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Advances in Fuzzy Systems
Volume 2016, Article ID 4058758, 8 pages
Research Article

Two-Stage Stratified Randomized Response Model with Fuzzy Numbers

1College of Science and Theoretical Studies, Saudi Electronic University, P.O. Box 93499, Riyadh 11673, Saudi Arabia
2Department of Statistics and Operations Research, Aligarh Muslim University, Aligarh 202002, India

Received 16 November 2015; Revised 3 February 2016; Accepted 4 February 2016

Academic Editor: Ashok B. Kulkarni

Copyright © 2016 Mohammad Faisal Khan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


We consider an allocation problem in two-stage stratified Warner’s randomized response model and minimize the variance subject to cost constraint. The costs (measurement costs and total budget of the survey) in the cost constraint are assumed as fuzzy numbers, in particular triangular and trapezoidal fuzzy numbers due to the ease of use. The problem formulated is solved by using Lagrange multipliers technique and the optimum allocation obtained in the form of fuzzy numbers is converted into crisp form using -cut method at a prescribed value of . An illustrative numerical example is presented to demonstrate the proposed problem.