MDP 308

Quality Management

Lecture #6

Control charts for variables and process capability analysis

Today’s lecture

How to develop a control chart for variables

Control charts for variables (cont.)

ҧ𝑥 − 𝑅 charts (revision)

ҧ𝑥 − 𝑠 charts

Process capability analysis

Control charts for variables

Variables are the measurable characteristics of a product or service.

Measurement data is taken and arrayed on charts.

The objectives of the variable control charts are:

1. For quality improvement

2. To determine the process capability

3. For decisions regarding product specifications

4. For current decisions on the production process

5. For current decisions on recently produced items

Control charts for variables

Types of control charts for variables

1. ҧ𝑥 and R chart: the mean and range are used.

2. ҧ𝑥 and s chart: uses the sample standard deviation instead of the

range. Of course the standard deviation is more accurate

measure, but this chart necessitates the use of a suitable

calculating tool for quick computation of sample standard

deviation.

3. x and MR chart: uses the individual measurements (x) directly

instead of calculating a measure for central tendency. The

moving range, which is the difference between two consecutive

measurements is used to get a sense of the dispersion. This

chart is only used when products take long time to be finished

and there is not enough time to collect samples with suitable

sizes.

ҧ𝑥 and R charts

The ҧ𝑥 chart is developed from the average of each

subgroup data.

used to detect changes in the mean between subgroups.

The R chart is developed from the ranges of each

subgroup data

used to detect changes in variation within subgroups

Control chart components

Centerline

shows where the process average is centered or the central

tendency of the data

Upper control limits (𝑈𝐶𝐿 ҧ𝑥 and 𝑈𝐶𝐿𝑅) and Lower

control limits (𝐿𝐶𝐿 ҧ𝑥 and 𝐿𝐶𝐿𝑅)

describes the process spread

The Control Chart Method

R Control Chart:

𝑈𝐶𝐿𝑅 = D4 x ത𝑅

𝐿𝐶𝐿𝑅 = D3 x ത𝑅

Center line (𝐶𝐿𝑅) = ത𝑅

ഥ𝒙 Control Chart:𝑈𝐶𝐿 ҧ𝑥 = ҧ𝑥0+ A2 x ത𝑅𝐿𝐶𝐿 ҧ𝑥 = ҧ𝑥0 − A2 x ത𝑅Center line (𝐶𝐿 ҧ𝑥) = ҧ𝑥0

How to develop a control chart for variables?

Define the problem

Use other quality tools to help determine the general problem that’s occurring and the process that’s suspected of causing it.

Select a quality characteristic to be measured Identify a characteristic to study - for example, part

length or any other variable affecting performance.

Choose a subgroup size to be sampled

Choose homogeneous subgroups

Homogeneous subgroups are produced under the same

conditions, by the same machine, the same operator, the same

mold, at approximately the same time.

Try to maximize chance to detect differences between

subgroups, while minimizing chance for difference with a

group.

Collect the data

Generally, collect 20-25 subgroups before calculating the

control limits.

Each time a subgroup of sample size 𝑛 is taken, an average

is calculated for the subgroup and plotted on the control

chart.

For subgroup 𝑖, the average of the subgroup’s

measurements ( ҧ𝑥𝑖) is calculated as:

ҧ𝑥𝑖 =σ𝑗=1𝑛 𝑥𝑗

𝑛

Determine trial centerline (CL)

The centerline should be the population mean,

Since it is unknown, we use Ӗ𝑥, or the grand average of the

subgroup averages.

Remember that Ӗ𝑥 is an unbiased estimator of according

to the central limit theorem

If 𝑘 is the number of subgroups, we get

Ӗ𝑥 =σ𝑖=1𝑘 ҧ𝑥𝑖

𝑘

Determine trial control limits - ҧ𝑥 chart

The normal curve displays the distribution of the sample

averages.

A control chart is a time-dependent pictorial

representation of a normal curve.

Processes that are considered under control will have

99.73% of their graphed averages fall within 6.

UCL & LCL calculation

𝑈𝐶𝐿 ҧ𝑥 = Ӗ𝑥 + 3𝜎 ҧ𝑥

𝐿𝐶𝐿 ҧ𝑥 = Ӗ𝑥 − 3𝜎 ҧ𝑥

Where

𝜎 ҧ𝑥 = 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 ҧ𝑥

Alternatively, as an approximation, let

3𝜎 ҧ𝑥 = 𝐴2 × ത𝑅Where 𝐴2 is a constant that depends on

the subgroup size 𝑛

Determine trial control limits - R chart

The range chart shows the spread or dispersion of the

individual samples within the subgroup.

If the product shows a wide spread, then the individuals within

the subgroup are not similar to each other.

Equal averages can be deceiving.

Calculated similar to x-bar charts;

Use D3 and D4

𝑈𝐶𝐿𝑅 = 𝐷4 ത𝑅𝐿𝐶𝐿𝑅 = 𝐷3 ത𝑅

Example: Control Charts for Variable Data

Slip Ring Diameter (cm)

Sample 1 2 3 4 5 X R

1 5.02 5.01 4.94 4.99 4.96 4.98 0.08

2 5.01 5.03 5.07 4.95 4.96 5.00 0.12

3 4.99 5.00 4.93 4.92 4.99 4.97 0.08

4 5.03 4.91 5.01 4.98 4.89 4.96 0.14

5 4.95 4.92 5.03 5.05 5.01 4.99 0.13

6 4.97 5.06 5.06 4.96 5.03 5.01 0.10

7 5.05 5.01 5.10 4.96 4.99 5.02 0.14

8 5.09 5.10 5.00 4.99 5.08 5.05 0.11

9 5.14 5.10 4.99 5.08 5.09 5.08 0.15

10 5.01 4.98 5.08 5.07 4.99 5.03 0.10

50.09 1.15

Calculation

From previous table:

k = 10

σ𝑖=1𝑘 ҧ𝑥𝑖= 50.09

σ𝑖=1𝑘 𝑅𝑖 = 1.15

Thus;

Ӗ𝑥 = 50.09/10 = 5.009 cm

ത𝑅 = 1.15/10 = 0.115 cm

Note: The control limits are only preliminary with 10 samples.

It is desirable to have at least 25 samples.

Trial control limit

𝑈𝐶𝐿 ҧ𝑥 = Ӗ𝑥 +𝐴2 × ഥ𝑅= 5.009 + (0.577)(0.115) = 5.075 cm

𝐿𝐶𝐿 ҧ𝑥 = Ӗ𝑥 − 𝐴2 × ഥ𝑅 = 5.009 - (0.577)(0.115) =

4.943 cm

𝑈𝐶𝐿𝑅 = 𝐷4 ത𝑅 = (2.114)(0.115) = 0.243 cm

𝐿𝐶𝐿𝑅 = 𝐷3 ത𝑅 = (0)(0.115) = 0 cm

For A2, D3, D4 taken from tables at n = 5

3-Sigma Control Chart Factors

Sample size x-chart R-chart

n A2 D3 D4

2 1.88 0 3.27

3 1.02 0 2.57

4 0.73 0 2.28

5 0.58 0 2.11

6 0.48 0 2.00

7 0.42 0.08 1.92

8 0.37 0.14 1.86

X-bar Chart

4.94

4.96

4.98

5.00

5.02

5.04

5.06

5.08

5.10

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

X b

ar

LCL

CL

UCL

R Chart

0.00

0.05

0.10

0.15

0.20

0.25

0 1 2 3 4 5 6 7 8 9 10 11

Subgroup

Range

LCL

CL

UCL

Another Example of ഥ𝒙 & 𝑹 chart

Subgroup x1

x2

x3

x4

ഥ𝒙 UCL-X

LCL-X-bar

R UCL-R

LCL-R

1 6.35 6.4 6.32 6.37 6.36 6.47 6.35 0.08 0.20 0

2 6.46 6.37 6.36 6.41 6.4 6.47 6.35 0.1 0.20 0

3 6.34 6.4 6.34 6.36 6.36 6.47 6.35 0.06 0.20 0

4 6.69 6.64 6.68 6.59 6.65 6.47 6.35 0.1 0.20 0

5 6.38 6.34 6.44 6.4 6.39 6.47 6.35 0.1 0.20 0

6 6.42 6.41 6.43 6.34 6.4 6.47 6.35 0.09 0.20 0

7 6.44 6.41 6.41 6.46 6.43 6.47 6.35 0.05 0.20 0

8 6.33 6.41 6.38 6.36 6.37 6.47 6.35 0.08 0.20 0

9 6.48 6.44 6.47 6.45 6.46 6.47 6.35 0.04 0.20 0

10 6.47 6.43 6.36 6.42 6.42 6.47 6.35 0.11 0.20 0

11 6.38 6.41 6.39 6.38 6.39 6.47 6.35 0.03 0.20 0

12 6.37 6.37 6.41 6.37 6.38 6.47 6.35 0.04 0.20 0

13 6.4 6.38 6.47 6.35 6.4 6.47 6.35 0.12 0.20 0

14 6.38 6.39 6.45 6.42 6.41 6.47 6.35 0.07 0.20 0

15 6.5 6.42 6.43 6.45 6.45 6.47 6.35 0.08 0.20 0

16 6.33 6.35 6.29 6.39 6.34 6.47 6.35 0.1 0.20 0

17 6.41 6.4 6.29 6.34 6.36 6.47 6.35 0.12 0.20 0

18 6.38 6.44 6.28 6.58 6.42 6.47 6.35 0.3 0.20 0

19 6.35 6.41 6.37 6.38 6.38 6.47 6.35 0.06 0.20 0

20 6.56 6.55 6.45 6.48 6.51 6.47 6.35 0.11 0.20 0

21 6.38 6.4 6.45 6.37 6.4 6.47 6.35 0.08 0.20 0

22 6.39 6.42 6.35 6.4 6.39 6.47 6.35 0.07 0.20 0

23 6.42 6.39 6.39 6.36 6.39 6.47 6.35 0.06 0.20 0

24 6.43 6.36 6.35 6.38 6.38 6.47 6.35 0.08 0.20 0

25 6.39 6.38 6.43 6.44 6.41 6.47 6.35 0.06 0.20 0

Given Data

Calculation

From previous table:

k = 25

σ𝑖=1𝑘 ҧ𝑥𝑖= 160.25

σ𝑖=1𝑘 𝑅𝑖 = 2.19

Thus;

Ӗ𝑥 = 160.25/25 = 6.41 mm

ത𝑅 = 2.19/25 = 0.0876 mm

Trial control limit

𝑈𝐶𝐿 ҧ𝑥 = Ӗ𝑥 +𝐴2 × ഥ𝑅= 6.41 + (0.729)(0.0876) = 6.47 mm

𝐿𝐶𝐿 ҧ𝑥 = Ӗ𝑥 − 𝐴2 × ഥ𝑅 = 6.41 – (0.729)(0.0876) =

6.35 mm

𝑈𝐶𝐿𝑅 = 𝐷4 ത𝑅 = (2.282)(0.0876) = 0.20 mm

𝐿𝐶𝐿𝑅 = 𝐷3 ത𝑅 = (0)(0.0876) = 0 mm

For A2, D3, D4 from table at n = 4.

X-bar Chart

R Chart

Revised CL & Control Limits

Calculation based on discarding subgroup 4 & 20 ( ҧ𝑥chart) and subgroup 18 for R chart:

= (160.25 - 6.65 - 6.51)/(25 - 2)

= 6.40 mm

= (2.19 - 0.30)/(25 – 1)

= 0.079 = 0.08 mm

dnew

d

R RR

k k

d

new

d

x xx

k k

−=

−

New Control Limits

New value:

Using standard value, CL & 3 control limit obtained using formula:

2

, , Oo new o new o

Rx x R R

d= = =

2 1

,

,

x o o x o o

R o R o

UCL x A LCL x A

UCL D LCL D

= + = −

= =

From table:

A = 1.500 for a subgroup size of 4,

d2 = 2.059, D1 = 0, and D2 = 4.698

Calculation results:

6.40o newx x mm= = mmd

RRR o

onewo 038.0059.2

079.0,079.0

2

=====

6.40 (1.500)(0.038) 6.46x o oUCL x A mm= + = + =

6.40 (1.500)(0.038) 6.34x o oUCL x A mm= − = − =

mmDUCL oR 18.0)038.0)(698.4(2 ===

mmDLCL oR 0)038.0)(0(1 ===

Trial Control Limits & Revised Control Limit

6.30

6.35

6.40

6.45

6.50

6.55

6.60

6.65

0 2 4 6 8

Subgroup

Mean

, X

-bar

0.00

0.05

0.10

0.15

0.20

0 2 4 6 8

Subgroup

Ran

ge, R

UCL = 6.46

CL = 6.40

LCL = 6.34

LCL = 0

CL = 0.08

UCL = 0.18

Revised control limits

Revise the charts

In certain cases, control limits are revised

because:

out-of-control points were included in the

calculation of the control limits.

the process is in-control but the within

subgroup variation significantly improves.

Revising the charts

Interpret the original charts

Isolate the causes

Take corrective action

Revise the chart Only remove points for which you can determine an

assignable cause

The ҧ𝑥 − 𝑅 charts (summary)

The initial control limits and center lines are then calculated using the following formulas:

𝑈𝐶𝐿 ҧ𝑥 = Ӗ𝑥 +𝐴2 × ഥ𝑅𝐿𝐶𝐿 ҧ𝑥 = Ӗ𝑥 −𝐴2 × ഥ𝑅

𝐶𝐿 ҧ𝑥 = Ӗ𝑥𝑈𝐶𝐿𝑅 = 𝐷4 ത𝑅𝐿𝐶𝐿𝑅 = 𝐷3 ത𝑅

𝐶𝐿𝑅 = ത𝑅 Then the ҧ𝑥 and 𝑅 charts are drawn after selecting an

appropriate scale.

The out of control points are identified for each chart individually. Those points are excluded from their corresponding charts and the new control limits are determined.

The ҧ𝑥 − 𝑅 charts (summary)

The new values of Ӗ𝑥 and ത𝑅 are evaluated after excluding the out of control points in each chart individually. The new values are denoted Ӗ𝑥𝑛𝑒𝑤 and ത𝑅𝑛𝑒𝑤

The new control limits and center lines are evaluated as follows:

ҧ𝑥0 = 𝐶𝐿 ҧ𝑥 = Ӗ𝑥𝑛𝑒𝑤𝑅0 = 𝐶𝐿𝑅 = ത𝑅𝑛𝑒𝑤

𝜎0 =𝑅0𝑑2

𝑈𝐶𝐿 ҧ𝑥 = ҧ𝑥0 + 𝐴 × 𝜎0𝐿𝐶𝐿 ҧ𝑥 = ҧ𝑥0 − 𝐴 × 𝜎0𝑈𝐶𝐿𝑅 = 𝐷2 × 𝜎0𝐿𝐶𝐿𝑅 = 𝐷1 × 𝜎0

If in the new charts, there are any out of control points, the re-evaluation process is repeated.

The ҧ𝑥 − 𝑠 charts

We follow very similar steps as in ҧ𝑥 − 𝑅 charts for constructing these charts but with different calculations as the sample standard deviation (𝑠𝑖) is used instead of the range.

The calculations used are:

Ӗ𝑥 =ҧ𝑥1 + ҧ𝑥2 +⋯+ ҧ𝑥𝑘

𝑘

ҧ𝑠 =𝑠1 + 𝑠2 +⋯+ 𝑠𝑘

𝑘𝑈𝐶𝐿 ҧ𝑥 = Ӗ𝑥 + 𝐴3 × ത𝑠𝐿𝐶𝐿 ҧ𝑥 = Ӗ𝑥 − 𝐴3 × ത𝑠

𝐶𝐿 ҧ𝑥 = Ӗ𝑥

𝑈𝐶𝐿𝑠 = 𝐵4 ҧ𝑠𝐿𝐶𝐿𝑠 = 𝐵3 ҧ𝑠𝐶𝐿𝑠 = ҧ𝑠

The ҧ𝑥 − 𝑠 charts

If out of control points exist, the are excluded. The new values of Ӗ𝑥 and ҧ𝑠 are evaluated after excluding the out of control points in each chart individually. The new values are denoted Ӗ𝑥𝑛𝑒𝑤 and ҧ𝑠𝑛𝑒𝑤

The new control limits and center lines are evaluated as follows:

ҧ𝑥0 = 𝐶𝐿 ҧ𝑥 = Ӗ𝑥𝑛𝑒𝑤𝑠0 = 𝐶𝐿𝑠 = ҧ𝑠𝑛𝑒𝑤

𝜎0 =𝑠0𝑐4

𝑈𝐶𝐿 ҧ𝑥 = ҧ𝑥0 + 𝐴 × 𝜎0𝐿𝐶𝐿 ҧ𝑥 = ҧ𝑥0 − 𝐴 × 𝜎0𝑈𝐶𝐿𝑠 = 𝐵6 × 𝜎0𝐿𝐶𝐿𝑠 = 𝐵5 × 𝜎0

Table of constants for ҧ𝑥 − 𝑅 and ҧ𝑥 − 𝑠

Process capability

Process capability is the ability of the process to meet the design specifications for a product.

Nominal value is a target for design specifications.

Tolerance is an allowance above or below the nominal value. It’s defined using specification limits:

Lower specification limit (LSL)

Upper specification limit (USL)

20 25 30

Upper

Specification (USL)

Lower

Specification (LSL)

Nominal

value

Process Capability

Process is capable

Process distribution

Process is not capable

20 25 30

Upper

specification (USL)

Lower

Specification (LSL)

Nominal

value Process distribution

Process Capability

Process capability analysis

Process control study - refers only to the “voice of the

process” - looking at the process using an agreed

performance measure to see whether the process forms

a stable distribution over time.

Process Capability study looks at short term capability

and long term performance of a process with regard to

customer specifications.

Process Capability study measures the “goodness of a

process” - comparing the voice of the process with the

“voice of the customer”. The voice of the customer here

is the specification range (tolerance) and/or the nearest

customer specification limit.

Process control study

Process Control

Out of Control

(Special Causes Present)

In Control

(Special Causes Eliminated)

Note - no reference to

specs !

Process capability study (analysis)

In Control but not Capable

(Variation from Common Causes

Excessive)

In Control and Capable

(Variation from Common

Causes Reduced)Lower Spec Limit

Upper Spec Limit

Process Capability

1. Predicting how well the process will hold the tolerances.

2. Assisting product developers/designers in selecting or modifying a process.

3. Assisting in Establishing an interval between sampling for process monitoring.

4. Specifying performance requirements for new equipment.

5. Selecting between competing vendors.

6. Planning the sequence of production processes when there is an interactive effect of processes on tolerances

7. Reducing the variability in a manufacturing process.

Uses of process capability study

Process capability measures

Two measures are commonly used for process capability analysis, they are:

1. Process capability ratio, Cp, is the tolerance width divided by 6 standard deviations (process variability).

𝐶𝑝 =𝑡𝑜𝑙𝑒𝑟𝑎𝑛𝑐𝑒

6 ො𝜎=𝑈𝑆𝐿 − 𝐿𝑆𝐿

6 ො𝜎2. Process Capability Index, Cpk, is an index that measures

the potential for a process to generate defective outputs relative to either upper or lower specifications.

𝐶𝑝𝑘 = minӖ𝑥 − 𝐿𝑆𝐿

3 ො𝜎,𝑈𝑆𝐿 − Ӗ𝑥

3 ො𝜎We take the minimum of the two ratios because it gives the worst-case situation.

What is ො𝜎?

ො𝜎 is the standard deviation of the process measured

output.

For R chart,

ො𝜎 =ഥ𝑅𝑑2

For s chart,

ො𝜎 =ത𝑠𝑐4