A linear programming formulation of the traveling Salesman problem (original) (raw)
A Flow-Based Formulation of the Travelling Salesman Problem with Penalties on Nodes
Sustainability
The travelling salesman problem (TSP) is one of combinatorial optimization problems of huge importance to practical applications. However, the TSP in its “pure” form may lack some essential issues for a decision maker—e.g., time-dependent travelling conditions. Among those shortcomings, there is also a lack of possibility of not visiting some nodes in the network—e.g., thanks to the existence of some more cost-efficient means of transportation. In this article, an extension of the TSP in which some nodes can be skipped at the cost of penalties for skipping those nodes is presented under a new name and in a new mathematical formulation. Such an extension can be applied as a model for transportation cost reduction due to the possibility of outsourcing deliveries to some nodes in a TSP route. An integer linear programming formulation of such a problem based on the Gavish–Graves-flow-based TSP formulation is introduced. This formulation makes it possible to solve the considered problem ...
A Small-Order-Polynomial-Sized Linear Program for Solving the Traveling Salesman Problem
arXiv (Cornell University), 2016
We present an O(n 6) linear programming model for the traveling salesman (TSP) and quadratic assignment (QAP) problems. The basic model is developed within the framework of the TSP. It does not involve the city-to-city variables-based, traditional TSP polytope referred to in the literature as "the TSP polytope." We do not model explicit Hamiltonian cycles of the cities. Instead, we use a time-dependent abstraction of TSP tours and develop a direct extended formulation of the linear assignment problem (LAP) polytope. The model is exact in the sense that it has integral extreme points which are in one-to-one correspondence with LAP assignment solutions and TSP tours. It can be solved optimally using any linear programming (LP) solver, hence offering a new (incidental) proof of the equality of the computational complexity classes "P " and "N P ." The extensions of the model to the time-dependent traveling salesman problem (TDTSP) as well as the quadratic assignment problem (QAP) are straightforward. The reasons for the non-applicability of existing negative extended formulations results for "the TSP polytope" to the model in this paper as well as our software implementation and the computational experimentation we conducted are briefly discussed.
Springer eBooks, 2013
This paper presents a self-contained introduction into algorithmic and computational aspects of the traveling salesman problem and of related problems, along with their theoretical prerequisites as seen from the point of view of an operations researcher who wants to solve practical problem instances. Extensive computational results are reported on most of the algorithms described. Optimal solutions are reported for instances with sizes up to several thousand nodes as well as heuristic solutions with provably very high quality for larger instances. This is a preliminary version of one of the chapters of the volume "Networks" edited by M.O. Ball, T.
The Travelling Salesman Problem in the Function of Transport Network Optimalization
Interdisciplinary Management Research, 2013
The fundamental objective and purpose of this paper is to analyze the Travelling Salesman Problem (TSP) as a function of forming and optimizing transport networks. There are several software solutions for solving such problems, based on a heuristic algorithm. In practical application, the starting point consists of algorithms with solutions close to the optimum, or at least those with one optimal solution. Accordingly, the basic assumption of this paper is to use object modelling and programming in the spreadsheet interface (VBA for Excel), of which detailed analysis shows more than one optimal solution that could be used to create a flexible and adaptive transport network.
Models and algorithms for the Asymmetric Traveling Salesman Problem: an experimental comparison
EURO Journal on Transportation and Logistics, 2012
This paper surveys the most effective mathematical models and exact algorithms proposed for finding the optimal solution of the well-known Asymmetric Traveling Salesman Problem (ATSP). The fundamental Integer Linear Programming (ILP) model proposed by Dantzig, Fulkerson and Johnson is first presented, its classical (assignment, shortest spanning r-arborescence, linear programming) relaxations are derived, and the most effective branch-and-bound and branch-andcut algorithms are described. The polynomial ILP formulations proposed for the ATSP are then presented and analyzed. The considered algorithms and formulations are finally experimentally compared on a set of benchmark instances.