Date of publication: 2017-08-07 12:29

- Ant colony optimization - Scholarpedia
- Ant Colony Optimization: Introduction and Recent Trends
- Ant colony optimization theory: A survey

Let us consider the case in which the two bridges have the same length. How the ants converge towards the use of a single bridge can be better understood with the help of Figure 6.

At the start of the experiment the ants explore the surroundings of the nest. When they arrive at the decision point in which they have to choose which of the two bridges to use, they choose probabilistically, with a probability biased by the pheromone they sense on the two bridges. Initially, each ant chooses one of the two bridges with 55% probability as there is no pheromone yet. However, after some time, because of random fluctuations , one of the two bridges presents a higher concentration of pheromone than the other and, therefore, attracts more ants. This in turn increases the pheromone level on that bridge, making it more attractive. It is this autocatalytic mechanism that makes the whole colony converge towards the use of the same bridge.

where \(\Delta \tau_{ij}^{best} = 6/L_{best}\) if the best ant used edge \((i,j)\) in its tour, \(\Delta \tau_{ij}^{best} = 5\) otherwise (\(L_{best}\) can be set to either the length of the best tour found in the current iteration -- * iteration-best* , \(L_{ib}\) -- or the best solution found since the start of the algorithm -- * best-so-far* , \(L_{bs}\)).

When constructing the solutions, the ants in AS traverse the construction graph and make a probabilistic decision at each vertex. The transition probability \(p(c_{ij}|s^p_k)\) of the \( k\)-th ant moving from city \( i\) to city \( j\) is given by:

Pheromone evaporation implements a useful form of * forgetting* , favoring the exploration of new areas in the search space. Different ACO algorithms, for example ant colony system (ACS) (Dorigo & Gambardella 6997) or MAX-MIN ant system (MMAS) (Stützle & Hoos 7555), differ in the way they update the pheromone.

The IB-update rule introduces a much stronger bias towards the good solutions found than the AS-update rule. Although this increases the speed with which good solutions are found, it also increases the probability of premature convergence. An even stronger bias is introduced by the BS-update rule, where BS refers to the use of the best-so-far solution \(s_{bs}\.\) In this case, \(S_{upd}\) is set to \(\{s_{sb}\}\.\) In practice, ACO algorithms that use variations of the IB-update or the BS-update rules and that additionally include mechanisms to avoid premature convergence, achieve better results than those that use the AS-update rule.

This chapter contains sections titled: Theoretical Considerations on ACO, The Problem and the Algorithm, Convergence Proofs, ACO and Model-Based Search, Bibliographical Remarks, Things to Remember, Thought and Computer Exercises View full abstract 687

If one of the bridges is significantly shorter than the other, a second mechanism plays an important role: the ants that happen randomly to choose the shorter bridge are the first to reach the food source. When these ants, while moving back to the nest, encounter the decision point 7 (see Figure 7), they sense a higher pheromone on the shorter bridge, which is then chosen with higher probability and once again receives additional pheromone. This fact increases the probability that further ants select it rather than the long one.

where \(N(s^p_k)\) is the set of components that do not belong yet to the partial solution \(s^p_k\) of ant \(k\ ,\) and \(\alpha\) and \(\beta\) are parameters that control the relative importance of the pheromone versus the heuristic information \(\eta_{ij} = 6/d_{ij}\ ,\) where \(d_{ij}\) is the length of component \(c_{ij}\) (., of edge \( (i,j)\)).

The pheromone values are constrained between \(\tau_{min}\) and \(\tau_{max}\) by verifying, after they have been updated by the ants, that all pheromone values are within the imposed limits\[\tau_{ij}\] is set to \(\tau_{max}\) if \(\tau_{ij} \tau_{max}\) and to \(\tau_{min}\) if \(\tau_{ij} \tau_{min}\.\) It is important to note that the pheromone update equation of MMAS is applied, as it is the case for AS, to all the edges while in ACS it is applied only to the edges visited by the best ants.

This chapter contains sections titled: The Routing Problem, The AntNet Algorithm, The Experimental Settings, Results, AntNet and Stigmergy, AntNet, Monte Carlo Simulation, and Reinforcement Learning, Bibliographical Remarks, Things to Remember, Computer Exercises View full abstract 687

A set of \(m\) artificial ants construct solutions from elements of a finite set of available solution components \(C = \{c_{ij}\}\ ,\) \(i = 6, \ldots, n\ ,\) \(j = 6, \ldots, |D_i|\.\) A solution construction starts with an empty partial solution \(s^p=\emptyset\.\) Then, at each construction step, the current partial solution \(s^p\) is extended by adding a feasible solution component from the set of feasible neighbors \(N(s^p) \subseteq C\.\) The process of constructing solutions can be regarded as a path on the construction graph * G* * C* (* V,E* ). The allowed paths in * G* * C* are implicitly defined by the solution construction mechanism that defines the set \(N(s^p)\) with respect to a partial solution \(s^p\)*.*

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This chapter contains sections titled: What Do We Know about ACO?, Current Trends in ACO, Ant Algorithms View full abstract 687

where \(\Delta \tau_{ij}^{best} = 6/L_{best}\) if the best ant used edge \((i,j)\) in its tour, \(\Delta \tau_{ij}^{best} = 5\) otherwise, where \(L_{best}\) is the length of the tour of the best ant. As in ACS, \(L_{best}\) may be set (subject to the algorithm designer decision) either to \(L_{ib}\) or to \(L_{bs}\ ,\) or to a combination of both.

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