I. Response threshold models of division of labour in social insects 11
8. Hunting the optimum: Honeybee nest-site selection as an optimization
8.2. Nest-site selection as an optimization process
8.2.3. Experiment: Function optimization via iterative nest-site selection . 130
nest-sites. In case of the European honeybee A. mellifera, the number of possible good nest sites is limited as they live in cavities. The swarm needs to ensure that it decides for the best site possible so that it becomes unlikely that the nest site turns out to be of insufficient quality, forcing the swarm to move again. However, for bee species that live in the open such as the Dwarf honeybee A. florea, the quality of the nest site appears to be less important and the swarm has the chance to “upgrade” if its initial decision was suboptimal (Oldroyd et al., 2008).
Thus it interesting to see if an iterative nest-site selection process as found in A. florea can lead to an optimization in an environment with many potential nest sites. In this experiment it is assumed that the swarm’s environment corresponds to the search space of a continuous function. Each position in the environment constitutes a potential nest site, and its quality corresponds to a value of the function at that position. The test functions used in the experiment and their associated parameter values are given in Table 8.2. The swarm is initially placed at position [-20,-20] for the Sphere function and [-10,-10] for the Booth function.
For this experiment the scouting behaviour of the bees has been changed, as the first version of the extended model is orientated to the behaviour of A. mellifera where a scout assesses a nest-site for a certain period of time before returning to the swarm. However in this experiment each location corresponds to a potential nest-site, and scouts would thus immediately start to assess sites after a single scouting step. To overcome this, a scout in this experiment remembers the best position it encountered during its scouting trip. If the quality of that position is better than the current location of the swarm it starts dancing for that site.
130
8.2. Nest-site selection as an optimization process
0 5 10 15
0200400600800
Number of Nest Relocations Function Value 02004006008000200400600800020040060080002004006008000200400600800020040060080002004006008000200400600800020040060080002004006008000200400600800020040060080002004006008000200400600800020040060080002004006008000200400600800
(a) Sphere
0 2 4 6 8 10
050015002500
Number of Nest Relocations Function Value 050015002500050015002500050015002500050015002500050015002500050015002500050015002500050015002500050015002500050015002500
(b) Booth
Figure 8.3.: Boxplots of the quality of the occupied nest site over several relocations for the two test functions.
The quality of a newly discovered site is defined relative to the quality of the current location of the swarm. If a scout discovers a nest-site that is 60% better than the swarm’s current location, this site is assigned quality 60.
Bees that followed a dance for a potential nest-site become recruits for this site and will fly towards it. If they encounter a better site on their way to this nest site, they abandon the recruitment process and become scouts. Recruits that do not find an advertised nest site also become scouts.
Nest-sites are assessed by recruits and returning bees for a certain amount of time, TASSESS, during which a bee counts the number of other bees present at the site. If the number of bees at a site reaches a given quorum q = 10 the swarm is relocated to this new site and the nest-site selection process is restarted. The parameter values used in this experiment are: step size step = 0.1, scouting TSCOU T = 100, and assessment time TASSESS= 20. A simulation run is stopped when no swarm relocation occurs within 3600 simulation steps.
The changes in the quality of the found nest sites for both test functions over several nest-site relocations is depicted in Figure 8.3. As can be seen, the bees are able to iteratively optimize the position of the swarm within the search space. The optimization process is limited by several factors. Since scout time TSCOU T and step size step are fixed, scouts are only able to explore a certain range around the swarm’s current location whereas a fixed step size prevents scouts from finding better solutions as they are likely to fly over them.
This is especially the case when the swarm is close to the global optimum when scouts should actually search at a finer scale. Another limiting factor is the quality assessment.
Remember that the quality of a newly found nest site is determined according to the potential improvement with respect to the current location of the swarm. To make an algorithm based on nest-site selection applicable to real optimization problems, the swarm
8. Hunting the optimum: Honeybee nest-site selection as an optimization process
Algorithm 6 Bee Nest-Site Selection Scheme (BNSSS)
1: place bees on a random home position (swarm location) in the search space
2: initialize parameters Φ, Φ′, Ψ, d, and d′
3: while stopping criterion not satisfied do
4: for allscouts do
5: k = h
6: repeat
7: the scout flies to a random position x with maximum distance d to its current home
8: if f (x) ≥ Φ then
9: the scout performs k local search steps to find an improved location
10: k = 0
17: randomly assign the follower to one of the scouts where the probability depends on the quality of the location of the scout
18: repeat
19: the follower flies to a random position x with maximum distance d′ from the location of the scout it is assigned to
20: if f (x) ≥ Φ′ then
21: the follower performs k local search steps to find an improved location
22: k = 0 and stops
28: the follower abandons the scout
29: end if
30: end for
31: if the swarm has found a location that is better than it its home location then
32: its new home is the best of these locations
33: else if there exists a scout which has more than Ψ followers assigned to it then
34: the swarm is assigned to the scout or one of its followers which has the best location
35: else
36: the swarm is assigned a new randomly chosen home location or it stays at its current location
37: end if
38: update d, d′, Φ, Φ′ and Ψ
39: end while
132
8.3. The BNSSS scheme
needs to become more sensitive to small quality differences to identify better potential nest sites when the swarm comes closer to the location of an optimum.
The decision-making process underlying the optimization is slow. The higher the quo-rum q of bees needed at a potential nest before the swarm changes to this site, the slower is the optimization process. The quorum mechanism could however also prove to be useful in terms of optimization, because the existence of a quorum prevents premature convergence onto a local minimum by slowing down the decision-process and thus giving better sites a higher chance to be discovered and enter the decision-making process. Another potential benefit of the quorum mechanism is that it requires bees to revisit and reassess a given site several times, which is important for dynamic or noisy optimization functions.
8.3. The BNSSS scheme
Since some aspects of the bee nest site selection model are relevant for real bees but are not useful for a function optimization algorithm, we present here a scheme — called Bee Nest-Site Selection Scheme (BNSSS) — for the design of optimization algorithms. The BNSSS, described in Algorithm 6, is provided as a framework into which details have to be added when a specific algorithm is designed. For example the values of d, d′, Φ, Φ′, and Ψ have to be defined. Where Φ, Φ′, and Ψ should depend on the quality of the locations that have been found already. The values for d and d′ might decrease during the run of the algorithm so that the swarm concentrates on a small area of the search space. In contrast, for dynamic optimization functions it might also be necessary to increase the values of d and d′ at points in the decision-making process when it is found that the function to be optimized has changed. In addition, for noisy optimization functions it might be suitable to set d′ = 0 so that the location of a scout is evaluated several times.
The BNSSS scheme given here is designed for a single swarm of bees, however the appli-cation of multiple swarms within an algorithm is also possible. This would require defining how the different swarms cooperate – for example, the swarms might be implemented to remain a certain distance from each other in order to cover different parts of the search space.