Goal programming and Lexicographic Goal Programming Apporches in bi-objective Stratified Sampling: An Integer Solution

Transcription

1 Goal programming and Lexicographic Goal Programming Apporches in bi-objective Stratified Sampling: An Integer Solution 5.1 Introduction Goal programming (GP) was proposed by Charnes and Cooper (1961) to solve multiple objective decision making problems (MODMP). It has been studied by many researchers and successfully applied to many diverse, real life problems. Now it has been accepted as a basic mathematical programming method for solving multiple objective decision making problems (MODMP). Preemptive goal programming is a special case of goal programming, in which the most important (upper level) goals are optimized with before least important goals. In non preemptive models, the goals are assigned weights and considered simultaneously. 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 goal programming, where the objective function is stated in such a way that the all goals must be attained. In the absence of the knowledge of the utility function of the decision-maker, there are several ways to find a set of (weakly) non-dominated solutions to the multiple objective optimization problems. One of the earliest ways was to find all nondominated solutions. This is impractical for a reasonable size problem. However, even if we can compute all non-dominated solutions, it can be an arduous task for a decision-maker to select a compromise solution from a multitude of solutions. The other methods are: -constraint method, weighted method, goal programming, and interactive methods. In the -constraint method, the decision maker specifies acceptable levels of all but one objective function; these values are used as constraints 140

2 and the problem is solved as a single criterion optimization problem. In the weightedmethod, the decision-maker specifies relative weights for each of the objectives and the problem is solved as a single criterion problem. In goal programming formulation, the decision-maker specifies the priority of the objective functions. The problem is solved for the top priority objective first, and then this value is never allowed to deteriorate. The problem is solved for the next priority and so on, until the problem is solved. In interactive methods, decision-maker is shown one or more non dominated solutions at a time and asks to choose one. If the decision-maker is satisfied with the solution, the process stops; otherwise, the decision-maker specifies the desired changes in the values or directions of the objective functions and the problem is resolved. The process continues until the decision-maker finds an acceptable solution. The goal programming approach allows a simultaneous solution of a system of complex objective rather than a single objective. In other words goal programming, is a technique that is capable of handling decision problems that deals with a single goal, with multiple sub goals. Moreover, the objective function of a goal programming model may be composed of non homogeneous units of measures such as pounds and dollars rather than one type of unit. In this chapter goal programming (GP) and Preemptive goal programming (PGP) approaches are used for allocating the sample size in stratified random sampling problem. We have discussed the methods how to solve the allocation problems in stratified sampling and compare both the method which one is the best one. 5.2 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 and Khan, 1967; Chatterjee, 1968, 1972; Bethal, 1985, 1989; and Megerson, Clark 141

3 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 stratum and, where is the number of strata into which the units are divided. Let be the size of the sample drawn from the 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 is the characteristic under study. Let be an unbiased estimate of the stratum mean ; that is, then is an unbiased estimate of the population mean. The variance of is given by (5.2.1) where and, Let be the cost of measuring one unit in the stratum. Then the cost function is considered as (5.2.2) where is the overhead cost and is the total cost for the survey. 142

4 (5.2.3) 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. 5.3 Goal programming in bi- objective Stratified Sampling Let and are the two goals of minimizing the cost as well as the total sample size respectively. Then these goals subject to the constraint can be attained by solving the following two sub problems as: (5.3.1) and (5.3.2) then the solution of non linear programming problem (NLPP) (5.3.1) gives the value of, (5.3.3) Similarly is calculated by solving the non linear programming problem (NLPP) (5.3.2) i.e., (5.3.4) 143

5 where is the maximum variance of the estimator. To find underachievement or overachievement in the target value (goal), let = negative deviation from goal (underachievement or amount below the target value) = positive deviation from goal (overachieved or amount above the target value) where are number of goals. We can formulate the discussed problem as a goal programming problem as follows: Now the objective is to find the optimal allocation of sample size, which is obtained by solving the following integer goal programming problem: (5.3.5) where will be minimized and. The solution of the non linear integer goal programming problem (NLIGPP) (5.3.5) gives the allocation which minimizes the cost as well as sample size. 5.4 Lexicographic goal programming in bi-objective stratified sampling The general lexicographic programming problem is considered as: (5.4.1) 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 144

6 each objective and priorities the order in which goals are to be achieved. Since the different objectives have their own importance in the problem. Then the use of lexicographic goal programming problem is considered where the goals are arranged in the lexicographic order. Here we consider the importance of cost is more than the importance of the sample size. Then the lexicographic integer goal programming problem is defined as: (5.4.2) Solving non linear programming problem (5.4.2), we get the optimum value of defined by. The importance of minimizing the sample size is least, so we have the following non linear integer programming problem (NLIPP):- (5.4.3) Now solve the non linear programming problem (5.4.3), we get the optimum value of defined by. Finally for obtaining the optimum values of allocation of sample sizes, we solve the following non linear integer programming problem:- (5.4.4) 145

7 After solving the non linear integer programming problem (5.4.4), we get the optimum allocation of sample sizes that will provide the minimum value of cost as well as total sample size. 5.5 Numerical illustration Consider the following data of Jessen (1942) for illustrating the proposed method to find out the optimum value of allocation. For this purpose, the data of one characteristic are used which are tabulated in the following Table 1. Table 1 The bi-objective formulation for the given data is given by (5.5.1) The non linear integer programming problem (5.5.1) can be rewritten as: 146

8 (5.5.2) [ ] Goal programming approach for allocating sample sizes To find the goal, the non linear integer programming problem (NLIPP) (5.3.1) for the data of Table 1 is written as: (5.5.3) Now we solve the non linear integer programming problem (NLIPP) (5.5.3) by LINGO SOFTWARE, we get the following solution in 79 steps. with For finding the goal, the non linear integer programming problem (NLIPP) (5.3.2) for the data of Table 1 is written as:- 147

9 (5.5.4) Now we solve the non linear integer programming problem (NLIPP) (5.5.4) by LINGO Software, we get the following solution in 816 steps. with Now, the objective is to find out the allocation of sample size which is obtained by solving the following non linear integer goal programming problem (NLIGPP): (5.5.5) Now we solve the NLIGPP (5.5.5) by LINGO Software, we get the following solution: with. 148

10 [ ] Preemptive goal programming approach for allocating sample sizes To use lexicographic goal programming problem, first define the lexicographic order of objectives, here the cost is preferred than the total sample size. Thus the lexicographic goal programming problem (5.4.2) for the data of Table 1 is written as: (5.5.6) we solve the non linear integer goal programming problem (5.5.6) by LINGO Software, we get the following solution: with objective function and (5.5.7) Solving non linear integer programming problem (5.5.7), we get the following solution: with objective function Now, we have the following non linear integer programming problem: 149

11 (5.5.8) Solving the non linear integer programming problem (5.5.8) by LINGO Software, we get the final optimum solution as: with. 5.6 Conclusion If we solve the bi-objective stratified sampling problem through goal programming and lexicographic goal programming then goal programming provide the best allocation with minimum cost as well as the total sample size. 150