Allocation of Sample Sizes in bi-objective Stratified
Sampling Using Lexicographic Goal Programming
6.1
Introduction
The allocation in stratified sampling is a procedure of assigning some portion of the
total samples into each stratum according to the prescribed criterion. The usual
criterion is to minimize the variance of the estimator of the population parameter
under the constraints for the total survey cost. Neyman (1934) suggested the concept
of optimum allocation, Mahalanobis (1944) introduced the cost function, and Stuart
(1954) proved the optimum allocation using Cauchy inequality. It is well known that
the Neyman allocation yields an estimate of the population parameter with maximum
precision when all the strata have the same value for the survey cost.
In practice, however, there are certain limitations to the use of the Neyman allocation.
In multivariate stratified sampling, the Neyman allocation on the basis of any one
variable may lead to loss of precision on the other variables. The problem of
allocation in multivariate stratified sampling has been studied by Melaku and
Sadasivan (1987), Kreienbrock (1993), and Khan et al. (1997). Neverthless, if the
variables are correlated and certain variable is more important than the others it can be
said that the efficiency of the estimator using the Neyman allocation may be secured
generally. On the contrary, the absence of the strata standard deviations causes the
situation that the efficiency of the estimator from the Neyman allocation is severely
reduced. The variance of an estimator based on Neman allocation using the estimated
strata standard deviations has been calculated by Evans (1951). Lower bound of the
size of the preliminary samples to estimate the strata standard deviations for the
Neyman allocation was derived by Sukhatme and Sukhatme (1970). Alternative
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methods using other measures, such as the range of strata and the strata total of survey
variable, instead of the standard deviation, have been considered by Bankier (1988).
In real world problem situation, several conflicting objectives arises in multi-objective
problems such as cost function, total sample size etc. So goal programming problem
was introduced by Charns and Cooper (1961) to deal with the problem of achieving a
set of conflicting goals. The objective function searches to minimize deviations from
the set of pre-assigned goals.
Narula and Wellington (2002) introduce some of the single sample statistical
problems; those can be formulated and solved as multiple optimization problems.
Decision makers sometimes set such goals, even when they are attainable within the
available resources. Such problems are tackled with the help of the techniques of nonlinear goal programming, where the objective function is stated in such a way that the
all goals must be attained.
In this chapter, we have discussed the problem of bi-objective stratified sampling
formulated as non-linear goal programming problem (NLGP) and discussed how the
proper priority structure of non-linear goal programming (NLGP) model can be
selected for obtaining the allocation of sample size in stratified random sampling
where the two objectives are to be attained. In the solution process priorities are given
to each objective one after other and a set of solution is obtained. From the solutions,
the ideal solution is identified. The
-distances of different solutions from the ideal
solution are calculated. The solution corresponding to the minimum
-distance gives
the best compromise solution.
6.2
Formulation of the problem
Many authors have considered the general problem of optimal design in stratified and
multistage sampling (see, for example, Hartley, 1965; Folks and Antle, 1965; Kokan
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and Khan, 1967; Chatterjee, 1968, 1972; Bethal, 1985, 1989; and Megerson, Clark
and Fenley, 1986). In stratified sampling the total population
is
first partitioned into several sub-populations (called strata). Population characteristics
can be inferred with samples from each stratum, exploiting the gain in precision in the
estimates, administrative convenience, and the flexibility of using different sampling
procedures in the different sub-populations.
Let
be the number of units in the
number of strata into which the
drawn from the
stratum and
units are divided. Let
, where
is the
be the size of the sample
stratum. Assume that the samples are drawn independently from
different strata.
The problem of optimally choosing the
is known as the optimal allocation
problem. The objectives in this problem are to minimize the variance of the estimate
of the population characteristic under study and minimize the total cost of sampling.
Let
denote an unbiased estimator of the population mean, where
characteristic under study. Let
be an unbiased estimate of the stratum mean
is the
; i.e.,
then
is an unbiased estimate of the population mean . The variance of
is given by
(6.2.1)
where
Let
and
,
be the cost of measuring one unit in the
stratum. Then the cost function is
considered as
(6.2.2)
where
is the overhead cost and
is the total cost for the survey.
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(6.2.3)
where
and
is the prefixed variance of the estimator of the
population mean. So, the approach is to minimize cost as well as sample size for fixed
variance.
6.3 Solution procedure
In conventional priority based non linear goal programming, the solution under the
decision makers imposed priority structure is considered as the optimal. But in
different complex decision making situation, desired solution may not be acceptable
underweight structure. Thus a better solution is always expected for which alternative
priority under the given weight structure may be considered.
To select the proper priority structure the following procedure is performed. If in the
formulation, priorities for
objective functions are considered. So involvement of
priorities indicates that
(Factorial) different solutions can be obtained from
problems arises for
number of different priority structures.
Let
,
solution obtained by giving priorities to
and
be the
number of
objective functions.
If priority is given to cost function then we have to solve the following lexicographic
programming problem:
(6.3.1)
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and solution of lexicographic programming problem (6.3.1) is given by
.
Similarly if priority is given to total sample then, we have to solve the following
lexicographic programming problem:
(6.3.2)
and solution of lexicographic programming problem (6.3.2) is given by
. Then the ideal solution is defined as:
(say).
But in practice ideal solution can never be achieved. The solution, which is closest to
the ideal solution, is accepted as the best compromise solution, and the corresponding
priority structure is identified as most appropriate priority structure in the planning
context.
To obtain the best compromise solution, following goal programming problem is to
be solved:
(6.3.3)
where
are the under and over deviational variable respectively.
Now,
(6.3.4)
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Eqn. (6.3.4) is defined as the
the
solution
-distance from the ideal solution
,
to
.
Therefore,
(6.3.5)
Hence,
is the best compromise solution.
6.4 Lexicographic goal programming in allocation problem
Preemptive goal programming is a mathematical programming method developed to
solve problems with conflicting linear or non linear objectives and linear and non
linear constraints. The user is able to provide levels, or targets, of achievement for
each objective and priorities the order in which goals are to be achieved. Since the
different objectives have their own importance in the problem. The use of
lexicographic goal programming problem is considered where the goals are arranged
in the lexicographic order. In this chapter there are only two objective functions (cost,
sample size).
If first priority is given to cost function, the lexicographic goal programming
problem is defined as
(6.4.1)
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Now, we solve the problem (6.4.1) using lexicographic goal programming and find
the allocation of sample sizes as
and
Similarly, if first priority is given to total sample size function, thus the
lexicographic programming problem is defined as:
(6.4.2)
Now, we solve the problem (6.4.2) using lexicographic goal programming and find
the allocation of sample sizes as
6.5
and
Numerical illustration
Consider the following data of Jessen (1942) for illustrating the proposed method to
find out the optimum values of allocations. For this purpose, the data of one
characteristic are used which are tabulated in the following Table 1.
Table 1
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Solution of the problem when priority given to cost function
The lexicographic goal programming problem (6.4.1) for the data of Table 1 is written
as:
(6.5.1)
Now, we solve the non linear integer lexicographic goal programming problem (6.5.1)
by LINGO Software, we obtained the following solution:
with
and
Solution of the problem when priority given to sample size
The lexicographic goal programming problem (6.4.2) for the data of Table 1 is written
as:
(6.5.2)
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Now, we solve the non linear integer lexicographic goal programming problem (6.5.2)
by LINGO Software, we obtained the following solution
with
and
Table-2
In table-
the
-distances of all possible solutions from the ideal solution are
calculated. From table- it is found that the minimum of the
solutions from the ideal solution is
-distances of possible
.
Table-
which corresponds to the priority sample size.
Therefore the best compromise
solution of the problem is
with
.
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6.6
Conclusion
This paper provided a profound study of allocation of sample size in stratified
sampling using
-distances. The solution corresponding to the minimum distance is
the best compromise solution.
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