Showing papers in "Transportation Science in 1998"

A Survey of Optimization Models for Train Routing and Scheduling

Abstract: The aim of this paper is to present a survey of recent optimization models for the most commonly studied rail transportation problems. For each group of problems, we propose a classification of models and describe their important characteristics by focusing on model structure and algorithmic aspects. The review mainly concentrates on routing and scheduling problems since they represent the most important portion of the planning activities performed by railways. Routing models surveyed concern the operating policies for freight transportation and railcar fleet management, whereas scheduling models address the dispatching of trains and the assignment of locomotives and cars. A brief discussion of analytical yard and line models is also presented. The emphasis is on recent contributions, but several older yet important works are also cited.

Shortest Path Algorithms: An Evaluation Using Real Road Networks

Abstract: The classic problem of finding the shortest path over a network has been the target of many research efforts over the years. These research efforts have resulted in a number of different algorithms and a considerable amount of empirical findings with respect to performance. Unfortunately, prior research does not provide a clear direction for choosing an algorithm when one faces the problem of computing shortest paths on real road networks. Most of the computational testing on shortest path algorithms has been based on randomly generated networks, which may not have the characteristics of real road networks. In this paper, we provide an objective evaluation of 15 shortest path algorithms using a variety of real road networks. Based on the evaluation, a set of recommended algorithms for computing shortest paths on real road networks is identified. This evaluation should be particularly useful to researchers and practitioners in operations research, management science, transportation, and Geographic Information Systems.

Flight String Models for Aircraft Fleeting and Routing

Abstract: Given a schedule of flight legs to be flown by an airline, the fleet assignment problem is to determine the minimum cost assignment of flights to aircraft types, called fleets, such that each scheduled flight is assigned to exactly one fleet, and the resulting assignment is feasible to fly given a limited number of aircraft in each fleet. Then the airline must determine a sequence of flights, or routes, to be flown by individual aircraft such that assigned flights are included in exactly one route, and all aircraft can be maintained as necessary. This is referred to as the aircraft routing problem. In this paper, we present a single model and solution approach to solve simultaneously the fleet assignment and aircraft routing problems. Our approach is robust in that it can capture costs associated with aircraft connections and complicating constraints such as maintenance requirements. By setting the number of fleets to one, our approach can be used to solve the aircraft routing problem alone. We show how to extend our model and solution approach to solve aircraft routing problems with additional constraints requiring equal aircraft utilization. With data provided by airlines, we provide computational results for the combined fleet assignment and aircraft routing problems without equal utilization requirements and for aircraft routing problems requiring equal aircraft utilization.

Railway Timetabling Using Lagrangian Relaxation

Abstract: We present a novel optimization approach for the timetabling problem of a railway company, i.e., scheduling of a set of trains to obtain a profit maximizing timetable, while not violating track capacity constraints. The scheduling decisions are based on estimates of the value of running different types of service at specified times. We model the problem as a very large integer programming problem. The model is flexible in that it allows for general cost functions. We have used a Lagrangian relaxation solution approach, in which the track capacity constraints are relaxed and assigned prices, so that the problem separates into one dynamic program for each physical train. The number of dual variables is very large. However, it turns out that only a sm all fraction of these are nonzero, which one may take advantage of in the dual updating schemes. The approach has been tested on a realistic example suggested by the Swedish National Railway Administration. This example contains 18 passenger trains and 8 freight trains to be scheduled during a day on a stretch of single track, consisting of 17 stations. The computation times are rather modest and the obtained timetables are within a few percent of optimality.

A Two-Stage Stochastic Network Model and Solution Methods for the Dynamic Empty Container Allocation Problem

Abstract: Containerized liner trades have been growing steadily since the globalization of world economies intensified in the early 1990s. However, these trades are typically imbalanced in terms of the numbers of inbound and outbound containers. As a result, the relocation of empty containers has become one of the major problems faced by liner operators. In this paper, we consider the dynamic empty container allocation problem where we need to reposition empty containers and to determine the number of leased con tainers needed to meet customers? demand over time. We formulate this problem as a two-stage stochastic network: in stage one, the parameters such as supplies, demands, and ship capacities for empty containers are deterministic; whereas in stage two, these parameters are random variables. We need to make decisions in stage one such that the total of the stage one cost and the expected stage two cost is minimized. By taking advantage of the network structure, we show how a stochastic quasi-gradient method and a stochastic hybrid approximation procedure can be applied to solve the problem. In addition, we propose some new variations of these methods that seem to work faster in practice. We conduct numerical tests to evaluate the value of the two-stage stochastic model over a rolling horizon environment and to investigate the behavior of the solution methods with different implementations.

A Decomposition Approach to the Inventory Routing Problem with Satellite Facilities

Abstract: This paper presents a comprehensive decomposition scheme for solving the inventory routing problem in which a central supplier must restock a subset of customers on an intermittent basis. In this setting, the customer demand is not known with certainty and routing decisions taken over the short run might conflict with the long-run goal of minimizing annual operating costs. A unique aspect of the short-run subproblem is the presence of satellite facilities where vehicles can be reloaded and customer deliveries continued until the closing time is reached. Three heuristics have been developed to solve the vehicle routing problem with satellite facilities (randomized Clarke-Wright, GRASP, modified sweep). After the daily tours are derived, a parametric analysis is conducted to investigate the tradeoff between distance and annual costs. This leads to the development of the efficient frontier from which the decision maker is free to choose the most attractive alternative. The proposed procedures are tested on data sets generated from field experience with a national liquid propane distributor.

An Exact Constraint Logic Programming Algorithm for the Traveling Salesman Problem with Time Windows

Abstract: This paper presents a constraint logic programming model for the traveling salesman problem with time windows which yields an exact branch-and-bound optimization algorithm without any restrictive assumption on the time windows Unlike dynamic programmi ng approaches whose performance relies heavily on the degree of discretization applied to the data, our algorithm does not suffer from such space-complexity issues The data-driven mechanism at its core more fully exploits pruning rules developed in opera tions research by using them not only a priori but also dynamically during the search Computational results are reported and comparisons are made with both exact and heuristic algorithms On Solomon's well-known test bed, our algorithm is instrumental in achieving new best solutions for some of the problems in set RC2 and strengthens the presumption of optimality for the best known solutions to the problems in set C2m

Queue Spillovers in Transportation Networks with a Route Choice

Abstract: This paper explores some of the traffic phenomena that arise when drivers have to navigate a network in which queues back up past diverge intersections. If a diverge provides two alternative routes to the same destination and the shorter route has a bottleneck that generates a queue, one would expect that queue to stabilize at an equilibrium level where the travel time on both routes is roughly equal. If the capacity of the alternative route is unlimited then this network can accommodate any demand level. However, if the bottleneck is so close to the upstream end of the link that the equilibrium queue cannot be contained in the link, then the trip time on the queued route cannot grow to match that on the alternate route. This means that the alternative route can never be attractive, even if the queue spills back past the diverge, and that drivers approaching the diverge will act as if the alternative route did not exist. As a result, a steady flow into the system greater than the capacity of the bottleneck will cause a queue to grow all the way back to the origin (blocking it). The final result is an "oversaturated static state" where the queue regulates the input flow into the system. Curiously, if the bottleneck capacity of this network is reduced below a critical level (or is eliminated altogether) then the alternative route becomes attractive again and the system cannot reach the saturation point. This phenomenon is explored in the paper, where it is also shown that: i) a network can become permanently oversaturated/undersaturated as a result of a temporary increase/decrease in link capacity, ii) even under the most favorable assumptions, and in contrast to the equivalent assignment problem with point queues, a network can be stable both in an oversaturated and an under-saturated state, and iii) temporary endogenous disturbances can permanently reverse the saturation state of a network. These findings suggest that in certain situations the time-dependent traffic assignment problem with physical queues is chaotic in nature and that (as in weather forecasting) it may be impossible to obtain input data with the required accuracy to make reliable predictions of cumulative output flows.

Modeling Train Delays in Urban Networks

Abstract: The reliability of urban passenger trains is a critical performance measure for passenger satisfaction and ultimately market share. A delay to one train in a peak period can have a severe effect on the schedule adherence of other trains. This paper presents an analytically based model to quantify the expected positive delay for individual passenger trains and track links in an urban rail network. The model specifically addresses direct delay to trains, knock-on delays to other trains, and delays at scheduled connections. A solution to the resultant system of equations is found using an iterative refinement algorithm. Model validation, which is carried out using a real-life suburban train network consisting of 157 trains, shows the model estimates to be on average within 8% of those obtained from a large scale simulation. Also discussed, is the application of the model to assess the consequences of increased scheduled slack time as well as investment strategies designed to reduce delay.

The Operational Airline Crew Scheduling Problem

Abstract: This paper describes the operational airline crew scheduling problem and represents a first published attempt to solve it. The problem consists of modifying, as necessary, personalized planned monthly assignments of airline crew members during day-to-day operations. It requires overing, at minimal cost, all flight segments from a given time period with available crew while minimizing the disturbances of crew members. To generate modified pairings for selected crew members, both the classical crew pairing problem and the problem of constructing personalized monthly assignments must be treated simultaneously. An optimization approach is proposed for the problem in which the flight schedule is fixed and represents input data. The problem is mathematically formulated as a Set Partitioning type problem, and a column generation method embedded in a branch-and-bound search tree has been implemented to solve it. Good results, from the point of view of both solution times and achieved objectives, have been obtained on generated test problems. Because the solution time is reasonable, several different scenarios of the same problem may be solved. A final decision can then be made by considering all scenarios and choosing the one whose solution is the best in the given situation.

The Four-Day Aircraft Maintenance Routing Problem

Abstract: Federal aviation regulations require that all aircraft undergo maintenance after flying a certain number of hours. Most major U.S. airlines observe the maintenance regulations by requiring that aircraft spend a night at a maintenance station after at most three or four days of flying. In addition, some airlines require that every aircraft goes through a special maintenance station for what is commonly called a balance check. Airlines usually schedule routine maintenance only at night so as not to cut into aircraft utilization. The maintenance routing problem is to find a routing of the aircraft that satisfies the short-term routine maintenance requirements. In Gopalan, R. and Talluri, K. T. ("The Aircraft Maintenance Routing Problem," Opns. Res. in press) we modeled this problem as one of generating an appropriate directed graph (called a line-of-flight-graph), and of finding a special Euler Tour called the k-day Maintenance Euler Tour (k-MET, for k = 3, 4,...) in that directed graph-for finding a maintenance routing in which the aircraft would spend at most k days of flying before overnighting at a maintenance station and have an opportunity for a balance-check. In the same paper we gave a polynomial-time algorithm for finding a 3-MET, if one exists, in the directed graph. In this paper we consider the routing problem when the requirement is to overnight at a maintenance station after at most four days of flying and to undergo the balance check every n days, where n is the number of planes in the fleet of the equipment type under consideration. We show that this problem is NP-complete; in fact, that the k-MET problem is NP-complete for all k ≥ 4. When the number of maintenance stations is exactly one, we show that the 4-MET problem can be solved by solving an appropriate bipartite matching problem; and hence in polynomial time. As a corollary to this result, we show that when there is no balance check station visit requirement, the four-day routing problem, in a given LOF-graph, can be solved (without any restrictions on the number of maintenance stations) in polynomial time. We show how our polynomial-time algorithms for the 3-MET problem and the restricted 4-MET problem can be used to design effective heuristics for the 4-MET problem.

Constructing Railroad Blocking Plans to Minimize Handling Costs

Abstract: On major domestic railroads, a typical general merchandise shipment may pass through many classification yards on its route from origin to destination. At these yards, the incoming traffic, which may consist of a number of individual shipments, is reclassified (sorted and grouped together) to be placed on outgoing trains. Each reclassification incurs costs due to handling and delay. To prevent shipments from being reclassified at every yard they pass through, several shipments may be grouped together to form a block. A block has associated with an origin destination pair that may or may not be the origin or destination of any of the individual cars contained in the block. The objective of the railroad blocking problem is to choose which blocks to build at each yard and to assign sequences of blocks to deliver each shipment to minimize total mileage, handling, and delay costs. We model the railroad blocking problem as a network design problem in which yards are represented by nodes and blocks by arcs. Our model is intended as a strategic decision-making tool. We develop a column generation, branch-and-bound algorithm in which attractive paths for each shipment are generated by solving a shortest path problem. Our solution approach is unique in constraining the classification resources of each yard and simultaneously solving for different priority classes of shipments. We implement our algorithm and find near-optimal solutions in about one hour for the blocking problem of a large domestic railroad, in which the paths that shipments may take in the physical network are restricted. The resulting network design problem has 150 nodes, 1300 commodities, and 6800 possible arcs (blocks). We test the robustness of our solution on 19 test instances that are variations of the data for the real-world problems. If shipments are restricted to following one of a limited number of paths in the rail network, then, in four hours or less, our algorithm finds solutions within 0.4% of optimal for all test cases. Furthermore, the solutions obtained are no more than 3.9% from optimal even if all possible paths are allowed.

Models and Algorithms for Container Allocation Problems on Trains in a Rapid Transshipment Shunting Yard

Abstract: Multimodal transport of containers can be an alternative to the road transportation but it requires to be competitive in terms of quality of service and price. In rail rail container terminals, new techniques are developed to facilitate rapid transfers of the containers between trains. In this article, we address the problem of the optimization of the operations management of rapid rail rail transshipment shunting yards. We are interested specifically in the optimization of containers allocation on tra ins (for the initial loading and their reloading after transshipment). We have developed a class of models with different levels of complexity and realism and we have proposed optimal and heuristic methods to solve them. The experimental results on realistic datasets are very promising in terms of the minimization of the container moves in a terminal as well as the use and sizing of the handling equipments.

Dynamic Control of Logistics Queueing Networks for Large-Scale Fleet Management

Abstract: Dynamic fleet management problems are normally formulated as networks over dynamic networks. Additional realism usually implies the inclusion of complicating constraints, typically producing exceptionally large integer programs. In this paper, we present for the first time the formulation of dynamic fleet management problems in an optimal control setting, using a novel formulation called a Logistics Queueing Network (LQN). This formulation replaces a single, large optimization problem with a series of very small problems that involve little more than solving a single sort at each point in space and time. We show that this approach can produce solutions that are within a few percent of a global optimum but provide for considerably more flexibility than standard linear programs.

The Preferential Bidding System at Air Canada

Abstract: This paper describes the Preferential Bidding Problem solved in the airline industry to construct personalized monthly schedules for pilots and officers. This problem consists in assigning to crew members pairings, days off, annual leaves, training periods, etc., while considering a set of weighted bids that reflect individual preferences. This assignment must be done under strict seniority restrictions: the construction of a maximum-score schedule for a particular crew member must never be done at the expense of a more senior employee. This research and development project has resulted in the Preferential Bidding System that has been used at Air Canada since May 1995. The solution process is summarized as follows. For each employee, from the most senior to the most junior, a so-called residual problem is solved: given an employee and a set of unassigned pairings, the solution to an integer linear program determines the employee's maximum-score schedule while taking into account all the remaining employees. The residual problem is solved by column generation embedded in a branch-and-bound tree. Integer solutions are obtained by using very efficient cutting planes, without which it would have been impossible to solve some of these residual problems.

Implicit Enumeration of Hyperpaths in a Logit Model for Transit Networks

Abstract: This paper investigates the application of a logit model to urban transit networks where every set of competitive transit lines is described by a particular graph structure called hyperpath. It shows that a sequential form of the logit model transcends the inherent limitations of the global form while retaining the algorithmic advantages similar to those obtained with the ordinary logit model for private vehicle networks.

Real-Time Optimization of Containers and Flatcars for Intermodal Operations

Abstract: We propose a dynamic model for optimizing the flows of flatcars that considers explicitly the broad range of complex constraints that govern the assignment of trailers and containers to a flatcar. The problem is formulated as a logistics queueing network which can handle a wide range of equipment types and complex operating rules. The complexity of the problem prevents a practical implementation of a global network optimization model. Instead, we formulate a global model with the specific goal of providing network information to local decision makers, regardless of whether they are using optimization models at the yard level. Thus, our approach should be relatively easy to implement given current rail operations. Initial experiments suggest that a flatcar fleet that is managed locally, without the benefit of our network information, can achieve the same demand coverage as a fleet that is 10 percent smaller, but is managed locally with our network information.

Improved Empty Freight Car Distribution

Abstract: In this paper we consider the problem of distributing empty freight cars in a railway company. We describe and analyze the current planning process, identify the shortcomings of the process, and stress the importance of a reliable distribution process for satisfying customer demand and reducing capital costs. We show how the process can be improved using an optimization model which includes capacity constraints on the trains and adheres explicitly to the arrival and departure times of the trains. The optimization model can be characterized as a multicommodity network flow model with integer requirements. Computational tests show that the model can be solved in acceptable time for real size problems, and indicate that the model generates distribution plans that can improve the quality of the planning process.

An Approximate Model and Solution Approach for the Long-Haul Crew Pairing Problem

Abstract: The crew pairing problem requires the coverage of a set of long-haul flights by a minimum cost set of crew pairings. A crew pairing is a sequence of flights flown by one crew, starting and ending at the same location, and satisfying a variety of work regulations and collective bargaining agreements. We present a new solution approach that solves first an approximate model of the problem and then uses its solution as an advanced start solution for conventional approaches. Using data provided by a long-haul airline, we demonstrate that our new approach can be used with a deadhead selector to identify deadheads quickly that might improve significantly the quality of the crew pairing solution. Deadheads, flights to which crews are assigned as passengers, reposition crews for better utilization. We speed up the solution process by using our advanced start solution and by quickly providing good lower bounds on the optimal solution values. Our experiments show that the lower bounds are on average within 0.85% of the optimal solution value. Further, we show that compared to existing methods, we reduce solution costs and run times by an average of 20% and over 80%, respectively.

Freight Handling Personnel Scheduling at Air Cargo Terminals

Abstract: More than ever before, the airline industry must manage its operations very carefully if it is to adapt to the changes in its environment. One important planning problem is the scheduling of freight handling employees at air cargo terminals. Indeed, this area is particularly critical for achieving cost reductions while maintaining customer service levels. This paper presents a new model and an appropriate solution approach for the work-scheduling problem. In particular, a demand leveling procedure is introduced to take advantage of the flexibility that is often encountered in such problems. Experimental results obtained with actual data from a major airline are also presented.

The Multiple Connections Multi-Airport Ground Holding Problem: Models and Algorithms

Abstract: In recent years air traffic has dramatically increased without a corresponding development of airports. Therefore, airports? limited capacity causes air traffic congestion and consequent expensive delays. The only strategy that can be applied in the short term with low investments aims at the optimal management of present resources; its principal device is Ground Holding, which consists of delaying an aircraft take off whenever it is foreseen it will not land in time because of congestion. We consider a traffic situation with "multiple connections" or "banking," i.e., the situation where some flights are assigned a set of "preceding" flights; no "successive" flight can start until all its preceding flights have landed. The problem consists of distributing delays to flights, so as to minimize the total delay cost, by respecting airport capacity, connections, and time constraints imposed by airlines. We construct an integer linear programming model and we solve it to optimality with CPLEX. Because the computation time is too high (hours) for real-world instances, we propose an alternative heuristic algorithm, which shows a very low computation time (seconds) and acceptable errors when tested on 30 realistic instances with strongly diversified data.

Scheduling of Time-Shared Jet Aircraft

Abstract: Motivated by a real application, we consider the following aircraft scheduling problem. At any time, the aircraft are at different locations or are serving a customer and new customer requests arrive, each consisting of a departure location, departure time, and destination. Our objective is to satify these requests (by subcontracting extra aircraft if necessary) at minimum cost under additional constraints of maintenance requirements and previously scheduled trips. We show that the jet aircraft schedul ing problem is NP complete and discuss three special cases. We show that the second and third special cases are also NP complete. We provide a polynomial time network flow based algorithm for the first special case and a pseudo-polynomial time dynamic programming algorithm for the second special case. We formulate the problem as a 0-1 integer program and observe that most small and medium size problems can be solved by Cplex. For larger and difficult instances, we provide a fast heuristic with good performance.

A Tactical Decision Support System for Empty Railcar Management

Abstract: In this paper, we present a tactical model to assist in the task faced by the railroad industry on a day-to-day basis of centrally managing the distribution and repositioning of empty railcars for shipping automobiles. The problem involves a group of eight principal automobile manufacturers (shippers) who have pooled their autorack resources (railcars for shipping automobiles) to improve utilization and reduce the number of empty miles logged. However, this consolidation gives rise to various equity and priority issues related to timeliness in service, particularly in the case of shortages. Accordingly, our model takes into account such practical issues, including uncertainties in transit times, priorities with respect to time and demand locations, multiple objectives related to minimizing different degrees of latenesses in delivery, and blocking considerations. We investigate the performance of two principal models that have been developed for this purpose. The first model, TDSS1 incorporates all the identified features of the problem except for blocking (a consolidation of shipments from any origin to only a limited number of destinations), and results in a network formulation of the problem. The second model, TDSS2 extends TDSS1 by further including blocking considerations, and results in a network flow problem with side constraints and discrete side variables. We then show how the resulting mixed-integer-programming formulation can be enhanced via some partial convex hull constructions. To accommodate the strict run-time limit requirements imposed in practice, 21 principal heuristics are developed and tested to solve this problem. By examining the performance of these procedures with respect to speed of operation and the quality of solutions produced on a test bed of real-world problem instances, we prescribe a solution strategy for implementation in making production runs.

Optimizing Auto-Carrier Loading

Abstract: Delivering automobiles, vans, and trucks to auto dealerships is a unique problem among the routing and scheduling problem class. A subproblem is that of loading the vehicles onto an auto-carrier (equipment specially designed for this purpose). Allocation of the vehicles to this equipment has typically been done by means of trial and error and heuristic methods. This paper shows how the loading aspect of the delivery problem can be solved using optimizing methods. The approach has the potential of replacing the empirical methods currently used in practice.

A Probabilistic Approach to Evaluate Strategies for Selecting a Parking Space

Abstract: Upon entering the parking lot of a facility, a driver must select a parking space. This paper proposes and evaluates strategies which are intended to find a "good" parking space. Three performance measures are defined for evaluating the "goodness" of a selected parking space. One of these is based on the amount of walking required of the customer, another is based on the amount of driving required, and the third is a combination of the first two and represents the time required for the customer to reach the front door of the facility. Two strategies are defined for selecting a parking space. The first is simple and somewhat conservative, whereas the second strategy is more complicated and aggressive. A probabilistic approach is used to evaluate the three performance measures for each strategy using a parking lot of somewhat "typical" configuration. This approach treats driver decisions and parking space availability as random experiments, and uses conditional probability to evaluate the performance of the strategies.

Acceptance and Dispatching Policies for a Distribution Problem

Abstract: A dynamic and stochastic distribution problem with a number of terminals and a fleet of vehicles is analyzed. Customers request the transportation of batches of loads between different origins and destinations. A request can be accepted or rejected; if the request is accepted, a reward is received. Holding costs for vehicles and loads at terminals as well as transportation costs are included in the model. The objective is to determine a policy for accepting transportation requests and for dispatching vehicles that maximizes the expected value (rewards minus costs) of operating the distribution system. A Markov decision process model is developed, optimal policies are characterized, and algorithms that exploit the structure of the problem are developed.

Combinatorial Issues in Air Traffic Optimization

Abstract: The current airway network used by aircraft is composed of a set of segments that intersect on special points defined by radio beacons emitting signals from the ground. This network leads to excess flight length, which for the European network is estimated to be 8%. In the near future, the Global Positioning System, which can determine precisely the location of aircraft, might allow the design of a network without using any ground fixed radio beacons. Therefore, we can project a new skyway network with straight airways between airports, allowing an airway to change level one or more times between its origin and destination to avoid potential conflict points. We present some segment set combinatorial issues to achieve such a network. In particular, we propose heuristics or algorithms for the problems of the maximum clique, the coloring, the N-coloring, and other more general problems of coloring of a set of segments. Finally, we discuss some results based on actual data analysis.