An analogy with the way ant colonies function has suggested the definition of a new computational paradigm, which we call ant system (AS). We propose it as a viable new approach to stochastic combinatorial optimization. The main characteristics of this model are positive feedback, distributed computation, and the use of a constructive greedy heuristic. Positive feedback accounts for rapid discovery of good solutions, distributed computation avoids premature convergence, and the greedy heuristic helps find acceptable solutions in the early stages of the search process. We apply the proposed methodology to the classical traveling salesman problem (TSP), and report simulation results. We also discuss parameter selection and the early setups of the model, and compare it with tabu search and simulated annealing using TSP. To demonstrate the robustness of the approach, we show how the ant system (AS) can be applied to other optimization problems like the asymmetric traveling salesman, the quadratic assignment and the job-shop scheduling. Finally we discuss the salient characteristics-global data structure revision, distributed communication and probabilistic transitions of the AS.

/pdf/ant-system-optimization-by-a-colony-of-cooperating-agents-gw00wix2tf.pdf

Ant system: optimization by a colony of cooperating agents

This paper introduces the ant colony system (ACS), a distributed algorithm that is applied to the traveling salesman problem (TSP). In the ACS, a set of cooperating agents called ants cooperate to find good solutions to TSPs. Ants cooperate using an indirect form of communication mediated by a pheromone they deposit on the edges of the TSP graph while building solutions. We study the ACS by running experiments to understand its operation. The results show that the ACS outperforms other nature-inspired algorithms such as simulated annealing and evolutionary computation, and we conclude comparing ACS-3-opt, a version of the ACS augmented with a local search procedure, to some of the best performing algorithms for symmetric and asymmetric TSPs.

http://www-igm.univ-mlv.fr/~lombardy/ens/JavaTTT0708/fourmis.pdf

Ant colony system: a cooperative learning approach to the traveling salesman problem

https://leeds-faculty.colorado.edu/glover/TS%20-%20Future%20Paths%20for%20Integer%20Programming.pdf

Future paths for integer programming and links to artificial intelligence

Ants colonies exhibit very interesting behaviours: even if a single ant only has simple capabilities, the behaviour of a whole ant colony is highly structured. This is the result of coordinated interactions. But, as communication possibilities among ants are very limited, interactions must be based on very simple flows of information. In this paper we explore the implications that the study of ants behaviour can have on problem solving and optimization. We introduce a distributed problem solving environment and propose its use to search for a solution to the travelling salesman problem.

Distributed Optimization by Ant Colonies

We describe an artificial ant colony capable of solving the travelling salesman problem (TSP). Ants of the artificial colony are able to generate successively shorter feasible tours by using information accumulated in the form of a pheromone trail deposited on the edges of the TSP graph. Computer simulations demonstrate that the artificial ant colony is capable of generating good solutions to both symmetric and asymmetric instances of the TSP. The method is an example, like simulated annealing, neural networks and evolutionary computation, of the successful use of a natural metaphor to design an optimization algorithm.

https://people.idsia.ch/~luca/acs-bio97.pdf

Ant colonies for the travelling salesman problem

: An O(n sup 3) heuristic algorithm is described for solving n-city travelling salesman problems (TSP) whose cost matrix satisfies the triangularity condition. The algorithm involves as substeps the computation of a shortest spanning tree of the graph G defining the TSP, and the finding of a minimum cost perfect matching of a certain induced subgraph of G. A worst-case analysis of this heuristic shows that the ratio of the answer obtained to the optimum TSP solution is strictly less than 3/2. This represents a 50% reduction over the value 2 which was the previously best known such ratio for the performance of other polynomial-growth algorithms for the TSP.

/pdf/worst-case-analysis-of-a-new-heuristic-for-the-travelling-2w6lcy5duq.pdf

Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem

The vehicle-scheduling problem involves the design of several vehicle tours to meet a given set of requirements for customers with known locations, subject to a capacity constraint for the vehicles...

https://link.springer.com/content/pdf/10.1057%2Fjors.1969.75.pdf

An Algorithm for the Vehicle-dispatching Problem

We present a tree-search algorithm for two-dimensional cutting problems in which there is a constraint on the maximum number of each type of piece that is to be produced The algorithm limits the size of the tree search by deriving and imposing necessary conditions for the cutting pattern to be optimal A dynamic programming procedure for the solution of the unconstrained problem and a node evaluation method based on a transportation routine are used to produce upper bounds during the search The computational performance of the algorithm is illustrated by tests performed on a large number of randomly generated problems with constraints of varying "tightness" The results indicate that the algorithm is an effective procedure for solving cutting problems of medium size

/pdf/an-algorithm-for-two-dimensional-cutting-problems-2gwhuf78sp.pdf

Nicos Christofides

Papers

Worst-Case Analysis of a New Heuristic for the Travelling Salesman Problem

An Algorithm for the Vehicle-dispatching Problem

An Algorithm for Two-Dimensional Cutting Problems

Distribution management : mathematical modelling and practical analysis

Project scheduling with resource constraints: A branch and bound approach