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In the above, f(zi) is a measure of average similarity of Zi with the other data vectors in the neighborhood. The constant a controls the scale of dissimilarity, by determining when two data vectors should be grouped together. The probabilities of picking up and dropping a data vector are expressed as
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A summary of the ant [Lumer and Faieta 1994]: 1. Initialization:
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(a) place each data vector zi randomly on the grid (b) place agents at randomly selected sites (c) set values for k1,k2 a, s and the maximum number of time steps 2. For t = 1 to tmax, for each agent: (a) If the agent is unladen, and the site is occupied by an item Zi, i. Compute f ( z i ) and p p (z i ). ii. If U(0,1) < Pp(zi), pick up data vector zi. (b) Otherwise, if the agent carries data vector zi and the site is empty: i. Compute f ( z i ) and Pd(zi). ii. If U (0,1) < Pd(z i ), drop data vector zi.
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(c) Move to a randomly selected neighboring site not occupied by another agent. Some remarks about the algorithm above are necessary: the grid should have more sites than the number of ants; and there should be more sites than data vectors. The algorithm also has the tendency to create more clusters than are necessary, basically overfitting the data. This problem can be addressed in the following ways: Having the ants move at different speeds. Fast-moving agents will form coarser clusters by being less selective in their estimation of the average similarity of a data vector to its neighbors. Slower agents are more accurate in refining the cluster boundaries. Having agents that move at different speeds prevents the clustering process from converging too fast. Different moving speeds are easily modeled using
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where v ~ U (l,v max ) and vmax is the maximum moving speed. Using short-term memory for each agent, which allows each agent to remember the last m data vectors dropped by the agent, and the locations of these drops. If the agent picks up another data vector similar to the last m data vectors, the agent will move in that direction. This approach ensures that similar data vectors are grouped into the same cluster.
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Applications of Ant Colony Optimization
The study of the behavior of ants has resulted in developing algorithms applied to a variety of problems. Algorithms that model the foraging behavior of ants (e.g. food collection) resulted in new combinatorial optimization approaches (with applications to network routing, job scheduling, etc); the ability of ants to dynamically distribute labor resulted in adaptive task allocation strategies; cemetery organization and brood sorting resulted in graph coloring and sorting algorithms; and the cooperative transport characteristics resulted in robotic implementations. This section reviews some of these applications with reference to the relevant literature.
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AGO has been used to solve the quadratic assignment problem (QAP) [Maniezzo et al. 1994]. The QAP concerns the optimal assignment of n activities to n locations. Formally, the QAP is defined as a permutation of assignments which minimizes
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where dij is the distance between locations i and j, and fhk characterizes the flow (e.g. data transfer) between activities h and k. ACO has also been used successfully for the job-scheduling problem (JSP) [Colorni et al. 1994]. For the JSP, operations of a set of jobs must be scheduled to be executed on M machines in such a way that the maximum completion times of all operations is minimized, and only one job at a time is processed by a machine. The graph coloring problem (GCP) is a well-known optimization problem also solved by AGO [Costa and Hertz 1997]. This problem involves coloring the nodes of a graph, using q colors, such that no adjacent nodes have the same color. Another NP-hard problem solved by AGO is the shortest common supersequence problem (SCSP) [Michel and Middendorf 1998]. The aim of the SCSP is to find a string of minimum length that is a supersequence of each string in a set of strings. ACO has also been used for routing optimization in telephone networks [Schoonderwoerd et al. 1996] and data communications networks [Di Caro and Dorigo 1998]. Robotics is a very popular field for the application of models of ants behavior, especially swarm-based robotics. Some of these applications include (see [Dorigo 1999] for lists of references) adaptive task allocation to groups of agents or robots; robots for distributed clustering of objects and sorting objects; self-assembling (or metamorphic) robots; and cooperative transport by a swarm of robots. These are just a few applications. For more information on these, and other applications of ACO, the reader is referred to [Dorigo 1999].