Optimal inspection policy for three-state systems monitored by control charts

This paper presents the formulations of the expected long-run cost per time unit for a system monitored by a static control chart and by an adaptive control chart respectively. The static chart has a fixed sampling interval and a fixed sample size. The adaptive chart has a fixed sample size but variable sampling intervals. The system is supposed to have three states, normal working state, failure delay time state, and failed state. Two levels of repair are used to maintain the system. A minor repair is used to restore the system if a detectable defect is confirmed by an inspection. A major repair will be performed if the system fails. The expected cost per time unit for maintaining such a system is obtained. The objective of such analysis is to find an optimal sampling policy for the inspection process. An artificially generated data example and a real data example are used to compare the expected cost per time unit for both the static and adaptive control charts.     

Hotelling’s T charts with variable sample size and control limit

The ideas of variable sampling interval (VSI), variable sample size (VSS), variable sample size and sampling interval (VSSI), and variable parameters (VP) in the univariate case have been successfully applied to the multivariate case to improve the efficiency of Hotelling’s T² chart with fixed sampling rate (FSR) in detecting small process shifts. However, the main disadvantage in using most of these control schemes is an increasing in the complexity due to the adaptive changes in sampling intervals. In this paper, retaining the lengths of sampling intervals constant, a variable sample size and control limit (VSSC) T² chart is proposed and described. The statistical efficiency of the VSSC T² chart in terms of the average time to signal a shift in process mean vector is compared with that of the VP, VSSI, VSS, VSI, and FSR T² charts. From the results of comparison, it shows that the VSSC T² chart for a (very) small shift in the process mean vector gives a better performance than the VSSI, VSS, VSI, and FSR T² charts; meanwhile, it presents a similar performance to the VP T² chart. Furthermore, from the viewpoint of practicability, it is more convenient for administrating the control chart than the VSI, VSSI, and VP T² chart. Thus, it may provide a good option for quick response to small shifts in a multivariate process.     

CUSUM control charts based on likelihood ratio for preliminary analysis

To detect and estimate a shift in either the mean and the deviation or both for the preliminary analysis, the statistical process control (SPC) tool, the control chart based on the likelihood ratio test (LRT), is the most popular method. Sullivan and woodall pointed out the test statistic lrt(n1, n2) is approximately distributed as x2(2) as the sample size n,n1 and n2 are very large, and the value of n1 = 2,3,..., n - 2 and that of n2 = n - n1. So it is inevitable that n1 or n2 is not large. In this paper the limit distribution of lrt(n1, n2) for fixed n1 or n2 is figured out, and the exactly analytic formulae for evaluating the expectation and the variance of the limit distribution are also obtained. In addition, the properties of the standardized likelihood ratio statistic slr(n1, n) are discussed in this paper. Although slr(n1, n) contains the most important information, slr(i, n)(i≠n1) also contains lots of information. The cumulative sum (CUSUM) control chart can obtain more information in this condition. So we propose two CUSUM control charts based on the likelihood ratio statistics for the preliminary analysis on the individual observations. One focuses on detecting the shifts in location in the historical data and the other is more general in detecting a shift in either the location and the scale or both. Moreover, the simulated results show that the proposed two control charts are, respectively, superior to their competitors not only in the detection of the sustained shifts but also in the detection of some other out-of-control situations considered in this paper.     

Comparison of control charts for Poisson count data in health‐care monitoring

Statistical surveillance is a noteworthy endeavor in many health‐care areas such as epidemiology, hospital quality, infection control, and patient safety. For monitoring hospital adverse events, the Shewhart u‐control chart is the most used methodology. One possible issue of the u‐chart is that in health‐care applications the lower control limit (LCL) is often conventionally set to zero as the adverse events are rare and the sample sizes are not sufficiently large to obtain LCL greater than zero. Consequently, the control chart loses any ability to signal improvements. Furthermore, as the area of opportunity (sample size) is not constant over time, the in‐control and out‐of‐control run length performances of the monitoring scheme are unknown. In this article, on the basis of a real case and through an intensive simulation study, we first investigate the in‐control statistical properties of the u‐chart. Then we set up several alternative monitoring schemes with the same in‐control performances and their out‐of‐control properties are studied and compared. The aim is to identify the most suitable control chart considering jointly: the ability to detect unexpected changes (usually worsening), the ability to test the impact of interventions (usually improvements), and the ease of use and clarity of interpretation. The results indicate that the exponentially weighted moving average control chart derived under the framework of weighted likelihood ratio test has the best overall performance.     

Joint economic design of EWMA control charts for mean and variance

Control charts with exponentially weighted moving average (EWMA) statistics (mean and variance) are used to jointly monitor the mean and variance of a process. An EWMA cost minimization model is presented to design the joint control scheme based on pure economic or both economic and statistical performance criteria. The pure economic model is extended to the economic-statistical design by adding constraints associated with in-control and out-of-control average run lengths. The quality related production costs are calculated using Taguchi’s quadratic loss function. The optimal values of smoothing constants, sampling interval, sample size, and control chart limits are determined by using a numerical search method. The average run length of the control scheme is computed by using the Markov chain approach. Computational study indicates that optimal sample sizes decrease as the magnitudes of shifts in mean and/or variance increase, and higher values of quality loss coefficient lead to shorter sampling intervals. The sensitivity analysis results regarding the effects of various inputs on the chart parameters provide useful guidelines for designing an EWMA-based process control scheme when there exists an assignable cause generating concurrent changes in process mean and variance.     

Analysis of ordered categorical data to develop control charts for monitoring customer loyalty

In the highly competitive business environment of today, the cost to attract new customers is much higher than the cost required to maintain the existing ones. To keep the balance between the acquisition rate and defection rate through executing offensive and defensive marketing policies, it is required to have real time information using an efficient method to monitor customer loyalty. The relationship between customer loyalty and customer satisfaction should be kept in mind when one develops a method for loyalty monitoring. This paper presents several control charts classified in two groups based on the scale used to assess customer loyalty. In the first group of control charts, customer loyalty is considered as a binary random variable modeled by Bernoulli distribution whilst in the second group, an ordinal scale is considered to report loyalty level. Performance comparison of the proposed techniques using ARL criterion indicates that chi‐square and likelihood‐ratio control charts developed based on Pearson chi‐square statistic and ordinal logistic regression model respectively are able to rapidly detect the significant changes in loyalty behavior. To show how to apply the procedures and how to interpret their results, two illustrative synthetic cases are also explained. Copyright © 2009 John Wiley & Sons, Ltd.     

Estimating the limits for statistical process control charts: A direct method improving upon the bootstrap

This paper considers the problem of finding limits for a statistical process control (SPC) chart for the process mean, when the process distribution is unknown. The bootstrap method estimates these limits relying on Monte Carlo methods, which are subject to simulation errors. Therefore this paper develops a computationally efficient enumeration method for exact calculations of the control limits.     

A matheuristic approach for solving a home health care problem

We deal with a Home Health Care Problem (HHCP) which objective consists in constructing the optimal routes and rosters for the health care staffs. The challenge lies in combining aspects of vehicle routing and staff rostering which are two well known hard combinatorial optimization problems. To solve this problem, we initially propose an integer linear programming formulation (ILP) and we tested this model on small instances. To deal with larger instances we develop a matheuristic based on the decomposition of the ILP formulation into two problems. The first one is a set partitioning like problem and it represents the rostering part. The second problem consists in the routing part. This latter is equivalent to a Multi-depot Traveling Salesman Problem with Time Windows (MTSPTW).     

A planning and routing model for patient transportation in health care

In this paper, a problem concerning both the planning of health care services and the routing of vehicles, for patients transportation is addressed. An integrated approach, based on the column generation technique, is proposed to solve the planning and routing problem. Preliminary results on real data show the effectiveness of the proposed approach.     

Mid-term and short-term planning support for home health care services

In this paper we are looking at routing and scheduling problems arising in the context of home health care services. Many small companies are working in this sector in Germany and planning is still done manually, resulting in long planning times and relatively inflexible solutions.     

Variable sampling interval Shewhart control charts for monitoring the multivariate coefficient of variation

In many industrial manufacturing processes, the ratio of the variance to the mean of a quantity of interest is an important characteristic to ensure the quality of the processes. This ratio is called the coefficient of variation (CV). A lot of control charts have been designed for monitoring the CV of univariate quantity in the literature. However, the CV control charts for multivariate quantity have not received much attention yet. In this paper, we investigate a variable sampling interval (VSI) Shewhart control chart for monitoring multivariate CV. The time between two consecutive samples is allowed to vary according to the previous value of the multivariate CV, which will help the chart to detect the process shifts faster. The comparison with the fixed sampling interval Shewhart chart is implemented to highlight the advantage of the VSI method. Finally, an illustrative example is demonstrated on real data.     

Maximum principles for discrete-time nonlinear stochastic control systems

In this paper we treat discrete-time stochastic control systems. Using corresponding results for systems, which are linear with respect to the state variables, we derive under convexity assumptions optimality conditions in form of maximum principles     

A new control chart in contaminated data of t‐Student distribution for individual observations

Control charts are the most popular tool for monitoring production quality. In traditional control charts, it is usually supposed that the observations follow a multivariate normal distribution. Nevertheless, there are many practical applications where the normality assumption is not fulfilled. Furthermore, the performance of these charts in the presence of measurement errors (outliers) in the historical data has been improved using robust control charts when the observations follow a normal distribution. In this paper, we develop a new control chart for t‐Student data based on the trimmed T² control chart () through the adaptation of the elements of this chart to the case of this distribution. Simulation studies show that a control chart performs better than T² in t‐Student samples for individual observations. Copyright © 2012 John Wiley & Sons, Ltd.     

Application of the analytical hierarchy process to establish health care waste management systems that minimise infection risks in developing countries

This paper focuses on the application of the analytical hierarchy process (AHP) technique in the context of sustainable development to establish and optimise health care waste management (HCWM) systems in rural areas of developing countries. This is achieved by evaluating the way in which the AHP can best be combined with a life cycle management (LCM) approach, and addressing a main objective of HCWM systems, i.e. to minimize infection of patients and workers within the system. The modified approach was applied to two case studies: the sub-Saharan African countries of South Africa and Lesotho. Quantitative weightings from the AHP are used to identify alternative systems that have similar outcomes in meeting the systems objective, but may have different cost structures and infection risks. The two case studies illustrate how the AHP can be used (with strengths and weaknesses) in environmental engineering decision support in developing countries.     

A control parametrization algorithm for optimal control problems involving linear systems and linear 1

Computational schemes based on control parametrization techniques are known to be very efficient for solving optimal control problems. However, the convergence result is only available for the case in which the dynamic system is linear and without the terminal equality and inequality constraints. This paper is to improve this convergence result by allowing the presence of the linear terminal inequality. For illustration, an example arising in the study of optimally one-sided heating of a metal slab in a furnace is considered.     

Bayesian statistical process control for Phase I count type data

Count data, most often modeled by a Poisson distribution, are common in statistical process control. They are traditionally monitored by frequentist c or u charts, by cumulative sum and by exponentially weighted moving average charts. These charts all assume that the in‐control true mean is known, a common fiction that is addressed by gathering a large Phase I sample and using it to estimate the mean. “Self‐starting” proposals that ameliorate the need for a large Phase I sample have also appeared. All these methods are frequentist, ie, they allow only retrospective inference during Phase I, and they have no coherent way to incorporate less‐than‐perfect prior information about the in‐control mean. In this paper, we introduce a Bayesian procedure that can incorporate prior information, allow online inference, and should be particularly attractive for short‐run settings where large Phase I calibration exercises are impossible or unreasonable.     

Necessary conditions for optimality for control problems with time delays appearing in both state and control variables

An integral maximum principle is developed for a class of nonlinear systems containing time delays in state and control variables. Its proof is based on the theory of quasiconvex families of functions, originally developed by Gamkrelidze and extended by Banks. This result is used to obtain a pointwise principle of the Pontryagin type.The authors wish to acknowledge Professor J. M. Blatt for suggesting this problem. Further, they also wish to acknowledge the referee of the paper for bringing to their attention the problems discussed in Section 6.     

A simulation-based multivariate Bayesian control chart for real time condition-based maintenance of complex systems

When complex systems are monitored, multi-observations from several sensors or sources may be available. These observations can be fused through Bayesian theory to give a posterior probabilistic estimate of the underlying state which is often not directly observable. This forms the basis of a Bayesian control chart where the estimated posterior probability of the state can be compared with a preset threshold level to assess whether a full inspection is needed or not. Maintenance can then be carried out if indicated as necessary by the inspection. This paper considers the design of such multivariate Bayesian control chart where both the transition between states and the relationship between observed information and the state are not Markovian. Since analytical or numerical solutions are difficult for the case considered in this paper, Monte Carlo simulation is used to obtain the optimal control chart parameters, which are the monitoring interval and the upper control limit. A two-stage failure process characterised by the delay time concept is used to describe the underlying state transition process and Bayesian theory is used to compute the posterior probability of the underlying state, which is embedded in the simulation algorithm. Extensive examples are shown to demonstrate the modelling idea.     

Multi-criteria diagnosis of control knowledge for cartographic generalisation

The development of interactive map websites increases the need of efficient automatic cartographic generalisation. The generalisation process, which aims at decreasing the level of details of geographic data in order to produce a map at a given scale, is extremely complex. A classical method for automating the generalisation process consists in using a heuristic tree-search strategy. This type of strategy requires having high quality control knowledge (heuristics) to guide the search for the optimal solution. Unfortunately, this control knowledge is rarely perfect and its evaluation is often difficult. Yet, this evaluation can be very useful to manage knowledge and to determine when to revise it. The objective of our work is to offer an automatic method for evaluating the quality of control knowledge for cartographic generalisation based on a heuristic tree-search strategy. Our diagnosis method consists in analysing the system’s execution logs, and in using a multi-criteria analysis method for evaluating the knowledge global quality. We present an industrial application as a case study using this method for building block generalisation and this experiment shows promising results.     

Second order sufficient conditions and sensitivity analysis for the optimal control of a container crane under state constraints

Stability and sensitivity analysis of parametric control problems has recently been elaborated for optimal control problems subject to pure state constraints. This paper illustrates the numerical aspects of sensitivity analysis for a complex practical example: the optimal control of a container crane with a state constraint on the vertical velocity. The multiple shooting method is used to determine a nominal solution satisfying first order necessary conditions. Second order sufficient conditions are checked by showing that an associated Riccati equation has a bounded solution. Sensitivity differentials of optimal solutions an computed with respect to variations in the swing angle