Assessment of Three Heuristics for Developing Large-Scale Spatial Forest Harvest Scheduling Plans
In this study, three heuristics were developed to assess
the quality of forest plans that can be developed and the time required
to develop them. The three heuristics include threshold accepting, 1-opt
tabu search and a combined heuristic which consisted of threshold accepting,
1-opt tabu search and 2-opt tabu search. The combined heuristic was developed
to capitalize on the unique search properties of both threshold accepting
and tabu search. Present hypothesis was that each of the three heuristics
would produce forest plans of approximately equal solution quality. The
three heuristics are assessed using forest plans developed from nine hypothetical
landownerships containing various ownership patterns and age class distributions.
The combined heuristic found the highest quality forest plans for most
problems with older and normal age class distributions. In problems with
younger age class distributions, the combined heuristic produced slightly
inferior solutions as compared to threshold accepting. The variation in
forest plan quality was lowest when using the combined heuristic or threshold
accepting, thus these two processes are of value for large-scale forest
plan development efforts.
The use and evaluation of spatial harvest scheduling techniques has increased
in recent years due to the need to develop forest plans that accommodate
multiple and often conflicting management objectives. In addition, there
has been a recognizable increased need for sustainable forest harvesting
practices that not only recognize economics, but also recognize the preservation
and maintenance of bio-diversity, aesthetic values and public recreation
areas (Bettinger and Chung, 2004). Further, federal and state regulations
and polices have resulted in increasingly complex objectives for the management
of forests in the North America (Bettinger and Sessions, 2003). And, in
many instances, compliance with regulatory restrictions, voluntary forest
certification programs and organizational goals and policies related to
landscape conditions are now as important as economic objectives in the
development of forest plans. Long-term forest planning allows people to
understand whether sustainable forestry is actually being practiced. Therefore,
locating efficient algorithms to assist with the development of forest
plans has become a very practical and import issue.
An optimization process seeks to maximize (or minimize) certain economical
and ecological objectives subject to various constraints, while assigning
forest management actions to timber stands over some lengthy period of
time and across a landscape. There are two general classes of forest planning
algorithms, one is based on traditional mathematical programming techniques
and the other is based on heuristics. The first class includes exact algorithms,
such as linear programming, mixed integer programming, integer programming
(Bevers and Hof, 1999; Hof et al., 1994; Hof and Joyce, 1992) and
dynamic programming (Snyder and ReVelle, 1997; Hoganson and Borges, 1998).
The appeal of exact algorithms is that the optimal solution to a problem
(if found) will be reported. However, as the size of a planning problem
increases, solving it may become computationally impractical (Lockwood
and Moore, 1993), particularly if integer variables are used. Although
computer hardware and software technology continues to advance, the use
of exact algorithms remains limited in application to small- and medium-sized
forest planning problems. The second class include heuristics such as
Monte Carlo simulation (Nelson and Brodie, 1990), simulated annealing
(Dahlin and Sallnas, 1993; Lockwood and Moore, 1993; Murray and Church,
1995), threshold accepting (Bettinger et al., 2003), tabu search
(Bettinger et al., 1997; Batten et al., 2005) and genetic
algorithms (Glover et al., 1995; Falcão and Borges, 2001). Although heuristics cannot guarantee that they can locate the optimum
solution to a problem, they can usually find good solutions to complex
planning problems, making them attractive for large-scale spatial forest
Bettinger et al. (2002) compared eight heuristics on three small,
but increasing difficult wildlife planning problems. They found that threshold
accepting, simulated annealing and tabu search with 2-opt moves worked
the best in most cases. Heinonen and Pukkala (2004) performed a comparison
of 1-opt and 2-opt compartment neighborhoods in the development of forest
plans for medium-sized forest planning problems and confirmed the notion
that 2-opt neighborhoods were important for tabu search. One question
that lingers is whether differences in forest plan quality arising from
the use of different search methods becomes negligible when relatively
large forests are considered, given the large number of choices available
and the greater amount of flexibility inherent in these problems. In this
research, we will compare how three heuristics perform in the development
of nine forest plans. One of the heuristics combined the unique search
properties of the other two. Each plan considers a different type of land
ownership situation. The goal is to determine which heuristic works best
for the management problem under consideration. Knowledge gained from
the previous research will inform the selection of test heuristics. For
example, it has been shown that threshold accepting is a relatively fast
heuristic and it can transition an inferior solution to a problem to a
very good solution in a fraction of the time that tabu search requires.
Tabu search, on the other hand, uses deterministic choices to refine the
quality of solutions. Present hypothesis is that a combination of these,
utilizing their unique search behavior, can produce higher quality forest
plans for larger landscapes then when each is used alone. Thus the contribution
of this study is in the testing of methods for the development of large,
relatively complex, yet realistic forest plans.
MATERIALS AND METHODS
This study was conducted in 2006. The scheduling problems we use in this
research involve three types of ownership patterns of parcels that are
typical in the Southeastern United States (clumped, random and dispersed
parcels). Three forest age class distributions are then assigned to each
timber stand randomly; these represent a young forest age class structure,
a normal forest and an older forest age class structure. The size of each
hypothetical forest is approximately 28,300 ha. Forest plans were then
developed for each of these 9 hypothetical forests. We describe next the
formulation of the forest planning problem and the heuristic techniques
that were tested. The planning problem described below was solved first
using linear programming, although the spatial harvest placement constraints
were ignored. Thus the results derived from linear programming are assumed
to come from a relaxed problem. The heuristics were employed to solve
the full spatial planning problem. Each were developed using the C-sharp
The forest planning problem uses a time horizon that is 20 years long;
each planning period is one year long. This tactical planning problem
seeks to maximize the net present value of timber harvested from the forest
and the problem formulation has been used in other research endeavors
(Zhu et al., 2007; Zhu and Bettinger, 2008). The objective function
and constraints are formulated as:
where, XT1it = 1, otherwise XT2it = 0
||Area of timber stand i
||Clearcut age for timber stand i
||Age when first thinning occurs for timber stand i
||Age when second thinning occurs for timber stand i
||Logging cost for timber stand i harvested in time period t
||maximum clearcut area
||The total number of timber stands
||Set of all timber stands adjacent to timber stand i
||The set of all timber stands adjacent to those timber stands adjacent
to timber stand i
||The total number of planning periods in the planning horizon
||The available clearcut timber harvest volume from timber stand i
during time period t
||The unscheduled timber harvest volume from timber stand i at the
end of period 20, whether or not a harvest had been applied during
the period represented by the plan
||The available first thinning timber harvest volume from timber stand
i during time period t
||The available second thinning timber harvest volume from timber
stand i during time period t
||A binary variable, which = 1 if timber stand i is clearcut harvested
in time period t and 0 otherwise
||A binary variable, which = 1 if timber stand i is first-thinned
in time period t and 0 otherwise
||A binary variable, which = 1 if timber stand i is second-thinned
in time period t and 0 otherwise
The objective function of this problem assesses the difference between
revenue generation and the cost of management activities (prices versus
logging costs) for each activity (clearcut, first thinning and second
thinning) applied to each timber stand over the time horizon. Decisions
are integer in nature (yes or no), therefore when an activity is applied
to a stand it is applied to the entire stand. Revenues and costs are discounted
from the mid-point of each planning period. The value of the ending inventory
(the timber that remains standing at the end of the time horizon) is included
to fully value the forest management enterprise. Equation
2 indicates that each stand can only be clearcut harvested one time
during the time horizon and Eq. 3 indicates that each
stand can only be first thinned once during the time horizon. Equation
4 indicates that each stand can only be second thinned once during
the time horizon, given that it was previously first-thinned. Equation
5 ensures that clearcuts do not exceed the maximum clearcut size assumed
and also assumes that the green-up period is 2 years. This set of constraints
represent a slight modification of the original ARM model presented by
Murray (1999). Here, Si is the subset of clearcut harvested
stands containing all stands adjacent to neighbors of stand i and stands
adjacent to the neighbors of the neighbors, etc. As Murray (1999) suggested,
this constraint involves a recursive function that senses a sprawling
cluster of stands clearcut harvested within the green-up period and that
the cluster depends on the contiguity of stands and their direct or indirect
relationship to stand i. As a result, the relationship is not easily described
by a linear equation. Equation 6 represents an ending
inventory constraint, where the standing volume at the end of the time
horizon must be at least 90% of the standing volume at the beginning of
the time horizon. Equations 7 and 8
ensure that the time period between a thinning activity and a clearcutting
activity is at least six years. Equations 9 and 10
are wood-flow constraints and maintain the volume scheduled for harvest
during each time period to a proportion of the standing volume at the
end of the time horizon. Equations 11 and 12
are also wood-flow constraints and limit deviations in scheduled harvest
volumes from one period to the next, as a form of stability. This problem
formation can be considered a Model I (Johnson and Scheurman, 1977), integer
We assumed an interest rate of 6% and derived timber stumpage prices by Mart-South
(2004). The costs of reasonable forest establishment practices for the Southeastern
United States were derived by Smidt et al. (2005). The maximum clearcut
size we assumed was 97 ha (240 acres) and the green-up period associated with
the adjacency constraints was assumed to be 2 years.
The heuristic methods we tested included threshold accepting and tabu
search. Threshold accepting, initially described by Dueck and Scheuer
(1990), has been successfully applied to forest planning problems (Bettinger
et al., 2002; Bettinger et al., 2003) and in general, can
produce solutions that are as good, if not better than, those produced
by simulated annealing. The parameters required for the threshold accepting
heuristic include the initial threshold, number of total iterations, the
rate of change in the threshold and the number of unsuccessful iterations
allowed per threshold. Threshold accepting is a stochastic local search
process. In our implementation, a timber stand and a management prescription
are selected at random and temporarily inserted into the current forest
plan. Then the clearcut adjacency constraints are assessed. If the addition
to the forest plan results in a constraint violation, then the addition
is rejected. If the addition to the forest plan results in no constraint
violations and improves the quality of the plan, then it is formally accepted.
If the addition to the forest plan does not result in a constraint violation,
but leads to a solution that is of lower quality than the best solution
stored in memory, then is formally accepted only if the quality of the
resulting solution is within the threshold (best solution-current threshold
value). The threshold accepting search process begins with a relatively
large threshold value, which decreases as the number of iterations progresses,
until it is so small that the search process terminates. This search process
allows large deviations in solution quality in the initial stages of the
search and small deviations later in the search. Through numerous trials,
we arrived at the following parameters: allow 500 iterations per threshold
value, use an initial threshold level of $1,000,000 (US) and reduce the
threshold value by $200 each time it is changed. If more than 500 unsuccessful
attempts to change a plan pass, the threshold changes as well.
Tabu search is a deterministic search process that was initially described
by Glover (1989). Tabu search has been applied extensively to forest planning
problems (e.g., Bettinger et al., 1997; Bettinger et al.,
2002). In its basic form, tabu search uses 1-opt moves, which change the
status of an individual decision variable. In present case, this would
involve changing the timing of a harvest applied to a timber stand. Once
a choice has been assigned to a stand, the stand is considered tabu for
a pre-defined number of iterations of the search process. During this
time, no other changes can be made to the stand unless the changes result
in a forest plan that is better (in solution quality) than any other forest
plan developed during the search. This is frequently considered the aspiration
Within tabu search, a neighborhood of potential changes is assessed and
the best choice of harvests is selected from this set. The neighborhood
in present case involves assessing all of the potential changes to a forest
plan, although region-limited approaches have shown value (Bettinger et
al., 2007). As with threshold accepting, the constraints are assessed
after selecting a choice from the neighborhood. If the choice results
in a violation of one or more constraints, it is disregarded and another
choice is made. Based on numerous trials, we determined that the best
parameters for tabu search were to allow the process to operate for 20,000
iterations and to use a tabu state of 500 iterations. While we considered
intensification and diversification techniques, such as the use of a frequency
list or strategic oscillation, we decided here to utilize this standard
version of tabu search as a standard basis for comparison. The number
of iterations was chosen to approximate the number of decisions considered
by the threshold accepting algorithm.
The search process within tabu search can be intensified using a 2-opt
neighborhood, where the harvest timing of two different timber stands
are switched. A 2-opt process has been shown as a valuable way to further
refine the quality of a forest plan (Bettinger et al., 1997; Bettinger
et al., 2002). Due to the computational burden of this process,
we used a region-limited approach and only assessed a neighborhood of
100 timber stands at one time. This neighborhood moved 50 stands forward
each time the 2-opt process was employed. Tabu search with 2-opt moves
cannot be used by itself, since no introduction of new opportunities is
possible (only the swapping of opportunities). As a result, this 2-opt
process is employed in conjunction with the standard 1-opt tabu search
process. The tabu state for 2-opt moves, however, is 400 iterations of
the search process, a parameter we determined again through numerous trials.
This combined form of tabu search represents a portion of the third heuristic
We developed a combined heuristic that utilized the strengths of threshold
accepting and tabu search. In this combined heuristic, threshold accepting
is enacted first, using similar parameters as described earlier, in order
to quickly move the development of a forest plan to a very good area of
the solution space. The parameters were adjusted slightly so that when
combined with the tabu search process, the total amount of potential decisions
made approximates those considered when using threshold accepting and
tabu search heuristics in isolation. Within the combined heuristic, after
threshold accepting is employed, a 1-opt tabu process is performed using
the result (best solution) from threshold accepting. A 2-opt tabu search
process is subsequently employed using result (best solution) of the 1-opt
tabu search process.
To compare the results that are generated, each of the three heuristics
generated 30 solutions (forest plans) for each of the nine hypothetical
planning problems, resulting in 27 sets of 30 solutions each. The forest
plans were developed using a Pentium III computer equipped with a 2.7
GHz processor. Each of the forest plans can be considered an independent
sample from a larger population because each began with a randomly-defined
forest plan, thereby inducing the development of statistically independent
samples (Golden and Alt, 1979; Los and Lardinois, 1982). A series of statistics
are then developed to evaluate the average, maximum, minimum, variation
in each set of 30 solutions.
RESULTS AND DISCUSSION
To begin, while each of the three heuristics attempts to locate the optimal
solution to the planning problem, the overall best solution generated
by each can vary by as much as 8 to 12% (Table 1). The
largest percentage variation is found when developing the forest plans
for the younger forest, the least variation is found when developing plans
for the normal forest. The combined heuristic located the best solutions
for the problems involving normal and old forest age classes. In the 2
of 9 instances where it did not locate the best solution (when considering
the younger forest), the best solution generated by the combined heuristic
was within about 0.2% of the best solution generated by the threshold
accepting heuristic. While the combined heuristic consisted of threshold
accepting and tabu search processes, one could argue that these differences
are minor, since randomly generated initial plans were used. Further,
one could argue that the addition of tabu search provided little assistance
in these cases. In general, the combined heuristic and threshold accepting
produced similar solutions, although in a few cases, the combined heuristic
produced forest plans that were $200,000 (US) or greater in net present
value. The time required to generate a solution using the combined heuristic,
however, was also about twice as long as when using the other heuristics,
mainly due to the inclusion of 2-opt tabu search procedure.
The threshold accepting heuristic performed reasonably well on its own
and much better than the 1-opt tabu search heuristic, when applied to
these forest planning problems. Tabu search using only 1-opt moves was
out-performed by both threshold accepting and the combined heuristic.
Again, this is a standard 1-opt tabu search procedure and others (Richards
and Gunn, 2000; Bettinger et al., 1999) have suggested ways in
which the tabu search process can be enhanced to facilitate the development
of higher quality forest plans. However, we used the standard tabu search
process as a point of reference for current and for future work in this
area and serves increasingly as a benchmark against which other heuristics
The worst-case performance for the threshold accepting and combined heuristics
(Table 2) was very similar.
|| Quality of the best solution generated by three heuristics
and associated time required to generate the solution
|| Worst-case performance of three heuristics, when applied
to nine hypothetical forest planning problems
|| Variation in performance of three heuristics, when
applied to nine hypothetical forest planning problems
However, the worst-case
performance of 1-opt tabu search was much lower and variation in forest
plan quality was much higher when using 1-opt tabu search (Table
3). This indicates that the other two heuristics were also significantly
better, on average, at producing high-quality forest plans than when using
1-opt tabu search. However, the variation in forest plan quality, when
using threshold accepting or the combined heuristic was generally higher
when the young age class distribution was considered. And, when the normal
forest was considered, variation among forest plan quality was lowest
when using threshold accepting. In the case of the younger forest, the
difficulty in meeting the wood-flow constraints during the first few time
periods of the plans likely contributed to the difficulty in locating
consistently high-quality solutions.
Another manner in which to evaluate the performance of the heuristics
in generating forest plans is to compare the results to the plan that
was developed using linear programming. One can use the reduction in solution
quality (percentage reduction) as an estimate of the cost of implementing
the adjacency and green-up policy, since these spatial constraints were
not included in the linear programming problem formulation. We found that
the percentage reductions when using threshold accepting or the combined
heuristic were 0.3 to 4.5% below the relaxed linear programming situation.
This is consistent with the results found by others who sought to determine
the cost of spatial restrictions in forestry operations in the southern
US (Boston and Bettinger, 2001).
One might also argue that the improvement in solutions generated by combined
heuristic is simply because it required a much longer length of time to
generate a single forest plan. However, because the parameters for threshold
accepting and tabu search were carefully selected, the comparison among
results seems valid. Roughly speaking, the amount of decisions made by
threshold accepting and the combined heuristic were about the same. Therefore,
even if we increased the number of iterations allowed by threshold accepting
and 1-opt tabu search, we hypothesized that the results would not change
significantly. The reason why the combined heuristic performs better in
most cases is that it uses 2-opt tabu search process to switch the schedule
for two units simultaneously.
The 2-opt tabu search intensification process works well when applied
to small and medium forest planning problems (Bettinger et al.,
2002; Heinonen and Pukkala, 2004) and with this study, shows promise for
larger forest management problems. What remains to be determined is whether
the sequence of scheduling techniques (threshold accepting, 1-opt tabu
search, 2-opt tabu search) needs adjustment to further enhance the performance
of a combined heuristic. Our method for developing the combined heuristic
consisted of acknowledging that threshold accepting could move the search
quickly to local optima of high quality and assuming that tabu search
could be used to refine the search, especially 2-opt tabu search. Further
assessments of the appropriate sequence (or collection) of heuristic processes,
based on the behavior of the search process, seems to be a worthwhile
area of exploration. In addition, while we concentrated here on the assessment
of a combined heuristic that used threshold accepting and tabu search
processes, other types of heuristics could provide processes to help diversify
(avoid becoming stuck in local optima) or intensify the search (carefully
search a good area of the solution space). Threshold accepting was selected
based on previous research (Bettinger et al., 2002). The 1-opt
tabu search process was also selected based on advantageous characteristics
(deterministic moves made, regardless of decrease in solution value).
And the 2-opt tabu search has been suggested as a worth-while addition
to tabu search algorithms even though the processing time requirements
may be high. Other heuristics, such as genetic algorithms, have been included
in combined forest planning heuristics (Boston and Bettinger, 2001). However,
the effect of the crossover routine on the maintenance of adjacency is
generally so high that a number of constraint violations must be attended
to (usually by unscheduled harvests) to maintain feasibility at each iteration
of the search.
The significance of this research to the scientific community is the
finding that when examining realistic forest management behavior in North
America and in attempting to optimize this behavior at the forest-level,
either a fast stochastic heuristic (threshold accepting) or one that uses
an intensive deterministic search process (tabu search with 2-opt moves)
are appropriate. Although this has been suggested in previous research,
we have shown here that this assertion is robust across a broad range
of landowner sizes, spatial configurations and forest age classes. Previous
study was not as complete and examined a smaller range of potential landowner
conditions. While heuristics such as threshold accepting can provide very
good solutions to a variety of forest planning problems and can produce
them relatively quickly, the addition of tabu search procedures (to form
a combined heuristic) may allow one to produce higher quality solutions,
on average, to many types of forest planning problems. The additional
time required (0.5 to 1 h) to generate a solution that has a $50,000-200,000
increase in net present value seems moot, given that the plan will be
implemented over a series of years. However, additional parameterization
(for both tabu search and threshold accepting) and additional programming
logic are the main disadvantages of the approach. This work has shown
that the cost of typical adjacency constraints (2 year green-up and 97
ha maximum clear cut size) is consistent with current knowledge, 1-5%
reduction in NPV from relaxed case. Recent advancements in spatial harvest
scheduling (Bettinger and Zhu, 2006) may be useful in developing even
better solutions. However, these new techniques have generally only been
tested on limited sets of small problems and may provide difficult to
integrate with other search processes.
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