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

Affiliation(s)

CSIRO Department of Mathematics, Integral University, Lucknow, India.

Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India.

CSIRO Department of Mathematics, Integral University, Lucknow, India.

Department of Statistics & Operations Research, Aligarh Muslim University, Aligarh, India.

ABSTRACT

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.

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.

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.

M. Khan, I. Ali, Y. Raghav and A. Bari, "Allocation in Multivariate Stratified Surveys with Non-Linear Random Cost Function,"

References

[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

[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