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## Chapter 15

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**Chapter 15**Statistical Quality Control (Revised 8/10/04) To Accompany Russell and Taylor, Operations Management, 4th Edition, 2003 Prentice-Hall, Inc. All rights reserved.**UCL**LCL Statistical Process Control • Take periodic samples from process • Plot sample points on control chart • Determine if process is within limits • Prevent quality problems**Variation**• Common Causes • Variation inherent in a process • Can be eliminated only through improvements in the system • Special Causes • Variation due to identifiable factors • Can be modified through operator or management action**Types of Data**• Attribute data • Product characteristic evaluated with a discrete choice • Good/bad, yes/no • Variable data • Product characteristic that can be measured • Length, size, weight, height, time, velocity**SPC Applied to Services**• Nature of defect is different in services • Service defect is a failure to meet customer requirements • Monitor times, customer satisfaction**Service Quality Examples**• Hospitals • Timeliness, responsiveness, accuracy of lab tests • Grocery Stores • Check-out time, stocking, cleanliness • Airlines • Luggage handling, waiting times, courtesy • Fast food restaurants • Waiting times, food quality, cleanliness, employee courtesy**Service Quality Examples**• Catalog-order companies • 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**Control Charts**• Graph establishing process control limits • Charts for variables • Mean (x-bar), Range (R) • Chart for attributes • P Chart**Out of control**Upper control limit Process average Lower control limit 1 2 3 4 5 6 7 8 9 10 Sample number Process Control Chart Figure 15.1**A Process is In Control if**No sample points outside limits Most points near process average About equal number of points above & below centerline Points appear randomly distributed**Development of Control Chart**• Based on in-control data • If non-random causes present, find the special cause and discard data • Correct control chart limits**Control Chart for Attributes**• p Charts • Calculate percent defectives in sample**UCL = p + zp**LCL = p - zp where z = the number of standard deviations from the process average p = the sample proportion defective; an estimate of the process average p = the standard deviation of the sample proportion p(1 - p) n p = p-Chart**95%**99.74% -3 -2 -1 =0 1 2 3 The Normal Distribution**Control Chart Z Values**• Smaller Z values make more sensitive charts • Z = 3.00 is standard • Compromise between sensitivity and errors**NUMBER OF PROPORTION**SAMPLE DEFECTIVES DEFECTIVE 1 6 .06 2 0 .00 3 4 .04 : : : : : : 20 18 .18 200 p-Chart Example 20 samples of 100 pairs of jeans Example 15.1**NUMBER OF PROPORTION**SAMPLE DEFECTIVES DEFECTIVE 1 6 .06 2 0 .00 3 4 .04 : : : : : : 20 18 .18 200 total defectives total sample observations p = = 200 / 20(100) = 0.10 p-Chart Example 20 samples of 100 pairs of jeans Example 15.1**NUMBER OF PROPORTION**SAMPLE DEFECTIVES DEFECTIVE p = 0.10 1 6 .06 2 0 .00 3 4 .04 : : : : : : 20 18 .18 200 0.10(1 - 0.10) 100 p(1 - p) n UCL = p + z = 0.10 + 3 UCL = 0.190 0.10(1 - 0.10) 100 p(1 - p) n LCL = p - z = 0.10 - 3 LCL = 0.010 p-Chart Example 20 samples of 100 pairs of jeans Example 15.1**0.20**UCL = 0.190 0.18 0.16 0.14 0.12 p = 0.10 0.10 Proportion defective 0.08 0.06 0.04 0.02 LCL = 0.010 2 4 6 8 10 12 14 16 18 20 Sample number p-Chart**Control Charts for Variables**• Mean chart ( x -Chart ) • Uses average of a sample • Range chart ( R-Chart ) • Uses amount of dispersion in a sample**UCL = D4R LCL = D3R**• R k R = where R = range of each sample k = number of samples Range ( R- ) Chart**nA2D3 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 0.44 0.18 1.82 10 0.11 0.22 1.78 11 0.99 0.26 1.74 12 0.77 0.28 1.72 13 0.55 0.31 1.69 14 0.44 0.33 1.67 15 0.22 0.35 1.65 16 0.11 0.36 1.64 17 0.00 0.38 1.62 18 0.99 0.39 1.61 19 0.99 0.40 1.61 20 0.88 0.41 1.59 SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART Range ( R- ) Chart Table 15.1**OBSERVATIONS (SLIP-RING DIAMETER, CM)**SAMPLE k1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 R-Chart Example Example 15.3**R**k 1.15 10 UCL = D4R = 2.11(0.115) = 0.243 LCL = D3R = 0(0.115) = 0 OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 R = = = 0.115 0.28 – 0.24 – 0.20 – 0.16 – 0.12 – 0.08 – 0.04 – 0 – UCL = 0.243 R = 0.115 Range LCL = 0 | 1 | 2 | 3 | 4 | 5 | 6 | 8 | 9 | 10 | 7 Sample number R-Chart Example Example 15.3**x1 + x2 + ... xk**k = x = = = UCL = x + A2R LCL = x - A2R where x = the average of the sample means = x-Chart Calculations**OBSERVATIONS (SLIP-RING DIAMETER, CM)**SAMPLE k1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 • x k 50.09 10 = x = = = 5.01 cm UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94 = = x-Chart Example Example 15.4**5.10 –**5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – UCL = 5.08 OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k1 2 3 4 5 x R 1 5.02 5.01 4.94 4.99 4.96 4.98 0.08 2 5.01 5.03 5.07 4.95 4.96 5.00 0.12 3 4.99 5.00 4.93 4.92 4.99 4.97 0.08 4 5.03 4.91 5.01 4.98 4.89 4.96 0.14 5 4.95 4.92 5.03 5.05 5.01 4.99 0.13 6 4.97 5.06 5.06 4.96 5.03 5.01 0.10 7 5.05 5.01 5.10 4.96 4.99 5.02 0.14 8 5.09 5.10 5.00 4.99 5.08 5.05 0.11 9 5.14 5.10 4.99 5.08 5.09 5.08 0.15 10 5.01 4.98 5.08 5.07 4.99 5.03 0.10 50.09 1.15 • x k 50.09 10 = x = = = 5.01 cm = x = 5.01 Mean UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94 = = LCL = 4.94 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 Sample number x-Chart Example Example 15.4**Using x- and R-Charts Together**• Each measures the process differently • Both process average and variability must be in control**Sample Size Determination**• Attribute control charts • 50 to 100 parts in a sample • Variable control charts • 2 to 10 parts in a sample**Process Capability**• Process limits (The “Voice of the Process” or The “Voice of the Data”) - based on natural (common cause) variation • Tolerance limits (The “Voice of the Customer”) – customer requirements • Process Capability – A measure of how “capable” the process is to meet customer requirements; compares process limits to tolerance limits**Process Capability**• Range of natural variability in process • Measured with control charts. • Process cannot meet specifications if natural variability exceeds tolerances • 3-sigma quality • Specifications equal the process control limits. • 6-sigma quality • Specifications twice as large as control limits**Design Specifications**(a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time. Process Design Specifications (b) Design specifications and natural variation the same; process is capable of meeting specifications most the time. Process Process Capability Figure 15.5**Design Specifications**(c) Design specifications greater than natural variation; process is capable of always conforming to specifications. Process Design Specifications (d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification. Process Process Capability Figure 15.5**=**x - lower specification limit 3 , Cpk = minimum = upper specification limit - x 3 Process Capability Measures Process Capability Index**=**x - lower specification limit 3 , Cpk = minimum = minimum , = 0.83 = upper specification limit - x 3 9.50 - 8.80 3(0.12) 8.80 - 8.50 3(0.12) Computing Cpk Net weight specification = 9.0 oz 0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Example 15.7**Interpreting the Process Capability Index**Cpk < 1 Not Capable Cpk > 1 Capable at 3 Cpk > 1.33 Capable at 4 Cpk > 1.67 Capable at 5 Cpk > 2 Capable at 6