Multi-commodity Network Flow Model Research Articles

Code and Data Repository for A New Multicommodity Network Flow Model and Branch and Cut for Optimal Quantum Boolean Circuit Synthesis

This directory contains the entire set of result files that are reported in our paper.

A New Multicommodity Network Flow Model and Branch and Cut for Optimal Quantum Boolean Circuit Synthesis

This study introduces a new optimization model and a branch-and-cut approach for synthesizing optimal quantum circuits for reversible Boolean functions, which are pivotal components in quantum algorithms. Although heuristic algorithms have been extensively explored for quantum circuit synthesis, research on exact counterparts remains relatively limited. However, the need to design quantum circuits with guaranteed optimality is increasing, especially for improving computational fidelity on noisy intermediate-scale quantum devices. This study presents mathematical optimization as a viable option for optimal synthesis, with the potential to accommodate practical considerations arising in fast-evolving quantum technologies. We set a demonstrative problem to implement reversible Boolean functions using high-level gates known as multiple control Toffoli gates while minimizing a technology-based proxy called quantum cost—the number of low-level gates used to realize each high-level gate. To address this problem, we propose a discrete optimization model based on a multicommodity network and discuss potential future variations at an abstract level to incorporate technical considerations. A customized branch and cut is then developed upon different aspects of our model, including polyhedron integrality, surrogate constraints, and variable prioritization. Our experiments demonstrate the robustness of the proposed approach in finding cost-optimal circuits for all benchmark instances within a two-hour time frame. Furthermore, we present interesting intuitions from these experiments and compare our computational results with relevant studies, highlighting newly discovered circuits with the lowest quantum costs reported in this paper. History: Accepted by Giacomo Nannicini, Area Editor for Quantum Computing. This paper has been accepted for the INFORMS Journal on Computing Special Issue on Quantum Computing. Funding: This research was supported by the Ministry of Science and Information and Communication Technology, South Korea [Grants 2017R1E1A1A0307098814 and 2020R1A4A307986411]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0562 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0562 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

An optimization framework for efficient and sustainable logistics operations via transportation mode optimization and shipment consolidation: A case study for GE Gas Power

General Electric (GE) Gas Power, a leading manufacturer of gas and steam turbines, manufactures and installs these turbines in power generation plants worldwide. They source components for these turbines from suppliers globally and transport these components to manufacturing and assembly locations in the United States using various modes of transportation, including ocean, air, and ground. These transportation options have different lead times and costs. The challenge lies in identifying the most cost-effective solution that meets the assembly requirements, given the high volume of shipments and the complexity of the intermodal freight network. To address this challenge, a customized, multi-period (dynamic), multi-commodity network flow model and a novel heuristic approach with a rolling time horizon are developed. This model incorporates consolidation and storage options at intermediate nodes, allowing the business to optimize its shipments. This research addresses a unique real-world logistics problem and offers an optimization model together with an efficient solution approach for industry. Computational experiments indicate that the heuristic approach proposed can yield results closely aligned with those produced by state-of-the art commercial solvers. Furthermore, the experiments show the criticality of full container load shipments in achieving substantial cost-saving in global shipments.

Liner shipping network vulnerability to component disruptions: A China-Europe container flow analysis

This study conducts a vulnerability analysis of the China-Europe liner shipping network, including all relevant liner shipping services and China Railway Express services. The standard multicommodity network flow model is extended to assess network vulnerability and the relative importance of ports and services by simulating container flow redistribution in various disruption scenarios. The results show the network is highly resilient to both individual port disruptions and individual service disruptions. Additionally, the distinct shipping network structures of the three shipping alliances result in varying degrees of vulnerability, highlighting the significance of diversified market suppliers in enhancing the overall network resilience. Specifically, the Ocean Alliance is vital in facilitating China-Europe trade, especially along the Mediterranean route, and is more resilient to individual port disruptions.

Aggregation formulation for on‐site multidepot vehicle scheduling scenario

AbstractThe multidepot vehicle scheduling problem (MDVSP) is a fundamental public transport challenge. To address the large‐scale model and inherent solution symmetry associated with the traditional trip‐to‐trip connection‐based approach for MDVSP, a new trip‐to‐route (T2R) connection‐based approach is proposed. Considering real‐world problem characteristics with numerous trips sharing common origin–destination stations and travel times on one route, this approach aggregates same vehicle possible trip sequences into a T2R connection. Two time‐space network aggregation (TSNA) flow formulation versions, route pair‐based TSNA and station pair‐based TSNA, were constructed. Furthermore, TSNA equivalence under any given decomposition strategy, including first‐in‐first‐out, with the multicommodity network flow (MCNF) model was demonstrated. Given the favorable separable TSNA structure, an alternating direction method of multipliers (ADMM)‐based procedure is proposed to decompose the MDVSP into multiple subproblems that can be linearized and readily solved using commercial solvers. The quality of the solutions was assessed using lower bounds obtained from the Lagrangian relaxation problem. The effectiveness and superiority of the proposed MDVSP models and algorithms were subsequently confirmed using random data sets and real‐world instances.

Joint location optimization of charging stations and segments in the space-time-electricity network: An augmented Lagrangian relaxation and ADMM-based decomposition scheme

Electric vehicles that contribute to better air quality, less noise, and low-carbon emissions are a promising selection for sustainable transportation. However, the development of electric vehicles is impeded by various factors, including driving range anxiety, long recharging period, and insufficient charging facilities. As the charging-while-driving techniques gradually mature, electric vehicles can recharge on charging stations stationarily or charging lanes (segments) dynamically. Additional charging facilities should be constructed to improve the level of charging services. Under a predefined construction budget, the difficulty is how to determine their numbers and distributions. This study jointly locates charging stations and segments by maximizing the accessibility of electric vehicles. In space-time-electricity networks, we formulate a multi-commodity network flow model with location and routing. A decomposition scheme based on augmented Lagrangian relaxation and alternating direction method of multipliers is developed to tackle this problem. The standard and augmented Lagrangian relaxed problems are decomposed into many solvable subproblems. Numerical experiments are conducted in three transportation networks, showing that the proposed method can achieve good integrality gaps.

Network periodic train timetabling with integrated stop planning and passenger routing: A periodic time–space network construct and ADMM algorithm

Designing a passenger-centric railway service is of great importance for railway companies to attract more passengers and increase their revenue. Focusing on the passenger-centric operational plans, this paper considers the problem of integrated periodic train timetabling and train stop planning in high speed railway networks. In view of the immediate dependency of passengers’ routes on the train timetable and stop plan, simultaneous passenger routing is included to directly shape the train stops and timetable in a passenger-friendly way. Besides travel time, ticket price is also considered in passenger utility to better depict passengers’ route choices in the railway network. A railway network-based time–space network with train stop-skip technique, periodic properties and detailed consideration of different train headways is designed, based on which a multi-commodity network flow model is developed. With a novel linearization technique of train capacity penalty term, the alternating direction method of multipliers (ADMM) is used to relax and decompose the track and train capacity constraints. Further, time deviation of regular train services is achieved by a new shortest path algorithm to design more flexible timetables, however its application is still limited to small instances due to computational efforts required. The algorithm is finally tested on the Shandong high-speed railway network in China, and demonstrates its capability of providing tighter lower bounds than Lagrangian relaxation method and generating good feasible solutions. Besides, the impacts of different cycle times on service quality and train cost are evaluated.

Multicommodity network flow model of a human resource allocation problem considering time periods

This article addresses the problem of finding work assignments for employees within a given time horizon in a company using a multicommodity network flow model. The problem of human resource allocation is defined by the actual manpower demands of different periods which may vary during different periods. The investigation focuses on when workers should be called in-house and for how long to satisfy demands, while also complying with labour standards and regulations. Additional targets may also be set up, such as minimising the overall number of labour, as well as meeting “comfort” expectations, i.e. the most even working time should be realised for every worker within the event horizon. The paper describes how the multicommodity network flow model is constructed and the corresponding MILP mathematical programming model is formulated in a simple situation where there is only one position for the labour. Finally, the article explains how to construct the multicommodity network flow model and the MILP model for the general case, where there are multiple positions for the labour requiring various skills and competences per position within the periods.

Customised bus route design with passenger-to-station assignment optimisation

As an emerging and innovative public transportation mode, the customised bus (CB) has drawn widespread attention. Existing studies of the CB mainly focus on issues regarding the bus station location, route design, timetabling, and fare setting, but rarely consider the passenger to bus station assignment problem which can significantly influence passengers’ benefit and the buses’ operating scheme. Thus, this study focuses on the customised bus routing problem with passenger-to-station assignment (CBRP-PSA) aiming to simultaneously minimise passengers’ CB service access cost and the bus route cost. A time-discretized multi-commodity network flow model is developed to jointly determine the CB routes, timetables, and passenger-to-station assignment by allowing split loads and mixed loads. Through the dualization, the developed model is decomposed into two solvable sub-problems, and a Lagrangian-based heuristic solution algorithm is proposed. The model and algorithm are implemented in illustrative, medium-scale, and large-scale transportation networks to demonstrate their effectiveness under different scenarios.

A stabilizing benders decomposition method for the accessibility-oriented charging station location problem

Electric vehicles are a promising selection to deal with climate change and energy crises. However, the development of electric vehicles is impeded by various barriers including shortened driving range, lack of charging facilities, and long recharging periods. Setting additional charging facilities is essential to construct an efficient charging network. Previous studies neglected accessibility-oriented indicators to locate charging stations. This study locates charging stations by maximizing the space-time-electricity accessibility of electric vehicle travelers. For the accessibility-oriented charging station location problem, we formulate a multi-commodity network flow model in the space-time-electricity network. Then, we reformulate this model into a two-stage integer program and develop a Benders decomposition method with stabilizing techniques. At each iteration, a relaxed Benders master problem (RMP) and a set of dual routing subproblems (DSPs) are solved. The RMP provides an ascending lower bound and a feasible charging station location scheme. These DSPs generate a set of Benders optimality cuts for RMP and a feasible solution to update the upper bound. All instances in three transportation networks are solved optimally, showing that the stabilizing techniques can significantly improve the performance of the standard Benders decomposition method.

Integrated electric bus timetabling and scheduling problem

Vehicle timetabling and scheduling in a public transit system are usually performed separately, with the output of timetabling serving as the input of scheduling. An obvious drawback of this sequential planning method is that the trade-off between bus timetables and vehicle schedules may be neglected when determining solutions, which in turn results in that the obtained solutions may be inferior to those produced using an integrated framework. For example, a well-planned timetable may result in a schedule that requires a large vehicle fleet size with more operational cost, while a well-planned schedule may reduce the quality of a bus timetable by limiting the use of vehicles. In this paper, we introduce a time-space network-based framework for integrating electric bus timetabling and scheduling, with minimum and maximum headway times, depot requirements, deadheading and vehicle battery capacities considerations. The underlying time-space network is constructed with well-designed inventory arcs that represent multiple operations a bus may execute, thus decreasing the network size. Using the constructed network, we formulate the considered problem with a multi-commodity network flow model and develop a Lagrangian relaxation heuristic that consists of three phases, including generating relaxed solutions, making relaxed solutions feasible, and improving feasible solutions, to solve the integrated model. Tests on a set of instances confirm that the proposed integrated solution method can efficiently produce bus timetables and schedules with valid bounds, indicate that the integrated method can produce better solutions where the profit is increased by 5.29%–20.28%, and show how the headway times, service trip profit and operating cost settings affect the solution.

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.

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.

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.

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.

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.

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.

Optimizing electric vehicle routing problems with mixed backhauls and recharging strategies in multi-dimensional representation network

Electric vehicles are environmental transportation modes that are widely applied in green logistics systems. To guarantee the energy efficiency, the impacts of customer service modes and recharging strategies need to be integrated into the optimization of electric logistics resource. This paper proposes an electric vehicle routing problem with mixed backhauls, time windows, and recharging strategies (EVRPMBTW-RS), minimizing the total travel cost with sophisticated constraints on the time-dependent pickup and delivery requests, limited recharging station capacity, and battery remaining capacity of electric vehicles. Mixed service sequences of linehaul and backhaul customers is allocated for the routing planning, with the synchronous optimization of recharging strategies including the selection of recharging stations and determination of recharging time. A time-discretized multi-commodity network flow model is constructed based on an extended space–time-state modeling framework, which is formulated as a quadratic 0–1 programming model by using the augmented Lagrangian relaxation technique. After the dualization and linearized transformation, we decompose the model into a sequence of least-cost path subproblems based on the alternating direction multiplier method (ADMM). The subproblems are alternately minimized and solved using the time-dependent forward dynamic programming algorithm. The solution quality can be guaranteed through calculating the optimality gap between the best lower bound and upper bound for each iteration. The proposed solution approach is examined on examples of a simple 7-node network and real-world Yizhuang road network. This paper provides a theoretical foundation for the route optimization method of electric logistics vehicles, and contributes to improve the operational efficiency of electric logistics systems.