Statistical Quality Control Chapter 15 Statistical Quality Control To Accompany Russell and Taylor, Operations Management, 4th Edition,  2003 Prentice-Hall, Inc. All rights reserved.

Statistical Process Control Take periodic samples from process Plot sample points on control chart Determine if process is within limits Prevent quality problems UCL LCL

Variation Common Causes Special 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 Variable 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) Charts for attributes p and c

Process Control Chart Out of control Upper control limit Process 1 2 3 4 5 6 7 8 9 10 Sample number Upper control limit Process average Lower Out of control 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 discard data Correct control chart limits

Control Charts for Attributes p Charts Calculate percent defectives in sample c Charts Count number of defects in item

p-Chart UCL = p + zp LCL = p - zp 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 = p(1 - p) n

The Normal Distribution =0 1 2 3 -1 -2 -3 95% 99.74%

Control Chart Z Values Smaller Z values make more sensitive charts Z = 3.00 is standard Compromise between sensitivity and errors

p-Chart Example 20 samples of 100 pairs of jeans 1 6 .06 2 0 .00 NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE 1 6 .06 2 0 .00 3 4 .04 : : : 20 18 .18 200 Example 15.1

total sample observations p-Chart Example 20 samples of 100 pairs of jeans NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE 1 6 .06 2 0 .00 3 4 .04 : : : 20 18 .18 200 p = = 200 / 20(100) = 0.10 total defectives total sample observations Example 15.1

p-Chart Example p = 0.10 20 samples of 100 pairs of jeans 1 6 .06 NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE 1 6 .06 2 0 .00 3 4 .04 : : : 20 18 .18 200 p = 0.10 UCL = p + z = 0.10 + 3 p(1 - p) n 0.10(1 - 0.10) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = 0.10 - 3 Example 15.1

p-Chart 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 Proportion defective Sample number 2 4 6 8 10 12 14 16 18 20 UCL = 0.190 LCL = 0.010 p = 0.10

c-Chart UCL = c + zc c = c LCL = c - zc where c = number of defects per sample

c-Chart The number of defects in 15 sample rooms 1 12 190 2 8 15 3 16 1 12 2 8 3 16 : : 15 15 190 SAMPLE NUMBER OF DEFECTS c = = 12.67 190 15 UCL = c + zc = 12.67 + 3 12.67 = 23.35 LCL = c + zc = 12.67 - 3 12.67 = 1.99 Example 15.2

c-Chart 3 6 9 12 15 18 21 24 Number of defects Sample number 2 4 8 10 14 16 UCL = 23.35 LCL = 1.99 c = 12.67

Control Charts for Variables Mean chart ( x -Chart ) Uses average of a sample Range chart ( R-Chart ) Uses amount of dispersion in a sample

Range ( R- ) Chart UCL = D4R LCL = D3R R k R = where R = range of each sample k = number of samples

Range ( R- ) Chart SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART n A2 D3 D4 SAMPLE SIZE FACTOR FOR x-CHART FACTORS FOR R-CHART 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 Range ( R- ) Chart Table 15.1

R-Chart Example OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 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 Example 15.3

R-Chart Example R k R = = = 0.115 1.15 10 UCL = D4R = 2.11(0.115) = 0.243 LCL = D3R = 0(0.115) = 0 OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 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 UCL = 0.243 LCL = 0 Range Sample number R = 0.115 | 1 2 3 4 5 6 7 8 9 10 0.28 – 0.24 – 0.20 – 0.16 – 0.12 – 0.08 – 0.04 – 0 – Example 15.3

x-Chart Calculations x1 + x2 + ... xk = k x = = UCL = x + A2R LCL = x - A2R = where x = the average of the sample means =

x-Chart Example UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 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 UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94 = x = = = 5.01 cm x k 50.09 10 Example 15.4

x-Chart Example UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 LCL = 4.94 Mean Sample number | 1 2 3 4 5 6 7 8 9 10 5.10 – 5.08 – 5.06 – 5.04 – 5.02 – 5.00 – 4.98 – 4.96 – 4.94 – 4.92 – x = 5.01 = OBSERVATIONS (SLIP-RING DIAMETER, CM) SAMPLE k 1 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 UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08 LCL = x - A2R = 5.01 - (0.58)(0.115) = 4.94 = x = = = 5.01 cm x k 50.09 10 Example 15.4

Using x- and R-Charts Together Each measures the process differently Both process average and variability must be in control

Control Chart Patterns LCL UCL Sample observations consistently below the center line UCL LCL Sample observations consistently above the center line Figure 15.3

Control Chart Patterns LCL UCL Sample observations consistently increasing UCL LCL Sample observations consistently decreasing Figure 15.3

Zones for Pattern Tests UCL LCL Zone A Zone B Zone C Process average 3 sigma = x + A2R = 3 sigma = x - A2R 2 sigma = x + (A2R) 2 3 2 sigma = x - (A2R) 1 sigma = x + (A2R) 1 1 sigma = x - (A2R) x Sample number | 4 5 6 7 8 9 10 11 12 13 Figure 15.4

Control Chart Patterns 8 consecutive points on one side of the center line. 8 consecutive points up or down across Zones. 14 points alternating up or down. 2 out of 3 consecutive points in Zone A but still inside the control limits. 4 out of 5 consecutive points in Zone A or B.

Performing a Pattern Test 1 4.98 B — B 2 5.00 B U C 3 4.95 B D A 4 4.96 B D A 5 4.99 B U C 6 5.01 — U C 7 5.02 A U C 8 5.05 A U B 9 5.08 A U A 10 5.03 A D B SAMPLE x ABOVE/BELOW UP/DOWN ZONE Example 15.5

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 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 Design Specifications Process Capability (a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time. Design Specifications Process (b) Design specifications and natural variation the same; process is capable of meeting specifications most the time. Design Specifications Process Figure 15.5

Design Specifications Design Specifications Process Capability (c) Design specifications greater than natural variation; process is capable of always conforming to specifications. Design Specifications Process (d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification. Design Specifications Process Figure 15.5

Process Capability Measures Process Capability Ratio Cp = = tolerance range process range upper specification limit - lower specification limit 6

Computing Cp Net weight specification = 9.0 oz  0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Cp = = = 1.39 upper specification limit - lower specification limit 6 9.5 - 8.5 6(0.12) Example 15.6

Process Capability Measures Process Capability Index Cpk = minimum x - lower specification limit 3 = upper specification limit - x ,

x - lower specification limit upper specification limit - x Computing Cpk Net weight specification = 9.0 oz  0.5 oz Process mean = 8.80 oz Process standard deviation = 0.12 oz Cpk = minimum = minimum , = 0.83 x - lower specification limit 3 = upper specification limit - x , 8.80 - 8.50 3(0.12) 9.50 - 8.80 Example 15.7