Quality control – history and recent development V. Jevremović, K. Veljković, H. Elfaghihe Faculty of Mathematics, University of Belgrade, Serbia Introductory remarks Quality – from latin “qualitas” introduced by Cicero (106-43,BC), term originated from Greek, given by Plato (428-348,BC) ... It is not possible to cover all the history for such wide topic It is not possible to enumerate all of its recent development… “Few important names in Quality control and few, old and new ideas” Working with process or systems “Anything that can go wrong will go wrong” – Murphy’s law How to nullify Murphy’s law and improve the process under consideration? One possibility: statistical control by monitoring and adjustment Synergistic control: statistical process control and engineering process control Speaking statisticaly – probability to have absolutely stable production process is ZERO “Change alone is unchanging” Heraklietos History - beginings The control chart - invented by Walter Shewhart How to improve the reliability of telephony transmission systems? Shewhart framed the problem in terms of common and special causes of variation In 1924, May 16, in an internal memo Shewhart introduced the control chart as a tool for distinguishing between the two causes of variation. Shewhart's boss, George Edwards: "Dr. Shewhart prepared a little memorandum only about a page in length… all of the essential principles and considerations which are involved in what we know today as process quality control." Not only control charts but also… Shewhart Cycle Learning and Improvement cycle, combining both creative management thinking with statistical analysis. This cycle contains four continuous steps: Plan, Do, Study and Act. These steps (commonly refered to as the PDSA cycle), Shewhart believed, ultimately lead to total quality improvement. From USA… Shewhart created the basis for the control chart and the concept of a state of statistical control by carefully designed experiments. William Edwards Deming after 1924 and over the next 50 years, became the foremost champion and proponent of Shewhart's work. … to Japan… W.E.Deming was statistical consultant to the Supreme Commander for the Allied Powers and worked in Japan, so he spread Shewhart's thinking, and the use of the control chart, widely in Japanese manufacturing industry from the 1950s and 1960s. “It is not enough to do your best; you must know what to do, and then do your best.” “If you can't describe what you are doing as a process, you don't know what you're doing.” … and then in Japan In 1974 Dr. Kaoru Ishikawa - collection of process improvement tools in his text Guide to Quality Control. Known around the world: Seven basic tools in SQC Ishikawa - citations 1. "Engineers who pass judgment based on their experimental data, must know statistical methods by heart. " 2. "… quality control and statistical quality control must be conducted with utmost care. " 3. "… by studying quality control, and by applying QC properly, the irrational behaviour of industry and society could be corrected…" " Old seven " Cause–and–effect analysis Check sheets/tally sheets Control charts Graphs Histograms Pareto analysis Scatter analysis Fishbone diagram The fishbone diagram was drawn by a manufacturing team to try to understand the source of periodic iron contamination. The team used the six generic headings to prompt ideas. Layers of branches show thorough thinking about the causes of the problem. For example, under the heading “Machines,” the idea “materials of construction” shows four kinds of equipment and then several specific machine numbers. Note that some ideas appear in different places. “Calibration” shows up under “Methods” as a factor in the analytical procedure, and also under “Measurement” as a cause of lab error. Quality control - definition Quality control (QC) is a procedure or set of procedures intended to ensure that a manufactured product or performed service adheres to a defined set of quality criteria or meets the requirements of the client or customer. http://whatis.techtarget.com/definition/quality-control-QC S(tatistical)Q(uality)C(ontrol) – a set of statistical tools SQC can be divided into three categories: traditional statistical tools acceptance sampling statistical process control (SPC). Descriptive statistics - describing quality characteristics (mean, range, variance) Acceptance sampling - randomly inspecting a sample of goods and deciding whether to accept or reject the entire lot. Statistical process control - inspecting a random sample of output from a process and deciding whether the process fall within preset specification limits. Causes of variation in the quality of a product or process: common causes assignable causes. Common causes of variation are random causes that we cannot identify. Assignable causes of variation are those that can be identified and eliminated. Control chart - a graph used in SPC that shows whether a sample of data falls within the normal range of variation. Control chart: central line (CL), upper (UCL) and lower control limits (LCL). Control limits separate common from assignable causes of variation Control charts for variables monitor characteristics that can be measured Control charts for attributes monitor characteristics that can be counted Control charts for variables: X-bar charts monitor the mean or average value of a product characteristic. R-charts monitor the range or dispersion of the values of a product characteristic. S-charts monitor sample variance Control charts for attributes: P-charts are used to monitor the proportion of defects in a sample, C-charts are used to monitor the actual number of defects in a sample. Process capability the ability of the production process to meet or exceed preset specifications. measured by the process capability index Cp the ratio of the specification width to the width of the process variable. UCL and LCL are set based on previous knowledge and then from each sample we calculate mean value: Collect as many subgroups as possible before calculating control limits. With smaller amounts of data, the X-bar and R chart may not represent variability of the entire system. The more subgroups you use in control limit calculations, the more reliable the analysis. Typically, twenty to twenty-five subgroups will be used in control limit calculations. Steps In Making the Xbar and R Chart Collect the data. It is best to have at least 100 samples. Divide the data into subgroups, (4 or 5 data points each): The data obtained should be from the same grouping of products produced. A subgroup should not include data from a different lot or different process. Record the data on a data sheet. Design the sheet so that it is easy to compute the values of X bar and R for each subgroup Find the mean value (Xbar), the range, R for each subgroup, the overall mean, or X double bar . Compute the average value of the range (R). Compute the Control Limit Lines, they are calculated based on properties of normal distribution R chart is examined before the Xbar chart; if the R chart indicates the sample variability is in statistical control, then the Xbar chart is examined to determine if the sample mean is also in statistical control. If the sample variability is not in statistical control, then the entire process is judged to be not in statistical control regardless of what the Xbar chart indicates Getting the most Without a control chart, there is no way to know if the process has changed or to identify sources of process variability. Only the values out of limits stop the process – is it possible to have some other informations from charts? R1 - Any single data point falls outside the 3σ limit from the centerline R2 - Two out of three consecutive points fall beyond the 2σ limit on the same side of the centerline R3 - Four out of five consecutive points fall beyond the 1σ limit on the same side of the centerline R4 - Nine consecutive points fall on the same side of the centerline R1-R4 - the justification for investigation if assignable causes are present Possibility of false positives: Assuming observations are normally distributed, one expects Rule R1 to be triggered by chance one out of every 370 observations on average. The false alarm rate rises to one out of every 91.75 observations when evaluating all four rules X-bar and R charts applications To assess the system’s stability To determine if there is a need to stratify the data. To analyze the results of process improvements For standardization Where is the main problem? Normality assumptions: The quality characteristic to be monitored has normal distribution The parameters μ and σ for the random variable are the same for each unit Independence each unit is independent of its predecessors or successors Average Run Length (ARL) The Average Run Length is the number of points that, on average, will be plotted on a control chart before an out of control condition is indicated If the process is in control: ARL=1/ α If the process is out of control: ARL=1/(1- β) α - the probability of a Type I error, β - the probability of a Type II error. CUSUM charts CUSUM is short for cumulative sums. As measurements are taken, the difference between each measurement and the bench mark value is calculated, and this is cumulatively summed up. If the processes are in control, measurements do not deviate significantly from the bench mark, so measurements greater than the bench mark and those less than the bench mark averaged each other out, and the CUSUM value should vary narrowly around the bench mark level. If the processes are out of control, measurements will more likely to be on one side of the bench mark, so the CUSUM value will progressively depart from that of the bench mark. CUSUM involves the calculation of a cumulative sum (which is what makes it "sequential"). Samples from a process xn are assigned weights wn , and summed as follows: So=0 Sn+1=max(0, Sn+xn-wn) When the value of S exceeds a certain threshold value, a change in value has been found. The above formula only detects changes in the positive direction. When negative changes need to be found as well, the min operation should be used instead of the max operation, and this time a change has been found when the value of S is below the (negative) value of the threshold value. CUSUM charts CUSUM and Shewhart charts for Poisson distributed counts are used when the measurements are counts of events within a defined environment. CUSUM and Shewhart charts for normally distributed means and standard deviations are used when the measurements are usually made with a gauge or a measureing instrument, and are continuous with a normal distribution. CUSUM for binomially distributed proportions based on the Benulli distribution can be used to evaluate proportions. Time series in QC univariate time series nonstationary time series models IMA – integrated moving average ARIMA(p,d,q) multivariate time series VAR EWMA charts Exponentially Weighted Moving Average is a statistic that averages the data in a way that gives less and less weight to data as they are further removed in time. EWMA(t)=λY(t)+(1−λ)EWMA(t−1) , t=1,2,…,n. EWMA(0) is the mean of historical data (target) Y(t) is the observation at time t n is the number of observations to be monitored including EWMA(0) 0<λ≤1 is a constant that determines the depth of memory of the EWMA. The equation is due to Roberts (1959). The EWMA chart is sensitive to small shifts in the process mean, but does not match the ability of Shewhart-style charts to detect larger shifts EWMA control procedure can be made sensitive to a small or gradual drift in the process, whereas the Shewhart control procedure can only react when the last data point is outside a control limit. Ideas… Generalization of the run rules for the Shewhart control charts - signal occurs when m successive observations exceed k1*sigma control limit - signal occurs when (m-1) out of m successive observations exceed k2* sigma control limit Ideas… The Frechet control chart A new cc based on Frechet distance for monitoring simultaneously the process level and spread Frechet distance, in particular case when the distributions are closed with the respect to changes of location and scale, between rv X and Y: D = sqrt((mx-my)*2+(sx-sy)*2) Ideas… Economical desing of control charts how to determine the sample size, the interval between samples, and the control limits that will yield approximately maximum average net income Different approaches One important name in SQC Genichi Taguchi (1924 - 2012), engineer and statistician From the 1950s onwards, Taguchi developed a methodology for applying statistics to improve the quality of manufactured goods. Taguchi methods have been controversial among some conventional Western statisticians, but others have accepted many of the concepts introduced by Taguchi. Taguchi loss function Taguchi Loss Function - includes assessing economic loss from a deviation in quality without having to develop the unique function for each quality characteristic. TLF for one piece of product TLF for a sample TLF for one piece: Loss = Constant*(quality characteristic – target value)^2 TLF for a sample is: Loss = Constant*(standard deviation^2+ (process mean–target value) ^2) ‘Constant’ is the coefficient of the Taguchi Loss, or the ratio of functional tolerance and customer loss. Functional tolerance is the value at which 50 percent of the customers view the product as defective, and customer loss is the average loss to the customer at this point. ‘Quality characteristic’ indicates the actual value of the characteristic ‘Target value’ is the specified ideal value for this quality characteristic. TLF The upper and lower specification limits (USL and LSL, respectively) The amount of loss is minimum for the target (or nominal value of a part) and as you deviate from the target the amount of loss increases even if you are within the specified limits of the process. Some other loss functions: Assymetric loss functions Statistical Control by Monitoring and Adjustment, George E.P. Box, A. Luceno, M. del Carmen Paniagua-Quinones Frontiers in Statistical Quality Control, series of books, editors: H.-J. Lentz, P.-Th. Wilrich And a lot of internet sites

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