Control Charts

Control Charts

UCL

Statistics

UWL Target LWL LCL

1

2

3

4

5

Sample Number

6

7

8

1. Introduction

Introduction Quality control charts, are graphs on which the quality of the product is plotted as manufacturing or servicing is actually proceeding. By enabling corrective actions to be taken at the earliest possible moment and avoiding unnecessary corrections, the charts help to ensure the manufacture of uniform product or providing consistent services which complies with the specification.

History of Control Chart

Mr. Shewart, an American, has been credited with the invention of control charts for variable and attribute data in the 1920s, at the Bell Telephone Industries. The term ‘Shewart Control Charts’ is in common use.

Dynamic Picture of Process

Plotting graph, charting and presenting the data as a picture is common to process control method, used throughout the manufacturing and service industries. Converting data into a picture is a vital step towards greater and quicker understanding of the process.

Confidence While Control Charting

Control charting enables everyone to make decision and to know the degree of confidence with which the decisions are made. There may be some margin of error. No technique, even 100% automated inspection, can guarantee the validity of the result; there is always some room to doubt.

Control Charts Statistically based control chart is a device intended to be used - at the point of operation - by the operator of that process - to asses the current situation - by taking sample and plotting sample result To enable the operator to decide about the process.

What Control Chart Does?

It graphically, represents the output of the process. And Uses statistical limits and patterns of plot, for decision making

Analogy to Traffic Signal A control chart is like a traffic signal, the operation of which is based on evidence from samples taken at random intervals. A green signal - Process be allowed to continue without adjustment A yellow signal - Wait and watch trouble is possible A red signal - Process has wandered Investigate and adjust

Analogy to Traffic Signal Stop Investigate/Adjust

Wait and Watch

Go No action on Process

Decision About The Process

Go To let the process continue to run without any adjustment. This means only common causes are present.

Decision About the Process

Wait and watch Be careful and seek for more information This is the case where presence of trouble is possible

Decision About the Process Stop Take action ( Investigate/Adjust ) This means that there is practically no doubt a special cause has crept in the system. Process has wandered and corrective actions must be taken, otherwise defective items will be produced.

2. Why control charts

Why Control Chart?

To ensure that the output of the process is-

Normal

Whether Output is Normal? Both histogram and control chart can tell us whether the output is normal? However, Histogram views the process as history , as the entire output together. Control chart views the process in real time, at different time intervals as the process progresses.

Frequency

Histogram a History of Process Output 16 14 12 10 8 6 4 2 0

47

48 49 50 51 52 53 54 kg

Control Chart Views Process in Real Time Output of the process in real time

Mean

Target UCLx Target LCLx Range

UCLr

Time Intervals

Why Control Chart?

It helps in finding is there any change in location of process mean in real time

Change in Location of Process Mean Process with mean at less than target

Process with mean at Target

43 44 45 46 47 48

Process with mean at more than target

49 50 51 52

53

Why Control Chart?

It helps in finding Is there any change in the spread of the process in real time?

Change in Spread of Process Spread due to common causes

Larger spread due to special causes

43

44

45

46

47

48

49

50

51

52

53

Why Control Chart? To keep the cost of production minimum Since the control chart is maintained in real time, and gives us a signal that some special cause has crept into the system, we can take timely action. Timely action enables us to prevent manufacturing of defective. Manufacturing defective items is non value added activity; it adds to the cost of manufacturing, therefore must be avoided. By maintaining control chart we avoid 100% inspection, and thus save cost of verification.

Why Control Chart? Pre-requisite for process capability studies Process capability studies, are based on premises that the process during the study was stable i.e. only common causes were present. This ensures that output has normal distribution. The stability of the process can only be demonstrated by maintaining control chart during the study.

Why Control Chart?

Decision in regards to production process Control chart helps in determining whether we should : - let the process to continue without adjustment - seek more information - stop the process for investigation/adjustment.

3. Basic steps for control charting

Basic Steps for Control Charts

Step No. 1 Identify quality characteristics of product or process that affects “fitness for use”. Maintaining control chart is an expensive activity. Control charts should be maintained only for critical quality characteristics. Design of Experiments is one of the good source to find the critical quality characteristics of the process.

Basic Steps for Control Charts

Step No . 2 Design the sampling plan and decide method of its measurement. At this step we decide, how many units will be in a sample and how frequently the samples will be taken by the operator.

Basic Steps for Control Charts Step No. 3 Take samples at different intervals and plot statistics of the sample measurements on control chart. Mean, range, standard deviation etc are the statistics of measurements of a sample. On a mean control chart, we plot the mean of sample and on a range control chart, we plot the range of the sample.

Basic Steps for Control Charts Step No. 4 Take corrective action - when a signal for significant change in process characteristic is received. Here we use OCAP (Out of Control Action Plan) to investigate, as why a significant change in the process has occurred and then take corrective action as suggested in OCAP, to bring the process under control.

Summary of Control Chart Techniques In ‘Control Chart Technique’ we have Quality characteristics Sampling procedure Plotting of statistics Corrective action

4. Typical control chart

Elements of Typical Control Chart 1. Horizontal axis for sample number 2. Vertical axis for sample statistics e.g. mean, range, standard deviation of sample. 3. Target Line 4. Upper control line 5. Upper warning line 6. Lower control line 7. Lower warning line 8. Plotting of sample statistics 9. Line connecting the plotted statistics

Elements of Typical Control Chart Upper control line

Sample Statistics

Upper warning line

Target

Lower warning line Lower control line

1

2 3 4 Sample Number

5

5. Types of control chart

Types of Control Chart

We have two main types of control charts. One for variable data and the other for attribute data. Since now world-wide, the current operating level is ‘number of parts defective per million parts produced’, aptly described as ‘PPM’; control charts for ‘attribute data’ has no meaning. The reason being that the sample size for maintaining control chart at the ‘PPM’ level, is very large, perhaps equal to lot size, that means 100% inspection.

Most Commonly Used Variable Control Charts Following are the most commonly used variable control charts:

To track the accuracy of the process - Mean control chart or x-bar chart

To track the precision of the process - Range control chart

Most Common Type of Control Chart for Variable Data For tracking Accuracy Mean control chart Variable Control Chart For tracking Precision Range control chart

6. Concepts behind control charts

Understanding effect of shift of process mean

Case When Process Mean is at Target Process Mean

Target L

U -3s

42

43

44

45

46

+3 s

47

48

49

U-L=6s

50

51

52

Chances of getting a reading beyond U & L is almost nil

53

Case - Small Shift of the Process Mean Small shift in process

Process Mean

Shaded area shows the probability of getting a reading beyond U

U

L Target U-L = 6 s

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44

45

46

47

48

49

50

51

Chances of getting a reading outside U is small

52

53

Case - Large Shift of the Process Mean Large shift in process Process Mean

Target

L

Shaded area shows the probability of getting a reading beyond U

U

U-L = 6 s

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43

44

45

46

47

48

49

50

51

Chances of getting a reading outside U is large

52

53

Summary of Effect of Process Shift

When there is no shift in the process nearly all the observations fall within -3 s and + 3 s. When there is small shift in the mean of process some observations fall outside original -3 s and +3 s zone. Chances of an observation falling outside original -3 s and + 3 s zone increases with the increase in the shift of process mean.

Our Conclusion from Normal Distribution

When an observation falls within original +3 s and -3 s zone of mean of a process, we conclude that there is no shift in the mean of process. This is so because falling of an observation between these limits is a chance. When an observation falls beyond original +3 s and -3 s zone of process mean, we conclude that there is shift in location of the process

7. Distribution of population vs Distribution of mean

Distribution of Mean of Samples

Since on the control charts for accuracy we plot and watch the trend of the means and ranges of the samples, it is necessary that we should understand the behaviour of

distribution of mean of samples.

Distribution of Averages of Samples

Suppose we have a lot of 1000 tablets, and let us say, weight of the tablets follows a normal distribution having a standard deviation, s. Let us take a sample of n tablets. Calculate mean of the sample and record it. Continue this exercise of taking samples, calculating the mean of samples and recording, 1000 times. The mean of samples shall have normal distribution with standard deviation, Sm = (s÷ n). Distribution of population and ‘means of sample’ shall have same means.

Distribution - Population Vs Sample Means Distribution of means of samples [standard deviation = (s÷ n)]

Distribution of population (standard deviation = s

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49

Quality Characteristics

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51

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Control and Warning Limits for Mean Control Chart

If we know the standard deviation of the population, say sand the number of units in a sample, say n; then the control and warning limits are calculated as follows: If desired target of the process is T, then Upper control limit, UCL = T + 3 (s÷ n) Upper warning limit. UWL = T + 2 (s÷ n) Lower control limit, LCL = T - 3 (s÷ n) Lower warning limit, LWL = T - 2 (s÷ n)

Control Limits for Mean Control Chart Distribution of mean of samples UCL UWL

3 (s ÷ n)

2 (s ÷ n) Target 2 (s ÷ n)

3 (s÷ n)

1

2

3

LWL LCL

4

5

Sample Number

6

7

8. Establishing Control Charts

Establishing Control Chart Step No.1 Select quality characteristics which needs to be controlled - Weight - Length - Viscosity - Tensile Strength - Capacitance

Establishing Control Chart Step No.2 Decide the number of units, n to be taken in a sample. The minimum sample size should be 2. As the sample size increases then the sensitivity i.e. the quickness with which the chart gives an indication of shift of the process increases. However, with the increase of the sample size cost of inspection also increases.

Generally, n can be 4 or 5.

Establishing Control Chart

Step No. 3 Decide the frequency of picking up of sample If the shift in the process average causes more loss, then take smaller samples more frequently. If the cost of inspection is high then take smaller samples at large interval.

Establishing Control Chart

As and general guidance, for deciding the frequency of taking a sample, we can use the table given in the next slide. If our lot size in a shift is say 3000, then in a shift we require 50 units. If the sample size n, is say 4 then Number of visits to the process is = 50÷4 = 12 The time of an 8-hour shift, be divided in 12 equal parts. Samples should be taken round about every 45 minutes.

Establishing Control Chart Lot Size

Total Number of items

66 - 100

10

101 - 180

15

181 - 300

25

301 - 500

30

501 - 800

35

801 - 1300

40

1301 - 3200

50

3201 - 8000

60

Establishing Control Chart Step No. 4 Collect data on a special control chart data collection sheet. ( Minimum 100 observations) The data collection sheet has following main portions: 1. General details for part, department etc. 2. Columns for date and time sample taken 3. Columns for measurements of sample 4. Column for mean of sample 5. Column for range of sample

Typical Data Collection Sheet Part SN

Operation Date

Measurement

Time X1

1 2 3 ….. 25

Other Details

X2

X3

X4

Mean Range

Establishing Control Chart Step No. 5 Fill up the control chart data sheet 1) As per the plan, visit the process and collect a sample of required number of units. 2) Measure the units and record. 3) Take requisite number of samples ( 20-25). 4) Calculate the mean of each of the sample. 5) Calculate the range of each of the sample.

9. Establishing Trial Control Limits

Example - Establishing Trial Control Limits

A supervisor decided to put his process under statistical control. For the purpose of establishing control chart he collected 10 samples (Normally it should be 20 samples) containing 5 units. The samples were measured and the same is shown in the next slide. The desired target of the process, T is 50. Establish control chart for monitoring the process.

Example - Data Collection Subgroup Reading

Subgroup No.

X1

X2

X3

X4

X5

1

47

45

48

52

51

2

48

52

47

50

50

3

49

48

52

50

49

4

49

50

52

50

49

5

51

50

53

50

48

6

50

50

49

51

47

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51

48

50

50

54

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50

48

50

50

52

9

48

48

49

50

51

10

49

50

50

52

51

Mean of subgroup

Range of subgroup

Example - Calculation of Subgroup No.1 Measurements are 47, 45, 48, 52 & 51 Mean of measurements of subgroup No. 1 = (47 + 45 + 48 + 52 + 51)/5 = 48.6 Range of measurements of subgroup No. 1 = ( largest reading - smallest reading ) = ( 52 - 45 ) =7

Example - Calculation of subgroup Mean & Range Subgroup Reading

Subgroup No.

X5

Mean of subgroup

Range of subgroup

X1

X2

X3

X4

1

47

45

48

52

51

48.6

7

2

48

52

47

50

50

49.4

5

3

49

48

52

50

49

49.6

4

4

49

50

52

50

49

50.0

3

5

51

50

53

50

48

50.4

5

6

50

50

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51

47

49.4

4

7

51

48

50

50

54

50.6

6

8

50

48

50

50

52

50.0

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9

48

48

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50

51

49.2

3

10

49

50

50

52

51

50.2

3

Establishing Control Chart Step No. 6

Calculate Mean Range, R R=

Sum of ranges of subgroups Total number of subgroups

In our case R=

(7 + 5 +4 3 + 5 + 4 + 6 + 4 + 3 + 3 ) Total number of subgroups

Establishing Control Chart Step No. 7 Using following table of constants find trial control limit for mean and range control chart’

Sub Group Size

A2

D4

D3

2

1.880

3.267

0

3

1.023

2.527

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

Establishing Control Chart Step No. 8 Calculate Trial control Limits with target value, T Trial control limits for mean control chart Upper Control Limit, UCLx = T + A2 x R Lower Control Limit, LCLx = T - A2 x R Trial control limits for range control chart Upper Control Limit, UCLr = D4 x R Lower Control Limit, LCLr = D3 x R

Calculation of Trial Control Limits Size of Subgroup, n = 5 Factor A2, when n is 5 = 0.577 Factor D4, when n is 5 = 2.115 Factor D3, when n is 5 = 0 Target value, T = 50 Mean Range, R = 4.4

Establishing Control Chart Step No. 8

Trial control Limits in our case For mean control chart Upper Control Limit, UCLx = 50 + 0.577 x 4.4=52.5 Lower Control Limit, LCLx = 50 - 0.577 x 4.4=47.5 For range control chart Upper Control Limit, UCLr = 2.115 x 4.4 = 9.3 Lower Control Limit, LCLr = 0 x 4.4 = 0

Establishing Control Chart

Step No. 9 Discard the outliers Outliers are those observations which do not belong to normal population. If Outliers are included in the calculation, then the information is distorted.

Checking for Outliers Checking for mean outliers Scan column of sample means. If any mean of sample is more than UCLx or less than LCLx then drop that sample. Checking for range outliers Scan column of sample range. If any range is more than UCLr then drop that sample.

Checking for Outliers If any sample(s) is dropped then recalculate the trial control limits using remaining sample(s). Continue this exercise till there is no further droppings. When there is no further dropping trial control limits becomes control limits for control chart. In all we can drop up to 25% of the samples

Checking for Outliers In our case - None of the subgroup mean is more than 52.5 - None of the subgroup mean is less than 47.5 - None of the range is more than 9.3 - None of the range is less than 0 Hence there is no revision of trial control limits is required. These limits can be used for maintaining the control charts.

Calculation of Control Limits for Mean Control Chart Step No. 10 Compute warning limits for mean control chart

Upper warning limit, UWLx = T +

Lower warning limit, LWLx = T -

2 x A2 x R 3 2 x A2 x R 3

Calculation of Control Limits for Mean Control Chart Warning limits for mean control chart in our example

Uwlx = 50 +

2 x 0.577 x 4.4 3

= 51.7 Lwlx = 50 -

2 x 0.577 x 4.4 3

= 48.3

Action and Warning Limits for Mean Control chart UCLx UWLx Mean

Target LWLx LCLx

1

2 3 4 5 Sample Number

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7

Action and Warning Limits for Mean Control Chart for Example UCLx=52.5 UWLx=51.7 Mean

Target=50 LWLx=48.3 LCLx= 47.5

1

2 3 4 5 Sample Number

6

7

Constants for Range Control chart Sample size, n

D4

D3

DWLR

DWUR

2

3.27

0

0.04

2.81

3

2.57

0

0.18

2.17

4

2.28

0

0.29

1.93

5

2.11

0

0.37

1.81

6

2.00

0

0.42

1.72

7

1.92

0.08

0.46

1.66

Calculation of Control Limits for Range Control Chart

Step No. 11 Compute warning limits for range control chart Upper Warning Limit, UWLr = DWUR x R

Lower Warning Limit, LWLr = DWLR x R

Action and Warning Limits for Control Chart UCLx UWLx Mean

Target LWLx

Range

LCLx UCLr UWLr R LWLr

1

2 3 4 5 Sample Number

6

7

Calculation of Warning Limits for Range Control Chart

In our case Size of sub group, n = 5 Mean range R = 4.4 DWUR when n is 5 = 1.81 DWLR when n is 5 = 0.37

Calculation of Warning Limits for Range Control Chart In our case warning limits for range control chart Upper Warning Limit, UWLr = DWUR x R = 1.81 x 4.4 =8 Lower Warning Limit, LWLr = DWLR x R = 0.37 x 4.4 = 1.6

Action and Warning Limits for Control Chart UCLx = 52.5

Mean

UWLx = 51.7 Target = 50 LWLx = 48.3 LCLx = 47.5 UCLr = 9.3

Range

UWLr = 8 R = 4.4 LWLr = 1.6

1

2 3 4 5 Sample Number

6

7

Flow Chart for Establishing Control Chart Start Decide subgroup size

Record observations

Find mean and range of each subgroup

Calculate mean range, R

Flow Chart for Establishing Control Chart UCLx = T + A2 x R LCLx = T - A2 x R UCLr = D4 x R LCLr = D3 x R

Is any sub-group mean or range out side the control limit ?

No

Yes

Drop that Group

Flow Chart for Control Chart

Select suitable scale for mean control chart and range control chart

Draw Lines for Target, UCL, UWL, LCL & LWL for mean Mean range, UCL , UWL, LCL & LWL for range

Stop

10. Interpreting control charts

Interpreting Control Chart The control chart gets divided in three zones. Zone - 1 If the plotted point falls in this zone, do not make any adjustment, continue with the process.

Zone - 2 If the plotted point falls in this zone then special cause may be present. Be careful watch for plotting of another sample(s). Zone - 3 If the plotted point falls in this zone then special cause has crept into the system, and corrective action is required.

Zones for Mean Control Chart Action

Sample Mean

Zone - 3

UCL UWL

Zone - 2

Warning

Zone - 1

Continue

Target Zone - 1

Continue

Zone - 2

Warning

Zone - 3

Action

1

2

3

4

LWL LCL

5

Sample Number

6

7

Interpreting Control Chart Because the basis for control chart theory follows the normal distribution, the same rules that governs the normal distribution are used to interpret the control charts. These rules include:

- Randomness. - Symmetry about the centre of the distribution. - 99.73% of the population lies between - 3 s of and + 3 s the centre line. - 95.4% population lies between -2 s and + 2 s of the centre line.

Interpreting Control Chart

If the process output follows these rules, the process is said to be stable or in control with only common causes of variation present. If it fails to follow these rules, it may be out of control with special causes of variation present. special causes must be found and corrected.

These

Interpreting Control Chart A single point above or below the control limits. Probability of a point falling outside the control limit is less than 0.14%. This pattern may indicate: - a special cause of variation from a material, equipment, method, operator etc. - mismeasurement of a part or parts. - miscalculated or misplotted data point.

Interpreting Control Chart One point outside control limit

Statistics

UCL UWL

Target LWL LCL

1

2

3 4 5 Sample Number

6

7

8

Interpreting Control Chart

Seven consecutive points are falling on one side of the centre line. Probability of a point falling above or below the centre line is 50-50. The probability of seven consecutive points falling on one side of the centre line is 0.78% ( 1 in 128) This pattern indicates a shift in the process output from changes in the equipment, methods, or material or shift in the measurement system.

Interpreting Control Chart Seven consecutive points on one side of the centre line

Statistics

UCL UWL

Target LWL LCL

1

2

3 4 5 Sample Number

6

7

8

Interpreting Control Chart Two consecutive points fall between warning limit and corresponding control limit. In a normal distribution, the probability of two consecutive points falling between warning limit and corresponding control limit is 0.05% (1 in 2000). This could be due to large shift in the process, equipment, material, method or measurement system.

Interpreting Control Chart Two consecutive points between warning limit and corresponding control limit

Statistics

UCL UWL

Target LWL LCL

1

2

3 4 5 Sample Number

6

7

8

Interpreting Control Chart

Two points out of three consecutive points fall between warning limit and corresponding control limit. This could be due to large shift in the process, equipment, material, method or measurement system.

Interpreting Control Chart Two points out of three consecutive points between warning limit and corresponding control limit

Statistics

UCL UWL

Target LWL LCL

1

2

3 4 5 Sample Number

6

7

8

Interpreting Control Chart A trend of seven points in a row upward or downward demonstrates nonrandomness. This happens when - Gradual deterioration or wear in equipment. - Improvement or deterioration in technique. - Operator fatigue.

Interpreting Control Chart Seven consecutive points having upward trend

Statistics

UCL UWL

Target LWL LCL

1

2

3 4 5 Sample Number

6

7

8

Interpreting Control Chart Seven consecutive points having downward trend

Statistics

UCL UWL

Target LWL LCL

1

2

3

4

Sample Number

5

6

7

8