Economic design of a Vp X̄ chart

We develop an economic model for X̄ control charts having all design parameters varying in an adaptive way, that is, in real time considering current sample information. In the proposed model, each of the design parameters can assume two values as a function of the most recent process information. The cost function is derived and it provides a device for optimal selection of the design parameters. Through a numerical example one can foresee the savings that the developed model possibly provides. © 2001 Elsevier Science B.V. All rights reserved.

The no-central chi-square chart with two-stage samplings

Throughout this article, it is assumed that the no-central chi-square chart with two stage samplings (TSS Chisquare chart) is employed to monitor a process where the observations from the quality characteristic of interest X are independent and identically normally distributed with mean μ and variance σ2. The process is considered to start with the mean and the variance on target (μ = μ0; σ2 = σ0 2), but at some random time in the future an assignable cause shifts the mean from μ0 to μ1 = μ0 ± δσ0, δ >0 and/or increases the variance from σ0 2 to σ1 2 = γ2σ0 2, γ > 1. Before the assignable cause occurrence, the process is considered to be in a state of statistical control (defined by the in-control state). Similar to the Shewhart charts, samples of size n 0+ 1 are taken from the process at regular time intervals. The samplings are performed in two stages. At the first stage, the first item of the i-th sample is inspected. If its X value, say Xil, is close to the target value (|Xil-μ0|< w0σ 0, w0>0), then the sampling is interrupted. Otherwise, at the second stage, the remaining n0 items are inspected and the following statistic is computed. Wt = Σj=2n 0+1(Xij - μ0 + ξiσ 0)2 i = 1,2 Let d be a positive constant then ξ, =d if Xil > 0 ; otherwise ξi =-d. A signal is given at sample i if |Xil-μ0| > w0σ 0 and W1 > knia:tl, where kChi is the factor used in determining the upper control limit for the non-central chi-square chart. If devices such as go and no-go gauges can be considered, then measurements are not required except when the sampling goes to the second stage. Let P be the probability of deciding that the process is in control and P 1, i=1,2, be the probability of deciding that the process is in control at stage / of the sampling procedure. Thus P = P1 + P 2 - P1P2, P1 = Pr[μ0 - w0σ0 ≤ X ≤ μ0+ w 0σ0] P2=Pr[W ≤ kChi σ0 2], (3) During the in-control period, W / σ0 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ0 = n0d2, i.e. W / σ0 2 - xn0 22 (λ0) During the out-of-control period, W / σ1 2 is distributed as a non-central chi-square distribution with n0 degrees of freedom and a non-centrality parameter λ1 = n0(δ + ξ)2 / γ2 The effectiveness of a control chart in detecting a process change can be measured by the average run length (ARL), which is the speed with which a control chart detects process shifts. The ARL for the proposed chart is easily determined because in this case, the number of samples before a signal is a geometrically distributed random variable with parameter 1-P, that is, ARL = I /(1-P). It is shown that the performance of the proposed chart is better than the joint X̄ and R charts, Furthermore, if the TSS Chi-square chart is used for monitoring diameters, volumes, weights, etc., then appropriate devices, such as go-no-go gauges can be used to decide if the sampling should go to the second stage or not. When the process is stable, and the joint X̄ and R charts are in use, the monitoring becomes monotonous because rarely an X̄ or R value fall outside the control limits. The natural consequence is the user to pay less and less attention to the steps required to obtain the X̄ and R value. In some cases, this lack of attention can result in serious mistakes. The TSS Chi-square chart has the advantage that most of the samplings are interrupted, consequently, most of the time the user will be working with attributes. Our experience shows that the inspection of one item by attribute is much less monotonous than measuring four or five items at each sampling.

A synthetic control chart for monitoring the process mean and variance

Purpose - The aim of this paper is to present a synthetic chart based on the non-central chi-square statistic that is operationally simpler and more effective than the joint X̄ and R chart in detecting assignable cause(s). This chart will assist in identifying which (mean or variance) changed due to the occurrence of the assignable causes. Design/methodology/approach - The approach used is based on the non-central chi-square statistic and the steady-state average run length (ARL) of the developed chart is evaluated using a Markov chain model. Findings - The proposed chart always detects process disturbances faster than the joint X̄ and R charts. The developed chart can monitor the process instead of looking at two charts separately. Originality/value - The most important advantage of using the proposed chart is that practitioners can monitor the process by looking at only one chart instead of looking at two charts separately. © Emerald Group Publishing Limted.

Double sampling X̄ control chart for a first order autoregressive process

In this paper we propose the Double Sampling X̄ control chart for monitoring processes in which the observations follow a first order autoregressive model. We consider sampling intervals that are sufficiently long to meet the rational subgroup concept. The Double Sampling X̄ chart is substantially more efficient than the Shewhart chart and the Variable Sample Size chart. To study the properties of these charts we derived closed-form expressions for the average run length (ARL) taking into account the within-subgroup correlation. Numerical results show that this correlation has a significant impact on the chart properties.

Preliminary results concerning the VSS X- chart with unknow in-control parameters

The VSS X- chart is known to perform better than the traditional X- control chart in detecting small to moderate mean shifts in the process. Many researchers have used this chart in order to detect a process mean shift under the assumption of known parameters. However, in practice, the process parameters are rarely known and are usually estimated from an in-control Phase I data set. In this paper, we evaluate the (run length) performances of the VSS X- control chart when the process parameters are estimated and we compare them in the case where the process parameters are assumed known. We draw the conclusion that these performances are quite different when the shift and the number of samples used during the phase I are small. ©2010 IEEE.

A single chart with supplementary runs rules for monitoring the mean vector and the covariance matrix of multivariate processes

The MRMAX chart is a single chart based on the standardized sample means and sample ranges for monitoring the mean vector and the covariance matrix of multivariate processes. User's familiarity with the computation of these statistics is a point in favor of the MRMAX chart. As a single chart, the recently proposed MRMAX chart is very appropriate for supplementary runs rules. In this article, we compare the supplemented MRMAX chart and the synthetic MRMAX chart with the standard MRMAX chart. The supplementary and the synthetic runs rules enhance the performance of the MRMAX chart. © 2013 Elsevier Ltd.

A side-sensitive synthetic chart combined with an X chart

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Some Comments Regarding the Synthetic (X)over-bar Chart

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A new sampling strategy to reduce the effect of autocorrelation on a control chart

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Double-objective economic statistical design of the VP T-2 control chart: Wald's identity approach

Research has shown that applying the T-2 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 T-2 control 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.

A new sampling strategy for the Shewhart control chart monitoring a process with wandering mean

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Radiographic and safety details of vertebral body stenting: results from a multicenter chart review

BACKGROUND Up to one third of BKP treated cases shows no appreciable height restoration due to loss of both restored height and kyphotic realignment after balloon deflation. This shortcoming has called for an improved method that maintains the height and realignment reached by the fully inflated balloon until stabilization of the vertebral body by PMMA-based cementation. Restoration of the physiological vertebral body height for pain relief and for preventing further fractures of adjacent and distant vertebral bodies must be the main aim for such a method. A new vertebral body stenting system (VBS) stabilizes the vertebral body after balloon deflation until cementation. The radiographic and safety results of the first 100 cases where VBS was applied are presented. METHODS During the planning phase of an ongoing international multicenter RCT, radiographic, procedural and followup details were retrospectively transcribed from charts and xrays for developing and testing the case report forms. Radiographs were centrally assessed at the institution of the first/senior author. RESULTS 100 patients (62 with osteoporosis) with a total of 103 fractured vertebral bodies were treated with the VBS system. 49 were females with a mean age of 73.2 years; males were 66.7 years old. The mean preoperative anterior-middle-posterior heights were 20.3-17.6-28.0 mm, respectively. The mean local kyphotic angle was 13.1[degree sign]. The mean preoperative Beck Index (anterior edge height/posterior edge height) was 0.73, the mean alternative Beck Index (middle height/posterior edge height) was 0.63. The mean postoperative heights were restored to 24.5-24.6-30.4 mm, respectively. The mean local kyphotic angle was reduced to 8.9[degree sign]. The mean postoperative Beck Index was 0.81, the mean alternative one was 0.82. The overall extrusion rate was 29.1%, the symptomatic one was 1%. In the osteoporosis subgroup there were 23.8% extrusions. Within the three months followup interval there were 9% of adjacent and 4% of remote new fractures, all in the osteoporotic group. CONCLUSIONS VBS showed its strengths especially in realignment of crush and biconcave fractures. Given that fracture mobility is present, the realignment potential is sound and increases with the severity of preoperative vertebral body deformation.