STATISTICAL QUALITY CONTROL IN TEXTILES

  STATISTICAL QUALITY CONTROL IN TEXTILES

Concept of Quality

Simply, quality refers to one or more desirable characteristics that a product should possess.

Quality is inversely proportional to (unwanted) variability.

Quality Characteristics:

Every product possesses a number of properties that jointly describe what the user or consumer thinks of as quality. These properties are known as quality characteristics.

For example, fiber length is known to be one of the important quality characteristics of a fiber.

Quality Cost :

Preventing, detecting and dealing with defects cause costs that are called quality costs or costs of quality.

Quality costs can be broken down into four broad groups.

(1). Prevention Costs: product/process design. Process control. Burn-in. Training. Quality data acquisition and analysis. (2)Appraisal Costs: Inspection and test of incoming material. Product inspection and test. Material and services consumed. Maintaining accuracy of test equipment. (3).Internal failure Costs: Scrap Rework Retest Failure analysis Downtime Yield losses Downgrading/ off-spacing (4). External failure costs: Complaint adjustment Returned product/material Liability costs External costs

Quality history

Year Event 1700-1900 Quality was largely determined by the efforts of an individual craftsman. 1915-1919 WWI- British government began a supplier certification program. 1919 Technical Inspection Association was formed in England, this later becomes the Institute of Quality Assurance. 1924 W A Shewhart introduced the concept of control charts in Bell Laboratories technical memorandum. 1928 Accepatance sampling techniques were developed and refined by H. F. Dodge and H. G. Romig at Bell Labs. 1931 W. A. Shewhart published Economic control of quality of manufactured product – outlining statistical methods for use in production and control chart methods. 1932-1933 British textile industry began use of statistical techniques for product/process development. 1944 Industrial Quality Control began publication. 1954 E. S. Page introduced CUSUM control chart. 1960 The concept of quality control circle was introduced in japan by K. Ishikawa. 1960 The zero defects (ZD) programs are introduced in US induatries.

1.2 Introduction to Quality Control : Concept of quality control : Quality control describes the directed use of testing to measure the achievement of a specified standard. Quality control is a superset of testing, although it often used synonymously with testing. The concept of quality, namely that the object, the service, the performance – whatever it is – should be “fit for purpose”. Quality does not mean being at the top of the range, but it does mean being efficient (so things happen on time and on cost), reliable (whatever the weather, every day of the week) and giving good value for money. “The goal of quality control should be to identify process steps that best predict outcomes” - A. Blanton Godfrey Link Among Quality, Productivity, and Cost : Take an example. A ring frame machine produces 1000 cops per day. The production cost per cop is Rs. 15. It is seen that 75% of the cops conform to specifications and 60% of the non-conforming cops can be reworked at an additional expense of Rs. 5 per cop, the rest 40% of the nonconforming cops is scrapped. The cost per good cop is then After implementation of process control program, it is seen that 80% of the cops conform to specifications and 60% of the non-conforming cops can be reworked at additional expense of Rs. 5 per cop, the rest is required to be scrapped. The cost per good cop is then Quality Control Methodology : Statistical process control Statistical product control Six sigma Total quality management (TQM) Total quality assurance (TQA) Note: TQM and TQA are beyond the scope of this course. Statistical Process Control : Statistical process control involves measurement and analysis of process variation by using statistical techniques namely control charts. This is most often used for manufacturing processes, in the intent of statistical process control is to monitor product quality and maintain processes to fixed targets. Statistical Product Control : Statistical Product Control involves in making dispositions on accepting or rejecting a lot (or batch) of product that has already been produced. It is most often used to evaluate products that are received from outside sources and where it is not possible to implement statistical process control. Six Sigma : Technical products with many complex components typically have many opportunities for failure or defects to occur. Motorola developed the six-sigma program as a response to the demand for such products. The focus of six sigma lies in reducing variability in key product quality characteristics to the level at which failure or defects are extremely unlikely. 2.1 Random Variable Random Variable: The science and application of statistics deal with quantities that can vary. Such quantities are called variables. Generally, there are two types of variables, depending on the origin and nature of the values that they assign. A variable whose values are obtained by comparing the variable with a standard measuring scale such that they can assign any number including whole numbers and fractional numbers is known as continuous variable. A variable whose values are obtained simply by counting such that they can be always whole numbers or integers, is known as discontinuous or discrete variable. Characteristics of Random Variable : A random variable is generally characterized by its statistical as well as probability characteristics. The basic statistical characteristics include mean (measure of central tendency of the data) and standard deviation (measure of variation in the data). In order to get more information on the data, the frequency distribution of the data need to be evaluated. The probability characteristics include probability distribution and its parameters. 2.2 Continuous Random Variable The Continuous Random Variable x : Let x be a continuous random variable, x ε (xmin,xmax). Let the number of measurements be n. Then x takes the values x1, x2, x3, …,xn. Alternately, we write x takes the values xj, where j=1, 2, 3, …, n. The measured values of x are different, because the variable x is a random variable. Actually, the number of measurements is limited mainly because of the time and the capacity of the measuring instrument. Here, we consider that the number of measurements is very large and it can be increased without any limitation of time. Statistical Characteristics of x : Distribution of x : Let us divide the data domain (xmin,xmax) into m number of classes of constant class interval δx as follows Let us mark the classes by serial number i=1, 2, …, m. Then, we get Histogram of x [1]: Observation : As the number of class increases, the width of class decreases. The contours of the histogram should have roughly similar shape, but the steps become smoother until they “diminish” and become “infinitely small”. This is valid if and only if the chosen “higher” number of class is very small as compared to the number of measurements, that is m< Statistical Characteristics : For a finite (limited) number m of classes, the statistical characteristics of the random variable x are described below. It is known that in a given class, the measured value xj does not differ by more than δx/2 from the class value xi. For simplicity, we consider that all values in a given class are of same value xi. Then, Discussion : Here, we used the class value xi for all calculations. This value may differ by δx/2 from the real measured value xj. As a result, the statistical characteristics obtained by using the class value are erroneous and this error decreases as the class width (δx) decreases. Let us now decrease the class width by (a) increasing the number of classes say twice and also (b) increasing the measurements say twice and repeat this procedure to infinity. As a result, intuitively, the class width becomes smaller and smaller until it becomes “infinitesimal”. Such a class with infinitely small width is defined as “elementary class”, its width is denoted by the differential symbol dx, instead of the symbol δx used to denote higher or finite value. Then, (1) The contours of the histogram should have roughly similar shape, but the steps become smoother until they “diminish” and become “infinitely small”. The contours of the histogram change to a continuous function called probability density function f(x). (3) The area under each elementary class of x is f(x)dx. This product expresses the relative frequency of x in an elementary class of lower limit x and upper limit x+dx. (4). The area under the probability density curve still remains one. Thus, In other words, the integration of all probabilities (“cumulative probability”) from xmin to xmax equals to one. It is possible to find out the cumulative probability of x from the following expression: Note: Here we use integral expression instead of summation expression Remark: For simplicity, we suggest that the domain of values x is finite (closed) interval (xmin,xmax). It can be proved that it is valid even when xmin =-∞ and xmax =∞. Statistical Characteristics : Probability : According to the classical definition of probability, it is the ratio of the number of successful outcomes to the number of all outcomes. If we have n measurements and only ni measurements belong to i-th class (i=1, 2, …,m) then, the probability that a randomly chosen value belongs to i-th class is We see that probability and relative frequency possess the same value, that is, probability is relative frequency and vice-versa. Relative frequency is used when we would like to characterize a value which is already measured. It means relative frequency is used as “ex post”. In opposite to this, probability is used to explore the future based on past investigation. Hence probability is used as “ex ante”. The earlier concept of “relative frequency” and “probability” for a class of certain width is also applicable for an “elementary class”. Thus, we understand the meaning of f(x)dx not only as the relative frequency of x in the elementary class of lower limit x and upper limit x+dx, but also as the “probability of occurrence” (future measured values) of x in the elementary class.   2.5 Discrete Random Variable The Discrete Random Variable x : Let x be a discrete random variable, such that it can only assign values with whole numbers or integers. Let the number of observations be n. Then x takes the values . Alternately, we write x takes the values xj, where j=1, 2, 3, …, n. The observed values of x are different, because the variable x is a random variable. Actually, the number of observations is limited mainly because of the time and the cost of the sample. Here, we consider that the number of observations is very large and it can be increased without any limitation of time. Statistical Characteristics of x : Mean: Variance: where is the mean of the square values of x. Standard deviation: Distribution of x : Let us divide the data domain (xmin, xmax) into m number of classes, each class corresponds to one single value. Let us mark the classes by serial number i=1, 2, …, m. Then, we get Histogram of x : Statistical Characteristics : For a finite (limited) number m of classes, the statistical characteristics of the random variable x are described below. Mean: Mean of square values: Variance: Standard deviation: 3.1 Introduction to Quality : Population & Sample  By population we mean the aggregate or totality of objects or individuals regarding which inferences are to be made. The number of objects or individuals present in the population is known as size of the population. By sample we mean a collection consisting of a part of the objects or individuals of a population which is selected as a basis for making inferences about certain population facts. The number of objects or individuals present in the sample is known as size of the sample. The technique of obtaining a sample is called sampling. Parameter and Statistic A parameter is a population fact which depends upon the values of the individuals comprising the population. For example, mean, variance, etc. associated with a population are known as population parameters. They are constants for a given population. A statistic is a sample fact which depends upon the values of the individuals comprising the sample. For example, mean, variance, etc. associated with a sample are known as sample statistics. Of course, many samples of a given size can be formed from a given population, accordingly the statistics will vary from sample to sample. So they are not constants, but are variables. The difference between the value of a population parameter and the value of the corresponding statistic for a particular sample is known as sampling error. Estimation of Population Parameters We can calculate the value of statistics for a sample, but, it is practically impossible to calculate the value of parameters for a population. Therefore, we often estimate the population parameters based on the sample statistics. The two methods of estimation are point estimation interval estimation Sampling Distribution Sampling distribution of a sample statistic is the relative frequency distribution of a large number of determinations of the value of this statistic, each determination being based on a separate sample of the same size and selected independently  but by the same sampling technique from the same population. Standard Error The standard error of any statistic is the standard deviation of its sampling distribution of a sample statistic. Bias If the mean of the sampling distribution of a statistic is equal to that of the corresponding population parameter, then the statistic is said to be unbiased. If, on the other hand, the mean of the sampling distribution of a statistic is not equal to that of the corresponding population parameter, then the statistic is said to be biased. Bias may arise from two sources: Technique of sample selection (troublesome, no way to assess its magnitude) Character of the statistic (less troublesome, possible to find out its magnitude and direction and then make allowance accordingly). 3.2 Sampling Technique : Simple Random Sample Assume a sample of a given size is selected from a given population in such a way that all possible samples of this size which could be formed from this population have equal chance of selection, then such a sample is called simple random sample. Simple Random Sampling Scheme Step 1: Assign some identification numbers to each individuals of the population. Step 2: Take out an individual randomly, using “Random Number Table” Step 3: Repeat Step 2 until you obtain the desired number of individuals in a sample. Note: No two individuals assigning the same random number can be taken to form this sample. Random Number Table This is a huge collection of ten-digit numbers such that the ten digits not only would occur with equal frequencies but also are arranged in a random order. An Assumption It is practically impossible to numerically identifying each individual of a population either because of the large size of the population or because of the inaccessibility or current non-existence of some of the individuals. In such situations, some of the available individuals may be used as a sample. When some are used, these should be selected at random from those available. Although such samples do not comply with the definition of simple random sample, they are often treated as such. That is, they are simply assumed to be random samples of the population involved. Note: This assumption has been widely used in sampling of textile materials.       3.3 Point Estimation of Population Parameters : Estimation of Population Mean Suppose from a population, we draw a number of samples each containing n random variables x1,x2,.....,xn Then, each sample has a mean which is a random variable. The mean (expected) value of this variable is where μ is the population mean. Thus, sample mean is an unbiased estimator of population mean μ The variance of the variable is where σ is the variance of the population. Note that the standard deviation of the variable is σ/√n , which is known as standard error of the variable Clearly, larger samples give more precise estimates of the population mean than do smaller samples. Estimation of Population Variance Suppose from a large population of mean μ; and variance σ2 , we draw a number of samples each containing n random variables x1,x2,.....,xn Let be the mean of these samples and S2 is the variance of these samples. Clearly, and S2 are variables. The expected value of sample variance is Since E(S2)≠ σ2, the estimator S2 is said to be biased estimate of the population variance. Estimation of Population Standard Deviation Suppose from a large population of mean μ; and variance σ2 , we draw a number of samples each containing n random variables x1,x2,.....,xn Let be the mean of these samples and S2 is the variance of these samples. Clearly, and S2 are variables. The expected value of sample variance is Estimation of Difference Between Two Population Means The variance of the difference of the sample means is Estimation of Population Proportion Assume a population consists of “good” items and “bad” items. Let the proportion of good items in this population be p. Hence, the proportion of bad items in this population is 1-p. Let us draw n items from this population such that it resembles a series of n independent Bernoulli trials with constant probability p of selection of good items in each trial. Then the probability of selection of x good items in n trials, as given by binomial distribution, is nCxPx(1-P)n-x, where Then, the mean (expected) number of good items in a n trials is np and the standard deviation of number of good items in n trials is √np(1-P). Suppose we draw n items from this population to form a sample. Assume x be the number of good items in this sample. Hence, the proportion of good items in this sample is P'=x/n The mean (expected) proportion of good items in this sample is Hence, the sample proportion is an unbiased estimator of population proportion. The variance of sample proportion is given by 3.4 Interval Estimation of Population Parameters : Probability Distribution of Sample Means If the population from which samples are taken is normally distributed with mean μ and variance σ2 then the sample means of size n is also normally distributed with mean μ and variance σ2/n. If the population from which samples are taken is not normally distributed, but has mean μ and variance σ2, then the sample means of size n is normally distributed with mean μ and variance σ2/n when n —>∞ (large sample). Estimation of Population Mean Assume population distribution is normal, regardless of sample size or take large samples n —>∞ , regardless of population distribution. The sample means follows normal distribution with mean μ and standard deviation σ/√n. Popular Confidence Intervals for Population Mean Often, the 95 percent confidence intervals of population mean μ are estimated, they are The 99 percent confidence intervals of population mean μ are Illustration Consider the theoretical population of yarn strength follows normal distribution with mean at 14.56 cN.tex-1 and standard deviation at 1.30 cN.tex-1. Then, for random samples of 450 yarns selected from this population, the probability distribution of sample means follow normal distribution with mean 14.56 cN.tex-1. and standard deviation 0.0613 cN.tex-1 (1.3/21.21 cN.tex-1 = 0.0613 cN.tex-1). This distribution is shown in the next slide. In the long run, 68.26 percent of the mean strength of random samples of 450 yarns selected from this population will involve sampling errors of less than 0.0613 cN.tex-1. Or, the probability of a sample mean being in error by 0.1201 (1.96×0.0613) cN.tex-1 or more is 0.05. Probability Distribution of Sample Means Assume that the population from which samples are taken is normally distributed, but mean μ and variance σ2 are unknown. Then, we consider a statistic T1 as defined below One can see here that as n ≥ 30 the t-distribution practically approaches to a normal distribution. Small sample (practically): n < 30 Large sample (practically): n ≥ 30 Note: For large sample, one can then find out the confidence interval of population mean based on normal distribution as discussed earlier. Estimation of Population Mean Illustration Consider a population of cotton fibers has mean length 25 mm and standard deviation of length is 0.68 mm. Then, for random samples of 10 fibers selected from this population, the probability distribution of sample means follow t-distribution with mean length 25 mm and standard deviation of length 0.23 mm, and degree of freedom 9. The probability of mean length of 10 fibers selected from the population being in error by 0.52 mm (2.262×0.23) or more is 0.05. Probability Distribution of Difference Between Two Sample Means Let x1,x2,.....,xn1 and y1,y2,.....,yn2 be two independent sample observations from two normal populations with means μx,μy and variances σx,σy, respectively. Or, let x1,x2,.....,xn1 and y1,y2,.....,yn2 be two independent large sample observations from two populations with means μx,μy and variances σx,σy, respectively. Then the variable u is a standard normal variable with mean zero and variance one. Estimation of Difference Between Two Populations Probability Distribution of Difference Between Two Sample Means Or, let x1,x2,.....,xn1 and y1,y2,.....,yn2 be two independent small sample observations from two populations with means μx,μy and variances σx,σy, respectively. Then the variable T2 Estimation of Difference Between Two Population Means Probability Distribution of Sample Variances Suppose we draw a number of samples each containing n random variables x1,x2,.....,xn from a population that is normally distributed with mean μ and variance σ2 then the sample means of size n is also normally distributed with mean μ and variance σ2/n. Then the variable Estimation of Population Variance Probability Distribution of Sample Variances Estimation of Population Variance Probability Distribution of Sample Proportions Take large samples n —> ∞, then we know that binomial distribution approaches to normal distribution. Then the variable V is a standard normal variable with mean zero and variance one. where p'=x/n, x is the number of successes in the observed sample, n being the sample size. Note: Earlier we have shown that the standard deviation of p’ is √[p(1-p)]/n. But, when p is not known, p’ can be taken as an unbiased estimator of p, then the standard deviation of p’ can be written as √[p'(1-p')]/n. Estimation of Population Proportion Illustration Consider a population consists of “good” garments and “bad” garments. A random sample of 100 garments selected from this population showed 20 garments were bad, hence the proportion of good garments in this sample was p'=0.80. Then, for random samples of 100 garments taken from this population, the probability distribution of follows normal distribution with mean 0.8 and standard deviation 0.04[√(0.8 * 0.2)/100]. In long run 68.26 percent of the means of random samples of 100 garments selected from this population will involve sampling errors of less than 0.04. Or, the probability of a sample mean being in error by 0.0784 (1.96×0.04) or more is 0.05.                                           3.5 Testing of Hypothesis : Need for Testing Testing of statistical hypothesis is a process for drawing some inference about the value of a population parameter from the information obtained in a sample selected from the population. Types of Test 1. One-tailed test 2. Two-tailed test Illustration Sometimes we may be interested only in the extreme values to one side of the statistic, i.e., the so-called one “tail” of the distribution, as for example, when we are testing the hypothesis that one process is better than the other. Such tests are called one-tailed tests or one-sided tests. In such cases, the critical region considers one side of the distribution, with the area equals to the level of significance. Sometimes we may be interested in the extreme values on both sides of the statistic, i.e., the so-called two “tails” of the distribution, as for example, when we are testing the hypothesis that one process is not the same with the other. Such tests are called two-tailed tests or two-sided tests. In such cases, the critical region considers two side of the distribution, with the area of both sides equals to the level of significance. Testing Procedure Step 1 : State the statistical hypothesis. Step 2 : Select the level of significance to be used. Step 3 : Specify the critical region to be used. Step 4 : Find out the value of the test statistic. Step 5 : Take decision. Statement of Hypothesis Suppose we are given a sample from which a certain statistic such as mean is calculated. We assume that this sample is drawn from a population for which the corresponding parameter can tentatively take a specified value. We call this as null hypothesis. This is usually denoted by This null hypothesis will be tested for possible rejection under the assumption that the null hypothesis is true. Alternative hypothesis is complementary to null hypothesis. This is usually denoted by HA For example if H: μ=μ0 is rejected then HA: μ ≠μ 0, μ <;μ 0, μ >μ 0 Selection of Level of Significance The level of significance, usually denoted by , is stated in terms of some small probability value such as 0.10 (one in ten) or 0.05 (one in twenty) or 0.01 (one in a hundred) or 0.001 (one in a thousand) which is equal to the probability that the test statistic falling in the critical region, thus indicating falsity of H. Specification of Critical Region A critical region is a portion of the scale of possible values of the statistic so chosen that if the particular obtained value of the statistic falls within it, rejection of the hypothesis is indicated. The phrase “test statistic” is simply used here to refer to the statistic employed in effecting the test of hypothesis. The Decision In this step, we refer the value of the test statistic as obtained in Step 4 to the critical region adopted. If the value falls in this region, reject the hypothesis. Otherwise, retain or accept the hypothesis as a tenable (not disproved) possibility. Illustration: A Problem A fiber purchaser placed an order to a fiber producer for a large quantity of basalt fibers of 1.4 GPa breaking strength. Upon delivery, the fiber purchaser found that the basalt fibers, “on the whole”, were weaker and asked the fiber producer for replacement of basalt fibers. The fiber producer, however, replied that the fibers produced met the specification of the fiber purchaser, hence, no replacement would be done. The matter went to court and a technical advisor was appointed to find out the truth. The advisor conducted a statistical test. Illustration: The Test Step 1 : Null hypothesis                 H:μ [GPa]=1.4 Alternative hypothesis       HA:μ [GPa]>1.4 where μ is the population mean breaking strength of basalt fibers as ordered by the fiber purchaser. Step 2 : The level of significance was chosen as α=0.01. Step 3: The advisor wanted to know the population standard deviation σ of strength. So he made a random sample with 65 fibers n=65 and observed the sample standard deviation s of breaking strength was 0.80 GPa. Then, he estimated the population standard deviation of strength as follows: Illustration: The Test Continued Then, the critical region for mean breaking strength is found as: Step 5: The advisor observed that the sample mean breaking strength was 1.12 GPa. Step 6 : The advisor referred to the observed value[GPa] to the critical region he established and noted that it fell in this region. Hence, he rejected the null hypothesis and thus accepted the alternative hypothesis μ[GPa] <1 .4.="" div=""> Errors Associated with Testing of Hypothesis Let us analyze the following situations: Possibilities Course of Action True H Accept (Desired correct action) Reject (Undesired erroneous action) False H Accept (Undesired erroneous action) Reject (Desired correct action) Type I Error Rejecting H when it is true. Type II Error Accepting H when it is false. In situations where Type I error is possible, the level of significance α represents the probability of such an error. Higher is the value of level of significance, higher is probability of Type I error. Type I Error α=0 means complete elimination of occurrence of Type I error. Of course, it implies that no critical region exists, hence H is retained always. In this case, in fact, there is no need to analyze or even collect any data at all. Obviously, while such a procedure would completely eliminate the possibility of making a Type I error, it does not provide a guarantee against error, for every time that the H stated is false, a Type II error would necessarily occur. Similarly, by letting α=1 it would be possible to eliminate entirely the occurrence of Type II error at the cost of committing a Type I error for every true H tested. Thus, the choice of a level of significance represents a compromise effect at controlling the two type of errors that may occur in testing statistical hypothesis Type II Error We see that for a given choice of α, there is always a probability for Type II error. Let us denote this probability by β This depends upon: (1) the value of α chosen (2) the location of critical region (3) the variability of the statistic (4) the amount by which the actual population parameter differs from the hypothesized value of it, stated in H. Because in any real situation, the actual value of a population parameter can never be known, the degree of control exercised by a given statistical test on Type II error can never be determined. Illustration: The beta value Let us assume that the actual population mean breaking strength of basalt fibers supplied by the fiber producer was 1.0 GPa. In this case, Type II error will occur when [GPa] >1.1670. The probability Β of this can be found as under Hence, the probability of Type II error is 0.0475. That is, if in this situation, this test were to be repeated indefinitely, 4.75 percent of the decisions would be of Type II error. Illustration: Effect of alpha on beta α Β As alpha increases, beta decreases 0.001 0.1814 0.005 0.0778 0.010 0.0475 0.050 0.0094 0.100 0.0033 Illustration:Effect of Location of Critical Region on Beta Value Illustration: Effect of Sample Variability on Critical Region on Beta Value Illustration: Effect of Difference Between Actual & Hypothesized Values of Population Parameter Power of A Statistical Test Suppose that the actual value of a population parameter differs by some particular amount from the value, H, hypothesized for it such that the rejection of H is the desired correct outcome. The probability when this outcome will be reached is the probability that the test statistic falls in the critical region. Let us refer to this probability as the power (P) of the statistical test. Since Β represents the probability that the test statistic does not fall in the critical region, then the probability P that the test statistic falls in the critical region is 1-Β=P. Hence, the power of a test is the probability that it will detect falsity in the hypothesis. Power Curve The power curve of a test of a statistical hypothesis, H, is the plot of P-values which correspond to all values that are possible alternatives to H. In other words, power curve may be used to read the probability of rejecting H for any given possible alternative value of μ . Power Curve: Illustration Let us draw the power curve for the previous example. There exists an infinite collection of μ - Values (μ[GPa]<1 .4="" 1.4="" accordingly="" alternative="" and="" are="" div="" exist.="" here.="" hypothesized="" of="" p-values="" possible="" shown="" some="" that="" the="" to="" value="" values=""> μ[GPa] μ[-] Β[-] P[-]=1-Β[-] 0.7 4.67 0 1 0.8 3.67 0.0001 09999 0.9 2.67 0.0038 0.9962 1.0 1.67 0.0475 0.9525 1.1 0.67 0.2514 0.7486 1.2 -0.33 0.6293 0.3707 1.3 -1.33 0.9082 0.0918 Power Curve: Illustration Inadequacy of Statistical Hypothesis Test We have seen that the decision of a statistical hypothesis test is based on whether the null hypothesis is rejected or is not rejected at a specified level of significance (α-value). Often, this decision is inadequate because it gives the decision maker no idea about whether the computed value of the test statistic is just barely in the rejection region or whether it is very far into this region. Also, some decision makers might be uncomfortable with the risks implied by a specified level of significance, say α=0.05. To avoid these difficulties, the P-value approach has been widely used. P-value Approach [2] The P-value is the probability that the test statistic takes on a value that is at least as extreme as the observed (computed) value of the test statistic when the null hypothesis H is true. Otherwise, it can be said that the P-value is the smallest level of significance that would lead to rejection of the null hypothesis with the given data. Testing Procedure Step 1 : State the statistical hypothesis. Step 2 : Find out the value of the test statistic. Step 3 : Find out the P-value. Step 4 : Select the level of significance to be used. Step 5 : Take decision. Illustration: The Test 4. Shewhart Control Charts 4.1 Introduction : Why Control Charts? [1] Basis of Control Charts The basis of control charts is to checking whether the variation in the magnitude of a given characteristic of a manufactured product is arising due to random variation or assignable variation. Random variation: Natural variation or allowable variation, small magnitude Assignable variation: Non-random variation or preventable variation, relatively high magnitude. If the variation is arising due to random variation, the process is said to be under control. But, if the variation is arising due to assignable variation then the process is said to be out of control. Types of Control Charts 4.2 Basics of Shewhart Control Charts : Major Parts of Shewhart Control Chart Central Line (CL): This indicates the desired standard or the level of the process. Upper Control Limit (UCL): This indicates the upper limit of tolerance. Lower Control Limit (LCL): This indicates the lower limit of tolerance. If m is the underlying statistic so that E(m)=μm & Var(m)=σ2m CL=μm UCL=μm+3σm LCL=μm-3σm Why 3σ? Let us assume that the probability distribution of the sample statistic m is (or tends to be) normal with mean μm and standard deviation σm Then, This means the probability that a random value of m falls in-between the 3-σ limits is 0.9973, which is very high. On the other hand, the probability that a random value of m falls outside of the 3-σ limits is 0.0027, which is very low. When the values of m fall in-between the 3-σ limits, the variations are attributed due to chance variation, then the process is considered to be statistically controlled. But, when one or many values of m fall out of the 3-σ limits, the variations are attributed due to assignable variation, then the process is said to be not under statistical control. Analysis of Control Chart: Process Out of Control The following one or more incidents indicate the process is said to be out of control (presence of assignable variation). Analysis of Control Chart: Process Out of Control… Note: Sometimes the 2σ limits are called as warning limits. Then, the 3 σ limits are called as action limits. Analysis of Control Chart: Process Under Control When all of the following incidents do not occur, the process is said to be under control (absence of assignable variation). (1) A point falls outside any of the control limits. (2) Eight consecutive points fall within 3σ limits. (3) Two out of three consecutive points fall beyond 2σ limits. (4) Four out of five consecutive points fall beyond 1σ limits. (5) Presence of upward or downward trend (6) Presence of cyclic trend 4.3 Shewhart Control Charts for Variables : The Mean Chart (-Chart) Let x ij, j=1,2,.....,n be the measurements on ith sample (i=1,2,…,k). The mean i , range Ri , and standard deviation si for ith sample are given by Then the mean of sample means, the mean of sample ranges, and the mean of sample standard deviations are given by Let us now decide the control limits for i. The Range Chart (R-Chart) Let xij, j=1,2,.....,n be the measurements on ith sample (i=1,2,…,k). The range Ri for ith sample is given by Then the mean of sample ranges is given by Let us now decide the control limits for Ri. The Standard Deviation Chart (s-Chart) Let xij, j=1,2,.....,n be the measurements on ith sample (i=1,2,…,k). The range si for ith sample is given by Then the mean of sample ranges is given by Let us now decide the control limits for si Table Illustration Illustration (-chart) Illustration (R-chart) Illustration (s-chart) Illustration (Overall Conclusion) Although the process variability is in control, the process cannot be regarded to be in statistical control since the process average is out of control. 4.4 Shewhart Control Charts for Attributes : Control Chart for Fraction Defective (p-Chart) The fraction defective is defined as the ratio of the number of defectives in a population to the total number of items in the population. Suppose the production process is operating in a stable manner, such that the probability that any item produced will not conform to specifications is p and that successive items produced are independent. Then each item produced is a realization of a Bernouli random variable with parameter p. If a random sample of n items of product is selected, and if D is the number of items of product that are defectives, then D has a binomial distribution with parameter n and p; that is P{D=x}=nCxp (1-p)n-x, x=0,1,...n. The mean and variance of the random variable D are np and np(1-p), respectively. The sample fraction defective is defined as the ratio of the number of defective items in the sample of size n; that is p'= D/n. The distribution of the random variable p' can be obtained from the binomial distribution. The mean and variance of p' are p and p(1-p)/n, respectively. When the mean fraction of defectives p of the population is not known. Let us select m samples, each of size n. If there are Di defective items in ith sample, then the fraction defectives in the ith sample is p'i= Di/n, i=1,2,....,m.The average of these individual sample fraction defectives is Control Chart for Number of Defectives (np-Chart) It is also possible to base a control chart on the number of defectives rather than the fraction defectives. Illustration [2] The following refers to the number of defective knitwears in samples of size 180. Control Chart for Defects (c-Chart) Consider the occurrence of defects in an inspection of product(s). Suppose that defects occur in this inspection according to Poisson distribution; that is Where x is the number of defects and c is known as mean and/or variance of the Poisson distribution. When the mean number of defects c in the population is not known. Let us select n samples. If there are ci defects in ith sample, then the average of these defects in samples of size n is Note: If this calculation yields a negative value of LCL then set LCL=0. Illustration [2] The following dataset refers to the number of holes (defects) in knitwears. 5.1 Introduction : Process Capability Analysis When the process is operating under control, we are often required to obtain some information about the performance or capability of the process. Basis of Control Charts The basis of control charts is to checking whether the variation in the magnitude of a given characteristic of a manufactured product is arising due to random variation or assignable variation. Random variation: Natural variation or allowable variation, small magnitude Assignable variation: Non-random variation or preventable variation, relatively high magnitude. If the variation is arising due to random variation, the process is said to be under control. But, if the variation is arising due to assignable variation then the process is said to be out of control. Process capability refers to the uniformity of the process. The variability in the process is a measure of uniformity of the output. There are two ways to think about this variability. (1) Natural or inherent variability at a specified time, (2) Variability over time. Let us investigate and assess both aspects of process capability. Natural Tolerance Limits The six-sigma spread in the distribution of product quality characteristic is customarily taken as a measure of process capability. Then the upper and lower natural tolerance limits are Upper natural tolerance limit = μ + 3σ Lowe natural tolerance limit = μ - 3σ Under the assumption of normal distribution, the natural tolerance limits include 99.73% of the process output falls inside the natural tolerance limits, that is, 0.27% (2700 parts per million) falls outside the natural tolerance limits. 5.2 Techniques for Process Capability Analysis : Techniques for Process Capability Analysis Histogram Probability Plot Control Charts Histogram It gives an immediate visual impression of process performance. It may also immediately show the reason for poor performance. Example: Yarn Strength (cN.tex-1) Dataset Frequency Distribution Histogram Probability Plot Probability plot can determine the shape, center, and spread of the distribution. It often produces reasonable results for moderately small samples (which the histogram will not). Generally, a probability plot is a graph of the ordered data (ascending order) versus the sample cumulative frequency on special paper with a vertical scale chosen so that the cumulative frequency distribution of the assumed type (say normal distribution) is a straight line. The procedure to obtain a probability plot is as follows. (1) The sample data x1,x2,......,xn is arranged as x(1), x(2), ...., x(n) where x(1) is the smallest observation, x(2) is the second smallest observation, and x(n)is the largest observation, and so forth. (2) The ordered observations x(j) are then plotted again their observed cumulative frequency (j-0.5)/n on the appropriate probability paper. (3) If the hypothesized distribution adequately describes the data, the plotted points will fall approximately along a straight line. Example: Yarn Strength (cN.tex-1) Dataset 5.3 Measures of Process Capability Analysis : Measure of Process Capability: Cp Illustration Measure of Process Capability: Cpu and Cpl The earlier expression of Cp assumes that the process has both upper and lower specification limits. However, many practical situations can give only one specification limit. In that case, the one-sided Cp is defined by Illustration Process Capability Ratio Versus Process Fallout [1] Assumptions: (1) The quality characteristic is normally distributed. (2) The process is in statistical control. (3) The process mean is centered between USL and LSL. Measure of Process Capability: Cpk We observed that Cp measures the capability of a centered process. But, all process are not necessarily be always centered at the nominal dimension, that is, processes may also run off-center, then the actual capability of non-centered processes will be less than that indicated by Cp. In the case when the process is running off-center, the capability of a process is measured by the following ratio Interpretations (1) When Cpk=Cp then the process is centered at the midpoint of the specifications. (2) When Cpk < Cp then the process is running off center. (3) When Cpk=0, the process mean is exactly equal to one of the specification limits. (4) When Cpk<0 lies="" limit.="" mean="" outside="" process="" specification="" td="" the="" then=""> (5) When Cpk <-1 entire="" lies="" limits.="" outside="" process="" specification="" td="" the="" then=""> Illustration Inadequacy of Cpk Measure of Process Capability: Cpm Measure of Process Capability: Cpmk Illustration Take the example of process A and process B. Here T=50 cN. Then, Note to Non-normal Process Output An important assumption underlying the earlier expressions and interpretations of process capability ratio are based on a normal distribution of process output. If the underlying distribution is non-normal then 1) Use suitable transformation to see if the data can be reasonably regarded as taken from a population following normal distribution. 2) For non-normal data, find out the standard capability index 3) For non-normal data, use quantile based process capability ratio 5.4 Inferential Properties of Process Capability Ratios : Confidence Interval on Cp Confidence Interval on Cpk Example Test of Hypothesis about Cp Many a times the suppliers are required to demonstrate the process capability as a part of contractual agreement. It is then necessary that Cp exceeds a particular target value say Cp0. Then the statements of hypotheses are formulated as follows. H: Cp=Cpo (The process is not capable.) HA: Cp>Cpo (The process is capable.) The Cp(high) is defined as a process capability that is accepted with probability 1-α and Cp(low) is defined as a process capability that is likely to be rejected with probability 1-β. Example A fabric producer has instructed a yarn supplier that, in order to qualify for business with his company, the supplier must demonstrate that his process capability exceeds Cp=1.33. Thus, the supplier is interested in establishing a procedure to test the hypothesis H: Cp=1.33 HA: Cp>1.33 The supplier wants to be sure that if the process capability is below 1.33 there will be a high probability of detecting this (say, 0.90), whereas if the process capability exceeds 1.66 there will be a high probability of judging the process capable (again, say 0.90). Then, Cp(low)=1.33, Cp(high)=1.66, and α=β=0.10. Let us first find out the sample size n and the critical value C. Then, from table, we get, n=70 and To demonstrate capability, the supplier must take a sample of n=70 and the sample process capability ratio Cp must exceed C=1.46. Note to Practical Application 6. Non-Shewhart Control Charts 6.1 Introduction : Non-Shewhart Control Charts Cumulative sum control chart (Cusum control chart) Moving average control chart (MA control chart) Exponentially weighted moving average control chart (EWMA control chart) 6.2 Cusum Control Chart : Overview [1 , 2 ] The Shewhart control charts are relatively insensitive to the small shifts in the process, say in the order of about 1.5 σ or less. Then, a very effective alternative is an advanced control chart: Cumulative sum (CUSUM) control chart. Illustration: Yarn Strength Data Consider the yarn strength (cN.tex-1) data as shown here. The first twenty of these observations were taken from a normal distribution with mean μ=10 cN.tex-1 and standard deviation σ=1cN.tex-1. The last ten observations were taken from a normal distribution with mean μ=11 cN.tex-1 and standard deviation σ=1cN.tex-1. The observations are plotted on a Basic Control Chart as shown in the next slide. Illustration: Basic Control Chart CUSUM: What is it? The cumulative sum (CUSUM) of observations is defined as When the process remains in control with mean at μ, the cumulative sum is a random walk with mean zero. When the mean shifts upward with a value μ0 such that μ > μ0 then an upward or positive drift will be developed in the cumulative sum. When the mean shifts downward with a value μ0 such that μ < μ0 then a downward or negative drift will be developed in the CUSUM. Illustration: CUSUM Tabular CUSUM The tabular CUSUM works by accumulating deviations from μ (the target value) that are above the target with one statistic C+ and accumulating deviations from μ (the target value) that are below the target with another statistic C-. These statistics are called as upper CUSUM and lower CUSUM, respectively. If the shift δ in the process mean value is expressed as where μ1 denotes the new process mean value and μ and σ indicate the old process mean value and the old process standard deviation, respectively. Then, K is the one-half of the magnitude of shift. If either Ci+ and Ci+ exceed the decision interval H, the process is considered to be out of control. A reasonable value for H is five times the process standard deviation, H=5σ. Illustration: Tabular CUSUM (Missing Eqn ) Illustration: Tabular CUSUM (Missing Eqn) Illustration: CUSUM Status Chart Concluding Remarks 6.3 MA Control Chart : Moving Average: What is it?  Control Limits If μ denotes the target value of the mean used as the center line of the control chart, then the three-sigma control limits for Mi are The control procedure would consists of calculating the new moving average Mi as each observation xi becomes available, plotting Mi on a control chart with upper and lower limits given earlier and concluding that the process is out of control if Mi exceeds the control limits. In general, the magnitude of the shift of interest and w are inversely related; smaller shifts would be guarded against more effectively by longer-span moving averages, at the expense of quick response to large shifts. Illustration The observations xi of strength of a cotton carded rotor yarn for the periods 1≤i≤30 are shown in the table. Let us set-up a moving average control chart of span 5 at time i. The targeted mean yarn strength is 4.5 cN and the standard deviation of yarn strength is 0.5 cN. Data Calculations The statistic Mi plotted on the moving average control chart will be for periods i≥5. For time periods i<5 1="" also="" are="" average="" averages="" div="" for="" i="" in="" is="" moving="" observations="" of="" periods="" plotted.="" shown="" table.="" the="" these="" values=""> Control Chart Conclusion Note that there is no point that exceeds the control limits. Also note that for the initial periods i Comparison Between Cusum Chart and MA Chart The MA control chart is more effective than the Shewhart control chart in detecting small process shifts. However, it is not as effective against small shifts as cusum chart. Nevertheless, MA control chart is considered to be simpler to implement than cusum chart in practice. 6.4 EWMA Control Chart : Exponentially Weighted Moving Average (EWMA): What is it?[2] Suppose the individual observations are x1,x2,x3,… The exponentially weighted moving average is defined as where 0<λ≤1 is a constant and z0=μ, where μ is process mean. What is it called “Exponentially Weighted MA”? The control limits are where L is known to be the width of the control chart. The choice of the parameters L and λ will be discussed shortly. Choice of λ and L The choice of λ and L are related to average run length (ARL). ARL is the average number of points that must be plotted before a point indicates an out-of-control condition. So, ARL=1/p, where p stands for the probability that any point exceeds the control limits. As we know, for three-sigma limit, p=0.0027, so ARL=1/0.0027=370. This means, even if the process is in control, an out-of-control signal will be generated every 370 samples, on the average. In order to detect a small shift in process mean, which is what is the goal behind the set-up of an EWMA control chart, the parameters λ and L are required to be selected to get a desired ARL performance. The following table illustrates this. As a rule of thumb, λ should be small to detect smaller shifts in process mean. It is generally found that 0.05≤λ≤0.25 work well in practice. It is also found that L=3 (3-sigma control limits) works reasonably well, particularly with higher values of λ. But, when λ is small, that is, λ≤0.1, the choice of L between 2.6 and 2.8 is advantageous. Illustration Let us take our earlier example of yarn strength in connection with MA control chart. Here, the process mean is taken as μ=4.5 cN and process standard deviation is taken as σ=0.5 cN. We choose λ=0.1 and L=2.7. We would expect this choice would result in an in-control average run length equal to 500 and an ARL for detecting a shift of one standard deviation in the mean of ARL=10.3. The observations of yarn strength, EWMA values, and the control limit values are shown in the following table. Table Graph Conclusion Note that there is no point that exceeds the control limits. We therefore conclude that the process is in control. 7. Acceptance Sampling Techniques 7.1 Introduction : Why Acceptance Sampling? [1] Acceptance Sampling: Attributes & Variables The input or output articles are available in lots or batches (population). It is practically impossible to check each and every article of a batch. So we randomly select a few articles (sample) from a batch, inspect them, and then draw conclusion whether the batch is acceptable or not. This is called acceptance sampling. Sometimes the articles inspected are merely classified as defective or non-defective. Then we deal with acceptance sampling of attributes. Sometimes the property of the articles inspected is actually measured, then we deal with acceptance sampling of variables. 7.2 Acceptance Sampling of Attributes : Definition Let us take a sample of size n randomly from a batch. If the number of defective articles in the sample is not greater than a certain number c, then the batch is accepted; otherwise, it is rejected. This is how we define acceptance sampling plan. Probability of Acceptance Let us assume that the proportion of defective articles in the batch is p. Then, when a single article is randomly chosen from a batch, the probability that it will be defective is p. Further assume that the batch size is sufficiently larger than the sample size n so that this probability is the same for each article in the sample. Thus, the probability of finding exactly r number of defective articles in a sample of size n is Now the batch will be accepted if r = c, i.e., if r=0 or 1 or 2 or….or c. Then, according to the addition rule of probability, the probability of accepting the batch is Operating Characteristic This tells that once n and c are known, the probability of accepting a batch depends only on the proportion of defectives in the batch. Thus, Pa(p) is a function of p. This function is known as Operating Characteristic (OC) of the sampling plan. Acceptable Quality Level (AQL) p=p1 This represents the poorest level of quality for the producer’s process that the consumer would consider to be acceptable as process average. Ideally, the producer should try to produce lots of quality better than p1. Assume there is a high probability, say 1-α, of accepting a batch of quality p1. Then, the probability of rejecting a batch of quality p1 is α, which is known as producer’s risk. when p=p1 then pa (p1)=1-α Lot Tolerance Proportion Defectives (LTPD) p=p2>p1 This represents the poorest level of quality that the consumer is willing to accept in an individual lot. Below this level, it is unacceptable to the consumer. In spite of this, let there will be a small chance (probability) β of accepting such a bad batch by the consumer, β is known as consumer’s risk. When p > p2 then pa(p2)=β. LTPD is also known as rejectable quality level (RQL). Finding of n and c Illustration [2] Design a sampling plan for which AQL is 2%, LTPD is 5%, and the producer’s risk and consumer’s risk are both 5%. Effect of Sample Size n on OC Curve Effect of Acceptance Number c on OC Curve 7.3 Acceptance Sampling of Variables : Problem Statement Consider a producer supplies batches of articles having mean value μ of a variable (length, weight, strength, etc.) and standard deviation σ of the variable. The consumer has agreed that a batch will be acceptable if   where μ0 denotes the critical (nominal) mean value of the variable and T indicates the tolerance for the mean value of the variable. Otherwise, the batch will be rejected. Here, the producer’s risk α is the probability of rejecting a perfect batch, the one for which μ= μ0 . The consumer’s risk β is the probability of accepting an imperfect batch, the one for which μ= μ0 ±T. Sampling Scheme Let us assume that the probability distribution of mean of samples, each of size n, taken from a batch, is (or tends to) normal with mean μ, where μ =μ0 and standard deviation σ/√n Then, the batch will be accepted if where t denotes the tolerance for sample mean . Otherwise, the batch will be rejected. Here, the producer’s risk α is the probability of rejecting a perfect batch and the consumer’s risk β is the probability of accepting an imperfect batch. The Producer’s Risk Condition The Consumer’s Risk Condition Finding n and t : Illustration A producer (spinner) supplies yarn of nominal linear density equal to be 45 tex. The customer (knitter) accepts yarn if its mean linear density lies within a range of 45 ± 1.5 tex. As the knitter cannot test all the yarns supplied by the spinner, the knitter would like to devise an acceptance sampling scheme with 10% producer’s risk and 5% consumer’s risk . Assume the standard deviation of count within a delivery is 1.2 tex. Assume the mean linear density of yarn samples, each of size n, follows (or tends to follow) normal distribution with mean 45 tex and standard deviation 1.2 tex. Then, the standard normal variable takes the following values Thus, the sampling scheme is as follows: Take a yarn sample of size nine and accept the delivery if the sample mean lies in the range of 45±0.68 tex, that is in-between 44.32 tex and 45.68 tex, otherwise, reject the delivery. 8. Six Sigma and Its Application to Textiles 8.1 What is Six Sigma? [1] Six Sigma as a Statistical Measure The lowercase Greek letter sigma—s—stands for standard deviation, which is known to be a measure of variation present in a set of data, a group of items, or a process. For example, if you weigh fabric pieces of many different sizes, you’ll get a higher standard deviation than if you weigh fabric pieces of all the same size. The sigma measurement develops to help (1) Focussed measurements of paying the customers of a business. Many of the measures, such as labour hours, costs, and sales volume, companies have traditionally used evaluate things that are not related to what the customer really cares about (2) Provide a consistent way to measure and to compare different processes. Using the sigma scale, we could assess and compare performance of, say, the cloth dying process with the cloth delivery process—two very different but critical activities. In the language of Six Sigma, customer requirements and customer expectations are called “critical to quality (CTQ)”. Six Sigma as a Goal “Even if you’re on the right track, you’ll get run over if you just sit there.” WILL ROGERS When a business violates important customer requirements, it is generating defects, complaints, and cost. The greater the number of defects that occur, the greater the cost of correcting them, as well as the risk of losing the customers. Ideally, your company wants to avoid any defects and the resulting cost in money and customer satisfaction. We use the sigma measure to see how well or poorly a process performs and to give everyone a common way to express that measure. Six Sigma as a System of Management A significant difference between Six Sigma and seemingly similar programs of past years is the degree to which management plays a key role in regularly monitoring program results and accomplishments. managers at all levels are held accountable for a variety of measures: • Customer satisfaction • Key process performance • Scorecard metrics on how the business is running • Profit-and-loss statements • Employee attitude These measures provide feedback on the performance of the regions. At regular meetings, managers review key measures within their hotels and select new Six Sigma projects that target those measures that have fallen off. In short, Six Sigma is a system that combines both strong leadership and grassroots energy and involvement. In addition, the benefits of Six Sigma are not just financial. People at all levels of a Six Sigma company find that better understanding of customers, clearer processes, meaningful measures, and powerful. 8.2 Six Sigma Quality [2] Limitation of Three Sigma Quality Let us consider that a product consists of 100 mutually independent parts and the probability of producing the product within 3-sigma control limit is 0.9973. Then, the probability that any specific part of the product is non-defective is 0.9973×0.9973×0.9973×…100 times = (0.9973)100 = 0.7631. If it is assumed that all 100 parts must be non-defective for the product to function satisfactorily, then 23.70% of the products produced under the three-sigma quality will be defective. This is not an acceptable situation, because there are many products that have many components assembled in them. Six-sigma Quality Under six-sigma quality, the probability that any specific unit of the hypothetical product above is non-defective is 0.9999998, or 0.002 parts per million (ppm), a much better situation. Statistics of Six-sigma Quality When six-sigma concept was 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 times standard deviations off target. Under this scenario. A six-sigma process would produce about 3.4 ppm defective.                                      8.3 Themes of Six Sigma [1] Theme One: Genuine Focus on Customer As mentioned, companies launching Six Sigma have often been appalled to find how little they really understand about their customers. In six sigma, customer focus becomes the top priority. For example, the measures of Six Sigma performance begin with the customer. Six Sigma improvements are defined by their impact on customer satisfaction and value. Theme Two: Data- and-Fact-Driven Management Six Sigma takes the concept of “management by fact” to a new, more powerful level. Despite the attention paid in recent years to improved information systems, knowledge management, and so on, many business decisions are still being based on opinions and assumptions. Six Sigma discipline begins by clarifying what measures are key to gauging business performance and then gathers data and analyzes key variables. Then problems can be much more effectively defined, analyzed, and resolved—permanently. At a more down-to-earth level, Six Sigma helps managers answer two essential questions to support data-driven decisions and solutions. 1. What data/information do I really need? 2. How do we use that data/information to maximum benefit? Theme Three: Processes Are Where Action is Whether focused on designing products and services, measuring performance, improving efficiency and customer satisfaction, or even running the business, Six Sigma positions the process as the key vehicle of success. One of the most remarkable breakthroughs in Six Sigma efforts to date has been convincing leaders an managers—particularly in service-based functions and industries—that mastering processes is a way to build competitive advantage in delivering value to customers. Theme Four : Proactive Management Most simply, being proactive means acting in advance of events rather than reacting to them. In the real world, though, proactive management means making habits out of what are, too often, neglected business practices: defining ambitious goals and reviewing them frequently, setting clear priorities, focusing on problem prevention rather than firefighting, and questioning why we do things instead of blindly defending them. Far from being boring or overly analytical, being truly proactive is a starting point for creativity and effective change. Six Sigma, as we’ll see, encompasses tools and practices that replace reactive habits with a dynamic, responsive, proactive style of management. Theme Five : Boundary Less Collaboration “Boundarylessness” is one of Jack Welch’s mantras for business success. Years before launching Six Sigma, GE’s chairman was working to break down barriers and to improve teamwork up, down, and across organizational lines. The opportunities available through improved collaboration within companies and with vendors and customers are huge. Billions of dollars are lost every day because of disconnects and outright competition between groups that should be working for a common cause: providing value to customers. Theme Six : Drive for Perfection & Tolerate Failure How can you be driven to achieve perfection and yet also tolerate failure? In essence, though, the two ideas are complementary. No company will get even close to Six Sigma without launching new ideas and approaches—which always involve some risk. If people who see possible ways to be closer to perfect are too afraid of the consequences of mistakes, they’ll never try. Fortunately, the techniques we’ll review for improving performance include a significant dose of risk management so the downside of setbacks or failures is limited. The bottom line, though, is that an company that makes Six Sigma its goal will have to keep pushing to be ever more perfect while being will in to accept—and manage—occasional setbacks. 8.4 Implementation of Six Sigma  On-ramp 1: The Business Transformation Employees and managers can often sense the need for a company to break away from old habits and to transform itself. For those organizations with the need, vision, and drive to launch Six Sigma as a full-scale change initiative, this first on-ramp, business transformation, is the right approach. Everywhere, management will be trying to drive results from the changes and to control their impact. As an employee, you may find yourself on a Six Sigma team challenged to improve a critical business process or a key product. Teams chartered along the business-transformation highway are often asked to look at key process areas and to make recommendations for change. These teams may scrutinize:- Distribution of the products by the company The effectiveness of the sales process New-product development Critical customer complaints Product defects and habitual problems Information systems critical to business decision making Large-scale cost reductions If your company chooses the business-transformation on ramp, you’ll know it! This approach will have an impact on your work, how you measure your work, how you interact with customers and peers, and how you and your job performance are evaluated. On-ramp 2: Strategic Improvement The middle on-ramp offers the most options. A strategic improvement effort can be limited to one or two critical business needs, with teams and training aimed at addressing major opportunities or weaknesses. Or, it may be a Six Sigma effort concentrated in limited business units or functional areas of the organization. In fact, to those directly involved, the strategic-improvement approach can seem as all-encompassing as the all-out corporate wide effort, but it simply is not as extensive or ambitious as the most aggressive efforts. On the other hand, a number of companies that have started with the more limited strategic focus have later expanded Six Sigma into a full-scale corporate change initiative, and yours may evolve that way, too. On-ramp 3: Problem Solving The problem-solving approach is best for companies that want to tap into the benefits of Six Sigma methods without creating major change ripples within the organization. If your business takes this approach, there’s a strong probability that only a few people will be significantly engaged in the effort—unless, of course, it gets ramped up later. The benefit of this approach is in focusing on meaningful issues and addressing their root causes, using data and effective analysis rather than plain old gut feel. As an example of this on-ramp, a major real estate company is running a few training classes and putting people to work on some key problems. Although the company will have a handful of Black Belts trained and some projects completed in a few months, that’s about all you can predict for now. This company, like most others taking a problem-solving route, is really just kicking the tires on the Six Sigma vehicle. 8.5 Process of Problem Solving Using Six Sigma  Phase 1: Identifying and Selecting Project(s) In this phase, management reviews a list of potential Six Sigma projects and selects the most promising to be tackled by a team. Setting good priorities is difficult but very important to making the team’s work pay off. We counsel leaders to pick projects based on the “two Ms”: meaningful and manageable. A project has to have real benefit to the business and customers and has to be small enough so the team can get it done. At the end of this phase, your leadership group should have identified high-priority problems and given them some preliminary boundaries. Phase 2: Forming The Team Hand-in-hand with problem recognition comes team and team leader (Black Belt or Green Belt) selection. Naturally, the two efforts are related. Management will try to select team members who have good working knowledge of the situation but who are not so deeply rooted in it that they may be part of the problem. Smart leaders realize that DMAIC team participation should not be handed to idle slackers. If you are chosen for a team, it means that you are viewed as someone with the smarts and the energy to be a real contributor! Phase 3: Developing The Charter The Charter is a key document that provides a written guide to the problem or project. The Charter includes the reason for pursuing the project, the goal, a basic project plan, scope and other considerations, and a review of roles and responsibilities. Typically, parts of the Charter are drafted by the Champion and added to and refined by the team. In fact, the Charter usually changes over the course of the DMAIC project. Phase 4: Training The team Training is a high priority in Six Sigma. In fact, some people say that “training” is a misnomer because a lot of “classroom” time is spent doing real work on the Black Belt’s or team’s project. The focus of the training is on the DMAIC process and tools. Typically, this training lasts one to four weeks. The time is spread out, though. After the first week, the team leader and/or team members go back to their regular work but budget a key portion of their time to working on the project. After a two- to five-week “intersession” comes the second training session, then another work period and another week of training. Phase 5: DMAIC and implementing Solutions Nearly all DMAIC teams are responsible for implementing their own solutions, not just handing them off to another group. Teams must develop project plans, training, pilots, and procedures for their solutions and are responsible for both putting them in place and ensuring that they work—by measuring and monitoring results—for a meaningful period of time. Phase 6: Handing Off The Solution Eventually, of course, the DMAIC team will disband, and members return to their “regular” jobs or move on to the next project. Because they frequently work in the areas affected by their solutions, team members often go forward to help manage the new process or solution and ensure its success. The hand-off is sometimes marked by a formal ceremony in which the official owner, often called “Process Owner,” of the solutions accepts responsibility to sustain the gains achieved by the team. (Dancing and fun may go on until the wee hours of the morning. . . .) Just as important, the DMAIC team members take a new set of skills and experience to apply to issues that arise every day.    8.6 Case Studies : Case Study I: Reduction of Shade Variation of Linen Fabrics Using Six Sigma [3] The objective of this work was to reduce the shade matching time in the fabric dyeing process by optimizing the effect of the controllable parameters. The problem was tackled using the DMAIC cycle of disciplined Six Sigma methodology. Initially, the process baseline sigma level was found as 0.81 and a target sigma level was set at 1.76. Actions taken on the critical activities led to the reduction in average excess time as 0.0125 h/m. The yield of the overall process has improved to 82% with an improved sigma level of 2.34. The estimated annual saving is to the tune of Rupees eighteen lakhs (over $40,000). Project Background In the weaving department, for some qualities, fabrics are woven from the dyed yarn (called yarn dyed quality) and for other qualities, fabrics are prepared from grey/bleached yarn and coloring is carried out in the dyeing department (called piece dyed quality) . These fabrics are single colored fabrics. Shade variation is found in the case of piece dyed quality of the fabrics, which literary means not meeting the customer’s expectation regarding that particular shade. Owing to variation of the shade as compared to the sample, color addition/ color stripping are carried out which unnecessarily increases the process cycle time, labor cost along with the wastage of dyes, chemicals, power, and sometimes fabrics also. The sole purpose of selection of this project was: I to improve the productivity by reducing the shade matching time. II to reduce the cost of poor dyeing quality. III to reduce the process cycle time. IV to deliver the product to the customer on time. V to impart moral strength to shop floor people regarding shade matching. Objective of The Project The core objective of the project is to reduce the shade matching time in the fabric dyeing process by optimizing the effect of the controllable parameters involved in the dyeing process through a disciplined Six Sigma methodology. Six Sigma Methodology (DMAIC Cycle) Define Phase of the Study The steps in this phase are: I Developing a project charter, where primarily the project team and the project boundary are identified. II Preparing a process diagram, which is shown in Table 1 in the form of supplier-input-process-output-customer (SIPOC). III Defining terms related to a Six Sigma like unit (each lot of fabric under the process of dyeing) and defect (a particular lot/unit generates a defect if its shade does not meet the customer’s requirement at the first attempt after the dyeing process). IV Estimating project completion time of four months including system standardization Measure Phase of The Study The steps in this phase are: 1) to identify the relevant parameters for measurement. 2) to develop proper data collection plans both for historical data and planned data. 3) to estimate the baseline process performance through the sigma level based on process yield and fixing the target sigma level. 4) to estimate the status quo of the dyeing process through the identified parameters. 5) to segregate the significant factors for the next course of study through analysis and improvement phase Performance of Fabric Quality Comparison of Defects Against Shades Olive, D.K. grey, teak, magenta, and navy were processed more frequently along with the frequent generation of defective lots. These five shades were contributing to 68% of the total defective lots. Machinewise Performance of Fabric Lots Case Study II: Reduction of Yarn Packing Defects Using Six Sigma [4] This project was aimed at reduction of rejection during packing of finished yarn cones through measuring the current performance level and initiating proper remedial action thereafter. Pareto Chart for No. of Defectives for various Defects Almost 65% of rejections were due to weight variation of cones (i.e., either overweight and underweight). Defectives due to Underweight The major counts in terms of packing rejection due to underweight were Ne 2/42sP, Ne 4/12sP, Ne 2/20sP, Ne 3/20sP, Ne 1/30sV, Ne 3/12sP. Defectives due to Overweight It is found that the major counts in terms of packing rejection due to overweight were Ne 2/42sP, Ne 1/30sV, Ne 4/12sP, Ne 2/57sLY, Ne 3/20sP.