Chance constrained compromise mixed allocation in multivariate stratified sampling

Abstract

Consider a multivariate stratified population with  strata and  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  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.