Pareto Optimal Paths Research Articles

Ripple spreading algorithm: a new method for solving multi-objective shortest path problems with mixed time windows

In emergency management, the transportation scheduling of emergency supplies and relief personnel can be regarded as the multi-objective shortest path problem with mixed time window (MOSPPMTW), which has high requirements for timeliness and effectiveness, but the current solution algorithms cannot simultaneously take into account the solution accuracy and computational speed, which is very unfavorable for emergency path decision-making. In this paper, we establish MOSPPMTW matching emergency rescue scenarios, which simultaneously enables the supplies and rescuers to arrive at the emergency scene as soon as possible in the shortest time and at the smallest cost. To solve the complete Pareto optimal surface, we present a ripple spreading algorithm (RSA), which determines the complete Pareto frontier by performing a ripple relay race to obtain the set of Pareto optimal path solutions. The proposed RSA algorithm does not require an initial solution and iterative iterations and only needs to be run once to obtain the solution set. Furthermore, we prove the optimality and time complexity of RSA and conduct multiple sets of example simulation experiments. Compared with other algorithms, RSA performs better in terms of computational speed and solution quality. The advantage is especially more obvious in the computation of large-scale problems. It is applicable to various emergency disaster relief scenarios and can meet the requirements of fast response and timeliness.

Optimal path planning of unmanned surface vehicle under current environment

Path planning for unmanned surface vehicles (USVs) in the presence of ocean currents is a challenging task. Several relevant methods have been proposed, but they all assume a constant surge velocity for the whole path. This hypothesis makes it impossible to achieve the simultaneous optimization of speed and energy consumption. To deal with this issue, in this paper an optimal control path planning method that considers the currents’ characteristics is proposed. Unlike traditional algorithms such as the A* algorithm and genetic algorithm, the proposed Multi-domain Coupling Collocation (MDCC) method does not require a constant thrust assumption. In addition, it can obtain the Pareto optimal path under various current environments and optimize the sailing parameters. Specifically, first a multi-objective optimal control path planning model considering the kinematic characteristics of USVs is established. Second, the MDCC method is proposed to solve it, which includes dividing the time interval, subdomain collocation, and transformation optimization. Last, the optimality and high efficiency of the proposed method are proved theoretically. Simulation experiments show that the proposed method can provide solutions to this problem where existing algorithms fail, and has strong robustness and high efficiency. Furthermore, the obtained path is superior to those obtained using the existing algorithms in various scenarios.

Economic Growth Models with Heterogeneous Discounting

A survey of theoretical economic growth models with agents having different subjective discount factors is proposed. The structure of equilibrium paths in such models, their dynamics and convergence to stationary equilibria, and the relationship with Pareto optimal paths are described. Models with socially determined discount factors in which time preferences are formed endogenously are discussed, and the basic difficulties associated with social choice in the case of heterogeneous discount factors are examined. The models presented in the paper shed light on internal mechanisms of a market economy that lead to the division of society into the rich and the poor.

A dynamic ripple-spreading algorithm for solving mean–variance of shortest path model in uncertain random networks

Combinatorial optimization involves more and more evaluation indicators, and some parameters cannot be accurately described. This paper considers a shortest path problem where arc costs include both uncertainty and randomness, and the decision-maker wishes to minimize both the expected cost and the variance of this cost. Firstly, a mean–variance model for the shortest path problem with uncertain arc cost and random arc cost is proposed, and the equivalent deterministic model of the model is deduced. Secondly, we develop a dynamic ripple spreading algorithm (DRSA) to solve the model, based on the ripple spreading patterns on the natural water surface. Then, the ripple spreading speed of the algorithm is simulated and predicted by hybrid prediction algorithm (HPA) on the basis of obtaining real urban traffic data, and it is verified by theoretical proof that DRSA can find the Pareto optimal path from the source node to the destination node within a single run. Finally, the proposed mean–variance shortest path problem model and DRSA are verified by numerical experiments.

A deterministic and nature-inspired algorithm for the fuzzy multi-objective path optimization problem

Increasing evaluation indexes have been involved in the network modeling, and some parameters cannot be described precisely. Fuzzy set theory becomes a promising mathematical method to characterize such uncertain parameters. This study investigates the fuzzy multi-objective path optimization problem (FMOPOP), in which each arc has multiple crisp and fuzzy weights simultaneously. Fuzzy weights are characterized by triangular fuzzy numbers or trapezoidal fuzzy numbers. We adopt two fuzzy number ranking methods based on their fuzzy graded mean values and distances from the fuzzy minimum number. Motivated by the ripple spreading patterns on the natural water surface, we propose a novel ripple-spreading algorithm (RSA) to solve the FMOPOP. Theoretical analyses prove that the RSA can find all Pareto optimal paths from the source node to all other nodes within a single run. Numerical examples and comparative experiments demonstrate the efficiency and robustness of the newly proposed RSA. Moreover, in the first numerical example, the processes of the RSA are illustrated using metaphor-based language and ripple spreading phenomena to be more comprehensible. To the best of our knowledge, the RSA is the first algorithm for the FMOPOP that can adopt various fuzzy numbers and ranking methods while maintaining optimality.

Multi-Objective Safe-Interval Path Planning With Dynamic Obstacles

Path planning among dynamic obstacles is a fundamental problem in Robotics with numerous applications. In this work, we investigate a problem called Multi-Objective Path Planning with Dynamic Obstacles (MOPPwDO), which requires finding collision-free Pareto-optimal paths amid obstacles moving along known trajectories while simultaneously optimizing multiple conflicting objectives, such as arrival time, communication robustness and obstacle clearance. Most of the existing multi-objective A*-like planners consider no dynamic obstacles, and naively applying them to address MOPPwDO can lead to large computation times. On the other hand, efficient algorithms such as Safe-Interval Path Planing (SIPP) can handle dynamic obstacles but for a single objective. In this work, we develop an algorithm called MO-SIPP by leveraging both the notion of safe intervals from SIPP to efficiently represent the search space in the presence of dynamic obstacles, and search techniques from multi-objective A* algorithms. We show that MO-SIPP is guaranteed to find the entire Pareto-optimal front, and verify MO-SIPP with extensive numerical tests with two and three objectives. The results show that the MO-SIPP runs up to an order of magnitude faster than the conventional alternates.

Patrol robot path planning in nuclear power plant using an interval multi-objective particle swarm optimization algorithm

This paper presents an interval multi-objective path planning (PP) scheme for patrol robot in nuclear power plant. The purpose of this PP scheme is to find collision-free paths with the shortest length and smallest risk degree. Firstly, a novel workspace modeling method is proposed to describe the static PP environment of patrol robot in nuclear power plant. Then considering the conflicts of the shortest length and smallest risk degree, an interval multi-objective particle swarm optimization (IMOPSO) method is used. In the IMOPSO, an ingenious interval update law for the particle’s global best position and local best position based on the crowding distance of each risk degree interval is used to increase the diversity of population, and an iterative procedure is adopted to update the particle’s position when the found paths are collided with some existing obstacles. Finally, three representative simulation tests are used to verify the validity of proposed IMOPSO method. Results show that comparing with other three well-known multi-objective evolutionary algorithms, our proposed method has the advantages of finding a better Pareto optimal paths.

Finding all Pareto optimal paths by simulating ripple relay race in multi-objective networks

This paper proposes a novel nature-inspired method, so-called ripple-spreading algorithm (RSA) for multi-objective path optimization problem (MOPOP). Unlike most existing methods mainly capable of finding partial or approximated Pareto front, this paper focuses on calculating the complete Pareto front. This is achieved by taking advantage of the optimality principle in natural ripple-spreading phenomenon. Basically, the proposed RSA carries out a one-off ripple relay race in the route network, and then the complete Pareto front will be identified with guaranteed optimality by backtracking those Pareto non-dominated ripples (PNDRs) which reached the destination node. Theoretical analyses and comprehensive experiments show that all complete Pareto fronts of a one-to-all MOPOP can also be found in just a single run of ripple relay race, and the reported method can be further extended to calculate all Pareto optimal paths in dynamical networks, which have rarely been touched by existing MOPOP methods. Since many real-world application problems can be converted into MOPOP, the reported method has a great potential of applications.

On computing Pareto optimal paths in weighted time-dependent networks

A weighted point-availability time-dependent network is a list of temporal edges, where each temporal edge has an appearing time value, a travel time value, and a cost value. In this paper we consider the single source Pareto problem in weighted point-availability time-dependent networks, which consists of computing, for any destination d, all Pareto optimal pairs (t,c), where t and c are the arrival time and the cost, respectively, of a path from the source to d (a pair (t,c) is Pareto optimal if there is no path with arrival time smaller than t and cost no worse than c or arrival time no greater than t and better cost). We design and analyse a general algorithm for solving this problem, whose time complexity is O(|E|log⁡P), where E is the set of temporal edges and P is the maximum number of Pareto optimal pairs for each node of the network. Our algorithm can be used to solve several different minimum cost path problems in weighted point-availability time-dependent networks with a vast variety of cost definitions, and it can be easily modified in order to deal with the single destination Pareto problem. All our results apply to directed networks, but they can be easily adapted to undirected networks with no edges with zero travel time.

Determining transmission line path alternatives using a valley-finding algorithm

Since new (Power) Transmission Lines (TLs) can have a long-term effect on the makeup of a landscape and on the human living space, it should be expected that the route of any new TL will be based on objective criteria that take into account the views of the public. Geographic Information Science (GIScience) provides powerful tools that assist in the determination of feasible locations for new TLs based on objective criteria and georeferenced data by combining methods from Multi-Criteria Decision Analysis (MCDA) and Least Cost Path (LCP) analysis. If such an approach is applied, the LCP analysis usually yields one optimal result. However, stakeholders and decision-makers prefer to compare multiple distinct path alternatives in order to find a solution that will be acceptable to as many stakeholders as possible.We have developed a method that calculates spatially distinct and Pareto optimal path alternatives based on the same cost surface using an algorithm for determining valleys on a Digital Elevation Model (DEM) to determine local low-cost points, which were then connected to a network graph by geometrical rules. Finally, we selected all non-dominated path alternatives that represented Pareto optimal conditions regarding a specific main objective. We then asked ten expert stakeholders to cross-compare the calculated path alternatives and assess our novel method. The concept of applying linear programming to obtain Pareto optimal path alternatives yielded routings that were mostly preferred over the LCP and had a greater likelihood of being realized than the results obtained by conducting the LCP analysis. The stakeholders determined the method's key concept to be useful and assert it a high potential to support planning, matter-of-fact argumentation, and discussions about TL routing.

SDN-Enabled Adaptive and Reliable Communication in IoT-Fog Environment Using Machine Learning and Multiobjective Optimization

The Internet-of-Things (IoT) devices, backed by resourceful fog computing, are capable of meeting the requirements of computationally-intensive tasks. However, many existing IoT applications are unable to perform well, due to different Quality-of-Service (QoS) requirements, while communicating with the fog server. Besides, constantly changing traffic demands of applications is another challenge. For example, the demand for real-time applications includes communicating over a path that is less prone to delay, and applications that offload computationally intensive tasks to the fog server need a reliable path that has a lower probability of link failure. This results in a tradeoff between conflicting objectives that are constantly evolving, i.e., minimizing end-to-end delay and maximizing the reliability of paths between IoT devices and the fog server. We propose a novel approach that takes advantage of machine learning (ML) and multiobjective optimization (MOO)-based techniques. The reliability of links is evaluated using an ML-based algorithm in an software-defined network (SDN)-enabled multihop scenario for the IoT-fog environment. By considering the two conflicting objectives, the MOO algorithm is used to find the Pareto-optimal paths. Our experimental evaluation considers two applications with different QoS requirements—a real-time application (App-1) using UDP sockets and a task offloading application (App-2) using TCP sockets. Our results show that: 1) the tradeoff between the two objectives can be optimized and 2) the SDN controller was able to make adaptive decision on-the-fly to choose the best path from the Pareto-optimal set. The App-1 communicating over the selected path finished its execution in 13% less time than communicating over the shortest path. The App-2 had 41% less packet loss using the selected path compared to using the shortest path.

Rotary unmanned aerial vehicles path planning in rough terrain based on multi-objective particle swarm optimization

This paper presents a path planning approach for rotary unmanned aerial vehicles (R-UAVs) in a known static rough terrain environment. This approach aims to find collision-free and feasible paths with minimum altitude, length and angle variable rate. First, a three-dimensional (3D) modeling method is proposed to reduce the computation burden of the dynamic models of R-UAVs. Considering the length, height and tuning angle of a path, the path planning of R-UAVs is described as a tri-objective optimization problem. Then, an improved multi-objective particle swarm optimization algorithm is developed. To render the algorithm more effective in dealing with this problem, a vibration function is introduced into the collided solutions to improve the algorithm efficiency. Meanwhile, the selection of the global best position is taken into account by the reference point method. Finally, the experimental environment is built with the help of the Google map and the 3D terrain generator World Machine. Experimental results under two different rough terrains from Guilin and Lanzhou of China demonstrate the capabilities of the proposed algorithm in finding Pareto optimal paths.

Pareto Optimal Path Generation Algorithm in Stochastic Transportation Networks

Routing problems play a crucial part in urban transportation network operation and management. This study addresses the problem of finding a set of non-dominated shortest paths in stochastic transportation networks. Instead of the previous practice of assuming the travel time variability to be tracked by a known probability density function, it is extracted from the existing correlation between the traffic flow and the corresponding links' time. The time horizon is divided into time intervals/slots in which the network is assumed to experience a static traffic equilibrium with different traffic conditions for each slot. Starting with Priori demand information, prior generated paths, and a chosen traffic assignment method, the proposed methodology conducts successive simulations to the network intervals. It manages to draw both links and paths probability distribution of their travel time considering the correlation among them. Then, multi-objective analysis is conducted on the generated paths to produce the Pareto-optimal set for each demand node pair in the network. Numerical studies are conducted to show the methodology efficiency and generality for any network. The expected travel time and the reliability could be drawn for each path in the network.

Online AUV Path Replanning Using Quantum-Behaved Particle Swarm Optimization with Selective Differential Evolution

This paper presents an online AUV (autonomous underwater vehicle) path planner that employs path replanning approach and the SDEQPSO (selective differential evolution-hybridized quantum-behaved particle swarm optimization) algorithm to optimize an AUV mission conducted in an unknown, dynamic and cluttered ocean environment. The proposed path replanner considered the effect of ocean currents in path optimization to generate a Pareto-optimal path that guides the AUV to its target within minimum time. The optimization was based on the onboard sensor data measured from the environment, which consists of a priori unknown dynamic obstacles and spatiotemporal currents. Different sensor arrangements for the forward-looking sonar and horizontal acoustic Doppler current profiler (H-ADCP) were considered in 2D and 3D simulations. Based on the simulation results, the SDEQPSO path replanner was found to be capable of generating a time-optimal path that offered up to 13% reduction in travel time compared to the situation where the vehicle simply followed a path with the shortest distance. The proposed replanning technique also showed consistently better performance over a reactive path planner in terms of solution quality, stability, and computational efficiency. Robustness of the replanner was verified under stochastic process using the Monte Carlo method. The generated path fulfilled the vehicle’s safety and physical constraints, while intelligently exploiting ocean currents to improve the vehicle’s efficiency.

Finding equitable risk routes for hazmat shipments

This paper develops a model to analyse hazmat shipments routing in the context of hazmat transportation between the specified origin-destination (OD) pair. A novel aspect of this model is the consideration of risk equity using the standard deviation, an established computation to assess equity. To solve the model, a two-phase method is developed, in which the multi-objective shortest path algorithm is used to obtain the Pareto-optimal paths set, and get the routes using estimation of distribution algorithm after paths choice. We then present a test problem of hazmat shipment with consideration of risk equity and discuss computational results.

A two-stage algorithm for generating a set of Pareto-optimal trajectories of an object

A two-stage algorithm is proposed for generating trajectories and parameters of the object's motion based on searching for a set of Pareto-optimal paths from the initial vertex to the terminal vertex. The features of graph construction in the first and second stages of the algorithm are described. At the first stage, the set of vertices of the graph uniformly cover the area of the object's motion, and the edges connect only the nearest neighboring vertices. At the second stage, the set of vertices of the graph includes only those vertices through which the paths constructed in the first stage pass. For this set of vertices, a complete graph is constructed: the edges connect each vertex to all the others. An algorithm for constructing a set of Pareto-optimal characteristics of graph paths is described. The two-stage approach allows to significantly reduce the calculation time in comparison with the application of the described algorithm for a complete graph in one stage. Two variants of the algorithm are considered. The first algorithm requires a longer calculation time, but allows to obtain more trajectories that are diverse. In particular, it is possible to obtain trajectories of different classes, for example, avoiding obstacles from different sides, etc. For the case when time for calculations is not enough, an abridged algorithm is proposed. The results of computational experiments are presented.

Multi-objective path planning for unmanned surface vehicle with currents effects

This paper investigates the path planning problem for unmanned surface vehicle (USV), wherein the goal is to find the shortest, smoothest, most economical and safest path in the presence of obstacles and currents, which is subject to the collision avoidance, motion boundaries and velocity constraints. We formulate this problem as a multi-objective nonlinear optimization problem with generous constraints. Then, we propose the dynamic augmented multi-objective particle swarm optimization algorithm to achieve the solution. With our approach, USV can select the ideal path from the Pareto optimal paths set. Numerical simulations verify the effectiveness of our formulated model and proposed algorithm.

Car-like mobile robot path planning in rough terrain using multi-objective particle swarm optimization algorithm

Abstract This paper presents a path planning algorithm for car-like mobile robots operating on a known static rough terrain environment. The purpose of this approach is to find collision free and feasible paths with minimum length and terrain roughness. First, a new workspace modeling method is proposed to model the rough terrain environment. Then, considering the nonholonomic constraints of car-like robots, a MOPSO (multi-objective particle swarm optimization) based method is used to solve the problem. In the proposed algorithm, a new updating method for particle’s global best position based on crowding radius is used to increase population diversity. And to improve the algorithm efficiency, a nonuniformity factor is adopted to update the particle’s position when the path collides with obstacles. Finally, two simulation tests are designed using Microsoft Robotics Developer Studio 4 and Matlab. Results show the advantages of the proposed algorithm in finding Pareto optimal paths.

Bi-objective path planning using deterministic algorithms

Path planning has become a central problem in motion planning. The classic version of the problem aims to find an obstacle-free path with the minimum length for a given workspace containing a set of obstacles and two sources and destination points. However, some real world applications consider maximizing the path clearance (i.e., the distance between the robot and obstacles) as the secondary objective. This bi-objective path planning problem has been studied using evolutionary and other heuristic algorithms which do not guarantee achieving Pareto optimal paths. In this paper, we first study this problem using deterministic algorithms. Next, we propose an efficient algorithm for the problem in the grid workspace. We then propose an O(n3) time algorithm for the problem under the Manhattan distance in a continuous workspace containing n vertical segments as obstacles. Finally, we show the obtained solutions are proper approximation for the problem under the Euclidean distance.

The multi-criteria constrained shortest path problem

In this study, we propose an exact method for finding all the Pareto-optimal paths for a multi-criteria constrained shortest path problem. We show that solving the special bi-criteria problem is equivalent to generating at most |P| constrained shortest paths with successive tightened constraints, where |P| is the total number of all Pareto-optimal paths. For the general multi-criteria case, we propose a decomposition procedure and theoretically prove that this method can identify all the Pareto-optimal paths from at most (u-1)!|P| candidate paths, where u is the number of criteria. Numerical studies demonstrate that our algorithm is highly efficient and robust.