Optimum Multivariate Stratified Sampling Designs with Travel Cost: A Multiobjective Integer Nonlinear Programming Approach

The problem of optimum allocation in stratified sampling and its solution is well known in sampling literature for univariate populations (see Cochran, 1977 Cochran , W. G. ( 1977 ). Sampling Techniques. , 3rd ed. New York : Wiley . [Google Scholar]; Sukhatme et al., 1984 Sukhatme , P. V. , Sukhatme , B. V. , Sukhatme , S. , Ashok , C. ( 1984 ). Sampling Theory of Surveys With Applications. , 3rd ed. Ames , and New Delhi : Iowa State University Press and Indian Society of Agricultural Statistics . [Google Scholar]). In multivariate populations where more than one characteristics are to be studied on every selected unit of the population the problem of finding an optimum allocation becomes more complex due to conflicting behaviour of characteristics. Various authors such as Dalenius (1953 Dalenius , T. ( 1953 ). The multivariate sampling problem . Skandinavisk Actuarietidskrift 36 : 92 – 102 . [Google Scholar], 1957 Dalenius , T. ( 1957 ). Sampling in Sweden. Contributions to the Methods and Theories of Sample Survey Practice . Stockholm : Almqvist and Wicksell . [Google Scholar]), Ghosh (1958 Ghosh , S. P. ( 1958 ). A note on stratified random sampling with multiple characters . Calcutta Statistical Association Bulletin 8 : 81 – 89 . [Google Scholar]), Yates (1960 Yates , F. ( 1960 ). Sampling Methods for Censuses and Surveys. , 3rd ed. London : Charles Griffin . [Google Scholar]), Aoyama (1963 Aoyama , H. ( 1963 ). Stratified random sampling with optimum allocation for multivariate populations . Annals of the Institute of Statistical Mathematics 14 : 251 – 258 .[Crossref], [Web of Science ®] , [Google Scholar]), Gren (1964 Gren , J. ( 1964 ). Some methods of sample allocation in multivariate stratified sampling . Przeglad Statystyczny 11 : 361 – 369 (in Polish) . [Google Scholar], 1966 Gren , J. ( 1966 ). Some application of non-linear programming in sampling methods . Przeglad Statystyczny 13 : 203 – 217 (in Polish) . [Google Scholar]), Folks and Antle (1965 Folks , J. L. , Antle , C. E. ( 1965 ). Optimum allocation of sampling units to the strata when there are r responses of interest . Journal of American Statistical Association 60 : 225 – 233 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Hartley (1965 Hartley , H. O. (1965). Multiple purpose optimum allocation in stratified sampling. Proc. Amer. Statist. Assoc. Social Statist. Sec. 258–261. [Google Scholar]), Kokan and Khan (1967 Kokan , A. R. , Khan , S. U. ( 1967 ). Optimum allocation in multivariate surveys: An analytical solution . Journal of Royal Statistical Society, Ser. B 29 : 115 – 125 . [Google Scholar]), Chatterjee (1972 Chatterjee , S. ( 1972 ). A study of optimum allocation in multivariate stratified surveys . Skandinavisk Actuarietidskrift 55 : 73 – 80 . [Google Scholar]), Ahsan and Khan (1977 Ahsan , M. J. , Khan , S. U. ( 1977 ). Optimum allocation in multivariate stratified random sampling using prior information . Journal of Indian Statistical Association 15 : 57 – 67 . [Google Scholar], 1982 Ahsan , M. J. , Khan , S. U. ( 1982 ). Optimum allocation in multivariate stratified random sampling with overhead cost . Metrika 29 : 71 – 78 .[Crossref] , [Google Scholar]), Chromy (1987 Chromy , J. R. ( 1987 ). Design optimization with multiple objectives. Proceedings of the Survey Research Methods, 194–199 . [Google Scholar]), Wywial (1988 Wywial , J. ( 1988 ). Minimizing the spectral radius of means vector from sample variance-covariance matrix sample allocation between strata. Prace Naukowe Akademii Ekonomicznej we Wroclawiu 404:223–235 (in Polish) . [Google Scholar]), Bethel (1989 Bethel , J. ( 1989 ). Sample allocation in multivariate surveys . Survey Methodology 15 : 47 – 57 . [Google Scholar]), Kreienbrock (1993 Kreienbrock , L. ( 1993 ). Generalized measures of dispersion to solve the allocation problem in multivariate stratified random sampling . Communication in Statistics—Theory and Methds 22 : 219 – 239 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]), Jahan et al. (1994 Jahan , N. , Khan , M. G. M. , Ahsan , M. J. ( 1994 ). A generalized compromise allocation . Journal of the Indian Statistical Association 32 : 95 – 101 . [Google Scholar]), Khan et al. (1997 Khan , M. G. M. , Ahsan , M. J. , Jahan , N. ( 1997 ). Compromise allocation in multivariate stratified sampling: An integer solution . Naval Research Logistics 44 : 69 – 79 .[Crossref], [Web of Science ®] , [Google Scholar]), Khan et al. (2003 Khan , M. G. M. , Khan , E. A. , Ahsan , M. J. ( 2003 ). An optimal multivariate stratified sampling design using dynamic programming . Australian & New Zealand J. Statist. 45 : 107 – 113 .[Crossref], [Web of Science ®] , [Google Scholar]), Ahsan et al. (2005 Ahsan , M. J. , Najmussehar, Khan , M. G. M. ( 2005 ). Mixed allocation in stratified sampling . Aligarh Journal of Statistics 25 : 87 – 97 . [Google Scholar]), Díaz-García and Ulloa (2006 Díaz-García , J. A. , Ulloa , C. L. ( 2006 ). Optimum allocation in multivariate stratified sampling: Multi-objective programming. Comunicación Técnica No. I-06-07/28-03-206 (PE/CIMAT), Guanajuato, México . [Google Scholar], 2008 Díaz-García , J. A. , Ulloa , C. L. ( 2008 ). Multi-objective optimization for optimum allocation in multivariate stratified sampling . Survey Methodology 34 : 215 – 222 .[Web of Science ®] , [Google Scholar]), Ahsan et al. (2009 Ansari , A. H. , Najmussehar, Ahsan , M. J. ( 2009 ). On multiple response stratified random sampling design . International Journal of Statistical Sciences , Kolkata, India, 1(1):1–11 . [Google Scholar]) etc. used different compromise criteria to work out a compromise allocation that is optimum for all characteristics in some sense.

Almost all the previous authors used some function of the sampling variances of the estimators of various characteristics to be measured as an objective that is to be minimized for a fixed cost given as a linear function of sample allocations. Because the variances are not unit free it is more logical to consider the minimization of some function of squared coefficient of variations as an objective. Previously this concept was used by Kozok (2006 Kozok , M. ( 2006 ). On sample allocation in multivariate surveys . Communication in Statistics—Simulation and Computation 35 : 901 – 910 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]).

Furthermore, investigators have to approach the sampled units in order to get the observations. This involves some travel cost. Usually this cost is neglected while constructing a cost function. This travel cost may be significant in some surveys. For example if the strata consist of some geographically difficult-to-approach areas.

The authors problem of optimum allocation in multivariate stratified sampling is discussed with an objective to minimize simultaneously the coefficients of variation of the estimators of various characteristics under a cost constraint that includes the measurement as well as travel cost. The formulated problem of obtaining an optimum compromise allocation turns out to be a multiobjective all-integer nonlinear programming problem. Three different approaches are considered: the value function approach, ∈ –constraint method, and Distance–based method, to obtain compromise allocations. The cost function considered also includes the travel cost within stratum to reach the selected units. Additional restrictions are placed on the sample sizes to avoid oversampling and ensure the availability of the estimates of the strata variances. Numerical examples are also presented to illustrate the computational details of the proposed methods.     

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