Biogeosciences An application of mathematical models to select the optimal alternative for an integral plan to desertification and erosion control ( Chaco Area – Salta Province – Argentina )

Multi-criteria Decision Analysis (MCDA) is concerned with identifying the values, uncertainties and other issues relevant in a given decision, its rationality, and the resulting optimal decision. These decisions are difficult because the complexity of the system or because of determining the optimal situation or behaviour. This work will illustrate how MCDA is applied in practice to a complex problem to resolve such us soil erosion and degradation. Desertification is a global problem and recently it has been studied in several forums as ONU that literally says: “Desertification has a very high incidence in the environmental and food security, socioeconomic stability and world sustained development” . Desertification is the soil quality loss and one of FAO’s most important preoccupations as hunger in the world is increasing. Multiple factors are involved of diverse nature related to: natural phenomena (water and wind erosion), human activities linked to soil and water management, and others not related to the former. In the whole world this problem exists, but its effects and solutions are different. It is necessary to take into account economical, environmental, cultural and sociological criteria. A multi-criteria model to select among different alternatives to prepare an integral plan to ameliorate or/and solve this problem in each area has been elaborated taking in account eight criteria and five alternatives. Six sub zones have been established following previous studies and in each one the initial matrix and weights have been defined to apply on different criteria. Three multicriteria decision methods have been used for the different sub zones: Correspondence to: J. B. Grau (j.grau@upm.es) ELECTRE, PROMETHEE and AHP. The results show a high level of consistency among the three different multicriteria methods despite the complexity of the system studied. The methods are fully described for La Estrella sub zone, indicating election of weights, Initial Matrixes, algorithms used for PROMETHEE, and the Graph of Expert Choice showing the AHP results. A brief schema of the actions recommended for each of the six different sub zones is discussed.


Introduction
The Salta Province (Fig. 1) has 155 000 km 2 .It is at NW of Argentina (NOA) having latitudes around 25 • S. Some winds from South or SE made climate less hot and bring rain from 400 to 800 mm/year (with peaks of 1200 mm in high altitude places in SO), and altitude has great ranges (at NE are areas at 200 m and at NW a PUNA region with summits higher than 6000 m).With 1 200 000 inhabitants it has a low density of population, and the city of Salta concentrates more or less fifty percent of the total.
The Study Area is the central part of "Chaco Salteño" (Figs. 1 and 2) that is at East of Salta Province, a bit at NE of it being a West part of large Chaco's region of South America, and has lower rains decreasing from NO to SE, as does altitude.It has a number of areas used for agriculture, but at East it contains large natural areas degraded and in them dispersed Indians live in small colonies.Apart from main links roads have low standards and in some parts environment is deteriorating progressively.
Rivers from elevations cross the area flowing into the important fluvial artery Bermejo River that comes from higher Published by Copernicus Publications on behalf of the European Geosciences Union.Bolivia at North flowing to distant great Paraná River far at SE.This river presents in Chaco Salteño a zone known as a meander digression area since due to low slopes and sediments its course is forming meanders and changes frequently.That area in rainy period is transformed into an immense sheet of water that completely isolates the communities living along the river 5 to 10 km from the riverside.It produces constant erosion that makes a great lot of sediments setting down at the Paraná River, generating an important cost in the continuous dredging.The majority and more important rivers drain towards the Paraná through this region, such as Pilcomayo and San Francisco that flows into Bermejo.The whole Chaco Salteño region has climate suffering from the lack of water, even worse at SE of it.Rainfall comes often from South and is concentrated in the summer time (from January to March).The groundwater resources are poor in volume and in quality (salty and with arsenic).It is possible to find good quality groundwater but in deep levels (100 m) with high operating costs and water is progressively taken from some rivers for irrigation.
Water is one of the most critical factors, as much for human and animal consumption, as for the production system in general and the flood events due to lack of appropriated infrastructures, and consequently is the main erosion factor.The area object of this paper is a central part of the Chaco Salteño (Figs. 1 and 2) and the severe problems for desertification and erosion are located in the North, Centre and East of this area.Besides the water, other factors linked to the human activities have an important influence in the erosion and progressive desertification of this region and environment degradation: -Historically the human handling of natural forest to use in the railway and other activities produced an environment degradation process.
-Later on the autochthonous population in large parts at East followed the irrational wood extraction and did an over pasture by letting to grown wild pigs and goats as "modus vivendi" contributing to make the situation worse.
-Actually the farms and extensive crops cultivation with single-crop in some locations do not give solution to the desertification problem.
The authors have studied these problems, considering the educational, economic, sanitary as well as social problems linked to the propriety of the lands.They believe that only one integral plan considering all factors involved and the differences among the subareas will be the starting point to change the direction of the desertification process and environment degradation.In the following lines as a synthesis a set of alternatives will be evaluated in view of relevant criteria using Multi-Criteria Decision Methods MCDM, procedure known as Multi-Criteria Decision Analysis MCDA, as an aid for posterior elaboration of an integral plan for the region.

Study sub zones
The zone has been divided in 6 sub zones as representative for study due to the environmental and socioeconomic diversities, as presented in the preliminary contributions (Anton, 2009b) and in a report for the Spanish Agency AECID (Anton, 2009a).The zones have received the name of a local center and are indicated in the Fig. 2 of the study area: Las Lajitas, La Estrella, Pichanal, Martin Hickmann, Rivadavia banda sur, Joaquín V. González.

Multi-criteria methods applied
The following discrete MCDM have been applied, they are all outranking methods: -ELECTRE I, see Roy (1985), Roy and Bouyssou (1993).
For ELECTRE and PROMETHEE methods MATH-CAD ® software was used (version MATHCAD8PRO that admits programming, with sheets legible with MATH-CAD14), from Romero (1993) for ELECTRE, and for AHP method EXPERT CHOICE ® software (from EXPERT CHOICE Inc., www.expertchoice.com).These methods got recently varied extensions and also software, but for the goals of this paper the adopted versions were robust and reliable.
These methods have been used by authors formerly in multiple applications such as: "Madrid-Valencia high-speed rail line: a route selection" (Anton and Grau, 2004a, b), "Election of water resources management entity using a multi-criteria decision (MCD) method in Salta province (Argentina)" (Grau et al., 2008), "Compromise Programming Calibration for financial analysis of firms of a common sector of business, case study for a set of Spanish banks in 1995" (Antón et al., 2004c(Antón et al., , 2007)), "Use of Decision Theory for qualification of the lands of the Community of Madrid" (Anton, 2008), "MCDM Methods for Waste Management Planning in a rural Area" (Grau et al., 2003(Grau et al., , 2007)).

Alternatives and criteria
Since MCDA involves a certain element of subjectiveness, the expertise and knowledge of the persons implementing MCDA in a particular area play a significant part in the accuracy and fairness of MCDA's conclusions.The authors were in relation with the local studies and policies and three of them live and work in Argentina.F. Colombo S. for environment has participated in books and papers about plants and forests of NOA, e.g.Colombo et al. (2001).L. de los Rios is consultant for agricultural businesses or associations such as PROGRANO, J. M. Cisneros has worked for agricultural planning and genie rural, e.g.Cisneros (1996).Numerous official documents, surveys and legislation for the agriculture of this area, such as in research center INTA Cerrillos (Instituto Nacional de Tecnología Agropecuaria just South of Salta) for agriculture were very useful in this study (INTA, 2002).First author is in relation with the administration of the province for development of water resources that is expanding as the province gets with more population and activity, in the scope of legislation evolving aiming for sustainable use of territory, e.g.Bonasso Law (2007) of Argentina, or see Salta (2010) for a web of the local government of Salta Province.The Paruelo (2009) from FAUBA (Facultad de Agronomía de la Universidad de Buenos Aires) is a survey on the accumulative effects on the forests of this region at East of Salta, connected with official decisions being then imposed for conservation of these forests.The planning for a sustainable development is in the reality as the region is evolving to a more intensely used large territory based on agriculture and forests, that into a structured argentine society originated by previous incorporation of Indians and immigrants and with modern institutions and techniques.
The five alternatives mentioned below have been selected taking into account in situ studies.

E. Erosion control crop with industrial use (biomass).
Eight criteria have been applied for the five alternatives in each sub zone following field research, expert panels, social investigation and personal interviews.For these studies about Chaco Area with these much independent alternatives for ELECTRE and PROMETHEE methods it seemed appropriated to obtain in a first stage for each j -criteria and for each i-alternative a qualifying valuation in the range (0-10), that is in itself of "more is better kind" for all, also for erosions that in name are bad by nature (more erosion is worse and thus got a worse valuation) or for water requirements (more requirements got worse valuation), and these valuations were put as Initial Matrix elements (Im(i,j )) for these methods, getting an especial Im matrix, with some differences due to diversities, for each sub zone and also for subcase variants of a sub zone.

Criterion 1: water erosion (WE)
The water erosion is very important because the interaction between natural and socioeconomic conditions.The relative water erosion indexes figure in the Initial Matrix at Table 1.The water erosion is in itself of "more is worst" kind, so the indexes in the table are valuations in inverse order of the erosion expected.

Criterion 2: eolian erosion (EE)
Winds erode, transport and deposit materials, and are effective agents in several areas of this region.Said also as "Wind Erosion" it is of "more is worst" kind for any measure of erosion intensity, but the indexes for it as in the example of Table 1 for a La Estrella case are quality evaluations and are thus in inverse order "more is better" referred to a range (0-10).

Criterion 3: implementation facility (IF)
The indexes in Table 1 were established taking into account actors' opinions, as in Anton et al. (2006 and2009a).It has been considered as of "more is better" kind.To implement crops (D and E) is considered much easier than to obtain a new autochthonous forest (A); that got indexes depending on the trees to implement, 1 to 5 in the example that follows in 4.

Criterion 4: water resources (WR)
The needs of water resources were considered alternative by alternative for each sub zone.The amounts of water needed are of kind "more is worst", but to have valuation indexes in range (0-10) that in themselves are "of kind more is better" we have considered the "Availability of obtaining

Criterion 5: economical benefits (EB)
The relative economical benefits using each alternative in a period of 25 years have been obtained.We have considered this criterion as of "more is better" kind.All alternatives are beneficial and have got a not bad 5 index in Table 1, the (C) and (E) have got 8, as (C) produces more valuable cattle also, and as (E) will produce usable crops.

Criterion 6: hand power (HP)
The social situation in that region of Chaco, including Indians Wichi at SE and along Bermejo river, results in that there is interest in giving employment to the majority of the population, and a valuation in rage (0-10) has been given for each i-alternative being higher if it reduces the existing not employed, considering anyhow this criterion as of "more is better" kind.E.g. the alternative (A) Autochtonous forest has got a low 2 in the example La Estrella of Table 1 as it requires less hand power in long periods.

Criterion 7: environmental impacts (EI)
They have been considered in each sub zone.The environmental impacts have been calculated according to Gomez Orea (1999).In itself impacts are considered as "more is worst" kind, the valuation by indexes in Table 1 is in inverse order, the autochthonous forest has got the best 8 index.

Criterion 8: social acceptance (SA)
The figures included in this criterion have been obtained from the results of different forums and meetings with institutions, organizations and native people, as put in Anton et al. (2009a).This criterion has been defined as of "more is better" kind, all alternatives were accepted and got no less than index 5, and the (C) Traditional Farms got 9 as preferred in these zones.
3 Multi-Criteria Decision Methods applied

Decisional matrix development
For each sub zone the 5 alternatives and the 8 criteria were considered obtaining an Initial Matrix, also Decisional Matrix, of valuating indexes in 1-10 scale, that was done for all the six sub zones and for the different methods, introducing these Matrixes in MATHCAD ® PC sheets (in form as in examples of Fig. 3 and of Fig. 6.These values are independent for each sub zone, due to diversities, and in the study some sub-cases were varied for some sub zones.As indicated later in 4.3 Comments the expertise of authors resulted for these global planning decision procedures in such qualifying valuations "of more is better kind", and hence the valuating indexes I j of ELECTRE and PROMETHEE have all the value +1.The weights of criteria for ELECTRE were also assessed from results from expert panels and local actors, and they were slightly different for each sub zone, and also for subcases in each sub zone.

Application of ELECTRE method
The ELECTRE method was applied following Romero (1993) using similar MATHCAD ® sheets for all the sub zones, and let explain it following the case for La Estrella sub zone with data in Table 1, as represented in Fig. 3.
A. Data, Im(i,j ) = Im ij were in Initial Matrix Im.All the j -criteria, called sometimes attributes and with (j = 1. . .8), result in all the present study of kind "more is better" and hence the ELECTRE criteria indexes I j are all 1.The i-alternatives are indicated and correspond to the five (A, B, C, D, E) presented in Sect.3.3, and are indexed with (i or k = 1. . .5).
The weights w j of criteria for ELECTRE, that have to quantify the strength of coalitions of criteria if added, were assessed representing the relative importance of each criterion for the authors, as containing their expert opinion about the effect of the criteria on the solution.The weights were normalized so as to add 1 getting W (j ) = W j .In this example the first criteria WE was given a higher value 0.2, the two following EE and IF value 0.15, the other 0.1, as relative strengths of the criteria for ELECTRE method attributed by the authors.
The values in Initial matrix Im or of weights were taken with differences for the other sub zones, in calculations done with similar Mathcad sheets, as the sub zones have some peculiarities in soil, rain, floods, slopes, sociology, etc.
B. To obtain preferences first the sheet obtains the Concordance Indexes Matrix C ik , represented as the 5 × 8matrix C in Fig. 3 obtained from that indicates how much alternative i is better than alternative k adding the weights of the criteria for which that occurs.
C. Now in ELECTRE the obstacles for these preferences, or discordances, are considered.Let first calculate the ranges D. Let calculate the Normalised Decisional Matrix Dm ij = Im ij •W j /R j , and let obtain a Discordance Indexes Matrix G.The diagonal elements of these dominance matrixes do not intervene and let take them as 0, and these obtained matrixes are in Fig. 3  H.An alternative that is better than some of the others and worse to none is considered in the kernel.This method selects as shown in Fig. 3  Q R S T U V W YX `a b Yc d e f g h i p q r s t u v w x y Y Y d e f g h i j k l m n o p q r s t corresponding ELECTRE dominance graph, that indicates the alternatives (A) and (B) are in the kernel.That is for Concordance because they have higher values in Im matrix for criteria 1 (Water Erosion Index) and 2 (Eolian erosion index) that have higher weights (0.2 and 0.15), and no very bad values in that matrix for the other criteria.

Application of PROMETHEE methods
The authors have used the Preference Ranking Organization Method (The PROMETHEE Method for Multiple Criteria Decision-Making) from Brans and Wincke (1985).Following Brans two techniques were adopted, the technique PROMETHEE I provides a partial preorder and PROMETHEE II a total preorder on the set of possible alternatives, in the previous reference total preorder is more for the ranking problem of alternatives and partial preorder is more for the choice problem.The criteria and the alternatives are the same as for ELECTRE, and for all the sub zones the Initial Matrixes Im ij , normalized weights W j and indexes I j were taken similarly from it, as that election is acceptable and as it gave reasonable results, easing the comparison of methods.
I. For all the cases the subtraction PROMETHEE formula was used as, where the non negative "preference function" P (i,j ,k) is positive if criteria j indicates preference of alternative i over alternative k and 0 if not, all the I j being 1 as before, using for each j -criteria a criteria preference function p(j ,x) as follows.
For each j -criteria a j -Type of preference function must be elected following Brans et al. (1985) between 6 Types, and Type I is the "Usual Criterion" and it has been adopted for the j -criteria with j = (4,5) that correspond to WR and EB criteria.It gives a strict preference for the criteria i with the best value index Im(i,j ), it is defined with p(j,x) = (0 if (x ≤ 0), otherwise 1). (5) For the other j -criteria the Type III "Criterion with Linear Preference" was adopted so as the decision-maker prefers progressively alternative i to k for larger deviations between Im(i,j ) and Im(k,j ), with p(j,x) = (|x|/m(j ) if (|x| ≤ m(j )), otherwise 1).( 6) Using it the preference increases linearly until deviation equals m(j ), after this value the preference is strict.For the thresholds m(j ) the value 2 was taken for j = (1,8) or WE and SA criteria, the value 4 for j = (2,3) or EE and IF criteria and the value 6 for j = (6,7) or criteria HP and EI, see Table 2.
II. Preference indexes were later defined for Initial PROMETHEE method as and for Weighted PROMETHEE method as incorporating the expert assessment of a relative importance of criteria in the values of the weights.
III.The method was applied for all sub zones, and in each for one or more sub-cases; in the Case 1 weights similar as those used in ELECTRE-I method have been adopted and other cases 2 slightly different weights, considering also some modifications in the data of the Initial matrixes and of weights, in view to study the result of varied hypothesis on the influence of the criteria.
A. Let show some cases that follow for La Estrella sub zone.
B. Case 1: the Initial Matrix was selected taking the same figures as in Table 1 and is shown in Table 2. Two procedures have been applied in order to obtain alternative pre-order: These sub zone La Estrella results for the technique PROMETHEE-II are represented in Fig. 5.
IV.To present the PROMETHEE method let take the Subcase 1.A and the Sub-case 1.B for sub zone La Estrella that uses Table 2 values of Initial Matrix and of weights, in a procedure resumed in Fig. 6, using thus Initial and Weighted PROMETHEE and for both giving results with PROMETHEE-II and I techniques.In that figure are the Initial Matrix Im(i,j ), and the weights W (j ), the indexes I j being also all 1 as the 8 criteria are of kind "more is better".Preference indexes were calculated, representing incoming flows (more unfavorable for alternative i if higher, all are positive) and out-coming flows (favorable for the alternative i, all are positive) were calculated, and from them net flows Sub-case 1.A, Table 2, using Initial PROMETHEE: Sub-case 1.B, Table 2, using Weighted PROMETHEE: Sub-case 2.A, Table 3, using Initial PROMETHEE: Sub-case 2.B, Table 3, using Weighted PROMETHEE: ! " Fig. 5. Pre-order of alternatives selected by PROMETHEE-II methods to sub zone La Estrella.
All the flows resulting for the alternatives (A, B, C, D, E) in these cases are indicated in Fig. 6.For PROMETHEE-II technique, "Ranking the Actions by a Total Preorder", the net flows, or total preorder flows, Tpf(i) are taken as valuations, they are higher if the ialternative is better, and a result vector P has been defined.For PROMETHEE-I, "Ranking the Actions by a Total Preorder", a combination of incoming and outcoming flows is considered obtaining a Partial preorder matrix of elements Cpp(i,k), shown in Fig. 6 for these cases, that are "the alternative i is indifferent to the alternative k", 3. −1 otherwise, indicating that "they are incomparable".
The results are in Fig. 6, first for Initial PROMETHEE and then for Weighted PROMETHEE, that are explained next: A. For Initial PROMETHEE, Technique PROMETHEE I total preorder, Alternative (E) (Erosion control crop with industrial use) is well preferred, due especially to the criteria (3 Implementation facility; 5 Economical benefits, 8 Social acceptance) that have higher values in matrix Im, next is (C) (Traditional farming), then come (A) (Autochthonous forest) and (B) (High value forest).With Technique PROMETHEE II partial preorder the alternative (E) outranks the (A, B, D) and is not outranked, the (C) is not outranked and outranks (B) and (D).The alternative (D) outranks none and is outranked by all.( & !(d e f g ¤h g i p q 1r ¦s t u v %w ¤x y ¦ ¤    D. The weights introduced in Weighted PROMETHEE give importance to erosion, and that is preferable than way because the higher weights for criteria 1 and 2 are due that the authors had given higher importance for erosion criteria when assessing the values of the weights.The Weighted PROMETHEE is preferable that Initial PROMETHEE because the preferences from experts about criteria are incorporated in it. V. The Weighted PROMETHEE gives preference to alternatives (A) and (B), has was done with ELECTRE, but also to alternative (E), and that is because they take differently into account the criteria for which an it is lower than some others, ELECTRE does by a measure of worst discordance with other criteria and PROMETHEE by adding the unfavorable incoming flows.

Application of AHP method
For the same sub zone La Estrella this method selects (A) and (B) alternatives as the best.For that the authors have followed the Expert Choice PC software guided following the Case 1 data and expertise, and in Fig. 7 the EXPERT CHOICE graphical interface is shown for this example.The computations for alternatives and criteria that result from AHP method application are summarized in Table 4, and in it the alternative (A) gets the higher total score, showing the partial score contributions.The data were in AHP introduced by pair-wise comparisons of the criteria by the authors with the same conceptual considerations as for elaboration of data in Tables 1 to  3. Some of them had prior experience with combination of these methods, e.g. for Anton (2006) with panels from Salta for AHP comparisons, they tend sometimes be slightly more favorable for environment or EI and less for EB.The Fig. 7 is very expressive about the results of AHP, that tend to the same results as with ELECTRE and PROMETHEE, especially to select good alternatives, but authors have more confidence in the results of ELECTRE methods and especially of "modified PROMETHEE" methods than in AHP, that is more subjective because it starts with decision analyst elicidating subjective degrees of preferences between alternatives.

Results and discussion
MCDM has been applied for the six sub zones, and in this work details were given for LA ESTRELLA sub zone.As a result of the whole the Table 5 summarizes the application of MCDM to aid to select alternatives to desertification control and erosion in Salta Province.The large scale results depend much on the sub zone, and the three methods ELEC-TRE, PROMETHEE and AHP gave results with a significant similarity.
For Las Lajitas sub zone, that is a relatively agricultural flat zone as seen in Fig. 2, the alternative (C) of farms and extensive agriculture is recommended and in fact they grow actually mostly cereals in that area in big plots.A bit at South in J. V. Gonzalez sub zone forest is also recommended, and in fact exist in Fig. 2, irrigation being possible in that area.In La Estrella sub zone the slopes are higher being near elevations at West, and natural or high value forest is recommended by all the methods.The authors have visited South of La Estrella the large farm La Moraleja (of about 40 000 ha), at East of Luis Burela and Apolinario Sarabia small towns, with irrigations from a river, where in alternative (B) they grow trees for wood as teak, cherry tree, walnut tree, in alternative (C) cereals and others including melons and lemons, bringing with buses indian wichi hand power for harvest, and about 30% of surface are original trees in alternative (A).
For Pichanal subzone, in terraces of Bermejo river, the alternative (C) of varied agriculture is found the best, but down river and at NE in M. Hickman natural forest is recommended, it exist being in part degraded.The sub zone Rivadavia is more down river, with less rains and greater danger of floods, primitive occupation with many indian wichi, and much degraded forest, conserving natural forest is recommended, the efforts to enhance quality are difficult in this very large non organized area around Rivadavia.
The design of alternatives and criteria are intended to global studies about the region of study representing Chaco Salteño.The five alternatives represent the real variety of possible farming types that are rather incompatible.The set of eight criteria represents the different consequences when a certain alternative is selected for handling a sub zone.The results depend on the elicitation on the Initial matrixes for ELECTRE and for PROMETHEE methods, and also of the weights.The running of the numerous trials comparing the election of alternatives with the behavior in existing parcels in the field has given reliability and robustness to the results.For a special area more detailed sub-alternatives and local properties of soil, rain, floods . . .will be more precise and can be included adjusting the initial matrixes.The criteria seem robust and include the real scope for decision.
The comparison of the MCDM used was in part done for the sub zone La Estrella case.The results of ELECTRE and PROMETHEE with similar initial data are mainly similar although the presentation is different.The AHP is more subjective in the introduction of data, that could enhance the effect of environmental criteria including erosion and desertification over the economic ones and that has occurred to authors in prior studies for more local especial decisions, but that has not happened in the present study where the environment criteria weights were higher and with decision analysts with deep experience.

Fig. 2 .
Fig. 2. Study area inside "Chaco Salteño" (Salta Province), with centers of sub zones.From Forest Map of Salta Province.State Secretary of Environmental and Sustainable Development, Argentine Republic, 2002; included in annex.
represented as the 5 × 5-matrix D in Fig. 3. E. Now let take for concordance and discordance thresholds ct and dt the averages of the non diagonal elements of the square Matrixes C ik and D ik respectively, obtaining ct = 0.5 and dt = 0.806 as shown in Fig. 3, F. and with them let have a. the Matrix of concordant dominance Mcd ik = (1 if (C ik ≥ ct), otherwise 0) and b. the Matrix of discordant dominance Mdd ik = (1 if (D ik ≤ dt), otherwise 0), getting with them the c.Matrix of aggregated dominance from Mad ik = Mcd ik • Mdd ik for each (i,k).

Fig. 3 .
Fig. 3. Mathematical elements of erosion and desertification integral control plan using ELECTRE-I.

Fig. 4 .
Fig. 4. ELECTRE graph and kernel showing the best alternatives to control of desertification in Salta, sub zone La Estrella.
( 8 9 @ A B %C D E F 1G ¦H I P Q ¤R ¦S T U V W X ¤W Y `a %b ¦a c

Fig. 6 .
Fig. 6.Mathematical elements of erosion and desertification integral control plan using Partial and Total Preorder for Initial PROMETHEE, and for Weighted PROMETHEE.

Fig. 7 .
Fig. 7. Graph of Expert Choice AHP application to sub zone La Estrella.

Table 1 .
Initial Matrix for La Estrella sub zone, for ELECTRE.
as the 5 × 5-matrixes Mad, Mcd and Mdd.If for a given (i,k) the Mcd ik is 1, that is an indication of dominance, and if the Mdd ik is 1 there is no discordance of alternative i over alternative k; and if both are 1, i.e. if Mad ik = 1, the alternative i is considered better than the k one.

Table 2 .
Case I, Initial Matrix for La Estrella sub zone, with weights, type of criterion and thresholds, for PROMETHEE.

Table 3 .
Case II, Initial Matrix for La Estrella sub zone, with weights, type of criterion and thresholds, for PROMETHEE.

Table 4 .
Application of AHP method to select alternatives to desertification control for La Estrella sub zone in Salta Province (Argentina).

Table 5 .
Summary of MCDM application to alternatives to desertification and erosion control in Salta province (Argentina).