Symmetric Traveling Salesman Research Articles

Angular bisector insertion algorithm for solving small-scale symmetric and asymmetric traveling salesman problem

Angular bisector insertion algorithm for solving small-scale symmetric and asymmetric traveling salesman problem

Minimization and maximization versions of the quadratic travelling salesman problem

The travelling salesman problem (TSP) asks for a shortest tour through all vertices of a graph with respect to the weights of the edges. The symmetric quadratic travelling salesman problem (SQTSP) associates a weight with every three vertices traversed in succession. If these weights correspond to the turning angles of the tour, we speak of the angular-metric travelling salesman problem (Angle TSP). In this paper, we first consider the SQTSP from a computational point of view. In particular, we apply a rather basic algorithmic idea and perform the separation of the classical subtour elimination constraints on integral solutions only. Surprisingly, it turns out that this approach is faster than the standard fractional separation procedure known from the literature. We also test the combination with strengthened subtour elimination constraints for both variants, but these turn out to slow down the computation. Secondly, we provide a completely different, mathematically interesting MILP linearization for the Angle TSP that needs only a linear number of additional variables while the standard linearization requires a cubic one. For medium-sized instances of a variant of the Angle TSP, this formulation yields reduced running times. However, for larger instances or pure Angle TSP instances, the new formulation takes more time to solve than the known standard model. Finally, we introduce MaxSQTSP, the maximization version of the quadratic travelling salesman problem. Here, it turns out that using some of the stronger subtour elimination constraints helps. For the special case of the MaxAngle TSP, we can observe an interesting geometric property if the number of vertices is odd. We show that the sum of inner turning angles in an optimal solution always equals . This implies that the problem can be solved by the standard ILP model without producing any integral subtours. Moreover, we give a simple constructive polynomial time algorithm to find such an optimal solution. If the number of vertices is even, the optimal value lies between 0 and and these two bounds are tight, which can be shown by an analytic solution for a regular n-gon.

On the core of traveling salesman games

We define excess rate to study the core of traveling salesman games from a perspective of optimization, propose a new variant of the traveling salesman problem, and build a link between the two problems. An exact formula for the lowest achievable excess rate is found, which explains the existence of core emptiness. We then develop an implementable method to check whether empty core exists in general case. The results apply for both symmetric and asymmetric traveling salesman games.

El problema del agente viajero: un algoritmo determinístico usando búsqueda tabú


 Se implementa un algoritmo correspondiente al método Búsqueda Tabú, llamado EraDeterminístico, experimentado con el algoritmo básico que explora el espacio de búsqueda e incorporando la diversificación como estrategia para explorar nuevas regiones. El algoritmo se desarrolla en el ambiente de programación Visual Basic 6.0 y la implementación se orienta a encontrar soluciones cercanas a la óptima del problema NP−Completo del Agente Viajero Simétrico (AVS). Para probar la funcionalidad, el modelo se compara con algunas instancias de la Travel Salesman Problem Library (TSPLIB), algunas instancias aleatorias y se aplica a tres situaciones reales. Finalmente, se presenta una sección donde se exponen recomendaciones y conclusiones, las cuales orientan posibles desarrollos futuros de ésta teoría y donde se pone en evidencian las bondades y la eficiencia de la implementación.

An interactive memetic algorithm for production and manufacturing problems modelled as a multi-objective travelling salesman problem

In this paper, a preference-based, interactive memetic random-key genetic algorithm (PIMRKGA) is developed and used to find (weakly) Pareto optimal solutions to manufacturing and production problems that can be modelled as a symmetric multi-objective travelling salesman problem. Since there are a large number of solutions to these kinds of problems, to reduce the computational effort and to provide more desirable and meaningful solutions to the decision maker, this research focuses on using interactive input from the user to explore the most desirable parts of the efficient frontier instead of trying to reproduce the entire frontier. Here, users define their preferences by selecting among five classes of objective functions and by specifying weighting coefficients, bounds, and optional upper bounds on indifference tradeoffs. This structure is married with the memetic algorithm – a random-key genetic algorithm hybridised by local search. The resulting methodology is an iterative process that continues until the decision maker is satisfied with the solution. The paper concludes with case studies utilising different scenarios to illustrate possible manufacturing and production related implementations of the methodology.

New facets of the STS polytope generated from known facets of the ATS polytope

While it had been known for a long time how to transform an asymmetric traveling salesman (ATS) problem on the complete graph with n vertices into a symmetric traveling salesman (STS) problem on an incomplete graph with 2 n vertices, no method was available for using this correspondence to derive facets of the symmetric polytope from facets of the asymmetric polytope until the work of E. Balas and M. Fischetti in [Lifted cycle inequalities for the asymmetric traveling salesman problem, Mathematics of Operations Research 24 (2) (1999) 273–292] suggested an approach. The original Balas–Fischetti method uses a standard sequential lifting procedure for the computation of the coefficient of the edges that are missing in the incomplete STS graph, which is a difficult task when addressing classes of (as opposed to single) inequalities. In this paper we introduce a systematic procedure for accomplishing the lifting task. The procedure exploits the structure of the tight STS tours and organizes them into a suitable tree structure. The potential of the method is illustrated by deriving large new classes of facet-defining STS inequalities.