Showing papers in "Transportation Science in 2004"
TL;DR: The objective of this paper is to review the current status of ship routing and scheduling and focus on literature published during the last decade, indicating both accelerating needs for and benefits from such systems.
Abstract: The objective of this paper is to review the current status of ship routing and scheduling. We focus on literature published during the last decade. Because routing and scheduling problems are closely related to many other fleet planning problems, we have divided this review into several parts. We start at the strategic fleet planning level and discuss the design of fleets and sea transport systems. We continue with the tactical and operational fleet planning level and consider problems that comprise various ship routing and scheduling aspects. Here, we separately discuss the different modes of operations: industrial, tramp, and liner shipping. Finally, we take a glimpse at naval applications and other related problems that do not naturally fall into these categories. The paper also presents some perspectives regarding future developments and use of optimization-based decision-support systems for ship routing and scheduling. Several of the trends indicate both accelerating needs for and benefits from such systems and, hopefully, this paper will stimulate further research in this area.
707 citations
TL;DR: A two-stage hybrid algorithm that minimizes the number of vehicles, using simulated annealing, and minimizes travel cost by using a large neighborhood search that may relocate a large number of customers is proposed.
Abstract: The vehicle routing problem with time windows is a hard combinatorial optimization problem that has received considerable attention in the last decades. This paper proposes a two-stage hybrid algorithm for this transportation problem. The algorithm first minimizes the number of vehicles, using simulated annealing. It then minimizes travel cost by using a large neighborhood search that may relocate a large number of customers. Experimental results demonstrate the effectiveness of the algorithm, which has improved 10 (17%) of the 56 best published solutions to the Solomon benchmarks, while matching or improving the best solutions in 46 problems (82%). More important perhaps, the algorithm is shown to be very robust. With a fixed configuration of its parameters, it returns either the best published solutions (or improvements thereof) or solutions very close in quality on all Solomon benchmarks. Very preliminary results on the extended Solomon benchmarks are also given.
369 citations
TL;DR: This paper develops a two-phase approach based on decomposing the set of decisions: a delivery schedule is created first, followed by the construction of a set of delivery routes.
Abstract: In this paper, we present a solution approach for the inventory-routing problem. The inventory-routing problem is a variation of the vehicle-routing problem that arises in situations where a vendor has the ability to make decisions about the timing and sizing of deliveries, as well as the routing, with the restriction that customers are not allowed to run out of product. We develop a two-phase approach based on decomposing the set of decisions: A delivery schedule is created first, followed by the construction of a set of delivery routes. The first phase utilizes integer programming, whereas the second phase employs routing and scheduling heuristics. Our focus is on creating a solution methodology appropriate for large-scale real-life instances. Computational experiments demonstrating the effectiveness of our approach are presented.
313 citations
TL;DR: A generic real-time multivehicle truckload pickup and delivery problem that captures most features of the operational problem of a real-world trucking fleet that dynamically moves truckloads between different sites according to customer requests that arrive continuously is introduced.
Abstract: In this paper we formally introduce a generic real-time multivehicle truckload pickup and delivery problem. The problem includes the consideration of various costs associated with trucks' empty travel distances, jobs' delayed completion times, and job rejections. Although very simple, the problem captures most features of the operational problem of a real-world trucking fleet that dynamically moves truckloads between different sites according to customer requests that arrive continuously.We propose a mixed-integer programming formulation for the offline version of the problem. We then consider and compare five rolling horizon strategies for the real-time version. Two of the policies are based on a repeated reoptimization of various instances of the offline problem, while the others use simpler local (heuristic) rules. One of the reoptimization strategies is new, while the other strategies have recently been tested for similar real-time fleet management problems.The comparison of the policies is done under a general simulation framework. The analysis is systematic and considers varying traffic intensities, varying degrees of advance information, and varying degrees of flexibility for job-rejection decisions. The new reoptimization policy is shown to systematically outperform the others under all these conditions.
306 citations
TL;DR: This work provides a bilevel programming formulation for this network design problem based on the nature of the relationship between the regulator and carriers and presents an application of the methodology in Western Ontario, Canada.
Abstract: Dangerous-goods shipments remain regulated despite the widespread deregulation of the transportation industry. This is mainly due to the societal and environmental risks associated with these shipments. One of the common tools used by governments in mitigating transport risk is to close certain roads to vehicles carrying hazardous materials. In effect, the road network available to dangerous goods carriers can be determined by the government. The associated transport risk, however, is determined by the carriers' route choices. We provide a bilevel programming formulation for this network design problem. Our approach is unique in terms of its focus on the nature of the relationship between the regulator and carriers. We present an application of our methodology in Western Ontario, Canada.
298 citations
TL;DR: It is shown that many large crossdocks in practice suffer from poor design that increases labor costs on the dock and on the pattern of freight flows inside.
Abstract: Within both retail distribution and less-than-truckload transportation networks crossdocks vary greatly in shape. Docks in the shape of an I, L, or T are most common, but unusual ones may be found, including those in the shape of a U, H, or E. Is there a best shape? We show that the answer depends on the size of the facility and on the pattern of freight flows inside. Our results suggest that many large crossdocks in practice suffer from poor design that increases labor costs on the dock.
288 citations
TL;DR: This work presents integrated models and solution algorithms that simultaneously optimize the selection of flight legs for and the assignment of aircraft types to the selected flight legs in the airline schedule planning process.
Abstract: Constructing a profitable schedule is of utmost importance to an airline because its profitability is critically influenced by its flight offerings. We focus our attention on the steps of the airline schedule planning process involving schedule design and fleet assignment. Airline schedule design involves determining when and where to offer flights such that profits are maximized, and fleet assignment involves assigning aircraft types to flight legs to maximize revenue and minimize operating cost. We present integrated models and solution algorithms that simultaneously optimize the selection of flight legs for and the assignment of aircraft types to the selected flight legs. Preliminary results, based on data from a major U.S. airline, suggest that significant benefits can be achieved.
285 citations
TL;DR: It is demonstrated that with careful implementation it is possible, in most cases, to maintain the O( n 3 ) complexity or, in a few cases, increase the time complexity toO( n3logn).
Abstract: Insertion heuristics have proven to be popular methods for solving a variety of vehicle routing and scheduling problems. In this paper, we focus on the impact of incorporating complicating constraints on the efficiency of insertion heuristics. The basic insertion heuristic for the standard vehicle routing problem has a time complexity ofO( n 3 ). However, straightforward implementations of handling complicating constraints lead to an undesirable time complexity ofO( n 4 ). We demonstrate that with careful implementation it is possible, in most cases, to maintain theO( n 3 ) complexity or, in a few cases, increase the time complexity toO( n 3logn). The complicating constraints we consider in this paper are time windows, shift time limits, variable delivery quantities, fixed and variable delivery times, and multiple routes per vehicle. Little attention has been given to some of these complexities (with time windows being the notable exception), which are common in practice and have a significant impact on the feasibility of a schedule as well as the efficiency of insertion heuristics.
253 citations
TL;DR: A general framework for the implementation of time-varying travel times in various vehicle-routing algorithms is presented and computational tests with travel time data obtained from a traffic information system in the city of Berlin are reported on.
Abstract: Models and algorithms for vehicle routing are usually based on known constant travel times between all relevant locations, an assumption that is far from reality, particularly for urban areas. But the consideration of travel times that vary with the time of day poses two serious problems: the adaptation of the algorithms and the procurement of reliable data about the behavior of the travel times in the road network. This article describes the derivation of travel time data from modern traffic information systems. It presents a general framework for the implementation of time-varying travel times in various vehicle-routing algorithms. Finally, it reports on computational tests with travel time data obtained from a traffic information system in the city of Berlin.
232 citations
TL;DR: A dynamic routing system that dispatches a fleet of vehicles according to customer orders arriving at random during the planning period, which describes three routing procedures for event-based dispatching, which differ in the length of the planning horizon per event.
Abstract: With the increasing availability of real-time information and communication systems in logistics, the need for appropriate planning algorithms, which make use of this technology, arises. Customers in transport markets increasingly expect quicker and more flexible fulfillment of their orders, especially in the electronic marketplace. This paper considers a dynamic routing system that dispatches a fleet of vehicles according to customer orders arriving at random during the planning period. Each customer order requires a transport from a pickup location to a delivery location in a given time window. The system disposes of online communication with all drivers and customers and, in addition, disposes of online information on travel times from a traffic management center. This paper presents a planning framework for this situation which, to our knowledge, has not yet been addressed in the literature. It then describes three routing procedures for event-based dispatching, which differ in the length of the planning horizon per event. We focus on the use of dynamic travel time information, which requires dynamic shortest path calculations. The procedures are tested and compared using real-life data of an urban traffic management center and a logistics service provider.
232 citations
TL;DR: Users experience lower trip costs when they travel with users unlike themselves than with an equal number of users like themselves, and that schedule delay costs decline at a rate smaller than the unit cost of travel time.
Abstract: Under relatively general assumptions a unique deterministic departure-time user equilibrium with a finite departure rate exists in the bottleneck model with drivers who differ in their unit costs of travel time, preferred times of arrival, and schedule delay cost functions. Existence requires that schedule delay cost functions be upper semicontinuous with respect to arrival time, and that schedule delay costs decline at a rate smaller than the unit cost of travel time. Uniqueness requires, more restrictively, that schedule delay cost functions be continuous.Several properties of equilibrium trip cost functions are derived forn groups of users withN iin groupi. The trip cost of a user in groupi is a nondecreasing function of eachN j , but typically rises more quickly with respect toN ithanN j ,j?i. Thus, users experience lower trip costs when they travel with users unlike themselves than with an equal number of users like themselves.
TL;DR: This work forms a Markov decision process model of the stochastic inventory routing problem and proposes approximation methods to find good solutions with reasonable computational effort and indicates how the proposed approach can be used for other Markov decisions involving the control of multiple resources.
Abstract: This work is motivated by the need to solve the inventory routing problem when implementing a business practice called vendor managed inventory replenishment (VMI). With VMI, vendors monitor their customers' inventories and decide when and how much inventory should be replenished at each customer. The inventory routing problem attempts to coordinate inventory replenishment and transportation in such a way that the cost is minimized over the long run. We formulate a Markov decision process model of the stochastic inventory routing problem and propose approximation methods to find good solutions with reasonable computational effort. We indicate how the proposed approach can be used for other Markov decision processes involving the control of multiple resources.
TL;DR: This study discusses how to dispatch AGVs by utilizing information about locations and times of future delivery tasks by utilizing the heuristic algorithm for overcoming the excessive computational time needed for solving the mathematical model.
Abstract: To reduce delay in ship operations in automated container terminals, it is important to make different types of container handling equipment to operate harmoniously during this operation. Delivery operations by automated guided vehicles (AGVs) play an important role for synchronizing operations of container cranes with yard cranes. This study discusses how to dispatch AGVs by utilizing information about locations and times of future delivery tasks. A mixed-integer programming model is provided for assigning optimal delivery tasks to AGVs. A heuristic algorithm is suggested for overcoming the excessive computational time needed for solving the mathematical model. Objective values and computational times of the heuristic algorithm are compared with those of the optimizing method. To test performances of the heuristic algorithm, a simulation study is performed by considering the uncertainties of various operation times and the number of future delivery tasks for looking ahead. Also, the performance of the heuristic algorithm is compared with those of other dispatching rules.
TL;DR: This work develops an alternative optimization solution approach for the multiple vehicle pickup and delivery problem (MVPDP) that does not require these constraints to be tight and was able to optimally solve problem instances of up to 5 vehicles and 17 customers on problems without clusters.
Abstract: We consider the multiple vehicle pickup and delivery problem (MVPDP) with the objective of minimizing the total travel cost and the fixed vehicle cost. Most of the optimization-based approaches for solving the MVPDP are developed for a restrictive hard time window or tight capacity environment that depend significantly on the reduction of the feasible solution space. We develop an alternative optimization solution approach for the MVPDP that does not require these constraints to be tight. The problem is formulated as a 0-1 integer-programming problem. A branch-and-cut algorithm is developed to optimally solve the problem. Four classes of valid inequalities for the MVPDP are proposed. By using the proposed solution approach, we were able to optimally solve problem instances of up to 5 vehicles and 17 customers on problems without clusters and up to 5 vehicles and 25 customers on problems with clusters within a stopping criterion of three CPU hours on a SUN Fire 4800 server.
TL;DR: This paper proposes a very general class of dynamic assignment models, and proposes an adaptive, nonmyopic algorithm that involves iteratively solving sequences of assignment problems no larger than what would be required of a myopic model.
Abstract: There has been considerable recent interest in the dynamic vehicle routing problem, but the complexities of this problem class have generally restricted research to myopic models. In this paper, we address the simpler dynamic assignment problem, where a resource (container, vehicle, or driver) can serve only one task at a time. We propose a very general class of dynamic assignment models, and propose an adaptive, nonmyopic algorithm that involves iteratively solving sequences of assignment problems no larger than what would be required of a myopic model. We consider problems where the attribute space of future resources and tasks is small enough to be enumerated, and propose a hierarchical aggregation strategy for problems where the attribute spaces are too large to be enumerated. Finally, we use the formulation to also test the value of advance information, which offers a more realistic estimate over studies that use purely myopic models.
TL;DR: This work considers a model formulation of the line-planning problem where total operating costs are to be minimized, and develops a branch-and-cut approach, for which a variety of valid inequalities and reduction methods are developed.
Abstract: An important strategic phase in the planning process of a railway operator is the development of a line plan, i.e., a set of routes (paths) in a network of tracks, operated at a given hourly frequency. We consider a model formulation of the line-planning problem where total operating costs are to be minimized. This model is solved with a branch-and-cut approach, for which we develop a variety of valid inequalities and reduction methods. A computational study of five real-life instances based on examples from Netherlands Railways (NS) is included.
TL;DR: A lower bound for the number of short cycles is determined using the hub connectivity of a fleet assignment, and fleet-assignment models (FAMs) that embed many short cycles and reduce hub connectivity within a solution are presented.
Abstract: Airline decision makers cancel flights in operations because of disruptions. When canceling a flight, they usually cancel a cycle, a sequence of flights that begins and ends at the same airport. Consequently, a fleet assignment and aircraft rotation with many short cycles is frequently less sensitive to a flight cancellation than one with only a few short cycles. In this paper, we determine a lower bound for the number of short cycles using the hub connectivity of a fleet assignment, and we present fleet-assignment models (FAMs) that embed many short cycles and reduce hub connectivity within a solution. We show that solutions to such models perform better in operations than those of traditional FAMs that minimize planned operating cost and passenger spill.
TL;DR: Two heuristic approaches are proposed for the well-knowntraveling salesman problem in which cities correspond to customers providing or requiring known amounts of a product, and the vehicle has a given upper limit capacity.
Abstract: This paper deals with a generalisation of the well-knowntraveling salesman problem (TSP) in which cities correspond to customers providing or requiring known amounts of a product, and the vehicle has a given upper limit capacity. Each customer must be visited exactly once by the vehicle serving the demands while minimising the total travel distance. It is assumed that any unit of product collected from a pickup customer can be delivered to any delivery customer. This problem is calledone-commodity pickup-and-delivery TSP (1-PDTSP). We propose two heuristic approaches for the problem: (1) is based on a greedy algorithm and improved with ak-optimality criterion and (2) is based on a branch-and-cut procedure for finding an optimal local solution. The proposal can easily be used to solve the classical "TSP with pickup-and-delivery," a version studied in literature and involving two commodities. The approaches have been applied to solve hard instances with up to 500 customers.
TL;DR: A complete analysis of the sensitivity of elastic demand traffic (Wardrop) equilibria, with clear advantage when applying sensitivity analysis within a bilevel application, such as for congestion pricing, OD estimation, or network design.
Abstract: The contribution of the paper is a complete analysis of the sensitivity of elastic demand traffic (Wardrop) equilibria. The existence of a directional derivative of the equilibrium solution (link flow, least travel cost, demand) in any direction is given a characterization, and the same is done for its gradient. The gradient, if it exists, is further interpreted as a limiting case of the gradient of the logit-based SUE solution, as the dispersion parameter tends to infinity. In the absence of the gradient, we show how to compute a subgradient. All these computations (directional derivative, (sub)gradient) are performed by solving similar traffic equilibrium problems with affine link cost and demand functions, and they can be performed by the same tool as (or one similar to) the one used for the original traffic equilibrium model; this fact is of clear advantage when applying sensitivity analysis within a bilevel (or mathematical program with equilibrium constraints, MPEC) application, such as for congestion pricing, OD estimation, or network design. A small example illustrates the possible nonexistence of a gradient and the computation of a subgradient.
TL;DR: In this article, a real-time traveling saleman problem with time windows for various degrees of dynamism is examined and the impact of this criterion choice on the distance traveled is examined.
Abstract: In this paper we examine the traveling saleman problem with time windows for various degrees of dynamism. In contrast to the static problem, where the dispatcher can plan ahead, in the dynamic version, part or all of the necessary information becomes available only during the day of operation. We seek to minimize lateness and examine the impact of this criterion choice on the distance traveled. Our focus on lateness is motivated by the problem faced by overnight mail service providers. We propose a real-time solution method that requires the vehicle, when idle, to wait at the current customer location until it can service another customer without being early. In addition, we develop several enhanced versions of this method that may reposition the vehicle at a location different from that of the current customer based on a priori information on future requests. The results we obtained on both randomly generated data and on a real-world case study indicate that all policies proved capable of significantly reducing lateness. Our results also show that this can be accomplished with only small distance increases. The basic policy outperformed the other methods primarily when lateness and distance were equally minimized and proved very robust in all environments studied. When only lateness was considered, the policy to reposition the vehicle at a location near the current customer generally provided the largest reductions in average lateness and the number of late customers. It also produced the least extra distance to be traveled among the relocation policies.
TL;DR: This work model and analyze the problem of constructing a minimum expected total cost route from an origin to a destination that anticipates and then responds to service requests, if they occur, while the vehicle is en route, and presents several structured results associated with the optimal expected cost-to-go function and an optimal policy for route construction.
Abstract: Mobile communication technologies enable communication between dispatchers and drivers and hence can enable fleet management based on real-time information. We assume that such communication capability exists for a single pickup and delivery vehicle and that we know the likelihood, as a function of time, that each of the vehicle's potential customers will make a pickup request. We then model and analyze the problem of constructing a minimum expected total cost route from an origin to a destination that anticipates and then responds to service requests, if they occur, while the vehicle is en route. We model this problem as a Markov decision process and present several structured results associated with the optimal expected cost-to-go function and an optimal policy for route construction. We illustrate the behavior of an optimal policy with several numerical examples and demonstrate the superiority of an optimal anticipatory policy, relative to a route design approach that reflects the reactive nature of current routing procedures for less-than-truckload pickup and delivery.
TL;DR: This paper presents a solution approach to the dynamic vehicle scheduling problem, where a "cluster-reschedule" heuristic is used, where trips are assigned to depots by solving the static problem and then solved dynamic single-depot problems.
Abstract: This paper presents a solution approach to the dynamic vehicle scheduling problem. This approach consists of solving a sequence of optimization problems, where we take into account different scenarios for future travel times. We discuss the potential benefit of our approach compared to the traditional one, where the vehicle scheduling problem is solved only once for a whole period and the travel times are assumed to be fixed. Because in the multiple-depot case we cannot solve the problem exactly within reasonable computation time, we use a "cluster-reschedule" heuristic where we first assign trips to depots by solving the static problem and then solve dynamic single-depot problems. We use new mathematical formulations of these problems that allow fast solution by standard optimization software. Results of a computational study with real-life data are presented, in which we compare different variants of our approach and perform a sensitivity analysis with respect to deviations of the actual travel times from estimated ones.
TL;DR: A cost-allocation problem that arises in a distribution-planning situation at the Logistics Department at Norsk Hydro Olje AB, Stockholm, Sweden is studied, allowing the use of vehicles with different capacities.
Abstract: In this paper, we study a cost-allocation problem that arises in a distribution-planning situation at the Logistics Department at Norsk Hydro Olje AB, Stockholm, Sweden. We consider the routes from one depot during one day. The total distribution cost for these routes is to be divided among the customers that are visited. This cost-allocation problem is formulated as a vehicle-routing game (VRG), allowing the use of vehicles with different capacities. Cost-allocation methods based on different concepts from cooperative game theory, such as the core and the nucleolus, are discussed. A procedure that can be used to investigate whether the core is empty or not is presented, as well as a procedure to compute the nucleolus. Computational results for the Norsk Hydro case are presented and discussed.
TL;DR: In this article, the authors describe a model that can be used to find an optimal allocation of train types and subtypes to the lines, which is more effective than the manually planned one.
Abstract: For a commercially operating railway company, providing a high level of service for the passengers is of utmost importance. The latter requires high punctuality of the trains and an adequate rolling stock capacity. Unfortunately, the latter is currently (in 2002) one of the bottlenecks in the service provision by the main Dutch railway operator NS Reizigers. Especially during the morning rush hours, many passengers cannot be transported according to the usual service standards because of a shortage of the rolling stock capacity. On the other hand, a more effective allocation of the available rolling stock capacity seems to be feasible, because there are also several trains with some slack capacity. The effectiveness of the rolling stock capacity is determined mainly by the allocation of the train types and subtypes to the lines. Therefore, we describe in this paper a model that can be used to find an optimal allocation of train types and subtypes to the lines. This optimal allocation is more effective than the manually planned one, which is accomplished by minimizing the shortages of capacity during the rush hours. The model is implemented in the modeling language OPL Studio 3.1, solved by CPLEX 7.0, and tested on several scenarios based on the 2001-2002 timetable of NS Reizigers. The results of the model were received positively, both by the planners and by the management in practice, because these results showed that a significant service improvement over the manually planned allocation can be achieved within a shorter throughput time of the involved part of the planning process.
TL;DR: This paper uses a time-dependent network to describe the possible car movements in time and space, and shows how this network can be transformed into a network with fixed costs on links representing movements of cars with identical origin and destination terminals, a capacitated network design model.
Abstract: In this paper, we consider empty freight car distribution in a scheduled railway system. We analyze the cost structure for the repositioning of empty cars, and conclude that the distribution cost shows an economy-of-scale behavior. In addition to the cost proportional to the number of cars sent from origin to destination, there is a cost related to car-handling operations at yards, which depends on the number of car groups that are handled. Thus, if we can find a transportation pattern in which fewer but larger groups of cars are built, the total distribution cost can be decreased.The objective of the paper is to propose an optimization model that explicitly takes this economy-of-scale effect into account. We use a time-dependent network to describe the possible car movements in time and space, and show how this network can be transformed into a network with fixed costs on links representing movements of cars with identical origin and destination terminals. The resulting optimization model is a capacitated network design model, where each capacity constraint limits the flow on several arcs. We describe a tabu heuristic for solving the model, and present computational results.
TL;DR: A new model and an efficient metaheuristic that determines the number and the location of TCs as well as the best transportation alternative-LTL, FTL, Parcel, or own fleet-on each segment accounting for both weight and volume metrics is presented.
Abstract: The fast development of transport activities and the introduction of shipment consolidation have considerably changed the logistics context over the last three decades. Consolidation terminals, also called transshipment centers (TC) or hubs, have justified their presence by improving the loading of trucks in terms of both volume and weight. In addition, the possibility of using external carriers, exclusively or in coordination with a private fleet, can reduce costs and increase customer service. The right combination of these strategies can dramatically impact the cost of transport. However, the complexity of the decisions has also increased and existing models have to be improved to tackle these new challenges. In this paper, after discussing the different formulations for distribution networks with transshipment centers existing in the literature, we present a new model and an efficient metaheuristic that determines the number and the location of TCs as well as the best transportation alternative-LTL, FTL, Parcel, or own fleet-on each segment accounting for both weight and volume metrics. The ability of our heuristic to solve this complex problem comes from a judicious combination of tabu search and variable neighborhood search. The performance of this approach is evaluated on several test data problems generated with real cost structures published by a U.S. carrier. The heuristic solutions are compared to optimal ones obtained by an exact method for small-sized instances of the simpler problems. Finally, we address issues in carrier price structure to achieve efficient shipment practices.
TL;DR: An alternative method of computing the equilibrium distribution is proposed, applicable to the class of Markov models with linear exponential learning filters, and the robustness of the approximations is discussed, and shown to be connected with the "volatility" of the traffic assignment process.
Abstract: Markov traffic-assignment models explicitly represent the day-to-day evolving interaction between traffic congestion and drivers' information acquisition and choice processes. Such models can, in principle, be used to investigate traffic flows in stochastic equilibrium, yielding estimates of the equilibrium mean and covariance matrix of link or route traffic flows. However, in general these equilibrium moments cannot be written down in closed form. While Monte Carlo simulations of the assignment process may be used to produce "empirical" estimates, this approach can be extremely computationally expensive if reliable results (relatively free of Monte Carlo error) are to be obtained. In this paper an alternative method of computing the equilibrium distribution is proposed, applicable to the class of Markov models with linear exponential learning filters. Based on asymptotic results, this equilibrium distribution may be approximated by a Gaussian process, meaning that the problem reduces to determining the first two multivariate moments in equilibrium. The first of these moments, the mean flow vector, may be estimated by a conventional traffic-assignment model. The second, the flow covariance matrix, is estimated through various linear approximations, yielding an explicit expression. The proposed approximations are seen to operate well in a number of illustrative examples. The robustness of the approximations (in terms of network input data) is discussed, and shown to be connected with the "volatility" of the traffic assignment process.
TL;DR: A tighter formulation of the PDPTWP based on Dantzig-Wolfe decomposition is proposed that makes use of the precedence constraints and makes it possible to solve the pricing problem in pseudopolynomial time through dynamic programming.
Abstract: In the classical vehicle-routing problem (VRP) the objective is to service some geographically scattered customers with a given number of vehicles at the minimal cost. In the present paper, we consider a variant of the VRP where the vehicles should deliver some goods between groups of customers. The customers have an associated time window, a precedence number, and a quantity. Each vehicle should visit the customers within their time windows, in nondecreasing order of precedence respecting the capacity of the vehicle. The problem will be denoted the pickup-and-delivery problem with time windows and precedence constraints (PDPTWP). The PDPTWP has applications in the transportation of live animals where veterinary rules demand that the livestocks are visited in a given sequence in order not to spread specific diseases. We propose a tighter formulation of the PDPTWP based on Dantzig-Wolfe decomposition. The formulation splits the problem into a master problem, which is a kind of set-covering problem, and a subproblem that generates legal routes for a single vehicle. The LP-relaxation of the decomposed problem is solved through delayed column generation. Computational experiments show that the obtained bounds are less than 0.24% from optimum for the considered problems. As solving the pricing problems takes up the majority of the solution time, a reformulation of the problem is proposed that makes use of the precedence constraints. By merging customers having the same precedence number into "super nodes," the pricing problem may be reformulated as a shortest-path problem defined on an acyclic layered graph. This makes it possible to solve the pricing problem in pseudopolynomial time through dynamic programming. The paper concludes with a comprehensive computational study involving real-life instances from the transportation of live pigs. It is demonstrated that instances with up to 580 nodes can be solved to optimality.
TL;DR: This work develops a linear time algorithm for determining a delivery schedule for a route, i.e., a given sequence of customer visits, that maximizes the total amount of product that is delivered on the route.
Abstract: This work is motivated by the need to solve the inventory routing problem when implementing a business practice called vendor managed inventory replenishment. With vendor managed inventory replenishment, vendors monitor their customers' inventories, and decide when and how much inventory should be replenished at each customer. The inventory routing problem attempts to coordinate inventory replenishment and transportation in such a way that the cost is minimized over the long run. In this paper, we develop a linear time algorithm for determining a delivery schedule for a route, i.e., a given sequence of customer visits, that maximizes the total amount of product that is delivered on the route. This problem is not as easy as it may seem at first glance because of delivery windows at customers and the two dueling effects of increased inventory holding capacity at customers as time progresses and increased delivery times as more product is delivered at customers. Efficiently constructing such delivery schedules is important because it has to be done numerous times in insertion heuristics and local search procedures employed in solution approaches for the inventory routing problem.
TL;DR: An analytical expression for the cumulative distribution function of travel time for a vehicle traversing a freeway link of arbitrary length is derived and a numerical inversion algorithm is presented to invert the transforms.
Abstract: We derive an analytical expression for the cumulative distribution function of travel time for a vehicle traversing a freeway link of arbitrary length. The vehicle's speed is assumed to be modulated by a random environment that can be modeled as a stochastic process. We first present a partial differential equation (PDE) describing the travel time distribution and obtain a solution in terms of Laplace transforms. Next, we present a numerical inversion algorithm to invert the transforms. The technique is demonstrated on two example problems. Numerical results indicate great promise for this approach to the link travel-time problem.