A mathematical model for agroforestry optimization

A mathematical model for agroforestry optimization Abstract: In the present article, we will describe some extensions of an agroforestry model that has been proposed and computationally implemented in [7]. Our generalizations consist of the inclusion of two additional species of tree, one culture, and a declaration of regeneration tours as variables definable by us as a parameter and the weight allocation by rentability of the treeless soil utilization as well as an exhaustive exploration of the different soil utilization scenarios in order to obtain the one who gives the best economic performance. KeyWords: Optimal control, Mathematical modeling, Agroforestry, Environmental sustainability Received: April 8, 2021. Revised: January 20, 2022. Accepted: February 10, 2022. Published: March 2, 2022. 1 Introduction The problem of forest regeneration in lands whose land use is shared between forest preservation, agri culture, or cattle grazing has been one of the great challenges of forest preservation. Such a situation is the case of the Dehesas in Spain where private landowners who make mixed use of the land are faced with the dilemma of making sustainable use of their lands in environmental terms and, on the other hand, exploiting whatever resource is economically attrac tive. In this last sense, various initiatives can be ob served in Europe to promote the forestry development of these multipleuse lands through government sub sidies to stimulate the planting of trees. The develop ment of wooded areas in these lands in private hands can turn them into spaces for ecotourism or even de pending on the type of tree planted; it could be used to produce cork such as cork oaks or produce acorns to feed pigs whose meat is highly valued for its qual ity as would be the case of holm oaks. In the review that we have carried out on this topic, we can men tion the work carried out in the case of Dehesas in Spain by the authors of [4, 5, 7]. In this article, we are inspired by the work carried out by [7] where the problem of economic and sustainable optimization of pastures in the Andalusian region in southern Spain is studied. In this article, he proposes a model of eco nomic and agroforestry optimization where land use for grazing, agriculture, and sowing of oaks and cork oaks is combined. In this mixed use of the land that they carry out in the article mentioned above, they define some rules of rotation of both pasture crops and tree renewal based on the lifetime of each tree species and also taking into account the natural refor estation specific to each species. The distribution of wooded and nonwooded areas from which the opti mization model is started is introduced as initial con ditions. Once some of these parameters have been defined, a linear optimization is carried out, taking into account the restrictions imposed by the needs of crop rotation as well as the search for obtaining eco nomic profit. Of course, one of these restrictions is not exceeding the limit of land available. We have included the details of the model exposed in [7] as well as a description of it in the section 9 of this ar ticle for the purpose of completeness of the exposi tion of our model. Based on the initial distribution of surfaces as well as nonnegativity restrictions of the variables involved, a MATLAB linprog toolbox for linear optimization routine is called that produces the decisions that will be made at each time step, and this information feeds the dynamic model that describes the behavior over time of the system from the appli cation of the decisions calculated by the linear opti mizer mentioned above. The time window used in both the [7] model and ours is 0to 100 years. The so lution to the forest optimization problem raised in [7] is approached as an optimal control problem in dis crete time and finite time horizon and that describes the agroforestry decision processes by social and pri vate agents in a pasture system. In this dynamic op timization model, laws of motion of the states are de fined during the time horizon, control variables and conditions of nonnegativity of these, global restric tion of resources, as well as artificial and natural re generation restrictions of adult trees that are useful for 1CARLOS RODRÍGUEZ LUCATERO, 2MARCELO OLIVERA VILLAROEL, 3PAOLA OVANDO 1Departamento de Tecnologías de la Información Universidad Autónoma Metropolitana Unidad Cuajimalpa, MEXICO 2Departamento de Teoría y Procesos del Diseño Universidad Autónoma Metropolitana Unidad Cuajimalpa, MEXICO 3Research Staff Member Spanish Council for Scientific Research, SPAIN WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 108 Volume 17, 2022 the definition of physical and legal restrictions in each case study addressed in our research. Some additional restrictions relative to the planting of holm oaks and cork oaks on bare ground are added for each specific case, which guides forest management to a steady state. Similarly, an objective function is defined that represents the variable of income at net present value at a fixed horizon for each of the Dehesa farms stud ied in [7]. The same authors of [7] define a gradi ent vector of the objective function that allows them to understand the operation of the proposed dynamic optimization program, and from the expressions that appear in said gradient vector, the linear nature of the decision problem can be observed. In the model that we have just described very quickly in the previous paragraphs, a very complete analysis of the income or earnings of a farm owner is made both in scenar ios where there is public intervention through subsi dies and incentives to friendly developments with the environment, as in scenarios where there is no pub lic intervention. In that sense, this work is very com plete. However, the rules that are defined for the use of the land for reforestation purposes or for agricul ture are carried out by defining very specific rules to the particular needs of each of the farms studied, which seems to us to be a limitation of this model. Another limitation of the model mentioned above is that it only works with two tree species (holm oak and cork oak). On the other hand, the optimization is car ried out from a single scenario of initial distribution of crops in bare soil. Due to these limitations in the model proposed in [7], we propose a generalization of the previous model by adding a greater number of tree species, discarding the Adhoc rules of crop dis tribution in bare soil and evaluating all possible crop distributions on bare ground to choose the scenario that maximizes income. The latter can generate de generate cases in which the global resource restriction is violated and that we eliminate after processing. On the other hand, this exhaustive test of the possible ini tial distributions can be costly in terms of calculation time, which we try to alleviate by means of a random sampling of the space of possible initial distributions of bare soil. This model expands and refines [5] by allowing a more flexible framework for land use re placeability and tree species selection. This approxi mation then allows us to examine in discrete time the composition and distribution of managed forest, aban doned forest, and nonforest uses at the farm level, which characterizes private land holdings in the re gion that is the empirical focus of our paper. This type of models has been previously applied to ana lyze efficient forest investment decisions through af forestation programs [22] forest management invest ments [5] and forest management decisions, such as thinning [20],[21]. Optimal control techniques in dis crete time have also been applied to solve farmland decision problems, whereas the farm owner or holder has a preference to conserve forest areas within the farm [3],[19]. The contribution of our paper is as fol lows. Our primary focus contributes to understand ing how different initial land use distributions and en vironmental conditions affect private actors’ invest ment decisions in forest management and afforesta tion. These changes in the distribution, in turn, throws light on how these decisions affect the longerterm provision of ecosystem services and (relatedly) for est ecosystem asset values over time [24],[25]. That is what our contribution in this article consists of, which will be described in the following sections of this work. 2 Optimal management of forest systems based on environmental economic behavior In this article we develop a framework of economic analysis for agroforestry systems by implementing of a bioeconomic mathematical model based on a dy namic optimization system inspired in the work de veloped by [7] and [22]. For this reason we use as object of evaluation of the optimization system, the rent at present value in perpetuity of a typical farm. The model allows to see the impact on the conser vation of forest areas through the implementation of policies of subsidies to forest products, the payment of environmental services, the subsidies of the gen eration of commercial plantations and the effects of livestock control without restrictions in the systems Forestry [8], being able to extrapolate to different types of ecosystems as long as the level of analysis is a farm type. The calculation of the annual income of the farm type at present value considers an infinite horizon time whereby it develops a process of nor malization for generating an average farm. The op timization model uses as methodology the resolution of an optimal control problem in discrete time follow ing the research line of [1],[3], [7], [19] considering the decision of the agents that manage a typical farm with forestry systems within their production system and the preference to conserve forested area on the farm. It is considered that the income of the farm in cludes a function of the price of the land determined by the age of the trees and the agricultural activi ties developed on the farm, being a substitute of the price estimated by the market of forest systems. It is assumed that decisionmakers are maximizers of the economic rent [4], [7]. In order to obtain the optimal income of the farm analyzed, different land use op tions are categorized in order to maximize the income at present value of the integrated management of the farm. In the model proposed in [7] it is assumed that WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 109 Volume 17, 2022 the owner decides between the different land use op tions from the initial distribution that faces as owner. That is, it decides which should be the percentage of land allocated to different forest and agricultural op tions. Such decisions are taken in concordance with the costs and longterm benefits that offer the farm size. One of the possible options is the abandonment of the farm in the long term. The model considers that the different uses of the land are replaceable be tween them, as long as the cost benefit of the chosen activity pays the replenishment and opportunity costs of the activities to be replaced, considering the sub stitution between more than two forest species, thus expanding the base model developed by [7] That is, the objective function of the analysis framework is a decision model where the renewal of the forest sys tem associated with the farm, that has as options the natural renewal with associated management, the ar tificial renewal with the development of plantations or the abandonment of the systems forestry with the introduction of livestock systems without associated forest management. Unlike [7] the model considers nonlinear cost systems in the development of the in come models of the type farms analyzed, consider ing the substitution of more than two forest species in the farm as well as pastures and agricultural sys tems. Moreover, in contrast to the works of [11] and [22] the farms abandonment is considered as a plau sible option to the agroforestry management model. Within the forest management process of the analysis framework is considered a decision already made in advance, optimal shifts according to the forest species existing in each farm, that it is considered an exoge nous decision to the process of optimal forest control, solving the problem of [10]. The model’s objective is to choose the type of renewal of forest systems as sociated with the farm through the selection of opti mal sequences of reforestation, forest plantations, and natural regeneration of existing adult trees, abandon ment of the forest or the implementation of new agri cultural areas of pastures. The final goal is to offer an optimal distribution of land use within the farm, each time one of the cutting shifts of the existing for est species is effective. The model takes into account a typical farm considering the following restrictions and assumptions to obtain the annual income of the enterprise according to the type of renewal of the for est systems associated with the farm: 1. The number of commercial forest species on the farm 2. Production cycles of each forest species. The oretical growth rate of the commercial forest species 3. The land use of the farm including conserva tion areas, agricultural (cereal crops, fruit trees, grassland among others), forest management and abandoned or dismantled areas 4. Age of the existing trees on the farm 5. Depending on the case, the management of ma ture forests (steady state) or forest plantations in their initial stages of growth can be considered. 6. Types of management of forest systems, if it is a natural regeneration or a commercial plantation 7. Profits from agricultural activities are considered part of the land use income 8. Livestock activities directly affect the processes of natural regeneration 9. The costs and benefits of the production of forestry and agricultural species in the long term are considered, as well as the provision of other types of services associated with forest systems and the farm as recreational and hunting activi ties. 10. Forestry activities include the collection of non timber and timber products 11. Subsidies are considered by productive systems, that is, forestry, agriculture and/or conservation 12. The costs and sale prices within the production system are normalized to a reference year to avoid short and medium term distortions such as inflationary shocks or the exchange market. That is, it is considered a constant price for the devel opment of the analysis. 13. The model considers both positive and negative price shocks 14. The model considers that the farm is taking mar ket prices and cannot affect them in the short or long term 15. Different discount rates are considered In our model we call this initial distribution of land use options scenario. Our model is a generalization of the model proposed in [7] because we generate each possible scenario or initial land distribution options and for each possible scenario we apply the same kind of decision optimization performed in [7]. We obtain and record for each scenario its economic utility. Af ter processing all the possible scenarios we select the one that has the higher economic utility. WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 110 Volume 17, 2022 3 Current dynamics of European forest systems The current dynamics of European forest systems cannot be understood without the interference of hu man activities in the forest environment; since the composition and structure of the landscape is condi tioned to the interactions of the existing conditions in space and the use of space by humans [23]. The use of European forest systems traditionally seeks to ef fectively use the resources available in the environ ment. Thus, the current distribution conditions of agricultural activities with the efficient use of pastures and land suitable for cultivation are associated with the composition of forest systems for the extraction of timber and nontimber resources; [12] generating cultural landscapes with its own dynamics of biodi versity and economic productivity [6] An example of this, the cork oak forests and the combination of ce real crops and pastures and hunting activities in the region. The composition and structure of forests are strongly associated with the profitability of the eco nomic and of each region’s economic and social dy namics so that the landscape units of these systems are always in constant movement. This is evident in the last 50 years with the demographic changes of ru ral systems in Europe, where there is an abandonment of rural activities and there is an ageing of population structures, an example is the abandonment of farm land, or the raising of sheep and goats [5], [15]. The change in demographic composition has had an im pact on the amount of energy contributed by humans to their environment, leaving aside the agricultural and forestry exploitation and with it a change in the structure of the landscape [23]. Changes in this artifi cial composition of the environment have many con sequences among the most relevant: a loss of value in the economic production of forest systems for their owners and therefore an abandonment of preexisting forest management; loss and spaces with very im portant cultural, economic and social values; losses of crops and pastures due to the increase in transi tional spaces of thickets and groves, although this has a lower erosion rate, higher water quality in rivers and reservoirs and an unclear result on the gain and loss of biodiversity according to the stage of forest suc cession in which the abandoned space is located [17], [18]. The longterm effect over outgoing is the gain inhomogeneity of forest structure and a lower risk of fires, although in the process of forest succession can be generated colonization of shrub species, generalist arboreal, and pyrophytes that make the landscape lose its economic and environmental value and increase their risk of fires [2] In this sense, the European Union develops a series of public policies for forest manage ment, which is summarized in Rural development pol icy, Protection against fire and air pollution, Conser vation of biodiversity, adaptation to climate change, Competitiveness of forestry investigation. The ac tions are on outgoing of these policies include sub sidy programs for forest plantations, payment of en vironmental services, incentives for the conservation of nontimber extractions such as cork and acorn col lection and nonextractive activities such as hunting activities among others [9]. The changes described both in the demographic composition of forest re gions, local economic incentives and regions make forest management a complex issue, which must be faced by decisionmakers of public policies as well as owners of farms with forestry systems throughout the country. In this sense, develop a framework of analysis that models the optimal behavior of a forest state, based on the economic benefits of forest man agement, is necessary to see the composition and dis tribution of forest systems. In the case of the study, the analysis will focus on optimizing the management of a typical forest state. We develop an economic analysis framework applied to agroforestry systems through the implementation of a bioeconomic model based on a dynamic optimization system. This type of models have been previously applied to analyze optimal forest investment decisions, though afforesta tion programs [22], and or forest management invest ments [5],[7]. A number of studies apply optimal con trol techniques in discrete time to solve farmland de cisions problems whereas the farm owner or holder has a preference to conserve forest areas within the farm [3]. This model expands and refines [5] optimal control model allowing for a more flexible optimiza tion framework in terms of land use replaceability and species selection. More specifically, our model in cludes two more forest species, considering in total four forest species that can be selected out of eight possible options at each farm. On the other hand, for est species can be substituted at the same land plot, for more profitable species as existing trees reach their rotation age. This latter change respect to [5] model is possible as our models allows for the competition of the four forest species for land that is available for afforestation/reforestation, which in turn accounts for both nonforest land uses (i.e. treeless shrubs, pas tures, crop lands and other) and abandoned forest. The management of forest represents a complex prob lem, since forest can be considered as renewable re sources, at the time they can be exhausted in order to satisfy other land use demands, such as food pro duction or land development. A way to achieve an equilibrium point between these opposites forces is to propose models that include other forest benefits, that are normally not considered in conventional eco nomic analysis. To this end we use mathematical opti mization tools based on the theory of optimal control. WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 111 Volume 17, 2022 4 Dynamic optimization We are going to model our forestry problem under the methodology of dynamic optimization. Our basic hypothesis is: Uniqueness: although each of the phenomena an alyzed ( land use on farms ) have similar behaviors and react in a similar way to external and internal stimuli as can be for instance, subsidies and taxes, change in relative prices of forest, agricultural and livestock products that determine the change in land use on farms, but different departure point. Since con ditions of distribution of land uses are very different in each farm, being conditioned by geographic, cli matic and historical factors, we consider that we can apply linear optimization processes to our problem. Our main goal is to optimize the incomes that are in fluenced by the changes in land uses. There are two main types of land use in the farms: • Woodlands • Treeless Woodlands are classified by commercial species that generate income to the owners of the farm. Commercial species have two handling options: • Natural character where species regenerate natu rally, which mean that the species meets its cy cle of natural reproduction and drive conditions that allow their reproduction of the species in a controlled manner, giving the conditions for the prevalence of the species being replaced by sys tems of secondary forests or thickets systems, which are extremely expensive to replace other land use. So, once the forest is replaced by scrub systems there is no change in land use. • The other option is artificial character where the species are planted and managed in a controlled manner within the farm There are some conditions that have to be taken into account within the process • We can not have more land than the one provided from the beginning. • There are no negative soil quantities • There are substitution between land uses , but this substitution is between woodlands and treeless uses depending on the profitability of each land use. • There is an additional limit given in handling land use scrub , scrub arising from the abandonment of trees that meet their natural cycle and are not given the conditions for the prevalence of the species being replaced by systems of secondary forests or thickets systems , which are extremely expensive to replace other land use. So , once the forest is replaced by scrub systems no change in land use. 5 Optimal control Model description In this section we are going to describe the mathemat ical tools used in the model proposed in [7] as well as in our model. Both models are discretetime fi nite temporal horizon optimal control models. Un der this decision process, the agents try to maximize the present value profit corresponding to the average hectare of grassland and can adapt each year his forest management schedule by means of artificial planta tions of trees on treeless soil and by natural regenera tion of existing adult trees. The Matlab code used for implementing both models admit a maximal temporal horizon lower than the critical regeneration age from artificial planted blockhead whose value is T < 142 years. In the following subsections we will describe the details of the mathematical model and the corre sponding implementation in Matlab. For the sake of completeness the datails of the model proposed in [7] are included in the section 9 of the present article. 5.1 Control vectors of our model The optimization process of the decisions taken by the agents involved in the agroforestal process follow the same procedure of the agents in model proposed in [7], but we have added in our model two more control vectors in order to include two new types of tree, the Stone Pine tree and eucalyptus. The control vectors are defined as follows: zt=             ni,t +na i,t 0 0 . . . −ni,t . . . 0 0             za,t =             ti,t 0 0 . . . −na i,t . . . 0 0             (1) vt=             ns,t +na s,t 0 0 . . . −ns,t . . . 0 0             va,t =             ts,t 0 0 . . . −na s,t . . . 0 0             (2) WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 112 Volume 17, 2022 wt=             np,t +na p,t 0 0 . . . −np,t . . . 0 0             wa,t =             tp,t 0 0 . . . −na p,t . . . 0 0             (3) rt=             ne,t +na e,t 0 0 . . . −ne,t . . . 0 0             ra,t =             te,t 0 0 . . . −na e,t . . . 0 0             (4) Where np,t and ne,t represent de Stone Pine tree and Eucalyptus surface naturally regenerated by natural planted trees of such species, na p,t and na e,t represent de Stone Pine tree and Eucalyptus surface naturally regenerated by artifially planted trees of such species, tp,t and te,t represent the surface reforested with Stone Pine trees and Eucalyptus by humans. The rest of the variables still represent the surfaces mentioned in [7] model. The details of this model can be consulted in the section 9 of the present article. Concerning the utilization of the deforested surface of land the previous model due to [7], the author imposed as a constraint just allowing one tree refor esting species for each deforested type of surface of a particular farm land. For avoiding these adhoc rules of reforestation that depended on the particular farm j. We eliminated the farm numbering unlike what is done in [7] model where there are 4different types of farm and in each case one or two species of tree can be sown and in some cases only one type of tree can be sown in a specific treeless terrain type. See the equation 23 in section 9 of the present article where we describe the precedent model of [7]. From our point of view this was a very unnatural constraint and we modified this criteria allowing to reforest these surface by any species of tree using a weighted asignation of surface to each species that depends on their corresponding economic rentability. ut=     ((ξt−(ti,t +ts,t +tp,t +te,t)/4, (ξt−(ti,t +ts,t +tp,t +te,t), ts,t)/4, (ξt−(ti,t +ts,t +tp,t +te,t)/4, (ξt−(ti,t +ts,t +tp,t +te,t)/4 (5) Where ξtis a transition function of soil utilization and is calculated as follows ξt={x320,t−1+xa 320,t−1+s196,t +sa 194,t +p320,t−1+pa 320,t−1+e320,t−1+ea 320,t−1 (6) 5.2 The state vector movement law of our model Introducing the new state variable as well as the new control vectors the movement law equations that de scribe the temporal evolution of the controlled state variables are the following: xt+1 =T xt+zt+1 xa,t+1 =T xa,t +za,t+1 pt+1 =Qpt+wt+1 pa,t+1 =Qpa,t +wa,t+1 et+1 =Ret+rt+1 ea,t+1 =Rea,t +ra,t+1 st+1 =Anst+vt+1 sa,t+1 =Aasa,t +va,t+1 dt+1 =dt+ut+1                            =∀t∈[0, T −1], T < ∞ (7) Where T, Anand Aahave the same meaning of the one in the model proposed in [7] and where the matrix Q, and Rare the transition matrix, corresponding to stone pine and eucalyptus reforesting land and where the elements are equal to zero excepted the elements just under the main diagonal whose values are equal to 1. The dimensions of Qand Rare 321 ×321. 5.3 The control variable constrains of our model The global resource restriction that establish that the sizes of each farm stay constant, is now expressed as follows: Sj=i(T2)·(xt+xa,t) + i(T6) ·(pt+pa,t) + i(T4)·st +i(T4)·sa,t +i(T8)·(et+ea,t)+ i(4) ·dt ∀j∈ {1,...,4}and ∀t∈ {1,...,T} (8) Where i(n)is a row vector of ones of dimension n and Sjis the productive agricultural surface of farm WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 113 Volume 17, 2022 jfor each j∈ {1, . . . 4}. Additionally some con strains regarding the natural regeneration process are imposed in such a way that the surface to be regen erated cannot be greater than the surface of trees in crital age. These constrains are expressed as follows: ni,t ≤x249,t−1 na i,t ≤xa 249,t−1 ns,t ≤s143,t−1 na s,t ≤sa 141,t−1 np,t ≤p249,t−1 na p,t ≤pa 249,t−1 ne,t ≤e249,t−1 na e,t ≤ea 249,t−1                            =∀t∈[0, T −1], T < ∞(9) By other side, the author of [7] mentioned that given the initial distribution of soil utilization, it can be ob served that very big portions of land in each farm are available for being reforested. Given this situation, it can happen that the reforestation resources can be exhausted in one period of decision. For avoiding this case some constraints imposed on the number of reforestation hectares should be compliant with the spanish forestry and public laws. The corresponding proposed upper bounds are represented by the follow ing inequalities: ni,t +na i,t +ti,t ≤Sn i,j 250 ∀t∈[0, T −1], T < ∞ ns,t +na s,t +ts,t ≤Sn i,j 144 ∀t∈[0, T −1], T < ∞ np,t +np i,t +ti,t ≤Sn i,j 250 ∀t∈[0, T −1], T < ∞ ∧ j∈ {1,...,4} ne,t +ne i,t +ti,t ≤Sn i,j 250 ∀t∈[0, T −1], T < ∞ ∧ j∈ {1,...,4} (10) 5.4 Control Optimization Objective function of our model The objective function of a control optimization model represents the base on which a deciding agent takes decisions. In the case of this agroforestry prob lem the function whose value is maximized is a cubic polynomial that evaluates the profit net present value of the reforested land depending on the soil utiliza tion. This function has the following mathematical form: fi=aiτ3+biτ2+ciτ+di∀τ∈ {0,...,320}Holm oak fs=asτ3+bsτ2+csτ+ds∀τ∈ {0,...,196}blockhead fp=apτ3+bpτ2+cpτ+dp∀τ∈ {0,...,320}stobe pine fe=aeτ3+beτ2+ceτ+de∀τ∈ {0,...,320}eucalyptus (11) Based on these functions it can be calculated the land price for hectare row vectors for each type of planta tion and stored in the following vectors: fi= (fi(0), . . . , fi(320)) fp= (fp(0), . . . , fp(320)) fs= (fs(0), . . . , fs(196)) fs,a = (fs(0), . . . , fs(196)) fd= (pct, pp, pm, po) (12) For enabling the agents to take decisions, it can be calculated the level of annual return vectors using diferent costing methods for each kind of soil utiliza tion. The following row vectors store this informa tion: yi,n = (yi,n 0, . . . , yi,n 320) yi,a = (yi,a 0, . . . , yi,a 320) yp,n = (yp,n 0, . . . , yp,n 320) yp,a = (yp,a 0, . . . , yp,a 320) ys,n = (ys,n 0, . . . , ys,n 196) ys,a = (ys,a 0, . . . , ys,a 194) ye,n = (ye,n 0, . . . , ye,n 320) ye,a = (ye,a 0, . . . , ye,a 320) ys,d = (ys ct, ys p, ys m, ys o) (13) Assuming that the deciding agent maximizes the finite horizon net present value of the profit associ ated to the average hectare of the farm j, the objective function gets the following form: Fj=1 Sj{∑T t=0 δt[yi,nxt +yi,axa,t +ys,nst+ys,asa,t +yp,npt +yp,apa,t +yq,ddt] −[fi(x0+xa,0) + fss0+fs,asa,0 +fp(p0+pa,0) + fdd0] +δT[fi(xT+xa,T ) + fssT +fs,asa,T +fp(pT+pa,T ) + fddT]} (14) q=     i(ilex) ∀j p(pine) ∀j e(eucalyptus) ∀j s(suber) ∀j 5.5 Our model control optimization Finally the control optimization problem can be writ ten as the maximization of the objective function as follows: max {ni,t,na i,t,ns,t,na s,t,np,t,na p,t,ti,t,ts,t}t=T t=1 Fj(15) subject to: equations (25)(26)(27)(28)(29) and the given intial conditions x0, xa,0, s0, sa,0, p0, pa,0, d0as well as the time horizon T. 6 Matlab implementation of our model As was mentioned in section 1 our model differs from the model proposed in [7] in two ways. The first is the WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 114 Volume 17, 2022 introduction of an additional tree species, the stone pines. The details of our mathematical model are de scribed in section 5. We also mentioned at the final part of the section 1 that our Matlab implementation generalizes the model of [7] in the sense while this model perform an optimization of the decision pro cess just over one fixed scenario, our proposal first generate all the possible scenarios, then process all them in order to calculate for each one their present value profit and finally extract the scenario with re spect to the obtained present value profit. In this way our model performs two kind of optimization, the first is over the decision process in each scenario and the second is on the selection of the scenario that has the best economic performance. The steps followed by our model can be expressed algorithmically as fol lows: 1) Table_of_scenarios = Generation_of_scenarios; 2) for i=1 to rows(Table_of_scenarios) 3) decision_optimization(Table_of_scenarios(i)); 4) end 5) Table_of_scenarios = Sort_by_present_value(Table_of_scenarios); 6) Best_Scenario = Table_of_scenarios(last_row); 7) Output(Best_Scenario); Each step of this process will be described in the following subsections. 6.1 Definition and representation of a scenario In our model we have four tree species oak, block head, stone pine, eucalyptus and four types of treeless land. Associated to each tree species we have four reforestation land when the trees become old. We have as well four types of treeless land. In the model proposed in [7] the land occupied by some type of tree can only be reforested by the same kind of tree. In our model we do not take into account this constraint. Then we can reforest a treeland piece that was occupied by oaks with stone pines. By other side, if we reforested some treeland piece with some type of tree, we cannot use this same type for reforesting another reforestation treeland piece. In what concerns the four types of utilizations of treeless land, we can repeat the utilization type in this kind of surfaces without restriction. Then ascenario is composed by three parts. The first part is associated with way we use the reforestation land, the second part is associated with way we use the treeless land, and third part is associated with the economic performance of the utilization of the land utilization of the reforestation land and the treeless land. The first vector position is associated with the oak reforestation land, the second vector position with the blockhead reforestation land, the third vector position is associated with stone pine reforestetion land, the fourth vector position with the eucalyptus reforestation land, etc. For representing a scenario in Matlab we use a numerical vector with nine places. The first four positions are used for the reforestation land, the following four positions for the treeless land, and the last position for the present value economic performance. We assigned numbers for codify each reforestation species as well as for each treeless utilization as follows: For reforestation land species code Oak 2 blockhead 4 stone pine 6 eucalyptus 8 For treeless land use code culture 9 Pastureland 10 scrub land 11 other 12 So an instance of scenario could be 4 6 8 2 10 10 9 9 1,000,000 This vector means that the oak reforestation land will be occupied by blockhead, the blockhead reforesta tion land will be occupied by stone pines, the stone pine reforestation land will be occupied by culture, the culture reforestation land will be occupied by oaks, the scrub land as well by the pastureland, the pastureland will be used for pastureland and the culture land and the scrub land will be used for culture. The last number means that the present value profit of this scenario is 1,000,000 Euros. 6.2 Generation of the table of all possible scenarios Sometimes we have to face problems where an ex haustive exploration of all the possible cases in the state spaces is necessary and not only to calculate the total number of possible configurations in such state space. For example, looking for permutations of a given set of objects. This kind of tasks are called by some authors as enumeration problems but this term can be mistaken as the calculation of the number of possibilities and for this reason it is more appropri ate to call them listing problems. Such is the kind of problem that we need to face for the generation of all possible scenarios of initial conditions in the land dis tribution of the model. In this section the generation of all the possible scenarios to be evaluated will be WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 115 Volume 17, 2022 described. For this purpose we will have to gener ate a table whose lines will be numerical vectors of 9positions where the 4first positions represent the assignment of tree species to areas for artificial refor estation, the next 4positions will correspond to the assignment of types of crops to areas of nonforest use and the last position will be used to record the evalua tion of the cost at present value of said area allocation. In the case of the available areas used for reforesta tion, if one area is occupied by a species of tree, then the other areas that remain available cannot be used to reforest with the same species of tree, that is, in this case we will be talking about permutations. no repe tition. On the other hand, in the areas for agricultural use, we can repeat crops, that is, we will be talking about permutations with repetition. The number of different ways of occupation of the reforestation land can be calculated as the number of permutations with out repetition of the four tree species. The concepts of combinatorial analysis that we will use to generate the different scenarios of land use and reforestation are those of the rule of product, per mutations without repetition and combinations with repetition. We will define these concepts below that are taken from [14]. Definition 1. (The combinatorial rule of product): If an action of choice can be split into a first and a second independent stages, and if there are mpos sible outcomes for the first stage and if, for each of these outcomes, there are npossible outcomes for the second stage, then the total procedure can be carried out, in the designated order, in m×nways. This principle is sometimes called the principle of choice. Based on the definition 1 we can be define the following concepts. Definition 2. (Variations): A variation of the kth class of n elements is an ordered kelement group formed from a set of nelements. The elements are not repeated and depend on the order of the group’s elements (therefore arranged). Vk(n) = n×(n−1) ×(n−2)× . . . ×(n−k+ 1) = n! (n−k)! (16) Definition 3. (Permutation with repetitions): Same as the definition 2 but when the objects can be re peated. The calculation of nobjects in kpossible positions in a linear array (using the definition1) is PR(n, k) = n×n×n×. . . ×n | {z } k =nk(17) Details about such notions of combinatorial anal ysis can be found in chapter 1 of book [14]. Thus, we can calculate the total number of possible permutations without repetition of nobjects making use of the formula of definition 2. In our case n= 4 and k= 4. So the number of possible land uses for reforestation would be V4(4) = 24. In addition, the number of different ways of oc cupation of the treeless land can be calculated as the number of permutations with repetitions of nobjects taken from kto kusing the formula of definition 3. In our case n= 4 and k= 4. So the number of possible land uses for non reforestation utilization of the land is 44= 256. From all the above and applying the product combinatorial principle, we have that the number of possible scenarios and therefore the num ber of rows in the table will be 24×256 = 6144. Once we know the total number of possible scenarios that correspond to the rows, we can proceed to the gen eration of the 6144 rows of the table. Combinatorial analysis allows us to be able to calculate the size that a problem space can have. However, sometimes this is not enough but it is necessary to explicitly obtain the complete list of these possibilities. That is exactly the problem that concerns us. To do this, we first generate the list of all the per mutations without repetition of the 4numbers that correspond to the codes used to denote each species of tree. For this, a Matlab program inspired by the Lex icographic order permutation generation algorithm that can be found in [16] (page 319), was imple mented. The algorithm such as it is described in in [16] (page 319) is •L1.[Visit] Visit the permutation a1a2. . . an •L2.[Find j] Set j←n−1. If aj≥aj+1 decrease jby 1repeatedly until aj< aj+1. Terminate the algorithm if j= 0. •L3.[Increase aj] Set l←n.If aj≥aldecrease l by 1repeatedly until aj< al. Then swap aj↔ al. •L4.[Reverse aj+1 . . . an] Set k←j+ 1 and l← n. Then while k < l interchange ak↔aland set k←k+ 1, l ←l−1. Return to L1 The algorithm correction is assured as step L2 has in the index jthe smallest index such that it has been visited to this point of processing all possible permu tations starting with the prefix a1a2. . . ajsuch that the following permutation in lexicographic order will make ajto have a bigger value, in L3 before the inter change aj↔alwe have that aj+1 ≥. . . ≥al−1≥ al> aj≥al+1 ≥. . . ≥anand after the interchange we have aj+1 ≥. . . ≥al−1≥aj> al≥al+1 ≥ . . . ≥an. According to D. Knuth in his book [16] this WSEAS TRANSACTIONS on SYSTEMS and CONTROL DOI: 10.37394/23203.2022.17.13 Carlos Rodríguez Lucatero, Marcelo Olivera Villaroel, Paola Ovando E-ISSN: 2224-2856 116 Volume 17, 2022