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Exact solution techniques for two-dimensional cutting and packing. (English) Zbl 1487.90556

Summary: We survey the main formulations and solution methods for two-dimensional orthogonal cutting and packing problems, where both items and bins are rectangles. We focus on exact methods and relaxations for the four main problems from the literature: finding a packing with minimum height, packing the items into the minimum number of bins, finding a packing of maximum value, and determining the existence of a feasible packing.

MSC:

90C27 Combinatorial optimization
90-02 Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming

References:

[1] Alvarez-Valdés, R.; Parajón, A.; Tamarit, J., A tabu search algorithm for large-scale guillotine (un)constrained two-dimensional cutting problems, Computers & Operations Research, 29, 7, 925-947 (2002) · Zbl 0995.90075
[2] Alvarez-Valdes, R.; Parreño, F.; Tamarit, J., A branch-and-cut algorithm for the pallet loading problem, Computers & Operations Research, 32, 11, 3007-3029 (2005) · Zbl 1071.90046
[3] Alvarez-Valdes, R.; Parreño, F.; Tamarit, J., A tabu search algorithm for a two-dimensional non-guillotine cutting problem, European Journal of Operational Research, 183, 3, 1167-1182 (2007) · Zbl 1138.90430
[4] Alvarez-Valdes, R.; Parreño, F.; Tamarit, J., Reactive GRASP for the strip-packing problem, Computers & Operations Research, 35, 4, 1065-1083 (2008) · Zbl 1179.90269
[5] Alvarez-Valdes, R.; Parreño, F.; Tamarit, J., A branch and bound algorithm for the strip packing problem, OR Spectrum, 31, 2, 431-459 (2009) · Zbl 1160.90696
[6] Alves, C.; Clautiaux, F.; Valério de Carvalho, J.; Rietz, J., Dual-feasible functions for integer programming and combinatorial optimization (2016), Springer International Publishing: Springer International Publishing Cham · Zbl 1354.90101
[7] Amor, H. B.; Desrosiers, J.; Valério de Carvalho, J., Dual-optimal inequalities for stabilized column generation, Operations Research, 54, 3, 454-463 (2006) · Zbl 1167.90529
[8] Arahori, Y.; Imamichi, T.; Nagamochi, H., An exact strip packing algorithm based on canonical forms, Computers & Operations Research, 39, 12, 2991-3011 (2012) · Zbl 1349.90704
[9] Baker, B.; Coffman Jr., E.; Rivest, R., Orthogonal packing in two dimensions, SIAM Journal on Computing, 9, 4, 846-855 (1980) · Zbl 0447.68080
[10] Baldacci, R.; Boschetti, M., A cutting-plane approach for the two-dimensional orthogonal non-guillotine cutting problem, European Journal of Operational Research, 183, 3, 1136-1149 (2007) · Zbl 1138.90433
[11] Balogh, J.; Békési, J.; Dósa, G.; Epstein, L.; Levin, A., Lower bounds for several online variants of bin packing, Proceedings of the revised selected papers 15th international workshop approximation and online algorithms, WAOA 2017, 102-117 (2018), Springer Verlag · Zbl 1436.68401
[12] Beasley, J., Algorithms for unconstrained two-dimensional guillotine cutting, Journal of the Operational Research Society, 36, 4, 297-306 (1985) · Zbl 0589.90040
[13] Beasley, J., An exact two-dimensional non-guillotine cutting tree search procedure, Operations Research, 33, 1 (1985) · Zbl 0569.90038
[14] Beasley, J., Or-library: distributing test problems by electronic mail, Journal of the Operational Research Society, 41, 11, 1069-1072 (1990)
[15] Bekrar, A.; Kacem, I.; Chu, C.; Sadfi, C., An improved heuristic and an exact algorithm for the 2D strip and bin packing problem, International Journal of Product Development, 10, 1-3, 217-240 (2010)
[16] Belov, G.; Kartak, V. M.; Rohling, H.; Scheithauer, G., One-dimensional relaxations and LP bounds for orthogonal packing, International Transactions in Operational Research, 16, 6, 745-766 (2009) · Zbl 1179.90273
[17] Belov, G.; Kartak, V. M.; Rohling, H.; Scheithauer, G., Conservative scales in packing problems, OR Spectrum, 35, 2, 505-542 (2013) · Zbl 1263.90069
[18] Belov, G.; Rohling, H., LP bounds in an interval-graph algorithm for orthogonal-packing feasibility, Operations Research, 61, 2, 483-497 (2013) · Zbl 1267.90118
[19] Belov, G.; Scheithauer, G., A branch-and-cut-and-price algorithm for one-dimensional stock cutting and two-dimensional two-stage cutting, European Journal of Operational Research, 171, 1, 85-106 (2006) · Zbl 1091.90080
[20] Benders, J., Partitioning procedures for solving mixed variables programming problems, Numerische Mathematik, 4, 1, 238-252 (1962) · Zbl 0109.38302
[21] Bennell, J.; Oliveira, J., The geometry of nesting problems: A tutorial, European Journal of Operational Research, 184, 2, 397-415 (2008) · Zbl 1136.90030
[22] Bennell, J.; Oliveira, J., A tutorial in irregular shape packing problems, Journal of the Operational Research Society, 60, supp 1, S93-S105 (2009) · Zbl 1168.90300
[23] Bennell, J. A.; Oliveira, J.; Wäscher, G., Cutting and packing, International Journal of Production Economics, 145, 2, 449-450 (2013)
[24] Berkey, J. O.; Wang, P. Y., Two dimensional finite bin packing algorithms, Journal of the Operational Research Society, 38, 423-429 (1987) · Zbl 0611.90079
[25] Bettinelli, A.; Ceselli, A.; Righini, G., A branch-and-price algorithm for the two-dimensional level strip packing problem, 4OR, 6, 4, 361-374 (2008) · Zbl 1168.90577
[26] Bezerra, V.; Leao, A.; Oliveira, J.; Santos, M., Models for the two-dimensional level strip packing problem - a review and a computational evaluation, Journal of the Operational Research Society, 71, 4, 606-627 (2020)
[27] Borgulya, I., A parallel hyper-heuristic approach for the two-dimensional rectangular strip-packing problem, Journal of computing and information technology, 22, 4, 251-265 (2014)
[28] Bortfeldt, A.; Wäscher, G., Constraints in container loading - A state-of-the-art review, European Journal of Operational Research, 229, 1, 1-20 (2013) · Zbl 1317.90172
[29] Boschetti, M.; Hadjiconstantinou, E.; Mingozzi, A., New upper bounds for the two-dimensional othogonal non guillotine cutting stock problem, IMA Journal of Management Mathematics, 13, 2, 95-119 (2002) · Zbl 1097.90546
[30] Boschetti, M.; Mingozzi, A., The two-dimensional finite bin packing problem. Part I: New lower bounds for the oriented case, 4OR, 1, 1, 27-42 (2003) · Zbl 1062.90051
[31] Boschetti, M.; Mingozzi, A., The two-dimensional finite bin packing problem. Part II: New lower and upper bounds, 4OR, 1, 2, 135-147 (2003) · Zbl 1097.90033
[32] Boschetti, M.; Montaletti, L., An exact algorithm for the two-dimensional strip-packing problem, Operations Research, 58, 6, 1774-1791 (2010) · Zbl 1228.90090
[33] Burke, E.; Hyde, M.; Kendall, G., A squeaky wheel optimisation methodology for two-dimensional strip packing, Computers & Operations Research, 38, 7, 1035-1044 (2011) · Zbl 1205.90240
[34] Burke, E.; Kendall, G.; Whitwell, G., A new placement heuristic for the orthogonal stock-cutting problem, Operations Research, 52, 4, 655-671 (2004) · Zbl 1165.90690
[35] Burke, E. K.; Hyde, M.; Kendall, G.; Woodward, J., A genetic programming hyper-heuristic approach for evolving 2-D strip packing heuristics, IEEE Transactions on Evolutionary Computation, 14, 6, 942-958 (2010)
[36] Caprara, A.; Monaci, M., On the two-dimensional knapsack problem, Operations Research Letters, 32, 1, 5-14 (2004) · Zbl 1056.90115
[37] Caprara, A.; Monaci, M., Bidimensional packing by bilinear programming, Mathematical Programming, 118, 1, 75-108 (2009) · Zbl 1169.90428
[38] Castro, P.; Grossmann, I., From time representation in scheduling to the solution of strip packing problems, Computers & Chemical Engineering, 44, 45-57 (2012)
[39] Castro, P.; Oliveira, J., Scheduling inspired models for two-dimensional packing problems, European Journal of Operational Research, 215, 1, 45-56 (2011) · Zbl 1242.90108
[40] Chazelle, B., The bottom-left bin-packing heuristic: An efficient implementation, IEEE Transactions on Computers, C-32, 8, 697-707 (1983) · Zbl 0526.68065
[41] Chen, C.; Lee, S.; Shen, Q., An analytical model for the container loading problem, European Journal of Operational Research, 80, 1, 68-76 (1995) · Zbl 0927.90087
[42] Christensen, H. I.; Khan, A.; Pokutta, S.; Tetali, P., Approximation and online algorithms for multidimensional bin packing: A survey, Computer Science Review, 24, 63-79 (2017) · Zbl 1398.68007
[43] Christofides, N.; Hadjiconstantinou, E., An exact algorithm for orthogonal 2-d cutting problems using guillotine cuts, European Journal of Operational Research, 83, 1, 21-38 (1995) · Zbl 0903.90134
[44] Christofides, N.; Mingozzi, A.; Toth, P., State-space relaxation procedures for the computation of bounds to routing problems, Networks, 11, 2, 145-164 (1981) · Zbl 0458.90071
[45] Christofides, N.; Whitlock, C., An algorithm for two-dimensional cutting problems, Operations Research, 25, 1, 30-44 (1977) · Zbl 0369.90059
[46] Cintra, G.; Miyazawa, F.; Wakabayashi, Y.; Xavier, E., Algorithms for two-dimensional cutting stock and strip packing problems using dynamic programming and column generation, European Journal of Operational Research, 191, 1, 61-85 (2008) · Zbl 1146.90499
[47] Clautiaux, F.; Alves, C.; Valério de Carvalho, J., A survey of dual-feasible and superadditive functions, Annals of Operations Research, 179, 1, 317-342 (2010) · Zbl 1229.90155
[48] Clautiaux, F.; Carlier, J.; Moukrim, A., A new exact method for the two-dimensional orthogonal packing problem, European Journal of Operational Research, 183, 3, 1196-1211 (2007) · Zbl 1135.90045
[49] Clautiaux, F.; Carlier, J.; Moukrim, A., New reduction procedures and lower bounds for the two-dimensional bin packing problem with fixed orientation, Computers & Operations Research, 34, 8, 2223-2250 (2007) · Zbl 1144.90465
[50] Clautiaux, F.; Jouglet, A.; Carlier, J.; Moukrim, A., A new constraint programming approach for the orthogonal packing problem, Computers & Operations Research, 35, 3, 944-959 (2008) · Zbl 1278.90147
[51] Clautiaux, F.; Jouglet, A.; Moukrim, A., A new graph-theoretical model for k-dimensional guillotine-cutting problems, (McGeoch, C. C., Experimental algorithms (2008), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 43-54
[52] Clautiaux, F.; Sadykov, R.; Vanderbeck, F.; Viaud, Q., Combining dynamic programming with filtering to solve a four-stage two-dimensional guillotine-cut bounded knapsack problem, Discrete Optimization, 29, 18-44 (2018) · Zbl 1506.90219
[53] Codato, G.; Fischetti, M., Combinatorial Benders’ cuts for mixed-integer linear programming, Operations Research, 54, 4, 756-766 (2006) · Zbl 1167.90601
[54] Coffman Jr., E.; Csirik, J.; Galambos, G.; Martello, S.; Vigo, D., Bin packing approximation algorithms: Survey and classification, Handbook of combinatorial optimization, 455-531 (2013), Springer: Springer New York, NY
[55] Costa, G.; Delorme, M.; Iori, M.; Malaguti, E.; Martello, S., Training software for orthogonal packing problems, Computers & Industrial Engineering, 111, 139-147 (2017)
[56] Côté, J.-F.; Dell’Amico, M.; Iori, M., Combinatorial Benders’ cuts for the strip packing problem, Operations Research, 62, 3, 643-661 (2014) · Zbl 1302.90173
[57] Côté, J.-F.; Gendreau, M.; Potvin, J.-Y., An exact algorithm for the two-dimensional orthogonal packing problem with unloading constraints, Operations Research, 62, 5, 1126-1141 (2014) · Zbl 1327.90254
[58] Côté, J.-F.; Iori, M., The meet-in-the-middle principle for cutting and packing problems, INFORMS Journal on Computing, 30, 4, 646-661 (2018) · Zbl 1528.90211
[59] Crainic, T.; Perboli, G.; Tadei, R., Recent advances in multi-dimensional packing problems, (Volosencu, C., New technologies - trends, innovations and research (2012), IntechOpen: IntechOpen Rijeka), 91-110
[60] Cui, Y.; Zhao, Z., Heuristic for the rectangular two-dimensional single stock size cutting stock problem with two-staged patterns, European Journal of Operational Research, 231, 2, 288-298 (2013) · Zbl 1317.90326
[61] Cui, Y.-P.; Zhou, Y.; Cui, Y., Triple-solution approach for the strip packing problem with two-staged patterns, Journal of Combinatorial Optimization, 34, 2, 588-604 (2017) · Zbl 1381.90070
[62] Cung, V.-D.; Hifi, M.; Le Cun, B., Constrained two-dimensional cutting stock problems a best-first branch-and-bound algorithm, International Transactions in Operational Research, 7, 3, 185-210 (2000)
[63] Dantzig, G., Discrete variable extremum problems, Operations Research, 5, 2, 266-277 (1957) · Zbl 1414.90353
[64] Dell’Amico, M.; Martello, S., Optimal scheduling of tasks on identical parallel processors, ORSA Journal on Computing, 7, 2, 191-200 (1995) · Zbl 0859.90081
[65] Delorme, M.; Iori, M.; Martello, S., Bin packing and cutting stock problems: Mathematical models and exact algorithms, European Journal of Operational Research, 255, 1, 1-20 (2016) · Zbl 1346.90700
[66] Delorme, M.; Iori, M.; Martello, S., Logic based Benders’ decomposition for orthogonal stock cutting problems, Computers & Operations Research, 78, 290-298 (2017) · Zbl 1391.90514
[67] Delorme, M.; Iori, M.; Martello, S., BPPLIB: a library for bin packing and cutting stock problems, Optimization Letters, 12, 2, 235-250 (2018) · Zbl 1401.90183
[68] Dolatabadi, M.; Lodi, A.; Monaci, M., Exact algorithms for the two-dimensional guillotine knapsack, Computers & Operations Research, 39, 1, 48-53 (2012) · Zbl 1251.90237
[69] Dowsland, K. A.; Dowsland, W. B., Solution approaches to irregular nesting problems, European Journal of Operational Research, 84, 506-521 (1995) · Zbl 0918.90116
[70] Dyckhoff, H., A new linear programming approach to the cutting stock problem, Operations Research, 29, 6, 1092-1104 (1981) · Zbl 0474.90059
[71] Dyckhoff, H., A typology of cutting and packing problems, European Journal of Operational Research, 44, 2, 145-159 (1990) · Zbl 0684.90076
[72] Fekete, S.; Schepers, J., New classes of fast lower bounds for bin packing problems, Mathematical Programming, 91, 1, 11-31 (2001) · Zbl 1051.90020
[73] Fekete, S.; Schepers, J., A combinatorial characterization of higher-dimensional orthogonal packing, Mathematics of Operations Research, 29, 2, 353-368 (2004) · Zbl 1082.90095
[74] Fekete, S.; Schepers, J., A general framework for bounds for higher-dimensional orthogonal packing problems, Mathematical Methods of Operations Research, 60, 2, 311-329 (2004) · Zbl 1076.90049
[75] Fekete, S.; Schepers, J.; van der Veen, J., An exact algorithm for higher-dimensional orthogonal packing, Operations Research, 55, 3, 569-587 (2007) · Zbl 1167.90483
[76] Ferreira, E.; Oliveira, J., Fekete and Schepers’ graph-based algorithm for the two-dimensional orthogonal packing problem revisited, (A., B.; J., H.; H., K.; G., P.; R., S., Intelligent decision support (2008), Gabler: Gabler Wiesbaden), 15-31
[77] Fleszar, K., An exact algorithm for the two-dimensional stage-unrestricted guillotine cutting/packing decision problem, INFORMS Journal on Computing, 28, 4, 703-720 (2016) · Zbl 1355.90080
[78] Friedow, I.; Scheithauer, G., Using contiguous 2D-feasible 1D cutting patterns for the 2D strip packing problem, Proceedings of the Operations research proceedings 2015, 71-77 (2017), Springer International Publishing · Zbl 1375.90249
[79] Friesen, D.; Langston, M., Variable sized bin packing, SIAM Journal on Computing, 15, 1, 222-230 (1986) · Zbl 0589.68036
[80] Furini, F.; Malaguti, E., Models for the two-dimensional two-stage cutting stock problem with multiple stock size, Computers & Operations Research, 40, 8, 1953-1962 (2013) · Zbl 1348.90487
[81] Furini, F.; Malaguti, E.; Durán, R.; Persiani, A.; Toth, P., A column generation heuristic for the two-dimensional two-staged guillotine cutting stock problem with multiple stock size, European Journal of Operational Research, 218, 1, 251-260 (2012) · Zbl 1244.90191
[82] Furini, F.; Malaguti, E.; Thomopulos, D., Modeling two-dimensional guillotine cutting problems via integer programming, INFORMS Journal on Computing, 28, 4, 736-751 (2016) · Zbl 1355.90081
[83] Gálvez, W.; Grandoni, F.; Heydrich, S.; Ingala, S.; Khan, A.; Wiese, A., Approximating geometric knapsack via l-packings, Proceedings of the 2017 IEEE 58th annual symposium on foundations of computer science (FOCS), 260-271 (2017)
[84] Garey, M.; Johnson, D., Computers and intractability: A guide to the theory of NP-completeness (1979), W.H. Freeman and Company: W.H. Freeman and Company San Francisco · Zbl 0411.68039
[85] Gendreau, M.; Iori, M.; Laporte, G.; Martello, S., A tabu search algorithm for a routing and container loading problem, Transportation Science, 40, 3, 342-350 (2006)
[86] Gendreau, M.; Iori, M.; Laporte, G.; Martello, S., A tabu search heuristic for the vehicle routing problem with two-dimensional loading constraints, Networks, 51, 1, 4-18 (2008) · Zbl 1146.90012
[87] Geoffrion, A., Generalized Benders decomposition, Journal of Optimization Theory and Applications, 10, 4, 237-260 (1972) · Zbl 0229.90024
[88] Gilmore, P. C.; Gomory, R. E., Multistage cutting stock problems of two and more dimensions, Operations Research, 13, 1, 94-120 (1965) · Zbl 0128.39601
[89] Gonçalves, J.; Resende, M., A biased random key genetic algorithm for 2D and 3D bin packing problems, International Journal of Production Economics, 145, 2, 500-510 (2013)
[90] Grandcolas, S.; Pinto, C., A sat encoding for multi-dimensional packing problems, (Lodi, A.; Milano, M.; Toth, P., Integration of ai and or techniques in constraint programming for combinatorial optimization problems (2010), Springer Berlin Heidelberg: Springer Berlin Heidelberg Berlin, Heidelberg), 141-146 · Zbl 1285.90048
[91] Hadjiconstantinou, E.; Christofides, N., An exact algorithm for general, orthogonal, two-dimensional knapsack problems, European Journal of Operational Research, 83, 1, 39-56 (1995) · Zbl 0903.90124
[92] Hadjiconstantinou, E.; Iori, M., A hybrid genetic algorithm for the two-dimensional single large object placement problem, European Journal of Operational Research, 183, 3, 1150-1166 (2007) · Zbl 1146.90481
[93] Han, X.; Iwama, K.; Ye, D.; Zhang, G., Approximate strip packing: Revisited, Information and Computation, 249, 110-120 (2016) · Zbl 1345.68275
[94] Henning, S.; Jansen, K.; Rau, M.; Schmarje, L., Complexity and inapproximability results for parallel task scheduling and strip packing, Theory of Computing Systems, 64, 120-140 (2020) · Zbl 1477.68119
[95] Herz, J., Recursive computational procedure for two-dimensional stock cutting, IBM Journal of Research and Development, 16, 5, 462-469 (1972) · Zbl 0265.90057
[96] Hifi, M., An improvement of Viswanathan and Bagchi’s exact algorithm for constrained two-dimensional cutting stock, Computers & Operations Research, 24, 8, 727-736 (1997) · Zbl 0914.90225
[97] Hooker, J.; Ottosson, G., Logic-based Benders decomposition, Mathematical Programming, 96, 1, 33-60 (2003) · Zbl 1023.90082
[98] Imahori, S.; Yagiura, M., The best-fit heuristic for the rectangular strip packing problem: An efficient implementation and the worst-case approximation ratio, Computers & Operations Research, 37, 2, 325-333 (2010) · Zbl 1175.90429
[99] Iori, M.; Martello, S., Routing problems with loading constraints, TOP, 18, 1, 4-27 (2010) · Zbl 1279.90146
[100] Iori, M.; Martello, S., An annotated bibliography of combined routing and loading problems, Yugoslav Journal of Operations Research, 23, 3, 311-326 (2013) · Zbl 1313.90026
[101] Iori, M.; Martello, S.; Monaci, M., Metaheuristic algorithms for the strip packing problem, Optimization and Industry: New Frontiers, 159-179 (2003), Springer US · Zbl 1051.90030
[102] Iori, M.; Salazar González, J.; Vigo, D., An exact approach for the vehicle routing problem with two-dimensional loading constraints, Transportation Science, 41, 2, 253-264 (2007)
[103] Jakobs, S., On genetic algorithms for the packing of polygons, European Journal of Operational Research, 88, 1, 165-181 (1996) · Zbl 0913.90229
[104] Jansen, K.; Rau, M., Closing the gap for pseudo-polynomial strip packing, Proceedings of the paper presented at ESA 2019, Munich) (2019) · Zbl 07525499
[105] Johnson, D. S., Near-optimal bin packing algorithms (1973), MIT: MIT Cambridge, MA, Ph.D. thesis.
[106] Joncour, C.; Pêcher, A., Consecutive ones matrices for multi-dimensional orthogonal packing problems, Electronic Notes in Discrete Mathematics, 36, 327-334 (2010) · Zbl 1237.90198
[107] Joncour, C.; Pêcher, A.; Valicov, P., MPQ-trees for the orthogonal packing problem, Electronic Notes in Discrete Mathematics, 11, 423-429 (2012) · Zbl 1237.90003
[108] Kellerer, H.; Pferschy, U.; Pisinger, D., Knapsack problems (2004), Springer, Berlin, Germany · Zbl 1103.90003
[109] Kenmochi, M.; Imamichi, T.; Nonobe, K.; Yagiura, M.; Nagamochi, H., Exact algorithms for the two-dimensional strip packing problem with and without rotations, European Journal of Operational Research, 198, 1, 73-83 (2009) · Zbl 1163.90803
[110] Kierkosz, I.; Luczak, M., A hybrid evolutionary algorithm for the two-dimensional packing problem, Central European Journal of Operations Research, 22, 4, 729-753 (2014) · Zbl 1339.90337
[111] Korf, R.; Moffitt, M.; Pollack, M., Optimal rectangle packing, Annals of Operations Research, 179, 1, 261-295 (2010) · Zbl 1201.90172
[112] Kwon, B.; Lee, G., Spatial scheduling for large assembly blocks in shipbuilding, Computers & Industrial Engineering, 89, 203-212 (2015)
[113] Leao, A.; Toledo, F.; Oliveira, J.; Carravilla, M.; Alvarez-Valdés, R., Irregular packing problems: A review of mathematical models, European Journal of Operational Research, 282, 3, 803-822 (2020) · Zbl 1431.90133
[114] Lesh, N.; Marks, J.; McMahon, A.; Mitzenmacher, M., Exhaustive approaches to 2D rectangular perfect packings, Information Processing Letters, 90, 1, 7-14 (2004) · Zbl 1170.90523
[115] Lodi, A.; Martello, S.; Monaci, M., Two-dimensional packing problems: A survey, European Journal of Operational Research, 141, 2, 241-252 (2002) · Zbl 1081.90576
[116] Lodi, A.; Martello, S.; Monaci, M.; Cicconetti, C.; Lenzini, L.; Mingozzi, E.; Eklund, C.; Moilanen, J., Efficient two-dimensional packing algorithms for mobile WiMAX, Management Science, 57, 12, 2130-2144 (2011)
[117] Lodi, A.; Martello, S.; Monaci, M.; Vigo, D., Two-dimensional bin packing problems, (Paschos, V. T., Paradigms of combinatorial optimization: Problems and new approaches (2014), John Wiley & Sons, Ltd: John Wiley & Sons, Ltd Hoboken, NJ, USA), 107-129 · Zbl 1204.90085
[118] Lodi, A.; Martello, S.; Vigo, D., Heuristic and metaheuristic approaches for a class of two-dimensional bin packing problems, INFORMS Journal on Computing, 11, 4, 345-357 (1999) · Zbl 1034.90500
[119] Lodi, A.; Martello, S.; Vigo, D., Recent advances on two-dimensional bin packing problems, Discrete Applied Mathematics, 123, 1, 379-396 (2002) · Zbl 1022.90020
[120] Lodi, A.; Martello, S.; Vigo, D., Models and bounds for two-dimensional level packing problems, Journal of Combinatorial Optimization, 8, 3, 363-379 (2004) · Zbl 1084.90031
[121] Lodi, A.; Martello, S.; Vigo, D., TSpack: a unified tabu search code for multi-dimensional bin packing problems, Annals of Operations Research, 131, 1-4, 203-213 (2004) · Zbl 1066.90142
[122] Lodi, A.; Monaci, M., Integer linear programming models for 2-staged two-dimensional knapsack problems, Mathematical Programming, 94, 2, 257-278 (2003) · Zbl 1030.90064
[123] Macedo, R.; Alves, C.; Valério de Carvalho, J., Arc-flow model for the two-dimensional guillotine cutting stock problem, Computers & Operations Research, 37, 6, 991-1001 (2010) · Zbl 1178.90292
[124] Martello, S., Knapsack, packing and cutting, INFOR, 32, 4, 217-218 (1994) · Zbl 0806.00019
[125] Martello, S., Two-dimensional packing problems in telecommunications, Pesquisa Operacional, 34, 1, 31-38 (2014)
[126] Martello, S.; Monaci, M., Models and algorithms for packing rectangles into the smallest square, Computers & Operations Research, 63, 161-171 (2015) · Zbl 1349.68321
[127] Martello, S.; Monaci, M.; Vigo, D., An exact approach to the strip-packing problem, INFORMS Journal on Computing, 15, 3, 310-319 (2003) · Zbl 1238.90116
[128] Martello, S.; Toth, P., Lower bounds and reduction procedures for the bin packing problem, Discrete Applied Mathematics, 28, 1, 59-70 (1990) · Zbl 0704.90074
[129] (free download at http://www.or.deis.unibo.it/knapsack.html)
[130] Martello, S.; Vigo, D., Exact solution of the two-dimensional finite bin packing problem, Management Science, 44, 3, 388-399 (1998) · Zbl 0989.90114
[131] Martin, M.; Birgin, E.; Lobato, R.; Morabito, R.; Munari, P., Models for the two-dimensional rectangular single large placement problem with guillotine cuts and constrained pattern, International Transactions in Operational Research, 27, 2, 767-793 (2020) · Zbl 07767413
[132] Martin, M.; Morabito, R.; Munari, P., A bottom-up packing approach for modeling the constrained two-dimensional guillotine placement problem, Computers & Operations Research, 115, 104851 (2020) · Zbl 1458.90553
[133] Matayoshi, M., The 2D strip packing problem: A new approach with verification by EA, Proceedings of the 2010 IEEE international conference on systems, man and cybernetics, 2492-2499 (2010)
[134] Melega, G.; Araujo, S.; Jans, R., Classification and literature review of integrated lot-sizing and cutting stock problems, European Journal of Operational Research, 271, 1, 1-19 (2018) · Zbl 1403.90038
[135] Messaoud, S.; Chu, C.; Espinouse, M.-L., Characterization and modelling of guillotine constraints, European Journal of Operational Research, 191, 1, 112-126 (2008) · Zbl 1145.90057
[136] Mesyagutov, M.; Mukhacheva, E.; Belov, G.; Scheithauer, G., Packing of one-dimensional bins with contiguous selection of identical items: An exact method of optimal solution, Automation and Remote Control, 72, 1, 141-159 (2011) · Zbl 1233.90242
[137] Mesyagutov, M.; Scheithauer, G.; Belov, G., LP bounds in various constraint programming approaches for orthogonal packing, Computers & Operations Research, 39, 10, 2425-2438 (2012) · Zbl 1251.90331
[138] Monaci, M.; Toth, P., A set-covering-based heuristic approach for bin-packing problems, INFORMS Journal on Computing, 18, 1, 71-85 (2006) · Zbl 1241.90191
[139] Morabito, R.; Arenales, M., Staged and constrained two-dimensional guillotine cutting problems: An AND/OR-graph approach, European Journal of Operational Research, 94, 3, 548-560 (1996) · Zbl 0947.90537
[140] Morabito, R.; Pureza, V., Heuristic approach based on dynamic programming and and/or-graph search for the constrained two-dimensional guillotine cutting problem, Annals of Operations Research, 179, 297-315 (2010) · Zbl 1201.90174
[141] Mrad, M., An arc flow-based optimization approach for the two-stage guillotine strip cutting problem, Journal of the Operational Research Society, 66, 11, 1850-1859 (2015)
[142] Mrad, M.; Meftahi, I.; Haouari, M., A branch-and-price algorithm for the two-stage guillotine cutting stock problem, The Journal of the Operational Research Society, 64, 5, 629-637 (2013)
[143] Nesello, V.; Delorme, M.; Iori, M.; Subramanian, A., Mathematical models and decomposition algorithms for makespan minimization in plastic rolls production, Journal of the Operational Research Society, 69, 3, 326-339 (2018)
[144] Ntene, N.; van Vuuren, J., A survey and comparison of guillotine heuristics for the 2D oriented offline strip packing problem, Discrete Optimization, 6, 2, 174-188 (2009) · Zbl 1159.90527
[145] Oliveira, J.; Júnior, A. N.; Silva, E.; Carravilla, M., A survey on heuristics for the two-dimensional rectangular strip packing problem, Pesquisa Operacional, 36, 2, 197-226 (2016)
[146] Oliveira, J.; Wäscher, G., Cutting and packing, European Journal of Operational Research, 183, 3, 1106-1108 (2007)
[147] Onodera, H.; Taniguchi, Y.; Tamaru, K., Branch-and-bound placement for building block layout, Proceedings of the 28th ACM/IEEE design automation conference, 433-439 (1991)
[148] Ortmann, F.; van Vuuren, J., Modified strip packing heuristics for the rectangular variable-sized bin packing problem, ORiON, 26, 1, 21-44 (2010)
[149] Parreño, F.; Alvarez-Valdes, R.; Oliveira, J.; Tamarit, J., A hybrid GRASP/VND algorithm for two- and three-dimensional bin packing, Annals of Operations Research, 179, 1, 203-220 (2010) · Zbl 1201.90176
[150] Pisinger, D.; Sigurd, M., The two-dimensional bin packing problem with variable bin sizes and costs, Discrete Optimization, 2, 2, 154-167 (2005) · Zbl 1077.90057
[151] Pisinger, D.; Sigurd, M., Using decomposition techniques and constraint programming for solving the two-dimensional bin-packing problem, INFORMS Journal on Computing, 19, 1, 36-51 (2007) · Zbl 1241.90118
[152] Pollaris, H.; Braekers, K.; Caris, A.; Janssens, G.; Limbourg, S., Vehicle routing problems with loading constraints: State-of-the-art and future directions, OR Spectrum, 37, 2, 297-330 (2015) · Zbl 1311.90032
[153] Puchinger, J.; Raidl, G., Models and algorithms for three-stage two-dimensional bin packing, European Journal of Operational Research, 183, 3, 1304-1327 (2007) · Zbl 1135.90029
[154] Queiroz, T.; Hokama, P.; Schouery, R.; Miyazawa, F., Two-dimensional disjunctively constrained knapsack problem: Heuristic and exact approaches, Computers & Industrial Engineering, 105, 313-328 (2017)
[155] Queiroz, T.; Miyazawa, F., Order and static stability into the strip packing problem, Annals of Operations Research, 223, 1, 137-154 (2014) · Zbl 1306.90124
[156] Rao, M., On the cutting stock problem, Journal of the Computer Society of India, 7, 35-39 (1976)
[157] Rietz, J.; Alves, C.; Valério de Carvalho, J., Theoretical investigations on maximal dual feasible functions, Operations Research Letters, 38, 3, 174-178 (2010) · Zbl 1192.90127
[158] Rietz, J.; Alves, C.; Valério de Carvalho, J., Worst-case analysis of maximal dual feasible functions, Optimization Letters, 6, 8, 1-19 (2011)
[159] Rietz, J.; Alves, C.; Valério de Carvalho, J., On the extremality of maximal dual feasible functions, Operations Research Letters, 40, 1, 25-30 (2012) · Zbl 1242.90202
[160] Russo, M.; Boccia, M.; Sforza, A.; Sterle, C., Constrained two-dimensional guillotine cutting problem: upper-bound review and categorization, International Transactions in Operational Research, 27, 2, 794-834 (2020) · Zbl 07767414
[161] Russo, M.; Sforza, A.; Sterle, C., An exact dynamic programming algorithm for large-scale unconstrained two-dimensional guillotine cutting problems, Computers & Operations Research, 50, 97-114 (2014) · Zbl 1348.90554
[162] Scheithauer, G., Equivalence and dominance for problems of optimal packing of rectangles, Ricerca Operativa, 83, 3-34 (1997)
[163] Scheithauer, G., Introduction to cutting and packing optimization (2018), Springer International Publishing · Zbl 1391.90002
[164] Serairi, M.; Haouari, M., A theoretical and experimental study of fast lower bounds for the two-dimensional bin packing problem, RAIRO-Operations Research, 52, 2, 391-414 (2018) · Zbl 1405.90027
[165] Silva, E.; Alvelos, F.; Valério de Carvalho, J., An integer programming model for two- and three-stage two-dimensional cutting stock problems, European Journal of Operational Research, 205, 3, 699-708 (2010) · Zbl 1188.90181
[166] Silva, E.; Oliveira, J.; Wäscher, G., 2DCPackGen: A problem generator for two-dimensional rectangular cutting and packing problems, European Journal of Operational Research, 237, 3, 846-856 (2014) · Zbl 1338.90355
[167] Silva, E.; Oliveira, J.; Wäscher, G., The pallet loading problem: a review of solution methods and computational experiments, International Transactions in Operational Research, 23, 1-2, 147-172 (2016) · Zbl 1338.90354
[168] Soh, T.; Inoue, K.; Tamura, N.; Banbara, M.; Nabeshima, H., A sat-based method for solving the two-dimensional strip packing problem, Fundamenta Informaticae, 102, 3-4, 467-487 (2010) · Zbl 1214.68374
[169] Strecker, T.; Hennig, L., Automatic layouting of personalized newspaper pages, (Fleischmann, B.; Borgwardt, K.-H.; Klein, R.; Tuma, A., Proceedings of the Operations research proceedings 2008 (2009), Springer: Springer Berlin, Heidelberg), 469-474 · Zbl 1209.90258
[170] Terno, J.; Lindemann, R.; Scheithauer, G., Zuschnittprobleme and ihre praktische lösung, Technical Report (1987), Verlag Harry Deutsch, Thun und FrankfurtMain · Zbl 0657.65089
[171] Valério de Carvalho, J., Exact solution of bin-packing problems using column generation and branch-and-bound, Annals of Operations Research, 86, 0, 629-659 (1999) · Zbl 0922.90113
[172] Valério de Carvalho, J., LP models for bin packing and cutting stock problems, European Journal of Operational Research, 141, 2, 253-273 (2002) · Zbl 1059.90095
[173] Valério de Carvalho, J., Using extra dual cuts to accelerate column generation, INFORMS Journal on Computing, 17, 2, 175-182 (2005) · Zbl 1239.90089
[174] Vanderbeck, F., A nested decomposition approach to a three-stage, two-dimensional cutting-stock problem, Management Science, 47, 6, 864-879 (2001) · Zbl 1232.90326
[175] Velasco, A.; Uchoa, E., Improved state space relaxation for constrained two-dimensional guillotine cutting problems, European Journal of Operational Research, 272, 1, 106-120 (2019) · Zbl 1403.90589
[176] Viswanathan, K.; Bagchi, A., Best-first search methods for constrained two-dimensional cutting stock problems, Operations Research, 41, 4, 768-776 (1993) · Zbl 0782.90076
[177] Wäscher, G.; Haußner, H.; Schumann, H., An improved typology of cutting and packing problems, European Journal of Operational Research, 183, 3, 1109-1130 (2007) · Zbl 1278.90347
[178] Wei, L.; Oon, W.-C.; Zhu, W.; Lim, A., A skyline heuristic for the 2D rectangular packing and strip packing problems, European Journal of Operational Research, 215, 2, 337-346 (2011) · Zbl 1237.90213
[179] Wei, L.; Qin, H.; Cheang, B.; Xu, X., An efficient intelligent search algorithm for the two-dimensional rectangular strip packing problem, International Transactions in Operational Research, 23, 1-2, 65-92 (2016) · Zbl 1338.90357
[180] Xu, Z.; Lee, C.-Y., New lower bound and exact method for the continuous berth allocation problem, Operations Research, 66, 3, 778-798 (2018) · Zbl 1443.90197
[181] Yanasse, H.; Katsurayama, D., Checkerboard pattern: proposals for its generation, International Transactions in Operational Research, 12, 1, 21-45 (2005) · Zbl 1060.90066
[182] Yanasse, H.; Katsurayama, D., An enumeration scheme to generate constrained exact checkerboard patterns, Computers & Operations Research, 35, 6, 2114-2128 (2008) · Zbl 1152.90583
[183] Yu, G.; Mao, Y.; Xiao, J., A new lower bound for online strip packing, European Journal of Operational Research, 250, 3, 754-759 (2016) · Zbl 1346.90724
[184] Yu, G.; Mao, Y.; Xiao, J., A new upper bound for the online square packing problem in a strip, Journal of Combinatorial Optimization, 33, 4, 1411-1420 (2017) · Zbl 1376.90055
[185] Yu, G.; Mao, Y.; Xiao, J., New upper bounds for online strip packing, Discrete Optimization, 23, 20-32 (2017) · Zbl 1387.90231
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