Foundations of Metaheuristics
3.3 Design Elements of Metaheuristics
Since heuristics are problem-specific, properties of the problem to be solved need to be incorporated into the design of the heuristic. If the heuristic does not fit the problem, its solution is likely to be not possible. Thus, the development of heuristic optimization techniques is a difficult process that needs to be performed very carefully. In contrast to heuristics, metaheuristic optimization techniques rep- resent search strategies that are problem independent and, thus, widely applicable (Glover, 1986). Often these techniques are inspired by search strategies from other domains (e.g. biology, physics etc.) and a huge variety of different types of meta- heuristics with slightly different properties and functionalities were developed (B¨ack et al., 1997; Glover & Kochenberger, 2003). In general, metaheuristics represent extended variants of improvement heuristics (Reeves, 1993; Rayward-Smith et al., 1996; Rayward-Smith, 1998; Michalewicz & Fogel, 2000; Rothlauf, 2006a).
In general, the application of a metaheuristic is straightforward, since only two conditions must be fulfilled (Rothlauf, 2006a):
• solutions of the problem have to be represented as a set (or string) of variables or symbols to allow its processing, and
• the quality of each solution must be quantifiable to allow pairwise fitness comparisons.
Although the basic search strategy of a metaheuristic is independent of the problem, leading to many different applications in various domains, there still is a conflict between ease of application and effectiveness.1 The broader the applicability of an optimization technique, the poorer are the results of these tech- niques on individual problems. This relationship is expressed by the No-Free- Lunch-Theorem (NFL) published by Wolpert and Macready (1997). In short, the NFL states that no heuristic can outperform other optimization methods if they do not inherit problem-specific knowledge. Thus, traditional heuristics exploiting problem-specific information might outperform standard metaheuristics, and there is a need to consider problem-specific knowledge in the design of metaheuristic op- timization techniques to achieve superior performance (Droste & Wiesmann, 2002;
Puchta & Gottlieb, 2002; Bonissone et al., 2006; Rothlauf, 2006a).
3.3 Design Elements of Metaheuristics
In his research, Rothlauf (2006a) examines the design of metaheuristics, the influ- ence of different design variants and the consideration of problem-specific knowl- edge on their performance. Four basic design elements are identified that all metaheuristics have in common and that characterize the different variants:
1. solution representation and variation operators, 2. fitness function,
3. initialization, and 4. search strategy.
1For overviews see for example Biethahn and Nissen (1995), Osman and Laporte (1996), B¨ack et al. (1997), Alander (2000), Blum and Roli (2003).
In the following, a short introduction to each element is given. Each element is subject of research on metaheuristics and many studies are available that can be used for further information.2
3.3.1 Solution Representation and Variation Operators
The objective of variation or search operators is to modify solutions of a problem during an optimization procedure. For this purpose, it is necessary to encode any given solution of a problem as a string that can be processed by these operators. The solution of a problem is denoted as the phenotype, its representation as the genotype, the mapping between the phenotype and the genotype is referred to as the represen- tation of a metaheuristic (Rothlauf, 2006b). Because the search operators work on the genotypes, the representation and the operators cannot be treated independently but must be considered as a joint element of a metaheuristic.
Although not selective, direct and indirect representations can be distinguished.
When using a direct representation, an explicit mapping is not specified. Instead, so- lutions are represented in their most natural search space and the variation operators are applied directly to the solutions. This calls for individual variation operators, since the solution representation is problem-specific. In contrast, an indirect rep- resentation has an explicit mapping: problem solutions are represented as standard data structures (for example binaries, integers etc.). This allows the application of standard search operators to the genotypes. An additional advantage of using indi- rect representations is that constraints or restrictions in the search space may be effi- ciently modeled by a specific encoding (only feasible solutions can be processed) or that an advantageous mapping might decrease problem difficulty (Rothlauf, 2006a).
However, finding a proper representation then is a challenge in metaheuristic design.
Search operators can be classified as local search operators and recombining search operators. In the search space, a local search step moves from one solution to a solution in its neighborhood. The concept of neighborhood aims at similarities be- tween solutions. If there is a way to quantify the similarities or common elements, a neighborhood of a solution can be defined that contains all solutions that are similar to the current solution to a specific extent (distance). A high locality exists if this distance corresponds to the distance between their objective values and the distance of the genotypes. If the genotype encodes continuous values, the real difference between the values can be used as distance. If there are discrete values (for exam- ple discrete choices), the quantification becomes difficult. However, the number of decision variables encoded and subject to variation allows a quantification of the differences and, thus, allows the definition of a distance and the neighborhood. The local search operator moves to a neighboring solution by modifying the current so- lution, thus, in fact, the distance of the genotype is defined by the search operator:
each solution that can be obtained by a single search step is a neighboring solution.
2See for example Holland (1975), Grefenstette (1985), Michalewicz and Fogel (1989), Goldberg et al. (1989), Liepins and Vose (1990), Storer et al. (1995), Michalewicz and Schoenauer (1996), B¨ack et al. (1997), Goldberg (2002), Rothlauf (2006b), Rothlauf (2006a) etc.
3.3 Design Elements of Metaheuristics 51 When using local search operators, it is assumed that the structure of the search space can guide the search because good solutions are grouped together. Then, the optimal solution can be obtained by applying only small changes to a solution and the objective value of solutions considered earlier is used to guide the future search process (Manderick et al., 1991). Local search operators can only yield optimal so- lutions if the search steps are sufficiently small and if the global optimum can be reached by neighboring steps. If multiple optima exist, there is the chance that a local search operator only yields a local optimum instead of the global optimum.
In contrast to local search operators, recombination operators require at least two solutions, because these operators recombine elements of solutions to construct one or more new solutions. The original solutions are commonly referred to as parents, the new solution as child or offspring. When applying recombination operators, it is assumed that the problem under investigation is decomposable: subproblems can be solved independently and the combination of the optimal partial solutions yields an overall optimal solution. Recombination operators represent global search oper- ators, because the resulting solution inherits properties of both parents and is not limited to the neighborhood of one parent. General design principles for recombi- nation search operators were formulated by Radcliffe (1991) and (1994).
When choosing a combination of representation and operators, the bias and locality have to be taken into account. A bias exists if the choice of representation/op- erators alone pushes the search of the metaheuristic into a specific direction (Caruana
& Schaffer, 1988). For example, the operators work not randomly but perform only selected modifications to the solutions. High locality is necessary to allow a guided search. If the locality is high, small changes to the genotype by a variation operator result in small changes of the phenotype and its objective value (Lohmann, 1993;
Rothlauf, 2006b). Thus, with each application of the operator, a neighboring solution in the search space is obtained. If the locality is low, the search process represents a random walk through the search space.
3.3.2 Fitness Function
A fitness function assigns a fitness value to each solution, which is used by the metaheuristic to compare the quality of solutions. In many cases, the fitness func- tion corresponds to the objective function of the problem behind it. This objective function expresses the quality of the current solution with respect to the goal that is to be achieved. In general, this is the primary objective of the optimization process.
However, sometimes the fitness or objective value is modified by the metaheuristic to improve the metaheuristic’s performance or to include additional characteristics of the corresponding solution. One example for performance improvements is to smooth or scale the fitness landscape for guided local search techniques. Beside any additional objectives, a very important characteristic of a solution that can be incor- porated in the fitness value is violations of given restrictions. A solution might be infeasible but could inherit some favorable properties that should be kept in the op- timization process. Thus, instead of removing the current solution from the search process, its fitness value can be decreased by a penalty function.
Metaheuristics represent iterative search procedures, and usually a large number of fitness evaluations is necessary. Thus, their application is only possible if the fitness calculation is not too complex and does not require too high an amount of computational time and effort. On the other hand, metaheuristics can work only with pairwise fitness comparisons, and especially in the beginning of the optimization process, large differences in the fitness value of different solutions exist. Thus, if necessary, fitness values can either be obtained by approximations (at the beginning of the optimization process) or can focus only on the differences of similar solutions (at the end of the optimization process).
3.3.3 Initialization
In the initialization step, a solution is presented to the metaheuristic and serves as an initial solution from where the optimization process starts. If the metaheuristic uses recombination operators, the initialization must provide a population of solutions to allow recombination between different solutions. If some problem-specific knowl- edge or information about high-quality solutions exists, the initial solutions can be constructed using this knowledge to lead to a good starting point for the optimiza- tion process. If there is no such knowledge, the initial solution has to be created randomly. All possible solutions then have to have the same selection probability and no specific solution or solution properties are favored. If a population needs to be initialized, the diversity of this population should be high to ensure an effective application of the recombination search operator.
3.3.4 Search Strategy
Decisions on the search strategy of a metaheuristic focus on the exploration and ex- ploiting phases (Blum & Roli, 2003; Rothlauf, 2006a). In exploitation, the search is focused on promising regions in the search space, whereas in exploration new areas in the search space are investigated. In general, two basic search strategies can be identified. They depend on the main search operator used during optimiza- tion, leading to local search and recombination-based search strategies. The control of exploitation and exploration, or intensification and diversification, is addressed differently in each strategy, and variants exist each focusing on different mecha- nisms for intensification and diversification (see Rothlauf (2006a) for a selection of representative examples).
In local search strategies, a new solution is iteratively chosen from the neighbor- hood of the current solution. The fitness of the solution is used to guide the search process to regions of the search space with high-quality solutions (intensification).
To escape from local optima, diversification is necessary. This can be accomplished by further varying the solution for example by changing the definition of a neigh- borhood of a solution. An alternative would be to start multiple instances of the optimization process, each with a different initial solution. A very common strategy to allow diversification is not to restrict search steps to intensification (improve- ment) steps but to allow inferior solutions during the search. Then such search steps