# Nonparametric Shewhart-type control charts with runs-type signaling rules: Case K and Case U

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## Introduction

Statistical process control (SPC) refers to the collection of statistical procedures and problem solving tools used to control and monitor the quality of the output of some production process, including the output of services (see e.g. Balakrishnan et al., (2006) p. 6678 and Montgomery, (2005) p. 148). The aim of SPC is to detect and eliminate or, at least reduce, unwanted variation in the output of a process. The benefits include saving time, increasing profits and an overall increase in the quality of products and services.

Control chart

A control chart is a statistical procedure (or scheme) that can be depicted graphically for on-lineprocess monitoring of a measurable characteristic (such as the mean measurement value or the percentage nonconforming items) with the objective to show whether the process is operating within the limits of expected variation (see e.g. Ruggeri, Kenett and Faltin (2007) p. 429) . The simplest and most widely used control chart is the Shewhart-type of chart; this chart is named after the father of quality control i.e. Dr. Walter A. Shewhart (1891-1967) of Bell Telephone Laboratories, who developed the chart in the 1930’s and laid the foundation of modern statistical process control in his book Economic Control of Quality of Manufactured Product that was originally published in 1931.The the wider use and popularity of control charts outside manufacturing, which lead to Quality Management and Six Sigma, can be attributed to Deming (1986).

### Multivariate control charts

Some practical situations require the simultaneous monitoring and control of two or more related(correlated) quality characteristics. The usual practice (see Ryan, (2000) p. 253) is to monitor each characteristic separately; this results in a univariate control chart for each variable but, may beinefficient or may lead to erroneous conclusions (see Ryan, (2000) p. 254 and Montgomery, (2005)p.486). Control charts to deal with multiple measurements (variables) were therefore developed.

#### Phase I and Phase II control charts

The statistical process control regime is typically implemented in two stages: Phase I (the so-called
retrospective phase) and Phase II (the prospective or the monitoring phase). In Phase I, the primary
interest is to better understand the process and to assess process stability; the latter step often consists of trying to bring a process in-control by analysing historical or preliminary data, locating and eliminating any assignable causes of variation.

Chapter 1 Introduction and research objectives
1.0 Introduction
1.1 Research objectives
1.1.1 Chapter 2
1.1.2 Chapter 3
1.1.3 Chapter 4
Chapter 2 Variables control charts: Phase I
2.0 Chapter overview
2.1 Phase I SPC
2.1.1 Design and implementation of two-sided Shewhart-type Phase I charts
2.2 Shewhart-type S2, S and R charts: Phase I
2.2.1 Phase I S2 chart
2.2.2 Phase I S chart
2.2.3 Phase I R chart
2.3 Literature review: Univariate parametric Shewhart-type Phase I variable charts for location and spread
2.3.1 Phase I charts for the normal distribution
2.3.2 Phase I charts for other settings
2.4 Concluding remarks: Summary and recommendations
2.5 Appendix 2A: SAS® programs
2.5.1 SAS® program to find the charting constants for the Phase I S2 chart
2.5.2 SAS® program to find the charting constants for the Phase I S chart
2.5.3 SAS® program to find the charting constants for the Phase I R chart
Chapter 3 Attributes control charts: Case K and Case U
3.0 Chapter overview
3.1 The p-chart and the c-chart for standards known (Case K)
3.1.1 Probability of a no-signal
3.1.2 Operating characteristic and the OC-curve
3.1.3 False alarm rate
3.1.4 Run-length distribution
3.1.5 Average run-length
3.1.6 Standard deviation and percentiles of the run-length
3.1.7 In-control and out-of-control run-length distributions
3.2 The p-chart and the c-chart for standards unknown (Case U)
3.2.1 Phase I of the Phase II p-chart and c-chart
3.2.2 Phase II p-chart and c-chart
3.2.3 Conditional Phase II run-length distributions and characteristics
3.2.3.1 Conditional characteristics of the p-chart
3.2.3.2 Conditional characteristics of the c-chart
3.2.4 Unconditional Phase II run-length distributions and characteristics
3.2.4.1 Unconditional characteristics of the p-chart
3.2.4.2 Unconditional characteristics of the c-chart
3.3 Concluding remarks: Summary and recommendations
3.4 Appendix 3A: Characteristics of the p-chart and the c-chart in Case K
3.4.1 The p-chart in Case K: An example
3.4.2 The p-chart in Case K: Characteristics of the in-control run-length distribution
3.4.3 The c-chart in Case K: An example
3.4.4 The c-chart in Case K: Characteristics of the in-control run-length distribution
Chapter 4 Nonparametric Shewhart-type control charts with runs-type signaling rules: Case K and Case U
4.0 Chapter overview
4.1 Runs-type signaling rules
4.1.1 The 1-of-1 charts
4.1.2 The k-of-k and k-of-w charts
4.2 Sign charts for the known πth quantile (Case K)
4.2.1 Run-length distributions of the sign charts
4.2.2 Transition probability matrices of the sign charts
4.2.3 The in-control run-length characteristics of the one-sided and two-sided sign charts
4.2.4 Design of the upper (lower) one-sided 1-of-1, 2-of-2 and 2-of-3 sign charts
4.2.5 Performance comparison of the one-sided sign charts
4.2.6 Design of the two-sided 2-of-2 DR, the 2-of-2 KL and the 2-of-3 sign charts
4.2.7 Performance comparison of the two-sided sign charts
4.3 Precedence charts for the unknown πth quantile (Case U)
4.3.1 Run-length distributions of the two-sided precedence charts
4.3.2 Unconditional ARL, VARL and FAR calculations
4.3.3 Run-length distributions of the one-sided precedence charts
4.3.4 Design and implementation of the two-sided precedence charts
4.3.5 Performance comparison of the two-sided precedence charts
4.4 Concluding remarks: Summary and recommendations
4.5 Appendix 4A: SAS® programs
4.5.1 SAS® programs to simulate the run-length distributions of the upper one-sided X-bar, sign and SR charts in Case K
4.5.2 SAS® programs to simulate the run-length distributions of the two-sided precedence charts in Case U
Chapter 5 Concluding remarks: Summary and recommendations for future research
References

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Univariate parametric and nonparametric statistical quality control techniques with estimated process parameters