Quality and OperationsManagement

Process Control and Capability Analysis

Process Control

• Recognizes that variance exists in all processes• Sources of variation

– systematic

– assignable

• Purpose – to detect and eliminate ‘out-of-control’ conditions

– to return a process to an ‘in-control’ state

• Basic tool -- the SPC chart(s)

Measuring A Process

• Types of measurements– variables data

• length, weight, speed, output, etc• discrete values

– attributes data• good vs bad, pass vs fail, etc• binary values

• Types of charts– variables -- X-R chart– attributes -- p, np, c and u

• Basic assumption -- sample means are normally distributed

Getting Started with SPCX-R Charts

• Determine sample size and frequency of data collection• Collect sufficient historical data• Ensure normality of distribution• Calculate factors for control charts

• Construct control chart• Plot data points• Determine outliers and eliminate assignable causes• Recalculate control limits with reduced data set• Implement new process control chart

X

R

UCLx LCLx

UCLrLCLr

Basic Properties

• x = std dev of sample mean = /n (where = process standard deviation)

• conventional approach uses 3 /n• limitations of control charts

– Type I Error: probability that an in-control value would appear as out-of-control

– Type II Error: probability that a shift causing an out-of-control situation would be mis-reported as in-control

– delays due to sampling interval– charting without taking action on assignable causes– over control actions

Type 1 and Type 2 Error

Type 1error

Type 2error

No error

No error

Alarm No Alarm

In Control

Out of Control

Suppose 1 > , thenType 2 Error = Z [( + 3 x - 1) / x ]

Type 1 Error = 0.0027 for 3 charts

Type 2 Error Example

Suppose: = 101= 10.2 = 4/3n = 9thus,x = 4/9

Then, Type 2 Error = Z [( + 3 x - 1) / x ]= Z [(10 + 12/9 - 10.2) / (4/9)]= Z [2.55] = 0.9946

if 1= 11.0, then Type 2 Error = Z[0.75] = .7734if 1= 12.0, then Type 2 Error = Z[-1.50] = .0668

Prob.{shift will be detected in 3rd sample after shift occurs}= 0.0668*0.0668*(1-0.0668) = 0.0042Average number of samples taken before shift is detected= 1/(1-0.0668) = 1.0716Prob.{no false alarms first 32 runs, but false alarm on 33rd}= (0.9973)32*(0.0027) = .0025Average number of samples taken before a false alarm= 1/0.0027 = 370

Tests for Unnatural Patterns

• Probability that “odd” patterns observed are not “natural” variability are calculated by using the probabilities associated with each zone of the control chart

• Use the assumption that the population is normally distributed

• Probabilities for X-chart are shown on next slide

Normal Distribution Applied to X-R Control Charts

A

A

B

B

C

C

+3

+2

+1

-1

-2

-3

Probability = .00135

Probability = .1360

Probability = .3413

Probability = .3413

Probability = .1360

Probability = .02135

Probability = .00135

Probability = .02135UCLx

LCLx

X

Outer 3rd

Outer 3rd

Middle 3rd

Middle 3rd

Inner 3rd

Inner 3rd

A Few Standard Tests

• 1 point outside Zone A

• 2 out of 3 in Zone A or above (below)

• 4 out of 5 in Zone B or above (below)

• 8 in a row in Zone C or above (below)

• 10 out of 11 on one side of center

Tests for Unnatural Patterns

• 2 out of 3 in A or beyond– .0227 x .0227 x (1-.0227) x 3 = .0015

• 4 out of 5 in B or beyond– .15874 x (1-.1587) x 5 = .0027

• 8 in a row on one side of center– .508 = .0039

Other Charts

• P-chart– based on fraction (percentage) of defective units in a

varying sample size

• np-chart– based on number of defective units in a fixed sample size

• u-chart– based on the counts of defects in a varying sample size

• c-chart– based on the count of defects found in a fixed sample size

SPC Quick Reference Card

• P-chart– based on fraction (percentage) of defective units in a varying sample size– UCL/LCLp = p 3(p)(1-p)/n

• np-chart– based on number of defective units in a fixed sample size– UCL/LCLnp = np 3(np)(1-p)

• u-chart– based on the counts of defects in a varying sample size– UCL/LCLu = u 3u/n

• c-chart– based on the count of defects found in a fixed sample size– UCL/LCLc = c 3c

• X-R chart– variables data

– UCL/LCLX = X 3 x = X 3 / n = X A2R where R/d2

– UCLR = D4R and A2 = 3/d2 n– LCLR = D3R

• for p, np, u, c and R chart the LCL can not be less than zero.

Process Capability

• Cp: process capability ratio– a measure of how the distribution compares to the width of

the specification– not a measure of conformance– a measure of capability, if distribution center were to match

center of specification range

• Cpk: process capability index– a measure of conformance (capability) to specification – biased towards “worst case”– compares sample mean to nearest spec. against distribution

width

How Good is Good Enough?

• Cp = 1.0 => 3 => 99.73% (in acceptance) => .9973 => 2700 ppm out of tolerance– PG&E operates non-stop

• 23.65 hours per year without electricity

– average car driven 15,000 miles per year• 41 breakdowns or problems per year

• Cp = 2.0 => 6 => 99.99983% (in acceptance) => .9999983 => 3.4 ppm out of tolerance– PG&E operates non-stop

• 1.78 minutes per year without electricity

– average car driven 15,000 miles per year• 0.051 breakdowns or problems per year or one every 20 years