Process Capability and Statistical Process Control

1. Explain what statistical quality control is.2. Calculate the capability of a process.3. Understand how processes are monitored

with control charts for both attribute and variable data

How many paint defects are there in the finish of a car?

How long does it take to execute market orders?

How well are we able to maintain the dimensional tolerance on our ball bearing assembly?

How long do customers wait to be served from our drive-through window?

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Assignable variation: caused by factors that can be clearly identified and possibly managed◦ Example: a poorly trained employee that creates

variation in finished product output Common variation: variation that is

inherent in the production process◦ Example: a molding process that always leaves

“burrs” or flaws on a molded item

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When variation is reduced, quality is improved

However, it is impossible to have zero variation◦ Engineers assign acceptable limits for variation◦ The limits are know as the upper and lower

specification limits Also know as upper and lower tolerance limits

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Traditional view is that quality within the range is good and that the cost of quality outside this range is constant

Taguchi views costs as increasing as variability increases, so seek to achieve zero defects and that will truly minimize quality costs

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Taguchi argues that tolerance is not a yes/no decision, but a continuous function

Other experts argue that the process should be so good the probability of generating a defect should be very low

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Process limits

Specification limits

How do the limits relate to one another?

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Capability index (Cpk) shows how well parts being produced fit into design limit specifications

Also useful to calculate probabilities

3

X-UTLor

3

LTLXmin=Cpk

XUTL

ZXLTL

Z UTLLTL

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Data◦ Designed for an average of 60 psi

Lower limit of 55 psi, upper limit of 65 psi◦ Sample mean of 61 psi, standard deviation of

2 psi Calculate Cpk

6667.06667.0,1min

23

6165,

23

5561min

3,

3min

xUSLLSLx

C pk

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02410.002275.000135.0)2or 3(

02275.0)2(

22

6165

psi 65 than More

00135.0)3(

32

6155

psi 55 than Less

ZZP

ZP

XXZ

ZP

XXZ

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We are the maker of this cereal. Consumer Reports has just published an article that shows that we frequently have less than 15 ounces of cereal in a box.

Let’s assume that the government says that we must be within ± 5 percent of the weight advertised on the box.

Upper Tolerance Limit = 16 + .05(16) = 16.8 ounces

Lower Tolerance Limit = 16 – .05(16) = 15.2 ounces

We go out and buy 1,000 boxes of cereal and find that they weight an average of 15.875 ounces with a standard deviation of .529 ounces.

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Specification or Tolerance Limits◦ Upper Spec = 16.8 oz◦ Lower Spec = 15.2 oz

Observed Weight◦ Mean = 15.875 oz◦ Std Dev = .529 oz

3

;3

XUTLLTLXMinC pk

)529(.3

875.158.16;

)529(.3

2.15875.15MinC pk

5829.;4253.MinC pk

4253.pkC

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An index that shows how well the units being produced fit within the specification limits.

This is a process that will produce a relatively high number of defects.

Many companies look for a Cpk of 1.3 or better… 6-Sigma company wants 2.0!

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Attribute (Go or no-go information)◦ Defectives refers to the acceptability of product

across a range of characteristics.◦ Defects refers to the number of defects per unit

which may be higher than the number of defectives.◦ p-chart application

Variable (Continuous)◦ Usually measured by the mean and the standard

deviation.◦ X-bar and R chart applications

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Statistical Process Control (SPC) Charts

UCL

LCL

Samples over time

1 2 3 4 5 6

UCL

LCL

Samples over time

1 2 3 4 5 6

UCL

LCL

Samples over time

1 2 3 4 5 6

Normal BehaviorNormal Behavior

Possible problem, investigatePossible problem, investigate

Possible problem, investigatePossible problem, investigate

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x

0 1 2 3-3 -2 -1z

Standard deviation units or “z” units.

Standard deviation units or “z” units.

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We establish the Upper Control Limits (UCL) and the Lower Control Limits (LCL) with plus or minus 3 standard deviations from some x-bar or mean value. Based on this we can expect 99.73% of our sample observations to fall within these limits.

xLCL UCL

99.73%

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Created for good/bad attributes Use simple statistics to create the

control limits

p

p

p

zspLCL

zspUCL

n

pps

p

1

size Sample samples ofNumber

samples all from defects ofnumber Total

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1 – 2- 5- 7 Rule 1 point above UCL or 1 point below LCL 2 consecutive points near the UCL or 2

consecutive points near the LCL 5 consecutive decreasing points or 5

consecutive increasing points 7 consecutive points above the center line

or 7 consecutive points below the center line

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00063.000990.0303033.03

06003.000990.0303033.03

00990.0300

03033.0103033.01

03033.0000,3

91

size Sample x samples ofNumber

samples all from defects ofnumber Total

p

p

p

spLCL

spUCL

n

pps

p

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In variable sampling, we measure actual values rather than sampling attributes

Generally want small sample size1. Quicker2. Cheaper

Samples of 4-5 are typical Want 25 or so samples to set up chart

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level confidence specific afor deviations standard ofNumber z

process for theset lue target vaaor means sample of Average X

size Sample n

ondistributi process theofdeviation Standard s

means sample ofdeviation Standard s

where

UCL

X

X

X

ns

zsXLCL

zsX

X

X

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RD

RD

R

RAX

RAX

X

3R

4R

2X

2X

LCL

UCL

Chart

LCL

UCL

Chart

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