1,721,072 research outputs found
Theoretical and computational study of several linearisation techniques for binary quadratic problems
Full text linkWe perform a theoretical and computational study of the classical linearisation techniques (LT) and we propose a new LT for binary quadratic problems (BQPs). We discuss the relations between the linear programming (LP) relaxations of the considered LT for generic BQPs. We prove that for a specific class of BQP all the LTs have the same LP relaxation value. We also compare the LT computational performance for four different BQPs from the literature. We consider the Unconstrained BQP and the maximum cut of edge-weighted graphs and, in order to measure the effects of constraints on the computational performance, we also consider the quadratic extension of two classical combinatorial optimization problems, i.e., the knapsack and stable set problems
Models for the two-dimensional two-stage cutting stock problem with multiple stock size
No full textWe 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. © 2013 Elsevier Ltd
A pseudo-polynomial size formulation for 2-stage 2-dimensional knapsack problems
No full textTwo dimensional cutting problems are about obtaining a set of rectangular items from a set of rectangular stock pieces and are of great relevance in industry, whenever a sheet of wood, metal or other material has to be cut. In this paper, we consider the 2-stage two-dimensional knapsack (2TDK) problem which requires finding the maximum profit subset of rectangular items obtainable through 2-stage guillotine cuts in a rectangular panel. We propose a formulation having a pseudopolynomial number of variables and constraints which can still be enumerated for the instances present in the literature. We compare the proposed formulation with the previous best known polynomial size one. Extensive computational experiments show that the new model is characterized by a stronger linear programming relaxation and can be effectively solved with a general-purpose MIP solver
Exact weighted vertex coloring via branch-and-price
No full textWe consider the Weighted Vertex Coloring Problem (WVCP), in which a positive weight is associated to each vertex of a graph. In WVCP, one is required to assign a color to each vertex in such a way that colors on adjacent vertices are different, and the objective is to minimize the sum of the costs of the colors used, where the cost of each color is given by the maximum weight of the vertices assigned to that color. This NP-hard problem arises in practical scheduling applications, where it is also known as Scheduling on a Batch Machine with Job Compatibilities. We propose the first exact algorithm for the problem, which is based on column generation and branch-and-price. Computational results on a large set of instances from the literature are reported, showing excellent performance when compared with the best heuristic algorithms from the literature. © 2012 Elsevier B.V. All rights reserved
A fast heuristic approach for train timetabling in a railway node
No full textWe 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. © 2013 Elsevier B.V
Approximated perspective relaxations: a project and lift approach
No full textThe perspective reformulation (PR) of a Mixed-Integer NonLinear Program with semi-continuous variables is obtained by replacing each term in the (separable) objective function with its convex envelope. Solving the corresponding continuous relaxation requires appropriate techniques. Under some rather restrictive assumptions, the Projected PR (Formula presented.) can be defined where the integer variables are eliminated by projecting the solution set onto the space of the continuous variables only. This approach produces a simple piecewise-convex problem with the same structure as the original one; however, this prevents the use of general-purpose solvers, in that some variables are then only implicitly represented in the formulation. We show how to construct an Approximated Projected PR (Formula presented.) whereby the projected formulation is “lifted” back to the original variable space, with each integer variable expressing one piece of the obtained piecewise-convex function. In some cases, this produces a reformulation of the original problem with exactly the same size and structure as the standard continuous relaxation, but providing substantially improved bounds. In the process we also substantially extend the approach beyond the original (Formula presented.) development by relaxing the requirement that the objective function be quadratic and the left endpoint of the domain of the variables be non-negative. While the (Formula presented.) bound can be weaker than that of the PR, this approach can be applied in many more cases and allows direct use of off-the-shelf MINLP software; this is shown to be competitive with previously proposed approaches in some applications
The Time Dependent Traveling Salesman Planning Problem in Controlled Airspace
No full textThe integration of drones into civil airspace is one of the most challenging problems for the automation of the controlled airspace, and the optimization of the drone route is a key step for this process. In this paper, we optimize the route planning of a drone mission that consists of departing from an airport, flying over a set of mission way points and coming back to the initial airport. We assume that during the mission a set of piloted aircraft flies in the same airspace and thus the cost of the drone route depends on the air traffic and on the avoidance maneuvers used to prevent possible conflicts. Two air traffic management techniques, i.e., routing and holding, are modeled in order to maintain a minimum separation between the drone and the piloted aircraft. The considered problem, called the Time Dependent Traveling Salesman Planning Problem in Controlled Airspace (TDTSPPCA), relates to the drone route planning phase and aims to minimize the total operational cost. Two heuristic algorithms are proposed for the solution of the problem. A mathematical formulation based on a particular version of the Time Dependent Traveling Salesman Problem, which allows holdings at mission way points, and a Branch and Cut algorithm are proposed for solving the TDTSPPCA to optimality. An additional formulation, based on a Travelling Salesman Problem variant that uses specific penalties to model the holding times, is proposed and a Cutting Plane algorithm is designed. Finally, computational experiments on real-world air traffic data from Milano Linate Terminal Maneuvering Area are reported to evaluate the performance of the proposed formulations and of the heuristic algorithms
Lower Bounding Techniques for DSATUR-based Branch and Bound
No full textGiven an undirected graph, the Vertex Coloring Problem (VCP) consists of assigning a color to each vertex of the graph such that two adjacent vertices do not share the same color and the total number of colors is minimized. DSATUR-based Branch-and-Bound is a well-known exact algorithm for the VCP. One of its main drawbacks is that a lower bound (equal to the size of a maximal clique) is computed once at the root of the branching scheme and it is never updated during the execution of the algorithm. In this article, we show how to update the lower bound and we compare the efficiency of several lower bounding techniques
A lexicographic pricer for the fractional bin packing problem
Full text linkWe propose an exact lexicographic dynamic programming pricing algorithm for solving the Fractional Bin Packing Problem with column generation. The new algorithm is designed for generating maximal columns of minimum reduced cost which maximize, lexicographically, one of the measures of maximality we investigate. Extensive computational experiments reveal that a column generation algorithm based on this pricing technique can achieve a substantial reduction in the number of columns and the computing time, also when combined with a classical smoothing technique from the literature
A branch-and-price algorithm for the temporal bin packing problem
Full text linkWe study an extension of the classical Bin Packing Problem, where each item consumes the bin capacity during a given time window that depends on the item itself. The problem asks for finding the minimum number of bins to pack all the items while respecting the bin capacity at any time instant. A polynomial-size formulation, an exponential-size formulation, and a number of lower and upper bounds are studied. A branch-and-price algorithm for solving the exponential-size formulation is introduced. An overall algorithm combining the different methods is then proposed and tested through extensive computational experiments
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