In this paper, we propose a probabilistic modeling clonal selection algorithm which combines the clonal selection algorithm (CSA) and the probabilistic modeling (PM) for traveling salesman problem (TSP). The clonal selection algorithm is employed by the natural immune system to define the basic features of an immune response to an antigenic stimulus, can initialize antibodies and maintain the p...

In this study, we try to implant chaotic features into the learning algorithm of self-organizing map. We call this concept as Chaotic SOM (CHAOSOM). As a first step to realize CHAOSOM, we consider the case that learning rate and neighboring coefficient of SOM are refreshed by chaotic pulses generated by the Hodgkin-Huxley equation. We apply the CHAOSOM to solve a traveling salesman problem and ...

We prove that both minimum and maximum traveling salesman problems on complete graphs with edge-distances 1 and 2 (denoted by min TSP12 and max TSP12, respectively) are approximable within 3/4. Based upon this result, we improve the standard approximation ratio known for maximum traveling salesman with distances 1 and 2 from 3/4 to 7/8. Finally, we prove that, for any 2 > 0, it is NP-hard to ap...

Local search heuristics are among the most popular approaches to solve hard optimization problems. Among them, Very Large Scale Neighborhood Search techniques present a good balance between the quality of local optima and the time to search a neighborhood. We develop a language to generate exponentially large neighborhoods for sequencing problems using grammars. We develop efficient generic dyn...

Graphs (1-skeletons) of Traveling-Salesman-related polytopes have attracted a lot of attention. Pedigree polytopes are extensions of the classical Symmetric Traveling Salesman Problem polytopes (Arthanari 2000) whose graphs contain the TSP polytope graphs as spanning subgraphs (Arthanari 2013). Unlike TSP polytopes, Pedigree polytopes are not “symmetric”, e.g., their graphs are not vertex trans...

This paper deals with evolutionary algorithms for solving the vehicle routing problem. More precisely, the paper is concerned with eight evolutionary crossover operators, which have originally been designed for the traveling salesman problem but can also be used for vehicle routing. The considered crossovers are tested on a set of well known benchmark problem instances. The obtained experimenta...

The Generalized Traveling Salesman Problem (GTSP) is a modification of the Traveling Salesman Problem in which nodes are partitioned into clusters and exactly one node from each cluster is visited in a cycle. It has numerous applications, including airplane routing, computer file sequencing, and postal delivery. To produce solutions to this problem, a genetic algorithm (GA) heuristic mimicking ...

In this work we investigate two variants of the Stochastic Vehicle Routing Problem: The Vehicle Routing Problem with Stochastic Demands and the Vehicle Routing Problem with Stochastic Demands and Customers. We show that under some moderate conditions there is an asymptotic equivalence between the Vehicle Routing Problem with Stochastic Demands and the Traveling Salesman Problem, as well as betw...

This paper studies the close-enough traveling salesman problem, a variant of the Euclidean traveling salesman problem in which the traveler visits a node if it passes through the neighborhood of that node. We introduce an improved version of the adaptive internal discretization scheme, recently proposed in the literature, and a heuristic that combines this scheme with to a second-order cone pro...

We present a Monte Carlo algorithm to find approximate solutions of the traveling salesman problem. The algorithm generates randomly the permutations of the stations of the traveling salesman trip, with probability depending on the length of the corresponding route. Reasoning by analogy with statistical thermodynamics, we use the probability given by the Boltzmann-Gibbs distribution. Surprising...