Chapter 15

Statistical Quality Control

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

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SPC Applied to Services 9 Nature of defect is different in services 9 Service defect is a failure to meet customer requirements 9 Monitor times, customer satisfaction

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Service Quality Examples 9 Hospitals 9Timeliness, responsiveness, accuracy of lab tests

9 Grocery Stores 9Check-out time, stocking, cleanliness

9 Airlines 9Luggage handling, waiting times, courtesy

9 Fast food restaurants 9Waiting times, food quality, cleanliness, employee courtesy To Accompany Russell and Taylor, Operations Management, 4th Edition, © 2003 Prentice-Hall, Inc. All rights reserved.

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Variation

Service Quality Examples 9 Catalog-order companies 9Order accuracy, operator knowledge and courtesy, packaging, delivery time, phone order waiting time 9 Insurance companies 9Billing accuracy, timeliness of claims processing, agent availability and response time

9 Common Causes 9Variation inherent in a process 9Can be eliminated only through improvements in the system

9 Special Causes 9Variation due to identifiable factors 9Can be modified through operator or management action

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Types of Data 9 Attribute data 9 Product characteristic evaluated with a discrete choice 9 Good/bad, yes/no

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Control Charts 9 Graph establishing process control limits 9 Charts for variables 9Mean (x-bar), Range (R)

9 Variable data 9 Product characteristic that can be measured 9 Length, size, weight, height, time, velocity To Accompany Russell and Taylor, Operations Management, 4th Edition, © 2003 Prentice-Hall, Inc. All rights reserved.

9 Chart for attributes 9P Chart

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Process Control Chart

A Process is In Control if

Out of control

Upper control limit

1. No sample points outside limits 2. Most points near process average 3. About equal number of points above & below centerline 4. Points appear randomly distributed

Process average

Lower control limit

1

2

3

4

5

6

7

8

9

10

Sample number

Figure 15.1

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Common Out-ofcontrol Signs One observation outside the limits

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Development of Control Chart 9 Based on in-control data 9 If non-random causes present, find the special cause and discard data

Sample observations consistently below or above the average

9 Correct control chart limits

Sample observations consistently decrease or increase

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Control Chart for Attributes

p-Chart UCL = p + zσp LCL = p - zσp

9 p Charts 9Calculate percent defectives in sample

where z = the number of standard deviations from the process average p = the sample proportion defective; an estimate of the process average σp = the standard deviation of the sample proportion σp =

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The Normal Distribution

99.74% -2σ

-1σ

µ=0

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Control Chart Z Values 9 Smaller Z values make more sensitive charts 9 Z = 3.00 is standard 9 Compromise between sensitivity and errors

95% -3σ

p(1 - p) n

1σ

2σ

3σ

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p-Chart Example

p-Chart Example

20 samples of 100 pairs of jeans

20 samples of 100 pairs of jeans

SAMPLE

1 2 3 : : 20

NUMBER OF DEFECTIVES

PROPORTION DEFECTIVE

SAMPLE

6 0 4 : : 18 200

.06 .00 .04 : : .18

1 2 3 : : 20

Example 15.1

NUMBER OF DEFECTIVES

PROPORTION DEFECTIVE

6 0 4 : : 18 200

.06 .00 total defectives p = .04 total sample observations : = 200: / 20(100) = 0.10 .18

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p-Chart Example

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p-Chart

PROPORTION DEFECTIVE

p = 0.10

6 .06 0 0.10(1 - 0.10) p(1.00 - p) UCL = p + z = 0.10 + 3 100 4 n.04 : UCL := 0.190 : : 0.10(1 - 0.10) p(1 - p) LCL = p z = 0.10 - 3 18 .18 100 n 200 LCL = 0.010

0.16 0.14 Proportion defective

1 2 3 : : 20

NUMBER OF DEFECTIVES

UCL = 0.190

0.18

20 samples of 100 pairs of jeans SAMPLE

0.20

0.12 0.10

p = 0.10

0.08 0.06 0.04 0.02

LCL = 0.010 2

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4

6

8 10 12 Sample number

14

16

18

20

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Control Charts for Variables

Range ( R- ) Chart UCL = D4R

9 Mean chart ( x -Chart ) 9Uses average of a sample

R=

9 Range chart ( R-Chart )

R = range of each sample k = number of samples

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FACTOR FOR x-CHART A2

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

1.88 1.02 0.73 0.58 0.48 0.42 0.37 0.44 0.11 0.99 0.77 0.55 0.44 0.22 0.11 0.00 0.99 0.99 0.88

FACTORS FOR R-CHART D3 D4

Range ( R- ) Chart 0.00 0.00 0.00 0.00 0.00 0.08 0.14 0.18 0.22 0.26 0.28 0.31 0.33 0.35 0.36 0.38 0.39 0.40 0.41

3.27 2.57 2.28 2.11 2.00 1.92 1.86 1.82 1.78 1.74 1.72 1.69 1.67 1.65 1.64 1.62 1.61 1.61 1.59

∑R k

where

9Uses amount of dispersion in a sample

SAMPLE SIZE n

LCL = D3R

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

1

2

3

4

5

x

R

1 2 3 4 5 6 7 8 9 10

5.02 5.01 4.99 5.03 4.95 4.97 5.05 5.09 5.14 5.01

5.01 5.03 5.00 4.91 4.92 5.06 5.01 5.10 5.10 4.98

4.94 5.07 4.93 5.01 5.03 5.06 5.10 5.00 4.99 5.08

4.99 4.95 4.92 4.98 5.05 4.96 4.96 4.99 5.08 5.07

4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99

4.98 5.00 4.97 4.96 4.99 5.01 5.02 5.05 5.08 5.03

0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10

Sum

50.09

1.15

Example 15.3

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R-Chart Example

UCL = D4R = 2.11(0.115) = 0.243 ∑R 1.15 = = 0.115 OBSERVATIONS DIAMETER, CM) k 10 LCL = D(SLIP-RING 3R = 0(0.115) = 0 SAMPLE0.28 k – 1 2 3 4 5 x R

x-Chart Calculations

R=

Range

1 2 3 4 5 6 7 8 9 10

0.24 – 0.20 – 0.16 – 0.12 – 0.08 – 0.04 – 0–

5.02 5.01 4.94 5.01 UCL 5.03 = 0.2435.07 4.99 5.00 4.93 5.03 4.91 5.01 0.115 5.03 4.95R =4.92 4.97 5.06 5.06 5.05 5.01 5.10 5.09 5.10 5.00 5.14 4.99 LCL =5.10 0 5.01 | | 4.98| 5.08 |

1

2

3

4.99 4.95 4.92 4.98 5.05 4.96 4.96 4.99 5.08 5.07 | |

4.96 4.96 4.99 4.89 5.01 5.03 4.99 5.08 5.09 4.99|

4 5 6 7 Sample number

4.98 0.08 5.00 0.12 4.97 0.08 4.96 0.14 4.99 0.13 5.01 0.10 5.02 0.14 5.05 0.11 5.08 0.15 5.03 | | 0.10| 8 91.1510 50.09

x1 + x2 + ... xk x= = k = UCL = x + A2R

= LCL = x - A2R

where = x = the average of the sample means

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x-Chart Example

x-Chart Example 5.10 – 5.08 –

OBSERVATIONS (SLIP-RING DIAMETER, CM) 1

2

3

4

50.09 = ∑x 4.94 4.99 x1= = 5.02 5.01= 5.01 cm 10 5.03 5.07 4.95 k 2 5.01

5

x

R

4.96 4.98 4.96 5.00 3 4.99 5.00 4.93 4.92 4.99 4.97 4 5.03 4.91 5.01 4.98 4.89 4.96 = 4.92+ (0.58)(0.115) 5.03 5.05 5.01 4.99 UCL5 = x + A2R4.95 = 5.01 = 5.08 6 4.97 5.06 5.06 4.96 5.03 5.01 = 5.01- (0.58)(0.115) 5.10 4.96 4.99 5.02 LCL7 = x - A2R5.05 = 5.01 = 4.94 8 5.09 5.10 5.00 4.99 5.08 5.05 9 5.14 5.10 4.99 5.08 5.09 5.08 10 5.01 4.98 5.08 5.07 4.99 5.03

0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10

50.09

1.15

Example 15.4

SAMPLE k

1

2

3

4

5.04 – 5.02 50.095.01 4.94 4.99 = ∑x x1= = = 5.01 cm k5.02 – 5.0110=5.03 5.07 4.95 2

5

x

4.96 4.98 4.96 5.00 x = 5.01 3 4.99 5.00 4.93 4.92 4.99 4.97 5.00 – 5.03 4.91 4 5.01 4.98 4.89 4.96 = 4.92+ (0.58)(0.115) 5.03 5.05 5.01 4.99 UCL5 = x 4.98 + A–2R4.95 = 5.01 = 5.08 6 4.97 5.06 5.06 4.96 5.03 5.01 = 5.01- (0.58)(0.115) 5.10 4.96 4.99 5.02 LCL7 = x -4.96 A2–R5.05 = 5.01 = 4.94 8 5.09 LCL 5.10 5.00 4.99 5.08 5.05 = 4.94 4.94 – 9 5.14 5.10 4.99 5.08 5.09 5.08 10 5.08 5.07 4.99 5.03 4.92 – 5.01 4.98 | 1 Example 15.4

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UCL = 5.08

OBSERVATIONS (SLIP-RING DIAMETER, CM)

Mean

SAMPLE k

5.06 –

| 2

| 3

| | | | 4 5 6 7 Sample number

R 0.08 0.12 0.08 0.14 0.13 0.10 0.14 0.11 0.15 0.10

| 50.09 | |1.15 8 9 10

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Using x- and R-Charts Together 9 Each measures the process differently 9 Both process average and variability must be in control

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Process Capability 9 Process limits (The “Voice of the Process” or The “Voice of the Data”) - based on natural (common cause) variation 9 Tolerance limits (The “Voice of the Customer”) – customer requirements 9 Process Capability – A measure of how “capable” the process is to meet customer requirements; compares process limits to tolerance limits

Sample Size Determination 9 Attribute control charts 950 to 100 parts in a sample

9 Variable control charts 92 to 10 parts in a sample

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Process Capability 9 Range of natural variability in process 9 Measured with control charts.

9 Process cannot meet specifications if natural variability exceeds tolerances 9 3-sigma quality 9 Specifications equal the process control limits.

9 6-sigma quality 9 Specifications twice as large as control limits To Accompany Russell and Taylor, Operations Management, 4th Edition, © 2003 Prentice-Hall, Inc. All rights reserved.

Process Capability

Process Capability

Design Specifications

Design Specifications

(a) Natural variation exceeds design specifications; process is not capable of meeting specifications all the time.

(c) Design specifications greater than natural variation; process is capable of always conforming to specifications. Process

Process Design Specifications

Design Specifications

(b) Design specifications and natural variation the same; process is capable of meeting specifications most the time.

(d) Specifications greater than natural variation, but process off center; capable but some output will not meet upper specification. Process

Figure 15.5

Process Figure 15.5

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Process Capability Measures: Cp Process Capability ratio Cp = =

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Process Capability Measures: Cpk Process Capability Index

Tolerance range Process range USL – LSL 6σ

Cp < 1

Not Capable

Cp ≥ 1

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Cpk = minimum

= x - lower specification limit , 3σ = upper specification limit - x 3σ

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Interpreting the Process Capability Index

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

= x - lower specification limit , 3σ = upper specification limit - x 3σ 8.80 - 8.50 9.50 - 8.80 , 3(0.12) 3(0.12)

= 0.83 Example 15.7

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Cpk < 1

Not Capable

Cpk > 1

Capable at 3σ

Cpk > 1.33

Capable at 4σ

Cpk > 1.67

Capable at 5σ

Cpk > 2

Capable at 6σ