Optimal Sampling Design Under Logistical Constraints with Mixed Integer Programming

The goal of survey design is often to minimize the errors associated with inference: the total of bias and variance. Random surveys are common because they allow the use of theoretically unbiased estimators. In practice however, such design-based approaches are often unable to account for logistical or budgetary constraints. Thus, they may result in samples that are logistically inefficient, or infeasible to implement. Various balancing and optimal sampling techniques have been proposed to improve the statistical efficiency of such designs, but few models have attempted to explicitly incorporate logistical and financial constraints. We introduce a mixed integer linear program (MILP) for optimal sampling design, capable of capturing a variety of constraints and a wide class of Bayesian regression models. We demonstrate the use of our model on three spatial sampling problems of increasing complexity, including the real logistics of the US Forest Service Forest Inventory and Analysis survey of Tanana, Alaska. Our methodological contribution to survey design is significant because the proposed modeling framework makes it possible to generate high-quality sampling designs and inferences while satisfying practical constraints defined by the user. The technical novelty of the method is the explicit integration of Bayesian statistical models in combinatorial optimization. This integration might allow a paradigm shift in spatial sampling under constrained budgets or logistics.