Non-dominated Paths Research Articles

The cross-entropy method for solving bi-criteria network flow problems in discrete-time dynamic networks

The present study tries to focus on the problem of finding the maximum flow along with the shortest path in a dynamic network. Networks basically state problems where the transfer is not instantaneous and time is required to traverse the arcs. For solving bi-criteria network problems, a two-phased exact algorithm and a cross-entropy (CE) algorithm based on bi-criteria are proposed, where the costs change as time functions. First, the two-phased complete enumeration algorithm was proposed to generate the non-dominated paths. Then, the efficiency and validation of the solutions generated by both the algorithms are compared. The computational results for 53 random instances showed that in problems with sizes of 50–300 nodes, the non-dominated paths obtained from the two algorithms are identical. The only difference is that the solution time of the CE algorithm is significantly less; however, for problems with more than 300 nodes, the non-dominated paths generated in the CE algorithm have sometimes a few path(s) less than the other one. In addition, the mean of the CPU time for the CE algorithm is about 0.073 s, whereas the mean of the CPU time of the other algorithm is far more, about 200 s. For greater size problems, CPU time has exponential growth compared with that of the complete enumeration algorithm.

Bi-objective decision support system for routing and scheduling of hazardous materials

This paper presents a Pareto-based bi-objective optimization of hazardous materials vehicle routing and scheduling problem with time windows and shows its application to a realistic hazardous material logistics instance. A meta-heuristic solution algorithm is also proposed, which returns a set of routing solutions that approximate the frontier of the Pareto optimal solutions based on total scheduled travel time and total risk of whole transportation process. It works in a single-step fashion simultaneously constructing the vehicle route and selecting the optimal paths connecting the routed locations from a set of non-dominated paths obtained in terms of travel time and risk value.

Two-step Solving Algorithm for Hazmat Transportation Routing and Dispatch Problem

This paper focus on the multi-objective routing and dispatch problem for a given hazardous materials shipment with multiple time windows and time varying attributes. Section 1 consists of background and purpose of the research. Section 2 proposes a mathematical model for this problem by introducing the punishing functions to convert time windows into penalty variables of model. Section 3 develops a two-step solving algorithm for model, of which the first step is to identify the non-dominated paths with specific dispatch time based on the improved dynamic programming method, and the second step is to choose the optimal alternatives from non-dominated paths by the application of the grey correlation analysis method. Section 4 carries out a case study to test the methodology proposed in the paper. The results show that the model and the two-step algorithm can effectively solve road hazardous materials transportation routing and dispatch problem.

Finding Reliable Shortest Path in Stochastic Time-dependent Network

This paper addresses the problem of finding reliable a priori shortest path to maximize the probability of arriving on time in a stochastic and time-dependent network. Optimal solutions to the problem can be obtained from finding non-dominated paths, which are defined based on first-order stochastic dominance. We formulate the problem of finding non-dominated paths as a general dynamic programming problem because Bellman's principle of optimality can be applied to construct non-dominated paths. A label-correcting algorithm is designed to find optimal paths based on the new proporty for which Bellman's Principle holds. Numerical results are provided using a small network.

A cost-time trade-off Königsberg bridge problem traversing all the seven bridges allowing repetition

An extended version of Konigsberg bridge problem is considered. After having split into two streams, Pregel River flows through the city of Konigsberg, now known as Kaliningrad, forming two islands. Seven bridges are built across the river providing links among the four land masses consisting of two islands, right and left banks of the river. Costs and times of traversing the bridges are given. The set of nondominated solutions of this problem is obtained providing the total cost and total time of nondominated paths starting from one land mass and returning to it after having traversed all the seven bridges allowing repetition. The problem is solved in two phases. In the first phase, the problem is represented by a graph whose nodes represent the land masses and edges represent the bridges. In the second phase, the set of nondominated solutions of the problem is obtained through generating and solving a sequence of prioritized bicriterion problems, all of whose nodes are of even degree. The generation of the sequence of prioritized bicriterion problems is done through augmenting the number of edges by duplicating some edges in the graph representing the problem in such a way that all the nodes in each of the prioritized bicriterion problems are of even degree; earlier all the nodes in the graph representing the problem were of odd degree. Now, since, each of the prioritized bicriterion problems has all the nodes of even degree, each of them is amenable to solution by Cycle finding and Fleury’s algorithms.

Model and Algorithm for Routing and Scheduling Problem in Hazardous Materials Transportation Network

To reduce the risks associated with hazardous materials transportation by optimizing routes and delivery schedules, a multiobjective routing and scheduling program with multiple time windows and time varying attributes in hazardous materials transportation network is studied. First, a mathematical model for this program is proposed and two punishing functions are introduced to the model for converting multiple time windows into the penalty variables of model. Second, a two-step solving algorithm for model is developed, and the first step of algorithm is to find the set of non-dominated paths with specific departure time from the origin based on the improved dynamic programming method, the second step is to identify the optimal alternatives from the non-dominated set by using the grey correlation analysis method. Finally, a case study is given to show that the model and algorithm can effectively solve the routing and scheduling program in hazmat transportation network, and can identify the optimal transportation paths for the decision maker.

Bicriteria path problem minimizing the cost and minimizing the number of labels

We address a bicriterion path problem where each arc is assigned with a cost value and a label (such as a color). The first criterion intends to minimize the total cost of the path (the summation of its arc costs), while the second intends to get the solution with a minimal number of different labels. Since these criteria, in general, are conflicting criteria we develop an algorithm to generate the set of non-dominated paths. Computational experiments are presented and results are discussed.

Finding Reliable Shortest Paths in Dynamic Stochastic Networks

A reliable a priori shortest path (RASP) offers the least time budget and ensures that a traveler can arrive at the destination on time with the desired probability. A RASP is equivalent to enumerating all nondominated paths under the first-order stochastic dominance rule. Compared with the problem in static networks, the RASP problem becomes more complex in dynamic networks because it is more difficult to compute path travel time distributions. Two modules of process are the keys to solving the RASP problem. One module is the convolution scheme (how to compute a path travel time distribution from its member links' travel time distributions), and the other module is the stochastic dominance scheme (how to determine nondominated paths). This study aims to find an efficient solution algorithm for this problem. An alternative convolution method is developed on the basis of the adaptive discretization approach, which was originally proposed to solve the RASP problem in static networks. In contrast, the higher-order stochastic dominance rule can be shown to reduce the number of nondominated paths; this rule promises to improve computational efficiency. Numerical experiments show that the second-order stochastic dominance rule offers good approximations, and the central processing unit time is reduced by at least 50%.

Finding Reliable Shortest Paths in Road Networks Under Uncertainty

The aim of this study is to investigate the solution algorithm for solving the problem of determining reliable shortest paths in road networks with stochastic travel times. The availability of reliable shortest paths enables travelers, in the face of travel time uncertainty, to plan their trips with a pre-specified on-time arrival probability. In this study, the reliable shortest path between origin and destination nodes is determined using a multiple-criteria shortest path approach when link travel times follow normal distributions. The dominance conditions involved in such problems are established, thereby reducing the number of generated non-dominated paths during the search processes. Two solution algorithms, multi-criteria label-setting and A* algorithms, are proposed and their complexities analyzed. Computational results using large scale networks are presented. Numerical examples using data from a real-world advanced traveller information system is also given to illustrate the applicability of the solution algorithms in practice.

Indirect and Direct Multicost Algorithms for Online Impairment-Aware RWA

We consider the online impairment-aware routing and wavelength assignment (IA-RWA) problem in transparent WDM networks. To serve a new connection, the online algorithm, in addition to finding a route and a free wavelength (a lightpath), has to guarantee its transmission quality, which is affected by physical-layer impairments. Due to interference effects, the establishment of the new lightpath affects and is affected by the other lightpaths. We present two multicost algorithms that account for the actual current interference among lightpaths, as well as for other physical effects, performing a cross-layer optimization between the network and physical layers. In multicost routing, a vector of cost parameters is assigned to each link, from which the cost vectors of the paths are calculated. The first algorithm utilizes cost vectors consisting of impairment-generating source parameters, so as to be generic and applicable to different physical settings. These parameters are combined into a scalar cost that indirectly evaluates the quality of candidate lightpaths. The second algorithm uses specific physical-layer models to define noise variance-related cost parameters, so as to directly calculate the <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Q</i> -factor of candidate lightpaths. The algorithms find a set of so-called nondominated paths to serve the connection in the sense that no path is better in the set with respect to all cost parameters. To select the lightpath, we propose various optimization functions that correspond to different IA-RWA algorithms. The proposed algorithms combine the strength of multicost optimization with low execution times, making them appropriate for serving online connections.

위험물 수송 최적경로 탐색 알고리즘 개발: Efficient Vector Labeling 방법으로

This paper deals with a methodology for searching optimal route of hazard material (hazmat) vehicles. When we make a decision of hazmat optimal paths, there is a conflict between the public aspect which wants to minimize risk and the private aspect which has a goal of minimizing travel time. This paper presents Efficient Vector Labeling algorithm as a methodology for searching optimal path of hazmat transportation, which is intrinsically one of the multi-criteria decision making problems. The output of the presented algorithm is a set of Pareto optimal paths considering both risk and travel time at a time. Also, the proposed algorithm is able to identify non-dominated paths which are significantly different from each other in terms of links used. The proposed Efficient Vector Labeling algorithm are applied to test bed network and compared with the existing k-shortest path algorithm. Analysis of result shows that the proposed algorithm is more efficient and advantageous in searching reasonable alternative routes than the existing one.

Optimal Routing with Multiple Objectives: Efficient Algorithm and Application to the Hazardous Materials Transportation Problem

Abstract: This article presents an efficient parametric optimization method for the biobjective optimal routing problem. The core process is a bounded greedy single‐objective shortest path approximation algorithm. This method avoids the computationally intensive dominance check with labeling methods and overcomes the deficiency with existing parametric methods that can only find extreme nondominated paths. Moreover, we propose a decomposition scheme to convert a multiobjective routing problem into a number of biobjective problems. We then compare its computational performance against the classic label‐correcting method over a set of synthetically generated random networks and illustrate its algorithmic advances and solution behaviors by an example application of routing hazardous materials in a U.S. northeastern highway network.

Solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs

In this paper, we present an algorithm for solving the bicriteria traffic equilibrium problem with variable demand and nonlinear path costs. The path cost function considered is comprised of two attributes, travel time and toll, that are combined into a nonlinear generalized cost. Travel demand is determined endogenously according to a travel disutility function. Travelers choose routes with the minimum overall generalized costs. The algorithm involves two components: a bicriteria shortest path routine to implicitly generate the set of non-dominated paths and a projection and contraction method to solve the nonlinear complementarity problem (NCP) describing the traffic equilibrium problem. Numerical experiments are conducted to demonstrate the feasibility of the algorithm to this class of traffic equilibrium problems.

Redundant multicast routing in multilayer networks with shared risk resource groups: Complexity, models and algorithms

This paper presents a redundant multicast routing problem in multilayer networks that arises from large-scale distribution of realtime multicast data (e.g., Internet TV, videocasting, online games, stock quotes). Since these multicast services commonly operate in multilayer networks, the communications paths need to be robust against a single router or link failure as well as multiple such failures due to shared risk link groups (SRLGs). The main challenge of this multicast is to ensure the service availability and reliability using a path protection scheme, which is to find a redundant path that is SRLG-disjoint (diverse) from each working path. The objective of this problem is, therefore, to find two redundant multicast trees, each from one of the two redundant sources to every destination, at a minimum total communication cost whereas two paths from the two sources to every destination are guaranteed to be SRLG-diverse (i.e., links in the same risk group are disjoint). In this paper, we present two new mathematical programming models, edge-based and path-based, for the redundant multicast routing problem with SRLG-diverse constraints. Because the number of paths in path-based model grows exponentially with the network size, it is impossible to enumerate all possible paths in real life networks. We develop three approaches (probabilistic, non-dominated and nearly non-dominated) to generate potentially good paths that may be included in the path-based model. This study is motivated by emerging applications of internet-protocol TV service, and we evaluate the proposed approaches using real life network topologies. Our empirical results suggest that both models perform very well, and the nearly non-dominated path approach outperforms all other path generation approaches.

Ant colony system based routing and scheduling for hazardous material transportation

This paper presents a new meta-heuristic algorithm using an ant colony system (ACS) for multi-objective optimisation of hazardous material (HAZMAT) transportation. We focus on the vehicle routing problem with time windows (VRPTW) aspect of HAZMAT transportation problem. A VRPTW formulation considering multiple attributes in application to HAZMAT transportation is provided. ACS in the proposed algorithm works in the framework of pareto-optimisation for routing and integrates a labelling algorithm for finding non-dominated paths for path choice purpose. Validity of the algorithm has been tested by applying it to several VRPTW benchmark problems. Results show that the proposed algorithm performs quite satisfactorily to the wide variety of VRPTW problems.

Upgrading Arc Median Shortest Path Problem for an Urban Transportation Network

In this paper, we propose an algorithm for an upgrading arc median shortest path problem for a transportation network. The problem is to identify a set of nondominated paths that minimizes both upgrading cost and overall travel time of the entire network. These two objectives are realistic for transportation network problems, but of a conflicting and noncompensatory nature. In addition, unlike upgrading cost which is the sum of the arc costs on the path, overall travel time of the entire network cannot be expressed as a sum of arc travel times on the path. The proposed solution approach to the problem is based on heuristic labeling and exhaustive search techniques, in criteria space and solution space, respectively. The first approach labels each node in terms of upgrading cost, and deletes cyclic and infeasible paths in criteria space. The latter calculates the overall travel time of the entire network for each feasible path, deletes dominated paths on the basis of the objective vector and identifies a set of Pareto optimal paths in the solution space. The computational study, using two small-scale transportation networks, has demonstrated that the algorithm proposed herein is able to efficiently identify a set of nondominated median shortest paths, based on two conflicting and noncompensatory objectives.

Shortest path problem considering on-time arrival probability

This paper studies the problem of finding a priori shortest paths to guarantee a given likelihood of arriving on-time in a stochastic network. Such “reliable” paths help travelers better plan their trips to prepare for the risk of running late in the face of stochastic travel times. Optimal solutions to the problem can be obtained from local-reliable paths, which are a set of non-dominated paths under first-order stochastic dominance. We show that Bellman’s principle of optimality can be applied to construct local-reliable paths. Acyclicity of local-reliable paths is established and used for proving finite convergence of solution procedures. The connection between the a priori path problem and the corresponding adaptive routing problem is also revealed. A label-correcting algorithm is proposed and its complexity is analyzed. A pseudo-polynomial approximation is proposed based on extreme-dominance. An extension that allows travel time distribution functions to vary over time is also discussed. We show that the time-dependent problem is decomposable with respect to arrival times and therefore can be solved as easily as its static counterpart. Numerical results are provided using typical transportation networks.

A bicriteria routing model for multi-fibre WDM networks

All-optical WDM networks are characterised by multiple metrics (hop-count, cost, delay, available band- width, loss probability, reliability), but generally routing algorithms only optimise one metric. Having in mind the inherent limitations of this type of approach, it seems potentially advantageous in the development of multicriteria models capable of explicitly representing the different performance objectives and enabling to treat in a consistent manner the trade off among the various criteria. A bicriteria model for obtaining a topological path (unidirectional or symmetric bidirectional) for each lightpath request in a WDM network is proposed. The first criterion is related to bandwidth usage in the links (or arcs) of the network. The second criterion is the number of links (hops) of the path. The model resolution approach uses a k-shortest path algorithm as well as preference thresholds defined in the objective function space, combined with a Chebyshev distance to a reference point (which changes with the analysed preference region). The solution of this bicriteria model is a non-dominated topological path. A heuristic procedure is then used to assign wavelengths to the links. The performance of the bicriteria model is analysed by comparing it with two monocriterion approaches, using two test networks.

MINIMIZING EXPOSURE RISK AND TRAVEL TIMES OF HAZARDOUS MATERIAL TRANSPORTATION IN URBAN AREAS

A HAZMAT vehicle routing problem considering multi-objective requirements and a meta-heuristic solution algorithm is presented in this paper. In contrast to existing studies, an attempt has been made to carry out routing and route choice processes together as a single step process. Pareto-optimal concept using non-dominated paths are utilized, dealing with multiple objectives during both of these processes. An application of the proposed algorithm for solving a test HAZMAT vehicle routing problem is also presented. The set of pareto-optimal solutions obtained provides a number of economical and safe routing solutions, identifying the alternative routing choices in decision-making.

Dynamic Pricing with Heterogeneous Users

The bicriterion dynamic user equilibrium (BDUE) model characterizes the dynamic user equilibrium (DUE) in a network resulting from the path choice interactions of a population of heterogeneous trip makers with different values of time (VOT). The BDUE model represents an attempt to accommodate greater behavioral and policy realism in applying DUE models to designing and evaluating dynamic pricing strategies. It also represents an advance in generalizing heterogeneous user equilibrium models from the static regime to the dynamic traffic-assignment context. To effectively obtain time-varying path-flow patterns satisfying the BDUE conditions, a study was done to adapt the gap-driven and simulation-based algorithmic framework to solve the DUE problem (with a constant VOT). Especially, the proposed BDUE algorithm is a column generation-based approach that integrates the following components: (a) a simulation-based dynamic network loading model that captures traffic dynamics and determines experienced path travel times for a given time-varying path-flow pattern, (b) a path generation scheme that partitions the entire range of VOT into many subintervals and accordingly determines the corresponding multiple user classes and the least-generalized cost (i.e., extreme nondominated) paths for each user class, and (c) a multiclass path flow equilibrating method for updating the current path assignment. The results of the experiments conducted on several real networks show that the convergence pattern of the proposed algorithm is not affected by different VOT assumptions, and it is able to find close-to-BDUE solutions.