A decision tool based on bilevel optimization for the allocation of water resources in a hierarchical system

This paper addresses the optimal allocation of water among competing stakeholders during a finite planning horizon. We focus on those water systems where there are two levels of decision making organized according to a hierarchical framework. At the upper level, a central authority allocates water to demand points having regard to environmental and sustainability issues as well as balancing water users' supply/demand. At the lower level of the hierarchy, demand point managers allocate water to users prioritizing economic strategies. On the other hand, when it comes to allocating limited resources that affect public welfare, the authority in charge can also use different political instruments such as fees to influence the decisions made at those levels of decision making that are not directly within its competence. We propose a multiobjective multifollower bilevel optimization problem that aims to fulfill the central authority goals while including the reaction of the demand point managers in terms of optimization problems as constraints. Using the well-known Karush-Kuhn-Tucker approach, we transform the bilevel model into an equivalent multiobjective mixed-integer single-level model for which we provide tight big-M values. For the purpose of showing the versatility of the model, extensive computational experiments on a set of instances have been carried out. The results show that the optimization problem can be solved to optimality in small computing times using off-the-shelf mixed-integer solvers even for complex water systems and long planning periods. In addition, they illustrate the effect of imposing fees on the achievement of the central authority's objectives.