SAMPLE ALLOCATION PROBLEM IN MULTI-OBJECTIVE MULTIVARIATE STRATIFIED SAMPLE SURVEYS UNDER TWO STAGE RANDOMIZED RESPONSE MODEL

SAMPLE ALLOCATION PROBLEM IN MULTI-OBJECTIVE MULTIVARIATE STRATIFIED SAMPLE SURVEYS UNDER TWO STAGE RANDOMIZED RESPONSE MODEL YASHPAL SINGH RAGHAV, M. FAISAL KHAN AND S. KHALIL Department of Mathematics, Jazan University, Jazan, KSA College of Science & Theoretical Studies, Saudi Electronic University, Riyadh, KSA Department of Community Medicine, J.N. Medical College, Aligarh Muslim University, Aligarh, India


Introduction
In a questionnaire survey, if a question is highly sensitive or personal, the person may refuse to answer or may give evasive answer.To get response on such question the interviewer must encourage the truthful answers without revealing the identity of the person interviewed.Warner [1965] has developed his randomized response technique which is designed to eliminate evasive answers bias by reducing the rate of non-response keeping the respondents confidentiality.The Warner's model requires the interviewee to give a "Yes" or "No" answer either to a sensitive question or to its negative, depending on the outcome of a randomizing device not disclosed to the interviewer.
Mangat and Singh [1990] proposed a two-stage randomized response model in which each interviewee (who is selected using simple random sampling with replacement) is provided with two randomization devices.Applicability of this model has been illustrated by Singh and Mangat [1996].Mangat and Singh [1994] proposed another randomized response model which has the benefit of simplicity over that of Mangat and Singh [1990].Hong et al. [1994] suggested a stratified randomized response technique using a proportional allocation.It may be easy to derive the variance of the proposed estimator.However, it may cause a high cost in terms of time, effort and money because of the difficulty in obtaining a proportional sample from some stratum.Kim and Warde [2004] presented a stratified randomized response technique using an optimal allocation which is more efficient than that using a proportional.Ghufran et al. [2012] discussed the applicability of Warner's technique [1965].
When a single sensitive question with a dichotomous response is under analysis, several randomised response models have been introduced in the literature, starting from the pioneering randomised response model introduced by Warner [1965].Non-exhaustive list of such randomized response models is given in Chaudhuri and Saha [2005], Diana and Perri [2009], Huang [2006] and others.
In many applications of the randomized response technique more than one sensitive issues are under analysis i.e. multiple sensitive question settings are to be considered.When information on more than one characteristic is to be obtained on each unit of the selected sample, it is not feasible to use the individual optimum allocations in various strata unless there is a strong correlation between the characteristics under study Cochran [1977].Thus one has to use an allocation that is optimum in some sense for all the characteristics.
For a population the coefficient of variation (CV) is represented by the ratio of population standard deviation to the population mean.The coefficient of variation is used to compare the precision of various estimates that are measured in different units.Ostle [1954] found that the population coefficient of variation is an ideal device for comparing the variation in two series of data that are measured in two different units.
In real life situations we face problems with multiple objectives.Generally, objectives are conflicting in nature, so simultaneous optimization of objectives is not possible.There are several approaches in the literature through which these can be converted to single objective problems.On solving this single objective problem a set of non-dominated solutions is obtained from which an optimal compromise solution is chosen.An optimal compromise solution is that feasible solution which is preferred by decision maker (DM) on all other feasible solution, taking into consideration all criteria contained in the multi-objective functions.[1961] introduced goal programming technique to solve multi objective problems.Haimes et al [1971] introduced   constraint technique which deals multi objective problems by selecting one of the objective functions to be optimized and the remaining objective functions are converted into constraints by setting an upper bound to each of them see Rios [1989] and Miettinen [1999].The weighting method by Gass and Saaty (1955) and Zadeh (1963) introduced the objective function with a weighting coefficient and minimize the weighted sum of objectives.The Tchebycheff Method proposed by Steuer (1986Steuer ( , 1989) is an interactive weighting vector space reduction method.Value function method can be very useful if the DM could reliably express the value function see Dyer and Sarin [1981] and Miettinen [1999].In many situations, sufficient information about a variable is not available, or it is difficult to decide most important characteristic of the survey.In such situations, the distance-based technique is very useful see Steuer [1986] and Rios [1989].Khuri and Cornell [1986] also proposed another distance based technique.Fishburn [1974] widely examined lexicographic orders and utilities.SAMPLE ALLOCATION PROBLEM Lexicographic ordering technique is applied by arranging the objective function according to their importance see Panda et al. [2005] and Ali [2011].

Charnes and Cooper
In this paper, a lexicographic goal programming integrated with fixed priority 1 D -distances method is suggested for obtaining compromise allocation for multiple characteristics Warner's randomized response model.This problem is also solved by various existing methods namelythe value function approach, goal programming techniques,  -constraint method, distance- based method and Khuri & Cornel distance based method.A numerical example is also presented to illustrate the computational details for all methods.

Formulation of the problem
Under two-stage randomized response model at stage 1, an individual respondent selected in the sample from ith stratum of a stratified population is instructed to use the randomization device i R 1 which consists of the following two statements: (i) "I belong to sensitive group" and (ii) "Go to the randomization device i R 2 at the second stage" with known probabilities The probabilities of a "Yes" answer for where sij  is the proportion of respondents belonging to the sensitive group for jth characteristics in the ith stratum.The maximum likelihood estimator of sij  is given as where ij Y ˆ is the estimated proportion of "Yes" answers which follows a binomial distribution and i n denote the sample size from ith stratum.Expression (1) and ( 2) are from Mangat and Singh [1990]  Since i n are drawn independently from each stratum, the estimators for individual strata can be added to obtain the estimator for the overall population parameter.This gives the unbiased estimator of sj  , which is the population proportion of respondents belonging to the sensitive group for the j th characteristics, as are the strata weights.
Coefficients of variation for p characteristics are written as We are given that sj ˆ is the unbiased estimator of sj  .We have where The Multi-objective Integer Nonlinear Programming Problem (MINLPP) with linear cost constraint is given as (see Ghufran et al. (2013Ghufran et al. ( , 2014) ) : When the travel cost between the units is substantial, the cost constraint also becomes non-linear then problem (8) define as problem ( 9)

Solution Methods for RR Model
In this section we are given the procedures for solving the RR model problem by using various approaches of multi-objective optimization namely, Goal programming, Lexicographic goal programming, D1-Distances and the proposed approach.

Goal Programming
The goal programming is based on the basic idea to determine a feasible solution that minimizes the deviations from the goals.
(Same procedure will be followed for solving the Problem 9).

Lexicographic Goal Programming
Lexicographic goal programming is a special case of goal programming, in which the most important goals are optimised before the least important goals.Since the different objectives have different importance, we arrange them in lexicographic order according to their importance.
Here we consider the Randomise Response problem of sampling (Problem 8) with P objectives functions those having different priority levels.Here P! priorities structure can be made.suppose if highest priority is given to the characteristic which has the maximum coefficient of variation i.e.   , be in decreasing order of magnitude.Lexicographic goal programming approach requires solving first Problem 11: In the above Problem (12) the highest priority goals and constraints are considered first.If more than one solution is found for Problem (12), another goal programming problem is then formulated which takes into account the second priority goals and so on.This procedure is repeated until a unique solution is found gradually considering decreasing priority levels.If the minimum of problem ( 12 For the other different priority structure same procedure will be followed.Same procedure will be followed for the Problem (9).

D1 Distance Algorithm
This method is an extension of lexicographic goal programming.In this method the objectives functions are arranged in order of their priorities in different manners to generate set of priorities structures.An idle solution is then identified from these set of priorities structure.

SAMPLE ALLOCATION PROBLEM
The stepwise procedure of D1-Distance method for solving RR Model (Problem 8) with P objective functions is as follows: Step 1: Let us we have P objective functions then P! set of problems of different priority structure will generate and hence P! different solutions are obtained after solving P! problems.
be the P! number of solutions obtained in step 1.
Out of these solutions an idle is identified as follows:- Step 3: To obtain the best compromise solution, the following procedure is to be followed -First we define a distance function to obtain the distances of solutions from ideal solution and the solution with minimum distance is considered as optimal solution.Let the 1 D -distance from the ideal solution ) ,..., , ( Therefore, the optimal 1 D -distance from the ideal solution is given as  (Same procedure will be followed for solving the Problem 9).

Proposed Method (Fixed priority Ideal 1 D -Distances Method)
In this method the priorities of extremes are fixed i.e. in our case we put the first priority to the objective which has worst value (maximum value) and give last priority to the objective which has best objective value (minimum value) and the rest (P-2) objectives are solved under all possible combination in between these two fixed extreme priorities.Out of these solutions, an ideal solution is identified.Let where  ir d and  ir d are the under and over deviational variable respectively, (Same procedure will be followed for solving the Problem 9).

Numerical illustrations
The following data is taken from Ghufran et al. (2012)

Comparisons of Optimum allocation
In this section, a comparative study of the optimum allocations in table 4 and 5 have been given for the various existing approaches and proposed approach.The trace value of coefficient of variation for linear cost and quadratic cost function is summarised in Table 6 and Table 7 respectively.
and (ii) respectively.At stage 2 the respondents are instructed to use the randomization device i R 2 which consists of the following two statements: (i) "I belong to the sensitive group" and (ii) "I do not belong to the sensitive group" ) and (ii) respectively.
compromise allocation for the given problem.
with suffix ' i ' to denote the sijˆ is unbiased.
This optimisation programming technique is used to handle multiple, normally conflicting objectives.The use of the goal programming for decision making problems with several conflicting objectives was first introduced by Charnes and Cooper in 1961.
Thereafter various versions of goal programming have been proposed in the literature.Here we use goal programming technique to solve the Randomise Response problem (Problem 8).For this purpose we first solve separately the following p given objectives of problem 8 subject to the given set of constraints of Problem (8) to obtain the individual optimum solution.
) is ,  jG CV then in the next stage the problem must be solved for minimum values.Thus, final next problem to be solved is Problem (13):

Table 1
Calculations for ideal solutions

Table 2 :
-Distances from the ideal solution P.S.

:
Solve all the objectives for the given set of constraints ignoring other objectives.Fix the first priority to the objective having worst value and last priority to the objective having best value.

:
Rest priorities are given to other objectives subsequently.
Step5: Obtain Ideal solutionStep6: Calculate -distances of different solutions from the ideal solution.

Table 3 : Data for four Strata and four characteristics
as the case may be.Table3presents the relevant information.

Table 4 :
Summary of Results for Linear Cost Function

Table 5 :
Summary of Results for Non-Linear Cost Function