Monitoring the covariance matrix with the VMAX chart

In this article, we propose a new statistic to control the covariance matrix of bivariate processes. This new statistic is based on the sample vat-lances of the two quality characteristics, shortly VMAX statistic. The points plotted on the chart correspond to the maximum of the values of these two variances. The reasons to consider the VMAX statistic instead of the generalized variance vertical bar S vertical bar are faster detection of process changes and better diagnostic feature, that is, with the VMAX statistic It is easier to identify the out-of-control variable.

The Variable Sample Size (X)over-bar Chart with Estimated Parameters

The VSS X chart, dedicated to the detection of small to moderate mean shifts in the process, has been investigated by several researchers under the assumption of known process parameters. 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 chart when the process parameters are estimated, we compare them in the case where the process parameters are assumed known and we propose specific optimal control chart parameters taking the number of Phase I samples into account.

X̄ chart with variable sample size and sampling intervals

Recent theoretical studies have shown that the X̄ chart with variable sampling intervals (VSI) and the X̄ chart with variable sample size (VSS) are quicker than the traditional X̄ chart in detecting shifts in the process. This article considers the X̄ chart with variable sample size and sampling intervals (VSSI). It is assumed that the amount of time the process remains in control has exponential distribution. The properties of the VSSI X̄ chart are obtained using Markov chains. The VSSI X̄ chart is even quicker than the VSI or VSS X̄ charts in detecting moderate shifts in the process.

AATS for the X̄ Chart with Variable Parameters

A Fortran computer program is given for the computation of the adjusted average time to signal, or AATS, for adaptive X̄ charts with one, two, or all three design parameters variable: the sample size, n, the sampling interval, h, and the factor k used in determining the width of the action limits. The program calculates the threshold limit to switch the adaptive design parameters and also provides the in-control average time to signal, or ATS.

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.

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.

Establishing erosion susceptibility: analytical hierarchical process and traditional approaches

This study evaluated alternatives for producing erosion susceptibility maps, considering different weight combinations for an environment's attributes, according to four different points of views. The attributes considered were landform, steepness, soils, rocks and land occupation. Considered alternatives were: (1) equal weights, more traditional approach, (2) different weights, according to a previous study in the area, (3) different weights, based on other works in the literature, and (4) different weights based on the analytical hierarchical process. The area studied included the Prosa Basin located in Campo Grande-Mato Grosso do Sul State, Brazil. The results showed that the assessed alternatives can be used together or in different stages of studies aiming at urban planning and decision-making on the interventions to be applied.

A side-sensitive synthetic chart combined with an X chart

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Integrating on-line process control and imperfect corrective maintenance: An economical design

Existing studies of on-line process control are concerned with economic aspects, and the parameters of the processes are optimized with respect to the average cost per item produced. However, an equally important dimension is the adoption of an efficient maintenance policy. In most cases, only the frequency of the corrective adjustment is evaluated because it is assumed that the equipment becomes "as good as new" after corrective maintenance. For this condition to be met, a sophisticated and detailed corrective adjustment system needs to be employed. The aim of this paper is to propose an integrated economic model incorporating the following two dimensions: on-line process control and a corrective maintenance program. Both performances are objects of an average cost per item minimization. Adjustments are based on the location of the measurement of a quality characteristic of interest in a three decision zone. Numerical examples are illustrated in the proposal. (c) 2012 Elsevier B.V. All rights reserved.

Shetland Islands, United Kingdom, Nautical Chart, 1788 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: A new hydrographical survey of the islands of Shetland : with many improvements and additions, and the sailing directions, by Captn. Thomas Preston. It was printed for Robert Sayer, chartseller No. 63 Fleet Street 1st. Jany., 1788. Scale [ca. 1:419,830].The image inside the map neatline is georeferenced to the surface of the earth and fit to the European Datum 1950, Universal Transverse Mercator (UTM) Zone 30N projected coordinate system. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. This map shows features such as rocks, channels, points, ports, coves, islands, anchorage points, and more. Includes also selected land features such as towns and villages, drainage, selected buildings, and more. Relief shown by hachures; depths shown by soundings. Includes also profile views, navigational notes, and insets: Directions for Valey Sound -- The Isles of Feroe. This layer is part of a selection of digitally scanned and georeferenced historic maps from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of originators, ground condition dates, scales, and map purposes.

Bombay Harbour, India, Nautical Chart, 1806 (Raster Image)

This layer is a georeferenced raster image of the historic paper map entitled: A plan of Bombay harbour : principally illustrative of the entrance, constructed from measured bases, and a series of angles, taken in 1803 & 4 by James Horsburgh. It was published by James Horsburgh in 1806. Scale [ca.1:37,820].The image inside the map neatline is georeferenced to the surface of the earth and fit to the Kalianpur 1975 India Zone III projected coordinate system. All map collar and inset information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, index maps, legends, or other information associated with the principal map. This map shows features such as drainage, cities and other human settlements, fortification, shoreline features (rocks, shoals, anchorage points, ports, inlets, lighthouses, etc.), and more. Relief shown by depth soundings. Includes also profile views and navigational notes.This layer is part of a selection of digitally scanned and georeferenced historic maps from the Harvard Map Collection. These maps typically portray both natural and manmade features. The selection represents a range of originators, ground condition dates, scales, and map purposes.

Coastal Massachusetts, Marblehead to Hampton Harbor, New Hampshire, Nautical Chart, 1776 (Image 1 of 2) (Raster Image)

This layer is a georeferenced raster image of the untitled, historic nautical chart: [A chart of the harbours of Hampton, Newbury, Ipswich, Jebeka, Squam, Cape Ann, Manchester, Beverly, Salem, Marble Head &c.] (sheet originally published in 1776). The map is [sheet 23] from the Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England, from surveys taken by Samuel Holland and published by J.F.W. Des Barres, 1781. Scale [ca. 1:50,000]. This layer is image 1 of 2 total images of the two sheet source map, representing the southern portion of the map. Covers coastal Massachusetts from Ipswich Harbor to Marblehead. The image is georeferenced to the surface of the earth and fit to the 'World Mercator' (WGS 84) projected coordinate system. All map collar information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, or other information associated with the principal map. This map shows coastal features such as harbors, inlets, rocks, channels, points, coves, shoals, islands, and more. Includes also selected land features such as cities and towns, buildings, and roads. Relief is shown by hachures; depths by soundings. This layer is part of a selection of digitally scanned and georeferenced historic maps from The Harvard Map Collection. The entire Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England has been scanned and georeferenced as part of this selection.

Coastal Massachusetts, Marblehead to Hampton Harbor, New Hampshire, Nautical Chart, 1776 (Image 2 of 2) (Raster Image)

This layer is a georeferenced raster image of the untitled, historic nautical chart: [A chart of the harbours of Hampton, Newbury, Ipswich, Jebeka, Squam, Cape Ann, Manchester, Beverly, Salem, Marble Head &c.] (sheet originally published in 1776). The map is [sheet 24] from the Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England, from surveys taken by Samuel Holland and published by J.F.W. Des Barres, 1781. Scale [ca. 1:50,000]. This layer is image 2 of 2 total images of the two sheet source map, representing the northern portion of the map. Covers coastal Massachusetts and New Hampshire from Ipswich Harbor, Massachusetts to Hampton Harbor, New Hampshire. The image is georeferenced to the surface of the earth and fit to the 'World Mercator' (WGS 84) projected coordinate system. All map collar information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, or other information associated with the principal map. This map shows coastal features such as harbors, inlets, rocks, channels, points, coves, shoals, islands, and more. Includes also selected land features such as cities and towns, buildings, and roads. Relief is shown by hachures; depths by soundings. This layer is part of a selection of digitally scanned and georeferenced historic maps from The Harvard Map Collection. The entire Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England has been scanned and georeferenced as part of this selection.

New England Coast : Newburyport, Massachusetts to Cape Elizabeth, Maine, Nautical Chart, 1776 (Image 1 of 2) (Raster Image)

This layer is a georeferenced raster image of the untitled, historic nautical chart: [A chart of the coast from Cape Elizabeth westwards to Newbury Harbour] (sheet originally published in 1776). The map is [sheet 25] from the Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England, from surveys taken by Samuel Holland and published by J.F.W. Des Barres, 1781. Scale [ca. 1:130,000]. This layer is image 1 of 2 total images of the two sheet source map, representing the western portion of the map. Covers the coast of New England from Newburyport, Massachusetts to Kittery, Maine. The image is georeferenced to the surface of the earth and fit to the 'World Mercator' (WGS 84) projected coordinate system. All map collar information is also available as part of the raster image, including any inset maps, profiles, statistical tables, directories, text, illustrations, or other information associated with the principal map. This map shows coastal features such as harbors, inlets, rocks, channels, points, coves, shoals, islands, and more. Includes also selected land features such as cities and towns, buildings, and roads. Relief is shown by hachures. This layer is part of a selection of digitally scanned and georeferenced historic maps from The Harvard Map Collection. The entire Atlantic Neptune atlas Vol. 3 : Charts of the coast and harbors of New England has been scanned and georeferenced as part of this selection.