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

Abstract

The problem of optimum allocation in stratified sampling and its solution is well known in sampling literature for univariate populations (see Cochran, 1977; Sukhatme et al., 1984). 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, 1957), Ghosh (1958), Yates (1960), Aoyama (1963), Gren (1964, 1966), Folks and Antle (1965), Hartley (1965), Kokan and Khan (1967), Chatterjee (1972), Ahsan and Khan (1977, 1982), Chromy (1987), Wywial (1988), Bethel (1989), Kreienbrock (1993), Jahan et al. (1994), Khan et al. (1997), Khan et al. (2003), Ahsan et al. (2005), Díaz-García and Ulloa (2006, 2008), Ahsan et al. (2009) 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). 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.