Inspection in industry serves the function of determining conformity to specification. There can be inspection of incoming materials, inspection of the processes at several stages and final inspection of the finished product. The problem usually is to decide how much to inspect? Is it possible to examine each and every product? To what extent is it necessary and to what extent feasible? This leads us to the possibility of partial inspection.
There are situations, where 100 per cent inspection cannot be employed for reasons like absence of technical feasibility (like life-cycle test of Electric Bulb or destructive test to verify chemical composition of a steel item), lack of time, huge cost involvement, etc. This calls “for partial inspection, observations which are generalized for all the units. Such inspection is known as ‘sampling inspection’. Its principal uses in industry are:
1. Determination of the quality and acceptability of incoming raw materials
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2. Decision as to the quality and acceptability of semi-finished product for further processing as it passes from one division to other within the factory
3. Determination or the quality of outgoing product
4. Improvement and control of quality
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A sampling plan gives the procedure to be followed for acceptance or rejection of products submitted for acceptance. The product is usually submitted in lots or consignments, and the consumer will accept the lot of high quality products and obviously reject the lot of low quality products. Therefore, it is necessary to decide how the quality of a lot should be measured. The many ways in which the quality of a lot can be described depend mainly on the following characteristics:
1. The proportion of defective items in the lot (percentile defective)
2. The average number of defects per item or constant area of the lot (defects per item)
3. The averages of an important measurable characteristic of the item in the lot and the variability or dispersion among the items of the lot with respect to this characteristic (i.e., the mean and standard deviation for the whole lot)
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Attribute sampling plans are plans where item quality is defined by attributes and lot quality by fraction defective. Plans, where item quality is an actual measurement and lot quality is specified by tolerances (mean dimension and dispersion), are called variable sampling plans.
Operating Characteristic Curve:
To judge the suitability of any sampling plan, we need to compare its performance over a range of possible quality levels of the submitted product. A comparison is provided by the Operating Characteristic Curve (or the OC Curve) of the plan. It gives clearly how a sampling plan discriminated lots of varying quality and evaluates the risks associated with any sampling plan.
The above operating characteristic (OC) curve shows the probability of acceptance, Pa for different levels of lot quality. The horizontal axis here plots the quality characteristic.
There are three important points that are often referred on the OC Curve. These are:
1. Acceptable Quality Level (AQL)
2. Lot Tolerance Per Cent/Fraction Defective (LTPD or LTFD)
3. Point of Indifference
Acceptable Quality Level (AQL) or Average Acceptable Quality Level (AAQL) is the level at which the consumer is willing to accept at, all the time. However, due to chance variations the batches may be rejected occasionally even through the quality is better. The probability of such a chance is called ‘Producer’s Risk’.
Lot Tolerance Fraction Defective (LTFD) or Lot Tolerance Per Cent Defective is the level at which the consumer would like to reject at, all the time. When it is expressed in terms of percentage it is called LTPD, and when it is expressed in terms of a fraction, it is called LTFD. The sampling plan involves some risk that will accept a lot having such a chance is called ‘Consumer’s Risk’.
Point of Indifference is the point where the producer and the consumer take the same risk in the application of the plan. So, this is the point where the probability of acceptance and rejection of the lot are equal.
Single Sampling Plan:
The decision to accept or reject a lot is based upon a single sample. So from a homogeneous group, for this type of sampling plan, it is necessary to:
i. Make a lot size of ‘N’
ii. Take a random sample size of ‘n’
iii. Inspect all the ‘n’ units
iv. If the number of defective items is within ‘C’ then accept the lot of size ‘N’
v. If the number of defective items is outside the ‘C’ limit, then reject the lot size of ‘N’.
vi. ‘C’ is called the acceptance number or critical number of the plan. If we consider an example with numerical values for the single sampling plan with:
N= 10,000
N= 300
C= 5
The process of operation is as follows:
Make a homogenous lot of size 10,000. Out of which choose a sample size of 300 randomly. If it is possible to use random number table to select these 300 units, it is better. Otherwise pick 300 units in an unbiased manner from the lot. Inspect all the 300 units. If the number of defective units is 0, 1, 2, 3, 4 or 5, accept the lot of 10,000 as good. If the number of defective units is more than 5, reject the lot of 10,000.
The selection of the plan for quality control set up depends upon the precision required, the economic aspect implied in the plan and the operational feasibility of the plan in the quality control set up. Operational feasibility changes from company to company and from product to product.
The level of precision being attained can be accessed through the OC Curve and Average Outgoing Quality Curve. The Average Outgoing Quality Curve (AOQL) tells us how many bad items we will get if we follow a specified plan. The ordinate of this curve is lot fraction defective p.
For a specified quality level, the number of defective units that will be accepted is the value of for the given p. This curve rises to a certain limit and drops down afterwards. The maximum point reached by it is called the Average Outgoing Quality Limit (AOQL) and it is denoted by pa. As far as the consumer is concerned, this point is very important because it gives the number of defective units the consumer will get whatever may be the quality of production.
It is always an unknown fact how good is the production or the value of p, so sampling plans can be formulated stipulating the value of AOQL. It is surprising to note that the average outgoing quality (pa) decreases when the real quality of production becomes bad, i.e., when p increases.