Fuzzy Geometric Programming Approach in Multi-objective Multivariate Stratified Sample Surveys in Presence of Non-Response

 In this paper, we have formulated the problem of non-response in multivariate stratified sample surveys as a Multi-Objective Geometric Programming problem (MOGPP). The fuzzy programming approach has described for solving the formulated MOGPP. The formulated MOGPP has been solved and the solution is obtained. The obtained solution is the dual solution corresponding to the multi-objective multivariate stratified sample surveys in presence of non-response. Afterward with the help of dual solution of formulated MOGPP and primal-dual relationship theorem the optimum allocation of sample sizes of respondents and non respondents are obtained. A numerical example is given to illustrate the procedure.


INTRODUCTION
In stratified sampling heterogeneous population is converted into a homogeneous population by dividing it into homogeneous stratum.The maximum precision will be obtained with the best choices of the sample sizes.The problem of optimum allocation in stratified random sampling for univariate population is well known in sampling literature; see for example Cochran (1977) and Sukhatme et al. (1984).In multivariate stratified sample survey the problem of non-response can appear when the required data are not obtained.The problem of non-response may occur due to the refusal by respondents or they are not at home making the information of sample inaccessible.The problem of non-response occurs in almost all surveys.The extent of non-response depends on various factors such as type of the target population, type of the survey and the time of survey.For dealing the problem of non-response the population is divided into two disjoint groups of respondents and non respondents.For the stratified sampling it may be assumed that every stratum is divided into two mutually exclusive and exhaustive groups of respondents and non respondents.Hansen and Hurwitz (1946) presented a classical non-response theory which was first developed for the survey in which the first attempt was made by mailing the questionnaires and a second attempt was made by personal interview to a sub sample of the non respondents.They constructed the estimator for the population mean and derived the expression for its variance and also worked out the optimum sampling fraction among the non respondents.El-Badry (1956) further extended the Hansen and Hurwitz's technique by sending waves of questionnaires to the non respondent units to increase the response rate.The generalized El-Badry's approach for different sampling design was given by Foradari (1961).Srinath (1971) suggested the selection of sub samples by making several attempts.Khare (1987) investigated the problem of optimum allocation in stratified sampling in presence of non-response for fixed cost as well as for fixed precision of the estimate.Khan et al. (2008) suggested a technique for the problem of determining the optimum allocation and the optimum sizes of subsamples to various strata in multivariate stratified sampling in presence of non-response which is formulated as a nonlinear programming problem (NLPP).Varshney et al. (2011) formulated the multivariate stratified random sampling in the presence of non-response as a multi-objective integer nonlinear programming problem and a solution procedure is developed using lexicographic goal programming technique to determine the compromise allocation.Fatima and Ahsan (2011) address the problem of optimum allocation in stratified sampling in the presence of non-response.Raghav et al. (2014) have discussed the various multi-objective optimization techniques in the multivariate stratified sample surveys in case of non-response Geometric programming (GP) is a smooth, systematic and an effective non-linear programming method used for solving problems of sample surveys and engineering design that takes the form of convex programming.The convex Shafiullah and Bari: Fuzzy Geometric Programming Approach in Multi-objective Multivariate Stratified Sample Surveys in Presence of Non -Response IJOR Vol. 12, No. 2, 021−035 (2015) 1813-713X Copyright © 2015 ORSTW programming problems occurring in GP are generally represented by an exponential or power function.GP has certain advantages over the other optimization methods because it is usually much simpler to work with the dual than the primal one.The degree of difficulty (DD) plays a significant role for solving a non-linear programming problem by GP method.Geometric Programming (GP) has been known as an optimization tool for solving the problems in various fields.Duffin, Peterson and Zener (1967) and also Zener (1971) have discussed the basic concepts and theories of GP with application in engineering in their books.Beightler, C.S., and Phililps, D.T., also published a famous book on GP and its application in (1976).Engineering design problems was also solved by Shiang (2008) and Shaojian et al. (2008) with the help of GP.Davis and Rudolph (1987) applied GP to optimal allocation of integrated samples in quality control.Ahmed and Charles (1987) applied geometric programming to obtain the optimum allocations in multivariate double sampling.Maqbool et al. (2011), Shafiullah et al. (2013) have discussed the geometric programming approach for obtaining the optimum allocations in multivariate two-stage and three-stage sample surveys respectively.
In many real-world decision-making problems of sample surveys, environmental, social, economical and technical areas are of multiple-objectives problems.Multi-objective optimization problems differ from single-objective optimization.It is significant to realize that multiple objectives are often non-commensurable and in conflict with each other in optimization problems.The fuzzy goal is defined as the objective which can be obtained within exact target value.The multi-objective models with fuzzy objectives are more realistic than deterministic of it.The concept of fuzzy set theory was firstly given by Zadeh (1965).Later on, Bellman and Zadeh (1970) used the fuzzy set theory to the decision-making problem.Tanaka (1974) introduces the objective as fuzzy goal over the α-cut of a fuzzy constraint set and Zimmermann (1978) gave the concept to solve multi-objective linear-programming problem.Biswal (1992) and Verma (1990) developed fuzzy geometric programming technique to solve multi-objective geometric programming (MOGP) problem.Islam (2005Islam ( , 2010) ) has discussed modified geometric programming problem and its applications and also another fuzzy geometric programming technique to solve MOGPP and their applications.Fuzzy mathematical programming has been applied to several fields.
In this paper, we have formulated the problem of non-response in multivariate stratified sample surveys as a multi-objective geometric programming problem (MOGPP).The fuzzy geometric programming approach has described for solving the formulated MOGPP and optimum allocation of sample sizes of respondents and non respondents are obtained.A numerical example is given to illustrate the procedure.

FORMULATION OF THE PROBLEM
In stratified sampling the population of N units is first divided into L non-overlapping subpopulation called strata, of sizes    A more careful second attempt is made to obtain information on a random subsample of size h r out of ( ) where given as: ( ) where jhi y denote the value of the th i unit of the th h stratum for th j characteristic.
S is the stratum variance of the th j characteristic in the th h stratum among non respondents, given by: ( )

S
are not known they can be estimated through a preliminary sample or the value of some previous occasion, if available, may be used.Furthermore, the variance of ( ) ( ) (3) where ) ( w j y is an unbiased estimate of the overall population mean j Y of the th j characteristic and ( ) Assuming a linear cost function the total cost C of the sample survey may be given as:  ) The problem therefore reduces to find the optimal values of sample sizes of respondents * h n and non-respondents * h r which are expressed as:

MOGPP FORMULATION OF SAMPLE SURVEYS PROBLEM IN PRESENCE OF NON-RESPONSE
Geometric programming always transforms the primal problem of minimizing a "posynomial" subject to "posynomial" constraints to a dual problem of maximizing a function of the weights on each constraint.Posynomial functions can be defined as polynomials in several variables with positive coefficients in all terms and the power to which the variables are raised can be any real number.
The mathematical formulation of problem ( 5) can be rewritten as: ) If q = 1, let the function qh ψ be define as,

(
), , where q is the number of functions in objective function.The above expression (6) can be expressed in the standard Primal GPP as follows: ( ) ( ) where ( ) and ( ) functions, where the posynomial function is given as: where qh ξ are normalized constants.If q = 1, let the function qh ξ be define as, where q is the total number of functions in the constraint.
If q be the number of terms in the problem.Then the number of posynomial terms in objective function can be denoted by qh.For the above problem of sample surveys, q = 2 as h h r and n are two different variables corresponding to the th h strata.Therefore, the total number of posynomial terms for the discussed problem will be 2h and h= 1, 2,…, L.
Similarly, the total numbers of posynomial terms corresponding to the primal constraint are denoted by 2h as q ih ih j p iii w w i q and h L iv are the dual variables corresponding to the objective functions and constraints functions.The above formulated MOGPP ( 9) can be solved in the following two-steps: Step 1: For the Optimum value of the objective function, the objective function always takes the form: The Multi-Objective objective function for our problem is: where and .are normalized constants.1, 2. ; 1, 2,..., .
Step 2: The equations that can be used for MOGPP for the weights are given below: ∑ ∑

FGP APPROACH IN SAMPLE SURVEYS IN PRESENCE OF NON-RESPONSE
The solution procedure to solve the problem (6) consists of the following steps: Step 1: Solve the MOPP as a single objective problem using only one objective at a time and ignoring the others.These solutions are known as ideal solution.
Step 2: From the results of step-1, determine the corresponding values for every objective at each solution derived.With the values of all objectives at each ideal solution, pay-off matrix can be formulated as follows:

=
Step 3: The membership function for the given problem can be define as:  The fuzzy multi-objective formulation of the problem can be defined as: ( ) integers, are , and 0 , The problem to find the optimal values of ( * * , r n ) for this there are two types of fuzzy decision operators and they (1) (i) Fuzzy decision based on max-min operator (like Zimmermann's approach (1978)).Therefore the problem ( 12) is reduced to the following problems according to max-min operator ( ) ( ) The above problem (14) The problem (15) The problem (16) has been solved with the help of steps (1-2) discuss in section (3) and the corresponding solutions * i w 0 is the unique solution to the dual constraints, it will also maximize the objective function for the dual problem.Next, the solution of the primal problem will be obtained using primal-dual relationship theorem which is given below: Primal-dual relationship theorem: If * i w 0 is a maximizing point for dual problem (9), each minimizing points ( )  (17) where L ranges over all positive integers for which . The optimal values of respondents * h n and non-respondents * h r can be calculated with the help of the primal -dual relationship theorem (17).

NUMERICAL ILLUSTRATION
A numerical example is given to demonstrate the proposed method.The values of Practically this ratio may vary from stratum to stratum and from characteristic to characteristic.Consider a population of size N = 3850 divided into four strata.The two characteristics are defined on each unit of the population and the population means are to be estimated.The available information is shown in the given table.
The dual of the above problem ( 18) is obtained as:      ,w ,w ,w ,w ,w ,w ; iv w ,w ,w ,w ,w ,w ,w ,w For orthogonality condition defined in expression 19(iii) are evaluated with the help of the payoff matrix which is defined below Using the primal dual-relationship theorem (17), we have the optimal solution of primal problem: i.e., the optimal sample sizes of respondents and non respondents are computed as follows:  ( ) The dual of the above problem ( 20) is obtained as follows: For orthogonality condition defined in expression 21(iii) are evaluated with the help of the payoff matrix which is defined below: On applying the max-addition operator, the MOGPP, the standard primal problem reduces to the crisp problem as:  In order to maximize the above problem, we have to minimize , subject to the constraints as described below:   w ,w ,w ,w ,w ,w ,w ,w ; w ,w ,w ,w ,w ,w ,w ,w For orthogonality condition defined in expression 24(iii) are evaluated with the help of the payoff matrix which is defined below: and the objective value of the primal problem is 42.06568.

CONCLUSION
This paper provides a profound study of fuzzy programming for solving the multi-objective geometric programming problem (MOGPP).The problem of non-response in multivariate stratified sample survey has been formulated as MOGPP and solution obtained.The obtained solution of MOGPP is dual solution corresponding to the problem of non-response in multivariate stratified sample surveys (primal problem).Therefore next, we obtained the optimum allocation of sample sizes of respondents and non respondents with the help of dual solutions MOGPP and primal-dual relationship theorem.To ascertain the practical utility of the proposed method in sample surveys problem in presence of non-response a numerical example is also given to illustrate the procedure.

Remark:
The authors are grateful to the Editor -in -Chief and to the learned referees for their highly constructive suggestions that brings the earlier manuscript in the present form.
the sizes of the respondents.

:
be the sizes of non respondents groups.

hn: 1 hn
Units are drawn from the th h stratum.Further let out of h n , units belong to the respondents group.

1 :
The total sample size.
per unit cost of making the first attempt, unit cost for processing the results of all the p characteristics on the 1 h n selected units from respondents group in the th h stratum in the first attempt and ∑

1 (
Normality condition ,see 9(ii)) and for each primal variable i i r n and having qh terms.

Figure 1 :
Figure 1: Membership function for minimization variance problem ) (ii) Convex-fuzzy decision based on addition operator (like Tewari et al. (1987)).Therefore the problem (12) is reduced according to max-addition operator as values.Therefore the problem (15) reduce into the problem (16) define as values on some previous occasion may be used.It is assumed that the relative values of the stratum variances among the non respondents at the second attempt to the corresponding over all stratum variances are 1, 2,…,L and j = 1,2,…, p.This ratio has been taken as 0.25 in the example for the sake of simplicity.

i
In expression(18), we first keep the r constant and calculate the values of n as: the expression (13), we keep the n constant and calculate the values of r as:The optimal values and the objective function value are given below: formulated dual problems, we have the corresponding solution as: sizes of the primal problems can be calculated with the help of the primal -dual relationship theorem (17) as we have calculated in the sub-problem 1are given as follows:

Figure 2 :
Figure 2: The figure illustrate the graph of the fuzzy membership function

Figure 2 :
Figure 2: The figure illustrate the graph of the fuzzy membership function formulated dual problem(24) using lingo software we obtain the following values of the dual variables which are given as: of the sample sizes of the primal problems can be calculated with the help of the primal -dual relationship theorem (17) as we have calculated in the sub-problem 1are given as follows: IJOR Vol. 12, No. 2, 021−035 (2015

Shafiullah and Bari:
Fuzzy Geometric Programming Approach in Multi-objective Multivariate Stratified Sample Surveys in Presence of Non -Response

Table 1 :
Data for four Strata and two characteristicsFor solving MOGPP by using fuzzy programming, we shall first solve the two sub-problems: Sub problem1: On substituting the table values in sub-problem 1, we have obtained the expressions given below:  On substituting the table values in sub-problem 2, we have obtained the expressions given below: Sub problem1: + Shafiullah and Bari: Fuzzy Geometric Programming Approach in Multi-objective Multivariate Stratified Sample Surveys in Presence of Non -Response