Marco A. Boschetti

The Two-Dimensional Finite Bin Packing Problem Part I: New lower bounds for the oriented case

Abstract.The Two-Dimensional Finite Bin Packing Problem (2BP) consists of determining the minimum number of large identical rectangles, bins, that are required for allocating without overlapping a given set of rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. In this paper we describe new lower bounds for the 2BP where the items have a fixed orientation and we show that the new lower bounds dominate two lower bounds proposed in the literature. These lower bounds are extended in Part II (see Boschetti and Mingozzi 2002) for a more general version of the 2BP where some items can be rotated by

A Set Partitioning Approach to the Crew Scheduling Problem

90^{\circ}

A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem

. Moreover, in Part II a new heuristic algorithm for solving both versions of the 2BP is presented and computational results on test problems from the literature are given in order to evaluate the effectiveness of the proposed lower bounds.

The Two-Dimensional Finite Bin Packing Problem. Part II: New lower and upper bounds

The crew scheduling problem (CSP) appears in many mass transport systems (e.g., airline, bus, and railway industry) and consists of scheduling a number of crews to operate a set of transport tasks satisfying a variety of constraints. This problem is formulated as a set partitioning problem with side constraints (SP), where each column of the SP matrix corresponds to a feasible duty, which is a subset of tasks performed by a crew. We describe a procedure that, without using the SP matrix, computes a lower bound to the CSP by finding a heuristic solution to the dual of the linear relaxation of SP. Such dual solution is obtained by combining a number of different bounding procedures. The dual solution is used to reduce the number of variables in the SP in such a way that the resulting SP problem can be solved by a branch-and-bound algorithm. Computational results are given for problems derived from the literature and involving from 50 to 500 tasks.

Benders decomposition, Lagrangean relaxation and metaheuristic design

Abstract The two-dimensional orthogonal non-guillotine cutting problem (NGCP) appears in many industries (like wood and steel industries) and consists in cutting a rectangular master surface into a number of rectangular pieces, each with a given size and value. The pieces must be cut with their edges always parallel or orthogonal to the edges of the master surface (orthogonal cuts). The objective is to maximize the total value of the pieces cut. In this paper, we propose a two-level approach for solving the NGCP, where, at the first level, we select the subset of pieces to be packed into the master surface without specifying the layout, while at a second level we check only if a feasible packing layout exists. This approach has been already proposed by Fekete and Schepers [S.P. Fekete, J. Schepers, A new exact algorithm for general orthogonal d-dimensional knapsack problems, ESA 97, Springer Lecture Notes in Computer Science 1284 (1997) 144–156; S.P. Fekete, J. Schepers, On more-dimensional packing III: Exact algorithms, Tech. Rep. 97.290, Universitat zu Koln, Germany, 2000; S.P. Fekete, J. Schepers, J.C. van der Veen, An exact algorithm for higher-dimensional orthogonal packing, Tech. Rep. Under revision on Operations Research, Braunschweig University of Technology, Germany, 2004] and Caprara and Monaci [A. Caprara, M. Monaci, On the two-dimensional knapsack problem, Operations Research Letters 32 (2004) 2–14]. We propose improved reduction tests for the NGCP and a cutting-plane approach to be used in the first level of the tree search to compute effective upper bounds. Computational tests on problems derived from the literature show the effectiveness of the proposed approach, that is able to reduce the number of nodes generated at the first level of the tree search and the number of times the existence of a feasible packing layout is tested.

New lower bounds for the three-dimensional finite bin packing problem

Abstract.This paper is the second of a two part series and describes new lower and upper bounds for a more general version of the Two-Dimensional Finite Bin Packing Problem (2BP) than the one considered in Part I (see Boschetti and Mingozzi 2002). With each item is associated an input parameter specifying if it has a fixed orientation or it can be rotated by

An Exact Algorithm for the Two-Dimensional Strip-Packing Problem

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A dual ascent procedure for the set partitioning problem

. This problem contains as special cases the oriented and non-oriented 2BP. The new lower bound is based on the one described in Part I for the oriented 2BP. The computational results on the test problems derived from the literature show the effectiveness of the new proposed lower and upper bounds.

Scatter Search Methods for the Covering Tour Problem

Abstract Large part of combinatorial optimization research has been devoted to the study of exact methods leading to a number of very diversified solution approaches. Some of those older frameworks can now be revisited in a metaheuristic perspective, as they are quite general frameworks for dealing with optimization problems. In this work, we propose to investigate the possibility of reinterpreting decompositions, with special emphasis on the related Benders and Lagrangean relaxation techniques. We show how these techniques, whose heuristic effectiveness is already testified by a wide literature, can be framed as a “master process that guides and modifies the operations of subordinate heuristics”, i.e., as metaheuristics. Obvious advantages arise from these approaches, first of all the runtime evolution of both upper and lower bounds to the optimal solution cost, thus yielding both a high-quality heuristic solution and a runtime quality certificate of that same solution.

Algorithms for nesting with defects

The three-dimensional finite bin packing problem (3BP) consists of determining the minimum number of large identical three-dimensional rectangular boxes, bins, that are required for allocating without overlapping a given set of three-dimensional rectangular items. The items are allocated into a bin with their edges always parallel or orthogonal to the bin edges. The problem is strongly NP-hard and finds many practical applications. We propose new lower bounds for the problem where the items have a fixed orientation and then we extend these bounds to the more general problem where for each item the subset of rotations by 90° allowed is specified. The proposed lower bounds have been evaluated on different test problems derived from the literature. Computational results show the effectiveness of the new lower bounds.