Multi-Variate-Attribute Quality Control (MVAQC)

Transcription

1 by Submitted in total fulfillment of the requirements of the degree of Doctor of Philosophy July 2014 Department of Mechanical Engineering The University of Melboune

2 Abstract When the number of quality characteristics, in the form of variables or attributes, exceeds unity and there exists a non-zero correlation between them, then one is dealing with either a multivariate or multi-attribute quality control problem. Monitoring these quality characteristics independently can be very misleading. Control charts, as one of the statistical quality control tools, are generally applied for both variable and attribute quality characteristics. In the variable domain, a measurable characteristic of a product or process that affects the quality of the process output is measured and controlled by using variable control charts. In the attribute domain, the number of, or percentage of, defects in products or defective products is calculated and controlled. In multivariate quality control, several dependent variable characteristics are measured and monitored simultaneously. Similarly, in multi-attribute quality control, more than one dependent attribute characteristic is considered simultaneously. From the literature review, all previous works have focused on either multivariate or multi-attribute quality control individually due to differences between the nature of these two concepts. Naturally, variable characteristics follow variable distributions such as the normal distribution. The appearance of these types of distributions is that of a smooth curve. However, attribute characteristics follow discrete distributions like the Binomial and Poisson. The appearance of discrete distributions is that of a series of vertical spikes with the height of each spike proportional to the probability. The pattern of these kinds of distributions is discrete and usually with skewness. Nevertheless, it is common for engineers to appeal to the continuous symmetric normal distribution for designing any control charts. Developing a model which is applicable in the multivariate-attribute case as a combination of multivariate and multi-attribute may fill a gap in the tool box available to quality professionals. As an example of where both variable and attribute characteristics may jointly determine the quality of a process, consider a metal forming process with punching and bending operations. Here, the weight and thickness of metal sheet may be i

3 variable characteristics and the shape of the bent sheet and crumpling of the punched hole may be attribute characteristics. The first of the attribute characteristics may be checked with a gauge and the latter checked visually. In this research a new model is developed for quality control in multi-variate-attribute situations. Through this model, the original correlated and skewed quality characteristics are transformed in two steps to a set of uncorrelated and non-skewed variables which can be monitored as individual variables and by using univariate control charts including Shewhart control charts, MCUSUM and EWMA. In the proposed model, the out-ofcontrol states for the transformed variables can be traced back to the original variables to find the responsible one, which is not the case with many of the existing multivariate and multi-attribute methods. Two simulation studies are carried out to analyse the performance of the proposed model and compare it with the leading alternative. These studies clearly show the advantages of this new method, such as being more predictable, more sensitive and easier to understand. ii

4 Declaration This is to certify that I. the thesis comprises only my original work towards the PhD except where indicated in the Preface, II. due acknowledgement has been made in the text to all other material used, III. the thesis is less than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices. Name: Signature: Date: 7 February 2014 iii

5 Acknowledgments I would like to thank my supervisor Dr. Alan Smith and The University of Melbourne for their support during my PhD study. I also appreciate the panel members: Professor Saman Halgamuge, and Dr. Colin Burvill for their guidance and support. I would like to express my deep gratitude to my beloved family. In particular, I am indebted to my wife, Tahmineh, because of her encouragement, patience, and unwavering support in every possible way during my PhD. iv

6 Table of Contents Abstract Declaration.. Acknowledgements. Table of contents. List of Figures. List of Tables... i iii iv v ix xi Chapter 1. Introduction Chapter 2. Literature Review Univariate / Uni-attribute Domain Univariate Quality Control X and R / and s Charts.... X Uni-attribute Quality Control p chart and np chart c chart and u chart Multivariate / Multi-attribute Domain Multivariate Quality Control Introduction Multivariate Normal Distribution Multivariate Control Charts Control Ellipse Chi-square Control Chart Hotelling T 2 Control Chart Multivariate Exponentially Weighted Moving Average (MEWMA) Control Chart v

7 Multivariate Cumulative Sum (MCUSUM) Control Chart Multivariate Shewhart chart Principal Components Analysis (PCA) Simulated MINMAX Control Chart Multi-attribute Quality Control General multi-attribute quality control charts Patel s method (Chi-square Control Chart / Hotelling T 2 Control Chart for attributes). X MNP Chart (Multivariate np Chart) Jolayemi Multi-Attribute Control Chart (MACC) Bootstrap method and confidence interval approach Skinner s Deleted-Y statistic approach Exact Multi-Attribute Control Chart (E-MACC) Skewness Reduction Approach Correlation Reduction Approach NORTA Inverse transformed vectors method Artificial Neural Network (ANN) Marcucci s Control Chart D 2 Control Chart Special cases of multi-attribute quality control Concluding remarks.. 53 Chapter 3. Research Model and Assumptions Problem Statement Research Objectives and Research Questions Research Objective Research Questions. 67 vi 36 43

8 Peoposed Research Methodology Chapter 4. Design and development of Multi-Variate-Attribute Quality Control (MVAQC) Data Generation Background NORmal-To-Anything (NORTA) Algorithm Proposed method Case Studies Discussion Multi-Variate-Attribute Control Chart Model Assumptions Correlation Reduction Skewness Reduction Normality Test Developing the control charts Performance evaluation Diagnosis of out-of-control observations Simulation study and performance evaluation Introduction Case Study Subgroup size n= Subgroup size n= Subgroup size n= Subgroup size n= Overall comparison for case vii

9 Case Study Subgroup size n= Subgroup size n= Subgroup size n= Subgroup size n= Overall comparison for case Concluding remarks 219 Chapter 5. Final Concluding Remarks Conclusions Future Work Bibliography 229 Appendices Literature review for normality test methods Program codes for data generation - Case Study 1 Section Program codes for data generation - Case Study 2 Section Program codes for data generation - Case Study 3 Section Program codes for model Development - Case Study 1 Section Program codes for model Development - Case Study 2 Section Published Papers as outcome of sections 4.1 and viii

10 List of Figures Figure 1 Model of a process-based quality management system. 3 Figure 2 The Juran trilogy diagram.. 4 Figure 3 the input-output diagram for the quality control process 5 Figure 4 A typical control chart.. 10 Figure 5 Control charts for inner ( x1) and outer ( x 2 ) bearing diameters 17 Figure 6 Control region using independent control limits for x1 and Figure 7 individual control charts vs. ellipse control area Figure 8 A bivariate normal distribution. 21 Figure 9 A control ellipse for two independent variables. 23 Figure 10 A control ellipse for two dependent variables.. 24 Figure 11 A chi-square control chart for two quality characteristics. 25 Figure 12 Research Methodology. 68 Figure 13 Generating a multi-variate-attribute random vector by using NORTA. 76 Figure 14 Combined simulation method in using NORTA algorithm for generating multi-variate-attribute random vector.... Figure 15 Flowchart of major steps for proposed monitoring of Multi-Variate- 88 Attribute data.. Figure 16 Pearson correlation coefficients for several relationships 91 Figure 17 Skewness. 93 Figure 18 Algorithm of the proposed method for MVACC (conclusion) Figure 19 Control charts for the transformed variables, Case 1, n=1, approach A Figure 20 Hotelling T 2 control chart, case 1, n=1 112 Figure 21 Control charts for the transformed variables, Case 1, n=1, approach B 120 Figure 22 Control charts for the transformed variables, case 1, n=2, approach A. 127 ix x 80

11 Figure 23 Hotelling T 2 control chart, case 1, n=2 128 Figure 24 Control charts for the transformed variables, case 1, n=2, approach B 133 Figure 25 Control charts for the transformed variables, case 1, n=4, approach A 139 Figure 26 Hotelling T 2 control chart, case 1, n=4 140 Figure 27 Control charts for the transformed variables, case 1, n=4, approach B 144 Figure 28 Control charts for the transformed variables, case 1, n=10, approach A 149 Figure 29 Hotelling T 2 control chart, case 1, n= Figure 30 Control charts for the transformed variables, case 1, n=10, approach B 154 Figure 31 Control charts for the transformed variables, Case 2, n=1, approach A Figure 32 Hotelling T 2 control chart, case 2, n=1 167 Figure 33 Control charts for the transformed variables, Case 2, n=1, approach B 174 Figure 34 Control charts for the transformed variables, case 2, n=2, approach A 183 Figure 35 Hotelling T 2 control chart, case 2, n=2 184 Figure 36 Control charts for the transformed variables, case 2, n=2, approach B 190 Figure 37 Control charts for the transformed variables, case 2, n=4, approach A 198 Figure 38 Hotelling T 2 control chart, case 2, n=4 198 Figure 39 Control charts for the transformed variables, case 2, n=4, approach B 203 x

12 List of Tables Table 1 Comparison of current multivariate and multi-attribute quality control methods.... Table 2a Comparison of correlation matrices (based on Niaki and Abbasi 84 Table 2b Comparison of correlation matrices (based on Niaki and Abbasi) - continued..... Table 3 Transformations results, case 1, n=1, approach A 110 Table 4 ARL 1 results, case 1, n=1, approach A 114 Table 5 ARL 1 results for diagnosing, case 1, n=1, approach A 116 Table 6 Transformations results, case 1, n=1, approach B. 119 Table 7 ARL 1 results, case 1, n=1, Approach B 121 Table 8 ARL 1 results for diagnosing, case 1, n=1, Approach B 122 Table 9 Comparison of approaches A and B, case 1, n= Table 10 Transformations results, case 1, n=2, approach A. 127 Table 11 ARL 1 results, case 1, n=2, approach A 129 Table 12 ARL 1 results for diagnosing, case 1, n=2, approach A 130 Table 13 Transformations results, case 1, n=2, approach B. 132 Table 14 ARL 1 results, Case 1, n=2, approach B 134 Table 15 ARL 1 results for diagnosing, case 1, n=2, approach B 136 Table 16 Comparison of approaches A and B, case 1, n=2 137 Table 17 Transformations results, case 1, n=4, approach A 139 Table 18 ARL 1 results, case 1, n=4, approach A 141 Table 19 ARL 1 results for diagnosing, case 1, n=4, approach A 142 Table 20 Transformations results, case 1, n=4, approach B 143 Table 21 ARL 1 results, Case 1, n=4, approach B 145 Table 22 ARL 1 results for diagnosing, case 1, n=4, approach B 146 xi 55 85

13 Table 23 Comparison of approaches A and B, case 1, n=4 147 Table 24 Transformations results, case 1, n=10, approach A 149 Table 25 ARL 1 results, case 1, n=10, approach A 151 Table 26 ARL 1 results for diagnosing, case 1, n=10, approach A 152 Table 27 Transformations results, case 1, n=10, approach B 153 Table 28 ARL 1 results, Case 1, n=10, approach B 155 Table 29 ARL 1 results for diagnosing, case 1, n=10, approach B 156 Table 30 Comparison of approaches A and B, case 1, n= Table 31 Summary of different combinations of subgroup sizes n=1,2,4,10 and approaches A and B for case Table 32 Transformations results, case 2, n=1, approach A. 165 Table 33 ARL 1 results, case 2, n=1, approach A. 169 Table 34 ARL 1 results for diagnosing, case 2, n=1, approach A 170 Table35 Transformations results, case 2, n=1, approach B Table 36 ARL 1 results, case 2, n=1, Approach B 175 Table 37 ARL 1 results for diagnosing, case 2, n=1, Approach B 177 Table 38 Comparison of approaches A and B, case 2, n=1 179 Table 39 Transformations results, case 2, n=2, approach A 182 Table 40 ARL 1 results, case 2, n=2, approach A 185 Table 41 ARL 1 results for diagnosing, case 2, n=2, approach A 186 Table 42 Transformations results, case 2, n=2, approach B 189 Table 43 ARL 1 results, Case 2, n=2, approach B 192 Table 44 ARL 1 results for diagnosing, case 2, n=2, approach B. 193 Table 45 Comparison of approaches A and B, case 2, n=2 195 Table 46 Transformations results, case 2, n=4, approach A 197 xii

14 Table 47 ARL 1 results, case 2, n=4, approach A 199 Table 48 ARL 1 results for diagnosing, case 2, n=4, approach A 200 Table 49 Transformations results, case 2, n=4, approach B 202 Table 50 ARL 1 results, case 2, n=4, approach A 204 Table 51 ARL 1 results for diagnosing, case 2, n=4, approach B 205 Table 52 Comparison of approaches A and B, case 2, n=4 207 Table 53 Transformations results, case 2, n=10, approach A 209 Table 54 ARL 1 results, case 2, n=10, approach A 210 Table 55 Transformations results, case 2, n=10, approach B 212 Table 56 ARL 1 results, case 2, n=10, approach A 213 Table 57 Comparison of approaches A and B, case 2, n= Table 58 Summary of different combinations of subgroup sizes n=1,2,4,10 and approaches A and B for case xiii

15 Chapter 1 - Introduction 1

16 1. Introduction There are a lot of different definitions for quality proposed by many influential quality experts like Shewhart, Deming, Juran, Crosby, Feigenbaum, Taguchi, Garvin, and Ishikawa depending on their different points of view in their era. Competitive pressures and technological developments have affected the core knowledge of operations and production management and introduced many new concepts in the literature. One of the latest definitions of quality is degree to which a set of inherent characteristics fulfils requirements [1]. In this definition, a requirement, including its qualitative and quantitative characteristics, is defined as a need or expectation that is stated, generally implied or obligatory [1]. This means that, for achieving an acceptable level of quality, a combination of both qualitative and quantitative needs and expectations need to be considered. In revising ISO 9001, the international standard for managing quality, the authors introduced the process approach in 2000 and have retained this approach in the latest 2008 version of the standard[2]. As is shown in figure 1, the application of a system of processes within an organization, together with the identification and interactions of these processes, and their management to produce the desired outcomes, is referred to as the process approach[2]. Quality control needs to be considered as one of an organization s processes for product realization in interaction with other processes. As is shown in figure 1, the starting point for developing a product is the requirements stated by customers and the finishing point is the fulfillment of these requirements. In the manufacturing area, the requirements that are stated or implied as inputs are some product characteristics that should be measured on products as output, to make sure that all are fulfilled. 2

17 Figure 1 Model of a process-based quality management system [2] Quality Control, defined as part of quality management focused on fulfilling quality requirements, is considered as one of the elements of quality management. The other elements are quality policy, quality objectives, quality planning, quality assurance and quality improvement[1]. As is shown in figure 2, Juran introduced quality control as a part of his trilogy besides quality planning and quality improvement.the term control of quality emerged early in the twentieth century [3, 4]. In addition to other quality control methods, the statistical quality control movement gave the impression that quality control consisted solely of using statistical methods [5]. 3

18 Figure 2 The Juran trilogy diagram[5] During the 1950s, the term total quality control was treated as an all encompassing term by experts such as Feigenbaum. This saw quality control as meaning just that, anything done to control quality. In the United States, the term quality control now often has the narrow meaning defined previously. The term total quality management (TQM) is now used as the allembracing term. In Europe, the term quality control is also acquiring a narrower meaning. Recently, the European umbrella quality organization changed its name from European Organization for Quality Control to European Organization for Quality. In Japan, the term quality control retains a broad meaning. Their total quality control is roughly equivalent to the U.S. term total quality management. Although, In 1997, the Union of Japanese Scientists and Engineers (JUSE) adopted the term total quality management 4

19 (TQM) to replace total quality control (TQC) to more closely align themselves with the more common terminology used in the rest of the world. [5] The quality control process of Juran is one of the steps in the overall quality management sequence. Figure 3 shows the input-output features of this step. In Figure 3, the input is operating process features developed to produce the product features required to meet customer needs. The output consists of a system of product and process controls which can provide stability to the operating process.[5] Figure 3 The input-output diagram for the quality control process[5] Statistical Quality Control (SQC) focuses on sampling for acceptance, statistical process control (SPC), design of experiment (DOE), and process capability analysis. SPC largly consists of applying control charts for controlling processes by detecting any non-random shifts and variations. Control charts were developed in 1931 by Shewhart to be utilized for process monitoring. He gave the control chart the following definition: The control chart may serve, first, to define the goal or standard for a process that management strives to attain; second, it may be used as an instrument for attaining that goal and third, it may serve as a means of judging whether the goal has been reached [6]. 5

20 The original control charts are generally classified as univariate charts that can only be utilized to monitor a single variable characteristic of a process. Advancements in technology and increased customer expectations have raised the need to monitor correlated quality characteristics simultaneously. The original work in multivariate quality control can be attributed to Hotelling [7]. His work led to a number of multivariate techniques presented in literature review. There are many situations in which the simultaneous monitoring or control of two or more related quality characteristics is necessary. Monitoring these quality characteristics independently can be very misleading [8]. It is necessary to use multivariate statistical process control. Furthermore, with the greatly increased availability of high-speed computers and multivariate software, many users can now apply multivariate techniques [9]. In this domain, the problem becomes more complex if there are some correlated continuous and discrete variables, which can be considered as Multi-Variate-Attribute Quality Control (MVAQC). When the number of quality characteristics, in the form of variables or attributes, exceeds unity and there exists a non-zero correlation between (some of) them, then one is dealing with either a multi-variable or multi-attribute quality control problem. In these problems, if the correlated quality characteristics are monitored separately, there will be some error (e.g. type I and type II errors) which is different to that expected if considering only one characteristic as an independent characteristic. In multivariate quality control, several dependent variable characteristics are measured and monitored simultaneously. Similarly, in multi-attribute quality control, more than one dependent attribute characteristic is considered simultaneously. There are several works published in four different aspects of quality control including multivariate and multi-attribute control but none can be found in multi-variate-attribute as a combination of multivariate and multi-attribute cases. As an example of where both variable and attribute characteristics may jointly determine the quality of a process, consider a metal forming process with punching and bending operations. Here, the weight 6

21 and thickness of the metal sheet may be variable characteristics and the shape of the bent sheet and crumpling of the punched hole may be attribute characteristics. The first of the attribute characteristics may be checked with gauge and the latter checked visually. In situation similar to the above, if there are some interacting quality characteristics including variables and attributes, quality engineers use separate multivariable quality control procedures for monitoring the variable quality characteristics and multi-attribute quality control methods for monitoring the attribute quality characteristics. But, in practice, by considering the natural difference between variable and attribute quality characteristics such as skewness, required sample size, applying a general method designed for multi-variable- attribute cases is required. Regarding this need, the objectives of this research can be considered as: a) Design and develop control charts for monitoring process that includes multi-variateattribute (MVA) characteristics. b) Compare the performance of the developed method with the current applicable ones from the point of view of type I and type II errors. In this thesis, chapter 2 discusses the existing literature concerning the past developments involving univariate, uni-attribute, multivariate and multi-attribute situations. It compares these different approaches and develops a check list for the desired features of a multi-variate-attribute method. In chapter 3, the research model and hypothesis are explained. This chapter includes the research objectives, research questions and research procedure. In order to carry out simulation studies on the performance of the proposed multi-variate-attribute quality control method, it is necessary to apply a data generation method for generating the multi-variate-attribute random vector. Section 4.1 of chapter 4 discusses the existing literature concerning past developments involving the multivariate and multi-attribute random data generating methods and compares previous researches in this area. A new method is proposed in this section for generating the multi-variateattribute random vectors and is compared with the relevant existing methods. In section 4.2, the proposed multi-variate-attribute quality control method is explained, in detail, 7

22 including the initial transformations of the quality characteristics for reducing the correlation and skewness of the original data, developing the control charts for each of the transformed quality characteristics and also explains a proposed method for diagnosing the out-of-control state. In section 4.3, two simulation studies are presented, each compromising 8 variants including two approaches to transforming the original data and four subgroup sizes in order to analyse the performance of the proposed model and compare with the relevant existing model. Chapter 5 includes a conclusion of the research and the findings as well as opportunities that can be considered as future work for other researchers. As an outcome of this research study, two papers have been already published / under review in the international journals and are available in appendix 7. 8

23 Chapter 2 Literature Review 9

24 2. Literature Review 2.1. Univariate / Uni-attribute Domain The Chapter begins with a brief review of conventional sigle characteristic, Shewhart - type control charts, and a passing reference to some other approaches. Following is more extensive survey of existing multi-characteristic process contril methods. The chapter concludes with a checklist for evaluating any practical methods for monitoring processes that have a nature of variable and attribute quality characteristics. Statistical quality control, in general, consists of the variable and attribute domains in which each has a different concept and application. In the variable domain, a measurable characteristics of product or processes are measured and controlled by using variable control charts. In the attribute domain, the number of or percentage of defects in products or defective products are calculated and controlled by using attribute control charts. A control chart (Figure 4), is a graphical display of the variation of a quality characteristic over time. The chart contains a centre line (CL) that represents the desired average value of the quality characteristic. Two other horizontal lines, called the upper control limit (UCL) and the lower control limit (LCL) are shown on the chart. These control limits are chosen so that if the process is in control, nearly all of sample points will fall between them. As long as the points plot within the control limits, the process is assumed to be in control[8]. Figure 4 A typical control chart [8] 10

25 Univariate Quality Control Many of quality characteristics can be measured and stated as numerical values. For instance, as a product quality characteristic, internal diameter of a bearing can be measured by using a micrometer and stated on the basis of millimetre. As another example, temperature of an injection machine is measured by a thermometer as a process quality characteristic. These types of quality characteristics are called variable characteristics. For controlling these, average and variation (standard deviation) are considered and if these two factors are in control, the process is considered in control. A general model for control charts was proposed by Shewhart [6] and control charts developed according to these principles for controlling average of measurements are often called Shewhart control charts. Let w be a sample statistic that measures some quality characteristic of interest, and suppose that thedesired mean of w is μ w and the accepted standard deviation of w is σ w. Then the centre line, the upper control limit, and the lower control limit become UCL = µ + Lσ CL = µ w w LCL = µ Lσ w w w where L is the distance of a control limit from the center line, expressed in standard deviation units. If it is assumed that the quality characteristic follows a normal distribution, L could be considered as Z, where, is the probability of type I error α 2 (false alarm rate) and Z is the standard normal value of α 2. It means that, if the α 2 process is in control, it is expected that 100(1 α)%, the confidence interval, of the values w will fall between LCL and UCL. For example, if the 99.7% confidence interval (CI) is supposed, the constant L = 3. This value is 2 and 1 if CI is supposed to be 95.4% and 68.2% respectively. The constant L is arbitrarily chosen to be 3 in following formulas but it does not restrict the user having different limits based on different confidence intervals. By considering the probability of type I error in using XX chart, the above result could be applied (by considering the error) for other control charts even if the underlying distribution is non-normal [8]. 11 (1) (2) (3) α

26 X and R / X and s Charts: The most common univariate control charts are X, R, S. In these charts, X chart is applied for control of the process average and R and s charts are utilized for control of variability. In the ideal case, it is supposed that a quality characteristic is normally distributed with mean µ and standard deviation σ, where both are known. But in real cases, they are usually unknown. So, X is applied as an estimator for µ and R or s is utilized for estimating σ. Although X and R charts are widely used, generally, X and s charts are preferable to their more familiar counterparts, X and R charts, when either the sample size is moderately large (more than 10) or the sample size is variable [8]. For X and R charts, the control limits for the X chart are: UCL = µ CL = µ X LCL = µ X X + 3σ 3σ X And control limits for the R chart are: UCL = µ + 3σ CL = µ R R LCL = µ 3σ R R R X (4) (5) (6) (7) (8) (9) For X and s chart, the control limits for the X chart are the same as eq. 4, 5 and 6 and for s chart are: UCL = µ + 3σ CL = µ s LCL = µ 3σ s s s s (11) (10) (12) (12) In addition to above control charts, there are some other univariate control charts for various and specific application such as IMR (Individual Moving Range) chart in case of having a sample size one and auto-correlated (dependent) data [10], or Cumulative Sum Control (CUSUM) [11] and Exponentially Weighted Moving Average (EWMA) chart 12

27 [12] for detecting any minor shifts in process in the case of auto-correlated data. EWMA appears more robust with a better ARL than other individual control charts for the nonnormal case [244]. Further different applications of univariate control charts are proposed by Marcellus [13], Shore [10], Weheba [14] and Montgomery [8] Uni-attribute Quality Control In the uni-attribute domain, a measure of a quality characteristic is not used. Products are categorized as conforming and nonconforming products or containing conformities and nonconformities in terms of any attribute quality characteristics such as appearance, shape and performance. In attribute control charts, after determining the conforming and nonconforming products, the number or fraction of nonconforming products or number of nonconformities in an inspection unit of product are counted to develop a related control chart on the basis of its application. Some of these control charts are discussed below p chart and np chart The statistical principles underlying the control chart for p chart, fraction nonconforming, are based on the Binomial distribution. It is supposed that the production process is operating in a stable manner and all successive units produced are independent [8]. On the p chart, the control limits and centrline are: p(1 p) (13) UCL = µ p + 3σ p = p + 3 n (14) CL = µ p = p p(1 p) (15) LCL = µ p 3σ p = p 3 n where P represents the observed average fraction of nonconfoming and n is the sample size. 13

28 It is possible to base a control chart on the number nonconforming rather than the fraction nonconforming. This is often called an np chart.[15], [8]. On the np chart, control limits are: UCL = µ np + 3σ np = np + 3 np(1 p) (16) CL = µ np = np (17) LCL = µ np 3σ np = np 3 np(1 p) (18) c chart and u chart Consider the occurrence of nonconformities in an inspection unit of product. In most cases, the inspection unit will be a single unit of product, although this is not necessarily always so. In c chart, the defects or nonconformities occurring in the inspection unit are assumed to follow the Poisson distribution [8]. On the c chart, control limits and centreline are: UCL = µ c + 3σ c CL = µ = c LCL = µ 3σ = c 3 where c represents the observed average fraction of nonconformities. Another approach is the u chart, which involves setting up a control chart based on the average number of nonconformities per inspection unit. [15], [8]. On the u chart, control limits and centre line are: where c c CL = µ = u represents the observed average number of nonconformities per unit in a preliminary set of data c = c + 3 UCL = µ + 3σ = u + 3 u u LCL = µ 3σ = u 3 u u u u c c u n u n 14 (19) (20) (21) (22) (23) (24)

29 Formulas 13 to 24 are generated based on Shewhart control chart and normal distribution. So, for a precise estimation, certain conditions are to be considered (e.g. np >10 for Binomial and λ 15 for Poisson distributions, where λ is the parameter of the distribution. In addition to the four above attribute control charts that are the most common ones, there are some other ones for some specific applications [16]. Marcellus [13] proposed a Probability Control Chart that applies the real distribution (Binomial or Poisson) rather than the estimation with normal distribution.the Cumulative Count Conforming - CCC chart is applied for some processes with high quality and low nonconforming rate by using the Geometrical distribution concept, and the Cumulative Quality Control - CQC chart is applied for some processes with a high quality output by using the Exponential distribution concept [15]. Cumulative Sum Control (CUSUM) and Exponentially Weighted Moving Average (EWMA) charts have been developed for both fraction nonconforming and number of nonconformities [17], [18]. In addition, there are many papers have been proposed improving and developing the application of uni-attribute control charts. For instance, in a paper published by Woodall [19] as a literature review, more than 200 different papers were reviewed. In general, many papers are focused on improving ARL 0 (Average Run Length- average number of points that must be plotted before a point indicate an out-of-control condition) and bringing it to its nominal value based on a predefined value of probability of type I error, α X α. For instance, in a Shewhart control chart using three sigma limits, is and the nominal ARL is 1/0.0027=370.4 [20-24]. Hamada [25] applied a Bayesian Tolerance interval for designing control limits in p, np, c, and u charts. Jackson [26] discussed dealing with a mixture of various types of nonconformities that can lead to situations in which the total number of nonconformities is not adequately modeled by the Poisson distribution. The use of negative binomial distribution to model count data in inspection units of varying size has been studied by Sheaffer and Leavenworth [27]. The dissertation by Gardiner [28] describes the use of various discrete distributions to model the occurrence of defects in integrated circuits. Kaminiski and 15

30 Benneyan [29] have proposed control charts for counts based on geometric distribution for count or event data. They refer to the control chart for the total number of events as a g chart and the control chart for the average number of events as an h chart. In using demerit systems for attribute data, Jones and Woodall [30] provide a comprehensive discussion of demerit-based control charts [8] Multivariate / Multi-attribute Domain When the number of quality characteristics, in the form of variables or attributes, exceeds unity and there exists a non-zero correlation between them, then one is dealing with either a multi-variable or multi-attribute quality control problem. In these problems, if the correlated quality characteristics are monitored separately as independent characteristics, there will be some error (e.g. type I and type II errors) which is different to that expected if considering only one characteristic Multivariate Quality Control Introduction Contrary to the univariate case where the control area is figured as an interval cases, in multivariate, if the observations are assumed to follow a multivariate normal distribution then the ellipse (in two dimensions) and ellipsoids (in higher dimensions) define the control regions (areas or volumes). In a general literature review of multivariate quality control methods done by Montgomery [8], the original work in multivariate quality control is attributed to Hotelling who applied his procedures to bombsight data during World War II [7]. Subsequent papers dealing with control procedures for several related variables include [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]. Some extensive overview papers for multivariate quality control methods have been developed by Lowry and Montgomery [39], Alt [43], Jackson [34], Bersimis et al. [44], MacGregor and Kourti [45] and Wierda [46]. Montgomery illustrates multivariate quality control with 16

31 the following example [8]. Suppose that a bearing has both an inner diameter (x 1 ) and an outer diameter (x 2 ) that together determine the usefulness of the part. Suppose that they have independent normal distributions. Because both quality characteristics are measurements, they could be monitored by applying the usual XX chart to each characteristic, as illustrated in Fig 5. The process is considered to be in control only if the sample means XX 1 and XX 2 fall within their respective control limits. This is equivalent to the pair of means (XX 1, XX 2 ) plotting within the shaded region in figure 6. [8] Figure 5 Control charts for inner (XX 1 ) and outer (XX 2 ) bearing diameters [8] 17

32 Figure 6 Control region using independent control limits for XX 1 and XX 2 [8] Monitoring these two quality characteristics independently can be misleading. For example, note from Fig 6 that one observation appears somewhat unusual with respect to the others. That point would be inside the control limits on both of the univariate XX chart for x 1 and x 2, yet when examining the two variables simultaneously, the unusual behavior of the point is fairly obvious. Furthermore, note that the probability that either XX 1 or XX 2 exceeds three-sigma control limits is However, the joint probability that both variables exceed their control limit simultaneously when they are both in control is (0.0027)*(0.0027)= , which is considerably smaller than Furthermore, the probability that both x1 and x 2 will simultaneously plot inside the control limits when the process is really in control is (0.9973)*(0.9973)= Therefore, the use of two independent x charts has distorted the simultaneous monitoring of XX 1 and XX 2, in that the type I error and probability of a point correctly plotting in control are not equal to their advertised levels for the individual control charts [8]. This distortion in the process-monitoring procedure increases as the number of quality characteristics increases. In general, if there are p statistically independent quality characteristics for a particular product (the symbol p in this section is different from 18

33 fraction nonconforming in p charts) and if an XX chart with the probability of type I error α is maintained on each, then the true probability of type I error for joint control procedure is α = 1 (1 α) and the probability that all p means will simultaneously plot inside their control limits when the process is in control is P (all p means plot in control) = p ( 1 α) p Clearly, the distortion in the joint control procedure can be severe, even for moderate values of p. Furthermore, if quality characteristics are not independent, which usually would be the case if they relate to the same product, then equation 25 and 26 do not hold, and there is no easy way to measure the distortion in the joint control procedure. The difficulty with using independent univariate control charts can be illustrated by reference to figure 7 for a two-variable case, where the same observations are plotted in the ellipse as well as the Shewhart control limits for each variable. Note that by inspection of each of the individual Shewhart control charts, the process appears to be clearly in a state of statistical control, and none of the individual observations gives any indication of a problem. The true situation is only revealed in the multivariate plot where it is seen that the observation marked as is clearly outside the ellipse. There are some different proposed methods in multivariate domain. But the concept of multivariate normal distribution as a base for most of current methods is explained below, before a more detailed description of those methods. (25) (26) 19

34 Figure 7 individual control charts vs. ellipse control area [45] Multivariate Normal Distribution In univariate statistical quality control, the normal distribution generally is used to describe the behavior of continuous quality characteristics. The univariate normal probability density function is f ( x) = 1 2 2πσ e x µ σ < x < (27) µ 2 The mean of the normal distribution is and the variance is σ. Note that (apart from the minus sign and the constant coefficient) the term in the exponent of the normal distribution can be written as: 2 1 ( x µ )( σ ) ( x µ ) (28) 20

35 This quantity measures the squared standardized distance from x to the mean µ, where by the term standardized it means that the distance is expressed in standard deviation units. The same approach can be used in the multivariate normal distribution case. Suppose that these are p variables, x 1, x 2,,x p. These variables are arranged in a p-component vector X = x, x,..., x ]. Let µ = µ, µ,..., µ ] be the vector of the means of x, and let the [ 1 2 p [ 1 2 p variance and covariance value of the random variables in X be contained in a p p covariance matrix. The main diagonal elements of are the variances of x s and the off-diagonal elements are covariance values. Now the squared standardized (generalized) distance from X to µ is 1 ( X µ ) ( ) ( X µ ) The multivariate normal density function is (29) f ( x) = where 1 p ( 2π ) 2 < x < 1 2 e ( X µ ) ( X µ ), j=1,2, p. A multivariate normal distribution for p=2 variables (called the bivariate normal) is shown in Fig. 8. It may be noted that density function is a surface [8]. (30) Figure 8 A bivariate normal distribution [8] 21

36 Multivariate Control Charts In a multivariate environment, the process is monitored using different types of variable control charts. In the following sections, all existing variable control charts for multivariate cases will be discussed Control Ellipse As noted above. The control area for the bivariate normal case can be reviewed graphically as an ellipse. The ellipse boundaries have been established by the following procedure. Suppose that two quality characteristics x 1 and x 2 are jointly distributed according to the bivariate normal distribution. Let and µ be the mean vectors of the µ 1 2 quality characteristics, and and σ be the standard deviation of x 1 and x 2 respectively. σ1 2 The covariance between x 1 and x 2 is denoted by. It is assumed that, and σ ( σ12 σ1 σ 2 12 ) are known. If and x are the sample average of the two quality characteristics = σ 21 x1 2 computed from a sample of size n, then the below statistics will have a chi-square distribution with 2 degrees of freedom [8]. χ n [ σ x µ ) + σ ( x µ ) 2σ ( x µ )( )] 2 0 = 2 ( x2 µ 2 σ 1σ 2 σ 12 Equation 31 can be used as the basis of a control chart for the process means and µ. If the process means remain at the values and µ, then values of should be less 2 2 χ α,2 α, 2 χ α µ 1 2 than, where is the upper percentage point of the chi-square distribution 2 χ 0 χ 2 α, 2 with 2 degrees of freedom. If at least one of the means shifts to some new (out-ofcontrol) value, then the probability that the statistic exceeds increases. The process-monitoring procedure may be represented graphically. Consider the case in which the two random variables x 1 and x 2 are independent; that is, σ 12 = 0. If σ = 12 0, then equation (31) defines an ellipse centered at (, µ ) with principal axes parallel to µ 1 2 the and x axes, as shown in figure 9. Taking in equation (31) equal to x1 2 µ 1 2 implies that a pair of sample averages (, x ) yielding a value of that plots inside x1 2 2 χ 0 2 χ 0 χ 2 α, 2 2 χ 0 (31) 22

37 the ellipse indicates that the process is in control, whereas if the corresponding value of 2 χ 0 plots outside the ellipse the process is considered out of control. In the case where the two quality characteristics are dependent, then σ 12 0, and the corresponding control ellipse is shown in Fig. 10. When the two variables are dependent, the principal axes of the ellipse are no longer parallel to the, x axes [8]. x1 2 Figure 9 A control ellipse for two independent variables [8] 23

38 Figure 10 A control ellipse for two dependent variables [8] A couple of disadvantages associated with the control ellipse have been identified in the literature. The first is that the time sequence of the plotted points is lost but this could be overcome by numbering the plotted points or by using special plotting symbols to represent the most recent observations. The second and more serious disadvantage is that it is difficult to construct the ellipse for more than two quality characteristics [8]. The third is the limitation on applying the method in the case that variables are jointly distributed according to the bivariate normal distribution. On the other hand, simplicity and graphical output are the main advantages of this method Chi-square Control Chart To avoid the disadvantages of the control ellipse referred to in last section, it is customary to plot the values of 2 χ 0 computed from equation 31 for each sample on a control chart with only an upper limit at UCL = χ 2 α,2, as shown in figure 11. It may be noted that the time sequence of the data is preserved by this control chart, so that runs or other 24

39 nonrandom patterns can be investigated. Furthermore, it has an advantage that the state of the process is characterized by a single number (the value of the statistic 2 χ 0 particularly helpful when there are two or more quality characteristics of interest. ). This is Figure 11 A chi-square control chart for two quality characteristics [8] It is possible to extend these results to the case where p (greater than 2) related quality characteristics are controlled jointly. It is assumed that the jointed probability distribution of the p quality characteristics is the p-variate normal distribution. The procedure requires computing the sample mean for each of the p quality characteristics from samples of size n. This set of quality characteristic means is represented by the p 1 vector X. The test statistic plotted on the chi-square control chart for each sample is 2 1 χ0 = n( X µ ) ( ) ( X µ ) (32) where µ = µ, µ,..., µ ] [ 1 2 p is the vector of in-control means for each quality characteristic UCL = χ 2 α, and is the covariance matrix. The upper limit on the control chart is p [8]. Monthgomery and others identify some disadvantages associated with this method. The first is that the ability to detect a shift in the mean vector only depends on the magnitude of the shift, and not on the direction it is a directionally invariant control chart)[8]. In the 25

40 other words, Unlike other control charts with bilateral control limits which shows out of control cases beyond the upper control limit or below the lower control limit, the proposed chart only has an upper control limit that makes it unable to be used for detection of both process deterioration and process improvement[47]. The second is that it is not able to identify the subset of process variables that are responsible for the out-ofcontrol state [8],[48]. In the other words, in case of having an out-of-control observation in the chi-square chart, the model is not able to trace it back to the original quality characteristics and determin which one(s) are associated with the out-of-control observation Hotelling T 2 Control Chart In applying the chi-square control chart, in some cases, µ and matrixes are unknown and they need to be estimated from an analysis of preliminary samples of size n, taken when the process is assumed to be in control. Suppose that m such samples are available. The sample means and variances are calculated from each sample as usual; that is, n j=1,2,,p 1 x jk = x ijk k=1,2,,m (33) n i= 1 n S 2 jk = 1 n 1 (x ijk x jk) 2 i=1 j=1,2,,p k=1,2,,m (34) where x ijk is the ith observation on the jth quality characteristic in the kth sample. The covariance between quality characteristic j and quality characteristic h in the kth sample is s jhk n = 1 ( x n 1 i= 1 ijk x jk )( x ihk x hk ) The statistics x, s and s are taken averaged over all m samples to obtain jk jk jhk k=1,2,,m (35) j h 26

41 x s and s = 1 m j x jk m k= 1 = m 2 1 j = 1 m k= 1 m s 2 jk jh s jhk m k= 1 j=1,2,,p j=1,2,,p j h (36) (37) (38) The ( x ) are the elements of the vector X, and the p p average of sample covariance j matrices S is formed by s S = 2 1 s s The matrix of average sample covariance, S, is an unbiased estimate of process is in control. s s s 1p 2 p 2 p (39) when the Now suppose that S from equation 39 is used to estimate and the vector X is taken as the in-control value of the mean vector of the process. If µ is replaced with X and with S in equation (32), the test statistic now becomes T 2 = n( X X ) ( S) 1 ( X X ) (40) In this form, the procedure is usually called the Hotelling T 2 control chart. This is a directionally invariant control chart [8]. Alt [49] has pointed out that in multivariate quality control applications one must be careful to select the control limits for Hotelling s T 2 statistics (equation 40) based on how the chart is being used. It has been noted that there are two distinct phases of control chart usage. Phase I is the use of the charts for establishing control; that is, testing whether the process was in control when the m preliminary subgroups were drawn and the sample statistics X and S computed. The objective in phase I is to obtain an in-control set of observations so that control limit can be established for phase II, which is the monitoring of future production. Phase I analysis is sometimes called a retrospective analysis. 27

42 The phase I control limits for the T 2 control chart are given by UCL p( m 1)( n 1) F mn m p + 1 = α, p, mn m p+ 1 LCL = 0 In phase II, when the chart is used for monitoring future production, the control limits are as follows: UCL LCL = 0 p( m + 1)( n 1) F mn m p + 1 = α, p, mn m p+ 1 When µ and are estimated from a large number of preliminary samples, it is (41) (42) customary to use UCL = χ 2 α, p as the upper control limit in both phase I and phase II. Retrospective analysis of preliminary samples to test for statistical control and establish control limits also occurs in the univariate control chart setting. For the XX chart, it is typically assumed that if m 20 or 25 preliminary samples are used, the distinction between phase I and II limits is usually unnecessary, because the phase I and II limits nearly coincide[8]. Jensen et al. [50] point out that even larger sample sizes are required to ensure that the phase II average run length (ARL) performance will actually be close to the anticipated values. They recommend using as many phase I samples as possible to estimate the phase II limits. Lowry and Montgomery [39] show that in many situations a large number of preliminary samples would be required before exact phase II control limits are well approximated by the chi-square limits. The recommended values of m are always greater than 20 preliminary samples, and often more than 50 samples. Jensen et al. [50] observe that these recommended sample sizes are probably too small. Sample sizes of at least 200 are desirable when estimating the phase II limits. In some industrial setting the subgroup size (n) is naturally one. This situation occurs frequently in the chemical and process industries. Since these industries frequently have multiple quality characteristics that must be monitored, multivariate control charts would be of interest there. In this situation, the Hotelling T 2 statistic in equation 40 becomes 28 (43)

43 T 2 = ( X X ) ( S) 1 ( X X ) The phase II control limits for this statistic are p( m + 1)( m 1) UCL F 2 m mp LCL = 0 When the number of preliminary samples m is large, say m>100, many practitioners use an approximate control limit, either UCL or = α, p, m p p( m 1) m p = Fα, p, m p UCL = χ 2 α, p The chi-square limit in equation (46) is only appropriate if the covariance matrix is known, but it is widely used as an approximation. Lowry and Montgomery show that the chi-square limit should be used with caution. If p is large ( p 10 ), then at least 250 samples must be taken ( m 250 ) before equation (46) is a reasonable approximation to the correct value [39]. Tracy et al. point out that if n = 1, the phase I limits should be based on a beta distribution. This would lead to phase I limits defined as UCL ( m 1) m 2 = β α, p, ( m p 1) 2 2 LCL = 0 β where α, p, ( m p 1) is the upper α 2 2 percentage point of a beta distribution with parameters p/2 and (m-p-1)/2. Approximations to the phase I limit based on the F and chi-square distributions are likely to be inaccurate [41]. In addition to the above-mentioned papers on this subject, subsequent papers deal with Hotelling T 2 control chart. Mason and Young [51] gave as extensive presentation of the application of the T 2 Hotelling control chart. Champ et al. [52] reviewed properties of the T 2 Hotelling control chart when parameters are estimated and the effects of sample size on its performance. They also proposed a corrected control limits when large sample sizes are not available. [52]. Mason and Young [53] introduced a modification procedure 29 (44) (45) (46) (47)

44 for the T 2 control charts in order to enhance sensitivity toward detecting a small process shift. Mason and Chou [54] studied the effectiveness of using the T 2 control charts for batch (sub-grouped) processes. Mason et al. [55] presented a multivariate profile chart by superimposing an XX chart of univariate statistics on top of the T 2 chart. Chou et al. [56] compare the power of Hotelling T 2 statistic in using different estimators of covariance matrix for individual observation case. Chou et al. [57] used a kernel smoothing technique to estimate the distribution of the T 2 statistic as well as of the UCL of the T 2 chart for individual observation. Sullivan and Woodall [58] recommended a method for retrospective stage 1, which is almost as effective as the benchmark MCUSUM and MEWMA control charts in detecting a step shift. Nedumaran and Pignatiello [59] considered the issue of constructing retrospective T 2 control chart limits to control the overall probability of a false alarm at a specified value. Furthermore, Mason et al. [54] used the T 2 control chart for monitoring batch processes in both phase I and II operations. A method to determine the optimal sample size, interval between samples and critical region parameter for the Hotelling T 2 control chart was proposed by Montgomery and Klatt [60]. Given that the Hotelling T 2 control chart is formed on the same basic concepts as the Chisqare control chart, the Hotelling T 2 method has similar advantages and disadvantages as these noted previously for the Chi-square method Multivariate Exponentially Weighted Moving Average (MEWMA) Control Chart The chi-square (ellipse control) and T 2 charts described in the previous sections are essentially Shewhart-type control charts. That is, they use information only from the current sample; so consequently, they are relatively insensitive to small and moderate shifts in the mean vector. As it is noted, the T 2 chart can be used in both phase I and II situations. Cumulative sum (CUSUM) and exponentially weighted moving average (EWMA) control charts were developed to provide more sensitivity to small shifts in the 30

45 univariate case, and they can be extended to multivariate quality control problems. As in the univariate case, the multivariate version of these charts is a phase II procedure. The EWMA control chart was introduced by Robert [12].See also [61], [62] and [63] for further discussion of EWMA. The exponentially weighted moving average is defined as z i = λxi + ( 1 λ) zi 1 Where 0 λ 1 is a constant and the starting value (required with the first sample at i = 1 ) is the process target, so that z =. Sometimes the average of preliminary data is used as the starting value of the EWMA, so that [8]. Lowry et al. [38] have developed a Multivariate EWMA (MEWMA) control chart. The MEWMA is considered a logical extension of the univariate EWMA and is defined as: Z i = λx i + ( 1 λ) Zi 1 where 0 λ 1 and Z 0 = 0. The quantity plotted on the control chart is T 2 1 i = Zi Z i Z where the covariance matrix is λ = [ 1 (1 λ) ] 2 λ Z i 2i i 0 µ 0 z = x 0 which is analogous to the variance of univariate the EWMA. Prabhu and Rubger [65] have provided a comprehensive analysis of the average run length performance of MEWMA control chart, using a modification of Brook and Evan s [64] Markov chain approach. They give tables and charts to guide selection of the upper control limit UCL=H for the MEWMA. It is assumed that the process is in-control where (49) (50) (51) (48) 2 T i H δ =. The shift size is reported in terms of a quantity ( µ µ) (52) δ is usually called the non-centrality parameter. Large values of shifts in the mean. The value δ = 0 is the in-control state. δ correspond to bigger 31

46 Since the MEWMA with λ =1 is equivalent to the T 2 (or Chi-square) control chart, the MEWMA is more sensitive to smaller shifts. This is analogous to the univariate case[8]. Alt and Smith [38] proposed three control charts for monitoring the covariance matrix, which is analogous to EWMA for the variance [66]. Lowry et al. also proposed a formula for calculating element (k,l) of the covariance matrix. MEWMA is generally used for individual observations; however, it can be used for controlling means rather than individual observations [38],[67]. Scranton et al. [68] show how the ARL Performance of the MEWMA control chart can be further improved by applying it to only the important principal components of the monitored variables [68]. Reynolds et al. develop MEWMA procedures for simultaneous monitoring of the mean vector and Covariance matrix [69]. An economic model for MEWMA is discussed by Linderman et al. [70, 71] and Molnau et al. [72]. MEWMA control charts like their univariate counterparts are robust to the assumption of normality. Some authors report that the small values of the parameter λ result in a MEWMA that is very insensitive to the form of the underlying multivariate distribution of the process data [73], [67]. Small values of λ also provide very good performance in detecting small shifts, and they would seem to be a good general choice for the MEWMA. A discussion of design strategies for the MEWMA control chart is proposed by Testik and Borror [74], Stoumbos and Sullivan [245] and Testik et al. [246]. Hawkins et al. have recently proposed a modification of the MEWMA control chart in which the use of a single smoothing constant λ is generalized to a smoothing matrix which has non-zero diagonal elements [75], [8]. Rigdon[76],[77] gave an integral and a double-integral equation for the calculation of incontrol and out-of-control ARLs, respectively. Moreover, Bodden and Rigdon[78] have developed a computer program for approximating the in-control ARL of the MEWMA chart. Molnau et al. [79] presented a program that enables the calculation of the ARL for the MEWMA when the values of the shift in the mean vector, the control limit and the smoothing parameter are known. Kramer and Schmid[80] proposed a generalization of the MEWMA control scheme of Lowry et al. [38] for multivariate time-dependent observations. Sullivan and Woodall[58] 32 Zi

47 recommended the use of a MEWMA for the preliminary analysis of multivariate observations. Fasso[81] has developed a one-sided MEWMA control chart based on the restricted maximum likelihood estimator. Yumin [82] proposed the construction of a MEWMA using the principal components of the original variables. Choi et al.[75] proposed a general MEWMA chart in which the smoothing matrix is full instead of one having only a diagonal. The performance of this chart appears to be better than that of the MEWMA proposed by Lowry et al. [38]. Choi and colleagues have also provided a computer program for the estimation of control limits (Hawkins et al.[83]). Yeh et al.[84] introduced a MEWMA which is designed to detect small changes in the variability of correlated multivariate quality characteristics, while Chen et al.[85] proposed a MEWMA control chart that is capable of monitoring simultaneously the process mean vector and process covariance matrix. Runger et al.[86] showed how the shift detection capability of the MEWMA can be significantly improved by transforming the original process variables to a lowerdimensional subspace through the use of the U-transformation. The U-transformation is similar to principal components transformation. Tseng et al.[87] proposed a MEWMA controller under a linear multiple-input multiple-output model, while Castillo and Rajagopal[88] gave a multiple-input multiple-output extension to the univariate double EWMA, which was first used by Butler and Stefani[89]. In general, there are several different approaches to the design of MEWMA control charts: (i) statistical design; (ii) economic-statistical design; and (iii) robust design. A review and a comparison of these design strategies is provided by Testik and Borror[90]. Yeh et al.[91] gave a likelihood-ratio-based EWMA control chart that effectively monitors small changes of variability of multivariate normal processes. Margavio and Conerly[92] have developed two alternatives to the MEWMA chart. The first of these is an arithmetic multivariate moving average; the second is a truncated version of the MEWMA. Sullivan and Jones[93] proposed a self-starting control chart for individual observations. The use of this chart is advantageous when production is slow. Reynolds and Kim[94] proposed MEWMA charts based on sequential sampling in which the total 33

48 sample size taken at a sampling point depends on current and past data, while Kim and Reynolds[95] discussed the use of the MEWMA control chart for monitoring the process mean when sample sizes are unequal[44]. In addition to the above-mentioned papers, MEWMA has been reviewed as one of the multivariate quality control methods in [39], [44], [45], and [46] as well Multivariate Cumulative Sum (MCUSUM) Control Chart Similar to exponentially weighted moving average (EWMA) control charts, Cumulative sum (CUSUM) was developed to provide more sensitivity to small shifts in the univariate case. Cumulative control charts were first proposed by Page [11] and have been studied by many authors including some other authors [96], [97], [98], [99], [100], [101], [102], [103]. The CUSUM chart directly incorporates all the information in the sequence of sample values by plotting the cumulative sums of deviations of the sample values from the target value. For example, suppose that samples of size n 1 are collected, and x is the j average of the jth sample. Then if µ 0 is the target for the process mean, the cumulative sum control chart is formed by plotting the quantity i C i = ( x j µ 0 ) j= 1 (53) against the sample number i. C i is called the cumulative sum up to and including the ith sample. Because they combine information from several samples, cumulative sum charts are more effective than Shewhart chart for detecting small process shift[8]. The multivariate CUSUM (MCUSUM) control chart is presented as a relatively straightforward extension of the univariate CUSUM [8]. Crosier proposed two MCUSUM procedures [35]. The one with the best ARL performance (faster detection of small shifts in the mean vector) is based on the statistic where {( 1 } 1/2 1 ) ( 1 ) C i = i + i Σ i + i S X S X (54) 34

49 0, if Ci k Si = (55) ( Si 1 + Xi)(1 k/ Ci), if Ci > k with S 0 = 0, and k > 0. An out of control signal is generated when Y = ( S Σ S ) > H 1 1/2 i i i (56) where k and H are the reference value and decision interval for the procedure, respectively. Two different forms of the MCUSUM were proposed by Pignatiello [40]. Their bestperforming control chart is based on the following vectors of cumulative sums: D i i = j= i li + 1 X j (57) and MC = D Σ D kl 1 1/2 i max{0, ( i i) i} (58) where k > 0, l = l + 1 if MC > 0 and l = 1 otherwise. An out of control signal is i i 1 i 1 i generated if MC i > H. Both of these MCUSUM procedures are claimed to have better ARL performance than the Hotelling T 2 or the chi-square control charts. [8]. Smith[104] developed a MCUSUM procedure based on the likelihood ratio test, which is used to study shifts in the mean vector of a multivariate normal process. The procedure is adapted to study shifts in the covariance matrix of a multivariate normal process and to study shifts in the probabilities of a multinomial process. Healy [105] proposed his MCUSUM concept for detecting shifts in mean. Generally, when a trend occurs in one direction of the target mean and a resulting shift occurs in the other direction, the MCUSUM chart will not detect the shift immediately. Lowry and Montgomery[39] proposed using a combination of the MCUSUM chart and the T 2 limits to improve the chart sensitivity to large shifts. 35

50 Ngai and Zhang[106] gave a natural multivariate extension of the two-sided cumulative sum chart for controlling the process mean. Also, Chan and Zhang[107] propose cumulative sum charts for controlling the covariance matrix. Runger and Testik[108] provided a comparison of the advantages and disadvantages of MCUSUM schemes, as well as performance evaluations and a description of their interrelationships. A derivation was also provided and extensive simulation results that include initial and steady-state conditions were presented. Geometric descriptions were used, and names were proposed based on these geometric characteristics. In addition to the above-mentioned papers, MCUCUM has been reviewed as one of the multivariate quality control methods in [34], [39], [44], [45], and [46] as well. The main disadvantage associated with this method is that it is not able to identify the subset of process variables that are responsible for the out-of-control state [8] Multivariate Shewhart XX chart By considering shortcomings of T 2 control chart in identifying the responsible variables leading to an out-of-control condition, Houshmand and Golnabi [109] proposed a method to simultaneously control the overall process quality characteristics and to identify the responsible variables leading to an out-of-control condition. They called this method, Multivariate Shewhart Chart (MS Chart)", because it reduces to the Shewhart Chart when the process involves only one variable. The same approach can be used in the multivariate normal distribution case. The procedure is stablished by the following steps: Suppose that there are p variables, x 1, x 2,,x p, arranged into a p-component vector X = x, x,..., x ]. Let µ = µ, µ,..., µ ] be [ 1 2 p [ 1 2 p the vector of the means of x s, and let the variance and covariance value of the random variables in X be contained in a p p covariance matrix. Given a sample of size n taken from a multivariate normal population, the data can be normalized by subtracting the mean of each characteristic and dividing by the corresponding standard deviation. Therefore it is assumed that the sample is taken from a 36

51 multivariate normal population with mean 0 and correlation matrix ρ x. To control the process mean, first set an overall type I error with individual α i s satisfying the condition 1 α = where, p i= 1 1 α α = 1 (1 α) i i 1 p (59) The procedure requires computing the sample mean for each of the p quality characteristics from a sample of size n. Then the original vector X = x, x,..., x ] is [ 1 2 p converted to vector Z = [ z, z2,..., z 1 p ] using below conversion function ZZ = ( nρ x Σ x ) 1 2(XX µ ) (60) If ZZ i < ZZ αi for all i's, the process will be in control. If on the other hand ZZ i > ZZ αi, the process will be out-of-control and the variable x i is responsible for this out-of-control condition. Houshmand and Golnabi compared their method with the T 2 control chart and showed that neither is more powerful than the other in detecting out-of-control situation. In the other words, some samples that were detected as out-of-control signal using Hotelling T 2 control chart, were not detected using their method and vice-versa. Furthermore, comparing their performance based on the in-control average run length (ARL 0 ) for the first numerical example given in their paper to show that their method with ARL 0 equal to 2.75 against Hotelling T 2 with ARL 0 equal 3.33 does not work more properly because of the large probability of having type-i error. Moreover, even after applying the z1 z2 conversion transformation equation 60, the and are still correlated with ρ z = 0.89, and applying separate Shewhart control charts may be misleading Principal Components Analysis (PCA) Conventional multivariate control-charting procedures have been found to be reasonably effective as long as p (the number of variables to be monitored) is not very large [8] The 37

52 use of traditional multivariate Shewhart charts, MCUSUM and MEWMA for highdimensional correlated variables may be impractical[44]. Other methods are sometimes more useful in situations where it is suspected that the variability in the process is not equally distributed among all p variables. That is, most of the motion of the process is in a relatively small subset of the original process variables. Methods for discovering the sub-dimensions in which the process moves about are sometimes called Latent Structure Methods [8] or Projection methods [44]. Principal Components Analysis (PCA) and Partial Least Squares (PLS) are two methods used for reducing the dimensionality of the variables space. The PCA approach for monitoring process variables is used when product quality data are not available in the historical data set. The PLS approach for monitoring process variables has been developed from historical data sets, with measurements from both the process and the quality variables obtained during in-control operation. PCA is usually more applicable in practice when the historical data of product quality characteristics are not available from an in-control operation and needs to have the phase I of the quality control procedure. Only the more common PCA method will be reviewed in this section. The principal components of a set of process variables x 1, x2,..., x p are just a particular set of linear combinations of these variables such as, z z 1 2 = c = c x 1 x 1 + c 12 + c 22 x 2 x c 1p c x 2 p p x p (61) z p = c p1 x 1 + c p2 x c pp x p where the c ij s are constants to be determined. Geometrically the principal component variables z z,..., 1, 2 z p are the axes of a new coordinate system obtained by rotating the axes of the original x system. The new axes represent the directions of maximum variability, which is in fact, the basic intent of principal components. It is expected that finding the new set of orthogonal directions that define the maximum variability in the 38

53 original data will lead to a description of the process requiring considerably fewer than the original p. The information in the complete set of all p principal components is exactly equivalent to the information in the complete set of all the original process variables, but hopefully it canbe used far fewer than p principal components to obtain a satisfactory description[8]. It turns out finding the c ij s that define the principal components is fairly easy. The procedure begins by considering random variables x x,..., 1, 2 x p to be represented by a vector X with covariance matrix and letting the eigenvalues of be λ1 λ... λp 2 0. Then constants c s are simply the elements of the ith eigenvector associated with the eigenvalues. Basically, if C be the matrix whose columns are eigenvectors, then C ΣC = Λ where Λ is a p p diagonal matrix with main diagonal elements equal to the eigenvalues ij λ i λ λ λ 1 2 p The variance of the ith principal component is the ith eigenvalues λ i. Consequently, the proportion of variability in the original data explained by the ith principal component is given by the ratio λi λ + λ λ p Therefore, one can see how much variability is explained by retaining just a few (say, r) of the principal components by computing the sum of eigenvalues for those r components and comparing that total to the sum of all p eigenvalues. Once the principal components have been calculated and a subset of them selected, new principal component observation zij 1 2 (62) can be obtained by submitting the original observations xij into the set of retained principal components. The z ij s are sometimes called the principal component scores. Thus, the value r must be decided. There are a number of criteria for choosing r including: (1) significant tests for equality of roots, (2) retaining enough principal components to explain a prescribed proportion of the total variance and (3) stopping 39

54 when the residual variances are equal to some prescribed amount. The last one isclaimed to be the most appropriate for quality control[34]. Runger and Alt[110] presented a method for choosing r specifically for process control problems. After applying PCA to reduce the dimension of the original space, the control eclipse (that is called a principal component trajectory plot) is used for monitoring the z ij. As long as all z ij remain inside the ellipse, there is no evidence that the process mean has shifted; otherwise, there is some evidence that the process is out-of-control. [8]. If r is more than two, then pair-wise scatter plots of the principal components would be used analogously in addition to trajectory plots. However, interpretation and use of the charts become cumbersome. As a potentially useful alternative to the trajectory plots, Scranton et al. [68] proposed applying MEWMA control chart to the r retained principal component score. Jackson[111] presented three types of principal components control charts: (1) a T 2 control chart obtained from principal components scores based on Hotelling T 2 statistic; (2) a control chart for principal components residuals where residual term exists because of the use of only the first r significant principal components instead of all p components; and (3) a control chart for each independent principal component s scores for sample subgroup of one (individual observations). The PCA method is also widely used in cases where data are auto-correlated. Ku et al.[112] extended the use of PCA models in process monitoring to account for autocorrelation. Likewise, Runger[113] proposed a model which allows autocorrelation and cross-correlation in the data. Mastrangelo et al.[114] explored the use of PCA in auto-correlated processes. The U 2 -chart (U transformation is similar to PCA transformation) was proposed by Runger[115]. Chen et al.[116] proposed a robust PCA approach via hybrid projections pursuit. Nijhuis et al.[117] proposed a new control chart based on PCA that is called the (TC) 2 -chart and applied it to gas chromatography. Wilkstrom et al.[118] applied multivariate statistical process control to an electrolysis process. A mixed control chart has been presented which permits the simultaneous monitoring of principal component scores and principal component residuals; it is called the SMART-chart (Simultaneous 40

55 Monitoring And Residuals Tracking). Wilkstrom et al.[119] applied multivariate statistical process control to an electrolysis process, incorporating PCA, PLS and ARMA (AutoRegressive Moving Average) techniques into the analysis. Tsung[120] presented a method focused on process control schemes that are based on a combination of the process outputs and automatic control actions using adaptive PCA. Chiang et al.[121] discussed the use of discriminant analysis, PCA and PLS for fault diagnosis in chemical processes. Norvilas et al.[122] have developed an intelligent process-monitoring and fault-diagnosis environment by interfacing multivariate statistical process control monitoring techniques and knowledge-based systems for monitoring multivariate process operation. Lane et al.[123]proposed an extension to PCA which enables the simultaneous monitoring of a number of product grades or recipes. Schippers[124] proposed an integrated process control model using statistical process control, total productivity management and automated process control. Kano et al.[125] proposed a novel statistical monitoring method which is based on PCA, called moving PCA, in order to improve process-monitoring performance. Chen and Liu[126] proposed on-line batch process monitoring using dynamic PCA and dynamic PLS models. Arteaga and Ferrer[127] dealt with the missing-data problem in the estimation of latent variables scores from an existing PCA model. Badcock et al.[128] proposed two alternative projection techniques that focus on the temporal structure of multivariate data. Ramaker et al.[129], using simulation, studied the effect of the size of the training set and number of principal components on the false-alarm rate in statistical process monitoring[44]. The great advantage of PCA method is the reduction of the dimensionality and it can be attractive for a problem with large number of quality characteristics, but this advantage can be a disadvantage if the number of principal components scores, r, chosen inappropriately that is a risk of missing valuable information of original variables[44]. As the main disadvantage of PCA, by using the principal components rather than the original variables in quality control, interpretation of the out-of control principal components can be difficult, because they are not the original set of process variables but instead linear combinations of them[8]. Furthermore, if after applying PCA to reduce the dimension of the original space, any of the current quality control methods like the 41

56 control eclipse or pair-wise scatter plots (CC > 2) [8], Hotelling T 2 control chart[130], MEWMA[82] are applied for monitoring the principal components scores, have the same disadvantage Simulated MINMAX Control Chart Sepulveda and Nachlas1[131] developed a control chart, called the simulated minimax control chart, that monitors the maximum and minimum standardized sample mean of samples taken from a multivariate process. The method assumes that the data are normally distributed and that the variance covariance matrix is known and constant over time. Hence, by monitoring the maximum and minimum standardized sample mean, an out-of-control signal is directly connected with the corresponding out-of-control variable. They also discussed the statistical properties and the out-of-control ARL performance of the MINMAX control chart and showed that it works better than the Chi-square control chart in terms of detecting shifts in the means. The method also gives evidence about which variable caused an out-of-control signal. The limitation on variables which follow multivariate normal distribution can be a disadvantage in the application of this model as a general method Multi-attribute Quality Control Despite multi-attribute monitoring having many applications, almost all researchers have focused on multivariate control charting and only a few methods have been proposed to monitor multi-attribute processes (see, for example, Bourke[132], Chiu and Kuo[133], Montgomery[8], Xie et al. [15]. In many cases, in order to reduce inspection costs, multivariable controls can attain multiattribute status by using gages (go-no go) for gathering information about quality characteristics. If there is a correlation between some of the variables characteristics, then there will be correlation between the (converted) attribute characteristics. Xie [15] and Larpkiattawom[134] provides many examples of the application of multi-attribute charts 42

57 in industry. Amongst there is the case of a plating process in which two correlated attribute characteristics ar the location of incomplete coverage and unaccepted thickness. These two characteristics were quantified using number/fraction nonconforming General multi-attribute quality control charts In a multi-attribute environment, the process is monitored using different types of attribute control charts. In the following sections, all existing attribute control charts for multi-attribute cases will be discussed Patel s method (Chi-square Control Chart / Hotelling T 2 Control Chart for attributes) Patel[135] was one of those who first made a step toward working on multi-attribute data and suggested a Hotelling-type T 2 chart for observations from multivariate binomial or multivariate Poisson distributions, namely a multivariate control chart for fraction nonconforming and a multivariate chart for number of defects or non-conformities. The idea was based on the assumption that, similar to Shewhart attribute control charts for the univariate case, with the appropriate sample size selection, the distribution of the correlated attributes will approximate the multivariate normal distribution, which will allow the use of the multivariate normal control charting concept such as the use of the T 2 control chart. The disadvantages of this method are the skewness of the distributions of the attributes and (like the Hotelling T 2 method) the difficulty to determine which variable has shifted. Moreover, the proposed chart only has an upper control limit that makes it unable to be used for detection of both process deterioration and process improvement[47] MNP Chart (Multivariate np Chart) 43

58 Lu et al. [136] developed the MNP chart, which is a Shewhart-type control chart, in order to deal with multivariate attribute processes. They also dealt with the case in which estimation of the parameters is required. They extended univariate Shewhart np-charts to a MNP chart based on an X statistic, which is the weighted sum of the counts of nonconforming items for each quality characteristic in a sample and which considers the correlation of attributes (and consequently reduces the probability of not detecting defects) better than individual np-charts. The MNP chart has the advantage that it not only improves the efficiency of the identification of the critical assignable cause(s) that are responsible for existing out-of-control signal(s) (if any) and is simple to implement and interpret when out-of-control signals occur, but it is also more effective and more sensitive than the separate univariate control charts in monitoring multi-attribute processes, since it takes into consideration the correlation between the quality attributes. In their work, however, there is no mention of the ARL of the MNP chart and the distribution of the statistic X used in the chart. Besides the lack of quantitative discussion about the performance of the MNP control chart, another drawback in their work is the normality assumption. Moreover according to their method, a product unit can be classified only as either conforming or non-conforming by each monitored quality characteristic, but this binary classification is not appropriate in many cases since the quality of a product cannot change suddenly from absolutely perfect to completely worthless[47] Jolayemi Multi-Attribute Control Chart (MACC) Jolayemi[21] also developed a statistical model for the design of Multi-Attribute Control Chart (MACC) based on an approximation he had introduced earlier (see Jolayemi[ ]) for the convolution of independent binomial variables, as well as on an extension of np-control charts. The J approximation was used to develop an MACC in order to monitor the m attributes simultaneously instead of using m separate np-control charts. Jolayemi[21] also developed some formulae for finding quick solutions to the model in order to obtain the values of the design parameters of the chart. In this aper, the model 44

59 was illustrated by the use of some numerical examples, which were also used in order to compare the performance of the proposed MACC to the one of the common np-charts. Additionally, a procedure was developed for identifying the attribute(s) responsible for the out-of-control signals of the chart, if any, based on the use of acceptance numbers. When an MACC gives an out-of-control signal, however, according to the procedure proposed by Jolayemi[21] for the identification of the responsible attribute(s), one should count the number of defects related to each attribute on all the defective items found in a sample of size n. As a result the MACC does not always work if the inspection is destructive or if the inspection with respect to one attribute influences the accuracy of subsequent inspections of other quality attributes on the same unit of the product (carryover effects), since the value of the number of defectives is needed for each attribute. Another drawback is that attributes are not necessarily independent yet are treated as if they were. Despite these drawbacks, the procedure has some attractive features. Once an acceptance number is calculated for an attribute, there is no need to calculate more acceptance numbers in all future process assessments, unless management indicates new values of α or the in-control and out-of-control proportions of defectives for each attribute. Jolayemi[140] developed a model for an optimal economic design of MACCs for processes depending on multiple assignable causes, which is based on the assumption of independent attributes and on the use of the J approximation (Jolayemi[137]) and Gibra s model [141] for univariate np-chart, and gave the formulae required for the construction of the model and the computation of the expected total cost per unit time and consequently the development of the whole (economic) design of the MACC Bootstrap method and confidence interval approach Niaki and Abbasi[142] used a bootstrap method and confidence interval approach to design control charts for monitoring multi attribute processes. The purpose was to monitor a number of different correlated defects in a process involving a multivariate Poisson distribution, creating a multi-attribute c control charts. They compared the performance of their suggested method of designing control charts with the methods that Bonferroni and Sidak had previously proposed (which were found to be inappropriate for 45

60 monitoring correlated quality attributes simultaneously) and showed that their proposed bootstrap method outperforms them. They compared the performance of their method with the T 2 approach, in terms of ARL (both in-control and out-of-control ARL). The results showed that the bootstrap method works better for monitoring correlated attributes. As an outcome of their method, there are some control limits for each attribute that can be used for monitoring each characteristic separately. It could be helpful to find out which one causes any out-of-control situation but could be misleading as well due to not considering the correlation between the original data. Another drawback of this method is the necessity of using historical data for an in-control process in order to estimate the initial parameters of model that can be a restriction in real practice. Furthermore, the model is limited to the correlated defects in the process involving a multivariate Poisson distribution such as expected for multi-attribute c control charts Skinner s deleted-y statistic approach Skinner et al. [143] proposed a new statistic, called deleted-y, to be computed for each variable. They applied a k-standard deviation Shewhart-type control chart for each statistic for which the k-value was obtained by simulation to ascertain a given in-control average run length criterion for all univariate charts. They showed that not always did the deleted-y chart detect shift(s) quickly. Two cases are presented where the deleted-y chart does not detect the shift quickly. In the first, when the means of all the counts are not equal, and the largest mean count shifts, c chart detects the shift slightly sooner than the deleted-y chart. Also, the strength of the shift is not visible to the deleted-y chart. In the second case, when all the counts shift, the deleted-y chart does not detect the shift in all cases better than multiple c charts[47]. Another disadvantage of their method is that only the instances of process deterioration (not improvement) can be detected. Furthermore, finding the upper control limits (UCL) of the individual probabilistic control charts is a time-consuming process[144]. 46

61 Exact Multi-Attribute Control Chart (E-MACC) Gadre and Rattihalli[145], using the assumption of multinomial distributions for multiattribute processes, developed the exact multi-attribute control chart (E-MACC) by using MP-test (Most Powerful Test) with computed design parameters to determine if the parameters of the distribution changed. They proposed a procedure for identification of responsible attribute for the process to be in the out-of-control state. In their method, to apply the MP-test, the magnitude of the parameters of interest must be defined in advance. Calculating the input parameters is not simple and that could be a drawback of their method. They proposed a computer program for calculating the input parameters for the case of a two-attribute problem. However this computer program has the drawback that it is capable of calculating the parameters only for sample having sizes smaller than 38. As another issue of this method, it is capable of detecting only the process deterioration and not capable of identifying any process improvement Skewness Reduction Approach Niaki and Abbasi[146] proposed a methodology for the monitoring of fraction nonconforming in the multi-attribute domain based on transforming multi-attribute data so as to make their marginal probability distributions have almost zero skewness (rth root transformation method, in which r may be different for each attribute of the qualitycharacteristic vector). They used simulated data in order to find the power of the (proper) root transformation for each attribute. Then they estimated the transformed covariance matrix and applied the Hotelling T 2 control chart, which assumes a multivariate normal distribution, thus causing the proposed method to be an approximate instead of an exact procedure. Through examples, Niaki and Abbasi showed that the proposed transformation method performed better than other general transformation methods, such as the square root proposed by Ryan[147], arcsin method proposed by Anscombe[148], and Q-transformation proposed by Quesenberry [185], in terms of various performance measures (such as skewness and kurtosis). They also showed that, when using the original data under the presence of skewness, the in-control ARL value in the Hotelling T 2 control chart for the multi-attribute case proposed by Patel[135], is very low. Transforming the data, however, and removing the skewness, the in-control ARL has an 47

62 appropriate value. Furthermore, comparing the out-of-control ARL values, they showed that the proposed method with the transformed data performs better than the MNP chart, proposed by Lu et al. [136], for all the shifts considered in their example. The main drawback of this method is applying Hotelling T 2 -based control chart proposed by Patel[135], for controlling the transformed data. So, all issues of Patel s method are applicable in this method including the inability of identify the responsible attribute of any out-of-control state, and the inability to detection of both process deterioration and process improvement Correlation Reduction Approach In the later paper to that considered above, Niaki and Abbasi[149], used the concept of the transformation matrix in multivariate proposed by Houshmand and Golnabi [109], to developa symmetric square root transformation function to eliminate the correlation of attributes and for the transformed data proposed symmetric multi-attribute c control charts for defect counts. The results of simulation studies show that their proposed method performs better in all cases than the Hotelling T 2 -based method for the multiattribute case used by Patel [135], in terms of the in-control ARL criterion. Niaki and Abbasi claimed that the correlation between the transformed attributes either vanishes or becomes very small, which is an advantage that could be applied in other applications NORTA Inverse transformed vectors method In an even moe recent paper, Niaki and Abbasi[144] have proposed using the NORTA Inverse Transformed Vectors method for the number of nonconformities in an inspection unit of product based on a Poisson distribution. With by the NORTA algorithm one can generate an arbitrary multidimensional random vector by transforming a multidimensional standard normal vector. Niaki and Abbasi initially transformed a multiattribute random vector so that the marginal probability distributions associated with the transformed random variables are approximately normal. Then, they estimated the 48

63 covariance matrix of the transformed vector via simulation. Finally they used the Hotelling T 2 -based method for multi-attribute cases proposed by Patel [135] to monitor the processes. They compared their results with Patel s method (applied to the original data) in terms of in-control ARL (ARL 0 ) and with deleted-y method proposed by Skinner [143] in terms of the out-of-control ARL (ARL 1 ). The results show that the proposed method works better. The main disadvantage of their method is having to apply Hotelling T 2 -based control chart proposed by Patel[135], for controlling the transformed data. So, all issues Patel s method are applicable in this method including the inability to identifying the responsible attribute of any out-of-control state, and the inability to detect both process deterioration and process improvement Artificial Neural Network (ANN) Larpkiattaworn[134] proposed a back propagation neural network (BPNN) for the twoattribute bivariate Binomial process control case using the assumption of a positive correlation and a large enough sample size. He detected the out of control condition by an artificial neural network where the output was one if the process is under control and zero otherwise. In this paper, Larpkiattaworn also discussed different values of correlation between the two quality attributes and gave some suggestions on using three-attribute control charts like 2 χ, Mnp, and BPNN methods. Niaki and Abbasi[150] developed a methodology in order to overcome the problems arising when monitoring the quality of a multi-attribute process, namely the occurrence of a large number of false alarms (type I error) and an increase in the probability of not detecting defects when the process is monitored by a set of independent uni-attribute control charts. As a result, Niaki and Abbasi[150] designed a perception neural network to monitor either the proportions of several types of product non-conformities or the number of different types of defects in a product (instead of using several np-charts or several c-charts, respectively). They illustrated their method (by the use of simulation 49

64 experiments) and compared its performance with the performances of the MNP method proposed by Ku et al[136] for the multinomial case and Hotelling T 2 -based method for the multi-attribute case proposed by Patel [135] for multi-attribute Poisson case. They showed that the proposed neural network method not only detects shifts better than the existing methods in most out-of-control conditions based on the out-of-control ARL, but also it is capable of finding which attribute(s) is in the out-of-control condition. Whilst acknowledging the advantages of this method, there are general demerits of ANN. Some of these disadvantages are[151]: Trial and error element to building good models (in training phase) It is hard to interpret what is happening in the model (Black Box) and people only see the inputs and outputs Over-fitting. The iterative model fitting will often over-fit on the training dataset, unless steps are taking to alleviate this. Model performance relates to starting input values and parameters. Variable scaling. Neural nets fit better if all variables are of a similar scale Marcucci s Control Chart Marcucci[152] proposed a control chart based on the use of a multinomial distribution in the cases in which the proportions in each quality category are known or estimated using a base period. However, this idea may not always be applicable because of the fact that not all multi-attribute processes follow the multinomial distribution D 2 Control Chart Mukhopadhyay[153] proposed a multivariate attribute control chart to exercise simultaneous control of all the categories of non-conformities. A traditional approach to this problem has been to apply several p-charts one for each category of defect which 50

65 leads to a false type I error and consequently to a false power for the test. The problem can be resolved by applying a D 2 control chart that controls simultaneously the proportion defectives falling in various categories of defects on a single chart without any distortion in the advertised level of type I error. Similar to the Hotelling T 2 control chart, the inability of identifying the responsible attribute of any out-of-control state is a drawback of this method Special cases of multi-attribute quality control In this section, the multi-attribute quality control methods that are applicable to specific cases with very limited generality are reviewed. For instance, quality control of a high quality process, dealing with attribute characteristics with positive correlation, and dealing with linguistic data. Niaki and Abbasi [154] proposed a new methodology to monitor multi-attribute high quality processes in which not only there exist more than one type of defect in the observed nonconforming item but also there is a dependence structure between the number of conforming items before finding the first nonconforming item and the number of defects in the nonconforming item. They showed that their proposed method was better performing than the CCC (cumulative count of conforming) and c-charts in all cases, especially in those situations in which there were both positive and negative shifts around the mean. As another work in high-quality processes and for processes producing many defect-free products, Li et al.[155] proposed types of MZIP (multivariate using a zero-inflated Poisson) models, investigated distributional properties of an MZIP model and used reallife examples from a major electronic equipment manufacturer in order to motivate their study and illustrate the impact and usefulness of the proposed MZIP models in a manufacturing environment for equipment-fault detection and quality control. Jones et al.[30] studied a demerit rating system for a complex product with several types of defects according to the degree of defects on the product s performance quality. They proposed a demerit statistic that is a linear combination of the counts of these different 51

66 types of defects and determined the control limits based on the exact distribution of linear combinations of independent Poisson random variables. Chiu and Kuo[133] constructed a control chart (called the MP chart) to monitor the multivariate Poisson count data with positive correlations, using an exact probability method based on the sum of defects or non-conformities for each quality characteristic in order to obtain the control limits. They illustrated the MP chart by the use of numerical examples and evaluated it by simulation using ARL (both in-control and out-of- control ARL values). The results showed that the MP chart is more appropriate than the Shewhart-type control chart when positive correlation between variables exists. Similar to this model, Jiang et al. [156] proposed a symmetric c-chart and obtained the control limits to minimize the absolute deviation of in control ARL from the nominal value in case of positive correlation of attributes[133]. There is another special type of multi-attribute quality control. The binary classification process is a sign of whether the monitored quality characteristics meet the specifications or not, but does not supply the degree to which the specifications are met, which is done by the use of linguistic terms, such as exceptional, fine, bad. As a result the binary classification used in the charts might not be appropriate in situations where the quality characteristics of a product do not change suddenly from conforming to non-conforming. In such cases the solution to the problem comes from the use of various linguistic terms for the description of each quality characteristic, since without fully utilizing such intermediate information, the chart has poorer performance. To supplement the binary classification, several intermediate levels may be expressed in the form of linguistic terms. For example, the quality of a product can be classified by one of the terms perfect, good, medium, poor and bad, depending on its deviation from specifications. As a result, it seems reasonable to use fuzzy sets for modeling vague or linguistic data and then to design control charts for these fuzzy data. Control charts for linguistic variables have been developed by Raz and Wang[157], Wang and Raz[158] and Kanagawa et al.[159]. Each linguistic term is characterized by an appropriate membership function. Appropriately selected continuous functions can then be used to describe the quality characteristic associated with each linguistic term. This can be done 52

67 by the use of fuzzy control charts. Woodall et al.[160] reviewed statistical and fuzzy control charts based on categorical data. For more works of fuzzy quality control and linguistic data, the interested reader is also referred to [ ]. Genarally such work is outside of the scope of this thesis. As another special case in multi-attribute quality control, Gadre and Rattihalli[178] proposed some group inspection-based MACCs to identify process deterioration. The proposed charts were the multi-attribute np-chart (MA-np-chart), the multi-attribute synthetic chart (MA-Syn chart) and the multi-attribute group runs chart (MA-GR chart) and were developed using a MP-test based on the exact distribution proposed by themselves [145]. The results of a numerical example showed that the MA-GR chart performs much better than both the MA-Syn and the MA-np-chart, with the MA-np-chart performing the worst. In addition, the MA-GR chart can be based not only on 100% inspection, but also on non-100% inspection schemes if uniform sampling is used. In the multi-attribute quality control domain, in the case where the measurements are recorded only as pass or fail the measurements are called binary, while in a case where there are more than two possible values, the recorded measurements are called polytomous. Shiau et al.[179], extended the work in the paper by Yousry et al.[180] to the monitoring of processes with polytomous data and developed a method for monitoring the fractions of tested items falling into different categories of pass/fail forms. They described the theory lying beneath an empirical Bayes approach to monitoring polytomous data, using the multinomial distribution with a Dirichlet prior distribution having unknown hyperparameters to model the fractions mentioned above. They proposed a monitoring scheme based on the marginal distributions of the observed fractions falling into the pass/fail categories and studied the ARL of the proposed method for various situations as well. They found that if the marginal distribution of the observed fractions is not very skewed, then the detecting power of the proposed control chart is fairly good. Steiner et al.[181] proposed a control chart based on gauging theoretically continuous observations into multiple groups. Their motivation was the fact that, many times, it is more economical, quicker and easier to categorize a continuous quality characteristic into 53

68 several groups than it is to measure its quantities exactly and in such cases control charts based on grouped observations are usually better than standard control charts based on variables. As a drawback of this concept, loss of information is a concern however they believed that when group limits are carefully selected, the loss of information due to grouping is small. Moreover, the usual classification of units as conforming or nonconforming is inefficient when the proportion of non-conforming units is small due to an inverse relation of sample size and size of the proportion non-conforming to be detected Concluding remarks The literature review has revealed a long history in the use of control charts for monitoring processes. Whilst originally intended to be used for manufactured goods, control charts are now used much more widely for monitoring services and even the blood pressure of patients. Initially, control charts were used for the monitoring of a single quality characteristic; but, as the literature review has shown, researchers extended the methods to account for situations where there is more than one correlated quality characteristics. The cited works have all been concerned with cases where these characteristics have a fundamental feature: they are either of the continuous type or the discrete type, but not both. Some of these multivariate and multi-attribute methods were also found to have other restrictions. In attempting to provide an overview of the features of the existing multivariate and multi-attribute methods, table 1 has been produced. In this table, the application scope of each method, together with their reported and observed main advantages and disadvantages, is depicted. 54

69 Method No. Name Multivariate Multi-Attribute Main Advantages Main Disadvantages 1 Control Ellipse * 2 3 Chi-square Control Chart / Hotelling T 2 Control Chart # * MEWMA Control Chart/MCUSUM Control Chart * Simplicity Graphical output Applicable for more than two variables Time sequence of the data is preserved More sensitivity to small shifts in comparison with Chi-square and Hotelling T 2 control charts Time sequence of the plotted points is lost Limited application for more than two variables due to the complexity of graphical analysis Limited application to the case that variables are jointly distributed according to the bivariate normal distribution. Unable to detect the direction of the shift in mean vector and to be used for distinguishing between increase and decrease in a process Unable to identify the subset of process variables that are responsible for the out-of-control state Unable to identify the subset of process variables that are responsible for the out-of-control state Worse performance than Hotelling T 2 control charts in 4 Multivariate Shewhart X chart (MS Chart) * terms of ARL 0 Misleading application of separate Shewhart control chart regarding the correlation of transformed variables Unable to identify the process variables that are responsible for the out-of-control state 55

70 Method No. Name Multivariate Multi-Attribute Main Advantages Main Disadvantages 5 PCA * Reduction of the dimensionality Interpretation of the out-of control principal components can be difficult, because they are not the original set of process variables but instead their linear combination of them In applying any of the above mentioned charts for principal components, their disadvantages are applicable for this method as well 6 Simulated MINMAX * Able to identify the process variables that are responsible for the out-of-control state Better performance than Hotelling T 2 control charts in terms of ARL 1 Similar to Chi-square Control Chart / Hotelling T 2 Control Chart (no. 2) The assumption of jointed probability distribution that is the multivariate normal distribution 7 Chi-square Control Chart / Hotelling T 2 Control Chart for attributes (Patel) * Similar to Chi-square Control Chart / Hotelling T 2 Control Chart (no. 2) Assumption of multivariate normal distribution for attributes are more critical due to skewness of attributes 8 MNP Chart * 9 Jolayemi s Multi- Attribute Control Chart (MACC) * Correlation between the quality attributes are considered in the model Able to identify the process variables that are responsible for the out-of-control state Able to identify the process variables that are responsible for the out-of-control state Once an acceptance number is calculated for an attribute, there is no need to calculate more acceptance numbers in all future process assessments 56 limited to binary classification (multinomial distribution) No mention of the ARL of the MNP chart and the distribution of the statistic X Assumption of multivariate normal distribution for attributes limited to binary classification (multinomial distribution) It does not always work if the inspection is destructive or if the inspection with respect to one attribute makes it impossible to inspect an item correctly with respect to another attribute (carryover effects), since the value of the number of defectives is needed for each attribute.

71 Method No. Name Multivariate Multi-Attribute Main Advantages Main Disadvantages bootstrap method and confidence interval approach Skinner s Deleted-Y statistic approach 12 E-MACC * 13 Skewness Reduction Approach * * * Better ARL 0 and ARL 1 than Hotelling T 2 Able to identify the process variables that are responsible for the out-of-control state regarding using separate control limits for each variables Able to identify the process variables that are responsible for the out-of-control state The proposed transformation method performed better than other general transformation methods skewness, kurtosis and p-value Better ARL 1 than MNP method Misleading use of separate control limits for each variables regarding correlation between them necessity of using historical data for an in-control process in order to estimate the initial parameters of model limited to multivariate Poisson distribution case When the means of all the counts are not equal, and the largest mean count shifts or when all the counts shift, deleted-y chart does not detect the shift more quickly than C chart Only the instances of process deterioration (not improvement) can be detected Finding the upper control limits (UCL) of the individual probabilistic control charts is a time-consuming process Not capable of identifying the process improvement and detect only the process deterioration Assumption of multinomial distribution Calculating the input parameters is not simple applying Patel s method (no.7) for controlling the transformed data that causes its disadvantages are applicable for this method as well (e.g. inability of detecting responsible attribute of any out-of-control state and distinguishing between increase and decrease in a process 14 Correlation Reduction Approach * The correlation between the transformed attributes either vanishes or becomes very small Better ARL 0 than Patel s method (using original data) 57

72 Table 1 Comparison of current multivariate and multi-attribute quality control methods Method No. Name Multivariate Multi-Attribute Main Advantages Main Disadvantages 15 NORTA Inverse transformed vectors method * Better ARL 0 than Patel s method (using original data) and Better ARL 1 than deleted-y method applying Patel s method (no.7) for controlling the transformed data that causes its disadvantages are applicable for this method as well (e.g. inability of detecting responsible attribute of any out-of-control state and distinguishing between increase and decrease in a process 16 Artificial Neural Network (ANN) * * In most of the cases, Better ARL 1 than MNP method (for multinomial case) and Patel s method (for multivariate Poisson case) Able to identify the process variables that are responsible for the out-of-control state Trial and error element to building good models (In training phase) It is hard to interpret what is happening in the model (Black Box) and people only see the inputs and outputs Over-fitting. The iterative model fitting will often over-fit on the training dataset, unless steps are taking to alleviate this. Model performance relates to starting input values and parameters. Variable scaling. Neural nets fit better if all variables are on a similar scale. 17 Marcucci s Control Chart * Limited to use of a multinomial distribution for multiattribute processes 18 D 2 Control Chart * Better performance than compared with individual P chart in terms of type I error Unable to identify the subset of process variables that are responsible for the out-of-control state # Chi-square Control Chart (Average and variance-covariance matrixes are known) & Hotelling T2 Control Chart (Average and variance- covariance matrixes are unknown and estimated preliminary samples) 58

73 From a study of table 1, it has been possible to identify eight issues that need to be considered when evaluating a method. These eight issues are used below to structure the comparison of the existing methods. For ease of referencing, the number of each method from the first column of table 1 is mentioned in brackets. Correlation of the quality characteristics In general, the main reason for applying multivariate/multi-attribute control charts instead of the univariate/uni-attribute ones is the correlation between quality characteristics but in all of the current models (except MNP chart (8)), the value of the correlation is not considered. This can lead to unknown and unexpected affects on the performance of the methods as the value of the correlation differs. In the more risky circumstances, some of the models like MS chart (4) for transformed data, bootstrap method and confidence approach (10) used individual control charts as output without eliminating the correlation between the quality characteristics. This may be very misleading in a similar manner to the use of univariate/uni-attribue control charts for correlated variables. In correlation reduction approach (14), the transformed attributes were claimed to either vanish or become very small. Skewness of the quality characteristics Some of the quality characteristics naturally follow a symmetric distribution like the normal distribution, but some others naturally follow an asymmetric distribution when they are bounded at one limit (e.g. response time can never be zero). In multivariate/multi-attribute quality control, the skewness effect may be more significant than uni-variable cases due to the effects of the variables on each other. Generally, the skewness phenomena is more critical in multi-attribute cases because of the nature of the attribute characteristics and this issue is observed in the poor performance of Patel s method (7) where he used a multivariate normal distribution as an assumption in developing his model for monitoring attribute characteristics. Almost none 59

74 of the current methods focused on the effect of skewness; a notable exception is the skewness reduction approach (13). Diagnosing the out-of-control states In the case of having an out-of-control observation, determining the variable(s) that are associated with the observation is a vital subject in multivariate/multiattribute quality control. Being able to interpret out-of-control situations is an advantage and is possible with some, but not all of the surveyed methods due to the nature of the developed models. For example, Simulated MINMAX (6), MNP Chart (8), Jolayemi s MACC (9), bootstrap method and confidence interval approach (10), E-MACC (12), and ANN (16) are able to identify the process variables that are responsible for the out-of-control state. On the other hand, MS chart (4), Chi-square control chart / Hotelling T 2 control chart (2), Chi-square control chart / Hotelling T 2 control chart for attributes (Patel) (7) and also skewness reduction approach (13) and NORTA inverse transformed vectors method (15) that apply Patel s method (7) for the transformed data, MEWMA control chart/mcusum control chart (3), PCA (5) D 2 control chart are not able to identify the subset of process variables that are responsible for an out-of-control signal. In order to resolve this issue for the models that have not a built-in part to diagnose any out-of-control cases, some supplementary methods have been developed [8], [39], [44], [48], [45], [46], [182], [183], [184], [185], [186], [187], [188]. Detecting both process increases and decreases 60

75 In Hotelling T 2 based methods like Chi-square/ Hotelling T 2 control chart (2) or Patel s method (7) the proposed chart only has an upper control limit that makes it unable to be used for distinguishing between increases and decreases in a process. This disadvantage is applicable for some other methods that use Hotelling T 2 based methods for the transformed data e.g. PCA (5), Skewness reduction approach (13), and NORTA inverse transformed vectors method (15). Furthermore, in Deleted-Y method (11), only the instances of process deterioration (not improvement) can be detected. Similarly, E- MACC (12) is not capable of distinguishing between an increase and decrease in a process. Time sequence of the plotted points The main issue of some methods like the control ellipse (1) is that the time sequence of the sample data is lost when it is transformed. The same issue occurs in the PCA method (5) if the number of principle components is more than two and pair-wise scatter plots of them is used analogously in addition to the control ellipse (trajectory plots). Most of the other surveyed methods like the Hotelling T 2 based methods (e.g. Chi-square/ Hotelling T 2 control chart (2) or Patel s method (7), and MEWMA / MCUSUM control charts (3)) and also other methods that use Hotelling T 2 based methods for the transformed data e.g. PCA (5), Skewness reduction approach (13), and NORTA inverse transformed vectors method (15)) preserve the order of data collection. Application scope Some of the reviewed methods are limited in application in some areas regarding the assumptions and concepts considered in developing them. The control ellipse (1) is limited for more than two variables due to the complexity of graphical analysis since for p characteristics, one is as a p dimensional ellipsoid. The assumption of a specific type of distribution for the input variables limits the application of some methods or makes their performance questionable when the assumptions are not met in practice. An example of limiting distribution assumption is that the variables are jointly distributed according to 61

76 the bivariate normal distribution in the control ellipse method (1). The variables are jointly distributed according to the multivariate normal distribution in the simulated MINMAX (6), Patel s methods (7) and MNP chart (8). The multinomial distribution is assumed for the variables in Jolayemi s MACC (9), E-MACC (12) and Marcucci s control chart (17). The variables are supposed to follow a multivariate Poisson distribution for the bootstrap method and confidence interval approach (10). Another limitation in this method is the necessity of using historical data for an in-control process for estimating the initial parameters of the model. As another example, when the means of all the counts are not equal, and the largest mean count shifts or when all the counts shift, the deleted-y chart (11) does not detect the shift more quickly than a uni-attribute c chart. As another example of limitations to the applicability of a method, Jolayemi s MACC (9) does not always work if the inspection is destructive or if the inspection with respect to one attribute makes it impossible to inspect an item correctly with respect to another attribute (carryover effects), since the value of the number of defectives is needed for each attribute. Performance In the quality control environment, ARL is a typical measure that is used in order to evaluate the performance of process control methods. The ARL of a control scheme is the average number of samples that must be taken before a sample gives a sign of an out-ofcontrol condition of the process, which means that in the case of an in-control process a long ARL (ARL 0 ) is preferred, while in the case of an out-of-control process a short ARL (ARL 1 ) is preferred. Some of the above-mentioned methods are compared with some other methods in terms of ARL 0 (in phase I) or ARL 1 (in phase II) for detecting shifts in mean value, based on numerical examples or simulation studies presented in the reviewed papers. For instance, MEWMA/MCUSUM (3) control charts are more sensitive than Chisquare/Hotelling T 2 control charts (2) to small shifts of mean. Simulated MINMAX (6) has smaller ARL 1 than Chi-square/Hotelling T 2 control charts (2). Bootstrap method and confidence interval approach (10) works better, in terms of ARL 0 and ARL 1, than 62

77 Hotelling T 2 control charts (2). The skewness reduction approach (13) has smaller ARL 1 than MNP chart (8). ARL 0 is more in the Correlation reduction approach (14) and NORTA inverse transformed vectors method (15) than the Patel s method (7) using on original data. NORTA inverse transformed vectors method (15) has less ARL 1 than the Deleted-Y method (11). In most of the cases, ANN (16) has a smaller ARL 1 than MNP method (8) for multinomial case and Patel s method (7) for multivariate Poisson cases. D 2 control chart (18), as a multi-attribute method, has better performance than P chart, as a uni-attribute control chart in terms of type I error. Performance of MS chart (4) is worse than Chi-square/Hotelling T 2 control charts (2) in terms of ARL 0. Some methods have not been compared with other methods, like MNP chart (8). For anyone considering the use of MNP charts, it might be useful to perform some benchmarking. Simplicity Generally speaking, simplicity of a method is always an advantage when considering a model for application in practice in industry. For example, the simplicity of analysis of graphical output of the bi-variable control ellipse (1) is significant; however other disadvantages limit its application. In Jolayemi s MACC (9), once an acceptance number is calculated for an attribute, there is no need to calculate more acceptance numbers for all future process assessments and this can make the model simple. For Skinner s Y-deleted statistic approach (11), finding the upper control limits (UCL) of the individual probabilistic control charts is a time-consuming process. Calculating the input parameters is not simple in E-MACC (12). ANN (16) can be considered as the hardest model amongst the reviewed methods due to: Trial and error element to building good models (during training phase) It is hard to interpret what is happening in the model (black box) and people only see the inputs and outputs Over-fitting. The iterative model fitting will often over-fit on the training dataset, unless steps are taking to alleviate this. Model performance depends on starting input values and parameters. 63

78 Variable scaling. Neural nets fit better if all variables are on a similar scale. The above comparison of the existing multivariate and multi-attribute process control methods has shown that they have a range of characteristics with no one method clearly superior in all aspects to the others. It has also highlighted that only the artificial neural network approach is capable of properly accounting for a mixture of continuous and discrete control variables. Given the classical difficulties with ANN, there is a need for a practical and effective method for the multivariate-attribute situation. The remainder of this thesis is concerned with constructing such a method and then testing it. 64

79 Chapter 3 Research Model and Assumption 65

80 3. Research Model and Assumptions 3.1. Problem Statement The literature review, has identified works in four different aspects of quality control including univariate, uni-attribute situation, multivariate and multi-attribute but none were found for the multivariate-attribute as a combination of multivariate and multiattribute. This may be due to the differences between natures of these two concepts. Naturally, variable characteristics follow continuous distributions such as normal distribution. The appearance of a these types of probability distributions are that of a smooth curve. However, attribute characteristics follow discrete probability distributions such as Binomial and Poisson. The appearance of discrete probability distribution looks like a series of vertical spikes with the height of each spike proportional to the probability. Pattern of these kinds of distribution are discrete with more skewness than the variable ones. The Normal distribution as the most common continuous one is symmetric. As an example of where both variable and attribute characteristics may jointly determine the quality of a process, consider a metal forming process with punching and bending operations. Here, the weight and thickness of metal sheet may be variable characteristics and the shape of the bent sheet and crumpling of the punched hole may be attribute characteristics. The former of the attribute characteristics may be checked with a gauge and the latter checked visually. Control charts, as one of the statistical quality control tools, are generally applicable to both variable and attribute quality characteristics. In the variable domain, a measurable characteristic of a product or process that affects the quality of the process output, is measured and controlled by using variable control charts. In the attribute domain, the number of, or percentage of, defects in products or defective products is calculated and controlled. When the number of quality characteristics, in the form of variables or attributes, exceeds unity and there exists a non-zero correlation between them, then the situation is either of a multi-variable or a multi-attribute quality control problem. In multivariate quality control, several dependent variable characteristics are measured and monitored simultaneously. Similarly, in multi-attribute 66

81 quality control, more than one dependent attribute characteristic is considered simultaneously. In the present, if there are some interacting quality characteristics including variables and attributes, quality engineers may use separate multivariable quality control procedures for monitoring the variable quality characteristics and multi-attribute quality control methods for monitoring the attribute quality characteristics. However, as noted in the literature review, where the monitored process characteristics are correlated, the approach of treating the variable and attribute data separately can lead to significant unexpected type I and II errors. Therefore, there is a need for a general method that considers both variable and attribute data that are correlated in a holistic manner Research Objectives and Research Questions Research Objective Addressing the above mentioned gap in the research should be of considerable appeal to industry. So the research objectives are to Design and develop MultiVariate-Attribute control method as an effective means for monitoring a process that includes multi-variate-attribute (MVA) quality characteristics, and compare performance of the developed method with the current applicable ones from an error point of view (type I / type II errors) Research Questions The specific questions that arise from the research objective are: 1) Is there any error, associated with using current multivariate and multi-attribute control charts for monitoring a process utilising multivariable-attribute characteristics? 2) If the answer to question 1 is yes, what is the reason and how can it be reduced? 3) Is it possible to develop a general quality control method for the multi-variateattribute case that is not affected by the issues of the current methods referred to in section 2.3? 67

82 4) If the answer to question 4 is yes, how is the nearly developed method s performance in comparison with the current applicable multivariate or multi-attribute methods? Proposed Research Methodology A flowchart of the approach to be taken to meet the research objectives is shown in figure 12. After literature review and problem definition, the model is developed. As data generation is a requirement for this research, a method for MVA data generation needs to be identified and programmed before the proposed MVA quality control method. After programming both the data generation and data monitoring in MATLAB software, some cases are simulated and the proposed model performance is determined. Literature review Univariate and Multivariate control charts Comparing current methods and determining the merits and demerits Uni-attribute and Multi-attribute control charts Problem definition Problem description (Multi-Variate-Attribute (MVA)) Developing the conceptual model Model Development Programming for data generation (MVA) Programming the model Simulation Study Apply the generated data for the current methods Apply the generated data for the proposed method 68