Abstract
Recently, research on exact methods has been undertaken to solve forest management problems subject to constraints on the maximum clearcut area by using the area restriction model approach. Three main basic integer programming models for these problems have been discussed in the literature: the so-called cluster, path and bucket formulations. Solving these models via branch-and-bound, where all variables and constraints are used a priori, is adequately suited for real problems of a small to medium size, but is not appropriate for larger problems. In this paper, we describe a branch-and-price approach for the cluster model, and we show that this formulation dominates the bucket model, by completing the results of the dominance relationships between the bounds of the three models. Branch-and-price was tested on real and hypothetical forests ranging from 45 to 2945 stands and temporal horizons ranging from three to twelve periods were employed. Results show that the solutions obtained by the proposed approach stood within 1% of the optimal solution and were achieved in a short computation time. It was found that branch-and-bound was unable to produce solutions for most forests from 850 stands with either eleven or an average number of stands per clearcut greater or equal than eight.
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References
Alvelos, F.: Branch-and-price and multicommodity flows. Ph.D. Thesis, Universidade do Minho, Portugal (2005)
Barnhart, C., Johnson, E.L., Nemhauser, G.L., Savelsbergh, M.W.P., Vance, P.H.: Branch-and-price: Column generation for solving huge integer programs. Oper. Res. 46(3), 316–329 (1998)
Barrett, T.M., Gilles, J.K., Davis, L.S.: Economic and fragmentation effects of clearcut restrictions. For. Sci. 44(4), 569–577 (1998)
Boston, K., Bettinger, P.: Combining tabu search and genetic algorithm heuristic techniques to solve spatial harvest scheduling problems. For. Sci. 48(1), 35–46 (2002)
Clark, M.M., Mueller, R.D., McDonald, T.P.: A three-stage heuristic for harvest scheduling with access road network development. For. Sci. 46, 204–218 (2000)
Constantino, M., Martins, I., Borges, J.G.: A new mixed integer programming model for harvest scheduling subject to maximum area restrictions. Oper. Res. 56(3), 542–551 (2008)
Crowe, K., Nelson, J., Boyland, M.: Solving the area-restricted harvest-scheduling model using the branch and bound algorithm. Can. J. For. Res. 33(9), 1804–1814 (2003)
Diestel, R.: Graph Theory, 2nd edn. Graduate Texts in Mathematics. Springer, Berlin (2000)
Epstein, R., Goycoolea, M., Murray, A.T., Weintraub, A.: An adjacency-modeling problem based on constructing harvesting areas. In: Arthaud, G.J., Barrett, T.M. (eds.) Systems Analysis in Forest Resources. Kluwer Academic, Dordrecht (2003)
Falcão, A.O., Borges, J.G.: Combining random and systematic search procedures for solving spatially constrained forest management scheduling models. For. Sci. 48(3), 608–621 (2002)
Gondran, M., Minoux, M.: Graphs and Algorithms. Wiley, New York (1984)
Goycoolea, M., Murray, A.T., Barahona, F., Epstein, R., Weintraub, A.: Harvest scheduling subject to maximum area restrictions: exploring exact approaches. Oper. Res. 53(3), 90–500 (2005)
Goycoolea, M., Murray, A.T., Vielma, J.P., Weintraub, A.: Evaluating approaches for solving the area restricted model in harvest scheduling. For. Sci. 55(2), 149–165 (2009)
Guy, D., et al. (eds.): Column Generation. Springer, Berlin (2005)
ILOG: ILOG CPLEX 11.0—User’s Manual (2007)
Lockwood, C., Moore, T.: Harvest scheduling with spatial constraints: a simulated annealing approach. Can. J. For. Res. 23, 468–478 (1993)
Lübbecke, M.E., Desrosiers, J.: Selected topics in column generation. Oper. Res. 53(6), 1007–1023 (2005)
Martins, I., Constantino, M., Borges, J.G.: Forest management models with spatial structure constraints. Working Paper No. 2/1999, C.I.O./Faculdade de Ciências de Lisboa (1999)
Martins, I., Constantino, M., Borges, J.G.: A column generation approach for solving a non-temporal forest harvest model with spatial structure constraints. Eur. J. Oper. Res. 161(2), 478–498 (2005)
McDill, M.E., Rebain, S.A., Braze, J.: Harvest scheduling with area-based adjacency constraints. For. Sci. 48(4), 631–642 (2002)
Murray, A.T.: Spatial restrictions in harvest scheduling. For. Sci. 45(1), 45–52 (1999)
Murray, A.T., Weintraub, A.: Scale and unit specification influences in harvest scheduling with maximum area restrictions. For. Sci. 48(4), 779–789 (2002)
Nemhauser, G.L., Wolsey, L.A.: Integer and Combinatorial Optimization. Wiley, New York (1989)
Richards, E.W., Gunn, E.A.: A model and tabu search to optimize stand harvest and road construction schedules. For. Sci. 49(4), 608–618 (2000)
Richards, E.W., Gunn, E.A.: Tabu search design for difficult forest management optimization problems. Can. J. For. Res. 33, 1126–1133 (2003)
Vielma, J.P., Murray, A.T., Ryan, D.M., Weintraub, A.: Improving computational capabilities for adressing volume constraints in forest harvest scheduling problems. Eur. J. Oper. Res. 176(2), 1246–1264 (2007)
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Martins, I., Alvelos, F. & Constantino, M. A branch-and-price approach for harvest scheduling subject to maximum area restrictions. Comput Optim Appl 51, 363–385 (2012). https://doi.org/10.1007/s10589-010-9347-1
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DOI: https://doi.org/10.1007/s10589-010-9347-1