## Futher investigations of group screening designs; step-wise designs

##### Abstract

The inspection of individual members of a large population is an expensive and tedious process. Often when testing the results of manufacture, th2 work can be reduced greatly by examining only a
sample of the population and rejecting the whole if
the proportion of defectives in the sample is unduly large. In many inspections however, the objective
is to eliminate all the defective members of the
population. This situation arises in manufacturing processes where the defect being tested for can result in disastrous failures. It also arises in certain inspections of human populations with say infectious diseases.
Where the objective 1S to weed out individual
defective units, a simple inspection will not suffice .
In this case, we need designs which will classify
all the items in the population as defective or
non-defective. Such designs are known as screening designs. Earlier work in this area was done by
Dorfman [3 J and sterret [26 J. Connor [ 1 ] 1
Watson [28] and Patel [13 J 1 . [ 14 J have. app r-oechsc
.tbe problem from the point of "view of designs of e~periments and called these designs "Group
screening designs". This thesis is along the linss of Sterret's paper [26J. The problem has been a~proached from the point of view of design of
experiments.
Chapter I defines the concept of group screening designs and describes briefly the work done in this and related areas by several authors in the past. The chapter also lays down the assumptions which are used in this thesis.
In chapter II, step-wise group screening
designs have been introduced and are studied assuming that all factors have the same a-prior probability of being defective. Optimum group sizes in the initial step have been determined considering only the expected total number of runs. A comparison of two-stage group screening design with step-wise group screening design is presented.
Chapter III extends the results of chapter
II to the case where factors are defective with
unequal a-prior probabilities. It is shown that
under certain conditions. the minimum expected number of runs when screening is done under the assumption that factors are defective with unequal a-priori probabilities is smaller than the minimum expected number of runs wh sn screening 1S done
under the assumption that all factors are defective
with the same a-priori probability.
In chapter IV, the optimum sizes of the group-factors for both the cases when we screen with equal and with unequal a-priori probabilities
have been determined taking into consideration both
the expected number of incorrect decisions and the expected number of runs. To balance the apparently opposite trends of the eX~3cted number of runs and the expected number of incor~ect decisions. a cost function has been defined and optimum sizes of the group-factors determined ~y minimizing the cost function.
At the end. are given a series of tables
which show some group screening plans resulting from the work that has bean done in chapters II through IV. This appears in appendices I. II and III.
Throughout this thesis. it is assumed that
the value of 'P'. i.e., a-priori probability
of a factor to be defectives
.is known heuristically .
Thus no attempt is made to estimate 'p' in this\
thesis. The work has been extended to the case
with more than one value of 'p'. For example in a manufacturing plant turning out hundreds of items every day; the probability of the plant producing defective items will vary from time to time due to assignable causes of variation which affect the production. Thus In such a case. it is reasonable to assume that items will be defective with unequal
a-priori probabilities. Again we shall assume that the values of these a-pri2ri probabilities are
known heuristically. However. the optimum sizes of
the group-factors will depend on the expected
number of runs and the expected number of incorr3ct decisions.
Familiar calculus methods have been used to
solve must of the problems in this thesis. The methods used include Newton - Raphson iterative method, the method of Lagrange's multipliers and ordinary differentiation.

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