Heuristics based on large neighborhood search have recently shown outstanding results in solving various transportation and scheduling problems. Large neighborhood search methods explore a complex neighborhood by use of heuristics. Using large neighborhoods makes it possible to find better candidate solutions in each iteration and hence traverse a more promising search path. Starting from the large neighborhood search method, we give an overview of very large scale neighborhood search methods and discuss recent variants and extensions like variable depth search and adaptive large neighborhood search.

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Large Neighborhood Search

In the pickup and delivery problem with time windows vehicle routes must be designed to satisfy a set of transportation requests, each involving a pickup and delivery location, under capacity, time window, and precedence constraints. This paper introduces a new branch-and-cut-and-price algorithm in which lower bounds are computed by solving through column generation the linear programming relaxation of a set partitioning formulation. Two pricing subproblems are considered in the column generation algorithm: an elementary and nonelementary shortest path problem. Valid inequalities are added dynamically to strengthen the relaxations. Some of the previously proposed inequalities for the pickup and delivery problem with time windows are also shown to be implied by the set partitioning formulation. Computational experiments indicate that the proposed algorithm outperforms a recent branch-and-cut algorithm.

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Branch and Cut and Price for the Pickup and Delivery Problem with Time Windows

This paper presents a survey of vehicle routing problems with multiple synchronization constraints. These problems exhibit, in addition to the usual task covering constraints, further synchronization requirements between the vehicles, concerning spatial, temporal, and load aspects. They constitute an emerging field in vehicle routing research and are becoming a “hot” topic. The contribution of the paper is threefold: (i) It presents a classification of different types of synchronization. (ii) It discusses the central issues related to the exact and heuristic solution of such problems. (iii) It comprehensively reviews pertinent literature with respect to applications as well as successful solution approaches, and it identifies promising algorithmic avenues.

Synchronization in Vehicle Routing---A Survey of VRPs with Multiple Synchronization Constraints

Heuristics for multi-attribute vehicle routing problems: A survey and synthesis

In this paper, we describe an effective exact method for solving both the capacitated vehicle routing problem (cvrp) and the vehicle routing problem with time windows (vrptw) that improves the method proposed by Baldacci et al. [Baldacci, R., N. Christofides, A. Mingozzi. 2008. An exact algorithm for the vehicle routing problem based on the set partitioning formulation with additional cuts. Math. Programming115(2) 351--385] for the cvrp. The proposed algorithm is based on the set partitioning (SP) formulation of the problem. We introduce a new route relaxation called ng-route, used by different dual ascent heuristics to find near-optimal dual solutions of the LP-relaxation of the SP model. We describe a column-and-cut generation algorithm strengthened by valid inequalities that uses a new strategy for solving the pricing problem. The new ng-route relaxation and the different dual solutions achieved allow us to generate a reduced SP problem containing all routes of any optimal solution that is finally solved by an integer programming solver. The proposed method solves four of the five open Solomon's vrptw instances and significantly improves the running times of state-of-the-art algorithms for both vrptw and cvrp.

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New Route Relaxation and Pricing Strategies for the Vehicle Routing Problem

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.

Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

This paper presents an exact method for solving the symmetric two-echelon capacitated vehicle routing problem, a transportation problem concerned with the distribution of goods from a depot to a set of customers through a set of satellite locations. The presented method is based on an edge flow model that is a relaxation and provides a valid lower bound. A specialized branching scheme is employed to obtain feasible solutions. Out of a test set of 93 instances the algorithm is able to solve 47 to optimality, surpassing previous exact algorithms.

A Branch-and-Cut Algorithm for the Symmetric Two-Echelon Capacitated Vehicle Routing Problem

This paper is concerned with the cargo allocation problem considering empty repositioning of containers for a liner shipping company. The aim is to maximize the profit of transported cargo in a network, subject to the cost and availability of empty containers. The formulation is a multi-commodity flow problem with additional inter-balancing constraints to control repositioning of empty containers. In a study of the cost efficiency of the global container-shipping network, Song et al. (2005) estimate that empty repositioning cost constitutes 27% of the total world fleet running cost. An arc-flow formulation is decomposed using the Dantzig–Wolfe principle to a path-flow formulation. A linear relaxation is solved with a delayed column generation algorithm. A feasible integer solution is found by rounding the fractional solution and adjusting flow balance constraints with leased containers. Computational results are reported for seven instances based on real-life shipping networks. Solving the relaxe...

Liner Shipping Cargo Allocation with Repositioning of Empty Containers

https://orbit.dtu.dk/files/6442746/muller.pdf

A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

This article presents a general framework for formulating cutting planes in the context of column generation for integer programs. Valid inequalities can be derived using the variables of an equivalent compact formulation (i.e., the subproblem variables) or the master problem variables. In the first case, cuts are added to the compact formulation, either at the master level or the subproblem level, and the decomposition process is reapplied. In the second case, we show that it is possible to model inequalities defined on the master problem variables by adding new variables and constraints to the subproblem formulation. The augmented subproblem indirectly indicates that there exists an augmented compact formulation that includes these new variables and constraints. Three examples on how to apply this framework are presented: the vehicle routing problem with time windows, the edge coloring problem, and the cutting stock problem. © 2011 Wiley Periodicals, Inc. NETWORKS, Vol. 58(4), 301–310 2011 © 2011 Wiley Periodicals, Inc.

Simon Spoorendonk

Papers

Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

A Branch-and-Cut Algorithm for the Symmetric Two-Echelon Capacitated Vehicle Routing Problem

Liner Shipping Cargo Allocation with Repositioning of Empty Containers

A hybrid adaptive large neighborhood search heuristic for lot-sizing with setup times

Cutting Planes for Branch-and-Price Algorithms