Ch 4 - 28© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Example R-Chart

0

0.05

0.1

0.15

0.2

0.25

0.3

1 2 3 4 5 6 7 8 9 10

Sample number

Rang

e

Ch 4 - 29© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

x Chart Calculations

xn

cm

UCL x R

LCL x R

x

R

x x x

AA

n

1 2

2

2

50 0910

5 01

5 01 0 58 115 5 08

5 01 0 58 115 4 94

..

. . . .

. . . .

average of sample means

average range value

Ch 4 - 30© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

x-Chart Example

x x

n

50.0910

5.01cm

UCL x 2A R

5.01 (0.58)(0.115) 5.08

LCL x 2A R

5.01 (0.58)(0.115) 4.94

Ch 4 - 31© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Using x- and R-Charts Together

• Each measures process differently

• Process average and variability must be in control

Ch 4 - 32© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Example x-Chart

4.924.944.964.985.005.025.045.065.085.10

1 2 3 4 5 6 7 8 9 1

Sample number

Sam

ple

ave

rag

e

Ch 4 - 34© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

UCL

LCL LCL

UCL

Sample observationsconsistently below thecenter line

Sample observationsconsistently above thecenter line

Control Chart Patterns

Ch 4 - 35© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Control Chart Patterns

LCL LCL

UCL UCL

Sample observationsconsistently increasing

Sample observationsconsistently decreasing

Ch 4 - 36© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Control Chart Patterns

UCL

LCL LCL

UCL

Sample observationsconsistently below thecenter line

Sample observationsconsistently above thecenter line

Ch 4 - 38© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Control Chart Patterns

1. 8 consecutive points on one side of the center line.2. 8 consecutive points up or down across zones.3. 14 points alternating up or down.4. 2 out of 3 consecutive points in zone A but still inside

the control limits.5. 4 out of 5 consecutive points in zone A or B.

Ch 4 - 41© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Sample Size Determination

• Attribute control charts–50 to 100 parts in a sample

• Variable control charts–2 to 10 parts in a sample

Ch 4 - 42© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Process Capability• Range of natural variability in process

– Measured with control charts.• Process cannot meet specifications if natural

variability exceeds tolerances• 3-sigma quality

– specifications equal the process control limits. • 6-sigma quality

–specifications twice as large as control limits

Ch 4 - 43© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Process Capability

Process cannot meet specifications Process can meet specifications

Process capability exceeds specifications

PR

OC

ES

S

PR

OC

ES

S

PR

OC

ES

S

Naturalcontrollimits

Naturalcontrollimits

Naturalcontrollimits

Des

ign

spec

ifica

tions

Des

ign

spec

ifica

tions

Designspecifications

Ch 4 - 44© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Acceptance Sampling

• Accept/reject entire lot based on sample results• Not consistent with TQM of Zero Defects• Measures quality in percent defective

Ch 4 - 45© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Sampling Plan• Guidelines for accepting lot• Single sampling plan

–N = lot size–n = sample size (random)–c = acceptance number–d = number of defective items in sample

• If d <= c, accept lot; else reject

Ch 4 - 46© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Producer’s & Consumer’s Risk

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

• TYPE II ERROR = P(accept bad lot) or consumer’s risk–10% is typical value

Ch 4 - 47© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Quality Definitions

• Acceptance quality level (AQL)–Acceptable fraction defective in a lot

• Lot tolerance percent defective (LTPD)–Maximum fraction defective accepted in a lot

Ch 4 - 48© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Operating Characteristic (OC) Curve

• Shows probability of lot acceptance• Based on

–sampling plan –quality level of lot

• Indicates discriminating power of plan

Ch 4 - 49© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Operating Characteristic Curve

AQL LTPD

= 0.10

= 0.05P

roba

bilit

y of

acc

epta

nce,

Pa

{

0.60

0.40

0.20

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.80

{

Proportion defective

1.00

OC curve for n and c

Ch 4 - 50© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Average Outgoing Quality (AOQ)

• Expected number of defective items passed to customer

• Average outgoing quality limit (AOQL) is–maximum point on AOQ curve

Ch 4 - 51© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

AOQ Curve

0.015

0.010

0.005

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

AOQL

AverageOutgoingQuality

(Incoming) Percent Defective

AQL LTPD

Ch 4 - 52© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Double Sampling Plans

• Take small initial sample–If # defective < lower limit, accept–If # defective > upper limit, reject–If # defective between limits, take second sample

• Accept or reject based on 2 samples• Less costly than single-sampling plans

Ch 4 - 53© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Multiple (Sequential) Sampling Plans

• Uses smaller sample sizes• Take initial sample

–If # defective < lower limit, accept–If # defective > upper limit, reject–If # defective between limits, resample

• Continue sampling until accept or reject lot based on all sample data

Ch 4 - 54© 2000 by Prentice-Hall IncRussell/Taylor Oper Mgt 3/e

Choosing A Sampling Method

• An economic decision• Single sampling plans

–high sampling costs• Double/Multiple sampling plans

–low sampling costs