Abstract
Quality is the core competence of enterprises and even countries. The control chart, one of the great inventions of the last century, is the primary tool for statistical quality management. Now we know that, the control chart is a graphical display of the value of a measured quality characteristic over a period of time, or through a series of samples, that can help establish and maintain statistical quality control of a process. To use a control chart, the user is required to determine the chart parameters, usually consisting of the sample size, sampling interval and control limits, to meet specified considerations. The selection of these parameters is directly responsible for the economic benefit of a process. Taking into account the fact that evaluation of a business’s performance is always from an economic point of view, the economic modeling and design approaches for control charts appear to be a natural choice.The economic modeling and design of control charts is, in general, complicated and difficult. Traditionally, optimal designs of control charts are mostly oriented towards monitoring typical processes that have ideal characteristics, such as the sampling resources are unlimited, the quality deterioration has a step shift mechanism, the exact process distribution is available, etc. But this is not really the case in the real world. Under nonideal conditions, following traditional economic modeling and design approaches for control charts would detract from the economic benefit of the processes. To date, the literature on control-chart methods is quite large. While the research on the economic modeling and design of control charts is of great importance, knowledge about this area is far less than what we should have, particularly when considering nonideal situations. This greatly limits the application and popularity of the economic design approaches in practice.
Proceeding from the research gaps and the practical needs, this thesis pays attention to a couple of open research issues on the economic modeling and design of control charts under nonideal conditions, involving individual observations, progressive quality deterioration, and unknown process distribution. To resolve these essential research issues, we develop new economic modeling and design approaches that can help quantitatively measure the economic performance of the relevant control charts for monitoring particular processes and optimally design their parameters, to improve the economic benefit of these processes.
In the first research issue, we deal with the economic modeling and design of the SPRT chart for individual observations. In some practical processes, it may not be feasible to take samples larger than one for various reasons. Many existing control charts are shaped up for rational subgroups and perform poorly with individual observations, since the basis for rational subgrouping is no longer valid. In such situations, the CUSUM charts or the EWMA charts are usually recommended for individual observations, for making efficient use of all historical information. This research indicates that the SPRT chart is an excellent alternative solution, particularly suitable for the situations that only a single observation is available at one sampling point. The SPRT charts have been found to be highly effective from and only from a statistical point of view. In this research, we figure out the rule of transitions between process states and using this rule, we realize the economic modeling for the SPRT chart by means of a two-stage Markov chain approach. Compared with two competing individuals control charts (namely, the Fp CUSUM and VSI CUSUM charts), we find that the SPRT chart maintains overwhelming economic superiority under various process scenarios with individual observations. Further, the SPRT chart is rather robust to misspecifications of process or cost parameter in some degree. Taken together, we highly recommend using the economically designed SPRT chart for individual observations, instead of traditional individuals control charts.
In the second research issue, we work on the economic modeling and design of the X control chart subject to a general quality deterioration. The detection of step shifts in the process mean is one of the core concerns of control chart design and has been intensively studied in the literature. In practice, however, not all shifts obey this ideal behavior, because changes are usually continuous, for example, variations in temperature. As a consequence, the common consideration of a shift as a step function cannot always adequately describe what actually happens in the real world, that necessitates more realistic assumptions for economic modeling and design of control charts. However, the research in this line has been rarely considered in the literature. In this research, we derive the exact distribution of the sample mean X in the situation that the rational subgroup concept is invalid and using the Markov chain approach, we construct an improved economic model used to quantitatively measure the economic performance of the X control chart under various patterned mean shifts, extending the detection of step shifts to a more general situation. The finite production run is also considered in this model. The result of extensive numerical experiments indicates that, in most of the cases using an idealistic step shift pattern as the approximation for actual shifts are unlikely to give rise to significant cost penalties but relatively significant evaluated errors. This finding shows the dual impacts and provides good support to the use of the step shift assumption when accurate production-related cost assessments are needless. This research makes up for the deficiency of the existing literature on the progressive quality deterioration.
The previous two research issues that constitute the first part of the thesis, are devoted to the economic modeling and design of parametric control charts. The design of parametric control charts is dependent on the assumption of normality or some other particular distribution for the underlying process. However, this assumption could be questionable in practice, because knowledge about the underlying process distribution is often unavailable. As a consequence, the performance of traditional parametric control charts is largely degraded when the actual distribution deviates from the assumed distribution, especially if the sample size is small. The nonparametric (or distribution-free) control charts are capable of detecting changes occurred in the underlying process distribution without information about the process distribution itself. The in-control run length distribution of a nonparametric control chart is obtainable and remains invariant for all continuous process distributions, and accordingly, researchers make use of this distribution robustness property to design control charts. The nonparametric control charts in the existing literature are all designed with respect to statistical criteria, while robust designs of nonparametric control charts from an economic viewpoint have not been addressed yet. Therefore, in the second part of the thesis, we turn to the economic modeling and design of nonparametric control charts to cope with the third research issue.
For the nonparametric part, we focus on three representative Shewhart-type nonparametric control charts, including for location changes the Sign chart based on the single sample sign statistic and the WRS chart using the Wilcoxon rank-sum statistic and for scale changes the AB chart with the Ansari-Bradley statistic. In this research, we develop a universal economic model for nonparametric control charts based on Duncan’s economic model, which has a suitable bound on the average probability of false alarms in order to maintain desirable statistical properties. The presented model blends the business benefit of economic design with the robustness advantage of nonparametric approach. Since the distribution-free characteristic of a nonparametric control chart is not valid any more under a process shift (in location and/or scale), we embed some estimation technique in the economic model for estimating the unknown type II error. The hypothetical concept called actual performance is specifically proposed and its relations with theoretical optimal performance and with estimated optimal performance are clarified. We further emphasize the rationality and necessity of using actual performance as the main economic performance metric for nonparametric control charts. The actual performance of these economically designed nonparametric control charts is extensively studied and shown. The effects of the reference sample size, the falsealarm constraint and the misspecified model parameters are also explored. Based on these results, some beneficial recommendations are provided for practical implementation of these charts. This research overcomes the drawback to traditional economic design approaches for parametric control charts that must assume the underlying process distribution is known, and fills the research gap on the economic modeling and design of nonparametric control charts.
Through the research of the two parts related to the economic modeling and design of parametric and nonparametric control charts, this thesis aims to bridge some research gaps in the literature and to respond to the practical needs for preferable economic design approaches. Those findings in the thesis will provide qualitative and quantitative insights to support managerial decision making.
| Date of Award | 21 Jun 2017 |
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| Original language | English |
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| Supervisor | Min XIE (Supervisor) |