Non-dominated Paths Research Articles

Backbone Index and GNN Models for Skyline Path Query Evaluation over Multi-cost Road Networks

Skyline path queries (SPQs) extend skyline queries to multi-dimensional networks, such as multi-cost road networks (MCRNs). Such queries return a set of non-dominated paths between two given network nodes. Despite the existence of extensive works on evaluating different SPQ variants, SPQ evaluation is still very inefficient due to the nonexistence of efficient index structures to support such queries. Existing index building approaches for supporting shortest-path query execution, when directly extended to support SPQs, use an unreasonable amount of space and time to build, making them impractical for processing large graphs. In this paper, we propose a novel index structure, backbone index , and a corresponding index construction method that condenses an initial MCRN to multiple smaller summarized graphs with different granularity. We present efficient approaches to find approximate solutions to SPQs by utilizing the backbone index structure. Furthermore, considering making good use of historical query and query results, we propose two models, S kyline P ath G raph N eural N etwork (SP-GNN) and T ransfer SP-GNN (TSP-GNN), to support effective SPQ processing. Our extensive experiments on real-world large road networks show that the backbone index can support finding meaningful approximate SPQ solutions efficiently. The backbone index can be constructed in a reasonable time, which dramatically outperforms the construction of other types of indexes for road networks. As far as we know, this is the first compact index structure that can support efficient approximate SPQ evaluation on large MCRNs. The results on the SP-GNN and TSP-GNN models also show that both models can help get approximate SPQ answers efficiently.

Originating multiple-objective portfolio selection by counter-COVID measures and analytically instigating robust optimization by mean-parameterized nondominated paths

The COVID-19 pandemic is unleashing crises of humanity, economy, and finance. Portfolio selection is widely recognized as the foundation of modern financial economics. Therefore, it is naturally crucial and inviting to utilize portfolio selection in order to counter COVID-19 in stock markets. We originate a counter-COVID measure for stocks, extend portfolio selection, and construct multiple-objective portfolio selection. Because of the uncertainty in measuring counter-COVID, we perform robust optimization. Specifically, we analytically compute the optimal solutions as a trail of an optimal portfolio due to the change of counter-COVID. We call the trail as mean-parameterized nondominated path. Moreover, the path is a continuous function of the change, so the portfolio relatively mildly varies for the change. In contrast, researchers typically still focus on 2-objective robust illustrations and infrequently explicitly compute the optimal solutions for multiple-objective portfolio optimization.To the best of our knowledge, there is limited research for multiple-objective portfolio selection of COVID and for the robust optimization of multiple-objective portfolio selection. In such an area, this paper contributes to the literature. The implications to fight COVID are that investors minimize risk, maximize return, and maximize counter-COVID in stock markets and that investors ascertain the multiple-objective portfolio selection as relatively robust.

Coordinated routing of electric commercial vehicles with intra-route recharging and en-route battery swapping

A primary challenge in goods distribution using Electric Commercial Vehicles (ECVs) pertains to tackling their limited driving range. This paper proposes a multi-faceted approach towards increasing the driving range of ECVs by coordinating the options of: (i) intra-route recharging at an intermediate Recharging Station (RS), with (ii) synchronised en-route battery swapping services performed by Battery Swapping Vans (BSVs) at a pre-planned rendezvous time and space. We introduce and solve a variant corresponding to an Electric Vehicle Routing Problem with Time Windows, RSs and Synchronised Mobile Battery Swapping (EVRPTW-RS-SMBS). In the proposed model, route planning is carried out synchronously for two interdependent fleets, i.e., ECVs and BSVs, which work in tandem to complete the delivery tasks. To address methodological complications arising from the simultaneous consideration of intra-route recharging at RSs and the synchronised battery swapping on-the-fly, the paper develops a pre-optimisation procedure based on a Non-Dominated Path Identification (NDPI) algorithm that is used in deriving a significantly strengthened path-based formulation of the problem, and an efficient dynamic programming based heuristic algorithm. To gain practical insights on the economic and environmental added value and viability of the proposed logistics model, we compare different scenarios for goods distribution using ECVs in urban and regional levels in London and Southeast England, respectively. A set of numerical experiments are further performed to demonstrate the efficiency of the proposed algorithms. Our results indicate significant cost and emissions savings and an opportunity for going beyond last mile local deliveries using ECVs with the proposed logistics model.

An Efficient Label-Correcting Algorithm for the Multiobjective Shortest Path Problem

This paper proposes an exact algorithm to solve the one-to-one multiobjective shortest path problem. The solution involves determining a set of nondominated paths between two given nodes in a graph that minimizes several objective functions. This study is motivated by the application of this solution method to determine cycling itineraries. The proposed algorithm improves upon a label-correcting algorithm to rapidly solve the problem on large graphs (i.e., up to millions of nodes and edges). To verify the performance of the proposed algorithm, we use computational experiments to compare it with the best-known methods in the literature. The numerical results confirm the efficiency of the proposed algorithm. Summary of Contribution: The paper deals with a classic operations research problem (the one-to-one multiobjective shortest path problem) and is motivated by a real application for cycling itineraries. An efficient method is proposed and is based on a label-correcting algorithm into which several additional improvement techniques are integrated. Computational experiments compare this algorithm with the best-known methods in the literature to validate the performance on large-size graphs (Center for Discrete Mathematics and Theoretical Computer Science (DIMACS) instances from the ninth DIMACS challenge). New instances from the context of cycling itineraries are also proposed.

The Bi-objective Long-haul Transportation Problem on a Road Network

In this paper we study a long-haul truck scheduling problem where a path has to be determined for a vehicle traveling from a specified origin to a specified destination. We consider refueling decisions along the path while accounting for heterogeneous fuel prices in a road network. Furthermore, the path has to comply with Hours of Service (HoS) regulations. Therefore, a path is defined by the actual road trajectory traveled by the vehicle, as well as the locations where the vehicle stops due to refueling, compliance with HoS regulations, or a combination of the two. This setting is cast in a bi-objective optimization problem, considering the minimization of fuel cost and the minimization of path duration. An algorithm is proposed to solve the problem on a road network. The algorithm builds a set of non-dominated paths with respect to the two objectives. Given the enormous theoretical size of the road network, the algorithm follows an interactive path construction mechanism. Specifically, the algorithm dynamically interacts with a geographic information system to identify the relevant potential paths and stop locations. Computational tests are made on real-sized instances where the distance covered ranges from 500 to 1500 km. The algorithm is compared with solutions obtained from a policy mimicking the current practice of a logistics company. The results show that the non-dominated solutions produced by the algorithm significantly dominate the ones generated by the current practice, in terms of fuel costs, while achieving similar path durations. The average number of non-dominated paths is 2.7, which allows decision-makers to ultimately visually inspect the proposed alternatives.

A more efficient cutting planes approach for the green vehicle routing problem with capacitated alternative fuel stations

The Green Vehicle Routing Problem with Capacitated Alternative Fuel Stations assumes that, at each station, the number of vehicles simultaneously refueling cannot exceed the number of available pumps. The state-of-the-art solution method, based on the generation of all feasible non-dominated paths, performs well only with up to 2 pumps. In fact, it needs cloning the paths between every pair of pumps. To overcome this issue, in this paper, we propose new path-based MILP models without cloning paths, for both the scenario with private stations (i.e., owned by the fleet manager) and that with public stations. Then, a more efficient cutting plane approach is designed for addressing both the scenarios. Numerical results, obtained considering a set of benchmark instances ad hoc generated for this work, show both the efficiency and the effectiveness of this new cutting plane approach proposed. Finally, a sensitivity analysis, carried out by varying the number of customers to be served and their distribution, shows very good performances of the proposed approach.

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.

Establishing an Evacuation Network: A Path Ranking Approach

The evacuation of population exposed to flood hazard requires the establishment of a specific transportation network. This is to compute escape routes to be taken by affected population. At evacuation time, inhabitants of affected area must be evacuated through transportation network to safety area. The main objective of this work is to provide assistance to rescue forces in terms of accessibility by providing itinieraries between buildings at risk of flooding and safety points equipped for this purpose. Technically, the problem of k-shortest paths between two nodes in network has been extensively studied in the literature offering several efficient algorithms for different applications. Those algorithms so-called ranking methods aim to compute the first shortest path then the second one and so on. It’s well known that the computation in ranking methods is based only on one criterion which is generally distance or time. Often a single criterion is not sufficient in some real-world problems (road network, internet, etc.,) where one or several supplementary criteria (such as cost, capacity, security, etc.,) must also be taken into consideration. This multi-criteria optimization so-called labeling method aims to compute a Pareto front that represents a set of non-dominated paths. However it is very difficult to take a decision on which k-paths to select among this set and especially when selected paths are served as a data input for a sensitive problem such as the evacuation. We contribute in this paper to establish an evacuation network dedicated to the evacuation of population exposed to natural hazards and more particularly to flood hazard. Two ranking methods to compute paths between each origin (building) - destination (shelter) pair are presented. The criteria free-flow travel time, capacity and number of lanes of road are considered in computing paths. Thus we aim in the proposed ranking approach to simulate a multi-criteria aspect by combining travel time and number of lanes as weight function. The establishment of network (determination of k-paths between each building and shelter associated) is then performed according to several measures we introduce in this paper.

Shortest paths with ordinal weights

We investigate the single-source-single-destination “shortest” path problem in directed, acyclic graphs with ordinal weighted arc costs. We define the concepts of ordinal dominance and efficiency for paths and their associated ordinal levels, respectively. Further, we show that the number of ordinally non-dominated path vectors from the source node to every other node in the graph is polynomially bounded and we propose a polynomial time labeling algorithm for solving the problem of finding the set of ordinally non-dominated path vectors from source to sink.

Skyline Path Queries for Location-based Services

A skyline query finds objects that are not dominated by another object from a given set of objects. Skyline queries help us to filter unnecessary information efficiently and provide us clues for various decision making tasks. In this paper, we consider skyline queries for location-based services and proposed a framework that can efficiently compute all non-dominated paths in road networks. A path p is said to dominate another path q if p is not worse than q in any of the k dimensions and p is better than q in at least one of the k dimensions. Our proposed skyline framework considers several features related to road networks and return all non-dominated paths from the road networks. In our work, we compute skylines considering two different perspectives: business perspective and individual user’s perspective. We have conducted several experiments to show the effectiveness of our method. From the experimental results, we can say that our system can perform efficient computation of skyline paths from road networks.

A path-based solution approach for the Green Vehicle Routing Problem

The Green Vehicle Routing Problem concerns routing alternative fuel vehicles, based at a single depot, to handle a subset of customers while minimizing the total travel distance. Due to limited fuel autonomy, each vehicle may need to stop at Alternative Fuel Stations (AFSs) during its trip. We propose a two-phase solution approach in which a route is seen as the composition of paths, each one handling a subset of customers without intermediate stops at AFSs. In the first phase, all feasible paths are generated limiting their number through dominance rules. In the second phase, the paths are selected and properly combined to generate the routes via Mixed Integer Linear Programming. Our approach, tested on small- to medium-sized benchmark instances, outperforms the existing exact methods obtaining always the optimal solution in a smaller average computational time. Furthermore, the approach was converted into a heuristic one considering in the first phase only a subset of feasible non-dominated paths. In this way, we can also address larger- sized instances outperforming, in terms of solution quality, the best heuristic approach in the literature.

Algorithms for non-linear and stochastic resource constrained shortest path

Resource constrained shortest path problems are usually solved thanks to a smart enumeration of all the non-dominated paths. Recent improvements of these enumeration algorithms rely on the use of bounds on path resources to discard partial solutions. The quality of the bounds determines the performance of the algorithm. The main contribution of this paper is to introduce a standard procedure to generate bounds on paths resources in a general setting which covers most resource constrained shortest path problems, among which stochastic versions. In that purpose, we introduce a generalization of the resource constrained shortest path problem where the resources are taken in a monoid. The resource of a path is the monoid sum of the resources of its arcs. The problem consists in finding a path whose resource minimizes a non-decreasing cost function of the path resource among the paths that respect a given constraint. Enumeration algorithms are generalized to this framework. We use lattice theory to provide polynomial procedures to find good quality bounds. These procedures solve a generalization of the algebraic path problem, where arc resources belong to a lattice ordered monoid. The practical efficiency of the approach is proved through an extensive numerical study on some deterministic and stochastic resource constrained shortest path problems.

Traffic assignment in a tolled network with heterogeneous values of time and budgets on toll

A new network equilibrium problem was proposed for evaluating tolled traffic networks with heterogeneous values of time (VoT) and budgets on toll (BoT). Both parameters are important socio-economic characteristics primarily determined by the income level, trip purpose, etc., which directly affect the route choice and network flow pattern. Simpler models that embrace either of the attributes appeared a few times in the literature, but no study has been devoted to including both of them in an integrated model. This paper presents a mathematical programming model and method to characterize and solve this problem. In the well-known Frank-Wolfe algorithm framework, the core solution complexity lies in how to load travel demand into the network in an efficient manner. For tackling this difficulty, we developed a three-stage network loading procedure: (1) a bicriterion label-correcting algorithm finds all nondominated paths in the first stage; (2) a Graham scan algorithm is used to quickly identify all extreme nondominated paths in the second stage; (3) a constrained parametric algorithm is used to assign appropriate demand proportions to all eligible extreme nondominated paths. The numerical result derived from an illustrative example clearly shows how heterogeneous BoT and VoT reshape the network flow pattern from the simple case characterized merely by user equilibrium.

An Exact Algorithm for the Green Vehicle Routing Problem

We propose an exact algorithm for solving the green vehicle routing problem (G-VRP). The G-VRP models the optimal routing of an alternative fuel vehicle fleet to serve a set of geographically scattered customers. Vehicles’ fuel autonomy and possible refueling stops en route are explicitly modeled and maximum duration constraints are imposed on each vehicle route. We model the G-VRP as a set partitioning problem in which columns represent feasible routes corresponding to simple circuits in a multigraph. Each node in the multigraph represents one customer and each arc between two customers represents a nondominated path through a set of refueling stations visited by a vehicle when traveling directly between the two customers. We strengthen the set partitioning formulation by adding valid inequalities including k-path cuts and describe a method for separating them. We provide computational results on benchmark instances showing that the algorithm can optimally solve instances with up to ∼110 customers. The online appendix is available at https://doi.org/10.1287/trsc.2016.0734 .

Reliable Mobile Ad-Hoc Network Routing Using Firefly Algorithm

Routing in Mobile Ad-hoc NETwork (MANET) is a contemporary graph problem that is solved using various shortest path search techniques. The routing algorithms employed in modern routers use deterministic algorithms that extract an exact nondominated set of solutions from the search space. The search efficiency of these algorithms is found to have an exponential time complexity in the worst case. Moreover this problem is a multi-objective optimization problem in nature for MANET and it is required to consider changing topology layout. This study attempts to employ a formulation incorporating objectives viz., delay, hopdistance, load, cost and reliability that has significant impact on network performance. Simulation with different random topologies has been carried out to illustrate the implementation of an exhaustive search algorithm and it is observed that the algorithm could handle small-scale networks limited to 15 nodes. A random search meta-heuristic that adopts the nature of firefly swarm has been proposed for larger networks to yield an approximated non-dominated path set. Firefly Algorithm is found to perform better than the exact algorithm in terms of scalability and computational time.

Skyline Path Queries With Aggregate Attributes

The skyline path query is a novel extension of skyline queries. A skyline path query retrieves a set of non-dominated paths from origin $s$ to destination $t$ . On a road network using multiple path criteria, such as the distance, travel time, and number of travelers on a path, this paper extends the concept of skyline path query by considering a new type of criteria referred to as the aggregate attribute of paths. The method used for calculating this type of criteria is very different from that of existing criteria, and this can have a notable effect on the processing of ordinary skyline path queries. This paper defines the aggregate attributes of paths, discusses the impact of aggregate attributes on skyline path queries, and proposes a novel index tree with an intelligent algorithm to find the skyline path while taking aggregate attributes into account. Experiments demonstrate the effectiveness and efficiency of the proposed algorithm.

Criteria for the a Priori Shortest Path Generation in Uncertain Time-Varying Transportation Networks

This paper proposes a new definition of uncertain time-varying network to capture the uncertain and dynamic characteristics of the network with discrete uncertain link travel times. To find the a priori non-dominated paths in this type of network, three comparison criteria based on the uncertain measure, namely, deterministic dominance rule, first-order uncertain dominance rule and uncertain expected value dominance rule, are proposed to generate non-dominated paths in a single time interval and a time period, as more than one path may exist between an origin and destination for a given departure time. The proposed comparison methods are then applied to solving a simple uncertain time-varying network. The computational results verify the efficiency of three dominance rules for finding non-dominated paths.

Study on mean-standard deviation shortest path problem in stochastic and time-dependent networks: A stochastic dominance based approach

This paper studies a mean-standard deviation shortest path model, also called travel time budget (TTB) model. A route’s TTB is defined as this route’s mean travel time plus a travel time margin, which is the route travel time’s standard deviation multiplied with a factor. The TTB model violates the Bellman’s Principle of Optimality (BPO), making it difficult to solve it in any large stochastic and time-dependent network. Moreover, it is found that if path travel time distributions are skewed, the conventional TTB model cannot reflect travelers’ heterogeneous risk-taking behavior in route choice. This paper proposes to use the upper or lower semi-standard deviation to replace the standard deviation in the conventional TTB model (the new models are called derived TTB models), because these derived TTB models can well capture such heterogeneous risk-taking behavior when the path travel time distributions are skewed. More importantly, this paper shows that the optimal solutions of these two derived TTB models must be non-dominated paths under some specific stochastic dominance (SD) rules. These finding opens the door to solve these derived TTB models efficiently in large stochastic and time-dependent networks. Numerical examples are presented to illustrate these findings.

Path based algorithms for metro network design

This paper proposes a practical methodology for the problem of designing a metro configuration under two criteria: population coverage and construction cost. It is assumed that a set of corridors defining a rough a priori geometric configuration is provided by the planners. The proposed algorithm consists of fine tuning the location of single alignments within each corridor. This is achieved by means of a bicriteria methodology that generates sets of non-dominated paths. These alignments are then combined to form a metro network by solving a bicriteria integer linear program. Extensive computational experiments confirm the efficiency of the proposed methodology.

Multi-objective path finding in stochastic time-dependent road networks using non-dominated sorting genetic algorithm

In this paper, a new multi-objective path-finding model is proposed to find optimal paths in road networks with time-dependent stochastic travel times. This study is motivated by the fact that different travelers usually have different route-choice preferences, often involving multiple conflicting criteria such as expected path travel time, variance of path travel time and so forth. However, most of the existing studies have only considered the expected value of path travel time as the sole decision criterion. In order to solve the multi-objective model, the non-dominated sorting genetic algorithm is employed and its parameters are tuned by the Taguchi method. Moreover, a dynamic n-point crossover operator is developed to enhance the search capability of the genetic algorithm. Experimental results on a grid network demonstrate that the proposed approach is able to provide a set of non-dominated paths from which travelers can choose their paths based on their attitudes toward travel time uncertainty. Statistical analysis confirms that the dynamic n-point crossover operator outperforms the traditional one-point crossover operator.