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CHAPTER Statistical Quality Control Reid & Sanders, Operations Management © Wiley 2002 6 Learning Objectives • Describe quality control methods • Understand the use of statistical process control • Describe & apply control charts • Distinguish x-bar, R, p and c-charts • Define process capability • Describe & apply capability indexes • Define six-sigma capability Reid & Sanders, Operations Management © Wiley 2002 Page 2 Quality Control Methods • Descriptive statistics: – Used to describe distributions of data • Statistical process control (SPC): – Used to determine whether a process is performing as expected • Acceptance sampling: – Used to accept or reject entire batches by only inspecting a few items Reid & Sanders, Operations Management © Wiley 2002 Page 3 Descriptive Statistics • Mean (x-bar): – The average or central tendency of a data set • Standard deviation (sigma): – Describes the amount of spread or observed variation in the data set • Range: – Another measure of spread – The range measures the difference between the largest & smallest observed values in the data set Reid & Sanders, Operations Management © Wiley 2002 Page 4 The Normal Distribution Reid & Sanders, Operations Management © Wiley 2002 Page 5 Equations • Mean: n x x i i 1 n • Standard deviation: x X n Reid & Sanders, Operations Management © Wiley 2002 i 1 2 i n 1 Page 6 Impact of Standard Deviation Reid & Sanders, Operations Management © Wiley 2002 Page 7 Skewed Distributions (One Form of Non-Normal Distribution) Reid & Sanders, Operations Management © Wiley 2002 Page 8 SPC Methods • Control charts – Use statistical limits to identify when a sample of data falls within a normal range of variation Reid & Sanders, Operations Management © Wiley 2002 Page 9 Setting Limits Requires Balancing Risks • Control limits are based on a willingness to think something’s wrong, when it’s actually not (Type I or alpha error), balanced against the sensitivity of the tool - the ability to quickly reveal a problem (failure is Type II or beta error) Reid & Sanders, Operations Management © Wiley 2002 Page 10 Types of Data • Variable level data: – Can be measured using a continuous scale – Examples: length, weight, time, & temperature • Attribute level data: – Can only be described by discrete characteristics – Example: defective & not defective Reid & Sanders, Operations Management © Wiley 2002 Page 11 Control Charts for Variable Data • Mean (x-bar) charts – Tracks the central tendency (the average value observed) over time • Range (R) charts: – Tracks the spread of the distribution over time (estimates the observed variation) Reid & Sanders, Operations Management © Wiley 2002 Page 12 x-Bar Computations x1 x 2 ... x n x n x n UCLx x z x LCLx x z x Reid & Sanders, Operations Management © Wiley 2002 Page 13 Example • • • Assume the standard deviation of the process is given as 1.13 ounces Management wants a 3-sigma chart (only 0.26% chance of alpha error) Observed values shown in the table are in ounces Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.0 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 15.975 15.9 Sample means 15.875 Reid & Sanders, Operations Management © Wiley 2002 Page 14 Computations • Center line (x-double bar): 15.875 15.975 15.9 x 15.92 3 • Control limits: Reid & Sanders, Operations Management © Wiley 2002 Page 15 2nd Method Using R-bar R1 R2 ... Rn R n UCLx x A2 R LCLx x A2 R Reid & Sanders, Operations Management © Wiley 2002 Page 16 Control Chart Factors S a m p l e S i z e (n ) F a c to r fo r x -C h a r t F a c to r s fo r R -C h a r t A2 D3 D4 2 1.88 0.00 3.27 3 1.02 0.00 2.57 4 0.73 0.00 2.28 5 0.58 0.00 2.11 6 0.48 0.00 2.00 7 0.42 0.08 1.92 8 0.37 0.14 1.86 9 0.34 0.18 1.82 10 0.31 0.22 1.78 11 0.29 0.26 1.74 12 0.27 0.28 1.72 13 0.25 0.31 1.69 14 0.24 0.33 1.67 15 0.22 0.35 1.65 Reid & Sanders, Operations Management © Wiley 2002 Page 17 Example Time 1 Time 2 Time 3 Observation 1 15.8 16.1 16.0 Observation 2 16.0 16.0 15.9 Observation 3 15.8 15.8 15.9 Observation 4 15.9 15.9 15.8 15.975 15.9 0.3 0.2 Sample means 15.875 Sample ranges Reid & Sanders, Operations Management © Wiley 2002 0.2 Page 18 Computations 0.2 0.3 0.2 R 2.33 3 UCLx x A2 R 15.92 0.732.33 17.62 LCLx x A2 R 15.92 0.732.33 14.22 Reid & Sanders, Operations Management © Wiley 2002 Page 19 Example x-bar Chart X-bar Chart 18 UCL 17 Ounces 16 CL 15 14 LCL 13 12 1 2 3 4 5 6 7 8 9 10 Time Reid & Sanders, Operations Management © Wiley 2002 Page 20 R-chart Computations (Use D3 & D4 Factors: Table 6-1) 0.2 0.3 0.2 R 2.33 3 UCLR RD4 2.332.28 6.71 LCLR RD3 2.330 0 Reid & Sanders, Operations Management © Wiley 2002 Page 21 Example R-chart R Chart 8 7 UCL 6 Ounces 5 4 3 CL 2 1 0 LCL 1 2 3 4 5 6 7 8 9 10 11 Time Reid & Sanders, Operations Management © Wiley 2002 Page 22 Using x-bar & R-charts • Use together • Reveal different problems Reid & Sanders, Operations Management © Wiley 2002 Page 23 Control Charts for Attribute Data • p-Charts: – Track the proportion defective in a sample • c-Charts: – Track the average number of defects per unit of output Reid & Sanders, Operations Management © Wiley 2002 Page 24 Process Capability • A measure of the ability of a process to meet preset design specifications: – Determines whether the process can do what we are asking it to do • Design specifications (a/k/a tolerance limits): – Preset by design engineers to define the acceptable range of individual product characteristics (e.g.: physical dimensions, elapsed time, etc.) – Based upon customer expectations & how the product works (not statistics!) Reid & Sanders, Operations Management © Wiley 2002 Page 25 Measuring Process Capability Compare the width of design specifications & observed process output Reid & Sanders, Operations Management © Wiley 2002 Page 26 Capability Indexes • Centered Process (Cp): specificat ion width USL LSL Cp process width 6 • Any Process (Cpk): C pk USL LSL min ; 3 3 Reid & Sanders, Operations Management © Wiley 2002 Page 27 Example • Design specifications call for a target value of 16.0 +/-0.2 microns (USL = 16.2 & LSL = 15.8) • Observed process output has a mean of 15.9 and a standard deviation of 0.1 microns Reid & Sanders, Operations Management © Wiley 2002 Page 28 Computations • C p: • Cpk: USL LSL 16.2 15.8 0.4 Cp 0.66 6 60.1 0.6 USL LSL C pk min or 3 3 16.2 15.9 15.9 15.8 min or 30.1 30.1 0.3 0.1 min or min 1 or 0.33 0.33 0.3 0.3 Reid & Sanders, Operations Management © Wiley 2002 Page 29 Three Sigma Capability • Until now, we assumed process output should be modeled as +/- 3 standard deviations • By doing so, we ignore the 0.26% of output that falls outside +/- 3 sigma range • The result: a 3-sigma capable process produces 2600 defects for every million units produced Reid & Sanders, Operations Management © Wiley 2002 Page 30 Six Sigma Capability • Six sigma capability assumes the process is capable of producing output where +/- 6 standard deviations fall within the design specifications (even when the mean output drifts up to 1.5 standard deviations off target) • The result: only 3.4 defects for every million produced Reid & Sanders, Operations Management © Wiley 2002 Page 31 3-Sigma versus 6-Sigma Reid & Sanders, Operations Management © Wiley 2002 Page 32 The End Copyright © 2002 John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that permitted in Section 117 of the 1976 United State Copyright Act without the express written permission of the copyright owner is unlawful. Request for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. The purchaser may make back-up copies for his/her own use only and not for distribution or resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein. Reid & Sanders, Operations Management © Wiley 2002 Page 33