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8/11/2019 Risksensitive planning support for forest enterprises The YAFO model.pdf
1/13
Risk-sensitive planning support for forest enterprises: The YAFO model
Fabian Hrtl , Andreas Hahn, Thomas Knoke
Institute of Forest Management, Center of Life and Food Sciences Weihenstephan, Technische Universitt Mnchen (TUM), Hans-Carl-von-Carlowitz-Platz 2, 85354 Freising, Germany
a r t i c l e i n f o
Article history:
Received 4 October 2012
Received in revised form 11 March 2013
Accepted 14 March 2013
Keywords:
Economic optimization
Risk integration
Operational planning
Forest management planning
Nonlinear programming
Long-term objectives
a b s t r a c t
YAFO is a planning-support tool for the development of management plans under uncertainty focusing on
the forest enterprise level. Based on existing stand data, the software provides the calculation of manage-
ment scenarios (felling plans) for single stands that are optimized with respect to financial considerationsand ecologica l constraints. Under these constraints, YAFO predicts timber stocks, harvest amounts and
financial returns for each simulation period. The YAFO package consists of an optimization module, that
has been programmed using the modell ing software AIMMS. In addition, it contains two Excel-based
spreadsheet files an import and evaluation module and a risk analysis module. The YAFO model calcu-
lates financially optimized management scenarios by means of the net present value development of sin-
gle stands. Optionally, the objective function can also consider risk s and uncertainties due to natural
calamities and timber price fluctuations, using the value at risk approach or risk utility functions. Non-
linear programming algorithms are used as solution techniques. As YAFO provides the additional flexibil-
ity to switch between two timber grading options on stand level, effects of timber price scenarios on
grading can be analyzed. Due to its modular design, it can be easily adopted to individual data bases.
2013 Elsevier B.V. All rights reserved.
1. Introduction
If one is to approach the problem of managing a forest enter-
prise in a sustainable1 way, it is necessary to tackle the question
of when to harvest which timber volume from which stand (forest
area with the same treatment). Due to the long production periods
in forests, especially in Central Europe, this decision is crucial in or-
der to avoid negative consequences that can potentially last for dec-
ades. It is then no surprise that there is a long tradition of planning
techniques in forestry to address this problem. Georg Ludwig Hartig
(Hartig, 1795) and Heinrich Cotta (Cotta, 1804) are generally consid-
ered to be the first forest scientists to have developed such applica-
ble solution techniques as regulation by forest area and harvested
volume respectively. These techniques are commonly known as
control techniques in forestry literature (Davis et al., 2001; Bettin-
ger, 2009). In the English forestry literature, there has been a contin-uous enhancement of these initial forest planning techniques,
culminating in the integration of methods from decision theory,
operations research and finance theory into forest enterprise man-
agement (Davis et al., 2001; Buongiorno and Gilless, 2003; Rauscher,
2005; Reynolds et al., 2008).Thus, planning/decision support systems (DSSs) correspond to
specific eras of forest management, starting with sustained yield,
and finally emerging in sustainable forest management (SFM)
(Mathey et al., 2005; Hahn and Knoke, 2010). Mendoza (2005) dif-
ferentiates two approaches for decision support in forestry one
prescriptive, algorithmic and highly structured, and the other
descriptive, soft and qualitative. He states, the latter has become
more popular and more widely applied, in part because of its affin-
ity to the participatory management approach (Mendoza, 2005, p.
252). Participatory decision making, and ecologically and socially
sound decisions are primarily related to intragenerational fairness
a cornerstone of SFM. Intergenerational fairness, however the
second cornerstone of the World Commission on Environment
and Developments (WCED, 1987) definition of a sustainable devel-opment, and the originally relevant criteria for sustainable forestry
(Hahn and Knoke, 2010) is less frequently addressed. Timber har-
vests are thus a matter of allocation where cuttings have to be car-
ried out in an efficient way with regard to future harvesting options.
Hence, our interest focuses on the producers perspective, as weas-
sume sustainable forest management to be best promoted if land-
owners personally benefit. At these scales, research activities in
recent years have developed spatially explicit and geographically
sensitive systems (Varma et al., 2000; Reynolds et al., 2008). A
second point of action led to an increased emphasis on the adapta-
tion options due to serious changes in the decision environment
(Eriksson, 2006; Heinimann, 2010; Mermet and Farcy, 2011).
0168-1699/$ - see front matter 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.compag.2013.03.004
Corresponding author. Tel.: +49 8161 71 4619; fax: +49 8161 71 4545.
E-mail address: haertl@forst.wzw.tum.de (F. Hrtl).
URL: http://www.waldinventur.wzw.tum.de(F. Hrtl).1 Following Speidel (1984) sustain able here means the abilit y of a forest
enterprise to provide timber, infrastructure and additional goods and services for
the benefit of present and future generations in a continuous and optimal manner
(Knoke et al., 2012). For definition problems concerning this freque ntly used term
refer for example to Hahn and Knok e (2010).
Computers and Electronics in Agriculture 94 (2013) 5870
Contents lists available at SciVerse ScienceDirect
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In general, the classic forest planning problem of allocating
areas to space and time can be considered as maximising an objec-
tive function. If one considers the single stands of a forest enter-
prise as options of a finance portfolio, one can build an objective
function that calculates the net present value (NPV) of all manage-
ment activities during the planning horizon. By means of linear
programming (LP) methods, it is possible to solve this optimization
problem by computer-assisted numerical algorithms (Felbermeieret al., 2007), which covers the highly structured and algorithmic
approach Mendoza, 2005 figured out.
LP is by far the most used method in forest planning as well as
in general land-use optimization (Bettinger and Chung, 2004;
Weintraub and Romero, 2006). Current research focuses on exten-
sions to LP like mixed integer (Fonseca et al., 2012) and goal (mul-
ti-objective) programming (Diaz-Balteiro and Romero, 2008; Rivaz
and Yaghoobi, 2012), on non-linear approaches (Hof and Kent,
1990; Roise, 1990), on heuristic (stochastic) approaches like simu-
lated annealing (Georgiou and Papamichail, 2008), tabu search and
genetic algorithms (Mosquera et al., 2011; Janov, 2012; Pukkala
and Kellomki, 2012), and on dynamic programming (Benjamin
et al., 2009), as well as on combining these techniques with spa-
tially explicit models (Seppelt and Voinov, 2002; Baskent and
Keles, 2005; Gustafson et al., 2006; Mathey et al., 2008; Wei and
Murray, 2012) and risk considerations (Martell et al., 1998; Kangas
and Kangas, 2004; Knoke et al., 2005; Mathey and Nelson, 2010;
Verderame et al., 2010).
Several papers (e. g. Gong, 1998; Knoke et al., 2001; Knoke and
Moog, 2005; Alvarez and Koskela, 2006; Beinhofer, 2009; Roessiger
et al., 2011) have shown that forest management decisions are se-
verely influenced by risks. Such uncertainties can be caused by tim-
ber price fluctuations (Brazee and Mendelsohn, 1988; Haight, 1990)
as well as calamities (Meilby et al., 2001; Kouba, 2002; Hahn and
Knoke, 2010; Forsell et al., 2011; Hanewinkel et al., 2011; Griess
et al., 2012). So systems need to incorporate these deviations in or-
der to set up sustainable solutions. The evaluation of these uncer-
tainties can be accomplished for example through Monte Carlo
simulation techniques (Styblo Beder, 1995; Dieter, 2001). If theaim is to integrate such risk effects into a model, a potential solu-
tion is to describe the resulting fluctuations of financial returns by
statistical values, such as mean and standard deviation (mean-var-
iance analysis). This statistical approach was used by Markowitz
(1952, 1959) in his portfolio theory (Mills and Hoover, 1982; Hilde-
brandt and Knoke, 2011). This approach assumes a Gaussian distri-
bution of the fluctuating net revenues, but it has been shown to be
robust to deviations from normality (e. g. Glawischnig and Seidl,
2011). Other approaches for overcoming this limitation are repre-
sented by models based on stochastic dominance, downside risk,
and information gap theory (Knoke et al., 2008). Due to their math-
ematical structure, models considering risk effects in general must
be treated as nonlinear in finding a solution (Pukkala and Kangas,
1996; Knoke and Moog, 2005; Hildebrandt and Knoke, 2009;Knoke et al., 2012). Nevertheless there are approaches to the inclu-
sion of uncertainties in linear programming techniques using ma-
trix models (Eriksson, 2006) or in stochastic integer programming
(Alonso-Ayuso et al., 2011), or different measurements of risk, such
as absolute deviations (Konno and Yamazaki, 1991).
Literature is, however, largely missing approaches which allow
an easy and quick parameterisation of this risk-sensitive planning
problem with abdication of assumptions concerning linearity
(Yousefpour et al., 2012). Many research projects about DSS are
based on forest growth simulators to which are added capabilities
to optimize the planning with regard to biophysical objectives. A
few examples are LMS/FVS (McCarter et al., 1998; Crookston and
Dixon, 2005), SAGALP (Chen and Gadow, 2002), HEUREKA (Lmas
and Eriksson, 2003), SADfLOR (Borges et al., 2003), DSD (Lexeret al., 2005), MOTTI (Salminen et al., 2005), NED-2 (Twery et al.,
2005), FTM (Andersson et al., 2005), HARVEST (Gustafson and Ras-
mussen, 2002), 4S TOOL (Kirilenko et al., 2007), AFFOREST (Gil-
liams et al., 2005), EMDS (Reynolds, 2006), ESC (Pyatt et al.,
2001), FSOS (Liu et al., 2000), and FORESTAR (Shao et al., 2005).
Other solutions like SIMO (Rasinmki et al., 2009) or Woodstock
(Remsoft Inc., 2012) try to go beyond biophysical objectives but
are acting more as a model development tool than a model itself.
Furthermore, modelling approaches integrating risks, like the FOR-EST OPTIMIZER project (Stang and Knoke, 2009) are scarce, and
also retain a linearisation of risks. For an overview of different ap-
proaches see Bjrndal et al., 2012
We therefore see the need for a further development of an algo-
rithmic approach to address the question of optimal risk-sensitive
management on the forest enterprise level over time using nonlin-
ear programming (NLP). Thus, the model presented here is aimed
at making a planning and decision tool available to forest scien-
tists, as well as practitioners, that can be used to solve a multitude
of problems without requiring any major adaptions.
The model considers not only the risk effects mentioned above,
but also the effects of different timber price scenarios. Climate
change mitigation policies as well as a fear of increasing scarcity
of fossil fuels provokes a growing demand for producing energy
from biomass. Due to that increase, the prices of fuel wood are ris-
ing, so that the competition between the material and thermal use
of wood is becoming more and more severe (Raunikar et al., 2010).
To analyse the effects of these competing lines of timber use, we
expand the model to include an option to decide simultaneously be-
tween two timber grading options during the allocation of stand
areas.2 The combination of risk analysis with Monte Carlo simula-
tions, grading options and NLP techniques is a new way to handle
the planning problem at the enterprise level. For that purpose, the
model generates probability distributions of the objective function
out of the original data, using timber price statistics and survival
functions for tree species. Finally, this combination of risk analysis
on enterprise level with Monte Carlo techniques and NLP is unique
so far and not available in the packages mentioned above.
In all of the model approaches mentioned here, it is possible toanalyse effects of constraint settings which simulate demands for
maintaining or providing ecological or social functions of forests.
Comparing constrained solutions for the objective with uncon-
strained ones gives us the opportunity to evaluate the costs of such
ecosystem services (Duraiappah, 2005), for example, how much
money a forest owner requires in exchange for providing such
functions. In this way we solve the problem of non-existent mar-
kets and prices for such ecosystem functions, at least from a pro-
viders perspective (Knoke et al., 2008).
2. Method
2.1. Basic model
YAFO3 is a modular nonlinear optimization model for forest
enterprises. It is based on a forest property that is spatially divided
into forest stands. Every stand i is an independent management unit
that is characterized, from a financial point of view, through the
development over time of its net revenues. The stands cannot split
up or merge within the model. The model consists of seven time
periods numbered from 0 to 6 the last of which is a recovery per-
iod that collects all remaining stand areas at the end of the investi-
gated time horizon. No thinning or felling is carried out in the last
period, instead, the remaining area of the stands not felled during
the simulated time horizon is stored which is then used as a factor
2
This is optional. The model can also be used for one-scenario optimizations.3 Yet Another Forest Optimi zer.
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in calculating the net present value of the stands. The remaining six
periods span a time horizon of 3060 years, as typical time steps in
forest growth models are 5 or 10 years. At every point in time t a
decision must be made to either thin or finally fell parts of the stand
area. To thin means that the stand remains, at least until the next
time period, and that only single trees will be removed for stand
improvement. The cutting intensity of these thinnings is normally
defined by silvicultural concepts, and is not decided by the model.Thinnings produce intermediate returns during the rotation period.
To fell means to cut the entire stand, or parts of it, at the end of
the rotation period and establish a new stand generation. Regenera-
tion costs must then be paid. These costs are determined based on
the dominant tree species as well as the stand age. The older a stand,
the lower the regeneration costs, to simulate possibilities for natural
regeneration. This cost reduction is done by a regeneration cost
moderation function that follows a Weibull function, and can be ad-
justed to various local situations by its parameters. For planned as
well as salvage fellings a new stand generation is simulated with
its ingrowth volumes for the following periods. These ingrowth vol-
umes are included in the calculation of thinning and felling volumes
only from period five onwards as it is assumed that there will be no
utilizable ingrowth volumes in stands younger than 25 years.
Furthermore, the model decides in each case which grading op-
tion is to be applied to that particular harvest. The model is free to
choose between these options. Additionally, in every period, a cer-
tain partial area,fzits , of each stand must be cut (salvage felling). This
mechanism simulates expected tree drop-outs caused by wind,
snow or insects, and is based on a hazard rate that is calculated
as a function of the leading tree species, the mixture conditions,
and the stand age (see Section 2.2.2). At every point in time, t,
the model decides for every stand, i, in addition to the determined
salvage returns,zits, whether it is more profitable to realise the re-
turns of an (intermediate) thinning, dits, or those of a (final) felling,
aits, as a portfolio option. The index, s, shows that the model must
also choose between two grading options for every individual
stand in every period, which have different revenues and costs.
The optimizer can realize the thinning data (volumes, revenuesand costs) by assigning the stand areas or parts of it to be thinned
to the variables fdits. The remainder of the stand area is then as-
signed to the variablesfaits that realizes the final felling of the resid-
ual stand (volumes, revenues and costs).
The sum of the net present values (NPV) of all these net reve-
nues is the objective function that is to be optimized by the area
control method. The objective function has therefore the following
form:
maxf
ZXi
Xt
Xs
ditsfdits aitsf
aits zitsf
zits
1 r
t 1
with the constraints,
Xs
fdit0s
Xt0
t0
Xs
faits fzits
fi 8i; t
0 2a
Xs
fzits fzit 8i; t 2b
fd;a;zits P 0 8i; t; s 2c
The meaning of the symbols is as follows: rinterest rate, ttime, i
stand, s grading option,fi area of stand i, dits revenues per area from
thinning (net-of harvesting costs) in stand i at time tusing grading
option s,aits revenues per area from felling (net-of harvesting costs),zits revenues per area from salvage felling (net-of harvesting costs),
fdits thinning area, faits felling area, f
zits area of salvage felling. Con-
straint (2a) assures that for every point in time, t0
, the sum of the
area felled to date plus the current area to be thinned is equal tothe stand area. This means that every area not yet felled is thinned
automatically. Constraint (2b) ensures that the salvage felling area
in each period cannot be used as a thinning or final felling option.
Constraint (2c) prohibits solutions with negative areas.
This area allocation problem itself is modelled as an area control
scheme that allows stand areas to be shifted in space and time
using the modelling software AIMMS (Paragon Decision Technol-
ogy B.V., 2011). For every timber grading option there exists a sep-
arate scheme. The combination of both schemes is accomplishedusing the constraints according to Eqs. (2a) and (2b). This approach
has the advantage that model and data are strictly separated, so
that it is quite simple to use data sets that do not rely upon the data
preparation and evaluation module YAFO-EX. AIMMS symbolizes
the model in a tree structure. All model components are placed
in this tree as single objects. The objective functions as well as
the optimization problems are placed as objects in the model.
The former are categorized as variables in the AIMMS language,
the latter as mathematical programs. The connections between
these objects are implemented through object declarations.
The area for each stand in all periods is defined by seven con-
straints. This set of constraints represents the side condition
according to Eq. (2a) in the model as follows. For each period t
there is the following constraint:Xs
fdits faits
fRit 8i; t 3
with the recursive defined remaining area
fRitn :fRitn1
Xs
faitn1sfzits
4
After expanding the recursion the right side of Eq. (4) can be
combined in a different way:
fRitn fRit0
Xn1x0
Xs
faitxsXnx1
Xs
fzitxs
fifzit0
Xn1x0
Xs
faitxsXnx1
Xs
fzitxs
fiXn1x0
Xs
faitxsXnx0
Xs
fzitxs 5
Relinquishing the counting index x for the different points in
time tin Eq. (5) leads to the simplified formulation
fRit0 fiXt01t0
Xs
faits Xt0
t0
Xs
fzits 6
Substituting Eq. (3) into Eq. (6) gives
Xs
fdit0sfait0s
fi
Xt01t0
Xs
faits Xt0
t0
Xs
fzits
and finally after rearrangement the structure of Eq. (2a):
Xs
fdit0s
fiXt0
t0
Xs
faits Xt0
t0
Xs
fzits fiXt0
t0
Xs
faits fzits
7
So Eq. (3) with (4) and Eq. (2a) are identical.
Six additional constraints represent the salvage felling area con-
trol of Eq. (2b) for each period 05. The non-negativity constraint
(2c) is incorporated directly into the variable declarations. Addi-
tionally the following five biophysical constraints can be defined
at the enterprise level:
Lower limit of standing volume in (m3/ha)
Upper limit of standing volume in (m3/ha)
Maximum final felling volume in (m3/ha/period)
Maximum final felling area in (ha/period) Maximum total felling volume in (m3/ha/period)
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Again, six constraints are defined for each of these enterprise le-
vel parameters, separately for each period 05, to force the vari-
ables to the interval between the constraints. Period 6, as the
recovery basket, is affected only by the first two constraints. For
this purpose the model updates and saves the biophysical develop-
ment of the stands. With the exception of the enterprise-level con-
straints, the biophysical data do not affect the optimization
process. The main function of the biophysical data is to give theuser additional facts for management planning and to check the re-
sults. Carrying biophysical data as well as financial data through
the model system enables the calculation of a timber amount that
follows the optimized planning solution for the forest enterprise.
The actual realised thinning volumes are calculated by multiplying
the growth model-based thinning volumes by the thinning areas.
Similarly, the felling volumes are calculated by multiplying the
simulated stand volume by the felling area. The volume of the
stand after thinning is computed by multiplying the simulated
stand volume by the difference between the total stand area and
area already used.
The spreadsheet calculation YAFO-EX prepares the data so that it
can be imported bythe optimizer model YAFO-A (Fig. 1). Section 3.1
describes the data YAFO-EX requires as an input. The stands are
assigned to one of four categories representing hardwood and soft-
woods in pure and mixed stands. Using survival functions, age
dependent drop-out partial areas due to calamities are calculated
for every stand in each period. The timber volume is classified for
further analysis by the main tree groups spruce, pine, beech and
oak, and by the main classes saw log, industrial wood, fuel wood
(from compact wood) and brushwood. The brushwood amounts
are reduced by a exploitation factor which is chosen by the user in order to calculate economically usable amounts. Regeneration
costs provided by a separate data sheet are added to each stand.
Finally, in order to calculate the NPVs, an interest rate must be
entered.
The data processing described above is controlled by the user
through buttons provided in a central control sheet (Fig. 2). The
buttons are associated with VBA codes that activate the pro-
grammed commands, so that the handling is quite straightforward.
Through a series of processing steps, the data set is rearranged to
match the matrix format used by the area control scheme in the
model, with the stands in rows, and the periods in columns. The
name manager of Excel is used to assign name spaces to the data.
The I/O interface of YAFO-A accesses these names to assign the
model parameters to the appropriate data as listed in Section 3.1
Fig. 1. Flowchart of YAFO-EX and YAFO-MC (dashed box).
F. Hrtl et al./ Computers and Electronics in Agriculture 94 (2013) 5870 61
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(see also Fig. 3). The central control sheet also provides additional
buttons to activate the Monte Carlo simulation, YAFO-MC, that cal-
culates the spreading of the potential NPV of each stand in each
period by simulating the variations described in Section 2.2.2,
and derives variation coefficents and correlation matrices. After it
is computed, the risk data is written back to YAFO-EX (see
Fig. 1). Section 3.1 gives a list of the data the solution contains.
In addition to performing these preparatory work, YAFO-EX
evaluates the optimized results (see Fig. 1). For that purpose,
YAFO-A uses the I/O interface to export its solution back to
YAFO-EX. Based on this, YAFO-EX provides summaries illustratingthe progress of the stock, the harvest volumes, the NPVs, the felling
values, value increments and area distribution of the stand devel-
opment classes.
2.2. Risk integration
2.2.1. Value at risk and risk utility
To incorporate the risk effects already mentioned, the objective
function (1) must be expanded. For this purpose, certainty equiva-
lents that are derived from utility functions (Gerber and Pafumi,
1998; Bamberg et al., 2008) can be used, for example. Alternatively,
minimum values according to the Maximin decision rule can be
optimized (Young, 1998; Hildebrandt and Knoke, 2009; Hilde-
brandt and Knoke, 2011). As the worst case scenario for a givenobjective is normally very unlikely and therefore can be considered
irrelevant, or as is the case of a continuous distribution function
the probability of this worst case approaches zero, it makes sense
to focus on a defined threshold that is exceeded with a given prob-
ability (Mowrer, 2000). Such a limit is nothing other than a certain
quantile of the risk-driven probability function of the objective. In
finance this concept is well known as the value at risk (VAR)
(Stambaugh, 1996; Jorion, 1997; Knoke et al., 2012). If the realisa-
ble net revenues d, a andzfrom thinnings and (salvage) fellings are
distributed by risk effects and interpreted as expected values with
statistical spreads, the expected value of the objectiveZis distrib-
uted as well. FZrepresents the distribution function of that risk-dri-
ven objective function Z. The related inverse function F1Z p then
defines the p-quantile ofFZ, the value, that is exceeded byZwitha probability of 1p. The new objective is as follows:
maxf
Z F1Z p 8
Thus, the objective no longer optimizes the uncertain expected
value ofZbut instead, the worst value forZthat can be expected
with a certain probability of 1p. The assumption of a Gaussian
distribution
Z N EZ; s2Z
FZ 9
defines this distribution by the expected value E(Z) and the variance
s2Z. Using this precondition, F1 can be calculated as the inverse of a
normal distribution.Another approach to handling risk effects is the use of utility
functions that reduce the expected return with a weighted vari-
ance according to the assumed risk aversion of the manager. In this
case the objective function can be written as a certainty
equivalent:
maxf
Z EZ a
2s2Z 10
Herea is a constant, representing the absolute risk aversion of thedecision-maker.
If the decision-maker behaves as a risk-seeker, this can be easily
modelled with both approaches. In the case of Eq. (8) it is possible
to optimise other p-quantiles. A p-quantile of 0.5 represents a risk-
neutral behavior, whereas p-quantiles above 0.5 correspond torisk-seeking management decisions. In Eq. (10) a negative a canbe used to simulate a risk-seeking behavior.
2.2.2. Risk simulation by Monte Carlo
The two parameters E(Z) and s2Z, that are needed for the deter-
mination of the distribution function can be estimated, for exam-
ple, from local experience. But the model presented uses a
different approach: The parameters are estimated based on real
data by use of the integrated Monte Carlo (MC) module, YAFO-
MC, prior to the optimization process. The Monte Carlo simulation
is implemented as Visual Basic (VBA) code in a separate Excel file
that is linked to YAFO-EX. For both grading options, a separate
MC module is provided. The modules generate, by default, 1,000
possible proceeds and costs for every stand and period each cal-culated as a NPV sum of discounted net revenues from the possible
Fig. 2. Screenshot of the YAFO-EX central control sheet.
62 F. Hrtl et al./ Computers and Electronics in Agriculture 94 (2013) 5870
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final felling in that single period and the thinnings and salvage fel-
lings done thus far, by randomly modifying the evaluated growth
simulator data (see Fig. 1) by a mechanism for timber price fluctu-
ation and another one for calamity occurence. Using the nomencla-
ture ofPritsker (1997), this approach is afull Monte Carlo method,
as each draw is based on exact pricing.Timber price fluctuations are treated as a random variable. For
each simulation step there is a randomly chosen year between
1975 and 2010 that is associated with that step. That year numberdefines factors that weight the returns in this period using simple
multiplication. There are two different factors for hard and soft-
wood. These factors are calculated from the timber price statistics
published for the Bavarian state forest, and denote the percent
deviation of that years timber prices from the average price during
the time horizon mentioned. All prices are adjusted for inflation.
The prices are derived from two chief timber grades average
quality spruce timber, diameter class 2529 cm, for softwood,
and average quality beech, diameter class 4049 cm, for hardwood.
Inflation is estimated based on the so-called Long Series of the
German consumer price index (DESTATIS, 2011).
To randomize the appearance of a calamity this aspect is mod-
elled here in a different way as in the optimizer model YAFO-A.
Not average ratios of salvage areas per period are used but a ran-dom number between 0 and 1 is picked for every stand. In each
period one calamity is possible in every stand, and there can be just
one calamity for each stand during the simulated time horizon.
This random number is compared to the hazard rate computed
for the single stand. If the random number exceeds the hazard rate,
the calamity occurs. The stand is then felled as a whole and a new
stand generation is planted. The hazard rate is calculated using
survival functions according to Griess et al. (2012). The change of
the survival function st etb
a
during a given period in time h,
with respect to the initial state, defines the hazard rate a(t):
at st st h
st h 11
Different empirically derived values for a andb are determinedfor each of four stand types pure and mixed softwood stands as
well as pure and mixed hardwood stands. Returns due to calami-
ties are reduced with a calamity factor which is chosen by the user.
The associated salvage fellings are considered prior to regular fel-
lings in each period.
2.2.3. Risk evaluation
The 1000 simulation runs generate 1000 possible NPVs for
every stand in each period and each grading option. These NPVs
are saved for each period. From this data, a correlation matrix iscalculated between the uncertain NPVs as well as a variation coef-
Fig. 3. Flowchart of YAFO-A.
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ficient for each stand. It is possible to do this either at the stand le-
vel or to combine the stands to groups for this purpose. In the latter
case, the distribution of the average values for each group is calcu-
lated. To do this, YAFO-A requires a group attribute for each stand
(see Section 3.1). As the standard option in YAFO-EX, nine groups
are used (spruce, fir, pine, larch, douglas fir, beech, oak, valuable
hardwood, other hardwood).
Let bgts represent a vector containing the possible realisations ofthe uncertain NPV in period tand grading option s for the defined
stand group g= 1,2, . . . ,n. Then every matrix Kts, with periods
t= 0,1,. . . ,6, contains the correlations between these vectors:
Kts :
corrb1ts;b1ts corrb1ts;bnts
.
.
...
...
.
corrbnts;b1ts corrbnts;bnts
0BB@
1CCA 12
The covariances Vxyts are calculated for each period tand grad-
ing option s by multiplying the correlation coefficient Kxyts between
two stands or groupsx undy by the areasf(d,a,z), used by the model
inx undy, by the variation coefficients v of the NPVs, and by the
NPVs d0t: dt(1 + r)t, a0t: at(1 + r)
t and z0t:zt(1 + r)t that
can be realised in the stands or groups,x andy. Thus, the covari-ance matricesVts have the following components:
Vxyts : Kxyts vxts vyts fdxtsd0xtsf
axtsa0xtsf
zxtsz0xts
fdytsd0ytsfaytsa0ytsf
zytsz0yts
13
These covariance matrices are summed up by elements to cal-
culate the total variance
s2ZXx;y;t;s
Vxyts 14
of the objective functionZ. The variance of the last period is divided
by five to account for the fact that the model algorithm cannot dis-
tribute its decisions in this period forward into the future as can be
done in reality, because the model does not cover future periods.
The model can spread felling areas from period six to five or four,
although the particular stand might not have reached the NPV peak.
Taking the full variance of period six into the model causes an over-
estimated felling area in the preceding period, whereas reducing the
variance to zero lets the model try to avoid the fellings and to reach
the risk-free period six. The parameterisation of this factor must
balance these two opposing decisions in a reasonable way.4
The expected value ofZ, defined in Eq. (1), and the variance just
calculated define the distribution function FZ, as shown in Eq. (9).
The inverse function F1Z p to be maximised according to Eq. (8),
is also defined. In the model, this function is not calculated as
the inverse ofFZ but instead, a reduction factor is used. Assuming
a Gaussian distribution according to Eq. (9), the difference between
the expected value of the objective function EZ F1Z 0:5 andthe value at risk F1Z p can be expressed in terms of multiples q
of the standard deviation sZ ofZ. This multiplication factor, that
is equivalent to the desired value at risk quantile p, is equal to
the quantile q of the standardised normal distributionU(q), so that
U(q) = 1p. Therefore, the objective function of Eq. (8) can be cal-
culated by
F1Z p EZ qsZ: 15
Knoke and Wurm (2006) have shown that the spreading of re-
turns from forests follows a Gaussian distribution only in an
imperfect manner. Therefore, the optimizing model presented here
uses the Monte Carlo simulation mentioned to generate a more
realistic spreading of the uncertainty factors as a first step. In the
following nonlinear objective function, this approach is then sim-
plified by describingthis simulated spreading like a Gaussian one.According to Beinhofer (2009), the quality of the predicted results
is not highly affected through this simplification, as long as confi-
dence levels 1p below 95% are used.
So, three optimization programs exist in YAFO-A (Fig. 3): A sim-
ple NPV maximisation (Eq. (1)) and two programs considering risk:
value at risk VaR (Eq. (15)) and certainty equivalent CE (Eq. (10)).
All three problems are classified in AIMMS as nonlinear, although
the NPV maximisation is actually linear. This is due to the fact that
AIMMS does not distinguish between variables5 that are part of the
chosen objective function and those which are not. For all three
cases, AIMMS uses CONOPT (Consulting and Development A/S, xxxx;
Drud, 1994), a solver algorithm for nonlinear programs that search
for a local optimum. Consequently, we define subsets of variables
and constraints to construct a mathematical program that is ableto solve the NPV maximisation as a linear problem. Using these sub-
sets, a simplex algorithm can be applied to determine a global opti-
mum. The algorithm used in this case is the ILOG CPLEX solver (IBM
Corp., 2011).
The model also contains procedures (program code) that help
the user to automatize the process of solution finding. These proce-
dures can be initiated through buttons on the user interface
(Fig. 4). There are three main procedures that carry out the three
mathematical programs described above. In addition, to find a glo-
bal optimum of the nonlinear problems, there is also a multi-start
option. In multi-start mode, YAFO-A uses the multi-start module
included in AIMMS to search for an optimum, starting from the
20 best solutions that are calculated by 100 randomly selected
starting points. This search is repeated 10 times. For further detailsabout the multi-start module see Rinnooy Kan and Timmer (1987)
and Roelofs and Bisschop (2011).
3. Application and example
3.1. Data preparation
The users of our model must prepare their data in an Excel or
Calc sheet. To do so, the modular design of the model offers two
options. The users have the option of using the evaluation tool
YAFO-EX. This tool is designed to read stand data produced by for-
est growth simulators, simulate uncertainties with the help of the
coupled Monte Carlo module YAFO-MC, and deliverthese data sets
to the optimizer model. Alternatively, they can use a data manipu-lation of their own, as done by Hahn et al. (submitted for publica-
tion), and import the data directly to the optimizer YAFO-A via the
defined I/O Interface.
The spreadsheet file YAFO-EX uses a single sheet for each of the
two possible grading options, that must provide for each stand of
the investigated enterprise the following data structure line by
line:
Stand identifier/number
Year or period
An identification for thinning data (year or period value) and
residual stand data (0)
Proceeds and cost per hectare (/ha)
4 A spruce dominated stand in Bavaria typically reaches the maximum NPV in
between 60 and 100 years. Forty years or eight periods are necessary to cover this
period. Therefore, seen from the point of period six, there are, eight future periods
missing in the model to determine the correc t point in time when the NPVs of the
stands existing in period six, are reaching the maximum. On average, eight of nine
stands in period six are not mature. The redu ction of the variance in period sixprevents the model to spread these non-mature stand areas into period five. 5 In AIMMS every object that can be changed in the model is a variable.
64 F. Hrtl et al./ Computers and Electronics in Agriculture 94 (2013) 5870
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Total volume of (compact) wood (m3/ha)
Volume of brushwood (m3/ha) (optional) Dominant tree species
Stand age (years)
Wood volume of each timber grade class (m3/ha)
This data set is required for each period for the thinning and the
residual stand. For seven points in time, this implies 14 rows for
every stand in the data sheet. By entering additional data for
brushwood, the tool is able to calculate the capabilities for provid-
ing fuel wood amounts from brushwood. In order for these
amounts to be considered financially, they must be included in
the expected proceeds and costs per ha.
The I/O interface of YAFO-A imports data from any Excel or Calc
sheet, and exports the solution as well. To ensure data is assigned
to the correct model objects, the cell areas in Excel or Calc must bemarked with defined names. YAFO-EX already provides these
name conventions. The following data is imported by YAFO-A:
Stand identifier/number
Initial area of each stand (ha)
Initial age of each stand (years)
List of groups for stands grouping
Assignment of each stand to the groups
Thinning volumes for each stand, period and grading option
(m3/ha)
Stock volumes for each stand, period and grading option (m3/
ha)
NPV of proceeds and costs for each stand, period and grading
option from thinnings, fellings and salvage fellings (
/ha) Hazard rate of each stand in each period (%)
Correlation matrix for each period
Variation coefficents for each stand/group in each period
The following values are exported to the target solution file
(YAFO-EX as standard):
NPV sum ()
Value at risk ()
Certainty equivalent ()
Lists of areas used in each period and grading option by thin-
ning, salvage and final felling (ha)
Covariance matrix for each period
3.2. Example
As an example we demonstrate the application of YAFO on adata set of inventory plots of the second German federal forest
inventory BWI 2 (BMVEL, 2005) that has been projected by the
growth simulator WEHAM (Bsch, 2004a,b). For this purpose, the
growth simulator was set so that there was a possibility to thin
the stands but not to fell them finally, as the timing of the final fell-
ing will be determined by our model. To calculate brushwood
amounts we used volume expansion factors according to Zell
(2008). The data set tested consists of 267 plots (satellite sample
plots as used in the inventory) that belong to the state forest of
the geographical region Tertires Hgelland in Bavaria. These
plots are considered as 267 stands of a forest enterprise, each rep-
resenting a stand of 1 ha. The data show a softwood-dominated
tree species distribution with high standing timber volumes
(410 m3
/ha) that are typical for this region. One hundred fifty-sixof these 267 ha are covered by spruce-dominated stands, and
Fig. 4. Graphical user interface (GUI) of YAFO-A.
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52 ha are covered by beech. The other stands are dominated by
pine or other hardwoods. The growth simulator WEHAM has an
integrated grading function, so that, without the use of any addi-
tional programs, we obtain a graded result of the biophysical
development of the stands. We compare two grading variants:
The first scenario is meant to represent the actual grading practices
used in forestry at present, and emphasizes the material use of tim-
ber, with only moderate fuel wood amounts from small-sizedwood. The second scenario emphasizes the thermal use of wood,
by having minimum diameters for saw log and industrial wood
that are 12 cm larger than those used in the first scenario. Using
a database of our own, these volumes are evaluated and trans-
formed to the data structure YAFO-EX requires. The assumed tim-
ber prices and costs of harvesting are shown in Table 1. The
harvesting costs for spruce are used for all softwood species, and
those for beech for all hardwoods. Fir prices are set 5 below,
and pine prices 20 below spruce. The prices for larch and douglas
fir are set 10 above spruce. Beech prices are used for all hard-
woods except oak and low value hardwoods. Prices for oak are gi-
ven in the table. Low value hardwoods are priced at 5 below
beech.
The assumed regeneration costs are shown in Table 2. The
regeneration costs modification is implemented as a Weibull func-
tion following the form etb
a
where tis the stand age and the two
parameters are defined as a : 70 and b: 5. The data used byYAFO-EX to simulate volume and thinning amounts from ingrowth
are given in Table 3.
The factor to reduce net revenues from calamities as well as the
brushwood exploitation factor are set to 0.5, the interest rate to 2%,
and the value at risk quantile to 5%. We do not introduce any fur-
ther constraints at the forest enterprise level. The risk simulation
through the Monte Carlo module is complete after approximately
10 min.
We optimize the 267-stand enterprise in YAFO-A with and
without risk aspects. The linear program gives a global optimum
of 17,081 /ha for the NPV. The nonlinear optimization with risk ef-
fects is calculated using the multi-start option, resulting in thesolution of 14,690 /ha for the VAR (NPV at 17,001 /ha). The sum-
mary results for the timber production of the model enterprise are
shown in Tables 4 and 5.
The data in the tables are aggregated to the enterprise level and
displays logging volumes, area development and financial results.
This data can be used for supporting the decisions of the forest
manager. It is possible to retrace this data to the single stands.
So an operational felling plan in terms of a stand list can be pro-
vided for the manager.
4. Discussion
4.1. Model
The main focus of YAFO is the economic analysis of felling sce-
narios when making decisions based primarily on financial values.
Many other approaches are also capable of calculating financial
values, but either do not provide the possibility to consider them
as decision variables (for example LMS/FVS, DSD, FTM, 4S TOOL,
AFFOREST), or do not integrate all risk aspects financial as well
as natural ones (for example HEUREKA, SIMO, DSD, NED-2).
YAFO, however, provides the consideration of (biophysical) restric-
tions in the solution process, so that it is possible to implement
ecological and social barriers, at least to the extent that they can
be expressed using such constraints. For example, in order to main-
tain a certain level of ecosystem services (e. g. recreation, water
conservation) there can be the additional objective to maintain aspecific minimal average timber volume within the forest enter-
prise. To do so, it is easily possible to formulate a final timber vol-
ume that must be remain at the end of the investigated
management period.
In contrast to most other approaches, YAFO does not use linear
programming or heuristic algorithms to solve the decision prob-
lem, but rather NLP techniques. The advantage of this method over
linear models is that the risk aspects mentioned above can be eas-
ily integrated. The uncertainties included in the YAFO model coverrisks due to timber price fluctuations, as well as calamity probabil-
ities, and their relationship to species mixture. Unlike further ex-
tended frameworks of uncertainty (Williams, 2012), the model
assumes that the objectives are known and accepted. In NLP it is
possible to determine an optimal solution by using solver algo-
rithms for global optima, or as in our case by calculating it with
the help of advanced multi-start techniques. In contrast, heuristic
approaches can achieve only approximate solutions.
The spatially implicit nature of the model is achieved through
the consideration of the stands of the forest enterprise in the area
control scheme (Turner et al., 2001; Perry and Enright, 2007). This
approach fulfills level 4 of spatial recognition, as defined by Davis
et al. (2001). Further spatial effects, such as direct interactions be-
tween stands are not considered, as the YAFO model does not in-
clude information about the spatial arrangement of the stands, as
for example, SAGALP or HARVEST do.
The YAFO model is not designed as a monolithic solution with
an integrated growth simulator. Instead it is a modular tool to sup-
port managers of private and public forest land in their decisions.
YAFO can be easily adapted to existing growth simulators, due to
its open data interface. According to Reynolds (2005), it is argu-
able if optimization systems are real DSS (Rauscher et al., 2007).
YAFO certainly can be considered as a DSS component, based on
the definition by Holsapple (2003) of DSS as problem-processing
systems supporting a decision-making process. According to the
definition given in Menzel et al. (2012), the tool can be recognized
either as a typical part of a DSS, or as a DSS in a wider sense. They
searched for criteria to merge the quantitative, analytical with the
qualitative, more participatory oriented approach, as differentiatedby Reynolds et al. (2008). YAFO as a new, innovative, and algorith-
mic model complies with the eight criteria Menzel et al. (2012) de-
rived for evaluating a DSS from a participatory planning
perspective.
One potential weakness of the modular approach is the lack of
interaction between optimization and growth simulation. The
growth prediction is not able to act in response to the thinning ac-
tions planned by the optimization. Thus, the thinnings used in the
financial model are determined by and limited to the decisions
made by the growth simulation. The provision of such interactions
is still a big advantage of the monolithic approach of other systems,
such as HEUREKA. However, due to the lack of a globally valid for-
est growth simulation, the ability of YAFO to use data from various
growth simulators represents a big advantage of the modular ap-proach. Existing growth simulators are parameterized for specific
regions. For example, SILVA has implemented growth functions
mainly for Bavaria, BWINPro for North West Germany, DSD for
southern Austria, LMS/FVS for the United States, SADfLOR for Por-
tugal, and HEUREKA for Sweden. Combining a universally valid
financial model with a specific local/regional growth simulator
solution would inhibit a broader use of our system, as such all-
in-one solutions are usually not easily adaptable to other regional
(growth) conditions. Our target is to provide an open tool that can
be linked to different growth models. Although this design means a
higher workload for the users, as they must take care to appropri-
ate data import and export from one program to the other, in the
end it provides greater flexibility (see e.g. Nute et al., 2005). The
example above shows that interaction with the WEHAM growthsimulator.
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Another advantage to the YAFO model is its speed. The risk sim-
ulation through the Monte Carlo module is complete after approx-
imately 10 min, and the calculation time of YAFO-A is only about
2 min.6 For two grading options, at six points in time across 267
stands, in which the model distinguishes between thinning and fell-
ing actions as well as grading options for the salvage fellings, the
optimization problem consists of 9612 independent decision vari-
ables and 6974 constraints.
7
In total then, the model must calculateabout 39,000 variables. Our previous attempts using the Excel Add-
in, Whats Best (LINDO Systems Inc., 2011), for nonlinear program-
ming required several hours of computation time and did not reach
feasible solutions in either case. The redesign of the model within
AIMMS is therefore a big step towards practicality.
4.2. Example
Analysing the solution for the risk-ignoring linear NPV optimi-
zation one can see that the high initial average stand volume of
410 m3/ha is immediately reduced by a heavy harvesting operation
of 19 m3/ha/a distributed across 33 ha of the enterprise area. This
decreases the timber increment to 11 m3/ha/a. The main reason
for this is the large percentage of high volume spruce stands in
age classes IV (6079 years) and V (8099 years) that have already
reached, or even exceeded, their financially optimal rotation age.
The average volume is reduced during the following periods down
to 304 m3/ha in period five. From this point it increases over the
next 5 years, ending at 349 m3/ha. A little bit more than one third
of the forest area (99 ha) is felled during the considered time
horizon.
In contrast, the harvests undertaken by the risk-sensitive vari-
ant are more equally distributed. The initial harvest amount is only
12 m3/ha/a, whereas the harvest volume in the subsequent periods
ranges between 11 and 14 m3/ha/a. In these periods, the risk-
ignoring variant harvested between 9 and 12 m3/ha/a. The total fi-
nal felling area of 90 ha is below the one in the first case, while the
minimum average stand volume rises to 320 m3/ha, finally ending
at 367 m3/ha in period 6. The main reason for these results is thatthe algorithm tries to arrange the harvesting activities in a more
evenly distributed fashion to avoid high risk effects. This is quite
clear in the development of the area in spruce of age class V. The
risk-ignoring variant reduces this age class through its first harvest
action from 39 ha to 26 ha. In contrast, the value at risk optimiza-
tion distributes this reduction across four consecutive harvests, not
reaching a comparable level of 28 ha remaining spruce until period
three 15 years later.
This equalizing tendency also affects the financial results. While
the risk-free optimization allows a drop in periodical revenues
from an initial 689/ha/a to 461 /ha/a including major fluctua-
tions, the results of the value at risk variant are more continuous:
The maximum of 433/ha/a in period zero is accompanied by a fi-
nal value of 533/ha/a. Although the revenues reduce to 376 /ha/a in period two, this minimum exceeds the minimum of the NPV
variant (294 /ha/a) by 82/ha/a. The results thus show that deal-
ing with risks in a forest enterprise planning process leads to a
more equally distributed logging plan, and therefore allows the
owner to benefit from a more balanced flow of net revenues.
Another result can be seen by analysing how the timber har-
vested is split up between the two grading options. In both optimi-
zation approaches, about 10% of the timber is graded according to
the fuel wood scenario. This timber comes from the younger hard-
wood stands, because it is more profitable to grade these small-
sized hardwoods as fuel wood and sell them, for example, via con-
tract felling than for the owner to conduct the harvest and sellthem as saw logs (refer to the assumed price scenario in Table 1).
Table 1
Income revenue over harvesting cost for spruce and beech.
Species Diameter
class (cm)
Avg. price
(/m3)
Harvesting costs
(/m3)
Income revenue
(/m3)
Spruce 614 40 21 19
1519 47 22 25
2024 59 20 39
2529 64 19 45
3034 64 18 463539 61 17 44
4044 60 16 44
P45 60 17 43
Industrial
wood
40 21 19
Fuel wood 10 0 10
Beech 614 40 26 14
1519 42 25 17
2024 44 22 22
2529 44 21 23
3034 49 19 30
3539 62 18 44
4044 72 16 56
4549 72 16 56
5054 80 18 62
P55 84 18 66
Industrialwood
40 26 14
Fuel wood 20 0 20
Oak 614 40 26 14
1519 40 25 15
2024 40 22 18
2529 58 21 37
3034 76 19 57
3539 103 18 85
4044 140 16 124
4549 140 16 124
5054 164 18 146
P55 174 18 156
Industrial
wood
40 26 14
Fuel wood 20 0 20
Table 2
Regeneration costs.
Tree species Costs (/ha)
Beech 6400
Valuable hardwood 5225
Other hardwood 4895
Oak 8250
Spruce 1600
Fir 2700
Douglas fir 3958
Pine 3630
Larch 1400
Table 3
Young stand data of thinning and volume growth.
Tree species Age (years) Thinning (m3) Volume (m3)
Softwood 20 15 40
25 20 60
30 30 100
Hardwood 20 0 10
25 5 25
30 10 40
6 We used a PC with an Intel Core i5-2400 CPU and 3.1 GHz.7 7 267 for every stand, plus 6 267 for the salvage felling, plus non-negativity
constraints of the same size, plus 32 optional biophysical constraints at the enterpriselevel.
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Having the flexibility to grade simultaneously in two ways is one of
the key strengths of the model. Of particular advantage in our ap-
proach is the opportunity to compare two grading variants not
just by means of their total NPV or value at risk but also to be
able to analyse such gradual shifts between the grading options
at the individual stand level.
5. Summary
The goal of the development of the YAFO optimization model is
to provide a planning-support tool that can be easily adopted for
risk neutral, risk averse and risk-seeking decision makers, as well
as to a wide variety of problems. Therefore it sets a high valueon the strict division between model representation and data man-
agement. We use commercially available software that can usually
be acquired with low costs at least for the research sector. The
modular concept including defined interfaces between the mod-
ules allows for the combination of the users data in different
ways within the model: Either a graded and valuated data set via
the Excel tool YAFO-EX (for example, for processed data from
growth simulators), or a set of coefficients with separate risk eval-
uation (correlation matrices and variation coefficients) for directimport into the optimizer, YAFO-A, can be used. The use of the
Monte Carlo module, YAFO-MC, is optional, as the system can han-
dle nonlinear problems with uncertainty evaluations as well as
simple linear problems. The model can also distinguish between
two grading options. The version of the tool presented here is inde-
pendent of the number of stands that need to be investigated. One
current limitation is the number of periods in time that can be con-
sidered. The ability to find local or global optima for nonlinear
problems is not a question of the presented model but of the solver
the users apply in their AIMMS system. As AIMMS provides an
open interface for that reason, it is possible to combine our model
with different solver algorithms.
The example shown performs as expected: Both the periodical
biophysical and financial results occur more smoothly when the
optimization considers uncertainties due to risk effects. Another
interesting result is the response to the two different grading op-
tions. As described, this decision is made by the model at the indi-
vidual stand level. In the future, it might be helpful to examine this
effect in greater detail. Thus, the model may enable us to analyse
the behaviour of forest owners in response to various timber price
scenarios. For example, it would be valuable to investigate how
timber amounts shift between the material and thermal-use tim-
ber grades under the assumption of increasing prices for fuel wood
that are likely given the expected continued increase in oil prices.
Acknowledgements
The study presented here is part of the Project G33 Competi-tion for wood: Ecological, social and economic effects of the mate-
rial and energetic utilization of wood funded by the Bavarian State
Ministry of Food, Agriculture and Forestry, and as Project
22009411 by the German Federal Ministry of Food, Agriculture
and Consumer Protection. The authors wish to thank Laura Carlson
and Yolanda Wiersma for the language editing of the manuscript
and two anonymous reviewers for valuable suggestions.
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Table 4
Results of the forest enterprise optimization without considering uncertainties.
Period 0 1 2 3 4 5 6 Avg.
Initial stock (m3/ha) 410 372 376 370 378 366 349 375
Remaining stock (m3/ha) 317 319 317 327 319 304 349 322
Regular felling (m3/ha/a) 18 10 11 8 11 12 0 12
Salvage felling (m3/ha/a) 1 1 1 1 1 1 0 1
Total felling (m3/ha/a) 19 11 12 9 12 12 0 12
Increment (m3/ha/a) 11 11 11 10 9 9 10
Areas (ha)
Intermediate felling 234 220 205 197 184 168
Final felling 29 12 13 7 12 14
Salvage felling 4 2 2 2 2 2
Regeneration 33 47 62 70 83
Sum 267 267 267 267 267 267
Logging volume (m3/ha/a)
Saw log 15 8 9 6 9 9 9
Industrial wood 2 2 2 2 2 2 2
Firewood 2 1 1 1 1 1 1
Brush firewood 2 1 1 1 1 1 1
Profit (/ha/a)
Intermediate felling 203 144 160 173 187 189
Final felling 473 217 260 115 248 263
Sum regular felling 676 362 420 288 434 452
Salvage felling 13 7 6 6 7 8Sum 689 368 426 294 441 461
Table 5
Results of the forest enterprise optimization maximising the value at risk.
Period 0 1 2 3 4 5 6 Avg.
Initial stock (m3/ha) 411 408 405 407 407 390 367 399
Remaining stock (m3/ha) 350 345 350 353 341 320 367 347
Regular felling (m3/ha/a) 12 12 10 10 12 13 0 11
Salvage felling (m3/ha/a) 1 1 1 1 1 1 0 1
Total felling (m3/ha/a) 12 13 11 11 13 14 0 12
Increment (m3/ha/a) 12 12 11 11 10 9 11
Areas (ha)
Intermediate felling 254 236 224 211 195 177Final felling 9 14 10 10 13 15
Salvage felling 4 3 3 3 3 2
Regeneration 13 31 43 56 72
Sum 267 267 267 267 267 267
Logging volume (m3/ha/a)
Saw log 9 10 8 8 10 11 9
Industrial wood 2 2 2 2 2 2 2
Firewood 1 1 1 1 1 1 1
Brush firewood 1 1 1 1 1 1 1
Profit (/ha/a)
Intermediate felling 210 152 172 188 201 199
Final felling 210 278 196 186 282 323
Sum regular felling 420 430 368 374 483 523
Salvage felling 13 11 8 8 9 10
Sum 433 441 376 383 491 533
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