#### Appendix B.3. Definition of Model Inputs

**Constants**

$T$ is the length of the planning horizon.

$t$ is the time period.

$L=T/t$ is the number of periods into which the planning horizon is divided.

$ts$ is the time span that defines the final age class.

$S=T/ts$ is the number of final age classes, taking into account the range of years in which regeneration can occur.

$B$ is the total forest area.

${B}_{i}$ is the area in stand $i$.

${b}_{BA{U}_{1}}$ is the minimum harvest area when ${x}_{BA{U}_{1}}$ prescriptions are selected.

${b}_{BA{U}_{2}}$ is the minimum harvest area when ${x}_{BA{U}_{2}}$ prescriptions are selected.

${b}_{GS}$ is the minimum harvest area when ${x}_{GS}$ prescriptions are selected.

${b}_{GT{R}_{1}}$ is the minimum harvest area when ${x}_{GT{R}_{1}}$ prescriptions are selected.

${b}_{GT{R}_{2}}$ is the minimum harvest area when ${x}_{GT{R}_{2}}$ prescriptions are selected.

${H}_{\mathrm{min}}$ is the minimum volume harvested per period.

$mAge$ and $MAge$ are the minimum and maximum harvest ages during the planning horizon, respectively.

${V}_{ijl}$ is the volume harvested per hectare in stand $i$, prescription $j$ at period $l$.

${F}_{z}^{i}$ is the initial forest inventory on $z$ site index.

$BVN$ is the Black Vulture Nest protection area.

$C{F}_{0}$ is the initial investment and $C{F}_{t}$ is the cash-flow per year.

$r$ is the discount rate.

$\gamma $ is the parameter that expresses the density of wood in tons per m^{3} (0.5) and the percentage of carbon content (50%) existing in dry biomass, whose value is obtained through $\gamma =0.5\cdot 0.5=0.25\mathrm{tC}/{\mathrm{m}}^{3}$.

**Index Sets**

$i$ is the number of stands.

$j$ is the number of prescriptions, defining a complete treatment schedule for each stand, following the Model I structure (Johnson and Scheurman, 1977).

$k$ is the number of initial age classes.

$l$ is the index of periods.

$z$ is the site index.

$v$ is the number of priority levels.

**Variables**

${x}_{ij}$ is the area harvested at prescription $j$ in stand $i$.

${x}_{BA{U}_{1}}$ is the area harvested by $BA{U}_{1}$ at prescription $j$ in stand $i$.

${x}_{BA{U}_{2}}$ is the area harvested by $BA{U}_{2}$ at prescription $j$ in stand $i$.

${x}_{GS}$ is the area harvested by $GS$ at prescription $j$ in stand $i$.

${x}_{GT{R}_{1}}$ is the area harvested by $GT{R}_{1}$ at prescription $j$ in stand $i$.

${x}_{GT{R}_{2}}$ is the area harvested by $GT{R}_{2}$ at prescription $j$ in stand $i$.

${x}_{TNP}$ is the area harvested by $TNP$ at prescription $j$ in stand $i$.

${x}_{NOM}$ is the area selected for $NOM$ at prescription $j$ in stand $i$.

$u{x}_{i}$ is a binary variable to force decision variable ${x}_{i}$ to take either a zero value or a value greater than, or equal to, the minimum harvest area designated by parameters ${b}_{BA{U}_{1}}$, ${b}_{BA{U}_{2}}$, ${b}_{GS}$, ${b}_{GT{R}_{1}}$ and ${b}_{GT{R}_{2}}$.

${A}_{s}$ is the area belonging to s final age class $s$ at the ending period.

${F}_{z}^{f}$ is the ending forest inventory from site index $z$.

$S{V}_{ij}$ is the total standing volume at prescription $j$ in stand $i$.

$P$ is the cumulative percentage of the number of trees per hectare for each stand $i$.

$Q$ is the cumulative percentage of volume.

$C{C}_{ij}$ is the carbon capture at prescription $j$ in stand $i$.

$E{C}_{ij}$ are the carbon emissions produced by cuttings at prescription $j$ in stand $i$.

${R}_{q}$ is the normalized vector (as the difference between ideal and anti-ideal values) for each criterion $q$.

#### Appendix B.4. Definition of LGP Model

**Criteria**

$NF$ is the normal forest as an aggregated normalized function of the even flow harvest volume per period (${H}_{l}$), the forest ending inventory for each site class (${F}_{z}$) and the area control for each age class (${A}_{s}$).

$IM$ is the ideal management to be applied in each stand $i$.

$G$ is the gini index as an inequaluty measure.

$VOL$ is the volume harvested at the end of the planning horizon.

$NPV$ is the Net Present Value at the end of the planning horizon.

$C$ is the carbon balance at the end of the planning horizon.

**Goals**

(2)

**Ideal Management** (

$IM$)

(5)

**Harvest volume** (

$VOL$)

(6)

**Net Present Value** (

$NPV$)

The following constraints were introduced into the LGP model. First, endogenous ones were introduced to ensure that the area chosen by the model cannot exceed the available area in each stand. Second, a minimum harvest area was proposed for each treatment, since, as the treatments propose cutting sequences at several periods (2–4), a minimum area necessary for the intervention (5 ha for BAU_{1} and BAU_{2}, and 10 ha for GTR_{1} and GTR_{2}) for each of them, was considered. In the case of GS, this minimum area refers to the size of the gaps in which the thinnings are applied. Finally, for TPN and NOM, no amount of minimum area was imposed for their application. The last constraint ensures a minimum harvested volume per period, according to the current forest management plan and the timber harvested in past years.

**Subject to:**

(8)

**Auxiliary variables domain**(9)

**Minimum harvest area per treatment**(10)

**Minimum and maximum harvest age**(11)

**Minimum harvested volume per period**(12)

**Non-negativity constraints**