porting a solution from a given set of alternatives, for the greater good of the situation at hand. In practice, the consensus process is typically a dynamic and iterative group discussion process, usually coordinated by a moderator, responsible for helping the members to bring their opinions closer to one another [65]. During each step of the process, the moderator keeps track of the ac- tual level of consensus between the decision makers, typically by means of some pre-determined consensus measure establishing the distance from the current consensus level to the ideal state of consensus (i.e. full and unanimous agreement of all members with respect to a specific alter- native). If the current consensus level is deemed unacceptable, that is, if it is lower than some pre-determined threshold level, indicating that there exists a considerable degree of discrepancy between the decision makers’ opinions, then the moderator asks the members to discuss their opinions further in an effort to bring them closer to consensus. The consensus attainment deci- sion making process is also sometimes able to function without the use of a moderator, whose inclusion in the process may be too time-consuming, but is rather controlled automatically by the group, receiving input from the various decision makers, assessing the consensus level at each step of the process, and providing feedback (output) on the current state and progress of the process back to the decision makers.
According to Bressen [14], while majority voting typically leads to a (technically) much simpler and faster decision making process than does consensus attainment, consensus attainment offers three major advantages over majority voting. First, consensus attainment leads to more effective implementation (when all decision makers’ ideas and concerns are accounted for, they are more likely to participate actively towards making something happen). In majority voting, on the other hand, members with a minority position are usually overruled and are naturally unwilling to participate in the decision process with much eagerness. Secondly, consensus attainment builds connections amongst decision makers (using consensus attainment suggests that some time is taken to achieve a certain level of agreement amongst the members on how to proceed with the decision making process before actually moving forward). Majority voting, on the other hand, creates winners and losers, which has a dividing effect amongst the decision makers [14]. Lastly, consensus attainment usually leads to higher quality decisions (integrating the wisdom and experience of all members into the decision making process typically generates better and smarter decisions than majority voting does, particularly when the decision makers are assumed to be rational).
2.2 Optimisation techniques
In this section, an overview is provided of general search techniques aimed at finding exact or approximate solutions to optimisation problems. This is followed by a more elaborate discussion on the class of stochastic search techniques, members of which are later implemented in this dissertation. There does, however, not seem to be full consensus on a standard classification for optimisation search methods in the literature.
According to Coello et al. [29], one way to differentiating between different types of solution search and optimisation techniques is to classify them as enumerative, deterministic or stochastic. Such techniques may further be classified as exact, heuristic or metaheuristic [153]. Examples of popular search techniques in these categories are listed in Figure 2.2. Other classifications of op- timisation techniques may also include partitions according to traditional versus non-traditional, local versus global or sequential versus parallel techniques.
Full enumeration techniques evaluate every possible solution in the solution space of the op- timisation problem, either explicitly or implicitly. If the problem complexity is relatively low
Stochastic Deterministic Enumerative Exact Heuristic Metaheuristic
Full Depth First Search Full Breadth First Search
Full Branch-and-Bound
The Simplex Algorithm Full Simulated Annealing
Rule-Based Partial Enumeration
Greedy Heuristics Gradient Methods
Random Walk Monte Carlo Simulation
Intelligent
Partial Enumeration Tabu Search
Truncated Simulated Annealing Particle Swarm Optimisation
Ant Colony Optimisation Evolutionary Algorithms
Figure 2.2: Global search categories and examples of techniques.
and the search can be performed within a reasonable amount of time, then these techniques are guaranteed to find a global optimum. Adopting this kind of approach when solving an optimisa- tion problem is typically extremely straightforward and easy to apply. Enumeration approaches may, however, be entirely insufficient when dealing with large optimisation problems.
Deterministic search techniques overcome the problem of a large search space, by incorporating the notion of problem domain knowledge to reduce the size of the search space. Here, the aim is to implement some means of guiding or restricting the search space in order to find good solutions within an acceptable time. Although proven to be successful methods for solving a wide variety of problems, the complex features associated with many real-world optimisation problems, such as multimodality3, highly dimensionality, discontinuous and/or NP-completeness, often cause deterministic methods to be ineffective for solving problems of this kind4 over large search
spaces, due to their dependence on problem domain knowledge for guiding the search process [29].
Stochastic search techniques were developed as an alternative for solving such irregular problems. Here, the subsequent state of the search process is induced probabilistically (i.e. randomly) instead of deterministically. These techniques are based on probabilistic sampling of a set of possible solutions and, throughout the search, maintain some form of record of good solutions found. A predetermined function assigning performance values to candidate/partial solutions as well as a mapping mechanism linking the problem domain to the algorithm is required. Given 3A problem is said to be multimodal if it contains an objective function with more than one optimum (i.e. one
or more global optimum in addition to one or more local optimum of inferior quality) [162].
2.2. Optimisation techniques 21
the level of complexity of some irregular multiobjective problems, stochastic methods are often able to yield satisfactory solutions to such problems for which the search space is not chaotic5.
Metaheuristics have recently become very popular for solving multiobjective optimisation prob- lems. A metaheuristic is a state-of-the-art search technique which may be defined as a higher- level procedure designed to strategically guide one or more lower-level procedures or heuristics to search for feasible solutions in spaces where the search task is hard [31]. Metaheuristics are char- acterised by their typically approximate, stochastic and non problem-specific nature. According to Suman and Kumar [138], the increasing acceptance of these search techniques is a result of their ability to: (1) find multiple candidate solutions in a single run, (2) function without the use of derivatives, (3) converge with great speed and accuracy towards Pareto optimal solutions, (4) accommodate both continuous and combinatorial optimisation problems with relative ease and (5) be less affected by the shape or continuity of the Pareto front.
One way of classifying metaheuristics is to refer to their search strategies, which, according to Blum and Roli [7], is either trajectory-based or population-based. These categories are typically characterised by the number of candidate solutions generated during every iteration of the search process. Trajectory-based metaheuristics start with a single initial candidate solution and, at every iteration, replace the current solution by a different, single candidate solution in its neigh- bourhood (examples of such metaheuristics include simulated annealing [138], tabu search [48] and variable neighbourhood search [123]). Population-based metaheuristics, on the other hand, start with an initial population of multiple candidate solutions, which are enhanced through an iterative process by replacing part of the population with carefully selected new solutions (examples of such metaheuristics include genetic algorithms [29], ant colony optimisation [47] and particle swarm optimisation [114]). Trajectory-based approaches are usually able to find locally optimal solutions quickly, and are thus often referred to as exploitation-oriented meth- ods. Population-based approaches, on the other hand, strongly promote diversification within the search space, and are thus often referred to as exploration-oriented methods. Additionally, population-based approaches often incorporate a learning component.
Other classes of stochastic search techniques include hybrid metaheuristics, parallel metaheuris- tics and hyperheuristics. A hybrid metaheuristic typically combines several optimisation ap- proaches (such as other metaheuristics, artificial intelligence, or mathematical programming techniques) with a standard metaheuristic. These approaches run concurrently with one an- other, and exchange information, in order to guide the search process. Parallel metaheuristics, on the other hand, use the techniques of parallel programming to implement multiple meta- heuristic searches in parallel so as to guide the search process more effectively. Hyperheuristics were initially introduced to devise new algorithms for solving problems by combining known search techniques in ways that allow each of them to compensate, to some extent, for the weak- nesses of others [17]. Their goal is to be generally applicable to a large range of problems by performing well in terms of computational speed, solution quality, repeatability and favourable worst-case behaviours, in cases where standard metaheuristic techniques fail to perform well on some of these counts.
In the following three sections, five metaheuristics are presented in varying amounts of detail. For reasons that will later become apparent, simulated annealing and evolutionary algorithm techniques are discussed in more detail and are reviewed in the context of both single- and multiobjective optimisation problems, while the remaining algorithms are only briefly discussed in the context of single-objective optimisation problems.
5A problem is said to be chaotic if small differences in initial conditions yield widely diverging outcomes in