ATTRIBUTE CONTROL CHARTS

In many cases the data are not a result of measuring a continuous variable, but are the result of counting how often a specific event or attribute, e.g. a failure has occurred. For these circumstances another type of control chart has been developed—the attribute control charts.

There are four types of attribute control chart as shown in Table 7.8.

As shown in Table 7.8 the attribute control charts are classified into the four groups depending on what kind of measurements are being used. The simplest control charts to use are the np chart and the c chart because the number of non-conforming units or non-comformities are charted. It is a drawback, however, that the sample size must be constant for these simple charts. The charts to measure and analyse proportions are a little bit more difficult to use but both can adjust for varying sample sizes. Some further explanations will be given below.

The p chart is a control chart to analyse and control the proportion of failures or defects in subgroups or samples of size n. This control chart, as well as the np chart, is based on assumptions such as the binomial distribution.

The attribute being looked at must have two mutually exclusive outcomes and must be independent from one sampled unit to another. For example the unit being looked at must either be good or bad according to some quality specification or standard. The unit may be a tangible product or it may be a non-tangible product (event). For example the suggestions analysed in section 7.7.2 could be analysed by a p chart because each response time either conformed to the standard (13 days) or not. Another assumption from the binomial distribution is that the probability for the specified event, e.g. defect, is constant from sample to sample, i.e. variation around the average is random. This and the other assumptions are analysed and tested by using the control chart.

The np chart is a control chart to analyse and control the number of failures or defects in subgroups or samples of size n. As mentioned above the assumptions of the binomial distribution are the theoretical foundation of this chart.

Table 7.8 The four types of attribute control chart

Data Non-conforming units Non-conformities

Numbers np chart c chart

Proportion p chart u chart

The c chart is a control chart to analyse and control the number of non-conformities (defects, failures) with a constant sample size. The difference from the np chart is that for each unit inspected there are more than two mutually exclusive outcomes.

The sample space for the number of non-conformities for each inspected unit has no limits, i.e. the number of non-conformities (failures) may in theory vary from zero to infinity. The c chart as well as the u chart is based on the poisson distribution. As with the assumptions for the binomial distribution it is assumed that the probability for the specified event (e.g. defect) is constant, i.e. variation around the distribution average is random.

Complex products, e.g. cars, computers, TV sets etc., require the use of c charts or u charts. The same is the case with continuous products, e.g. cloth, paper, tubes etc. For random events occurring in fixed time intervals, e.g. the number of complaints within a month, the poisson distribution is also the correct distribution to apply and hence the control charts to apply should be the c chart or the u chart.

The u chart is a control chart to analyse and control the proportion of non-conformities (defects, failures) with a varying sample size. As with the c chart the theoretical foundation and hence the assumptions behind the u chart is the poisson distribution.

To construct the control charts use the following formulas:

(a) The p chart

For each sample (subgroup) the failure proportion (p) is calculated and charted in the control chart. The failure proportion is calculated as shown below:

(7.12) where:

NF=number of failures in the sample

n=sample size (number inspected in sub group)

Construction of control limits is done as follows:

(7.13)

(7.14)

where:

(7.15) and:

TNF=Total Number of Failures in all the samples inspected TNI=Total Number Inspected (the sum of all samples).

For varying sample sizes the control limits vary from sample to sample. If varying control limits may give problems to the users then plan for fixed sample sizes. For small variations (±20%) using the average sample size is recommended. The benefit of using the average sample size is that the control limits are constant from sample to sample.

(b) The np chart

For each sample the number of failures (np=the number of non-conforming units) is counted and charted in the control chart. Construction of the control limits is done as follows:

(7.16) (7.17)

(c) The c chart

For each sample (subgroup) the number of non-conformities (c) is counted and charted on the c chart.

Construction of the control limits is done as follows:

(7.18) (7.19) where:

(d) The u chart

For each sample the number of non-conformities (failures) is counted and measured relatively (u) to the number of units inspected and chartered in the control chart.

The average number of non-conformities per inspected unit is calculated as follows:

where:

TNc=total number of non-conformities (c) in all samples TNI=total number of units inspected

Construction of the control limits is done as follows:

As with the p chart the control limits vary if the sample size (=n) varies. Fundamentals of total quality management 100

(7.21)

(7.22)

(e) A case example

A company decided to choose the number of credit notes per week as a checkpoint because they realized that credit notes were a good indicator for customer dissatisfaction. The historical data for the last 20 weeks were collected and the data are shown in Table 7.9. The number of sales invoices per week was relatively constant in this period.

Table 7.9 The number of credit notes per week in 20 weeks

Week Number 1 2 2 0 3 11 4 5 5 3 6 4 7 4 8 1 9 3 10 7 11 1 12 1 13 3 14 0 15 2 16 10 17 3 18 5 19 3 20 4

Fig. 7.13 Control chart (c chart) for the

number of credit notes per week. Using the c chart the following control limits can be computed.

(7.23) (7.24) (7.25) The control chart is shown in Figure 7.13.

From Figure 7.13 we can see that the process is not in statistical control. The specific causes behind the out of control data in weeks 3 and 16 should be identified and controlled. Under these assumptions the control chart should be revised by deleting the data from these two weeks.

The revised control chart is shown in Figure 7.14 and as we see the process is now in statistical control. The control chart can now be used to control future credit note data.

Fig. 7.14 Revised control chart for the number

of credit notes per week.