An Optimal Multivariate Stratified Sampling Design Using Auxiliary Information: An Integer Solution Using Goal Programming Approach

In multivariate stratified random sampling, for practical purposes we need an allocation which is optimum in some sense for all characteristics because the individual optimum allocations usually differ widely unless the characteristics are highly correlated. Cochran (1977) suggested the use of characteristicwise average of individual optimum allocations as a compromise allocation for correlated characteristics. For uncorrelated ones, many authors have suggested various criteria to work out an optimum compromise allocation. For example a compromise criterion may be to minimize the total loss in efficiency of the estimate due to not using the individual optimum allocations. When auxiliary information is also available, it is customary to use it to increase the precision of the estimates. Moreover, for practical implementation of an allocation, we need integer values of the sample sizes. The present article addresses the problem of determining the integer optimum compromise allocation when the population means of various characteristics are of interest and auxiliary information is available for the separate and combined ratio and regression estimates. The problem is formulated as a multiobjective nonlinear programming problem and a solution procedure is developed using goal programming technique. The goal is to minimize the weighted sum of the increases in the variances due to not using the individual optimum allocations subject to budgetary and other constraints.