Quality - 1

QUALITY ON THE SHOP FLOOR

We must start from the premise that people care about the quality of the work produced.  A few people may “say” they are not interested but in practice only a very small minority actually do not care.  

The first is the length of the pen and the second the diameter.  To a certain degree it does not matter to the customer, how long the pen is, within reason.  It does matter what the diameter is because if it’s too small the cap will keep falling off and if it’s too big the cap will not fit.  Similarly the bore of the cap is important. From this we can see that some dimensions are more critical than others.  Design and Process Engineers will take a long hard look at each dimension and will assign a tolerance or latitude to them.  The degree of latitude usually indicates how important a dimension really is.  If there were little latitude, a tight tolerance, the dimension would probably be considered critical.  While a large tolerance normally indicates that it is not so important.  By this stage you may well be thinking, well that is OK but if I make something on a machine it will be the same all the time, so why worry.  Years ago, it was just this idea that caused so many problems.

Many people in industry failed to recognize that in every process there is a degree of variability.  Everybody knew that you had to make parts within a given tolerance for the part to be OK.  For example a machine may well have been cutting rods to length, say 6 inches long.  The person working the machine would have set it up and measured the length of the rod that had been cut.  Provided the measurement was within the tolerance the production run was started.  After a while the person would check another and find that it was either too big or too small, and so adjust the machine to correct the fault.  Later another check might reveal that the latest part measured was again out of specification.  Yet again the machine would be altered.  At the end of the normally long run the parts would be used in an assembly and quiet a few would not fit properly.  What had gone wrong?  Care had been taken at the start to set up the machine correctly.  Parts were checked at intervals and the machine was put right.  When bad parts were found, we looked back through the previous parts until we found the last one that was OK.  All the parts made since the last good part were scrapped.  This went on day in and day out, setters, inspectors and people working the machines used to argue all the time.

What was happening?  People would take one single part and measure it and if it were within the specification would say OK to run.  The machines made different sized parts.  They may, by chance, have selected a part that was close to the bottom limit of the tolerance but was in fact one of the largest of the parts the machine was making.  Most of the other parts the machine made were smaller and some of those were so small that they were unusable.  The result was despite taking great care many bad parts were made.  When we now look at processes we consider variability and apply Statistical Process Control (SPC).

STATISTICAL PROCESS CONTROL (SPC)

You can see that there are some very tall, some very small and the majority somewhere in between with a lot reasonably close to the average height.  If the maximum height was 2.0m and the minimum 1.0 m, we could estimate how many people would be between 1.4 m and 1.7 m, but we won’t for the moment.  All we need to know is there are ways of doing it.  What we do need to know is that the average height is at the centre of the curve and in this case it is 1.5 m (statistically we call this average, X-bar).  Extending each side of the centre line of the curve is distances measured in sigma.  If you measure one sigma each side of the centre line you will encapsulate 68% of the people.  If you measure two sigma each side of the centre line you will encapsulate 95% of the people and three sigma will give you 99.73% which is nearly all the people.  So “rough and dirty” we know that the group ranged from 1.0 m to 2.0 m and that 6 equal divisions of sigma span the range so one sigma is 0.167 m.  We will take a bit off because 6 sigma only covers 99.73% and so make sigma = 1.65 m.  Now we know that the average was 1.5 m so if we measure out one sigma either side of the centre we get 1.5 – 0.165 = 1.335 m on the shorter end and 1.5 + 0.165 = 1.665 m on the taller end.  So we now know that there are about 68% of the people ranging in height from 1.335 m to 1.665 m.  Similarly there are 95% of the people ranging from 1.17 m to 1.83 m (2 sigma distances either side of the centre line).  Finally there are 99.73% of the people between 1.005 m and 1.995 m.  This is not exact but it’s near enough for you and me.  Mr Stat, he’s the short one on the end, would disagree because he is the inspector and he knows a thing or two about maths.  He is always nit picking!  The advantage that Mr Stat has over us is that he knows if we take measurements of a relatively small group of “things” he can get a computer to work out the curve, the centre average and the sigma.  From that information he will be able to tell what the whole population will be like.  And we will think he is a genius.

Turning back to the rod cutting machine we would find that if we measured 500 rods and put the result into a graph we would get a pattern very similar to the bell shaped curve that resulted from measuring the height of the people.  Mr Stat would not need to measure all 500 rods because he could measure 50 and get an extremely good estimate of the spread of results that measuring 500 rods would give.  In this case Mr Stat would be conducting a machine capability study to see how much variation the cutting machine would produce.  He may well find that the computed variation is as much as 0.15 inches either way so the smallest rod would be 5.85 inches and the largest 6.15 inches.  If the drawing specification were 5.9 inches to 6.1 inches, some scrap parts would be made all the time despite all the care taken.  Today, the machine would be declared incapable of making parts to the tolerance set by Engineering, even though it could make quite a large number within the tolerance.  If the machine were found to be producing rods with far less variation, say making rods from 5.96 to 6.04 inches the machine would be declared capable.  Once we know that providing everything is OK, the machine is capable of producing all the parts within specification, more than half the quality battle has been won.  We now need to guard against things going wrong.  To do this we will not fall into the trap of measuring just one part every now and again, as we did in the past.  Instead we will measure 5 consecutively produced parts; say every 30 minutes or 60 minutes.  This is on-going SPC using average and range charts.

AVERAGE AND RANGE CHARTS

Most people on the shop floor will not be involved with machine capability studies but will need to use average and range charts (called X-bar & R charts).  The first aspect of X-bar and R charts is to appreciate why we take measurements and calculate an average and also a range.  The reason is these two factors, used together, accurately describe the measurement of the part.  Let’s look at a couple of examples.

You can see looking at sample 1 and 2 that the average of the five parts is the same for sample 1 as it is for sample 2.  But the range is much greater on sample 1 than sample 2.  So if we were checking parts coming off the machine and simply calculated the average measurement and compared that with the allowed part tolerance we would still not know enough to satisfy ourselves that everything was OK.  The average is important but so is the range.

Wow! This look complicated, but it’s not.  How do you eat an elephant? One bit at a time.  First let’s look at the Average Chart.  It consists of a series of point’s zig-zaging across the chart.  These are the averages of each sample of five parts that we measure each time we conduct a check of the process.  If we measured 5 parts and the results were 20; 18, 19; 20; and 18 the average would be 19.  We would plot a point on the chart corresponding to 19 at position 1.  At position 2 we would plot the average of the next sample of 5 parts that we measured 30 minutes later.  There are two red dotted lines one above and one below the zig-zag.  These are called the Upper and Lower Control Limits that Mr Stat calculated for us.  When we take a sample of 5 parts, calculate the average and it is either above the Upper Limit or below the Lower Limit, we must stop the job and take some sort of action.  In the middle of the zig-zag is a dark line, which is an average of some of the preceding averages that we have plotted.  Mr Stat uses that and other data to figure out where to put the red control limits on the chart.

CALCULATION OF CONTROL LIMITS

Before we look at the calculations lets look at a few of the symbols –

X-bar (usually written as an X with a minus sign above it) = average of a sample.

X-bar-bar (usually written as an X with an equals sign above it) = average of a number of averages.

R-bar (usually written as a R with a minus sign above it) = average of a number of ranges

UCLx = Upper Control Limit used on the average chart.

LCLx = Lower Control Limit used on the range chart.

UCLR = Upper Control Limit used on the range chart.  

Lets assume the results are as follows -

First we need to calculate the average of the averages and the average range-

X-bar-bar = (20+21+19+22+19+20+21) / 7 samples = 20.3 (average of the 7 averages)  

R-bar        = (4+2+2+4+5+3+2) / 7 samples               = 3.1 (average of the 7 range values)

Next we need to calculate the control limits

UCLx      = X-bar-bar + (A2 x R-bar)  =  20.3 + (0.577 x 3.1)  =  22.1

UCLx      = X-bar-bar - (A2 x R-bar)   =  20.3 -  (0.577 x 3.1)  =  18.5

UCLR      = D4 x R-bar                         =  2.114 x 3.1                =    6.6

            Where A2 = 0.577 and D4 = 2.114 for a sample size of 5 parts (obtained from a table of factors).

And finally we need to plot the control limits on the chart -

Note:-  LCLR not used on sample size of 5.

Formulae for calculating Control Limits

UCLx = X-bar-bar + A2R-bar   UCLR = D4R-bar

LCLx = X-bar-bar - A2R-bar    LCLR = D4R-bar

Well let’s leave Mr Stat to his work of calculating control limits, meanwhile you need to be able to recognize what is happening to the process as you monitor it.  In Statistical Instability you can take a look at different results and try to figure out what is wrong with the process.  

SAMPLE SIZE  

In the above example we used a sample size of 5 parts.  As you can see you can use any number of sample sizes but you only need to take enough readings to obtain the information to tell you how the process is running.  Little and often is the golden rule and 5 parts per sample is convenient.  Whatever sample size you use, try to use it throughout the factory.  But if a particular process is using more than 5 parts make sure the details are highlighted on the chart.  If you were monitoring a multi station process, injection moulding etc then there could be a case for using more than 5 parts.  If you had a 32-cavity gasket die for example you might wish to sample 8 parts each time.  The decision on what to set up should be made upon the degree of variability that you find when completing the machine capability study.