Integer Multi-commodity Network Flow Research Articles

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.

A Dynamic and Flexible Berth Allocation Model with Stochastic Vessel Arrival Times

This study proposes a berth-flow network modeling approach to deal with the dynamic berth allocation problem (DBAP) with stochastic vessel arrival times. In this approach, uncertain vessel arrival times are represented using discrete probability distributions and a flexible berth allocation scheme based on the blocking plan concept is incorporated into the model to effectively utilize wharf space. This new model is referred to as the stochastic dynamic (vessel arrival) and flexible (berth space) berth allocation problem (SDFBAP) model. The aim is to minimize the sum of the expected values of unanticipated schedule delay costs and the penalties for being unable to service all vessels within the planning horizon. The proposed model is formulated as an integer multi-commodity network flow problem which can be solved with off-the-shelf solvers. Computational experiments are conducted using a real example to demonstrate the effectiveness and efficiency of the SDFBAP model. A simulation-based approach is adopted to evaluate the SDFBAP model. A number of scenario analyses are also conducted to gain insight into important model parameters.

Optimal scheduling of a taxi fleet with mixed electric and gasoline vehicles to service advance reservations

This study addresses the problem of scheduling a fleet of taxis that are appointed to solely service customers with advance reservations. In contrast to previous studies that have dealt with the planning and operations of a taxi fleet with only electric vehicles (EVs), we consider that most taxi companies may have to operate with fleets comprised of both gasoline vehicles (GVs) and plug-in EVs during the transition from GV to (complete) EV taxi fleets. This paper presents an innovative multi-layer taxi-flow time-space network which effectively describes the movements of the taxis in the dimensions of space and time. An optimization model is then developed based on the time-space network to determine an optimal schedule for the taxi fleet. The objective is to minimize the total operating cost of the fleet, with a set of operating constraints for the EVs and GVs included in the model. Given that the model is formulated as an integer multi-commodity network flow problem, which is characterized as NP-hard, we propose two simple but effective decomposition-based heuristics to efficiently solve the problem with practical sizes. Test instances generated based on the data provided by a Taiwan taxi company are solved to evaluate the solution algorithms. The results show that the gaps between the objective values of the heuristic solutions and those of the optimal solutions are less than 3%, and the heuristics require much less time to obtain the good quality solutions. As a result, it is shown that the model, coupled with the algorithms, can be an effective planning tool to assist the company in routing and scheduling its fleet to service reservation customers.

Stochastic fleet deployment models for public bicycle rental systems

ABSTRACTThis paper presents two stochastic bike deployment (SBD) models that determine the optimal number of bicycles allocated to each station in a leisure-oriented public bicycle rental system with stochastic demands. The SBD models represent the stochastic demands using a set of scenarios with given probabilities. A multilayer bike-flow time-space network is constructed for developing the models, where each layer corresponds to a given demand scenario and effectively describes bicycle flows in the spatial and temporal dimensions. As a result, the models are formulated as the integer multi-commodity network flow problem, which is characterized as NP-hard. We propose a heuristic to efficiently obtain good quality solutions for large-size model instances. Test instances are generated using real data from a bicycle rental system in Taiwan to evaluate the performance of the models and the solution algorithm. The test results show that the models can help the system operator of a public bicycle system make effective fleet deployment decisions.

Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems

We consider the piecewise linear multicommodity network flow problem with the addition of a constraint specifying that the total flow on each arc must be an integer. This problem has applications in transportation and logistics, where total flows might represent vehicles or containers filled with different products. We introduce formulations that exploit this integrality constraint by adapting to our problem a technique known as discretization that has been used to derive mixed-integer programming models for several combinatorial optimization problems. We enhance the discretized models either by adding valid inequalities derived from cut-set inequalities or by using flow disaggregation techniques. Since the size of the formulations derived from discretization and flow disaggregation rapidly increases with problem dimensions, we develop an efficient and effective Lagrangian relaxation method to compute lower and upper bounds. We perform computational results on a large set of randomly generated instances that allow us to compare the relative efficiency of the different modeling alternatives (flow disaggregation plus addition of cut-set inequalities with or without discretization), when used within the Lagrangian relaxation approach.

A planning model and solution algorithm for multi-trip split-delivery vehicle routing and scheduling problems with time windows

This study proposes a daily vehicle routing model for minimizing the total cost of replenishing inventory within a supply chain. The first major contribution of this research is to allow multiple use of vehicles in a split delivery vehicle routing problem with time windows (SDVRPTW), which is more realistic for various real-life applications. The multi-trip SDVRPTW (MTSDVRPTW) is formulated using the time–space network technique, which provides greater flexibility for formulating the complicated interactions between vehicles and products when multi-trip, split delivery, and delivery time windows are simultaneously considered. The resulting formulation of the MTSDVRPTW can be categorized as an integer multi-commodity network flow problem with side constraints. A two-step solution algorithm is proposed to solve this NP-hard problem, which is the second major contribution of this research. Finally, a real-world scale numerical example is performed to demonstrate and to test the methodology. The results indicate that these vehicle routing problems can be solved effectively and efficiently and that the proposed methodology has great potential for inventory replenishment scheduling where split deliveries and multiple trips for a single vehicle are allowed and time window constraints are imposed.

A network flow model for the dynamic and flexible berth allocation problem

This paper presents a new model for the dynamic berth allocation problem (BAP). The model is developed using a berth-flow network modeling approach and is formulated as an integer multi-commodity network flow problem. In addition, an innovative flexible berth-space utilization scheme, based on blocking plans, is incorporated into the proposed model. This is referred to as the dynamic (vessel arrivals) and flexible (berth space) BAP model (or DFBAP), and is designed to better utilize wharf space in a container port. Computational experiments conducted on an instance generated using actual data show that the DFBAP model is more effective and efficient than the method currently used by port authorities. A set of scenario analyses is also performed to obtain insights into important model parameters.

Algorithms for an Integer Multicommodity Network Flow Problem with Node Reliability Considerations

In this paper, we consider the problem of sending a set of multiple commodities from their origin to destination nodes via intermediate hubs. Each hub node is associated with a reliability function, which depends on the total flow that crosses that hub. The probability that each commodity is successfully relayed from its origin to its destination is given by the product of hub reliabilities on the commodity's path. The problem we consider seeks to find minimum-cost commodity paths such that each commodity reaches its destination with a sufficiently large probability. We first formulate the problem as a nonlinear multicommodity network-flow problem and prove that it is strongly NP-hard. We then present two linearization techniques for this formulation, and propose a pair of lower- and upper-bounding formulations, which can then be used within an exact cutting-plane algorithm to solve the problem. Finally, we analyze the computational effectiveness of our proposed strategies on a set of randomly generated instances.

Per-Seat, On-Demand Air Transportation Part I: Problem Description and an Integer Multicommodity Flow Model

The availability of relatively cheap small jet planes has led to the creation of on-demand air transportation services in which travelers call a few days in advance to schedule a flight. A successful on-demand air transportation service requires an effective scheduling system to construct minimum-cost pilot and jet itineraries for a set of accepted transportation requests. We present an integer multicommodity network flow model with side constraints for such dial-a-flight problems. We develop a variety of techniques to control the size of the network and to strengthen the quality of the linear programming relaxation, which allows the solution of small instances. In Part II, we describe how this core optimization technology is embedded in a parallel, large-neighborhood, local search scheme to produce high-quality solutions efficiently for large-scale real-life instances.

Per-Seat, On-Demand Air Transportation Part II: Parallel Local Search

The availability of relatively cheap small jet aircrafts suggests a new air transportation business: dial-a-flight, an on-demand service in which travelers call a few days in advance to schedule transportation. A successful on-demand air transportation service requires an effective scheduling system to construct minimum-cost pilot and jet itineraries for a set of accepted transportation requests. In Part I, we introduced an integer multicommodity network flow model with side constraints for the dial-a-flight problem and showed that small instances can be solved effectively. Here, we demonstrate that high-quality solutions for large-scale real-life instances can be produced efficiently by embedding the core optimization technology in a local search scheme. To achieve the desired level of performance, metrics were devised to select neighborhoods intelligently, a variety of search diversification techniques were included, and an asynchronous parallel implementation was developed.

A dynamic logistics coordination model for evacuation and support in disaster response activities

This paper describes an integrated location-distribution model for coordinating logistics support and evacuation operations in disaster response activities. Logistics planning in emergencies involves dispatching commodities (e.g., medical materials and personnel, specialised rescue equipment and rescue teams, food, etc.) to distribution centres in affected areas and evacuation and transfer of wounded people to emergency units. During the initial response time it is also necessary to set up temporary emergency centers and shelters in affected areas to speed up medical care for less heavily wounded survivors. In risk mitigation studies for natural disasters, possible sites where these units can be situated are specified according to risk based urban structural analysis. Logistics coordination in disasters involves the selection of sites that result in maximum coverage of medical need in affected areas. Another important issue that arises in such emergencies is that medical personnel who are on duty in nearby hospitals have to be re-shuffled to serve both temporary and permanent emergency units. Thus, an optimal medical personnel allocation must be determined among these units. The proposed model also considers this issue. The proposed model is a mixed integer multi-commodity network flow model that treats vehicles as integer commodity flows rather than binary variables. This results in a more compact formulation whose output is processed to extract a detailed vehicle route and load instruction sheet. Post processing is achieved by a simple routing algorithm that is pseudo-polynomial in the number of vehicles utilized, followed by the solution of a linear system of equations defined in a very restricted domain. The behavior and solvability of the model is illustrated on an earthquake scenario based on Istanbul’s risk grid as well as larger size hypothetical disaster scenarios.

Periodic airline fleet assignment with time windows, spacing constraints, and time dependent revenues

Given the sets of flights and aircraft of an airline carrier, the fleet assignment problem consists of assigning the most profitable aircraft type to each flight. In this paper we propose a model for the periodic fleet assignment problem with time windows in which departure times are also determined. Anticipated profits depend on the schedule and the selection of aircraft types. In addition, short spacings between consecutive flights which serve the same origin–destination pair of airports are penalized. We propose a non-linear integer multi-commodity network flow formulation. We develop new branch-and-bound strategies which are embedded in our branch-and-price solution strategy. Finally, we present computational results for periodic daily schedules on three real-world data sets.

Routing of Railway Carriages

In the context of organizing timetables for railway companies the following railway carriage routing problem occurs. Given a timetable containing rail links with departure and destination times/stations and the composition of the trains, find a routing of railway carriages such that the required carriages are always available when a train departs. The problem is formulated as an integer multi-commodity network flow problem with nonlinear objective function. We will present a local search approach for this NP-hard problem. The approach uses structural properties of the integer multi-commodity network flow formulation of the problem. Computational results for a real world instance are given.