S6-1

Operations Operations ManagementManagement

Statistical Process ControlStatistical Process ControlSupplement 6Supplement 6

S6-2

OutlineOutline Statistical Process Control (SPC).

Mean charts or X-Charts.

Range chart or R-Charts.

Control charts for attributes.

Managerial issues and control charts.

Acceptance Sampling.

S6-3

Statistical technique to identify when non-random variation is present in a process.

All processes are subject to variability. Natural causes: Random variations.

Assignable causes: Correctable problems. Machine wear, unskilled workers, poor materials.

Uses process control charts.

Statistical Process Control (SPC)Statistical Process Control (SPC)

S6-4

Produce GoodProvide Service

Stop Process

No

Yes

Is process in control?

Take Samples

Find Out Why

CreateControl Chart

Start

Statistical Process Control StepsStatistical Process Control Steps

Take Sample

Inspect Sample

S6-5

Process Control ChartsProcess Control Charts

Plot of Sample Data Over Time

0

20

40

60

80

1 5 9 13 17 21

Time

Sam

ple

Val

ue

Upper control limit

Lower control limit

S6-6

Process is not in control if: Sample is not between upper and lower control

limits.

A non-random pattern is present, even when between upper and lower control limits.

Based on sample being normally distributed.

Control ChartsControl Charts

S6-7

Distribution of Sample MeansDistribution of Sample Means

x means sample of Mean

n

xx

Standard deviation of

the sample means

(mean)

x2 withinfall x all of 95.5%

x3 withinfall x all of 99.7%

x3 x2 x x x1 x2 x3

S6-8

X

As sample size gets large enough,

distribution of mean values becomes approximately normal for any population distribution.

Central Limit Theorem

XX

Central Limit TheoremCentral Limit Theorem

S6-9

ControlCharts

RChart

VariablesCharts

AttributesCharts

XChart

PChart

CChart

Continuous Numerical Data

Categorical or Discrete Numerical Data

Control Chart TypesControl Chart Types

S6-10

Characteristics for which you focus on defects.

Categorical or discrete values. ‘Good’ or ‘Bad’. # of defects.

AttributesAttributesVariablesVariables

Quality CharacteristicsQuality Characteristics

Characteristics that you measure, e.g., weight, length.

Continuous values.

S6-11

Shows sample means over time.

Monitors process average.

Example: Weigh samples of coffee.

Collect many samples, each of n bags. Sample size = n.

Compute mean and range for each sample.

Compute upper and lower control limits (UCL, LCL).

Plot sample means and control limits.

XX Chart Chart

S6-12

X Chart Control Limits - Std. Dev. of Process Is Known

sample mean at time i

xx zσxxLCLzσxxUCL

n

ix

x

n

i 1

nσ

xσ

= known process standard deviation

S6-13

Each sample is 4 measurements.

Process mean is 5 lbs.

Process standard deviation is 0.1 lbs.

Determine 3 control limits.

XX Chart - Example 1 Chart - Example 1

85.441.0

35

15.541.0

35

xLCL

xUCL

S6-14

X Chart Control Limits - Std. Dev. of Process is Not Known

sample range at time i

A2 is from Table S6.1

RAxxLCLRAxxUCL 22

n

iRn

iR 1

sample mean at time i

n

ix

x

n

i 1

S6-15

Factors for Computing Control Factors for Computing Control Chart LimitsChart Limits

SampleSize, n

MeanFactor, A2

UpperRange, D4

LowerRange, D3

2 1.880 3.268 0

3 1.023 2.574 0

4 0.729 2.282 0

5 0.577 2.115 0

6 0.483 2.004 0

7 0.419 1.924 0.076

8 0.373 1.864 0.136

9 0.337 1.816 0.184

10 0.308 1.777 0.223

S6-16

Each sample is 4 measurements.

Determine 3 control limits. sample mean range. 1 5.02 .12 4.96, 5.03, 5.01, 5.08 2 4.99 .08. 3 4.97 .13. 4 5.03 .18. 5 4.99 .14.

XX Chart - Example 2 Chart - Example 2

905.413.0729.05095.513.0729.05

xLCLxUCL

13.00.5 Rx

S6-17

XX Chart - Example 2 Chart - Example 2

4.9

5.0

5.1

Time

Sam

ple

Mea

n

Upper control limit

Lower control limit

S6-18

sample values mean range 6 5.05, 5.00, 4.80, 4.95 4.95 0.25 7 5.00, 5.10, 5.10, 5.00 5.05 0.10 8 4.80, 5.20, 5.10, 5.00 5.025 0.40

4.9

5.0

5.1

Time

Sam

ple

Mea

n

Upper control limit

Lower control limit

Example 2 – New SamplesExample 2 – New Samples

S6-19

Shows sample ranges over time. Sample range = largest - smallest value in sample.

Monitors process variability.

Example: Weigh samples of coffee. Collect many samples, each of n bags.

Sample size = n.

Compute range for each sample & average range.

Compute upper and lower control limits (UCL, LCL).

Plot sample ranges and control limits.

RR Chart Chart

S6-20

sample range at time i

From Table S6.1

RR Chart Control Limits Chart Control Limits

n

R R

R D LCL

R D UCL

i

n

1i

3R

4R

S6-21

Each sample is 4 measurements.

Determine 3 control limits. sample mean range 1 5.02 .12 2 4.99 .08 3 4.97 .13 4 5.03 .18 5 4.99 .14

RR Chart - Example 2 Chart - Example 2

013.00297.013.0282.2

R

R

LCLUCL

13.00.5 Rx

4.96, 5.03, 5.01, 5.08

S6-22

RR Chart - Example 2 Chart - Example 2

0

0.2

0.3

Time

Sam

ple

Ran

ge

Upper control limit

Lower control limit

0.1

S6-23

sample values mean range 6 5.05, 5.00, 4.80, 4.95 4.95 0.25 7 5.00, 5.10, 5.10, 5.00 5.05 0.10 8 4.80, 5.20, 5.10, 5.00 5.025 0.40

Example 2 – New SamplesExample 2 – New Samples

0

0.2

0.3

Time

Sam

ple

Ran

ge

Upper control limit

Lower control limit

0.1

S6-24

Control Chart StepsControl Chart Steps Collect 20 to 25 samples of n=4 or n=5 from a

stable process & compute the mean and range.

Compute the overall mean and average range.

Calculate upper and lower control limits.

Collect new samples, and plot the means and ranges on their respective control charts.

S6-25

Control Chart Steps - ContinuedControl Chart Steps - Continued

Investigate points or patterns that indicate the process is out of control. Assign causes for the variations.

Collect additional samples and revalidate the control limits.

S6-26

Use of Control ChartsUse of Control Charts

S6-27

sample values mean range 1 4.9, 5.0, 5.1 5.0 0.2 2 5.2, 5.3, 5.4 5.3 0.2 3 5.5, 5.6, 5.7 5.6 0.2

4 5.8, 5.9, 6.0 5.9 0.2

Example 3Example 3

2454.52.0023.145.5

6546.52.0023.145.5

xLCLxUCL

2.045.5 Rx

02.00

5148.02.0574.2

RLCL

RUCL

S6-28

Example 3 – Control ChartsExample 3 – Control Charts

5.0

5.5

6.0

Time

Sam

ple

Mea

n

Upper control limit = 5.6546

Lower control limit = 5.2454

0.0

0.5

1.0

Time

Sam

ple

Ran

ge

Upper control limit = 0.5148

Lower control limit = 0

S6-29

sample values mean range 1 5.0, 5.0, 5.0 5.0 0.0 2 4.5, 5.0, 5.5 5.0 1.0 3 4.0, 5.0, 6.0 5.0 2.0

4 3.0, 5.0, 7.0 5.0 4.0

Example 4Example 4

20975.375.1023.10.5

79025.675.1023.10.5

xLCLxUCL

75.10.5 Rx

075.10

5045.475.1574.2

RLCL

RUCL

S6-30

Example 4 – Control ChartsExample 4 – Control Charts

3.0

5.0

7.0

Time

Sam

ple

Mea

n

Upper control limit = 6.79025

Lower control limit = 3.20975

0.0

3.0

6.0

Time

Sam

ple

Ran

ge

Upper control limit = 4.5045

Lower control limit = 0

S6-31

Attributes control chart.

Shows % of nonconforming items.

Example: Count # defective chairs & divide by total chairs inspected. Chair is either defective or not defective.

pp Chart Chart

S6-32

Attributes control chart.

Shows number of defects in a unit. Unit may be chair, steel sheet, car, etc. Size of unit must be constant.

Example: Count # defects (scratches, chips etc.) in each chair of a sample of 100 chairs.

cc Chart Chart

S6-33

Quality testing for incoming materials or finished goods.

Procedure: Take one or more samples at random from a lot

(shipment) of items. Inspect each of the items in the sample. Decide whether to reject the whole lot based on

the inspection results.

Acceptance SamplingAcceptance Sampling

S6-34

Inspecting all items is too expensive. The larger the sample inspected:

The greater the cost for inspection. The less likely you are to accept a “bad” lot or to

reject a “good” lot.

Key questions: How many should be inspected in each lot? How confident are you in the accept/reject

decision?

Acceptance SamplingAcceptance Sampling