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Statistical Process Control Chapter 4 Chapter Outline Foundations of quality control Product launch and quality control activities Quality measures and control charts Transformation processes and variation Statistical process control (SPC) Variation and conformance quality SPC in services Chapter Outline (2) SPC overview Objectives of SPC Control chart format Hypothesis testing Terminology (what is n?) Chapter Outline (3) Control charts for variables x-bar charts R charts Control chart patterns Control charts for attributes p charts Chapter Outline (4) Process capability re-visited Control limits vs. specification limits Process capability ratio, Cp Cp does not work when the mean and the target are not equal Process capability index, Cpk Product launch activities: Revise periodically Customer Requirements Product Specifications Process Specifications Statistical Process Control: Measure & monitor quality Ongoing Activities Meets Specifications? Yes Conformance Quality No Fix process or inputs Quality Measures and Control Charts Discrete measures Good/bad, yes/no (p charts) Count of defects (c charts) Variables – continuous numerical measures Length, diameter, weight, height, time, speed, temperature, pressure Controlled with x and R charts Variation in a Transformation Process Inputs • Facilities • Equipment • Materials • Energy Transformation Process Outputs Goods & Services •Variation in inputs create variation in outputs • Variations in the transformation process create variation in outputs Types of Variation Common Cause Variation Common cause (random) variation: systematic variation in a process. Results from usual variations in inputs, output rates, and procedures Usually results from a poorly designed product or process, poor vendor selection, or other management issues If the amount of common cause variation is not acceptable, it is management's responsibility to take corrective action. Types of Variation Special Cause Variation Special cause (non-random or assignable cause) variation: a short-term source of variation in a process. Results from changes or abnormal variations in inputs, outputs, or procedures. Usually results from errors by workers, first-line supervisors, or vendors The cause can and should be identified. Corrective action should be taken. Statistical Process Control (SPC) A process is in control if it has no assignable cause variation. The process is consistent or predictable. SPC distinguishes between common cause and assignable cause variation Measure characteristics of goods or services that are important to customers Make a control chart for each characteristic The chart is used to determine whether the process is in control Specification Limits The target is the ideal value Example: if the amount of beverage in a bottle should be 16 ounces, the target is 16 ounces Specification limits are the acceptable range of values for a variable Example: the amount of beverage in a bottle must be at least 15.8 ounces and no more than 16.2 ounces. Range is 15.8 – 16.2 ounces. Lower specification limit = 15.8 ounces or LSPEC = 15.8 ounces Upper specification limit = 16.2 ounces or USPEC = 16.2 ounces Specifications and Conformance Quality A product which meets its specification has conformance quality. Capable process: a process which consistently produces products that have conformance quality. Must be in control and meet specifications Capable Transformation Process Inputs • Facilities • Equipment • Materials • Energy Capable Transformation Process Outputs Goods & Services that meet specifications If the process is capable and the product specification is based on current customer requirements, outputs will meet customer requirements. Applying SPC to Services Nature of defect is different in services Service defect is a failure to meet customer requirements Monitor times, customer satisfaction, quality of work, product availability Copyright 2006 John Wiley & Sons, Inc. 4-15 Applying SPC to Services (2) Hospitals Grocery Stores timeliness and quickness of care, staff responses to requests, accuracy of lab tests, cleanliness, courtesy, accuracy of paperwork, speed of admittance and checkouts waiting time to check out, frequency of out-of-stock items, quality of food items, cleanliness, customer complaints, checkout register errors Airlines flight delays, lost luggage and luggage handling, waiting time at ticket counters and check-in, agent and flight attendant courtesy, accurate flight information, passenger cabin cleanliness and maintenance Copyright 2006 John Wiley & Sons, Inc. 4-16 Applying SPC to Services (3) Fast-Food Restaurants Catalogue-Order Companies waiting time for service, customer complaints, cleanliness, food quality, order accuracy, employee courtesy order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time Insurance Companies billing accuracy, timeliness of claims processing, agent availability and response time Copyright 2006 John Wiley & Sons, Inc. 4-17 Objectives of Statistical Process Control (SPC) Determine Whether the process is in control Whether the process is capable Whether the process is likely to remain in control and capable Control Chart Format Measure Upper Control Limit (UCL) Process Mean Lower Control Limit (LCL) Sample Hypothesis Test H0: The process mean (or range) has not changed. (null hypothesis) H1: The process mean (or range) has changed. (alternative hypothesis). If the process has only random variations and remains within the control limits, we accept H0. The process is in control. Terminology We take periodic random samples n = sample size = number of observations in each sample X and R Charts for Variables X = Sample mean Measure of central tendency Central Limit Theorem: X is normally distributed. R = Sample range Measure of variation R has a gamma distribution (not normal) Data for Examples 4.3 and 4.4 Sample 1 2 3 … 10 Slip-ring diameter (cm) 1 2 3 4 5 5.02 5.01 4.94 4.99 4.96 5.01 5.03 5.07 4.95 4.96 4.99 5.00 4.93 4.92 4.99 … … … … … 5.01 4.98 5.08 5.07 4.99 X R 4.98 0.08 5.00 0.12 4.97 0.08 … … 5.03 0.10 50.09 1.15 Note: n = number in each sample = 5 Calculate X and R for Each Sample Sample 1: X = 5.02 + 5.01 + 4.94 + 4.99 + 4.96 5 = 4.98 R = range = maximum - minimum = 5.02 - 4.94 = 0.08 Repeat for all samples Calculate X and R X = 4.98 + 5.00 + 4.97 + … + 5.03 = 5.01 10 R = 0.08 + 0.12 + 0.08 + … + 0.10 = 0.115 10 The Normal Distribution 95% 99.74% -3s -2s -1s m=0 1s 2s 3s Control Limits for X 99.7% confidence interval for X: (X - 3s, X + 3s). This may be approximated as (X - A2R, X + A2R). A2 is a factor which depends on n and is obtained from a table. 3s Control Chart Factors Sample size n 2 3 4 5 6 7 8 x-chart A2 1.88 1.02 0.73 0.58 0.48 0.42 0.37 R-chart D3 0 0 0 0 0 0.08 0.14 D4 3.27 2.57 2.28 2.11 2.00 1.92 1.86 Control Limits for X and R For X: LCL = X - A2R = 5.01 - 0.58 (0.115) = 4.94 UCL = X + A2R = 5.01 + 0.58 (0.115) = 5.08 For R: LCL = D3R = 0 (0.115) = 0 UCL = D4R = 2.11 (0.115) = 0.243 5.10 – 5.08 – UCL = 5.08 5.06 – Mean 5.04 – 5.02 – x= = 5.01 5.00 – 4.98 – 4.96 – LCL = 4.94 4.94 – 4.92 – | 1 | 2 | 3 | | | | 4 5 6 7 Sample number | 8 | 9 | 10 R Chart 0.28 – 0.24 – Range 0.20 – 0.16 – UCL = 0.243 R = 0.115 0.12 – 0.08 – 0.04 – 0– LCL = 0 | | | 1 2 3 | | | | 4 5 6 7 Sample number | 8 | 9 | 10 Control Chart Pattern – Change in Mean UCL UCL LCL Sample observations consistently below the center line LCL Sample observations consistently above the center line Copyright 2006 John Wiley & Sons, Inc. 4-32 Control Chart Patterns: Trend UCL UCL LCL Sample observations consistently increasing LCL Sample observations consistently decreasing Copyright 2006 John Wiley & Sons, Inc. 4-33 Control Charts for Attributes p-charts uses portion defective in a sample c-charts uses number of defects in an item Copyright 2006 John Wiley & Sons, Inc. 4-34 p-Chart UCL = p + zsp LCL = p - zsp z = number of standard deviations from process average p = sample proportion defective; an estimate of process average sp = standard deviation of sample proportion sp = Copyright 2006 John Wiley & Sons, Inc. p(1 - p) n 4-35 p-Chart Example SAMPLE NUMBER OF DEFECTIVES PROPORTION DEFECTIVE 6 0 4 : : 18 200 .06 .00 .04 : : .18 1 2 3 : : 20 20 samples of 100 pairs of jeans Copyright 2006 John Wiley & Sons, Inc. 4-36 p-Chart Example (cont.) p= total defectives = 200 / 20(100) = 0.10 total sample observations UCL = p + z p(1 - p) = 0.10 + 3 n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = p - z p(1 - p) = 0.10 - 3 n 0.10(1 - 0.10) 100 LCL = 0.010 Copyright 2006 John Wiley & Sons, Inc. 4-37 0.20 UCL = 0.190 0.18 p-Chart Example Proportion defective 0.16 0.14 0.12 0.10 p = 0.10 0.08 0.06 0.04 0.02 LCL = 0.010 2 Copyright 2006 John Wiley & Sons, Inc. 4 6 8 10 12 14 Sample number 16 18 20 4-38 Process Capability Revisited A process must be in control before you can decide whether or not it is capable. Control charts measure the range of natural variability in a process (what the process is actually producing) Specification limits are set to meet customer requirements. Process cannot meet specifications if one or both control limits is outside specification limits Process Meets Customer Requirements Upper specification limit UCL X LCL Lower specification limit Process Does Not Meet Customer Requirements UCL Upper specification limit X Lower specification limit LCL Process Capability Ratio Cp For a product characteristic, let LSL = lower specification limit USL = upper specification limit m = mean, s = standard deviation If (1) The process is in control and (2) m = target (or mean = target) we can use the process capability ratio, Cp to determine whether the process is capable Computing Cp Given: LSL = 8.5, USL = 9.5, target = 9, m = 9, s = 0.12 Note that m = target USL LSL 9.5 8.5 1 Compute: Cp 6s 6(0.12) 0.72 1.39 If Cp > 1, the process is capable. If Cp < 1, the process is not capable Conclusion: Cp = 1.39 > 1 process is capable. Check on the Accuracy of Cp LCL = m - 3s = 9 – 3(0.12) = 8.64 UCL = m + 3s = 9 + 3(0.12) = 9.36 The specification limits are 8.5 – 9.5 The control limits are within the specification limits. The process is capable. Computing Cp – Another Example Given: LSL = 8.5, USL = 9.5, target = 9, m = 8.8, s = 0.12 Note that m does not equal the target. USL LSL 9.5 8.5 1 Compute: Cp 6s 6(0.12) 0.72 1.39 Conclusion: Cp = 1.39 > 1 process is capable. Wrong! LCL = m - 3s = 8.8 – 3(0.12) = 8.44 < LSL Cp can give the wrong answer if m does not equal the target. Use Cpk Computing Cpk Given: LSL = 8.5, USL = 9.5, target = 9, m = 8.8, s = 0.12 Compute: 8.8 - 8.5 9.5 8.8 m LSL USL m C pk minimum , min , 3s 3s 3(0.12) 3(0.12) min 0.83,1.94 0.83 Cpk < 1 the process is not capable Cpk always tells you whether the process is capable. Note: If m is not given, use x instead of m