Practical comparison of approximation algorithms for scheduling problems

In this paper we consider an experimental study of approximation algorithms for scheduling problems in parallel machines minimizing the average weighted completion time. We implemented approximation algorithms for the following problems: P|r j|sigmaCj, Psigmaw jCj, P|r j|sigmaw jCj, Rsigmaw jCj and R|r j|sigmaw jCj. We generated more than 1000 tests over more than 200 different instances and present some practical aspects of the implemented algorithms. We also made an experimental comparison on two lower bounds based on the formulations used by the algorithms. The first one is a semidefinite formulation for the problem Rsigmaw jCj and the other one is a linear formulation for the problem R|r j|sigmaw jCj. For all tests, the algorithms obtained very good results. We notice that algorithms using more refined techniques, when compared to algorithms with simple strategies, do not necessary lead to better results. We also present two heuristics, based on approximation algorithms, that generate solutions with better quality in almost all instances considered.Neste artigo consideramos um estudo experimental de alguns algoritmos aproximados para problemas de escalonamento em máquinas paralelas onde se deve minimizar o tempo de término ponderado das tarefas. Foram implementados algoritmos aproximados para os seguintes problemas: P|r j|sigmaCj, Psigmaw jCj, P|r j|sigmaw jCj, Rsigmaw jCj and R|r j|sigmaw jC j . Foram gerados mais de 1000 testes sobre mais de 200 instâncias diferentes e com isso apresentamos aspectos práticos dos algoritmos implementados. Também fizemos um estudo experimental sobre dois limitantes inferiores baseados em formulações usadas pelos algoritmos. A primeira é uma formulação semidefinida para o problema Rsigmaw jCj e a outra é uma formulação linear para o problema R|r j|sigmaw jCj. Em todos os testes os algoritmos obtiveram resultados muito bons. Notamos que algoritmos usando técnicas mais refinadas, quando comparados com algoritmos que usam estratégias simples, não necessariamente geram soluções melhores. Também apresentamos duas heurísticas, baseadas nos algoritmos aproximados, que geram soluções de melhor qualidade em quase todas as instâncias consideradas.227252Fundação de Amparo à Pesquisa do Estado de São Paulo (FAPESP)Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES

Mathematical Models and Exact Algorithms for the Colored Bin Packing Problem

This paper focuses on exact approaches for the Colored Bin Packing Problem (CBPP), a generalization of the classical one-dimensional Bin Packing Problem in which each item has, in addition to its length, a color, and no two items of the same color can appear consecutively in the same bin. To simplify modeling, we present a characterization of any feasible packing of this problem in a way that does not depend on its ordering. Furthermore, we present four exact algorithms for the CBPP. First, we propose a generalization of Val\'erio de Carvalho's arc flow formulation for the CBPP using a graph with multiple layers, each representing a color. Second, we present an improved arc flow formulation that uses a more compact graph and has the same linear relaxation bound as the first formulation. And finally, we design two exponential set-partition models based on reductions to a generalized vehicle routing problem, which are solved by a branch-cut-and-price algorithm through VRPSolver. To compare the proposed algorithms, a varied benchmark set with 574 instances of the CBPP is presented. Results show that the best model, our improved arc flow formulation, was able to solve over 62% of the proposed instances to optimality, the largest of which with 500 items and 37 colors. While being able to solve fewer instances in total, the set-partition models exceeded their arc flow counterparts in instances with a very small number of colors

Algorithms for the Bin Packing Problem with Scenarios

This paper presents theoretical and practical results for the bin packing problem with scenarios, a generalization of the classical bin packing problem which considers the presence of uncertain scenarios, of which only one is realized. For this problem, we propose an absolute approximation algorithm whose ratio is bounded by the square root of the number of scenarios times the approximation ratio for an algorithm for the vector bin packing problem. We also show how an asymptotic polynomial-time approximation scheme is derived when the number of scenarios is constant. As a practical study of the problem, we present a branch-and-price algorithm to solve an exponential model and a variable neighborhood search heuristic. To speed up the convergence of the exact algorithm, we also consider lower bounds based on dual feasible functions. Results of these algorithms show the competence of the branch-and-price in obtaining optimal solutions for about 59% of the instances considered, while the combined heuristic and branch-and-price optimally solved 62% of the instances considered