CONTROL CHARTS FOR NONCONFORMITIES WHEN UNIT INTERVAL VARIES: U CHARTS

The c chart assumes that the inspection unit (area, time or volume) from period to period is constant in size. However, in many applications of the c chart, the inspection unit can vary from period to period. In this case, we use a U chart to monitor the number of nonconformities. The inspection unit is often chosen for operational or data-collection simplicity. One approach used to set up a control chart in this case is based on the average number of nonconformities per inspection unit. That is, we calculate [pic], where c is the total nonconformities in a sample of n inspection units.

EXAMPLE: In the fabrication of printed circuit boards, let us say we choose as our basic sample size (or inspection unit) n = 100 boards. However, we may produce different lots of board from one production run to the next. From example,

Day 1: We make 100 boards, Day 2: We make 250 boards, Day 3: We make 50 boards and so on. In terms of our chosen inspection unit of n = 100 boards, we can express these daily productions as

Day 1: n1 = 1 unit; Day 2: n2 = 2.5 (inspection) units; Day 3: n3 = 0.5 units, etc.

In general, let ni = number of inspection units for sample (or subgroup) i, i = 1,2, …, n and ci = number of nonconformities found in subgroup i, then the average number of nonconformities per unit for sample i is ui = ci/ni. Note ni doesn’t have to be a whole number as shown in the above example.

The center line for the U chart is estimated by:

[pic]
[pic]estimates the average nonconformities per unit.

As in p charts, there are three different ways to set up the U chart control limits, each has its pros and cons.

1. Variable control limits. That is, the control limits depend on the number of inspection units ni.
[pic]
The major disadvantage of this chart is that we cannot apply the run rules for this chart. This will lead to too many false alarms.

2. The other method is to calculate the control limits based on average inspection size, where [pic]. The control limits are:
[pic]
3. The third method is to use a standardized control chart (this is the preferred option). This alternative involves plotting a standardized statistic zi calculated as follows:
[pic]

The control chart is UCL = +3, CL = 0 and LCL = -3. We plot the zi values on this chart. This method is preferred because we can apply the sensitizing (run or pattern) rules safely in this case.

EXAMPLE: LASERBOND: A manufacturer of Paper (see handout given in class).

Note: Here the inspection unit is defined as one roll of paper. a. We compute [pic]:

[pic] The ui values are simply obtained by dividing ci by ni: ui = ci/ni. The u-chart limits are:

[pic] [pic]

For example, for the first subgroup, the control limits are: [pic] Now u1 = 10/52 = 5.2. So u1 falls within its corresponding limits and, therefore, is in statistical control. Doing the other points and looking at the control chart (see handout given in class), we see that the process appears in control with no indications of special causes.

b. Because the sample size (number of rolls) is variable, we can’t simply project horizontal limits into the future. But we “extend” (use the same) formulas into the future.

c. We now need to compute the average inspection (interval) size: [pic] We found that [pic] Therefore:

[pic] Looking at the control chart (see Handout), the process appears in control with no indications of special causes. When this chart is used, for points that plot close to the control limits, we may need to calculate the exact limits for these points to see if they are in control or not.

d. To compute the standardized values, we use the following formula: [pic], where [pic] for example, for n1 = 52, we calculate z1 = -0.50128. Since -3 < -0.50128 < +3, the first point is in control. See the entire control chart in the handout given in class. We find when the supplementary runs rules are applied, rule 6 is violated (4 out of 5 observations are beyond 1σ). If special-cause explanation can be found, these observations should be removed and the limits updated.