2.2. Simulation Scenarios Using ModisPinaster
Growth simulators provide useful information about stand development and production under traditional or user specified silvicultural scenarios, which can be used to support decision making. In this work, the Model with Diameter Distribution for
Pinus pinaster (ModisPinaster) was selected as the appropriate simulator as it was developed to be used in the overall region of Tâmega’s Valley, where the case study is located. ModisPinaster [
21,
22] is a model developed for maritime pine, that simulates stand growth and yield, predicts the occurrence of mortality associated with competition or damages from extreme weather events, such as windrow and uprooting, and presents flexibility to select alternative silvicultural procedures to support forest management. It is a model that is available for free use, in a friendly interface, on the Computer Aided Projection of Strategies in Silviculture (CAPSIS) platform (
http://www.inra.fr/capsis accessed on 8 March 2021) [
31]. A comprehensive explanation of ModisPinaster’s components and examples of its use can be seen in a reference by Fonseca et al. [
22].
The minimum input data required to the initialization of ModisPinaster is information about the variables stand age (t, yr); number of trees per hectare (N, trees·ha
−1); basal area per hectare (G, m
2·ha
−1); dominant diameter (d
dom, cm) and dominant height (h
dom, m), as defined in
Section 2.1.
After initializing ModisPinaster with input data, simulations are run on an annual basis from the initialization point up to a maximum value of 65 years. The model allows testing alternative approaches to density regulation (manual or automatic prescriptions based on spacing factors and density indices, in addition to the usual density measures of number of trees or basal area per hectare) and provides diverse information as output, such as stem volume, biomass (aboveground, per component, and root), carbon, and energy, among other characteristics. The output data is accessible at any age within the simulation period. The authors selected as output information the standing volume and the volume of material that is removed in thinning. These volume values serve as input data in the proposed DSS. We considered a stand density regulation model based on the stand density index (SDI), as defined for maritime pine by Luis and Fonseca (2004) [
32], ranging from 35% (the lower limit of complete occupation) to 55% (a value close to the lower limit of self-thinning, or natural mortality by competition). That is, when the stand reaches a maximum stipulated value of SDI equal to 55%, the model simulates thinning from below, so that the remaining stand has an SDI equal to 35%.
The simulations were performed for two different scenarios, presented in
Figure 2. Scenario 1 consists of maintaining the stand until 25 years of age and, at that age, performing a clear-cut. For stands older than 25 years at the first period, Scenario 1 assumes that a clear-cut is performed in the first year of simulation, and this simulates the growth of a new regenerated stand for another 25 years and a clear-cut is performed again. Scenario 2 consists of letting the stand develop to 40 years of age and then performing a clear-cut at that age. Thinning is only carried out on stands that were managed according to Scenario 2 because, in the case of Scenario 1, as the age of rotation age is 25 years, the inventory data showed that the density of the stands was below the stipulated threshold for thinning.
2.4. Optimization Model
This section presents the optimization model to obtain the management plan for a given study area aiming to maximize the harvested timber volume, considering constraints that limit the areas of clearings, and also sustainability, operational and silvicultural constraints. The practices that are considered are thinning and clear-cuts following one of the two scenarios, Scenario 1 and Scenario 2, presented in
Section 2.2. The problem consists of determining which set of stands are harvested under Scenario 1 and which are harvested under Scenario 2. In addition, the year/period in which the first clear-cut is performed is also identified.
Let be the planning horizon with n one-year periods, be the stands set, and be the set of age classes.
We assign the following parameters for each stand : —area of stand j (ha); —age (years) of stand j in the first period; and —timber volume obtained by, respectively, clearcutting and thinning stand j in period l (m3); —set of periods to perform a thinning at stand j.
It is also considered parameter A
max = 10 ha which is the prescribed value in Portugal for the maximum area that can be cut in the same year and the following two years. To formulate constraints on the maximum clear-cut area and the green-up constraints, consider set
of all possible clusters that cannot be harvested as a whole at period
l, with
, and which are minimal [
27,
33]. Each of these clusters is a contiguous group of stands, whose total area is greater than A
max and that does not contain any similar cluster in the sense that if a stand is excluded from a cluster then the remaining could be cut at once. If
r is the number of stands in such a cluster then at most
r-1 of them can be harvested during the green-up period (3 years). It should be noted that two stands are adjacent when both share a boundary that is not a discrete set of points. The minimal infeasible clusters are obtained recursively, starting from clusters with only one stand, and growing them one stand at each time, until the maximum area is met.
We define the following variables: (clearcutting Scenario 1 binary variable) taking value 1 if a clearcutting is performed, adopting Scenario 1, in stand j in period l and 0 otherwise; (clearcutting Scenario 2 binary variable) taking value 1 if a clearcutting is performed, adopting Scenario 2, in stand j in period l and 0 otherwise; (Scenario 1 binary variable) taking value 1 if stand j is harvested according Scenario 1 and 0 otherwise; (Scenario 2 binary variable) taking value 1 if stand j is harvested according Scenario 2 and 0 otherwise.
In the following is presented the base forest management optimization model, FMO, which aims to maximize the volume of wood harvested during the planning horizon considering only two groups of constraints: the ones that impose that each stand is managed by one of the two scenarios with the suitable ages, constraints (2)—(8), and the ones that limit the area of clearings, constraints (9).
The objective function (1) corresponds to maximization of the volume of timber removed over the planning horizon.
Constraints (2) guarantee that each stand is harvested under one of the two scenarios, 1 or 2. Constraints (3) identify, for each stand, the harvested scenario adopted. Constraints (3) link variables and , ensuring that if variable takes the value one then there is at least one clear cut of stand j under Scenario 1, otherwise, no clear-cutting takes place over the planning horizon, under this scenario. Moreover, if variable takes the value one then there is one clear-cut of stand j under Scenario 2, otherwise, no clear-cutting takes place over the planning horizon under this scenario. Note that a stand can be subjected to a maximum of two clear-cuts whether Scenario 1 or Scenario 2 is adopted.
Constraints (4)—(7) impose conditions on the date of the first clear-cut, considering a tolerance tol years. For a stand with initial age greater or equal to 25 and less than 40, constraints (4), impose that if Scenario 1 is adopted the first clear-cut must take place in the first period or in the following tol periods. Constraints (5) imposes that if a stand reaches 25 years for an intermediate period and is managed under Scenario 1, then it must be cut in that period or in the following tol periods, as long as it is within the planning horizon. Constraints (6) impose that if a stand reaches 40 years in period l of the planning horizon and is managed according to scenario 2, then it must be cut in that period or the following tol periods. Constraints (7) impose that if the initial age is greater or equal to 40 years then the first clear-cut must take place in the first period or in the following periods within the tolerance tol, independently of being managed according to scenarios 1 or 2.
Constraints (8) impose an interval of at least four years between interventions, that is, it has to be at least four years between a thinning and a clear-cut.
Constraints (9) are known as path constraints. These constraints prevent the formation of clear-cuts whose areas are greater than 10 ha and ensure the green-up period requirement [
25,
26]. Constraints (10) establish the variables domain.
Besides constraints (2)–(10), after some computational tests, it was concluded that, although the green-up constraints contribute to ensuring sustainability, they are not sufficient, it would be necessary considering a group of sustainability constraints, which will be described next.
Additional constraints on the final average age and age structure will be included. Let
be an integer age variable representing the age, in years, of stand
j in period
l. It holds:
The following constraint imposes that the average age of the forest stands at the end of the planning horizon should be at least Ft years:
where
is the integer variable corresponding to the age of stand
j in the last year of the planning horizon. The obtained model, including this final age constraint, is denoted by FMO_FA.
To ensure sustainability, constraints on the age structure of the forest will be also included. Constraints (15) ensure that, on a period
l, the area in each age class is within
and
times the target area for each class,
Ta:
where
are age class binary variable taking value 1 if stand
j in period
l belongs to age class
k and 0 otherwise. The obtained model, including the class age constraint, is denoted by FMO_CA.
The following constraints, (16)–(19), should also be included to state the age class that stand
j belongs at each period
l. Considering the age class
be given by
, we have:
As operational constraints, a balanced income over the planning horizon is guaranteed. To do so, the planning horizon is divided into several consecutive periods,
, with equal length and it is ensured that the relative difference in the removed volume between two consecutive periods does not exceed
:
where
VolumePk is the volume of timber removed in the
k-th period,
Tk, given by:
The obtained model, including balanced revenue of volume constraints, is denoted by FMO_RV.
To demonstrate the dynamics of a DSS, the original model was adjusted according to the achieved results. In a first approach, model FMO was considered, where neither the age constraints, nor the revenue regularity were admitted. These sets of constraints were gradually included, resulting in the five models presented in
Table 2.
In order to solve the models, we used the optimization software Xpress 7.2 (available at
http://www.fico.com/Xpress accessed on 11 January 2021). Computations were performed on a computer equipped with an i5–2500 3.3 GHZ/3.7 Turbo CPU and 8 GB of RAM. The branch-and-bound algorithm was allowed to run for two hours at the most.