Multicommodity Network Flow Problem Research Articles

ECONOMIC BENEFITS OF DEPLOYING FOLDABLE CONTAINERS: REDUCING BUNKER AND CONTAINER MANAGEMENT COSTS IN A MULTI-PORT SHIPPING NETWORK

This study seeks to explore the effectiveness of employing foldable containers (FLDs) in liner shipping to reduce relocation and the empty containers and bunker costs (BCs) associated with ship operations. This resolves a minimum-cost multi-commodity network flow problem by optimizing container fleet size and empty container relocation in a multi-port shipping service network. Port handling time and sailing speed provided by obtained optimal solutions enable the determination of ship BCs as a secondary step. The numerical experiments demonstrate the comparative effect of FLDs against standard ones on the reduction of the costs of empty containers and containership bunker oil.

A novel method for solving the multi-commodity flow problem on evolving networks

The minimum cost multi-commodity flow (MMCF) problem on evolving networks is the latest challenge to solving the min-cost network flow problem of multi-commodity flowing from multi-source and multi-sink on an evolving network. Previous researches primarily focus on the solution of MMCF problem within static networks. However, researchers ignored the MMCF problem within networks where the structure dynamically changes over time. Evolving networks have a dynamically changing and unpredictable network structure. The existing methods must recalculate when the network structure changes, leading to high computational complexity. This manuscript proposes a dynamic graph computing model for adaptive dynamic path selection and optimization. First, the model selects the path with the least cost for each commodity, assuming the route can transport a specified number of such commodities to find the optimal path faster. Second, we design path selection and flow update strategies for each node in the graph to adapt to changes in the graph structure. At the same time, we can prevent multiple commodity conflicts from exceeding the marginal capacity limit. Finally, the optimal path of each commodity is dynamically pruned based on the original optical path to obtain the tree structure and determine the absolute path to meet the specified transportation quantity of each commodity. Performance analysis results show that our method can efficiently cope with the dynamic changes of the network structure, which provides ideas for solving the multi-commodity network flow problem in dynamic networks.

Redesigning large-scale multimodal transit networks with shared autonomous mobility services

This study addresses a large-scale multimodal transit network design problem, with Shared Autonomous Mobility Services (SAMS) as both transit feeders and an origin-to-destination mode. The framework captures spatial demand and modal characteristics, considers intermodal transfers and express services, determines transit infrastructure investment and path flows, and generates transit routes. A system-optimal multimodal transit network is designed with minimum total door-to-door generalized costs of users and operators, satisfying transit origin–destination demand within a pre-set infrastructure budget. Firstly, the geography, demand, and modes in each zone are characterized with continuous approximation. The decisions of network link investment and multimodal path flows in zonal connection optimization are formulated as a minimum-cost multi-commodity network flow (MCNF) problem and solved efficiently with a mixed-integer linear programming (MILP) solver. Subsequently, the route generation problem is solved by expanding the MCNF formulation to minimize intramodal transfers. The model is illustrated through a set of experiments with the Chicago network comprised of 50 zones and seven modes, under three scenarios. The computational results present savings in traveler journey time and operator cost demonstrating the potential benefits of collaboration between multimodal transit systems and SAMS.

Planning container inspection and repair: A case study

The aim of this paper is to introduce a real-life problem of optimal planning of container inspection and repair on several facilities over several time periods assuring effective supply of empty containers indispensable for seamless global maritime transportation. The containers are of several types and quality levels. The objective is to minimize the total holding, inspection, repair, transportation and rejection costs. We formulate a deterministic min-cost multi-commodity network flow problem and prove NP-hardness for two special cases. Values of the rejection costs are determined such that no container is rejected in an optimal solution if there exists a feasible solution with no container rejected. An algorithmic mechanism is proposed to support a negotiation of periodic container demands between the container provider and the container users with the aim to reduce the overall costs. Computer experiments with instances obtained by random deviation from industrial data demonstrate that the majority of instances with up to 4 facilities, 4 container types, 7 quality levels, and 7, 14 or 30 time periods can be solved with an acceptable quality using an academic version of CPLEX in two hours on a standard PC.

Optimization Approach to Rolling Stock Scheduling for Timetable Temporarily Changed due to Disaster

When railway operators devise a rolling stock schedule plan, it is desirable that the schedule plan for the new timetable is similar to the previous schedule. In this study, we focus on rolling stock scheduling following a large-scale natural disaster. Generally, partial suspensions occur some sections in the line from the damage caused by the disaster, so a temporary timetable is generated. Subsequently, a rolling stock schedule that is close to the previous one must be created. We propose a two-phase rolling stock scheduling algorithm for the temporarily changed timetable. The proposed scheduling algorithm creates a set of duties based on the multi-commodity network flow problem in the first phase. Following that, the algorithm provides a set of rosters by solving the travelling salesman problem. Finally, we confirm that the proposed algorithm can produce a good solution in terms of evaluation criteria and computational time.

Joint chance-constrained multi-objective multi-commodity minimum cost network flow problem with copula theory

This study focuses on a specific problem in network optimization, namely the minimum cost multi-commodity network flow (MCNF) problem. The problem is complicated by the presence of uncertain parameters, including various types of costs associated with each arc in the network. The study presents a multi-objective approach to solving this problem, where the coefficients of the capacity constraints are modelled as random variables with a normal distribution, and the dependence between them is modelled using an Archimedean copula. The capacity constraints are presented as joint chance constraints, and a multi-objective problem is formulated to deal with the uncertainty. This uncertain multi-objective problem is then converted into a certain multi-objective problem using fuzzy programming. The resulting certain multi-objective MCNF problem is converted to a certain single objective problem using second-order cone programming (SOCP), which is solved using either piecewise tangent approximation or piecewise linear approximation methods. The proposed approaches are tested using numerical examples and experimental tests to demonstrate their effectiveness in solving large-scale network problems efficiently. The results show that the proposed approaches are a useful tool for solving uncertain multi-objective MCNF problems in real-world applications.

The Truck Platooning Routing Optimization Model Based on Multicommodity Network Flow Theory

Truck platooning has been identified as an emerging and promising operational technology with the advantages of fuel consumption savings and carbon emissions reductions. We formulate the truck platooning routing optimization problem as a multi-commodity network flow problem from a transportation optimization and scheduling perspective. Based on fuel consumption savings generated through the reduction of aerodynamic drag by the formation of truck platooning, the route of each truck is also set to be a decision variable needing settlement to facilitate the formation of truck platooning to maximize fuel consumption savings. Considering fuel consumption and detour costs, we construct a truck platooning routing optimization model with minimum overall system fuel consumption as the optimization objective. The output of the routing optimization model could both reflect the composition of each truck platooning on each link and directly show the routings of each truck. To explore the impact of the restrictions on the number of trucks in truck platooning on overall fuel consumption savings, road networks are constructed and the truck platooning routing optimization model is solved by the commercial solver CPLEX. Compared to individual trucks, 8% or 12% fuel consumption savings are achieved, respectively, with the number of trucks being restricted or not restricted in truck platooning. Considering the different fuel reduction rates of the following trucks in platooning on the system performance in terms of the total fuel cost, a sensitivity analysis is also conducted. The results also show that the ideal truck platooning routing plan can be obtained by the proposed model, and the study provides a theoretical reference for the promotion and application of truck platooning.

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.

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.

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.

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

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

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.

Train timetabling with stop-skipping, passenger flow, and platform choice considerations

In conventional railway planning processes, stop-skipping decisions are often made at the line planning stage, which is executed prior to train timetabling and platform assignment. However, stop-skipping can shorten passenger journey time and also save on train operating costs. Hence, integrating train timetabling, stop-skipping, and platform choice decisions can help generate train timetables with improved passenger convenience and higher train operating efficiency. Integrating these decisions is a challenging task, as these decisions affect passenger train transfer behavior, which in turn affects the entire passenger flow. This study is a first attempt at integrating these decisions while simultaneously taking into account the passenger flow. We consider a train timetabling problem on a single, one-way track with stop-skipping, platform choice, and passenger flow considerations, and we formulate it as a constrained minimum-cost multi-commodity network flow problem on a time–space network. We analyze the problem’s complexity and develop a Lagrangian relaxation heuristic to solve the problem. We conduct a computational study with randomly generated data that captures the characteristics of the Beijing–Shanghai high-speed railway line. The computational results report the effectiveness of our Lagrangian relaxation heuristic and how the railway’s service capacity and passenger traffic intensity affect the solution.

An analytical solution to the multicommodity network flow problem with weighted random routing

We derive an analytical expression for the mean load at each node of an arbitrary undirected graph for the non-uniform multicommodity flow problem under weighted random routing. We show the mean load at each node, net of its demand and normalized by its (weighted) degree, is a constant equal to the trace of the product of two matrices: the Laplacian of the demand matrix and the generalized inverse of the graph Laplacian. For the case of uniform demand, this constant reduces to the sum of the inverses of the non-zero eigenvalues of the graph Laplacian. We note that such a closed-form expression for the network capacity for the general multicommodity network flow problem has not been reported before, and even though (weighted) random routing is not a practical procedure, it enables us to derive a (tight) upper bound for the capacity of the network under more standard routing policies. Using this new expression, we compute network capacity for a sample of demand matrices for some prototypical networks, including uniform demand (one unit between all node pairs) and broadcast demand (one unit between a source node and each other node as destination), and finally derive estimates of the mean load in some asymptotic cases.

The multi-commodity network flow problem with soft transit time constraints: Application to liner shipping

The multi-commodity network flow problem (MCNF) consists in routing a set of commodities through a capacitated network at minimum cost and is relevant for routing containers in liner shipping networks. As commodity transit times are often a critical factor, the literature has introduced hard limits on commodity transit times. In practical contexts, however, these hard limits may fail to provide sufficient flexibility since routes with even tiny delays would be discarded. Motivated by a major liner shipping operator, we study an MCNF generalization where transit time restrictions are modeled as soft constraints, in which delays are discouraged using penalty functions of transit time. Similarly, early commodity arrivals can receive a discount in cost. We derive properties that distinguish this model from other MCNF variants and adapt a column generation procedure to efficiently solve it. Extensive numerical experiments conducted on realistic liner shipping instances reveal that the explicit consideration of penalty functions can lead to significant cost reductions compared to hard transit time deadlines. Moreover, the penalties can be used to steer the flow towards slower or faster configurations, resulting in a potential increase in operational costs, which generates a trade-off that we quantify under varying penalty functions.

Integrated optimization approach to metro crew scheduling and rostering

Due to its complexity, the metro crew management problem is usually decomposed into a crew scheduling problem and a crew rostering problem. These two decomposed problems are often performed separately in a sequential manner, which may result in solutions that are inferior to those obtained via an integrated framework. Accordingly, this study seeks to develop an integrated optimization approach to model and solve the metro crew scheduling problem and the crew rostering problem simultaneously. To this end, we firstly construct a multi-layer time–space network, in which each layer represents a duty. We then formulate the considered problem as a minimum-cost multi-commodity network flow problem in which each crew is regarded as a commodity and present a Lagrangian relaxation heuristic to solve the proposed network flow problem. Finally, we conduct a computational study using the real-life data to test the effectiveness of our Lagrangian relaxation heuristic, compare the performance of our heuristic with the benchmark solution, and report the benefits obtained via the proposed integrated optimization approach.

Machine Learning-Based Multipath Routing for Software Defined Networks

Network softwarization has recently been enabled via the software-defined networking (SDN) paradigm, which separates the data plane from control plane allowing for a flexible and centralized control of networks. This separation facilitates implementation of machine learning techniques for network management and optimization. In this work, a machine learning-based multipath routing (MLMR) framework is proposed for software-defined networks with quality-of-service (QoS) constraints and flow rules space constraints. The QoS-aware multipath routing problem in SDN is modeled as multicommodity network flow problem with side constraints, that is known to be NP-hard. The proposed framework utilizes network status estimates, and their corresponding routing configurations available at the network central controller to learn a mapping function between them. Once the mapping function is learned, it is applied on live-inputs of network status and routing requests to predict a multipath routing solutions in real-time. Performance evaluations of the MLMR framework on real traces of network traffic verify its accuracy and resilience to noise in training data. Furthermore, the MLMR framework demonstrates more than 98.99% improvement in computational efficiency.

A Hybrid Fuzzy Stochastic Model for Fractional Multi-commodity Network Flow Problems

A Hybrid Fuzzy Stochastic Model for Fractional Multi-commodity Network Flow Problems

Variable fixing heuristics for the capacitated multicommodity network flow problem with multiple transport lines, a heterogeneous fleet and time windows

ABSTRACT In this paper, we investigate a new variant of the multi-commodity network flow problem, taking into consideration multiple transport lines and time windows. This variant arises in a city logistics environment, more specifically in a long-haul passenger transport system that is also used to transport urban freight. We propose two mixed integer programming models for two objective functions: minimization of network operational costs and minimization of travel times. Since the problems under study are NP-hard, we propose three size reduction heuristics. In order to assess the performance of the proposed algorithms, we carried out computational experiments on a set of synthetic problem instances. We use the relative percentage deviation as performance criterion. For the cost objective function, a LP-and-Fix algorithm outperforms other methods in most tested instances, but for the travel time, a hybrid method (size reduction with LP-and-Fix algorithm) is, in general, better than other approaches.

An online optimization-based procedure for the assignment of airplane seats

Due to the large number of air flights these days, all procedures involved in their operational management should be carefully optimized. This work presents a novel approach to the seat assignment problem, which focuses on deciding where to seat the passengers of different online purchases. This problem is currently solved by most airlines with a set of simple pre-defined rules that do not take into account future sales. Instead, the approach in this work is based on solving an integer multicommodity network flow problem, where different commodities are associated with expected future demands of different types of passengers. One feature of the developed optimization model is that it has to be solved online (that is, in real time), thus it must be both effective and fast, which prevented the use of more sophisticated (but also more time consuming, as it was experimentally observed) models based on stochastic programming. Using a real database of flights by Vueling Airlines S.A., we generated a set of synthetic online purchases simulating a pseudo-real flight. Applying our approach to this synthetic data, we observed that (1) the optimization model could be satisfactorily solved in real-time using the state-of-the-art CPLEX solver; (2) and the seat assignment obtained was of higher quality than that obtained by the simple pre-defined rules used by airlines.