therefore, a geology map has been used in order to estimate K. An experienced researcher-geologist, who has repeatedly worked on this particular island, taking into consideration certain classes of the geological map, and considering the soil existing above, has proposed 5 classes for K factor as displayed in Table 4.
Thus, a raster GIS layer for K factor has been created and entered as input to the algorithm (Figure 2).
The LS factor has been created with SAGA GIS using a method proposed by Moore et al.  and was used as another raster GIS layer into the algorithm.
Cultivation terraces, is one of the oldest means to reduce soil loss  . This cultivation practice is extremely popular in islands due to their efficacy to stop or reduce soil loss. Taking this into consideration, C and P factors have been defined according to relevant literature   (Table 5).
In order to obtain a balanced land use allocation pattern, six hard constraints were introduced to the algorithm:
・ The first three constraints were related to allowable slopes. The maximum acceptable slopes for Dry farming land, Irrigated farming land, and Urban land were 15%, 8%, and 25% respectively  . For this constraint to be realized, a slope map has been created as raster GIS layer which was used as input to the algorithm (Figure 3). This constraint can be defined with Formula (9):
where: Slopei is the slope of land-use parcel I;
is the maximum allowable slope for land-use.
Figure 2. The K factor raster GIS layer.
Table 4. Estimation of soil erodibility factor (K) based on geological map classes.
Table 5. The C and P factor values which have been used.
・ A fourth constraint regarding the maximum land that can be transformed into Urban land was implemented. Since the model was applied to a Greek island, the relative results of the Greek Censuses held in the last 40 years were considered. Based on these data an upper limit of 100% increase was set for the Urban land parcels.
where: Urbi shows the number of land-use parcels which are transformed to urban;
Urbinit is the initial urban land-use parcels;
q is the desired percent for maximum urban increase;
P stands for the total population of the units.
Figure 3. Slope map of Naxos island.
As the model developed concerns a Greek island, the relevant provisions of the Greek legislation were also considered. Urban land was not allowed to be allocated to wildlife sanctuaries, small island wetlands, and Natural 2000 network areas. Furthermore, certain constraints related to Urban land were introduced as shown in Table 6. All these partial constraints have been combined as the fifth constraint by creating a mask GIS raster layer that defined areas where Urban land class category should not be introduced (Figure 4) because of institutional provisions.
The following Formula (11) applies the urban restriction constraint:
where: Urban_maski is the urban restriction mask value for land-use parcel i. In restricted areas Uran_mask equals 0.
・ Finally, a sixth constraint has been applied for the “Other” category, not allowing this category to be generated in areas where it did not already exist. For this constraint to be realized, the Landsat TM classification of 1987 has also been used as GIS raster layer reference and used in the problem definition of the algorithm. The following Formula (12) defines this restriction:
where: lui is the class assigned to land-use unit I;
Classi is the class value of the reference raster;
K is an auxiliary variable which should be greater or equal to 0.
5. Results and Discussion
The proposed NSGA-II algorithm was set to run for 15,000 generations in order to handle the described objectives, variables, and constraints. The first feasible solution has been achieved in generation 166 and Figure 5 presents four graphs which show the progress of this iteration process for certain combinations of objective functions. In these graphs blue points represent the 170th generation solutions, cerise points represent the 350th generation solutions, orange points represent the 2500th generation solutions, and grey points represent the final solutions belonging to the last generation of the algorithm. The graphs in Figure 5 reveal that better solutions are progressively achieved with respect to the objectives while the scatter of the solutions among the objectives constantly becomes narrower and ultimately forms a Pareto Front. This stepwise graph clearly demonstrates the
Figure 4. Urban restriction mask.
(a) (b)(c) (d)
Figure 5. NSGA-II algorithm convergence process.
Table 6. Institutional constraints related to urban land.
converging process of the proposed algorithm.
The examination of the objectives in two (Figure 5(a), Figure 5(b)) or three (Figure 5(c), Figure 5(d)) dimensions reveals that the algorithm performs sufficiently across all objectives. This trend is also demonstrated in Figure 6 that relates the per pixel average value of the objective functions to the generations created. Since, the problem has been formulated in such way that all objective functions are being minimized the average value of global solutions should decrease as the generation number increases. It can be noticed that a vast decrease has been achieved by the 4000th generation, and the algorithm could have been terminated in generation 5000 providing good solutions. Nevertheless, somewhat better solutions seem to have been achieved around the 10,000th generation while no further improvement was achieved after that point.
A large number of solutions (1279 solutions) lie on the Pareto front and “can be used to derive a suitable solution when considered against the qualitative requirements of different users”  . Different scenarios can be drawn as to which solution would be best for a specific area and whether the user should prefer an equally weighted solution or a solution that would favor certain objective function extremes. The “equal weight preferred solution possibly has the most balanced land use distribution”  and is the one that we have selected as the best solution in this study.
Table 7 summarizes land-use changes between the initial classification (1987) and the NSGA-II best solution. The modified NSGA-II model proposed to reduce the Irrigated farming land by 16.2% (552 ha decrease) mainly by transforming it into Dry farming land, and to increase Dry farming land by 131.5% (2003 ha increase). The reduction of the irrigated farming land seems to be caused by the slope constraints and the soil erosion objective. With this change, the multiobjective optimization algorithm managed to decrease the soil erosion from 1948 t/y to 1843 t/y. Besides the slope and erosion factors, since water reserves in Cyclades islands are rather low, the reduction of the irrigated farming land could be useful although it would decrease economic return. An impressive increase is proposed by the algorithm regarding Dry farming, and this is mainly caused by the economic return objective. The NSGA-II model also proposed the maximum allowed by the constraints increase for Urban land (100%) mostly on the eastern and central part of the island. Theoretically, the more urban areas are built, the higher economic return is achieved, taking into account an upper limit, above which degradation of the environment will occur. Within this framework, as mentioned in the constraints section, Urban land increase constraint was set to 100%, close to reality (Table 1). It has to be noted that executing the algorithm with different parameters (i.e. 5000 generations) or different number of locked categories (i.e. Urban land, Irrigated farming land and Dry farming land) the proposed new Urban land was more or less located in the same areas. Economic return after optimization increased by 18%.
Concerning agricultural land, the NSGA-II algorithm, given the defined objectives and constraints, proposed (Figure 7) that some parts of Naxos island used for Dry farming in the central part of the island in the classification of 1987 should be turned into Grassland, due to incompatible slopes. According to the NSGA-II best solution the Dry farming practices should be allocated mainly to the central and western part of the island because of its geomorphology.
Moreover, according to the NSGA-II best solution, many land parcels used for Irrigated farming land in the eastern part of Naxos island should be turned into Dry farming land because of the sharp slopes of the area. Farmers though, prefer irrigated farming which is more profitable, despite water deficiency, higher soil erosion and incompatible slopes.
Figure 6. The average per pixel objective function value of each generation.
Figure 7. The initial classification (1987) and the optimum result of the NSGA-II algorithm (after 15,000 generations).
Cyclades islands are touristic, and high pressure for Urban land increase occurs, which was not seriously considered during the NSGA-II model definition. This can be clearly seen in the 2010 classification, as the Urban land has increased by 112%, in comparison to the 1987 classification, and most of it have been constructed at the southwest part close to popular beaches of Naxos island. Thus, in order for the multiobjective optimization algorithm to provide more realistic results, the need for touristic development, which constitutes the main income of Naxos population, should also have been formulated as another objective function.
This analysis reveals that the optimal development of Naxos island should be based on agriculture, especially dry farming, together with balanced and not unilateral urban/touristic development, uniformly over the island
Table 7. Land use/cover changes between the initial classification (1987) and the best NSGA-II result generated within 15,000 generations.
and not just located around the coastline. Focusing on the word “balanced”, different scenarios and weights for every objective function should be evaluated during the best solution selection, while in this paper all objectives were equally weighted. In a later stage, weights could also be imposed to the current objectives in order to improve the proposed model and produce more realistic results.
Resource allocation problems necessitate addressing various and usually conflicting objectives. Multi-objective heuristic Pareto-front-based methodologies provide the mechanism to resolve this challenge. In land-use allocation, these methodologies aim to allocate certain land-uses to each land unit.
A heuristic algorithm which aims to support land-use allocation problems on Mediterranean islands has been presented in this paper. The proposed algorithm is a modified version of the NSGA-II, adjusted to allocate land-uses in Mediterranean islands taking into consideration legislation, geological characteristics, economic and environmental parameters.
The effectiveness of the modified NSGA-II algorithm was validated in a land-use allocation problem, which included four objectives, namely economic return growth, transformation suitability, maximum compactness, and least possible soil erosion, as well as six constraints, concerning geomorphology, legislation and urban development limitations. The results showed that the algorithm performed sufficiently, and could be possibly used to address even more objectives and constraints.
Comparing the development proposed by the GA with the actual development of the island, it should be noted that, in order to get more realistic results for land use planning in Mediterranean islands, the multi-objective optimization algorithm should also involve the interaction of touristic development, existing infrastructure, land ownership and environmental resources. Moreover, aiming at a more balanced development, different scenarios and weights for every objective function should be evaluated during the best solution selection, while in this paper all objectives were equally weighted.
NSGA-II is a widely used algorithm and a comparison standard, thus the authors use this algorithm as a starting point for their research on land-use allocation in Mediterranean islands. This research will be extended to other relevant multi-objective optimization algorithms for land-use planning, some of which can better handle larger number of objectives. Weights, interdependencies and uncertainties could also be imposed to the objective functions and constraints in order to improve the multi-objective optimization model and produce more balanced results.
This research was generously supported by ΙΚΥ Fellowships of Excellence for Postgraduate Studies in Greece-Siemens Program. The authors would like to acknowledge gratefully Dr Irene Galanos for her support on geological issues, and the National Cadastre and Mapping Agency of Greece for providing the preliminary borderline of the coastline of Naxos island.
Conflicts of Interest
The authors declare no conflicts of interest.
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