Discrete Network Design Research Articles

Solving discrete network design problem using disjunctive constraints

AbstractThis paper introduces a deterministic algorithm to solve the discrete network design problem (DNDP) efficiently. This non‐convex bilevel optimization problem is well‐known as an non deterministic polynomial (NP)‐hard problem in strategic transportation planning. The proposed algorithm optimizes budget allocation for large‐scale network improvements deterministically and with computational efficiency. It integrates disjunctive programming with an improved partial linearized subgradient method to enhance performance without significantly affecting solution quality. We evaluated our algorithm on the mid‐scale Sioux Falls and large‐scale Chicago networks. We assess the proposed algorithm's accuracy by examining the objective function's value, specifically the total travel time within the network. When tested on the mid‐scale Sioux Falls network, the algorithm achieved an average 46% improvement in computational efficiency, compared to the best‐performing method discussed in this paper, albeit with a 4.17% higher total travel time than the most accurate one, as the value of the objective function. In the application to the large‐scale Chicago network, the efficiency improved by an average of 99.48% while the total travel time experienced a 4.34% increase. These findings indicate that the deterministic algorithm proposed in this research improves the computational speed while presenting a limited trade‐off with solution precision. This deterministic approach offers a structured, predictable, and repeatable method for solving DNDP, which can advance transportation planning, particularly for large‐scale network applications where computational efficiency is paramount.

PANDA: A physarum-inspired algorithm to solve the multi-objective discrete network design problem

The Multi-Objective Discrete Network Design Problem (MO-DNDP) describes the optimisation of a transport network across multiple metrics. Due to the impractical processing times of exact solutions, heuristics have been harnessed to solve this problem. However, the scale of the networks, the set of candidate links and the metrics assessed have been limited. This research proposes a MO-DNDP that encapsulates 8 metrics across three real networks used as benchmarks scaling from 17 km2 to 41,850 km2 to encapsulate the complexity of optimising transport networks at different scales, traffic demands and constraint densities. To solve this more comprehensive MO-DNDP, this paper proposes the Physarum-inspired algorithm PANDA that improves upon the classical Physarum Solver algorithm by harnessing the attractant sensing and merging capabilities of the biological Physarum Polycephalum slime mould. PANDA is a first attempt at solving the complex MO-DNDP using the Physarum foraging model. PANDA consistently scored better than the classical Solver in the land use rating by circumventing landforms, built-up areas, and environmentally sensitive regions. Additionally, PANDA constructed networks that were on average 30 % better than the existing benchmark networks and 32 % better than the Physarum Solver networks. PANDA improved upon the benchmarks by constructing lower capacity roads that were optimised for the actual traffic demand between each origin–destination pair instead of shared arterials.

Mathematical program with equilibrium constraints approach with genetic algorithm for joint optimization of charging station location and discrete transport network design

ABSTRACT This paper focuses on the joint optimization of the charging station location problem (CSLP) and discrete network design problem (DNDP) in a transportation network. We present a variational inequality (VI) formulation to describe the user equilibrium (UE) state of gasoline vehicles (GVs) and electric vehicles (EVs). Based on the mixed-UE model, a mathematical program with equilibrium constraints (MPEC) model is formulated for integrating the decisions of deploying EV charging stations (EVCSs) and adding new links to minimize the total travel cost (TTC) of all vehicles. A modified genetic algorithm is developed to tackle the MPEC model with an adaptive path generation procedure to address the mixed-UE model. Finally, we conduct numerical experiments to identify the efficacy of the proposed models and algorithms. Specifically, we propose a two-step optimization model and explore a performance comparison between the joint and two-step optimization approaches, while the joint optimization exhibits superiority in minimizing the TTC.

A hybrid bacterial foraging – simulated annealing framework for improving road networks

Effective techniques for reducing traffic congestion include adding new edges to the road network or increasing the capacity of existing edges. To determine the optimal set of improvements that can be made to an existing road network, it is standard practise to solve a Discrete Network Design Problem. Due to the bi-level nature of its formulation, it is one of the most difficult problems to solve to optimality. The Bacterial Foraging Optimization technique has been investigated in a previous work to gauge its effectiveness in determining the n-optimal solution to the Discrete Network Design Problem. This concept is further developed in this paper by hybridizing the Bacterial Foraging Optimization algorithm with Simulated Annealing. In the elimination step, the metropolis criterion from Simulated Annealing ensures a more diverse search of the solution space. We also employ a modified candidate set of improvements that incorporates both edge and capacity additions. The effectiveness of the hybrid bacterial foraging optimization algorithm in determining n-optimal solutions to the Discrete Network Design Problem has been demonstrated by testing it on the 24-node, 76-arc Sioux Falls network.

Solving Discrete Network Design Problems Using Repairing Path Flows

Solving Discrete Network Design Problems Using Repairing Path Flows

A modified bacterial foraging algorithm for improving road networks

One of the challenges that transportation planners face is deciding which improvements to make to existing road networks. A solution methodology for this problem must consider multiple factors, such as motorist travel behaviour and budget constraints. This is commonly represented as a bi-level optimization problem called the Discrete Network Design Problem (DNDP). In this paper, we devise and implement a Modified Bacterial Foraging Optimization (MBFO) algorithm that provides the decision maker with n-optimal solutions to the DNDP. This enables transportation planners to evaluate and compare various improvements before deciding which ones to implement. The use of a binary mapping function facilitates the implementation of Bacterial Foraging Optimization (BFO) to solve the DNDP. The MBFO employs a modified chemotaxis step with a single mutation operation to represent the tumbling motion and a revised swimming operation to ensure a diverse search of the solution space. To rank and eliminate the solutions, a gain index that measures improvement in fitness function value is used. The MBFO is tested on the 24-node, 76-arc Sioux Falls network, and the results show that it can converge to the n-optimal solution.

Economic topology optimization of District Heating Networks using a pipe penalization approach

In the presented study, a pipe penalization approach for the economic topology optimization of District Heating Networks is proposed, drawing inspiration from density-based topology optimization. For District Heating Networks, the upfront investment is a crucial factor for the rollout of this technology. Today, the pipe routing is usually designed relying on a linearization of the underlying heat transport problem. This study proposes to solve the optimal pipe routing problem as a non-linear topology optimization problem, drawing inspiration from density-based topology optimization. The optimization problem is formulated around a non-linear heat transport model and minimizes a detailed net present value representation of the heating network cost. By relaxing the combinatorial problem of pipe placement, this approach remains scalable for large-scale applications. In a design study on a realistic medium-sized network with 160 houses, a strong influence of economic parameters on the optimal network topology was observed. For this case, the optimization algorithm converges to a discrete network topology and near-discrete pipe design in about 10 min by using the proposed intermediate pipe penalization strategy. The optimal discrete network design found by the algorithm showed to outperform simple rounding post-processing steps by up to 2.8% of their respective net present value.

Bi-objective design-for-control for improving the pressure management and resilience of water distribution networks

This paper investigates control and design-for-control strategies to improve the resilience of sectorized water distribution networks (WDN), while minimizing pressure induced pipe stress and leakage. Both evolutionary algorithms (EA) and gradient-based mathematical optimization approaches are investigated for the solution of the resulting large-scale non-linear (NLP) and bi-objective mixed-integer non-linear programs (BOMINLP). While EAs have been successfully applied to solve discrete network design problems for large-scale WDNs, gradient-based mathematical optimization methods are more computationally efficient when dealing with large search spaces associated with continuous variables in optimal network control problems. Considering the advantages of each method, we propose a sequential hybrid method for the optimal design-for-control of large-scale WDNs, where a refinement stage relying on gradient-based mathematical optimization is used to solve continuous optimal control problems corresponding to design solutions returned by an initial EA search. The proposed method is applied to compute the Pareto front of a bi-objective design-for-control problem for the operational network BWPnet, where we consider reopening closed connections between isolated supply areas. The results show that the considered design-for-control strategy increases the resilience of BWPnet while minimizing pressure induced leakage. Moreover, the refinement stage of the proposed hybrid method efficiently improves the coarse approximation computed by the initial EA search, returning a continuous and even Pareto front approximation.

Balancing traffic flow in the congested mass self-evacuation dynamic network under tight preparation budget: An Australian bushfire practice

Disaster preparation budget plays a key role in the success of an emergency response operation. This research presents a novel bi-level model for mass self-evacuation discrete network design problems to find the best balance between the preparation budget and evacuation congestion. In this model, the importance of including non-compliance behavior in evacuation is considered. The proposed model aims to design an evacuation network with the smoothest traffic flow by using the maximum capacity of a dynamic transportation network where a number of roads become inaccessible as the bushfire spreads. The output of this research aids the emergency authorities to decide the best roads and shelters to equip in the evacuation process considering the access orders of the roads. The proposed approach covers the last four key phases of the entire evacuation process. Benders Decomposition and heuristics accelerators have been employed to help generate the outputs in less computational time for large-scale instances. The model and the solution approach have been validated through numerical experiments in different sizes. In addition, the model has been tested on the real case of the Churchill bushfire, Victoria. The results show that altering a number of factors such as budget and the number of allowable shelters will significantly reduce the overall evacuation time and makes the roads less congested. A sensitivity analysis is also conducted which demonstrates that it is possible to reduce traffic congestion without raising the budget.

BO-B&B: A hybrid algorithm based on Bayesian optimization and branch-and-bound for discrete network design problems

<abstract> <p>A discrete network design problem (DNDP) is conventionally formulated as an analytical bi-level programming problem to acquire an optimal network design strategy for an existing traffic network. In recent years, multimodal network design problems have benefited from simulation-based models. The nonconvexity and implicity of bi-level DNDPs make it challenging to obtain an optimal solution, especially for simulation-related models. Bayesian optimization (BO) has been proven to be an effective method for optimizing the costly black-box functions of simulation-based continuous network design problems. However, there are only discrete inputs in DNDPs, which cannot be processed using standard BO algorithms. To address this issue, we develop a hybrid method (BO-B&B) that combines Bayesian optimization and a branch-and-bound algorithm to deal with discrete variables. The proposed algorithm exploits the advantages of the cutting-edge machine-learning parameter-tuning technique and the exact mathematical optimization method, thereby balancing efficiency and accuracy. Our experimental results show that the proposed method outperforms benchmarking discrete optimization heuristics for simulation-based DNDPs in terms of total computational time. Thus, BO-B&B can potentially aid decision makers in mapping practical network design schemes for large-scale networks.</p> </abstract>

Optimization of Multimodal Discrete Network Design Problems Based on Super Networks

In this paper, we investigate the multimodal discrete network design problem that simultaneously optimizes the car, bus, and rail transit network, in which inter-modal transfers are achieved by slow traffic modes including walking and bike-sharing. Specifically, a super network topology is presented to signify the modal interactions. Then, the generalized cost formulas of each type of links in the super network are defined. And based on the above formulas a bi-objective programming model is proposed to minimize the network operation cost and construction cost with traffic flow equilibrium constraints, investment constraints and expansion constraints. Moreover, a hybrid heuristic algorithm that combines the minimum cost flow algorithm and simulated annealing algorithm is presented to solve the proposed model. Finally, the effectiveness of the proposed model and algorithm is evaluated through two numerical tests: a simple test network and an actual multimodal transport network.

Multi-period transportation network investment decision making and policy implications using econometric framework

Transportation infrastructure projects take numerous years of planning before they are scheduled for construction. Prioritization of such projects over a multi-period planning horizon (under a limited budget) is a difficult task, as it is usually formulated as a bilevel network design problem (NDP). Although multi-period network investment is studied in the literature, its application by public agencies is limited because of the complexities involved in network design problems and the computational time needed to analyze large scale networks (i.e., performing the traffic assignment at the lower level of the NDP). The contribution of this research is two-fold. First, it extends a previously published single year discrete network design formulation to a multi-period discrete network design problem (MPNDP) to capture both the spatial and temporal patterns of multi-period network investment decisions. Second, using the MPNDP investment results; and the network characteristics, this research develops and evaluates a new econometric model the Multi-Period Econometric Network Investment Model (MENIM). MENIM can be used by agencies in place of MPNDP to approximate network investments. The proposed model is calibrated and validated using medium to large scale networks and results show that it provides comparable results to MPNDP within acceptable computational times. Patterns of multi-period network investments from these numerical experiments are also extensively discussed along with policy recommendations for public agencies.

Benders Decomposition for the Profit Maximizing Capacitated Hub Location Problem with Multiple Demand Classes

This paper models the profit maximizing capacitated hub location problem with multiple demand classes to determine an optimal hub network structure that allocates available capacities of hubs to satisfy demand for commodities from different market segments. A strong deterministic formulation of the problem is presented, and a Benders reformulation is described to optimally solve large-size instances of the problem. A new two-phase methodology is developed to decompose the Benders subproblem, and two effective separation routines are derived to strengthen the Benders optimality cuts. The algorithm is enhanced by the integration of improved variable-fixing techniques. The deterministic model is further extended by considering uncertainty associated with the demand to develop a two-stage stochastic program. To solve the stochastic version, a Monte Carlo simulation–based algorithm is developed that integrates a sample average approximation scheme with the proposed Benders decomposition algorithm. Novel acceleration techniques are presented to improve the convergence of the algorithm proposed for the stochastic version. The efficiency and robustness of the algorithms are evaluated through extensive computational experiments. Computational results show that large-scale instances with up to 500 nodes and three demand classes can be solved to optimality, and that the proposed separation routines generate cuts that provide significant speedups compared with using Pareto-optimal cuts. The developed two-phase methodology for solving the Benders subproblem as well as the variable-fixing and acceleration techniques can be used to solve other discrete location and network design problems.

Computational benchmarking of exact methods for the bilevel discrete network design problem

The discrete network design problem (DNDP) is a well-studied bilevel optimization problem in transportation. The goal of the DNDP is to identify the optimal set of candidate links (or projects) to be added to the network while accounting for users’ reaction as governed by a traffic equilibrium. Several approaches have been proposed to solve the DNDP exactly using single-level, mixed-integer programming reformulations, linear approximations of link travel time functions, relaxations and decompositions. To date, the largest DNDP instances solved to optimality remain of small scale and existing algorithms are no match to solve realistic problem instances involving large numbers of candidate projects. In this work, we examine the literature on exact methodologies for the DNDP and attempt to categorize the main approaches employed. We introduce a new set of benchmarking instances for the DNDP and implement three solution methods to compare computational performance and outline potential directions for improvement. For reproducibility purposes and to promote further research on this challenging bilevel optimization problem, all implementation codes and instance data are provided in a publicly available repository.

Bio-inspired metaheuristics for the generalised discrete cost multicommodity network design problem

We investigate a new variant of network design problems (NDPs) called the generalised discrete cost multicommodity network design problem (GDCMNDP) that arises in a wide variety of real-life situations such as transportation, telecommunication and logistics. The problem consists on identifying the optimal capacitated network by choosing the connections to be installed in order to satisfy partially or totally the multicommodity demands. The objective is to minimise the sum of installation costs and penalty costs due to the unrouted demands. For the GDCMNDP, we propose three basic greedy heuristics and three bio-inspired metaheuristics: a basic genetic algorithm, a hybrid genetic algorithm via a variable neighbourhood search procedure and a biogeography-based optimisation heuristic. To assess the performance of the proposed approaches, computational results are reported using real-world and benchmark instances from the literature. Computational results show that our hybrid genetic algorithm performs well by obtaining very good final solutions in reasonable times.

Land use oriented bi-level discrete road network design

Abstract Although there is a broad consensus that integrating land use and transport will facilitate sustainable development, land use element is barely considered in network design problem. This paper proposes accessibility to quantitatively describe transport characteristics of a location. With assumptions about the relationship between land use/land cover types and accessibility, this paper introduces a bi-level programming model for land use oriented discrete network design problem (DNDP) and develops a genetic algorithm (GA)-based solution procedure which incorporates Frank-Wolf algorithm for user equilibrium (UE) assignment and Dijkstra algorithm for accessibility measurement. A numerical example is provided to demonstrate the applicability of the model and algorithm. The results indicate that the accessibility of the target traffic zone is improved to meet the demands of commercial areas.

Bio-inspired metaheuristics for the generalised discrete cost multicommodity network design problem

Bio-inspired metaheuristics for the generalised discrete cost multicommodity network design problem

System optimal relaxation and Benders decomposition algorithm for the large-sized road network design problem

Given a set of candidate road projects with associated costs, finding the best subset with respect to a limited budget is known as the discrete network design problem (DNDP). The DNDP is often characterised as a bilevel programming problem which is known to be NP-hard. Despite a plethora of research, due to the combinatorial complexity, the literature addressing this problem for large-sized networks is scarce. To this end, we first transform the bilevel problem into a single-level problem by relaxing it to a system-optimal traffic flow. As such, the problem turns to be a mixed integer nonlinear programming (MINLP) problem. Secondly, we develop an efficient Benders decomposition algorithm to solve the ensuing MINLP problem. The proposed methodology is applied to three examples, a pedagogical network, Sioux Falls and a real-size network representing the City of Winnipeg, Canada. Numerical tests on the network of Winnipeg at various budget levels demonstrate promising results.

Road Maintenance Optimization Model Based on Dynamic Programming in Urban Traffic Network

Urban road maintenance is an important part of urban traffic management. However, in modern cities, road maintenance work needs to occupy some traffic resources; therefore, unreasonable road maintenance schemes often lead traffic networks to unexpected large-scale congestion. In this paper, a dynamic programming model is proposed in order to minimize the delay caused by road maintenance scheme. This model can obtain a globally optimal maintenance scheme which contains the decisions and sequence for every stage of maintenance. Each stage of this model can be boiled down to a discrete network design problem. This model helps make suggestions for the traffic managers with the request of minimizing the delay caused by the maintenance scheme. This paper uses two examples to illustrate this method, one is a small-scale Nguyen-Dupuis network, and the other one is a larger scale Sioux-Falls network.

Integrating link-based discrete credit charging scheme into discrete network design problem

In this paper, we present an optimization model for integrating link-based discrete credit charging scheme into the discrete network design problem, to improve the transport performance from the perspectives of both transport network planning and travel demand management. The proposed model is a mixed-integer nonlinear bilevel programming problem, which includes an upper level problem for the transport authority and a lower level problem for the network users. The lower level sub-model is the traffic network user equilibrium (UE) formulation for a given network design strategy determined by the upper level problem. The network user at the lower level tries to minimize his/her own generalized travel cost (including both the travel time and the value of the credit charged for using the link) by choosing his/her route. While the transport authority at the upper level tries to find the optimal number of lanes and credit charging level with their locations to minimize the total system travel time (or maximize the transportation system performance). A genetic algorithm is used to solve the proposed mixed-integer nonlinear bilevel programming problem. Numerical experiments show the efficiency of the proposed model for traffic congestion mitigation, reveal that interaction effects across the tradable credit scheme and the discrete network design problem which amplify their individual effects. Moreover, the integrated model can achieve better performance than the sequential decision problems.