A truck scheduling problem arising in intermodal container transportation

We address a truck scheduling problem that arises in intermodal container transportation, where containers need to be transported between customers (shippers or receivers) and container terminals (rail or maritime) and vice versa. The transportation requests are handled by a trucking company which operates several depots and a fleet of homogeneous trucks that must be routed and scheduled to minimize the total truck operating time under hard time window constraints imposed by the customers and terminals. Empty containers are considered as transportation resources and are provided by the trucking company for freight transportation. The truck scheduling problem at hand is formulated as Full-Truckload Pickup and Delivery Problem with Time Windows (FTPDPTW) and is solved by a 2-stage heuristic solution approach. This solution method was specially designed for the truck scheduling problem but can be applied to other problems as well. We assess the quality of our solution approach on several computational experiments.     

Maritime crude oil transportation - A split pickup and split delivery problem

The maritime oil tanker routing and scheduling problem is known to the literature since before 1950. In the presented problem, oil tankers transport crude oil from supply points to demand locations around the globe. The objective is to find ship routes, load sizes, as well as port arrival and departure times, in a way that minimizes transportation costs. We introduce a path flow model where paths are ship routes. Continuous variables distribute the cargo between the different routes. Multiple products are transported by a heterogeneous fleet of tankers. Pickup and delivery requirements are not paired to cargos beforehand and arbitrary split of amounts is allowed. Small realistic test instances can be solved with route pre-generation for this model. The results indicate possible simplifications and stimulate further research.     

Dynamic control of multicommodity fleet management problems

Dynamic fleet management problems with multiple equipment types and limited substitution can be modeled as dynamic, multicommodity network flow problems. These problems are further complicated by the presence of time windows on task arcs (a task, or load, can be handled at different points in time) and the need for integer solutions. In this paper, we formulate the problem as a dynamic control problem, and show that we can produce solutions within four to five percent of a linear relaxation. In addition, we can solve the ultra-large problems that arise in certain applications; these problems are beyond the capabilities of state-of-the-art linear programming solvers.     

Fleet assignment and routing with schedule synchronization constraints

This paper introduces a new type of constraints, related to schedule synchronization, in the problem formulation of aircraft fleet assignment and routing problems and it proposes an optimal solution approach. This approach is based on Dantzig–Wolfe decomposition/column generation. The resulting master problem consists of flight covering constraints, as in usual applications, and of schedule synchronization constraints. The corresponding subproblem is a shortest path problem with time windows and linear costs on the time variables and it is solved by an optimal dynamic programming algorithm. This column generation procedure is embedded into a branch and bound scheme to obtain integer solutions. A dedicated branching scheme was devised in this paper where the branching decisions are imposed on the time variables. Computational experiments were conducted using weekly fleet routing and scheduling problem data coming from an European airline. The test problems are solved to optimality. A detailed result analysis highlights the advantages of this approach: an extremely short subproblem solution time and, after several improvements, a very efficient master problem solution time.     

The hub location and network design problem with fixed and variable arc costs: formulation and dual-based solution heuristic

Many air, less-than-truck load and intermodal transportation and telecommunication networks incorporate hubs in an effort to reduce total cost. These hubs function as make bulk/break bulk or consolidation/deconsolidation centres. In this paper, a new hub location and network design formulation is presented that considers the fixed costs of establishing the hubs and the arcs in the network, and the variable costs associated with the demands on the arcs. The problem is formulated as a mixed integer programming problem embedding a multi-commodity flow model. The formulation can be transformed into some previously modelled hub network design problems. We develop a dual-based heuristic that exploits the multi-commodity flow problem structure embedded in the formulation. The test results indicate that the heuristic is an effective way to solve this computationally complex problem.     

A time–space network based exact optimization model for multi-depot bus scheduling

The vehicle scheduling problem, arising in public transport bus companies, addresses the task of assigning buses to cover a given set of timetabled trips with consideration of practical requirements, such as multiple depots and vehicle types as well as depot capacities. An optimal schedule is characterized by minimal fleet size and minimal operational costs including costs for unloaded trips and waiting time. This paper discusses the multi-depot, multi-vehicle-type bus scheduling problem (MDVSP), involving multiple depots for vehicles and different vehicle types for timetabled trips. We use time–space-based instead of connection-based networks for MDVSP modeling. This leads to a crucial size reduction of the corresponding mathematical models compared to well-known connection-based network flow or set partitioning models. The proposed modeling approach enables us to solve real-world problem instances with thousands of scheduled trips by direct application of standard optimization software. To our knowledge, the largest problems that we solved to optimality could not be solved by any existing exact approach. The presented research results have been developed in co-operation with the provider of transportation planning software PTV AG. A software component to support planners in public transport was designed and implemented in context of this co-operation as well.     

Transportation planning in freight forwarding companies: Tabu search algorithm for the integrated operational transportation planning problem

The integrated operational transportation planning problem extends the traditional vehicle routing and scheduling problem by the possibility of outsourcing a part of the requests by involving subcontractors. The purpose of this paper is to present the integrated planning problem and to propose an approach for solving it by a tabu search heuristic. Existing approaches from literature which discuss vehicle routing combined with outsourcing regard only one specific type of subcontracting. This paper describes and explores the complex situation where an own fleet and several types of subcontracting are used for request fulfillment. As the approach contains new aspects, unknown to the literature so far, tabu search is extended to special types of moves. On the basis of computational results the cost structure is analyzed in order to investigate the long-term planning question whether and to what extend it is profitable to maintain an own fleet.     

Multi-depot vehicle scheduling problems with time windows and waiting costs

The multi-depot vehicle scheduling problem with time windows (MDVSPTW) consists of scheduling a fleet of vehicles to cover a set of tasks at minimum cost. Each task is restricted to begin within a prescribed time interval and vehicles are supplied by different depots. The problem is formulated as an integer nonlinear multi-commodity network flow model with time variables and is solved using a column generation approach embedded in a branch-and-bound framework. This paper breaks new ground by considering costs on exact waiting times between two consecutive tasks instead of minimal waiting times. This new and more realistic cost structure gives rise to a nonlinear objective function in the model. Optimal and heuristic versions of the algorithm have been extensively tested on randomly generated urban bus scheduling problem (UBSP) and freight transport scheduling problem (FTSP). The results show that such a general solution methodology outperforms specialized algorithms when minimal waiting costs are used, and can efficiently treat the case with exact waiting costs.     

Converging upon basic feasible solutions through Dantzig–Wolfe decomposition

We derive an important property for solving large-scale integer programs by examining the master problem in Dantzig–Wolfe decomposition. In particular, we prove that if a linear program can be divided into subproblems with affinely independent corner points, then there is a direct mapping between basic feasible solutions in the master and original problems. This has implications for integer programs where the feasible region has integer corner points, ensuring that integer solutions to the original problem will be found even through the decomposition approach. An application to air traffic flow scheduling, which motivated this result, is highlighted.     

A combined ship scheduling and allocation problem

We present a bulk ship scheduling problem that is a combined multi-ship pickup and delivery problem with time windows (m-PDPTW) and multi-allocation problem. In contrast to other ship scheduling problems found in the literature, each ship in the fleet is equipped with a flexible cargo hold that can be partitioned into several smaller holds in a given number of ways. Therefore, multiple products can be carried simultaneously by the same ship. The scheduling of the ships constitutes the m-PDPTW, while the partition of the ships' flexible cargo holds and the allocation of cargoes to the smaller holds make the multi-allocation problem. A set partitioning approach consisting of two phases is proposed for the combined ship scheduling and allocation problem. In the first phase, a number of candidate schedules (including allocation of cargoes to the ships' cargo holds) is generated for each ship. In the second phase, we minimise transportation costs by solving a set partitioning problem where the columns are the candidate schedules generated in phase one. The computational results show that the proposed approach works, and optimal solutions are obtained on several cases of a real ship planning problem.     

The Vehicle Scheduling Problem with Multiple Vehicle Types

The vehicle scheduling problem is specified in terms of a set of tasks to be executed with a fleet of multiple vehicle types. The purpose of this paper is to formulate the problem and to show that the heuristic and exact methods developed for the vehicle scheduling problem with time windows and with a single type of vehicle can be extended in a straightforward fashion to the multiple-vehicle-types problem.     

Cargo ships routing and scheduling: Survey of models and problems

When a ship costs thousands of dollars per day, significant savings can be achieved by proper fleet routing and scheduling. In contrast to vehicle scheduling, relatively little work has been done in ship routing and scheduling. This paper discusses briefly the differences between vehicle and ship routing and scheduling and the reasons for the low attention to ship scheduling in the past. The various modes of operation of cargo ships are described and a classification scheme for ship routing and scheduling models and problems is proposed. A review of ship routing, scheduling and related models is provided. The review is broken down into the following categories: transportation system models, liner operations, tramp shipping, industrial operations and other models. Finally, recent trends in ship scheduling, shortcomings in existing models and requirements from realistic models are discussed.     

Continuous management of airlift and tanker resources: A constraint-based approach

Efficient allocation of aircraft and aircrews to transportation missions is an important priority at the USAF Air Mobility Command (AMC), where airlift demand must increasingly be met with less capacity and at lower cost. In addition to presenting a formidable optimization problem, the AMC resource management problem is complicated by the fact that it is situated in a continuously executing environment. Mission requests are received (and must be acted upon) incrementally, and, once allocation decisions have been communicated to the executing agents, subsequent opportunities for optimizing resource usage must be balanced against the cost of solution change. In this paper, we describe the technical approach taken to this problem in the AMC barrel allocator, a scheduling tool developed to address this problem and provide support for day-to-day allocation and management of AMC resources. The system utilizes incremental and configurable constraint-based search procedures to provide a range of automated and semi-automated scheduling capabilities. Most basically, the system provides an efficient solution to the fleet scheduling problem. More importantly to continuous operations, it also provides techniques for selectively reoptimizing to accommodate higher priority missions while minimizing disruption to most previously scheduled missions, and for selectively “merging” previously planned missions to minimize nonproductive flying time. In situations where all mission requirements cannot be met, the system can generate and compare alternative constraint relaxation options. The barrel allocator technology is currently transitioning into operational use within AMC's Tanker/Airlift Control Center (TACC). A version of the barrel allocator supporting airlift allocation was first incorporated as an experimental module of the AMC's Consolidated Air Mobility Planning System (CAMPS) in September 2000. In May 2003, a new tanker allocation module is scheduled for initial operational release to users as part of CAMPS Release 5.4.     

An interactive,computer-aided ship scheduling system

This paper is concerned with a fleet scheduling and inventory resupply problem faced by an international chemical operation. The firm uses a fleet of small ocean-going tankers to deliver bulk fluid to warehouses all over the world. The scheduling problem centers around decisions on routes, arrival/departure times, and inventory replenishment quantities. An interactive computer system was developed and implemented at the firm, and was successfully used to address daily scheduling issues as well as longer range planning problems. The purpose of this paper is to first present how the underlying decision problem was analyzed using both a network flow model and a mixed integer programming model, and then to describe the components of the decision support system developed to generate schedules. The use of the system in various decision making applications is also described.     

A sweep-based algorithm for the fleet size and mix vehicle routing problem

This paper presents a new sweep-based heuristic for the fleet size and mix vehicle routing problem. This problem involves two kinds of decisions: the selection of a mix of vehicles among the available vehicle types and the routing of the selected fleet. The proposed algorithm first generates a large number of routes that are serviced by one or two vehicles. The selection of routes and vehicles to be used is then made by solving to optimality, in polynomial time, a set-partitioning problem having a special structure. Results on a set of benchmark test problems show that the proposed heuristic produces excellent solutions in short computing times. Having a fast but good solution method is needed for transportation companies that rent a significant part of their fleet and consequently can take advantage of frequent changes in fleet composition. Finally, the proposed heuristic produced new best-known solutions for three of the test problems; these solutions are reported.     

A conic quadratic formulation for a class of convex congestion functions in network flow problems

In this paper we consider a multicommodity network flow problem with flow routing and discrete capacity expansion decisions. The problem involves trading off congestion and capacity assignment (or expansion) costs. In particular, we consider congestion costs involving convex, increasing power functions of flows on the arcs. We first observe that under certain conditions the congestion cost can be formulated as a convex function of the capacity level and the flow. Then, we show that the problem can be efficiently formulated by using conic quadratic inequalities. As most of the research on this problem is devoted to heuristic approaches, this study differs in showing that the problem can be solved to optimum by branch-and-bound solvers implementing the second-order cone programming (SOCP) algorithms. Computational experiments on the test problems from the literature show that the continuous relaxation of the formulation gives a tight lower bound and leads to optimal or near optimal integer solutions within reasonable CPU times.     

Implementing the Dantzig-Fulkerson-Johnson algorithm for large traveling salesman problems

 Dantzig, Fulkerson, and Johnson (1954) introduced the cutting-plane method as a means of attacking the traveling salesman problem; this method has been applied to broad classes of problems in combinatorial optimization and integer programming. In this paper we discuss an implementation of Dantzig et al.'s method that is suitable for TSP instances having 1,000,000 or more cities. Our aim is to use the study of the TSP as a step towards understanding the applicability and limits of the general cutting-plane method in large-scale applications. Received: December 6, 2002 / Accepted: April 24, 2003 Published online: May 28, 2003 RID="*" ID="*" Supported by ONR Grant N00014-03-1-0040     

A heuristic approach to the multi-period multi-commodity transportation problem

This paper describes an approach to solving a real-world problem which involves the transportation of multiple types of commodities from a number of sources to a number of destinations in discrete time periods, using a capacitated heterogeneous fleet of vehicles. The preliminary objective is to minimize the total number of discrete periods needed to complete the entire operation. The problem is first formulated as a mixed integer programme and its tractability is then greatly improved by reformulating it through backward decomposition into two separate models and solved iteratively. A heuristic approach harnessing specific features of the second approach is developed for solving large size problems to obtain near-optimal solutions within reasonable time. The design of the heuristic also takes into consideration the secondary objectives of minimizing the total vehicle capacity used and minimizing the total capacity of sources needed to satisfy the demands at the destinations. Computational results are provided for a variety of randomly generated problems as well as problems from the literature. The approach described here may be applied to the multi-period transportation of personnel and goods from multiple starting points to multiple destinations in both military and civilian applications.     

The Size and Composition of a Road Transport Fleet

The paper starts with a discussion of the simple fleet size problem. It is shown that this simple problem can be formulated as a linear program.The second part of the paper consists of an actual case study. The fleet concerned is faced with highly seasonal demand which can be met by the firm's own vehicles or by outside hire. There are two types of vehicle, both of which are available in six different sizes. Linear programming was used to find the optimum size and composition of the company fleet. The results, which were substantially implemented, recommended a smaller company fleet and concentration on larger and more flexible vehicles.     

计及多重阻塞的电动汽车移动储能特性建模

为实现城市交通电力耦合系统在城市道路、充电设施、输电线路阻塞环境下的优化运行,提出了计及多重阻塞的动态交通电力流联合优化方法。首先,基于时空网络模型,提出了计及电动汽车移动、静止、充电、排队模式的队列时空网络模型,构建了适用于电动汽车的车辆调度模型,进而形成动态交通分配模型,以减少交通出行损失。其次,通过优化发电机组、储能等的出力和备用计划,计及城市电网安全、备用约束,构建了安全约束动态经济调度模型,以降低碳排放及发电成本。随后,形成多目标动态优化模型,并将其转换为混合整数凸二次规划问题。最后,在耦合IEEE-30、Sioux Falls系统中验证了所提模型的有效性。