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**Statistical Quality Control**

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

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**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

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**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

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**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

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**SPC Applied to Services**

Nature of defect is different in services Service defect is a failure to meet customer requirements Monitor times, customer satisfaction

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**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

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**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

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**Control Charts Graph establishing process control limits**

Charts for variables Mean (x-bar), Range (R) Charts for attributes p and c

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**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

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**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

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**Development of Control Chart**

Based on in-control data If non-random causes present discard data Correct control chart limits

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**Control Charts for Attributes**

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

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**p-Chart 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 = p(1 - p) n

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**The Normal Distribution**

=0 1 2 3 -1 -2 -3 95% 99.74%

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**Control Chart Z Values Smaller Z values make more sensitive charts**

Z = 3.00 is standard Compromise between sensitivity and errors

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**p-Chart Example 20 samples of 100 pairs of jeans 1 6 .06 2 0 .00**

NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE : : : 200 Example 15.1

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**total sample observations**

p-Chart Example 20 samples of 100 pairs of jeans NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE : : : 200 p = = 200 / 20(100) = 0.10 total defectives total sample observations Example 15.1

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**p-Chart Example p = 0.10 20 samples of 100 pairs of jeans 1 6 .06**

NUMBER OF PROPORTION SAMPLE DEFECTIVES DEFECTIVE : : : 200 p = 0.10 UCL = p + z = p(1 - p) n 0.10( ) 100 UCL = 0.190 LCL = 0.010 LCL = p - z = Example 15.1

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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

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**c-Chart UCL = c + zc c = c LCL = c - zc where**

c = number of defects per sample

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**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 + zc = = 23.35 LCL = c + zc = = 1.99 Example 15.2

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**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

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**Control Charts for Variables**

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

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**Range ( R- ) Chart UCL = D4R LCL = D3R R k R = where**

R = range of each sample k = number of samples

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**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 Range ( R- ) Chart Table 15.1

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**R-Chart Example OBSERVATIONS (SLIP-RING DIAMETER, CM)**

SAMPLE k x R Example 15.3

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**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 x R 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

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**x-Chart Calculations x1 + x2 + ... xk = k x = =**

UCL = x + A2R LCL = x - A2R = where x = the average of the sample means =

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**x-Chart Example UCL = x + A2R = 5.01 + (0.58)(0.115) = 5.08**

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

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**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 x R UCL = x + A2R = (0.58)(0.115) = 5.08 LCL = x - A2R = (0.58)(0.115) = 4.94 = x = = = 5.01 cm x k 50.09 10 Example 15.4

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**Using x- and R-Charts Together**

Each measures the process differently Both process average and variability must be in control

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**Control Chart Patterns**

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

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**Control Chart Patterns**

LCL UCL Sample observations consistently increasing UCL LCL Sample observations consistently decreasing Figure 15.3

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**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

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**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.

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**Performing a Pattern Test**

B — B B U C B D A B D A B U C — U C A U C A U B A U A A D B SAMPLE x ABOVE/BELOW UP/DOWN ZONE Example 15.5

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**Sample Size Determination**

Attribute control charts 50 to 100 parts in a sample Variable control charts 2 to 10 parts in a sample

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**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

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**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

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**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

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**Process Capability Measures**

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

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**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 6(0.12) Example 15.6

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**Process Capability Measures**

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

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**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 , 3(0.12) Example 15.7

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