k-delivery traveling salesman problem on tree networks

In this paper we study the k-delivery traveling salesman problem (TSP)on trees, a variant of the non-preemptive capacitated vehicle routing problem with pickups and deliveries. We are given n pickup locations and n delivery locations on trees, with exactly one item at each pickup location. The k-delivery TSP is to find a minimum length tour by a vehicle of finite capacity k to pick up and deliver exactly one item to each delivery location. We show that an optimal solution for the k-delivery TSP on paths can be found that allows succinct representations of the routes. By exploring the symmetry inherent in the k-delivery TSP, we design a 5/3-approximation algorithm for the k-delivery TSP on trees of arbitrary heights. The ratio can be improved to (3/2 - 1/2k) for the problem on trees of height 2. The developed algorithms are based on the following observation: under certain conditions, it makes sense for a non-empty vehicle to turn around and pick up additional loads

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

Dial a Ride from k-forest

The k-forest problem is a common generalization of both the k-MST and the dense-$k$-subgraph problems. Formally, given a metric space on $n$ vertices $V$, with $m$ demand pairs $\subseteq V \times V$ and a ``target'' $k\le m$, the goal is to find a minimum cost subgraph that connects at least $k$ demand pairs. In this paper, we give an $O(\min\{\sqrt{n},\sqrt{k}\})$-approximation algorithm for $k$-forest, improving on the previous best ratio of $O(n^{2/3}\log n)$ by Segev & Segev. We then apply our algorithm for k-forest to obtain approximation algorithms for several Dial-a-Ride problems. The basic Dial-a-Ride problem is the following: given an $n$ point metric space with $m$ objects each with its own source and destination, and a vehicle capable of carrying at most $k$ objects at any time, find the minimum length tour that uses this vehicle to move each object from its source to destination. We prove that an $\alpha$-approximation algorithm for the $k$-forest problem implies an $O(\alpha\cdot\log^2n)$-approximation algorithm for Dial-a-Ride. Using our results for $k$-forest, we get an $O(\min\{\sqrt{n},\sqrt{k}\}\cdot\log^2 n)$- approximation algorithm for Dial-a-Ride. The only previous result known for Dial-a-Ride was an $O(\sqrt{k}\log n)$-approximation by Charikar & Raghavachari; our results give a different proof of a similar approximation guarantee--in fact, when the vehicle capacity $k$ is large, we give a slight improvement on their results.Comment: Preliminary version in Proc. European Symposium on Algorithms, 200

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

Towards the solution of variants of Vehicle Routing Problem

Some of the problems that are used extensively in -real life are NP complete problems. There is no any algorithm which can give the optimal solution to NP complete problems in the polynomial time in the worst case. So researchers are applying their best efforts to design the approximation algorithms for these NP complete problems. Approximation algorithm gives the solution of a particular problem, which is close to the optimal solution of that problem. In this paper, a study on variants of vehicle routing problem is being done along with the difference in the approximation ratios of different approximation algorithms as being given by researchers and it is found that Researchers are continuously applying their best efforts to design new approximation algorithms which have better approximation ratio as compared to the previously existing algorithms

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

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

The Position-Aware-Market: Optimizing Freight Delivery for Less-Than-Truckload Transportation

The increasing competition faced by logistics carriers requires them to ship at lower cost and higher efficiency. In reality, however, many trucks are running empty or with a partial load. Bridging such residual capacity with real time transportation demand enhances the efficiency of the carriers. We therefore introduce the Position-Aware-Market (PAM), where transportation requests are traded in real time to utilize transportation capacities optimally. In this paper we mainly focus on the decision support system for the truck driver, which solves a profit- maximizing Pickup and Delivery Problem with Time Windows (PM-PDPTW). We propose a novel Recursive Branch-and-Bound algorithm that solves the problem optimally, and apply it to a Tabu-Search heuristic for larger problem instances. Simulations show that problems with up to 50 requests can be solved optimally within seconds. Larger problems with 200 requests can be solved approximately by Tabu-Search in seconds, retaining 60% of the optimal profit

전기 마이크로 모빌리티 공유 시스템에서의 배터리 교체와 재배치 작업 최적화

학위논문 (석사) -- 서울대학교 대학원 : 공과대학 산업공학과, 2021. 2. 박건수.In this thesis, we consider a battery swapping and mobility inventory rebalancing problem arising in electric micro-mobility sharing systems. Vehicles are equipped with swappable batteries and they are managed by staffs' visiting each vehicle and changing depleted batteries. With the free-floating property of the system, vehicles can locate anywhere in a service area without designated stations, which increases the difficulty to visit and collect every single vehicle. In order to successfully meet user demand during the daytime, operators have to redistribute the vehicles with the right number in the right place and swap batteries with insufficient levels into fully charged ones overnight. Therefore, it is essential that operators take battery charging(swapping), staff routing, rebalancing problem all together into consideration. We aim to satisfy demand as much as possible and at the same time minimize routing and swapping costs. We formulate this problem in a mixed integer linear programming. Target inventory level for rebalancing, an important parameter used in the system, is suggested by analyzing a stochastic process that incorporates demand changes. Being a special case of vehicle routing problem with pickup and delivery, it shares the difficulty and complexity of VRP in practically large size. So as to give efficient solutions in large size problems, we develop a Cluster-first Route-second heuristic where a set partitioning problem considers inventory imbalances and approximates routing distances. We benchmark our heuristic approach on a pure MLIP formulation. The experimental result confirms that the heuristic is good at decomposing a large problem and gives efficient solutions even in practically large instances.본 연구는 교체형 배터리를 이용하는 전기 마이크로 모빌리티 공유 시스템에서의 배터리 교체 및 차량 재배치를 효율적으로 수행하는 방법을 제시하고자 한다. 수요를 성공적으로 충족시키기 위해선 모빌리티의 공급과 이용자의 수요를 맞춰주기 위한 차량 재고 차원에서의 재배치 작업과 배터리 수준을 유지시켜주는 배터리 관리 차원에서의 교체 작업이 필수적이다. 또한 충전소로 차량을 옮길 필요 없이 바로 교체할 수 있으므로 담당 직원이 산발적으로 위치한 각 모빌리티들을 순회하며 위 작업들을 진행해야 한다. 이동하며 작업하는 비용과 시간이 대부분이기 때문에 이동 순서를 함께 최적화하는 것이 비용 개선에 필수적이다. 따라서 작업 결정과 경로 결정을 동시에 고려하는 충전 및 재배치 모형을 제시한다. 이때 free-floating 모빌리티 공유시스템의 이용 수요를 효과적으로 반영하고자 수요를 stochastic process로 모델링하고 이를 이용하여 재배치 목표 수량을 구한다. 문제의 크기가 큰 경우 효율적으로 본 충전 및 재배치 모형의 좋은 해를 얻기 위한 방법으로, 해당 서비스지역의 각 구역들을 클러스터링하고 그 뒤에 스태프들의 경로와 작업을 결정하는 휴리스틱을 제안한다. 여러 스태프를 순회시키는 복잡한 형태를 클러스터링으로써 작은 크기의 문제들로 분해하여 빠르게 문제를 풀고자 한다. 이를 위해 각 클러스터에는 한 명의 스태프가 배정되고, 한 클러스터 내에서 소속된 구역들이 필요로 하는 작업들을 한 명의 스태프가 모두 진행하도록 구성한다. 최소걸침나무 근사법을 적용한 set partitioning 문제를 풀어 클러스터링을 진행한다. 계산실험 결과, 고안된 휴리스틱은 차량의 수가 많아 크기가 큰 상황에서도 빠른 시간내에 더 좋은 해를 냈다.Chapter 1. Introduction 1 1.1 Background 1 1.2 Related literature 6 1.2.1 Rebalancing in bike sharing systems 6 1.2.2 Charging and rebalancing in free-floating electric vehicle(FFEV) sharing 9 1.2.3 Charging of electric micro-mobility with swappable batteries 10 1.3 Motivation and contributions 12 1.4 Organization of the thesis 14 Chapter 2. Mathematical formulations 15 2.1 Basic assumptions and problem description 15 2.2 Demand Modeling and Target Inventory 18 2.3 Mixed integer linear programming formulation 23 Chapter 3. Heuristic approach 30 3.1 Cluster-first route-second approach 31 3.2 Clustering problem with routing cost approximation 33 3.2.1 Minimum spanning tree approximation 33 3.2.2 Clustering problem 35 3.2.3 Cluster-first Route-second heuristic 41 Chapter 4. Computational experiments 42 4.1 Design of experiment 42 4.2 Comparative Analysis 47 Chapter 5. Conclusion 52Maste