Michele Monaci

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.

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.

Passenger Railway Optimization

Publisher Summary Railway transportation can be split into passenger transportation and cargo transportation. This chapter discusses the European situation, where the major part of railway transportation consists of passenger transportation without addressing important problems in cargo transportation—such as car blocking, train makeup, train routing, and empty car distribution. The chapter describes several mathematical models and optimization techniques that have been developed for effectively supporting traditional planning processes in passenger railway transportation. A lot of research has been carried out in this area, both of a practical and theoretical nature. The results of this research are starting to be applied in practice. Real-time control is at the other side of the planning spectrum. The current trend in the railway industry is a shift from “planning in detail” to “effective real-time control.” Disturbances and disruptions in the railway operations are inevitable. Therefore, large parts of the operational plans are never carried out.

Light Robustness

We consider optimization problems where the exact value of the input data is not known in advance and can be affected by uncertainty. For these problems, one is typically required to determine a robust solution, i.e., a possibly suboptimal solution whose feasibility and cost is not affected heavily by the change of certain input coefficients. Two main classes of methods have been proposed in the literature to handle uncertainty: stochastic programming (offering great flexibility, but often leading to models too large in size to be handled efficiently), and robust optimization (whose models are easier to solve but sometimes lead to very conservative solutions of little practical use). In this paper we investigate a heuristic way to model uncertainty, leading to a modelling framework that we call Light Robustness. Light Robustness couples robust optimization with a simplified two-stage stochastic programming approach, and has a number of important advantages in terms of flexibility and ease to use. In particular, experiments on both random and real word problems show that Light Robustness is sometimes able to produce solutions whose quality is comparable with that obtained through stochastic programming or robust models, though it requires less effort in terms of model formulation and solution time.

A Lagrangian heuristic algorithm for a real-world train timetabling problem

The train timetabling problem (TTP) aims at determining an optimal timetable for a set of trains which does not violate track capacities and satisfies some operational constraints.In this paper, we describe the design of a train timetabling system that takes into account several additional constraints that arise in real-world applications. In particular, we address the following issues: • Manual block signaling for managing a train on a track segment between two consecutive stations. • Station capacities, i.e., maximum number of trains that can be present in a station at the same time. • Prescribed timetable for a subset of the trains, which is imposed when some of the trains are already scheduled on the railway line and additional trains are to be inserted. • Maintenance operations that keep a track segment occupied for a given period. We show how to incorporate these additional constraints into a mathematical model for a basic version of the problem, and into the resulting Lagrangian heuristic. Computational results on real-world instances from Rete Ferroviaria Italiana (RFI), the Italian railway infrastructure management company, are presented.

A Metaheuristic Approach for the Vertex Coloring Problem

Given an undirected graph G = (V, E), the vertex coloring problem (VCP) requires to assign a color to each vertex in such a way that colors on adjacent vertices are different and the number of colors used is minimized. In this paper, we propose a metaheuristic approach for VCP that performs two phases: the first phase is based on an evolutionary algorithm, whereas the second one is a postoptimization phase based on the set covering formulation of the problem. Computational results on a set of DIMACS instances show that the overall algorithm is able to produce high-quality solutions in a reasonable amount of time. For four instances, the proposed algorithm is able to improve the best-known solution while for almost all the remaining instances, it finds the best-known solution in the literature.

On the two-dimensional Knapsack Problem

We address the two-dimensional Knapsack Problem (2KP), aimed at packing a maximum-profit subset of rectangles selected from a given set into another rectangle. We consider the natural relaxation of 2KP given by the one-dimensional KP with item weights equal to the rectangle areas, proving the worst-case performance of the associated upper bound, and present and compare computationally four exact algorithms based on the above relaxation, showing their effectiveness.

Integer linear programming models for 2-staged two-dimensional Knapsack problems

Abstract. We are given a unique rectangular piece of stock material S, with height H and width W, and a list of m rectangular shapes to be cut from S. Each shapes type i (i = 1, ..., m) is characterized by a height , a width , a profit , and an upper bound ubi indicating the maximum number of items of type i which can be cut. We refer to the Two-Dimensional Knapsack (TDK) as the problem of determining a cutting pattern of S maximizing the sum of the profits of the cut items. In particular, we consider the classical variant of TDK in which the maximum number of cuts allowed to obtain each item is fixed to 2, and we refer to this problem as 2-staged TDK (2TDK). For the 2TDK problem we present two new Integer Linear Programming models, we discuss their properties, and we compare them with other formulations in terms of the LP bound they provide. Finally, both models are computationally tested within a standard branch-and-bound framework on a large set of instances from the literature by reinforcing them with the addition of linear inequalities to eliminate symmetries.

A Set-Covering-Based Heuristic Approach for Bin-Packing Problems

Several combinatorial optimization problems can be formulated as large set-covering problems. In this work, we use the set-covering formulation to obtain a general heuristic algorithm for this type of problem, and describe our implementation of the algorithm for solving two variants of the well-known (one-dimensional) bin-packing problem: the two-constraint bin-packing problem and the basic version of the two-dimensional bin-packing problem, where the objects cannot be rotated and no additional requirements are imposed. In our approach, both the column-generation and the column-optimization phases are heuristically performed. In particular, in the first phase, we do not generate the entire set of columns, but only a small subset of it, by using greedy procedures and fast constructive heuristic algorithms from the literature. In the second phase, we solve the associated set-covering instance by means of a Lagrangian-based heuristic algorithm. Extensive computational results on test instances from the literature show that, for the two considered problems, this approach is competitive, with respect to both the quality of the solution and the computing time, with the best heuristic and metaheuristic algorithms proposed so far.

Metaheuristic Algorithms for the Strip Packing Problem

Given a set of rectangular items and a strip of given width, we consider the problem of allocating all the items to a minimum height strip. We present a Tabu search algorithm, a genetic algorithm and we combine the two into a hybrid approach. The performance of the proposed algorithms is evaluated through extensive computational experiments on instances from the literature and on randomly generated instances.