statistical process control used to determine whether the output of a process conforms to product or...

Statistical Process ControlStatistical Process Control

Used to determine whether the output of a process conforms to product or service specifications. We use control charts to detect production of defective products or services or to indicate that the production or service process has changed. We can measure items such as:

•Increases in defective production•Decrease in service complaints

•Consistently low measure of units•Decline in re-work or scrap

Common Causes of VariationCommon Causes of Variation

x xi

i1

n

n

xi x 2

n 1

Purely random, unidentifiable sources of variation that are unavoidable with the current process. For example, weigh 100 bags of M&M’s. The results on a scatter gram tend to have the shape of a bell curve. Symmetric: same number of points above and below the mean. Skewed: preponderance of observations either above or below the mean.

Common Causes

Characteristics of distributions Mean—the average observation Spread—the dispersion of observations

around the mean Shape—whether the observations are

symmetrical or skewed

Common cause variation is normally distributed (symmetrical) and stable (the mean and spread do not change over time).

Assignable Causes of Variation

Any cause of variation that can be identified and eliminated. Such as an employee that needs training or a machine that needs repair.

Change in the mean, spread, or shape of a process distribution is a symptom that an assignable cause of variation has developed.

After a process is in statistical control, SPC is used to detect significant change, indicating the need for corrective action.

The Inspection Process

Use of inspection to simply remove defectives is improper. It does nothing to prevent defects. We cannot “inspect” quality into a part.

Quality measurements Variables—a characteristic measured on a

continuous scale: such as weight, length, volume, or time.

Advantage: if defective, we know by how much—the direction and magnitude of corrections are indicated.

Disadvantage: precise measurements are required.


Quality measurements Attributes—a characteristic that can be counted (yes-no,

integer number). Used to determine conformance to complex specifications, or when measuring variables is too costly

Advantages - Quickly reveals when quality has changed, provides an integer number of how many are defective. Requires less effort and fewer resources than measuring variables.

Disadvantages: Doesn’t show by how much they were defective, the direction and magnitude of corrections are not indicated, Requires more observations, as each observation provides little information


SamplingComplete inspection

Used when costs of failure are high relative to costs of inspection

Inspection is automatedSome defects are not detected because of

Inspector fatigue or Imperfect testing methods


Sampling plans Used when Inspection costs are high or Inspection

destroys the product Some defectives lots may be purchased and some

good lots may be rejected when The sample does not perfectly represent the population Testing methods are imperfect

Sampling plans include Sample size, n random observations Time between successive samples Decision rules that determine when action should be taken


Sampling distributions Sample means are usually dispersed about the

population mean according to the normal probability distribution (reference the central limit theorem described in statistics texts).

Control charts Used to judge whether action is required A sample characteristic measured above the upper

control limit (UCL) or below the lower control limit (LCL) indicates that an assignable cause probably exists.

Inspection station location

Purchased input materials Could use acceptance sampling

Work in process Not after every process

Before it is covered up Before costly, irreversible, or bottleneck operations

so that resources are used efficiently Final product or service

Before stocking or shipping to the customer Customers often play a major role in final

inspection of services

Normal DistributionNormal Distribution

–3–2–1+1+2+3Mean

68.26%95.44%99.74%

= Standard deviation

Control ChartsControl Charts

UCL

Nominal

LCL

Assignable causes likely – may also indicate

improvement

1 2 3Samples

Common Causes

Using Control Charts for Process Using Control Charts for Process ImprovementImprovement

Measure the process When changes are indicated,

find the assignable cause Eliminate problems, incorporate

improvements Repeat the cycle

Indicators of out of control conditions

A trend in the observations (the process is drifting)

A sudden or step change in the observations A run of five or more observations on the

same side of the mean (If we flip a coin and get “heads” five times in a row, we become suspicious of the coin or of the coin flipping process.)

Several observations near the control limits (Normally only 1 in 20 observations are more than 2 standard deviations from the mean.)

Control Chart ExamplesControl Chart Examples

Nominal

UCL

LCL

Sample number

Var

iati

on

s

In Control


Nominal

UCL

LCL

Sample number

Var

iati

on

s

Run – generally 5 or more, take action, even if not outside control limits


Nominal

UCL

LCL

Sample number

Var

iati

on

s

Sudden change – last 4 are unusual


Nominal

UCL

LCL

Sample number

Var

iati

on

s

Upward run


Nominal

UCL

LCL

Sample number

Var

iati

on

s

Out of control

Control Limits and ErrorsControl Limits and Errors

LCL

Processaverage

UCL

Three-sigma limits – the cost of searching for assignable cause is large relative to the cost of not detecting a shift in the process average.

Type I error:Probability of searching for a cause when none exists

Type II error:Probability of concludingthat nothing has changed

Shift in process average

Control Limits and ErrorsControl Limits and Errors

Type I error:Probability of searching for a cause when none exists

Two-sigma limits – the cost of not detecting a shift in the process exceeds the cost of searching for assignable causes. With smaller control limits, a shift in process average will be detected sooner.

Type II error:Probability of concludingthat nothing has changed

Shift in process average

UCL

LCL

Processaverage

Sample Sample

Number 1 2 3 4

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5047

Special Metal Screw

Control Charts for VariablesControl Charts for Variables

Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5039

0.5027 – 0.50090.5027 – 0.5009 == 0.00180.0018(0.5014 + 0.5022 +(0.5014 + 0.5022 + 0.5009 + 0.5027)/40.5009 + 0.5027)/4 == 0.50180.5018

Special Metal Screw

_



Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

Special Metal Screw

_

Sample Sample

Number 1 2 3 4 R x

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

R = 0.0021

x = 0.5027

Special Metal Screw

=

_


Control Charts – Special Metal Screw

R-Charts R = 0.0021

UCLR = D4RLCLR = D3R


Control Chart FactorsControl Chart Factors

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 0 0 3.2673.26733 1.0231.023 0 0 2.5752.57544 0.7290.729 0 0 2.2822.28255 0.5770.577 0 0 2.1152.11566 0.4830.483 0 0 2.0042.00477 0.4190.419 0.076 0.076 1.9241.924


Control Charts—Special Metal Screw

R-Charts R = 0.0021 D4 = 2.282D3 = 0

UCLR = 2.282 (0.0021) = 0.00479 in.LCLR = 0 (0.0021) = 0 in.

UCLR = D4RLCLR = D3R



X-Charts

UCLx = x + A2RLCLx = x - A2R

==

R = 0.0021x = 0.5027=



x-Charts

UCLx = 0.5027 + 0.729 (0.0021) = 0.5042 in.LCLx = 0.5027 – 0.729 (0.0021) = 0.5012 in.

UCLx = x + A2RLCLx = x - A2R

==

R = 0.0021 A2 = 0.729x = 0.5027=


Measure the process Find the assignable cause Eliminate the problem Repeat the cycle


Control Charts for Variables Using Control Charts for Variables Using

UCLx = 5.0 + 1.96(1.5)/ 6 = 6.20 min

UCLx = 5.0 – 1.96(1.5)/ 6 = 3.80 min

UCLUCLxx = = xx + + zzxx

LCLLCLxx = = xx – – zzxx

x = /n

==

==

Sunny Dale Bank

x = 5.0 minutes = 1.5 minutesn = 6 customersz = 1.96

=

Control Charts for AttributesControl Charts for Attributes

Hometown BankHometown Bank

UCLUCLpp = = pp + + zzpp

LCLLCLpp = = pp – – zzpp

pp = = pp(1 – (1 – pp))//nn


Sample WrongNumber Account

Number 1 15 2 12 3 19 4 2 5 19 6 4 7 24 8 7 9 1010 1711 1512 3

Total 147

Total defectives

Total observationsp =

n = 2500

147

12(2500)p =

p = 0.0049


UCLUCLpp = = pp + + zzpp

LCLLCLpp = = pp – – zzpp

pp = 0.0049(1 – 0.0049)/2500 = 0.0049(1 – 0.0049)/2500

n = 2500 p = 0.0049



pp = 0.0014 = 0.0014

n = 2500 p = 0.0049

UCLUCLpp = 0.0049 + 3(0.0014) = 0.0049 + 3(0.0014)

LCLLCLpp = 0.0049 – 3(0.0014) = 0.0049 – 3(0.0014)


p-ChartWrong Account Numbers

Measure the process Find the assignable cause Eliminate the problem Repeat the cycle

p-Chart comments

Two things to note:The lower control limit cannot be negative.When the number of defects is less than the

LCL, then the system is out of control in a good way. We want to find the assignable cause. Find what was unique about this event that caused things to work out so well.

c = 20 z = 2

UCLc = c + z c

LCLc = c – z c


c = 20 z = 2

UCLc = 20 + 2 20

LCLc = 20 – 2 20


UCLc = 28.94

LCLc = 11.06

Process CapabilityProcess CapabilityNominal

value

800 1000 1200 Hours

Upperspecification

Lowerspecification

Process distribution

(a) Process is capable

Process CapabilityProcess Capability

Nominalvalue

Hours

Upperspecification

Lowerspecification

Process distribution

(b) Process is not capable

800 1000 1200


Lowerspecification

Mean

Upperspecification

Six sigma

Four sigma

Two sigma

Nominal value

Upper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours = 48 hours

Process Capability RatioProcess Capability Ratio

Cp =

Upper specification - Lower specification

6

Cp = 1200 – 800

6(48)Cp = 1.39


Lightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours = 48 hours

Cp = 1.39

Cpk = Minimum of

ProcessProcessCapabilityCapabilityIndexIndex

Upper specification – x

3

x – Lower specification

3,

=

=


Lightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours = 48 hours

1200 – 900

3(48)

900 – 800

3(48)


,Cpk = Minimum of

Cp = 1.39

Process CapabilityProcess CapabilityLightbulb ProductionUpper specification = 1200 hoursLower specification = 800 hoursAverage life = 900 hours = 48 hours

Cp = 1.39Cpk = 0.69


ProcessProcessCapabilityCapabilityRatioRatio

Taguchi's Quality Loss FunctionTaguchi's Quality Loss Function

Lo

ss (

do

llar

s)

Lower Nominal Upperspecification value specification

statistical process control used to determine whether the output of a process conforms to product or...

Documents

costs of inspection

inspection costs

service process

process distribution

current process

statistical process

scrap slide

common causes of variation