Integrated optimization on train scheduling and preventive maintenance time slots planning

We address the problem of simultaneously scheduling trains and planning preventive maintenance time slots (PMTSs) on a general railway network. Based on network cumulative flow variables, a novel integrated mixed-integer linear programming (MILP) model is proposed to simultaneously optimize train routes, orders and passing times at each station, as well as work-time of preventive maintenance tasks (PMTSs). In order to provide an easy decomposition mechanism, the limited capacity of complex tracks is modelled as side constraints and a PMTS is modelled as a virtual train. A Lagrangian relaxation solution framework is proposed, in which the difficult track capacity constraints are relaxed, to decompose the original complex integrated train scheduling and PMTSs planning problem into a sequence of single train-based sub-problems. For each sub-problem, a standard label correcting algorithm is employed for finding the time-dependent least cost path on a time-space network. The resulting dual solutions can be transformed to feasible solutions through priority rules. Numerical experiments are conducted on a small artificial network and a real-world network adapted from a Chinese railway network, to evaluate the effectiveness and computational efficiency of the integrated optimization model and the proposed Lagrangian relaxation solution framework. The benefits of simultaneously scheduling trains and planning PMTSs are demonstrated, compared with a commonly-used sequential scheduling method.     

Capacitated maximum covering location models: Formulations and solution procedures

This paper presents two formulations and two solution procedures for a capacitated maximum covering location problem. In the first formulation, the problem is presented as a mixed-interger linear programming model which maximizes covered demand. In the second model, the objective function maximizes the weighted covered demand while at the same time minimizing the average distance from the uncovered demands to the located facilities. The second formulation attempts to account for the assignment of the demand which is not “covered” to located facilities which have excess capacity. This assignment is very important, especially for locating emergency service facilities. Two heuristic procedures are proposed to solve these models. These are based on greedy adding technique and Lagrangian relaxation. At each iteration, the demands are allocated to the facilities using an out-of-kilter method. The performance of the solution techniques are compared to the optimal solutions in a variety of test problems.     

Solution methods for the taxi pooling problem

In Taiwan, taxi pooling is currently performed by some taxi companies using a trial-and-error experience-based method, which is neither effective nor efficient. There is, however, little in the literature on effective models and solution methods for solving the taxi pooling problem. Thus, in this study we employ network flow techniques and a mathematical programming method to develop a taxi pooling solution method. This method is composed of three models. First, a fleet routing/scheduling model is constructed to produce fleet/passenger routes and schedules. A solution algorithm, based on Lagrangian relaxation, a sub-gradient method and a heuristic to find the upper bound of the solution, is proposed to solve the fleet routing/scheduling model. Then, two single taxi-passenger matching models are constructed with the goals of decreasing number of passenger transfers and matching all passengers and taxis. These two taxi-passenger matching models are directly solved using a mathematical programming solver. For comparison with the solution method, we also develop another heuristic by modifying a heuristic recently proposed for solving a one-to-many taxi pooling problem. The performance of the solution method and the additional heuristic are evaluated by carrying out a case study using real data and suitable assumptions. The test results show that these two solution methods could be useful in practice.     

Simultaneous passenger train routing and timetabling using an efficient train-based Lagrangian relaxation decomposition

This paper focuses on the simultaneous passenger train routing and timetabling problem on the rail network consisting of both unidirectional and bidirectional tracks using an efficient train-based Lagrangian relaxation decomposition. We first build an integer linear programming model with many 0–1 binary and non-negative integer decision variables, after then reformulate it as a train path-choice model for providing an easier train-based Lagrangian relaxation decomposition mechanism based on the construction of space-time discretized network extending from node-cell-based rail network. Moreover, through reformulating safety usage interval restrictions with a smaller number of constraints in this reformulated model, the train-based decomposition needs fewer Lagrangian multipliers to relax these constraints. On the basis of this decomposition, a solving framework including a heuristic algorithm is proposed to simultaneously optimize both the dual and feasible solutions. A set of numerical experiments demonstrate the proposed Lagrangian relaxation decomposition approach has better performances in terms of minimizing both train travel time and computational times.     

Trial-and-error method for congestion pricing scheme under side-constrained probit-based stochastic user equilibrium conditions

A toll pattern that can restrict link flows on the tolled links to some predetermined thresholds is named as effective toll solution, which can be theoretically obtained by solving a side-constraint traffic assignment problem. Considering the practical implementation, this paper investigates availability of an engineering-oriented trial-and-error method for the effective toll pattern of cordon-based congestion pricing scheme, under side-constrained probit-based stochastic user equilibrium (SUE) conditions. The trial-and-error method merely requires the observed traffic counts on each entry of the cordon. A minimization model for the side-constrained probit-based SUE problem with elastic demand is first proposed and it is shown that the effective toll solution equals to the product of value of time and optimal Lagrangian multipliers with respect to the side constraints. Then, employing the Lagrangian dual formulation of the minimization method, this paper has built a convergent trial-and-error method. The trial-and-error method is finally tested by a numerical example developed from the cordon-based congestion pricing scheme in Singapore.     

Truck backhauling on networks with many terminals

Truck backhauling reduces empty truck-miles by having drivers haul loads on trips back to their home terminal. This paper develops a model to help coordinate backhauling between many (more than two) terminals. Two mathematical programming formulations of this backhauling problem are given. One formulates it as a “matching” problem that leads to a heuristic for solving the very large backhauling problems that arise in practice. Using Lagrangian relaxation, the other formulation allows a very tight bound on the optimal solution to be calculated. The quality of the heuristic solution can be determined by comparison with this bound. A large scale example based on actual truck shipments demonstrates how the model might be used in planning truck backhauling. For this example, the heuristic yields a solution within 1% of optimal.     

On the existence of pricing strategies in the discrete time heterogeneous single bottleneck model

In this paper, we study the pricing strategies in the discrete time single bottleneck model with general heterogeneous commuters. We first prove that in the system optimal assignment, the queue time must be zero for all the departures. Based on this result, the system optimal problem is formulated as a linear program. The solution existence and uniqueness are discussed. Applying linear programming duality, we then prove that the optimal dual variable values provide an optimal toll with which the system optimal solution is also an equilibrium solution. Extensive computational results are reported to demonstrate the insights gained from the formulations in this paper. These results confirm that a system optimal equilibrium can be found using the proposed approach.     

A solution algorithm for a dynamic deterministic user equilibrium assignment model with departure time choice

Abstract

A route-based combined model of dynamic deterministic route and departure time choice and a solution method for many origin and destination pairs is proposed. The divided linear travel time model is used to calculate the link travel time and to describe the propagation of flow over time. For the calculation of route travel times, the predictive ideal route travel time concept is adopted. Solving the combined model of dynamic deterministic route and departure time choice is shown to be equivalent to solving simultaneously a system of non-linear equations. A Newton-type iterative scheme is proposed to solve this problem. The performance of the proposed solution method is demonstrated using a version of the Sioux Falls network. This shows that the proposed solution method produces good equilibrium solutions with reasonable computational cost.     

Equity-oriented skip-stopping schedule optimization in an oversaturated urban rail transit network

In this study, we focus on improving system-wide equity performance in an oversaturated urban rail transit network based on multi-commodity flow formulation. From the system perspective, an urban rail transit network is a distributed system, where a set of resources (i.e., train capacity) is shared by a number of users (i.e., passengers), and equitable individuals and groups should receive equal shares of resources. However, when oversaturation occurs in an urban rail transit network during peak hours, passengers waiting at different stations may receive varying shares of train capacity leading to the inequity problem under train all-stopping pattern. Train skip-stopping pattern is an effective operational approach, which holds back some passengers at stations and re-routes their journeys in the time dimension based on the available capacity of each train. In this study, the inequity problem in an oversaturated urban rail transit network is analyzed using a multi-commodity flow modeling framework. In detail, first, discretized states, corresponding to the number of missed trains for passengers, are constructed in a space-time-state three-dimensional network, so that the system-wide equity performance can be viewed as a distribution of all passengers in different states. Different from existing flow-based optimization models, we formulate individual passenger and train stopping pattern as commodity and network structure in the multi-commodity flow-modeling framework, respectively. Then, we aim to find an optimal commodity flow and well-designed network structure through the proposed multi-commodity flow model and simultaneously achieve the equitable distribution of all passengers and the optimal train skip-stopping pattern. To quickly solve the proposed model and find an optimal train skip-stopping pattern with preferable system-wide equity performance, the proposed linear programming model can be effectively decomposed to a least-cost sub-problem with positive arc costs for each individual passenger and a least-cost sub-problem with negative arc costs for each individual train under a Lagrangian relaxation framework. For application and implementation, the proposed train skip-stopping optimization model is applied to a simple case and a real-world case based on Batong Line in the Beijing Subway Network. The simple case demonstrates that our proposed Lagrangian relaxation framework can obtain the approximate optimal solution with a small-gap lower bound and a lot of computing time saved compared with CPLEX solver. The real-world case based on Batong Line in the Beijing Subway Network compares the equity and efficiency indices under the operational approach of train skip-stopping pattern with those under the train all-stopping pattern to state the advantage of the train skip-stopping operational approach.     

General stochastic user equilibrium traffic assignment problem with link capacity constraints

This paper addresses a general stochastic user equilibrium (SUE) traffic assignment problem with link capacity constraints. It first proposes a novel linearly constrained minimization model in terms of path flows and then shows that any of its local minimums satisfies the generalized SUE conditions. As the objective function of the proposed model involves path‐specific delay functions without explicit mathematical expressions, its Lagrangian dual formulation is analyzed. On the basis of the Lagrangian dual model, a convergent Lagrangian dual method with a predetermined step size sequence is developed. This solution method merely invokes a subroutine at each iteration to perform a conventional SUE traffic assignment excluding link capacity constraints. Finally, two numerical examples are used to illustrate the proposed model and solution method.     

Joint inventory-location problem under the risk of probabilistic facility disruptions

This paper studies a reliable joint inventory-location problem that optimizes facility locations, customer allocations, and inventory management decisions when facilities are subject to disruption risks (e.g., due to natural or man-made hazards). When a facility fails, its customers may be reassigned to other operational facilities in order to avoid the high penalty costs associated with losing service. We propose an integer programming model that minimizes the sum of facility construction costs, expected inventory holding costs and expected customer costs under normal and failure scenarios. We develop a Lagrangian relaxation solution framework for this problem, including a polynomial-time exact algorithm for the relaxed nonlinear subproblems. Numerical experiment results show that this proposed model is capable of providing a near-optimum solution within a short computation time. Managerial insights on the optimal facility deployment, inventory control strategies, and the corresponding cost constitutions are drawn.     

Optimal speed detector density for the network with travel time information

In the expressway network, detectors are installed on the links for detecting the travel time information while the predicted travel time can be provided by the route guidance system (RGS). The speed detector density can be determined to influence flow distributions in such a way that the precision of the travel time information and the social cost of the speed detectors are optimized, provided that each driver chooses the minimum perceived travel time path in response to the predicted travel time information. In this paper, a bilevel programming model is proposed for the network with travel time information provided by the RGS. The lower-level problem is a probit-based traffic assignment model, while the upper-level problem is to determine the speed detector density that minimizes the measured travel time error variance as well as the social cost of the speed detectors. The sensitivity analysis based algorithm is proposed for the bilevel programming problem. Numerical examples are provided to illustrate the applications of the proposed model and of the solution algorithm.     

Personalized real-time traffic information provision: Agent-based optimization model and solution framework

The advancement of information and communication technology allows the use of more sophisticated information provision strategies for real-time congested traffic management in a congested network. This paper proposes an agent-based optimization modeling framework to provide personalized traffic information for heterogeneous travelers. Based on a space–time network, a time-dependent link flow based integer programming model is first formulated to optimize various information strategies, including elements of where and when to provide the information, to whom the information is given, and what alternative route information should be suggested. The analytical model can be solved efficiently using off-the-shelf commercial solvers for small-scale network. A Lagrangian Relaxation-based heuristic solution approach is developed for medium to large networks via the use of a mesoscopic dynamic traffic simulator.     

A tailored branch-and-price approach for a joint tramp ship routing and bunkering problem

This paper deals with a practical tramp ship routing problem while taking into account different bunker prices at different ports, which is called the joint tramp ship routing and bunkering (JSRB) problem. Given a set of cargoes to be transported and a set of ports with different bunker prices, the proposed problem determines how to route ships to carry the cargoes and the amount of bunker to purchase at each port, in order to maximize the total profit. After building an integer linear programming model for the JSRB problem, we propose a tailored branch-and-price (B&P) solution approach. The B&P approach incorporates an efficient method for obtaining the optimal bunkering policy and a novel dominance rule for detecting inefficient routing options. The B&P approach is tested with randomly generated large-scale instances derived from real-world planning problems. All of the instances can be solved efficiently. Moreover, the proposed approach for the JSRB problem outperforms the conventional sequential planning approach and can incorporate the prediction of future cargo demand to avoid making myopic decisions.     

A chance-constrained based stochastic dynamic traffic assignment model: Analysis, formulation and solution algorithms

This paper is concerned with the system optimum-dynamic traffic assignment (SO-DTA) problem when the time-dependent demands are random variables with known probability distributions. The model is a stochastic extension of a deterministic linear programming formulation for SO-DTA introduced by Ziliaskopoulos (Ziliaskopoulos, A.K., 2000. A linear programming model for the single destination system optimum dynamic traffic assignment problem, Transportation Science, 34, 1–12). The proposed formulation is chance-constrained based and we demonstrate that it provides a robust SO solution with a user specified level of reliability. The model provides numerous insights and can be a useful tool in producing robust control and management strategies that account for uncertainty in applications where SO-DTA is relevant (e.g. evacuation modeling, computing alternate routes around freeway incidents and establishing lower bounds on network performance).     

Improved bush-based methods for network contraction

Calculating equilibrium sensitivity on a bush can be done very efficiently, and serve as the basis for a network contraction procedure. The contracted network (a simplified network with a few nodes and links) approximates the behavior of the full network but with less complexity. The network contraction method can be advantageous in network design applications where many equilibrium problems must be solved for different design scenarios. The network contraction procedure can also be used to increase the accuracy of subnetwork analysis. This method requires calculating travel time derivatives between two nodes, with respect to the demand between them, assuming that the flow distributes in a way that equilibrium is maintained. Previous research describes two methods for calculating these derivatives. This paper presents a third method, which is simpler, faster, and just as accurate. The method presented in this paper reformulates the linear system of equations defining these sensitivities as the solution to a convex programming problem, which can be solved by making minor modifications to static user equilibrium algorithms. In addition, the model is extended to capture the interactions between the path travel times and network flows, and a heuristic is proposed to compute these interactions. The accuracy and complexity of the proposed methodology are evaluated using the network of Barcelona, Spain. Further, numerical experiments on the Austin, Texas regional network validate its performance for subnetwork analysis applications.     

A model and a solution algorithm for airport common use check-in counter assignments

In this research we develop an integer programming model to assist airport authorities to assign common use check-in counters. Due to the many complicated factors that have to be considered in such a model, the problem size is expected to be huge, making its solution difficult and therefore not applicable to the real world. Therefore, we develop a heuristic method, containing three heuristic models, to solve the model. These heuristic models are formulated as zero–one integer programs that can be solved using the simplex method and the branch-and-bound technique. To test how well the proposed models and the solution method may be applied in the real world, we perform a case study concerning the operations of a major airport in Taiwan.     

Biofuel refinery location and supply chain planning under traffic congestion

This research focuses on planning biofuel refinery locations where the total system cost for refinery investment, feedstock and product transportation and public travel is minimized. Shipment routing of both feedstock and product in the biofuel supply chain and the resulting traffic congestion impact are incorporated into the model to decide optimal locations of biofuel refineries. A Lagrangian relaxation based heuristic algorithm is introduced to obtain near-optimum feasible solutions efficiently. To further improve optimality, a branch-and-bound framework (with linear programming relaxation and Lagrangian relaxation bounding procedures) is developed. Numerical experiments with several testing examples demonstrate that the proposed algorithms solve the problem effectively. An empirical Illinois case study and a series of sensitivity analyses are conducted to show the effects of highway congestion on refinery location design and total system costs.     

Finding the most reliable path with and without link travel time correlation: A Lagrangian substitution based approach

Path travel time reliability is an essential measure of the quality of service for transportation systems and an important attribute in travelers’ route and departure time scheduling. This paper investigates a fundamental problem of finding the most reliable path under different spatial correlation assumptions, where the path travel time variability is represented by its standard deviation. To handle the non-linear and non-additive cost functions introduced by the quadratic forms of the standard deviation term, a Lagrangian substitution approach is adopted to estimate the lower bound of the most reliable path solution through solving a sequence of standard shortest path problems. A subgradient algorithm is used to iteratively improve the solution quality by reducing the optimality gap. To characterize the link travel time correlation structure associated with the end-to-end trip time reliability measure, this research develops a sampling-based method to dynamically construct a proxy objective function in terms of travel time observations from multiple days. The proposed algorithms are evaluated under a large-scale Bay Area, California network with real-world measurements.     

An approximate dynamic programming approach for the empty container allocation problem

The objective of this study is to demonstrate the successful application of an approximate dynamic programming approach in deriving effective operational strategies for the relocation of empty containers in the containerized sea-cargo industry. A dynamic stochastic model for a simple two-ports two-voyages (TPTV) system is proposed first to demonstrate the effectiveness of the approximate optimal solution obtained through a simulation based approach known as the temporal difference (TD) learning for average cost minimization. An exact optimal solution can be obtained for this simple TPTV model. Approximate optimal results from the TPTV model utilizing a linear approximation architecture under the TD framework can then be compared to this exact solution. The results were found comparable and showed promising improvements over an existing commonly used heuristics. The modeling and solution approach can be extended to a realistic multiple-ports multiple-voyages (MPMV) system. Some results for the MPMV case are shown.