Textile Engineering

Statistical Techniques, Design of Experiments and Stochastic Modeling

Anindya Ghosh, Bapi Saha, Prithwiraj Mal

• Publisher: CRC Press
• Year: 2021
• Print ISBN: 9781032600000
• eBook ISBN: 9781000455434
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

Focusing on the importance of the application of statistical techniques, this book covers the design of experiments and stochastic modeling in textile engineering. Textile Engineering: Statistical Techniques, Design of Experiments and Stochastic Modeling focuses on the analysis and interpretation of textile data for improving the quality of textile processes and products using various statistical techniques.

FEATURES

• Explores probability, random variables, probability distribution, estimation, significance test, ANOVA, acceptance sampling, control chart, regression and correlation, design of experiments and stochastic modeling pertaining to textiles
• Presents step-by-step mathematical derivations
• Includes MATLAB® codes for solving various numerical problems
• Consists of case studies, practical examples and homework problems in each chapter

This book is aimed at graduate students, researchers and professionals in textile engineering, textile clothing, textile management and industrial engineering. This book is equally useful for learners and practitioners in other scientific and technological domains.

1Introduction

DOI: 10.1201/9781003081234-1

1.1 Introduction

Statistics is dealing with the collection of data, their subsequent description, summarization and analysis, which often leads to the drawing of conclusions. In order to draw a conclusion from the data, we must consider the possibility of chance or probability. For example, suppose that a finishing treatment was applied to a fabric to reduce its bending rigidity. The average reduction in bending rigidity was found to be lower for the fabric samples receiving the finishing treatment than that of untreated fabric samples. Can we conclude that this result is due to the finishing treatment? Or is it possible that the finishing treatment is really ineffective and that the reduction in bending rigidity was just a chance occurrence?

In statistics, we are interested in finding information about a total collection of elements, which refers to population. Nevertheless, the population is often too huge to inspect each of its members. For example, suppose that a garment manufacturer marks the size of men's trousers according to the waist size. For the purpose of designing the trousers, the manufacturer needs to know the average waist size of the men in the population to whom the trousers will be sold. In this example, in order to find out the accurate average, the waist size of every man in the population would have to be measured. However, there may be millions of men in the population and hence, it is quite impracticable to measure the waist size of every man in the population. In such cases, we try to learn about the population by examining a subgroup of its elements. This subgroup of a population is called a sample.

Definition 1.1: The total collection of all the elements is called a population. A subgroup of the population is called a sample.

The sample must be representative as well as informative about the population. For example, suppose that we are interested about the length distribution of a given variety of cotton fibre and we examine the lengths of the 300 fibres from a combed sliver. If the average length of these 300 fibres is 23.4 mm, are we justified in concluding that this is approximately the average length of the entire population? Probably not, for we could certainly argue that the sample chosen in this case is not representative of the total population because usually combed sliver contains a smaller number of short fibres. The representation does not mean that the length distribution of fibre in the sample is exactly that of the total population, but the sample should be chosen in such a way that all parts of the population had an equal chance to be included in the sample. In general, a given sample cannot be considered to be representative of a population unless that sample has been chosen in a random manner. This is because any specific non-random selection of a sample often results in biasing toward some data values as opposed to others.

Definition 1.2: A sample of k members of a population is said to be a random sample, if the members are chosen in such a way that all possible choices of the k members are equally likely.

It is unrealistic and impossible to examine every member of the population. Once a random sample is chosen, we can use statistical inference to say something meaningful about the entire population by studying the elements of the sample.

It is inevitable that the natural and man-made products vary one from another. For example, a cotton plant produces fibres with varying lengths. In a mass-producing manufacturing process, it is impossible to produce all the articles which are absolutely identical. Had there been no variation in a population, then it would have been sufficient to examine only one member to know everything about the population. But if there is variation in a population, the examination of only a few members contained in a random sample may provide incomplete and uncertain information about the population. The random or chance sources of variation give rise to this uncertainty. For example, the growth habit of the cotton plant, genetics and environmental conditions during cotton fibre development are the probable sources of random variation of cotton fibre length. One of the important tasks of statistics is to measure the degree of this uncertainty. Although, the values of a random variable may vary arbitrarily, there is generally an underlying pattern in the variation. The presence of such pattern in the data enables us to draw some meaningful conclusions about the population from the results of the sample. The recognition and description of patterns in the data are of fundamental importance in statistics.

Statistics is dealing with the quantities that vary. Such quantities are known as variables. The variables that are affected by random sources of variation are called random variables. Random variables are of two kinds, viz., discrete and continuous. A random variable is called discrete random variable if the set of its possible values is either finite or countable infinite. The number of warp breaks in a loom shed per shift, the number of defective ring frame bobbins produced in a day and the number of defects per 100 square meters of a fabric are some examples of discrete random variables. A random variable is called a continuous random variable if the set of its possible values is not finite and uncountable. A continuous random variable takes on value in an interval. The fineness of cotton fibre, tenacity of a yarn and bending rigidity of a fabric are some examples of continuous random variables.

1.2 Organization of the Book

This book is divided into 12 chapters which discuss various statistical methods for the analysis of the data pertaining to the domain of textile engineering. A brief view of these chapters is given below.

This chapter is an elementary overview of statistics. A brief idea of the population, sample, random sample, random variation, uncertainty and variables is given in this chapter.

Frequency distribution and histograms are among the few techniques which are largely used in summarizing the data. In addition, a suitable representation of the data which is simple in nature is helpful in many situations. Chapter 2 discusses the representation and summarization of the data. It deals with the frequency distribution, relative frequency, histogram, probability density curve, mean, median, mode, range, mean deviation and variance.

Chapter 3 deals with the concept of probability. Firstly, some basic terminologies, viz., random experiment, sample space and events have been introduced. It is followed by the various definitions of probability, set theoretic approach to probability and conditional probability. A brief idea of random variable, probability mass function, probability density function and probability distribution function has been furnished in this chapter. It then explains the expectation, variance, moment, moment generating function, characteristic function, multivariate distribution and transformation of random variables.

Chapter 4 covers discrete probability distribution. Some important discrete probability distributions, viz., Bernoulli distribution, binomial distribution, Poisson distribution and hypergeometric distribution have been explained in this chapter with numerical examples and MATLAB^® coding.

Chapter 5 contains the discussion on continuous probability distribution. Some important univariate continuous probability distributions, viz., uniform distribution, exponential distribution, Gaussian or normal distribution and lognormal distribution have been explained in this chapter with numerical examples and MATLAB^® coding. Normal approximation to the binomial distribution as well as Poisson distribution has also been explained in this chapter. In addition, this chapter provides a brief view of bivariate normal distribution.

Statistical inference is an important decision-making tool to say something meaningful about population on the basis of sample information. The statistical inference is divided into two parts, viz. estimation and hypothesis testing. Chapter 6 is dealing with the sampling distribution and estimation. It begins with the explanation of distribution of sample mean, central limit theorem, chi-square distribution, Student's t-distribution and F-distribution. It then discusses on the point estimation and interval estimation. The section of point estimation is comprised of the discussion of unbiased estimator, consistency, minimum variance unbiased estimator, sufficiency and maximum likelihood estimator. The interval estimation of mean, difference between two means, proportion, difference between two proportions, variance and the ratio of two variances have been explained in the section of interval estimation with numerical examples and MATLAB^® coding.

Chapter 7 treats the statistical test of significance. At the outset, the concept of null hypothesis, alternative hypothesis, type-I and type-II errors have been explained. Then various statistical tests concerning mean and difference between two means with small and large sample sizes, proportions, variance and difference between two variances, difference between expected and observed frequencies have been described with numerical examples and MATLAB^® coding.

Analysis of variance (ANOVA) is a common procedure for comparing multiple population means across different groups while the number of groups is more than two. In Chapter 8, one-way ANOVA, two-way ANOVA with and without replication have been explained with numerical examples and MATLAB^® coding. This chapter also discusses the multiple comparisons of treatment means.

A key objective in many statistical investigations is to establish the inherent relationships among the variables. This is dealt with the regression and correlation analysis. In Chapter 9, firstly, simple linear regression, coefficient of determination, correlation coefficient and rank correlation coefficient have been explained. It then explains quadratic regression and multiple linear regression. In addition, a matrix approach of regression has been discussed with reference to simple linear regression, quadratic regression, multiple linear regression and multiple quadratic regression. Finally, the test of significance of regression coefficients has been discussed. A substantial number of numerical examples and MATLAB^® coding have been furnished in this chapter.

It is of utmost importance to conduct experiments in a scientific and strategic way to understand, detect and identify the changes of responses due to change in certain input factors in complex manufacturing process. Hence, a statistical design of experiment (DOE) is required to layout a detailed experimental plan to understand the key factors that affect the responses and optimizing them as required. Chapter 10 discusses the need of designing such experiments to investigate and find the significant and insignificant factors affecting the responses. Initially, the basic principle of the design of experimentations is discussed. The importance of randomization, replication, different types of blocking and their effects are explained along with examples. The need of factorial design, fractional factorial design and response surface designs are explicated in details with examples. However, none of these designs of experiments consider the noise factor, which might have a significant role affecting the response. The Taguchi design of experiment considers noise factors and helps to overcome such limitations, which is explained towards the end of this chapter along with examples. The advantages and limitations of each design are also thoroughly discussed. Lastly, numerical examples along with MATLAB^® coding are furnished in this chapter.

Chapter 11 includes sampling inspection and various means to check whether a lot is acceptable or rejected. Acceptance sampling, which is a statistical technique to deal with inspection of raw materials or products and to conclude whether to accept or reject in context to consumer's and producer's risk along with examples is initially discussed. Acceptance sampling for both the attributes and variance, average outgoing quality, average total inspection, assurance about a minimum/maximum/mean value are explained with suitable examples. Various control charts, which are a statistical technique of dealing with sampling inspection of materials at the conversion stage, are also described with appropriate examples. The need and concepts of control charts are explained initially. The centre line, warning limits and action limits and their interpretation with regard to control charts for mean, range, fraction defectives and number of defects are discussed with suitable examples of each. Finally, numerical examples with MATLAB^® coding are given in this chapter.

In many situations, when a system is time dependent, a sequence of random variables which are functions of time is termed as a stochastic process. Chapter 12 deals with the stochastic modelling and its application in textile manufacturing. It begins with a brief discussion on the Markov chain followed by an application of it on the mechanism of a carding machine. The process of formation of the stochastic differential equation for a random process is discussed subsequently with its application in resolving the fibre breakage problem in the yarn manufacturing process. Finally, a numerical example and corresponding MATLAB^® coding is given in this chapter.

Table of contents

• Cover Page
• Half-Title Page
• Title Page
• Copyright Page
• Contents
• Foreword
• Preface
• Authors
• Chapter 1 Introduction
• Chapter 2 Representation and Summarization of Data
• Chapter 3 Probability
• Chapter 4 Discrete Probability Distribution
• Chapter 5 Continuous Probability Distributions
• Chapter 6 Sampling Distribution and Estimation
• Chapter 7 Test of Significance
• Chapter 8 Analysis of Variance
• Chapter 9 Regression and Correlation
• Chapter 10 Design of Experiments
• Chapter 11 Statistical Quality Control
• Chapter 12 Stochastic Modelling
• Appendix A: Statistical Tables
• Appendix B: MATLAB® Coding for Statistical Tables
• Appendix C: Answers to Exercises
• Bibliography
• Index