lecture 17 control charts

Control Charts for Variables

Control charts were invented to serve one purpose—to identify process changes as quickly as possible after

the change occurs. They do nothing more and nothing less.

Quality Control in Textiles

Engr. Fareha Asim


Selection of Characteristics for Investigation

• A single component has several quality characteristics.

• It is not feasible to maintain a control chart for each characteristic.

• Base selection of quality characteristics on priority:

1. Those that cause more nonconforming items

2. Those that increase cost.

• Select “vital few” from “trivial many”: Use Pareto Analysis.


Engr. Fareha Asim


Control Chart for Variables

X and R charts: for sample averages and ranges.

X and s charts: for sample means and standard deviations.


Engr. Fareha Asim


Control Charts for Mean and Range

The two types of charts go together when monitoring variables, because they measure the two critical parameters: central tendency and dispersion.

1.An x chart is used if the quality of the output is measured in terms of a variable such as length, weight, temperature, and so on.

2.An R chart is used to monitor the range of the measurements in the sample.


Engr. Fareha Asim



Development of Charts

Step 1

For each sample, calculate the sample mean and range using the formula:

X = Σ Xi and R = Xmax - Xmin

n

Where:

Xi is the ith observation

n is the number of sample size

Xmax is the largest observation

Xmin is the smallest observation


Engr. Fareha Asim




Step 2

Obtain and draw the centre line and the trial control limits for each chart.

For the X-bar chart, the centre line is given by:

X = Σ Xi / g

For the R chart, the centre line is found from:

R = Σ R /g

Where g represents the number of samples.


Engr. Fareha Asim




Step 3

Conceptually:

the 3 σ control limits for X-bar charts are X +/- 3σx

the 3 σ control limits for R-charts are R +/- 3σR


Engr. Fareha Asim



• For an x-bar chart, the mean of each sample is computed and plotted on the chart; the points are sample means. The samples tend to be small, usually around 4 or 5.

n is the sample size (or number of observations)

k is the number of samples


Engr. Fareha Asim



R-chart• In an R-chart, the range is the difference between the

smallest and largest values in a sample. This range reflects the process variability instead of the tendency toward a mean value.

• R is the range of each samplek is the number of samples.


Engr. Fareha Asim



Development of Charts X bar charts control limits

Step 3

Range for sample i

# Samples

Mean for sample i

From Table

RAxxLCL

RAxxUCL

n

R R

i

n

1i

n

xi

n

ix


Engr. Fareha Asim



Development of Charts – R charts control limits

Step 3

Range for Sample i

# Samples

From Table

n

R R

R D LCL

R D UCL

i

n

1i

3R

4R


Engr. Fareha Asim



n d2 A2 d3 D3 D45 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .

n d2 A2 d3 D3 D45 2.326 0.577 0.864 0 2.1146 2.534 0.483 0.848 0 2.0047 2.704 0.419 0.833 0.076 1.9248 2.847 0.373 0.820 0.136 1.8649 2.970 0.337 0.808 0.184 1.81610 3.078 0.308 0.797 0.223 1.777. . . . . .. . . . . .


Engr. Fareha Asim

[Tabulate A2, D3 and D4 from Table A.23, Probability & Statistics for Engineers and Scientists, 8 th Edition. Walpole Myers}




Engr. Fareha Asim

Special Metal Screw

Sample

Number

Sample

1 2 3 4

1 0.5014 0.5022 0.5009 0.5027

2 0.5021 0.5041 0.5024 0.5020

3 0.5018 0.5026 0.5035 0.5023

4 0.5008 0.5034 0.5024 0.5015

5 0.5041 0.5056 0.5034 0.5047



Special Metal Screw

Sample Sample

Number 1 2 3 4 Range Mean

1 0.5014 0.5022 0.5009 0.5027 0.0018 0.5018

2 0.5021 0.5041 0.5024 0.5020 0.0021 0.5027

3 0.5018 0.5026 0.5035 0.5023 0.0017 0.5026

4 0.5008 0.5034 0.5024 0.5015 0.0026 0.5020

5 0.5041 0.5056 0.5034 0.5047 0.0022 0.5045

Average 0.0021 0.5027


Engr. Fareha Asim



Special Metal ScrewControl Charts - Special Metal Screw

R - Charts R = 0.0021

UCLR = D4R = 2.282(0.0021) = 0.00479LCLR = D3R = 0(0.0021) = 0


Engr. Fareha Asim



Special Metal ScrewControl Charts - Special Metal Screw

R - Charts R = 0.0021

UCLR = D4R = 2.282(0.0021) = 0.00479LCLR = D3R = 0(0.0021) = 0

Control Charts - Special Metal Screw

R - Charts R = 0.0020 D4 = 2.2080

Control Chart Factors (Three Sigma Limits)Control Chart Factors (Three Sigma Limits)

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 00 3.2683.26833 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924


Engr. Fareha Asim



Range Chart - Special Metal Screw

0.005

0.004

0.003

0.002

0.001

0 1 2 3 4 5 6

Ran

ge

(in

.)

Sample number

UCLR = 0.00479

LCLR = 0

R = 0.0021


Engr. Fareha Asim




x - Charts R = 0.0021x = 0.5027

UCLx = x + A2R = 0.5027 + 0.729(0.0021)LCLx = x - A2R = 0.5027 - 0.729(0.0021)UCL = 0.5042LCL = 0.5012


Engr. Fareha Asim




R = 0.0020x = 0.5025

x - Charts

UCLx = x + A2RLCLx = x - A2R

Control Chart Factors (Three Sigma Limits)Control Chart Factors (Three Sigma Limits)

Factor for UCLFactor for UCL Factor forFactor for FactorFactorSize ofSize of and LCL forand LCL for LCL forLCL for UCL forUCL forSampleSample xx-Charts-Charts RR-Charts-Charts RR-Charts-Charts

((nn)) ((AA22)) ((DD33)) ((DD44))

22 1.8801.880 00 3.2683.26833 1.0231.023 00 2.5752.57544 0.7290.729 00 2.2822.28255 0.5770.577 00 2.1152.11566 0.4830.483 00 2.0042.00477 0.4190.419 0.0760.076 1.9241.924


Engr. Fareha Asim



X bar Control Charts - Special Metal Screw

0.5050

0.5040

0.5030

0.5020

0.5010

1 2 3 4 5

Ave

rag

e (i

n.)

Sample number

x = 0.5027

UCLx = 0.5042

LCLx = 0.5012

0.5045


Engr. Fareha Asim




Step 4

• An R chart is analyzed before the X bar chart to determine out of control situations.

• A R-chart reflects process variability, which should be brought to control first.

• If R-chart shows an out of control situation, the limits on X- bar chart may not be meaningful.


Engr. Fareha Asim




Step 5

Delete the out of control points for which remedial actions have been taken to remove special causes and use the remaining samples to determine the revised centre line and control limits for the X and R charts.

These are revised control limits.

These are trial control limits for immediate future until the limits are revised again.


Engr. Fareha Asim


It is possible that an out-of-control signal will appear on one kind of chart and not the other.


Engr. Fareha Asim


Control Charts for the Mean and Standard Deviation

X-bar cart is exactly the same

Standard deviation instead of range for the spread

Preferred over x-bar and R-charts since standard deviation is a better measure of variation than the range.

Also requires slight modification of control limit formulas


Engr. Fareha Asim



• Control limits for X-bar chart


Engr. Fareha Asim



• Control limits for s-bar chart


Engr. Fareha Asim

• Example 17.2

[Tabulate A3, B3 and B4 from Table A.23, Probability & Statistics for Engineers and Scientists, 8 th Edition. Walpole Myers}

lecture 17 control charts

Documents