2.5.1
Heuristics
It has been noted above that MOOPs are often hard to solve as they tend not to be amenable to analytical methods due to their usual non-linearity and multi- dimensionality (both in decision and objective space), and also often or usually, having very large search spaces from which solutions must be picked. The sizes of the search spaces of such problems tends to preclude the use of exhaustive and complete searching, which would take too long to be practical, or even possible (since they may in theory take many years).
Moreover, multimodality tends to occur as soon as the number of OFs exceed one, for non-trivial competing objectives, even where each OF by itself is of single modality and is a convex function (Fonseca and Fleming, 1995b).
The inherent characteristics of these problems leads to the adoption of other methods which are determinedly less than complete explorers of the problem space, with the corollary that solutions found may not always be the best, but either the method returns the best solution often enough, or approximate solutions are deemed “good enough”.
Heuristics then, are “criteria, methods, or principles for deciding which among several alternative courses of action promises to be the most effective in order to achieve some goal.”, (Pearl, 1984). Heuristic methods require knowledge about the problem domain, and may be of a stochastic nature, but tend to have sim- ple, incomplete or unreliable information about the exact problem. There is an essential dichotomy at the heart of these methods: to be simple, yet effective. A common phrase used to describe them is “rule of thumb”, which in turn may
imply some sort of summation or scoring of elements or steps in the problem. The so-called Greedy heuristic (Atallah and Blanton, 2009) is a good example of such a method: it adopts the general strategy of adopting the locally optimal choice at each decision point with the hope that this will eventually find the global op- timum, while accepting that this will not always be the case; yet it needs specific knowledge about the problem in order to be used. The book of Polya(2004) is it- self a heuristic guide to solving problems generally, even though aimed specifically at mathematical problems.
The term meta-heuristic was introduced byGlover (1986) in his discussion of the Tabu search algorithm, which uses a heuristic, as above, but upon which is im- posed a further strategy - that of penalizing moves that take a path already taken, and an organised memory mechanism. Various metaheuristic implementations have found use in the real-world, tackling necessary tasks such as job scheduling and vehicle or network routing (S¨orensen and Glover,2013), in which any solution that works is “good enough” even if such may not be the global optimum.
It is frequently the case that in pursuing difficult problems, especially real- world engineering ones, that the Pareto-optimal set, PFopt, is neither known, nor indeed knowable analytically (Veldhuizen and Lamont, 2000), therefore until proven otherwise, it is usually necessary to assume that a given set of solutions obtained in a front, is an approximation to the global: PFapprox.
2.5.2
Meta-heuristics
The term metaheuristic is often found in the literature and as it is by nature inher- ently general, the definitions given for it tend to vary, nevertheless the following set of common properties are fundamental in describing it (Blum and Roli, 2003) :
Have a goal of exploring the search space efficiently to find (near) optimal solutions.
Use a variety of techniques from local searches to complex learning strategies.
Are often, or usually, non-deterministic, and approximate.
May possess mechanisms to avoid becoming trapped in local minima.
Are often described at a level of abstraction from a specific problem.
Are not problem-specific.
May be hybridised with some domain-specific knowledge as heuristics that are controlled by a higher level strategy.
May use organised memory to leverage search experience to determine the direction of the search.
A metaheuristic thus is both a framework and a set of concepts, strategies and mechanisms that are consistent, for the design of heuristic algorithms. S¨orensen and Glover (2013) refine the definitionmetaheuristic to be “a high-level problem- independent algorithmicframework that provides a set of guidelines or strategies to develop heuristic optimization algorithms”. They also note that problem-specific implementations of heuristic algorithms within a metaheuristic framework, are also metaheuristics.
Figure 2.11 depicts the broad categories of metaheuristcs that have been in- spired by nature, with the common thread that living organisms in some way necessitate optimisation in the struggle to survive long enough to reproduce, ex- cept for the simulated annealing (SA) algorithm (Kirkpatrick et al., 1983), which uses the metaphor of the annealing process of metallurgy, as a system having multiple potential energy states therefore being amenable to statistical mechanics (analysis of aggregate properties of atoms in solids or liquids).
Particle Swarm optimisation (PSO) introduced by Kennedy and Eberhart
(1995) is based on velocities of moving particles in 2, 3 or n-dimensional (hyper- )spaces, and acceleration towards solutions nearer to the global optima, where behaviour was inspired by flocking and foraging of birds and fish, but also the desire to model human social behaviour. However, there are similarities with evo- lutionary algorithms (EA), in that: it is stochastic; there is an operator (agent velocity adjustment) similar in effect to that of crossover in genetic algorithms (GA); the concept of fitness of a solution is used; and the group ‘knowledge’ of the best position found so far (gbest), being analogous to elitism. Each particle, termed agent, stores the coordinates of the best solution location it has visited (pbest) and tracks other agents in its vicinity, while maintaining a minimum sep- aration, that it is informed are the current local optima (lbest). At each step, the agent evaluates its fitness and is accelerated towardspbest andlbest positions, with weighted random adjustments to its acceleration vector. A significant difference to GAs is that a PSO agent only has access tolbest in other agents, whereas GAs swap chromosomes with many individuals. Another significant difference is that a PSO can be considered to be a “directed mutation” method, in that individu- als may be modified, related to their fitness, but no re-sampling occurs, unlike in evolutionary methods. Re-sampling is the creation of new solutions from existing ones; in canonical PSOs the population is fixed.
Ant colony optimization (ACO) (Dorigoet al.,1996), (Dorigo and Blum,2005) and its derivatives, share some common themes with PSO of animal behaviour and foraging, in which colonies of the same species attempt to maximise their net energy gain (food consumption versus the energy expended to acquire it) over time. In ACO, the agents are metaphorical ants that move stochastically over a landscape, communicating with each other by assigning values to the paths they have taken that are relative to their fitness (the metaphor has it that the fitness is indirectly proportional to the distance to the food and the path value is a pheromone trail, for example when used for the Travelling Salesman problem (TSP), shorter paths are travelled by more ants in a given time, thus acquire higher values). The path values are adjusted by the number of agents travelling
them, and by a deterministic degradation function (a metaphorical pheromone evaporation) that helps to prevent premature convergence, thus the path value provides a bias guiding the agents in their decision to take it.
Various comparisons have been performed to investigate the relative perfor- mance of these approaches, but as specific algorithm details vary (for example, whether a GA is real or binary encoded, if elitism is used, and so on), and as the NFL theorem necessitates that a given algorithm will always be out-performed by another on some problems, it cannot be said of any specific metaheuristic algo- rithm that it is “better” than the others, generally.
Figure 2.11: Nature-inspired metaheuristics.
However, so many derivative metaheuristics have arisen that are inspired by nature, from EAs through various insect and bacterial foraging, to flocking or swarming, shark hunting strategies using L´evy flights (Chechkin et al., 2008) and even music-based metaphors, that it becomes a matter of intrigue to wonder how much such strategies really differ fundamentally, given that they are all attempting essentially the same thing: to both search widely in a very large space and yet to home in on regions containing good solutions in the hope of finding better ones; also known as the diversify/intensify or explore/exploit dichotomy. For example,
B¨ack (1995, p. 257) declares that “Evolutionary Algorithms are representatives of the mathematical concept of a Markov process (respectively chain, in discrete
spaces)”. Although diversify and explore both have the meaning of searching more widely in the search space, while intensify and exploit have the meaning of focusing in on promising regions, the pair diversify/intensify now are largely used in the context of some deliberate strategy, often involving specific memory caches or divisions, whereas explore/exploit tend to be more associated with stochastic undirected approaches (Blum and Roli, 2003).
Along withS¨orensen(2013), the conclusion could be reached that it would be better to research existing approaches to refine them or define them analytically, in the sense of analysing how they work, rather than introduce new derivative methods that are not really novel in their fundamentals unless they can provably be shown to contribute to the field.
Therefore, the following section considers the Evolutionary Algorithm class of metaheuristics that has shown itself to be particularly useful in tackling MOOPs, and the Genetic Algorithm in particular, as a refinement was identified that was novel and potentially of use for MOOPs in general and complex engineering sys- tems in particular.