Nature-Inspired Optimization Algorithms || Other Algorithms and Hybrid Algorithms

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  • 15 Other Algorithms and HybridAlgorithms

    There are many other algorithms in the literature, but we have not covered them in detailin this book. One of the reasons is that some of these algorithms, such as ant colonyoptimization and harmony search, have extensive literature and books devoted to them,so readers can easily find information about them elsewhere. While for some otheralgorithms, such as some of the hybrid algorithms, their efficiency and performancemay be preliminary, and further research is needed to verify these results.

    However, for completeness, here we briefly introduce a few other widely used algo-rithms and briefly touch on some of the hybrid algorithms.

    15.1 Ant Algorithms

    15.1.1 Ant Behavior

    Ants are social insects that live together in organized colonies whose population sizecan range from about 2 million to 25 million. When foraging, a swarm of ants ormobile agents interact or communicate in their local environment. Each ant can layscent chemicals or pheromone so as to communicate with others, and each ant is alsoable to follow the route marked with pheromone laid by other ants. When ants find afood source, they mark it with pheromone as well as marking the trails to and fromit. From the initial random foraging route, the pheromone concentration varies. Theants follow the routes with higher pheromone concentrations, and the pheromone isenhanced by the increasing number of ants. As more and more ants follow the sameroute, it becomes the favored path. Thus, some favorite routes emerge, often the shortestor more efficient. This is actually a positive feedback mechanism.

    Emerging behavior exists in an ant colony; such emergence arises from simpleinteractions among individual ants. Individual ants act according to simple and localinformation (such as pheromone concentration) to carry out their activities. Althoughthere is no master ant overseeing the entire colony and broadcasting instructions tothe individual ants, organized behavior still emerges automatically. Therefore, suchemergent behavior can be similar to other self-organized phenomena that occur inmany processes in nature, such as the pattern formation in some animal skins (e.g.,tiger and zebra skins).

    The foraging pattern of some ant species (such as army ants) can show extraordinaryregularity. Army ants search for food along some regular routes with an angle of about

    Nature-Inspired Optimization Algorithms. http://dx.doi.org/10.1016/B978-0-12-416743-8.00015-4 2014 Elsevier Inc. All rights reserved.

    http://dx.doi.org/10.1016/B978-0-12-416743-8.00015-4
  • 214 Nature-Inspired Optimization Algorithms

    123 apart. We do not know how they manage to follow such regularity, but studiesshow that they could move into an area and build a bivouac and start foraging. On thefirst day, they forage in a random direction, say, the north, and travel a few hundredmeters, then branch to cover a large area. The next day they will choose a differentdirection, which is about 123 from the direction on the previous day, and so coveranother large area. On the following day, they again choose a different direction about123 from the second days direction. In this way, they cover the whole area over abouttwo weeks and then move out to a different location to build a bivouac and forage inthe new region [40].

    The interesting thing is that ants do not use the angle of 360/3 = 120 (this wouldmean that on the fourth day, they will search the empty area already foraged on thefirst day). The beauty of this 123 angle is that after about three days, it leaves an angleof about 10 from the direction of the first day. This means the ants cover the wholecircular region in 14 days without repeating or covering a previously foraged area. Thisis an amazing phenomenon.

    15.1.2 Ant Colony Optimization

    Based on these characteristics of ant behavior, scientists have developed a number ofpowerful ant colony algorithms, with important progress made in recent years. MarcoDorigo pioneered the research in this area in 1992 [6,7]. Many different variants haveappeared since then.

    If we use only some of the features of ant behavior and add some new characteristics,we can devise a class of new algorithms.

    There are two important issues here: the probability of choosing a route and the evap-oration rate of pheromone. There are a few ways of solving these problems, althoughit is still an area of active research. Here we introduce the current best method.

    For a network routing problem, the probability of ants at a particular node i to choosethe route from node i to node j is given by

    pi j =i j d

    i jn

    i, j=1 i j di j

    , (15.1)

    where > 0 and > 0 are the influence parameters, and their typical values are 2. i j is the pheromone concentration on the route between i and j , and di jis the desirability of the same route. Some a priori knowledge about the route, such asthe distance si j , is often used so that di j 1/si j , which implies that shorter routes willbe selected due to their shorter traveling time, and thus the pheromone concentrationson these routes are higher. This is because the traveling time is shorter, and thus lessof the pheromone has been evaporated during this period.

    This probability formula reflects the fact that ants would normally follow the pathswith higher pheromone concentrations. In the simpler case when = = 1, theprobability of choosing a path by ants is proportional to the pheromone concentrationon the path. The denominator normalizes the probability so that it is in the range between0 and 1.

  • Other Algorithms and Hybrid Algorithms 215

    The pheromone concentration can change with time due to the evaporation ofpheromone. Furthermore, the advantage of pheromone evaporation is that the systemcould avoid being trapped in local optima. If there is no evaporation, the path randomlychosen by the first ants will become the preferred path due to the attraction of otherants by their pheromone. For a constant rate of pheromone decay or evaporation, thepheromone concentration usually varies with time exponentially:

    (t) = 0e t , (15.2)where 0 is the initial concentration of pheromone and t is time. If t 1, then wehave (t) (1 t)0. For the unitary time increment t = 1, the evaporation canbe approximated by t+1 (1 )t . Therefore, we have the simplified pheromoneupdate formula:

    t+1i j = (1 )ti j + ti j , (15.3)

    where [0, 1] is the rate of pheromone evaporation. The increment ti j is theamount of pheromone deposited at time t along route i to j when an ant travels adistance L . Usually ti j 1/L . If there are no ants on a route, then the pheromonedeposit is zero.

    There are other variations on this basic procedure. A possible acceleration scheme isto use some bounds of the pheromone concentration, and only the ants with the currentglobal best solution(s) are allowed to deposit pheromone. In addition, some ranking ofthe solution fitness can also be used. These are still topics of active research.

    Because a complex network system is always made of individual nodes, this algo-rithm can be extended to solve complex routing problems reasonably efficiently. In fact,ant colony optimization and its variants have been successfully applied to the Internetrouting problem, the traveling salesman problem, and other combinatorial optimizationproblems.

    15.1.3 Virtual Ant Algorithms

    Since we know that ant colony optimization has successfully solved combinatorialproblems, it can also be extended to solve the standard optimization problems of mul-timodal functions. The only problem now is to figure out how the ants will move on ad-dimensional hypersurface. For simplicity, we discuss the 2D case, which can easilybe extended to higher dimensions. On a 2D landscape, ants can move in any direction,or 0 360, but this will cause some problems. The question is how to update thepheromone at a particular point, since there are an infinite number of points. One solu-tion is to track the history of each ants moves and record the locations consecutively,and the other approach is to use a moving neighborhood or window. The ants smellthe pheromone concentration of their neighborhood at any particular location.

    In addition, we can limit the number of directions the ants can move by quantizingthe directions. For example, ants are only allowed to move left and right, and up anddown (only four directions). We use this quantized approach here, which makes theimplementation much simpler. Furthermore, the objective function or landscape can

  • 216 Nature-Inspired Optimization Algorithms

    be encoded into virtual food so that ants will move to the best locations where the bestfood sources are. This will make the search process even more simpler. This simplifiedalgorithm, called the virtual ant algorithm (VAA) and developed by Xin-She Yangand his colleagues in 2006 [40], that has been successfully applied to topologicaloptimization problems in engineering.

    15.2 Bee-Inspired Algorithms

    Bee algorithms form another class of algorithms that are closely related to the antcolony optimization. Bee algorithms are inspired by the foraging behavior of honey-bees. Several variants of bee algorithms have been formulated, including the honeybeealgorithm (HBA), the virtual bee algorithm (VBA), the artificial bee colony (ABC)optimization, the honeybee-mating algorithm (HBMA), and others [1,19,20,29,41].

    15.2.1 Honeybee Behavior

    Honeybees live in constructed colonies where and they store honey they have foraged.Honeybees can communicate by pheromone and waggle dance. For example, analarming bee may release a chemical messenger (pheromone) to stimulate an attackresponse in other bees. Furthermore, when bees find a good food source and bring somenectar back to the hive, the

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