## Statistical Process Control: Theory and practice

There is always a certain amount of variation in every process and if we only have common cause variation and we are trying to adjust for this variation we will actually cause more variation on the output. Look at the display of Landing positions at the right of the launcher. Each landing position is different and the variation may be due to common causes which are always present in this process.

We now have more information to get an idea how the process is performing. We can calculate the average of the shots which is Rounded. If we fire 45 more shots we will get a better estimate of the real average of the process. After 50 shots the process average is Rounded.

We can now move the launcher so future shots will center around Assuming that nothing changes in this process, future output should now be centered on At this stage we do not know very much about our process and we do not know if things are likely to change over time. We fire off another shots then we will create a control chart.

The landing position is a continuous scale. We are actually using complete numbers for the results , etc. This type of data is called variable data. We will use an Xbar and Range chart as the control chart for this process. In an Xbar and Range chart, the data is arranged into subgroups. The number in brackets means that the column is part of a subgroup.

For each row of the table, the data in these five columns represents one subgroup. Look at subgroup Look at the row with the number 25 in the grey column at the left. This subgroup starts at shot number The subgroup columns contain the results of shots , , , and An Xbar and range chart contains two graphs. For each row in the data table, the subgroup average is plotted Xbar along with the largest value in the subgroup minus the smallest value Range.

The point at which the launcher was moved is shown on the chart. This is the equivalent of a written note on a paper chart and this is the sort of information an operator should record on a control chart. Next we will calculate the control limits and draw control lines on the chart. The purpose of these lines is to show when we should suspect that something has changed which affects the process in other words, a special cause of variation has occurred. We calculate the control limits from a section of the data. Of course we know that the launcher has been moved and moving the launcher is a special cause of variation, so we should calculate the limits with results which come after the launcher move.

For now, we will not worry about how the calculations are made. This means that all the variation comes from common causes. Common cause variation is just the normal random variation which is inherent in the process. If we want to be in full control of a process we must use the charts to identify when special cause variation occurs, determine if things were better before or after the change, then make one of these situations permanent. We fire off another shots. Look at the control chart. All we need to know now is whether there has been any change in the process since the lines were calculated.

Is the output stable? Look to see if any of the points are outside the control limits. It looks as if something unusual happened around subgroup 40 launcher shot Subgroup average drops below the control line so a special cause of variation has occurred. As an operator, your job is to produce results as close as possible to but the average landing position has suddenly changed.

You could, of course re-centre the process move the launcher. This might help in the short term, but you have no idea whether things might suddenly change back to normal. The only really satisfactory solution is to carry out an investigation, find the source of the special cause of variation, learn from what happened, then make sure that this kind of change does not occur again. A word now about specification or tolerance limits.

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In most industrial processes, the operator is given specification or tolerance limits as well as the target value. World class quality does not come from treating everything within the specification limits as equally acceptable. We must try to produce as close as we can to the target value. This is what the customer really wants. In our bouncing ball process, an unknown special cause of variation made the subgroup average fall at around subgroup 40 shot It might be that the individual results are still within the specified tolerance limits, but our customer would prefer the results to be So we must make efforts to produce with the average output at and the minimum variation that our process is capable of.

So we must investigate and remove special causes of variation even if we are still producing within the specification or tolerance limits. When we did the investigations we found out that one batch of balls has slightly less bounce than normal. We discard this batch and demand from our supplier that they supply us with statistically stable product they can only be sure of doing this by using control charts. We have removed this special cause of variation so things should return to normal from shot We fire off another 50 shots. The control chart should show clearly that a change occurred around subgroup 40 and things returned to normal around subgroup Now we will look at how the control limits were calculated.

The mathematics are not difficult. You need to find the average Xbar average subgroup average and the average range for the section of the data that you use to calculate the limits. By distinguishing between special cause variation and common cause variation, control charts can help operators and managers to run processes which produce on-target with minimum variation.

If special cause variation is present, we must find the root cause and stop this from occurring again in the future. We ask the questions:. To reduce common cause variation we might need better machinery, more frequent maintenance or less common cause variation within raw materials. In this histogram, the measurements are the landing positions of the balls from the launcher simulation.

The possible landing positions are set out on the horizontal scale and this is divided into a number of sections. For each section, a column is drawn and the height of the column represents the number of balls which have fallen within that section of landing positions. Now we will fire more balls and look at the histogram contain more results. This is how the histogram looks after 60 shots. Lets put a still more results in the histogram. Here you see the histogram after shots. It should become clear after we have fired this many shots that the highest columns are near the middle of the histogram.

This means that most of the balls land near the middle of the range of possible results. It occurs frequently in nature and is common in industrial processes. We know a great deal about normal distributions and this helps us to make some general statements about the outcome of processes. We will now produce a normal distribution starting with a new set of shots. Notice that the histogram shape looks like a bell with a high middle and tails at each end. This box shows some figures for the data which is used to make the histogram.

The Average is calculated in the normal way from the individual results: The individual results are added up then divided by the number of results. Standard Deviation gives us a figure for how much the individual values in a set of measurements are spread around the Average. A set of measurements where most of the values are near the Average has a low Standard Deviation, a set of measurements where most of the values are far away from the Average has a high Standard Deviation.

You can create and use control charts without knowing how to calculate Standard Deviation. For those who want to know, here is how to calculate a Standard Deviation:. First, find the Average of all the values For every individual value, find the distance from the Average. Square this number multiply it by itself. Add all the squares together Divide by the number of measurements minus one. Find the square root of this number. I repeat that you do not have to remember how to calculate Standard Deviation to draw control charts or to use the charts to improve processes.

All you need to know is that Standard Deviation is a measurement of spread. For Normal distributions, we can use Standard Deviation to make some useful statements about a set of measurements. We can also make some predictions about future measurements from the same process if it is reasonable to assume that the process will not change it will only be reasonable to make this assumption if the process is stable. If past measurements show a normal distribution, and the process is stable, then we can say :.

The green zone is up to 1 Standard Deviation either side of the Average. The yellow zone is more than 1 but less than 2 Standard Deviations from Average The purple zone is more than 2 but less than 3 Standard Deviations from Average The red zone is more than 3 Standard Deviations from Average The blue lines show the specification limits for the launcher simulation. If we drag the specifications to put them at the border of yellow and purple the percentage OK is recalculated. The percentage OK figure now shows how many of the results are within 2 Standard Deviations from Average. You see it is So, we now know that for a normal distribution, the majority of results will be less than one Standard Deviation from Average.

However, we also know that there will be a small number of results more than 3 times Standard Deviation from Average. We cannot tell WHEN these extreme results will happen, but we know that they will happen sometime. These percentages tell us approximately what has happened in the past. Also, the percentages given above for the normal distribution are only true over the very long term. It would be wrong to suggest that we can tell with confidence what the measurements will be in any one particular batch of goods.

In lesson 2, we used a section of data to calculate control limits for a process. In that example, the process was stable in the early stages and we used data from that period to calculate the control limits.

In this lesson we are going to investigate what happens if the process is unstable while producing the data which is used to calculate control limits. We ran a simulation of a process which is unstable in the early stages. You now see that the process is indicating instability even though the data used to calculate the control limits contains instability. This ability of Shewhart control charts to detect special causes of variation, even when these special causes are present in the data used to calculate the control limits, is very important.

Most industrial processes are not naturally in a state of statistical control. The control limits are set at 3 times sigma from the average. Sigma is similar to standard deviation. This sigma is calculated based on the average range of the subgroups range is the maximum value minus the minimum value or in other words based on the within subgroup variation. The reason that the Xbar chart detects special variation is because the control limits are calculated using an estimate of standard deviation based on the average subgroup range. Since the subgroups are taken from consecutive products, this means that all the variation between subgroups is filtered out.

When using control charts it is important to ensure that subgroups contain mostly common cause variation. Normally this can be done by measuring a small number of consecutive products for each subgroup, and having a time gap between the subgroups. Sometimes it is not possible to take consecutive measurements from a process which can be grouped in a subgroup. For example, there is no variation if we take consecutive measurements eg temperature or the pH value of a bath.

In this type of chart we plot the individual measurements on one graph and the differences between the consecutive measurements on the other graph, this is called the Moving Range sometimes only the individual points are shown, the moving range chart is omitted. If we run a process with is unstable in the early stages and we chart the individual values we see the control chart below.

We see that the chart is able to detect both disturbances in the average as well as disturbances in the range. The chart shows some instability, both by having some points outside the control limits and because there are long runs in the data. A run is where a number of consecutive results are all above average or all below average. Lets look at how control limits for individual value chart are calculated:. During an implementation we will also implement control charts where removing instability is not the highest priority because it is not the most critical characteristic.

In that case we may use different ways to calculate limits. This advanced subject is outside the scope of this training. Lesson 5 — Binomial control charts. In lesson 2 we looked at Xbar and Range control charts. In lesson 4 the X individual value chart was introduced. In both these cases, we used variable or measurement data. This is data which comes from a continuous scale. Attribute data comes from discrete counts. For example:. With attribute type data, in order to choose the correct type of control chart, we have to look at the way the data was generated.

If we know in advance that the set of data will exhibit the characteristics of Binomial data or Poisson data then these types of charts should be used. Binomial data is where individual items are inspected and each item either possesses the attribute in question or it does not. Each bead scooped is either blue or it is not blue — so if we create a stream of samples taken from the box and we count the number of blue beads in the samples, then we can assume that the resulting data will be Binomial type data.

The random variation of Binomial data acts in a particular way, because of this we can calculate where to put the control limits. All we need to know is the average of the data set and the sample size. It is used when we know we have Binomial data and the sample size does not change. If we have binomial data but the sample size is not constant, then we cannot use a np chart. We will now use the simulation to add new samples to the data we have already started, but we will change the sample size:.

When the sample size is not constant for every scoop we have to convert counts to a rate or proportion. We convert to a rate by dividing the attribute count by the sample size. You will notice that there is a step in the control limit lines at the point where the sample size changed. The purpose of the control limits is to show the maximum and minimum values that we can put down to random common cause variation. Any points outside the limits indicate that something else has probably occurred to cause the result to be further from the average. As we have said before, the random common cause variation of Binomial data acts in a particular way.

The variation with large sample sizes is smaller than the variation with small sample sizes. We can use the simulation to demonstrate this. Look at the results in the Data Table and keep in mind that the proportion of red beads in the box has not changed. In rare cases like in this simulation we can even have 4 and we have a false alarm. Look again at the results in the Data Table. Look at the way the points which correspond to the small sample size samples 60 — 90 vary up and down, then compare this with the variation with the large sample size after Keep in mind that we are not looking at absolute numbers here, we are looking the proportion of the sample which is red.

Look at the position of the control limits for the small subgroupsize and the large subgroupsize. This illustrates one of the basic points about using control charts for attributes. Small subgroupsizes produce control charts which are not sensitive because there is so much random common cause variation in small sample sizes. Large sample sizes produce more sensitive control charts. What this means is that if a process has a special cause of variation acting on it from time to time, it may not produce any points outside the control limits if the sample size is small.

The same special cause of variation is more likely to produce points outside the control limits if we use a large sample size. Notice that the limits have to be separately calculated for each subgroupsize. The example given is for sample number 1 subgroups 1 to Criteria for binomial data:. We can only use an np chart or a p chart if we know in advance that the data produced will be binomial data.

The full conditions which have to be satisfied before we can consider a set of data to be Binomial are:. For example, we might want to count the number of blemishes on a surface. The only difference is the way the control limits are calculated: Look at how the limits are calculated. Notice that the sample size is not used anywhere in these calculations.

The rate is simply the attribute count divided by the sample size or area of opportunity for the sample. Look at how the limits are calculated.

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Notice that the control limits are tighter for larger areas of opportunity. Statistical Process Control : Theory and practice. Barrie Wetherill , Don W. Statistical process control SPC is now recognized as having a very important role to play in modern industry. If we cannot be confident that the data we have fulfills the conditions to be binomial or Poisson data, then we can usually rely on an X chart to do a pretty good job. We now have a non constant sample size. Sometimes X charts should be rate charts when the sample size is not constant and sometimes they should not — it depends on what the measurement represents.

In our case the number of red beads scooped is definitely dependent on the sample size so we should look at an X chart based on rates. The p chart and the X rate chart are both showing proportions and the control limits have been calculated using scoops 1 — Compare the two charts. Look at the data and the control limits before and after the change of sample size the change was at subgroup number Because we have not changed the number of beads in the box, we are looking at the results of a stable process so in theory control charts should not show any points outside the control limits.

## Educational web tool for Statistical Process Control - IEEE Conference Publication

There is always more random common cause variation with small sample sizes and you can see that the points on both charts jump up and down more after we change to a smaller sample size. Because the control limits on a binomial chart are based on a theoretical knowledge of the way binomial data behave, the control limits change to accommodate the different sample sizes.

On X charts, the control limits are based on the variation between successive points in the data stream. When this variation changes due to altering the sample size, this can be misinterpreted as a process change. X charts with low average:. When the average count is very small, another problem prevents us from using X charts. With attribute counts, the data can only take integer values such as 6, 12, 8 etc. Values such as 1. The discreteness of the values is not a problem when the average is large, but when the average is small less than 1 then the only values which are likely to appear are 0, 1, 2 and occasionally 3.

The whole idea of control charts is that we want to gain insight into the physical variations which are happening in a process by looking at the variation of some measurement at the output of the process. When the measurements are constrained to a few discrete values then the results are not likely to reflect subtle physical changes within the process. For this reason X charts should not be used for attribute counts when the average count is low. Lesson 9 gives more information about using attribute control charts when the average count is low.

Lesson 7 — Pareto chart. A Pareto chart helps us to identify priorities for tackling problems. The columns in the table represent 10 types of non-conformity or imperfection which can occur in Assembly M Each of the 25 rows contains the results of one inspection. In a Pareto chart, the categories of data are shown as columns and the height of each column represents the total from all the samples.

The order of the columns is arranged so that the largest is shown on the left, the second largest next and so on. Since these counts usually represent defects or non-conformities, the biggest problems are therefore the categories on the left of the chart. Our Pareto chart makes it immediately obvious that the most frequent problem is Smidgers appearing on the assembly.

It is common practice on Pareto charts to superimpose a cumulative percentage curve. At each point on this curve you can see the percentage of the overall number of non-conformities or imperfections which are caused by the categories to the left of the point. This is best illustrated by example. We have added a red line for the second defect Scrim pitted. If we concentrate our efforts on reducing the number of smidgers and pitted scrims, then even if we are only partially successful, we are likely to make a substantial difference to the number of assembles which we have to send for rework.

Of course, all types of problems do not have an equal impact in terms of cost or importance. So if we know the cost of putting right each type of problem, then it is better to draw the Pareto chart with the column heights representing the total cost. We are now looking at the total cost associated with each column total number multiplied by unit cost. When the chart is showing costs, we get a different picture from the picture we get when it is showing numbers Although smidgers are the most common imperfection in assembly M, they are easy to remove — a quick wipe with a cloth is all that is needed.

A pitted scrim, on the other hand, needs the assembly to be dismantled. A leaking gear housing is the big nightmare — but fortunately they are not very common. Although we do not get many gear leaks, they are actually the second biggest problem in terms of costs. Smidgers do not cost the company a lot of money despite the fact that they occur in large numbers. So looking at Pareto chart for M costs, we should concentrate our efforts in eliminating pitted scrims and gear leaks. Although a Pareto clearly identifies the major cause of problems you also have to consider the amount of efforts required to solve a problem.

It might be that an issue is very easy to fix, so make sure you always briefly review all issues before you start to solve the most important problems. Pareto are used in a lot of different situations and can be adapted to get the right information. This Pareto is shown with the bars horizontally. With downtime analysis the total downtime is important but more information is required. In this Pareto we see that a label is added with the total downtime in minutes but also the number of downtimes is given.

One long downtime might require a different approach than a large number of short downtimes. In this Pareto the color of the bar indicates a downtime category. Lesson 7 summary:. Return to the index Lesson 8 — Scatter chart. When we want to reduce or eliminate a problem, we will need to come up with ideas or theories about what is causing the problem.

One way to check if a theory should be taken seriously is to use a scatter chart, also called regression analysis. To use a scatter chart, we first have to take a series of measurements of two things over a period of time. The two things that we would measure are the problem itself, and the thing that we think may be causing the problem.

We then plot the measurements on a scatter chart. The scatter chart will help us to see whether there is a mathematical relationship between two sets of measurements. Analysis using a Pareto chart showed that the problem of surface flaking of the plugs was costing the company a lot of money. The team quickly found that everyone had a different opinion of what was OK and what was a flaker.

The first job, therefore, was to come up with a good definition of a flaker which everyone could use. The process operators were shown how to use control charts and they started keeping a chart of the number of flakers produced in each batch. This chart showed that the process was unstable.

Mary, one of the process operators on the team, said she always feels cold on days that they have a lot of flakers. The process operators started keeping records of the air temperature at the time the plugs were made.

Statistical Quality Control - 4 - Imp Exam Problem - Range Chart from Samples

At one of the team meetings Jack pointed out that on at least two occasions when the number of flakers was outside the control limits, it was raining. The team asked the lab for help to test the theory that rain was a factor. One of the engineers pointed out that it was actually raining that very day but there were very few flakers. Nevertheless he still suggested that it might be a good idea to measure the moisture content of the main ingredient. Because each plug is either a flaker or it is not a flaker, the chart we should use is a binomial chart. The data is out of control because some points are outside the control limits.

There are also runs of 10 consecutive points above and below the average line — these also indicates instability. On this chart, the number of flakers is on the vertical axis and the air temperature is on the horizontal axis. For each row in the data table, a dot is put where the two values meet. In a scatter chart, if the measurements on the horizontal axis are not related in any way to the measurements on the vertical axis, then the dots will appear at random, with no pattern visible. If there is a mathematical relationship between them then the dots will tend to group into a fuzzy line or curve. In this case there does not seem to be any pattern to the points on the scatter chart. We can conclude, therefore, that there is no correlation between air temperature and the number of flakers produced.

This means that we can say that the air temperature is not a factor in producing flakers. On this scatter chart we see flakers on the vertical axis and moisture content on horizontal axis. There appears to be a correlation between the two sets of numbers because we can see the dots have formed into a fuzzy line. This chart is showing that flakers increase when the moisture content increases. This still does not prove that one causes the other.

There could be a third factor which causes BOTH to change at the same time. Still, we seem to have a clue here. The equation for this line is shown at the top right of the chart. The R-squared figure is a measure of how well the data fits the line. If R-squared is 0 or near 0 then there is no correlation between the data on the two axes so the line and the equation has no relevance. Now look again at the scatter of the temperature. You can now see the best fit line through these points.

The R-squared value is low showing that there is no correlation between the two sets of data. A few remarks are important when using scatter charts. When looking at scatter charts it might be important to include all other relevant information. It might be important to look simultaneously at control charts, scatter charts and data table to get a better understanding what is exactly going on.

This analysis is beyond the scope of this training. Another important aspect of a scatter analysis is that the results are strongly influenced by an outlier. If we look at the scatter chart with temperature and add an outlier 18 flakers with 35 degrees we get the following result:. You see that one outlier is drastically changing the R-squared value. So always look at the chart and ask yourself what is happening exactly. Lesson 8 summary:. Return to the index Lesson 9 — Attribute control charts with low average.

We are now going to look at a particular problem you can encounter with attribute control charts. We will generate four streams of data from a process and create a special cause of variation in each data stream. For all four colours, the number of beads in the box doubles after shot 20, therefore we would expect to see a clear signal that the special type of variation has occurred.

The control limits are calculated using all the data so scoops should give results below average and scoops 21 — 40 should give results above average. Look at the chart for red beads. The chart is showing points outside the control limits and there are long runs below and above average — so the special variation is clearly visible on this chart. Now look at the charts for Green, Yellow and Blue beads. The special cause of variation is not so obvious on these charts, especially on the chart for blue beads. Now we will look at the position of the upper control limit for each chart: Look at the average Avg.

Look at the Upper control limit UCL figure for red beads chart. The upper control limit is probably about 1. Look at the Average and Upper control limit values for the other three charts. Work out approximately how many times greater the UCL is than the Average. As the average figure gets lower, the upper control limit gets further from the average.

In the Blue bead chart, the Upper control limit is many times greater than the average. The only thing that is different between the four charts is the average number of beads scooped. This demonstration shows one of the inherent problems with control charts for attributes. If the average of the samples is low, then attribute control chart are not sensitive at detecting special variation.

Because we usually count problems or failures, this means that as we get more successful at removing problems, the charts become less good at separating special cause variation from common cause variation. For example, you are trying to produce a product or service within a given specification of time, weight or length, then make control charts from the time, weight or length measurement. This will indicate special variation much better than an attribute chart showing the number of out-of-specification products.

With extreme low defect percentages you can also use the Cumulative Count Control chart which uses an exponential scale but this goes beyond the scope of this training. An important part of any SPC implementation is the use of process capability indices. In this lesson we explain the most common used indices Cp, Cpk, Pp and Ppk.

There is some confusion about the use of these indices. In this lesson we will try and remove some of the confusion and explain the differences between the indices and how they can be used in a practical way. This lesson is not following the tutorial like the other lessons. The tutorial lesson is summarized in a video at the end of this lesson. First we will provide the definitions of the indices and give some historic insight in the development of these indices which will explain some of the confusion.

What is important to know before we will explain the definitions of the indices is that the definitions in the past have changed. Ppk was defined under the Q system of Ford as the preliminary capability index and the Cpk was defined as the long term capability index. In some cases the Cpk value on the histogram was calculated differently from the Cpk calculations on the control chart.

When the big three Ford , GM and Chrysler merged their quality manuals into the QS system the definitions where changed and these definitions are still the standard nowadays in the TS manual and will be used and explained in this lesson. Cp sometimes also named Cpi stands for the capability index of the process. The formula for the calculation is:. The refers to the estimated standard deviation. The estimated standard deviation is calculated using the following formula:. In normal words it means the Cp index is calculated based on the within subgroup variation. So if the variation within the subgroup is very small you will have a good Cp index no matter how much the process average is drifting or what the location of the process is so the Cp index shows you how capable your machine is to produce consecutive products within the required variation Tolerance.

If we now report both Cp and Cpk index we know how capable the process is to produce within the required variation tolerance and if the process is producing in the middle of the tolerance. Is the information about Cp and Cpk enough to indicate if the process is running within specifications. The answer is no because these 2 indices are calculated based on the within subgroup variation and it is still possible there is a large amount of between subgroup variation which is not taken into account.

## Statistical Process Control (SPC)

Let us try to explain this with an example. The chart shows we had a lot of variation between subgroups Xbar chart but the variation with the subgroup was much better in control Range chart. The Cp index for this process is 1. We see that these 2 indices are not enough and we need more information to know if the process is producing within specification limits. If we only use Cp and Cpk we need to add the requirement that the process must be in control. If the average chart is in control it indicates the process is stable and the process average is not fluctuating.

In that case we could indicate the percentage of subgroups out of control but there is also another possibility. The Pp index is calculated in the same way as the Cp index but now using the real standard deviation instead of the estimated standard deviation. So the formula is:. So the Pp index uses both within subgroup variation and between subgroup variation in the calculation and indicates how well the process was capable to produce within specification limits over the reported time period. Cp indicates how well a process is capable to produce consecutive products within the required variation.

The difference between Cp and Cpk indicates if the process is producing in the middle of the tolerance. The difference between Cpk and Ppk indicates if the process is stable or in other words if there are special causes of variation which are influencing the average of the process even if control limits are not properly set. If the Ppk value is below 1. Let me explain with an example:. Process 1: Unstable, Long term not capable, Short term capable , On target This process is out of control and has assignable causes. There is more between subgroup variation than within subgroup variation.

Process 2: Stable, Short term capable, Long term capable, Not on target This process has a wrong process setting and if the process is brought on target the Ppk will be acceptable. Process 3: Stable, Short term not capable, Long term not capable, On target This process is not capable to produce consecutive products within the allowed tolerance so this process needs to be altered.

### Introduction and Background

There is also a tutorial for this lesson but because interactivity is required we have a recording of this sessions. Please view process capability video training. For a training about process capability indices click on training. The information on this website and the free training is offered by DataLyzer International.

When implementing SPC it is important that people are properly trained. To improve the quality of training and to reduce the cost of training DataLyzer International has developed a training module using process simulations. If you like what you see please link to this site and refer the site to others. Please send your feedback to mschaeffers datalyzer. Statistical Process Control Dear visitor, this site aims at informing you about statistical process control and also offers you a full SPC training.

Create constancy of purpose toward improvement of product and service with the aim to become competitive, stay in business and provide jobs. Adopt the new philosophy. We are in a new economic age, created by Japan. A transformation of Western style of management is necessary to halt the continued decline of industry.