tpk4161 supply chain analytics statistical process control

Norwegian University of Science and Technology

PhD candidate: Oluyisola, Olumide E. (CSSBB)Supervisors: Prof. Strandhagen, J. O. and Asso. Prof. Semini, M. G.

The Production Management GroupDepartment of Mechanical and Industrial Engineering

NTNU-Trondheim

18.Oct.2017

TPK4161 Supply Chain Analytics

-

Statistical Process Control

Norwegian University of Science and Technology 2

Learning objectives

• To understand when control charts can be applied

• To understand different types of control charts and there

application areas

• To understand how to collect data for control charts

(rational subgrouping)

• To understand how to construct control charts

• To understand how to analyze control chart behavior


Outline

• Introduction

• Types of variation

• Control charts selection

• Rational subgrouping and control limits

• Xbar– R, Xbar – s, IMR charts with examples

• Attribute charts: p, np, c and u charts with examples

• Short-Run control charts and MAMR chart

• Analyzing control chart behavior with many examples

• Summary


What is SPC?

• The ‘control’ of processes using statistical principles and tools.

• SPC entails the use of statistical tools to monitor (usually

production) a process for ‘significant’ deviations, that indicate the

likelihood of a rejected product

• Pioneered by W. A. Shewhart in the early 1920s.

• Later applied by W. E. Deming during the 2nd World War to improve

quality in the manufacture of weaponries

• After the war, Deming supported the adoption of SPC by the

Japanese Industry, with remarkable results in product quality.


The Central Limit Theorem (CLT)

The central limit theorem states that:• regardless of the shape of the population, the sampling distribution

of the mean is approximately normal if the sampling size is

sufficiently large. The approximation improves as the sample size

gets larger.

▪ In general, nearly normal population will have a nearly normal

sampling distribution of the mean for small sample sizes.

▪ However, for non-normal populations, when the sample size is

increased (typically, ≥ 30), the sampling distribution of the mean

tends towards a normal distribution.


The Central Limit Theorem (CLT)


Types of variation

Common-cause Special-cause

Also referred to as natural variation.▪

Inherent to the process▪

Generally not controllable by process ▪

operatorsExamples: ▪

natural variation in raw materials,•

variation in ambient temperature,•

variation in electrical or •

pneumatic sources, etc• .

Also referred to as assignable cause▪

Events that the operator, when ▪

properly alerted, can usually remove or adjustExamples: tool wear, large changes in ▪

raw materials, broken equipment, etc.

Process in control (stable) Process out of control (unstable)


Selecting a control chart


Rational Subgrouping

• The variable selected for monitoring will be a “leading indicator” of special causes –

one that detects special causes before others do.

• In the case of an Xbar – R chart,

– any process shift should be detected by the Xbar chart,

– the R chart should capture only common cause variation.

• Thus, you want a high probability of capturing variation between samples (subgroups)

while the variation within samples is kept low.

• To minimize within-sample variation, it is vital that samples consists of parts that are

produced successively by the same process.

• The next sample data is collected somewhat later so that any process shifts which

may have occurred will be displayed on the chart as within sample variation.


Rational subgrouping – example 1

Which of these two options leads to more accurate results?

Option A

Option B

Machine 1

Machine 2

Machine 3

Machine 4

Machine 5

Machine 1

Machine 2

Machine 3

Machine 4

Machine 5


Rational subgrouping – example 1

explanation• In this example, in a workstation with five machines or robots performing the same

function and producing the same output, sampling can be tricky.

• Choosing the five products from each machine (option A) would have the five

elements of the sample being produced by five different processes.

– This also skews the measured values in an unpredictable way, dampening or amplifying them.

• A better choice would be to select from the same machine, in five consecutive arrays

(option B).

– This option ensures that the same process is being monitored, and minimizes the ‘within-sample’ variation,

thus allowing the subgroup-to-subgroup variation to become more prominent. However, in this chase there

might be a need for more control charts, for all five machines in an extreme case.

• Ultimately, the choice of sample size depends to some extent on the resources

available to do the measuring, as well as how obstructive or destructive the

measurement activity is. However, the larger the sample size, the more sensitive the

chart.


Control limits

• Calculated from process data, and represents the ‘voice of the process’

• Set at ±3𝜎; the upper control limit (UCL) at +3𝜎 and the lower control limit

(LCL) at -3𝜎.

• When calculating the control limits, it is better to collect as much data as

practical. Many authors suggest at least 25 subgroups.

– NB: The examples used in this note use fewer examples for simplicity.

# With variable control charts, subgroup sizes are generally held constant, but

may vary with some of the attribute control charts.


• When variable, continuous data is to be controlled, a carriable chart

is used.

• The ‘property’ to be controlled is continuous (as opposed to being

discrete).

• There are three common types of variable control charts:

– ത𝑋 - s chart (or Xbar – s chart)

– ത𝑋 - R chart (also referred to as Xbar – R chart)

– Individual and moving range chart (IMR)

Variable charts


Xbar – R Chart

• Typically used when the sample size of each subgroup is

‘approximately’ <10, since the range is a better estimate of

dispersion for smaller sample sizes.

• Control limits are given by the following formulas:

ഥ𝑋𝑖 =

𝑗=1

𝑛

𝑥𝑖𝑗

𝑛= average of the 𝑖𝑡ℎ subgroup; plot points

തത𝑋 =

𝑖=1

𝑘 ത𝑋𝑖

𝑘= center line of the ഥ𝑋 𝑜𝑟 𝑋𝑏𝑎𝑟 chart

𝑘 = number of subgroups 𝑛 = sample size of each subgroup

ത𝑅 =σ𝑖=1

𝑘 𝑅𝑖

𝑘= center line of the 𝑅 chart

𝑅𝑖 = range of the 𝑖𝑡ℎ subgroup

LCL ҧ𝑋= തത𝑋 - A2

ത𝑅

UCL ҧ𝑋= തത𝑋 + A2

ത𝑅

LCL𝑅 = D3ത𝑅

UCL𝑅 = D4ത𝑅


Example 2

Sample7:00 a.m.

8:00 a.m.

9:00 a.m.

10:00 a.m.

11:00 a.m.

12:00 p.m.

1:00 p.m.

2:00 p.m.

3:00 p.m.

4:00 p.m.

1 7.127 7.125 7.123 7.127 7.128 7.125 7.126 7.126 7.127 7.128

2 7.123 7.126 7.129 7.127 7.125 7.125 7.123 7.126 7.129 7.123

3 7.123 7.121 7.129 7.124 7.126 7.127 7.123 7.127 7.128 7.122

4 7.126 7.122 7.124 7.125 7.127 7.128 7.125 7.128 7.129 7.124

Data are collected in a face-and-plunge operation done on a lathe.

The dimension being measured is the groove inside diameter (ID),

which has a tolerance of 7.125 ± 0.010. Four parts are measured

every hour. Thee values have been entered in the table below:


Example 2 – Solution

തത𝑋 = 7.12565

ത𝑅 = 0.00401

LCL ҧ𝑋= 7.12565 – (0.729)(0.00401) = 7.122727

UCL ҧ𝑋= 7.12575 + (0.729)(0.00401) = 7.128573

LCL𝑅 = D3ത𝑅 = (0)(0.00401) = 0

UCL𝑅 = D4ത𝑅 = (2.282)(0.00401) = 0.00915

• The values of A2, D3, and D4 can be found in the Appendix (Control chart constants) for n = 4.

LCL ҧ𝑋= തത𝑋 - A2

ത𝑅

UCL ҧ𝑋= തത𝑋 + A2

ത𝑅

LCL𝑅 = D3ത𝑅

UCL𝑅 = D4ത𝑅


Xbar – s chart

• Typically used when the sample size of each subgroup is

‘approximately’ ≥ 10, since the standard deviation, s, is a better

estimate of dispersion for larger sample sizes.

• Control limits are given by the following formulas:

ഥ𝑋𝑖 =

𝑗=1

𝑛

𝑥𝑖𝑗

𝑛= average of the 𝑖𝑡ℎ subgroup; plot points

തത𝑋 =

𝑖=1

𝑘 ത𝑋𝑖

𝑘= center line of the ത𝑋 chart

𝑘 = number of subgroups 𝑛 = sample size of each subgroup

ҧ𝑠 =σ𝑖=1

𝑘 𝑠𝑖

𝑘= center line of the 𝑠 chart

𝑠𝑖 = standard deviation of the 𝑖𝑡ℎ subgroup

LCL ҧ𝑋= തത𝑋 - A3 ҧ𝑠

UCL ҧ𝑋= തത𝑋 + A3 ҧ𝑠

LCL𝑠 = B3 ҧ𝑠

UCL𝑠 = B4 ҧ𝑠


Xbar – s chart example


Individual and Moving Range chart (IMR)

• Although larger sample sizes produces more representative charts,

there are some situations when a sample size of one is the only

feasible option.

• Examples: very slow processes; processes in which the

measurements are very expensive to obtain – for instance with

destructive tests.

• In this cases, the IMR (a.k.a., the XmR) is appropriate.

• The range is calculated as the absolute value of the difference

between each measurement and the previous one.


Individual and Moving Range chart (IMR)

ത𝑋 =σ𝑖=1

𝑛 𝑥𝑖

𝑛= center line of the X chart

𝑚 ത𝑅 =σ𝑖=1

𝑛−1 𝑚𝑅𝑖

𝑛 − 1= center line of the 𝑚𝑅 chart (moving range)

𝑥𝑖 = individual data measurements; plot points

𝑛 = number of individual data measurements

𝑚𝑅𝑖 = ȁ𝑥𝑖 − ȁ𝑥𝑖 − 1 for i = 2, 3, …, n

LCL ҧ𝑋= ഥ𝑋 - E2𝑚 ത𝑅

UCL ҧ𝑋= ഥ𝑋 + E2𝑚 ത𝑅

LCL𝑚𝑅 = D3𝑚 ത𝑅

UCL𝑚𝑅 = D4𝑚 ത𝑅


Example 3

• Using the data recorded for a milling process in at a thruster

manufacturer, construct an individual and moving average chart

Reading Individual data element

1 290

2 288

3 285

4 290

5 291

6 287

7 284

8 290

9 290

10 288



ത𝑋 = 288.3

𝑚 ത𝑅 = 2.89

LCL ҧ𝑋= 288.3 – (2.660)(2.89) = 280.62

UCL ҧ𝑋= 288.3 + (2.660)(2.89) = 295.98

LCL𝑚𝑅 = (0)(2.89) = 0

UCL𝑚𝑅 = (3.267)(2.89) = 9.44

• The values of E2, D3, and D4 can be found in the Appendix (Control chart constants) for n = 2.

LCL ҧ𝑋= ഥ𝑋 - E2𝑚 ത𝑅

UCL ҧ𝑋= ഥ𝑋 + E2𝑚 ത𝑅

LCL𝑚𝑅 = D3𝑚 ത𝑅

UCL𝑚𝑅 = D4𝑚 ത𝑅


Attribute charts

• Attributes charts are used to control processes where each process output being

monitored can be classified only in binary forms, such as good or bad.

• p and np charts are used to control proportion defective

• c and u charts are used to control counts of defects.

• c and np charts require constant sample sizes.

• A defective is also knows as a “nonconformance.” A defect is also known as a

“nonconformity.”

# When a lower control limit computes to a negative value, it is artificially set to zero. This

is because it is impossible to have a negative proportion defective or defect counts.


p Chart

• Used when sample size varies

• Used to control the proportion or percentage of defectives.

NB: The control limits on the chart are not straight since it reflects each subgroup’s individual sample

size

𝑝𝑖 =𝐷𝑖

𝑛𝑖

= plot points

ҧ𝑝 =σ𝑖=1

𝑘 𝐷𝑖

σ𝑖=1𝑘 𝑛𝑖

= center line of the 𝑝 chart

𝑛𝑖 = sample size of the 𝑖𝑡ℎ subgroup

𝐷𝑖 = number of defective units nonconforming units in the 𝑖𝑡ℎ subgroup

LCL𝑝 =ഥ𝑝 - 3ҧ𝑝(1− ҧ𝑝)

𝑛𝑖

UCL𝑝 = ҧ𝑝 + 3ҧ𝑝(1− ҧ𝑝)

𝑛𝑖

𝑘 = number of subgroups


Example 4

• A test was conducted to

determine the presence of

an undesirable bacteria in

samples collected over 13

days from milk supplied by

dairy farms in Sør-

Trondelag.

Date No. of defectives

Sample size

8-Sept 12 115

9-Sept 14 125

10-Sept 18 111

11-Sept 13 133

12-Sept 17 120

13-Sept 15 118

14-Sept 15 137

15-Sept 15 108

16-Sept 16 110

17-Sept 11 124

18-Sept 14 128

19-Sept 13 144

20-Sept 14 141


Example 4 - Solution LCL𝑝 =ഥ𝑝 - 3ҧ𝑝(1− ҧ𝑝)

𝑛𝑖

UCL𝑝 = ҧ𝑝 + 3ҧ𝑝(1− ҧ𝑝)

𝑛𝑖


np Chart

• Used when sample size is constant

• Used to control the proportion or percentage of defectives.

𝑛 ҧ𝑝 = 𝑛σ𝑖=1

𝑘 𝐷𝑖


= center line of the 𝑛𝑝 chart


𝐷𝑖 = number of defective units nonconforming units in the 𝑖𝑡ℎ subgroup; plot points

LCL𝑛𝑝 = 𝑛 ҧ𝑝 - 3 𝑛 ҧ𝑝(1 − ҧ𝑝)

UCL𝑛𝑝 = 𝑛 ҧ𝑝 + 3 𝑛 ҧ𝑝(1 − ҧ𝑝)


ҧ𝑝 =σ𝑖=1

𝑘 𝐷𝑖



Example 5

Packages containing 1000

lightbulbs are randomly

selected, and all 1000 bulbs

are light-tested. The test

results are summarized in

the table:

Date No. of defectives

Sample sizes

8-Sept 9 1000

9-Sept 12 1000

10-Sept 13 1000

11-Sept 12 1000

12-Sept 11 1000

13-Sept 9 1000

14-Sept 7 1000

15-Sept 0 1000

16-Sept 12 1000

17-Sept 8 1000

18-Sept 9 1000

19-Sept 7 1000

20-Sept 11 1000


Example 5 - Solution LCL𝑛𝑝 = 𝑛 ҧ𝑝 - 3 𝑛 ҧ𝑝(1 − ҧ𝑝)

UCL𝑛𝑝 = 𝑛 ҧ𝑝 + 3 𝑛 ҧ𝑝(1 − ҧ𝑝)

𝑛 ҧ𝑝 = 1000120

13000= 9.23

ҧ𝑝 =120

13000= 0.00923

LCL𝑛𝑝 = 9.23 - 3 1000 0.00923 1 − 0.00923 = 0.16

UCL𝑛𝑝 = 9.23 + 3 1000 0.00923 1 − 0.00923 = 18.30


c Chart

• Used when sample size is constant

• Use to control the number of defects.

𝑛 = sample size of each subgroup

𝐷𝑖 = number of defects nonconformities in the 𝑖𝑡ℎ subgroup; plot points

LCL𝑐 = ҧ𝑐 - 3 ҧ𝑐

UCL𝑐 = ҧ𝑐 + 3 ҧ𝑐


ҧ𝑐 =σ𝑖=1

𝑘 𝑐𝑖

𝑘


Example 6

Glass panes are inspected

for defects such as bubbles,

scratches, chips, inclusions,

waves and dips. The data

gathered from are

documented in the following

table:

Date No. of defects Sample size

15-May 19 150

16-May 12 150

17-May 13 150

18-May 12 150

19-May 18 150

20-May 19 150

21-May 17 150

22-May 20 150

23-May 22 150

24-May 18 150

25-May 19 150

26-May 17 150

27-May 11 150



LCL𝑐 = ҧ𝑐 - 3 ҧ𝑐

= 16.69 - 3 16.69= 4.44

ҧ𝑐 =

𝑖=1

𝑘𝑐𝑖

𝑘=

217

13= 16.69

UCL𝑐 = ҧ𝑐 + 3 ҧ𝑐

= 16.69 + 3 16.69= 28.95


u Chart

• Used when sample size varies

• Use to control the number of defects per unit

# The control limits in the chart appear ragged because they reflect each subgroup’s individual sample

size

𝑢𝑖 =𝑐𝑖

𝑛𝑖

= number of defects per unit in the 𝑖𝑡ℎ subgroup;

plot points

ത𝑢 =σ𝑖=1

𝑘 𝑐𝑖


= center line of the 𝑢 chart


𝑐𝑖 = number of defects nonconformities in the 𝑖𝑡ℎ subgroup

LCL𝑢 = ഥ𝑢 - 3ഥ𝑢

𝑛𝑖

UCL𝑢 = ഥ𝑢 + 3ഥ𝑢

𝑛𝑖



Example 7

Glass panes are inspected

for defects such as bubbles,

scratches, chips, inclusions,

waves and dips. The data

gathered from are

documented in the following

table:

Date No. of defects Sample size

1-Dec 4 125

2-Dec 8 111

3-Dec 3 133

4-Dec 7 120

5-Dec 5 118

6-Dec 5 137

7-Dec 6 108

8-Dec 10 110

9-Dec 4 124

10-Dec 3 128

11-Dec 4 144

12-Dec 7 138

13-Dec 11 150



LCL𝑢 = ഥ𝑢 - 3ഥ𝑢

𝑛𝑖

= 0 (-ve value)

UCL𝑐 = ഥ𝑢 + 3ഥ𝑢

𝑛𝑖

= 0.0998

ത𝑢 =

𝑖=1

𝑘𝑐𝑖

𝑖=1

𝑘𝑛𝑖

=77

1646= 0.0468


Short-Run Control Charts

• Not commonly used, but suitable when data is collected

infrequently or irregularly

• Can be used with historical target values, attribute and

variable data, and individual or subgroup-ed averages

# Not within the scope of this lecture. If interested, see Griffith (1996) and Oakland (2007)


Moving Average and Moving Range

Control Charts (MAMR)

• Suitable when

– Data is collected periodically, or when it takes time to produce a single item

– It is desirable to dampen the effects of overcontrol

– It may be necessary to detect even smaller shifts in the process

• Key considerations

– The selected moving average length significantly affects the overall sensitivity of the chart. The longer the length, the less sensitive the chart is.

– The specific selection of the length should depend on the “out-of-control” detection rules being used in each case.

# Not within the scope of this lecture. If interested, see listed references


Analyzing control charts

• The use of ±3𝜎 in the control limits formulas constitutes an generally accepted

economic trade-off between looking for special causes when it does not exist

and not looking for it when it exists. (This relates to the power of a test of

statistical significance.)

• In addition, the ±3𝜎 limits covers approximately 99.73% of the data.

– Points within the ±3𝜎 control limits are due to common cause variation

– Points outside are attributed to special causes;

• In this case, investigate immediately so that the root cause(s) can be determined

before the process strays to far away from the target, i.e., goes out of control.

NB: Adjusting a process when it is not warranted by out-of-control conditions is

referred to as “process tampering.” This usually results in destabilizing a process,

causing it to go out of control.


Analyzing control charts – The Minitab

Rules

1. One point more that ±3𝜎 from the center line (either side).

2. Nine points in a row on the same side of the center line

3. Six points in a row, all increasing or all decreasing

4. Fourteen points in a row, alternating up and down

5. Two out of three points more than 2𝜎 from the center line (same side)

6. Four out of five points more than 1𝜎 from the center line (same side)

7. Fifteen point in a row within 1𝜎 of the center line (either side)

8. Eight points in a row more than 1𝜎 from the center line (either side)


Analyzing control charts – The AIAG

Rules

1. Points beyond the control limits

2. Seven points in a row on one side of the average

3. Seven points in a row that are consistently increasing (equal or greater than the preceding points) or consistently decreasing

4. Over 90% of the plotted points are in the middle of the control limit region (for 25 or more subgroups)

5. Less than 40% of the plotted points are in the middle of the control limit region (for 25 or more subgroups)

6. Non-random patterns such as cycles

# AIAG = Automotive Industry Action Group


Example 8a

Example 8(a): Out-of-control condition #1 from AIAG. This is the same as Minitab out-of-control

condition #1


Example 8b

Example 8(b): Out-of-control condition #2 from AIAG.


Example 8c

Example 8(c): Out-of-control condition #3 from The Minitab Rules

UCL

LCL


Example 8d

Example 8(d): Out-of-control condition #6 from AIAG


Example 8e

Example 8(e): Out-of-control condition #4 from The Minitab Rules


Summary

• A control chart is the equivalent graphical hypothesis test.

– The null hypothesis is that the process has not changed;

– The alternative hypothesis is that it has changed;

• As each point is plotted, the chart is examined to see if there is

sufficient evidence to reject the null hypothesis, and conclude that

the process may have changed.

• Finally, in certain situations, for example, if an increase in values

represent a safety hazard, it would not be necessary to wait for the

specified number of successively increasing data points before

taking action.


Learning objectives

To understand when control charts can be applied

To understand different types of control charts and there

application areas

To understand how to collect data for control charts

(rational subgrouping)

To understand how to construct control charts

To understand how to analyze control chart behavior


Further reading

• Griffith, G. 1996. Statistical Process Control Methods for Long and Short Runs. ASQ

Quality Press, Milwaukee

• Kubiak, T. M. and Benbow, D. W. 2009. The certified six sigma black belt handbook.

ASQ Quality Press, Milwaukee

• Oakland, J.S., 2007. Statistical process control. Routledge. (downloadable from

through Oria.no)

• Thompson, J.R., 2002. Statistical process control: The Deming paradigm and beyond.

CRC Press.

• Wheeler, 1990. Advanced Topics in Statistical Process Control: The Power of

Shewhart’s Charts. SPC Press, Knoxville


Appendix - Control Chart Constants page 1

Source: Kubiak & Benbow (2009)


Appendix - Control Chart Constants page 2

Source: Kubiak & Benbow (2009)

tpk4161 supply chain analytics statistical process control

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