Reflections on the number of independent solutions for forest spatial harvest scheduling problems: a case of simulated annealing

To assess the quality of results obtained from heuristics through statistical procedures, a number of independently generated solutions to the same problem are required, however the knowledge of how many solutions are necessary for this purpose using a specific heuristic is still not clear. Therefore, the overall aims of this paper are to quantitatively evaluate the effects of the number of independent solutions generated on the forest planning objectives and on the performance of different neighborhood search techniques of simulated annealing (SA) in three increasing difficult forest spatial harvest scheduling problems, namely non-spatial model, area restriction model (ARM) and unit restriction model (URM). The tested neighborhood search techniques included the standard version of SA using the conventional 1-opt moves, SA using the combined strategy that oscillates between the conventional 1-opt moves and the exchange version of 2-opt moves, and SA using the change version of 2-opt moves. The obtained results indicated that the number of independent solutions generated had clear effects on the conclusions of the performances of different neighborhood search techniques of SA, which indicated that no one particular neighborhood search technique of SA was universally acceptable. The optimal number of independent solutions generated for all alternative neighborhood search techniques of SA for ARM problems could be estimated using a negative logarithmic function based on the problem size, however the relationships were not sensitive (i.e., 0.13 < p < 0.78) to the problem size for non-spatial and URM harvest scheduling problems, which should be somewhat above 250 independent runs. The types of adjacency constraints did moderately affect the number of independent solutions necessary, but not significantly. Therefore, determining an optimal number of independent solutions generated is a necessary process prior to employing heuristics in forest management planning practices.