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

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

실시간 동적 계획법 및 강화학습 기반의 공공자전거 시스템의 동적 재배치 전략

학위논문 (박사) -- 서울대학교 대학원 : 공과대학 건설환경공학부, 2020. 8. 고승영.The public bicycle sharing system is one of the modes of transportation that can help to relieve several urban problems, such as traffic congestion and air pollution. Because users can pick up and return bicycles anytime and anywhere a station is located, pickup or return failure can occur due to the spatiotemporal imbalances in demand. To prevent system failures, the operator should establish an appropriate repositioning strategy. As the operator makes a decision based on the predicted demand information, the accuracy of forecasting demand is an essential factor. Due to the stochastic nature of demand, however, the occurrence of prediction errors is inevitable. This study develops a stochastic dynamic model that minimizes unmet demand for rebalancing public bicycle sharing systems, taking into account the stochastic demand and the dynamic characteristics of the system. Since the repositioning mechanism corresponds to the sequential decision-making problem, this study applies the Markov decision process to the problem. To solve the Markov decision process, a dynamic programming method, which decomposes complex problems into simple subproblems to derive an exact solution. However, as a set of states and actions of the Markov decision process become more extensive, the computational complexity increases and it is intractable to derive solutions. An approximate dynamic programming method is introduced to derive an approximate solution. Further, a reinforcement learning model is applied to obtain a feasible solution in a large-scale public bicycle network. It is assumed that the predicted demand is derived from the random forest, which is a kind of machine learning technique, and that the observed demand occurred along the Poisson distribution whose mean is the predicted demand to simulate the uncertainty of the future demand. Total unmet demand is used as a key performance indicator in this study. In this study, a repositioning strategy that quickly responds to the prediction error, which means the difference between the observed demand and the predicted demand, is developed and the effectiveness is assessed. Strategies developed in previous studies or applied in the field are also modeled and compared with the results to verify the effectiveness of the strategy. Besides, the effects of various safety buffers and safety stock are examined and appropriate strategies are suggested for each situation. As a result of the analysis, the repositioning effect by the developed strategy was improved compared to the benchmark strategies. In particular, the effect of a strategy focusing on stations with high prediction errors is similar to the effect of a strategy considering all stations, but the computation time can be further reduced. Through this study, the utilization and reliability of the public bicycle system can be improved through the efficient operation without expanding the infrastructure.공공자전거 시스템은 교통혼잡과 대기오염 등 여러 도시문제를 완화할 수 있는 교통수단이다. 대여소가 위치한 곳이면 언제 어디서든 이용자가 자전거를 이용할 수 있는 시스템의 특성상 수요의 시공간적 불균형으로 인해 대여 실패 또는 반납 실패가 발생한다. 시스템 실패를 예방하기 위해 운영자는 적절한 재배치 전략을 수립해야 한다. 운영자는 예측 수요 정보를 전제로 의사결정을 하므로 수요예측의 정확성이 중요한 요소이나, 수요의 불확실성으로 인해 예측 오차의 발생이 불가피하다. 본 연구의 목적은 공공자전거 수요의 불확실성과 시스템의 동적 특성을 고려하여 불만족 수요를 최소화하는 재배치 모형을 개발하는 것이다. 공공자전거 재배치 메커니즘은 순차적 의사결정 문제에 해당하므로, 본 연구에서는 순차적 의사결정 문제를 모형화할 수 있는 마르코프 결정 과정을 적용한다. 마르코프 결정 과정을 풀기 위해 복잡한 문제를 간단한 부문제로 분해하여 정확해를 도출하는 동적 계획법을 이용한다. 하지만 마르코프 결정 과정의 상태 집합과 결정 집합의 크기가 커지면 계산 복잡도가 증가하므로, 동적 계획법을 이용한 정확해를 도출할 수 없다. 이를 해결하기 위해 근사적 동적 계획법을 도입하여 근사해를 도출하며, 대규모 공공자전거 네트워크에서 가능해를 얻기 위해 강화학습 모형을 적용한다. 장래 공공자전거 이용수요의 불확실성을 모사하기 위해, 기계학습 기법의 일종인 random forest로 예측 수요를 도출하고, 예측 수요를 평균으로 하는 포아송 분포를 따라 수요를 확률적으로 발생시켰다. 본 연구에서는 관측 수요와 예측 수요 간의 차이인 예측오차에 빠르게 대응하는 재배치 전략을 개발하고 효과를 평가한다. 개발된 전략의 우수성을 검증하기 위해, 기존 연구의 재배치 전략 및 현실에서 적용되는 전략을 모형화하고 결과를 비교한다. 또한, 재고량의 안전 구간 및 안전재고량에 관한 민감도 분석을 수행하여 함의점을 제시한다. 개발된 전략의 효과를 분석한 결과, 기존 연구의 전략 및 현실에서 적용되는 전략보다 개선된 성능을 보이며, 특히 예측오차가 큰 대여소를 탐색하는 전략이 전체 대여소를 탐색하는 전략과 재배치 효과가 유사하면서도 계산시간을 절감할 수 있는 것으로 나타났다. 공공자전거 인프라를 확대하지 않고도 운영의 효율화를 통해 공공자전거 시스템의 이용률 및 신뢰성을 제고할 수 있고, 공공자전거 재배치에 관한 정책적 함의점을 제시한다는 점에서 본 연구의 의의가 있다.Chapter 1. Introduction １ 1.1 Research Background and Purposes １ 1.2 Research Scope and Procedure ７ Chapter 2. Literature Review １０ 2.1 Vehicle Routing Problems １０ 2.2 Bicycle Repositioning Problem １２ 2.3 Markov Decision Processes ２３ 2.4 Implications and Contributions ２６ Chapter 3. Model Formulation ２８ 3.1 Problem Definition ２８ 3.2 Markov Decision Processes ３４ 3.3 Demand Forecasting ４０ 3.4 Key Performance Indicator (KPI) ４５ Chapter 4. Solution Algorithms ４７ 4.1 Exact Solution Algorithm ４７ 4.2 Approximate Dynamic Programming ５０ 4.3 Reinforcement Learning Method ５２ Chapter 5. Numerical Example ５５ 5.1 Data Overview ５５ 5.2 Experimental Design ６１ 5.3 Algorithm Performance ６６ 5.4 Sensitivity Analysis ７４ 5.5 Large-scale Cases ７６ Chapter 6. Conclusions ８２ 6.1 Conclusions ８２ 6.2 Future Research ８３ References ８６ 초 록 ９２Docto

Models and Algorihtm for the Optimization of Real-World Routing and Logistics Problems

Logistics involves planning, managing, and organizing the flows of goods from the point of origin to the point of destination in order to meet some requirements. Logistics and transportation aspects are very important and represent a relevant costs for producing and shipping companies, but also for public administration and private citizens. The optimization of resources and the improvement in the organization of operations is crucial for all branches of logistics, from the operation management to the transportation. As we will have the chance to see in this work, optimization techniques, models, and algorithms represent important methods to solve the always new and more complex problems arising in different segments of logistics. Many operation management and transportation problems are related to the optimization class of problems called Vehicle Routing Problems (VRPs). In this work, we consider several real-world deterministic and stochastic problems that are included in the wide class of the VRPs, and we solve them by means of exact and heuristic methods. We treat three classes of real-world routing and logistics problems. We deal with one of the most important tactical problems that arises in the managing of the bike sharing systems, that is the Bike sharing Rebalancing Problem (BRP). We propose models and algorithms for real-world earthwork optimization problems. We describe the 3DP process and we highlight several optimization issues in 3DP. Among those, we define the problem related to the tool path definition in the 3DP process, the 3D Routing Problem (3DRP), which is a generalization of the arc routing problem. We present an ILP model and several heuristic algorithms to solve the 3DRP

El problema del viajante de comercio con recogida y entrega de una mercancía con demanda dividida.

This thesis focuses on a kind of problems thoroughly studied in operational research, in particular in Vehicle Routing Problems. Despite there are a lot of studies about them, the most basic are still difficult to solve optimally, even for small size and the lacking practical use. Because of that, this thesis develops exact and heuristic algorithms to solve a particular problem known as The split-demand one-commodity pickup-and-delivery travelling salesman problem (SD1PDTSP). This combines three vehicle routing problems. The first one is the Capacitated Vehicle Routing Problem (CVRP), aiming at designing the routes for a vehicle fleet to deliver a commodity from the depot to a set of customers. Each route starts and ends at the depot, and the load of a vehicle through a route should never exceed the vehicle capacity. A fundamental assumption is that each customer cannot be visited more than once. The aim of the CVRP is to satisfy the demand of each customer while minimizing the total distance travelled by the fleet. The second problem is the Split Delivery Vehicle Routing Problem (SDVRP), which allows a customer to be visited more than once. The third problem still moves one commodity while it allows more than one pickup location. It is the One Commodity Pickup-and-Delivery Travelling Salesman Problem (1-PDTSP). Using the elements of these three problems, the SD1PDTSP is defined as follows. Let us consider a finite set of locations. The travel distances (or costs) between the locations are assumed to be known. Each location is related to a customer, with a known positive or negative demand of a commodity (e.g. bicycles). We assume that the sum of all demands is equal to zero. Customers with negative demands correspond to pickup locations, and customers with positive demands correspond to delivery locations. It is assumed that there is one vehicle with a given capacity that must visit each location at least once through a route to move the commodity and satisfy all the customer demands. Each visit may partially satisfy a customer, and all the visits to that customer must end up with exactly its full demand. The SD1PDTSP consists of finding a minimum-cost route for the vehicle such that it satisfies the demand of all customers. Note that such a route may not exist, and in general checking whether a feasible solution exists is a NP-complete problem due to the limitation in the number of visits to each location and the vehicle capacity. The importance of these kind of problems is not due to the computational complexity only, but also the variety of practical applications. In fact, the idea of Operational Research appears formally in 1938 in the SecondWorldWar, in the frame in collaborative researches between soldiers and scientists about planning of fight military operations. A part of the most obvious examples in logistic (distribution of commodities, scholar routes,etc.), there are more examples as operational control of traffic light, those that can be found in robotic systems that allow to solve production problems, in genetic, etc. Other kind of practical applications arise due to the growing worry about environment. This thesis studies the case of the management of bicycle sharing systems. These are increasingly in demand due to the need of space, the high traffic density and the high emission of noise and CO2

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