Mating-based optimization algorithms

Honeybee Mating Optimization Algorithm One example of a genetically-based opti-mization algorithm is the honeybee mating optiopti-mization algorithm (HBMO) developed by Abbass (2001a,b) for discrete optimization problems. The HBMO algorithm contains four main (artificial) entities called queens, drones, brood, and workers. The algorithm operates in two stages: maiden flight and brood development. Both stages are executed alternately until a stopping criterion is satisfied.

The HBMO operates with several queens. At the beginning of the maiden flight each queen is equipped with a single randomly generated reference solution. In addition a queen has a flight speed s, an energy e, and a limit for the amount of sperm (i.e., sample solutions from drones) she can store in her spermatheca (i.e., a pool of new sample solutions). A queen stops her maiden flight either when her energy is depleted or when a maximum

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number of drone solutions has been collected. During each step of her maiden flight a queen Q encounters a drone D. She will absorb the drone’s solution with a probability of

p(Q, D) = e^−d/s (8.1)

where d represents the fitness difference between the queen’s reference solution and the drone’s solution and s represents the flight speed of the queen. As can be seen from p(Q, D) a queen is very likely to accept a drone’s solution if the solution is better than the queen’s reference solution or if her speed is high. After each step the queen’s flight speed and energy are decreased, which results in the queen becoming more selective with respect to absorbing potential drone solutions over the course of her flight.

The maiden flight is completed when a maximum number of drone solutions has been collected, or when the queen’s energy is depleted. At this point, a queen will mate with a drone solution randomly selected from those in her spermatheca. Mating involves the application of a crossover operation to the selected drone solution and the queen’s reference solutions, and results in a single offspring. In addition a mutation operation might be applied. The survival of the offspring depends on the quality of this offspring reference solution.

The offspring solutions of the queens are nursed by the workers during the brood devel-opment stage. A worker represents a local search heuristic and nursing corresponds to the application of this heuristic to try and improve the offspring solution. Then, before a new maiden flight stage is started the least fit queens are replaced with the fittest offspring until no offspring is fitter than the least fit queen. Again, this process is repeated until a stopping criterion is satisfied.

Initially the HBMO algorithm was proposed for solving the Boolean satisfiability prob-lem (Abbass, 2001a,b) and has since been adapted for several other probprob-lems such as water reservoir management (Afshar et al., 2007; Haddad et al., 2008; Mohan and Babu, 2010), data clustering (Fathian and Amiri, 2008; Marinakis et al., 2008) and vehicle routing (Marinakis et al., 2009a).

Bumble bees mating optimization The Bumble Bees Mating Optimization algorithm (BBMO) was introduced by Marinakis et al. (2009b) and is closely related to HBMO.

Here the mating behaviour of bumble bees is used as a template for optimization. The algorithm is initialized by creating a number of solutions to a given problem, with each solution corresponding to the genotype of a bumblebee. The best solution to the initial population becomes the queen, while the rest of the population are considered to be drones.

As in the HBMO algorithm the queen will select drones in terms of fitness and mate with them until her spermatheca is full.

8. Hunting the optimum: Honeybee nest-site selection as an optimization process

After this initial mating the queen will start to lay eggs. A queen is able to produce two kinds of bees: workers and drones. Workers are the result of a point-wise crossover between the queen’s genotype and one of the drones (i.e., given a certain crossover rate C_r it is decided for each position in the genome if the queen’s genetic information is kept rand(0, 1) < C_r or the drone’s genetic information will be used). In contrast, drones are created via random mutations of the queens genotype.

The fittest individuals of the worker population are considered as new queen candidates.

Via a feeding process (which corresponds to local search) the new queens are fed by the old queen as well as a number of workers. The local search is applied multiple times to single positions of the candidate queens genotype (i.e., single dimensions of the problem) and realized via the following equation:

nq_i= nq_i+ (b_max−(b_max− b_min) · lsi

Here nq_i corresponds to the genotype of a new queen candidate, q denotes the solution of the old queen and w_k corresponds to a worker bee. M denotes the number of workers that will feed the candidate queen. The impact of the workers versus the impact of the old queen on the local search is steered via the parameters b_max, b_min, lsi and lsi_max, where lsi_max corresponds to the maximum number of local search steps which are performed, lsi corresponds to the current local search step and b_max and b_min are parameters in the range of (0,1). Marinakis et al. choose b_max to be close to 1 and b_min to be close to 0, which leads to the local search incorporating the old queen’s solution in early local search iterations, while focusing on the workers in the later phase.

After the feeding the candidates, queens as well as the drones leave the hive and mate and the fittest fertilized candidate queens survive and continue the reproduction process.

The BBMO algorithm was initially introduced as a hybrid approach to clustering (Marinakis et al., 2009b) and has since been applied to the vehicle routing prob-lem (Marinakis and Marinaki, 2010) as well as unconstrained optimization probprob-lems (Marinakis et al., 2010).

Honeybee inspired evolutionary computation Other approaches that are based on hon-eybee mating utilise bee-inspired operators within existing evolutionary computation al-gorithms, see for example Sato and Hagiwara’s bee system (Sato and Hagiwara, 1997), Jung’s queen-bee evolution (Jung, 2003) or Karci’s bee-inspired genetic crossover operator (Karci, 2004). As these methods extend well-known optimization methods we will not go into further detail here.

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