On-line single server dial-a-ride problems

In this paper results on the dial-a-ride problem with a single server are presented. Requests for rides consist of two points in a metric space, a source and a destination. A ride has to be made by the server from the source to the destination. The server travels at unit speed in the metric space and the objective is to minimize some function of the delivery times at the destinations. We study this problem in the natural on-line setting. Calls for rides come in while the server is travelling. This models e.g. the taxi problem, or, if the server has capacity more than 1 a minibus or courier service problem. For two versions of this problem, one in which the server has infinite capacity and the other in which the server has capacity 1, both having as objective minimization of the time the last destination is served, we will design algorithms that have competitive ratio's of 2. We also show that these are best possible, since no algorithm can have competitive ratio better than 2 for these problems. Then we study the on-line problem with objective minimization of the sum of completion times of the rides. We prove a lower bound on the competitive ratio of any algorithm of 1 + \sqrt{2} for a server with any capacity and of 3 for servers with capacity 1

An asymptotically optimal algorithm for pickup and delivery problems

Pickup and delivery problems (PDPs), in which objects or people have to be transported between specific locations, are among the most common combinatorial problems in real-world operations. One particular PDP is the Stacker Crane problem (SCP), where each commodity/customer is associated with a pickup location and a delivery location, and the objective is to find a minimum-length tour visiting all locations with the constraint that each pickup location and its associated delivery location are visited in consecutive order. The SCP is a route optimization problem behind several transportation systems, e.g., Transportation-On-Demand (TOD) systems. The SCP is NP-Hard and the best know approximation algorithm only provides a 9/5 approximation ratio. We present an algorithm for the stochastic SCP which: (i) is asymptotically optimal, i.e., it produces a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n[superscript 2+ϵ]), where n is the number of pickup/delivery pairs and ϵ is an arbitrarily small positive constant. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations.Singapore-MIT Alliance for Research and Technology Cente

Tabu Search Heuristics for the Crane Sequencing Problem

Determining the sequence of relocating items (or resources) moved by a crane from existing positions to newly assigned locations during a multiperiod planning horizon is a complex combinatorial optimisation problem, which exists in power plants, shipyards, and warehouses. Therefore, it is essential to develop a good crane route technique to ensure efficient utilisation of the crane as well as to minimize the cost of operating the crane. This problem was defined as the Crane Sequencing Problem (CSP). In this paper, three construction and three improvement algorithms are presented for the CSP. The first improvement heuristic is a simple Tabu Search (TS) heuristic. The second is a probabilistic TS heuristic, and the third adds diversification and intensification strategies to the first. The computational experiments show that the proposed TS heuristics produce high-quality solutions in reasonable computation time

Asymptotically Optimal Algorithms for Pickup and Delivery Problems with Application to Large-Scale Transportation Systems

The Stacker Crane Problem is NP-Hard and the best known approximation algorithm only provides a 9/5 approximation ratio. The objective of this paper is threefold. First, by embedding the problem within a stochastic framework, we present a novel algorithm for the SCP that: (i) is asymptotically optimal, i.e., it produces, almost surely, a solution approaching the optimal one as the number of pickups/deliveries goes to infinity; and (ii) has computational complexity O(n^{2+\eps}), where $n$ is the number of pickup/delivery pairs and \eps is an arbitrarily small positive constant. Second, we asymptotically characterize the length of the optimal SCP tour. Finally, we study a dynamic version of the SCP, whereby pickup and delivery requests arrive according to a Poisson process, and which serves as a model for large-scale demand-responsive transport (DRT) systems. For such a dynamic counterpart of the SCP, we derive a necessary and sufficient condition for the existence of stable vehicle routing policies, which depends only on the workspace geometry, the stochastic distributions of pickup and delivery points, the arrival rate of requests, and the number of vehicles. Our results leverage a novel connection between the Euclidean Bipartite Matching Problem and the theory of random permutations, and, for the dynamic setting, exhibit novel features that are absent in traditional spatially-distributed queueing systems.Comment: 27 pages, plus Appendix, 7 figures, extended version of paper being submitted to IEEE Transactions of Automatic Contro

Ride Sharing with a Vehicle of Unlimited Capacity

A ride sharing problem is considered where we are given a graph, whose edges are equipped with a travel cost, plus a set of objects, each associated with a transportation request given by a pair of origin and destination nodes. A vehicle travels through the graph, carrying each object from its origin to its destination without any bound on the number of objects that can be simultaneously transported. The vehicle starts and terminates its ride at given nodes, and the goal is to compute a minimum-cost ride satisfying all requests. This ride sharing problem is shown to be tractable on paths by designing a O(h*log(h)+n) algorithm, with h being the number of distinct requests and with n being the number of nodes in the path. The algorithm is then used as a subroutine to efficiently solve instances defined over cycles, hence covering all graphs with maximum degree 2. This traces the frontier of tractability, since NP-hard instances are exhibited over trees whose maximum degree is 3

Heuristic for the preemptive asymmetric stacker crane problem

International audienceIn this paper, we deal with the preemptive asymmetric stacker crane problem in an heuristic way. We first present some theoretical results which allow us to turn this problem into a specific tree design problem. We next derive from this new representation a simple, efficient local search heuristic, as well as an original LIP model. We conclude by presenting experimental results which aim at both testing the efficiency of our heuristic and at evaluating the impact of the preemption hypothesis

Algorithms for the on-line travelling salesman

In this paper the problem of efficiently serving a sequence of requests presented in an on-line fashion located at points of a metric space is considered. We call this problem the On-Line Travelling Salesman Problem (OLTSP). It has a variety of relevant applications in logistics and robotics. We consider two versions of the problem. In the first one the server is not required to return to the departure point after all presented requests have been served. For this problem we derive a lower bound on the competitive ratio of 2 on the real line. Besides, a 2.5-competitive algorithm for a wide class of metric spaces, and a 7/3-competitive algorithm for the real line are provided. For the other version of the problem, in which returning to the departure point is required, we present an optimal 2-competitive algorithm for the above mentioned general class of metric spaces. If in this case the metric space is the real line we present a 1.75-competitive algorithm that compares with a \approx 1.64 lower bound

Collision-free path planning for robots using B-splines and simulated annealing

This thesis describes a technique to obtain an optimal collision-free path for an automated guided vehicle (AGV) and/or robot in two and three dimensions by synthesizing a B-spline curve under geometric and intrinsic constraints. The problem is formulated as a combinatorial optimization problem and solved by using simulated annealing. A two-link planar manipulator is included to show that the B-spline curve can also be synthesized by adding kinematic characteristics of the robot. A cost function, which includes obstacle proximity, excessive arc length, uneven parametric distribution and, possibly, link proximity costs, is developed for the simulated annealing algorithm. Three possible cases for the orientation of the moving object are explored: (a) fixed orientation, (b) orientation as another independent variable, and (c) orientation given by the slope of the curve. To demonstrate the robustness of the technique, several examples are presented. Objects are modeled as ellipsoid type shapes. The procedure to obtain the describing parameters of the ellipsoid is also presented

Solution techniques for a crane sequencing problem

In shipyards and power plants, relocating resources (items) from existing positions to newly assigned locations are costly and may represent a significant portion of the overall project budget. Since the crane is the most popular material handling equipment for relocating bulky items, it is essential to develop a good crane route to ensure efficient utilization and lower cost. In this research, minimizing the total travel and loading/unloading costs for the crane to relocate resources in multiple time periods is defined as the crane sequencing problem (CSP). In other words, the objective of the CSP is to find routes such that the cost of crane travel and resource loading/unloading is minimized. However, the CSP considers the capacities of locations and intermediate drops (i.e., preemptions) during a multiple period planning horizon. Therefore, the CSP is a unique problem with many applications and is computationally intractable. A mathematical model is developed to obtain optimal solutions for small size problems. Since large size CSPs are computationally intractable, construction algorithms as well as improvement heuristics (e.g., simulated annealing, hybrid ant systems and tabu search heuristics) are proposed to solve the CSPs. Two sets of test problems with different problem sizes are generated to test the proposed heuristics. In other words, extensive computational experiments are conducted to evaluate the performances of the proposed heuristics