Abstract
“Real world” decision-making often involves complex problems that are riddled with incompatible and inconsistent performance objectives. These problems typically possess competing design requirements which are very difficult – if not impossible – to quantify and capture at the time that any supporting decision models are constructed. There are invariably unmodelled design issues, not apparent during the time of model construction, which can greatly impact the acceptability of the model's solutions. Consequently, when solving many practical mathematical programming applications, it is generally preferable to formulate numerous quantifiably good alternatives that provide very different perspectives to the problem. This solution approach is referred to as modelling-to-generate-alternatives (MGA). This study demonstrates how the nature-inspired, Firefly Algorithm can be used to efficiently create multiple solution alternatives that both satisfy required system performance criteria and yet are maximally different in their decision spaces.
TopIntroduction
Decision-making in the “real world” involves complex problems that tend to be riddled with competing performance objectives and possess requirements which are very difficult to incorporate into any underlying decision support models (Brugnach, Tagg, Keil, De Lange, 2007; Janssen, Krol, Schielen, Hoekstra, 2010; Mowrer, 2000; Walker, Harremoes, Rotmans, Van der Sluis, Van Asselt, Janssen, Krayer von Krauss, 2003). While an optimal solution might provide the theoretically best answer to a mathematical model, in general, it will not be the best solution to the fundamental “real” problem since there are invariably unmodelled objectives and unquantifiable issues not incorporated in the problem formulation (Brugnach et al., 2007; Gunalay, Yeomans, 2012; Gunalay, Yeomans, Huang, 2012; Janssen et al., 2010; Loughlin, Ranjithan, Brill, Baugh, 2001). Consequently, it is preferable to generate a number of different alternatives that provide multiple, disparate perspectives to any particular problem (Imanirad, Yeomans, 2014; Matthies, Giupponi, Ostendorf, 2007; Yeomans, Gunalay, 2011). Preferably these alternatives should all possess good (i.e. near-optimal) objective measures with respect to the modelled objective(s), but be as fundamentally different as possible from each other in terms of the system structures characterized by their decision variables (Yeomans, 2011).
To address this option creation need, several approaches collectively referred to as modelling-to-generate-alternatives (MGA) have been developed (Loughlin et al., 2001; Yeomans, Gunalay, 2011; Yeomans, 2012). The principal motivation for MGA is to create a small set of alternatives that are as maximally different from each other in the decision space as possible, yet are still considered “good” with respect to all of the modelled objective(s) (Yeomans, 2011; Yeomans, 2012). By adopting a maximally different method, the resulting alternative solution set is likely to provide very different perspectives with respect to any unmodelled issues, while simultaneously providing different choices that all perform somewhat similarly with respect to the modelled objectives (Gunalay, Yeomans, 2012; Gunalay et al., 2012; Walker et al., 2003; Yeomans, 2011).
In this chapter, it is shown how a modified version of the metaheuristic Firefly Algorithm (FA) of Yang (2009; 2010) can be used to efficiently generate a set of maximally different solution alternatives. Yang (2010) has demonstrated that, for optimization and calculational purposes, the FA is more computationally efficient than the more commonly-employed enhanced particle swarm, genetic algorithm, and simulated annealing metaheuristic procedures. Thus, this FA-based MGA procedure can be considered very computationally efficient (Imanirad, Yeomans, 2014). This demonstrates the MGA proficiencies of the FA-based approach for constructing multiple, maximally different solution alternatives to the highly non-linear optimization problem of Loughlin et al. (2001).
Key Terms in this Chapter
Firefly Algorithm: The Firefly Algorithm is a computationally efficient, nature-inspired, population-based metaheuristic that derives its solution approach based upon the characteristics of fireflies.
Maximally Different Solutions: “Good” solution alternatives should possess near-optimal objective measures with respect to all of the known modelled objectives, but be fundamentally different from each other in terms of the system structures characterized by their decision variables. A difference model is employed to generate alternatives that are as far apart in the decision space as possible. The resulting alternative solution set of MGA provides disparate choices that all perform well with respect to the known modelled objectives, yet very differently with respect to any unknown, unmodelled and/or unquantified issues. Hence, these solutions will provide entirely different perspectives to the original problem.
Meta-Heuristics: High-level, overarching heuristic approaches that have wide-ranging applicability to many different mathematical programming problems.
Heuristics: Approximation schemes used in problem-solving to generate good, though not necessarily optimal, solutions to mathematical programming problems.
Unmodelled Issues: Aspects of a mathematical problem not captured during the construction and formulation of its corresponding mathematical/computer model.
Stochastic MGA: Modelling to Generate Alternatives in which some or all of the parameters, objectives, constraints and/or other problem characteristics are expressed in some form of uncertainties, probability distributions or some other stochastic representation.
Modelling to Generate Alternatives (MGA): A modelling approach to systematically provide a set of “good” alternatives with respect to all of the problem’s modelled objectives. The primary motivation for MGA is to produce a manageably small set of alternatives that are good with respect to the known modelled objectives yet as different as possible from each other in the decision space – namely the solution set should provide maximally different alternatives.
Unquantified Issues: System objectives and requirements that are neither explicitly apparent nor included in the problem formulation stage – it is impossible to express these aspects quantitatively.
Nature-Inspired Meta-Heuristics: Meta-heuristics whose fundamental solution characteristics have been motivated by phenomena occurring in the natural environment.