Control charts for monitoring the mean vector and the covariance matrix of bivariate processes

In this article, we propose new control charts for monitoring the mean vector and the covariance matrix of bivariate processes. The traditional tools used for this purpose are the T (2) and the |S| charts. However, these charts have two drawbacks: (1) the T (2) and the |S| statistics are not easy to compute, and (2) after a signal, they do not distinguish the variable affected by the assignable cause. As an alternative to (1), we propose the MVMAX chart, which only requires the computation of sample means and sample variances. As an alternative to (2), we propose the joint use of two charts based on the non-central chi-square statistic (NCS statistic), named as the NCS charts. Once the NCS charts signal, the user can immediately identify the out-of-control variable. In general, the synthetic MVMAX chart is faster than the NCS charts and the joint T (2) and |S| charts in signaling processes disturbances.

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.

Technical and economic evaluation of different types of control of Brevipalpus phoenicis (Acari: Tenuipalpidae) in citrus leprosies management

The objective of this work was to evaluate the effect of acaricide applications and pruning of symptomatic branches in citrus leprosis management in Brazil. It was conducted in an orange plantation of the 'Pera' variety, grafted onto the 'Cleopatra' tangerine, in two seasons (2006-2007 and 2007-2008). The experimental design was randomized blocks in a factorial scheme consisting of the following factors: (A) acaricide, in three levels: spirodiclofen and cyhexatin applied in rotation, lime sulphur; no acaricides; (B) pruning to remove branches that showed symptoms of leprosis, with two levels: with pruning, without pruning. We carried out periodic assessments of Brevipalpus phoenicis (Geijskes) populations (vector of the leprosis virus), leprosis incidence and severity, fruit yield, and the economic feasibility of the applied strategies. Based on the results, we concluded that spirodiclofen and cyhexatin were more effective than lime sulphur in B. phoenicis control. Control with lime sulphur required more applications than spirodiclofen and cyhexatin in rotation, making it more expensive. Pruning of symptomatic branches used in isolation was not sufficiently effective to control leprosis and significantly increased control costs. Profits were higher when the control involved sprayings of spirodiclofen and cyhexatin in alternation, with or without pruning.

Joint economic design of X̄ and R control charts for processes subject to two independent assignable causes

A model for the joint economic design of X̄ and R control charts is developed. This model assumes that the process is subject to two assignable causes. One assignable cause shifts the process mean; the other shifts the process variance. The occurrence of the assignable cause of one kind does not block the occurrence of the assignable cause of another kind. Consequently, a second process parameter can go out-of-control after the first process parameter has gone out-of-control. A numerical study of the cost surface to the model considered has revealed that it is convex, at least in the interest region.

Effect of intensification on grow out of the Amazon River prawn, Macrobrachium amazonicum

The effects of intensification on growth, survival, productivity, population structure, and distribution of harvested biomass in individual size classes of Macrobrachium amazonicum in semi-intensive culture were evaluated. Postlarvae (0.01 g) were stocked in 12 ponds at densities of 10, 20, 40, and 80/m(2) (three replicates per treatment) and raised for 5.5 mo. Average individual weight significantly decreased and productivity significantly increased as stocking density increased (P < 0.001), while survival was not affected (P > 0.05). Prawn mean weight at harvest ranged from 3.6 (80/m(2)) to 7.0 g (10/m(2)). Average survival ranged from 65.5% (40/m(2)) to 72.8% (20/m(2)), while productivity ranged from 508 (10/m(2)) to 2051 kg/ha (80/m(2)). Harvested biomass showed a clear bimodal distribution in individual size classes indicating the occurrence of heterogeneous growth, which may affect management and market strategies. Harvested biomass of prawns weighing more than 7 g (the best market size) increases for stocking densities up to 40/m(2) and stabilizes between 40 and 80/m(2). Growth reduction was associated with a decreasing frequency and average weight of green claw 1 and green claw 2 male morphotypes and adult females as density increased. Thus, the distribution of male morphotypes and sexually mature females are affected by density-dependent factors. Results suggest that prawn density plays an important role on M. amazonicum grow-out phase, as has been demonstrated for other species of the genus Macrobrachium. M. amazonicum tolerates grow-out intensification and may be raised in both semi-intensive and intensive systems stocked at very high densities yielding high productivity.

A new chart based on sample variances for monitoring the covariance matrix of multivariate processes

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

A new chart for monitoring the covariance matrix of bivariate processes

In this article we consider a control chart based on the sample variances of two quality characteristics. The points plotted on the chart correspond to the maximum value of these two statistics. The main reason to consider the proposed chart instead of the generalized variance |S| chart is its better diagnostic feature, that is, with the new chart it is easier to relate an out-of-control signal to the variables whose parameters have moved away from their in-control values. We study the control chart efficiency considering different shifts in the covariance matrix. In this way, we obtain the average run length (ARL) that measures the effectiveness of a control chart in detecting process shifts. The proposed chart always detects process disturbances faster than the generalized variance |S| chart. The same is observed when the size of the samples is variable, except in a few cases in which the size of the samples switches between small size and very large size.

The Synthetic Control Chart Based on Two Sample Variances for Monitoring the Covariance Matrix

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Bivariate control charts with double sampling

In this article, we consider the T(2) chart with double sampling to control bivariate processes (BDS chart). During the first stage of the sampling, n(1) items of the sample are inspected and two quality characteristics (x; y) are measured. If the Hotelling statistic T(1)(2) for the mean vector of (x; y) is less than w, the sampling is interrupted. If the Hotelling statistic T(1)(2) is greater than CL(1), where CL(1) > w, the control chart signals an out-of-control condition. If w < T(1)(2) <= CL(1), the sampling goes on to the second stage, where the remaining n(2) items of the sample are inspected and T(2)(2) for the mean vector of the whole sample is computed. During the second stage of the sampling, the control chart signals an out-of-control condition when the statistic T(2)(2) is larger than CL(2). A comparative study shows that the BDS chart detects process disturbances faster than the standard bivariate T(2) chart and the adaptive bivariate T(2) charts with variable sample size and/or variable sampling interval.

Optimised placement of control and protective devices in electric distribution systems through reactive tabu search algorithm

Optimised placement of control and protective devices in distribution networks allows for a better operation and improvement of the reliability indices of the system. Control devices (used to reconfigure the feeders) are placed in distribution networks to obtain an optimal operation strategy to facilitate power supply restoration in the case of a contingency. Protective devices (used to isolate faults) are placed in distribution systems to improve the reliability and continuity of the power supply, significantly reducing the impacts that a fault can have in terms of customer outages, and the time needed for fault location and system restoration. This paper presents a novel technique to optimally place both control and protective devices in the same optimisation process on radial distribution feeders. The problem is modelled through mixed integer non-linear programming (MINLP) with real and binary variables. The reactive tabu search algorithm (RTS) is proposed to solve this problem. Results and optimised strategies for placing control and protective devices considering a practical feeder are presented. (c) 2007 Elsevier B.V. All rights reserved.

Constrained optimization model for the design of an adaptive (X)over-bar chart

An economic-statistical model is developed for variable parameters (VP) (X) over bar charts in which all design parameters vary adaptively, that is, each of the design parameters (sample size, sampling interval and control-limit width) vary as a function of the most recent process information. The cost function due to controlling the process quality through a VP (X) over bar chart is derived. During the optimization of the cost function, constraints are imposed on the expected times to signal when the process is in and out of control. In this way, required statistical properties can be assured. Through a numerical example, the proposed economic-statistical design approach for VP (X) over bar charts is compared to the economic design for VP (X) over bar charts and to the economic-statistical and economic designs for fixed parameters (FP) (X) over bar charts in terms of the operating cost and the expected times to signal. From this example, it is possible to assess the benefits provided by the proposed model. Varying some input parameters, their effect on the optimal cost and on the optimal values of the design parameters was analysed.

X charts with supplementary samples to control the mean and variance

A standard X chart for controlling a process takes regular individual observations, for instance every half hour. This article proposes a modification of the X chart that allows one to take supplementary samples. The supplementary sample is taken (and the (X) over bar and R values computed) when the current value of X falls outside the control limits. With the supplementary sample, the signal of out-of-control is given by an (X) over bar value outside the (X) over bar chart's control limits or an R value outside the R chart's control limit. The proposed chart is designed to hold the supplementary sample frequency, during the in-control period, as low as 5% or less. In this context, the practitioner might prefer to verify an out-of-control condition by simply comparing the (X) over bar and R values with the control limits. In other words, without plotting the (X) over bar and R points. The X chart with supplementary samples has two major advantages when compared with the standard (X) over bar and A charts: (a) the user will be plotting X values instead of (X) over bar and R values; (b) the shifts in the process mean and/or changes in the process variance are detected faster.

Economic design of two-stage (X)over-bar charts: the Markov chain approach

When the (X) over bar chart is in use, samples are regularly taken from the process, and their means are plotted on the chart. In some cases, it is too expensive to obtain the X values, but not the values of a correlated variable Y. This paper presents a model for the economic design of a two-stage control chart, that is. a control chart based on both performance (X) and surrogate (Y) variables. The process is monitored by the surrogate variable until it signals an out-of-control behavior, and then a switch is made to the (X) over bar chart. The (X) over bar chart is built with central, warning. and action regions. If an X sample mean falls in the central region, the process surveillance returns to the (Y) over bar chart. Otherwise. The process remains under the (X) over bar chart's surveillance until an (X) over bar sample mean falls outside the control limits. The search for an assignable cause is undertaken when the performance variable signals an out-of-control behavior. In this way, the two variables, are used in an alternating fashion. The assumption of an exponential distribution to describe the length of time the process remains in control allows the application of the Markov chain approach for developing the cost function. A study is performed to examine the economic advantages of using performance and surrogate variables. (C) 2003 Elsevier B.V. All rights reserved.

Economic design of (X)over-bar charts with variable parameters: the Markov chain approach

This paper presents an economic design of (X) over bar control charts with variable sample sizes, variable sampling intervals, and variable control limits. The sample size n, the sampling interval h, and the control limit coefficient k vary between minimum and maximum values, tightening or relaxing the control. The control is relaxed when an (X) over bar value falls close to the target and is tightened when an (X) over bar value falls far from the target. A cost model is constructed that involves the cost of false alarms, the cost of finding and eliminating the assignable cause, the cost associated with production in an out-of-control state, and the cost of sampling and testing. The assumption of an exponential distribution to describe the length of time the process remains in control allows the application of the Markov chain approach for developing the cost function. A comprehensive study is performed to examine the economic advantages of varying the (X) over bar chart parameters.

Optimized Allocation of Control and Protective Devices in Electric Distribution Systems

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)