Efficiency in forest management: A multiobjective harvest scheduling model

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Efficiency in forest management: A multiobjective harvest scheduling model M. Hernandez a,∗, T. Gómez a, J. Molina a, M.A. León b, R. Caballero a a Department of Applied Economics (Mathematics), Universidad de Málaga, Campus El Ejido s/n, 29071 Málaga, Spain b Department of Mathematics, University of Pinar del Río, Pinar del Río, Cuba

a r t i c l e

i n f o

Article history: Received 15 March 2013 Accepted 28 June 2014 JEL classification: C02. Mathematical methods C61. Optimization techniques, Programming models Q23. Forestry Keywords: Forest planning Multiobjective programming Adjacency constraints CO2 captured Nonlinear binary models Metaheuristic

a b s t r a c t This paper presents a new forest harvest scheduling model taking into account four conflicting objectives. The economic factor of timber production is considered and also aspects related to environmental protection. We also incorporate adjacency constraints to limit the maximum contiguous area where clear-cutting can be applied. The model proposed is applied to a timber production plantation in Cuba located in the region of Pinar del Río. One factor to be taken into account in Cuban plantations is that the forest has a highly unbalanced age distribution. Therefore, in addition to the classical objectives of forest planning, we have the objective of rebalancing age distribution by the end of the planning horizon. Explicitly, the four objectives considered in the model are: (a) obtaining a balance-aged forest; (b) minimizing the area with trees older than the rotation age; (c) maximizing the NPV of the forest over the planning horizon; and (d) maximizing total carbon sequestration over the whole planning horizon. The solution to the proposed model provides a set of efficient management plans that are of assistance in analysing the tradeoffs between the economic and ecological objectives. The model is also applied to randomly generated simulated forests to compare its performance

∗ Corresponding author. Tel.: +34 952131169; fax: +34 952132061. E-mail addresses: m [email protected] (M. Hernandez), [email protected] (T. Gómez), [email protected] (J. Molina), [email protected] (M.A. León), r [email protected] (R. Caballero). http://dx.doi.org/10.1016/j.jfe.2014.06.002 1104-6899/© 2014 Department of Forest Economics, Swedish University of Agricultural Sciences, Umeå. Published by Elsevier GmbH. All rights reserved.

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in other contexts. As the problem is a multiobjective binary nonlinear model, a metaheuristic procedure is used in order to solve it. © 2014 Department of Forest Economics, Swedish University of Agricultural Sciences, Umeå. Published by Elsevier GmbH. All rights reserved.

Introduction The management of forest resources has become a complex issue that has shifted from its early focus on industrial needs to include other objectives such as environmental protection, recreational value and social demands. This has led to increasingly complex decision-making procedures requiring decision models that have to meet and support the new requirements of the decision-making process. Among the more relevant models are the multi-objective optimization models (Steuer and Schuler, 1978; Bare and Mendoza, 1988; Kazana et al., 2003; Tóth and McDill, 2008, etc.) that try to simultaneously combine several conflicting objectives. Many techniques are available to address forest management planning with multiple criteria (DiazBalteiro and Romero, 2008), depending on the problem to be solved and the data available. In this paper, we focus on harvest planning which involves identifying the stands to be treated, the kind of treatment to be applied, and the schedule. Management planning endeavours to simultaneously fulfil different types of objectives, while taking into account certain environmental considerations such as the maximum adjacent area in which clear-cutting can be conducted. This involves using a multiobjective model whose resolution provides a set of efficient solutions (an approximation of the Pareto frontier) to the problem. The approach selected does not demand too much information from the decision-maker (DM), and the analysis of the efficient set allows us to compare tradeoffs between different objectives to gain greater understanding of the situation being addressed. As Tóth and McDill (2008) stated, better decisions can be made if the DM understands the tradeoff structure between competing objectives. The model presented in this study includes economic as well as silvicultural and ecological objectives. Given the key role of forests as climate regulators, it is relevant to include ecological objectives in forestry management (Bateman and Lovett, 2000; Couture and Reynaud, 2011). In this regard, the Kyoto Protocol was a significant step forward that recommended forestry as a means to offset industrial carbon dioxide emissions (Platinga et al., 1999). This was emphasized at subsequent Kyoto Protocol meetings (Montreal (December 2005), Nairobi (November 2006), Bali (December 2007), Copenhagen (December 2009), Mexico (December 2010)). Some studies have included carbon sequestration as an additional objective when planning forest harvesting. Hoen and Solberg (1994) suggested a twocriteria model that analysed the trade-off between the net present value (NPV) of the harvest and the present carbon sequestration value over the planning horizon. On the other hand, Díaz-Balteiro and Romero (2003) developed different goal programming models that included an operational measure of carbon sequestration together with other economic and silvicultural criteria. In this line, the present study includes maximizing net carbon sequestration during the planning horizon as an objective. The model proposed is applied to a Cuban plantation for timber production. One feature of Cuban plantations is that their age distribution is very unbalanced. Therefore, in addition to the classic objectives of forest planning, the age distribution of these plantations has to be balanced by the end of the planning horizon to obtain a constant flow of timber. This objective has been modelled in the present study by following the fractional formulation provided by Gómez et al. (2006). In the literature, usually, rebalancing the forest area by ages is modelled as a set of constraints that must be satisfied (Buongiorno and Gilless, 2003; Díaz-Balteiro and Romero, 2003; Tóth et al., 2006). But in lots of cases, mostly when the forest is organized by units or stands, if the forest is not initially age balanced, to impose these constraints might be very restricted and might lead to unfeasible solutions. With the new formulation proposed in this work, this age balance requirement is treated in a more flexible way, because we add this requirement as an objective instead of a set of constraints. So, the unfeasibility problems are

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eliminated. Furthermore, the proposed objective adopts a fractional formulation that modelizes the requirement using a relative manner instead of the classic formulation (absolute manner). Basically, the aim was to minimize differences between areas covered by each age class at the end of the planning. But when the objective is not possible to satisfy a linear formulation can lead to a solution that is not the most “balanced” possible. In order to prevent this situation, the fractional formulation is adopted. Besides, our model also includes adjacency constraints aimed at limiting the maximum contiguous area to which clear-cutting can be applied. Using spatial constraints in forest planning is a common practice in the management of private and public forests. They help to preserve the beauty and biodiversity of the forest environment (Weintraub and Murray, 2006), and are based on dividing the forest into basic management units, which vary in surface area but have similar characteristics. Consequently, in order to incorporate these spatial constraints, binary variables must be used in the model. The literature describes at least two different approaches to adjacent constraints. In the Unit Restriction Model (URM), two or more neighbouring basic units cannot be cleared simultaneously. However, in the Area Restriction Model (ARM), the area adjacent to a clear-cutting area cannot exceed a given threshold (maximum opening size) (Murray, 1999). Roise (1990) and Snyder and Revelle (1997) describe multiobjective problems with adjacency constraints based on the URM model. On the other hand, Tóth et al. (2006) developed a two-criteria model with adjacency constraints based on ARM that was later increased to three criteria by Tóth and McDill (2008). However, the authors themselves pointed out that the exact procedures used to solve the problem can be computationally expensive when the problems are large. In this paper, we use adjacency constraints under the ARM approach, which, according to the literature, is more powerful and complex than the URM (Murray and Weintraub, 2002; McDill et al., 2002; Goycoolea et al., 2005; Weintraub and Murray, 2006). Thus, our problem can be defined as a multiobjective nonlinear programming model with binary variables. The following criteria are considered relevant to the strategic management: (a) obtaining a balance-aged forest, i.e., the area covered by each age-class should be roughly the same by the end of the planning horizon; (b) minimizing the area with trees older than the rotation age; (c) maximizing the NPV of the forest over the planning horizon; and (d) maximizing total carbon sequestration over the whole planning horizon. As such, it is very complex and requires metaheuristic procedures to solve it. Heuristic methods are being increasingly used to solve forest management problems because exact methods are not sufficiently powerful given the complexity of the problems (Pukkala and Heinonen, 2006). The literature has described several methods, such as simulated annealing, taboo search, and genetic algorithms, among others (Jones et al., 2002; Borges et al., 2002; Falcão and Borges, 2002; Caro et al., 2003; Liu et al., 2006). The metaheuristic algorithm used to solve the present problem is an adaptation of the evolutionary method SSPMO (Scatter Search Procedure for Multiobjective Optimization; Molina et al., 2007). In this study, a new multiobjective model is proposed and resolved for the operational planning of a timber production forest. The model is applied to a forest in Cuba and finally, we investigated to what extent the model is valid to other contexts and different starting conditions by planning the harvest in a series of randomly generated simulated forests that had characteristics similar to the Cuban plantation. The paper is structured as follows: Section “The mathematical model” develops the model proposed; in Section “Optimization method” we refer to the optimization method used in the resolution of the problem; Section “Case study area” describes the Cuban plantation; and Section “Results and discussion” shows the results obtained in the application of the model in the real plantation described and in some hypothetical scenarios for purposes of comparison. Finally, Section “Conclusions” presents the conclusions.

The mathematical model In this section we develop a model to plan the harvest schedule of a plantation over a planning horizon of T years. We assume that the forest area is divided into U different management units. The aim is to determine the treatment (first thinning, second thinning, clear-cut, etc.) to be applied to each

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entire unit in a specific year of the planning horizon, in such a way that the objectives are fulfilled while taking into account technical, physical and environmental constraints. Each unit u (u = 1, 2,. . ., U) is characterized by: • Su , which is the area of unit u measured in hectares. After a clear-cut, the units are replanted, so this area is constant over the entire planning horizon. • Au,t , which is the age of unit u in year t of the planning horizon, expressed in years. Au,0 is the starting age of this unit. We denote by N* the rotation age (suitable age for clear-cutting). • SIu , which is the site index of unit u. This is a productivity indicator, and we suppose SIu = 1, 2,. . ., M. Let us assume that J different silvicultural treatments can be applied to the management units, apart from clear-cutting. These treatments depend on the age of the unit. Thus, we assume that it is possible to apply treatment j (j = 1, 2,. . ., J) to unit u (u = 1, 2,. . ., U) in year t if its age ranges from Lbj to Ubj years, that is, if Au,t−1 ∈ [Lbj , Ubj ]. All of these age intervals are disjoint sets, so it is not possible to apply more than one treatment to the same unit in the same year. In the model, we use two different types of decision variables: binary variables xu,t which represent whether clear-cutting is applied (xu,t = 1) or not (xu,t = 0) to management unit u (u = 1, 2,. . ., U) in year t (t = 1, 2,. . ., T); and binary variables yu,t which show whether a unit u (u = 1, 2,. . ., U) receives the corresponding treatment in year t (t = 1, 2,. . ., T) (yu,t = 1) or not (yu,t = 0). Note that if yu,t = 1, then unit u will receive the treatment j (j = 1, 2,. . ., J) such that Au,t−1 ∈ [Lbj , Ubj ]. Let be zu = (xu,1 , xu,2 , . . ., xu,T , yu,1 , yu,2 , . . ., yu,T ) the vector that indicates the harvesting and thinning plan for the management unit u along the planning horizon, and z = (z1 , z2 , . . ., zU ) the joint vector of management plans for all the units in the forest. Feasible harvesting schedules are limited by different constraints, as described below. Firstly, the age of each unit is changing during the planning horizon. Thus, this age increases by 1 in relation to the previous year, or, if the management unit has been clear-cut, its age becomes 1. That is: Au,t = (Au,t−1 + 1) (1 − xu,t ) + xu,t

u = 1, 2, . . ., U; t = 1, 2, . . ., T

(1)

On the other hand, it is obvious that clear-cutting and an intermediate treatment cannot be applied at the same period to the same management unit: xu,t + yu,t ≤ 1 u = 1, 2, . . ., U; t = 1, 2, . . ., T

(2)

A treatment j (j = 1, 2,. . ., J) can also be applied at most once during the period associated with the treatment, that is:



yu,t ≤ 1 u = 1, 2, . . ., U; j = 1, 2, . . ., J

(3)

t/Au,t−1 ∈ [Lbj Ubj ]

As described in Section “Introduction”, we address the problem of adjacency constraints within the framework of the ARM, whereby the areas adjacent to clear-cutting cannot exceed a certain size. Thus, we assume that the units are grouped depending on adjacency, such that there are K adjacency groups G1 , G2 , . . ., GK . These groups are calculated in such a way that each Gk is made up of two or more adjacent units, whose joint area exceeds a permitted threshold (i.e., maximum opening size) (McDill et al., 2002). Thus, all the units belonging to the same adjacency group cannot be clear-cut at the same time during the same planning period. However, groups are calculated in such a way that the joint area of all units but one does not exceed the permitted area limit. Thus, if the combined area of neighbouring basic units (within a particular group, Gk ) does not exceed that maximum area limit, then the decision on whether or not to clear-cut these units in the same period is determined by the model’s solution. Furthermore, given that this model deals with yearly periods, when dealing with adjacencies we take into account green-up requirements (Weintraub and Murray, 2006) that prevent clear-cutting

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management units belonging to the same adjacency group until g years or planning periods have passed. Consequently:

 t+g−1 

xu,s ≤ |Gk | − 1 k = 1, 2, . . ., K; t = 1, 2, . . ., T − g

(4)

u ∈ Gk s=t

Sustainability constraints are also included. On the one hand, young units cannot be clear-cut, that is, units with trees younger than n* years: T 

⎛ ⎝

t=1

⎞



xu,t ⎠ = 0

(5)

u/Au,t−1
where n* < N* and has to be set by the DM. On the other hand, the area to which clear-cutting is applied during each period is constrained. Firstly, it cannot exceed the area that ensures the perpetuation of the forest (Se) which is given by the total area in the plantation divided by the rotation age. Second, we have to establish minimum bounds for the area felled as a whole in each period which is a percentage of the area Se. Thus, if  is the cited percentage, the constraint that formalizes this requirement is as follows: Se ≤

U 

Su xu,t ≤ Se

0 ≤  ≤ 1, t = 1, 2, . . ., T

(6)

u=1

In addition, we endeavour to maintain harvest levels at the maximum sustainable yield. Thus, if Vt cc represents the maximum volume that can be extracted without degrading the forest in year t, u,t−1

ha the volume obtained by clear-cutting basic unit u in year t, and u,t−1 the volume obtained from the corresponding treatment of unit u, in year t; then this constraint can be expressed by the following equation: U 

ha t vcc t = 1, 2, . . ., T u,t−1 xu,t + vu,t−1 yu,t ≤ V

(7)

u=1

Lets us now see the objectives set in the plan. As mentioned, one of the objectives of the DM is to reorganize the plantation structure by ages during the planning horizon. In order to do this, age-classes have been defined in the following way: The first age-class is comprised of management units with ages ranging between 1 and m years (m being a constant divisor of N*), the second age-class includes management units with ages between m + 1 and 2 m years, and so on until the last age-class, which is comprised by units older than N* − m years. Note that there will be I = N*/m age-classes in total. The constant m has to be set by the DM. Bearing this in mind, let Sit denotes the total area available of age-class i at period t, which is expressed as Au,t =im

Sit =



Su

i = 1, 2, . . ., I − 1

(8)

u/Au,t =(i−1)m+1

and SIt =



Su

(9)

u/Au,t >N∗−m

Instead of the traditional linear formulation, where each age-class is forced to occupy the same area at the end of the planning horizon, in this work we use the fractional formulation developed by Gómez et al. (2006). This formulation is more flexible in order to reach the balance among age-classes, avoiding unfeasibility issues. Fractional programming is commonly used in different management

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problems that require the relative comparison of two magnitudes (Frenk and Schaible, 2005). Thus, we compare the areas occupied by two age-classes in a relative manner (by the quotient between these areas), instead of the absolute approach that is the traditional scheme. The objective will be to maximize the ratio S1T /SIT that shows the relative difference between the surface area covered by trees in the first age-class and the area covered by the last age-class. Proposition 1. Let be z∗ = (z ∗ 1 , z ∗ 2 , . . ., z ∗ U ) a feasible solution of the model. Let be T ≥ N* − m and U Se = S/N ∗ where S is the total forest area, that is S = S . u=1 u If S∗Ti (i = 1, . . ., I = N*/m) denote the total surface area available at the i age-class at period T for this solution, then S∗T1 /S∗TI ≤ 1. Also, if S∗T1 /S∗TI = 1 then S∗T1 = S∗T2 = ... = S∗TI . Proof.

See Appendix A.

Bearing in mind this proposition 1, then the objective will be to maximize this ratio because it has been proved that this ratio is lower than or equal to 1. Also, if this ratio reaches value 1 then a total balance-aged forest is obtained. If a total balance-aged forest is not possible to obtain, with this formulation, the minimum relative difference between this two surfaces is reached, which fits in with our balance objective. This provides an alternative way of balancing the forest, making the analysis more flexible in relation to the imposition of constraints. Thus, the first objective is Max f 1 (z) =

S1T

(10)

SIT

A further objective is to reduce the area of units with trees over the rotation age, N*, that is: Min f 2 (z) =



SuT

(11)

u/Au,t >N∗

We also define the economic objective of the plan; that is, the net present value (NPV) obtained through the planning horizon has to be maximized. Max f 3 (z) =

U T  

cc ha (NPVu,t−1 xu,t + NPVu,t−1 yu,t ) + LVT

(12)

t=1 u=1 cc ha where NPVu,t−1 represents the NPV obtained after clear-cutting unit u in year t, and NPVu,t−1 represent the NPV obtained after applying the corresponding treatment to unit u in year t. Also, LVT is the potential liquidation value (Seidl et al. (2007)) of the forest at the end of the planning, discounted to the present. This is a variable term that depends on the final forest situation. Finally, the total carbon sequestered by the plantation over the planning horizon should be maximized. The carbon sequestered during each period is expressed as the difference between the carbon sequestered due to the growth of the timber biomass plus the harvest, and the amount of carbon released to the atmosphere at harvest. We assume that the carbon sequestrated in a certain proportion of the harvest is never released, following the line of Creedy and Wurzbacher (2001). This proportion is related with the wood harvest that is destined to long-lived products which store carbon for long periods of time (construction, poles). Let  represents the proportion of carbon contained in timber biomass (tonnes carbon/m3 timber), cc , ˇha denote, respectively, the proportion of fixed carbon released during clear-cutting and and let ˇu,t u,t the corresponding treatment of unit u in year t. The equation that measures the balance of net carbon in a generic t period is

Ct =

U 

cc cc cc ha ha ((vcc u,t − vu,t−1 ) + xu,t (1 − ˇu,t )vu,t−1 + yu,t (1 − ˇu,t )vu,t−1 )

(13)

u=1

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So, the fourth objective, i.e., maximizing the total net carbon sequestered by the plantation, is formulated as follows: Max f 4 (z) =

T 

Ct

(14)

t=1

Thus, the model can be expressed as Max f (z) = (f 1 (z), −f 2 (z), f 3 (z), f 4 (z)) Subject to :

Constraints (1) − (7)

(15)

z ∈ {0, 1} This is a multiobjective non-linear programming problem with binary variables, and once solved provides a set of efficient forest management plans that combine technical as well as economic and environmental factors. Optimization method The model proposed is framed within multiobjective mathematical programming. The basic purpose of multiobjective programming (MOP) techniques is to generate or approximate the set of efficient solutions. Let us consider the multiobjective problem (15). A forest management feasible plan z* is said to be a Pareto optimal or an efficient solution for problem (15) if no feasible plan is strictly better than z* for at least one criterion and not worse in the remaining criteria. If z* is efficient then f (z∗) is called non-dominated. The set of forest management efficient plans is called the efficient set, and these plans, in the objective space, constitute the efficient frontier. Ideal values can be calculated by optimizing each of the objective functions individually within the feasible set, and the obtained vector is called the ideal point. In general, this point is unattainable, that is, there is no a feasible plan which simultaneously optimize all the objectives, given the usual conflict among them. But the components of this ideal point provide upper bounds for the values of the objectives in the efficient frontier. Also, another tool commonly used in the MOP is the payoff matrix. This matrix gives us information about the ranges of variability for each objective under consideration and the degree of conflict among them. To this end, the ideal point is calculated, and then it is determinated the value of each objective function at each of the optimal solutions. In this matrix, the worst result obtained for each objective constitutes a component of the called anti-ideal point. Two common approaches to generate or approximate the efficient set are the weighting method and the constraint method. In both cases, the initial multiobjective problem is reformulated into a single objective problem. In the first case, positive weights are assigned to each objective, and then the weighted objectives are added. Varying these weights in a systematic way, the efficient set can be approximated. In the second case, one of the objectives is optimized and the others are incorporated as constraints. Under certain conditions, an efficient point can be generated for each set of bounds on the additional constraints. Some works dealing with the application of these approaches for forest management are Roise (1990), Snyder and Revelle (1997), and Tóth et al. (2006). However, in an integer problem (that is our case) there can be some efficient points which are not possible to obtain by the weighting approach (because the efficient set is not a convex set), and the constraint method implies a huge computational burden. In this context, metaheuristic algorithms are gaining greater popularity given that most exact techniques – designed for linear problems – are costly or even unfeasible for solving fairly difficult problems (such as those with nonlinear functions or constraints, absence of explicit formulations or with binary or integer variables). These difficulties are common in actual MOP problems, as in our case. Evolutionary methods are the most widely applied to these kinds of problems, and predominate in the field of multiobjective metaheuristics (Coello et al., 2007). Among all the different methods available in the literature, we have used the evolutionary method SSPMO (scatter search procedure

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for multiobjective optimization, Molina et al., 2007) based on scatter search (SS) (Glover et al., 2000) to solve our forest planning problem. As mentioned in Section “Introduction”, we applied SSPMO method adapted to the forestry context since it has demonstrated competitive problem-solving abilities (Gómez et al., 2011). SSPMO has some interesting properties for our case. It is devoted to obtain a dense and welldistributed sample of the efficient set, more than solving the problem as fast as possible. Where most of the algorithms have a stopping condition based on computational cost, as for example number of generations (the case of the most popular in the literature: NSGA-II), SSPMO has a stopping condition based on quality of the approximation obtained. The algorithm does not stop till obtaining a dense and well-distributed approximation, this is, a set of efficient points covering all the area among the optimal points of each objective without empty areas. This kind of method is thus devoted for problems where computational time consumption is not a problem and quality and density of the approximation obtained is quite more important, as ours. Case study area Our forest management problem is situated in Cuba, where the industrial use of forest resources is one of the pillars of the economy. However, the forestry area of Cuba has suffered dramatically due to indiscriminate exploitation and natural disasters which has lead to remain old growth forests with subsequent financial losses and other problems. This also means that Cuban forests have a highly unbalanced age structure and, thus, an important objective in the Cuban context is to plan the redistribution of the forest in order to get a balance aged distribution. Thus, one of the main objectives of the Cuban Forestry Law (published July 1998) is to regulate the sustainable and multiple use of forests as well as to promote the rational use of forestry products. So, 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. Also, another main aim in this context is to promote the rational 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. The province of Pinar del Rio in Cuba contains the largest forests in the country. The EFI Macurije (“Empresa Forestal Integral Macurije”) company manages one of the largest forested surface areas in the sector (91,036.2 ha) located in the west of the province, in the municipalities of Guane and Mantua, as seen in Fig. 1. The main production activities of the company are silviculture, sawn timber production, and pine charcoal and resin. This company owns the only wood impregnation plant that exists in Cuba. This highlights the significant role that this company has in the country’s economy.

Fig. 1. Location of EFI Macurije.

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63.3 ha. (S1)

9

182.7 ha. 26 ha. (S2) (S3) 42.6 ha. (S4) 581.5 ha. (S5)

Age-class 0-5 (S1) 6-10 (S2) 11-15 (S3)

2451.6 ha. (S6)

16-20 (S4) 21-25 (S5) > 25 (S6) Fig. 2. Area covered by each age-class at baseline.

The model described in Section “The mathematical model” has been applied to a plantation owned by this company, EFI Macurije, which has 3347.7 ha of Pinus caribaea. This planted area was divided into 305 management units according to physical characteristics (accessibility, altitude, slopes, etc.) and with ages ranging from 3 to 41 years. Fig. 2 shows the initial distribution of the plantation’s area by age-classes, where units with 1–5year-old trees are the first age-class, and units with trees over 25 years old are the last age-class. As it can be seen, the plantation has a rather unbalanced distribution. 73.23% of the total area contained trees older than 25 years, and only 1.89% of the total area is occupied of units with ages under 5 years old. Also, the relative difference between the area of basic units less than 5 years old and those older than 25 is 2.58%. Consequently, one of the objectives in the model proposed has been to balance the plantation by age-classes, but also taking into account economic and environmental aspects. The values of the model parameters for this application are shown in Table 1. The volume coefficients used in the model depend on the treatment applied. On one hand, for clear-cutting, functions that link timber volume (m3 /ha) and age, by index class were estimated by regression analysis. The data used come from the updated inventory of forest management plan of the company. Polynomial models were selected for their simplicity and good fits obtained. On the other Table 1 Model parameters. Planning horizon Number of m. units Number of site indexes Rotation age Number of adjacency groups Number of silvicultural treatments Age intervals (years) where each intermediate treatment can be apply Number of years for the green-up requirement Minimum age for clear cutting Time span that defines the age-class Area that ensures the perpetuation of the forest Percentage of Se that establishes minimum bounds for the area to be felled in each period Maximum sustainable volume

T = 30 years U = 305 M=4 N* = 30 years K = 103 J=3 [Lb1 , Ub1 ] = [11,15] [Lb2 , Ub2 ] = [16,25] [Lb3 , Ub3 ] = [25,100] g = 1 year n* = 15 years m = 5 years 1 Se = 30 × 3, 347.7 = 111.6 ha  = 0.75 Vt = 34,869 m3 ∀t

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hand, there are volume coefficients (m3 /ha) for intermediate treatments. They appear in Appendix B, Tables 5 and 6. The atmospheric carbon conversion rate in biomass is 47.54%, i.e.,  = 0.4754 (Alvarez et al., 2005). The ˇ values were determined from Table 7 of the Forest Research Institute (1990) where the different uses of logging activities, by percentages, are described. They are showed in Appendix B in Tables 7 and 8. Timber prices (in cuban pesos/m3 ) depend on the wood assortment: 73.30 for bole, 47.5 for roundwood and 7.43 for firewood. Also, a discount rate of 8% has been used in accordance with the company. Finally, this problem includes 18,300 binary variables, 4 objective functions and over 22,000 constraints. Results and discussion The case studied in Section “Optimization method” was resolved by the evolutionary metaheuristic method described previously, using a Pentium IV processor (3.2 GHz) computer with a resolution time of 48,346 s. 134 efficient management plans over 30 years were attained. The efficient points obtained by SSPMO were shown in a single graph (Fig. 3). In order to do that, we normalized the four functions. In this way, and regardless of the measuring unit used for each objective, all values obtained were normalized to lie between 0 and 1. Normalization was performed as follows: fˆ r =

r f r − fmin

r fmax

r − fmin

f2 −f2 fˆ 2 = 2max 2 fmax − fmin

123 121 119 117 115 113

r = 1, 3, 4 (for criteria where more is better)

(16)

(for criterion where less is better)

(17)

125

127 129

111 109 107 105 103 101

131 133

1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 0

1

3

5

7

9

11

13 15 17 19 21 23 25 27 29

99

31

Normalized f1

33

Normalized f2

35

Normalized f3

37

97

39

95

Normalized f4

41

93

43

91

45

89 87

47 49

85

51

83 81

53 79

77

75

73 71 69

61 67 65 63

59

57

55

Fig. 3. Distribution of efficient points for the different normalized objectives.

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Table 2 Pay-off matrix. f1 (z*i ) z*1 z*2 z*3 z*4

0.870 0.835 0.736 0.593

f2 (z*i ) (ha.)

f3 (z*i ) (Cuban pesos)

132.2 109.8 389.4 206.3

4,206,050 4,214,117 4,225,670 4,161,465

f4 (z*i ) (tonnes) 279,895 282,689 265,158 284,295

Total volume harvested (m3 ) 907,110.9 906,368.9 889,222.8 885,483.4

Total area cut (ha.) 3,278 3,265.2 3,310.4 3,161.4

r r and fmin are the maximum and minimum values, respectively, of the function fr (r = 1, 2, 3 where fmax and 4) over the efficient frontier. Thus, all normalized functions have been set to be maximized. Fig. 3 displays the attainment of each objective function along a separate axis for each efficient point. Each axis starts in the centre of the chart (value 0) and ends on the outer ring (value 1). For example, in the efficient point 71 the values of the normalized objectives are 0.25, 0.75, 0.45 and 0.92, respectively. These records show that the attainment of balance objective (f1 ) is in the 25% of its variability range, and for the economic objective (f3 ) that percentage is 45%. From Fig. 3, it can be noted that the best value for the economic objective (f3 , solution 1) is obtained at the expense of worse values for the objectives 2 and 4. Along the points 1–17, where the normalized NPV ranges from 0.82 to 1, the normalized values for the carbon objective take their worst values. From points 18 to 23, the values for these two functions are more or less the same and from point 24 onwards the carbon objective take better values than the economic one. Anyway, in general, the normalized values for the captured carbon objective are very stable and high, but a trade off between this and the economic objective is easily seen. This figure provides a visual tool to better understanding the approximation of the efficient frontier obtained, and the different possibilities that the DM has in order to take the final decision. Our four objectives are in conflict each other, so actually all these solutions are potentially attractive to DM. There is no a priori justification for choosing any solution in the efficient frontier. However, the analysis of such solutions, supported by Fig. 3, provides DM with greater knowledge and understanding of the problem. After this, the DM can refine the efficient solution space and/or Efficient Frontier by incorporating his/her own preferences and professional judgments. In order to assist him/her, we also can use the pay-off matrix that analyses the degree of conflict among the objectives. In our case, this matrix obtained is shown in Table 2. In order to provide more information, two additional columns have been added to display the total volume of timber harvested and the total area to which clear-cutting is applied at each solution. Each column of that matrix represents the values that each objective function takes at each solution z*i . Each row represents the optimal solution of each particular objective. Thus, the first value in the first column (0.870) corresponds to the maximum value for the balance objective function (f1 ). The second value (0.835) in the first column corresponds to the value of the function f1 evaluated at the point that minimizes the area of units with trees over the rotation age (f2 ). On the other hand, the elements of the first row mean that the solution with the maximum balance (f1 = 0.870) corresponds to leave a surface of 132.2 ha occupied by trees over the rotation age, obtaining an NPV of 4,206,050 cuban pesos, and capturing 279,895 tonnes of carbon. The items shown in the main diagonal (in bold characters) represents the optimal values of each objective, the ideal point, as mentioned above. Also, the worst value of each objective (anti-ideal) is represented in the matrix by italic characters. From the analysis of the information provided in the pay-off matrix, the conflict among the objectives considered is again confirmed. It is noteworthy between the economic and the carbon sequestration objectives. In fact, the best carbon sequestration corresponds to the solution where the NPV has the smallest value, and vice versa. This fact can be also observed in Fig. 3 (with normalized values). Similar results have been found in literatures (see Díaz-Balteiro and Romero, 2003; Backéus et al., 2005). Moreover, the solution that maximizes carbon sequestration (z*4 ) also yields the worst balance regarding age distribution within the plantation. It is also the solution that less hectares cut along the planning horizon (3,161.4 ha), because when carbon capture is maximized it is tried to avoid clear cutting (that is the management option that releases more carbon to the atmosphere). Besides,

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629.2 ha. (S6)

547.5 ha. (S1)

Age-classes 0-5 (S1)

544.4 ha. (S2)

546.7 ha. (S5)

537 ha. (S4)

542.9 ha. (S3)

6-10 (S2) 11-15 (S3) 16-20 (S4) 21-25 (S5) > 25 (S6)

Fig. 4. Final age distribution in the plantation (solution 45).

in solution z*3 , the optimal from the economic point of view, the trees with ages above 30 represent the highest value, 389.4 ha (11.6% of the total area), and it is also the solution in which more hectares are cut during the planning (3.310.4 ha), because when you try to maximize NPV the units selected to be cut are not too old in order to obtain the best economic value. As it is derived from the data shown above, it is necessary to search for a suitable harvest schedule which represents a compromise among the objectives, and this issue depends on DM’s preferences, as previously commented on. Note that the principal use of MOP techniques is to provide a systematic and analytical approach to help identify and evaluate new alternatives, so that the decisions the managers take are more supported. In this line, our interest is to provide tools to assist forest managers to develop a more efficient forest management plan. Thus, a range of choices is offered to them and then, using the information that identifies the structure of the tradeoffs among the competing objectives, they may choose the better solution according on their preferences. For example, let us focus on the solution that gets the maximum balanced age-distribution at the end of the planning (solution 45). Note that the maximum value of 1 has not been achieved, so the total balance among age-classes is not reached in this problem. However, the relative difference between the areas covered by first age-class units and last age-class units increases from 2.58% in the first year to 87% in year 30. Although a fully balanced age distribution is not achieved, there has been a striking improvement compared to the baseline, as shown in Fig. 4 (also see Fig. 2). In the final year, the number of hectares occupied by trees older than 25 years is 18.8% of the total plantation area, compared to 73.23% at the initial situation. Finally, to test whether these results were due to the features of the case studied, we randomly generated simulated plantations with characteristics similar to the real case. Software was used that generated forests similar to the real plantation (in terms of soil type and species) but with different sizes, initial age distributions and site indexes. We generated seven plantations with 100, 200, 300 and 400 basic units. In each simulated forest, the programme randomly assigned to each one a given age and surface area, with an average of 10 ha. The clustering of units into adjacency groups was also randomized, but with the restriction that those in the same adjacency group had to belong to the same site index. Table 3 shows the mean size (in hectares), the number of adjacency groups, and the initial ratio (i.e., the ratio between the number of hectares of first age-class trees and last age-class at baseline) for each simulated plantation group. Each simulated forest problem was solved by applying the model proposed in this paper using the heuristic method described. The results are summarized in Table 4. The results obtained for the simulations confirm the good performance of the model because it is not influenced by the baseline conditions of the real plantation. In fact, the problem solved for the

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Table 3 Mean data values for the randomly generated forests. No. units

100

200

300

400

Size (ha.) No. adjacency groups Initial ratio S10 /SI0

1105.93 10 0.203

2073.98 20 0.398

3133.97 30 0.488

4085.21 40 0.483

Table 4 Mean results for the simulated forests. No. units 1

f f2 (ha) f3 (Cuban pesos) f4 (tonnes) No. efficient points Solving time (s)

100 0.7174 109.73 878,112 119,268 126 86,348.93

200

300

400

0.7844 154.98 1,870,021 234,200 120 72,443.86

0.8439 162.37 2,435,306 316,767 136 54,950.29

0.9029 188.014 3,622,241 479,711 126 56,356.00

real case could be included in the data shown in column 300 in Tables 3 and 4, and if these data are compared to those shown in Table 2 it can be seen that the results are similar. The number of basic units has an effect on the values of some objectives. Thus, in all cases, a more balanced age distribution is achieved for the plantation, as shown by the fact that the mean values of objective f1 are higher than those shown in the row “Initial ratio” of Table 3. This objective improves as the number of basic units increases, which indicates that the greater the number of units, the better the distribution of hectares by age-class achieved by the model. On the other hand, by the end of the planning horizon, the number of hectares of old trees (f2 ) did not exceed 10% of the total plantation in any instance, although by the end of planning horizon a completely balanced age distribution was not achieved in any simulation. Finally, the results corroborate that f3 (NPV) and f4 (carbon) are two objectives which are in conflict in all the simulations generated because as one objective improves the other deteriorates.

Conclusions This study addresses the operational planning of a plantation taking into account economic factors, such as the revenue from timber harvests, and environmental factors, such as its role in climate regulation. Regulating the age distribution of trees was added to these two objectives to obtain more sustainable production and also reducing the number of hectares with trees aged above the rotation age by the end of the horizon planning. Thus, the aim was to generate the set of efficient plans based on these objectives. By using this approach, the information required from DM is minimized, and the analysis of the efficient set allows us to compare the level of conflict between the different objectives. This provides a more comprehensive picture of the problem and helps managers take more reliable decisions. The real case used to validate the model was a Cuban plantation with baseline conditions characterized by uneven age distribution. A total of 124 efficient management plans were obtained and the level of conflict between objectives was compared using a pay-off matrix with special emphasis on NPV and carbon sequestration. After analysing other randomly generated forest simulations, we concluded that the model could be applied to other single-species plantations where the aim is timber production, regardless of the number of hectares and baseline ages, obtaining similar results. We consider that applying multi-objective decision models to forestry may be of great use in improving both the quality and the management of available resources given that these models can combine technical criteria with environmental or economic criteria, among others.

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Acknowledgments The authors wish to express their gratitude to the referees for their valuable and helpful comments, which have contributed to improving the quality of the paper. This research has been partially funded by the Regional Government of Andalusia (Excellence Research Project P10-TIC-06618). Appendix A. Appendix A Proof of Proposition 1

U

Let us denote the total clearcut area at each period of the planning horizon as C t = S x∗ , t = u=1 u u,t 1, 2, . . ., T . Taking into account that T ≥ N* − m, being I = N*/m, then T − (I − 1)m ≥ 0, and so the following relations hold: S∗T1 ≤ C T + C T −1 + · · · + C T −m+1 S∗T2 ≤ C T −m + C T −m−1 + · · · + C T −2m+1 ... S∗TI−1 ≤ C T (I−2)m + · · · + C T −(I−1)m+1 S∗TI = S − (S∗T1 + S∗T2 + · · · + S∗TI−1 Besides this, from (6) and from hypothesis Set = S/N∗, C t ≤ Set = S/N∗ (t = 1, 2, . . ., T ) so S∗Ti ≤ S/I (i = 1, ..., I − 1) and thus, S∗T1 + S∗T2 + · · · + S∗TI−1 ≤ ((I − 1)S)/I. Therefore, S∗T1 S∗TI

≤

S/I =1 S − ((I − 1)/I)S

Also, if S∗T1 /S∗TI = 1, then S∗T1 = S − (S∗T1 + S∗T2 + · · · + S∗TI−1 ) or S = 2S∗T1 + S∗T2 + · · · + S∗TI−1 being S∗Ti ≤ S/I (i = 1, . . ., I − 1), so necessarily, S∗Ti = S/I (i = 1, . . ., I − 1). And also S∗TI = S/I. That is, S∗T1 = S∗T2 = . . . = S∗TI = (1/I)S.  Appendix B. Appendix B Tables 5 and 6 show volume per hectare harvested from each site index, age class, and treatment, in accordance with the data provided by the company. Table 5 Timber volume functions (m3 /ha) for clear-cutting, being a the age of the unit. Site index

Timber yield function

R2

p value

1 2 3 4

v = −58.347 + 16.4639a − 0.1916a2 v = −45.842 + 12.486a − 0.129a2 v = −27.969 + 6.571a − 0.001a3 v = −8.097 + 3.179a − 0.025a2 + 0.001a3

0.991 0.992 0.976 0.970

p < 0.001 p < 0.001 p = 0.002 p = 0.002

Table 6 Timber volume harvested (m3 /ha) for each intermediate treatment. Site index

1 2 3 4

Treatments j=1

j=2

j=3

32.523 25.668 20.589 20.13

70.467 55.614 44.607 43.614

81.3096 64.1691 51.471 50.325

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Tables 7 and 8 show beta parameters values per hectare harvested from each site index, age and treatment. Table 7 Beta parameters for clear-cutting. Age

1–5 6–10 11–15 16–20 >21

Site index 1

2

3

4

0.415 0.333 0.303 0.279 0.261

0.446 0.346 0.315 0.291 0.273

0.453 0.357 0.327 0.309 0.291

0.519 0.404 0.40 0.321 0.309

Table 8 Beta parameters for intermediate treatments. Treatments

j=1 j=2 j=3

Site index 1

2

3

4

0.544 0.375 0.340

0.631 0.410 0.359

0.655 0.446 0.410

0.673 0.544 0.519

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