A Multiobjective bidimensional bin packing problem N. Dahmani1 , F. Clautiaux2 , S. Krichen1 , and E. -G. Talbi2 1
2
Laboratoire de Recherche Op´erationnelle, de D´ecision et de Contrˆ ole de Processus, Institut Sup´erieur de Gestion de Tunis
[email protected] [email protected] Laboratoire d’Informatique Fondamentale de Lille, Bat M3 Cit´e Scientifique , 59655 Villeneuve d’Ascq fran¸
[email protected] [email protected]
The problem addressed in this research is a new variant of multiobjective bidimensional bin packing problem (Mo2-DBPP). Given N items i = {1, . . . , N }, each one is characterized by two sizes in two independent dimensions: a weight ci and a volume di and an infinite number of identical bins with a weight and a volume capacity C and D respectively. The weight capacity constraint corresponds to the “hard” constraint that cannot be violated while the volume capacity constraint is related to a “soft” constraint. The main objectives are to simultaneously minimize the number of bins and the maximum volume loaded into the bins. The goal is to find a compromise between the number of bins and the maximum volume of the bins. The problem is closely related the m-Dimensional Vector Packing Problem (m-DVPP) [3]. It can model several real world applications in the fields of loading, scheduling, and layout design. In practice, the first dimension refers to the weight and the second one refers to the volume (or the length, the load, the time for loading, . . . ). To our knowledge, only few works on multiobjective bin packing problem have been proposed [7, 4, 8, 5]. We first state a biobjective linear programming formulation for the problem. As Mo2-DBPP is an NP-hard problem, we use approximation methods to generate the set of potentially efficient solutions. As a first attempt to validate the biobjective model, we present two different methodologies for designing and implementing grounded multiobjective metaheuristics namely population based and single-solution based metaheuristics. These algorithms are implemented using ParadisEO [6]. Regarding the population based metaheuristics, we design three multiobjective evolutionary algorithms. For the single-solution based metaheuristics, we devise an iterated tabu search algorithm (ITS) and an evolutionary algorithm (EA) based on ǫ-Constraint methods. For ITS and EA, the generation process of the potentially efficient solutions starts by initializing the first objective with a lower bound on the number of bins. Then, iteratively incremented until it reaches the number of items. Each time the first objective is fixed, the algorithm is executed and the second objective is optimized. For both methodologies, we implement the solution encoding, the solution evaluation and the genetic operators (for the genetic algorithms). 1. Solution encoding A permutation σ = (σ(1), . . . , σ(n)) of n items is applied. Moreover, we define a binary variable x ∈ {0, 1} that informs about the procedure used for decoding a solution in addition to an integer variable lb that represents a lower bound on the number of bins. The used lower
Fig. 1. Chromosome encoding
bound is the simplest one, introduced by Eilon and Christofides (1971) [2] P ci LB1 = ⌈ i∈N ⌉ c
(1)
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N. Dahmani, F. Clautiaux, S. Krichen, and E. -G. Talbi
2. Solution evaluation A corresponding Mo2-DBPP solution is evaluated using two procedures which are a variant of the so-called first-fit (FF) algorithm and the least loaded (LL) algorithm depending on the binary variable x. First, we assume that we have a predefined ordered list of q bins w = (w(1), . . . , w(q)) where q := lb. • If x = 0, FF is applied. This algorithm consists on packing iteratively each item σ(i) into the partially filled bin with the smallest index and which has sufficient residual capacity. Starting from the first item σ(1) and the first bin w(1). If no such a bin is available, then item σ(i) is packed into a new bin. The process is stopped when there is no more items to pack. • If x = 1, LL is applied. This heuristic attempts to balance the load between the bins by assigning the incoming items to the least loaded bin (i.e. the largest residual capacity). Starting from the item σ(1), LL tries to iteratively pack item σ(i) into the current m bins by choosing each time the least loaded one. If it failed to pack an existing item into the m bins, it starts repacking after a new bin is opened. The process is stopped when there no more items to pack. 3. Genetic operators • The crossover operator We investigated the performance of the Two points crossover.A pair of crossing points is randomly selected. The generated offspring preserves the items outside the selected two points from the first parent chromosome. The remaining items are inserted from the second parent respecting the order of their appearance. For the part that determines the decoding function to use (i.e. x), if it is the same for the two parents then this information is kept for the generated offspring. If not, the offspring has 50% of chance to inherit the corresponding information from either from the first parent or the second one. The same strategy is used for defining the portion dedicated to the lower bound lb. • Mutation operators We adapted the shift mutation operator that shifts a pair of components from the chromosome and the swap mutation operator that swap a pair of randomly choosing components from the chromosome. To illustrate the effectiveness of our algorithmic approaches, an experimental investigation is performed on various benchmarks [1]. An extensive comparative study is developed to observe the behavior of the proposed approaches and derive the best algorithms regarding the problem settings.
References 1. A. Caprara and P. Toth. Lower bounds and algorithms for the 2-dimensional vector packing problem. Discrete Applied Mathematics, 111(3):231–262, 2001. 2. S. Eilon and N. Christofides. The loading problem. Management Science, 17:259–267, 1971. 3. M. R. Garey, R. L. Graham, D. S. Johnson, and Andrew. Resource constrained scheduling as generalized bin packing. Journal of Combinatorial Theory, 21:257–298, 1976. 4. M. J. Geiger. Bin packing under multiple objectives — a heuristic approximation approach. In The Fourth International Conference on Evolutionary Multi-Criterion Optimization: Late Breaking Papers, pages 53–56, Matsushima, Japan, M¨ arz 2007. 5. A. Khanafer, F. Clautiaux, and E.-G. Talbi. New lower bounds for bin packing problems with conflicts. European Journal of Operational Research, 206(2):281–288, 2010. 6. A. Liefooghe, M. Basseur, L. Jourdan, and E.-G. Talbi. ParadisEO-MOEO: A framework for evolutionary multi-objective optimization. In S. Obayashi, C. Poloni, and K. Deb, editors, Evolutionary Multi-Criterion Optimization, Fourth International Conference (EMO 2007), volume 4403, pages 386– 400, Matsushima, Japan, 2007. Springer-Verlag. 7. D. Liu, K. Tan, S. Huang, C. Goh, and W. Ho. On solving multiobjective bin packing problems using evolutionary particle swarm optimization. European Journal of Operational Research, 190(2):357–382, 2008. 8. M. Sathe, O. Schenk, and H. Burkhart. Solving bi-objective many-constraint bin packing problems in automobile sheet metal forming processes. In EMO ’09: Proceedings of the 5th International Conference on Evolutionary Multi-Criterion Optimization, pages 246–260, Berlin, Heidelberg, 2009. Springer-Verlag.
