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

Smart energy pricing for demand-side management in renewable energy smart grids

Smart grids are expected to provide various benefits to society by integrating advances in power engineering with recent developments in the field of information and communications technology. One of the advantages is the support to efficient demand-side management (DSM), for example, changes in consumer demands for energy based on using incentives. Indeed, DSM is expected to help grid operators balance time-varying generation by wind and solar units, and the optimization of their usage. This paper focuses on DSM considering renewable energy generation and proposes an auction, in which consumers submit bids to renewable energy usage plans. An additional model is introduced to allow consumers to compute their bid for a given usage plan. Both models have been extended to include energy storage devices. The proposed model is compared to a system with time-varying pricing for energy, where it is shown to allow consumers to use more appliances, to lead to a larger profit, and to reduce the peak-to-average ratio of energy consumption. Finally, the impact of the use of energy storage in households and in the energy provider is also consideredCONSELHO NACIONAL DE DESENVOLVIMENTO CIENTÍFICO E TECNOLÓGICO - CNPQCOORDENAÇÃO DE APERFEIÇOAMENTO DE PESSOAL DE NÍVEL SUPERIOR - CAPESFUNDAÇÃO DE AMPARO À PESQUISA DO ESTADO DE SÃO PAULO - FAPESP311499/2014-7; 425340/2016-3; 477692/20125; 308689/2017-8não tem2013/21744-8; 2015/11937-9; 2016/01860-1; 2016/23552-