Testing the use of lazy constraints in solving area-based adjacency formulations of harvest scheduling models

Spatially explicit harvest scheduling models to enforce maximum harvest opening size restrictions often lead to combinatorial problems that are hard to solve. This article shows that the inequalities required by one of the three existing formulations, the Path model are typically lazy. In other words, these constraints are rarely binding during optimization, especially if the maximum opening size is large relative to the average management unit size. By solving 60 hypothetical and 8 real forest problems with varying maximum clearcut sizes and to varying target optimality gaps, we confirm that applying the path constraints only when they are violated during optimization leads to shorter solution times. Although the Lazy Path constraints performed better than the other formulation/solution approaches, the relative superiority of the method was more obvious at larger optimality gaps. Nearly 95% of the problem instances solved fastest with the "lazy" method at a target gap of 1%, and almost 92% solved fastest at 0.05%. At 0.01%, the Lazy Path approach was still superior in the majority of cases, but the percentage was much lower (57%). This is a significant improvement compared with the 14, 10, and 19% shares of the other approaches. If only the real instances are considered, the Lazy Path approach performed best in 68% of the instances with 1 and 0.01% optimality gaps and in 61% of the instances with 0.05% gap. A closer analysis of the results suggests that the relative superiority of the approach increases with problem size and maximum clearcut size.