Forest landscape management modeling using simulated annealing

Abstract

This paper presents a new landscape management model using a simulated annealing approach. The model is capable of achieving target landscape structure, in the form of composition and configuration objectives, in a near optimal fashion by spatially and temporally scheduling treatment interventions. Management objectives and constraints are identified in an objective function. Penalty cost functions for each objective establish common non-monetary units, and a mechanism for making trade-offs among different objectives. Management strategies, as well as alternative solutions as combinations of treatment scheduling of each stand, are formulated around treatment regimes, including varying intensities of planting, pre-commercial thinning, commercial thinning, two-stage harvesting and clear-cutting. The model then examines alternative solutions using a heuristic process, and evaluates their effects on the objective over an entire planning horizon.

The model was tested on a 20,000 ha (987 stands) hypothetical forest landscape with four replicates, differing in initial age class composition and spatial configuration. Management objectives included: (i) maximizing harvest volume, (ii) minimizing deviations in harvest flow, (iii) maintaining harvest block size between 40 and 100 ha, (iv) maintaining a one period adjacency delay, and (v) achieving an inverse-J distribution of harvest opening patches. Objective accomplishment, when compared to an aspatial optimal solution, varied from 72% for even flow harvest, to 99.9% for adjacency delay. These results generally reflect the objective priorities established for the test.

Results also suggested that the achievement of an inverse-J distribution of harvest opening patches depended not only upon the spatial harvest pattern, but initial forest conditions as well. In the case of the test forests, however, the effects of different initial age class structure and spatial configuration lasted a relatively short time. We conclude that simulated annealing allows a great deal of flexibility in designing landscape management in a near optimal fashion.

Introduction

Forest management design is a challenging part of the management planning process. Perhaps the most significant aspect of that challenge is developing a sound forest modeling approach that accommodates multiple, often conflicting, management objectives, such as wood supply, wildlife habitat, water quality, and biodiversity.

Multiple forest objectives are difficult to integrate in a model. Most often they do not share common measurement units and are described with different methods. Furthermore, the need in landscape management to include spatial configuration of forest conditions, as well as their aspatial composition, increases difficulty (Baskent and Jordan, 1995a). Management objectives that incorporate spatial configuration preclude using a simple forest description with an a priori stratification (Nur et al., 2000). As a result, traditional modeling approaches are inefficient and ineffective in landscape management design. Finding a better approach, however, is not straightforward.

So far, a variety of modeling approaches, involving a variety of forest descriptions and management objectives, have been developed using mathematical optimizing and simulation techniques. Simulation involves a heuristic approach whereby important lessons in forest dynamics, including configuration, may be learned on the way to finding a solution, i.e. an intervention schedule. It does not, however, produce an optimal solution due to its sequential search nature and failure to make inter-temporal trade-offs. Nor, is simulation effective where multiple management objectives exist. Landscape management, however, involves multiple objectives (composition and configuration), most of which are conflicting and spatial in nature, and often an optimal or near optimal solution is desired. Optimizing approaches, on the other hand, have the appeal of guaranteeing an optimal schedule, even where multiple objectives exist. They are almost impossible to formulate, though, where management objectives involve spatial configuration of forest conditions, as well as their composition (Murray, 1999, Nur et al., 2000). While some relaxed optimization techniques, such as integer or mixed integer programming (MIP), have been used in accommodating spatial constraints, such as block size and adjacency delay (Nelson and Brodie, 1990, Hof et al., 1994), MIP has shown little promise in solving real problems in a reasonable time (Kirby et al., 1986, Bettinger et al., 1999). Several limitations, directly related to problem size and the non-linear nature of configuration objectives (Lockwood and Moore, 1993, Bettinger et al., 1998, Murray, 1999) limit the utility of MIP approaches. For example, Bettinger et al. (1999) used MIP to solve a simple 700-unit management problem with a single harvest choice over five periods, but failed to obtain a feasible solution in a reasonable time—it took several days to reach an optimal solution for even a 40-unit, hypothetical management problem. Optimization techniques do not look promising where configurational objectives, such as patch size distribution, are involved, even in a relatively small management problem. Perhaps, that explains why no studies to date have shown a mathematical formulation involving patch size and distribution objectives.

Neither simulation nor mathematical optimizing approaches are capable of solving the forest landscape management design problem. A new approach is needed. Like other types of combinatorial optimization techniques, such as hill climbing and tabu search (Yoshimoto et al., 1994, Murray and Church, 1995, Bettinger et al., 1998) simulated annealing (Kirkpatrick, 1984) is such an approach (Lockwood and Moore, 1993, Murray and Church, 1995, Ohman and Eriksson, 1998). It is a modified simulation approach that uses a smart search technique to find the best combination of management interventions for achieving management objectives. The technique starts with an initial intervention solution, i.e. a schedule, and then seeks a better one for achieving management objectives by making and evaluating gradual changes.

Lockwood and Moore (1993) first applied simulated annealing to the problem of finding a harvest intervention schedule that maximized sustainable wood supply while adhering to harvest block size limits and an harvest adjacency delay. They demonstrated that simulated annealing could handle such spatial constraints with reasonable speed and, at the same time, provide a near optimal solution. Nelson and Liu (1994) developed an SA algorithm to solve a similar problem and showed that SA was able to generate solutions superior to the hill climbing algorithm. Murray and Church (1995) and Boston and Bettinger (1999) compared simulated annealing to other meta-heuristics, e.g. tabu search and MCIP, and found out that SA was generally able to locate the best solution values to simple problems. Ohman and Eriksson (1998) demonstrated SA potential in maintaining core areas, i.e. contiguous old growth, as defined by Baskent and Jordan (1995a), using a small forest of 200 stands, a single treatment, a single rotation period, and a limited set of objectives. For landscape management problems, however, a large number of stands, a large set of management objectives and constraints, a large array of silvicultural treatments, and a long planning horizon exist. To date, simulated annealing potential in landscape management design has not been explored to that extent. Simulated annealing studies involving configuration, as well as composition, objectives are not evident.

Given the difficulty of applying traditional modeling approaches to landscape management design, and the potential of simulated annealing, the main objective of the research reported here was to demonstrate the utility of simulated annealing in providing solutions where both forest composition and configuration objectives exist. To this end, SA was used to formulate a landscape management problem and explain the spatial dynamics of four forests differing in age class and spatial structure.