Abstract: This paper presents a survey of the research on the vehicle routing problem with time windows (VRPTW). The VRPTW can be described as the problem of designing least cost routes from one depot to a set of geographically scattered points. The routes must be designed in such a way that each point is visited only once by exactly one vehicle within a given time interval, all routes start and end at the depot, and the total demands of all points on one particular route must not exceed the capacity of the vehicle. Both traditional heuristic route construction methods and recent local search algorithms are examined. The basic features of each method are described, and experimental results for Solomon's benchmark test problems are presented and analyzed. Moreover, we discuss how heuristic methods should be evaluated and propose using the concept of Pareto optimality in the comparison of different heuristic approaches. The metaheuristic methods are described in the second part of this article.

Abstract: Classical facility location models like the P-median problem (PMP) and the uncapacitated fixed-charge location problem (UFLP) implicitly assume that, once constructed, the facilities chosen will always operate as planned. In reality, however, facilities "fail" from time to time due to poor weather, labor actions, changes of ownership, or other factors. Such failures may lead to excessive transportation costs as customers must be served from facilities much farther than their regularly assigned facilities. In this paper, we present models for choosing facility locations to minimize cost, while also taking into account the expected transportation cost after failures of facilities. The goal is to choose facility locations that are both inexpensive under traditional objective functions and also reliable. This reliability approach is new in the facility location literature. We formulate reliability models based on both the PMP and the UFLP and present an optimal Lagrangian relaxation algorithm to solve them. We discuss how to use these models to generate a trade-off curve between the day-to-day operating cost and the expected cost, taking failures into account, and we use these trade-off curves to demonstrate empirically that substantial improvements in reliability are often possible with minimal increases in operating cost.

Abstract: For each point of a road network, let there be given the number of cars starting from it, and the destination of the cars. Under these conditions one wishes to estimate the distribution of traffic flow. Whether one street is preferable to another depends not only on the quality of the road, but also on the density of the flow. If every driver takes the path that looks most favorable to him, the resultant running times need not be minimal. Furthermore, it is indicated by an example that an extension of the road network may cause a redistribution of the traffic that results in longer individual running times.

Abstract: Traveling salesman problems with profits (TSPs with profits) are a generalization of the traveling salesman problem (TSP), where it is not necessary to visit all vertices. A profit is associated with each vertex. The overall goal is the simultaneous optimization of the collected profit and the travel costs. These two optimization criteria appear either in the objective function or as a constraint. In this paper, a classification of TSPs with profits is proposed, and the existing literature is surveyed. Different classes of applications, modeling approaches, and exact or heuristic solution techniques are identified and compared. Conclusions emphasize the interest of this class of problems, with respect to applications as well as theoretical results.

Abstract: Traffic operations in public walking spaces are to a large extent determined by differences in pedestrian traffic demand and infrastructure supply. Congestion occurs when pedestrian traffic demand exceeds the capacity. In turn, the latter is determined by a number of factors, such as the width of the bottleneck and the wall surface, as well as the interaction behavior of the pedestrians passing the bottleneck.This article discusses experimental findings of microscopic pedestrian behavior in case of bottlenecks. Results for both a narrow bottleneck and a wide bottleneck are discussed and compared to the results of an experiment without a bottleneck. It is shown how pedestrians inside bottlenecks effectively form layers or trails, the distance between which is approximately 45 cm. This is less than the effective width of a single pedestrian, which is around 55 cm. The layers are thus overlapping, a phenomenon which is referred to as the "zipper" effect. The pedestrians within these layers follow each other at 1.3 seconds, irrespective of the considered experiment. For the narrow bottleneck case (width of one meter) two layers are formed; for the wide bottleneck case (width of two meters), four or five layers are formed, although the life span of these layers is rather small.The zipper effect causes the capacity of the bottleneck to increase in a stepwise fashion with the width of the bottleneck, at least for bottlenecks of moderate width (less than 3 m). This has substantial implications for the design of walking facilities.

Abstract: In the berth-allocation problem (BAP) the aim is to optimally schedule and assign ships to berthing areas along a quay. The objective is the minimization of the total (weighted) service time for all ships, defined as the time elapsed between the arrival in the harbor and the completion of handling. Two versions of the BAP are considered: the discrete case and the continuous case. The discrete case works with a finite set of berthing points. In the continuous case ships can berth anywhere along the quay. Two formulations and a tabu search heuristic are presented for the discrete case. Only small instances can be solved optimally. For these sizes the heuristic always yields an optimal solution. For larger sizes it is always better than a truncated branch-and-bound applied to an exact formulation. A heuristic is also developed for the continuous case. Computational comparisons are performed with the first heuristic and with a simple constructive procedure.

Abstract: Many companies with consumer direct service models, especially grocery delivery services, have found that home delivery poses an enormous logistical challenge due to the unpredictability of demand coupled with strict delivery windows and low profit margin products. These systems have proven difficult to manage effectively and could benefit from new technology, particularly to manage the interaction between order capture and order delivery. In this article, we define routing and scheduling problems that incorporate important features of this emerging business model and propose algorithms, based on insertion heuristics, for their solution. In the proposed home delivery problem, the company decides which deliveries to accept or reject as well as the time slot for the accepted deliveries so as to maximize expected profits. Computational experiments reveal the importance of an approach that integrates order capture with order delivery and demonstrates the quality and value of the proposed algorithms.

Abstract: We propose local search algorithms for the vehicle routing problem with soft time-window constraints. The time-window constraint for each customer is treated as a penalty function, which is very general in the sense that it can be nonconvex and discontinuous as long as it is piecewise linear. In our algorithm, we use local search to assign customers to vehicles and to find orders of customers for vehicles to visit. Our algorithm employs an advanced neighborhood, called the cyclic-exchange neighborhood, in addition to standard neighborhoods for the vehicle routing problem. After fixing the order of customers for a vehicle to visit, we must determine the optimal start times of processing at customers so that the total penalty is minimized. We show that this problem can be efficiently solved by using dynamic programming, which is then incorporated in our algorithm. We report computational results for various benchmark instances of the vehicle routing problem. The generality of time-window constraints allows us to handle a wide variety of scheduling problems. As an example, we mention in this paper an application to a production scheduling problem with inventory cost, and report computational results for real-world instances.

Abstract: Airline crew scheduling algorithms widely used in practice assume no disruptions. Because disruptions often occur, the actual cost of the resulting crew schedules is often greater. We consider algorithms for finding crew schedules that perform well in practice. The deterministic crew scheduling model is an approximation of crew scheduling under uncertainty with the assumption that all pairings will operate as planned. We seek better approximate solution methods for crew scheduling under uncertainty that still remain tractable. We give computational results from three fleets that indicate that the crew schedules obtained from our method perform better in a model of operations with disruptions than the crew schedules found via deterministic methods. Under mild assumptions we provide a lower bound on the cost of an optimal crew schedule in operations, and we demonstrate that some of the crew schedules found using our method perform very well relative to this lower bound.

Abstract: This paper presents two different models and algorithms for integrated vehicle and crew scheduling in the multiple-depot case. The algorithms are both based on a combination of column generation and Lagrangian relaxation. Furthermore, we compare those integrated approaches with each other and with the traditional sequential one on randomly generated, as well as real-world, data instances for a suburban/extraurban mass transit system. To simulate such a transit system, we propose a new way of randomly generating data instances such that their properties are the same as for our real-world instances.

Abstract: In the locomotive-scheduling problem (or the locomotive-assignment problem), we must assign a consist (a set of locomotives) to each train in a preplanned train schedule so as to provide each train with sufficient locomotive power to pull the train from its origin to its destination. Locomotive-scheduling problems are among the most important problems in railroad scheduling. In this paper, we report the results of a study on the locomotive-scheduling problem as it is faced by CSX Transportation, a major U.S. railroad company. We consider the planning version of the locomotive-scheduling model (LSM) in which multiple types of locomotives exist, and we need to decide which set of locomotives should be assigned to each train. We present an integrated model that determines: the set of active and deadheaded locomotives for each train; the light-traveling locomotives from power sources to power sinks; and train-to-train connections (for which we specify which inbound trains and outbound trains can directly connect). An important feature of our model is that we explicitly consider consist bustings and consistency. A consist is said to be busted when a set of locomotives coming on an inbound train is broken into subsets to be reassigned to two or more outbound trains. A solution is consistent over a week if a train receives the same locomotive assignment each day that it runs. We will provide a mixed-integer programming (MIP) formulation of the locomotive-assignment problem. However, an MIP of this size cannot be solved to optimality or near optimality in acceptable running times using commercially available software. Using problem decomposition, integer programming, and very large-scale neighborhood search, we have developed a solution technique to solve this problem within 30 minutes of computation time on a Pentium III computer. Our solution obtained a potential savings of over 400 locomotives over the solution obtained by the in-house software developed by CSX.

Abstract: Passengers on a transit network with common lines are often faced with the problem of choosing between either to board the arriving bus or to wait for a faster one. Many assignment models are based on the classical assumption that at a given stop passengers board the first arriving carrier of a certain subset of the available lines, often referred to as the attractive set. In this case, it has been shown that, if the headway distributions are exponential, then an optimal subset of lines minimizing the passenger travel time can be easily determined. However, when online information on future arrivals of buses are posted at the stop, it is unlikely that the above classical assumption holds. In this case, passengers may choose to board a line that offers the best combination of displayed waiting time and expected travel time to their destination once boarded. In this paper, we propose a general framework for determining the probability of boarding each line available at a stop when online information on bus waiting times is provided to passengers. We will also show that the classical model without online information may be interpreted as a particular instance of the proposed framework, this way achieving an extension to general headway distributions. The impact of the availability of information regarding bus arrivals and that of the regularity of transit lines on the network loads, as well as on the passenger travel times, will be illustrated with small numerical examples.

Abstract: Train units need regular preventive maintenance. Given the train units that require maintenance in the forthcoming one to three days, the rolling stock schedule must be adjusted so that these urgent units reach the maintenance facility in time. Maroti and Kroon (2004) propose a model that requires a large amount of input data. In this paper we describe a less involved multicommodity flow type model for this maintenance routing problem. We study the complexity of the problem. It turns out that the feasibility problem for a single urgent train unit is polynomially solvable but the optimization version is NP-hard. Finally, we report our computational experiments on practical instances of NS Reizigers, the main Dutch operator of passenger trains.

Abstract: The aim of this paper is to present a model structure that analyzes the hub-spoke network design issue within a competitive framework. Under deregulation, airlines have developed hub-and-spoke networks, enabling them to increase frequency by aggregating demand and to prevent entry into the marketplace by reducing airfares. While liberalization in the United States and Europe was undertaken to increase competition, the results in this direction are unclear. This research evaluates airline profits based on a microeconomic theory of airline behavior under deregulation and the effect on hub-and-spoke networks. Through a two-stage, Nash best-response game, we search for equilibria in the air transportation industry. The game is applied to Western Europe, where profitable hubs and monopolistic equilibria are clearly identifiable, and duopolistic equilibria are potentially viable, given sufficient demand.

Abstract: In this article, we introduce a new arc routing problem that we call the profitable arc tour problem. This problem is defined on a graph in which profits and travel costs are associated with the arcs. The objective is to find a set of cycles in the graph that maximizes the collection of profit minus travel costs, subject to constraints limiting the number of times that profit is available on arcs and the maximal length of cycles. The problem is related both to constrained flow problems and to vehicle-routing problems. We tackle it from this standpoint and propose a branch-and-price algorithm for its solution. In the column-generation phase, the issue of the collection decisions while traveling through the arcs is addressed. In the branching phase, the fact that viewing solutions in terms of flow variables regularly induces an integer flow matrix leads us to introduce a branching method called the flow-splitting method. Finally, the relationships of this problem with constrained flow optimization are taken into account in an initial phase of the algorithm.

Abstract: This paper addresses a fleet-sizing problem in the context of the truck-rental industry. Specifically, trucks that vary in capacity and age are utilized over space and time to meet customer demand. Operational decisions (including demand allocation and empty truck repositioning) and tactical decisions (including asset procurements and sales) are explicitly examined in a linear programming model to determine the optimal fleet size and mix. The method uses a time-space network, common to fleet-management problems, but also includes capital cost decisions, wherein assets of different ages carry different costs, as is common to replacement analysis problems. A two-phase solution approach is developed to solve large-scale instances of the problem. Phase I allocates customer demand among assets through Benders decomposition with a demand-shifting algorithm assuring feasibility in each subproblem. Phase II uses the initial bounds and dual variables from Phase I and further improves the solution convergence without increasing computer memory requirements through the use of Lagrangian relaxation. Computational studies are presented to show the effectiveness of the approach for solving large problems within reasonable solution gaps.

Abstract: We consider a system in which multiple items are transferred from a warehouse or a plant to a retailer through identical capacitated vehicles, or by identical freight wagons. Any mixture of the items may be loaded onto a vehicle. The retailer is facing dynamic deterministic demand for several items, over a finite planning horizon. A vehicle incurs a fixed cost for each trip made from the warehouse to the retailer. In addition, there exist item-dependent variable shipping costs and inventory holding costs at the retailer, which are both constant over time. The objective is to find a shipment schedule that minimizes the total cost, while satisfying demand on time.We address and partially resolve the question regarding the problem's complexity by introducing a dynamic programming algorithm whose complexity is polynomial for a fixed number of items, but exponential otherwise. Our dynamic programming formulation is based on properties satisfied by the optimal solution, and uses an innovative way for partitioning the problem into subproblems.

Abstract: An optimization-based approach is developed for routing a just-in-time (JIT) supply pickup and delivery system. The approach defines routes among suppliers serving a large JIT assembly plant, the timing of these routes, and the frequency that they are run (implicitly defining parts quantities picked up each visit). The solutions satisfy various operational constraints at the JIT facility, including requirements for high-frequency/small-quantity deliveries and limits on space for parts storage. The solution space we consider here is restricted by an operational discipline that the industry calls common frequency routing. Under this system we only consider routing designs where each part source is being served by a single route run at a fixed daily frequency instead of designs where multiple routes visit that supplier, each potentially run at a different frequency. This dramatically reduces dimensionality of the problem and is also known to provide both management and operational advantages in practice. In solving the formulation, column generation and tabu search strategies have been developed, the latter suitable for realistic-sized problems. The utility of the approach is illustrated through a number of examples.

Abstract: In the skip delivery problem (SDP), a fleet of vehicles must deliver skips to a set of customers. Each vehicle has a maximum capacity of two skips, and has to start and end its tour at a central depot. The demand of each customer can be greater than the capacity of the vehicles. The objective is to minimize the cost of the total distance traveled by the vehicles to serve all the customers. We show that the SDP is solvable in polynomial time, while its generalization to the case where all vehicles have a capacity greater than two, known as the split delivery vehicle routing problem (SDVRP), is shown to be NP-hard, even under restricted conditions on the costs. We also show that, if the costs are symmetrical and satisfy the triangle inequality, the SDP is reducible in polynomial time to a problem of possibly smaller size, where each customer has unitary demand. This property allows a remarkable simplification of the problem.

Abstract: We present and analyse a model of the combined choice of departure time and route in a congested road network. Using the property of equilibrium solutions that for each origin-destination pair the total cost associated with travel is identical for all travellers, we establish a general result that relates assignments to various components of cost. The analysis is developed to include time-varying tolls and to establish a formula that will induce any specified inflow profile as an equilibrium. We introduce an iteration operator to calculate equilibrium inflow profiles, and present the results of example calculations for a range of test problems.

Abstract: The current airline practice in conducting fleet assignments is to begin assigning aircraft capacity to scheduled flights well in advance of departures. However, the accuracy of the passenger demand forecast improves markedly over time, and revisions to the initial fleet assignment become naturally pertinent when the observed demand differs considerably from the assigned aircraft capacities. The demand-driven refleeting (DDR) approach proposed in this paper offers a dynamic reassignment of aircraft capacities to the flight network, when improved demand forecasts become available, so as to maximize the total revenue. Because of the need to preserve the initial crew schedule, this reassignment approach is limited within a single family of aircraft types and to the flights assigned to this particular family. This restriction makes it computationally tractable to include more relevant path-level demand information into the DDR model. Accordingly, we construct a mixed-integer programming model for this enhanced problem context and study its polyhedral structure to explore ways for tightening its representation and for deriving certain classes of valid inequalities. Various schemes for implementing such reformulation techniques are investigated and tested using a set of simulated and real instances obtained from United Airlines.

Abstract: Motivated by a real application that requires effective medium-term planning in intermodal transportation, we develop a mathematical programming model that jointly considers pricing and operations planning. Building on available work from marketing and transportation literature, we develop a new method for considering pricing, as well as develop distinctly different operations submodels that are appropriate for this context. We solve the resulting nonlinear optimization problem (with continuous and 0-1 variables) to optimality through a decomposition that exploits the structure of subproblems. Computational testing on industrial-sized problems shows that our algorithm is very quick, allowing for effective scenario testing and "what-if" analysis.

Abstract: The single-sink, fixed-charge transportation problem has a variety of applications, including supplier selection, product distribution/fleet selection, and process selection. In this paper we present a dynamic programming algorithm for solving this important problem that is very easy to implement and that improves considerably in terms of computational attractiveness on the best methods in the literature.

Abstract: The multiple pairs shortest path problem (MPSP) arises in many applications where the shortest paths and distances between only some specific pairs of origin-destination (OD) nodes in a network are desired. The traditional repeated single-source shortest path (SSSP) and all pairs shortest paths (APSP) algorithms often do unnecessary computation to solve the MPSP problem. We propose a new shortest path algorithm to save computational work when solving the MPSP problem. Our method is especially suitable for applications with fixed network topology but changeable arc lengths and desired OD pairs. Preliminary computational experiments demonstrate our algorithm's superiority on airline network problems over other APSP and SSSP algorithms.

Abstract: In network models for dynamic traffic assignment (DTA), the travel time on a link is often treated as a function of the number of vehicles on the link. Instead of applying this model to the whole link, we divide the link into segments, apply the model (suitably adjusted) sequentially to these segments, and investigate how the solution is affected by various levels of discretisation (as the discretisation is refined, the solution converges to the solution of the Lighthill-Whitham-Richards (LWR) model). We also restrict the link (and segment) travel-time function to ensure that it satisfies a first-in-first-out (FIFO) property and explore how this affects and restricts the form of the flow-density functions used in the LWR model. We numerically illustrate the solution of the discretised model for various travel-time functions and patterns of inflows, for both homogeneous and inhomogeneous links. Subject to the above restriction on the flow-density function, the numerical results suggest that dividing "long" links into even a few segments can make the model solution closely approximate the LWR solution, while retaining tractability in the network model. We also observe, for example, that the whole-link (undescretised) travel-time model has a "flattening" effect on the profiles of flows and travel times (this effect can be reduced to any desired extent by using discretisation); and that the travel time and outflow for an inhomogeneous link can be approximated very closely by treating the link as homogeneous, with capacity parameters set equal to the average capacity from the inhomogeneous link.

Abstract: Many transportation applications, including applications in intelligent transportation systems, require the solution of a series of shortest path problems in which only the travel time along a set of arcs of the network change from one problem instance to the next One could use an existing path algorithm to solve each problem instance independently as it arises However, significant savings in computation time can often be achieved through the use of a reoptimization algorithm that would begin from the prior solution in determining the updated optimal solution for the given arc travel-time changes Such quick solution is critical for providing routing instructions to travelers in real time as travel-time information is retrieved from the traffic network Numerous works have presented reoptimization techniques for use in updating shortest path trees in deterministic and static networks; however, it appears that no reoptimization technique exists in the literature for updating paths where future travel times in time-varying networks change In this paper, such procedures are proposed The proposed techniques can provide updated solutions given simultaneous and arbitrary changes (increasing and decreasing in value) in any number of network arcs Further, this technique can be extended for use in stochastic networks

Abstract: This article is a preface to the translation by Braess, Nagurney, and Wakolbinger of the 1968 paper by Braess, "ii¾œber ein Paradoxon aus der Verkehrsplanung" (Unternehmensforschung12 258-268).

Abstract: The travel time I„(t) on a link has often been treated in dynamic traffic assignment (DTA) as a function of the number of vehicles x(t) on the link, that is, I„(t) = f(x(t)). In earlier papers, bounds on the gradient of this travel time function f(x) have been introduced to ensure that the model, and in particular the exit times and outflows, have various desirable properties, including a first-in-first-out (FIFO) property. These gradient conditions can be restrictive, because most commonly used travel time functions do not satisfy the conditions for all inflow rates. However, in this paper we extend the earlier results to show that the same properties (including FIFO) can be achieved by instead assuming f(x) is convex, convex about a point, or has certain weaker properties that are satisfied by most travel time functions f(x) proposed or used in practice. These results hold under the conditions in which the travel time function I„(t) = f(x(t)) has generally been applied in the DTA literature, that is, with each link being homogeneous (uniform capacity along the link) and without obstructions or traffic lights. In that case, even if f(x) does not satisfy the above gradient condition, the range in which it is violated is not attainable and hence cannot cause a problem.