Statistical Process Control

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

Outline Statistical Process Control (SPC) Control Charts for Variables
The Central Limit Theorem Setting Mean Chart Limits (x-Charts) Setting Range Chart Limits (R-Charts) Using Mean and Range Charts Control Charts for Attributes Managerial Issues and Control Charts © 2011 Pearson Education, Inc. publishing as Prentice Hall

Outline – Continued Process Capability Acceptance Sampling
Process Capability Ratio (Cp) Process Capability Index (Cpk ) Acceptance Sampling Operating Characteristic Curve Average Outgoing Quality © 2011 Pearson Education, Inc. publishing as Prentice Hall

Learning Objectives When you complete this supplement you should be able to: Explain the use of a control chart Explain the role of the central limit theorem in SPC Build x-charts and R-charts List the five steps involved in building control charts © 2011 Pearson Education, Inc. publishing as Prentice Hall

Learning Objectives When you complete this supplement you should be able to: Build p-charts and c-charts Explain process capability and compute Cp and Cpk Explain acceptance sampling Compute the AOQ © 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Statistical Process Control (SPC)
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 Points which might be emphasized include: - Statistical process control measures the performance of a process, it does not help to identify a particular specimen produced as being “good” or “bad,” in or out of tolerance. - Statistical process control requires the collection and analysis of data - therefore it is not helpful when total production consists of a small number of units - While statistical process control can not help identify a “good” or “bad” unit, it can enable one to decide whether or not to accept an entire production lot. If a sample of a production lot contains more than a specified number of defective items, statistical process control can give us a basis for rejecting the entire lot. The issue of rejecting a lot which was actually good can be raised here, but is probably better left to later. © 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” © 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Each of these represents one sample of five boxes of cereal
Samples To measure the process, we take samples and analyze the sample statistics following these steps Each of these represents one sample of five boxes of cereal (a) Samples of the product, say five boxes of cereal taken off the filling machine line, vary from each other in weight Frequency Weight # Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

The solid line represents the distribution
Samples To measure the process, we take samples and analyze the sample statistics following these steps The solid line represents the distribution (b) After enough samples are taken from a stable process, they form a pattern called a distribution Frequency Weight Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Samples To measure the process, we take samples and analyze the sample statistics following these steps (c) There are many types of distributions, including the normal (bell-shaped) distribution, but distributions do differ in terms of central tendency (mean), standard deviation or variance, and shape Figure S6.1 Weight Central tendency Variation Shape Frequency © 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 Prediction Weight Time Frequency Figure S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Samples To measure the process, we take samples and analyze the sample statistics following these steps Prediction ? (e) If assignable causes are present, the process output is not stable over time and is not predicable Weight Time Frequency Figure S6.1 © 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 Students should understand both the concepts of natural and assignable variation, and the nature of the efforts required to deal with them. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process Control (a) In statistical control and capable of producing within control limits Frequency Lower control limit Upper control limit (b) In statistical control but not capable of producing within control limits This slide helps introduce different process outputs. It can also be used to illustrate natural and assignable variation. (c) Out of control (weight, length, speed, etc.) Size Figure S6.2 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Types of Data Variables Attributes
Characteristics that can take any real value May be in whole or in fractional numbers Continuous random variables Defect-related characteristics Classify products as either good or bad or count defects Categorical or discrete random variables Once the categories are outlined, students may be asked to provide examples of items for which variable or attribute inspection might be appropriate. They might also be asked to provide examples of products for which both characteristics might be important at different stages of the production process. © 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 The mean of the sampling distribution (x) will be the same as the population mean m x = m This slide introduces the difference between “natural” and “assignable” causes. The next several slides expand the discussion and introduce some of the statistical issues. The standard deviation of the sampling distribution (sx) will equal the population standard deviation (s) divided by the square root of the sample size, n s n sx = © 2011 Pearson Education, Inc. publishing as Prentice Hall

Population and Sampling Distributions
Three population distributions Beta Normal Uniform Distribution of sample means Standard deviation of the sample means = sx = s n Mean of sample means = x | | | | | | | -3sx -2sx -1sx x +1sx +2sx +3sx 99.73% of all x fall within ± 3sx 95.45% fall within ± 2sx Figure S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Sampling Distribution
Sampling distribution of means Process distribution of means x = m (mean) It may be useful to spend some time explicitly discussing the difference between the sampling distribution of the means and the mean of the process population. Figure S6.4 © 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

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

Setting Control Limits
Hour 1 Sample Weight of Number Oat Flakes 1 17 2 13 3 16 4 18 5 17 6 16 7 15 8 17 9 16 Mean 16.1 s = 1 Hour Mean Hour Mean n = 9 For 99.73% control limits, z = 3 UCLx = x + zsx = (1/3) = 17 ozs LCLx = x - zsx = (1/3) = 15 ozs © 2011 Pearson Education, Inc. publishing as Prentice Hall

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

Setting Chart Limits For x-Charts when we don’t know s
Upper control limit (UCL) = x + A2R Lower control limit (LCL) = x - A2R where R = average range of the samples A2 = control chart factor found in Table S6.1 x = mean of the sample means © 2011 Pearson Education, Inc. publishing as Prentice Hall

Control Chart Factors Sample Size Mean Factor Upper Range Lower Range
n A2 D4 D3 Table S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process average x = 12 ounces Average range R = .25 Sample size n = 5 UCL = Mean = 12 LCL = UCLx = x + A2R = 12 + (.577)(.25) = = ounces LCLx = x - A2R = = ounces © 2011 Pearson Education, Inc. publishing as Prentice Hall

Restaurant Control Limits
For salmon filets at Darden Restaurants Sample Mean x Bar Chart UCL = x – LCL – | | | | | | | | | 11.5 – 11.0 – 10.5 – Sample Range Range Chart UCL = R = LCL = 0 | | | | | | | | | 0.8 – 0.4 – 0.0 – © 2011 Pearson Education, Inc. publishing as Prentice Hall

Restaurant Control Limits
Capability Histogram LSL USL 10.2 10.5 10.8 11.1 11.4 11.7 12.0 Specifications LSL USL 12 Capability Mean = Std.dev = 1.88 Cp = 1.77 Cpk = 1.7 © 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 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Setting Chart Limits For R-Charts Upper control limit (UCLR) = D4R
Lower control limit (LCLR) = D3R where R = average range of the samples D3 and D4 = control chart factors from Table S6.1 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Average range R = 5.3 pounds Sample size n = 5 From Table S6.1 D4 = 2.115, D3 = 0 UCL = 11.2 Mean = 5.3 LCL = 0 UCLR = D4R = (2.115)(5.3) = 11.2 pounds LCLR = D3R = (0)(5.3) = 0 pounds © 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) x-chart (x-chart detects shift in central tendency) UCL LCL R-chart (R-chart does not detect change in mean) UCL LCL Figure S6.5 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Mean and Range Charts (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 R-chart (R-chart detects increase in dispersion) UCL LCL Figure S6.5 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Steps In Creating Control Charts
Take samples from the population and compute the appropriate sample statistic Use the sample statistic to calculate control limits and draw the control chart Plot sample results on the control chart and determine the state of the process (in or out of control) Investigate possible assignable causes and take any indicated actions Continue sampling from the process and reset the control limits when necessary © 2011 Pearson Education, Inc. publishing as Prentice Hall

Manual and Automated Control Charts
© 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) © 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 UCLp = p + zsp ^ p(1 - p) n sp = ^ LCLp = p - zsp ^ Instructors may wish to point out the calculation of the standard deviation reflects the binomial distribution of the population where p = mean fraction defective in the sample z = number of standard deviations sp = standard deviation of the sampling distribution n = sample size ^ © 2011 Pearson Education, Inc. publishing as Prentice Hall

p-Chart for Data Entry p = = .04 sp = = .02 1 6 .06 11 6 .06
Sample Number Fraction Sample Number Fraction Number of Errors Defective Number of Errors Defective Total = 80 p = = .04 80 (100)(20) (.04)( ) 100 sp = = .02 ^ © 2011 Pearson Education, Inc. publishing as Prentice Hall

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

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

Control Limits for c-Charts
Population will be a Poisson distribution, but applying the Central Limit Theorem allows us to assume a normal distribution for the sample statistics UCLc = c + 3 c LCLc = c - 3 c Instructors may wish to point out the calculation of the standard deviation reflects the Poisson distribution of the population where the standard deviation equals the square root of the mean where c = mean number defective in the sample © 2011 Pearson Education, Inc. publishing as Prentice Hall

c-Chart for Cab Company
c = 54 complaints/9 days = 6 complaints/day UCLc = c + 3 c = = 13.35 | 1 2 3 4 5 6 7 8 9 Day Number defective 14 – 12 – 10 – 8 – 6 – 4 – 2 – 0 – UCLc = 13.35 LCLc = 0 c = 6 LCLc = c - 3 c = = 0 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Managerial Issues and Control Charts
Three major management decisions: Select points in the processes that need SPC Determine the appropriate charting technique Set clear policies and procedures © 2011 Pearson Education, Inc. publishing as Prentice Hall

Which Control Chart to Use
Variables Data Using an x-Chart and R-Chart Observations are variables Collect samples of n = 4, or n = 5, or more, each from a stable process and compute the mean for the x-chart and range for the R-chart Track samples of n observations each. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Attribute Data Using the p-Chart Observations are attributes that can be categorized as good or bad (or pass–fail, or functional–broken), that is, in two states. We deal with fraction, proportion, or percent defectives. There are several samples, with many observations in each. For example, 20 samples of n = 100 observations in each. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Attribute Data Using a c-Chart Observations are attributes whose defects per unit of output can be counted. We deal with the number counted, which is a small part of the possible occurrences. Defects may be: number of blemishes on a desk; complaints in a day; crimes in a year; broken seats in a stadium; typos in a chapter of this text; or flaws in a bolt of cloth. Table S6.3 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Patterns in Control Charts
Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Normal behavior. Process is “in control.” Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. One plot out above (or below). Investigate for cause. Process is “out of control.” Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Trends in either direction, 5 plots. Investigate for cause of progressive change. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Two plots very near lower (or upper) control. Investigate for cause. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Run of 5 above (or below) central line. Investigate for cause. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Upper control limit Target Lower control limit Ask the students to imagine a product, and consider what problem might cause each of the graph configurations illustrated. Erratic behavior. Investigate. Figure S6.7 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process Capability The natural variation of a process should be small enough to produce products that meet the standards required A process in statistical control does not necessarily meet the design specifications Process capability is a measure of the relationship between the natural variation of the process and the design specifications © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process Capability Ratio
Cp = Upper Specification - Lower Specification 6s A capable process must have a Cp of at least 1.0 Does not look at how well the process is centered in the specification range Often a target value of Cp = 1.33 is used to allow for off-center processes Six Sigma quality requires a Cp = 2.0 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Insurance claims process Process mean x = minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s © 2011 Pearson Education, Inc. publishing as Prentice Hall

Insurance claims process Process mean x = minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 6(.516) © 2011 Pearson Education, Inc. publishing as Prentice Hall

Insurance claims process Process mean x = minutes Process standard deviation s = .516 minutes Design specification = 210 ± 3 minutes Cp = Upper Specification - Lower Specification 6s = = 1.938 6(.516) Process is capable © 2011 Pearson Education, Inc. publishing as Prentice Hall

Process Capability Index
Cpk = minimum of , Upper Specification - x Limit 3s Lower x - Specification Limit A capable process must have a Cpk of at least 1.0 A capable process is not necessarily in the center of the specification, but it falls within the specification limit at both extremes © 2011 Pearson Education, Inc. publishing as Prentice Hall

New Cutting Machine New process mean x = .250 inches Process standard deviation s = inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches © 2011 Pearson Education, Inc. publishing as Prentice Hall

New Cutting Machine New process mean x = .250 inches Process standard deviation s = inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cpk = minimum of , (.251) (3).0005 © 2011 Pearson Education, Inc. publishing as Prentice Hall

New Cutting Machine New process mean x = .250 inches Process standard deviation s = inches Upper Specification Limit = .251 inches Lower Specification Limit = .249 inches Cpk = minimum of , (.251) (3).0005 (.249) Both calculations result in New machine is NOT capable Cpk = = 0.67 .001 .0015 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Interpreting Cpk Cpk = negative number Cpk = zero
Cpk = between 0 and 1 Cpk = 1 Cpk > 1 Figure S6.8 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Acceptance Sampling Form of quality testing used for incoming materials or finished goods Take samples at random from a lot (shipment) of items Inspect each of the items in the sample Decide whether to reject the whole lot based on the inspection results Only screens lots; does not drive quality improvement efforts Rejected lots can be: Returned to the supplier Culled for defectives (100% inspection) Here again it is useful to stress that acceptance sampling relates to the aggregate, not the individual unit. You might also discuss the decision as to whether one should take only a single sample, or whether multiple samples are required. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Operating Characteristic Curve
Shows how well a sampling plan discriminates between good and bad lots (shipments) Shows the relationship between the probability of accepting a lot and its quality level You can use this and the next several slides to begin a discussion of the “quality” of the acceptance sampling plans. You will find additional slides on “consumer’s” and “producer’s” risk to pursue the issue in a more formal manner in subsequent slides. © 2011 Pearson Education, Inc. publishing as Prentice Hall

The “Perfect” OC Curve Keep whole shipment P(Accept Whole Shipment)
% Defective in Lot P(Accept Whole Shipment) 100 – 75 – 50 – 25 – 0 – | | | | | | | | | | | Return whole shipment Cut-Off © 2011 Pearson Education, Inc. publishing as Prentice Hall

Probability of Acceptance Consumer’s risk for LTPD
An OC Curve Figure S6.9 Probability of Acceptance Percent defective | | | | | | | | | 100 – 95 – 75 – 50 – 25 – 10 – 0 –  = 0.05 producer’s risk for AQL LTPD AQL  = 0.10 Consumer’s risk for LTPD Bad lots Indifference zone Good lots © 2011 Pearson Education, Inc. publishing as Prentice Hall

AQL and LTPD Acceptable Quality Level (AQL)
Poorest level of quality we are willing to accept Lot Tolerance Percent Defective (LTPD) Quality level we consider bad Consumer (buyer) does not want to accept lots with more defects than LTPD Once the students understand the definition of these terms, have them consider how one would go about choosing values for AQL and LTPD. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Producer’s and Consumer’s Risks
Producer's risk () Probability of rejecting a good lot Probability of rejecting a lot when the fraction defective is at or above the AQL Consumer's risk (b) Probability of accepting a bad lot Probability of accepting a lot when fraction defective is below the LTPD This slide introduces the concept of “producer’s” risk and “consumer’s” risk. The following slide explores these concepts graphically. © 2011 Pearson Education, Inc. publishing as Prentice Hall

OC Curves for Different Sampling Plans
n = 50, c = 1 n = 100, c = 2 This slide presents the OC curve for two possible acceptance sampling plans. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Average Outgoing Quality
AOQ = (Pd)(Pa)(N - n) N where Pd = true percent defective of the lot Pa = probability of accepting the lot N = number of items in the lot n = number of items in the sample It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Average Outgoing Quality
If a sampling plan replaces all defectives If we know the incoming percent defective for the lot We can compute the average outgoing quality (AOQ) in percent defective The maximum AOQ is the highest percent defective or the lowest average quality and is called the average outgoing quality level (AOQL) It is probably important to stress that AOQ is the average percent defective, not the average percent acceptable. © 2011 Pearson Education, Inc. publishing as Prentice Hall

SPC and Process Variability
Lower specification limit Upper specification limit Process mean, m (a) Acceptance sampling (Some bad units accepted) (b) Statistical process control (Keep the process in control) This may be a good time to stress that an overall goal of statistical process control is to “do it better,” i.e., improve over time. (c) Cpk >1 (Design a process that is in control) Figure S6.10 © 2011 Pearson Education, Inc. publishing as Prentice Hall

Printed in the United States of America.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. © 2011 Pearson Education, Inc. publishing as Prentice Hall

Statistical Process Control

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