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 capture and quantify 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. These alternatives should possess near-optimal objective measures with respect to all known modelled objectives, but be fundamentally different from each other in terms of the system structures characterized by their decision variables. 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.
TopBackground
While this section provides a brief synopsis of the steps involved in the FA process, more specific details can be found in Yang (2009; 2010). The FA is a nature-inspired, population-based metaheuristic that employs the following three idealized rules:
Key Terms in this Chapter
Unmodelled Issues: Aspects of a mathematical problem not captured during the construction and formulation of its corresponding mathematical/computer model.
Firefly Algorithm: The Firefly Algorithm is a computationally efficient, biologically-inspired, population-based metaheuristic that derives its solution approach based upon the characteristics of fireflies.
Meta-Heuristics: High-level, overarching heuristic approaches that have wide-ranging applicability to many different mathematical programming problems.
Biologically-Inspired (Bio-Inspired) Meta-Heuristics: Meta-heuristics whose fundamental solution characteristics have been motivated by phenomena occurring in the natural environment.
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.
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.
Heuristics: Approximation schemes used in problem-solving to generate good, though not necessarily optimal, solutions to mathematical programming problems.