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Stigmergy as an Approach to Metaheuristic Optimization - Computer ...

22 2 **Optimization** with **an**t colonies 2.4.5 The number of **an**ts A question often arises: why use a colony of **an**ts instead of a single **an**t? The fact is, although a single **an**t is capable of generating a solution, the convergence **an**d quality of solutions obtained with a colony is often much better. This is most obvious when ACO is used on geographically distributed problems, where the differential length effect exploited by **an**ts in the solution c**an** only arise in the presence of a colony of **an**ts. For example, in routing problems **an**ts solve m**an**y shortest-path problems in parallel, **an**d this requires a colony of **an**ts for each of these problems (finding a path between two vertices). But in combina**to**rial optimization problems the differential length effect is usually not exploited. This me**an**s that m > 1 **an**ts building r solutions in l iterations is equivalent **to** one **an**t building r solutions in lm iterations. In this c**as**e the number of **an**ts is not so import**an**t. But this holds only in theory. In practice there are a lot of ACO algorithms that use m > 1 **to** successfully solve hard combina**to**rial optimization problems. In general, the best value for m is different for every individual algorithm, **an**d in most c**as**es it h**as** **to** be set experimentally. Fortunately, most ACO algorithms are quite robust in terms of the number of **an**ts used. 2.4.6 C**an**didate lists If **an** ACO algorithm is applied **to** a problem where **an**ts have a large neighborhood **to** choose from, the solution construction is signific**an**tly slowed down **an**d the probability of m**an**y **an**ts visiting the same state is very small. This problem c**an** be reduced by the use of a c**an**didate list. A c**an**didate list consists of a small set of promising neighbors of the current state. This promising neighbors are usually created considering a priori available information about the problem or some dynamically generated information. When it is applied the ACO c**an** concentrate more on interesting parts of the search space. So far, the use of c**an**didate lists or similar approaches in ACO algorithms is still rather unexplored. Inspiration from other techniques like TS [39] or Greedy R**an**domized Adaptive Search Procedure (GRASP) [30], where extensive use is made of c**an**didate lists, could be useful for the development of effective c**an**didate-list strategies for ACO.

2.5 ACO-b**as**ed algorithms for combina**to**rial optimization 23 2.5 ACO-b**as**ed algorithms for combina**to**rial optimization Here we present characteristic examples of a very large group of ACO-b**as**ed algorithms that solve hard combina**to**rial optimization problems. The Ant System (AS) [15, 22] w**as** the first **an**t-b**as**ed algorithm used **to** solve a hard combina**to**rial problem, i.e., the Traveling Salesm**an** Problem (TSP). The main characteristics of this algorithm are positive feedback, distributed computation, **an**d the use of a constructive greedy heuristic [26]. The Ant-Q (AQ) [24, 35] is a distributed approach **to** combina**to**rial optimization b**as**ed on reinforcement learning. The AQ finds its b**as**is in the AS **an**d the Q-learning algorithm [127]. The fundamental difference between the AS **an**d AQ is that in AQ only the **an**t that found the “best” path gets **to** deposit pheromone on its trail. The Ant Colony System (ACS) [25, 110] algorithm is a successor of the AS **an**d AQ. The main improvements are: pheromone trail updates are done offline, **an**ts use a pseudo-r**an**dom-proportional rule decision rule, **an**d step by step updates are done online. For that re**as**on it is simpler **an**d more efficient th**an** the AS **an**d AQ. The MAX-MIN Ant System (MMAS) [113] is also **an** extension of the AS with the following differences. Like in ACS, pheromone trail updates are done offline. To avoid stagnation, pheromone values are bounded by **an** upper **an**d lower limit. Trails are initialized with the maximum possible amount of pheromone. The r**an**k-b**as**ed Ant System (ASr**an**k) [12] is **an** elitist variation of the AS. Here a r**an**k of “best” **an**ts is kept at all times **an**d only they are allowed **to** deposit pheromone. At the end of each iteration the currently best solution is used **to** update pheromone. The Hybrid Ant System (HAS) [36, 37] is the ACS updated with local search. Here local search is applied every time the **an**ts build their solutions. Pheromone trails are updated according **to** these newly acquired locally optimal solutions. The Approximate Nondeterministic Tree Search (ANTS) [86] algorithm is similar in structure **to** the tree-search algorithm [91]. The main difference is in its lack of a complete backtracking mech**an**ism.

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