Manufacturing Technology (ME461) Lecture25

8/12/2019 Manufacturing Technology (ME461) Lecture25

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Manufacturing Technology

(ME461)

Instructors: Shantanu Bhattacharya


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Review of previous lectures

Acceptance sampling Occurrence ratio and fraction defective.

Binomial distribution.

How acceptance sampling leads to

formulation of a binomial distribution?

Mean and standard deviation of thebinomial distribution.


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Mean and standard Deviation of a

Binomial distribution

Where many sets of trial are made of an event with constant probability of occurrence p,

the expected average number of occurences in the long run is np.

So np is the expected value, or mathematical expectation, of x where x=0 if an item is

acceptable or x=1 if it is rejectable.

Mathematical expectation is an operator expressed as follows:


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An experimental example of the meaning of average and

standard deviation of Binomial distribution

No. of non

conforming

items

Frequency

4 3

3 7

2 9

1 160 5

Total 40

The table on the left may be used to clarify certain

aspects of the mean and standard deviation of the

Binomial distribution.Column on the extreme left gives the number of

non conforming items in each sample of 10.

The result of 40 samples may be arranged into a

frequency distribution .

The average calculated by normal method results

in 1.675 and the standard deviation by a similarmethod result in 1.127.

If we use the formulae for the mean and deviation

of a binomial distribution the expected average


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The expected

standard

deviation

So, the

observed values

seem fairly close

to the expectedvalue.


10/15

Control charts for attributes

There are several different types of control charts that may be used. These are the

following:

1. The p chart, the chart for fraction rejected as non conforming to specifications.2. The np chart, the control chart for number of non conforming items.

3. The c chart, the control chart for non conformities.

4. The u chart, the control chart corresponding to the non conformities per unit.

Control charts for fraction rejected

The most versatile and widely used attributes control chart is the p chart. This is the

chart for the fraction rejected as non conforming to specifications. The principle

advantage that these charts provide is that only 1 chart is sufficient to describe 10s,

100s or 1000s of quality characteristics by classifying the item as accepted or rejected.

Control limits for the p-chart


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Problems introduced by variable

subgroup sizeOne practical difference between the X chart and p chart is the variability in subgroupsize in the p chart.

P charts typically use data taken for other purposes than the control chart; where

subgroups consist of daily or weekly production and tend to vary a lot.

Whenever subgroup size is expected to vary, a decision must be made as to the way in

which control limits are to be shown on the p chart. There are three solutions to all these

problems.

1. Compute new control limits for every subgroup, and show these fluctuating limits on

the control chart. This is illustrated in the first 2 months of the 4 months.

2. Estimate the average subgroup size for the immediate future. Compute one set of

limits for this average, and draw them on the control chart. Whenever the actual

subgroup size is substantially different from this estimated average, separate limits

may be computed for individual subgroups.

3. Draw several sets of control limits on the chart corresponding to different subgroup

sizes. A good plan is to use three sets of limits, one for expected average, one close

to the expected minimum and one close to the expected maximum.


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This example shows a 4 month

record of daily 100% inspection

of a single critical quality

characteristics of a part of an

electrical device.

When after a change in design,the production of this part was

started early in June, the daily

fraction rejected was computed

and plotted on a chart.

At the end of the month the

average fraction rejected P wascomputed.

Trial control limits were

computed for each point. A

standard value of fraction

rejected p0 was then established

to apply to future production.During July new control limits

were computed and plotted daily

on the basis of the no. of parts

n inspected during the day.

Table 1


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A single set of control limits

was established for August,

based on the estimated average

daily production.

At the end of August, a revised

p0 was computed to apply to

September, and the control

chart was continued during

September with this revised

value.

Calculation of the trial control

limits:

Table 1 shows the number

inspected and no. rejected on

each day. The fraction rejected

on each day is the no. of partsrejected each day divided by the

number inspected that day.

For example: on June 6th, Pi =

31/3350 = 0.0093.

At the end of the month the

average P is computed

The standard deviation is calculated on the basis of thisobserved value P

Table 2


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Determination of P0: If all

the points fall within the

trial control limits, the

standard value P0 may be

assumed to be equal to P.

Here point fell outside thecontrol limit. Thus a

revised value of P is

calculated.

With the days June 7, 12,

13 and 22 eliminated, the

remainder no. of rejects is290 and no. inspected is

46399. The revised P0 =

0.0063.

After considering this and

the previous record on

similar parts of slightlydifferent design, it was

decided to assume p0 =

0.0065.

Table 2 is calculated on

the basis of p0.

Table 3

bl h f l l


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Establishment of control limit

based on expected average

subgroup size:

Although the correct position of

3-sigma control limits on a p chartdepends on subgroup size, the

calculation of new limits for each

new subgroup consumes some

time and effort.

Where the variation in subgroup

size is not too great , thecalculation of each new subgroup

consumes time and effort.

Where the variation of subgroup

size is not more than 25%, it may

be good enough for practical

purposes to establish a single setof control limits based on the

expected average subgroup size.

For the month of august (Table

3) the estimated average daily

production was 2600.

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Manufacturing Technology (ME461) Lecture25

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