Silvano Martello

Two-dimensional packing problems: A survey

We consider problems requiring to allocate a set of rectangular items to larger rectangular standardized units by minimizing the waste. In two-dimensional bin packing problems these units are finite rectangles, and the objective is to pack all the items into the minimum number of units, while in two-dimensional strip packing problems there is a single standardized unit of given width, and the objective is to pack all the items within the minimum height. We discuss mathematical models, and survey lower bounds, classical approximation algorithms, recent heuristic and metaheuristic methods and exact enumerative approaches. The relevant special cases where the items have to be packed into rows forming levels are also discussed in detail.

The Three-Dimensional Bin Packing Problem

The problem addressed in this paper is that of orthogonally packing a given set of rectangular-shaped items into the minimum number of three-dimensional rectangular bins. The problem is strongly NP-hard and extremely difficult to solve in practice. Lower bounds are discussed, and it is proved that the asymptotic worst-case performance ratio of the continuous lower bound is ?. An exact algorithm for filling a single bin is developed, leading to the definition of an exact branch-and-bound algorithm for the three-dimensional bin packing problem, which also incorporates original approximation algorithms. Extensive computational results, involving instances with up to 90 items, are presented: It is shown that many instances can be solved to optimality within a reasonable time limit.

Heuristic and Metaheuristic Approaches for a Class of Two-Dimensional Bin Packing Problems

Two-dimensional bin packing problems consist of allocating, without overlapping, a given set of small rectangles (items) to a minimum number of large identical rectangles (bins), with the edges of the items parallel to those of the bins. According to the specific application, the items may either have a fixed orientation or they can be rotated by 90°. In addition, it may or not be imposed that the items are obtained through a sequence of edge-to-edge cuts parallel to the edges of the bin. In this article, we consider the class of problems arising from all combinations of the above requirements. We introduce a new heuristic algorithm for each problem in the class, and a unified tabu search approach that is adapted to a specific problem by simply changing the heuristic used to explore the neighborhood. The average performance of the single heuristics and of the tabu search are evaluated through extensive computational experiments.

The selective travelling salesman problem

Abstract Given a weighted graph with profits associated with the vertices, the selective travelling salesman problem (or orienteering problem) consists of selecting a simple circuit of maximal total profit, whose length does not exceed a prespecified bound. This paper provides integer linear programming formulations for the problem. Upper and lower bounds are then derived and embedded in exact enumerative algorithms. Computational results are reported.

New trends in exact algorithms for the 0-1 knapsack problem

Abstract While the 1980s were focused on the solution of large sized “easy” knapsack problems (KPs), this decade has brought several new algorithms, which are able to solve “hard” large sized instances. We will give an overview of the recent techniques for solving hard KPs, with special emphasis on the addition of cardinality constraints, dynamic programming, and rudimentary divisibility. Computational results, comparing all recent algorithms, are presented.

Recent advances on two-dimensional bin packing problems

We survey recent advances obtained for the two-dimensional bin packing problem, with special emphasis on exact algorithms and effective heuristic and metaheuristic approaches.

Lower bounds and reduction procedures for the bin packing problem

The bin packing problem, in which a set of items of various sizes has to be packed into a minimum number of identical bins, has been extensively studied during the past fifteen years, mainly with the aim of finding fast heuristic algorithms to provide good approximate solutions. We present lower bounds and a dominance criterion and derive a reduction algorithm. Lower bounds are evaluated through an extension of the concept of worst-case performance. For both lower bounds and reduction algorithm an experimental analysis is provided.

An Exact Approach to the Strip-Packing Problem

We consider the problem of orthogonally packing a given set of rectangular items into a given strip, by minimizing the overall height of the packing. The problem is NP-hard in the strong sense, and finds several applications in cutting and packing. We propose a new relaxation that produces good lower bounds and gives information to obtain effective heuristic algorithms. These results are used in a branch-and-bound algorithm, which was able to solve test instances from the literature involving up to 200 items.

A Tabu Search Algorithm for a Routing and Container Loading Problem

This article considers a combination of capacitated vehicle routing and three-dimensional loading, with additional constraints frequently encountered in freight transportation. It proposes a tabu search algorithm that iteratively invokes an inner tabu search procedure for the solution of the loading subproblem. The algorithm is experimentally evaluated both on instances adapted from vehicle routing instances from the literature and on new real-world instances.

An MILP Approach for Short-Term Hydro Scheduling and Unit Commitment With Head-Dependent Reservoir

The paper deals with a unit commitment problem of a generation company whose aim is to find the optimal scheduling of a multiunit pump-storage hydro power station, for a short term period in which the electricity prices are forecasted. The problem has a mixed-integer nonlinear structure, which makes very hard to handle the corresponding mathematical models. However, modern mixed-integer linear programming (MILP) software tools have reached a high efficiency, both in terms of solution accuracy and computing time. Hence we introduce MILP models of increasing complexity, which allow to accurately represent most of the hydroelectric system characteristics, and turn out to be computationally solvable. In particular we present a model that takes into account the head effects on power production through an enhanced linearization technique, and turns out to be more general and efficient than those available in the literature. The practical behavior of the models is analyzed through computational experiments on real-world data.