Cost optimal sampling for the integrated observation of different populations
Section 3. Problem

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Given the above framework, we are interested in finding the vector ${\pi }^A={\left({\pi }_1^A,\dots ,{\pi }_j^A,\dots , {\pi }_{M^A}^A\right)}^{\prime }$ of inclusion probabilities that minimizes the expected survey cost bounding the sampling variances, $V\left({\widehat{Y}}_v^A|m^A\right)$ $\left(v=1,\dots ,V\right)$ and $V\left({\widehat{Y}}_r^B|m^A\right)$ $\left(r=1,\dots ,R\right)$ under given variance constraints:

$$\{\begin{array}{l}\text{min}{\displaystyle \sum _{j\in U^A}}{c}_j{\pi }_j^A\hfill \\ V\left({\widehat{Y}}_v^A|m^A\right)\le V_v^{*}\forall v=1,\dots ,V\hfill \\ V\left({\widehat{Y}}_r^B|m^A\right)\le V_r^{*}\forall r=1,\dots ,R\hfill \\ 0<{\pi }_j^A\le 1\forall j=1,\dots ,M^A\hfill \end{array}(3.1)$$

where $m^A$ is given by the ${\pi }^A$ vector minimizing the cost function, $V_v^{*}$ $\left(v=1,\dots ,V\right)$ and $V_r^{*}$ $\left(r=1,\dots ,R\right)$ are the variance thresholds fixed by the sampling designer and $c_j$ is the variable cost for observing the unit $j$ in the population $U^A$ and the $L_j^A={\displaystyle {\sum }_{i=1}^{N^B}{L}_{j,i}^B}$ linked units in the population $U^B.$ In other words, we want to obtain the optimal selection probabilities that will minimize the variance of estimates obtained for both $U^A$ and $U^B.$ For the agricultural example, this would translate to developing optimal selection probabilities that will lead to estimates for the population of farms, as well as the population of rural households, with specified precision.

A reasonable expression of $c_j$ is

$$c_j=f_c\left(C^A,L_j^A;C^B\right),(3.2)$$

where $f_c$ is a known monotone non-decreasing function, $C^A$ is the per unit cost for observing a unit in the population $U^A$ and $C^B$ is the cost for observing the elementary unit in the population $U^B.$ Brewer and Gregoire (2009) propose an extensive analysis of different forms of costs functions.

The minimization problem (3.1) is a generalization of the univariate precision constrained optimization approach (Cochran, 1977). The problem (3.1) assumes that all the values $y_{j,v},$ $y_{i,r},$ $L_j^A,$ $L_{j,i}^B,$ $L_i^B,$ ${\beta }_v$ and ${\beta }_r$ are known. In this case, problem (3.1) becomes a classical Linear Convex Separate Problem (LCSP) (Boyd and Vanderberg, 2004) and it can be solved by the algorithm proposed Chromy (1987), originally developed for multivariate optimal allocation in an SSRSWOR design and implemented in standard software tools. (See for example the Mauss-R software available at: http://www3.istat.it/strumenti/metodi/software/campione/mauss_r/.) Alternatively, the LCSP can be dealt with by the SAS procedure NLP as suggested by Choudhry et al. (2012). The vectors ${\beta }_v$ and ${\beta }_r$ depend on the vector ${\pi }^A.$ Falorsi and Righi (2015) define a new algorithm which finds the optimal solution taking into account the dependence between ${\beta }_v$ and ${\beta }_r$ with the optimal vector ${\pi }^A.$

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