Quantifying uncertainty in Pareto fronts arising from spatial data

Multi-objective spatial optimization problems require spatial data input that can contain uncertainties. Via the validation of constraints and the computation of objective values this uncertainty propagates to the Pareto fronts. Here, we develop a method to quantify the uncertainty in Pareto fronts by finding the extreme lower and upper bound of the range of optimal values in the objective space, i.e. the Pareto interval. The method is demonstrated on a land use allocation problem with initial land use (for objectives and constraints) and soil fertility (for one objective) as uncertain input data. Pareto intervals resulting from uncertain land use data were wide and irregularly shaped, whereas the ones from uncertain soil data were narrow and regularly shaped. Furthermore, in some objective-space regions, optimal land use patterns remained relatively stable under uncertainty, while elsewhere they were clouded. This information can be used to select solutions robust to spatial input data uncertainty.