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Statistical Process Control Processes that are not in a state of statistical control show excessive variations or exhibit variations that change with time Control charts are used to detect whether a process is statistically stable. Control charts differentiates between variations that is normally expected of the process due chance or common causes that change over time due to assignable or special causes

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Common cause variation Variations due to common causes are inherent to the process because of: ◦ the nature of the system ◦ the way the system is managed ◦ the way the process is organized and operated can only be removed by ◦ making modifications to the process ◦ changing the process are the responsibility of higher management

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Special Cause Variation Variations due to special causes are localized in nature exceptions to the system considered abnormalities often specific to a ◦ certain operator ◦ certain machine ◦ certain batch of material, etc. Investigation and removal of variations due to special causes are key to process improvement

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Causes of Variation Two basic categories of variation in output include common causes and assignable causes. Common causes are the purely random, unidentifiable sources of variation that are unavoidable with the current process. ◦ If process variability results solely from common causes of variation, a typical assumption is that the distribution is symmetric, with most observations near the center. Assignable causes of variation are any variation-causing factors that can be identified and eliminated, such as a machine needing repair.

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Statistical Process Control (SPC) Charts Statistical process control (SPC) charts are used to help us distinguish between common and assignable causes of variation. We will cover 2 types: Variables Control Charts: Service or product characteristics that is continuous and can be measured, such as weight, length, volume, or time. Attributes Control Charts: Service or product characteristics that can be counted...pass/fail, good/bad, rating scale. Color, inspection, test results (pass/fail), number of defects, types of defects SPC charts Explained

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Control Charts Control Charts are run charts with superimposed normal distributions

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Purpose of Control Charts Control charts provide a graphical means for testing hypotheses about the data being monitored.

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Control and warning limits The probability of a sample having a particular value is given by its location on the chart. Assuming that the plotted statistic is normally distributed, the probability of a value lying beyond the: warning limits is approximately 0.025 or 2.5% chance (plus or minus 2-sigma from the mean) control limits is approximately 0.001 or 0.1% chance (plus or minus 3 sigma from the mean), this is rare and indicates that ◦ the variation is due to an assignable cause ◦ the process is out-of-statistical control

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Out of Control Processes Run rules are rules that are used to indicate out-of-statistical control situations. Typical run rules for Shewhart X-charts with control and warning limits are: a point lying beyond the control limits 2 consecutive points lying beyond the warning limits (0.025x0.025x100 = 0.06% chance of occurring) 7 or more consecutive points lying on one side of the mean ( 0.5 7 x100 = 0.8% chance of occurring and indicates a shift in the mean of the process) 5 or 6 consecutive points going in the same direction (indicates a trend) Other run rules can be formulated using similar principles

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Nominal UCL LCL Variations Sample number Control Charts

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Nominal UCL LCL Variations Sample number Control Charts

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Nominal UCL LCL Variations Sample number Control Charts

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Nominal UCL LCL Variations Sample number Control Charts

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Sampling Sampling plan: A plan that specifies a sample size, the time between successive samples, and decision rules that determine when action should be taken. Sample size: A quantity of randomly selected observations of process outputs. Why do I need to sample? The case of potato chips

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Means and Ranges The mean of the sample is the sum of all observations in a sample divided by the number of observations in the sample. The range of the sample is the difference between the largest observation in a sample and the smallest. The mean of the process is the sum of all sample means divided by the number of samples The mean range for the process is the sum of all ranges divided by the number of samples

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Statistical Process Control Methods Control Charts for variables are used to monitor the mean and variability of the process distribution. R-chart (Range Chart) is used to monitor process variability. - x-chart is used to see whether the process is generating output, on average, consistent with a target value set by management for the process or whether its current performance, with respect to the average of the performance measure, is consistent with past performance. ◦ If the standard deviation of the process is known, we can place UCL and LCL at “z” standard deviations from the mean at the desired confidence level.

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Another way to develop control charts so for small sample sizes The control limits for the x-chart are: UCL x = x + A 2 R and LCL x = x - A 2 R Where X = central line of the chart, which can be either the average of past sample means or a target value set for the process. A 2 = constant to provide three-sigma limits for the sample mean. The control limits for the R-chart are UCL R = D 4 R and LCL R = D 3 R where R = average of several past R values and the central line of the chart. D 3,D 4 = constants that provide 3 standard deviations (three- sigma) limits for a given sample size. – = – = =

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Control Chart Factors TABLE 5.1|FACTORS FOR CALCULATING THREE-SIGMA LIMITS FOR |THE x -CHART AND R -CHART Size of Sample ( n ) Factor for UCL and LCL for x -Chart ( A 2 ) Factor for LCL for R -Chart ( D 3 ) Factor for UCL for R -Chart ( D 4 ) 21.88003.267 31.02302.575 40.72902.282 50.57702.115 60.48302.004 70.4190.0761.924 80.3730.1361.864 90.3370.1841.816 100.3080.2231.777

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YearQuarterExam 1Exam 2Exam 3 2009Quarter 1858290 2009Quarter 2736777 2009Quarter 385 2009Quarter 4907383 2010Quarter 1708398 2010Quarter 2898183 2010Quarter 3959391 2010Quarter 4728395 Over the past 2 years, Professor Matta has been asked to teach one section of Process Analytics in each quarter (4 Quarters/ Year). Each time he taught, he would give 3 exams. The class average grade on these exams over the last 8 quarters have been as follows:

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Exam Grades in Prof Matta’s PA class during the past 2 years with Averages: YearQuarterExam 1Exam 2Exam 3Average 2009Quarter 185829085.67 2009Quarter 273677772.33 2009Quarter 385 85.00 2009Quarter 490738382.00 2010Quarter 170839883.67 2010Quarter 289818384.33 2010Quarter 395939193.00 2010Quarter 472839583.33 Overall Average83.67

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Exam Grades in Prof Matta’s PA class during the past 2 years with Averages and Ranges: YearQuarterExam 1Exam 2Exam 3AverageRange 2009Quarter 185829085.678.00 2009Quarter 273677772.3310.00 2009Quarter 385 85.000.00 2009Quarter 490738382.0017.00 2010Quarter 170839883.6728.00 2010Quarter 289818384.338.00 2010Quarter 395939193.004.00 2010Quarter 472839583.3323.00 Overall Average83.6712.25

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For Prof. Matta’s Case: The sample size = 3 (3 exams/quarter) A21.02 D30 D42.57 X-Bar83.67 R-Bar12.25 UCLx = X-Bar + A2 * R-Bar96.16 LCLx = X-Bar - A2 * R-Bar71.17 UCLr = D4* R-Bar31.48 LCLr = D3 * R-Bar0.00

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Control Charts for Prof Matta’s Exams:

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Control Charts Example: At Quikie Car Wash, the wash process is advertised to take less than 7 minutes. Consequently, management has set a target average of 390 seconds for the wash process. Suppose that the average range for a sample of 9 cars is 10 seconds. Establish the means and ranges control limits using this data.

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Solved Example X = 390 sec, n = 9, R= 10 sec From Table 5.1 in your book, A 2 = 0.337, D 3 = 0.184, D 4 = 1.816 UCL R = D 4 R= 1.816(10 sec) = 18.16 sec LCL R = D 3 R= 0.184(10 sec) = 1.84 sec UCL x = x + A 2 R= 390 sec + 0.337(10 sec) = 393.37 sec LCL x = x - A 2 R = 390 sec – 0.337(10 sec) = 386.63 sec

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Marlin Bottling Company Problem 5 Marlin Bottling Company Problem 5 in the book chapter 2 Marlin Bottling Company Problem 5 Marlin Company Sample1234 X-BARR 10.600.610.590.60 0.02 20.60 0.610.60 0.01 30.580.570.59 0.580.02 40.620.610.600.590.600.03 50.590.61 0.60 0.02 60.590.580.620.580.590.04

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Marlin Co. Bottling – In class Calculations Factor for UCLFactor forFactor Size ofand LCL forLCL forUCL for Samplex-ChartsR-ChartsR-Charts (n)(A 2 )(D 3 )(D 4 ) 21.880 03.267 31.023 02.575 40.729 02.282 50.577 02.115 60.483 02.00

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Control Charts for Attributes p-chart: A chart used for controlling the proportion of defective services or products generated by the process. p = p (1 – p )/ n Where n = sample size p = central line on the chart, which can be either the historical average population proportion defective or a target value. z = normal deviate (number of standard deviations from the average) p Control limits are: UCL p = p+z p and LCL p = p−z p ––

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Hometown Bank Hometown BankExample The operations manager of the booking services department of Hometown Bank is concerned about the number of wrong customer account numbers recorded by Hometown personnel. Each week a random sample of 2,500 deposits is taken, and the number of incorrect account numbers is recorded. The results for the past 12 weeks are shown in the following table. Is the booking process out of statistical control? Use three-sigma control limits.

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SampleWrongProportion NumberAccount #Defective 1150.006 2120.0048 3190.0076 4 20.0008 5190.0076 6 40.0016 7240.0096 8 70.0028 910 0.004 10170.0068 1115 0.006 12 30.0012 Total 147 Hometown Bank Using a p-Chart to monitor a process n = 2500 p = 147 12(2500 ) = 0.0049 p = p (1 – p )/ n p = 0.0049(1 – 0.0049)/2500 p = 0.0014 UCL p = 0.0049 + 3(0.0014) = 0.0091 LCL p = 0.0049 – 3(0.0014) = 0.0007

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Hometown Bank Using a p-Chart to monitor a process Example

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In class Problem

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In Class Problem

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Control Charts Two types of error are possible with control charts A type I error occurs when a process is thought to be out of control when in fact it is not A type II error occurs when a process is thought to be in control when it is actually out of statistical control These errors can be controlled by the choice of control limits

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