Chance Constrained Compromise Mixed Allocation in Multivariate Stratified Sampling

Consider a multivariate stratified population with L strata and 1  p characteristics. Let the estimation of the population means be of interest. In such cases the traditional individual optimum allocations may differ widely from characteristic to characteristic and there will be no obvious compromise between them unless they are highly correlated. As a result there does not exist a single set of allocations ) ,..., , ( 2 1 L n n n that can be practically implemented on all characteristics. Assuming the characteristics independent many authors worked out allocations based on different compromise criterion such allocations are called compromise allocation. These allocations are optimum for all characteristics in some sense. Ahsan et al. (2005) introduced the concept of ‘Mixed allocation’ in univariate stratified sampling. Later on Varshney et al. (2011) extended it for multivariate case and called it a ‘Compromise Mixed Allocation’. Ahsan et al. (2013) worked on mixed allocation in stratified sampling by using the ‘Chance Constrained Programming Technique’, that allows the cost constraint to be violated by a specified small probability. This paper presents a more realistic approach to the compromise mixed allocation by formulating the problem as a Chance Constrained Nonlinear Programming Problem in which the per unit measurement costs in various strata are random variables. The application of this approach is exhibited through a numerical example assuming normal distributions of the random parameters.


Introduction
In stratified sampling, the use of any particular type of allocation depends on the nature of the population, objectives of survey, the available budget, etc. Practically, there are situations where all strata of a stratified population do not allow the use of a single type of allocation.For example, in the absence of the strata weight Similarly, if strata sizes L h N h ,..., 2 , 1 ;  are unknown proportional allocation can not be used.Thus, when no information about some strata of the population is available, 'equal' allocation may be used for a given total sample size for that strata.If the only information available for some strata is h N , 'proportional' allocation may be used, that is, may be used.As the range h R of a stratum provides an approximation to the standard deviation in the absence of the knowledge of the stratum standard deviations h S , the allocation may be taken as Murthy (1967)).When full information about the population is available, the obvious choice is the 'optimum allocation'.Ahsan et al. (2005) divided the strata into disjoint groups and used different allocations for different groups.They called their allocation as 'Mixed Allocation'.Later on Varshney and Ahsan (2011) extended the work of Ahsan et al. (2005) for multivariate stratified sampling.Ahsan et al. (2013) worked out mixed allocation using chance constraint that allows the cost constraint to be violated by a specified small probability.
In this paper the work of Ahsan and Naz (2013) 2013) are summarized for the sake of continuity.

Let the
L strata of a multivariate stratified population be divided into k groups , where the group The strata are grouped according to the information available for them, that is, full information partial information or zero information (See Kozak (2006(a), 2006(b))).Or according to some other measure of information as desired in the 'Introduction' of this manuscript.
Without loss of generality we can assume that the first It is to be noted that under the above scheme

Chance constrained mixed allocation
Ahsan and Naz (2013) formulated the mixed allocation that minimizes the variance of the stratified sample mean for a fixed cost as the following CCNLPP because in a survey if the costs for enumerating a characteristic in various strata are not known exactly and these are being estimated from sample costs that may be subjected to random variations.They may increase to a level where the cost constraint is violated.
The chance constraint (3.2) may be expressed as Since h q are normally distributed and j  are unknown constants, 0 d will also be normally distributed with a mean Using (3.5), (3.6) and (3.7) the constraint (3.4) can now be expressed as Thus the probability of realizing 0 d smaller than or equal to 0 C can be written as Inequality (3.10) will be satisfied if and only if Inequality (3.11) gives the deterministic equivalent to the linear chance constraint (3.4).
Thus the solution of the stochastic nonlinear programming problem (3.1)-(3.3)can be obtained by solving the equivalent deterministic programming problem

Chance constrained compromise mixed allocation
Using the results of sections 2 and 3, the deterministic equivalent of the CCNLPP (3.12)-(3.14)for multivariate case may be expressed as the following NLPP: , are the usual restrictions to avoid oversampling and to estimate the strata variances.
When the numerical data are available the NLPP (4.1)-(4.3)may be solved using a suitable Nonlinear Programming Technique.In the next section an application of the proposed formulation is given for an artificial data.The solution to the NLPP (4.1)-(4.3) is obtained by the optimization software LINGO (2001).1.

Varshney et al. (2011) gave a numerical illustration using an artificial data given in Table
The values of 2  ˆh c  are assumed by authors.

Table 2 Expected cost with their estimated variances for the two characteristics
The authors assumed that the probability of violation of the cost constraint is 0.01, that is, the cost of constraint should be satisfied with probability 0.99.This gives the value of 0  as 2.33 from standard normal area table.
The strata are so numbered that: (i) Strata 1, 2 and 3 constitute group 1 G in which equal allocation is to be used, that is } . The trace value is 6.501194863.Cochran (1977)  Chatterjee formulated the problem as  Sukhatme et al. (1984) obtained the compromise allocation by minimizing the sum of the variances for the p characteristics under linear cost constraints.The NLPP for this allocation is given as:

Sukhatme's compromise allocation
Using the values given in Table 1 and with the objective value as 4.240085924 which is also the trace value.

Summary of the Results
Table 3 gives the summary of the results of the numerical illustration.

Table 3 Summary of the results
The last column of Table 3 provides the Relative Efficiencies (R.E.) of the four compromise allocations as compared to the proportional allocation.It can be seen that the proposed allocation is the most efficient among the considered allocations.
can not be used.

2 hS
for some strata are available, an allocation h h h S W n  may be used.When there is evidence to believe that the relative standard error of the stratum mean h Y based on one sample unit does not vary considerably over some strata, an allocation


Using compromise criterion ofYates (1960),Varshney and Ahsan (2011) formulated the problem of finding a compromise mixed allocation as the following Nonlinear Programming Problem (NLPP) for multivariate stratified sampling as to the th l characteristics according to its relative importance, h c is the cost of measuring all the p characteristics on a selected unit of the th h stratum, that is, unit cost of measuring the th l characteristic in the th h stratum.Without loss of generality we can assume that


represents the cumulative distribution function of the standard normal variate evaluated at x .If 0  denotes the value of the standard normal variable at which 0 0 Rao S. S. (2010)).Using (3.6) and (3.7), we get 0 12)-(3.14) is a nonlinear programming problem and could be solved by a suitable nonlinear programming technique.Ahsan and Naz (2013) used the optimization software LINGO (2001) to obtain a solution.
is extended for multivariate stratified sampling, where in cost constraint, a small probability of violation is allowed.the course of the survey due to random causes in practice it becomes a random variable.Thus the problem of compromise mixed allocation can be viewed as a Chance Constrained Nonlinear Programming Problem (CCNLPP).In section 2 and 3 the work of Varshney et al. (2011) and Ahsan et al. (

Table 1
Data for seven strata and two characteristics Using the values given in the Table1and Table2and the values of gave the compromise criteria by averaging the individual optimum allocations *