Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function
Affiliation(s)
Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India. Department of Computing and Informatics Saudi Electronics University, Riyadh, Saudi Arabia.
Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India. Department of Computing and Informatics Saudi Electronics University, Riyadh, Saudi Arabia.
ABSTRACT
In this paper, the problem of non-response with significant travel costs in multivariate stratified sample surveys has been formulated of as a Multi-Objective Geometric Programming Problem (MOGPP). The fuzzy programming approach has been described for solving the formulated MOGPP. The formulated MOGPP has been solved with the help of LINGO Software and the dual solution is obtained. The optimum allocations of sample sizes of respondents and non respondents are obtained with the help of dual solutions and primal-dual relationship theorem. A numerical example is given to illustrate the procedure.
KEYWORDS
Geometric Programming, Fuzzy Programming, Non-Response with Travel Cost, Optimum Allocations, Multivariate Stratified Sample Surveys
Geometric Programming, Fuzzy Programming, Non-Response with Travel Cost, Optimum Allocations, Multivariate Stratified Sample Surveys
Cite this paper
, S. , Khan, M. and Ali, I. (2014) Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function. American Journal of Operations Research, 4, 173-188. doi: 10.4236/ajor.2014.43017.
, S. , Khan, M. and Ali, I. (2014) Fuzzy Geometric Programming in Multivariate Stratified Sample Surveys in Presence of Non-Response with Quadratic Cost Function. American Journal of Operations Research, 4, 173-188. doi: 10.4236/ajor.2014.43017.
References
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http://dx.doi.org/10.1080/01621459.1946.10501894 [4] El-Badry, M.A. (1956) A Sampling Procedure for Mailed Questionnaires. Journal of the American Statistical Association, 51, 209-227.
http://dx.doi.org/10.1080/01621459.1956.10501321 [5] Fordori, G.T. (1961) Some Non-Response Sampling Theory for Two Stage Designs. Mimeograph Series No. 297, North Carolina State University, Raleigh. [6] Srinath, K.P. (1971) Multiple Sampling in Non-Response Problems. Journal of the American Statistical Association, 66, 583-586.
http://dx.doi.org/10.1080/01621459.1971.10482310 [7] Khare, B.B. (1987) Allocation in Stratified Sampling in Presence of Non-Response. Metron, 45, 213-221. [8] Khan, M.G.M., Khan, E.A. and Ahsan, M.J. (2008) Optimum Allocation in Multivariate Stratified Sampling in Presence of Non-Response. Journal of the Indian Society of Agricultural Statistics, 62, 42-48. [9] Varshney, R., Ahsan, M.J. and Khan, M.G.M. (2011) An Optimum Multivariate Stratified Sampling Design with NonResponse: A Lexicographic Goal Programming Approach. Journal of Mathematical Modeling and Algorithms, 65, 291-296. [10] Fatima, U. and Ahsan, M.J. (2011) Nonresponse in Stratified Sampling: A Mathematical Programming Approach. The South Pacific Journal of Natural and Applied Sciences, 29, 40-42. [11] Varshney, R., Najmussehar and Ahsan, M.J. (2012) An Optimum Multivariate Stratified Double Sampling Design in Presence of Non-Response. Optimization Letters, 6, 993-1008. [12] Raghav, Y.S., Ali, I. and Bari, A. (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in Case of Non-Response. Journal of Statistical Computation and Simulation, 84, 22-36.
http://dx.doi.org/10.1080/00949655.2012.692370 [13] Duffin, R.J., Peterson, E.L. and Zener, C. (1967) Geometric Programming: Theory & Applications. John Wiley & Sons, New York. [14] Zener, C. (1971) Engineering Design by Geometric Programming. John Wiley & Sons, New York. [15] Beightler, C.S. and Philips, D.T. (1976) Applied Geometric Programming. Wiley, New York. [16] Davis, M. and Schwartz, R.S. (1987) Geometric Programming for Optimal Allocation of Integrated Samples in Quality Control. Communications in Statistics—Theory and Methods, 16, 3235-3254.
http://dx.doi.org/10.1080/03610928708829568 [17] Ahmed, J. and Bonham, C.D. (1987) Application of Geometric Programming to Optimum Allocation Problems in Multivariate Double Sampling. Applied Mathematics and Computation, 21, 157-169.
http://dx.doi.org/10.1016/0096-3003(87)90024-5 [18] Ojha, A.K. and Das, A.K. (2010) Multi-Objective Geometric Programming Problem Being Cost Coefficients as Continuous Function with Weighted Mean. Journal of Computing, 2, 2151-9617. [19] Maqbool, S., Mir, A.H. and Mir, S.A. (2011) Geometric Programming Approach to Optimum Allocation in Multivariate Two-Stage Sampling Design. Electronic Journal of Applied Statistical Analysis, 4, 71-82. [20] Shafiullah, Ali, I. and Bari, A. (2013) Geometric Programming Approach in Three—Stage Sampling Design. International Journal of Scientific & Engineering Research (France), 4, 2229-5518. [21] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.
http://dx.doi.org/10.1016/S0019-9958(65)90241-X [22] Bellman, R.E. and Zadeh, L.A. (1970) Decision-Making in a Fuzzy Environment. Management Sciences, 17, B141B164.
http://dx.doi.org/10.1287/mnsc.17.4.B141 [23] Tanaka, H., Okuda, T. and Asai, K. (1974) On Fuzzy-Mathematical Programming. Journal of Cybernetics, 3, 37-46.
http://dx.doi.org/10.1080/01969727308545912 [24] Zimmermann, H.J. (1978) Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems, 1, 45-55.
http://dx.doi.org/10.1016/0165-0114(78)90031-3 [25] Biswal, M.P. (1992) Fuzzy Programming Technique to Solve Multi-Objective Geometric Programming Problems. Fuzzy Sets and Systems, 51, 67-71.
http://dx.doi.org/10.1016/0165-0114(92)90076-G [26] Verma, R.K. (1990) Fuzzy Geometric Programming with Several Objective Functions. Fuzzy Sets and Systems, 35, 115-120.
http://dx.doi.org/10.1016/0165-0114(90)90024-Z [27] Islam, S. and Roy, T.K. (2005) Modified Geometric Programming Problem and Its Applications. Journal of Applied Mathematics and Computing, 17, 121-144. [28] Islam, S. (2010) Multi-Objective Geometric Programming Problem and Its Applications. Yugoslav Journal of Operations Research, 20, 213-227.
http://dx.doi.org/10.2298/YJOR1002213I [29] Tiwari, R.N., Dharman, S. and Rao, J.R. (1987) Fuzzy Goal Programming—An Additive Model. Fuzzy Sets and Systems, 24, 27-34.
http://dx.doi.org/10.1016/0165-0114(87)90111-4 [30] Lindo Systems Inc. (2013) LINGO User’s Guide. Lindo Systems Inc., Chicago.
[1] Cochran, W.G. (1977) Sampling Techniques. 3rd Edition, John Wiley and Sons, New York. [2] Sukhatme, P.V., Sukhatme, B.V., Sukhatme, S. and Asok, C. (1984) Sampling Theory of Surveys with Applications. Iowa State University Press, Ames and Indian Society of Agricultural Statistics, New Delhi. [3] Hansen, M.H. and Hurwitz, W.N. (1946) The Problem of Non-Response in Sample Surveys. Journal of the American Statistical Association, 41, 517-529.
http://dx.doi.org/10.1080/01621459.1946.10501894 [4] El-Badry, M.A. (1956) A Sampling Procedure for Mailed Questionnaires. Journal of the American Statistical Association, 51, 209-227.
http://dx.doi.org/10.1080/01621459.1956.10501321 [5] Fordori, G.T. (1961) Some Non-Response Sampling Theory for Two Stage Designs. Mimeograph Series No. 297, North Carolina State University, Raleigh. [6] Srinath, K.P. (1971) Multiple Sampling in Non-Response Problems. Journal of the American Statistical Association, 66, 583-586.
http://dx.doi.org/10.1080/01621459.1971.10482310 [7] Khare, B.B. (1987) Allocation in Stratified Sampling in Presence of Non-Response. Metron, 45, 213-221. [8] Khan, M.G.M., Khan, E.A. and Ahsan, M.J. (2008) Optimum Allocation in Multivariate Stratified Sampling in Presence of Non-Response. Journal of the Indian Society of Agricultural Statistics, 62, 42-48. [9] Varshney, R., Ahsan, M.J. and Khan, M.G.M. (2011) An Optimum Multivariate Stratified Sampling Design with NonResponse: A Lexicographic Goal Programming Approach. Journal of Mathematical Modeling and Algorithms, 65, 291-296. [10] Fatima, U. and Ahsan, M.J. (2011) Nonresponse in Stratified Sampling: A Mathematical Programming Approach. The South Pacific Journal of Natural and Applied Sciences, 29, 40-42. [11] Varshney, R., Najmussehar and Ahsan, M.J. (2012) An Optimum Multivariate Stratified Double Sampling Design in Presence of Non-Response. Optimization Letters, 6, 993-1008. [12] Raghav, Y.S., Ali, I. and Bari, A. (2014) Multi-Objective Nonlinear Programming Problem Approach in Multivariate Stratified Sample Surveys in Case of Non-Response. Journal of Statistical Computation and Simulation, 84, 22-36.
http://dx.doi.org/10.1080/00949655.2012.692370 [13] Duffin, R.J., Peterson, E.L. and Zener, C. (1967) Geometric Programming: Theory & Applications. John Wiley & Sons, New York. [14] Zener, C. (1971) Engineering Design by Geometric Programming. John Wiley & Sons, New York. [15] Beightler, C.S. and Philips, D.T. (1976) Applied Geometric Programming. Wiley, New York. [16] Davis, M. and Schwartz, R.S. (1987) Geometric Programming for Optimal Allocation of Integrated Samples in Quality Control. Communications in Statistics—Theory and Methods, 16, 3235-3254.
http://dx.doi.org/10.1080/03610928708829568 [17] Ahmed, J. and Bonham, C.D. (1987) Application of Geometric Programming to Optimum Allocation Problems in Multivariate Double Sampling. Applied Mathematics and Computation, 21, 157-169.
http://dx.doi.org/10.1016/0096-3003(87)90024-5 [18] Ojha, A.K. and Das, A.K. (2010) Multi-Objective Geometric Programming Problem Being Cost Coefficients as Continuous Function with Weighted Mean. Journal of Computing, 2, 2151-9617. [19] Maqbool, S., Mir, A.H. and Mir, S.A. (2011) Geometric Programming Approach to Optimum Allocation in Multivariate Two-Stage Sampling Design. Electronic Journal of Applied Statistical Analysis, 4, 71-82. [20] Shafiullah, Ali, I. and Bari, A. (2013) Geometric Programming Approach in Three—Stage Sampling Design. International Journal of Scientific & Engineering Research (France), 4, 2229-5518. [21] Zadeh, L.A. (1965) Fuzzy Sets. Information and Control, 8, 338-353.
http://dx.doi.org/10.1016/S0019-9958(65)90241-X [22] Bellman, R.E. and Zadeh, L.A. (1970) Decision-Making in a Fuzzy Environment. Management Sciences, 17, B141B164.
http://dx.doi.org/10.1287/mnsc.17.4.B141 [23] Tanaka, H., Okuda, T. and Asai, K. (1974) On Fuzzy-Mathematical Programming. Journal of Cybernetics, 3, 37-46.
http://dx.doi.org/10.1080/01969727308545912 [24] Zimmermann, H.J. (1978) Fuzzy Programming and Linear Programming with Several Objective Functions. Fuzzy Sets and Systems, 1, 45-55.
http://dx.doi.org/10.1016/0165-0114(78)90031-3 [25] Biswal, M.P. (1992) Fuzzy Programming Technique to Solve Multi-Objective Geometric Programming Problems. Fuzzy Sets and Systems, 51, 67-71.
http://dx.doi.org/10.1016/0165-0114(92)90076-G [26] Verma, R.K. (1990) Fuzzy Geometric Programming with Several Objective Functions. Fuzzy Sets and Systems, 35, 115-120.
http://dx.doi.org/10.1016/0165-0114(90)90024-Z [27] Islam, S. and Roy, T.K. (2005) Modified Geometric Programming Problem and Its Applications. Journal of Applied Mathematics and Computing, 17, 121-144. [28] Islam, S. (2010) Multi-Objective Geometric Programming Problem and Its Applications. Yugoslav Journal of Operations Research, 20, 213-227.
http://dx.doi.org/10.2298/YJOR1002213I [29] Tiwari, R.N., Dharman, S. and Rao, J.R. (1987) Fuzzy Goal Programming—An Additive Model. Fuzzy Sets and Systems, 24, 27-34.
http://dx.doi.org/10.1016/0165-0114(87)90111-4 [30] Lindo Systems Inc. (2013) LINGO User’s Guide. Lindo Systems Inc., Chicago.