One of the main reasons for stratifying the population is to produce a gain in precision of the estimates, in the sample surveys. For achieving this, one of the problem is determination of optimum strata boundaries. The strata boundaries should be obtained in such a way, so that it can reasonably expect to reduce the cost of the survey as much as possible without sacrificing the accuracy or alternatively, reducing the margin of error to the greatest possible extent for the same expected cost. In this paper, we have discussed the way of obtaining optimum strata boundaries when the cost of every unit varies in the whole strata. The problem is formulated as non-linear programming problem which is solved by using Bellman's principle of optimality. For numerical illustration an example is presented for uniformly distributed study variable.

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Optimum stratification is the method of choosing the best boundaries that make the strata internally homogenous. Many authors have attempted to determine the optimum strata boundaries (OSB) when a study variable is itself a stratification variable. However, in many practical situations fetching information regarding the study variable is either difficult or sometimes not available. In such situations we find help in the variable (s) closely related to the study variable. Using auxiliary information many authors have formulated the problem as a MPP by redefining the problem as the problem of optimum strata width, and developed a solution procedure using dynamic programming technique. By using many distributions they worked out the optimum strata boundary points for the population under different allocation. In this paper, under proportional allocation OSBs are determined for the study variable using two auxiliary variables as the basis of stratification with uniform, right-triangular, exponential and lognormal frequency distribution by formulating the problems which are executed by using dynamic programming. Empirical studies are presented to illustrate the computation details of the solution procedure and its comparison with the existing literature.

In the present investigation, some theory has been developed for optimum stratification, when two auxiliary variables treated as the basis of stratification with one study variable under study. The problem has been formulated as mathematical programming problem and then solved by dynamic programming. Empirical studied have been made to illustrate the proposed method with the comparisons of other existing methods.

The current study discusses the solution for obtaining stratification points under Neyman allocation having one study variable and two auxiliary variables. Using dynamic programming approach non-linear programming problem has been solved. The proposed technique has gained in precision rather than using only one auxiliary variable. Numerical illustration has been given in which each of the auxiliary variable is supposed to follow different distribution. Through the empirical study, the proposed method has been compared with the Ravindra and Sukhatme (1969) and Khan et al.(2005) methods with the conclusion of having its more relative efficiency.