A hybrid algorithm for two-dimensional guillotine cutting problems. Brazil. Brazil.

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1 A hybrid algorithm for two-dimensional guillotine cutting problems David Álvarez Martínez 1;a, Luis Miguel Escobar Falcón 2;b, Eliana Mirledy Toro 3;c and Rubén Augusto Romero Lázaro 4;d 1 São Paulo State University, Deparment of Engineering of Ilha Solteira, São Paulo, Brazil 2 Technological University of Pereira, Risaralda, Colombia, 3 Technological University of Pereira, Risaralda, Colombia, 4 São Paulo State University, Deparment of Engineering of Ilha Solteira, São Paulo, Brazil a david.unesp@gmail.com, b luismescobarf@gmail.com, c elianam@utp.edu.co, d rube n@dee.feis.unesp.br ABSTRACT Cutting and packing problems are considered classic problems in operations research that are of high interest due to their large spectrum of application in the industry, and their solution requires a treatment of high mathematical complexity and complex computational structures. This study presents the constrained and unconstrained k-staged two-dimensional guillotineable single knapsack problem of rectangular items, with and without associated weights to the items, with and without ninety degrees rotations to the items, and with guillotine cuts. An appropriate encoding of the problem is presented to be employed using a metaheuristic hybrid algorithm of variable neighborhood search and simulated annealing. To check the eficiency of the presented methodology, test cases were taken from specialized literature in order to analyze and compare the presented solution method with the state-of-the-art handling of the problems. Results of good quality, some of which have not previously been reported before in the literature were obtained in this work. KEY WORDS: Constrained and unconstrained k-staged two-dimensional knapsack problem, variable neighborhood search, simulated annealing. The tables 1-19 present the obtained results with the proposed algorithm in this study and the times used to achieve these answers. From the specialized literature (references from [1] to [11]) was taken the best quality solution for each case, besides of the used time to reach each solution. In the cases were the reported solution presents the same quality, are only taken in count the ones with the smallest computational time. The solution units presented on the tables 1-19 for the unweighted instances are the percentage given by the relation: solution

2 obtained divided over the total area of the knapsack, and for the weighted instances the units represent the associated profit with each piece. Unconstrained Two-staged Fixed Problem Unweighted Instances Weighted Instances Instance Best Proposed Instance Best Proposed ID Used ID Used UU UW UU UW11 15, UU UW UU UW UU UW UU UW UU UW UU UW7 10, UU UW UU UW UU UW LUU LUW1 158,363, LUU LUW2 257,372, LUU LUW3 261,252, LUU LUW4 473,754, Table 1: Results of the Unconstrained two-staged fixed weighted and unweighted problems.

3 Unweigthed Instances Weighted Instances Instance ID Unconstrained Two-staged Rotated Problem Best Best Known Proposed Used UU UU UU UU UU UU UU UU UU UU UU LUU LUU LUU LUU , UW UW11 17, , UW UW UW UW UW UW7 11, , UW UW UW LUW1 171,079, ,079, LUW2 325,725, ,725, LUW3 433,859, ,859, LUW4 568,436,545 13, ,436, Table 2: Results of the Unconstrained two-staged rotated weighted and unweighted problems. All Best Known s reported by Hifi, (2001) [6].

4 Unweigthed Instances Weighted Instances Instance ID Unconstrained Three-staged Fixed Problem Best Best Known Proposed Used UU1 96, UU2 99, UU3 97, UU4 98, UU5 99,15 10, UU6 98,79 10, UU7 98,84 11, UU8 98,98 10, UU9 99,20 10, UU10 99,01 11, UU11 99,76 15, LUU1 N/A N/A LUU2 N/A N/A LUU3 N/A N/A LUU4 N/A N/A UW UW11 15, , UW UW UW UW UW , UW7 10,464 11,101 10, UW , UW , UW , LUW1 N/A N/A 158,363, LUW2 N/A N/A 257,372, LUW3 N/A N/A 261,252, LUW4 N/A N/A 473,754, Table 3: Results of the Unconstrained three-staged fixed weighted and unweighted problems All Best Known s reported by Hifi, (2001) [6].

5 Unweighted Instances Weighted Instances Instance ID Unconstrained Three-staged Rotated Problem Best Best Known Proposed Used UU1 98, UU2 99, UU3 99, UU4 99, UU5 99,64 11, UU6 98,81 11, UU7 99,46 11, UU8 99,37 11, UU9 99,36 11, UU10 99,22 13, UU11 99,88 19, LU1 N/A N/A LU2 N/A N/A LU3 N/A N/A LU4 N/A N/A UW UW11 18, , UW UW UW UW UW , UW ,900 11, UW , UW , UW , LUW1 N/A N/A 171,836, LUW2 N/A N/A 326,399, LUW3 N/A N/A 433,882, LUW4 N/A N/A 568,690, Table 4: Results of the Unconstrained three-staged rotated weighted and unweighted problems. All Best Known s reported by Hifi, (2001) [6].

6 Unconstrained Non-staged Weighted Fixed Problem Instance Best Known Best Proposed ID Used W1 162, a 162, W2 35, ª 35, W3 234, a 234, UW b UW a UW b UW b UW b UW b UW7 10, b 10, UW b UW b UW b UW11 15, a 15, Table 5: Results of the Unconstrained non-staged weighted fixed problem. a DPH algorithm, Song et al., (2010) [10]; b Young-Gun et al., (2003) [11]. Instance ID Unconstrained Non-staged Weigthed Rotated Problem Best Proposed Used UW UW11 18, UW UW UW UW UW UW7 11, UW UW UW LUW1 171,836, LUW2 326,399, LUW3 433,882, LUW4 568,690, Table 6: Results of the Unconstrained non-staged weighted rotated problem.

7 Unconstrained Non-staged Unweigthed Fixed Problem Instance Best Known Best Proposed ID Used GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT a GCUT b Table 7: Results of the Unconstrained non-staged unweigthed fixed problem. a SDP Algorithm, Cintra et al., (2008) [2] b DPH algorithm, Song et al., (2010) [10] Unconstrained Non-staged Unweigthed Rotated Problem Instance Best Known Best Proposed ID Used GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT GCUT Table 8: Results of the Unconstrained non-staged unweigthed rotated problem. All Best Known s reported by SDP Algorithm, Cintra et al. (2008) [2].

8 Constrained Two-Staged Weighted Fixed Problem Instance ID Best Known Best Proposed Used HH 10, ª 10, ª a A <0.01 c A ª STS c STS ª CHL ª CHL ª CW c CW ª Hch ª Hch b Table 9: Results of the Constrained two-staged weighted fixed problem. a ALGO1_ESGA algorithm, Hifi and M'Hallah, (2005) [7]; b ALGO1_SGA algorithm, Hifi and M'Hallah, (2005) [7]; c Hifi and Roucairol, (2001) [8]. Instance ID Constrained Two-Staged Weighted Rotated Problem Best Proposed Used HH 10, A A STS STS CHL CHL CW CW Hch Hch Table 10: Results of the Constrained two-staged weighted rotated problem.

9 Constrained Two-Staged Unweigthed Fixed Problem Instance ID Best Known Best Proposed Used 2SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < Table 11.a: Results of the Constrained two-staged unweighted fixed problem.

10 Constrained Two-Staged Unweigthed Fixed Problem ID Best Known Best Proposed Used 2SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < SCUI < Table 11.b: Results of the Constrained two-staged unweighted fixed problem. All Best Known s reported by Cui, (2007) [3].

11 Constrained Two-Staged Unweigthed Rotated Problem Instance ID Best Known Best Proposed Used 2SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI Table 12.a: Results of the Constrained two-staged unweighted rotated problem.

12 Constrained Two-Staged Unweigthed Rotated Problem Instance ID Best Known Best Proposed Used 2SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI SCUI Table 12.b. Results of the Constrained two-staged unweighted rotated problem. All Best Known s reported by Cui, (2007) [3].

13 Constrained Three-Staged Weighted Fixed Problem Instance Best Known Best Proposed ID Used 3SCUI1 631,863 6ª 631, SCUI2 4,544,395 68ª 4,544, SCUI3 1,833, c 1,833, SCUI4 4,339,083 9 c 4,339, SCUI5 2,676,651 4 b 2,676, SCUI6 4,245,663 4 b 4,245, SCUI7 4,305, c 4,305, SCUI8 3,610, ª 3,608, SCUI9 7,446,330 9 b 7,446, SCUI10 6,038, a 6,038, SCUI11 1,168,284 4 a 1,162, SCUI12 2,567,171 4 c 2,567, SCUI13 2,795,973 4 b 2,795, SCUI14 3,152,403 3 c 3,152, SCUI15 5,018, b 5,018, SCUI16 2,157,408 9 b 2,157, SCUI17 6,881, a 6,854, SCUI18 7,548, a 7,548, SCUI19 5,704, b 5,704, SCUI20 5,726, a 5,676, Table 13: Results of the Constrained three-staged weighted fixed problem. a VEA, Cui, (2008) [4]; b VHA1, Cui, (2008) [4]; c VHA2, Cui, (2008) [4].

14 Constrained Three-Staged Unweigthed Fixed Problem Instance ID Best Known Best Proposed Used 3SCUI b SCUI b SCUI c SCUI b SCUI c SCUI b SCUI b SCUI b SCUI b SCUI b SCUI ª SCUI ª SCUI b SCUI c SCUI b SCUI b SCUI b SCUI b SCUI b SCUI b Table 14: Results of the Constrained three-staged unweighted fixed problem. a VEA algorithm, Cui, (2008) [4]; b VHA1algortihm, Cui, (2008) [4]; c VHA2 algorithm, Cui, (2008) [4].

15 Constrained Three-Staged Rotated Problem Instance ID Unweighted Weighted Best Proposed Best Proposed Used Used 3SCUI , SCUI ,545, SCUI ,833, SCUI ,339, SCUI ,678, SCUI ,249, SCUI ,306, SCUI ,610, SCUI ,447, SCUI ,040, SCUI ,169, SCUI ,567, SCUI ,796, SCUI ,153, SCUI ,021, SCUI ,158, SCUI ,885, SCUI ,552, SCUI ,709, SCUI ,731, Table 15: Results of the Constrained weighted and unweigthed three-staged rotated problem. Constrained Non-staged Weighted Fixed Problem Instance ID Best Known Best Proposed Used CW a CW a CW b CW b CW5 11, b 11, CW6 12, b 12, CW c CW b CW9 10, b 10, CW b CW b TH N/A d TH N/A d CHW N/A d CHW N/A d Table 16: Results of the Constrained non-staged weighted fixed problem. a FHZ98 algorithm, Fayard et al., (1998) [5]; b MP_AOG algorithm, Morabito and Pureza, (2007) [9]; c MP algorithm, Morabito and Pureza, (2007) [9]; d ATP2 algorithm, Alvarez-Valdes et al., (2002) [1].

16 Instance ID Constrained Non-staged Weighted Rotated Problem Best Proposed Used CW CW CW CW CW CW CW CW CW CW CW TH TH CHW CHW Table 17. Results of the Constrained non-staged weighted rotated problem. Instance ID Constrained Non-staged Unweighted Fixed Problem Best Proposed Best Known Used CU1 98,64 <1 a 98, CU2 99,43 <1 a 99, CU3 99,84 6 b 99, CU4 98,62 35 b 98, CU5 98, a 98, CU6 99,65 10 a 99, CU7 95, b 95, CU8 97,65 21 a 97, CU9 97,78 20 b 97, CU10 98, b 98, CU11 99, b 99, Table 18. Results of the Constrained non-staged weighted fixed problem. a MP_AOG algorithm, Morabito and Pureza, (2007) [9]; b MP algorithm, Morabito and Pureza, (2007) [9].

17 Instance ID Constrained Non-staged Unweighted Rotated Problem Best Proposed Used CU1 98, CU2 99, CU3 99, CU4 99, CU5 98, CU6 99, CU7 97,5 570 CU8 99,2 611 CU9 98, CU10 98, CU11 99, Table 19. Results of the Constrained non-staged weighted rotated problem. REFERENCES [1] Alvarez-Valdés, R.; Parajón, A.; Tamarit, J. M.; A tabu search algorithm for large-scale guillotine (un)constrained two-dimensional cutting problems. Computers & Operations Research. Vol. 29, pp , (2002) [2] Cintra, G.F.; Miyazawa, F.K.; Wakabayashi, Y.; Xavier E.C.; Algorithms for twodimensional cutting stock and strip packing problems using dynamic programming and column generation, European Journal of Operational Research, Vol. 191, pp , (2008) [3] Cui, Y.; An exact algorithm for generating homogenous two-segment cutting patterns, Engineering Optimization, Vol. 39, pp , (2007) [4] Cui, Y.; Heuristic and exact algorithms for generating homogenous constrained three-staged cutting patterns, Computers & Operations Research, Vol. 35, pp , (2008) [5] Fayard, D.; Hifi, M.; Zissimopoulos, V.; An efficient approach for large-scale two-dimensional guillotine cutting stock problems, Journal of the Operational Research Society, Vol. 49, pp , (1998) [6] Hifi, M; Exact Algorithms for Large-Scale Unconstrained Two and Three Staged Cutting Problems, Computational Optimization and Applications, Vol. 18, pp , (2001) [7] Hifi, M; M'Hallah, R.; An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems, Operations Research, Vol. 53, pp , (2005) [8] Hifi, M.; Roucairol, C.; Approximate and Exact Algorithms for Constrained (Un) Weighted Two-dimensional Two-staged Cutting Stock Problems, Journal of Combinatorial Optimization, Vol. 5, pp , (2001) [9] Morabito, R.; Pureza, V.; Geração de padrões de cortes bidimensionais guilhotinados restritos via programação dinâmica e busca em grafo-e/ou, Produção, Vol. 17, pp , (2007)

18 [10] Song, X.; Chu, C.B.; Lewis, R.; Nie, Y.Y.; Thompson J.; A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting, European Journal of Operational Research, 202, , (2010) [11] Young-Gun, G.; Kang, M.-K.; Seong, J.; A best-first branch and bound algorithm for unconstrained two-dimensional cutting problems. Operations Research Letters, 31, , (2003)