Chebyshev Approximate Solution to Allocation Problem in Multiple Objective Surveys with Random Costs 3

In this paper, we consider an allocation problem in multivariate surveys as a convex programming problem with non-linear objective functions and a single stochastic cost constraint. The stochastic constraint is converted into an equivalent deterministic one by using chance constrained programming. The resulting multi-objective convex programming problem is then solved by Chebyshev approximation technique. A numerical example is presented to illustrate the computational procedure.


Introduction
Let us again consider the problem defined in chapter 2 of allocating the sample to various strata for a given budget when several characters are under study.The variances of various characters are minimized under the condition of fixed budget.
So we consider the following p convex programming problems given in (2.8) of chapter 2. the following p problems where all the objective functions are linear:

Chapter 3
If the costs i c in the various strata are assumed random with independent normal distributions, the problem (3.2) are transformed the following chance constrained programming form: is a specified probability.

Deterministic equivalent using Chance Constrained Programming
We have assumed that the costs i c , L i ,.

^2 ^i i and σ µ
Thus Thus, an equivalent deterministic constraint to the stochastic constraint { } The equivalent deterministic non-linear programming problem to the chance constrained programming problem (3.3) is obtained as

Convex Chebyshev Approximation Problem
Consider p convex smooth functions (3.10) and a region Ω defined by q inequalities Corresponding to the points Ω ∈ ) ,..., ( , where j a are some constants. The problem (3.12) then is equivalent to

Solutions Using Chebyshev Approximation Technique
The p objective functions in { } ) ( 9 . 3 i are linear.The single constraint { } Kokan and Khan (1967)] and { } 3 iii are upper and lower bounds on i x .So (3.9) represents p convex programming problems.Let us denote the feasible region defined by 3.9 (ii) and (iii) by Ω .Suppose that the feasible region is not void.
Let us introduce an auxiliary variable p j x L ,..., 1 , 1 = + .From (3.10) to (3.12) it follows that the problem (3.9) is equivalent to the convex Chabyshev's approximation problem of finding where j a are the weights assigned to the p variances according to their importance.
The problem (3.10) is then equivalent to the following problem with a linear objective function: The non-linear programming problem in (3.15) is convex as the objective function { } ii are linear.Further, the left hand side in { } ii is convex.So it is possible to solve the convex programming problem (3.15) by using any standard convex programming algorithm.The optimal sample numbers thus obtained may turn out to be fractional.However, it is known that the variance functions are flat at the optimum solution.So for large or even moderate sample size it is enough to round the fractional values to the nearest integers.However, for small the branch and bound method should be applied for finding the optimal integer solution.The approach of this chapter has been accepted for publication in American Journal of Computational Mathematics.
See Khan,M.Faisal et al.

Numerical Illustration
Let us consider the data of numerical illustration considered in section 2.8.
In order to demonstrate the procedure the following are also assumed.The per unit of measurement in various strata are independently normally distributed with the following means and variances The total amount available for the survey C is assumed as 600 units including an expected overhead cost 0 t = 100 units.
Let the chance constraint { } 3 ii be required to be satisfied with 99% probability.
. The value of standard normal variable α K corresponding to 99% confidence limits is 2.33.Thus, the problem (3.15) is obtained as:

σ
are the estimated means and variances from the sample.

Fig
Fig 3.1: Convexity of the function .) ( max x f i i We may compare these results with the compromise solution of chapter 2 which is obtained by solving the following NLP problem: