Enhancing the ϵ-constraint method through the use of objective reduction and random sequences: Application to environmental problems

Interactive Optimization With Parallel Coordinates: Exploring Multidimensional Spaces for Decision Support

“…Conversely, exact methods, by relying on scalarization techniques and efficient deterministic single objective approaches can quickly find optima of even large scale multiobjective problems [Branke et al, 2008;Williams, 2013;Schüler et al, 2018b, (p. 61, 64)]. This efficiency is critical because the number of solutions required to explore the solution space grows exponentially with the number of objectives (Cohon, 1978;Copado-Méndez et al, 2016). Ultimately however, the question of whether exact or heuristic methods are most efficient is debatable, and depends on the problem considered.…”

“…Despite the advantages of the ǫ-constraint method, Chankong and Haimes (2008, p. 285) noted that it can be inefficient when perturbating the values of the ǫ n,p bounds in the incremental fashion described above. As such, and especially when many dimensions are involved, the generation of solutions using the ǫ-constraint method can be time-consuming and uneven across the objective space when interrupted prematurely, leading to a poor representation of the Pareto front (Collette and Siarry, 2004;Chankong and Haimes, 2008;Copado-Méndez et al, 2016). This lack of efficiency is particularly problematic in interactive methods, as the user should be presented with an overview of the Pareto optimal solutions as fast as possible in order to know which areas lead to preferred alternatives.…”

“…i. smart ordering of axes, e.g., by exhaustive pairwise depiction (Heinrich and Weiskopf, 2013), by conflicting objectives, prioritizing those with most convex Pareto fronts (Unal et al, 2017), by similarity-based reordering (Lu et al, 2016), by use of 3D parallel coordinates (Johansson and Forsell, 2016), or by visual (Keeney, 1992, p. 57) or automatic dimensionality reduction techniques (Copado-Méndez et al, 2016), ii. improved visualization of polylines and patterns, e.g., visually bundling clusters into compact polygons (Palmas et al, 2014), or using polynomial curves and other line styles to emphasize various data properties (Franken, 2009;Heinrich and Weiskopf, 2013), FIGURE 11 | Comparer dashboard containing three representative solutions from a cluster analysis on the chosen criteria.…”

Temporal sustainability efficiency analysis of urban areas via Data Envelopment Analysis and the hypervolume indicator: Application to London boroughs

“…Originally introduced by Zitzler and Thiele (1998), the hypervolume indicator has been traditionally applied to assess the numerical performance of multi-objective algorithms (Beume et al, 2007;Copado-Méndez et al, 2016;Zitzler et al, 2007) given its appealing theoretical properties (see Zitzler et al (2003)). The hypervolume measures the size of the dominated d-dimensional space between a Pareto frontier defined by a set of points q' and a reference (dominated) point qref (Chatterjee, 2003) (see Fig.…”

“…The second distinction of the proposed approach is the use of a pseudorandom sequence, instead of a random one as in Everson's case, to generate the points used in the analysis, as previously demonstrated by Copado-Méndez et al (2016). The motivation for this is that pseudorandom sequences show higher discrepancy compared to random ones (Bratley et al, 1992).…”