蚁群算法在解决旅行商问题(TSP)中的应用

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"应用蚁群算法解决旅行商问题 (TSP)，包括48个城市，并提供了算法详解及相关文献。标签涉及蚁群、TSP和Java编程。" 本文介绍了一种分布式算法——蚁群系统（Ant Colony System, ACS），该算法被应用于解决经典的旅行商问题（Traveling Salesman Problem, TSP）。在蚁群系统中，一组合作的智能体，即“蚂蚁”，共同协作寻找TSP问题的优质解。这些蚂蚁通过在TSP图的边路上释放信息素这种间接通信方式来协同工作，构建解决方案。在蚁群算法中，每只蚂蚁随机地选择路径并访问城市，同时在路径上留下信息素痕迹。信息素的浓度会随着时间逐渐蒸发，但蚂蚁在经过的路径上会加强信息素的沉积，尤其是那些较短的路径。这种正反馈机制使得算法倾向于发现更优的路径。作者通过实验研究了蚁群系统的运作机制，结果显示ACS在解决TSP问题上的性能优于模拟退火算法和进化计算等其他自然启发式算法。进一步的，作者对比了增强版的ACS-3-opt，它结合了局部搜索策略，与一些针对对称和非对称TSP问题的最佳算法进行比较。实验表明，这种改进的蚁群系统在求解效率和解的质量上都有显著提升。关键词：自适应行为、蚁群、涌现行为、旅行商问题。 I. 引言旅行商问题是一个经典的组合优化问题，其自然界的类比激发了各种算法的设计。蚁群算法就是其中一种受到生物界，特别是蚂蚁寻路行为启发的智能优化方法。蚂蚁通过信息素的沉积和感知来找到从巢穴到食物源最短路径的行为，为解决TSP问题提供了新的思路。蚁群算法的优势在于其分布式特性和自组织能力，能够有效地探索复杂问题的解决方案空间。此外，由于信息素的动态更新机制，算法具有一定的全局优化能力和抗早熟收敛的能力。总结来说，蚁群算法在解决旅行商问题上表现出色，尤其在处理大量城市时，能够找到接近最优解的路径。结合局部搜索策略的版本更是在效率和精度上进行了优化，这使得蚁群算法在实际应用中，如路线规划、物流配送等领域有着广泛的应用前景。同时，这种算法的设计思路也为其他复杂优化问题的解决提供了借鉴。

DORIGO AND GAMBARDELLA: ANT COLONY SYSTEM 55

Fig. 3. The ACS algorithm.

probability with which ant in city chooses to move to the

city

otherwise

(1)

where

is the pheromone, is the inverse of the

distance

, is the set of cities that remain to be

visited by ant

positioned on city (to make the solution

feasible), and

is a parameter which determines the relative

importance of pheromone versus distance

In (1) we multiply the pheromone on edge

by the

corresponding heuristic value

. In this way we favor the

choice of edges which are shorter and which have a greater

amount of pheromone.

In ant system, the global updating rule is implemented as

follows. Once all ants have built their tours, pheromone is

updated on all edges according to

(2)

where

if tour done by ant

otherwise

is a pheromone decay parameter, is the length

of the tour performed by ant

, and is the number of ants.

Pheromone updating is intended to allocate a greater amount

of pheromone to shorter tours. In a sense, this is similar to

a reinforcement learning scheme [2], [26] in which better

solutions get a higher reinforcement (as happens, for exam-

ple, in genetic algorithms under proportional selection). The

pheromone updating formula was meant to simulate the change

in the amount of pheromone due to both the addition of new

pheromone deposited by ants on the visited edges and to

pheromone evaporation.

Pheromone placed on the edges plays the role of a dis-

tributed long-term memory: this memory is not stored locally

within the individual ants, but is distributed on the edges of

the graph. This allows an indirect form of communication

called stigmergy [9], [23]. The interested reader will ﬁnd a

full description of ant system, of its biological motivations,

and computational results in [12].

Although ant system was useful for discovering good or

optimal solutions for small TSP’s (up to 30 cities), the time

required to ﬁnd such results made it infeasible for larger

problems. We devised three main changes to improve its

performance which led to the deﬁnition of the ACS, presented

in the next section.

III. ACS

The ACS differs from the previous ant system because

of three main aspects: i) the state transition rule provides a

direct way to balance between exploration of new edges and

exploitation of a priori and accumulated knowledge about the

problem, ii) the global updating rule is applied only to edges

which belong to the best ant tour, and iii) while ants construct

a solution a local pheromone updating rule (local updating

rule, for short) is applied.

Informally, the ACS works as follows:

ants are initially

positioned on

cities chosen according to some initialization

rule (e.g., randomly). Each ant builds a tour (i.e., a feasible

solution to the TSP) by repeatedly applying a stochastic greedy

rule (the state transition rule). While constructing its tour, an

ant also modiﬁes the amount of pheromone on the visited

edges by applying the local updating rule. Once all ants have

terminated their tour, the amount of pheromone on edges is

modiﬁed again (by applying the global updating rule). As

was the case in ant system, ants are guided, in building their

tours, by both heuristic information (they prefer to choose

short edges) and by pheromone information. An edge with

a high amount of pheromone is a very desirable choice. The

pheromone updating rules are designed so that they tend to

give more pheromone to edges which should be visited by

ants. The ACS algorithm is reported in Fig. 3. In the following

we discuss the state transition rule, the global updating rule,

and the local updating rule.

A. ACS State Transition Rule

In the ACS the state transition rule is as follows: an ant

positioned on node

chooses the city to move to by applying

the rule given by (3)

if exploitation

otherwise biased exploration

(3)

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