COMPUTER PROGRAM FOR DETERMINING AN OPTIMUM SOLUTION 
IN LONG-TERM FOREST EXPLOITATION PROCESS 
INFORMATICA 4/1988 
UDK 519.854/.857 
Janez Barle, Janez Grad 
Ekonomska fakulteta Borisa Kidriča, Ljubljana 
One can define the long-term forest exploitation process in different ways. Methods of operations 
research can be used successfully for this purpose. A group of experts at the University of Ljub­
ljana, Vugoslavia developed a specific tree-growing function and, based on that, prescribed a set 
of procedures, relations and states which could represent mathematical model for optimizing the 
forest exploitation process. The mathod was defined as a descrete dynamic programming process, based 
upon Bellmanls principle. In this paper we briefly describe the model and computer program prototy-
pe for solving it. They both can easily be extended by introducing more complexity into them. The 
program was written in FORTRAN and tested on DEC-10 computer. 
1. INTRODUCTION 
In the paper we describe a possible usage of 
dynamic programming, based upon Bellman's 
principle, in order to obtain the optimum so-
lution of a predefined long-term forest explo-
itation process. In this pro­
cess measures and activities are introduced at 
each step within the iterative solution process 
and their effects on the intermediate results 
and the final solution are stated in'the form 
of mathematical relations. The whole forest 
area which is taken into consideration is divi-
ded into a number of homogeneous parts -^ seg-
ments. For each of these segments, five special 
functions are imposed by means of which "we di-
rect (guide) the process within a number of 
time intervals, either years or decades. The 
functions are: (Fl) the maximum possible tree 
growth capacity, (F2) exploitation capacity, 
(F3) the quality of the existing tree speci-
mens, (F4) level of administration - čare for 
improved growing conditiond, and (F5) stage 
in segment development, based upon the age 
(oldness) of the tree specimens in it. The 
functions help to optimize exploitation ende-
avours. At each step of the interative soluti­
on process four possible activities can be 
imposed: (Al) no activity at ali, (A2) exchange 
of the existing tree specimen with a new one, 
(A3) rarefying, and (A4) restoration. The ex-
ploitation policy is defined by a seguence of 
chosen activities during the iterative pro­
cess. The stated functions, activities and 
time intervals are part of the mathematical 
model and dynamic programming process respec-
tively. The problem and the corresponding com­
puter program can be extended by incorporating 
some sdditional functions and activities. Pro­
gram prototype in FORTRAN was written and tes­
ted on DEC-10 computer at the University of 
Ljubljana, Yugoslavia. In the paper we also 
comment the problems which are associated with 
dynamic programming applied to such type of 
problem. 
2. A PREDEFINED LONG-TERM FOREST EKPLOITATION 
PROCESS 
We mean by that a stated seguence of policies 
and activities that should be pursued in fo­
rest exploitation within a life-time long peri­
od of some tree specimen in order to achieve 
optimum results in accordance with predefined 
goals and criterions. The possible and necess-
ary actions and the most siuitable time for 
these actions to be carried out depend on the 
values of some chosen and defined functions 
which describe sufficiently the tree-growing 
process. It is these functions, which must 
be predefined by a group of experts in forest 
exploitation process - both practicians and 
theoricians, by means of which more or less 
complexity and reality is introduced in the 
basic model that is to be solved. After seve-
ral years of analytic and experimental studies 
of a group of experts in Ljubljana, Yugoslavia 
(VADNAL, KOTAR, ZADNIK, STIRN, GAŠPERŠIČ /2/, 
VADNAL, ZADNIK, STIRN /3/), the following 
suggestions and basic ideas have been made: 
(i) The paradigm is bounded both geographicaly 
and in time. The whole area considered is di-
vided into smaller regions which are further 
partitioned into smaller parts with specific 
environmental conditions and characteristics. 
Some parts within different regions can have 
similar or nearly equal environmental condi­
tions and characteristics. They can or can not 
be exploited (treated) independently from each 
other. The area is exploited for a specific 
number of time periods, measured in years or 
decedes. 
(ii) In order to evaluate (quantify) observed 
characteristics of each part of the forest un-
der consideration, time dependant analytical 
growth functions have been developed: 
1) Function of growth 
Y(t) = a(l - (1+T)e X p(-T)), 
where T = (2n - 1) t"/(np"), for a > O, 
p > O, and n > 1. a defines the asymptotic 
value of Y(t), p is the value of t 

40 
from where on Y(t)' starts to decline, and n 
determines the speed of convergence y(t) towa-
rds value a. The value of t = t,, when the fo­
rest restoration procedures start can be def-
ined by solving the equation 
l + T + T^-exp(T) =0 
which is obtained from the relation 
t,Y'(t,) = VCt,), or can be stated on the base 
ot experiences. 
2) Function of increase (by growth) y(t) = 
Y(t)' where y(t2)' = y(p)' = 0. 
3) Function of average growth f(t) = Y(t)/t, 
where f(t,) = y(t-), and therefore t,Y(t,)= 
Y(t3). -5 J J J 
4) Function of uniform growth s(t) = mt, 
where sCt^) = Vtt^), and s{t3) ' = Y(t3)'. 
(iii) Five functions have been introduced 
which describe five possible states of forest 
exploitation process: 
1) The maximum possible tree growth capacity -
Fl. Fl is constant during the exploitation 
process. 
2) Exploitation capacity - F2. F2 is a conti-
nouous function, where 1 £ F2 £ 10. Its 
value shows the degree in forest exploita-
tion. 
3) The quality of the existing tree specimens 
- F3. 
This function enables us to make an assess-
ment about the quality of the trees and 
whether to begin the exploitation process of 
some particular part of the forest or not. 
The parts are divided into five gualitative 
groups, 1 £ F3 £ 5. 
4) Level of administration - čare for better 
growing conditions - F 4. 
The value of F4, where 1 £ F4 £ 5, shows a 
degree of obstructing influence of unwanted 
tree species on growth of the wanted ones. 
5) The segment development stage - F5, based 
upon the age (oldness) of the tree speci­
mens in it. Tree species are divided into 
four stages, accordingly: 
O < F5 < t. 
t^ 1 F5 < t^ 
4 1 
F5 < t 
3' 
and 
tj _< F5 
1^4^ 
Tree species of different stages differ both 
in size and quality and can therefore be 
used for different purposes. 
(iv) The forest exploitation process is carried 
out by performing a seguence of prescribed ac-
tivities. The seguence order of these activit-
ies helps to optimize the process. The follo-
wing four activities have been taken into 
consideration: 
1) No activity at ali. - Al. 
In this čase the forest develops in accor-
dence with the laws of nature. The value of 
F2 changes only within the first two stages 
of segment development; it is diminished for 
an empirically defined quantity pr(stage, 
F2, Al). Similarly, the values of F3 and F4 
change within the first three stages of de­
velopment. They too are diminished for some 
empirically defined guantities pr(stage, F3, 
where t. indicates the upper time limit of 
the stated tree growth cycle. t, is defined 
Al) and pr(stage, F4,A1), respectively. Each 
time the value of the increment (1 for a 
year or 10 for a decade) is added to the pre-
vious value of F5. At the boundary points 
this causes a change of the stage of deve­
lopment. 
2) Exchange of the existing tree specimen with 
a new one - A2. We can pursue this activity 
only within the first two segment development 
stages. The reason for doing that is the una-
dequate existing tree species. The exchange 
is carried out there - in those parts (seg-
ments) where the effects are most visible. 
A2 exercises the following influence upon 
F2, ..., F5: Their values are changed only 
within the first two stages of segment de­
velopment. F2, F3 and F4 are increased by 
some empirically defined quantities pr(sta­
ge, F2, A2),pr(stage, F3, A2) and pr(stage, 
F4, A2), respectively, while the value of F5 
is reduced to zero and the process starts 
from the beginning. 
3) Rerefying - A3. 
The activity can only take plače within the 
first three segment development stages. Wit-
hin the first stage it is exercised by cutt-
ing dovifn the unwanted species only what helps 
in guicker growth of the wanted ones. Here 
the activity imposes additional expenses. 
Within the second and third stage, in addi-
tion to the cutting down of the unwanted 
species, we also cut down some trees of the 
wanted species what helps in guicker growth 
of the most qualita"tive samples. Here the 
activity brings some profit. Due to A3 the 
values of F2, F3 and F4 are changed by some 
empirically obtained guantities pr(stage, F2, 
A3), pr(stage, F3, A3), and pr(stage, F4,A3), 
respectively. A3 has no particular impact on 
F5. The time increment (1 or 10) is added to 
the value of F5 after the time period ex-
pires. 
4) Restoration - A4. 
Restoration starts at t = t3, ends at t = t4, 
and can take different length of time. It is 
characterized by wood-cutting on higher scale. 
After restoration is done, the forest segment 
under consideration passes over into the 
first stage. The way in which the stage of 
restoration is being accomplished has a great 
influence on the results of the forest explo-
itation process. The span of time t^ - t3 is 
generally divided into more steps. At each 
step the increments of F2 and F3 are compu-
ted and added to the previous values of F2 
and F3, respectively. F4 does not change 
within A4, while the increment 1 (10) is 
added to F5 after each year (decade) that is 
passing by. F5 reduces to zero or some higher 
value after the restoration is over. 
Ak). Each measure 
ny step of the exploit-
can be expressed in 
The resulting inco-
say I(stage, Ak), 
s upon Fi, for i = 1, 
the undertaken acti-
can be divided into 
variable costs 
as t^ = a/10. 
(v) The outcomes - R(stage 
- activity undertaken at a 
ation process, results and 
a form of costs and income 
me of some particular step 
for k = 1, 2, 3, 4, depend 
2, ..., 5, and the size of 
vity Ak. The entire costs 
fixed costs FIX(stage) and 
V(stage, Ak). Accordingly, 
R(stage, Ak) = I(stage, Ak) -FIX(stage) -
- V(stage, Ak) 
1) R(stage, Al). 
In this čase we have no income and no vari­
able costs. Therefore R(stage. Al) = -
- FIX(stage) 
where FIX(stage) is some empirically obtained 

41 
quantity. 
2) R(stage, A2). 
Activity A2 is carried out only within the 
first two stages of the exploitation pro-
cess. We have some possible income only wit-
hin the secohd stage, which can be expressed 
as qi (stage, A2)Y(t) where q.^(2,A2) is 
some empirically stated weight (factor), and 
qi(l, A2) = 0. Fixed costs FIX(stage) are 
defined (empirically). Variable costs are 
obtained as a total of the costs of removal 
the unwanted species, say qc(stage, A2)Y(t), 
and the costs of plariting new samples, 
say NT(stage), where g^ and NT are defined 
empirically. Accordingly, 
RCstage, A2) = qi(stage, A2)Y(t) -
-FIX(stage) - q5.(stage, A2)Y(t) - NT(stage) 
for stage =1,2. 
3) .R(stage, A3). 
Activity A3 is carried out within the first 
three stages. The outcome R(stage, A3) can 
be expressed as 
R(stage, A3) = qi(stage, A3)Y(t) - FIX(sta-
ge) - qc(stage, A3)Y(t) for stage = 1,2,3, 
where g^, FIX, and g^ are obtained empiri-
cally and qi(l, A3) = O . qi and g^ have a 
similar meaning as in čase of R(stage, A2). 
4) R(4, A4). 
Activity A4 is carried out only within the 
fourth stage. We differentiate two possibi-
lities: 
I. t3 = t4, when ali work is done in a very 
short tirne. In this čase we deal with one 
outcome only, defined as 
R(4, A4) = qi(4, A4)Y(t3) 
qc(4, A4)Y(t3) - NT(4) 
FIX(4) 
II.t3 + t4, and assume that there are d4 steps 
of the exploitation process within the fo­
urth step. At each step n4, where n4 = 1,2, 
...., d4, we compute R(4, A4, n4) as 
R(4,A4,n4) = I— qi(4, A4, n4)Y(t) -
-FIX(4) - —- qc(4, A4, n4)Y(t) - NT(4, n4) 
where again g-^, 
rically. 
and NT are defined empi-
(vi) Managing the exploitation process is poss­
ible before and after the process being in 
progress. By this we mean some further pres-
cribed conditions and rules which should be 
or should not be taken into account within 
the problem solving process, in accordanoe 
with the type of optimization process that 
we pursue. We distinguish among the follow-
ing possible alternatives: 
No alternations of the prescribed exploita-
tion process are possible while the process 
being in progress. 
Some alternations of the originally prescri­
bed exploitation process are possible while 
the process being in progress. We may inte-
rrupt the process, insert some new input data 
and proceed the process from this point on 
or start the program from the beginning. 
The final result of the process is prescri­
bed in advance as well as the starting 
conditions. 
The final result is the optimum value that 
can be obtained by the prescribed starting 
conditions. 
3. DYNAMIC PROGRAMMING PROCESS ALGORITHM 
PROTOTYPE 
3.1. General description of the descrete dyna-
mic programming 
Descrete dynamic programming process is an i-
terative process (BELLMAN, DREYFUS /1/). At ea­
ch step i, for i = O, 1, ..., N, we define a 
certain number of possible points Xij, for j = 
1.2, ..., Mj^. For each point we compute the 
function value fj^j which is involved in the pro­
cess of optimization. This value is the optimum 
value among function values fi-i k/ for k = 1, 
2,... Mi_i, incremented by the computed outco-
mes between Xj__i Y. ="^'^ this particular point 
Xij. After computing f^j for ali i = O, 1, ..., 
Nand j = 1, 2, . . . , Mj^ we define the optimum • 
seguence of the operations and decisions that 
were made during the exploitation process for 
ali steps i = N, N-1, ..., 0. Sometimes two or 
more alternatives are possible which ali give 
the same optimum solution. 
3.2. Forest exploitation dynamic programming 
algorithm 
In order to start the process we need the foll-
owing input data: 
a, p, n, ti, t2/ t3, t4; in the prototype we 
don't compute the values of t^, t2 and t3 
but we read them as input data instead. 
For each segment, i.e. for each čase of the 
exploitation process: 
ali empirically defined values of FIX(stage), 
.NT(stage), qi(stage, Ak), and g^(stage, Ak), 
for stage =1, 2, 3, 4 and k = 1, 2, 3, 4. 
ali empirically defined values of pr(stage, 
Fi, Ak) , for stage = 1, .'. . , 4, i = 2, 3, 4 
and k =1, ..., 4, that represent the incre-
ment of Fi due to the activity Ak. 
F2, F3, F4, F5 which define the starting con­
ditions of the forest exploitation process. 
The whole experiment can be repeated several 
times for different starting values of F2, 
..., F5. 
The process is as follows: 
For each t, where t.= F5 + 10, F5 + 20, ..., in 
accordance. with the stage to which t belongs 
(stage = 1, 2, 3 or 4), the possible activities 
(Al, A2, A3 and/or A4) at that stage take plače 
for ali existing (active) points xt-io s (see 
3.1.), where s=l, 2, 3, .... 
For each activity that takes plače at some 
''t-10,s the following happens: 
new values of F2, F3, F4 and F5 are computed, 
defining a new point x^^ j and a new step. 
There is: new value of Fi = old value of Fi + 
pr(stage, Fi, Ak), for i = 2, 3, 4, where "-" 
sign appears for Al, and "+" sign appers for 
A2, A3 or A4, respectively. The value of F5 is 
increased by 10 or reduced to zero (for A2 or 
when F5 > t4). 
the outcomes R(stage, Ak), see chapter 2.(v), 
aro computed and addcd to ft-10 s (see 3.1) 
in order to obtain ft,j-
Some of the 250 possible different points xtj, 
for j = 1,2, ..., 250, are encountered at each 
step t of the iterative process. They are 
defined by different values of F2, F3 and F4 
(10x5x5 = 250). The computer program builds two 
arrays X(250, 6) and FX{250) at each step of 
the iterative process in order to save ali the 
the necessary intermediate data. The elements 

42 
in row j of array X contain the following data 
about the point xt,j: 
xji = F2, Xj2 = F3, Xj3 = F4, Xj4 = k, Xj5 = 
= s, Xj6 = F5 
where 
k (=1, 2, 3 or 4) definds the activity Ak ta-
ken at step t-10 that caused a transition to 
xt,j, and 
s definds the point xt-io,s fJ^om which the 
transition to xt,j was made. 
s stands for the'row number of array X at pre-
vious step (t-10) in which the data about 
xt-io,s''^^ stored. 
The elements FXj, for j = 1, 2, ..., 250, re-
present the values of the computed ftj-
II. 
After completing the first part of the algori-
thm we locate the optimum value fT,j i" the 
last step T of the interative process, where 
, 250) 
optimum (fT,s' ^°^ s = 1, 2, 
Afterwards we trace back to the beginning ali 
actions that vere carried out during the explo-
itation process. We do this by means of data 
stored in arrays X and FX. 
If the number of steps in the iterative process 
is fixed and defined, let say by t4/10, then 
'^ '^ t4 in čase when the starting value of F5 
= O, and T < t4 otherwise. 
4. CONCLUSIONS ON THE APPROACH 
Obtained experiences show that a method for op-
timising the forest exploitation process, based 
upon descrete dynamic programming is reasona-
ble and adeguate. More complexity and necessary 
modifications can easily be introduced after 
obtaining and analyzing some experimental re-
sults. Large number of input data and interme-
diate results which are storage demanding may 
be regarded as the only inconvenience. Diffe-
rent strategies may reduce this problem by 
applying the secondary disk storage at each 
step of the interative process. The empirica-
lly obtained input data can also be stored in 
files in advance and kept there for as long as 
necessarv. 
REFERENCES 
1. Bellman, R., E., Dreyfus, S., Applied Dyna-
mic Programming, Princeton University Press, 
Princeton, New Yersey, 1962. 
2. Vadnal, A., Kotar, M., Zadnik Stirn, L., 
Gašperšič, F., "Uporaba rastnih funkcij v 
gozdarstvu". Zbornik gozdarstva in lesar­
stva 23, str. 149-179, Ljubljana 1983, 
Yugoslavia. 
3. Vadnal, A., Zadnik Stirn, L., Matheraatical 
Model for Long-term Silvicultural and Uti-
lization Planning, SYM OP IS'86, Herceg 
Novi, 7-10 October 1986, Proceedings, p.p. 
183-189, FON, Belgrade 1986, Yugoslavia. 
Fig.i"- AnQlytical grovvth functions[2.3] 

43 
RAČUNALNIŠKI PROGRAM ZA DOLOČITEV OPTIMALNE 
REŠITVE V DOLGOROČNEM IZKORIŠČANJU GOZDOV. Pro­
ces dolgoročnega izkoriščanja gozdov moremo o-
predeliti na več načinov. Primerne v ta namen 
so tudi metode operacijskega raziskovanja. Sku­
pina strokovnjakov Univerze Edvarda Kardelja v 
Ljubljani je za določitev matematičnega modela 
optimizacije postopka izkoriščanja gozdov raz­
vila posebno rastno funkcijo in na temelju le­
te definirala zaporedje potrebnih postopkov, 
relacij in postulatov. Metoda je bila definira­
na kot postopek, ki temelji na diskretnem dina­
mičnem programiranju. V članku zgoščeno opiše­
mo prototipa modela in računalniškega programa 
za njegovo rešitev. Model je možno razširiti z 
vgraditvijo nadaljnjih zahtev in pogojev. Prog­
ram je napisan v programskem jeziku FORTRAN in 
je bil testiran na računalniku DEC-10.