on in control and out of control ARL because, with this approach, there is a substantial probability that the actual run length in control may be too small. In addition, using the ARL approach may result in an actual out of control run length that is too large. Some numerical examples illustrate the efficacy of this design method. Copyright