4 Ceramic Substrate

EXAMPLE 15-4 Ceramic Substrate

  Suppose we wish to construct a fraction-defective control

  The control chart is shown in Fig. 15-16. All samples are

  chart for a ceramic substrate production line. We have 20 pre-

  in control. If they were not, we would search for assignable

  liminary samples, each of size 100; the number of defectives

  causes of variation and revise the limits accordingly. This

  in each sample is shown in Table 15-4. Assume that the sam-

  chart can be used for controlling future production.

  ples are numbered in the sequence of production. Note

  Practical Interpretation: Although this process exhibits

  that p (800 兾2000) 0.40; therefore, the trial parameters

  statistical control, its defective rate ( p 0.40 ) is very poor.

  for the control chart are

  We should take appropriate steps to investigate the process to determine why such a large number of defective units is be-

  ing produced. Defective units should be analyzed to determine

  UCL

  the specific types of defects present. Once the defect types are known, process changes should be investigated to determine

  0.25 ful in this regard.

  their impact on defect levels. Designed experiments may be use-

  Table 15-4 Number of Defectives in Samples of 100 Ceramic Substrates

  Sample

  No. of Defectives

  Sample

  No. of Defectives

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  CHAPTER 15 STATISTICAL QUALITY CONTROL

  p 0.6 UCL 0.5 0.4 p

  0.3 LCL 0.2 0.1

  Sample fraction defective,

  Figure 15-16 P chart

  for a ceramic substrate.

  Sample number

  Computer software also produces an NP chart. This is just a control chart of nP ⫽D , the number of defectives in a sample. The points, center line, and control limits for this chart are simply multiples (times n) of the corresponding elements of a P chart. The use of an NP chart avoids the fractions in a P chart but it is otherwise equivalent.

  15-6.2 U Chart (Control Chart for Defects per Unit)

  It is sometimes necessary to monitor the number of defects in a unit of product rather than the fraction defective. Suppose that in the production of cloth it is necessary to control the number of defects per yard or that in assembling an aircraft wing the number of missing rivets must be controlled. In these situations we may use the control chart for defects per unit, or the U chart. Many defects-per-unit situations can be modeled by the Poisson distribution.

  If each sample consists of n units and there are C total defects in the sample,

  C U ⫽ n

  is the average number of defects per unit. A U chart may be constructed for such data.

  If the number of defects in a unit is a Poisson random variable with parameter ␭, the mean and variance of this distribution are both ␭. Each point on the chart is an observed value of U, the average number of defects per unit from a sample of n units. The mean of U is ␭ and the variance of U is ␭兾n. Therefore, the control limits for the U chart with known ␭ are:

  ␭

  UCL ⫽␭⫹3

  B n ␭

  LCL ⫽␭⫺3 (15-25)