1. Introduction
Mediterranean Forests are ecosystems of great ecological value and although they represent less than 2% of the continental area, 20% of the world’s flora is included in these ecosystems [
1,
2]. This diversity, compared to Nordic forests, reveals a relatively high genetic variability due to the survival of many conifers and broadleaf species [
3,
4].
In these ecosystems, one of the major threats is forest fires. In fact, losses caused by fires are frequent and some authors [
5,
6] consider fires as one of the most perceptible risks faced by farmers, mainly due to the special situations that increase this phenomenon. The dry, hot summers that characterize the Mediterranean climate lead to increased fire risk and ecosystem fragility [
2].
Fire is a natural phenomenon, being part of the development strategy of some species and landscape renovation, it models the forests and is prior to Man’s attempts to confront it [
7]. Nevertheless, society’s development and human settlement have increased the occurrence of this phenomenon. Growing human populations have led to an increasing demand for several forest products, while there is a need to maintain the integrity of a multi-functional forest [
8]. In this context, the relevance of fire as an element in the forest’s evolution is considerable. Fire has multiple environmental impacts, which depend essentially on several factors regarding its characteristics, such as size and intensity [
5,
9].
Forest fires are one of the most important agents of land use change in Portugal, being a threat to forests [
10] and one of the most important natural risks affecting the country, especially in summer, with adverse economic impacts and consequences [
11].
Sustainable forest management is concerned with the enhancement of various forest-related functions (biodiversity conservation, soil maintenance, habitat, etc.) and whatever the purpose given to a forest and its products and services it is undeniable that the end result should generate maximum utility. Managers’ choices are conditioned by economic issues and often change the ability of forests to produce resources for future generations, to protect the flora and fauna [
12] and to be more resilient to forest fires. Therefore, forest management models must consider fire risk and fire losses.
Several studies carried out in the forest sector provide a considerable number of fire risk indicators [
13,
14,
15,
16,
17]. However, integration of such approaches in a management model always excludes a holistic approach, since they are only focused on forest stands [
6,
10] or do not connect existing approaches in a systemic way, as an optimization model [
14,
15,
16]. One question remains: What are the main aspects of risk evaluation that should be considered when the objective is to manage areas where forest coexists with agriculture, husbandry, shrubs, agricultural facilities, and villages?
The objective of this paper is to propose a methodological approach that integrates the fire risk in Mediterranean forest management models. This approach allows to assess the susceptibility to forest fire and fire damage and the subsequent economic variability. All components in the system are considered simultaneously (agriculture, forest, buildings, roads) and the climatic and structural determinants of fire are considered in the management of forest areas. To illustrate the approach proposed, we applied it in the Algarve region of southern Portugal, with its results being promising.
The remainder of this paper is presented as follows: in Section two, the theoretical framework is presented and existent studies analysed; in Section three, the formulation of the proposed approach is analysed; in Section four, the empirical implementation is described; and in Section five, the results are presented and discussed. Finally, Section six presents the main conclusions of this work.
3. The Methodological Approach
3.1. General Design of the Approach
The methodological approach was developed, considering previous studies and the specificity of our problem in which all the farms’ activities are accounted in an interconnected way.
Given the special nature of risk related to forest fires, the methodology used must allow risk quantification considering land use, biophysical characteristics and type of years associated with different climatic situations that favour forest fires and their effects on income variability.
Stochastic mathematical programming models are very useful to analyse the variability of farm income, allowing the incorporation of risk relative to uncertainty and the adjustment of the input–output coefficients [
19]. A general guideline of how to model risk in forestry and the use of stochastic models are provided by the authors of [
62]. In this type of model, each type of year is represented by a state of nature, in which conditions are described by a specific set of input–output coefficients that determine the respective income for each alternative plan. To each state of nature, an occurrence probability can be assigned [
63].
Stochastic programming was developed by the authors of [
64,
65] and requires the problem to be formulated as a constrained optimization problem. Stochastic programming models allow consideration of different risk sources that influence the objective function and constraints. As the decisions related to farm planning and management consider several levels, discrete stochastic programming is the best approach [
19].
Mathematical programming models easily consider the variability inherent to risk in the model parameters [
18]. One of the most used methods is the minimization of total absolute deviations (MOTAD) model [
66].
Thus, inspired by [
23], a model based on a combination of the discrete stochastic programming approach of [
64] and the approach to calculating fire damage [
16] is proposed. Its simplified mathematical formulation is presented as follows:
Subject to
where
UT is the objective function,
E is the expected farm income,
RSK is a variable that computes the total expected deviation of income,
θ is a coefficient of risk aversion,
p are product prices,
TCsn are the costs in each state of nature
sn,
DTsn is the total damage of forest fires in each state of nature
sn,
prsn is the probability of occurrence of each state of nature
sn,
is a variable that represents the area or livestock heads of
j agro-forest activities (forest, permanent and temporary crops and livestock) according to production technology, farm type, biophysical unit and state of nature,
Ssn and
Usn are the selling and in farm use variables of agro-forest products,
qj is the upper bound of land allocated to each
agro-forest activity, and
A is a matrix of the productivity coefficients for each agro-forestry activity, respectively.
Equation (1) represents a utility function, which comprises the total expected income less the expected risk. The expected income is calculated as the weighted farm income, considering the different states of nature and the respective probability of occurrence. The expected risk is the resulting total absolute deviation of farm income, in all states of nature, scaled by a coefficient of risk aversion, which represents the marginal rate for replacing a given production plan by other riskier. This procedure of considering risk in the objective function allows the trade-off problem between income and risk in a multi-criteria framework to be solved.
The objective function is subject to the set of constraints (3) and (4) established per state of nature. The former is associated with the allocation of activities according to the different biophysical and historical restrictions and the latter with the agro-forest and livestock production for sale or in farm use. Farm income per state of nature is calculated in Equation (2).
Significant questions still remain, such as: (1) How can we integrate risk? (2) How should the states of nature be considered?
3.2. Fire Damage
Fire damage is calculated according to the methodological approach proposed by [
16], which considers two main components: the hazard indicator and the potential damage. These are structural indicators of fire risk.
This approach is used by all Portuguese municipalities, and the authors of [
67] state that despite some questions about its operational application in the field, this approach is able to properly differentiate the actual loss for different elements that have the same vulnerability.
Thus, the expected total damage map shows the reader the potential loss of each charted place, and its value is calculated as follows:
where,
FRsn and
DPsn are the fire hazard and the potential damage in each state of nature
sn, and result from the sum of the values of all biophysical units, with different land uses and characteristics. The calculation is explained and presented in detail next.
The hazard indicator reflects the product of probability and susceptibility, sometimes called the fire risk. Hazard is the probability of occurrence in a given period of time and within a given area, of a potentially damaging phenomenon or a potentially damaging event that may cause loss or economic interference or environmental degradation [
16]. The Forest Fire Hazard Map shows a territory’s potential for fire occurrence and this territory’s suitability for a preventive action [
16,
68].
This is the hazard indicator chosen since it may represent the territorial susceptibility to forest fires. Thus, considering the territory is composed of several different biophysical units (different soils and slopes) the hazard indicator, in each state of nature
sn, for a given biophysical unit
b (
FRb) was calculated as [
68]:
where
is a variable that represents the area of
j agro-forest activity (forest, permanent and temporary crops and livestock) under each production technology and each farm type, in biophysical unit
b;
are the technical coefficients used to calculate the fire risk indices; and
EXP,
V,
DEC,
POP are exogenous structural fire risk parameters related to the aspect, roads network, slopes and population density respectively;
The contributions of the various indicators/parameters to the hazard indicator are presented in the next table (
Table 1).
The hazard indicator is classified according to [
70] as: class I–Low (1–103); class II–Moderate-Low (103–301); class III–Moderate (301–538); class IV–High (538–702); class V–Very high (702–1000).
Although the hazard indicator is a structural indicator, for the same location its value is different depending on the land use.
The second main component for calculating the expected fire damage is the potential damage resulting from the combination of vulnerability and economic value. For each state of nature
sn, it is calculated as the product of economic value by vulnerability:
where
VRb,j is the vulnerability of activity
j in biophysical unit
b;
EVb,j is the economic value of activity
j in biophysical unit
b. As a result,
DPb,j is the potential damage for activity
j in biophysical unit
b.
Vulnerability expresses the degree to which an element is exposed to risk and designates its resilience to the phenomenon and recovery after it. Vulnerability is analysed on a scale of zero (0) to one (1), where zero (0) means the element is resistant to the phenomenon (no damage occurrence) and one (1) means the element is completely destroyed [
16,
68].
Vulnerability indicators were defined by analysing several Municipal Plans for Defence Against Fires (MPDAF) and consulting [
16]. The National Forests authority provides in its guideline [
16] several standard values for Portugal, regarding the several land uses, which are the main reference to define the municipal vulnerability values in more than 300 Portuguese municipalities. For constructing the vulnerability indicators, each municipality carries out an analysis using a team of experts and information of the area available in their databases.
Table 2 shows the default values regarding vulnerability and economic value, based on Agriculture Economic Accounts, municipalities’ information and public information.
The economic value in euros allows calculation of the investment required to restore an element, or in our case the revenue lost due to forest fires.
Thus, the expected total damage combines the hazard components with the potential damage to indicate the potential loss due to the phenomenon. This means the expected total damage, or risk, depends on the hazard, vulnerability and economic value, and if any of the three risk factors changes, then the risk will change too.
3.3. Climatic Variability and States of Nature
In the case of Mediterranean forests, there is a marked annual climate variability that determines the risk of different potential damage. Under these conditions, the stochastic programming model is clearly superior to deterministic approaches.
Stochastic programming allows adaptive decisions to address the stochastic events and therefore may represent a more realistic and flexible way to take into account the impact of possible severe adverse events [
16]. Stochastic models are still very useful to analyse income variability as a result of uncertain resource availability and the adjustment of input–output coefficients [
19]. In Portugal, there are several examples of these models [
71,
72].
Decisions on planning and management of agriculture and forestry require the use of several decision points. For this reason, the discrete stochastic programming approach is particularly suited to this type of planning and management problem. As represented in decision trees, these problems tend to have many ramifications. Thus, to solve them using discrete stochastic programming, it is often necessary to restrict the number of sequential decision states and limit the number of events or states of nature in each of them.
The stochastic approach implies the definition of states of nature according to the favourable period for the occurrence of fires (over a historical sequence of 30 years’ data), classifying the types of years according to the risk index for forest fire progression (
IRFFPLL), given the conditions for the outbreak of forest fires and the conditions that favour the spread [
73,
74]:
where
T is the air temperature in °C;
U is the relative humidity in%;
V is the wind speed in km/h, when its course is between quadrants 350–360° and 0–180°. This indicator can be simplified in a deflagration index eliminating the wind component and may also be used to identify the days when forest fires tend to occur by relating the maximum temperature and the minimum relative humidity (
IRMAXLL):
To define the types of years corresponding to states of nature, an average daily risk of fire (using the
IRMAX and adapted
IRMAX, for which there is a more complete series), for the fire season, is considered and transformed in values from the qualitative scale proposed by the authors of [
73]. This scale considers the following five degrees of risk: (1) reduced risk (0 to 0.49); (2) moderate risk (0.5 to 0.99); (3) high risk (1 to 1.49); (4) very high risk (1.50 to 1.99); (5) maximum risk (>1.99). Then, an empirical analysis of these results is carried out, for adjustments in the states of nature to be considered. Several steps followed, including: (1) Implementation of the indicators; (2) Empirical analysis and comparison with other climatic variables; (3) Validation, by comparing with burnt areas in each year. This leads to the identification of three states of nature [
23].
Having defined types of years, it is essential to relate them with potential damage. This was done by determining allocation coefficients
with the use of cartographical information on land use, burnt area and propagation and outbreak of fires [
73] for each year individually. The coefficients are calculated using historical information of land use and burnt areas over a 20-year period for which information is available. Therefore, a coefficient by
sn, i.e.,
, considering the average or maximum values of the various burnt land uses by state of nature is built. Thus, Equation (7) can be rewritten as:
This stochastic approach was tested in a Forest Intervention Zone (FIZ) in the Algarve, southern Portugal, which is a delimitated area with common management, composed mainly of forest spaces, and subject to a forest management plan and a specific plan of forest intervention [
69]. This FIZ has management problems associated with the integration of agricultural, forestry and livestock breeding activities and is located in the inland Algarve in a demographically declining area [
75].
Figure 1 presents the location of the FIZ selected for this study in the context of Portugal and of the Algarve region (the grey piece in the map and the first yellow map).
For this FIZ, it was possible to identify and characterize the following three states of nature: SN1—years with a moderate low risk (less than 0.8 IRMAXLL) and 53% of occurrence probability; SN2—years with a moderate to high fire risk (IRMAXLL between 0.8 and 0.949) and 30% of occurrence probability; and SN3—years with a high risk of fire (IRMAXLL greater than or equal to 0.95) and 17% of occurrence probability.
The establishment of these states of nature has been validated using information on the 1993 to 2010 burned areas in the Algarve region and in the municipalities of Loulé and Silves (where the FIZ is located) according to data from the National Statistics Institute (INE) [
23]. A correlation coefficient for these municipalities of 0.56 was obtained. Also, a comparison of the burned area for the FIZ using data of the fire occurrences was done for the FIZ for a lower sequence of years (2001–2009).
Table 3 presents the burnt area per state of nature for the municipalities and for the FIZ.
The allocation coefficients are obtained by a yearly analysis from 1990 to 2009, using land use data and the burned area cartography as a basis, which has a lower resolution for the landscape homogeneous area in which the FIZ is located. Therefore, for the set of years corresponding to each state of nature, the burnt area per land use was calculated. The weight of the burned area for each land use was calculated. The final allocation coefficient is the average and the maximum weight of burned area per land use and state of nature. These final allocation coefficients represent the general situation for the average biophysical conditions of the FIZ.
3.4. Reduced Income Risk
One of the main interests of the proposed approach is the possibility of minimizing fire risk losses within states of nature (
sn), since agro-forestry producers generally have an attitude of risk aversion. In this model, minimizing the “risk of fire” (total damage) is a critical aspect. A review on the use of various risk assessment methods in mathematical programming models applied to agriculture are presented in [
63].
The mean-variance model [
76], the most common risk model, assumes that preferences rely on the expected yield (E) and the respective variance (V). The EV decision rule assumes a quadratic utility function, which was hard to compute [
66]. Thus, various approaches were developed using linear programming [
18].
The MOTAD model was presented by the authors of [
66]. In this model, the quadratic programming restriction is replaced by a constraint in the mean absolute deviation. The advantage of this approach is that it can be developed as a linear expression and can be solved using only linear programming models.
This formulation allows calculation of the efficient income/risk threshold. The model generates an income/risk frontier that comes close to the expected value/variance boundary, but is slightly less likely to contain the solution that maximizes the expected utility [
19]. In the absence of a real utility function, the final business plan selection should be left to the farmer [
66].
Other developments occurred: models that integrate a predominant perspective of security [
63] such as that proposed by the authors of [
77] who presented the Target MOTAD introducing a restriction in performance variances from a given target level. Others integrate multi-criteria techniques, such as a mean variance model based on the Target MOTAD formulations [
78]; an approach in which risk is considered a multi-criteria approach [
79]; and an approach based on compromise programming [
80].
The approach selected was the MOTAD [
66], considering that it can be further integrated in a multi-criteria framework which uses the total absolute deviation as a linear estimator of variance and allows calculation of a Risk/Income efficient frontier using linear programming [
19,
23,
71].
In our approach, the income risk is measured by the total absolute income deviation (
RSK), which is scaled by a coefficient of risk aversion θ in the objective function (1). Thus, the
RSK variable is endogenous in the model and its calculation is as follows:
where
Nsn are endogenous variables that account for the negative deviations of farm income in each state of nature (
Esn) from its average expected value
E. 4. Results and Discussion
Model results were obtained considering two alternative scenarios for the allocation coefficient used to calculate potential damage (DP). The first scenario (Ace) regards the average value of this allocation coefficient, and the second scenario (Mce) considers the maximum value that this allocation coefficient can take.
For both scenarios, the model was run considering first the maximization of expected farm income as the optimization criterion (Model 1) and then the minimization of risk as the optimization criterion (Model 2).
Thus, the results were analysed in terms of farm income, risk, fire damage and hazard indicator. In this analysis, it is important not only to show the expected figures, but also the values of variables that are obtained in each state of nature.
As stated before, allocation values for each state of nature were calculated using cartographical information on land use, burnt area and propagation and outbreak of fires.
Table 4 presents the final allocation coefficients for the two scenarios considered (Ace and Mce).
In scenario Ace, the average value of allocation coefficients is 0.005176 in state SN1, 0.004647 in state SN2 and 0.07529 in state SN3. As expected, the average value of allocation coefficients is higher in state of nature SN3. In scenario Mce, the allocation coefficients are generally larger than in scenario Ace, but are less scattered.
Table 5 presents the expected and stochastic farm income by state of nature obtained under scenarios Ace and Mce for Model 1 and Model 2.
In scenario Ace, the expected farm income is €469,379 in model 1. However, due to fire risk, this result can be higher or lower according to the pay-off of each state of nature.
In state of nature SN1, farm income is €484,240, which represents 3% more than the expected value. This state of nature has a moderate low fire risk, average relative humidity for the fire season being 70% and average temperature (from May to October) 21.2 °C. The fire season also presents 4.1 dry months, an average monthly rainfall of 33 mm and the months preceding the time of fires are very dry.
In state of nature SN2, farm income (€484,240) is higher than the expected value by more than 4%. In this case, fire risk conditions are less favourable than in state SN1, since average relative humidity is 60%, average temperature from May to October is 21.3 °C, average rainfall in these months is 29.4 mm and during the critical period of fires 4.2 dry months were recorded. Moreover, in some years, there is a dry period preceding the critical fire season. However, due to the damage allocation coefficient, farm income is greater than in state SN1.
State of nature SN3 presents the lowest farm income, which is 17.6% smaller than the expected value and is 20% less than farm income in state SN1 and state SN2. These worst results are related to the riskier conditions of state SN3. This state is characterized by over 50 days of high fire risk. Average relative humidity is 55.7%, average monthly temperature from May to October is 21.6 °C and average total monthly precipitation for these months is 28.6 mm.
In scenario Ace in model 2, instead of maximizing farm income, the objective is the minimization of fire risk. In this case, the expected farm income drops to €399,678, that is, almost 15% less than in model 1. Here, farm income does not vary according to the state of nature considered and is equal to its expected value. Please note that when minimizing fire risk, the model will allocate more inputs, in cleaning and other fire prevention activities which have influences on the amount of shrubs, thus leading to a lower fire risk. It is the cost of these cleaning and fire prevention activities that lead to a lower farm income, but that also lead to a null economical variability of income among states of nature (i.e., a stable income). Thus, it represents 17.5% and 18.5% less than the farm income obtained in model 1 for states SN1 and SN3. In state of nature SN3, the farm income of model 2 is 3.3% more favourable than in model 1.
In scenario Mce, model 1 presents a farm income of €418,911. This value is 14% and 6.6% greater than that in states SN1 and SN2. However, in state SN3, where fire risk is the highest, farm income only attains 56% of its expected value (€183,662). These results are above those obtained with model 1 under scenario Ace. The expected farm income is 10% less and stochastic values vary between 1.1% less in state SN1 and 52% less in state SN3.
Thus, farm income considers the damage caused by fire and its variability is due to the different expected fire damage in the states of nature. For both scenarios considered, farm income is higher in SN1 and SN2, while SN3 presents the lowest results because fire risk is more intense in this state of nature. Changes in damage allocation coefficients do not affect the distribution of risk consequences among states of nature. They only influence the amplitude of farm income.
As stated before, the expected damage greatly influences farm income and it is an indicator of the economic damage from fire that reflects actual and potential losses in the territory.
Table 6 shows the expected and stochastic values of fire damage obtained for model 1 under Ace and Mce scenarios.
In scenario Ace, the expected value of fire damage is €18,581. Regarding distribution among states of nature, the damage indicator tends to be low in states SN1 and SN2 but it is high in state SN3. For instance, while the damage indicator represents only 5% and 6% of its expected value in states SN1 and SN2, it is more than five times greater in state SN3. In scenario Mce, as was expected, the fire damage indicator is higher than in scenario Ace. Its expected value is almost three times greater and the stochastic value in state SN3 is almost four times higher. Thus, it can be concluded that fire damage presents residual values in scenario Ace, being concentrated in state SN3. In scenario Mce, the damage is also concentrated in state SN3, but state SN2 presents considerably higher damage than state SN1.
Figure 2 shows the average damage situation and
Figure 3 shows damage distribution among states of nature.
The spatial distribution shows that the areas with higher values of expected fire damage tend to be urban areas. In scenario Ace, the areas with higher damage (50 to 100 euros/ha) are concentrated in specific places that may easily be controlled by introducing, for instance, firebreaks alongside built-up areas. In order to prevent a worse scenario, such as that of Mce, damage will be more spread out and a more detailed strategy, such as a detailed set of prevention measures, is needed.
In
Figure 3, we can observe that allocation of the damage does not depend on the state of nature. Moreover, we can conclude that in both scenarios (Ac
e and Mce),
SN1 presents a similar distribution of damage among the biophysical units. There is a concentration of damage in urban areas, due to their value, but also in areas with high slopes that are difficult to access.
As stated before, fire damage greatly depends on the hazard indicator, which allows guiding preventive interventions in the most susceptible areas. The average value was 579 in scenario Ace and 559 in scenario Mce, which means great danger according to the CRIF methodology classification.
The hazard indicator does not show relevant differences among states of nature, given its structural nature and the residual nature of annual activities in this FIZ. Although the slight differences computed are not significant, they show that the model is sensitive to changes in the annual activities which may influence the hazard indicator.
The average distribution of the hazard indicator in biophysical units is represented in
Figure 4. In scenario
Mce, the fire hazard is reduced in the areas with steeper slopes, due to the fire hazard strategies adopted, with a consequent reduction in expected damage.
The approach used for the hazard indicator allows consideration of different damage scenarios and results in a detailed analysis by land use, which may provide important guidelines for management.
5. Conclusions
An integrated approach to including fire damage and risk of economic losses due to forest fires in a sustainable forest management model is presented in this paper.
The approach proposed is based on a stochastic programming model that allows endogenous incorporation of fire risk in the decision-making process. This is a sustainable forest management model where the decision-making process considers the maximization of farm income and minimization of economic risk simultaneously. Farm income is obtained for different states of nature which are related to different fire risk conditions and hence fire damages and economic losses. Farm income is weighted according to the probability of occurrence of the states of nature of fire risk. For fire risk, a MOTAD structure was considered, to minimize the effect of fire damage on farm income variability. Thus, the model allows the decision-making process to incorporate different paths representing different fire risk conditions and hence different responses by forest management.
Moreover, the model developed for the approach proposed considers all territorial land uses and the main factors that determine fire risk simultaneously (climatic and structural) in an integrated approach.
The results showed that the model is sensitive to the different components of total damage, giving important insights into sustainable forest management.
Several models in the literature provide manifold indicators to assess fire risk, but not in an integrated and holistic way that allows the decision-making process to incorporate the effect of all land uses, as well as fire risk conditions.
The calculation of fire damage and its integration in a forest management model is an important innovation that allows the definition of the best management strategy. This definition is enabled by risk considerations—considering different states of nature of fire damage and scenarios of damage allocation—which is also important for management.
Finally, this approach is also important to help design forest management policies. The integration of risks and technical considerations is also an important point and may be an added value for policy-making. The question is how to value this information with that purpose? The information provided by this approach has spatial references and allows development strategies in specific sites. Moreover, the damage calculated may be of great value for defining firebreaks and community management. This gathered information may be used to improve policy measures, leading to better fire-fighting strategies.