Column Generation-based Algorithm Research Articles

Mathematical Model to Evaluate the Impact of Power Supply Constraint for Electric Vehicles on Transportation Network

AbstractThis paper proposes the UE-based mathematical model to evaluate driver's choice of vehicle types and paths, explicitly considering the supply power constraint in addition to the charging station capacity constraints. Because the flows of EVs in the proposed model are represented in a path-based manner, we apply a column generation-based algorithm to avoid enumerating all of the possible paths. The contributions of this study are that 1) we consider both pre-trip charging and charging during a trip, and that 2) we focus on the difference in the impact of power supply constraints on vehicles that need to charge during a trip and those that do not. The proposed model is applied to a hypothetical network. As a result, we confirmed that 1) while the share of EVs in the middle distance tends to be higher, the share of EVs in the short and long distances tends to be lower, and that 2) in case of inadequate power supply, the share of EVs in the short and middle distances decreases significantly, whereas the share of EVs remains almost unchanged in long OD.

Ship deployment problem with green technology adoption for an inland river carrier under non-identical streamflow and speed limits

The maritime industry is currently experiencing a shift towards green corridors. The inland waterway is the main transportation corridor with a relatively low adoption of green technologies, which produces heavy air pollution in inland cities. This paper investigates the ship deployment problem for an inland river carrier considering the adoption of green fuel and green technologies for a given ship fleet. A non-linear programming (NLP) model is proposed to jointly optimize green technology adoption, sailing speed, and routing problems while considering two unique characteristics of inland river shipping, including the non-identical streamflow and the speed limits on each shipping leg. The analytical propositions reveal the optimal operation strategy of the inland river ship fleet, and a column generation-based algorithm is further designed to solve the proposed model. By jointly investigating the ship operation strategy and sulphur emissions, we find the following management insights: Firstly, the ship would have a lower willingness to invest in shore power (scrubber) if it installed scrubber (shore power). The higher streamflow and looser speed limits would reduce the spillover effect of green technology adoption. Secondly, emissions will spillover from ports to river legs if only shore power is provided. Our findings provide valuable insights for policymakers to promote complete green technology adoption to achieve comprehensive abatement of the entire inland river.

Column generation-based algorithm for fragment allocation: minimizing query splitting in distributed databases

Column generation-based algorithm for fragment allocation: minimizing query splitting in distributed databases

Joint rolling stock rotation planning and depot deadhead scheduling in complicated urban rail transit lines

This study investigates the joint rolling stock rotation planning and depot deadhead scheduling in complicated urban rail transit lines with multiple depots, multiple line services, and multiple compositions. The rotation planning aims at connecting the given train trips into train sequences (each of which is served by an individual rolling stock) and determining the routing plan of rolling stocks to pass through visited turnback stations while connecting train trips. The task of the depot deadhead scheduling is to determine conflict-free deadhead routes and deadhead timetables of rolling stocks between the origin/destination of each train sequence and corresponding depot. We formulate the studied problem as a generalized set partitioning-type model containing an exponential number of variables, by using a new time-space network representation and by proposing a novel modelling method for the departure–arrival headway requirement to control the number of constraints. Owing to the complexity of this model, a column generation-based algorithm is adopted to solve efficiently practical-size problems. We enhance a standard column generation (to compute tight lower bound) by further incorporating procedures of variable rounding and halted column generation, to strengthen the capability of searching for better quality solutions. Customized acceleration mechanisms are also explored to speed up the convergence of column generation and the computation of best-known integer solution. We compare our approach with three benchmark approaches and the trial-and-error-based empirical method used by rail dispatchers in practice. Computational results reveal that our approach outperforms these benchmark approaches by computing (near-)optimal solutions. Our optimized solution for a large real-world instance (computed within a practically reasonable computation time) is better than the empirical solution, in terms of all the considered objective function parts.

Column generation for service assignment in cloud-based manufacturing

Cloud-based manufacturing has gained significant attention in academia and industry, presenting opportunities for enhanced operational efficiency. This paper addresses two critical challenges in cloud manufacturing: service assignment and transportation within hybrid hub-and-spoke networks. We propose a novel approach to minimize the total cost, encompassing manufacturing and transportation costs, through a mixed integer programming model and a column generation-based algorithm. Numerical experiments are conducted to validate the efficacy of the proposed model and algorithm, considering different system sizes. The results demonstrate the tangible benefits of our approach. By introducing a hybrid hub-and-spoke network, we achieve a substantial average reduction of 7.46% in the total cost of the cloud manufacturing system. Furthermore, our proposed algorithm outperforms existing methods such as Particle Swarm Optimization (PSO) and CPLEX. Particularly, in large-scale instances where CPLEX fails to solve, our algorithm outperforms PSO by an additional 3.43% in terms of cost reduction. Moreover, sensitivity analysis provides valuable insights for cloud manufacturing, contributing to its effective implementation.

A column generation-based algorithm for gate assignment problem with combinational gates

The gate assignment is a complicated real-world problem occurring in airports. The present paper studies the gate assignment problem with combinational gates. The problem is multi-objective by nature, concerning four objectives. Among those objectives, the number of passengers allocated to nearby gates is aimed to be maximized. The remaining objectives are targeted to be minimized, i.e., the total passenger walking distance, the amount of carbon dioxide emitted by the aircraft assigned to remote gates, and the total distance traveled by the shuttle bus. To solve this problem, we formulate an integer programming model. A column generation-based algorithm based on a set-partitioning model is proposed to solve the problem effectively. In addition, the Pareto Local Search algorithm is adopted to obtain non-dominated solutions. Computational experiments show that the effectiveness of our model is verified through tests with instances of different sizes. The column generation-based algorithm can obtain the optimal solution or the approximately optimal solution with a small gap in a reasonable time. The results of the Pareto Local Search algorithm show the interaction between multiple objectives.

Path-Based Approach for Expanding Rail Transit Network in a Metropolitan Area

Rail transit network design is an important strategic problem in determining the layout of infrastructure and improving operating performance. A core transit network with multiclass rail transit systems has been constructed in many metropolitan areas worldwide. In this study, we aimed to expand an existing network to shorten travel time and improve service quality under the restriction of limited transport supply. We formulate the studied problem as a mixed-integer linear model to obtain optimal construction links, the number of trains required on each link, and the path selected by each traveler such that the weighted sum of total costs from the perspective of travelers, operators, and investors is minimized. The formulated model is path-based, where feasible paths for each traveler are generated to describe the full door-to-door journey, including the first/last mile, transfers, and multiclass transit modes. Owing to the complexity of the network design problem and because it is impractical to enumerate all feasible paths for each traveler in real-size problems, we propose a column generation-based algorithm to find both tight lower bounds and good-quality solutions efficiently by considering only a subset of feasible paths. We prove that the pricing subproblem in column generation can be decomposed into multiple shortest path problems, which can be solved efficiently and separately, based on O/D pairs instead of individual travelers. A rail transit network along a metropolitan corridor was studied as an example. Multiple computational experiments were conducted, and the results illustrate the validity and practicality of the proposed methodology for solving the problem.

A column generation approach for operational flight scheduling and aircraft maintenance routing

Aircraft is one of the most expensive resources owned by an airline which should be properly planned. The aircraft maintenance routing problem (AMRP) generates aircraft routes to serve scheduled flights, while satisfying the strict maintenance requirements. However, in operations, the pre-determined aircraft routes are usually disrupted due to unplanned maintenance requirements or insufficient remaining legal flying time to maintenance stations. Thus, airlines often have to re-route aircraft in real time. This study proposes a new aircraft re-routing approach to fulfil the maintenance requirements arising in the operational stage. Specifically, maintenance stations are capacity-constrained, while airlines could allocate maintenance resources (like staff and equipment) to other airports with additional costs. Besides, flights could be re-scheduled (i.e., cancelled with a high penalty), while the model endeavors to minimize the impact of recovery actions on the original plan. To achieve this, specialized flight networks are constructed, and a column generation-based algorithm is developed to obtain high-quality solutions within short computational times. Computational experiments show that the solutions obtained by the proposed algorithm are optimal or near-optimal with an optimality gap of 0.3% on average. In addition, some managerial insights on allocating maintenance resources to other airports to fulfil aircraft maintenance demands in operations are discussed.

Stochastic Production Planning with Flexible Manufacturing Systems and Uncertain Demand: A Column Generation-based Approach

The ongoing pandemic, namely COVID-19, has rendered widespread economic disorder. The deficiencies have delayed production at manufacturers in several industries on the supply side. The effects of disruption were more notable for industries with longer supply chains, especially reaching East Asia. Regarding the demand, sectors can be divided into three categories: i) the ones, like e-commerce companies, that experienced augmented demand; ii) the ones with a plunged demand, like what hotels and restaurants experience; iii) the companies experiencing a roller-coaster-ride business. After mitigation efforts, the economy started recovering, resulting in increased demand. However, regardless of their struggles, the companies have not fully returned to their pre-pandemic levels. One of the strategies to gain resilience in its supply chain and manage the disruptions is to employ flexible/hybrid manufacturing systems. This paper considers a flexible/hybrid manufacturing production setting with typically dedicated machinery to satisfy regular demand and a flexible manufacturing system (FMS) to handle surge demand. We model the uncertainty in demand using a scenario-based approach and allow the business to make here-and-now and wait-and-see decisions exploiting the cost-effectiveness of the standard production and responsiveness of the FMS. We propose a column generation-based algorithm as the solution approach. Our computational analysis shows that this hybrid production setting provides highly robust response to the uncertainty in demand, even with high fluctuations.

A column generation-based algorithm for midterm nurse scheduling with specialized constraints, preference considerations, and overtime

This paper presents an augmented mixed-integer linear programming (MILP) formulation for the nurse scheduling problem designed to accommodate overtime and several practical considerations that are part of the rostering process but have been absent in previous work. The goal is to minimize the sum of weighted uncovered demand and nurse preference violations over a one-month period. Model features include maximum number of consecutive working days and days off, minimum number of rest hours between shifts, vacations and birthdays. Preference violations are measured by the number of times certain days-off and days-on patterns appear in a nurse’s schedule, and the number of times an assignment contains a shift transition on two consecutive days. In addition, the formulation accounts for each nurse’s working status in the last few days of the previous month to ensure that constraints are not violated when new schedules are developed, and that preference violations are properly counted. The problem is solved in two stages due to the complexity of dealing with regular time and overtime in an integrated model. In Stage 1, a mixed-integer linear program is used to assign shifts to nurses without exceeding their contracted regular hours. Column generation is applied to derive solutions. In Stage 2, a heuristic is developed to add overtime to the rosters found in Stage 1. This is done with the goal of minimizing total overtime hours while trying to balance each nurse’s workload. Experimental results are provided for instances with up to 60 nurses based on data obtained from a 1000-bed hospital. We conclude with a sensitivity analysis of the uncovered demand weight to determine its impact on solution quality and computational effort.

Order batching and picker scheduling in warehouse order picking

This article focuses on the integration of order batching and picker scheduling decisions while taking into account two objectives that have been considered in the literature, namely the minimization of both total travel time to collect all items and makespan of the pickers. This integrated problem not only occurs naturally in wave picking systems in which the latest picking time of orders becomes the key performance metric, but also arises when there is a limit on the picker operating time. We present models that result from combining these objectives and analyze their relationship through bounds. We propose a column generation-based exact algorithm for the integrated problem. The novelty of the proposed approach lies in the ability of efficiently solving the integrated order batching and picker scheduling problem to optimality by designing a column generation subproblem based on the set of batches, which makes it a challenging optimization problem due to its size. We alleviate this difficulty by reformulating this subproblem, which allows efficient implicit enumeration of its variables. We have also devised a Variable Neighborhood Search algorithm used as a subprocedure within the proposed exact solution algorithm. Finally, we conduct experiments on randomly generated instances and show that the proposed algorithms are capable of solving instances with up to 100 orders.

Integrated rolling stock deadhead routing and timetabling in urban rail transit lines

This paper investigates an integrated rolling stock deadhead routing and timetabling problem from the rolling stock scheduling in a complicated urban rail transit line. Given the train sequences composed of passenger train trips within an operation day, the deadhead routing sub-problem aims at selecting the rolling stock deadhead routes between the depots and the first/last stations of each train sequence. The task of the deadhead timetabling sub-problem is to determine the arrival and departure times of rolling stocks along these selected routes. By means of a time-space network representation, we formulate the studied problem as a new (generalized set partitioning type) binary linear model, to minimize the weighted sum of total deadhead distance and total deadhead running time of rolling stocks during the depot exiting and entering operations. Owing to large numbers of constraints and decision variables in our model, a row and column generation-based algorithm is developed to solve practical-size problems efficiently. A row generation and a column generation are executed iteratively to reduce the numbers of considered constraints and decision variables in our algorithm, respectively. The pricing sub-problem (to identify new variables) in column generation is decomposed into multiple simplified problems, each of which is equivalent to a shortest path problem in the constructed time-space sub-network for any candidate route of each train sequence. These shortest path problems can be solved efficiently by using an existing shortest path algorithm. Computational experiments on a set of hypothetical, realistic, and real-world instances demonstrate that our approach can compute both tight lower bounds and (near-)optimal solutions (with a maximum relative optimality gap of 1.07%) for all the tested instances within a maximum computation time of approximately 3 hours for the studied tactical problem. Furthermore, our best-known solution for the real-world instance is better than the empirical solution designed by the rail managers mainly based on experience.

Column generation algorithms for mother plate design in steel plants

This paper investigates the mother plate design (MPD) problem which is a typical problem in the production planning of steel plants. The MPD problem refers to a variant of the two-dimensional variable-sized bin packing problem in which the height of a bin is in a given finite set and the width of each bin is continuous in an interval. The problem is first formulated as a mixed integer nonlinear programming model and then linearized into a mixed integer programming model. We present a column generation-based (CG-based) algorithm to approximately solve the MPD problem, in which a height combination scheme is devised to decrease the dimensions of the pricing sub-problem. We present an accelerated CG (ACG) algorithm with improved performance which adopts two heuristic accelerating strategies in the pricing process. We use practical instances collected from a Chinese steel plant to test the performance of the proposed algorithms. The results indicate that the ACG algorithm is suitable for solving real-world instances.

Simultaneous node and link districting in transportation networks: Model, algorithms and railway application

Partitioning a large network into multiple subnetworks is adopted extensively in various practical applications involving the architecture of distributed management. In this study, we consider strategic simultaneous node and link districting to partition the nodes and links of a given transportation network into a fixed number of mutually disjoint districts, based on the districting criteria of integrity, contiguity, balance, and independence. We first formulate the resulting problem as a mixed integer linear programming model to minimize the weighted sum of the total size deviation of districts and total cooperation between districts. An improved network flow-based technique is proposed to incorporate the complicated contiguity criterion by using a polynomial number of constraints. Valid inequalities, which break the symmetry within and between districts, are designed to strengthen the model. To solve this challenge problem, we reformulate it as a binary linear programming model, and develop a column generation-based algorithm to find tight lower bounds and good-quality solutions. Then, an iterative search algorithm is designed to obtain good-quality solutions rapidly. Furthermore, a more efficient hybrid algorithm is proposed by using the results of the iterative search algorithm to initialize the column generation-based algorithm. We assess the proposed model and algorithms by using various scales of instances derived from the train dispatcher desk districting problem, which is a practical application of the investigated problem in the context of railway. The computational results reveal that our approaches outperform existing approaches and a commercial solver, and our best algorithm can solve almost all the investigated instances to optimality within a considerably short average computation time. The districting solutions of our approaches are also better than the empirical solutions designed by railway managers mainly based on experience.

Column generation-based stochastic school bell time and bus scheduling optimization

Given a trip plan (a sequence of bus stops), the integrated school bell time and bus scheduling problem simultaneously optimizes the school bell time and bus routes (the assignment of buses to serve trips at a specific time) while maintaining the minimum level-of-service requirements. The objective is to minimize the total number of buses and the total vehicle time. The previous efforts focused on the deterministic problem. However, uncertainty is unavoidable in a real-world problem. Thus, we consider this integrated problem with stochastic travel times. Two Mixed Integer Programming (MIP) models and a Column Generation-based algorithm are proposed. The algorithm incorporates the merits of the Column Generation, Simulated Annealing, Insertion Algorithm, and Greedy Randomized Adaptive Search Procedure and gains the computational power that the existing methods do not have. The experiments on a real-world problem show that, after the bell time optimization, up to 41% of current buses can be saved with even better service in terms of the higher punctuality and shorter student ride time.

Column Generation Accelerated Algorithm and Optimisation for a High-Speed Railway Train Timetabling Problem

With the rapid development of high-speed railway (HSR) systems, the increasing demand for passenger traffic has put forward higher requirements for HSR train timetabling problems (HSRTTPs). This paper establishes two mathematical optimisation models with different optimisation objectives for an HSRTTP and solves these models through a column generation-based algorithm. However, the column generation-based algorithm has the disadvantage of a slow convergence rate, thus we put forward corresponding acceleration strategies for five stages of the algorithm: preprocessing, restricted master problem, pricing problem, branch-and-bound and postprocessing from a symmetry point between the computation efficiency and the accuracy. The effectiveness of the acceleration strategies was validated by a case study of the Beijing–Shanghai HSR. The results show that the proposed optimal acceleration strategies can increase the computation efficiency of the algorithm by 11.8× on average while ensuring the accuracy.

Vehicle scheduling problem with loss in bus ridership

This paper presents an optimization approach for the bus scheduling problem in transit systems, considering the loss in ridership. This research work is motivated by the considerable loss in the demand for bus transportation in Brazil in the last few decades. The main objective of the optimization approach is to allow transit operators to adjust the fleet given the loss of passengers. The approach combines services trips within a small time interval in the timetabling, being the peak demand of the grouped trips adjusted to the heterogeneous fleet, defining a lower cost assignment for a fallen demand of passengers. The grouping strategy was formulated as an integer linear programming model. Due to the complexity of the formulation, a column generation-based algorithm has been designed. We carried out several experiments with the developed approach, using real-world data from a transportation company in southern Brazil and large randomly generated instances, considering different demand profiles and acceptable grouping trip intervals. The results of model allowed savings in the vehicle scheduling, while keeping good service levels for passengers. The model proved to support decision making in the planning of public transport, considering different operational scenarios.

On Link Scheduling Under Blockage and Interference in 60-GHz Ad Hoc Networks

In this paper, we tackle the problem of minimum time length link scheduling in 60-GHz ad hoc wireless networks using directional antennas with directional beamforming, under both traffic demand and signal to interference and noise ratio constraints. Both single-hop and multi-hop cases are considered. For the single-hop scenario, a binary integer programming problem is formulated by incorporating a general interference model for directional transmissions and a Markov chain-based blockage model. Two effective solution algorithms are proposed, including a greedy algorithm that maximizes the instant throughput for each time slot, and a column generation-based algorithm that iteratively improves the current link schedule. For the multi-hop scenario, we develop a more complicated problem formulation incorporating both route selection and flow conservation constraints. We also develop an effective algorithm to solve the multi-hop problem. The performance of the proposed algorithms is validated with simulations.

Two-Stage Semi-Distributed Resource Management for Device-to-Device Communication in Cellular Networks

In cellular networks, the device-to-device (D2D) communication increases the network capacity by spatial reuse of radio resources and prolongs the battery life of devices by reducing the transmission power. In this paper, we propose a two-stage semi-distributed resource management framework for the D2D communication. At the first stage of the framework, the base station (BS) allocates resource blocks (RBs) to BS-to-user device (B2D) links and D2D links, in a centralized manner. At the second stage, the BS schedules the transmission using the RBs allocated to B2D links, while the primary user device of each D2D link carries out link adaptation on the RBs allocated to the D2D link, in a distributed fashion. The proposed framework has the advantages of both centralized and distributed design approaches, i.e., high network capacity and low control/computational overhead, respectively. We formulate the problems of RB allocation to maximize the radio resources efficiency, taking account of two different policies on the spatial reuse of RBs. To solve these problems, we suggest a greedy algorithm and a column generation-based algorithm. By simulation, it is shown that the proposed scheme achieves the near-optimal performance while reducing the control/computational overhead.

A Column Generation-based Algorithm for Multi-Class Dynamic User Equilibrium Problem

The multi-class dynamic user equilibrium (MDUE) model can be adopted in the lower level of a bi-level formulation of dynamic network design problems to predict heterogeneous travelers’ path choices in response to transportation authority’s decisions in the upper level. While a number of previous studies were devoted to the finite-dimensional MDUE (FMDUE) problem with a given number of user classes, the infinite-dimensional MDUE (IMDUE) problem was rarely addressed. This work proposes a column generation-based algorithm that solves the IMDUE problem as a series of FMDUE sub-problems. Each sub-problem is formulated as a nonlinear minimization program via a gap function and solved by a feasible descent direction method that is able to circumvent the needs to calculate partial derivatives for determining search direction, while maintaining the mechanisms of searching along feasible descent directions. Numerical results show this approach is superior to the algorithm based on the method of successive averages.