These various parameters may also be initialized in a dynamic or adaptive way to deal with the trade-off between intensification and diversification during the search. Typically, the value of the inertia weightwis initialized to 0.9 to decrease to 0.4. The number of particles can vary during the search. For instance, a particle can be removed when its performance is the worst one [138].
3.7 OTHER POPULATION-BASED METHODS
Other nature-inspired P-metaheuristics such as bee colony and artificial immune sys-tems may be used for complex optimization problem solving.
3.7.1 Bee Colony
The bee colony optimization-based algorithm is a stochastic P-metaheuristic that be-longs to the class of swarm intelligence algorithms. In the last decade, many studies based on various bee colony behaviors have been developed to solve complex com-binatorial or continuous optimization problems [77]. Bee colony optimization-based algorithms are inspired by the behavior of a honeybee colony that exhibits many fea-tures that can be used as models for intelligent and collective behavior. These feafea-tures include nectar exploration, mating during flight, food foraging, waggle dance, and division of labor.
Bee colony-based optimization algorithms are mainly based on three different models: food foraging, nest site search, and marriage in the bee colony. Each model defines a given behavior for a specific task.
3.7.1.1 Bees in Nature Bee is social and flying insect native to Europe, the Middle East, and the whole of Africa and has been introduced by beekeepers to the rest of the world [689,707]. There are more than 20,000 known species that inhabit the flowering regions and live in a social colony after choosing their nest called ahive. There are between 60,000 and 80,000 living elements in a hive. The bee is characterized by the production of a complex substance, the honey, and the construction of its nest using the wax. Bees feed on the nectar as energy source in their life and use the pollen as protein source in the rearing of their broods. The nectar is collected in pollen baskets situated in their legs.
Generally, a bee colony contains one reproductive female calledqueen, a few thousand males known asdrones, and many thousand sterile females that are called theworkers. After mating with several drones, the queen breeds many young bees calledbroods. Let us present the structural and functional differences between these four honeybee elements:
• Queen: In a bee colony, there is a unique queen that is the breeding female
the day criterion as follows: cell cleaning (day 1–2), nurse bee (day 3–11), wax production (day 12–17), guard honeybees (day 18–21), and foraging honeybees (day 22–42). The worker ensures the habitual activities of the bee colony such as honey sealing, pollen packing, fanning honeybees, water carrying, egg moving, queen attending, drone feeding, mortuary honeybees, and honeycomb building.
• Broods:The young bees are named broods. They born following the laying of
eggs by the queen in special honeycomb cells called: the brood frames. There-after, the workers add royal jelly on the brood heads. Few female larvae are selected to be future queens. In this case, they are flooded by royal jelly. The unfertilized eggs give birth to the broods. The young larvae are spinning by co-coon, capping the cell by the older sisters; it is the pupa stage. Then, they reach the development stage in which they receive nectar and pollen from foragers until leaving the beehive and spending its life as forager.
The foraging behavior (nest site selection, food foraging) and the marriage behavior in a bee colony are the main activities in the life of a bee colony that attract researchers to design optimization algorithms.
3.7.1.2 Nest Site Selection In late spring or early summer, bees belonging to a colony are divided into two sets: the queen and half of the workers and the daughter queen and the rest half of the workers. The former aims to establish a new colony and the latter perpetuate the old colony. For selecting a new nest site, few hundreds of scout bees among several thousands explore some nest sites. The other bees remain quiescent, probably to conserve the swarms energy supply, until decision making and then migrate to the selected nest site. The foraging bees indicate various nest sites by several dances having eight patterns calledwaggle dances16(Fig. 3.38). The
16The waggle dance of the honeybee was elucidated by Karl von Frisch, an Austrian ethologist. For this
FIGURE 3.38 The waggle dance. The direction is indicated by the angle from the sun; the distance is defined by the duration of the waggle part of the dance. A waggle run oriented 40◦
to the right indicates a food source 40◦
to the right of the direction of the sun outside the hive. The bee runs through a figure-eight pattern on a vertical comb. It passes through the central, and performs the waggle run with vibrating her body laterally. The waggle run duration is related to the food source distance with a rate of increase of about 75 ms [691].
speed and the orientation of the dance are related with the nest sites quality. Various attributes are used to characterize a nest site (e.g., entrance area, entrance direction, entrance height, cavity volume). Within time, the advertising sites decline until the bee’s dance focus on one site. In other words, scouts vote by dancing for their favorite site and then they make a group decision via a quorum sensing and not via consensus sensing. Thereafter, scouts inform their nest mates by waggle dances. Finally, the entire bee cluster migrates toward the new nest site [690,691].
The waggle dance is used to inform other bees about nest site location or even food sources. The quality of the nest site is related to cavity volume and entrance hole, perched several meters off the ground, facing south, and located at the cavity floor.
Searching for a nest site is more or less similar to searching food source. In-deed, this behavior starts with the environment exploration. The food source is discovered via waggle dance and then the exploitation of the nectar is performed. The quality (profitability) of a food source depends on many indicators such as the richness of the energy, the distance to the nest, and the difficulty of extracting the energy.
source (food). They have a sufficient knowledge about the food source. They find and exploit food source, memorize its location, load a portion of nectar to the beehive, and unload it to the food area in the hive. One can mention three possible states related to the amount of nectar; if it is in low level or exhausted, the recruit honeybees abandon food source and become unemployed honeybees. If the amount of nectar is still sufficient, they forage and do not share food source with the nest mates. Finally, if an important amount of nectar is found, they exert a waggle dance for recruiting the other bees of the nest.
Searching a food source is based on two strategies (see Table 3.8 that illustrates the analogy between natural and artificial bee colonies):
• Exploration of food sources:A scout bee explores the neighboring area to find
a food source. In the positive case, it returns to the dancing floor of the hive and informs the nest mates by a waggle dance. The onlookers become recruit bees and then they become employed foragers.
• Exploitation of a food source:An employed forager calculates the amount of
food sources and takes a decision according to the quality of nectar. Either it continues the exploitation by the memorization of the best food found so far or it abandons. In that case, the employed bee becomes an unemployed one, either a scout or an onlooker bee using a probabilistic dividing.
A colony of bees can extend itself over long distances (up to 14 km) and in multiple directions in parallel to exploit a large number of food sources. A colony prospers by deploying its foragers to food sources of good quality. In principle, flower patches with plentiful amounts of nectar or pollen that can be collected with less effort should be visited by more bees, whereas patches with less nectar or pollen should receive
fewer bees. The foraging process begins in a colony by scout bees being sent to search for promising flower patches. Scout bees move randomly from one patch to another. During the harvesting season, a colony continues its exploration, keeping a percentage of the population as scout bees. When they return to the hive, those scout bees that found a patch that is rated above a certain quality threshold (measured as a combination of some constituents, such as sugar content) deposit their nectar or pollen and go to the “dance floor” to perform a waggle dance (Fig. 3.39). This dance is primordial for communication between bees and contains three types of information regarding a flower patch: the direction in which it will be found, its distance from the hive, and its fitness (i.e., quality). This information guides the colony to send its bees to flower patches precisely. Each beeÂ’s knowledge of the neighboring environment
Hive
B Waggle dance for A
Waggle dance for B S
S
A
Unloading nectar from A
Unloading nectar from B A
A
R
R
F
O
O
A R
R
F R
R A
F
F
Then, the algorithm conducts searches in the neighborhood of the selected sites, as-signing more onlooker bees to search in the neighborhood of the bestesites. The bees can be selected directly according to the fitness associated with the sites they are visiting. Alternatively, the fitness values are used to determine the probability of the bees being selected. Searches in the neighborhood of the bestesites that rep-resent more promising solutions are made more detailed by recruiting more bees to follow them than the other selected bees. Together with scouting, this differen-tial recruitment is a key operation of the bee algorithm (BA). For each patch, only the bee with the highest fitness will be selected to form the next bee population. In nature, there is no such restriction. This restriction is introduced here to reduce the number of solutions to be explored. Then, the remaining bees in the popula-tion are assigned randomly around the search space scouting for new potential so-lutions. This process is repeated until a given stopping criterion is met. At the end of each iteration, the colony will have two groups for its new population, the repre-sentatives from each selected patch and the scout bees assigned to conduct random searches.
Algorithm 3.15 Template of the bee algorithm.
Random initialization of the whole colony of bees ; Evaluate the fitness of the population of bees ; Repeat/∗
Forming new population∗
/
Select sites for neighborhood search ; Determine the patch size ; Recruit bees for selected sites and evaluated their fitness ; Select the representative bee from each patch ;
Assign remaining bees to search randomly and evaluate their fitness ; UntilStopping criteria
search, the number of best sites out ofmselected sites (e), the number of bees recruited for bestesites (nep), the number of bees recruited for the other (m−e) selected sites (nsp), and the initial size of patches (ngh), which includes site and its neighborhood and a stopping criterion.
Example 3.22 Bee colony optimization for the job-shop scheduling problem. Let us consider the application of the bee foraging model to solve the job-shop scheduling problem (JSP) (see Example 1.33 for the definition of the problem). A feasible solution in a JSP is a complete schedule of operations specified in the problem. One can think of each solution as a path from the hive to the food source.
The makespan of the solution is analogous to the profitability of the food source in terms of distance and sweetness of the nectar. Hence, the shorter the makespan, the higher the profitability of the solution path. One can thus maintain a colony of bees, where each bee will traverse a potential solution path (i.e., disjunctive graph). The foragers move along branches from one node to another node in the disjunctive graph (Fig. 3.40). A forager must visit every node once, starting from the source node
O22
Source (hive)
Sink (Nectar)Sink (nectar)
Sink (nectar)
Sink (nectar)
O12 O22
O12
O11
O11
O12
O22
O21 O11
O21
O21
the queen starts to lay her eggs, which are produced in her ovaries. A good queen lays 1500–2000 eggs per day. Broods are made up of fertilized eggs that give birth to queens or workers and unfertilized eggs that generate drones.
The MBO (marriage in honeybees optimization) algorithm is inspired from the marriage behavior in a bee colony [4]. It is based on the mating flight that can be visualized as a set of transitions in a state space (the environment) where the queen moves between the different states in the space at some speed and mates with the drone encountered at each state probabilistically. At the beginning of the flight, the queen is initialized with some energy and returns to her nest when the energy is below some threshold or when her spermatheca is full. A drone mates with a queen probabilistically using the following formula:
prob(Q, D)=e −(f)
S(t)
where prob(Q, D) is the probability of adding the sperm of droneDto the spermatheca of queen Q, that is, the probability of a successful mating, (f) is the absolute difference between the fitness ofD(i.e.,f(D)) and the fitness ofQ(i.e.,f(Q)), and S(t) is the speed of the queen at timet. This function is an annealing function, where the probability of mating is high when either the queen is still in the start of her mating flight and therefore her speed is high or when the fitness of the drone is as good as the queen’s. After each transition in the space, the queen’s speed and energy decrease using the following equations:
S(t+1)=α
·
S(t)E(t+1)=E(t)−γ
Algorithm 3.16 shows the template for the MBO algorithm [4]. The algorithm starts with initializing the queen’s genotype at random. After that, workers (i.e., S-metaheuristics) are used to improve the queen’s genotype, thereby preserving the assumption that a queen is usually a good bee. Afterward, a set of mating flights are performed. In each mating flight, the queen’s energy and speed are initialized randomly. The queen then moves between different states (i.e., solutions) in the space according to her speed and mates with the drone she encounters at each state using the previously presented function. The transition made by the queen in the space is based on her speed. Therefore, at the beginning of the process, the speed is usually high and the queen makes very large steps in the space. If a drone is successfully mated with the queen, his sperm is added to the queen’s spermatheca (i.e., a set of partial solutions). Then, the speed and the energy of the queen are reduced. Once the queen finishes her mating flight, she returns to the nest and starts breeding by selecting a sperm from her spermatheca at random followed by crossover with the queen’s genome that complements the chosen sperm. This crossover operation results in a brood. This is the haploid crossover [4]. The mutation operator then acts on the brood. Therefore, if the same sperm is used once more to generate a brood, the resultant brood will be different because of mutation. This process is followed by applying the worker to improve the broods. The fitness of each worker is then updated based on the amount of improvement performed by the worker to the drone. Afterward, the queen is replaced with the fittest brood if the brood is better than the queen. The remaining broods are then killed and a new mating flight is launched [4].
Algorithm 3.16 Template of the MBO algorithm.
Random initialization of the queen’s ;
Improve the queen with workers (S-metaheuristic) ; Forpredefined maximum number of mating-flights Do
Initialize energy and speed ; While queen’s energy>0 Do
The queen moves between states and probabilistically chooses drones ; If a drone is selectedThen
Add its sperm to the queen’s spermatheca ; Update the queen’s internal energy and speed ; Endwhile
Generate broods by haploid-crossover and mutation ; Use the workers (S-metaheuristic) to improve the broods ; Update workers’ fitness ;
If The best brood is fitter than the queenThen
Replace the queen’s chromosome with the best brood’s chromosome ; Kill all broods ;
Endfor
![FIGURE 3.39 Bee colony behavior for food source (nectar) discovering. We assume two discovered food sources A and B [445]](https://thumb-eu.123doks.com/thumbv2/123dok_br/16370898.723315/5.892.196.692.462.925/figure-colony-behavior-source-nectar-discovering-discovered-sources.webp)
