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1. A forest planning problem solved via a linear fractional
goal programming model
T. Gómez a
, M. Hernández a
, M.A. León b
, R. Caballero a,*
a
Department of Applied Economics (Mathematics), University of Málaga, Campus El Ejido s/n, 29071 Málaga, Spain
b
Department of Mathematics, University of Pinar del Rı́o, Pinar del Rı́o, Cuba
Received 12 May 2005; received in revised form 2 February 2006; accepted 7 February 2006
Abstract
We present a linear fractional goal programming model to a timber harvest scheduling problem in order to obtain a balanced age class
distribution of a forest plantation in Cuba. The forest area of Cuba has been severely reduced due to indiscriminate exploitation and natural
disasters (fires, hurricanes, etc.). Thus, in this particular case, the main goal is to organize and regulate the forest. This involves a significant change
from its current distribution by ages to obtain a more even-aged structure over a planning horizon of 25 years which coincides with the rotation age.
This has been formalized as fractional goals which take into account the evolution of the forest and ensure attaining a balanced age class
distribution in a progressive and flexible way. The proposed model aims at achieving this new distribution while bearing in mind the economic
aspects of the forest as well as other factors. In order to test its potential we have applied the model to a Cuban plantation belonging to the forestry
company ‘‘Empresa Forestal Integral Pinar del Rı́o’’. We obtained several solutions that provided a regulated forest while respecting the economic
and other targets of the decision-makers.
# 2006 Elsevier B.V. All rights reserved.
Keywords: Forest management; Goal programming; Fractional programming
1. Introduction
Decision-making in forest planning has currently become a
multidimensional decision context, concerned with multiple and
sustainable use of the forests. They are not envisaged simply as a
source of goods and services; rather, the preservation of
biodiversity and environmental protection are also factors to
be taken into account. In fact, as Diaz-Balteiro and Romero
(2004) pointed out ‘‘the modernview of sustainability comprises
not only the classic timber production persistence but also the
sustainability of many attributes demanded by society and
produced by the forest’’. Therefore, we need multiple criteria
decision-making models to manage any forest system.
Field (1973) was a pioneer in this area who analysed a forest
planning problem using a multicriteria framework. He
considered three objectives to be relevant: income, timber
production, and recreational activities. From this time onwards
many other works applying multicriteria techniques to forestry
problems were published. Some authors have used goal
programming for timber production planning (Kao and Brodie,
1979; Field et al., 1980; Hotvedt, 1983). This technique proves
most suitable when we deal with a set of conflicting objectives
that need to verify some given thresholds or target values
chosen by the decision-maker who must also provide his/her
preferences regarding the achievement of such targets. On the
other hand, there are some works which do not require a priori
information about the decision-maker’s preferences regarding
the relative significance of the goals; this is the case of Dı́az-
Balteiro and Romero (1998, 2003) who designed a multigoal
programming model and obtained the best-compromise
solutions validated in terms of optimal utility. Other authors,
such as Steuer and Schuler (1978), De Kluyver et al. (1980),
Hallefjord et al. (1986), Bare and Mendoza (1988), and Kazana
et al. (2003), have used interactive multiobjective models,
where the preferences of the decision-maker are included
throughout the solving process in order to explore the set of
non-dominated solutions.
From this brief review of the literature, we can see that there
is no single multicriteria technique able to solve all forest
management problems and that none of the available methods is
www.elsevier.com/locate/foreco
Forest Ecology and Management 227 (2006) 79–88
* Corresponding author. Tel.: +34 952 131168; fax: +34 952 132061.
E-mail address: r_caballero@uma.es (R. Caballero).
0378-1127/$ – see front matter # 2006 Elsevier B.V. All rights reserved.
doi:10.1016/j.foreco.2006.02.012

2. superior to the rest. Therefore, the selection of a particular
method is driven by the type of information available and the
specific characteristics of the problem.
In this work, based on the information provided by the
decision-maker, we opted fora goal programmingmodel to solve
the forest planning problem provided. Futhermore, bearing in
mind the decision-maker’s preferences regarding goals, the
lexicographical approach was used to classify the different goals
according to priority levels. On the other hand, we used a
fractional goal to express the wish of the decision-maker to
regulate the plantation,in order to measure the relative difference
between the areas of two different age classes. We have no record
of harvest scheduling problems solved by using fractional
programming in the literature, perhaps due to the complexity of
the model required to solve them. Fractional programming is
commonly used in different management problems that require
the relative comparison of two magnitudes (for example cost/
time, cost/volume or output/input). Thus, we consider this
approach to be of great interest in forest management problems
that require this kind of comparison. In addition, fractional
programming is used when the efficiency of a system is to be
measured (Charnes et al., 1978). This approach has received
special attention in the last 4 decades in relation to different
management areas (Schaible, 1995). The first papers on
fractional programming were published in the 1960s, and from
then on multiple applications were developed in the literature
(Aggarwal, 1969; Kornbluth, 1983; Eichhorn, 1990).
Our forest management problem is situated in Cuba, where
forestry products are of high importance to the national
economy. However, the forestry area of Cuba has suffered
dramatically due to indiscriminate exploitation and natural
disasters (fires, hurricanes, etc.), as can be seen in Fig. 1. This
situation, together with the fear of greater ecological disasters,
has given rise to conservationist policies which lead to preserve
old growth forests with subsequent financial losses and other
problems. This also means that Cuban forests have a highly
imbalanced age structure and, thus, an important objective in
the Cuban context is to plan the redistribution of the forest into
even-aged stands.
However, Cuba is making great efforts regarding reforesting
and caring for its natural forests. The forestry law passed in July
1998, states that one of its core objectives is regulating the
multiple and sustainable use of our forests and promoting the
rational exploitation of forest products. Thus, as forestry policy,
Cuba isplanning toincreasethe numberofplantationswhich will
cover the timber needs of the country, and so decrease the
pressure on natural forests. By the end of 2003, the surface
covered by forest in Cuba was 2618.7 thousand ha (23.6% of the
archipelago surface). Of the total, 2254.8 thousand ha were
natural forests and 332.4 thousand ha were plantations, mainly
conifers, eucalyptus and other special species. The Cuban
Forestry Economic Development Program (Ministry of Agri-
culture, 1996) projects that by 2015 there will be around
700,000 ha of plantations with different productive purposes and
356,000 ha of natural forest will be improved and restored for
forestry production. As mentioned before, another main aim in
this context is the management of the existing plantations. Thus,
some Cuban forestry companies have focused on achieving an
almost completely regulated structure for their plantations in
order to ensure a sustainable flow of timber. In this line, the Pinar
del Rı́o University has been authorized to carry out this kind of
work with the forestrycompaniesintheregion.Thepresentstudy
is framed within this approach, and is preceded by the work of
León et al. (2003), where a goal programming model with linear
goals was formulated for the management planning process of a
Pinus caribaea plantation in this province. This model did not
attempt to obtain a balanced even-aged distribution over the
planning horizon and thus, in our work, we include fractional
goals which take into account the evolution of the plantation to
formalize this factor. On the other hand, once the existence of
solutions that verify the target values have been confirmed, we
apply an efficiency restoration technique. We have chosen the
interactive restoration technique (Tamiz and Jones, 1997) from
among the most well-known restoration techniques (Hannan,
1980; Caballero et al., 1998, etc.).
The lexicographic GP model with fractional goals proposed is
applied to a plantation within the San Juan y Martı́nez Forestry
Unit, which belongs to the forestry company ‘‘Empresa Forestal
Integral Pinar del Rı́o’’ from the town San Juan y Martı́nez, in the
southwest of the Pinar del Rı́o province (Cuba). This plantation
comprises 44.8% of the total area owned by this company and its
assets are oriented to different objectives: to provide raw
materials and to protect coastal ecosystem. Pine is the main
species occupying 77.26% of the area (Pinus caribaea, Pinus
tropicallis). In the planted forest there are 3984.3 ha of Pinus
caribaea whose main function is to provide small-sized timber
used directly by the tobacco industry. This is very important in
this area which is one of the best areas in the world for tobacco
cultivation, and also pulp production.
The structure of this paper is as follows: in the next section,
we review and summarize linear fractional goal programming,
and highlight the most relevant results that will be used later to
solve the problem. In Section 3 we develop the model
proposed. In Section 4 the model is applied to a specific case,
the San Juan and Martı́nez Management Unit. We then analyse
the results obtained. Finally, in Section 5 we draw some
conclusions followed by Appendix A, Appendix B and
References.
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88
80
Fig. 1. Evolution of forest area in Cuba.

3. 2. Goal programming with fractional goals
The main interest in fractional programming was generated
by the fact that various optimization problems from engineering
and economics require the optimization of a ratio between
physical and/or economic functions. Such problems, where
the objective function appears as a ratio or quotient of other
functions, constitute a fractional programming problem.
Thus, if these quotients have to verify certain target values,
we would have a set of fractional goals. In such a case, the
formulation of the problem to be solved is quite complex
because of the non-linearity of these goals. Thus, in this
section we look into the formulation of these types of
problems, called fractional goal programming problems, and
see the difficulties inherent in their resolution and how they
can be overcome.
In order to explain the general framework of a fractional GP
problem, let us assume that we have a problem such that q of its
attributes are linear fractional functions and such that its vector
of decision variables, x, has to verify a set of r linear constraints,
that is,
’1ðxÞ ¼
ct
1x þ a1
dt
1x þ b1
; . . . ; ’qðxÞ ¼
ct
qx þ aq
dt
qx þ bq
(1)
Ax b; x 0
where cm, dm, 2 Rn
, am, bm 2 R for m = 1, . . ., q; A 2 MrxnðRÞ
and b 2 Rr
. Let X be the feasible set for this problem; that is,
X ¼ fx 2 Rn
=Ax b; x 0g. It is assumed that dt
mx þ bm,
m = 1, . . ., q are strictly positive for every x 2 X.
Assuming that the decision-maker wants the attributes
wm(x) to surpass a target value, um (m = 1, . . ., q), the
problem is to determine whether there is any solution x that
verifies the constraint set and the q goals (also called soft-
constraints):
Ax b; x 0; ’mðxÞ ¼
ct
mx þ am
dt
mx þ bm
um; m ¼ 1; . . . ; q
(2)
The procedure followed in goal programming is to transform
the system of inequalities (2) into an optimization problem,
where some of the decision-maker’s preferences can be
included in terms of lexicographical priority levels. In order
to include the goals in the formulation of the optimization
problem, deviation variables, usually denoted as nm and pm, are
used. These variables, which are not negative, measure the
difference existing between the target values and the result
actually obtained for each of the attributes. In this case, where
we want all the target values to be surpassed, the deviation
variables to be minimized are the negative ones (nm). For more
details on goal programming, we recommend Chapter 1 of
Romero (1991).
In this way, if we establish priority levels to satisfy the q
goals, and we are in a certain level s (where the index set of the
goals in s will be denoted by Ns), the optimization problem that
has to be solved is the following:
min
X
m 2 Ns
wmnm
s:t: x 2 Xs
ct
mx þ am
dt
mx þ bm
þnm pm ¼ um m 2 Ns
nm; pm 0
(3)
where Xs = {x 2 X/wL(x) uL, L 2 N1, . . ., Ns1}, and wm is the
weight of the mth goal and nm, pm are the negative and positive
deviation variables for the goals in this level. If its solution
nullifies the objective function, then this is the solution that
satisfies all the goals at this priority level and we move on to the
next level. The disadvantage of this model is that some of the
constraints for problem (3) are non-linear, due to the previous
fractional goals. If we multiply such non-linear constraints by
factor dt
mx þ bm (by hypothesis, this is always positive in X), we
reach the following linear problem:
min
X
k
m 2 Ns
wmn0
m
s:t: x 2 Xs
ct
mx þ am ðdt
mx þ bmÞum þ n0
m p0
m ¼ 0
n0
m; p0
m 0 m 2 Ns
(4)
Obviously, for problems (3) and (4) to be equivalent, the
relationship between their variables has to be the following:
p0
m ¼ pmðdt
mx þ bmÞ; n0
m ¼ nmðdt
mx þ bmÞ m 2 Ns:
Although there is a close relationship between problems (3)
and (4), they are not equivalent (Awerbuch et al., 1976; Soyster
and Lev, 1978). However, if we focus on the search for solutions
of Xs that verify all the goals at the current priority level, we
only need to solve the linear problem (4) to deduce the
existence or non-existence of such solutions, as is shown in the
following theorem (Caballero and Hernández, 2006):
Theorem 1. Given problems (3) and (4) the following asser-
tions are valid:
(i) If, when solving (4) the solution is ðx
; n0
m; p0
mÞm¼1;...;k such
that Smwmn0
m ¼ 0, then there is at least one solution that
satisfies the goals in level s of the linear fractional problem
(2), which is identical to x*.
(ii) If, when solving (4), the solution ðx
; n0
m; p0
mÞm¼1;...;k, is such
that Smwmn0
m 0, then there is no solution that satisfies the
goals of the linear fractional problem (2) for the priority
level s.
Consequently, to solve the original problem (2) in a level s,
we move from problem (3) to its associated linear problem (4),
and resolve it. If the solution is such that the value of the
objective function in the optimum is zero we can be sure that the
point obtained is a solution that satisfies all the goals in level s
of the fractional problem (2). Otherwise we can guarantee that
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88 81

4. there will not be a point in X that satisfies all of these goals given
the current target values. In such a case, in order to find the point
that minimizes the nonachievement in Xs to the given target
values, we solve problem (3) directly by applying an algorithm
that calculates the point which minimizes the weighted sum of
the angular distances to the feasible set of the goals that cannot
be satisfied.
In this work, we use a fractional goal to measure the relative
difference between two areas. One of the objectives of the DM
is to reach a balance-aged structure by the end of the planning
horizon. As we will see in the next section, this desire for
balance has been modeled by using fractional programming by
comparing the areas occupied by two age classes in a relative
way (by the quotient between these areas), in order to get this
relative difference (this quotient) as close to one as possible at
the end of the last period. All this is done in a progressive way
during the planning periods, forcing the corresponding ratio to
be greater period by period.
3. The model
The model is initially formalized in a general way and then
applied to the specific case of a Cuban plantation which belongs
to the ‘‘Empresa Forestal Integral Pinar del Rı́o’’ forestry
company.
Let us assume that the plantation area to reorganize is
managed for wood production (pulpwood and small-sized
timber) and is classified according to productivity (site class)
and by the age of the stands (age class). Thus, the starting
situation is given by the following matrix:
S0
¼
s0
11 s0
12 . . . s0
1I
s0
21 s0
22 . . . s0
2I
. . . . . . . . . . . .
s0
H1 s0
H2 . . . s0
HI
0
B
B
B
@
1
C
C
C
A
where s0
hi is the total number of hectares of the site class h (h = 1,
2,. . ., H) withintheageclassi (i = 1,2,. . .,I) atthe starting point.
The sum of the column elements of the matrix shows the avai-
lable area at the starting point in each age class (S0
i ¼
PH
h¼1 s0
hi),
whereas the sum by rows gives the available area in each site
class (S0
h ¼
PI
i¼1 s0
hi). In this model, we want the number of age
classes to be constant throughout the planning process. For
each site class, the last age class is formed by the rotation age
stands, and this rotation age is the same for all the site classes. In
any other case, the model could be applied by each site class.
The planning horizon (T) has been divided into periods, so
that, when a period has elapsed, the trees in age class i become
age class i + 1. Thus, if t is the number of years in each class (for
reasons of simplicity we assume this number is constant), the
number of periods under consideration, denoted by P, is equal
to the number of years of the planning horizon divided by t. If
the plantation evolves without intervention and mortality or
disaster, the actual development of the stands would lead us to
move from matrix S0
to matrix S1
, and so on.
Therefore, the decision variables of our model represent the
number of hectares of a specific site class h (h = 1, 2, . . ., H) and
age class i (i = 1, 2, . . ., I) with intermediate treatment or final
cutting j (j = 1, 2, . . ., J) at period p ( p = 1, 2, . . ., P), denoted
by xp
hi j. The treatment to apply depends on age, and so the value
of the subscript j depends on the value of i, j 2 N(i), where
NðiÞ ¼ f j=ði; jÞ 2 Ng and N = {(i, j)/j is the treatment
corresponding to age class i}. Clearcutting is denoted by J,
the last value of the subscript j.
Due to the evolution of the forest, sp
hi depends on the area of
the previous period in the following way:
sp
h1 ¼
X
I
i¼1
xp
hiJ; h ¼ 1; 2; . . . ; H (5)
sp
hi ¼ s
ð p1Þ
hði1Þ xp
hði1ÞJ; i ¼ 2; . . . ; I 1; h ¼ 1; 2; . . . ; H
(6)
sp
hI ¼ s
ð p1Þ
hðI1Þ xp
hðI1ÞJ þ s
ð p1Þ
hI xp
hIJ; h ¼ 1; 2; . . . ; H (7)
In other words, the total area of age class 1 at the end of
period p is the total number of hectares harvested during that
period. The total area of age i (greater than 1) by the end of
period p is equal to the area occupied the stands of the previous
age which has not been harvested during this period. On the
other hand, the total at age I (last age class) is made up of what
was already in this age class plus what there was in age class
I 1 and which has not been felled in either case.
In our context, the following premises summarize the wishes
of the decision-maker:
The total harvested volume should be sustained for each
period into which the time horizon is divided.
The area covered by each age class should be roughly the
same by the end of the planning horizon.
Whenever possible avoid clearcutting at early ages.
The net present value (NPV) must be higher than a certain
threshold throughout the planning period.
We formalize the previous premises as goals, that is, as soft
constraints and thus our model becomes a goal programming
problem. The preferences regarding the satisfaction of the goals
are modelled by using the lexicographic approach according to
their priority and taking into account that they are the same in
each period p ( p = 1, 2, . . ., P).
3.1. First priority level
The area to which clearcutting is applied (j = J) should not
exceed the percentage of the forest area that would ensure the
replacement of the forest. Thus, the area which ensures the
perpetuation of the forest harvest in site class h, for period p,
Sep
h should not be exceeded. This area Sep
h is given by the total
area in site class h divided by the rotation age and multiplied by
the number of years in each class (the time span which defines
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88
82

5. the age class). Therefore, in each period, we have the following
H goals:
X
I
i¼1
xp
hiJ þ np
1h pp
1h ¼ Sep
h ; h ¼ 1; . . . ; H (G1)
where n1 and p1are the negative and positive deviation vari-
ables, respectively, and for each site class the positive ones are
unwanted.
In addition, given that all goals have the same relevance, the
function to be minimized in this level is the sum of the positive
deviation variables multiplied by the normalizing coefficient
1=Sep
h in order to prevent bias (see Romero, 1991).
3.2. Second priority level
We aim at keeping harvest levels up to the maximum
sustained yield. Thus, if Vp
represents this maximum sustained
volume at period p and vp
hi j is the volume per hectare harvested
from each site class, age, treatment and period (where due to the
aims of the plantation we assume that there are no differences in
volume from previously treated areas or non-treated areas), this
goal can be expressed by the following equation:
X
H
h¼1
X
ði; jÞ 2 N
vp
hi jxp
hi j þ np
2 pp
2 ¼ V p
(G2)
As before, the positive deviation variable is the one to be
minimized.
3.3. Third priority level
The area covered by each age class should be roughly the
same by the end of the planning horizon. This is expressed by a
goal establishing that the ratio between the number of hectares
in the first age class and the last age class in each period must be
above a target value. Thus, this is a fractional goal formulated
as follows:
Sp
1
Sp
I
þ np
3 pp
3 ¼
1
P
p; p ¼ 1; . . . ; P (G3)
where Sp
1 ¼
PH
h¼1
PI
i¼1 xp
hiJ and Sp
I ¼
PH
h¼1 s
ð p1Þ
hðI1Þ xp
hðI1ÞJ
þs
ð p1Þ
hI xp
hIJ (see (5) and (7)).
The target values increase within each period in such a way
that in the last period the target value is 1. If this last value is
reached, a balanced age class distribution by the end of the last
planning period is ensured (see Appendix A). In this case, the
unwanted deviation variable is the negative one.
In order to ensure such regulation is possible, we have to
assume that P I, but this assumption does not imply that
regulation must be achieved in one rotation, since expression
(G3) can reach value 1 before the last period.
3.4. Fourth priority level
We try to regulate the forest without having to sacrifice
young stands in the process, so no stand under age class I 1
should be cut. Consequently, this goal is formulated as
follows:
X
H
h¼1
X
I2
i¼1
xp
hiJ þ np
4 pp
4 ¼ 0 (G4)
where the positive deviation variable is the one to be mini-
mized.
3.5. Fifth priority level
Finally, the following goal reflects the economic objective of
the model. We want to exceed a value requested by the
decision-makers in each period NPVp
,
X
H
h¼1
X
ði; jÞ 2 N
NPVp
hi jxp
hi j þ np
5 pp
5 ¼ NPVp
(G5)
where NPVp
hi j is the net present value per each hectare har-
vested from site class h, age class i, and treatment j at period p.
The negative deviation variable is the one to be minimized.
These priority levels are applied to each period of the
planning horizon. Therefore, the objective function of the
model is as follows:
LexMinð f1
;...; fP
Þ
¼
X
H
h¼1
p1
1h
Se1
h
; p1
2;n1
3; p1
4;n1
5
;...;
X
H
h¼1
pp
1h
SeP
h
; pP
2 ;nP
3 ; pP
4 ;nP
5
(8)
On the other hand, the feasible set of the model is defined by
the following constraints.
We have area accounting constraints per site class and per
age class during each period p ( p = 1, 2, . . ., P):
X
j 2 NðiÞ
xp
hi j s
ð p1Þ
hi ; h ¼ 1; 2; . . . ; H;
i ¼ 1; . . . ; I; p ¼ 1; 2; . . . ; P
(9)
We also impose constraints to control some of the model’s
key values. To avoid excessive clearcutting in age class I 1,
we establish the following upper bound:
xp
hðI1ÞJ as
ð p1Þ
hðI1Þ; h ¼ 1; 2; . . . ; H;
p ¼ 1; 2; . . . ; P; 0 a 1 (10)
Similarly, we establish constraints to control the lower
bound of the total cutting area and thus guarantee the
regeneration of the stands:
X
I
i¼1
xp
hiJ bSep
h ; h ¼ 1; 2; . . . ; H;
p ¼ 1; 2; . . . ; P; 0 b 1 (11)
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88 83

6. Finally, we establish lower bounds for net present value:
X
H
h
X
ði; jÞ 2 N
NPVp
hi jxp
hi j gNPVp
;
p ¼ 1; 2; . . . ; P; 0 g 1 (12)
The values of the parameters a, b and g are calculated when
the model is applied to a particular situation and depend on the
decision-makers’ requests.
The complete formulation of the proposed model is set out in
Appendix A.
4. Results and discussion
This model has been applied to the San Juan y Martı́nez
Management Unit. The initial forest configuration is as follows:
S0
¼
0:0 0:0 198:0 188:0 83:2
32:2 344:6 405:9 79:0 759:6
33:5 236:8 266:7 102:0 692:4
30:6 78:9 130:5 174:4 148:0
0
B
B
@
1
C
C
A
As indicated, the sum by columns corresponds to the number
of hectares available in each age class at the starting situation,
S0
hði ¼ 1; 2; . . . ; 5Þ
ð 96:3 660:3 1001:1 543:4 1683:2 Þ
and the sum by rows refers to the availability of each site class,
S0
hðh ¼ 1; 2; . . . ; 4Þ
ð 469:2 1621:3 1331:4 562:4 Þ
There are four site classes in this plantation (H = 4) and five
age classes (I = 5). The planning horizon, T, coincides with the
rotation age which is defined by the type of species and the
objectives of the plantation. In our context, the rotation age is
equal to 25 years and is the same across site classes (León,
1999). The time unit for each planning period is 5 years and
thus, we have a total of five periods (P = 5).
Besides applying clearcutting (treatment 4) in all age
classes, the other intermediate treatments to be applied by age
class are as follows (as established by the Instrucción para la
Ordenación del Patrimonio Forestal en Cuba, Ministry of
Agriculture, Norma Ramal 595 (1982)): thinning 1 (j = 1) in
age class 2, thinning 2 (j = 2) in age class 3 and thinning 3
(j = 3) in age class 4. Therefore, the problem has a total of 160
decision variables. That is, thinning: 3 age classes 4 site
classes = 12 variables plus clearcutting: 5 age classes 4 site
classes = 20 variables leading to a total of 32 variables for one
period and 160 variables for the 5 periods.
For the first priority level, the target values are given by:
Sep
h ¼ Se0
h ¼
1
5
S0
h; h ¼ 1; . . . ; 4:
Regarding the second priority level, Vp
is 138,328 m3
for
every period and, as we pointed out, this corresponds to the
maximum sustained timber yield. For the third and fourth
priority levels, the target values have already been specified in
the model. Finally, for the fifth priority level, and in line with
the decision-makers’ requests, the minimum desired level of
NPV is 790,000 pesos1
for the first two periods and 760,000
pesos for the last three. Appendix B shows the matrix of
coefficients vp
hi j and NPVp
hi j (Table 2).
On the other hand, also in line with the decision-makers’
requests, the values for the parameters a, b and g have been
established as follows: in order to guarantee the regeneration of
stands, the value of parameter b takes the value 0.9. In this way,
we make sure that clearcutting (j = 4) will be applied to a
minimum of 90% of the area which ensures the perpetuation of
the forest harvest in site class h, for period p, Sep
h .
Similarly, the value of g is set to 0.9 to guarantee that the
values of NPV in each period are always more than or equal to
90% of the set target values.
The value of the a parameter establishes the percentage of
age class 4 that will be clearcut in each site class per period. In
order to balance the age class distribution of the forest, and
given the initial imbalanced distribution, we have to establish
this parameter with a value higher than 0. Initially, a was given
a value of 1, which meant that the constraint associated with this
parameter is redundant in the model and, therefore, clearcutting
in the total area of age class 4 was allowed. However, the
obtained solution involved excessive final cutting of stands for
this age class, and given that the decision-makers wanted
clearcutting to be applied to a small percentage of the total
available area, in a second resolution of the model a was given
the value 0.15 which was later reduced to 0.05.
The resolution of all the cases mentioned was done with the
program PFLMO (Caballero and Hernández, 2003) using the
resolution method described in Section 2. Given the high level
of initial imbalanced age structure in the plantation we were
forced to relax the target values of the fractional goal for period
3, from 0.6 to 0.5, which had no effect on the final equilibrium
achieved. After this adjustment, PFLMO found solutions that
satisfied all the goals and, therefore, balanced solutions by the
end of the planning horizon. In other words, all the optimal
solutions for all the values of a under consideration lead to a
balanced distribution of the area occupied by each age class,
that is, S5
i ¼ 796:8 ha ði ¼ 1; 2; . . . ; 5Þ.
As described in the introduction, once the existence of
solutions verifying all the target values was established, the
efficiency of the solution obtained was restored. In this case the
restoration technique used was the Interactive Restoration
Method that allows the decision-makers to work with several
options at this new stage of problem resolution. Once all the
sustainability goals and the balance-goal had been satisfied, the
decision-makers chose NPV, the economic objective, to be the
one to maximize within the set of solutions verifying all the
problem goals. Thus, the solution obtained after restoration
achieved a balanced age class distribution by the end of the
planning horizon and satisfied all the other goals of the problem
while yielding the greatest NPV for the company.
The model was solved with an initial value for a equal to 1.
The solution obtained is shown in Appendix B as Solution 1
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88
84
1
25 Cuban pesos ffi 1$.

7. (Table 3). As shown in Table 1, the NPV for the company is
4,151,784 pesos—which is quite high if we take into account
that the target value was around 3,860,000 pesos. However, the
decision-makers did not consider this solution to be acceptable
because it meant applying clearcutting to a large number of
hectares of age class 4 in all the periods.
The decision-makers wanted to impose a stricter constraint
on clearcutting in age class 4. Therefore, we solved the problem
again with a value for the parameter a = 0.15, which meant that
only a maximum of 15% of the total age class 4 area available
for each site class and in each period was available for
clearcutting. The solution obtained, for this value of a, is given
in Appendix B as Solution 2 (Table 4). Total NPV is 4,067,495
pesos and the total number of hectares cut in age class 4 is
398.77 (Table 1).
If the maximum area to be cut is further restricted to 5% of
the total area available (a = 0.05), the solution obtained (i.e., a
solution where the value of NPV has the highest value while
fulfilling all the goals of the problem) is given in Appendix B as
Solution 3 (Table 5). In this solution the NPVis 4,025,710 pesos
which is lower than in Solution 2 and 1 (Table 1). However, this
is the most suitable one because the area of stands to be cut from
age class 4 is considerably lower, and the financial value is still
valid for the decision-makers as it is above their target value.
The decision-makers also wanted to obtain the solution
which, while satisfying all the target values, involved the least
amount of cutting of age class 4, in order to compare such a
solution with the previous ones. This solution is shown in
Appendix B as Solution 4 (Table 6). In this case, the
clearcutting of age class 4 stands is only done during the
first period and, as shown in Table 1, only a very small
percentage of the total age class 4 area is involved, i.e., 0.04%.
However, the NPV obtained with this solution is lower than in
previous solutions, i.e., 4,000,371 pesos. Fig. 2 compares the
different solutions showing the tradeoffs between total NPV
achieved and the forest area harvested of age class 4.
Bearing these solutions in mind, the decision-makers
evaluated the different alternatives provided and chose Solution
3. This solution satisfies all the target values and only a
maximum of 5% of age class 4 underwent clearcutting. In
addition, the NPV in this solution is 4,025,710 pesos. The
decision-makers were fully satisfied with this solution and so
the resolution process ended.
We have shown that the model proposed enabled the
decision-maker to explore different and interesting options
within the goal of obtaining a balanced age distribution in the
plantation. This was not possible in the previous work of León
et al. (2003) where regulating the plantation was not formalized
as a goal, and therefore the solutions found did not achieve such
target. Including this factor in the current model and doing so in
a non-restrictive way has enabled the decision-maker to analyse
the trade off between this and other factors, such as the
economic one.
Fig. 3 shows the evolution of each age class during the
different planning periods for the solution chosen by the
decision-makers. As we can see, the area covered by each age
class has been balanced by the last period of the planning
horizon.
5. Conclusions
This model has achieved a solution that allows us to
calculate the area to be harvested in each site class during each
period with profits as large as possible, given that harvests are
constrained by the need to limit adverse impacts on the
ecosystem. It ensures a balanced age class distribution in the
plantation by the end of the planning horizon, which fully
satisfies the wishes of the decision-makers, thereby solving the
company’s requirements.
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88 85
Fig. 2. Comparing solutions.
Table 1
Comparison between solutions
Cutting in age 4 (ha) Cutting in age 4 (%) Cutting in age 5 (ha) Cutting in age 5 (%) NPV (pesos)
Solution 1 791.94 25.91 3,150.55 44 4,151,784
Solution 2 398.77 13.16 3,517.092 48 4,067,495
Solution 3 137.8 4.56 3,771.21 51 4,025,710
Solution 4 1.26 0.04 3,903.354 53 4,000,371
Fig. 3. Area covered by each age class during the different periods.

8. Thus, the fractional goal models the decision-makers’ desire
for a balanced age class distribution in a way that takes into
account the dynamic aspect of the problem, also ensuring that
those solutions which satisfy the goals fulfil this desire. All this
is achieved while taking into account the financial objectives,
among others. Thus, the model we offer not only achieves an
even-aged distribution of the forest, but also enables its efficient
exploitation.
Furthermore, the model allows us to calculate the number of
hectares undergoing different treatments (indicating the timber
volume to be extracted in each planning period), to know the net
present value generated by such management planning, and
also to reduce clearcutting during the planning horizon.
On the other hand, the model can be applied to pure
plantations of other species managed for wood production.
Acknowledgments
The authors wish to express their gratitude to the referees for
their valuable and helpful comments, which have contributed to
improve the quality of the paper. This research has been
partially founded by the research projects of Andalusian
Regional Government, CENTRA and Spanish Ministry of
Educacion y Ciencia.
Appendix A
The linear fractional goal model proposed is as follows:
LexMinð f1
;...; fP
Þ
¼
X
H
h¼1
p1
1h
Se1
h
; p1
2;n1
3; p1
4;n1
5
; ;
X
H
h¼1
pP
1h
SeP
h
; pP
2 ;nP
3 ; pP
4 ;nP
5
s.t.
X
I
i¼1
xp
hiJ þ np
1h pp
1h ¼ Sep
h ; h ¼ 1; . . . ; H; p ¼ 1; . . . ; P
(G1)
X
H
h¼1
X
ði; jÞ 2 N
vp
hi jxp
hi j þ np
2 pp
2 ¼ V p
; p ¼ 1; . . . ; P (G2)
Sp
1
Sp
I
þ np
3 pp
3 ¼
1
P
p; p ¼ 1; . . . ; P (G3)
X
H
h¼1
X
I2
i¼1
xp
hiJ þ np
4 pp
4 ¼ 0; p ¼ 1; . . . ; P (G4)
X
H
h¼1
X
ði; jÞ 2 N
NPVp
hi jxp
hi j þ np
5 pp
5 ¼ NPVp
; p ¼ 1; . . . ; P
(G5)
X
j 2 NðiÞ
xp
hi j s
ð p1Þ
hi ; h ¼ 1; 2; . . . ; H; i ¼ 1; . . . ; I;
p ¼ 1; 2; . . . P
xp
hðI1ÞJ as
ð p1Þ
hðI1Þ; h ¼ 1; 2; . . . ; H; p ¼ 1; 2; . . . ; P;
0 a 1
X
I
i¼1
xp
hiJ bSep
h ; h ¼ 1; 2; . . . ; H; p ¼ 1; 2; . . . ; P;
0 b 1
X
H
h
X
ði; jÞ 2 N
NPVp
hi jxp
hi j gNPVp
; p ¼ 1; 2; . . . ; P;
0 g 1
xp
hi j 0; h ¼ 1; . . . ; H; i ¼ 1; . . . ; I; j ¼ 1; . . . ; J;
p ¼ 1; 2; . . . ; P
np
1h; pp
1h; np
2 ; pp
2 ; np
3 ; pp
3 ; np
4 ; pp
4 ; np
5 ; pp
5 0; h ¼ 1; . . . ; H;
p ¼ 1; 2; . . . ; P
where Sp
1 ¼
PH
h¼1
PI
i¼1xp
hiJ and Sp
I ¼
PH
h¼1s
ð p1Þ
hðI1Þ xp
hðI1ÞJ þ
s
ð p1Þ
hI xp
hIJ
Proposition 1. All the solutions satisfying all the goals of the
proposed model achieve an even-aged structure by the end of
the last planning period.
Proof. Assume that x p
hi jðh ¼ 1; . . . ; H; i ¼ 1; . . . ; I; j ¼
1; . . . ; J; p ¼ 1; 2; . . . ; PÞ is a feasible solution satisfying
all the goals of the model, and let us prove that, for this
solution, SP
i¼1 ¼ SP
i¼2 ¼ ¼ SP
i¼I.
Let us denote the total clearcut area for each period of the
planning horizon as Cp
¼
PH
h¼1
PI
i¼1x p
hiJ ð p ¼ 1; 2; . . . ; PÞ
and the total forest area as S ¼
PH
h¼1 S0
h ¼
PI
i¼1 S0
i . As x p
hi j is a
solution that satisfies the goals, then from the fourth goal (G4),
Cp
¼
PH
h¼1
PI
i¼I1 x p
hi j ð p ¼ 1; 2; . . . ; PÞ and the following
relations hold:
SP
i¼1 ¼ CP
SP
i¼2 ¼ SP1
i¼1 ¼ CP1
SP
i¼3 ¼ SP1
i¼2 ¼ SP2
i¼1 ¼ CP2
SP
i¼I1 ¼ SP1
i¼I2 ¼ SP2
i¼I3 ¼ ¼ SPIþ2
i¼1 ¼ CPðI2Þ
SP
i¼I ¼ S ðCP
þ CP1
þ þ CPðI2Þ
Þ
Besides this, from (G1), Cp
PH
h¼1 Sep
h S=I ( p = 1, 2, . . .,
P) and thus, CP
þ þ CPðI2Þ
ðI 1ÞS=I. Therefore,
SP
i¼1
SP
i¼I
¼
CP
S ðCP þ CP1 þ þ CPðI2ÞÞ
S=I
S ððI 1Þ=IÞS
¼ 1:
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88
86

9. As x p
hi j satisfies (G3), SP
i¼1=SP
i¼I 1, thus, it is obvious that
SP
i¼1=SP
i¼I ¼ 1. It follows from this that CP
¼ S ðCP
þ
CP1
þ þ CPðI2Þ
Þ or S ¼ 2CP
þ CP1
þ þ CPðI2Þ
and, taking into account (G1), Cp
¼ S=Ið 8 p ¼ P; . . . ;
P ðI 2ÞÞ.
Consequently, SP
i¼1 ¼ SP
i¼2 ¼ ¼ SP
i¼I1 ¼ ð1=IÞS and
also SP
i¼I ¼ S ðI 1=IÞS ¼ ð1=IÞS.
Appendix B
Table 2 shows volume per hectare (second column)
harvested from each site class, age class, and treatment in
period p ( p = 1, . . ., 5), vp
hi j. The third column shows the net
present value per hectare harvested from each site class, age
class, and treatment in period p, ( p = 1, . . ., 5), NPVp
hi j. These
coefficients are assumed to be constant across all the periods, in
accordance with the data provided by the decision-maker’s
company.
The selected solutions are shown below. The rows
represent the periods. The first column in each table (named
FRACT) shows the value of each solution for the fractional
goal (goal G3) at each period. Columns 2–5 show the number
of hectares undergoing different management treatments:
column T1 shows the total number of hectares undergoing
treatment 1 in each period, and the same for columns T2
(treatment j = 2), T3 (treatment j = 3) and T4 (treatment
j = 4, that is, clearcutting). In column 6 (named T4 age 4),
we specifically show the number of hectares for
clearcutting in age class 4 for each period. Finally, in
column NPV we show the NPV generated by the solutions in
each period as well as the total NPV achieved, expressed in
Cuban pesos.
T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88 87
Table 2
Matrix of coefficients vp
hi j and NPVp
hi j
Variable vp
hi j (m3
) NPVp
hi j (pesos)
xp
114 7.27 16.7
xp
121 8 40.59
xp
124 21.89 108.87
xp
132 13 100.09
xp
134 25.03 117
xp
143 15 109.6
xp
144 51.31 140.7
xp
154 71.7 191.38
xp
214 11.95 27.46
xp
221 8 64.18
xp
224 25.03 262.49
xp
232 13 139
xp
234 69.61 373
xp
243 15 151.8
xp
244 103.6 420
xp
254 130 540.2
xp
314 18.14 41.67
xp
321 8 79.9
xp
324 57.47 779.24
xp
332 13 191
xp
334 113.03 815
xp
343 15 202.2
xp
344 154.5 928
xp
354 190 990
xp
414 25.37 58.28
xp
421 8 118.4
xp
424 75.71 1,026.6
xp
432 13 251
xp
434 145.45 1,052
xp
443 15 250
xp
444 201 1,348
xp
454 234 1,419.78
Table 3
Solution 1
FRACT T1 (ha) T2 (ha) T3 (ha) T4 (ha) T4 age 4 (ha) NPV (pesos)
P.1 0.51309 660.3 1001.1 335.5131 755.0499 207.8869 857,945
P.2 0.47551 96.3 660.3 738.5976 796.8596 93.84 848,004
P.3 0.5177 670.594 96.3 581.314 796.8601 78.9861 787,365
P.4 0.95014 703.02 755.05 2.9515 796.86 93.3485 792,759
P.5 1 703.02 796.86 437.1714 796.8604 317.879 865,711
Total 4,151,784
Table 4
Solution 2
FRACT T1 (ha) T2 (ha) T3 (ha) T4 (ha) T4 age 4 (ha) NPV (pesos)
P.1 0.486205 660.3 1001.1 431.83 728.422 81.51 854,400
P.2 0.468075 30.6 660.3 532.183 796.86 110.16 823,388
P.3 0.508897 643.966 96.3 576.905 796.8598 83.3948 783,699
P.4 0.920908 423.7923 643.966 81.855 796.86 14.445 783,686
P.5 1 112.48 703.02 382.6762 796.8599 109.2633 822,322
Total 4,067,495

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T. Gómez et al. / Forest Ecology and Management 227 (2006) 79–88
88
Table 5
Solution 3
FRACT T1 (ha) T2 (ha) T3 (ha) T4 (ha) T4 age 4 (ha) NPV (pesos)
P.1 0.479437 660.3 1001.1 436.7881 721.5678 27.17 852,725
P.2 0.466198 30.6 619.525 377.34 796.86 36.72 805,314
P.3 0.506679 359.5694 96.3 627.285 796.86 33.015 778,397
P.4 0.913671 378.76 637.112 91.485 796.86 4.815 781,821
P.5 1 112.48 598.715 328.014 796.86 36.07839 807,453
Total 4,025,710
Table 6
Solution 4
FRACT T1 (ha) T2 (ha) T3 (ha) T4 (ha) T4 age 4 (ha) NPV (pesos)
P.1 0.47513 660.3 1001.1 456.05 717.174 1.256 852,291
P.2 0.465003 30.6 486.889 397.2 796.86 0 795,441
P.3 0.505267 171.994 96.3 660.3 796.86 0 772,549
P.4 0.909091 357.815 632.718 96.3 796.86 0 780,418
P.5 1 112.48 501.481 340.884 796.86 0 799,672
Total 4,000,371