Water use optimization through alternative water depths in the Formoso Irrigation District

Otimização do uso da água no Perímetro Irrigado Formoso aplicando lâminas alternativas de água R E S U M O O objetivo do presente trabalho consistiu em propor um plano ótimo de cultivo utilizando um modelo de programação linear separável, com lâminas alternativas de água, que proporcione a maximização do retorno líquido do Perímetro Irrigado Formoso (PIF), especificamente quanto a área de lotes familiares. O modelo utilizado neste estudo foi baseado em dados do Relatório Anual Agrícola da 2a Superintendência Regional da CODEVASF em 2010, no Plano de Serviço e extensão para o PIF para 2011 e informações adicionais apresentadas por este departamento governamental. Com base nas culturas consideradas e suas respectivas funções de resposta à água, nas restrições de área cultivada, nos preços e nos custos de produção, a maximização da receita líquida no Perímetro Irrigado Formoso foi de R$ 68.384.956,53, utilizando o seguinte padrão de cultivo: 30 ha de abóbora, 30 ha de feijão Phaseolus, 977 ha de melancia, 1868 ha de banana, 1200 ha de mamão e 300 ha de limão Tahiti.A solução ótima encontrada pelo modelo indicou que a disponibilidade mensal de água no PIF não constituiu restrição efetiva à produção das culturas, uma vez que em todos os meses o volume de água necessário foi inferior ao volume máximo que o PIF pode disponibilizar (10.833.500 m3). Para os volumes mensais utilizados na solução ótima, o volume anual disponível não será restritivo se a capacidade anual de bombeamento for superior a 79.649.000 m3.


Introduction
Water is the main limiting factor in the agricultural production of semiarid regions. When water resources are a limiting factor for the production, irrigation planning must be used in order to allow the maximization of the production per unit of irrigation water. Deficit irrigation is one option to maximize water use efficiency for high yields per unit of irrigation water applied (Bekele & Tilahun, 2007;Figueiredo et al., 2007;Detomini et al., 2009).
In the current situation of irrigation districts, irrigation planning is essential in order to make the various water uses compatible, making different production sectors viable, monitoring quantity and quality of water resources and improving global wateruse efficiency levels (Frizzone, 2007).
An efficient irrigation management requires information related to crop water demand and crop-water production functions. The use of response functions allows finding useful solutions for the optimization of the use of water and fertilizers, obtaining the maximum of the product with a certain production cost (Kim & Schaible, 2000;Soares et al., 2002;Huang et al., 2004).
In order to develop optimal irrigation strategies, it is necessary to use relations between the water applied and yield, which are known as crop-water production functions. In irrigated areas, where various crops in different irrigation regimes are competing for a limited amount of water, one way of selecting an economically viable water depth, among the different water depth options available, is the use of techniques that help the decision-making process, and linear programming (LP) is one of the tools used in the optimal allocation of these resources (Dantas Neto, 1997).
Most problems involving water resources have non-linear functions. However, these functions can be linearized through the technique of piecewise linearization and the model can be treated as a separable linear programming problem (Frizzone et al., 1994).
This study aimed to propose an optimal cultivation plan, using a separable linear programming model, with alternative water depths (WD), which allows the maximization of the net revenue of the Formoso Irrigation District, especially with respect to the area of family plots, differently from the study conducted by Santos Júnior et al. (2014) for optimization, applying maximum water depths.

Material and Methods
The Formoso Irrigation District (FID) is located in the western part of Bahia state (BA), Brazil, approximately 30 km away from the city of Bom Jesus da Lapa -BA, in an area comprehended between the left bank of the São Francisco river, the right bank of the Corrente river and the Federal Highway BR -349, connecting Bom Jesus da Lapa to Santa Maria da Vitória -BA.
The FID has a total area of 19,471.5 ha, with 12,100 ha of irrigable land, in the final stages of implementation and occupation, distributed as follows: -Family plots: 913, totaling 4,700 ha; and -Business plots: 249, totaling 7,400 ha. In this study, it was considered an available area of 4,405 ha, representing the area for the family plots, already in operation in the Formoso Irrigation District.
According to Köppen's classification, the climate of the region is BSwh', semiarid hot, characterized by rains in the summer and a well-defined dry period during the winter, and absence of water surplus. The annual averages of temperature, rainfall and potential evapotranspiration are respectively 25.3 °C, 831 mm and 1,418 mm, and Latosols and Cambisols prevail in the region, with transitional vegetation between Caatinga and seasonal forest (Borges et al., 2010).
In this study, seven crops traditionally cultivated by the colonizers of the Formoso Irrigation District were considered: pumpkin, Phaseolus bean, Vigna bean, watermelon, maize, banana, papaya, besides Tahiti lime, which is under study. In the occasion, fruit cultivation represented 96.6% of the area, with banana representing 96.2% and the annual crops 3.4%, notably maize and bean (CODEVASF, 2010).
The general model is an adaptation proposed by Frizzone (1994), which aims to simulate an optimal cultivation pattern, in order to maximize the profit of the project, resulting from many crops (Table 1), considering the minimum and maximum water depths for each crop and their respective estimated yields.
The response functions of all studied crops were divided into linear segments. The criteria to select the zone for resource allocation for each crop were: water depth corresponding to maximum yield (w0); 2) minimum water depth below which *W -Water depth; **Y -Yield Table 1.Minimum and maximum water depths for each studied crop and their respective yields plants, in the studied area, suffer water stress and yield is greatly reduced (w9). Minimum and maximum amounts of water in order to obtain reasonable yields, as well as minimum and maximum yields for each crop are shown in Table 1.
The crop-water response functions that best fitted the available data are described in Table 2. Only the functions for the crops Phaseolus bean and maize were developed in the Formoso Irrigation District.

The objective function
The model used in this study was based on data from the 2010 Annual Agricultural Report of the 2 nd Regional Superintendency of CODEVASF (São Francisco and Parnaíba Valley Development Company), the 2011 Technical Assistance and Rural Extension Plan for the Formoso Irrigation District (CODEVASF, 2011) and additional information provided by the mentioned government department.
The coefficients of the objective function represent the net profit per unit area for each crop (R$ ha -1 ). The obtention of the coefficients was based on product prices, yield levels and production costs throughout the period considered in the analysis. Therefore, the aim was to design the model, using the characteristics of the land use, which allowed determining the best plans, suiting the goals of the study, through a matrix layout specific to the model used.
Production cost data for 5-year and 3-year cultivations were used for the crops Tahiti lime and banana, respectively, corresponding to the periods in which these crops were in full production.
The objective function for the irrigated area was specified as the maximization of the annual net return, NR, subjected to constraints of water availability and other inputs. It can be schematically represented as follows: The model is subjected to the following constraints: 1. Area: Y: Yield (kgha -1 ); W: Water depth (mm) where: NR -net return from the cultivation of n crops, in R$; Pi -unitary price of the product, in R$ kg -1 ; A ij -amount of the input j demanded by the crop i; Ci -unitary cost of the input j used by the crop i; X ik -area cultivated with the i-th crop, in ha; and Yi -f(w i ), yield obtained for the crop i, when applied w i water unities, in kg ha -1 .
3. Market and industrial processing capacity: where: WM -monthly amount of water available; WT -total amount of water available; S -maximum area available for cultivation in the month j, in ha; X ij -cultivated area in the month j, in ha; X i0 -total cultivated area with the crop I (ha); and X max , X min -constraint in the cultivated area of the crop i, in the period k.

The applied model
The objective function applied in the problem was: The eight studied crops were represented by the indices (1, 2, ..., 12), where X 1 is the area planted with pumpkin in February, X 2 the area planted with pumpkin in June and X 12 the area planted with Tahiti lime in its 5 th year. In the model with alternative water depths, crops can be irrigated with different water depths, which are represented by the indices (9, 8, ..., 0). The highest numbers designate the lowest water depths and vice versa. Thus, X 19 means area planted with pumpkin in February and irrigated with minimum water depth, that is, 210 mm, and X 129 means area planted with Tahiti lime and irrigated with minimum water depth, that is, 1400 mm. On the other hand, X 10 means area planted with pumpkin in February and irrigated with maximum water depth of 384 mm and X 120 means area planted with Tahiti lime and irrigated with maximum water depth of 2100 mm.
Area constraints correspond to the combination of crops in the 12 months of the year and determine that the occupied area must be lower than or equal to the available area. An available area of 4405 ha was considered in this study, which represents the area of the family plots in the Formoso Irrigation District. The equations referring to area constraints(C.1) are numbered as Eq. 10 and 11.

Area Constraint:
Water constraints ensure that monthly crop water demand is not higher than the water volume available in the district during each month. The monthly volume used in the calculation of the available volume in the most critical month (October) was equal to 10,833,500 m 3 , and the highest annual water volume offered to the users was equal to 79,649,300 m 3 , in 2009. Water constraint equations are numbered as Eq. 14, 15 and 16. where: AJAN -area irrigated in January; and ADEZ -area irrigated in December.

Water
Equations 12 and 13 ensure that, considering each crop separately, the area irrigated with certain irrigation depth must be less or equal to area irrigated with an immediately lower water depth. Constraints of minimum and maximum areas to be cultivated were incorporated into the model, using market conditions, internal consumption, processing capacity or regional problems, based on market research. The values of each constraint were estimated as a function of the mean production expected per hectare, and per cultivated area, data provided by CODEVASF and PLENA Consultoria e Projetos, which is the company administering the Formoso Irrigation District.

Market Constraints
AREAMIN3 -minimum area to be planted with Phaseolus beans in March; AREAMIN7 -minimum area to be planted with watermelon in April; AREAMAX7 -maximum area to be planted with watermelon in April; AREAMIN10 -minimum area to be planted with banana in the 3 rd year of the crop; AREAMIN11 -minimum area to be planted with papaya in the 1 st year of the crop; AREAMAX11 -maximum area to be planted with papaya in the 1 st year of the crop; and AREAMAX12 -maximum area to be planted with Tahiti lime in the 5 th year of the crop.
In the resolution of the linear programming model, the software LP -88 (LINEAR PROGRAMMING -88) was used, which solves linear equations using the revised simplex method, an iterative algorithm, for the optimization of net profit and for the study of sensitivity analysis.

Results and Discussion
In the Formoso Irrigation District (FID), the highest annual water volume offered to the users was equal to 79,649,300 m 3 and the highest monthly volume was 10,833,500 m 3 , in 2009. Using these water constraints, the model found the optimal cultivation plan (Table 3), with the respective sensitivity analysis of the objective function.
Pumpkin and Phaseolus beans cultivations were only indicated for planting in February and March, respectively, with areas of 30 ha, defined by the production demand to meet internal consumption.
For watermelon, the financial return per produced unity of this crop can be altered to values ranging from R$ 4,103.33 to R$ 4,415.89 in the model, without changing the optimal solution (cultivation of 917 ha).
All crops in the optimal cultivation plan, except for papaya and Tahiti lime, used water depths lower than the maximum water depth, which shows that not always the highest water depth and, consequently, the highest yield leads to the highest financial return. Santos Júnior et al. (2014) studied the allocation of water in the Formoso Irrigation District considering the irrigation water depths for maximum yield. In the study, they observed a financial return of R$ 67,643,676.78 for an optimal cultivation pattern that includes pumpkin 1, Phaseolus bean 1, watermelon 1, papaya and lemon, with cultivated areas equal to the ones obtained in this study. The crops watermelon 2 and banana had cultivated areas different from the ones obtained in this study: 1243 ha against 917 ha and 1542 ha against 1868 ha, respectively. This difference promoted a financial return 1.1% higher for the model with alternative water depths, representing a difference of R$ 741,279.75.
Marginal costs of crops not recommended for cultivation, called non-basic variables in the model, are shown in Table  4. Pumpkin and Phaseolus bean cultivations, with planting in June, were not recommended; in this case, there is a marginal cost associated with these activities, i.e., for each hectare cultivated with pumpkin during this period, there will be a reduction of R$ 531.52; for Phaseolus beans, the reduction will be equal to R$ 1,350.37.
The minimum value of the contribution to the profit that each crop, irrigated without deficit, must provide in order to be indicated for cultivation is shown in Table 4. Maize cultivation must not be recommended while its contribution to the profit is lower than R$ 3,820.27. Phaseolus beans, planted in June, will only participate in the optimal solution if its contribution to the profit is higher than R$ 3,830.37; this crop planted in March became a basic solution only because of the necessity to cultivate at least 30 ha to satisfy the internal demand.
Analysing Table 5, it is noticed that the resource land was not restrictive; in all months of the year, the occupied areas did not reach the limit of the available area, i.e., 4405 ha. The greatest occupation corresponds to the months of July, August and September, the period of the watermelon cycle (Table 5). Table 3.Cultivated areas (ha), water depths with their respective yields and financial return for each studied crop, and sensitivity analysis of the model's objective function Table 4.Marginal costs associated with the non-basic activities and minimum value of the contribution to the profit for a volume Similarly, land was not restrictive in several months of the year in a study conducted by Santos et al. (2009) in the Baixo Acaraú Irrigation Project -CE, also with analysis by linear programming tools. Andrade et al. (2008), seeking to establish an optimal cultivation pattern in the Gorutuba Irrigation District, MG, found that land-related constraints were not that pronounced in comparison to the effects of water and labor constraints.
The sensitivity analysis related to market and internal consumption constraints is shown in Table 6. The constraints of minimum area of pumpkin and Phaseolus beans have negative shadow prices, which means that the entry of these crops into the optimal solution, forced by the constraints, leads to reduction in the net return.
If the constraint AREAMIN3 (minimum area cultivated with Phaseolus beans planted in March), which has a minimum limit of 30 ha, had its value increased to 31 ha, the economic return of the objective function would be reduced in R$ 1,350.37. On the other hand, each hectare of land not cultivated with this crop would provide gain of R$ 1,350.37. The same reasoning applies to the constraint AREAMIN1.
The crops papaya and Tahiti lime, and the constraints AREAMAX11 and AREAMAX12 had positive shadow prices, which indicates that, if the maximum planting areas increase, the economic return will also increase. Thus, for each additional hectare of papaya, in the interval from 344 to 2291 ha, there would be a gain of R$ 9,042.13.
The constraint imposed to the banana crop AREAMIN10 has a shadow price equal to zero in the studied model. Thus, if the banana area (1868 ha) is increased or reduced, the basic variables composing the optimal solution will not be altered, but the values of the objective function will.
The sensitivity analysis for monthly and annual water volumes are shown in Table 7, considering a maximum availability of 79,649,300 m 3 per year and 10,833,500 m 3 per month.
The annual water availability in the Formoso Irrigation District presents itself as an effective constraint for the production system. The optimal cultivation pattern determined in the solution of the models resulted in the consumption of the entire volume available. Since the annual volume proved to be a scarce resource, a shadow price is associated to it (opportunity cost of using a certain water volume), which corresponds to the expected reduction in the value of the objective function if this volume becomes more restrictive in one unity. In this case, a unitary reduction in total water (1 m 3 ) from 79,649,300 to 72,997,000 m 3 would reduce the value of the objective function in R$ 234.89, without changing the basic variables of the optimal solution. On the other hand, each additional water unit from 79,649,300 m 3 to 96,255,000 m 3 would increase the value of the objective function in the same amount of monetary unit.
Monthly water availability was not limiting, since the volume of water used per month was always lower than the available value (10,833,500 m 3 ). Therefore, it has slack and its shadow price is equal to zero, indicating that the resource maximum monthly water volume that can be pumped to the cultivated area is not restrictive when the annual available volume is 79,649,300 m 3 . The models only present minimum volume values that can be used in the constraints of water per month, without changing the basic variables (crops) of the optimal solution. Santos et al. (2009), using linear programming for economic optimization in the Baixo Acaraú Irrigation Project -CE, corroborate with this study, as the water availability estimated for the Project was not limiting to the total use of land resources in six of the seven cultivation plans analysed.
Minimum and maximum water volume values represent the water availability limits for which the shadow price is valid and the current optimal solution is not altered; however, the values of cultivated areas can be changed. The annual water availability can range from 72,997,000 m 3 to 96,255,000 m 3 without changing the optimal solution and the shadow price.
The cultivated area of each crop, total area and the financial return for six water availability levels are shown in Table 8. Table 5.Sensitivity analysis of the resource land occupied per month Table 7.Sensitivity analysis of the water volume in the project Table 6. Sensitivity analysis of market and internal consumption constraints The minimum value for the annual water volume available was equal to 45,273,000 m 3 . Thus, for volumes lower than this one, the solution of the model was impossible, indicating that the constraints were not satisfied, specifically those limiting minimum cultivated area. The maximum value was determined when the annual water availability was not limiting, with shadow price equal to zero, and this volume was 97,218,000 m 3 .
It is observed that the same crops (pumpkin 1, Phaseolus beans 1, watermelon 1, banana, papaya and Tahiti lime) are part of the cultivation pattern predicted by the model. Pumpkin 1, Phaseolus beans 1 and watermelon 1 remained with constant cultivated areas at all water availability levels, according to the imposed constraints.
Tahiti lime only appears in the solution from the water availability of 67,198,000 m 3 . From this water availability on, the area cultivated with Tahiti lime was equal to 300 ha, maximum limit for the cultivation of this crop, which is being tested as an option in cultivation, substituting banana in areas affected by Panama disease. Table 9 presents the water depths used in the crops that are part of the cultivation pattern predicted by the model, with their respective yields, at the six levels of available water volume.
Analyzing Table 9, it is observed that pumpkin 1, Phaseolus beans 1 and watermelon 1 had an increase in water depth when water availability increased from 47,790,000 m 3 to 79,649,300 m 3 . At the level of 97,218,000 m 3 , the highest one, there was reduction of water depths for Phaseolus beans 1 and watermelon 1, with yield reductions from 2,667 to 2,612 kg ha -1 and from 33,238 to 28,474 kg ha -1 , respectively. This water depth reduction provided an additional water volume to the model, which was used by the banana crop.
Papaya cultivation was not influenced by the increase in availability. For its high marginal revenue, the model attributes the highest water depth available to this crop (1400 mm) for a yield of 94,784 kg ha -1 .

Conclusion
1. The maximization of the net revenue in the Formoso Irrigation District obtained with the model was equal to R$ 68,384,956.53, using the following cultivation pattern: 30 ha of pumpkin, 30 ha of Phaseolus beans, 977 ha of watermelon, 1868 ha of banana, 1200 ha of papaya and 300 ha of Tahiti lime, for the annual water volume of 79,649,300 m 3 .
2. The monthly water availability in the Formoso Irrigation District does not constitute effective restriction to the production of the crops if the monthly water volume provided is 10,833,500 m 3 .
3. Considering the monthly water volumes used in the optimal solution, with the maximum volume of 9,449,000 m 3 in September and a minimum of 4,687,000 m 3 in July, the annual volume available will not be restrictive if the annual pumping capacity is higher than 79,649,000 m 3 , which represents the highest water volume offered to the users in 2009.  Table 8.Cultivated area per crop (ha), total area (ha) and financial return (R$) for six water availability levels Table 9.Water depth (W) in mm and yield (Y) in kg ha -1 for six water availability levels