Mikkel M. Sigurd

Using Decomposition Techniques and Constraint Programming for Solving the Two-Dimensional Bin-Packing Problem

The two-dimensional bin-packing problem is the problem of orthogonally packing a given set of rectangles into a minimum number of two-dimensional rectangular bins. The problem is NP-hard and very difficult to solve in practice as no good mixed integer programming (MIP) formulation has been found for the packing problem. We propose an algorithm based on the well-known Dantzig-Wolfe decomposition where the master problem deals with the production constraints on the rectangles while the subproblem deals with the packing of rectangles into a single bin. The latter problem is solved as a constraint-satisfaction problem (CSP), which makes it possible to formulate a number of additional constraints that may be difficult to formulate as MIP models. This includes guillotine-cutting requirements, relative positions, fixed positions and irregular bins. The CSP approach uses forward propagation to prune inferior arrangements of rectangles. Unsuccessful attempts to pack rectangles into a bin are brought back to the master model as valid inequalities. Hence, CSP is used not only to solve the pricing problem but also to generate valid inequalities in a branch-and-cut system. Using delayed column-generation, we obtain lower bounds of very good quality in reasonable time. In all instances considered, we obtain similar or better bounds than previously published. Several instances with up to n = 100 rectangles are solved to optimality through the developed branch-and-price-and-cut algorithm.

The two-dimensional bin packing problem with variable bin sizes and costs

The two-dimensional variable sized bin packing problem (2DVSBPP) is the problem of packing a set of rectangular items into a set of rectangular bins. The bins have different sizes and different costs, and the objective is to minimize the overall cost of bins used for packing the rectangles. We present an integer-linear formulation of the 2DVSBPP and introduce several lower bounds for the problem. By using Dantzig-Wolfe decomposition we are able to obtain lower bounds of very good quality. The LP-relaxation of the decomposed problem is solved through delayed column generation, and an exact algorithm based on branch-and-price is developed. The paper is concluded with a computational study, comparing the tightness of the various lower bounds, as well as the performance of the exact algorithm for instances with up to 100 items.

A service flow model for the liner shipping network design problem

Global liner shipping is a competitive industry, requiring liner carriers to carefully deploy their vessels efficiently to construct a cost competitive network. This paper presents a novel compact formulation of the liner shipping network design problem (LSNDP) based on service flows. The formulation alleviates issues faced by arc flow formulations with regards to handling multiple calls to the same port. A problem which has not been fully dealt with earlier by LSNDP formulations. Multiple calls are handled by introducing service nodes, together with port nodes in a graph representation of the problem, and by introducing numbered arcs between a port and a novel service node. An arc from a port node to a service node indicate whether a service is calling the port or not. This representation allows recurrent calls of a service to a port, which previously could not be handled by LSNDP models. The model ensures strictly weekly frequencies of services, ensures that port-vessel draft capabilities are not violated, respects vessel capacities and the number of vessels available. The profit of the generated network is maximized, i.e. the revenue of flowed cargo subtracted operational costs of the network and a penalty for not flowed cargo. The model can be used to design liner shipping networks to utilize a container carrier’s assets efficiently and to investigate possible scenarios of changed market conditions. The model is solved as a Mixed Integer Program. Results are presented for the two smallest instances of the benchmark suite LINER-LIB-2012 presented in Brouer, Alvarez, Plum, Pisinger, and Sigurd (2013).

Single liner shipping service design

The design of container shipping networks is an important logistics problem, involving assets and operational costs measured in billions of dollars. To guide the optimal deployment of the ships, a single vessel round trip is considered by minimizing operational costs and flowing the best paying demand under commercially driven constraints. This paper introduces the Single Liner Shipping Service Design Problem. Arc-flow and path-flow models are presented using state-of-the-art elements from the wide literature on pickup and delivery problems. A Branch-and-Cut-and-Price algorithm is proposed, and implementation details are discussed. The algorithm can solve instances with up to 25 ports to optimality, a very promising result as real-world vessel roundtrips seldom involve more than 20 ports.

Ship Scheduling with Recurring Visits and Visit Separation Requirements

This chapter discusses an application of advanced planning support in designing a sea-transport system. The system is designed for Norwegian companies who depend on sea-transport between Norway and Central Europe. They want to achieve faster and more frequent transport by combining tonnage. This requires the possible construction of up to 15 new ships with potential investments of approximately 150 mill US dollars. The problem is a variant of the general pickup and delivery problem with multiple time windows. In addition, it includes requirements for recurring visits, separation between visits and limits on transport lead-time. It is solved by a heuristic branch-and-price algorithm.

Construction of Minimum-Weight Spanners

Spanners are sparse subgraphs that preserve distances up to a given factor in the underlying graph. Recently spanners have found important practical applications in metric space searching and message distribution in networks. These applications use some variant of the so-called greedy algorithm for constructing the spanner — an algorithm that mimics Kruskal’s minimum spanning tree algorithm. Greedy spanners have nice theoretical properties, but their practical performance with respect to total weight is unknown. In this paper we give an exact algorithm for constructing minimum-weight spanners in arbitrary graphs. By using the solutions (and lower bounds) from this algorithm, we experimentally evaluate the performance of the greedy algorithm for a set of realistic problem instances.

Optimization of the drayage problem using exact methods

Abstract Major liner shipping companies offer pre- and end-haulage as part of a door-to-door service, but unfortunately pre- and end-haulage is frequently one of the major bottlenecks in efficient liner shipping due to the lack of coordination between customers. In this paper, we apply techniques from vehicle routing problems to schedule pre- and end-haulage of containers, and perform tests on data from a major liner shipping company. The paper considers several versions of the scheduling problem such as having multiple empty container depots, and having to balance the empty container depot levels. The influence of the side constraints on the overall cost is analysed. By exploring the fact that the number of possible routes in the considered case is quite limited, we show that the model can be solved within a minute by use of column enumeration. Alternative constraints and problem formulations, such as balancing empty container storage level at depots, are considered. Computational results are reported on real-life data from a major liner shipping company.

Koenderink Corner Points

Koenderink characterizes the local shape of 2D surfaces in 3D in terms of the shape index and the local curvedness. The index characterizes the local type of surface point: concave, hyperbolic, or convex. The curvedness expresses how articulated the local shape is, from flat towards very peaked. In this paper we define corner points as point on a shape of locally maximal Koenderink curvedness. These points can be detected very robustly based on integration indices. This is not the case for other natural corner points like extremal points. Umbilici can likewise be detected robustly by integral expressions, but does not correspond to intuitive corners of a shape. Furthermore, we show that Koenderink corner points do not generically coincide with other well-known shape features such as umbilici, ridges, parabolic lines, sub-parabolic lines, or extremal points. This is formalized through the co-dimension of intersection of the different structures.