1813-713X Copyright © 2014 ORSTW International Journal of Operations Research Vol. 11, No. 4, 100111 (2014) A Fuzzy Chance Constraint Programming Approach for Optimal Allocation in Multivariate Stratified Surveys: A Compromise Solution Shamsher Khan * and M.M. Khalid Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh 202002, India Received May 2014; Revised October 2014; Accepted October 2014 Abstract  Optimal allocation of sample size among various strata is an important step to get the precise estimates for population parameters and to reduce the cost of the survey. A reasonable criterion for optimal allocation is the minimization of the variances of the estimates for a specified cost or to minimize the cost of survey for desired precision of the estimates. The total cost of survey is a function of sample sizes allocated to various strata and the unitary cost of collecting information/measurement associated to particular stratum. The measurement cost h c which vary from stratum to stratum and affected by some factors such as nature of climate, weather conditions which occurs randomly is considered as fuzzy random variable (FRV). The survey is taken as multivariate in which we want to study more than one characteristic. Thus, in this paper the problem of optimum allocation in multivariate stratified sampling is formulated as a multiobjective fuzzy chance constrained programming (MOFCCP) problem with measurement cost h c as a normally distributed FRV. Compromise solution of its deterministic equivalent is obtained by goal programming technique. In addition, an illustrative example is also given to demonstrate the correctness of the proposed approach. Keywords  Multivariate stratified surveys, Optimum allocation, Compromise allocation, Stochastic programming, Multiobjective fuzzy chance constraint programming, Goal programming technique 1. INTRODUCTION Variability or heterogeneity is inherent in the population units. The problem of obtaining a good estimator for population mean or total one should attempt either to increase the sample size or to devise certain method of sampling by which variability or heterogeneity can be reduced. One such method is stratified sampling. It consists in dividing the population into subpopulations called strata and each subpopulation as stratum. The problem of allocating the appropriate sample size to the respective stratum is known as the problem of optimum allocation. Yates (1960) suggested two useful approaches for optimal allocation. One approach is to “Minimize the variances of the estimates subject to a cost function or to a given sample size” and another approach is to “minimize the total cost for the desired precision of the estimates”. In large scale sample surveys one is generally concerned with estimation of more than one population characteristic. If these characteristics are highly correlated then optimal allocation may differ little among themselves, Swain (2003). But if not so, then optimal allocation for one character may not be optimal for others. In such situations we obtain a compromise allocation, optimal for all characters in some sense. The problem of optimum allocation of sample sizes to various strata is treated as mathematical programming problem by Kokan (1663) and a solution is proposed using nonlinear programming technique. Kokan and Khan (1967) have given an analytical solution for the optimum allocation in multivariate surveys. Many authors such as Chatterjee (1967, 1968), Ahsan and Khan (1977), Khan et al. (1997), Khan and Ahsan (2003) either suggested new approaches or explored exiting approaches further. As we know that in real life situations we face uncertainty. Uncertainty arises due to lack of knowledge or due to inherent vagueness. To make decisions in such type of situations, we use probability theory and fuzzy set theory respectively. Many problems in statistics such as regression analysis, sample surveys, cluster analysis, estimation and so on can be viewed as a mathematical programming problem, Arthanari and Dodge (1981). The mathematical programming problems in which some or all of the parameters are described by stochastic (or random or probabilistic) variables are called stochastic programming or probabilistic programming problems, (see Dantzig (1955), Rao (1978), Prekopa (1995), Uryasev and * Corresponding author’s email: shamsherstats@gmail.com International Journal of Operations Research