Optimal Control for Forest Management in the Czech Republic
Jitka Janov
´
a
1
and Ji
ˇ
r
´
ı Kadlec
2
1
Department of Statistics and Operational Analysis, Mendel University in Brno, Zemedelska 1, Brno, Czech Republic
2
Department of Forest and Forest Products Technology, Mendel University in Brno, Zemedelska 1, Brno, Czech Republic
Keywords:
Optimal Control, Forest Structure, Forestation, Biodiversity.
Abstract:
This contribution presents initial qualitative results and discussions when addressing the particular dynamic
optimization problems in Czech forestry management. First, we analyze the deterministic infinite time horizon
optimal control model aimed to determine the optimal paths for plantations of various mixed forests in the
Morava region in the Czech Republic. Second, the problem of optimal dynamic path for the subsidy rates is
established and its solution via optimal control using the simulated data is suggested. The at foremost aim of
the presentation is to present the research topic itself and to discuss the optimization and solution techniques
suggested.
1 INTRODUCTION
For 200 years the artificially planted spruce forests
have covered the majority of forest land in the Czech
Republic. For several last decades the problems with
exhausted soil and lack of biodiversity in forests have
escalated to intensive need for modifying the estab-
lished forest structure. The state forestry authorities
have introduced a subsidy policy to support the refor-
estation by original mixed forests and presented the
target optimal forest structure in Czech forests. Al-
though the state intentions and the economical sup-
port are clear, the particular optimal long run strate-
gies for each forest area are unknown. There are
number of possible particular combination of mixed
forests to involve, various rotation scenarios to choose
from and number of different forestation strategies to
adopt.
The decision process appears to consist from two
stages: first, the optimal rotation for each particular
forest type is identified and then the particular areas of
forest land are assigned to each forest type according
to selected criterion.
In this contribution we suppose the optimal tree
rotation for each forest type is known as well as the
production and profit functions for particular forest
types and the subject of our research is to address
the second stage of the decision process. Concerning
the appropriate methods of operations research that
would provide the support for this decision making
we can distinguish the static and dynamic approach.
The first one is represented by the linear program-
ming (LP) that is a common broadly used technique
in the forestry problems (see e.g. review (D’Amours
et al., 2008)). Using LP we obtain the optimal for-
est structure given the particular area and decision
criterion. However, there is no information on how
to arrive from current situation to the desired (opti-
mal) one. For this reason we consider the dynamic
optimization techniques to be more appropriate when
identifying the optimal long run strategy for the forest
owners in the Czech Republic.
In our research we propose to employ the opti-
mal control (OC) for finding the optimal time path for
the state variables - the areas forested by the particu-
lar forest types. We expect that for the infinite hori-
zon problem with (current value) profit maximization
criterion we obtain a steady state solution represent-
ing the optimal target forest structure and the opti-
mal state and control paths determining the appropri-
ate strategies of forestation. Optimal control tech-
niques have been steadily employed in the forestry
management research problems during the several last
decades for analyzing the potential optimal strategies
(for recent contributions mostly related to our prob-
lem see (Caparros et al., 2013), (Caparros and Jacque-
mont, 2003) (Bach, 1999) or (Cerda and Martin Bar-
roso, 2013)). However, due to the complex structure
of the dynamic decision problems in forestry some
theoretically oriented papers have appeared address-
ing the particular specifics of forestry decision prob-
lems (Sahashi, 2002).
466
Janová J. and Kadlec J..
Optimal Control for Forest Management in the Czech Republic.
DOI: 10.5220/0004924504660470
In Proceedings of the 3rd International Conference on Operations Research and Enterprise Systems (ICORES-2014), pages 466-470
ISBN: 978-989-758-017-8
Copyright
c
 2014 SCITEPRESS (Science and Technology Publications, Lda.)

Apart from the problem discussed above, we in-
tend to solve another dynamic decision problem: to
find the optimal subsidy strategy for reaching the de-
sired forest structure in the Czech Republic within the
least time. While the first problem concerns the forest
owner decision making under given (constant) subsi-
dies, the latter problem searches for the dynamic paths
of the values of particular subsidies to optimize the
forestation process in the Czech Republic as whole.
After a brief description of the situation and re-
lated data in Sec. 2, we introduce the two above men-
tioned decision problems: the forest owner foresta-
tion strategy in Sec. 3 and subsidy strategy in Sec.
4. We present the concept of the optimization models
and discuss the possibilities to identify the state equa-
tions. We suggest the form of the functions involved
and for the theoretical models suggested we provide a
qualitative solution and interpretations.
2 SITUATION
The current composition of forests in the Czech Re-
public together with natural and recommended distri-
bution is summarized in Table1.
In our research, we focus on the Morava region
in the east part of the Czech Republic. According
to the factors decisive for the forest type choice we
will define the homogeneous subregions as highlands
and lowlands and assign them the suitable forest types
(see Table 2). Subregions were defined in respect
to state policy of tree species change from spruce
to broadleaves and mixed forest types. Spruce is
out of optimal growing conditions in lowlands and
highlands where is negatively influenced by fungi,
pest and abiotic factors. Suitable forests types were
defined with respect to production and tree optimal
growing conditions. Mountains subregion is not de-
fined because spruce has optimal conditions there and
other tree species are out of production optimum.
3 FORESTATION STRATEGY
The problem is to identify the optimal forestation
strategy from the point of view of the forest owners
in the Morava region given the current forest struc-
ture and the subsidies: 12000 CZK per ha of natural
regeneration of desired tree species plus 20000 CZK
per ha in the fifth year after successful reforestation.
In our model, the criterion of decision making is the
economic profit from forests (including the income
from selling the timber and obtaining the subsidies
for the forest structure changes)
1
.
To address the decision problem we introduce in-
finite horizon OC problem with free terminal points
to be solved for each particular subregion.
max V =
∫
∞
0
Π(t)e
−ρt
(1)
Π(t) =
n
∑
i=1
R
i
(x
i
(t), u
i
(t)) − (2)
−K
i
(x
i
(t), u
i
(t)) (3)
s.t. (4)
u
i
(t) = ˙x
i
(t), (5)
n
∑
i=1
x
i
(t) = L, (6)
x
i
(t) ≥ 0, (7)
u
l
(t) ≥ 0, 1 ≤ l ≤ s < n (8)
x
i
(0) given, 1 ≤ i ≤ n. (9)
Here, n denotes the number of forest types appropriate
for the given subregion, x
i
(t) are the state variables
representing the area of land forested by type i in time
t. Setting
x
n
(t) = L −
n−1
∑
i=1
x
i
(t), (10)
where L denotes the total area of the subregion, we
can exclude the constraint (6) from the model. In (8)
s is the number of supported forest types and the con-
straint reflects the fact that once the piece of land is
reforested by the new (supported) forest type, it stays
in the new status.
Further notations: ρ = discount rate, control vari-
ables: u
i
(t) = total area reforested at time t of forest
type i (in ha per year), s is the number of supported
forest types, Π(t) = the current profit from the forests
in the subregion, R
i
(t) = total revenues and K
i
(t) =
total costs from growing and logging at time t. We as-
sume the revenue function can be split into two terms:
R
i
(t) = G
i
(x
i
(t)) + σ
i
u
i
(t),
where G
i
represents a known function of revenues
from logging in the particular forest type while
σ
i
u
i
(t) calculates the subsidy from increasing the area
forested by type i, where
σ
i
=
{
σ
i
> 0 for supported forest types
0 for the others
1
Note that apart from economic criterion we should
mention other benefits from forest planting, that could enter
the model. In our research we start with the simple eco-
nomic (financial) decision criterion, that mostly reflect the
objective of running the forest-business in the Czech Re-
public. Once the basic model is established, it could appear
useful to incorporate further criteria and/or constraints.
OptimalControlforForestManagementintheCzechRepublic
467

Table 1: Perceptual composition of tree species in Czech forests.
Composition Natural Current Recommended
Spruce 11.2 51.4 36.5
Fir 19,8 1,0 4,4
Pine 3,4 16,7 16,8
Larch 0,0 3,9 4,5
Other conifers 0,3 0,3 2,2
total conifers 34,7 73,2 64,4
Oak 19,4 7,0 9,0
Beech 40,2 7,7 18,0
Hornbeam 1,6 1,3 0,9
Ash 0,6 1,4 0,7
Maple 0,7 1,3 1,5
Elm 0,3 0,0 0,3
Birch 0,8 2,7 0,8
Linden 0,8 1,1 3,2
Alder 0,6 1,6 0,6
Other broadleaves 0,3 1,6 0,6
Total broadleaves 65,3 25,6 35,6
Unstock area 0,0 1,2 0,0
Table 2: Suitable forests in subregions: S= Spruce, SB= Spruce with beech,SF= Spruce with fir, SDG= Spruce with douglas
fir, B= Beech, BL= Beech with larch, BDG= Beech with douglas fir, BO=Beech with oak, OB=Oak with beech, BH=Beech
with hornbeam, BF=Beech with fir, Br= Birch, P= Pine.
Subregion Forest type current Forest type suitable
The higlands S SB, SF, SDG, B, BL, BO, Br
The lowlands S OB, BH, BL, BDG, BF, P
For the purpose of further estimation of the func-
tions G
i
(x
i
), K
i
(x
i
, u
i
), we adapt the assumptions
made in (Caparros et al., 2013) and set
G
i
= g
i0
+ g
i1
x
i
+
1
2
g
i2
x
2
i
, (11)
K
i
= k
i
+ a
i
x
i
+ b
i1
u
i
+
1
2
b
i2
u
2
i
. (12)
where the total cost function in (12) is composed from
the farming costs and reforestation costs. Note that
we assume b
i1
= b
i2
= 0 for the spruce type forest
- supposed to be reduced - i.e. decreasing the area
of the forest type (after logging the current stand) is
costless.
3.1 The Qualitative Solution
We analyze the small scale problem considering just
three forest types: x
1
, x
2
= areas forested by de-
sired (finantially supported) types and x
3
= the area
forested by spruce (supposed to be decreased). Ac-
cording to (10) we apply the substitution x
3
= L −
x
1
− x
2
and the OC problem is of the form
max V =
∫
∞
0
Π(x
1
, x
2
, u
1
, u
2
)e
−ρt
(13)
˙x
1
(t) = u
1
(t), (14)
˙x
2
(t) = u
2
(t), (15)
u
1
(t) ≥ 0, (16)
u
2
(t) ≥ 0, (17)
x
1
(0) ≥ 0, x
2
(0) ≥ 0 given. (18)
The non-negativity constraints (16-17) reflect the fact,
that the area forested by desired types may only in-
crease, i.e. once converting after clearcut the for-
est area to a desired type it remains in the new sta-
tus. This requirement stems from the aim of the opti-
mization - to permanently change the forest structure
and is also supported by the subsidy policy. These
constraints together with (18) ensures also the non-
negativity of state variables x
1
(t), x
2
(t).
Having
Π = R
1
− K
1
+ R
2
− K
2
+ R
3
− K
3
where
ICORES2014-InternationalConferenceonOperationsResearchandEnterpriseSystems
468

R
1
= g
10
+ g
11
x
1
+
1
2
g
12
x
2
1
+ σ
1
u
1
, (19)
K
1
= k
1
+ a
1
x
1
+ b
11
u
1
+
1
2
b
12
u
2
1
, (20)
R
2
= g
20
+ g
21
x
2
+
1
2
g
22
x
2
2
+ σ
2
u
2
, (21)
K
2
= k
2
+ a
2
x
2
+ b
21
u
2
+
1
2
b
22
u
2
2
, (22)
R
3
= g
30
+ g
31
(L − x
1
− x
2
) + (23)
+
1
2
g
32
(L − x
1
− x
2
)
2
, (24)
K
3
= k
3
+ a
3
(L − x
1
− x
2
). (25)
Using current value Hamiltonian
H
c
= Π + m
1
u
1
+ m
2
u
2
, (26)
m
i
= λ
i
e
ρt
, i = 1, 2 (27)
the maximum principle conditions are:
∂H
c
∂u
i
≤ 0 (28)
u
i
∂H
c
∂u
i
= 0 (29)
˙x
i
= u
i
(30)
˙m
i
= ρm
i
−
∂H
c
∂x
i
, i = 1, 2. (31)
The condition (29) is satisfied either with u
i
= 0 or
∂H
c
∂u
i
= 0. The first one represents the situation
∂H
c
∂u
i
= σ
i
− b
i1
+ m
i
≤ 0
which means that the profit from increasing the area
of type i is smaller than the profit from preserving the
additional area unit and using it otherwise. In this
situation the area forested by type i remains constant,
i.e. ˙x
i
= u
i
= 0.
The latter solution branch
∂H
c
∂u
i
= σ
i
− b
i1
− b
i2
u
i
+ m
i
= 0 (32)
is the case of increasing the area forested by type i by
u
i
ha per year.
To obtain the control and state paths the particular
forms of the functions involved are needed. We intend
to obtain these by regression analysis using the real
world data.
4 SUBSIDY STRATEGY
In the previous section the subsidy rates were given
constants. Now we assume these rates can be changed
during time and we state the question how to control
the subsidies rates to obtain the desired forests struc-
ture in the least time possible.
Using optimal control formulation we obtain:
min W =
∫
∞
0
1dt (33)
˙x
i
(t) = f
i
(u
1
(t), . . . , u
n
(t), x
1
(t), . . . , x
n
(t)),(34)
u
i
(t) ≥ 0, (35)
x
i
(0) ≥ 0, given (36)
where the control variable u
i
(t) is the subsidy rate for
the forest type i, while the state variable x
i
(t) is as in
the previous problem the forest area of type i.
The state equations (34) reflect the influence of
the subsidy rates and present composition of the for-
est onto the area change for particular forest type.
Reflecting the particular decision making of the for-
est owner, these equations are uneasy to be obtained.
Since there is a lack of real data on how the deci-
sion maker react to change of the subsidies, we can
not simply use regression analysis. Here, we suggest
that interesting theoretical information could be ob-
tained using the data from simulation of the decision
maker’s behavior. These simulation are supposed to
be based on the linear programming approach that ap-
peared to be a valid tool to support the profit criterion
optimization problems in forestry (e.g. (D’Amours
et al., 2008)). Particularly, we formulate a linear pro-
gramming problem of profit maximization given the
present values of future streams of subsidies s
i
, rev-
enues r
i
and costs c
i
per 1 ha of forest type i:
max
n
∑
i=1
x
i
· (r
i
+ s
i
− c
i
) (37)
s.t. (38)
n
∑
i=1
x
i
= L, (39)
k
∑
i=1
a
i j
x
i
≤ b
j
(40)
k
∑
i=1
d
i j
x
i
≥ e
j
, 1 ≤ k ≤ n (41)
x
i
(t) ≥ 0, 1 ≤ i ≤ n. (42)
The decision variables are the areas forested by partic-
ular types and constraints (40-41) reflect existing area
limits for the particular forest types and their combi-
nations.
Solving the linear programming model (37-42) re-
peatedly for different subsidies rates s
i
, we generate
the simulated data set which can be used to estimate
the relation among rate of subsidies and the change of
the forest structure, and in this way, to help establish
the state equation (34).
OptimalControlforForestManagementintheCzechRepublic
469

5 CONCLUSIONS
Although the operations research techniques offer
promising decisions support help within the forestry
industry, its real applications by forest managers and
authorities is infrequent in the Czech Republic. Par-
tially this is due to the natural unwillingness of the
decision makers to deal with mathematical models or
even their results which is supported by often failures
of theoretical models’ when addressing the real world
problems.
In our research we focused on identification of
particular decision problems in forestry that are of im-
portance for Czech forestry industry and that are com-
plex enough that the decision support is desirable. We
suppose that our results will not provide the ready-
made solution, but may be of help when making the
final decision. In this contribution we focused on pre-
senting the selected problems and the first approaches
to deal with them.
ACKNOWLEDGEMENTS
The research is supported by the Czech Science Foun-
dation grant nr. 13-25897S.
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