mdp 308 quality management control charts for variables. types of control charts for variables 1....
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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