An Optimal Control Theory for the Traveling Salesman Problem and Its Variants

We show that the traveling salesman problem (TSP) and its many variants may be modeled as functional optimization problems over a graph. In this formulation, all vertices and arcs of the graph are functionals; i.e., a mapping from a space of measurable functions to the field of real numbers. Many variants of the TSP, such as those with neighborhoods, with forbidden neighborhoods, with time-windows and with profits, can all be framed under this construct. In sharp contrast to their discrete-optimization counterparts, the modeling constructs presented in this paper represent a fundamentally new domain of analysis and computation for TSPs and their variants. Beyond its apparent mathematical unification of a class of problems in graph theory, the main advantage of the new approach is that it facilitates the modeling of certain application-specific problems in their home space of measurable functions. Consequently, certain elements of economic system theory such as dynamical models and continuous-time cost/profit functionals can be directly incorporated in the new optimization problem formulation. Furthermore, subtour elimination constraints, prevalent in discrete optimization formulations, are naturally enforced through continuity requirements. The price for the new modeling framework is nonsmooth functionals. Although a number of theoretical issues remain open in the proposed mathematical framework, we demonstrate the computational viability of the new modeling constructs over a sample set of problems to illustrate the rapid production of end-to-end TSP solutions to extensively-constrained practical problems.Comment: 24 pages, 8 figure

AN EFFECTIVE METAHEURISTIC FOR TOURIST TRIP PLANNING IN PUBLIC TRANSPORT NETWORKS

The Time-Dependent Orienteering Problem with Time Windows (TDOPTW) is a combinatorial optimization problem defined on graphs. Its real life applications are particularly associated with tourist trip planning in trans-port networks, where travel time between two points depends on the moment of travel start. In the paper an effective TDOPTW solution (evolutionary algorithm with local search operators) was presented and applied to gen-erate attractive tours in real public transport networks of Białystok and Athens. The method achieved very high-quality solutions in a short execution time

Distributed Services with Foreseen and Unforeseen Tasks: The Mobile Re-allocation Problem

In this paper we deal with a common problem found in the operations of security and preventive/corrective maintenance services: that of routing a number of mobile resources to serve foreseen and unforeseen tasks during a shift. We define the (Mobile Re-Allocation Problem) MRAP as the problem of devising a routing strategy to maximize the expected weighted number of tasks served on time. For obtaining a solution to the MRAP, we propose to solve successively a multi-objective optimization problem called the stochastic Team Orienteering Problem with Multiple Time Windows (s-TOP-MTW) so as to consider information about known tasks and the arrival process of new unforeseen tasks. Solving successively the s-TOP-MTW we find that considering information about the arrival process of new unforeseen tasks may aid in maximizing the expected proportion of tasks accomplished on time.location;reliability;routing;distributed services