Annals of Operations Research 95 (2000) 159–175
159
Even-aged restrictions with sub-graph adjacency
T.M. Barrett a and J.K. Gilless b
a
School of Forestry, University of Montana, Missoula, MT 59812, USA
E-mail: tara@forestry.umt.edu
b
Department of Environmental Science, Policy, and Management, University of California at Berkeley,
207 Giannini, Berkeley, CA 94720-3310, USA
E-mail: gilless@nature.berkeley.edu
Restrictions on the size and proximity of clearcuts have led to the development of a variety
of exact and heuristic methods to optimize the net present value of timber harvests, subject
to adjacency constraints. Most treat harvest units as pre-defined, and impose adjacency
constraints on any two units sharing a common border. By using graph theory notation to
define sub-graph adjacency constraints, opening size can be considered variable, which may
be more appropriate for landscape-level planning. A small example data set is used in this
paper to demonstrate the difference between the two types of adjacency constraints for both
integer programming and heuristic solution methods.
Keywords: harvest scheduling, integer programming, tactical planning, heuristic methods
1.
Background
Restrictions on the size of even-age timber harvest units have become common
in recent years. Typically, these restrictions set a maximum size limit on a harvest
unit, and specify a “green-up” or “exclusion” time period required before an adjacent
unit can be harvested. In general, these restrictions may be viewed as a compromise
between a public that dislikes clearcutting, and a forest industry that considers even-age
management to be an efficient and effective method of forest regeneration.
In the United States, restrictions on the maximum size of even-age timber harvest
units were established for the National Forests under the National Forest Management
Act of 1976 [16 USCA 1604(g)(3)]. Within the last decade, even-age harvesting on
private forest lands in the states of California, Maine, Oregon, and Washington has
been limited to maximum sizes ranging from 8 to 101 ha [6,17,25,28]. The Canadian
provinces of British Columbia, Nova Scotia, and Ontario, the countries of Finland and
Sweden, and some Australian states also have policies or regulations regarding the
size of even-age harvest units [30].
An extensive literature now exists on the problem of scheduling even-aged harvest
units that are subject to size and adjacency restrictions [12,19,26,29]. In a typical
scheduling problem, harvest units (small areas of forested land that are managed as a
single unit) are assigned silvicultural prescriptions (descriptions of when and how units
will be harvested, replanted, and managed in the future) to meet a specified objective
 J.C. Baltzer AG, Science Publishers
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
(often maximizing net present value) without violating adjacency constraints for any
time period.
Two approaches characterize the adjacency literature: (1) exact solution methods
utilizing integer programming for small problems, and (2) heuristic methods for larger
problems [29]. In most previous work, both exact and heuristic approaches have
utilized “simple adjacency” constraints to rule out scheduling two harvest units within
a specified time period if the units share a common border (examples: [8,9,12,20–
22,24,26]). Simple adjacency constraints are best suited to modeling situations where
the timber harvest units are all near the legal size limit.
In the real world, harvest units are variable in size and shape, having been
delineated using a complex set of criteria including stand (vegetation) type, site quality,
topography, ownership boundaries, stream boundaries, and roads. Legal limits do not
constrain scheduling an individual unit; they constrain the total size of opening created
within a specified time limit – the exclusion period. With a large limit on opening size
and a long exclusion period, some adjacent units may very well be harvested without
violating the legal requirements. A more realistic representation of the scheduling
problem in this case may be obtained by using “sub-graph adjacency” constraints
on the total size of opening created within the exclusion period. Lockwood and
Moore [16] used sub-graph adjacency, assigning penalty costs for violations of the
maximum opening size and violations of the minimum exclusion period.
We expand on Lockwood and Moore’s work [16] by using graph theory notation to present a formulation of the harvest scheduling problem with sub-graph adjacency constraints. A small example problem is used to demonstrate the difference
between the optimal solutions identified using simple adjacency and sub-graph constraints. For the latter approach, several simple heuristic methods find solutions near
the known optimum. The paper concludes with a discussion of the characteristics of
the harvest scheduling problem that affect the relative merits of the different solution
methods.
2.
Methods
2.1. Definitions
Harvest scheduling can be thought of as a silvicultural prescription assignment
problem, where stands (or polygons) in a planning area (a planar graph) are assigned a
prescription (or color) in such a way that there is no adjacency conflict. Graph theory
thus provides a useful medium for problem description. For readers unfamiliar with
graph theory notation, we provide some basic definitions based upon the treatment of
the topic in [7,11,14,18,23].
The topology of a GIS polygon layer can be represented as a graph, where
vertices represent individual polygons and edges represent relations (adjacencies) between polygons. Define a graph G as a finite non-empty set V (the vertex set),
together with a set E (the edge set) of unordered pairs of vertices. For exam-
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
161
Figure 1. GIS polygon layer represented as vertices and edges.
ple, the graph G (figure 1) can be described as G: V = {v1 , v2 , v3 , v4 }, E =
{(v1 , v2 ), (v1 , v3 ), (v2 , v3 ), (v2 , v4 ), (v3 , v4 )}.
Two vertices vi and vj are adjacent if (vi , vj ) ∈ E. The order of G is |V |. A
path is a sequence of edges e1 , . . . , en where vertices could be numbered so that for
each 1 6 i 6 n − 1, ei is (vi , vi+1 ). A graph G is connected if for every pair of
distinct vertices {vi , vj } there is a path between vi and vj . A connected sub-graph of
G refers to Gs such that Vs ⊆ V , Es ⊆ E and Gs is connected; that is, Gs is a graph
generated by the vertices in Vs with the edges that connect vertices in Vs . A tree is a
graph such that there is a unique path between each pair of vertices. A tree Gt is a
spanning tree of graph G if Vt = V .
S
A partition of a set A is S such that S = {A1 , A2 , . . . , An }, where i Ai = A
and Ai ∩ Aj = ∅ for all i 6= j.
The graph G used here to represent the topology of the polygon layer for a
harvest scheduling problem differs from such familiar graph problems in forestry as
the optimal thinning regime problem [10] or project scheduling [5] because the graph
is undirected. In addition, G is a planar graph, which is a graph that can always be
drawn in such a way so that no two edges intersect geometrically except at a vertex
at which they are both incident. This gives a limit on the number of simple adjacency
relationships that must be considered in the harvest scheduling problem, because the
number of edges in a planar graph is bounded by the following:
|E| 6 3|V | − 6 (See [7] or [18] for a proof).
(1)
Following is a problem formulation of NPV optimization for the harvest scheduling
problem with sub-graph adjacency, assuming even-age prescription types and no harvest flow constraints.
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
2.2. Problem specification
Choose Xhj to maximize net present value (NPV), where NPV equals
XX
Chj Aj Xhj
h
(2)
j
with specification that the assignment of a prescription is binary:
Xhj ∈ {0, 1} ∀h, j,
and only one prescription is assigned to each polygon:
X
Xhj = 1 ∀j,
(3)
h
subject to the adjacency constraints
X
Aj 6 L if
j∈Vs
XX
Uht Xhj = |Vs |, ∀t, t = 1, . . . , T , ∀Vs , s = 1, . . . , S,
(4a)
h j∈Vs
or, alternatively,
XX
Uht Xhj < |Vs | if
h j∈Vs
X
Aj > L, ∀t, t = 1, . . . , T , ∀Vs , s = 1, . . . , S,
(4b)
j∈Vs
either of which must operate over all the ways that harvest units could be blocked
together:
∀s, Vs = vs1 , . . . , vsMs | ∃Es ⊆ E to form Gs as a connected sub-graph
where
n
Xhj =
(
Uht =
1 if polygon j is assigned silvicultural prescription h,
0 otherwise.
1 if silvicultural prescription h results in a clearcut in any period from
t − e + 1 to t, where e is the exclusion length in planning periods,
0 otherwise.
Chj is the per land area NPV associated with applying prescription h to polygon j,
Aj is the land area in polygon j,
G = (V , E) are, respectively, the graph, vertex set and edge set of the harvest
scheduling problem,
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
163
L is the limitation (in land-area units) on contiguous clearcut size,
H is the number of silvicultural prescriptions,
J is the number of polygons in the forest map,
T is the number of planning periods,
S is the size of the set of possible connected sub-graphs in G,
Ms is the number of vertices (polygons) in set Vs .
Expression (2) is the net present value summed over the landscape: it is the total
value resulting from assignment of prescriptions to polygons (vertices). The assignment
of prescriptions to polygons is binary: either a polygon gets a particular prescription
or it does not get that prescription. Constraint (4a) ensures that if polygons forming
a connected sub-graph are assigned prescriptions contributing to a forest opening, the
total size of the opening must be less than the allowed limit. As an alternative, this can
be expressed as constraint (4b): polygons forming a connected sub-graph with total
area greater than the allowed limit can not all have prescriptions which contribute to a
forest opening. Any harvest contributes to an opening for the length of the exclusion
period; if the exclusion period is set at 30 years, any harvest within the past 30 years
still contributes to the total opening size.
One method of solving a sub-graph problem would be to examine the set of all
possible assignments, reject those that violate adjacency constraints, and choose the
optimal solution(s) out of the remaining. The first part of this approach, complete
enumeration of possible assignments, is on the order of H J , and would require a
very long time to solve even a simple problem of 100 polygons and 5 silvicultural
prescriptions.
Using an integer programming method usually decreases solution time substantially. For the simple adjacency version of this problem, limiting clearcuts of neighboring polygons, the adjacency restrictions reduce to
XX
Uht Xhj 6 1, ∀t, t = 1, . . . , T , ∀Pζ ,
(5)
h j∈Pζ
Pζ is a pair of adjacent polygons.
We know the maximum number of possible pairs of adjacent polygons in any
planar graph from equation (1), so we know the maximum number of expressions in
(5) for any problem is T (3J − 6). So, for example, a problem with 100 polygons, five
silvicultural prescriptions, and five planning periods would have at most 500 integer
variables and 1470 adjacency constraints. The number of adjacency constraints in
relation to the number of polygons is on the order of |J|, or a first degree polynomial.
Adjacency constraints can be recombined to create fewer equations, but doing so may
not decrease the total time required to solve a problem because the formulation of
adjacency constraints itself requires some computing time.
The number of constraints needed is an important indication of how easy a
problem will be to formulate and to solve. The formulation above ensures constraint
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
equations are possible for every connected sub-graph. Depending on the size of the
polygons, all of these constraints may not be needed. The graph G in figure 1,
considered for a single time period, would need at most 10 constraints: 5 constraints
with 2 vertices, 4 constraints with 3 vertices, and 1 constraint with all 4 vertices. But
if all of the polygons represented by G were of a size such that no two of them could
be combined without violating the size limit, only the 5 pairwise constraints would be
needed.
Using a solution technique such as the branch-and-bound algorithm to solve the
integer programming formulation of the sub-graph adjacency constraint may actually take less time than solving the simple adjacency version. However, writing the
adjacency constraints is of greater difficulty. Regardless of the type of adjacency constraints, most real planning problems are complex enough to prevent the use of exact
solution methods.
2.3. Solution methods
In order to compare simple adjacency with sub-graph adjacency, and to compare
exact and heuristic solution methods for both types of adjacency, we investigate the
following solution methods:
(a) Integer programming (IP) optimization with simple adjacency constraints.
(b) Integer programming optimization with sub-graph adjacency constraints.
(c) A heuristic method that schedules polygons in order of decreasing NPV/ha with
simple adjacency constraints.
(d) A heuristic method that schedules polygons in order of decreasing NPV/ha with
sub-graph adjacency constraints.
(e) A heuristic method that schedules polygons in order of decreasing NPV with simple
adjacency constraints.
(f) A heuristic method that schedules polygons in order of decreasing NPV with subgraph adjacency constraints.
2.4. Heuristic solution method with sub-graph adjacency constraints
The heuristic methods (c), (d), (e) and (f) are biased sample search methods,
similar to the approach taken by O’Hara et al. [24]. Heuristic methods (d) and (f)
differ from O’Hara et al.’s approach by the use of sub-graph adjacency constraints; a
polygon’s eligibility for harvest is determined by tracing the connected sub-graphs of
neighboring polygons that contribute to total opening size. Following is the algorithm
for biased sample search with sub-graph adjacency constraints, prioritized by NPV/ha
(method (d)):
(I) Set Xhj = 0 ∀h, j.
(II) Find h∗ , j ∗ corresponding to the maximum Chj .
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
165
(III) If Ch∗ j ∗ is equal for multiple j ∗ (which would be the case if NPV is homogenous within vegetation strata) then randomly pick a j ∗ from that set (without
replacement). Check that Xhj ∗ = 0 ∀h. For the selected j ∗ :
(A) For every time period t∗ = 1, . . . , T do
(1) For each j 0 such that (vj 0 , vj ∗ ) ∈ E and Uh∗ t∗ Xh∗ j 0 = 1
(a) Find the set Vj 0 which maximizes |Vj 0 | subject to:
(i) ∃Ej 0 ⊆ E which spans Vj 0
(ii) vj 0 ∈ Vj 0
(iii) ∀va ∈ Vj 0 (a 6= j), Uht∗ Xha = 1 for some h
S
(2) Let the set Vj ∗ = {vj ∗ ∪ { j 0 Vj 0 }} = {vj ∗ 1 , . . . , vj ∗ M }
P
(3) If a∈Vj∗ Aa > L then vj ∗ cannot be assigned prescription h∗
(B) If no adjacency restrictions exist in time periods t∗ = 1, . . . , T , then set
Xh∗ j ∗ = 1
(C) Increment to the next j ∗ and loop to (III.A). If all j ∗ have been considered,
go to (IV).
(IV) Find h∗ , j ∗ for the next highest Chj and branch to (III). If all Chj have been
considered, stop.
Step (I) initializes the polygons and step (II) ensures that polygons are assigned
in order of decreasing NPV/ha. For every value of NPV/ha, step (III) randomly picks
a polygon that has that NPV/ha value and then checks that it has not yet been assigned
a prescription. For each time period (step III.A), step III.A.1 finds each first order
neighbor that has a potentially conflicting prescription. (If Uh∗ t∗ Xh∗ j 0 = 1 then the
first order neighbor j 0 has been assigned a prescription that contributes to an opening
in time period t∗ .) Steps (III.A.1.a.i–iii) assemble the connected sub-graph of all the
other polygons that contribute to that opening in the time period t∗ . Step (III.A.2)
adds in the polygon under consideration, and if the area for the entire sub-graph is
greater than the limit (III.A.3) then the polygon under consideration is not eligible. If
no conflicts exist throughout the planning horizon, then the polygon is assigned the
prescription (step III.B).
A variation of this solution method was implemented as an ANSI standard Pascal program. For each value of Chj the program randomly chooses a polygon and
checks its eligibility based on its context (i.e., its spatial relationship to previously
scheduled harvest units). If the polygon is eligible, it is scheduled for harvest. If not
eligible, the next consecutively numbered polygon is checked for eligibility, and so on.
Once a polygon is successfully scheduled, a new polygon is chosen randomly. The
process for each Chj is finished when completely assigned or when all polygons are
ineligible.
Eligibility of a polygon for assignment depends on its spatial context, its topological relationship with other polygons. An “effective clearcut” (EC) is defined as a
connected sub-graph of polygons which all have been assigned a prescription with an
even-age harvest within an exclusion length of the current time period. For example,
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
Figure 2. A polygon map (left) and the graph representation of its topology (right). Shaded polygons
have been scheduled for harvest.
figure 2 shows two EC’s, one formed by polygons #2, #3 and #6, and the other formed
by polygons #7 and #8. To determine whether polygon #1 is eligible for harvest, the
potential effective clearcut size (PECS) is compared to the clearcut size limit. PECS is
defined as the sum of the effective clearcut size (ECS) of all neighboring EC’s plus the
land area of the polygon under consideration. The potential effective clearcut (PEC)
is also a connected sub-graph, and a polygon is eligible for harvest if constraint (4) is
met for that sub-graph.
In the solution algorithm, for each of the polygon’s neighbors scheduled for a
conflicting harvest, the edge set of the graph representation of the EC is reduced to
form a minimum spanning tree. Potential inefficiencies exist in the process of graph
reduction. Within the network of polygons contributing to an EC, there can be many
paths to any single polygon. This can result in “loops” which can significantly increase
solution time, or result in a program that does not end. To avoid these inefficiencies,
the eligibility subroutine (figure 3) finds potential contributors to an EC by visiting
them in an order equivalent to constructing a depth-first spanning tree. The depth-first
implementation is efficient, because each polygon is only examined once, and can be
programmed as a recursive function.
A few further modifications to the program increase efficiency. After a polygon
has been assigned for harvesting, each polygon contributing to its EC is assigned the
ECS land area as an attribute value. Determining whether all polygons contributing to
an EC have been found can then be done by comparing the ECS to the sum of land
area in the tree being constructed, so that an exhaustive search of all non-harvested
neighbors is not necessary. Another modification is to exit the algorithm with a decision
of ineligibility anytime the PECS exceeds the clearcut limit.
Taking out the steps in III.A.1.a of the heuristic is like using simple instead
of sub-graph adjacency constraints; this modification was made to the program to
correspond with solution method (c).
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
167
Figure 3. Allocation algorithm to check for eligibility of a polygon for harvest based on its context.
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
Figure 4. Data set of 30 polygons, with size of polygons shown in hectares.
3.
A case example
3.1. Data set
A data set of 30 polygons comprising 65 ha (figure 4) was used to demonstrate the
different methods. The determination of economic values and the method of polygon
delineation was typical of what would be used for landscape-level modeling, and as a
result was much coarser than would be expected with small area planning. Polygons in
the larger landscape from which this data set had been extracted had been delineated
by ownership, slope, riparian buffers, and vegetation type, with further division of
polygons by a variation on Voronoi tessellation [3] to be less than a size limit of 4 ha.
Edges of polygons appear square because vegetation was classified from remote sensing
data, and the top and right sides of the map are linear because of ownership boundaries.
Very small units, such as those at the top of the map, resulted from variations in
vegetation type; to better demonstrate the differences between the solution methods,
it was assumed that even these very small polygons would be economically feasible
to harvest by themselves. The map contained one instance where polygons touch
at a point rather than an edge; these polygons were not considered to be adjacent.
Harvesting was assumed to be clearcutting, followed by 60 year rotations for the
regeneration. A 4% real discount rate was used to compute NPV. The 30 polygon area
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
169
consisted of red fir vegetation types with varying size and density for which immediate
clearcutting would result in NPV estimates ranging from 10,290 to 20,300 US$/ha.
As a number of polygons share the same NPV/ha, the solution methods (c) and
(d) have a random component depending on the order of assignment. For this reason,
five random runs were used to show the range of total NPV produced by these methods.
The models allowed for harvesting to occur in any of six ten-year periods (i.e.,
there were 6 prescriptions for each polygon). This meant that without adjacency
constraints there would be 630 feasible solutions. The limitation on maximum opening
size (L) was set at 4.05 ha, with a ten year exclusion period, so that the 67 polygon
adjacencies resulted in 402 pairwise adjacency constraints. For the sub-graph version
of the problem, there were many ways of partitioning the map: one partition with
an order of 30, 67 partitions with an order of 29, and so on down to one partition
of order one. Counting up the number of partitions for all orders would have been a
burdensome task, but most of the polygons were not much smaller than the size limit of
4 ha, so that the number of ways that polygons can be combined into blocks that were
still less than 4 ha was relatively small. In fact, there were only 17 such combinations
that could be made: this resulted in 3,455 ways that the map could be partitioned with
all sub-graphs less than 4 ha. In other words, there were 3,455 possible ways we could
pre-block polygons together in the manner described by Jamnick and Walters [13].
To create the necessary adjacency constraints for the sub-graph IP problem, each
of the 17 ways polygons could be combined required a set of constraints for neighboring
harvests that could exceed L. For example, since polygons #1 and #4 (see figure 4)
could be combined, expression (4b) constraints needed would include
X.1 + X.4 + X.2 < 3,
X.1 + X.4 + X.3 < 3,
X.1 + X.4 + X.5 < 3
with each duplicated six times for the six different possible prescriptions. Compared
to the 402 adjacency constraints needed by the IP method with simple adjacency
(method (a)), the IP method with sub-graph adjacency (method (b)) required 636
constraints. Both models required an additional 30 constraints to ensure only one
prescription was assigned for every polygon.
3.2. Results
All of the methods came close to optimizing NPV (table 1): even the worst
method, (c), proposed an assignment capturing 90.1% of the true optimum identified
by method (b). Simple adjacency constraints produced lower NPV values than did subgraph adjacency constraints, but the difference between the constraint types was less
for the optimization methods than it was for the heuristic methods: method (a) NPV
was 98.3% of method (b), while method (c) NPV was 92.0% of the best method (d),
and method (e) NPV was 95.6% of method (f). Almost all of the solution methods
found a 4-color solution, including the worst, method (c).
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
Table 1
NPV and area harvested by solution method.
Method
(a) IP with simple adjacency
(b) IP with sub-graph adjacency
(c) Heuristic ordered by NPV/ha
with simple adjacency
(d) Heuristic ordered by NPV/ha
with sub-graph adjacency
(best of 5 simulations)
(e) Heuristic ordered by NPV
with simple adjacency
(f) Heuristic ordered by NPV
with sub-graph adjacency
Without adjacency constraints
NPV
Percent (%)
US$ (thousands) of method (b)
Area harvested (ha) by entry
1
2
3
4
5
853.0
868.1
782.1
98.3
100.0
90.1
22.12 23.83 13.77 4.91 0.00
24.57 23.83 13.77 2.45 0.00
16.40 16.54 19.08 12.60 0.00
850.5
98.0
21.42 22.03 12.56
8.62
0.00
815.5
93.9
23.08 15.89 12.51
9.11
4.03
853.0
98.3
24.52 19.15 15.89
5.06
0.00
1037.1
119.5
64.62
0.00
0.00
0.00
0.00
Higher NPV values corresponded with more area cut in the 1st time period,
and less land area cut in the 4th or 5th time period (see table 1). Even though
the NPV values and harvest area by time period were similar for all methods, the
assignment of harvest units to entries varied widely in some cases (figure 5). For
example, the IP optimization with simple adjacency (method (a)) and the heuristic
method of prioritizing by NPV with sub-graph adjacency constraints (method (f)) had
nearly identical NPV values, but the units harvested in the 3rd and 4th entries were
completely different. The simple and sub-graph IP optimizations had very similar
harvest schedules, with only 4 of the 30 polygons assigned differently, but the NPV
was $15,000 higher for the latter.
The solution values obtained by the 6 methods ranged from 75.4% to 83.7%
of the harvest schedule without adjacency constraints. If we increased the maximum
opening size to 32 ha for our 30 polygon example, with simple adjacency constraints
the solution would remain the same as for the 4 ha limit, but with sub-graph adjacency
constraints and heuristic method (d) the solution would harvest most of the area in
the first time period (figure 6). This would result in a moderate increase in NPV to a
value of US$980,200. However, the resulting spatial pattern of forest patches would
be very different. After the first time period, the simple adjacency IP solution would
indicate 66% forested with scattered openings of around 2 to 3 ha in size, while subgraph adjacency would indicate 30% forested with two large openings each greater
than 20 ha.
Solution times of the IP problems using Ketron’s MIPIII [15] on a Pentium
130 with 32 MB of RAM were very sensitive to the style in which the problem
was written: an initial formulation of the 30 polygon simple adjacency IP problem
ran overnight without finding a solution; reordering and reformulating the constraints
by using special ordered sets [15] produced an optimal solution in 71 minutes. In
comparison, a computer program that implemented a heuristic algorithm (method (d))
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
171
Figure 5. Harvest schedules (with a 4 ha limit) for IP with simple adjacency (method (a)), IP with
sub-graph adjacency (method (b)), simulation with simple adjacency (method (c)), and simulation with
sub-graph adjacency (method (f)).
typically scheduled a landscape of 8000 polygons in 10 to 60 minutes, even with
multiple prescriptions and a 200 year time horizon [4].
4.
Discussion
The harvest scheduling problem with simple adjacency constraints is closely
related to what is known as the “4-color” problem. It was hypothesized as early as
1852 that any planar map can be colored using no more than 4 colors in such a way that
any pair of adjacent polygons will be different colors; this is equivalent to saying that
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T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
Figure 6. Harvest schedules (with a 32 ha limit) for IP with simple adjacency (method (a)) and simulation
with sub-graph adjacency (method (d)).
with simple adjacency constraints, an area can be completely clearcut in no more than
four harvest entries. Mathematicians were unable to prove the “4-color” hypothesis as
a theorem [1], until the problem was solved by Kenneth Appel and Wolfgang Haken
in 1976, using a proof based on large-scale computer computation [2].
It was known in advance that a 4-color solution existed, as the graph for the
polygon map was planar. This was true for both the simple and sub-graph adjacency
constraint models, as adjoining polygons harvested within the same time period can be
viewed as a single polygon, and the new graph as planar. Although the two IP optimal
solutions were both 4-color solutions, this will not always be the case. Finally, the
polygon map is planar only because we have defined adjacency as sharing a common
edge. If adjacency is defined by distance, as is true in some state regulations, then
the corresponding graph is not necessarily planar and more than four harvest entries
might be needed to schedule all harvest units.
All of the six solution methods presented here did a reasonable job of scheduling
harvest units for this small data set. The integer programming optimization methods,
however, can only be applied to very small problems. Heuristic methods that use subgraph adjacency constraints can be used to find solutions for large data sets and are
amenable to sensitivity analysis. Even for large landscapes, the algorithm presented
requires only a single small modification to change the size limit on clearcut openings
or the exclusion period. Reformulating an integer programming problem to change
total opening size or exclusion period would be more difficult because constraints
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
173
would need to be rewritten. Although simple greedy algorithms can have limitations
in application to natural resource problems (e.g., [27]), in this case they produced
solutions reaching 90 to 98% of the optimum.
Although the example data set had only a few instances where polygons could
be combined without violating the 4 ha limit, using sub-graph adjacency constraints
consistently improved NPV, regardless of solution method. The importance of using
sub-graph rather than simple adjacency constraints depends on the size of potential
harvest units relative to maximum clearcut size. Using sub-graph constraints for a 4 ha
limit increased the integer programming optimal NPV by less than 2%; however, with a
32 ha limit, NPV increased by more than 15%. In addition, using sub-graph constraints
affects the spatial patterning of harvesting, and thus potential wildlife habitat. If
polygons are small compared to the limit, using simple adjacency constraints may
significantly misrepresent the problem. Modellers should examine the size distribution
of the harvest unit polygons before deciding whether simple adjacency constraints are
an adequate representation of the problem. Additional research, using either contrived
or real landscapes, would be helpful in understanding how the two types of adjacency
constraints may affect predicted outputs.
Limits on the size and proximity of even-age harvest units are now common on
public land, and have recently been imposed on private land in some states. Although
these limits address the size of the opening created, most previous research has focused
on constraining the harvest of any two units that share a common border. We have
presented a formulation of the harvest scheduling problem with sub-graph adjacency
constraints on the total size of opening. These constraints are determined by the
topology of the spatial data set. Sets of harvest units which could potentially violate the
maximum opening limitations are connected sub-graphs within the graph representation
of the data set. Graph theory notation thus provides an excellent medium for description
of this problem.
As both private and public forestland owners attempt to address ecological impacts, the spatial pattern of forest structure over the landscape becomes an important
concern. While sub-graph adjacency constraints do not directly capture the spatial
effects of harvests on wildlife populations, they do allow consideration of different
harvesting patterns. The heuristic solution method presented here has been used to
look at the effect of differing size limits, rotation lengths, and exclusion periods on
fragmentation of late-seral forest habitat over a 200 year time horizon [4]. The problem of spatially scheduling harvesting, once concerned only with evaluating economic
outputs, now has a much wider realm of application.
Acknowledgements
We owe many thanks to Larry S. Davis. Two anonymous reviewers also provided
substantial help, as did comments from Reg Barrett and John Radke on an earlier
version of this work.
174
T.M. Barrett, J.K. Gilless / Even-aged restrictions with sub-graph adjacency
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