16603138

A forest planning problem solved via a linear fractional

goal programming model

T. Go´mez

, M. Herna´ndez

, M.A. Leo´n

, R. Caballero

*

Department of Applied Economics (Mathematics), University of Ma´laga, Campus El Ejido s/n, 29071 Ma´laga, Spain b

Department of Mathematics, University of Pinar del Rı´o, Pinar del Rı´o, 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 Rı´o’’. 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 modern view ofsustainabilitycomprises 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, andrecreational 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 Dı´az-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 asSteuer and Schuler (1978),De Kluyver et al. (1980), Hallefjord et al. (1986),Bare and Mendoza (1988), andKazana 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

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* Corresponding author. Tel.: +34 952 131168; fax: +34 952 132061.

E-mail address:[email protected](R. Caballero).

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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 for a goal programming model 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 inFig. 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 is planning to increase the number of plantations which 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 Rı´o University has been authorized to carry out this kind of work with the forestry companies in the region. The present study is framed within this approach, and is preceded by the work of Leo´n et al. (2003), where a goal programming model with linear goals was formulated for the management planning process of a Pinus caribaeaplantation 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 theSan Juan y Martı´nez Forestry Unit, which belongs to the forestry company ‘‘Empresa Forestal Integral Pinar del Rı´o’’ from the town San Juan y Martı´nez, in the southwest of the Pinar del Rı´o 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 caribaeawhose 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 Section4the model is applied to a specific case, theSan Juan and Martı´nez Management Unit. We then analyse the results obtained. Finally, in Section 5 we draw some conclusions followed by Appendix A, Appendix B and References.

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 thatqof its attributes are linear fractional functions and such that its vector of decision variables,x, has to verify a set ofrlinear constraints, that is,

’₁ðxÞ ¼c

t

1xþa1

dt

1xþb1

;. . .;’_qðxÞ ¼c

t qxþaq dt

qxþbq

(1)

Axb; x0

wherecm, dm, 2Rn,am,bm2Rform= 1,. . .,q;A2MrxnðRÞ andb2Rr_{. LetXbe the feasible set for this problem; that is,} X¼ fx2Rn_{=Axb;x0g. It is assumed that d}t

mxþbm,

m= 1, . . .,q are strictly positive for everyx2X.

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 solutionx that verifies the constraint set and the q goals (also called soft-constraints):

Axb; x0; ’mðxÞ ¼ ct

mxþam dt

mxþbm

um; m¼1;. . .;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 asnmandpm, 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 theq goals, and we are in a certain levels(where the index set of the

goals inswill be denoted byNs), the optimization problem that has to be solved is the following:

min X

m2Ns

wmnm

s:t: x2Xs ct_mxþam dt

mxþbm

þnmpm¼um m2Ns

nm;pm0

(3)

whereXs= {x2X/wL(x)uL,L2N1,. . .,Ns1}, andwmis the weight of themth goal andnm, pmare 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 factordt

mxþbm(by hypothesis, this is always positive inX), we reach the following linear problem:

min X

k

m2Ns

wmn0m

s:t: x2Xs

ct_mxþam ðdmtxþbmÞumþn0mp

0

m¼0 n0_m;p0_m0 m2Ns

(4)

Obviously, for problems (3) and(4) to be equivalent, the relationship between their variables has to be the following:

p0_m¼ pmðdt_mxþbmÞ; n0m¼nmðd t

mxþbmÞ m2Ns: 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 Xsthat 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 Herna´ndez, 2006):

Theorem 1. Given problems(3) and(4)the following asser-tions are valid:

(i)If,when solving(4)the solution isðx;n_m0;p0_mÞ_m¼1;...;ksuch

that Smwmn0_m¼0, 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 solutionðx;n0_m; p0_mÞm¼1;...;k,is such

thatSmwmn0_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 levels, 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 levels of the fractional problem(2). Otherwise we can guarantee that

there will not be a point inXthat satisfies all of these goals given the current target values. In such a case, in order to find the point that minimizes the nonachievement inXs 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 Rı´o’’ 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:

S0¼

s0₁₁ s0

12 . . . s 0 1I s0₂₁ s0₂₂ . . . s0_2I

. . . .

s0_H1 s0_H2 . . . s0_HI 0

B B B @

1 C C C A

wheres0

hiis the total number of hectares of the site classh(h= 1, 2,. . .,H) within the age classi(i= 1, 2,. . .,I) at the 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 ¼PHh¼1s0hi), whereas the sum by rows gives the available area in each site class (S0_h¼PIi¼1s0hi). 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 classibecome age classi+ 1. Thus, iftis the number of years in each class (for reasons of simplicity we assume this number is constant), the number of periods under consideration, denoted byP, is equal to the number of years of the planning horizon divided byt. If the plantation evolves without intervention and mortality or disaster, the actual development of the stands would lead us to move from matrixS0to matrixS1, and so on.

Therefore, the decision variables of our model represent the number of hectares of a specific site classh(h= 1, 2,. . .,H) and

age classi(i= 1, 2,. . .,I) with intermediate treatment or final cuttingj(j= 1, 2,. . .,J) at periodp(p= 1, 2,. . .,P), denoted byx_{hi j}p . The treatment to apply depends on age, and so the value of the subscript j depends on the value of i,j2N(i), where NðiÞ ¼ fj=ði;jÞ 2Ng 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,s_hip depends on the area of the previous period in the following way:

s_hp₁¼X

I

i¼1

x_hiJp ; h¼1;2;. . .;H (5)

s_hip ¼sð_hðp_i1Þ_1Þx_hp_ði1ÞJ; i¼2;. . .;I1; h¼1;2;. . .;H

(6)

s_hIp ¼sð_hðp_I1Þ_1Þx_hp_ðI1ÞJþsð_hIp1Þx_hIJp ; h¼1;2;. . .;H (7)

In other words, the total area of age class 1 at the end of periodp is the total number of hectares harvested during that period. The total area of age i(greater than 1) by the end of periodpis 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 ageI(last age class) is made up of what was already in this age class plus what there was in age class I1 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 periodp(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 classh, for periodp, Se_hpshould not be exceeded. This areaSe_hpis given by the total area in site classhdivided by the rotation age and multiplied by the number of years in each class (the time span which defines

the age class). Therefore, in each period, we have the following Hgoals:

XI

i¼1

x_hiJp þn₁p_h p₁p_h ¼Se_hp; h¼1;. . .;H (G1)

wheren1 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=Se_hp in order to prevent bias (seeRomero, 1991).

3.2. Second priority level

We aim at keeping harvest levels up to the maximum sustained yield. Thus, ifVprepresents this maximum sustained volume at periodpandv_{hi j}p 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:

XH

h¼1

X

ði;jÞ 2N

v_{hi j}p x_{hi j}p þn₂pp₂p ¼Vp (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:

S₁p S_Ipþn

p

3 p

p

3 ¼

1

Pp; p¼1;. . .;P (G3)

whereS₁p¼PHh¼1

PI

i¼1x

p hiJ andS

p I ¼

PH

h¼1s ðp1Þ

hðI1Þx

p hðI1ÞJ

þsð_hIp1Þx_hIJp (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 (seeAppendix 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 PI, 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 classI1

should be cut. Consequently, this goal is formulated as follows:

XH

h¼1

XI2

i¼1

x_hiJp þn₄pp₄p ¼0 (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,

XH

h¼1

X

ði;jÞ 2N

NPV_{hi j}p x_{hi j}p þn₅pp₅p¼NPVp (G5)

where NPV_{hi j}p is the net present value per each hectare har-vested from site classh, age classi, and treatmentjat periodp. 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:

LexMinðf1;. . .;fPÞ

¼X

H

h¼1

p1 1h Se1

h

;p1₂;n1₃;p1₄;n1₅

;. . .;X

H

h¼1

p₁p_h SeP h

;pP₂;nP₃;pP₄;nP₅

(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 periodp (p= 1, 2,. . .,P):

X

j2NðiÞ

x_{hi j}p sð_hip1Þ; h¼1;2;. . .;H;

i¼1;. . .;I; p¼1;2;. . .;P

(9)

We also impose constraints to control some of the model’s key values. To avoid excessive clearcutting in age classI1, we establish the following upper bound:

x_hp_ðI1ÞJas_hð_ðp_I1Þ_1Þ; h¼1;2;. . .;H;

p¼1;2;. . .;P; 0a1 (10)

Similarly, we establish constraints to control the lower bound of the total cutting area and thus guarantee the regeneration of the stands:

XI

i¼1

x_hiJp bSe_hp; h¼1;2;. . .;H;

Finally, we establish lower bounds for net present value:

XH

h X

ði;jÞ 2N

NPV_{hi j}p x_{hi j}p gNPVp;

p¼1;2;. . .;P; 0g<1 (12)

The values of the parametersa,bandgare 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 Martı´nez Management Unit. The initial forest configuration is as follows:

S0¼

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_hði¼1;2;. . .;5Þ

ð96:3 660:3 1001:1 543:4 1683:2Þ

and the sum by rows refers to the availability of each site class,

S0_hðh¼1;2;. . .;4Þ

ð469:2 1621:3 1331:4 562:4Þ

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 (Leo´n, 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 theInstruccio´n para la Ordenacio´n 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 classes4 site classes = 12 variables plus clearcutting: 5 age classes4 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:

Se_hp¼Se0_h¼1

5S

0

h; h¼1;. . .;4:

Regarding the second priority level, Vpis 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 pesos1for the first two periods and 760,000 pesos for the last three. Appendix B shows the matrix of coefficientsv_{hi j}p and NPV_{hi j}p (Table 2).

On the other hand, also in line with the decision-makers’ requests, the values for the parametersa,b andg have been established as follows: in order to guarantee the regeneration of stands, the value of parameterbtakes 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 classh, for periodp, Se_hp.

Similarly, the value ofg 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 theaparameter 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,awas 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 modelawas 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 Herna´ndez, 2003) using the resolution method described in Section2. 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 ofaunder consideration lead to a balanced distribution of the area occupied by each age class, that is,S5_i ¼796:8 ha ði¼1;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 foraequal to 1. The solution obtained is shown in Appendix Bas Solution 1

1

(Table 3). As shown inTable 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 parametera= 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 ofa, is given inAppendix Bas 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 inAppendix Bas Solution 3 (Table 5). In this solution the NPV is 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. 2compares 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 ofLeo´n 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.

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

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:

LexMinðf1;. . .;fPÞ

¼X

H

h¼1

p1_1h Se1

h

;p1₂;n1₃;p1₄;n1₅

; ;X

H

h¼1

pP_1h SeP h

;pP₂;nP₃;pP₄;nP₅

s.t.

XI

i¼1

x_hiJp þn₁p_hp₁p_h¼Se_hp; h¼1;. . .;H; p¼1;. . .;P

(G1)

XH

h¼1

X

ði;jÞ 2N

v_{hi j}p x_{hi j}p þn₂pp₂p¼Vp; p¼1;. . .;P (G2)

S₁p S_Ipþn

p

3 p

p

3 ¼

1

Pp; p¼1;. . .;P (G3)

XH

h¼1

XI2

i¼1

x_hiJp þn₄pp₄p¼0; p¼1;. . .;P (G4)

XH

h¼1

X

ði;jÞ 2N

NPV_{hi j}p x_{hi j}p þn₅pp₅p¼NPVp; p¼1;. . .;P

(G5) X

j2NðiÞ

x_{hi j}p sð_hip1Þ; h¼1;2;. . .;H; i¼1;. . .;I;

p¼1;2;. . .P

x_hp_ðI1ÞJ asð_hp_ðI1Þ_1Þ; h¼1;2;. . .;H; p¼1;2;. . .;P;

0a1

XI

i¼1

x_hiJp bSe_hp; h¼1;2;. . .;H; p¼1;2;. . .;P;

0b1

XH

h X

ði;jÞ 2N

NPV_{hi j}p x_{hi j}p gNPVp; p¼1;2;. . .;P;

0g<1

x_{hi j}p 0; h¼1;. . .;H; i¼1;. . .;I; j¼1;. . .;J;

p¼1;2;. . .;P

n₁p_h;p₁p_h;n₂p;p₂p;n₃p;p₃p;n₄p;p₄p;n₅p;p₅p0; h¼1;. . .;H;

p¼1;2;. . .;P

whereS₁p¼PHh¼1

PI

i¼1x

p hiJandS

p I ¼

PH

h¼1s ðp1Þ

hðI1Þx

p hðI1ÞJþ sð_hIp1Þx_hIJp

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_{hi j}pðh¼1;. . .;H; i¼1;. . .;I; j¼

1;. . .;J; p¼1;2;. . .;PÞ is a feasible solution satisfying all the goals of the model, and let us prove that, for this solution,S_i¼1P ¼S_i¼2P ¼ ¼S_i¼P_I.

Let us denote the total clearcut area for each period of the planning horizon as Cp _¼PH

h¼1

PI

i¼1x p

hiJ ðp¼1;2;. . .;PÞ and the total forest area asS¼PH_h¼1S0_h ¼PI_i¼1S0_i. Asx_{hi j}pis a solution that satisfies the goals, then from the fourth goal(G4), Cp_¼PH

h¼1

PI

i¼I1x p

hi j ðp¼1;2;. . .;PÞand the following relations hold:

S_i¼1P ¼CP

S_i¼2P ¼S_i¼1P1¼CP1

S_i¼3P ¼S_i¼2P1¼S_i¼1P2¼CP2

S_i¼P_I1 ¼S_i¼P1_I2¼S_i¼P2_I3¼ ¼S_i¼1PIþ2¼CPðI2Þ

S_i¼P_I ¼S ðCPþCP1þ þCPðI2ÞÞ

Besides this, from(G1),CpPHh¼1Se

p

h S=I(p= 1, 2,. . .,

P) and thus, CPþ þCPðI2Þ ðI1ÞS=I. Therefore,

S_i¼1P S_i¼P_I ¼

CP

S ðCP_þCP1_{þ þC}PðI2Þ_Þ

S=I

Asx_{hi j}p satisfies(G3),S_i¼1P=S_i¼P_I1, thus, it is obvious that

S_i¼1P =S_i¼P_I ¼1. It follows from this that CP_{¼S ðC}P_þ CP1_{þ þC}PðI2Þ_ÞorS¼2CP_þCP1_{þ þC}PðI2Þ

and, taking into account (G1), Cp_{¼S=Ið 8 p¼P;. . .;} P ðI2ÞÞ.

Consequently, S_i¼1P ¼S_i¼2P ¼ ¼S_i¼P_I1¼ ð1=IÞS and alsoS_i¼P_I ¼S ðI1=IÞS¼ ð1=IÞS. &

Appendix B

Table 2 shows volume per hectare (second column) harvested from each site class, age class, and treatment in periodp (p=1,. . ., 5),v_{hi j}p . The third column shows the net present value per hectare harvested from each site class, age class, and treatment in periodp, (p= 1,. . ., 5), NPV_{hi j}p . 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.

Table 2

Matrix of coefficientsvhi jp and NPV p hi j

Variable vhi jp (m 3

) NPVhi jp (pesos)

x₁₁₄p 7.27 16.7

x121p 8 40.59

x₁₂₄p 21.89 108.87

x₁₃₂p 13 100.09

x₁₃₄p 25.03 117

x₁₄₃p 15 109.6

x₁₄₄p 51.31 140.7

x₁₅₄p 71.7 191.38

x214p 11.95 27.46

x₂₂₁p 8 64.18

x₂₂₄p 25.03 262.49

x₂₃₂p 13 139

x234p 69.61 373

x₂₄₃p 15 151.8

x₂₄₄p 103.6 420

x₂₅₄p 130 540.2

x₃₁₄p 18.14 41.67

x₃₂₁p 8 79.9

x₃₂₄p 57.47 779.24

x₃₃₂p 13 191

x334p 113.03 815

x₃₄₃p 15 202.2

x₃₄₄p 154.5 928

x₃₅₄p 190 990

x₄₁₄p 25.37 58.28

x₄₂₁p 8 118.4

x424p 75.71 1,026.6

x₄₃₂p 13 251

x₄₃₄p 145.45 1,052

x₄₄₃p 15 250

x₄₄₄p 201 1,348

x₄₅₄p 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

References

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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