The economic impact of green-up constraints inthe southeastern United States
Kevin Bostona,*, Pete Bettingerb
aWarnell School of Forest Resources, University of Georgia, Athens, GA 30602, USAbDepartment of Forest Resources, Oregon State University, Corvallis, OR 97331, USA
Received 27 December 1999; accepted 21 March 2000
Abstract
Green-up, or adjacency, requirements are a common constraint in forestry. The American Forest and Paper Association has
developed a Sustainable Forestry Initiative that includes a green-up constraint which limits the average clearcut opening to
48 ha for 3 years or until the average height of the regenerated trees is >1.4 m. In addition to constraining the average clearcut
size, many forestry companies in the southeastern USA voluntarily limit their maximum clearcut size to between 60 and 90 ha.
In this research, a heuristic algorithm was used to develop tactical forest plans that consider both the maximum and average
clearcut sizes. Economic effects of the green-up constraints were estimated for situations where intensive management can
reduce the length of the green-up time from 3 to 2 years on a 21 600 ha ownership in Georgia (USA). For a 60-ha maximum
opening size, this reduction in green-up time from 3 to 2 years resulted in an additional US$ 66 600 in present net worth
(PNW) over a 10-year analysis period. This corresponds to a US$ 10 per harvested ha, or a 0.8% increase in PNW. The bene®t
gained by reducing the length of the green-up period is less with a 90-ha maximum clearcut size, where PNW increases by
US$ 45 600, or US$ 6.70 per harvested ha, a 0.5% increase. While the total volume per period was near the volume goal
produced by a strategic forest plan, the spatial restrictions and the desire to maximize net present value resulted in lower
volume of timber products (sawlogs and chip-and-saw logs) from older forest stands. A sensitivity analysis showed that an
increase in price or yield further reduced the economic incentive for the reduction of the length of the green-up constraint. As
price or volume decreased below expectations, however, the incentive to use intensive forest management practices to reduce
the length of the green-up constraint became more attractive, since the differences between a 2-year and 3-year green-up time
requirement may be large enough to pay for more intensive management practices. # 2001 Elsevier Science B.V. All rights
reserved.
Keywords: Green-up constraints; Economic effects; Forest management
1. Introduction
Increasing competition from international pine-pro-
ducing regions is forcing forest industry organizations
in the southeastern United States to adopt intensive
forestry practices, including extensive site prepara-
tion, weed control, and fertilization. It is well accepted
that these practices can accelerate forest growth rates
(Knowe et al., 1985; Bacon and Zedaker, 1987), but
they also add to the operational costs of growing trees.
Forestry ®rms are under increasing pressure to pro-
duce higher returns for their shareholders and may
only adopt practices that improve their overall pro®t-
ability. Activities are often selected on their ability to
Forest Ecology and Management 145 (2001) 191±202
* Corresponding author.
E-mail addresses: [email protected] (K. Boston),
[email protected] (P. Bettinger).
0378-1127/01/$ ± see front matter # 2001 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 8 - 1 1 2 7 ( 0 0 ) 0 0 4 1 7 - 5
improve pro®ts, not just improve growth rates and
reach sustainability.
In addition to increased competition from interna-
tional producers, most forest products companies in
the southeastern United States have adopted the Amer-
ican Forest and Paper Association (AF&PA) Sustain-
able Forestry Initiative (SFI). The initiative is a
voluntary agreement among AF&PA members to
promote sustainable forestry practices. One compo-
nent of the SFI guidelines is a green-up constraint that
restricts the average clearcut size to <48 ha for either 3
years or until the average height of the regenerated
trees is >1.4 m. Additionally, many companies have
developed an internal policy to voluntarily limit their
maximum clearcut size to 60±90 ha. Estimating the
effects of the combination of these land manage-
ment constraints poses a challenging combinatorial
problem.
The ®nancial impact of various green-up con-
straints has generally been estimated by using rela-
tively long periods of time for adjacent stands to
`green-up'. For example, in Canada, three different
green-up periods (10, 20, and 30 years), three differ-
ent maximum opening sizes (20, 40, and 80 ha), and
three different rotation lengths (80, 100, and 120
years) were used to compare the cost of various
harvesting restrictions (Daust and Nelson, 1993).
The smallest maximum size and the longest green-
up time resulted in a 29% reduction in harvest volume
compared with an optimal solution from a linear
programming effort, which did not contain the spatial
constraints. In Oregon (USA), increasing the green-
up period from 20 to 30 years showed a 34±40%
reduction in the present net worth (PNW) (Yoshimoto
and Brodie, 1994). In California (USA), increasing
the green-up period from 10 to 20 years reduced PNW
by 13% for a 4-ha maximum opening size. As the
minimum patch size increased to 32 ha, there was
smaller difference for increased exclusion period
(Barrett et al., 1998).
A variety of heuristic techniques can be used to
solve problems that consider management options
with green-up size constraints, or spatially con-
strained harvest scheduling problems. Van Duesen
(1999) was able to solve numerically-large spatially
constrained problems by using a simulated annealing
algorithm, but did not compare the quality of his
solutions. Two problems based on a smaller spatial
scale showed no signi®cant difference in the objec-
tive function values for a random starting location
hill-climbing algorithm, Monte-Carlo integer pro-
gramming, simulated annealing, and tabu search
(Murray and Church, 1995). Four problems of larger
spatial scales were used to compare Monte-Carlo
integer programming, simulated annealing, and tabu
search. Simulated annealing found the best solution
to three of the four problems while tabu search found
the best solution to the fourth (Boston and Bettinger,
1999).
Most of the tabu search algorithms used to solve
harvest scheduling problems have a single search
neighborhood. Bettinger et al. (1999) describe a tech-
nique that expands this by allowing two search neigh-
borhoods to change their status simultaneously. This
technique produces signi®cant improvements in the
objective function value, but at the cost of longer
processing times. Tabu search also tends to ®nd solu-
tions that are concentrated in one portion of the
solution space (Glover et al., 1995). The development
of a technique that can combine several local optimal
solutions into a single new solution may yield better
results (Glover et al., 1995).
This paper begins by describing a heuristic tech-
nique that combines tabu search and genetic algo-
rithm techniques. Then, the heuristic technique is
used to develop several tactical forest management
plans and, subsequently, estimate the ®nancial effects
of reducing the green-up time and maximum clearcut
size constraint. The economic effects are estimated
for situations where the green-up time is reduced
from 3 to 2 years, and where the maximum clearcut
size ranges between 60, 70, 80, and 90 ha. The change
in revenue, if positive, could provide additional
®nancial incentive for companies to adopt intensive
forest management practices. The goal is to deter-
mine whether variations in green-up parameters mat-
ter ®nancially in forest planning. Obviously, this can,
and has been, shown to be true for long green-up
times and small maximum clearcut sizes. However,
it has not been shown for shorter differences in
green-up times and larger maximum clearcut sizes.
If there is an economic bene®t for reducing the length
of the green-up time in areas where tree species can
be more intensively managed, this will provide
further incentive for using intensive management
techniques.
192 K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
2. Methods
We ®rst develop the problem formulation for eval-
uating green-up constraint parameters, then present
the heuristic technique used to develop the tactical
forest plans. Then, a database from a southeastern
United States (Georgia) forest industry organiza-
tion, representing �21 600 ha, is used to evaluate
the economic effects of variations in green-up
requirements.
2.1. Model formulation
Two models were used to solve this problem, one
for a strategic 30-year plan, and one for a tactical 10-
year plan. The strategic plan provides the target
harvest goals for the tactical plan, and it can be
considered a `relaxed' problem, since the spatial
constraints for green-up are absent. The results from
this plan provide a theoretical upper-bound on the
solution. The objective function consists of maximiz-
ing the net present value of revenue from harvests less
logging costs:
maximizeXJ
j�1
XN
n�1
XT
t�1
�Revnt ÿ Lcnt�VjntXnt (1)
where J is the number of products, j the product type,
N the number of harvest units, n the harvest unit; T
the number of time periods, t the time period; Revnt
the revenue per cubic meter for unit n harvested in
time period t; Lcnt the logging cost per cubic meter
for unit n harvested in time period t, Vjnt the volume
per hectare of product j in unit n harvested
during time period t, Xnt the continuous variable
indicating whether unit n is harvested during time
period t.
A minimum harvest age of 19 years was assumed
for each timber stand, and individual product volumes
could not change by >5% per time period, except
sawlog volumes, which were set to 10% per period.
1ÿPN
n�1�VjntXnt� ÿPN
n�1�Vjnt�1Xnt�1�PNn�1�VjntXnt�
����������
� deviation j 8j; t � 1; . . . ; t ÿ 1 (2)
The formulation for the tactical planning problem is
similar to the strategic plan, and to the objective
function found in Boston and Bettinger (1999). The
objective function for the problem is:
maximizeXJ
j�1
XN
n�1
XT
t�1
�Revnt ÿ Lcnt�VjntXnt
ÿXJ
j�1
XT
t�1
�Vpjtdujt� ÿXJ
j�1
XT
t�1
�Vpjtdljt� (3)
where Xnt�0,1 is a variable indicating whether unit n
is harvested during time period t; Vpjt the volume
penalty per cubic meter of product j during time period
t, dujt the positive deviation from volume goal of
product j during time period t, and dljt the negative
deviation from volume goal of product j during time
period t. The volume goals we used were those that
resulted from the strategic plan.
Penalties were used to force the heuristic search
process towards solutions which would emulate the
strategic planning solution. A three-part step function
was used to penalize deviations from the volume goal
for each product. For deviations <10% from a volume
goal, the penalty value is 50% of the product price
(Table 1). For deviations between 10 and 25% from a
volume goal, the penalty value is equal to the price.
For deviations >25% from the volume goal, the pen-
alty value is 150% of the price.
Three main constraints were considered. The ®rst is
a volume constraint, which is an accounting constraint
used to sum the volume harvested in each period and
determine the deviations from the goal for each pro-
duct:XN
i�1
�VjntXnt� ÿ dunt � dlnt � volume goal 8j; t (4)
The second constraint is a singularity constraint,
which limits each unit to one treatment during the
planning horizon:XT
t�1
Xnt � 1 8n (5)
Table 1
Prices assumed for forest products
Products Price ($/m3)
Sawlogs 46.24
Chip-and-saw logs 32.12
Pulpwood 12.00
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202 193
The third constraint defines a maximum opening size
for each logging unit n and its set of adjacent neigh-
bors (Un). For a 3-year green-up constraints, we define
a set (Tm) of near-time periods (mz, where m1�tÿ3,
m2�tÿ2, m3�tÿ1, m4�t, m5�t�1, m6�t�2, and
m7�t�3; all mz�0, otherwise not in Tm; all mz�T,
otherwise not in Tm). We define a similar set for a 2-
year green-up constraint (mz, where m1�tÿ2,
m2�tÿ1, m3�t, m4�t�1, m5�t�2; all mz�0, other-
wise not in Tm; all mz�T, otherwise not in Tm). There-
fore, an opening is not just the harvests that occur in
time period t, but also includes those around each unit
n that have occurred during the near-time periods. The
maximum opening size constraint is:XUn
k�1
XTm
m�1
�AkXkm�" #
� �AnXnt�
� Maximum opening size 8n where Xnt � 1; t
(6)
where Un is the set of adjacent neighbors to unit n, k
the adjacent neighbors to unit n, Tm the set of near-
time periods, m a near-time period, Ak the area of unit
k; Xkm is 0,1, a variable indicating whether unit k is
harvested during near-time period m; An the area of
unit n; and Xnt is 0,1, a variable indicating whether unit
n is harvested during time period t.
A maximum average opening size can be de®ned
for each time period. If each opening is centered
around a focal unit (f) during a time period (t), we
can de®ne the size of the opening (Oft) as:
XUn
k�1
XTm
m�1
�AkXkm�" #
� �Af Xft� � Oft (7)
where Un is the set of adjacent neighbors to focal unit
f, k the adjacent neighbors to focal unit f, Tm the set of
near-time periods, m a near-time period, Ak the area of
unit k, Xkm the 0,1 variable indicating whether unit k is
harvested during near-time period m, Af the area of
focal unit f, and Xft is 0,1, a variable indicating whether
focal unit f is harvested during time period t.
As we are attempting to calculate the average open-
ing size, we do not wish to count openings more than
once. This mis-counting could occur if we allow each
unit (n) in an opening (composed of multiple units n)
to be considered the `center'. Therefore, only one unit
(n) can be delineated as the focal center (f) of the
opening in any time period, and the total number of
openings equals the number of focal centers of open-
ings. Thus,Xf �
Xn and
XXft �
XXnt (8)
The average opening size for a set of openings in a
time period (t) can then be constrained with the
following equation:PFf�1Oft
h iF
� average opening size 8t (9)
where F is the total number of openings, f the focal
center of an opening, or an opening itself, and Oft an
opening centered around a focal unit f during time
period t.
Unlike our strategic plan, the tactical forest plans
include the spatial constraints noted in equations (6)±
(9). For Eq. (6), we examined maximum opening size
constraints of 60, 70, 80, and 90 ha. For Eq. (9), we
used the AF&PA SFI maximum average clearcut size
of 48 ha. The strategic plan was developed using linear
programming techniques, and the tactical forest plans
were developed using a heuristic technique.
2.2. Sensitivity analysis
Two types of sensitivity analysis were completed
when developing the tactical forest plans, one for price
and one for volume. Prices for sawlogs and chip-and-
saw logs were increased by 5%, then decreased by 5%.
A similar approach was used to test the impact of
volume sensitivity where a uniformly distributed ran-
dom number generator was used to assign an adjust-
ment factor for each unit to ®rst increase, then
decrease the volume harvested in a unit between 0
and 5%. In each of these analyses, we examined
maximum opening size constraints of 60, 70, 80,
and 90 ha, and used the AF&PA SFI maximum aver-
age clearcut size of 48 ha.
2.3. Heuristic technique
The heuristic technique is a hybrid algorithm con-
sisting of ®ve components (Fig. 1). The ®rst is a
Monte-Carlo integer programming algorithm that
randomly develops an initial solution. This process
selects a logging unit, determines if it meets the
194 K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
minimum harvest age, and assures that its incorpora-
tion into the solution does not violate the green-up
constraints. This continues until 10% of the sawlog
volume has been achieved in all periods. Because
each new run of the heuristic technique uses a new
seed for a random number generator, this component
will allow an increase in the proportion of the
solution space explored when the program is executed
repeatedly.
The second component is the core tabu search
routine, similar to the algorithms described in Murray
and Church (1995), Bettinger et al. (1997), and Boston
and Bettinger (1999), and is composed of two ele-
ments: (1) a tabu list that maintains a record of the
recent moves; and (2) the aspiration criteria. After
experimentation, 100 iterations were selected as the
tabu list length. For this application, the aspiration
criterion was assumed to be the overall best objective
function value. The best move from the neighborhood
of moves will be considered ®rst, whether or not it
improves the current solution. If the move is not tabu,
it will be accepted into the solution. If the move is
tabu, yet exceeds the aspiration criteria, it too will be
accepted into the solution. If a move is tabu, yet does
not exceed the aspiration criteria, it is not accepted.
The third component is the intensi®cation routine.
The objective of an intensi®cation routine is to explore
a portion of the neighborhood that has already yielded
a good solution to search for better solutions. This
intensi®cation routine begins by recalling the current
best solution from the core tabu search routine. By
using a 2-opt neighborhood search routine described
in Bettinger et al. (1999), two units can simultaneously
change their status. the intensi®cation routine has the
same short-term memory features as the core tabu
search routine, but the tabu list has been reduced from
100 to 20 iterations.
Tabu search ®nds good solutions to large combina-
torial problems, but they tend to be concentrated in a
small portion of the solution space (Glover et al., 1995).
Thus, the fourth and ®fth components of the heuristic
technique have the goal of forcing the search to new
parts of the solution space. The fourth component
(Fig. 1) is a diversi®cation routine that schedules those
units with the lowest frequency of entering the solution,
while maintaining the minimum harvest age and not
violating the green-up constraints. This diversi®cation
will force the algorithm to the least explored portion of
the solution space. The resulting solution becomes the
starting point for the core tabu search routine.
The ®fth component has the goal of combining two
neighborhoods where good solutions were found with
the hope of ®nding a superior solution. It is based on a
crossover routine used in genetic algorithms. The
genetic crossover routine treats the solution to the
forest planning problems as if they were chromo-
somes, with each unit being a gene on a chromosome.
Fig. 1. Flow chart of the heuristic technique used to develop
tactical forest plans.
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202 195
The values for the genes, the alleles, become the
periods when the unit is harvested. By using a random
number generator that selects the crossover point, the
two `chromosomes' are recombined into two new
solutions. The solution with the highest objective value
survives the crossover and becomes the starting solu-
tion for a continuation of the core tabu search routine.
This heuristic technique has been shown to produce
good results to similar scheduling problems which
have 3000±5000 0±1 integer variables (companion
paper regarding validation of the heuristic technique
is in preparation), although the heuristic technique
described here uses 2-opt moves. Extending the use of
this solution method to larger scheduling problems
therefore seems reasonable, although the performance
of the solution method cannot be directly compared
against known optimal solutions.
2.4. Data description
The data set, from a forest products ®rm in Georgia
(USA), represents a typical industrial ownership in the
southeastern United States. The data set contains
mostly pine plantations, as well as a mixture of hard-
wood areas, which for this analysis are not managed
for timber production. It contains �1300 logging
units, resulting in �10 000 0±1 decision variables.
Yields were estimated with equations developed by
Harrison and Borders (1996) for three types of wood
products: sawlogs, chip-and-saw logs, and pulpwood.
The product prices that are assumed are shown in
Table 1. We assumed a logging cost of US$ 7.41 per
m3, and an 8% real discount rate. All penalty values
were discounted by using an 8% real discount rate.
3. Results
Production possibility curves for the 3- and 2-year
green-up time when used in conjunction with the
various maximum clearcut sizes are illustrated in
Fig. 2. When applied to the 21 600 ha forest from
the southeastern United States, the heuristic technique
shows that reducing the green-up time from 3 to 2
years had the largest improvement in the objective
function value when used in conjunction with the
smallest maximum opening size, or an improvement
of about US$ 66 600 compared to US$ 45 600 for the
largest opening size. These improvements range from
approximately US$ 10 per harvested ha for the 60-ha
maximum clearcut size to US$ 6.70 per harvested ha
for the 90-ha maximum clearcut size. Again, in all
these analyses, the average opening size was con-
strained to the SFI guidelines of 48 ha.
As expected, the larger maximum opening size
allows the harvested volume to be closer to volume
Fig. 2. Production possibilities over a range of maximum clearcut sizes.
196 K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
goals generated from our 30-year strategic plan (the
relaxed LP problem formulation). Each maximum
opening size (60, 70, 80, 90 ha) produces similar
results for timber volume in the ®rst years of each
10-year tactical plan, but the larger openings allow for
more volume to be harvested in the later periods.
Figs. 3 and 4 show that the chip-and-saw volume is
>90% of the target volumes in the ®rst few years of the
Fig. 3. Chip-and-saw volume produced in the first 10 years with a 3-year green-up time requirement.
Fig. 4. Chip-and-saw volume produced in the first 10 years with a 2-year green-up time requirement.
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202 197
strategic plan. The deviations from the sawlog volume
targets are larger (Figs. 5 and 6), and often the tactical
plan is able to only schedule about 75% of the target
volume. The deviations are also consistently larger for
the 3-year green-up constraint than for the 2-year
constraint. The opposite pattern is observed for pulp-
wood volume, where the tactical plan meets or
exceeds the goals of the strategic plan in all time
Fig. 5. Sawlog volume produced in the first 10 years with a 3-year green-up time requirement.
Fig. 6. Sawlog volume produced in the first 10 years with a 2-year green-up time requirement.
198 K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
periods for both the 3- and 2-year green-up constraints
(Figs. 7 and 8).
The results from the sensitivity analysis showed that
as prices or volumes are decreased the impact of the
green-up constraint becomes more signi®cant (Figs. 9
and 10). As prices are reduced, the magnitude of the
difference between the 3- and 2-year green-up con-
straint increases to US$ 106 per harvested ha for a
Fig. 7. Pulpwood volume produced in the first 10 years with a 3-year green-up time requirement.
Fig. 8. Pulpwood volume produced in the first 10 years with a 2-year green-up time requirement.
K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202 199
60-ha maximum clearcut size and US$ 92 per har-
vested ha for 90 ha maximum clearcut size. A similar
pattern is shown for a reduction in volume yield with
US$ 157 per harvested ha for 60-ha maximum clearcut
size. It is slightly reduced to US$ 153 per harvested ha
for a 90 ha maximum clearcut size. When volume or
price was increased there was no signi®cant difference
between the 3- and 2-year green-up constraints.
4. Discussion and conclusions
The goal of this analysis was to determine whether
variations in green-up parameters matter ®nancially in
forest planning efforts. Clearly, for relatively short-
rotation pine plantations in the southeastern United
States, shortening the green-up time assumption from
3 to 2 years does make a small difference in the PNW
Fig. 9. Production possibilities when prices are modified.
Fig. 10. Production possibilities when volumes are modified.
200 K. Boston, P. Bettinger / Forest Ecology and Management 145 (2001) 191±202
of the resulting solutions. While other researchers
(Daust and Nelson, 1993; Yoshimoto and Brodie,
1994; Barrett et al., 1998) have also shown that the
length of the green-up constraint is ®nancially impor-
tant, they do so at much longer temporal scales than
are required by the AF&PA SFI. To reduce the length
of the AF&PA green-up constraint will require the
adoption of intensive forest management techniques
(chemical site preparation, herbaceous weed control,
etc.). From this data set, the increase in PNW repre-
sents only 5±10% of the cost of the average herbicide
treatment in the southern United States (Dubois et al.,
1997). The use of intensive forest management
techniques cannot be justi®ed solely based on the
economic gain that results from a reduction in the
green-up constraint. However, many ®rms in the SE
region are currently using some intensive management
techniques and may require only a marginal increase
in expense to achieve the growth rate that would allow
a reduction of the green-up constraint from 3 to 2
years. Others may have to adopt a full suite of
intensive practices. We recommend that both forest
and stand level analysis should be completed when
evaluating the use of intensive forest management
practices.
We have also shown, at least for one forested area
and ownership in the southeastern United States, that
the maximum opening size assumed does matter.
Variations in the distribution of forest conditions
and the spatial arrangement of the age classes of other
forests may result in different conclusions, but larger
maximum opening sizes should allow more ¯exibility
in the spatial arrangement and timing of harvests. If
forest management organizations are in a position to
develop ¯exible forest policies, such as the maximum
allowable clearcut size, they should examine the costs
and bene®ts of various green-up times as well as
varying the maximum size of forest openings.
The impact of the green-up constraint on the volume
harvest was twofold. First, the products that are
derived from longer rotations, such as sawlogs and
chip-and-saw logs, did not meet the target goals that
we developed in our 30-year strategic plan. Second, a
surplus of pulpwood volume would be produced.
Because the area of older forest stands is limited at
the beginning of the planning horizon, the adjacency
constraint prevents harvesting them at the time indi-
cated by the strategic plan. Thus, stands younger than
desirable will be harvested in order to meet the overall
volume goals of the forest plan, yet they will not be old
enough to produce the desired mix of sawlog or chip-
and-saw log volume.
Sensitivity analysis shows that an increase in price
or yield will further reduce the economic incentive for
using intensive management to reduce green-up time
requirements. However, as prices decrease or the
yields do not meet their expected value, the incentive
to use intensive forest management practices to reduce
the greenup constraint can be justi®ed, as the PNW
differences for this analysis come closer to the cost of
herbaceous weed control.
Organizations that attempt to match forest harvest
scheduling to the needs of wood processing facilities
may need to incorporate the required green-up con-
straints into their planning process if they are to
accurately anticipate the amount and types of products
that will be available. With regard to the heuristic
technique we used to generate solutions to these forest
planning problems, we feel that the hybrid tabu
search-genetic algorithm method can produce better
results than a traditional tabu search technique,
because it allows for both an intensi®cation and a
diversi®cation in the search process, and because it
uses a genetic algorithm to combine local optimal
solutions. We recommend that others developing tabu
search applications should consider combining multi-
ple search strategies into heuristic harvest scheduling
models and move beyond the basic approaches heur-
istics offer.
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