Chapter 1Introduction
1.1 Statistical Process Control (SPC)
The history of process monitoring is very old and it had always been thereinformally in one form or the other. Formally on 16 May 1924，Dr, Walter A.Shewhart introduced control charting procedures for process monitoring. Later,Statistical Proems Control (SPC) became an important branch of process mon?itoring. In SPC, processes are monitored for their stability with respect to dif?ferent parameters using statistical techniques like Control Charts, Check Sheets.Pareto Diagrams, Histograms etc (cf. Montgomery (2012)). The statistical tech?niques for process monitoring are now referred to as the SPC tool-kit. Naturaland un-natural variations are two major types of variability which, may exist inany process. Natural variations axe considered to be inherent in a process evenif the process is established and maintained carefully while the un-natural vaxia-tions are control-lable and can be detected. The excessive amount of variationsmay cause change in process parameters i.e. location and dispersion parame?ters. The number of nonconformities items may be increased due to change inlocation parameter while increase in the amount of dispersion parameter maydisturb the quality of process and become reason of lower uniformity in process.Moreover, the quick detection of special causes of variation may decrease theprocess variability and in result the quality of a process can be improved. A pro?cess in the presence of natural variations is considered to be in-control otherwiseout-of-control and need to monitor by control chart. Initially, the control chartswas developed for monitoring the production process but later these have beenextensively used in all other areas which are increasing day by day, for examplesanalytical laboratories； animal production systems, education, electronics, epi?demiology, health care, pharmaceutical, nuclear engineering, military, mineralsetc (cf_ Woodall (2006), Thor et al. (2007), Hwang et al. (2008), Wang andLiang (2008), Santos (2009), Abbasi (2010), Shah et al. (2010)，Vries and Re-neau (2010)，Stewart et al. (2011), Tasdemir (2012)，Cora.in and Salmaso (2013)and Tasdemir and Kowalczuk (2014)).
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1.2 Control Charts
SPC is a collection of different methods that are used to examine a processand to improve the quality of its products (cf. Montgomery (2009), and Vriesand Reneau (2010)). Among these methods, the control chart is the most im?portant and the most commonly used tool. Control chart is a graphical displayof a quality characteristic that has been measured or computed from a sampleversus the sample number or time (cf. Montgomery (2012)). It contains a centerline, that represents the average value of the quality characteristic correspond?ing to the in-control state, and the two horizontal lines called upper and lowercontrol limits. The lines mentioned are abbreviated by CL, UCL and LCL. Thecharting statistics are plotted over the sample number vs. LCL and UCL to ob?serve the stability of the process. These control limits are chosen such that theassociated false alarm rate (the probability of falling the in-control points to theoutside of LCL and UCL. usually denoted by a*) is very small. The probabilitythat the out-of-control points fall beyond the control limits is called signalingprobability (power) of a control chart and it is generally used as performancemeasure. Another performance measure is average run length (ARL) which isaverage of a random variable run length (RL). The RL is defined as the num?ber of samples after that first oiit-of-coutrol signal shown. The average numberof samples to signal (ANOS) out-of-control status is another performance mea?sure used in this thesis.
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Chapter 2Probability Limits of the COM-Poisson Charts
2.1 Introduction
Statistical Process Control (SPC)’ many situations arise in whichproblems are modeled by discrete probability distributions. Discrete probabilitymodels are used to describe processes where the output of interest is a count,Shapiro and Zahedi (1990) defined a process in terms of the Bernoulli randomvariables and discussed applications of some discrete distributions in quality con?trol. The use of c and u charts as an appropriate tool for monitoring discrete outcome in production or administrative processes has been considered in manyarticles including Duncan (1986)，Banks (1989), Mimoz and Nielsen (1991), andMontgomery (2009). These charts are based on the assumption that the discreteoutcome is a Poisson proems. In many real life situations, this assumption is notvalid and one has to seek a better alternative. The Poison distribution dependson a single parameter, A, which is the mean as well as the variance. In manypractical problems, however, the mean and the variance are not same. In sudisituations； one plausible approach is to use the binomial distribution when themean is greater than variance (a case of under-dispersion) or the negative bino?mial distribution (which has the geometric distribution as a special case) whenthe mean is smaller than variance (a case of over-dispersion).
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2.2 The COM-Poisson Distribution
In this section, we discuss the COM-Poisson distribution, the distributionof sum of COM-Poisson random variables and maximum likelihood estimates(MLEs) of the unknown parameter of the COM-Poisson distribution.The COM-Poisson distribution (introduced by Conway and Maxwell (1962)，and revisit by Shmueli et al. (2005)) is a viable count distribution that generalizesthe Poisson distribution in light of associated data dispersion. In this section, we propose the exact fc-sigma and true probabilitycontrol limits for the generalized control charts propc^ed by Sellers (2012), forthe monitoring of total number of counts and average number of counts per unit.The use of true probability limits will improve the efficiency of the proposedchart and overcome the problems associated with fe-sigma control limits that willbe discussed later.
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Chapter 3 An EWMA Control Chart for Count Data....... 27
3.1 Introduction ....... 27
3.2 The Proposed GEWMA Chart ....... 30
3.3 The Performance Evaluation ....... 32
3.4 The Sensitivity of the Poisson EWMA chart....... 39
3.5 Generalization of the proposed chart....... 45
3.6 Conclusions and Discussions ....... 46
Chapter 4 CUSUM Charts for the COM-Poisson Processes....... 49
4.1 Introduction .......50
4.2 CUSUM Charts for the COM-Poisson Processes....... 53
4.2.1 The -CUSUM chart ....... 53
4.2.2 The Z/-CUSUM chart ....... 56
4.2.3 The 5-CUSUM chart ....... 56
4.3 Design of the CUSUM charts ....... 57
4.3.1 Markov chain approach ....... 58
4.3.2 The Choice of ....... 60
4.4 The Performance Evaluation ....... 62
4.4.1 In-Control Performance ....... 62
4.4.2 Out-of-Control Performance....... 65
4.5 Illustrative Examples....... 75
4.6 Summary and Discussions ....... 79
Chapter 5 Multivariate Chart for the COM-Poisson Attributes....... 81
5.1 Introduction ....... 81
5.2 Construction of the Multivariate COM-Poisson Chart....... 84
5.3 Performance Evaluation ....... 88
5.4 Generalization to MP and MNP charts....... 93
5.5 Applications of the MCP Chart....... 95
5.6 Conclusions and Discussion ....... 100
Chapter 7Designing of Gini-Chart to Non-Normal Processes
7.1 Introduction
Statistical techniques in the process control are popularly known as controlcharts. The literature on process control provides a variety of charts to monitordispersion and location parameters of any process. These range from Shewhart-type control charts, Moving Average (MA) charts. Cumulative Sum (CUSUM)charts and Exponentially Weighted Moving Average (EWMA) charts. EWMAand CUSUM control charts address smaller shifts whereas Shewhart type controlcharts address larger shifts in process parameters. The commonly used Shewhart- type control charts for monitoring the process dispersion include the R chart, Schart, and chart, while for average include X chart, X chart and Median Chart(see Ryan (1989), Grant and Leavemvorth (1996) and Montgomery (2009)). Thischapter focused on Shewhart type control charts.
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Conclusion
Statistical process control (SPC) is a collection of different tools that areused to examine a process and to improve the quality of its products. Controlchart (pioneered by Dr. Walter Shewhart in early 1920's) is the most famous andcommonly used monitoring tool in order to inv^tigate un-natural variations inprocess parameters. There are two types of control charts, namely, attributescontrol charts and variables control charts. Variables control charts deals withthe monitoring of quantities quality characteristics such as weight, volume, tem?perature etc. In SPC, many situations arise in which quality characteristicscannot be easily measured on a numerical scale. However, each inspected itemis usually classified as either conforming or nonconforming to the specificationsof that quality characteristic. The terminology ’defective’ or 'non-defective’ isused frequently in order to identify these two classifications of a product, or theterminology 'conforming' and 'noii-confonning respectively. These quality char?acteristics are known as attributes. The current thesis deals with attributes andvariables charts.
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