Allocation in Multivariate Stratified Surveys with Non-Linear Random Cost Function

In this paper, we consider an allocation problem in multivariate surveys with non-linear costs of enumeration as a problem of non-linear stochastic programming with multiple objective functions. The solution is obtained through Chance Constrained programming. A different formulation of the problem is also presented in which the non-linear cost function is minimised under the precision constraints on estimates of various characters. The solution is then obtained by using Modified E-model. A numerical example is solved for both the formulations.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

M. Khan, I. Ali, Y. Raghav and A. Bari, "Allocation in Multivariate Stratified Surveys with Non-Linear Random Cost Function," American Journal of Operations Research, Vol. 2 No. 1, 2012, pp. 100-105. doi: 10.4236/ajor.2012.21012.

 [1] J. Neyman, “On the Two Different Aspects of the Representative Method: The Method of Stratified Sampling and the Method of Purposive Selection,” Journal of the Royal Statistical Society, Vol. 97, No. 4, 1934, pp. 558-625. doi:10.2307/2342192 [2] T. Dalenius, “Sampling in Sweden: Contributions to the Methods and Theories of Sample Survey Practice,” Almqvist Och Wiksell, Stockholm, 1957. [3] S. P. Ghosh, “A Note on Stratified Random Sampling with Multiple Characters,” Calcutta Statistical Association Bulletin, Vol. 8, 1958, pp. 81-89. [4] F. Yates, “Sampling Methods for Censuses and Surveys,” 3rd Edition, Charles Griffin and Co., London, 1960. [5] H. Aoyama, “Stratified Random Sampling with Optimum Allocation for Multivariate Populations,” Annals of the Institute of Statistical Mathematics, Vol. 14, No. 1, 1963, pp. 251-258. doi:10.1007/BF02868647 [6] J. L. Folks and C. E. Antle, “Optimum Allocation of Sampling Units to the Strata When There Are R Responses of Interest,” Journal of the American Statistical Association, Vol. 60, No. 309, 1965, pp. 225-233. doi:10.2307/2283148 [7] A. R. Kokan and S. U. Khan, “Optimum Allocation in Multivariate Surveys: An Analytical Solution,” Journal of the Royal Statistical Society, Series B, Vol. 29, 1967, pp. 115-125. [8] S. Chatterji, “Multivariate Stratified Surveys,” Journal of the American Statistical Association, Vol. 63, No. 322, 1968, pp. 530-534. doi:10.2307/2284023 [9] M. J. Ahsan and S. U. Khan, “Optimum Allocation in Multivariate Stratified Random Sampling Using Prior Information,” Journal of Indian Statistical Association, Vol. 15, 1977, pp. 57-67. [10] N. Jahan, M. G. M. Khan and M. J. Ahsan, “A Generalized Compromise Allocation,” Journal of Indian Statistical Association, Vol. 32, No. 2, 1994, pp. 95-101. [11] M. G. M. Khan, M. J. Ahsan and N. Jahan, “Compromise Allocation in Multivariate Stratified Sampling: An Integer Solution,” Naval Research Logistics, Vol. 44, No. 1, 1997, pp. 69-79. doi:10.1002/(SICI)1520-6750(199702)44:1<69::AID-NAV4>3.0.CO;2-K [12] A. Prekopa, “Stochastic Programming,” Series Mathematics and Its Applications, Kluwer Academic Publishers, Berlin, 1995. [13] J. A. Diaz Garcia and M. M. Garay Tapia, “Optimum Allocation in Stratified Surveys: Stochastic Programming,” Computational Statistics and Data Analysis, Vol. 51, No. 6, 2007, pp. 3016-3026. doi:10.1016/j.csda.2006.01.016 [14] S. Javed, Z. H. Bakhshi and M. M. Khalid, “Optimum Allocation in Stratified Sampling with Random Costs,” International Review of Pure and Applied Mathematics, Vol. 5, No. 2, 2009, pp. 363-370. [15] Z. H. Bakhshi, M. F. Khan and Q. S. Ahmad, “Optimal Sample Numbers in Multivariate Stratified Sampling with a Probabilistic Cost Constraint,” International Journal of Mathematics and Applied Statistics, Vol. 1, No. 2, 2010, pp. 111-120. [16] J. Beardwood, J. H. Halton and J. M. Hammersley, “The Shortest Path through Many Points,” Mathematical Proceedings of the Cambridge Philosophical Society, Vol. 55, No. 4, 1959, pp. 299-327. doi:10.1017/S0305004100034095 [17] A. Charnes and W. W. Cooper, “Chance Constrained Programming,” Management Science, Vol. 6, No. 1, 1959, pp. 73-79. doi:10.1287/mnsc.6.1.73 [18] E. A. Khan, M. G. M. Khan and M. J. Ahsan, “On Compromise Allocation in Multivariate Stratified Sampling,” Aligarh Journal of Statistics, Vol. 23, 2003, pp. 31-47. [19] S. S. Rao, “Optimization-Theory and Applications,” Wily Eastern Limited, New Delhi, 1979. [20] S. Uryasev and P. M. Pardalos, “Stochastic Optimization,” Kluwer Academic Publishers, Dordrecht, 2001. [21] P. V. Sukhatme, B. V. Sukhatme, S. Sukhatme and C. Asok, “Sampling Theory of Surveys with Applications” 3rd Edition, Iowa State University Press, Ames, 1984. [22] Lindo Systems Inc., “LINGO User’s Guide,” Lindo Systems Inc., Chicago, 2001