Enrico Malaguti

Overview of Optimization Problems in Electric Car-Sharing System Design and Management

Car-sharing systems are increasingly employing environmentally-friendly electric vehicles. The design and management of Ecar-sharing systems poses several additional challenges with respect to those based on traditional combustion vehicles, mainly related with the limited autonomy allowed by current battery technology. We review the main optimization problems arising in Ecar-sharing systems at strategic, tactical and operational levels, and discuss the existing approaches often developed for similar problems, for example in car-sharing systems with traditional vehicles. We also outline open problems and fruitful research directions.

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.

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.

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.

Modeling Two-Dimensional Guillotine Cutting Problems via Integer Programming

We propose a framework to model general guillotine restrictions in two-dimensional cutting problems formulated as mixed-integer linear programs (MIPs). The modeling framework requires a pseudopolynomial number of variables and constraints, which can be effectively enumerated for medium-size instances. Our modeling of general guillotine cuts is the first one that, once it is implemented within a state-of-the-art MIP solver, can tackle instances of challenging size. We mainly concentrate our analysis on the guillotine two-dimensional knapsack problem (G2KP), for which a model, and an exact procedure able to significantly improve the computational performance, are given. We also show how the modeling of general guillotine cuts can be extended to other relevant problems such as the guillotine two-dimensional cutting stock problem and the guillotine strip packing problem (GSPP). Finally, we conclude the paper discussing an extensive set of computational experiments on G2KP and GSPP benchmark instances from the literature.

A Branch-and-Bound Algorithm for the Knapsack Problem with Conflict Graph

We study the knapsack problem with conflict graph (KPCG), an extension of the 0-1 knapsack problem, in which a conflict graph describing incompatibilities between items is given. The goal of the KPCG is to select the maximum profit set of compatible items while satisfying the knapsack capacity constraint. We present a new branch-and-bound approach to derive optimal solutions to the KPCG in short computing times. Extensive computational experiments are reported showing that, for instances with graph density of 10% and larger, the proposed method outperforms a state-of-the-art approach and mixed-integer programming formulations tackled through a general purpose solver. The online supplement is available at https://doi.org/10.1287/ijoc.2016.0742.

Solving the Temporal Knapsack Problem via Recursive Dantzig-Wolfe Reformulation

The Temporal Knapsack Problem (TKP) is a generalization of the standard Knapsack Problem where a time horizon is considered, and each item consumes the knapsack capacity during a limited time interval only. In this paper we solve the TKP using what we call a Recursive Dantzig-Wolfe Reformulation (DWR) method. The generic idea of Recursive DWR is to solve a Mixed Integer Program (MIP) by recursively applying DWR, i.e., by using DWR not only for solving the original MIP but also for recursively solving the pricing sub-problems. In a binary case (like the TKP), the Recursive DWR method can be performed in such a way that the only two components needed during the optimization are a Linear Programming solver and an algorithm for solving Knapsack Problems. The Recursive DWR allows us to solve Temporal Knapsack Problem instances through computation of strong dual bounds, which could not be obtained by exploiting the best-known previous approach based on DWR. We solve the Temporal Knapsack Problem using what we call a Multilevel Dantzig-Wolfe Reformulation (DWR) method.The idea of Multilevel DWR is to solve a Mixed Integer Program by recursively applying DWR.The method allows fast computation of strong bounds.

A branch-and-price algorithm for the k,c-coloring problem

In this article, we study the k,c-coloring problem, a generalization of the vertex coloring problem where we have to assign k colors to each vertex of an undirected graph, and two adjacent vertices can share at most c colors. We propose a new formulation for the k,c-coloring problem and develop a Branch-and-Price algorithm. We tested the algorithm on instances having from 20 to 80 vertices and different combinations for k and c, and compare it with a recent algorithm proposed in the literature. Computational results show that the overall approach is effective and has very good performance on instances where the previous algorithm fails.

Mathematical formulations for the Balanced Vertex k-Separator Problem

Given an indirected graph G = (V;E), a Vertex k-Separator is a subset of the vertex set V such that, when the separator is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. In this paper we focus on the Balanced Vertex k-Separator Problem, i.e., the problem of finding a minimum cardinality separator such that the sizes of the resulting disconnected subsets are balanced. We present a compact Integer Linear Programming formulation for the problem, and present a polyhedral study of the associated polytope. We also present an Exponential-Size formulation, for which we derive a column generation and a branching scheme. Preliminary computational results are reported comparing the performance of the two formulations on a set of benchmark instances.

Training software for orthogonal packing problems

Abstract An open source architecture for the interactive solution of packing problems in two dimensions is presented. Although primarily developed for helping engineering students to understand the algorithmic approaches to the solution of difficult combinatorial optimization problems, the application can be useful to practitioners and developers thanks to its visual tools. The paper gives intuitive and formal definitions of the problems at hand, discusses two natural heuristic approaches, provides technical information on the application, and reports the results of classroom experimental testings.