7QC Tools: Why do we Require to Plot X-bar and R-charts Simultaneously

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Abstract:

The main purpose of the control charts is to monitor the health of the process and this is done by monitoring both, accuracy and the precision of the process. The control charts is a tool that helps us in doing so by plotting following two control charts simultaneously for accuracy and precision.
Control chart for mean (for accuracy of the process)
Control chart of variability (for Precision of the process)
E.g. X-bar  and R chart (also called averages and range chart) and X-bar  and s chart

The Accuracy and the precision

We all must be aware of the following diagram that explains the concept of precision and accuracy in that analytical development.

Case-1:

If you are hitting the target all the time at the bull’s eye is called as  accuracy and if all your shots are concentrated at the same point then it is called as Precision.

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Figure-1: Accuracy and precision

Case-2:

You are off the target (inaccurate) all the time but your shots are concentrated at the same point i.e. there is not much variation (Precision)

Case-2:

It is an interesting case. Your shots are scattered around the bull’s eye but, on an average your shots are on the target (Accuracy), this is because of the average effect. But your shots are wide spread around the center (Imprecision).

Case-4

In this case all your shots are off target and precision is also lost.

Before we could correlate the above concept with the manufacturing process, we must have a look at the following diagram that explains the characteristics of a given manufacturing process.

Figure-2: Precision and Accuracy of a manufacturing process

Figure-2: Precision and Accuracy of a manufacturing process

The distance between the average of the process control limits and the target value (average of the specification limits) represents the accuracy of the process or how much the process mean is deviating from the target value.

Whereas the spread of the process i.e. the difference between LCL and UCL of the process represents the precision of the process or how much variation is there in the process.

Having understood the above two diagrams, it would be interesting to visualize the control chart patterns in all of the four cases discussed above. But, before that let’s have a look at the effect of time on a given process i.e. what happens to the process with respect to the time?

As the process continue to run, there will be wear and tear of machines, change of operators etc. and because of that there will be shift and drift in the process as represented by four scenarios described in the following diagram.

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Figure-3: Process behavior in a long run

A shift in the process mean from the target value is the loss of accuracy and change in the process control limits is the loss of precision. A process shift of ±1.5σ is acceptable in the long run.

If we combine figure-1 and figure-3, we get the figure-4, which enable us to comprehend the control charts in a much better way. This gives picture of the manufacturing process in the form of control charts in four scenarios discussed above.

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Figure-4: Control chart pattern in case of precision and accuracy issue

Above discussion is useful in understanding the reasons behind the importance of the control charts.

  1. Most processes don’t run under statistical control for long time. There are drifts and shift in the process with respect to the time, hence process needs adjustment at regular interval.
  2. Process deviation is caused by assignable and common factors/causes. Hence a monitoring tool is required to identify the assignable causes. This tool is called as control charts
  3. These control charts helps in determining whether the abnormality in the process is due to assignable causes or due to common causes
  4. It enables timely detection of abnormality prompt us to take timely corrective action
  5. It provides an online test of hypothesis that the process is under control
    1. Helps in taking decision whether to interfere with process or not.
      1. H0: Process is under control (common causes)
      2. Ha: Process is out of control (assignable causes)

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6.  Helps in continuous improvement:

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Figure-5: Control Charts provide an opportunity for continuous improvement

 

 

7QC Tools: Basis of Western Electric Rules of Control Charts

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We all are aware of these famous rule, for beginners let’s understand the basis of these rule. All rules are applied to one half of the control chart. The probability of getting a reaction to the test is ~0.01.

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  1. A single point outside 3σ control limits or beyond zone A.
    • Probability of finding a point in this region = 0.00135 if caused by the normal process. Anything in this region is a case of assignable cause.
  2. Two out of three consecutive points in zone A (beyond 2σ).
    • Probability of getting 2 consecutive points in zone A = 0.0227*0.0227 = 0.00052
    • Probability of 2 out of 3 points in zone A = 0.0227*0.0227*0.9773*3 = 0.0015
  3. Four out of 5 consecutive points in zone B (beyond 1σ)
    • Probability of getting one points in zone B = 0.1587
    • Probability of 4 points in zone B and 1 point in other part of the control chart = 0.1587*0.1587*0.1587*0.1587*0.8413*5 = 0.0027
  4. Eight consecutive points on one side of the central line.
    • Probability of getting one points in beyond central line = 0.5
    • Probability of 8 points in in succession on one side of the central line = 8*0.5 = 0.0039

7QC Tools: Case Study on Interpreting the Control Charts

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A process was running in a chemical plant. The final stage of the process was the crystallization, which gave the pure product. There were two crystallizer used for the purpose, each operated by a different individual. The SOP says that crystallizer has to be maintained between 30-40°C and for 110 to 140 minutes. The data for a month is captured below

picture109 In order to understand the process, I-MR control chart was plotted (for simplicity, R-chart is not captured).

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As we have learned from the earlier blog, the alternate points above and below the central line represents some short of stratification (see the short connecting arms and the concentration of data points in zone B and C).

We plotted the histogram of the above data set and kept on increasing the number of classes. What we saw was the emergence of a bimodal distribution as we kept on increasing the number of classes.

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So, one thing was sure, there were two processes running in the plant. Now question that was to be answered was “What is causing this stratification?”

We started with crystallizer, as soon as we plotted the simple run chart of the process with groups using Minitab®, we could see the difference. Crystallizer-2 was always giving better yield. This should not happen because both the crystallizer were identical and were connected to same utilities. Then we thought about the different operators might be the reason for this behavior, as this was the only factor that was different for both the crystallizer.

picture114When we plotted the same run chart with grouping, but this time operator was used for the purpose of grouping. We got the same result as was found with the crystallizers, the operator-2 working on the crystallizer-2 was producing more quantity of the product. This run chart is not shown here.

We further grilled down to the operating procedure adopted by the two operators. We studied temperature and the maintenance time using scatter plot. The results are shown below

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Finally, it was found that operator-2 was maintaining the crystallizer-2 at the lower end of the prescribed temperature and for longer duration. Hence, specification for temperature and the maintenance time was revised.

7QC Tools: Interpretation of Control Charts Made Easy

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Visual Inspection of the Control Charts for Unnatural Patterns

Besides above famous rules, there are patterns on the control charts that needs to be understood by every quality professionals. Let’s understand these patterns using following examples. It would be easier to understand them if we can imagine the type of distribution of the data displayed on the control chart.

Case-1: Non-overlapping distribution

As a production-in-charge, I am using two different grades of raw material with different quality attributes (non-overlapping but at the edge of the specification limits) and I am assuming that the quality attributes of the final product will be normally distributed i.e. I am assuming that most of final product will hit the center of the process control limits.

If the quality of the raw material is detrimental to the quality of the final product then my assumption about the output is wrong. Because the distribution of the final product quality would take a bimodal shape with only few data at the junction of the distribution. Same information would be reflected onto the control chart with high concentration of data points near the control limits and fewer or no points near the center. Here is the control chart of the final product

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In this completely non-overlapping distribution, there will be unusual long connecting arms in the control charts. There will be absence of points near the central line.

If we plot the histogram of this data set and go on increasing the number of classes, the two distribution would get separated.

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So, whenever we see a control charts with the data points concentrated towards the control limits and no points at the center of the control charts, immediately we should assume that it is a mixture of two non-overlapping distribution. Remember long connecting arms and few data points at the center of the control chart.

Case-2: Partially overlapping distribution

Assume this scenario: A product is being produced in my facility in two shifts by two different operators. Each day I have two batches, one in each shift. There is a well written batch manufacturing record indicating that the temperature of the reactor should be between 50 to 60 °C. The control chart of a quality attribute of the product is represented by following control chart.

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We can see that the data points on the control chart are arranged in an alternate fashion around the central line. The first batch (from the 1st shift) is below the central line and next batch (from the 2nd shift) is above the central line. This control chart shows that even we are following the same manufacturing process, there is a slight difference in the process. It was found that the 1st shift in-charge was operating towards 50 °C and the 2nd shift in-charge was operating towards 60 °C. This type of alternate arrangement is indication of stratification (due to operators, machines etc.) and is characterized by short connecting arms.

There are the cases of partially overlapping distribution resulting in a bimodal distribution, which means that there will be few points in the central region of the control charts but, majority of the data points would be distributed in zone C or B. In such cases, it would be appropriate to plot the histogram with groups (like operator, shift etc).

Case-3: Significant Overlapping distribution

If there is significant overlap between the two input distributions then it would be difficult to differentiate them in the final product and the combined distribution would give a picture of a single normal distribution. Suppose the operators in the above case-2 were performing the activity at 55 °C and 60 °C respectively. This would result in an overlapping distribution as shown below

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Case-4: Mixture of unequal proportion

As a shift-in-charge, I am running short of the production target. What I did to meet the production target was to mix the current batch with some of the material produced earlier for some other customer with slightly different specification. I hoped that it wouldn’t be caught by the QA!. The final control chart of the process looked like

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We can see from the control chart that if two distributions are mixed in an unequal proportions then the combined distribution would be an unsymmetrical distribution. In this case one-half of the control chart (in present case the lower half) would have maximum data points and other half would have less data points.

Case-5: Cyclic trends

If one observe a repetition of the trend on the control chart, then there is a cyclic effect like sales per month of the year. Sales in some of the specific months are higher than the sales in some other months.

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Case-6: Gradual shift in the trend

A gradual change in the process is indicated by the change in the location of the data points on the control charts. This chart is most commonly encountered during the continuous improvement programs when we compare the process performance before and after the improvement program.

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If it is observed that this shift is gradual on the control charts, then there must be a reason for the same, like wear and tear of machine, problem with the calibration of the gauges etc.

Case-7: Trend

If one observe that the data points on the control charts are gradually moving up or down, then it is a case of trend. This is usually cause by gradual shift in the operating procedure due to wear and tear of machines, gauges going out of calibration etc.

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Summary of unnatural pattern on the control charts
Unnatural pattern Pattern Description Symptom in control chart
Large shift (strays, freaks) Sudden and high change Points near and or beyond control limits
Smaller sustained shift Sustained smaller change Series of points on the same side of the central line
Trends A continuous changes in one direction Steadily increasing or decreasing run of points
Stratification Small differences between values in a long run, absence of points near the control limits A long run of points near the central line on the both sides
Mixture Saw-tooth effect, absence of points near the central line A run of consecutive points on both sides of central line, all far from the central line
Systematic Variation or stratification Regular alternation of high and low values A long run of consecutive points alternating up and down
Cycle Recurring periodic

movement

Cyclic recurring patterns of points

For the case study see next blog