Minimum Makespan Multi-vehicle Dial-a-Ride

Dial a ride problems consist of a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs, where each object requires to be moved from its source to destination vertex. We consider the multi-vehicle Dial a ride problem, with each vehicle having capacity k and its own depot-vertex, where the objective is to minimize the maximum completion time (makespan) of the vehicles. We study the "preemptive" version of the problem, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log^3 n)-approximation algorithm for preemptive multi-vehicle Dial a ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint. We also show that the approximation ratios improve by a log-factor when the underlying metric is induced by a fixed-minor-free graph.Comment: 22 pages, 1 figure. Preliminary version appeared in ESA 200

Minimum Makespan Multi-vehicle Dial-a-Ride *

Abstract Dial-a-Ride problems consist of a set V of n vertices in a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs {( , where each object requires to be moved from its source to destination vertex. In the multi-vehicle Dial-aRide problem, there are q vehicles each having capacity k and where each vehicle j ∈ [q] has its own depot-vertex r j ∈ V . A feasible schedule consists of a capacitated route for each vehicle (where vehicle j originates and ends at its depot r j ) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e. makespan) of vehicles, where the completion time of vehicle j is the time when it returns to its depot r j at the end of its route. We study the preemptive version of multi-vehicle Dial-a-Ride, where an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O(log 3 n)-approximation algorithm for preemptive multi-vehicle Diala-Ride, and an improved O(log t)-approximation for its special case when there is no capacity constraint (here t ≤ n is the number of distinct depot-vertices). There is an Ω(log 1/4− n) hardness of approximation known even for single vehicle capacitated Dial-a-Ride We also consider the special class of metrics induced by graphs excluding any fixed minor (eg. planar metrics). In this case, we obtain improved guarantees of O(log 2 n) for capacitated multivehicle Dial-a-Ride, and O(1) for the uncapacitated problem

Optimization of a city logistics transportation system with mixed passengers and goods

International audienceIn this paper, we propose a mathematical model and an adaptive large neighborhood search to solve a two{tiered transportation problem arising in the distribution of goods in congested city cores. In the rst tier, goods are transported in city buses from a consolidation and distribution center to a set of bus stops. The main idea is to use the buses spare capacity to drive the goods in the city core. In the second tier, nal customers are distributed by a eet of near{zero emissions city freighters. This system requires transferring the goods from buses to city freighters at the bus stops. We model the corresponding optimization problem as a variant of the pickup and delivery problem with transfers and solve it with an adaptive large neighborhood search. To evaluate its results, lower bounds are calculated with a column generation approach. The algorithm is assessed on data sets derived from a eld study in the medium-sized city of La Rochelle in France

Vehicle routing and location routing with intermediate stops:A review

This paper reviews the literature on vehicle routing problems and location rout-8 ing problems with intermediate stops. We classify publications into different categories from both an application-based perspective and a methodological perspective. In addition, we analyze the papers with respect to the algorithms and benchmark instances they present. Furthermore, we provide an overview of trends in the literature and identify promising areas for further research.</p

Vehicle routing and location routing with intermediate stops:A review

This paper reviews the literature on vehicle routing problems and location rout-8 ing problems with intermediate stops. We classify publications into different categories from both an application-based perspective and a methodological perspective. In addition, we analyze the papers with respect to the algorithms and benchmark instances they present. Furthermore, we provide an overview of trends in the literature and identify promising areas for further research.</p

Minimum Makespan Multi-Vehicle Dial-a-Ride

Dial-a-Ride problems consist of a set V of n vertices in a metric space (denoting travel time between vertices) and a set of m objects represented as source-destination pairs {( s i ,t i )} m i =1 , where each object requires to be moved from its source to destination vertex. In the multi-vehicle Dial-a-Ride problem, there are q vehicles, each having capacity k and where each vehicle j ∈ [ q ] has its own depot-vertex r j ∈ V . A feasible schedule consists of a capacitated route for each vehicle (where vehicle j originates and ends at its depot r j ) that together move all objects from their sources to destinations. The objective is to find a feasible schedule that minimizes the maximum completion time (i.e., makespan ) of vehicles, where the completion time of vehicle j is the time when it returns to its depot r j at the end of its route. We study the preemptive version of multi-vehicle Dial-a-Ride, in which an object may be left at intermediate vertices and transported by more than one vehicle, while being moved from source to destination. Our main results are an O (log 3 n )-approximation algorithm for preemptive multi-vehicle Dial-a-Ride , and an improved O (log t )-approximation for its special case when there is no capacity constraint (here t ≤ n is the number of distinct depot-vertices). There is an Ω (log 1/4-ϵ n ) hardness of approximation known even for single vehicle capacitated Dial-a-Ride [Gørtz 2006]. For uncapacitated multi-vehicle Dial-a-Ride, we show that there are instances when natural lower bounds (used in our algorithm) are ˜Ω(log t ) factor away from the optimum. We also consider the special class of metrics induced by graphs excluding any fixed minor (e.g., planar metrics). In this case, we obtain improved guarantees of O (log 2 n ) for capacitated multi-vehicle Dial-a-Ride, and O (1) for the uncapacitated problem. </jats:p

Approximation algorithms for regret minimization in vehicle routing problems

In this thesis, we present new approximation algorithms as well as hardness of approximation results for NP-hard vehicle routing problems related to public transportation. We consider two different problem classes that also occur frequently in areas such as logistics, robotics, or distribution systems. For the first problem class, the goal is to visit as many locations in a network as possible subject to timing or cost constraints. For the second problem class, a given set of locations is to be visited using a minimum-cost set of routes under some constraints. Due to the relevance of both problem classes for public transportation, a secondary objective must be taken into account beyond a low operation cost: namely, it is crucial to design routes that optimize customer satisfaction in order to encourage customers to use the service. Our measure of choice is the regret of a customer, that is the time comparison of the chosen route with the shortest path to a destination. From the first problem class, we investigate variants and extensions of the Orienteering problem that asks to find a short walk maximizing the profit obtained from visiting distinct locations. We give approximation algorithms for variants in which the walk has to respect constraints on the regret of the visited vertices. Additionally, we describe a framework to extend approximation algorithms for Orienteering problems to consider also a second budget constraint, namely node demands, that have to be satisfied in order to collect the profit. We obtain polynomial time approximation schemes for the Capacitated Orienteering problem on trees and Euclidean metrics. Furthermore, we study variants of the School Bus problem (SBP). In SBP, a given set of locations is to be connected to a destination node with both low operation cost and a low maximum regret. We note that the Orienteering problem can be seen as the pricing problem for SBP and it often appears as subroutine in algorithms for SBP. For tree-shaped networks, we describe algorithms with a small constant approximation factor and complement them by showing hardness of approximation results. We give an overview of the known results in arbitrary networks and we prove that a general variant cannot be approximated unless P = NP. Finally, we describe an integer programming approach to solve School Bus problems in practice and present an improved bus schedule for a private school in the lake Geneva region