Variant Of TSP Research Articles

Dealing with time in the multiple traveling salespersons problem with moving targets

The multiple traveling salespersons problem with moving targets is a generalization of the classical traveling salespersons problem, where the targets (nodes or objects) are moving over time. Additionally, for each target a visibility time window is given. The task is to find routes for several salespersons so that each target is reached exactly once within its visibility time window and the sum of all traveled distances of all salespersons is minimal. We present different modeling formulations for this TSP variant. The time requirements are modeled differently in each approach. Our goal is to examine what formulation is most suitable in terms of runtime to solve the multiple traveling salespersons problem with moving targets with exact methods. Computational experiments are carried out on randomly generated test instances to compare the different modeling approaches. The results for large-scale instances show, that the best way to model time requirements is to directly insert them into a formulation with discrete time steps.

Hybrid genetic algorithm with variable neighborhood search for multi-scale multiple bottleneck traveling salesmen problem

The paper gives an optimization model named multiple bottleneck traveling salesmen problem (MBTSP), which can model the optimization problems where there are multiple salesmen and tasks. In intelligent transport systems and multiple tasks cooperation, some real-world problems can be modeled by MBTSP, the scale of constructed model usually tends to multiple scales, therefore it is significant to study multiple scales MBTSP and its solving algorithms. The relevant literatures have proved that genetic algorithm and its versions can show good performance for the variants of TSP, thus this paper proposes a novel hybrid genetic algorithm (VNSGA) with variable neighborhood search (VNS) for multi-scale MBTSP. For VNSGA, the feasible solutions are constructed by dual-chromosome coding, then they are updated by the crossover operator, mutation operator and variable neighborhood search. During this process, the VNS can be carried out by the deleting and reinserting operator of the cities for optimization. The experiments show that VNSGA can demonstrate better solution quality than the state-of-the-art algorithms for MBTSP problem.

English

English

MODELING AND SOLVING THE TRAVELING SALESMAN PROBLEM WITH PRIORITY PRIZES

This paper addresses the Traveling Salesman Problem with Priority Prizes (TSPPP), an extension of the classical TSP in which the order of the node visits is taken into account in the objective function. A prize p ki is received by the traveling salesman when node i is visited in the k-th order of the route, while a travel cost c ij is incurred when the salesman travels from node i to node j . The aim of the TSPPP is to find the maximum profit n-node tour. The problem can be seen as a TSP variant with a more general objective function, aiming at solutions that in some way consider the quality of customer service and the delivery priorities and costs. A natural representation for the TSPPP is here grounded in the point of view of Koopmans and Beckmann approach, according to which the problem is seem as a special case of the quadratic assignment problem (QAP). Given the novelty of this TSP variant, we propose different mixed integer programming models to appropriately represent the TSPPP, some of them based on the QAP. Computational experiments are also presented when solving the MIP models with a well-known optimization software, as well as with a tabu search algorithm.

Optimal deterministic algorithms for some variants of Online Quota Traveling Salesman Problem

This paper is concerned with the Online Quota Traveling Salesman Problem. Depending on the symmetry of the metric and the requirement for the salesman to return to the origin, four variants are analyzed. We present optimal deterministic algorithms for each variant defined on a general space, a real line, or a half-line. As a byproduct, an improved lower bound for a variant of Online TSP on a half-line is also obtained.

Complexity and approximation for Traveling Salesman Problems with profits

In TSP with profits we have to find an optimal tour and a set of customers satisfying a specific constraint. In this paper we analyze three different variants of TSP with profits that are NP-hard in general. We study their computational complexity on special structures of the underlying graph, both in the case with and without service times to the customers. We present polynomial algorithms for the polynomially solvable cases and fully polynomial time approximation schemes (FPTAS) for some NP-hard cases.

Online and Offline Algorithms for the Time-Dependent TSP with Time Zones

The time-dependent traveling salesman problem (TDTSP) is a variant of TSP with time-dependent edge costs. We study some restrictions of TDTSP where the number of edge cost changes are limited. We find competitive ratios for online versions of TDTSP. From these we derive polynomial time approximation algorithms for graphs with edge costs one and two. In addition, we present an approximation algorithm for the orienteering problem with edge costs one and two.

Approximating Capacitated Routing and Delivery Problems

We provide approximation algorithms for some capacitated vehicle routing and delivery problems. These problems can all be viewed as instances of the following k-delivery TSP: given n source points and n sink points in a metric space, with exactly one item at each source, find a minimum length tour by a vehicle of finite capacity k to pick up and deliver exactly one item to each sink. The only known approximation algorithm for this family of problems is the 2.5-approximation algorithm of Anily and Hassin [ Networks, 22 (1992), pp. 419--433] for the special case k=1. For this case, we use matroid intersection to obtain a 2-approximation algorithm. Based on this algorithm and some additional lower bound arguments, we devise a factor-approximation for k-delivery TSP with arbitrary finite k. We also present a 2-approximation algorithm for the case $k = \infty$. We then initiate the study of dynamic variants of k-delivery TSP that model problems in industrial robotics and other applications. Specifically, we consider the situation where a robot arm (with finite or infinite capacity) must collect n point-objects moving in the Euclidean plane, and deliver them to the origin. The point-objects are moving in the plane with known, identical velocities---they might, for instance, be on a moving conveyor belt. We derive several useful structural properties that lead to constant-factor approximations for problems of this type that are relevant to the robotics application. Along the way, we show that maximum latency TSP is implicit in the dynamic problems, and that the natural "farthest neighbor" heuristic produces a good approximation for several notions of latency.

Magic labeling in graphs: Bounds, complexity, and an application to a variant of TSP

Let G be an undirected graph with n vertices and m edges. A natural number λ is said to be a magic labeling, positive magic labeling, and fractional positive magic labeling, if the edges can be labeled with nonnegative integers, naturals, and rationals ≥1, respectively, so that for each vertex the sum of the labels of incident edges is λ. G is said to be regularizable if it has a positive magic labeling. Denoting the minimum positive magic labeling, the minimum fractional positive magic labeling, and the maximum vertex degree by λ*, λ * , and 6, respectively, we prove that λ* ≤ min{[n/2]δ, 2m, 2λ * ). The bound 2m is also derivable from a characterization of regularizable graphs stated by Pulleyblank, and regularizability for graphs with nonbipartite components can be tested via the Bourjolly-Pulleyblank algorithm for testing 2-bicriticality in O(nm) time. We show that using the above bounds and maximum flow algorithms of Ahuja-Orlin-Tarjan regularizability (or 2-bicriticality) can be tested in T(n,m) = O(min(nm + n 2 √log n, nm log((n/m)√logn + 2)}), and λ * , as well as a 2-approximates solution to λ* can be computed in O(T(n, m)log n) time. For dense graphs, T(n, m) can be improved using parallel maximum flow algorithms. We exhibit a family of graphs for which [n/2]δ = 2λ* = 2λ * . Finally, given that the edges in G have nonnegative weights satisfying the triangle inequality, using a capacitated magic labeling solution, we construct a 2-approximate algorithm for the problem of covering all the vertices with optimal set of disjoint even cycles, each covering at least four vertices.