MBA 8452 Systems and Operations ManagementMBA 8452 Systems and Operations Management

Statistical Quality Statistical Quality ControlControl

2

Objective: Quality Analysis Process Variation

Be able to explain Taguchi’s View of the cost of variation.

Statistical Process Control Charts and Process Capability Be able to develop and interpret SPC charts. Be able to calculate and interpret Cp and Cpk

Be able to explain the difference between process control and process capability

Sample Size Be able to explain the importance of sample

size

3

Statistical Quality Control Approaches

Statistical Process Control (SPC) Sampling to determine if the process is

within acceptable limits (under control)

Acceptance Sampling Inspects a random sample of a product to determine if the lot is acceptable

4

Line graph shows plot of dataand variation from the average

1 2 3 4 5 6 7 8 9 10

Sample number

Processtarget or average

5

Why Statistical Quality Control?

Variations in Manufacturing/Service Processes Any process has some variations: common and/or

special Variations are causes for quality problems

If a process is stable (no special variation), it is able to produce product/service consistently

As variation is reduced, quality is improved

Statistics is the only science that is dedicated to dealing with variations.

Measure, monitor, and reduce variations in the process

6

Types of Variation

Natural (common) Assignable (special) Exogenous to process Not random Controllable Preventable Examples

tool wear human factors (fatigue) poor maintenance

Inherent to process Random Cannot be controlled Cannot be prevented Examples

weather accuracy of measurements capability of machine

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Cost of Variation: Traditional vs. Taguchi’s View

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Traditional View

IncrementalCost of Variability

High

Zero

LowerSpec

TargetSpec

UpperSpec

Taguchi’s View

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Statistical Process Control On-line quality control tool used when the

product/service is being produced Purpose: prevent systematic quality

problems Procedure

Take periodic random samples from a process Plot the sample statistics on control chart(s) Determine if the process is under control

If the process is under control, do nothing If the process is out of control, investigate and fix the

cause

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Statistical Process Control Types Of Data

Attribute data (discrete values) Quality characteristic evaluated about

whether it meets the required specifications Good/bad, yes/no

Variable data (continuous values) Quality characteristic that can be measured

Length, size, weight, height, time, velocity

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Statistical Process Control Control Charts

Charts for attributes p-chart (for proportions) c-chart (for counts)

Charts for variables R-chart (for ranges) -chart (for means) X

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Control Chart General Structure

1 2 3 4 5 6 7 8 9 10

Sample number

Uppercontrollimit (UCL)

Processtarget or average

Lowercontrollimit (LCL)

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A Process Is In Control If ...

No sample points outside control limits

Most points near the process average

About an equal # points above & below the centerline

Points appear randomly distributed

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Common Out-of-control Signs

One observation outside the limits

Sample observations consistently below or above the average

Sample observations consistently decrease or increase

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Issues In Building Control Charts

Number of samples: around 25

Size of each sample: large (100) for attributes and small (25) for variables

Frequency of sampling: depends

Control limits: typically 3-sigma away from the process mean

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Control Limits: The Normal Distribution

99.74 %

95 %

1 +2 -1-2-3

If we establish control limits at +/- 3 standard deviations (), then we would expect 99.74% of observations (X) to fall within these limits.

X

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Control Limits: General Formulas

UCL = mean + z (stand dev)

LCL = mean – z (stand dev)

z is the # of standard deviations z = 3.00 is the most commonly used value

with 99.7% confidence level Other z values can be used (e.g. z=2 for

95% confidence and z=2.58 for 99% confidence)

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Control Charts for Attributes p-charts

p

p

z- p = LCL

z + p = UCL

s

s

ns

)p-(1 p = p

nsObservatio ofNumber Total

Defectives ofNumber Total=

sample in the defectives of proportion Average =p

size sample n

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p-Chart

Example 20 Samples of 100 pairs of jeans each were randomly selected from the Western Jean

Company’s production line.

.03= 100

.10)-.10(1=

)p-(1 p = p n

s

0.01 = 3(.03) - .10 =z - p = LCL

0.19 = 3(.03) .10 =z + p = UCL

p

p

s

s

total defectives total sample observations

200 20 (100)

p =

= = 0.10

n=100 jeans in each sample

Proportion Sample Defect Defective

1 6 .06 2 12 .12 3 4 .04

. . .

. . .

. . .

20 18 .18Total 200

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p-Chart Example

. .

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0 2 4 6 8

10 12 14 16 18 20

Pro

port

ion

defe

ctiv

e

Sample number

UCL

LCL

p

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Control Charts For Variables

X-bar charts and R-charts

Where X = average of sample means = Xi / mR = average of sample ranges = Ri / mXi = mean of sample i, i = 1,2,…,mRi = range of sample i, i = 1,2,…,mm = total number of samples

A2, D3, and D4 are constants from Exhibit TN7.7

RA - x = LCL

RA + x = UCL

2

2

Limits ControlChart x R Chart Control Limits

UCL = D R

LCL = D R

4

3

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Example

If a company makes jeans, there are a specifications that must be met. The back pockets of the jeans can’t be

too small or too large. The control chart can be established to

monitor the measurements of the back pocket

Given 15 samples with 5 observations each, we can determine the Upper and Lower control limits for the range and x-bar charts.

X-bar and R Charts Example

Sample SampleSample 1 2 3 4 5 Mean Range

1 10.682 10.689 10.776 10.798 10.714 10.732 0.1162 10.787 10.860 10.601 10.746 10.779 10.755 0.2593 10.780 10.667 10.838 10.785 10.723 10.759 0.1714 10.591 10.727 10.812 10.775 10.730 10.727 0.2215 10.693 10.708 10.790 10.758 10.671 10.724 0.1196 10.749 10.714 10.738 10.719 10.606 10.705 0.1437 10.791 10.713 10.689 10.877 10.603 10.735 0.2748 10.744 10.779 10.110 10.737 10.750 10.624 0.6699 10.769 10.773 10.641 10.644 10.725 10.710 0.13210 10.718 10.671 10.708 10.850 10.712 10.732 0.17911 10.787 10.821 10.764 10.658 10.708 10.748 0.16312 10.622 10.802 10.818 10.872 10.727 10.768 0.25013 10.657 10.822 10.893 10.544 10.750 10.733 0.34914 10.806 10.749 10.859 10.801 10.701 10.783 0.15815 10.660 10.681 10.644 10.747 10.728 10.692 0.103

10.728 0.220

Observation

Overall Averages

(Xi) (Ri)

X R

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X-bar and R Charts

Example

n A2 D3 D42 1.88 0 3.273 1.02 0 2.574 0.73 0 2.285 0.58 0 2.116 0.48 0 2.007 0.42 0.08 1.928 0.37 0.14 1.869 0.34 0.18 1.8210 0.31 0.22 1.7811 0.29 0.26 1.74

Exhibit TN7.7Exhibit TN7.7

Since n=5, from Exhibit TN7.7 (also right table), we find

A2=0.58

D3=0

D4=2.11

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X-bar and R Charts: ExampleR chart

R

0

0.464

)220.0)(0(RD = LCL

)220.0)(11.2(RD = UCL

3

4

UCL

LCL0.000

0.100

0.200

0.300

0.400

0.500

0.600

0.700

0.800

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

R

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X-bar and R Charts

Example

10.600

10.856

=.58(0.220)-10.728RA - x = LCL

=.58(0.220)10.728RA + x = UCL

2

2

X-bar chart

10.550

10.600

10.650

10.700

10.750

10.800

10.850

10.900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

Sample

Mea

ns

UCL

LCL

X

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Process Capability

The ability of a process to meet product design/technical specifications Design specifications for products

(Tolerances) upper and lower specification limits (USL, LSL)

Process variability in production process natural variation in process (3 from the mean)

Process may not be capable of meeting specifications if natural variation in a process exceeds allowable variation (tolerances)

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Process Capability Illustrations

specification specification

specification specification

natural variation natural variation

(a) (b)

natural variation natural variation

(c) (d)

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Process Capability

Further Illustrations

Capable process

LSL USLTarget USLTargetLSL

Process variation

Highly capable process

Process not capable Process not capable

Tolerance variation

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Specification Limits Control Limits

Specification limits are pre-established for products before production

Control limits are used to monitor the actual production process performance It is possible that a process is under control,

but not capable to meet specifications It is also possible that a process that is

within specifications is out-of-control

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Control Limits Vs. Specification Limits

IllustrationsUSL

LSLLCL

UCL

(1) In control and within specifications

USL

LCL

UCL

LSL

(2) In control but exceeds specifications

USL

LSL

UCL

LCL

(3) Out-of-control and within specifications

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Process Capability Index:Cp -- Measure of Potential Capability

6 variationprocess

variationprocess LSLUSL

actual

allowableC p

Cp =

1

Cp <

1

Cp >

1

LSL USL

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Process Capability Index:Cpk -- Measure of Actual Capability

3,

3min

XUSLLSLXC pk

is the standard deviation of the production process

Cpk considers both process variation () and process location (X)

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Process Capability Index

ExampleA manufacturing process produces a certain part with a mean diameter of 2 inches and a standard deviation of 0.03 inches. The lower and upper engineering specification limits are 1.90 inches and 2.05 inches.

56.0]56.0,11.1min[

)03.0(3

205.2,

)03.0(3

90.12min

3,

3min

XUSLLSLX

C pk

83.0)03.0(6

90.105.2

6

LSLUSL

C p

Therefore, the process is not capable (the variation is too much and the process mean is not on the target)

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Impact of Process Location on Process Capability

= 2

5044 56 6238

5044 56 6238

5044 56 62 38

5044 56 6238

LSL USL

53

Cp = 2.0Cpk = 2.0

Cp = 2.0Cpk = 1.5

Cp = 2.0Cpk = 1.0

Cp = 2.0Cpk = 0

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Acceptance Sampling

Determines whether to accept or reject an entire lot of goods based on sample results

Measures quality in percent defective

Usually applied to incoming raw materials or outgoing finished goods

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Sampling Plan Guidelines for accepting or rejecting a lot Single sampling plan

N = lot size n = sample size c = max acceptance number of defects d = number of defective items in sample

If d <= c, accept lot; else reject

Sampling plan is developed based on the tradeoff between producer’s risk and consumer’s risk

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Producer’s & Consumer’s Risk Producer’s Risks

reject a good lot (TYPE I ERROR) = producer’s risk = P(reject good lot) 5% is common

Consumer’s Risks accept a bad lot (TYPE II ERROR) = consumer’s risk = P(accept bad lot) 10% is typical

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Quality Definitions Acceptable quality level (AQL)

Acceptable proportion of defects on average

“good lot” = the proportion of defects of the lot is less than or equal to AQL

Lot tolerance percent defective (LTPD) Maximum proportion of defects in a lot “bad lot” = the proportion of defects of

the lot is greater than LTPD

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Operating Characteristic Curve

n = 99c = 4

AQL LTPD

00.10.20.30.40.50.60.70.80.9

1

1 2 3 4 5 6 7 8 9 10 11 12

Pro

bab

ilit

y of

acc

epta

nce

=.10(consumer’s risk)

= .05 (producer’s risk)

Percent defective in a lot