Technical details of the optimization procedure • orce

We are faced with an optimal allocation problem, where we need to assign Primary Sampling Units (PSUs) to agencies in a way that minimizes total costs, including travel costs and fixed costs associated with each agency. This optimization model is largely based on “The Warehouse Location Problem”, by Dirk Schumacher. It’s a common problem in logistics that seeks to determine the best location for facilities to minimize costs.

Given the locations of the PSUs and the potential agencies, the task is to decide which agencies to include/train/hire, and which PSUs will be allocated to which agencies.

In other words, we have to decide simultaneously: (1) which agencies to activate, and (2) the distribution of PSUs per agency.

We start with a set of PSUs $U = \{1 \ldots n\}$ and a set of potential agencies $A = \{1 \ldots m\}$ that could be activated. We also have a cost function that provides the travel cost from an agency to a PSU. In addition, there is a fixed cost (including training costs, among others) associated with each agency if it is selected for data collection. Agencies with a small number of PSUs may be unfeasible. Agencies in the countryside with a large number of PSUs may also be unfeasible. The solution must have at least min_upas and at most max_upas per activated agency. Note that when allowing “semi-centralized” collection, there is no limit to the number of PSUs in the listed agencies.

To model this situation, we use two decision variables:

$x_{i,j}$ : a binary variable that is 1 if PSU $i$ is allocated to agency $j$ , and 0 otherwise.
$y_j$ : a binary variable that is 1 if agency $j$ is selected to perform the collection, and 0 otherwise.

$$ \begin{equation*} \begin{array}{ll@{}ll} \text{minimize} & \displaystyle\sum\limits_{i=1}^{n}\sum\limits_{j=1}^{m}\operatorname{travel\_cost}_{i,j} \cdot x_{i, j} + \sum\limits_{j=1}^{m}\operatorname{fixed\_cost}_{j} \cdot y_{j}& &\\ \text{subject to} & \displaystyle\sum\limits_{j=1}^{m} x_{i, j} = 1 & i=1 ,\ldots, n&\\ & \displaystyle x_{i, j} \leq y_j, & i=1 ,\ldots, n & j=1 ,\ldots, m&\\ & x_{i,j} \in \{0,1\} &i=1 ,\ldots, n, & j=1 ,\ldots, m \\ & y_{j} \in \{0,1\} &j=1 ,\ldots, m& \\ & \operatorname{(optional)} \sum\limits_{i=1}^{n}{x}_{i,j} >= ( \operatorname{min\_upas} \cdot y_{j}) & j=1 ,\ldots, m& \\ & \operatorname{(optional)} \sum\limits_{i=1}^{n}{x}_{i,j} <= \operatorname{max\_upas}_{j} & j=1 ,\ldots, m& \end{array} \end{equation*} $$ The optimization procedure is implemented in the alocar_uc function.