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S6 - 1© 2011 Pearson Education, Inc. publishing as Prentice Hall S6 Statistical Process Control PowerPoint presentation to accompany Heizer and Render Operations Management, 10e Principles of Operations Management, 8e PowerPoint slides by Jeff Heyl

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S6 - 2© 2011 Pearson Education, Inc. publishing as Prentice Hall Statistical Process Control The objective of a process control system is to provide a statistical signal when assignable causes of variation are present

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S6 - 3© 2011 Pearson Education, Inc. publishing as Prentice Hall Variability is inherent in every process Natural or common causes Special or assignable causes Provides a statistical signal when assignable causes are present Detect and eliminate assignable causes of variation Statistical Process Control (SPC)

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S6 - 4© 2011 Pearson Education, Inc. publishing as Prentice Hall Natural Variations Also called common causes Affect virtually all production processes Expected amount of variation Output measures follow a probability distribution For any distribution there is a measure of central tendency and dispersion If the distribution of outputs falls within acceptable limits, the process is said to be “in control”

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S6 - 5© 2011 Pearson Education, Inc. publishing as Prentice Hall Assignable Variations Also called special causes of variation Generally this is some change in the process Variations that can be traced to a specific reason The objective is to discover when assignable causes are present Eliminate the bad causes Incorporate the good causes

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S6 - 6© 2011 Pearson Education, Inc. publishing as Prentice Hall Samples To measure the process, we take samples and analyze the sample statistics following these steps (d)If only natural causes of variation are present, the output of a process forms a distribution that is stable over time and is predictable Weight Time Frequency Prediction Figure S6.1

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S6 - 7© 2011 Pearson Education, Inc. publishing as Prentice Hall Samples To measure the process, we take samples and analyze the sample statistics following these steps (e)If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Prediction ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? Figure S6.1

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S6 - 8© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Charts Constructed from historical data, the purpose of control charts is to help distinguish between natural variations and variations due to assignable causes

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S6 - 9© 2011 Pearson Education, Inc. publishing as Prentice Hall Process Control Figure S6.2 Frequency (weight, length, speed, etc.) Size Lower control limit Upper control limit (a) In statistical control and capable of producing within control limits (b) In statistical control but not capable of producing within control limits (c) Out of control

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S6 - 10© 2011 Pearson Education, Inc. publishing as Prentice Hall Types of Data Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables VariablesAttributes Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables

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S6 - 11© 2011 Pearson Education, Inc. publishing as Prentice Hall Central Limit Theorem Regardless of the distribution of the population, the distribution of sample means drawn from the population will tend to follow a normal curve 1.The mean of the sampling distribution (x) will be the same as the population mean x = n x = 2.The standard deviation of the sampling distribution ( x ) will equal the population standard deviation ( ) divided by the square root of the sample size, n

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S6 - 12© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Charts for Variables For variables that have continuous dimensions Weight, speed, length, strength, etc. x-charts are to control the central tendency of the process R-charts are to control the dispersion of the process These two charts must be used together

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S6 - 13© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we know Upper control limit (UCL) = x + z x Lower control limit (LCL) = x - z x wherex=mean of the sample means or a target value set for the process z=number of normal standard deviations x =standard deviation of the sample means = / n =population standard deviation n=sample size

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S6 - 14© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Hour 1 SampleWeight of NumberOat Flakes 117 213 316 418 517 616 715 817 916 Mean16.1 =1 HourMeanHourMean 116.1715.2 216.8816.4 315.5916.3 416.51014.8 516.51114.2 616.41217.3 n = 9 LCL x = x - z x = 16 - 3(1/3) = 15 ozs For 99.73% control limits, z = 3 UCL x = x + z x = 16 + 3(1/3) = 17 ozs

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S6 - 15© 2011 Pearson Education, Inc. publishing as Prentice Hall 17 = UCL 15 = LCL 16 = Mean Setting Control Limits Control Chart for sample of 9 boxes Sample number |||||||||||| 123456789101112 Variation due to assignable causes Variation due to natural causes Out of control

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S6 - 16© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For x-Charts when we don’t know Lower control limit (LCL) = x - A 2 R Upper control limit (UCL) = x + A 2 R whereR=average range of the samples A 2 =control chart factor found in Table S6.1 x=mean of the sample means

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S6 - 17© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Chart Factors Table S6.1 Sample Size Mean Factor Upper Range Lower Range n A 2 D 4 D 3 21.8803.2680 31.0232.5740 4.7292.2820 5.5772.1150 6.4832.0040 7.4191.9240.076 8.3731.8640.136 9.3371.8160.184 10.3081.7770.223 12.2661.7160.284

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S6 - 18© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits Process average x = 12 ounces Average range R =.25 Sample size n = 5

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S6 - 19© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 From Table S6.1

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S6 - 20© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL x = x + A 2 R = 12 + (.577)(.25) = 12 +.144 = 12.144 ounces LCL x = x - A 2 R = 12 -.144 = 11.857 ounces Process average x = 12 ounces Average range R =.25 Sample size n = 5 UCL = 12.144 Mean = 12 LCL = 11.857

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S6 - 21© 2011 Pearson Education, Inc. publishing as Prentice Hall R – Chart Type of variables control chart Shows sample ranges over time Difference between smallest and largest values in sample Monitors process variability Independent from process mean

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S6 - 22© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Chart Limits For R-Charts Lower control limit (LCL R ) = D 3 R Upper control limit (UCL R ) = D 4 R where R=average range of the samples D 3 and D 4 =control chart factors from Table S6.1

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S6 - 23© 2011 Pearson Education, Inc. publishing as Prentice Hall Setting Control Limits UCL R = D 4 R = (2.115)(5.3) = 11.2 pounds LCL R = D 3 R = (0)(5.3) = 0 pounds Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D 4 = 2.115, D 3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0

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S6 - 24© 2011 Pearson Education, Inc. publishing as Prentice Hall Mean and Range Charts (a) These sampling distributions result in the charts below (Sampling mean is shifting upward but range is consistent) R-chart (R-chart does not detect change in mean) UCL LCL Figure S6.5 x-chart (x-chart detects shift in central tendency) UCL LCL

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S6 - 25© 2011 Pearson Education, Inc. publishing as Prentice Hall Mean and Range Charts R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5 (b) These sampling distributions result in the charts below (Sampling mean is constant but dispersion is increasing) x-chart (x-chart does not detect the increase in dispersion) UCL LCL

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S6 - 26© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Charts for Attributes For variables that are categorical Good/bad, yes/no, acceptable/unacceptable Measurement is typically counting defectives Charts may measure Percent defective (p-chart) Number of defects (c-chart)

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S6 - 27© 2011 Pearson Education, Inc. publishing as Prentice Hall Control Limits for p-Charts Population will be a binomial distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCL p = p + z p ^ LCL p = p - z p ^ wherep=mean fraction defective in the sample z=number of standard deviations p =standard deviation of the sampling distribution n=sample size ^ p(1 - p) n p = ^

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S6 - 28© 2011 Pearson Education, Inc. publishing as Prentice Hall p-Chart for Data Entry SampleNumberFractionSampleNumberFraction Numberof ErrorsDefectiveNumberof ErrorsDefective 16.06116.06 25.05121.01 30.00138.08 41.01147.07 54.04155.05 62.02164.04 75.051711.11 83.03183.03 93.03190.00 102.02204.04 Total = 80 (.04)(1 -.04) 100 p = =.02 ^ p = =.04 80 (100)(20)

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S6 - 29© 2011 Pearson Education, Inc. publishing as Prentice Hall.11 –.10 –.09 –.08 –.07 –.06 –.05 –.04 –.03 –.02 –.01 –.00 – Sample number Fraction defective |||||||||| 2468101214161820 p-Chart for Data Entry UCL p = p + z p =.04 + 3(.02) =.10 ^ LCL p = p - z p =.04 - 3(.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04

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S6 - 30© 2011 Pearson Education, Inc. publishing as Prentice Hall.11 –.10 –.09 –.08 –.07 –.06 –.05 –.04 –.03 –.02 –.01 –.00 – Sample number Fraction defective |||||||||| 2468101214161820 p-Chart for Data Entry UCL p = p + z p =.04 + 3(.02) =.10 ^ LCL p = p - z p =.04 - 3(.02) = 0 ^ UCL p = 0.10 LCL p = 0.00 p = 0.04 Possible assignable causes present

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S6 - 31© 2011 Pearson Education, Inc. publishing as Prentice Hall Patterns in Control Charts Normal behavior. Process is “in control.” Upper control limit Target Lower control limit Figure S6.7

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S6 - 32© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7

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S6 - 33© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7

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S6 - 34© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7

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S6 - 35© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Run of 5 above (or below) central line. Investigate for cause. Figure S6.7

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S6 - 36© 2011 Pearson Education, Inc. publishing as Prentice Hall Upper control limit Target Lower control limit Patterns in Control Charts Erratic behavior. Investigate. Figure S6.7

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