The KIPS Transactions:PartB (정보처리학회논문지B)

Korea Information Processing Society (한국정보처리학회)
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A Heuristic Algorithm for the Two-Dimensional Bin Packing Problem Using a Fitness Function

적합성 함수를 이용한 2차원 저장소 적재 문제의 휴리스틱 알고리즘

  • 연용호 (목원대학교 공학교육혁신센터) ;
  • 이선영 (충북대학교 컴퓨터교육과) ;
  • 이종연 (충북대학교 컴퓨터교육과)
  • Published : 2009.10.31
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Abstract

The two-dimensional bin packing problem(2D-BPP) has been known to be NP-hard, and it is difficult to solve the problem exactly. Many approximation methods, such as genetic algorithm, simulated annealing and tabu search etc, have been also proposed to gain better solutions. However, the existing approximation algorithms, such as branch-and-bound and tabu search, have shown the low efficiency and the long execution time due to a large of iterations. To solve these problems, we first define the fitness function to simplify and increase the utility of algorithm. The function decides whether an item is packed into a given area, and as an important information for a packing strategy, the number of subarea that can accommodate a given item is obtained from the variant of the fitness function. Then we present a heuristic algorithm BF for 2D bin packing, constructed by the fitness function and subarea. Finally, the effectiveness of the proposed algorithm will be expressed by the comparison experiments with the heuristic and the metaheuristic of the literatures. As comparing with existing heuristic algorithms and metaheuristic algorithms, it has been found that the packing rate of algorithm BP is the same as 97% as existing heuristic algorithms, FFF and FBS, or better than them. Also, it has been shown the same as 86% as tabu search algorithm or better.

2차원 저장소 적재는 NP-hard 문제로서 그 문제의 정확한 해를 구하는 것이 어려운 것으로 알려져 있으며, 이의 더 좋은 해를 얻기 위해 유전자(genetic) 알고리즘, 시뮬레이티드 어닐링(simulated annealing), 타부서치(tabu search)등과 같은 근사적 접근법이 제안되어 왔다. 하지만 분지한계(branch-and-bound)나 타부서치 기법들을 이용한 기존의 대표적인 근사 알고리즘들은 휴리스틱 알고리즘의 해에 기반을 둠으로 효율성이 낮고 반복수행에 의한 계산시간이 길다. 따라서 본 논문에서는 이러한 근사 알고리즘의 복잡성을 간소화하고, 알고리즘의 효율성을 높이기 위해 적재가능성을 판단하는 적합성 함수(fitness function)를 정의하고 이를 이용하여 어떤 특정 개체의 적재영역을 판단하는데 영향을 주는 적재영역의 수를 계산한다. 또한, 이들을 이용한 새로운 휴리스틱 알고리즘을 제안하였다. 끝으로 기존의 휴리스틱 또는 메타휴리스틱 기법과의 비교실험을 통해 기존의 휴리스틱 알고리즘인 FFF와 FBS에 비해 97%의 결과가 같거나 우수하였으며, 타부서치 알고리즘에 비해 86%의 결과가 같거나 우수한 것으로 나타났다.

Keywords

References

  1. B. E. Bengtsson, Packing rectangular pieces-a heuristic approach, The Computer Journal Vol.25, No.3, pp.353-357, 1982 https://doi.org/10.1093/comjnl/25.3.353
  2. J. E. Beasley, Algorithms for unconstrained two-dimensional guillotine cutting, Journal of Operational Research Society, Vol.36, pp.297-306. https://doi.org/10.1057/jors.1985.51
  3. J. E. Beasley, An exact two-dimensional non-guillotine cutting tree search procedure, Operations Research, Vol.33, pp.49-64, 1985. https://doi.org/10.1287/opre.33.1.49
  4. J. O. Berkey and P. Y. Wang, Two dimensional finite bin packing algorithms Journal of the Operational Research Society 38, pp.423-429, 1987. https://doi.org/10.1057/jors.1987.70
  5. N. Christofides and C. Whitlock, An algorithm for twodimensional cutting problrems, Operations Research, Vol.25, pp.30-44, 1977. https://doi.org/10.1287/opre.25.1.30
  6. F. K. R. Chung, M. R. Garey and D. S. Johnson, On Packing two-dimensional bins, SIAM Journal of Algebraic and Discrete Methods, Vol.3, No.1, pp.66-76, 1982. https://doi.org/10.1137/0603007
  7. M. Dell'Amico and S. Martello, Optimal scheduling of tasks on identical parallel processors, ORSA Journal on Computing, Vol.7, No.2, pp.191-200, 1995. https://doi.org/10.1287/ijoc.7.2.191
  8. M. Dell'Amico, S. Martello and D. Vigo, A lower bound for the non-oriented two-dimensional bin packing problem, Discrete Applied Mathematics, Vol.118, pp.13-24, 2002. https://doi.org/10.1016/S0166-218X(01)00253-0
  9. H. Dyckhoff, A typology of cutting and packing problems, European Journal of Operational Research, Vol.44, No.2, pp. 145-159, 1990. https://doi.org/10.1016/0377-2217(90)90350-K
  10. H. Dyckhoff and G. Finke, Cutting and packing in production and distribution, Physica Verlag, Heidelberg, 1992.
  11. A. El-Bouri, N. Popplewell, S. Balakrishnan and A. Alfa, A search based heuristic for the two-dimensional bin-packing problem, INFOR, Vol.32, No.4, pp.265-274, 1994.
  12. P. C. Gilmore and R. E. Gomory, Multistage cutting problems of two and more dimensions, Operations Research, Vol.13, No.1, pp.94-120, 1965 https://doi.org/10.1287/opre.13.1.94
  13. E. Hadjiconstantinou and N. Christofides, An Exact algorithm for general, orthogonal, two-dimensional knapsack problem, European Journal of Operational Research, Vol.83, pp.39-56, 1995. https://doi.org/10.1016/0377-2217(93)E0278-6
  14. A. Lodi, S. Martello and M. Monaci, Two-dimensional packing problems: a survey, European Journal of Operational Research, Vol.141, pp.241-252, 2002. https://doi.org/10.1016/S0377-2217(02)00123-6
  15. A. Lodi, S. Martello, and D. Vigo, Neighborhood search algorithm for the guillotine non-oriented two-dimensional bin packing problem, in: S. Martello, I. H. Osman, C. Roucairol (Eds.), Meta-Heuristics: Advances and Trends in Local Search Paradigms for Optimization, Kluwer Academic Publishers, Boston, pp.125-139, 1998.
  16. A. Lodi, S. Martello and D. Vigo, Approximation algorithms for the oriented two-dimensional bin packing problem, European Journal of Operational Research, Vol.112, 158-166, 1999. https://doi.org/10.1016/S0377-2217(97)00388-3
  17. A. Lodi, S. Martello and D. Vigo, Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems, INFORMS Journal of Computing, Vol.11, No.4, pp. 345-357, 1999. https://doi.org/10.1287/ijoc.11.4.345
  18. A. Lodi, S. Martello, D. Vigo, Ricent advances on twodimensional bin packing problems, Discrete Applied Mathematics 123, pp.373-390, 2002. https://doi.org/10.1016/S0166-218X(01)00347-X
  19. S. Martello, M. Monaci and D. Vigo, An exact approach to the strip packing problem, Informs Journal on Computing, Vol.15, 310-319, 2003. https://doi.org/10.1287/ijoc.15.3.310.16082
  20. S. Martello and P. Toth, Lower bounds and reduction procedures for the bin packing problem, Discrete Applied Mathematics, Vol.28, pp.59-70, 1990. https://doi.org/10.1016/0166-218X(90)90094-S
  21. S. Martello and D. Vigo, Exact solution of the two dimensional finite bin packing problem, Management Science, Vol.44, pp. 388-399, 1998. https://doi.org/10.1287/mnsc.44.3.388