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COMPLETE BUSINESS STATISTICS by AMIR D. ACZEL & JAYAVEL SOUNDERPANDIAN 7th edition. Prepared by Lloyd Jaisingh, Morehead State University Chapter 13 Quality Control and Improvement McGraw-Hill/Irwin Copyright © 2009 by The McGraw-Hill Companies, Inc. All rights reserved. 13-2 13 Quality Control and Improvement • Using Statistics • W. Edwards Deming Instructs • Statistics and Quality • The x-bar Chart • The R Chart and the s Chart • The p Chart • The c Chart • The x Chart 13-3 13 LEARNING OBJECTIVES After studying this chapter you will be able to: • Determine when to use control charts • Create control charts for sample means, ranges and standard deviations • Create control charts for sample proportions • Create control charts for the number of defectives • Draw Pareto charts using spreadsheet templates • Draw control charts using spreadsheet templates 13-4 13-3 Statistics and Quality A control chart is a time plot of a statistic, such as a sample mean, range, standard deviation, or proportion, with a center line and upper and lower control limits. The limits give the desired range of values for the statistic. When the statistic is outside the bounds, or when its time plot reveals certain patterns, the process may be out of control. Value This point is out of the control limits UCL 3 3 Center Line LCL Time A process is considered in statistical control if it has no assignable causes, only natural variation. 13-5 Control Charts Value Process is in control Time Value Process mean varies over time: process is out of control Time 13-6 Control Charts (Continued) Value Process variance changes over time: process is out of control. Process mean and variance change over time: process is out of control. Time Value Time 13-7 Pareto Diagrams – Using the Template A Pareto diagram is a bar chart of the various problems in production and their percentages, which must add to 100%. A Pareto chart helps to identify the most significant problems and thus one can concentrate on their solutions rather than waste time and resources on unimportant causes. 13-8 Acceptance Sampling •Finished products are grouped in lots before being shipped to customers. •The lots are numbered, and random samples from these lots are inspected for quality. •Such checks are made before the lots are shipped and after the lots arrive at their destination. •The random samples are measured to find out which and how many items do not meet specifications •A lot is rejected whenever the sample mean exceeds or falls below some pre-specified limit. 13-9 Acceptance Sampling • For attribute data, the lot is rejected when the number of defectives or non-conforming items in the sample exceeds a pre-specified limit. • Acceptance sampling does not improve quality by itself. • It simply removes bad lots. • To improve quality, it is necessary to control the production process itself, removing any assignable causes and striving to reduce the variation in the process. 13-10 Six Sigma • Six Sigma is a further innovation, beyond Deming’s work, in the field of quality assurance and control. • The purpose of Six Sigma is to push defect levels below a certain specified threshold. • Six Sigma helps to improve quality. • The key to Six Sigma is a precise definition of the production process with accurate measurements and valid collection of data. 13-11 Six Sigma • It also involves detailed analysis to measure the relationships and causality of key factors in the production process. • Experimental Design is used to identify these key factors. • Strict control of the production process is exercised. Any variations are corrected, and the process is further monitored as it goes on line. • The essence of Six Sigma is the statistical methods described in this chapter. 13-12 13-4 The X-Bar Chart: A Control Chart for the Process Mean Elements of a control chart for the process mean: k xi Center line: x i 1 k LCL: x A2 R UCL: x A2 R where: k = number of samples, each of size n xi = Sample mean for sample i R Range of sample i i k R i i=1 R = k If the sample size in each group is more than 10: s / c4 s / c4 LCL = x - 3 UCL = x + 3 n n where s is the average of the standard deviations of all groups. n A2 2 3 4 5 6 7 8 9 10 15 20 25 1.880 1.023 0.729 0.577 0.483 0.419 0.373 0.337 0.308 0.223 0.180 0.153 c4 0.7979 0.8862 0.9213 0.9400 0.9515 0.9594 0.9650 0.9693 0.9727 0.9823 0.9869 0.9896 13-13 The X-Bar Chart: A Control Chart for the Process Mean (Continued) • Tests for assignable causes: One point beyond 3 (3s) Nine points in a row on one side of the center line Six points in a row steadily increasing or decreasing Fourteen points in a row alternating up and down Two out of three points in a row beyond 2 (2s) Four out of five points in a row beyond 1 (1s) Fifteen points in a row within 1 (1s) of the center line Eight points in a row on both sides of the center line, all beyond 1 (1s) 13-14 Tests for Assignable Causes Value 3 2 1 1 2 3 Test 1: One value beyond 3 (3s) Time Value 3 2 1 1 2 3 Test 2: Nine points in a row on one side of the center line. Time 13-15 Tests for Assignable Causes (Continued) Value 3 2 1 1 2 3 Test 3: Six points in a row steadily increasing or decreasing. Time Value 3 2 1 1 2 3 Test 4: Fourteen points in a row alternating up and down. Time 13-16 Tests for Assignable Causes (Continued) Value 3 2 1 1 2 3 Test 5: Two out of three points in a row beyond 2 (2s) Time Value 3 2 1 1 2 3 Test 6: Four out of five points in a row beyond 1 (1s) Time 13-17 Tests for Assignable Causes (Continued) Value 3 2 1 1 2 3 Test 7: Fifteen points in a row within 1 (1s) of the center line. Time Value 3 2 1 1 2 3 Test 8: Eight points in a row on both sides of the center line, all beyond 1 (1s) Time 13-18 X-bar Chart: Example 13-1 – Using the Template 13-19 X-bar Chart: Example 13-1(continued) – Using the Template Note: The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control. 13-20 X-bar Chart: Example 13-1(continued) – Using Minitab Xbar Chart of Concentration 10.8 UCL=10.784 Sample Mean 10.6 10.4 _ _ X=10.257 10.2 10.0 9.8 LCL=9.731 9.6 1 2 3 4 5 6 Sample 7 8 9 10 Note: The X-bar chart cannot be interpreted unless the R or s chart has been examined and is in control. 13-21 13-5 The R Chart and s Chart Elements of a control chart for the process range: Center line: R LCL: D3 R UCL: D4 R k R i i=1 where: R = k Elements of a control chart for the process standard deviation: Center line: s LCL: B3 s UCL: B4 s k s i i=1 where: s = k n 2 3 4 5 6 7 8 9 10 15 20 25 D3 0 0 0 0 0 0.076 0.136 0.184 0.223 0.348 0.414 0.459 D4 3.267 2.575 2.282 2.115 2.004 1.924 1.864 1.816 1.777 1.652 1.586 1.541 B3 0 0 0 0 0.030 0.118 0.185 0.239 0.284 0.428 0.510 0.565 B4 3.267 2.568 2.266 2.089 1.970 1.882 1.815 1.761 1.716 1.572 1.490 1.435 13-22 R Chart: Example 13-1 using the Template The process range seems to be in control. 13-23 s Chart: Example 13-1 using the Template The process standard deviation seems to be in control. 13-24 Example 13-2 using the Template 13-25 Example 13-2 using the Template Continued 13-26 Example 13-2 using the Template Continued Based on the x-bar, R, and s charts, the process seems to be in control. 13-27 Example 13-2 using Minitab Xbar Chart of Delivery Times UCL=7.196 7 Sample Mean 6 _ _ X=4.8 5 4 3 LCL=2.404 2 1 2 3 4 5 6 Sample 7 8 9 10 13-28 Example 13-2 using Minitab R Chart of Delivery Times UCL=6.029 6 Sample Range 5 4 3 _ R=2.342 2 1 0 LCL=0 1 2 3 4 5 6 Sample 7 8 9 10 13-29 Example 13-2 using Minitab S Chart of Delivery Times 3.5 UCL=3.149 3.0 Sample StDev 2.5 2.0 1.5 _ S=1.226 1.0 0.5 0.0 LCL=0 1 2 3 4 5 6 Sample 7 8 9 10 Based on the x-bar, R, and s charts, the process seems to be in control. 13-30 13-6 The p Chart: Proportion of Defective Items Elements of a control chart for the process proportion: Center line: p p(1 - p) p(1 - p) LCL: p - 3 UCL: p + 3 n n where: n is the number of elements in each sample p is the proportion of defectives in the combined, overall sample 13-31 13-6 The p Chart: Proportion of Defective Items – Using the Template for Example 13-3 Process is out of control – Two points fall outside the control limit 13-32 13-6 The p Chart: Proportion of Defective Items – Using Minitab for Example 13-3 P Chart of Defectives 0.4 1 1 0.3 Proportion UCL=0.2624 0.2 _ P=0.1125 0.1 0.0 LCL=0 1 2 3 4 5 6 7 Sample 8 9 10 11 12 Process is out of control – Two points fall outside the control limit 13-33 13-7 The c Chart: (Defects Per Item) Elements of a control chart for the number of imperfections per item, c: Center line: c LCL: c - 3 c UCL: c + 3 c where: c is the average number of defects or imperfections per item (or area, volume, etc. ) 13-34 The c Chart: Example 13-4 using the Template Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general downward trend should be investigated. 13-35 The c Chart: Example 13-4 using Minitab C Chart of Nonconformaties 18 1 16 UCL=15.81 Sample Count 14 12 10 _ C=7.56 8 6 4 2 0 LCL=0 1 3 5 7 9 11 13 15 Sample 17 19 21 23 25 Observe that one observation is outside the upper control limit, indicating that the process may be out of control. The general downward trend should be investigated. 13-36 13-8 The x Chart Sometimes we are interested in controlling the process mean, but our observations come so slowly from the production process that we cannot aggregate them into groups. In such case we may consider an x chart. An x-chart is a chart for the raw values of the variable in question. The chart is effective if the variable has an approximate normal distribution. The bounds are 3 standard deviations from the mean of the process. 13-37 13-8 The x Chart for Example 13-3 – Using Minitab I Chart of Defectives 20 UCL=17.80 Individual Value 15 10 _ X=4.5 5 0 -5 LCL=-8.80 -10 1 2 3 4 5 6 7 8 Observation 9 10 11 12 NOTE: The X-Chart Is same as the Individual chart in Minitab 13-38 13-8 The x Chart for Example 13-4 – Using Minitab I Chart of Nonconformaties 20 UCL=18.75 Individual Value 15 10 _ X=7.56 5 0 LCL=-3.63 -5 1 3 5 7 9 11 13 15 Observation 17 19 21 23 25