Andrea Lodi

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 algorithmic framework for convex mixed integer nonlinear programs

This paper is motivated by the fact that mixed integer nonlinear programming is an important and difficult area for which there is a need for developing new methods and software for solving large-scale problems. Moreover, both fundamental building blocks, namely mixed integer linear programming and nonlinear programming, have seen considerable and steady progress in recent years. Wishing to exploit expertise in these areas as well as on previous work in mixed integer nonlinear programming, this work represents the first step in an ongoing and ambitious project within an open-source environment. COIN-OR is our chosen environment for the development of the optimization software. A class of hybrid algorithms, of which branch-and-bound and polyhedral outer approximation are the two extreme cases, are proposed and implemented. Computational results that demonstrate the effectiveness of this framework are reported. Both the library of mixed integer nonlinear problems that exhibit convex continuous relaxations, on which the experiments are carried out, and a version of the software used are publicly available.

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 feasibility pump

In this paper we consider the NP-hard problem of finding a feasible solution (if any exists) for a generic MIP problem of the form min{cTx:Ax≥b,xj integer ∀j ∈ }. Trivially, a feasible solution can be defined as a point x* ∈ P:={x:Ax≥b} that is equal to its rounding , where the rounded point is defined by := x*j if j ∈ and := x*j otherwise, and [·] represents scalar rounding to the nearest integer. Replacing “equal” with “as close as possible” relative to a suitable distance function Δ(x*, ), suggests the following Feasibility Pump (FP) heuristic for finding a feasible solution of a given MIP.We start from any x* ∈ P, and define its rounding . At each FP iteration we look for a point x* ∈ P that is as close as possible to the current by solving the problem min {Δ(x, ): x ∈ P}. Assuming Δ(x, ) is chosen appropriately, this is an easily solvable LP problem. If Δ(x*, )=0, then x* is a feasible MIP solution and we are done. Otherwise, we replace by the rounding of x*, and repeat.We report computational results on a set of 83 difficult 0-1 MIPs, using the commercial software ILOG-Cplex 8.1 as a benchmark. The outcome is that FP, in spite of its simple foundation, proves competitive with ILOG-Cplex both in terms of speed and quality of the first solution delivered. Interestingly, ILOG-Cplex could not find any feasible solution at the root node for 19 problems in our test-bed, whereas FP was unsuccessful in just 3 cases.

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.

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.

MIPLIB 2010 - Mixed Integer Programming Library version 5

This paper reports on the fifth version of the Mixed Integer Programming Library. The miplib 2010 is the first miplib release that has been assembled by a large group from academia and from industry, all of whom work in integer programming. There was mutual consent that the concept of the library had to be expanded in order to fulfill the needs of the community. The new version comprises 361 instances sorted into several groups. This includes the main benchmark test set of 87 instances, which are all solvable by today’s codes, and also the challenge test set with 164 instances, many of which are currently unsolved. For the first time, we include scripts to run automated tests in a predefined way. Further, there is a solution checker to test the accuracy of provided solutions using exact arithmetic.

Cost-Based Domain Filtering

Constraint propagation is aimed at removing from variable domains combinations of values which cannot appear in any consistent solution. Pruning derives from feasibility reasoning. When coping with optimization problems, pruning can be performed also on the basis of costs, i.e., optimality reasoning. Propagation can be aimed at removing combination of values which cannot lead to solutions whose cost is better then the best one found so far. For this purpose, we embed in global constraints optimization components representing suitable relaxations of the constraint itself. These components provide efficient Operations Research algorithms computing the optimal solution of the relaxed problem and a gradient function representing the estimated cost of each variable-value assignment. We exploit these pieces of information for pruning and for guiding the search. We have applied these techniques to a couple of ILOG Solver global constraints (a constraint of difference and a path constraint) and tested the approach on a variety of combinatorial optimization problems such as Timetabling, Travelling Salesman Problems and Scheduling Problems with setup. Comparisons with pure Constraint Programming approaches and related literature clearly show the benefits of the proposed approach. By using cost-based filtering in global constraints, we can optimally solve problems that are one order of magnitude greater than those solved by pure CP approaches, and we outperform other hybrid approaches integrating OR techniques in Constraint Programming.

Heuristic algorithms for the three-dimensional bin packing problem

Abstract The Three-Dimensional Bin Packing Problem (3BP) consists of allocating, without overlapping, a given set of three-dimensional rectangular items to the minimum number of three-dimensional identical finite bins. The problem is NP-hard in the strong sense, and finds many industrial applications. We introduce a Tabu Search framework exploiting a new constructive heuristic for the evaluation of the neighborhood. Extensive computational results on standard benchmark instances show the effectiveness of the approach with respect to exact and heuristic algorithms from the literature.

Local Search and Constraint Programming

Real-world combinatorial optimization problems have two main characteristics which make them difficult: they are usually large, and they are not pure, i.e., they involve a heterogeneous set of side constraints. Hence, in most cases, exact approaches cannot be applied to solve real-world problems, whereas incomplete algorithms, and among them Local Search and Metaheuristic methods, have proved to obtain very good results in practice. Moreover, real-world applications typically lead to frequent update/addition of constraints, thus the algorithmic ap-proach requires flexibility, and this flexibility can be guaranteed by Constraint Programming.