A method for solving ship routing problemswith inventory constraints

Column generation is an iterative method used to solve a variety of optimization problems. It decomposes the problem into two parts: a master problem and one or more pricing problems (PP). The total computing time taken by the method is divided between these two parts. In routing or scheduling applications, the problems are mostly defined on a network, and the PP is usually an NP-hard shortest path problem with resource constraints. In this work, we propose a new heuristic pricing algorithm based on machine learning. By taking advantage of the data collected during previous executions, the objective is to reduce the size of the network and accelerate the PP, keeping only the arcs that have a high chance to be part of the linear relaxation solution. The method has been applied to two specific problems: the vehicle and crew scheduling problem in public transit and the vehicle routing problem with time windows. Reductions in computational time of up to 40% can be obtained. Funding: The authors thank Giro, Inc., and the Natural Sciences and Engineering Research Council of Canada for financial support [Grant CRDPJ 520349-17].

The purpose of this paper is to present an exact solution method for an important planning problem faced by many shipping companies dealing with transportation of bulk cargoes. A bulk shipping company usually has a set of contract cargoes that it is committed to carry and will try to derive additional revenue from optional spot cargoes. Each cargo, either it is a contract or spot cargo, consists of a given quantity to be picked up in a given loading port and delivered in a given unloading port within specified time windows. The shipping company controls a fixed fleet for the purpose of transporting the cargoes. In most of the literature on ship routing and scheduling problems, a cargo cannot be transported by more than one ship. By introducing split loads, this restriction is removed and each cargo can be transported by several ships. The resulting planning problem can be denoted as a maritime pickup and delivery problem with time windows and split loads (PDPTWSL). We present an arc flow formulation of the PDPTWSL. We also suggest a solution method based on a priori generation of single ship schedules and two alternative path flow models that deal with the selection of ship schedules and assignment of cargo quantities to the schedules. Computational results show that the solution method can provide optimal solutions to small realistic planning instances. We also believe this paper provides an important starting point for research on other exact solution methods for the maritime PDPTWSL.

This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with time windows. The standard Dantzig-Wolfe decomposition of the arc flow formulation leads to a set-partitioning problem as the master problem and an elementary shortest-path problem with resource constraints as the pricing problem. We introduce the subset-row inequalities, which are Chvatal-Gomory rank-1 cuts based on a subset of the constraints in the master problem. Applying a subset-row inequality in the master problem increases the complexity of the label-setting algorithm used to solve the pricing problem because an additional resource is added for each inequality. We propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step-like structure of the objective function of the pricing problem. Computational experiments have been performed on the Solomon benchmarks where we were able to close several instances. The results show that applying subset-row inequalities in the master problem significantly improves the lower bound and, in many cases, makes it possible to prove optimality in the root node.

This chapter describes a stochastic ship routing problem with inventory management. The problem involves finding a set of least cost routes for a fleet of ships transporting a single commodity when the demand for the commodity is uncertain. Storage at supply and consumption ports is limited and inventory levels are monitored in the model. Consumer demands are at a constant rate within each time period, and in the stochastic problem, the demand rate for a period is not known until the beginning of that period. The demand situation over the time periods is described by a scenario tree with corresponding probabilities. A decomposition formulation is given and it is solved using a Branch and Price framework. A master problem (set partitioning with extra inventory constraints) is built, and the subproblems, one for each ship, are solved by stochastic dynamic programming and yield the columns for the master problem. Each column corresponds to one possible tree of actions for one ship giving its schedule loading/unloading quantities for all demand scenarios. Computational results are given showing that medium sized problems can be solved successfully.

We propose a solution method for the log‐truck scheduling problem, which is a generalisation of the pick‐up and delivery problem with time windows. Our approach is based on column generation and pseudo branch and price. Each column in the proposed mathematical model represents one feasible route for one truck. We start by designing a priori an initial set of routes. Then, the subproblem, which is a constrained shortest path problem, is solved by applying a k‐shortest path algorithm. Numerical results from a case study are presented.

Granular tabu search for the pickup and delivery problem with time windows and electric vehicles

Logic-based Benders decomposition for the heterogeneous fixed fleet vehicle routing problem with time windows

A shift scheduling problem for a call centre is solved by applying a column generation method. The objective is to find an employee schedule within the planning horizon that meets staffing requirements provided by the customer and that minimises the costs of excess-staffing and under-staffing. To solve the problem, two iterative models were used: the first being a mixed integer-linear programming model, the master problem, and the second being a column generation model. In a first approach, the master problem was solved considering a fixed number of shift patterns. The second approach was to use both models iteratively to generate shift patterns to be used in the master problem. Both approaches met all constraints relatively fast, but the column generation model decreased the cost considerably when cheaper shift patterns were generated. The models contribute to the current literature by providing flexibility to consider under-coverage and over-coverage at two levels.

In recent decades, the rapid development of international economy has led to significant inter-regional trade imbalances. In marine transportation, the voluminous empty containers accumulated at ports in import dominated countries have to be re-circulated back to ports in export dominated countries for re-deployment. This repositioning of empty containers is crucial for shipping companies to provide effective transportation services. At present, empty container repositioning has proved to be highly expensive, contributing to 27% of the overall operation cost of shipping companies. Consequently, major stakeholders including the shipping companies have strived to reduce the repositioning cost. Unfortunately, few major improvements have been made over the years and there are still lingering research issues. The objective of this research, therefore, is to develop efficient and effective methodologies to obtain empty container repositioning plans that can significantly reduce transportation cost while fulfilling customer demands of empty and laden containers.
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\nIn this research, the empty container repositioning problem under two related real marine transportation scenarios has been studied. The first scenario considers multi-vessel transportation for both laden and empty containers with split vessel routing (denoted as ECRSVR). A mathematical model of ECRSVR is developed based on the formulation of the Pickup and Delivery Problem with Time Windows (PDPTW). However, a major research issue arises due to such integrated consideration of split vessel routing and transportation of empty and laden containers. The resulting model will encompass many more new decision variables and constraints. Thus, it cannot be readily solved by existing algorithms. To address this issue, a new Branch-and-Price (B&P) algorithm incorporating B&B and column generation is developed. Specifically, a new dynamic programming algorithm with new label structure and dominance rules is established to solve the pricing problem in B&P, and a new multi-layer branching scheme is developed for implementing the algorithm. The effectiveness and robustness of the proposed methodology are then tested on a set of randomly generated problems.
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\nThe second scenario extends the first by including transshipment activities in the ECRSVR (denoted as ECRSVRTS). Apparently, the inclusion of transshipment activities will significantly increase the size of the solution space of feasible routing and distribution schemes. Also, more constraints have to be included in the model to account for the interdependence between the distribution schemes of separate vessels. To tackle this problem, two mathematical models are developed. The first is a port-based model that explicitly characterizes the routing plans and distribution volumes for each vessel at each time interval as decision variables. The resulting integer program is non-linear and difficult to solve. Therefore, an equivalent event-driven node-based model inspired by the formulation of PDPTW is constructed. In this model, the pickup/delivery tasks and transshipment events are represented as nodes in a generated topology. The B&P scheme is then modified to solve the node-based model and a Hybrid Scatter Tabu Search (HSTS) is developed to solve the pricing problems. Results obtained from the numerical experiments using randomly generated test problems show that the proposed HSTS algorithm outperforms conventional scatter search and tabu search algorithms.

Pickup and delivery problem is commonly exist in logistics system, and its optimization considering time windows is essential in practical application. Meanwhile, with the increasing awareness of environmental protection, carbon emissions should also be considered during the routing design. In this paper, a low-carbon model is introduced for the pickup and delivery problem with time windows; a division method is proposed for the different delivery costs of logistics customers in different regions; a capacity constraint processing approach is proposed to solve the capacity of the vehicle during driving. Since the requirements for the logistic system are described by several objective formulations, the multi-objective approach is adopted to solve the optimization problem. The experiment selects two sets of Shanghai logistics data at different scales. Feasible solutions with advantages of different expectations can be obtained from the experiment result.

A column generation approach to capacitated p-median problems

Adaptive large neighborhood search for the pickup and delivery problem with time windows, profits, and reserved requests

Strong cutting planes for the capacitated multi-pickup and delivery problem with time windows

SYNOPTIC ABSTRACTWe apply Benders' decomposition procedure to the single-vehicle routing and scheduling problem with time windows, partial loads, and dwell times. We provide a formulation and demonstrate that the scheduling subproblem is the dual of a network flow problem. We describe an exact algorithm which exploits its structure, and construct a route improvement heuristic based on the master problem. A heuristic for building an initial route is also presented.

A column generation and Combinatorial Benders Decomposition algorithm for the Selective Dial-A-Ride-Problem