{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,2,21]],"date-time":"2025-02-21T14:44:32Z","timestamp":1740149072452,"version":"3.37.3"},"reference-count":22,"publisher":"Wiley","license":[{"start":{"date-parts":[[2016,1,1]],"date-time":"2016-01-01T00:00:00Z","timestamp":1451606400000},"content-version":"unspecified","delay-in-days":0,"URL":"http:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001501","name":"University Grants Commission","doi-asserted-by":"publisher","award":["F.30-90\/2015 (BSR)"],"award-info":[{"award-number":["F.30-90\/2015 (BSR)"]}],"id":[{"id":"10.13039\/501100001501","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Advances in Fuzzy Systems"],"published-print":{"date-parts":[[2016]]},"abstract":"<jats:p>We consider an allocation problem in two-stage stratified Warner\u2019s 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<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M1\"><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:math>-cut method at a prescribed value of<mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\" id=\"M2\"><mml:mrow><mml:mi>\u03b1<\/mml:mi><\/mml:mrow><\/mml:math>. An illustrative numerical example is presented to demonstrate the proposed problem.<\/jats:p>","DOI":"10.1155\/2016\/4058758","type":"journal-article","created":{"date-parts":[[2016,3,10]],"date-time":"2016-03-10T16:02:23Z","timestamp":1457625743000},"page":"1-8","source":"Crossref","is-referenced-by-count":1,"title":["Two-Stage Stratified Randomized Response Model with Fuzzy Numbers"],"prefix":"10.1155","volume":"2016","author":[{"ORCID":"https:\/\/orcid.org\/0000-0001-5053-5028","authenticated-orcid":true,"given":"Mohammad Faisal","family":"Khan","sequence":"first","affiliation":[{"name":"College of Science and Theoretical Studies, Saudi Electronic University, P.O. 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