Statistical Process Control 5 Part 5 ppt

They pointed out that, regardless of the ease or difficulty of grouping thedata from a particular process, the forming of subgroups is an essential step inthe investigation of stability

They pointed out that, regardless of the ease or difficulty of grouping thedata from a particular process, the forming of subgroups is an essential step inthe investigation of stability and in the setting up of control charts.Furthermore, the use of group ranges to estimate process variability is so

widely accepted that ‘the mean of subgroup ranges’ R may be regarded as the

central pillar of a standard procedure

Many people follow the standard procedure given on page 116 and achievegreat success with their SPC charts The short-term benefits of the methodinclude fast reliable detection of change which enables early corrective action

to be taken Even greater gains may be achieved in the longer term, however,

if charting is carried out within the context of the process itself, to facilitategreater process understanding and reduction in variability

In many processes there is a tendency for observations that are made over

a relatively short time period to be more alike than those taken over a longerperiod In such instances the additional ‘between group’ or ‘medium-term’variability may be comparable with or greater than the ‘within group’ or

‘short-term’ variability If this extra component of variability is random theremay be no obvious way that it can be eliminated and the within groupvariability will be a poor estimate of the natural random longer term variation

of the process It should not then be used to control the process

Caulcutt and Porter observed many cases in which sampling schemes based

on the order of output or production gave unrepresentative estimates of the

random variation of the process, if R/d nwas used to calculate  Use of thestandard practice in these cases gave control lines for the mean chart whichwere too ‘narrow’, and resulted in the process being over-controlled.Unfortunately, not only do many people use bad estimates of the processvariability, but in many instances sampling regimes are chosen on an arbitrarybasis It was not uncommon for them to find very different sampling regimesbeing used in the preliminary process investigation/chart design phase and thesubsequent process monitoring phase

Caulcutt and Porter showed an example of this (Figure 6.12) in which meanand range charts were used to control can heights on a can-making productionline (The measurements are expressed as the difference from a nominal valueand are in units of 0.001 cm.) It can be seen that 13 of the 50 points lie outsidethe action lines and the fluctuations in the mean can height result in theprocess appearing to be ‘out-of-statistical control’ There is, however, nosimple pattern to these changes, such as a trend or a step change, and theadditional variability appears to be random This is indeed the case for the

process contains random within group variability, and an additional source of random between group variability This type of additional variability is

frequently found in can-making, filling and many other processes

A control chart design based solely on the within group variability isinappropriate in this case In the example given, the control chart would

mislead its user into seeking an assignable cause on 22 occasions out of the

50 samples taken, if a range of decision criteria based on action lines, repeatpoints in the warning zone and runs and trends are used (page 118) As thisadditional variation is actually random, operators would soon becomefrustrated with the search for special causes and corresponding correctiveactions

To overcome this problem Caulcutt and Porter suggested calculating thestandard error of the means directly from the sample means to obtain, in thiscase, a value of 2.45 This takes account of within and between groupvariability The corresponding control chart is shown in Figure 6.13 Theprocess appears to be in statistical control and the chart provides a basis foreffective control of the process

Stages in assessing additional variability

1 Test for additional variability

As we have seen, the standard practice yields a value of R from k small samples of size n This is used to obtain an estimate of the within sample

standard deviation :

 = R/d

Figure 6.12 Mean and range chart based on standard practice

The standard error calculated from this estimate (/n) will be appropriate if

 describes all the natural random variation of the process A differentestimate of the standard error, e, can be obtained directly from the sample

means, X i:

i = 1 (X i – X )2/(k – 1)

X is the overall mean or grand mean of the process Alternatively, all the

sample means may be entered into a statistical calculator and the n – 1keygives the value of edirectly

The two estimates are compared If eand /n are approximately equal

there is no extra component of variability and the standard practice for controlchart design may be used If e is appreciably greater than /n there is

Figure 6.13 Mean and range chart designed to take account of additional random variation

(A formal significance test for the additional variability can be carried out

by comparing ne2/2with a required or critical value from tables of the F

distribution with (k – 1) and k(n – 1) degrees of freedom A 5 per cent level of

significance is usually used See Appendix G.)

2 Calculate the control lines

If stage 1 has identified additional between group variation, then the meanchart action and warning lines are calculated from e:

Action lines X ± 3 e;

Warning lines X ± 2 e

These formulae can be safely used as an alternative to the standard practice

even if there is no additional medium-term variability, i.e even when  = R/d n

is a good estimate of the natural random variation of the process

(The standard procedure is used for the range chart as the range isunaffected by the additional variability The range chart monitors the withinsample variability only.)

In the can-making example the alternative procedure gives the followingcontrol lines for the mean chart:

Upper Action Line 7.39

Lower Action Line –7.31

Upper Warning Line 4.94

Lower Warning Line –4.86

These values provide a sound basis for detecting any systematic variationwithout over-reacting to the inherent medium-term variation of the process.The use of e to calculate action and warning lines has important

implications for the sampling regime used Clearly a fixed sample size, n, is

required but the sampling frequency must also remain fixed as e takesaccount of any random variation over time It would not be correct to usedifferent sampling frequencies in the control chart design phase andsubsequent process monitoring phase

6.6 Summary of SPC for variables using X and R charts

If data is recorded on a regular basis, SPC for variables proceeds in three mainstages:

1 An examination of the ‘State of Control’ of the process (Are we incontrol?) A series of measurements are carried out and the results plotted

on X and R control charts to discover whether the process is changing due

to assignable causes Once any such causes have been found and removed,the process is said to be ‘in statistical control’ and the variations then resultonly from the random or common causes

2 A ‘Process Capability’ Study (Are we capable?) It is never possible toremove all random or common causes – some variations will remain Aprocess capability study shows whether the remaining variations areacceptable and whether the process will generate products or serviceswhich match the specified requirements

3 Process Control Using Charts (Do we continue to be in control?) The X and R

charts carry ‘control limits’ which form traffic light signals or decision rulesand give operators information about the process and its state of control

Control charts are an essential tool of continuous improvement and greatimprovements in quality can be gained if well-designed control charts areused by those who operate processes Badly designed control charts lead toconfusion and disillusionment amongst process operators and management.They can impede the improvement process as process workers andmanagement rapidly lose faith in SPC techniques Unfortunately, the authorand his colleagues have observed too many examples of this across a range ofindustries, when SPC charting can rapidly degenerate into a paper or computerexercise A well-designed control chart can result only if the nature of theprocess variation is thoroughly investigated

In this chapter an attempt has been made to address the setting up of meanand range control charts and procedures for designing the charts have beenoutlined For mean charts the standard error estimate ecalculated directly

from the sample means, rather than the estimate based on R/d n, provides asound basis for designing charts that take account of complex patterns ofrandom variation as well as simpler short-term or inter-group randomvariation It is always sound practice to use pictorial evidence to test thevalidity of summary statistics used

Chapter highlights

 Control charts are used to monitor processes which are in control, using

means (X ) and ranges (R).

 There is a recommended method of collecting data for a process capability

study and prescribed layouts for X and R control charts which include

warning and action lines (limits) The control limits on the mean and rangecharts are based on simple calculations from the data

 Mean chart limits are derived using the process mean X, the mean range

R, and either A2constants or by calculating the standard error (SE) from

R The range chart limits are derived from R and D1constants

 The interpretation of the plots are based on rules for action, warning andtrend signals Mean and range charts are used together to control theprocess

 A set of detailed rules is required to assess the stability of a process and

to establish the state of statistical control The capability of the process can

be measured in terms of , and its spread compared with the specifiedtolerances

 Mean and range charts may be used to monitor the performance of aprocess There are three zones on the charts which are associated withrules for determining what action, if any, is to be taken

 There are various forms of the charts originally proposed by Shewhart.These include charts without warning limits, which require slightly morecomplex guidance in use

 Caulcutt and Porter’s procedure is recommended when short- andmedium-term random variation is suspected, in which case the standardprocedure leads to over-control of the process

 SPC for variables is in three stages:

1 Examination of the ‘state of control’ of the process using X and R

charts,

2 A process capability study, comparing spread with specifications,

3 Process control using the charts

References

Bissell, A.F (1991) ‘Getting more from Control Chart Data – Part 1’, Total Quality Management,

Vol 2, No 1, pp 45–55.

Box, G.E.P., Hunter, W.G and Hunter, J.S (1978) Statistics for Experimenters, John Wiley &

Sons, New York, USA.

Caulcutt, R (1995) ‘The Rights and Wrongs of Control Charts’, Applied Statistics, Vol 44, No 3,

pp 279–88.

Caulcutt, R and Coates, J (1991) ‘Statistical Process Control with Chemical Batch Processes’,

Total Quality Management, Vol 2, No 2, pp 191–200.

Caulcutt, R and Porter, L.J (1992) ‘Control Chart Design – A review of standard practice’,

Quality and Reliability Engineering International, Vol 8, pp 113–122.

Duncan, A.J (1974) Quality Control and Industrial Statistics, 4th Edn, Richard D Irwin IL,

USA.

Grant, E.L and Leavenworth, R.W (1996) Statistical Quality Control, 7th Edn, McGraw-Hill,

New York, USA.

Owen, M (1993) SPC and Business Improvement, IFS Publications, Bedford, UK.

Pyzdek, T (1990) Pyzdek’s Guide to SPC, Vol 1 – Fundamentals, ASQC Quality Press,

Milwaukee WI, USA.

Shewhart, W.A (1931) Economic Control of Quality of Manufactured Product, Van Nostrand,

New York, USA.

Wheeler, D.J and Chambers, D.S (1992) Understanding Statistical Process Control, 2nd Edn,

SPC Press, Knoxville TN, USA.

(a) Design a decision rule whereby one can be fairly certain that the ballbearings constantly meet the requirements

(b) Show how to represent the decision rule graphically

(c) How could even better control of the process be maintained?

3 The following are measures of the impurity, iron, in a fine chemical which

is to be used in pharmaceutical products The data is given in parts permillion (ppm)

4 You are responsible for a small plant which manufactures and packsjollytots, a children’s sweet The average contents of each packet should

be 35 sugar-coated balls of candy which melt in your mouth

Every half-hour a random sample of five packets is taken, and the contentscounted These figures are shown below:

5 Plot the following data on mean and range charts and interpret the results.The sample size is four and the specification is 60.0 ± 2.0.

6 You are a Sales Representative of International Chemicals Your managerhas received the following letter of complaint from Perplexed Plastics,now one of your largest customers.

To: Sales Manager, International Chemicals

From: Senior Buyer, Perplexed Plastics

Subject: MFR Values of Polymax

We have been experiencing line feed problems recently which we suspectare due to high MFR values on your Polymax We believe about 30 percent of your product is out of specification

As agreed in our telephone conversation, I have extracted from our recordssome MFR values on approximately 60 recent lots As you can see, thevalues are generally on the high side It is vital that you take urgent action toreduce the MFR so that we can get our lines back to correct operating speed

Do you agree that their complaint is justified?

Discuss what action you are going to take

(See also Chapter 10, Discussion question 3)

7 You are a trader in foreign currencies The spot exchange rates of allcurrencies are available to you at all times The following data for onecurrency were collected at intervals of one minute for a total period of 100minutes, five consecutive results are shown as one sample.

8 The following data were obtained when measurements of the zincconcentration (measured as percentage of zinc sulphate on sodiumsulphate) were made in a viscose rayon spin-bath The mean and rangevalues of 20 samples of size 5 are given in the table.

(See also Chapter 10, Discussion question 4)

9 Conventional control charts are to be used on a process manufacturingsmall components with a specified length of 60 mm ± 1.5 mm Twoidentical machines are involved in making the components and processcapability studies carried out on them reveal the following data:

(See also Chapter 10, Discussion question 5)

10 The following table gives the average width in millimetres for each oftwenty samples of five panels used in the manufacture of a domesticappliance The range of each sample is also given.

± 5 mm, comment on the capability of the process

(See also Chapter 9, Discussion question 4 and Chapter 10, Discussionquestion 6)

Worked examples

1 Lathe operation

A component used as a part of a power transmission unit is manufacturedusing a lathe Twenty samples, each of five components, are taken at half-

hourly intervals For the most critical dimension, the process mean (X ) is

found to be 3.500 cm, with a normal distribution of the results about the mean,

and a mean sample range (R) of 0.0007 cm.

(a) Use this information to set up suitable control charts

(b) If the specified tolerance is 3.498 cm to 3.502 cm, what is your reaction?Would you consider any action necessary?

(See also Chapter 10, Worked example 1)

(c) The following table shows the operator’s results over the day Themeasurements were taken using a comparitor set to 3.500 cm and areshown in units of 0.001 cm The means and ranges have been added to theresults What is your interpretation of these results? Do you have anycomments on the process and/or the operator?

Record of results recorded from the lathe operation

Process mean X = 3.5000 cm

Mean sample range R = 0.0007 cm

Mean chart

From Appendix B for n = 5, A2= 0.58 and 2/3 A2= 0.39

Mean control chart is set up with:

Upper action limit X + A2R = 3.50041 cm

Upper warning limit X + 2/3 A2R = 3.50027 cm

Lower warning limit X – 2/3 A2R = 3.49973 cm

Lower action limit X – A2R = 3.49959 cm.

Range chart

From Appendix C D.999 = 0.16 D.975 = 0.37

D.025 = 1.81 D.001 = 2.34Range control chart is set up with:

Upper action limit D.001R = 0.0016 cm

Upper warning limit D.025R = 0.0013 cm

Lower warning limit D.975R = 0.0003 cm

Lower action limit D.999R = 0.0001 cm.

(b) The process is correctly centred so:

From Appendix B d n = 2.326

 = R/d n = 0.0007/2.326 = 0.0003 cm

The process is in statistical control and capable If mean and range chartsare used for its control, significant changes should be detected by the firstsample taken after the change No further immediate action is suggested.(c) The means and ranges of the results are given in the table above and areplotted on control charts in Figure 6.14