(393d) A Systematic Method for Reducing the Number of Objectives in Multi-Objective Optimization: Application to Environmental Problems

Multi-objective optimization is a useful technique in sustainability, as it can help in exploring and analyzing optimal trade-off solutions that balance economic, social and environmental criteria [1]. These methods provide as output a set of Pareto optimal alternatives from which decision-makers can choose the best ones according to their preferences. These so called Pareto solutions feature the property that it is not possible to find another feasible solution that is better in one of the objectives without necessarily worsening at least one of the others. One of the main advantages of this method is that it allows to articulate the decision-makers' preferences in the post-optimal analysis of the solutions found. By doing so, it is possible to indentify alternatives that achieve large environmental improvements at a marginal increase in the cost.

The main limitation of multi-objective optimization as applied to environmental problems is that its computational burden grows exponentially in size with the number of environmental objectives considered in the analysis. To circumvent this issue, it has been proposed to omit some of them, so the problem is kept in a manageable size. In this context, it is possible to define aggregated environmental indicators that can account for several environmental metrics without explicitly including them in the problem. These aggregated indicators are defined by assigning weights to the individual impact categories (e.g. global warming, acidification, ozone layer depletion, etc.) considered in the analysis (see for instance [2]). These weights, which should represent the views of the society or a group of stake holders, allow translating the whole range of environmental concerns into a single metric. This indicator can then be incorporated into a bi-criteria optimization framework (i.e. cost vs environmental impact) thus leading to two-dimensional Pareto sets that are easy to calculate and analyze.

To this end, the reduction and aggregation of objectives discussed above have relied on the decision-makers' preferences rather than on a systematic mathematical approach. However, it is well known that by omitting objectives one can change the dominance structure of the multi-objective problem. As a result, certain environmental problems might be given less importance or even left out of the analysis. Hence, at this point the question that arises is whether it is strictly necessary to include all the environmental metrics in the optimization problem or some of them can be omitted without losing information.

In this work we address this question and propose a systematic method for reducing the number of objectives in multi-objective optimization with emphasis on environmental problems. Particularly, given a specific environmental problem where a set of environmental metrics are to be minimized, we aim at computing a subset of objectives of given size such that the error of neglecting the remaining ones is minimized. We present both: (i) a mixed-integer linear programming (MILP) model that minimizes the error of projecting a set of Pareto solutions onto a space of lower dimension; and (ii) a rigorous procedure for obtaining upper bounds on the error associated with omitting objectives in any type of multi-objective optimization problem. We apply these techniques to two different environmental problems, discussing how they can help in identifying redundant environmental objectives thereby facilitating the calculation and analysis of the Pareto alternatives.

References

[1] Grossmann, I.E., Guillén-Gosálbez, G. Scope for the application of mathematical programming techniques in the synthesis and planning of sustainable processes. Computers & Chemical Engineering (2009), doi:10.1016/j.compchemeng.2009.11.012

[2] PRé-Consultants, The Eco-indicator 99, A damage oriented method for life cycle impact assessment. Methodology Report and Manual for Designers. Technical Report, PRé Consultants, Amersfoort, The Netherlands, 2000.