Travelling salesman problem ant system algorithm pheromone updating

The travelling purchaser problem and the vehicle routing problem are both generalizations of TSP.In the theory of computational complexity, the decision version of the TSP (where, given a length L, the task is to decide whether the graph has any tour shorter than L) belongs to the class of NP-complete problems.The TSP has several applications even in its purest formulation, such as planning, logistics, and the manufacture of microchips.Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing.In these applications, the concept city represents, for example, customers, soldering points, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments.The TSP also appears in astronomy, as astronomers observing many sources will want to minimize the time spent moving the telescope between the sources.Dantzig, Fulkerson and Johnson, however, speculated that given a near optimal solution we may be able to find optimality or prove optimality by adding a small number of extra inequalities (cuts).They used this idea to solve their initial 49 city problem using a string model.

As the algorithm was so simple and quick, many hoped it would give way to a near optimal solution method.

It is used as a benchmark for many optimization methods.

Even though the problem is computationally difficult, a large number of heuristics and exact algorithms are known, so that some instances with tens of thousands of cities can be solved completely and even problems with millions of cities can be approximated within a small fraction of 1%.

In many applications, additional constraints such as limited resources or time windows may be imposed.

The origins of the travelling salesman problem are unclear.

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