Multicommodity Network Flow Research Articles

Lagrangian relaxation-based decomposition approaches for the capacitated arc routing problem in the state-space-time network

ABSTRACT We study the capacitated arc routing problem in the state-space-time network. A multi-commodity network flow model is formulated in the state-space-time network. Two Lagrangian relaxation-based decomposition methods are developed to tackle this problem. By dualizing the coupling constraints to the objective function, the Lagrangian relaxed problem (LR) can be decomposed into several routing subproblems in the state-space-time network. Then we extend quadratic penalty terms to the LR and obtain an augmented Lagrangian relaxed problem (ALR). By applying the alternating direction method of multipliers and linearization techniques, we decompose the ALR into a sequence of routing subproblems. We solve these subproblems of LR and ALR by a time-dependent dynamic programming algorithm. Feasible solutions are produced according to the result of the LR or ALR. The Lagrangian multipliers are updated by the subgradient optimization method. We implement the proposed methods in three networks, showing that the ALR-based method has better performance.

A Time-Space Network-Based Optimization Method for Scheduling Depot Drivers

The driver scheduling problem at Chinese electric multiple-unit train depots becomes more and more difficult in practice and is studied in very little research. This paper focuses on defining, modeling, and solving the depot driver scheduling problem which can determine driver size and driver schedule simultaneously. To solve this problem, we first construct a time-space network based on which we formulate the problem as a minimum-cost multi-commodity network flow problem. We then develop a Lagrangian relaxation heuristic to solve this network flow problem, where the upper bound heuristic is a two-phase method consisting of a greedy heuristic and a local search method. We conduct a computational study to test the effectiveness of our Lagrangian relaxation heuristic. The computational results also report the significance of the ratio of driver size to task size in the depot.

An Exact Algorithm for Multi-Task Large-Scale Inter-Satellite Routing Problem with Time Windows and Capacity Constraints

In the context of a low-orbit mega constellation network, we consider the large-scale inter-satellite routing problem with time windows and capacity constraints (ISRPTWC) with the goal of minimizing the total consumption cost, including transmission, resource consumption, and other environmentally impacted costs. Initially, we develop an integer linear programming model for ISRPTWC. However, a difficult issue when solving ISRPTWC is how to deal with complex time window constraints and how to reduce congestion and meet transmission capacity. Along this line, we construct a three-dimensional time-space state network aiming to comprehensively enumerate the satellite network state at any moment in time and a task transmission route at any given time and further propose a time-discretized multi-commodity network flow model for the ISRPTWC. Then, we adopt a dynamic programming algorithm to solve the single-task ISRPTWC. By utilizing a Lagrangian relaxation algorithm, the primal multi-task routing problem is decomposed into a sequence of single-task routing subproblems, with Lagrangian multipliers for individual task route nodes and links being updated by a subgradient method. Notably, we devise a novel idea for constructing the upper bound of the ISRPTWC. Finally, a case study using illustrative and real-world mega constellation networks is performed to demonstrate the effectiveness of the proposed algorithm.

Consensus-based Dantzig-Wolfe decomposition

Dantzig-Wolfe decomposition (DWD) is a classical algorithm for solving large-scale linear programs whose constraint matrix involves a set of independent blocks coupled with a set of linking rows. The algorithm decomposes such a model into a master problem and a set of independent subproblems that can be solved in a distributed manner. In a typical implementation, the master problem is solved centrally. In certain settings, solving the master problem centrally is undesirable or infeasible, such as in the case of decentralized storage of data, or when independent agents who are responsible for the subproblems desire privacy of information. In this paper, we propose a fully distributed DWD algorithm which relies on solving the master problem using a consensus-based Alternating Direction Method of Multipliers (ADMM) method. We derive error bounds on the optimality gap and feasibility violation of the proposed approach. We provide preliminary computational results for our algorithm using a Message Passing Interface implementation on a delivery planning problem, the multi-commodity network flow problem, and synthetic instances where we obtain high quality solutions. An open-source implementation of the algorithm is available.

Multi-Objective Gate Allocation Problem Based on Multi-Commodity Network Flow Model

Gate allocation has always been a fundamental but critical issue in the daily operation of airports, which is related to service quality and schedule efficiency. In order to obtain reasonable and efficient gate allocation results, in this paper, a multi-commodity network flow model is proposed to describe the gate allocation process in flight flow, based on which a multi-objective optimization model is constructed. It not only comprehensively considers the flight information of aircraft arrivals and departures, but also integrates the broader interests of passengers, airlines, and airports. To solve it, a linear weighting technique is applied. In addition, K-means cluster analysis is used to explore different weight combinations, and on this basis, the idle time of the gate is introduced as a performance evaluation index to guide the selection of the final weight. By analyzing the optimization results of actual operation data, the proposed model significantly balances the interests of multiple parties and the number of flights at each gate and has a relatively high gate-utilization rate. It can provide rich decision support and a reasonable allocation scheme for airport management to a certain extent.

Train shunting with service scheduling in a high-speed railway depot

At high-speed railways, trains cover services during the day and are required to undergo maintenance at depots each night. A low-quality train schedule in the depot may result in delays in the availability of trains during the day which influences the reliability of the train timetables. Accordingly, this study examines the problem of train shunting with service scheduling in a depot where daily maintenance, cleaning operation, and safety operational requirements are considered. To cope with this complex problem, we first construct a two-layer time–space network in which each layer can only be used by trains traveling in the same direction. We then formulate the considered problem as a minimum-cost multi-commodity network flow model with incompatible arc sets and operational constraints. To solve the network flow problem, we present a Lagrangian relaxation heuristic. Finally, several computational experiments with practical data based on the Hefei–Nan depot and randomly generated data on trains’ arrival and departure times at the depot are conducted to confirm the effectiveness of our model and the efficiency of the proposed heuristics.

Customized bus scheme design of large transport terminals with jointly optimization of departure time, vehicle allocation and routing

The customized bus (CB) service of large transport terminals can provide passengers with convenient transfers and door-to-door services, which has the potential to help ease the pressure of arriving passenger flow in large transport terminals. Considering the costs of operators and passengers, this paper establishes a multi-commodity network flow optimization model to simultaneously obtain departure schedule, passenger-to-vehicle assignment and routing under the condition of multiple vehicle types. It provides more passengers with high-quality CB service and obtains higher vehicle capacity utilization. Accordingly, the solution is proposed that combines a branch-and-cut algorithm with grid-density-based clustering. An illustrative example verifies the effectiveness and tests the impact of the departure time-window. The case study of Chengdu city compares the results under different types of vehicle schemes and conducts the sensitivity analysis of penalty and minimum load ratio. The experiments conclude: (i) The proposed algorithm stably improves computation speed (76.38%) without affecting the optimal results; (ii) a loose time-window can serve more people, but the excessively loose one no longer has impacts on the results. (iii) The multi-type of vehicle scheme performs best compared to the single type of vehicle schemes.

A dynamic scheduling process and methodology using route deviations and synchronized passenger transfers for flexible feeder transit services

The general lack of first/last mile connectivity is one of the main challenges encountered by intercity transport systems. New transportation alternatives called flexible transit systems, which combine on-demand service adjustment capabilities to the regular route and master schedule characteristics of the conventional transit, are therefore increasingly proposed. Feeder transit services can run as one type of flexible transit services, connecting a service area to a major intercity transport network through a transfer hub. Considering the unstable and constantly changing characteristics of daily passenger demand, a dynamic scheduling methodology for feeder transit services is proposed to minimize passenger crowding in vehicles. Deviations in the base route are allowed for accommodating optional requests or visiting potential transfer locations, which further offer a new type of flexible scheduling operation called synchronized passenger transfer. Operators can dynamically expand the spatial coverage of vehicles out of their regular routes to meet each other at appointed locations and times. Then, passengers can transfer between vehicles to minimize or even avoid passenger crowding. In this study, a multi-commodity network flow optimization model with side constraints in a three-dimensional space–time–state network is proposed for this type of dynamic scheduling problem. Additionally, a coordinate descent-based solution framework consisting of three steps (augmentation, decomposition, and linearization) is developed. The proposed model and solution approach are investigated using an eight-node simple network and the Sioux-Falls network, and the Chicago sketch network was adopted to demonstrate the practicability of the proposed model.

Optimal Decision Making for Fractional Multi-commodity Network Flow Problem in a Multi-choice Fuzzy Stochastic Hybrid Environment

In this paper, a fractional multi-commodity network flow problem with multi-choice parameters is studied under hybrid fuzzy-stochastic conditions. In this problem, coefficients of the objective function in both the nominator and denominator take the form of multi-choice parameters, with the alternative choices for the nominator and denominator of the fraction being represented by fuzzy-stochastic and fuzzy variables, respectively. The arc capacities are also considered as fuzzy-stochastic variables. The main goal of the present research is to provide the decision-maker with a model by the help of which he/she can manage unknown factors across a multi-commodity network. Given that this problem is herein investigated in a hybrid fuzzy-stochastic environment and includes multi-choice parameters, we use the probability-possibility approach, Lagrange interpolating polynomial, and the Charnes–Cooper variable transformation technique to convert the problem into a deterministic one. Finally, efficiency of the proposed model is evaluated by presenting a couple of numerical instances.

Integrated electric logistics vehicle recharging station location–routing problem with mixed backhauls and recharging strategies

The widespread application of electric vehicles for city last-mile logistics has been enhanced by the emerging trend of urban sustainable mobility, which intends to reduce vehicle emissions and the dependence on fossil fuels. The recharging facility location is critical for electric logistics network planning, because it significantly affects future operation costs and efficiency. This study proposes an integrated electric logistics vehicle recharging station location–routing problem with mixed backhauls and recharging strategies, which is formulated as a time-discretized multicommodity network flow optimization model based on a space–time–state–energy representation network. This study aims to optimize the selection of the recharging station location considering route planning with complicated constraints of recharging capacity, facility construction budget, vehicle loading capacity, battery remaining capacity, spatial structure of real road networks, mixed pickup and delivery requests, and service time windows. A hybrid Lagrangian relaxation and alternating direction method of multipliers (LR-ADMM) decomposition solution framework is constructed to decouple the proposed integrated problem into a recharging station location problem for strategic planning and an electric vehicle routing problem with mixed backhauls, time windows, and recharging strategies for operational decisions. These two subproblems are solved alternately by time-dependent forward dynamic programming algorithms embedded into the LR-ADMM framework. The solution quality is guaranteed by calculating the optimality gap between the best lower and upper bounds for each iteration. The experimental results based on the Sioux-Falls network and real-world West Jordan network prove the computational effectiveness and optimization quality of the proposed solution approach.

Network flow based approaches for the pipelines routing problem in naval design

In this paper we propose a general methodology for the optimal automatic routing of spatial pipelines motivated by a recent collaboration with Ghenova, a leading Naval Engineering company. We provide a minimum cost multicommodity network flow based model for the problem incorporating all the technical requirements for a feasible pipeline routing. A branch-and-cut approach is designed and different matheuristic algorithms are derived for solving efficiently the problem. We report the results of a battery of computational experiments to assess the problem performance as well as a case study of a real-world naval instance provided by our partner company.

A Multi-Commodity Mathematical Modelling Approach—Hazardous Waste Treatment Infrastructure Planning in the Czech Republic

This paper presents an analysis of infrastructure for the processing of Czech hazardous waste and pays attention to predictions of waste management development in the upcoming years. For this purpose, a unique complex approach to modelling future waste management changes is applied. The method uses a multi-commodity network flow model with reverse flows between treatment facilities to consider complete waste management of hazardous waste. The future outlook (2030) for the forecasted generation of different types of hazardous waste in the Czech Republic requires decisions on waste treatment facility infrastructure. The uniqueness lies in using real data for such a wide scope of a task, further enhanced by concurrent analysis of more types of waste interconnected through limited processing capacities. The results indicate the insufficiency in hazardous waste thermal treatment and stabilization. A suggestion is to extend the incineration capacity because it influences the stabilization units, which must process the remaining waste. The recommended increase is 100% with different proportions in individual regions.

Deep Reinforcement Learning-Based Charging Pricing for Autonomous Mobility-on-Demand System

The autonomous mobility-on-demand (AMoD) system plays an important role in the urban transportation system. The charging behavior of AMoD fleet becomes a critical link between charging system and transportation system. In this paper, we investigate a strategic charging pricing scheme for charging station operators (CSOs) based on a non-cooperative Stackelberg game framework. The Stackelberg equilibrium investigates the pricing competition among multiple CSOs, and explores the nexus between the CSOs and AMoD operator. In the proposed framework, the responsive behavior of AMoD operator (order-serving, repositioning, and charging) is formulated as a multi-commodity network flow model to solve an energy-aware traffic flow problem. Meanwhile, a soft actor-critic based multi-agent deep reinforcement learning algorithm is developed to solve the proposed equilibrium framework while considering privacy-conservation constraints among CSOs. A numerical case study with city-scale real-world data is used to validate the effectiveness of the proposed framework.

Flight Scheduling and Fleet Sizing for an Airport Shuttle Air Taxi Service

A multicommodity network flow framework for optimal flight scheduling for an envisioned airport shuttle air taxi service is presented. The framework includes a trip generation model that simulates requests from individuals who travel to and from the airport and are willing to pay for the travel time savings associated with an airport shuttle air taxi service. The framework can be used to determine the fleet size required to satisfy a given user demand and to assess how an operator’s profitability can be influenced by multiple factors, including the population density and income distribution in a metropolitan area, fleet size, ticket price, and air taxi operating costs. The results for an example business airport shuttle service to and from Atlanta Hartsfield Jackson International Airport suggest that a viable business case is possible using electric vertical take-off and landing aircraft. For example, the results for this Atlanta case study indicate that it is possible to have profitable business use case with positive cash flow with a four-seat aircraft with a 60-mile design range and a 150 mph cruise speed using a fleet size of five aircraft at a price point of $5 per passenger-mile during near-term small-scale operations.

A hybrid fuzzy stochastic model for fractional multi-commodity network flow problems

A hybrid fuzzy stochastic model for fractional multi-commodity network flow problems

Risk-averse two-stage distributionally robust optimisation for logistics planning in disaster relief management

Relief logistics is vital to disaster relief management. Herein, a risk-averse two-stage distributionally robust programming model is proposed to provide decision support for planning disaster relief logistics. It is distinct from the conventional disaster relief logistics planning problem in that (i) the facility location-inventory model and the multi-commodity network flow formulation are integrated; (ii) the probability distribution information of the supply, demand, and road link capacity is partially known, and (iii) the two-stage distributionally robust optimisation (DRO) method based on the worst-case mean-conditional value-at-risk criterion is developed. For tractability, we reformulate the proposed DRO model as equivalent mixed-integer linear programs for box and polyhedral ambiguity sets, which can be directly solved to optimality using the CPLEX software. To evaluate the validity of the proposed DRO model, we conduct numerical experiments based on a real-world case study addressing hurricane threats in the Gulf of Mexico region of the United States. Furthermore, we compare the performance of the proposed DRO model with that of the conventional two-stage stochastic programming model. Finally, we report the managerial implications and insights of using the risk-averse two-stage DRO approach for disaster relief management.

Integrated line planning and train timetabling through price-based cross-resolution feedback mechanism

Railway line planning and train timetabling are two key planning steps that determine the operating cost and passenger service quality of a railway operator under the infrastructure capacity limitations. Traditionally, the line planning and train timetabling problems are solved sequentially at the strategic and tactical level, respectively. In this study, by introducing two types of binary decision variables, we first propose a unified integer linear programming (ILP) model for the integrated optimization of line planning and train timetabling. The line planning problem is modeled using ILP to satisfy passenger demand, whereas the cyclic train timetabling problem is formulated as a multi-commodity network flow model with a side track capacity constraint. The two types of binary decision variables are coupled by a cross-resolution consistency constraint, which ensures the conformity of the line planning and train timetabling decisions. Furthermore, a dual decomposition mechanism based on the Alternating Direction Method of Multipliers (ADMM) is developed to dualize the cross-resolution consistency and track capacity constraints, such that the original ILP model is decomposed into a line planning sub-problem and a set of train-specific sub-problems. After the linearization of the quadratic penalty terms in the ADMM, each sub-problem contains the Lagrangian relaxation price information based on the cross-resolution consistency constraint. Moreover, the primal and dual solutions are obtained by iteratively and efficiently solving the line planning sub-problem using a commercial solver, and each train-specific sub-problem through a tailored forward dynamic programming algorithm. Furthermore, a real-life case study is conducted based on the Beijing–Shanghai high-speed railway corridor to verify the efficiency and effectiveness of the proposed model and algorithm. The results of the numerical experiments demonstrate that the ADMM can achieve significantly smaller optimality gaps than Lagrangian relaxation, and the integrated optimization approach can improve the objective value by 5.78% on average compared with the sequential optimization approach.

Integrated Optimization of Port Rotation Direction and Fleet Deployment for Container Liner Shipping Routes

This paper provides an integrated planning methodology for the optimization of port rotation direction and fleet deployment for container liner shipping routes with consideration of demand uncertainty. We first consider a special case that demand is deterministic. A multicommodity flow network model is developed via minimizing the total network-wide cost. Its decisions are the selection of port rotation direction and fleet deployment and container routings in the shipping network. Afterward, we address the generic case that uncertain demand is considered, which is represented by potentially realizable demand scenarios. We develop a minimax regret model to procure the least maximum regret across all the demand scenarios. The proposed models are applied to an Asia-Europe-Oceania liner shipping network with 46 ports and 12 ship routes. Results could provide the liner company with a comprehensive decision tool to simultaneously determine port rotation direction and fleet deployment when tackling uncertain demand.

New partial aggregations for multicommodity network flow problems: An application to the fixed-charge network design problem

When solving hard multicommodity network flow problems using an LP-based approach, the number of commodities is a driving factor in the speed at which the LP can be solved, as it is linear in the number of constraints and variables. The conventional approach to improve the solve time of the LP relaxation of a Mixed Integer Programming (MIP) model that encodes such an instance is to aggregate all commodities that have the same origin or the same destination. However, the bound of the resulting LP relaxation can significantly worsen, which tempers the efficiency of aggregating techniques. In this paper, we introduce the concept of partial aggregation of commodities that aggregates commodities over a subset of the network instead of the conventional aggregation over the entire underlying network. This offers a high level of control on the trade-off between size of the aggregated MIP model and quality of its LP bound. We apply the concept of partial aggregation to two different MIP models for the multicommodity network design problem. Our computational study on benchmark instances confirms that the trade-off between solve time and LP bound can be controlled by the level of aggregation, and that choosing a good trade-off can allow us to solve the original large-scale problems faster than without aggregation or with full aggregation.

A multi-commodity network model for optimal quantum reversible circuit synthesis.

Quantum computing is a newly emerging computing environment that has recently attracted intense research interest in improving the output fidelity, fully utilizing its high computing power from both hardware and software perspectives. In particular, several attempts have been made to reduce the errors in quantum computing algorithms through the efficient synthesis of quantum circuits. In this study, we present an application of an optimization model for synthesizing quantum circuits with minimum implementation costs to lower the error rates by forming a simpler circuit. Our model has a unique structure that combines the arc-subset selection problem with a conventional multi-commodity network flow model. The model targets the circuit synthesis with multiple control Toffoli gates to implement Boolean reversible functions that are often used as a key component in many quantum algorithms. Compared to previous studies, the proposed model has a unifying yet straightforward structure for exploiting the operational characteristics of quantum gates. Our computational experiment shows the potential of the proposed model, obtaining quantum circuits with significantly lower quantum costs compared to prior studies. The proposed model is also applicable to various other fields where reversible logic is utilized, such as low-power computing, fault-tolerant designs, and DNA computing. In addition, our model can be applied to network-based problems, such as logistics distribution and time-stage network problems.