Fabio Furini

Improved rolling horizon approaches to the aircraft sequencing problem

In a scenario characterized by a continuous growth of air transportation demand, the runways of large airports serve hundreds of aircraft every day. Aircraft sequencing is a challenging problem that aims to increase runway capacity in order to reduce delays as well as the workload of air traffic controllers. In many cases, the air traffic controllers solve the problem using the simple “first-come-first-serve” (FCFS) rule. In this paper, we present a rolling horizon approach which partitions a sequence of aircraft into chunks and solves the aircraft sequencing problem (ASP) individually for each of these chunks. Some rules for deciding how to partition a given aircraft sequence are proposed and their effects on solution quality investigated. Moreover, two mixed integer linear programming models for the ASP are reviewed in order to formalize the problem, and a tabu search heuristic is proposed for finding solutions to the ASP in a short computation time. Finally, we develop an IRHA which, using different chunking rules, is able to find solutions significantly improving on the FCFS rule for real-world air traffic instances from Milano Linate Airport.

Automatic Dantzig---Wolfe reformulation of mixed integer programs

Dantzig–Wolfe decomposition (or reformulation) is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs). However, the method is not implemented in any state-of-the-art MIP solver as it is considered to require structural problem knowledge and tailoring to this structure. We provide a computational proof-of-concept that the reformulation can be automated. That is, we perform a rigorous experimental study, which results in identifying a score to estimate the quality of a decomposition: after building a set of potentially good candidates, we exploit such a score to detect which decomposition might be useful for Dantzig–Wolfe reformulation of a MIP. We experiment with general instances from MIPLIB2003 and MIPLIB2010 for which a decomposition method would not be the first choice, and demonstrate that strong dual bounds can be obtained from the automatically reformulated model using column generation. Our findings support the idea that Dantzig–Wolfe reformulation may hold more promise as a general-purpose tool than previously acknowledged by the research community.

Aircraft sequencing problems via a rolling horizon algorithm

Aircraft sequencing on the runway is a challenging optimization problem that aims to reduce the delays and the air traffic controllers workload in a scenario characterized by a continuous growth of the air transportation demand. In this paper we consider the problem of sequencing both arrivals and departures on a single runway airport. We formalize the problem using a Mixed Integer Programming Model and we propose a rolling horizon solution approach. Computational results on real-world air traffic instances from the Milano Linate Airport are reported. The results show that the proposed approach is able to significantly improve on the First Come First Served (FCFS) sequence.

Models for the two-dimensional two-stage cutting stock problem with multiple stock size

We consider a Two-Dimensional Cutting Stock Problem (2DCSP) where stock of different sizes is available, and a set of rectangular items has to be obtained through two-stage guillotine cuts. We propose and computationally compare three Mixed-Integer Programming models for the 2DCSP developing formulations from the literature. The first two models have a polynomial and pseudo-polynomial number of variables, respectively, and can be solved with a general-purpose MIP solver. The third model, having an exponential number of variables, is solved via branch-and-price techniques. We conclude the paper describing the results of extensive computational experiments on a set of benchmark instances from the literature.

Partial convexification of general MIPs by Dantzig-Wolfe reformulation

Dantzig-Wolfe decomposition is well-known to provide strong dual bounds for specially structured mixed integer programs (MIPs) in practice. However, the method is not implemented in any state-of-the-art MIP solver: it needs tailoring to the particular problem; the decomposition must be determined from the typical bordered block-diagonal matrix structure; the resulting column generation subproblems must be solved efficiently; etc. We provide a computational proof-of-concept that the process can be automated in principle, and that strong dual bounds can be obtained on general MIPs for which a solution by a decomposition has not been the first choice. We perform an extensive computational study on the 0-1 dynamic knapsack problem (without block-diagonal structure) and on general MIPLIB2003 instances. Our results support that Dantzig-Wolfe reformulation may hold more promise as a generalpurpose tool than previously acknowledged by the research community.

Heuristic and Exact Algorithms for the Interval Min-Max Regret Knapsack Problem

We consider a generalization of the 0-1 knapsack problem in which the profit of each item can take any value in a range characterized by a minimum and a maximum possible profit. A set of specific profits is called a scenario. Each feasible solution associated with a scenario has a regret, given by the difference between the optimal solution value for such scenario and the value of the considered solution. The interval min-max regret knapsack problem MRKP is then to find a feasible solution such that the maximum regret over all scenarios is minimized. The problem is extremely challenging both from a theoretical and a practical point of view. Its decision version is complete for the second level of the polynomial hierarchy hence it is most probably not in 𝒩𝒫. In addition, even computing the regret of a solution with respect to a scenario requires the solution of an 𝒩𝒫-hard problem. We examine the behavior of classical combinatorial optimization approaches when adapted to the solution of the MRKP. We introduce an iterated local search approach and a Lagrangian-based branch-and-cut algorithm and evaluate their performance through extensive computational experiments.

Generation of Antipodal Random Vectors With Prescribed Non-Stationary 2-nd Order Statistics

A Look-Up-Table-based method is proposed to generate random instances of an antipodal n-dimensional vector so that its 2-nd order statistics are as close as possible to a given specification. The method is based on linear optimization and exploits column-generation techniques to cope with the exponential complexity of the task. It yields a LUT whose storage requirements are only O(n3) and thus are compatible with hardware implementation for non-negligible n. Applications are shown in the fields of Compressive Sensing and of Ultra Wide Band systems based on Direct Sequence - Code Division Multiple Acces.

Uncommon Dantzig-Wolfe Reformulation for the Temporal Knapsack Problem

We study a natural generalization of the knapsack problem, in which each item exists only for a given time interval. One has to select a subset of the items as in the classical case, guaranteeing that for each time instant, the set of existing selected items has total weight no larger than the knapsack capacity. We focus on the exact solution of the problem, noting that prior to our work, the best method was the straightforward application of a general-purpose solver to the natural integer linear programming formulation. Our results indicate that much better results can be obtained by using the same general-purpose solver to tackle a nonstandard Dantzig-Wolfe reformulation in which subproblems are associated with groups of constraints. This is also interesting because the more natural Dantzig-Wolfe reformulation of single constraints performs extremely poorly in practice.

A fast heuristic approach for train timetabling in a railway node

Abstract We consider a conflict-free scheduling problem which arises in railway networks, where ideal timetables have been provided for a set of trains, but where these timetables may be conflicting. We use a space-time graph approach from the railway scheduling literature in order to develop a fast heuristic which resolves conflicts by adjusting the ideal timetables while attempting to minimize the deviation from the ideal timetable. Our approach is tested on realistic data obtained from the railway node of Milan.

Solving vertex coloring problems as maximum weight stable set problems

In Vertex Coloring Problems, one is required to assign a color to each vertex of an undirected graph in such a way that adjacent vertices receive different colors, and the objective is to minimize the cost of the used colors. In this work we solve four different coloring problems formulated as Maximum Weight Stable Set Problems on an associated graph. We exploit the transformation proposed by Cornaz and Jost (2008), where given a graph G , an auxiliary graph G ź is constructed, such that the family of all stable sets of G ź is in one-to-one correspondence with the family of all feasible colorings of G . The transformation in Cornaz and Jost (2008) was originally proposed for the classical Vertex Coloring and the Max-Coloring problems; we extend it to the Equitable Coloring Problem and the Bin Packing Problem with Conflicts. We discuss the relation between the Maximum Weight Stable formulation and a polynomial-size formulation for the VCP, proposed by Campelo etźal. (2008) and called the Representative formulation. We report extensive computational experiments on benchmark instances of the four problems, and compare the solution method with the state-of-the-art algorithms. By exploiting the proposed method, we largely outperform the state-of-the-art algorithm for the Max-coloring Problem, and we are able to solve, for the first time to proven optimality, 14 Max-coloring and 2 Equitable Coloring instances.