Spanning the Pareto Frontier of Environmental Problems

Today, many complex multi-objective environmental problems are dealt with using genetic algorithms (GAs). They apply the development and adaptation mechanism of a natural population to a "numerical" population of solutions to optimize a fitness function. Such mechanisms, namely: selection, mutation, and cross-over, are all based on a form of random search. In this respect, GAs have to solve a multi-objective problem themselves since they must find a compromise between the breath of the search (to avoid being trapped into a local minimum) and its depth (to avoid a too rough approximation of the optimal solution). To deal with this dilemma, most algorithms use "elitism", which allows preserving some of the current best solutions in the successive generations. If the initial population is randomly selected, as in many GA packages, the elite may concentrate in a limited part of the Pareto frontier, where the objectives are continuous and monotonic with respect to the decision variables. This is not always the case in environmental problems so that this setting prevents a complete spanning of all the alternatives. A complete view of the frontier is possible if one, first, solves the single objective problems that correspond to the extremes of the Pareto boundary, and then uses such solutions as elite members of the initial population. The paper compares this approach with other more conventional initializations, fixing the same number of function evaluations. For this purpose, we use some classical test sets, the solution of which is known analytically, with two, three, and four objectives. Then we show the results of the proposed algorithm in the optimization of the releases of a real multi-reservoir system, contrasting its performances again with those of available packages. Using this example, we also briefly discuss the issues related to the classical performance measures in multi-objective optimization.