Routing Optimization Problem Research Articles

Discovering Lin-Kernighan-Helsgaun heuristic for routing optimization using self-supervised reinforcement learning

Vehicle routing optimization is a crucial responsibility of transportation service providers, which can significantly reduce operating expenses and improve client satisfaction. Learning to tackle routing optimization problems automatically can be the next significant step forward in optimization technology. Despite recent advancements in automatically learned heuristics for routing optimization problems, state-of-the-art traditional methods such as Lin-Kernighan-Helsgaun (LKH) still outperform machine learning-based approaches. To narrow this gap, we propose a novel technique called self-supervised reinforcement learning (SSRL), which combines self-supervised learning with the LKH heuristic. We provide a node decoder and an edge decoder corresponding to reinforcement learning and self-supervised learning for learning node penalties and edge scores, respectively. The self-supervised part with cross-entropy loss offers strong gradient signals for parameter updates. At the same time, the reinforcement learning component functions as a regularizer to drive the supervised part, which focuses on particular rewards. SSRL learns and replicates all of the LKH’s significant components, improving the original LKH’s generalization and performance. Through experiments on multiple vehicle routing problems, SSRL has demonstrated superior accuracy and efficiency compared to existing methods. Our results provide empirical evidence of SSRL’s effectiveness and potential as a promising solution for optimizing complex routing problems.

Delay-Sensitive Service Provisioning in Software-Defined Low-Earth-Orbit Satellite Networks

With the advancement of space technology and satellite communications, low-Earth-orbit (LEO) satellite networks have experienced rapid development in the past decade. In the vision of 6G, LEO satellite networks play an important role in future 6G networks. On the other hand, a variety of applications, including many delay-sensitive applications, are continuously emerging. Due to the highly dynamic nature of LEO satellite networks, supporting time-deterministic services in such networks is challenging. However, we can provide latency guarantees for most delay-sensitive applications through data plane traffic shaping and control plane routing optimization. This paper addresses the routing optimization problem for time-sensitive (TS) flows in software-defined low-Earth-orbit (LEO) satellite networks. We model the problem as an integer linear programming (ILP) model aiming to minimize path handovers and maximum link utilization while meeting TS flow latency constraints. Since this problem is NP-hard, we design an efficient longest continuous path (LCP) approximation algorithm. LCP selects the longest valid path in each topology snapshot that satisfies delay constraints. An auxiliary graph then determines the routing sequence with minimized handovers. We implement an LEO satellite network testbed with Open vSwitch (OVS) and an open-network operating system (ONOS) controller to evaluate LCP. The results show that LCP reduces the number of path handovers by up to 31.7% and keeps the maximum link utilization lowest for more than 75% of the time compared to benchmark algorithms. In summary, LCP achieves excellent path handover optimization and load balancing performance under TS flow latency constraints.

Integrated scheduling problems of open shop and vehicle routing using an ensemble of group teaching optimization and simulated annealing

Production and distribution are two parts of importance in supply chains. Yet, seldom attention is given to an integrated production and distribution in the case of open shop scheduling and vehicle routing problems. This article addresses an integrated scheduling problem of open shop and vehicle routing optimization, in which the jobs are processed in an open shop, and subsequently are delivered to customers at diverse places. First, a mixed integer programming model is developed to define the problem with minimizing maximum completion time. Second, an ensemble of group teaching optimization and simulated annealing methods is devised to optimally handle the considered problem. A heuristic rule is designed to generate a high-quality initial solution, while a simulated annealing method considering key machines is proposed to reinforce exploitation abilities. A nearest insertion method is used to make delivery decisions of vehicles. At last, the sensitivity analysis of user parameters in the developed approach is executed. Via solving a set of benchmark instances, we carry out extensive comparison experiments. The formulated model is verified by a mathematical programming optimizer CPLEX and the devised approach is compared with four meta-heuristics. The results and discussions confirm the strong competitiveness of the developed model and method for solving the concerned problems.

Multi-Active Multi-Passive (MAMP)-IRS Aided Wireless Communication: A Multi-Hop Beam Routing Design

Prior studies on intelligent reflecting surface (IRS) have mostly considered wireless communication systems aided by a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single passive</i> IRS, which, however, has limited control over wireless propagation environment and suffers severe product-distance path-loss. To address these issues, we propose in this paper a new <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">multi-active multi-passive (MAMP)</i> -IRS aided wireless communication system, where a number of active and passive IRSs are deployed to assist the communication between a base station (BS) and a remote user in complex environment, by establishing a multi-hop reflection path across active and passive IRSs. In particular, the active IRSs enable to opportunistically amplify the reflected signal along the multi-reflection link, thus effectively compensating for the severe product-distance path-loss. For the new MAMP-IRS aided system, an optimization problem is formulated to maximize the achievable rate of a typical user by designing the active-and-passive IRS routing path as well as the joint beamforming of the BS and selected active/passive IRSs. To draw useful insights into the optimal design, we first consider a special case of the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">single-active multi-passive (SAMP)</i> -IRS aided system. For this case, we propose an efficient algorithm to obtain its optimal solution by first optimizing the joint beamforming given any SAMP-IRS routing path, and then optimizing the routing path by using a new path decomposition method and graph theory. Moreover, we show that the active IRS should be selected to establish the beam routing path when its amplification power and/or number of active reflecting elements are sufficiently large. Next, for the general MAMP-IRS aided system, we show that its challenging beam routing optimization problem can be efficiently solved by a new two-phase approach. Its key idea is to first optimize the inner passive-IRS beam routing between each two active IRSs for effective channel power gain maximization, followed by an outer active-IRS beam routing optimization for rate maximization. Last, numerical results are provided to validate our analytical results and demonstrate the effectiveness of the proposed MAMP-IRS beam routing scheme as compared to various benchmark schemes.

A revisited branch-and-cut algorithm for large-scale orienteering problems

The orienteering problem is a route optimization problem which consists of finding a simple cycle that maximizes the total collected profit subject to a maximum distance limitation. In the last few decades, the occurrence of this problem in real-life applications has boosted the development of many heuristic algorithms to solve it. However, during the same period, not much research has been devoted to the field of exact algorithms for the orienteering problem. The aim of this work is to develop an exact method which is able to obtain the optimum in a wider set of instances than with previous methods, or to improve the lower and upper bounds in its disability.We propose a revisited version of the branch-and-cut algorithm for the orienteering problem which includes new contributions in the separation algorithms of inequalities stemming from the cycle problem, in the separation loop, in the variables pricing, and in the calculation of the lower and upper bounds of the problem. Our proposal is compared to three state-of-the-art algorithms on 258 benchmark instances with up to 7397 nodes. The computational experiments show the relevance of the designed components where 18 new optima, 76 new best-known solutions and 85 new upper-bound values were obtained.

Quantum computing for a profusion of postman problem variants

In this paper we study the viability of solving the Chinese Postman Problem, a graph routing optimization problem, and many of its variants on a quantum annealing device. Routing problem variants considered include graph type, directionally varying weights, number of parties involved in routing, among others. We put emphasis on the explanation of how to convert such problems into quadratic unconstrained binary optimization (QUBO) problems. QUBO is one of two equivalent natural paradigms for quantum annealing devices, the other being the Ising Model. We also expand upon a previously discovered algorithm for solving the Chinese Postman Problem on a closed undirected graph to decrease the number of constraints and variables used in the problem. Optimal annealing parameter settings and constraint weight values are discussed based on results from implementation on the D-Wave 2000Q and Advantage. Results from classical, purely quantum, and hybrid algorithms are compared.

Pointerformer: Deep Reinforced Multi-Pointer Transformer for the Traveling Salesman Problem

Traveling Salesman Problem (TSP), as a classic routing optimization problem originally arising in the domain of transportation and logistics, has become a critical task in broader domains, such as manufacturing and biology. Recently, Deep Reinforcement Learning (DRL) has been increasingly employed to solve TSP due to its high inference efficiency. Nevertheless, most of existing end-to-end DRL algorithms only perform well on small TSP instances and can hardly generalize to large scale because of the drastically soaring memory consumption and computation time along with the enlarging problem scale. In this paper, we propose a novel end-to-end DRL approach, referred to as Pointerformer, based on multi-pointer Transformer. Particularly, Pointerformer adopts both reversible residual network in the encoder and multi-pointer network in the decoder to effectively contain memory consumption of the encoder-decoder architecture. To further improve the performance of TSP solutions, Pointerformer employs a feature augmentation method to explore the symmetries of TSP at both training and inference stages as well as an enhanced context embedding approach to include more comprehensive context information in the query. Extensive experiments on a randomly generated benchmark and a public benchmark have shown that, while achieving comparative results on most small-scale TSP instances as state-of-the-art DRL approaches do, Pointerformer can also well generalize to large-scale TSPs.

Heterogeneous instant delivery orders scheduling and routing problem

Scheduling a large number of time-sensitive orders of multiple types in a short time period is a major challenge for instant delivery platforms. This study introduces an integrated problem of heterogeneous order assignment and routing optimization in the real-world setting of third-party instant delivery platforms. A mixed integer programming model is presented to optimize the assignment of instant delivery orders of multiple types and the scheduling of couriers with different grades. We develop a column generation model and a heuristic solution procedure to yield a solution for the integrated problem and we design an exact sub-procedure to accelerate the pricing problem. Taking Songjiang District, Shanghai as an example, numerical experiments are performed to evaluate the effectiveness of the proposed model and the efficiency of the proposed algorithm. The results show that the algorithm can obtain near-optimal solutions within a reasonable computing time. Several managerial implications are obtained from sensitivity analysis. The proposed methodology can be recommended for platform operators to improve dispatch efficiency and optimize workforce resources.

Energy-efficient and delay-guaranteed routing algorithm for software-defined wireless sensor networks: A cooperative deep reinforcement learning approach

Wireless sensor networks (WSNs) have undergone rapid development but still suffer from energy shortage issues. Meanwhile, existing energy-efficient routing approaches lack a cross-layer cooperation mechanism and cannot offer intelligent decision-making capabilities for resource-constrained nodes. In this paper, we design an intelligent routing algorithm with adaptive duty cycling for application in software-defined WSNs (SDWSNs) to maximize the network lifetime and data transmission reliability subject to a long-term data queue stability constraint. In particular, we introduce Lyapunov optimization to decouple the multistage routing optimization problem into per-frame deterministic problems with delay guarantee. To optimize the routing algorithm and adaptively adjust the duty-cycling mechanism, we propose a cooperative deep reinforcement learning (CDRL) model. Moreover, we develop a centralized training and distributed execution framework to decouple the model training and inference processes, which provides the capability of intelligent routing decision-making for resource-constrained nodes with low computational complexity. In addition, we design a new hybrid routing metric and a unique feature construction and extraction scheme to capture fine-grained information from our hybrid routing metric. The proposed CDRL algorithm adopts a cooperative and iterative model optimization strategy to achieve highly reliable data transmission and optimal long-term network performance. Simulation results demonstrate the effectiveness of the proposed algorithm in terms of network lifetime, packet delivery ratio, and statistical delay performance under various scenarios compared to the EBR-RL, EA-AGG, and QLRP algorithms.

A Deep Reinforcement Learning Based Real-Time Solution Policy for the Traveling Salesman Problem

The rapid development of logistics and navigation has led to increasing demand for solving route optimization problems in real-time. The traveling salesman problem (TSP) tends to require fast and reliable online solutions, which may not be met by traditional iterative optimization algorithms. In this work, a real-time solution policy is proposed for TSP. The idea is to build a mapping between city information and optimal solutions using deep neural networks. Therefore, when given a new set of city coordinates, the optimal route can be directly and quickly calculated without iteration. Considering the recent advancement in computer vision with deep convolutional neural networks (DCNNs), an image representation is proposed to convert TSP to a computer vision problem. A problem decomposition method is introduced to reduce the mapping complexity. Taking advantage of the powerful fitting capabilities of DCNN, a deep reinforcement learning method is designed without any labeling requirement. The proposed method is superior for real-time applications compared with other algorithms.

An exact algorithm for the multi-trip container drayage problem with truck platooning

Container Drayage Problem (CDP) refers to the optimization problem of routing and scheduling a set of container trucks around a container terminal. Conventionally, a driver is stuck to one container truck and allowed to perform multiple trips (multi-trip) to the terminal within their working time. The recent development of automation technologies enables semi-autonomous trucks to follow the leading human-driven truck as a platoon on the road; therefore, truck platooning can save human labor and reduce the fuel cost of following trucks through aerodynamic drag reduction. In this paper, we study a multi-trip container drayage problem with truck platooning (MT-CDP-TP), where multi-trip, truck platooning, and fuel cost reduction are simultaneously considered in a CDP. Despite the operational benefits brought by the MT-CDP-TP, the problem is challenging to solve due to its NP-hardness when formulated as a multi-trip pickup and delivery problem with load-dependent cost. We propose a Branch-and-Price-and-Cut (BPC) algorithm, with a route-based set partitioning model and tight linear relaxations, to yield the exact solutions. Valid inequalities are generated based on a graph structure, where each node represents a feasible route, and each arc stands for the conflict between two routes. Moreover, we design a tailored pulse propagation algorithm with novel pruning procedures based on the dual information from the master problem and valid inequalities to solve the pricing problem efficiently. Extensive numerical experiments are conducted for performance validation, and the computational results show that the proposed exact algorithm can solve instances with up to 100 task nodes (i.e., 50 customers) and facilitate reducing the labor cost and fuel consumption by a wide margin in container drayage operations.

Joint dynamic routing and resource allocation in satellite-terrestrial integrated networks

The Satellite-Terrestrial Integrated Networks (STINs) is considered as a reliable and agile next-generation communication network scheme, because it is flexible to deploy and robust to disasters (e.g., earthquakes, floods, and volcanic eruptions). The routing and resource allocation directly affect the power consumption of satellites in the STINs. However, most of existing research works study these two issues separately, and neglect the impact of path selection on the satellite power consumption. This paper investigates the joint optimization problem of routing, bandwidth allocation, user association, channel allocation, and power allocation of the STINs. We aim at minimizing satellites’ power consumption while satisfying user's Quality of Service (QoS). Because the power consumption of satellites for routing in the space segment is involved with the data rate through satellite gateways in the user segment, this optimization problem is a Multi-Objective Optimization (MOO) problem. To this end, we convert the MOO problem to a Single-Objective Optimization (SOO) problem by linear weighted method. Considering that the SOO problem is a Mixed Integer Nonlinear Programming (MINLP) problem, we decompose it into the resource allocation sub-problem in the user segment and the routing optimization sub-problem in the space segment. And we solve these two sub-problems iteratively. Specifically, we convert the first sub-problem into a convex optimization problem and obtain the current user association and channel allocation. Then we formulate the second sub-problem into a multi-commodity flow problem and solve it based on the obtained user association and channel allocation. The SOO problem is solved by iteratively optimizing these two sub-problems. Experimental results demonstrate that our proposed algorithm is able to get an approximate optimal solution quickly and effectively reduce the power consumption of satellites compared with benchmark algorithms in the STINs.

Heuristics for Multi-Vehicle Routing Problem Considering Human-Robot Interactions

Autonomous mobile robots (AMRs) are being used extensively in civilian and military applications for applications such as underground mining, nuclear plant operations, planetary exploration, intelligence, surveillance and reconnaissance (ISR) missions and manned-unmanned teaming. We consider a multi-objective, multiple-vehicle routing problem in which teams of manned ground vehicles (MGVs) and AMRs are deployed respectively in a leader-follower framework to execute missions with differing requirements for MGVs and AMRs while considering human-robot interactions (HRI). HRI studies highlight the costs of managing a team of follower AMRs by a leader MGV. This paper aims to compute feasible visit sequences, replenishments, team compositions and number of MGV-AMR teams deployed such that the requirements for MGVs and AMRs for the missions are met and the routing, replenishment, HRI and team deployment costs are at minimum. The problem is first modeled as a a mixed-integer linear program (MILP) that can be solved to optimality by off-the-shelf commercial solvers for small-sized instances. For larger instances, a variable neighborhood search algorithm is offered to compute near optimal solutions and address the challenges that arise when solving the combinatorial multi-objective routing optimization problem. Finally, computational experiments that corroborate the effectiveness of the proposed algorithms are presented.

A hybrid approach to enhanced genetic algorithm for route optimization problems

Shortest path problem has emerged to be one of the significant areas of research and there are various algorithms involved in it. One of the successful optimization techniques is genetic algorithm (GA). This paper proposes an efficient hybrid genetic algorithm where initially we use a map reduction technique to the graph and then find the shortest path using the conventional genetic algorithm with an improved crossover operator. On comparing this hybrid algorithm with other algorithms, it has been detected that the performance of the modified genetic algorithm is better as comparison to the other methods in terms of various metrics used for the evaluation.

On the multi-period combined maintenance and routing optimisation problem

This paper focuses on the combined maintenance and routing optimisation problem in a multi-period environment that consists of scheduling maintenance operations for a set of geographically distributed machines, subject to non-deterministic failures with a set of technicians that perform preventive maintenance and corrective operations. This study uses a two-step approach based on column generation. The first step consists of a maintenance model that determines the optimal time until the next preventive maintenance operation, and each machine’s maintenance frequency while minimising the total expected maintenance costs. The second step involves a routing model that assigns and schedules maintenance operations for each technician over the planning horizon while minimising the routing and maintenance costs. We formulate the second step problem as a periodic vehicle routing problem and to solve it, we propose a column generation approach. The master problem is a set partitioning formulation that indicates which machines must be visited in the planning horizon. This decomposition has two auxiliary problems: the first one is an elementary shortest-path problem to find a feasible path for each period and the second one finds a set of visiting periods for each machine. Our approach balances the maintenance cost, routing cost, and failure probabilities.

Radial distribution network planning under uncertainty by implementing robust optimization

This paper presents a two-level robust optimal feeder routing model for the planning of radial distribution networks where power demand is uncertain. The robust model is solved by implementing the column and constraint generation strategy. The optimization goal is to find routes that provide minimal total annual costs that satisfy parameters for quality of supply and technical constraints under the worst uncertain power demand realization. The model incorporates annual capital recovery, energy loss, and undelivered energy costs. The column and constraint generation strategy was developed using a mixed-integer quadratically constraint programming model capable of obtaining a deterministic solution to the optimal feeder routing problem.

Multimodal Transportation Route Optimization of Cold Chain Container in Time-Varying Network Considering Carbon Emissions

To solve the multimodal transport route optimization problem considering carbon emission, the vehicle speed has time-varying characteristics, and the customer has a time window limit. The carbon emission of multimodal transport system is affected by the energy consumption of transport vehicles in the time-varying network. The time-varying network is uncertain, and carbon emissions may continue to rise after a gradual decline. Based on this, this study established the sum of the carbon emission cost, transportation cost, penalty cost for exceeding the time window, and the damage cost of the cold chain cargo as the objective function. A route optimization model of cold chain container multimodal transportation was established. Static and dynamic optimization scenarios were designed and a hummingbird evolutionary genetic algorithm was used to solve the model. The effectiveness of the model and the practical value of the study are verified by the empirical analysis of the multimodal transport network of the Yangtze River Delta economic group. Results show that the dynamic model of the time-varying network can more truly reflect the transportation of the multimodal transport network and meet the efficiency requirements for the cold chain container multimodal transport. This study aims to solve the time-varying network under cold chain route optimization of container intermodal transportation, provides new insights for related businesses and a theoretical basis for reasonable multimodal transport route decisions.

Optimization Model and Solution Algorithm for Rural Customized Bus Route Operation under Multiple Constraints

In order to improve the operational efficiency of public transportation systems in rural areas, we investigated the demand-responsive rural customized bus vehicle route optimization problem. First, a two-stage planning model describing the problem in the reservation phase and real-time phase was constructed with the objectives of minimizing the operating cost of the operator and the travel time cost of the passenger, and the passenger time window, vehicle characteristics, rated passenger capacity and the running time of the route were considered in the constraints. Second, a hybrid algorithm solution model combining bat algorithm and adaptive particle swarm algorithm was designed to obtain a more optimal solution. Finally, the effectiveness of the hybrid algorithm on the optimization model was verified by using the actual road network in some townships of Jing’an County, Jiangxi Province, China, and the obtained objective function value was reduced by 5.5%. The results show that the optimization model and hybrid algorithm designed in this paper can be used to provide theoretical references for opening demand-responsive customized bus route operation schemes in rural areas.

드론을 포함한 3종 운송수단의 배송경로 최적화

This study addresses the vehicle routing optimization problem with three nested transportation modes, consisting of ships, trucks, and drones. A typical nested-three-mode vehicle routing problem appears in delivery systems that include areas of sparsely scattered, varying-size islands. In the problem, drones play an important role in achieving high efficiency in the entire delivery system, of their high-speed accessibility. In this paper, we present a mathematical model for the problem and explore its potential usability as a heuristic. For the usability check-up, a geographic data set from the island area of Tongyeong in Korea is used in an illustrative scenario for disaster relief goods distribution. Our limited computational experiment shows promise in the proposed optimization model as a heuristic with a preset computation time limit when compared with greedy heuristics.

Improving Bundle Routing in a Space DTN by Approximating the Transmission Time of the Reliable LTP

Because the operation of space networks is carefully planned, it is possible to predict future contact opportunities from link budget analysis using the anticipated positions of the nodes over time. In the standard approach to space delay-tolerant networking (DTN), such knowledge is used by contact graph routing (CGR) to decide the paths for data bundles. However, the computation assumes nearly ideal channel conditions, disregarding the impact of the convergence layer retransmissions (e.g., as implemented by the Licklider transmission protocol (LTP)). In this paper, the effect of the bundle forwarding time estimation (i.e., the link service time) to routing optimality is analyzed, and an accurate expression for lossy channels is discussed. The analysis is performed first from a general and protocol-agnostic perspective, assuming knowledge of the statistical properties and general features of the contact opportunities. Then, a practical case is studied using the standard space DTN protocol, evaluating the performance improvement of CGR under the proposed forwarding time estimation. The results of this study provide insight into the optimal routing problem for a space DTN and a suggested improvement to the current routing standard.