Allocation of Sample Sizes in bi-objective Stratified Sampling Using Lexicographic Goal Programming 6.1 Introduction The allocation in stratified sampling is a procedure of assigning some portion of the total samples into each stratum according to the prescribed criterion. The usual criterion is to minimize the variance of the estimator of the population parameter under the constraints for the total survey cost. Neyman (1934) suggested the concept of optimum allocation, Mahalanobis (1944) introduced the cost function, and Stuart (1954) proved the optimum allocation using Cauchy inequality. It is well known that the Neyman allocation yields an estimate of the population parameter with maximum precision when all the strata have the same value for the survey cost. In practice, however, there are certain limitations to the use of the Neyman allocation. In multivariate stratified sampling, the Neyman allocation on the basis of any one variable may lead to loss of precision on the other variables. The problem of allocation in multivariate stratified sampling has been studied by Melaku and Sadasivan (1987), Kreienbrock (1993), and Khan et al. (1997). Neverthless, if the variables are correlated and certain variable is more important than the others it can be said that the efficiency of the estimator using the Neyman allocation may be secured generally. On the contrary, the absence of the strata standard deviations causes the situation that the efficiency of the estimator from the Neyman allocation is severely reduced. The variance of an estimator based on Neman allocation using the estimated strata standard deviations has been calculated by Evans (1951). Lower bound of the size of the preliminary samples to estimate the strata standard deviations for the Neyman allocation was derived by Sukhatme and Sukhatme (1970). Alternative 151 methods using other measures, such as the range of strata and the strata total of survey variable, instead of the standard deviation, have been considered by Bankier (1988). In real world problem situation, several conflicting objectives arises in multi-objective problems such as cost function, total sample size etc. So goal programming problem was introduced by Charns and Cooper (1961) to deal with the problem of achieving a set of conflicting goals. The objective function searches to minimize deviations from the set of pre-assigned goals. Narula and Wellington (2002) introduce some of the single sample statistical problems; those can be formulated and solved as multiple optimization problems. Decision makers sometimes set such goals, even when they are attainable within the available resources. Such problems are tackled with the help of the techniques of nonlinear goal programming, where the objective function is stated in such a way that the all goals must be attained. In this chapter, we have discussed the problem of bi-objective stratified sampling formulated as non-linear goal programming problem (NLGP) and discussed how the proper priority structure of non-linear goal programming (NLGP) model can be selected for obtaining the allocation of sample size in stratified random sampling where the two objectives are to be attained. In the solution process priorities are given to each objective one after other and a set of solution is obtained. From the solutions, the ideal solution is identified. The -distances of different solutions from the ideal solution are calculated. The solution corresponding to the minimum -distance gives the best compromise solution. 6.2 Formulation of the problem Many authors have considered the general problem of optimal design in stratified and multistage sampling (see, for example, Hartley, 1965; Folks and Antle, 1965; Kokan 152 and Khan, 1967; Chatterjee, 1968, 1972; Bethal, 1985, 1989; and Megerson, Clark and Fenley, 1986). In stratified sampling the total population is first partitioned into several sub-populations (called strata). Population characteristics can be inferred with samples from each stratum, exploiting the gain in precision in the estimates, administrative convenience, and the flexibility of using different sampling procedures in the different sub-populations. Let be the number of units in the number of strata into which the drawn from the stratum and units are divided. Let , where is the be the size of the sample stratum. Assume that the samples are drawn independently from different strata. The problem of optimally choosing the is known as the optimal allocation problem. The objectives in this problem are to minimize the variance of the estimate of the population characteristic under study and minimize the total cost of sampling. Let denote an unbiased estimator of the population mean, where characteristic under study. Let be an unbiased estimate of the stratum mean is the ; i.e., then is an unbiased estimate of the population mean . The variance of is given by (6.2.1) where Let and , be the cost of measuring one unit in the stratum. Then the cost function is considered as (6.2.2) where is the overhead cost and is the total cost for the survey. 153 (6.2.3) where and is the prefixed variance of the estimator of the population mean. So, the approach is to minimize cost as well as sample size for fixed variance. 6.3 Solution procedure In conventional priority based non linear goal programming, the solution under the decision makers imposed priority structure is considered as the optimal. But in different complex decision making situation, desired solution may not be acceptable underweight structure. Thus a better solution is always expected for which alternative priority under the given weight structure may be considered. To select the proper priority structure the following procedure is performed. If in the formulation, priorities for objective functions are considered. So involvement of priorities indicates that (Factorial) different solutions can be obtained from problems arises for number of different priority structures. Let , solution obtained by giving priorities to and be the number of objective functions. If priority is given to cost function then we have to solve the following lexicographic programming problem: (6.3.1) 154 and solution of lexicographic programming problem (6.3.1) is given by . Similarly if priority is given to total sample then, we have to solve the following lexicographic programming problem: (6.3.2) and solution of lexicographic programming problem (6.3.2) is given by . Then the ideal solution is defined as: (say). But in practice ideal solution can never be achieved. The solution, which is closest to the ideal solution, is accepted as the best compromise solution, and the corresponding priority structure is identified as most appropriate priority structure in the planning context. To obtain the best compromise solution, following goal programming problem is to be solved: (6.3.3) where are the under and over deviational variable respectively. Now, (6.3.4) 155 Eqn. (6.3.4) is defined as the the solution -distance from the ideal solution , to . Therefore, (6.3.5) Hence, is the best compromise solution. 6.4 Lexicographic goal programming in allocation problem Preemptive goal programming is a mathematical programming method developed to solve problems with conflicting linear or non linear objectives and linear and non linear constraints. The user is able to provide levels, or targets, of achievement for each objective and priorities the order in which goals are to be achieved. Since the different objectives have their own importance in the problem. The use of lexicographic goal programming problem is considered where the goals are arranged in the lexicographic order. In this chapter there are only two objective functions (cost, sample size). If first priority is given to cost function, the lexicographic goal programming problem is defined as (6.4.1) 156 Now, we solve the problem (6.4.1) using lexicographic goal programming and find the allocation of sample sizes as and Similarly, if first priority is given to total sample size function, thus the lexicographic programming problem is defined as: (6.4.2) Now, we solve the problem (6.4.2) using lexicographic goal programming and find the allocation of sample sizes as 6.5 and Numerical illustration Consider the following data of Jessen (1942) for illustrating the proposed method to find out the optimum values of allocations. For this purpose, the data of one characteristic are used which are tabulated in the following Table 1. Table 1 157 Solution of the problem when priority given to cost function The lexicographic goal programming problem (6.4.1) for the data of Table 1 is written as: (6.5.1) Now, we solve the non linear integer lexicographic goal programming problem (6.5.1) by LINGO Software, we obtained the following solution: with and Solution of the problem when priority given to sample size The lexicographic goal programming problem (6.4.2) for the data of Table 1 is written as: (6.5.2) 158 Now, we solve the non linear integer lexicographic goal programming problem (6.5.2) by LINGO Software, we obtained the following solution with and Table-2 In table- the -distances of all possible solutions from the ideal solution are calculated. From table- it is found that the minimum of the solutions from the ideal solution is -distances of possible . Table- which corresponds to the priority sample size. Therefore the best compromise solution of the problem is with . 159 6.6 Conclusion This paper provided a profound study of allocation of sample size in stratified sampling using -distances. The solution corresponding to the minimum distance is the best compromise solution. 160
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