Statistical Quality Control: W. Edwards Deming and Key Concepts, Study notes of Quality Management

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Quality Engineering - Class Notes (experimental)

Jonathan Rosenblatt

July 10, 2016

Preface

This text accompanies my Quality Engineering course at the Dept. of Industrial Engineering at the Ben-Gurion University of the Negev. It has several purposes:

• Help me organize and document the course material.
• Help students during class so that they may focus on listening and not writing.
• Help students after class, so that they may self-study.

At its current state it is experimental. It can thus be expected to change from time to time, and include mistakes. I will be enormously grateful to whoever decides to share with me any mistakes found. I also ask for the readers’ forgiveness for my Wikipedia quoting style. It is highly unorthodox to cite Wikipedia as one would cite a peer reviewed publication. I do so, in this text, merely for technical convenience. I hope the reader will find this text interesting and useful.

• 1 Introduction
  • 1.1 Terminology and Concepts
    • 1.1.1 Basic Terminology
    • 1.1.2 Statistical Terminology
  • 1.2 Some History
  • 1.3 Management Aspects of Improving Quality
  • 1.4 Programs and Initiatives
    • 1.4.1 Zero Defects Program (ZD)
    • 1.4.2 Quality is Free Initiative
    • 1.4.3 Value Engineering Program (VE)
    • 1.4.4 Total Quality Management (TQM)
    • 1.4.5 Six-Sigma
    • 1.4.6 Lean Systems
    • 1.4.7 Design for Six-Sigma (DFSS)
    • 1.4.8 Quality Systems and Standards
  • 1.5 DMAIC
  • 1.6 Bibliographic Notes
• 2 Exploratory Data Analysis
  • 2.1 Summary Statistics
    • 2.1.1 Summarizing Categorical Data
    • 2.1.2 Summarizing Continuous Data
  • 2.2 Visualization
    • 2.2.1 Visualizing Categorical Data
    • 2.2.2 Visualizing Continuous Data
    • 2.2.3 On-Line Visualization
• 3 Statistical Inference
  • 3.1 Goodness of Fit (GOF)
    • 3.1.1 QQplot and QQnorm
    • 3.1.2 Chi-Square GOF Test
    • 3.1.3 Kolmogorov–Smirnov GOF Test
• 4 System Capability Analysis
  • 4.1 Process Capacity Indexes
    • 4.1.1 Non-Conformance for a Non-Gaussian CTQ
    • 4.1.2 Process Capability of a Non-Centred Process
    • 4.1.3 Interval Estimation for Capability Indexes
    • 4.1.4 Testing Hypotheses on Capability Ratios
    • 4.1.5 Other estimators of process capability
  • 4.2 Bibliographic Notes
• 5 Statistical Process Control CONTENTS CONTENTS
  • 5.1 The Run Chart
  • 5.2 The X-bar Chart
    • 5.2.1 Control Limits and the Alarm Rate
    • 5.2.2 Rational Groupings
    • 5.2.3 Other Stopping Rules
  • 5.3 Shewhart Charts With Other Test Statistics
    • 5.3.1 R Chart
    • 5.3.2 s Chart
    • 5.3.3 s^2 Chart
    • 5.3.4 Regression Control Chart
    • 5.3.5 Derivative Chart
    • 5.3.6 p and np Chart
    • 5.3.7 c Chart
    • 5.3.8 u Chart
  • 5.4 Pooling Information Over Periods
    • 5.4.1 Moving Average Chart (MA)
    • 5.4.2 Exponentially Weighted Moving Average Chart (EWMA)
    • 5.4.3 CUSUM
  • 5.5 Multivariate
    • 5.5.1 Mass Univariate Control
    • 5.5.2 Hotteling’s T2
    • 5.5.3 Drill Down
  • 5.6 Multivariate extensions
    • 5.6.1 Mutivariate s chart
    • 5.6.2 Temporal Pooling of Multivariate Charts
  • 5.7 Economical Design of Control Charts
  • 5.8 Bibliographic Notes
• 6 Design of Experiments
  • 6.1 Terminology
  • 6.2 Dealing with Variability
    • 6.2.1 Gage R&R Studies
    • 6.2.2 Completely Randomized Designs
    • 6.2.3 Randomized Block Designs
  • 6.3 Factorial Designs
    • 6.3.1 Full Factorial Designs
    • 6.3.2 Fractional Factorial
  • 6.4 Continuous Factors
    • 6.4.1 Central Composite Design
    • 6.4.2 Response Surface Methodology
  • 6.5 The Industrial Emphasis
    • 6.5.1 Plackett-Burman design
    • 6.5.2 Taguchi Methods
  • 6.6 Optimal Designs
    • 6.6.1 Space Filling Design
    • 6.6.2 Covariance Optimality
  • 6.7 Sequential Designs
    • 6.7.1 Pooled Designs
  • 6.8 AB testing
  • 6.9 Computer Experiments
  • 6.10 Bibliographic Notes CONTENTS CONTENTS
• 7 Acceptance Sampling
  • 7.1 Acceptance Sampling Terminology
  • 7.2 Single Sampling Scheme
    • 7.2.1 Double Sampling Scheme
  • 7.3 Sequential Scheme
  • 7.4 Bibliographic Notes
• 8 Reliability Analysis
  • 8.1 Probabilistic Analysis
    • 8.1.1 A Static View
    • 8.1.2 A Time Dynamic View
  • 8.2 Statistical Analysis
    • 8.2.1 Identifiability
    • 8.2.2 Censored Events
    • 8.2.3 Accelerated Life Models
    • 8.2.4 Proportional Hazard Models
    • 8.2.5 Choosing the Base Failure Rate
    • 8.2.6 The Parametric Case
    • 8.2.7 The Semi Parametric Case
  • 8.3 Collecting the pieces
  • 8.4 Repairable systems
    • 8.4.1 Single component systems
    • 8.4.2 Multiple component systems
  • 8.5 Bibliographic Notes
• 9 Revisiting System Capability Analysis
  • 9.1 System Capability with Control Charts
  • 9.2 System Capability with Designed Experiments
• A Notation
• B R
• Bibliography
• 1.1 3-sigma probability of failure List of Figures
• 1.2 6-sigma probability of failure
• 1.3 Defects PPM at different productions sigmas.
• 1.4 DMAIC
• 2.1 Bar Plot
• 2.2 Mosaic Plot
• 2.3 Dot Plot
• 2.4 Histogram
• 2.5 BoxPlot
• 2.6 Stem and Leaf Pot
• 2.7 Scatter Plot
• 2.8 HexBin Plot
• 2.9 Covariance Matrix
• 2.10 Dashboard
• 3.1 Confusion Table
• 3.2 QQnorm- Gaussian
• 3.3 QQnorm- non Gaussian
• 3.4 Kolmogorov-Smirnov Test
• 4.1 Cpk and Cp
• 5.1 x¯-chart
• 5.2 Power Function
• 5.3 ARL 0 for EWMA
• 5.4 Geometry of Spectral Matrix Norms
• 6.1 Full Factorial Design
• 6.2 Interactions plot
• 6.3 One-at-a-time optimization
• 6.4 Design for Linear Models
• 6.5 Design for Non Linear Models
• 8.1 Series system.
• 8.2 Parallel system.
• 8.3 Bridge Structure
• 8.4 Failure rate of the parallel exponential component system.
• 8.5 Bathtub empirical hazard curve
• 8.6
• 8.7 Piecewise Exponential aproximation of the Weibull distribution.
• 1 Definition (The Mean) List of Definitions
• 2 Definition (The Median)
• 3 Definition (α-Trimmed Mean)
• 4 Definition (The Standard Deviation)
• 5 Definition (α Quantile)
• 6 Definition (The Range)
• 7 Definition (The Inter Quantile Range- IQR)
• 8 Definition (The Median Absolute Deviation- MAD)
• 9 Definition (Yule Skewness Measure)
• 10 Definition (Covariance)
• 11 Definition (Pearson’s Correlation Coefficient)
• 12 Definition (Spearman’s Correlation Coefficient)
• 13 Definition (Covariance Matrix)
• 14 Definition (Correlation Matrix)
• 15 Definition (Chi-Square GOF Test)
• 16 Definition (Kolmogorov–Smirnov GOF Test)
• 17 Definition (Cp)
• 18 Definition ( Cˆp)
• 19 Definition (Cp(q))
• 20 Definition (Cpk)
• 21 Definition (Cpm)
• 22 Definition (MA)
• 23 Definition (EWMA)
• 24 Definition (Hotteling’s T 2 statistic)
• 25 Definition (Frobenius norm)
• 26 Definition (Spectral norm)
• 27 Definition (Moving Sum Hotelling)
• 28 Definition (A-Optimality)
• 29 Definition (D-Optimality)
• 30 Definition (Structure Function)
• 31 Definition (Series System)
• 32 Definition (Parallel System)
• 33 Definition (k-out-of-p System)
• 34 Definition (Monotone System)
• 35 Definition (Reliability)
• 36 Definition (Improvement potential)
• 37 Definition (Birenbaum’s measure)
• 38 Definition (CDF) LIST OF DEFINITIONS LIST OF DEFINITIONS
• 39 Definition (PDF)
• 40 Definition (Survival Function)
• 41 Definition (Failure Rate)
• 42 Definition (Cumulative Risk)
• 43 Definition (IFR)
• 44 Definition (IFRA)
• 45 Definition (NBU)
• 46 Definition (NBUE)
• 47 Definition (Point availability)
• 48 Definition (Interval reliability)
• 49 Definition (Interval downtime)
• 50 Definition (MTTF)
• 51 Definition (MTTR)

1.1. TERMINOLOGY AND CONCEPTS CHAPTER 1. INTRODUCTION

14. Genichi Taguchi: “Uniformity around a target value.”
15. Genichi Taguchi: “The loss a product imposes on society after it is shipped.”
16. Gerald M. Weinberg: “Value to some person”.
17. Jonathan D. Rosenblatt: “The efficient fulfilment of a promise”.

Collecting ideas

1. Quality is not only about production.
2. Quality is the means, not the end.
3. Quality may deal with the design or with conformance to a given design.

Almost all of the above definitions, may apply to different characteristics, we call dimensions of quality. Following Wikipedia (2015b) Dimen-sions of Quality

Performance Performance refers to a product’s primary operating characteristics. This dimen- sion of quality involves measurable attributes; brands can usually be ranked objectively on individual aspects of performance.

Features Features are additional characteristics that enhance the appeal of the product or service to the user.

Reliability Reliability is the likelihood that a product will not fail within a specific time period. This is a key element for users who need the product to work without fail.

Conformance Conformance is the precision with which the product or service meets the specified standards.

Durability Durability measures the length of a product’s life. When the product can be repaired, estimating durability is more complicated. The item will be used until it is no longer economical to operate it. This happens when the repair rate and the associated costs increase significantly.

Serviceability Serviceability is the speed with which the product can be put into service when it breaks down, as well as the competence and the behavior of the service person.

Aesthetics Aesthetics is the subjective dimension indicating the kind of response a user has to a product. It represents the individual’s personal preference.

Perceived Quality Perceived Quality is the quality attributed to a good or service based on indirect measures.

1.1 Terminology and Concepts

1.1.1 Basic Terminology

1.1. TERMINOLOGY AND CONCEPTS CHAPTER 1. INTRODUCTION

Quality Characteristics A.k.a. Critical to Quality Characteristics (CTQs). May be physical, sensory, or temporal properties of a process/product. Obviously related to the dimensions of quality. In the BI world, these are typically known as key performance indicators (KPI). KPI

Quality Engineering “The set of operational, managerial, and engineering activities that a company uses to ensure that the quality characteristics of a product are at the nomi- nal or required levels and that the variability around these desired levels is minimum.” (Montgomery, 2007)

Variables Continuous measurements of some CTQ.

Attributes Discrete measurements of some CTQ.

Target Value The desired level of a particular CTQ. A.k.a. nominal value.

USL & LSL Largest and smallest allowable values of a CTQ.

Specifications The set of permissible values for all CTQs. Either a set of target values, or USL-LSL intervals.

Non-conformity A non conforming product is one that fails to meet the specification.

Fallout The same as non-conformity.

Defect A non-conformity that is serious enough to affect the use of the product.

DPMO Defect per million opportunities.

PPM Parts per million. The same as DPMO.

1.1.2 Statistical Terminology

Exploratory Data Analysis (EDA) An assumption free quantitative inspection of data; “Story telling”; no inference.

Inference Data analysis with the intention of generalizing from a sample to a population. Inclues hypothesis testing, parameter estimation, confidence estimation, prediction, and others.

Causal Inference Inference, with the intention of claiming causal relations between quan- tities under study.

Predictive Analytics Data analysis with the intention of making predictions with future data. Can be seen as inference, without aiming at causality.

Design of experiments (DOE) By far the best and most established way for causal inference. The random assignment of units to groups allows to interpret statistical correlations as causal.

Statistical Process Control (SPC) Data analysis with the intention of identifying anomalous behaviour with respect to history. A.k.a. anomaly detection, or novelty detection, in the machine learning literature..

1.3. MANAGEMENT ASPECTS OF IMPROVING QUALITYCHAPTER 1. INTRODUCTION

1951

2000

A. V. Feigenbaum publishes the first edition of his book, Total Quality Con- trol. G. E. P. Box and K. B. Wilson publish fundamental work on designed experi- ments; focus is on chemical industry. Applications of designed experiments in the chemical industry grow steadily after this. Joseph M. Juran is invited by the Japanese to lecture on quality manage- ment and improvement.

1954

Courses in statistical quality control become widespread in industrial engi- neering academic programs.

1960

ISO publishes the first quality systems standard.

1987

Motorola’s six-sigma initiative begins.

Motorola’s six-sigma approach spreads to other industries.

1997

ISO 9000:2000 standard is issued. Emphasis on supply-chain management and supplier quality. Expansion beyond the traditional industrial setting into financial services, health care, insurance.

Table 1.2: Adapted from (Montgomery, 2007, Table 1.1).

1.3 Management Aspects of Improving Quality

The founding fathers of QC have many do’s-and-don’ts for managers. See Montgomery (2007, Sec 1.4) for details. As usual, we collect recurring ideas:

1. The responsibility for quality rests with management.
2. QC is not a one-time project, but an on-going process. It may advance continuously, or incrementally.
3. QC is (or should be) manifested in organizational structure, training, recruitment, incentives, knowledge management, to name a few.

1.4 Programs and Initiatives

1.4.1 Zero Defects Program (ZD)

Quoting Wikipedia (2015h):

... a management-led program to eliminate defects in industrial production that en- joyed brief popularity in American industry from 1964 to the early 1970’s. Quality expert Philip Crosby later incorporated it into his “Absolutes of Quality Management” and it enjoyed a renaissance in the American automobile industry, as a performance

1.4. PROGRAMS AND INITIATIVES CHAPTER 1. INTRODUCTION

goal more than as a program, in the 1990s. Although applicable to any type of en- terprise, it has been primarily adopted within supply chains wherever large volumes of components are being purchased (common items such as nuts and bolts are good examples).

1.4.2 Quality is Free Initiative

Quoting Montgomery (2007):

... in which management worked on identifying the cost of quality (or the cost of nonquality, as the Quality is Free devotees so cleverly put it). Indeed, identification of quality costs can be very useful, but the Quality is Free practitioners often had no idea about what to do to actually improve many types of complex industrial processes.

1.4.3 Value Engineering Program (VE)

Quoting Wikipedia (2015g):

Value engineering (VE) is systematic method to improve the “value” of goods or products and services by using an examination of function. Value, as defined, is the ratio of function to cost. Value can therefore be increased by either improving the function or reducing the cost. It is a primary tenet of value engineering that basic functions be preserved and not be reduced as a consequence of pursuing value improvements.

1.4.4 Total Quality Management (TQM)

TQM originates in the 1980’s with the ideas of Deming and Juran. It is a very wide framework that attempts at capturing the company-wide efforts required for QC. According to Montgomery (2007, p.23):

TQM has only had moderate success for a variety of reasons, but frequently because there is insufficient effort devoted to widespread utilization of the technical tools of variability reduction. Many organizations saw the mission of TQM as one of training. Consequently, many TQM efforts engaged in widespread training of the workforce in the philosophy of quality improvement and a few basic methods. This training was usually placed in the hands of human resources departments, and much of it was ineffective. The trainers often had no real idea about what methods should be taught, and success was usually measured by the percentage of the workforce that had been “trained,” not by whether any measurable impact on business results had been achieved.

... Another reason for the erratic success of TQM is that many managers and execu- tives have regarded it as just another “program” to improve quality. During the 1950’s and 1960’s, programs such as Zero Defects and Value Engineering abounded, but they had little real impact on quality and productivity improvement.

1.4.5 Six-Sigma

Quoting Montgomery (2007):

1.4. PROGRAMS AND INITIATIVES CHAPTER 1. INTRODUCTION

0 200 400 600 800 1000

0.^ 0.^ 0.^ 0.^

Components

Failure Probability

Figure 1.1: The probability of failure as a func- tion of components under the 3-sigma standard.

0 200 400 600 800 1000

0.^ 0.^ 0.^ 0.^ 0.^

Components

Failure Probability

Figure 1.2: The probability of failure as a func- tion of components under the 6-sigma standard.

Remark 1 (PPM of Six-Sigma). In the six-sigma literature you will find the following table:

Figure 1.3: Defect PPM of varying sigma processes. Source: Souce: http://www.whatissixsigma.net/10-things-about-six-sigma/

As you may notice, the 3-sigma and the 6-sigma defect PPMs in the table are different than the ones in Eq.(1.2). This is explained in (Montgomery, 2007, p.29):

When the six-sigma concept was initially developed, an assumption was made that when the process reached the six-sigma quality level, the process mean was still subject to disturbances that could cause it to shift by as much as 1.5 standard deviations off target.

This means that the convention is to compute Eq.(1.2) while replacing the E [CT Q] = T assump- tion by |E [CT Q] − T | ≤ 1. 5 σ.

1.4.6 Lean Systems

Quoting Wikipedia (2015c) (my own emphasis in bold):

Essentially, lean is centered on making obvious what adds value by reducing ev- erything else. Lean manufacturing is a management philosophy derived mostly from the Toyota Production System (TPS) (hence the term Toyotism is also prevalent) and identified as “lean” only in the 1990s.

1.4.7 Design for Six-Sigma (DFSS)

Quoting Wikipedia (2015a) (my own emphasis in bold):

1.5. DMAIC CHAPTER 1. INTRODUCTION

It is based on the use of statistical tools like linear regression and enables empirical research similar to that performed in other fields, such as social science. While the tools and order used in Six Sigma require a process to be in place and functioning, DFSS has the objective of determining the needs of customers and the business, and driving those needs into the product solution so created. DFSS is relevant for relatively simple items / systems. It is used for product or process design in contrast with process improvement.

1.4.8 Quality Systems and Standards

The first quality standard was issued by the International Standards Organization (ISO) in 1987. Current quality standards are known as the ISO9000 series. These include: ISO

ISO9000:2000 Quality Management System-Fundamentals and Vocabulary.

ISO9001:2000 Quality Management System-Requirements.

ISO9004:2000 Quality Management System-Guidelines for Performance Improvement.

In Israel, it is the Standards Institute of Israel^1 that may give ISO9000 (like any ISO) certifications upon inspecting the candidate organization. As emphasized by Montgomery (2007, p.24), ISO is a set of rules and best practices, mostly oriented at knowledge management. It may help to preserve quality, but it does not, nor does it aim to, improve quality. As such, it will not be the focus of our course, which will focus on statistical tools.

Extra Info 1. [TODO: Just-in-Time, Poka-Yoke]

1.5 DMAIC

There are many names for the process of quantitative re-evaluations of performance against a given target: data driven decision making (DDD), Shewart cycle, etc. We will focus on one such framework, illustrated in Figure 1.4 known as DMAIC: Define, Measure, Analyze, Improve, Control. Here are some general observations on DMAIC:

1. It is aimed at promoting improvement and creative thinking.
2. It is not part of the six-sigma methodology, but will typically take part in its implementation.

What do the stages of DMAIC mean 2?

Define the problem, improvement activity, opportunity for improvement, the project goals, and customer (internal and external) requirements.

Measure process performance.

Analyze the process to determine root causes of variation, poor performance (defects).

Improve process performance by addressing and eliminating the root causes.

Control the improved process and future process performance.

In the following chapter we give a set of statistical tools required for measuring,analyzing and controlling a process.

(^1) https://portal.sii.org.il/heb/qualityauth/certificationtypes/qualitylinks/iso9001/ (^2) http://asq.org/learn-about-quality/six-sigma/overview/dmaic.html

Chapter 2

Exploratory Data Analysis

In this chapter, we give a short review of methods for exploratory data analysis (EDA), a.k.a. descriptive statistics. Descrip-tive Statistics

Recall that our goal is an assumptions-free description of our data. EDA thus consist of computing interpretable summaries of the data, called summary statistics, and visualizations.

2.1 Summary Statistics

We now distinguish between summary statistics that apply to attributes, categorical by definition, and variables, continuous by definition.

2.1.1 Summarizing Categorical Data

Univariate

Summarizing a vector of categorical data can naturally be done by tabulating it, i.e., computing the frequency and relative frequency of each category. Clearly averages, medians, and the likes are incomputable, since categorical data has no ordering, nor does it admit simple operations such as summation.

Extra Info 2. Variability of categorical data can clearly not be measured by its variance, since it does not admit a summation operation. It is, however, possible to define different measures of variability that do apply. The entropy is such an example. Entropy

Bivariate

Generalizing the univariate case to bivariate, or multivariate, one can keep tabulating. I.e., compute the frequency, and relative frequency, of combinations of categories.

2.1.2 Summarizing Continuous Data

Continuous variables admit many more mathematical manipulations than categorical attributes.

Univariate

We start by presenting the most natural summaries of the data. Without going into the formal definition, we refer to them as summary of location. LocationSum-

maries

These include:

Definition 1 (The Mean). The mean, or average, is defined as

x¯ :=

n

∑^ n

i=

xi (2.1)

2.1. SUMMARY STATISTICS CHAPTER 2. EDA

Definition 2 (The Median). The median is the observation that is smaller than half of the sample and larger than half of the sample.

Definition 3 (α-Trimmed Mean). The α-trimmed mean is the average of the observations left after ignoring the largest and the smallest (100α)% of them.

The na¨ıve average is the 0-trimmed mean, and the median is the 0.5-trimmed mean. From summaries of location, we move to summaries of scale. Summaryof Scale

Definition 4 (The Standard Deviation).

s(x) :=

n − 1

∑^ n

i=

(xi − x¯)^2 (2.2)

For the following, we require the definition of the sample quantiles, themselves not a scale summary.

Definition 5 (α Quantile). The α-quantile of the sample is the observation that is larger than (100α)%, and smaller then (100(1 − α))% of the sample.

The empirical maximum and minimum are then x 1. 0 and x 0. 0 , respectively.

Definition 6 (The Range).

Range(x) := max i

{xi} − min i

{xi} = x 1. 0 − x 0. 0 (2.3)

Definition 7 (The Inter Quantile Range- IQR).

IQR(x) := x 0. 75 − x 0. 25 (2.4)

Definition 8 (The Median Absolute Deviation- MAD).

M AD(x) := c{|xi − x 0. 5 |} 0. 5 (2.5)

where c is some constant. In the R function mad(), c is set at 1.4826 so that it estimates σ in a Gaussian population.

After summaries of scale, we move to summaries of skewness, or asymmetry.

Definition 9 (Yule Skewness Measure).

Y U LE(x) :=

1 2 (x^0.^75 +^ x^0.^25 )^ −^ x^0.^5 1 2 IQR(x)^

Bivariate

From univariate data x, we move to bivariate x, y. Clearly we can apply univariate summaries component-wise. We want, however, to summarize the joint behaviour of the data. For this purpose, we assume that data comes in pairs, implying that x and y are of same length.

Definition 10 (Covariance). The sample covariance, or empirical covariance is defined as

Cov(x, y) :=

∑n i=1(xi^ −^ x¯)(yi^ −^ ¯y) n − 1