The Multi-Vehicle Probabilistic Covering Tour Problem

This paper introduces the Multi-Vehicle Probabilistic Covering Tour Problem (MVPCTP) which extends the Covering Tour Problem (CTP) by incorporating multiple vehicles and probabilistic coverage. As in the CTP, total demand of customers is attracted to the visited facility vertices within the coverage range. The objective function is to maximize the expected customer demand covered. The MVPCTP is first formulated as an integer non-linear programming problem, and then a linearization is proposed, which is strengthened by several sets of valid inequalities. An effective branch-and-cut algorithm is developed in addition to a local search heuristic based on Variable Neighborhood Search to obtain upper bounds. Extensive computational experiments are performed on new benchmark instances adapted from the literature

The Dynamic Multi-objective Multi-vehicle Covering Tour Problem

This work introduces a new routing problem called the Dynamic Multi-Objective Multi-vehicle Covering Tour Problem (DMOMCTP). The DMOMCTPs is a combinatorial optimization problem that represents the problem of routing multiple vehicles to survey an area in which unpredictable target nodes may appear during execution. The formulation includes multiple objectives that include minimizing the cost of the combined tour cost, minimizing the longest tour cost, minimizing the distance to nodes to be covered and maximizing the distance to hazardous nodes. This study adapts several existing algorithms to the problem with several operator and solution encoding variations. The efficacy of this set of solvers is measured against six problem instances created from existing Traveling Salesman Problem instances which represent several real countries. The results indicate that repair operators, variable length solution encodings and variable-length operators obtain a better approximation of the true Pareto front

The In-Transit Vigilant Covering Tour Problem of Routing Unmanned Ground Vehicles

The routing of unmanned ground vehicles for the surveillance and protection of key installations is modeled as a new variant of the Covering Tour Problem (CTP). The CTP structure provides both the routing and target sensing components of the installation protection problem. Our variant is called the in-transit Vigilant Covering Tour Problem (VCTP) and considers not only the vertex cover but also the additional edge coverage capability of the unmanned ground vehicle while sensing in-transit between vertices. The VCTP is formulated as a Traveling Salesman Problem (TSP) with a dual set covering structure involving vertices and edges. An empirical study compares the performance of the VCTP against the CTP on test problems modified from standard benchmark TSP problems to apply to the VCTP. The VCTP performed generally better with shorter tour lengths but at higher computational cost

The multi-vehicle covering tour problem: building routes for urban patrolling

In this paper we study a particular aspect of the urban community policing: routine patrol route planning. We seek routes that guarantee visibility, as this has a sizable impact on the community perceived safety, allowing quick emergency responses and providing surveillance of selected sites (e.g., hospitals, schools). The planning is restricted to the availability of vehicles and strives to achieve balanced routes. We study an adaptation of the model for the multi-vehicle covering tour problem, in which a set of locations must be visited, whereas another subset must be close enough to the planned routes. It constitutes an NP-complete integer programming problem. Suboptimal solutions are obtained with several heuristics, some adapted from the literature and others developed by us. We solve some adapted instances from TSPLIB and an instance with real data, the former being compared with results from literature, and latter being compared with empirical data.Comment: 28 pages, 8 figures, 7 tables, Brazilian Operations Research Society; Printed version ISSN 0101-7438 / Online version ISSN 1678-514

The Subtour Centre Problem

The subtour centre problem is the problem of finding a closed trail S of bounded length on a connected simple graph G that minimises the maximum distance from S to any vertex ofG. It is a central location problem related to the cycle centre and cycle median problems (Foulds et al., 2004; Labbé et al., 2005) and the covering tour problem (Current and Schilling, 1989). Two related heuristics and an integer linear programme are formulated for it. These are compared numerically using a range of problems derived from tsplib (Reinelt, 1995). The heuristics usually perform substantially better then the integer linear programme and there is some evidence that the simpler heuristics perform better on the less dense graphs that may be more typical of applications

Heuristic solution approaches for the covering tour problem

Diese Arbeit beschäftigt sich mit dem Covering Tour Problem (CTP) und verschiedenen heuristischen Lösungsmethoden. Dieses Problem der Tourenplanung zählt zu den kombinatorischen Optimierungsproblemen, welche sehr oft im Bereich der Distributionslogistik international agierender Großunternehmen auftreten und durch deren Lösung man entsprechend Kosten einsparen und Gewinne maximieren kann. Im Zuge der Globalisierung der Weltwirtschaft rückt das Problem der Distributionskosten immer mehr in den Mittelpunkt. Das CTP kann auf einem ungerichteten Graphen G=(V U W, E)definiert werden. V U W ist eine Menge von Knoten. V sind jene Knoten, die von der zu konstruierenden Tour besucht werden können. T ist eine Teilmenge von V und beinhaltet jene Knoten, die von der Tour besucht werden müssen. W ist die Menge jener Knoten, welche von der Tour abgedeckt werden müssen, also in einer vorgegebenen Entfernung zur Tour liegen müssen. Das Kantenset E beinhaltet die Verbindungen zwischen sämtlichen Knoten. Ziel ist es nun, eine möglichst kurze Tour zu finden, die im Punkt v0 beginnt, alle Knoten aus T besucht, sämtliche Knoten aus W abdeckt und wieder in v0 endet. Um das Problem zu lösen, wurde das CTP gemäß einer bereits angewandten Methode in zwei Subprobleme, nämlich das Traveling Salesman Problem (TSP) und das Set Covering Problem (SCP) unterteilt und diese wurden vorgestellt. Nach einer kurzen Einführung der Ant Colony Optimierung wurden die Algorithmen GENI, GENIUS und GENI Ant Colony System für den TSP Teil und PRIMAL1 sowie ein Set Covering Ant Colony System für den SCP Teil detailliert beschrieben. In weitere Folge wurde erklärt, wie man die Algorithmen kombinieren kann, um das CTP zu lösen. Sämtliche Algorithmen wurden mit Hilfe der Programmiersprache C++ simuliert und getestet. Zunächst wurden die Algorithmen an Instanzen einer Datenbank getestet und mit bereits vorhandenen Lösungen verglichen, um ihre Funktionalität und Konkurrenzfähigkeit zu überprüfen. Da für das CTP keine Vergleichsinstanzen vorhanden sind, wurden stochastische Probleme entworfen und mit dem H-1-CTP Algorithmus und der von mir entworfenen Metaheuristik Covering Tour Ant Colony System bestehend aus GENI Ant Colony System und Set Covering Ant Colony System gelöst und die Ergebnisse verglichen, um dann die beiden Lösungsansätze zu bewerten.This thesis deals with the Covering Tour Problem (CTP) and different heuristic solution approaches. It can be classified as a combinatorial optimization problem. Logistics and distribution departments of economic global players have to handle this sort of problems to reduce costs and maximize profit. Distribution costs enjoy increasing importance due to the globalization of world economy. The CTP is defined on a complete undirected graph G=(V U W, E) with a set of vertices V U W where V is a set of vertices that can be visited, W defines the set of vertices that have to be covered by the tour and E is the set of edges. “Covered by the tour” means that any vertex v has to lie within a predefined distance of a vertex on the tour. The set V includes the subset T which holds the vertices that have to be visited by the tour. The solution to the CTP is a minimum length tour. The tour starts and ends at the depot and is defined by a certain subset so that all vertices that have to be visited are visited by the tour and all vertices that have to be covered lie within a predetermined distance of a vertex belonging to the tour. In order to solve the problem, it was classified as a combination of the Traveling Salesman Problem (TSP) and the Set Covering Problem (SCP) and the components were introduced. After a short description of Ant Colony Optimization, algorithms GENI, GENIUS and GENI Ant Colony System for the TSP part and PRIMAL1 as well as Set Covering Ant Colony System for the SCP part were introduced in detail. Then the combinations of these algorithms for solving the CTP were described. All algorithms were simulated and tested with the help of C++ programming language. First, algorithms were tested individually on instances from data libraries to ensure their functionality and competitiveness. Then stochastic instances were developed for the CTP because no comparable benchmarks exist and the H-1-CTP algorithm as well as the Covering Tour Ant Colony System, that I created myself, were run on these instances and results were compared

Covering tour problem with varying coverage: Application to marine environmental monitoring

In this paper, we present a novel variant of the Covering Tour Problem (CTP), called the Covering Tour Problem with Varying Coverage (CTP-VC). We consider a simple graph = ( ,), with a measure of importance assigned to each node in . A vehicle with limited battery capacity visits the nodes of the graph and has the ability to stay in each node for a certain period of time, which determines the coverage radius at the node. We refer to this feature as stay-dependent varying coverage or, in short, varying coverage. The objective is to maximize a scalarization of the weighted coverage of the nodes and the negation of the cost of moving and staying at the nodes. This problem arises in the monitoring of marine environments, where pollutants can be measured at locations far from the source due to ocean currents. To solve the CTP-VC, we propose a mathematical formulation and a heuristic approach, given that the problem is NP-hard. Depending on the availability of solutions yielded by an exact solver, we evaluate our heuristic approach against the exact solver or a constructive heuristic on various instance sets and show how varying coverage improves performance. Additionally, we use an offshore CO2 storage site in the Gulf of Mexico as a case study to demonstrate the problem’s applicability. Our results demonstrate that the proposed heuristic approach is an efficient and practical solution to the problem of stay-dependent varying coverage. We conduct numerous experiments and provide managerial insights.publishedVersio