INTRODUCTION
Irrigation efficiency is important information in the planning of water resource management in limited water resource area. Generally, the irrigation efficiency is the multiplication of conveyance, distribution and field application efficiencies. Often, most previous planning considered the irrigation efficiency as a constant value for all seasons (Brown, 1999; Ali et al., 2000; RID, 2004). However, it is likely that the efficiencies tend to vary due to the uncertainty of the water resources (Burke et al., 1999). Therefore, they may use the erroneous irrigation efficiency which unsuitable for the seasonal available water. Moreover, the participation of stakeholder in water resource management is the main effect of irrigation efficiency (Yoshida et al., 2004). Hence, farmer participation of water resource management is important factor for estimating irrigation efficiency.
A Fuzzy set is mathematical theory for describing the interested variables from uncertain factors or variables like seasonal inflows and the participation of stakeholder in water resource management. The relationship between input and output variables is defined from fuzzy rule, according to human processes in thinking and decision. In addition, fuzzy rules are relatively easy to explain and understand. Recently, the fuzzy model was accepted to describe the relationship of the uncertain variables (Ross, 1995; Shrestha et al., 1996; Jairaj and Vedula, 2000; Panigrahi and Mujumda, 2000; Umamahesh and Chandramouli, 2004). Often, the calibration processes of the fuzzy model were performed by manual adjusting (trial and error) the membership functions and rule bases. Therefore, depending on the result of the human adjustment, it does not guarantee to yield the optimal solution.
A Genetic Algorithm (GA) is search and optimization techniques based on the principles of national selection and genetics. GA is a robust method for searching for the optimum solution of a complex problem. It can provide the near global optimal solution. The GA was applied to solve the optimal solution of water resource problems (Goldberg, 1989; Wardlaw and Sharif, 1999; Chang et al., 2003; Kangrang and Chaleeraktrakoon, 2007). The best part of GA is that they can handle any type of objective function describing decision variables.
This study thus proposes the fuzzy set model for finding the irrigation efficiency which corresponding seasonal inflow and farmer participation in water resource management. The genetic algorithm technique is applied to calibrate the membership of the fuzzy model. The farmer participation will be calculated via the relationship between seasonal requiredarea and the whole area of irrigation project.
MODEL FORMULATION
In order to account for any uncertainty on seasonal inflow and farmer participation,
the fuzzy sets theory and its rulebased system were applied for estimating
irrigation efficiency. System inputs include the seasonal inflow and the farmer
participation in water resource management (F). Output is seasonal irrigation
efficiency. The total seasonal requiredarea is the summation of each farmer’s
area. It varies with the farmer decision of each season. Hence, the farmer participation
is substituted by the proportion of seasonal requiredarea and the overall area
of irrigation project. The participation in water resource management of each
season can be calculated by Eq. 1.
where:
X_{j} 
= 
The total requiredarea during season j (j = 1, 2, 3,…,
m). 
T 
= 
The whole area of irrigation project and m is the number of yearly historic
data. 
There are four steps in developing fuzzy model as the following. First step of creating a fuzzy inference system is to transform the crisp inputs into fuzzy variable through the membership function. The number and type of membership functions were constructed based on statistical data and experience of engineers, generally upon the considering problem (Jang et al., 1997; Saruwatari and Yomota, 1995). These types of membership function; trapezoidal, generalized bell, sigmoid, triangular and Gaussian were used to describe the input and output valuables. The optimal conditions of membership function for each type will be searched in the next section. Because there are many types of membership function, the high efficiency of optimization technique was required to search their optimal condition.
Then the fuzzy rule bases were created using the characteristic of seasonal historical data and fuzzy operator. The actual historical data of irrigation efficiency will be presented in the next section. These fuzzy operators; AND and OR were applied to combine the input variables.
Next step was to apply the input membership functions and the constructed rule bases to obtain the output membership functions. This step was done by the implication method which obtaining a fuzzy set of output when given a single number of each inputs. Then the output membership functions of each rule were jointed to one output fuzzy set.
Finally, the fuzzy set of output was converted into a single crisp value. The most common method was the centroid evaluation, which returns the center of area under the curve.
The adequacy of the fuzzy model was evaluated by considering the coefficient
of determination (R^{2}) which defined based on the actual irrigation
efficiency and the estimated irrigation efficiency as:
where:
φ_{j} 
= 
The estimated irrigation efficiency of the scenario during
season j (j = 1, 2, 3,…,m) which determined by fuzzy model 

= 
The actual irrigation efficiency of the scenario during season j which
calculated from cultivated area. 

= 
The average irrigation efficiency of above mentions and m is the number
of yearly historic data. 
The fuzzy model was calibrated by adjusting the membership functions and rule bases using the genetic algorithm technique. These five types of membership function; including trapezoidal, generalized bell, sigmoid, triangular and Gaussian were used to describe the input and output valuables for finding the suitable type. For each type of membership function, the optimal conditions were searched by the genetic algorithm technique; these performances will be stopped when obtained the highest coefficient of determination (closed to 1.0).
The calibration processes using the genetic algorithm were described as follows.
The genetic algorithm requires encoding schemes that transform the decision
variables into chromosome. This study, the decision variables were the typical
membership function of each type. Figure 1 and Eq.
3 show the typical membership function of triangular type. They present
that the decision variables of each membership function for 1 group are a, b
and c. These variables were transferred into the chromosome for searching in
the process of genetic algorithm.
where:
μ_{r} 
= 
Membership function of value r for input or output variable
(i.e., seasonal inflow, farmer participation and irrigation efficiency). 
μ 
= 
Membership value of the variable, r is the value of input or output variable.

Then, the genetic operations (reproduction, crossover and mutation) were performed.
These genetic operations generated new sets of chromosomes. The objective function
of the search was to maximize the coefficient of determination (R^{2}).
This study used population size = 80, crossover probability = 0.9, mutation
probability = 0.01 (Goldberg, 1989). The search will be stopped when obtained
the highest coefficient of determination, hence the optimal value of a, b and
c was met.
Generally, an irrigation efficiency is the overall system efficiency
which affecting by conveyance, distribution and field application (Brown, 1999;
Ali et al., 2000; Yoshida et al., 2004; RID, 2004; Ali and Shui,
2001). The actual irrigation efficiency of the system can be computed for each scenario by the following equation:
where:
Vr 
= 
The net volume of crop water requirement. 
Vd 
= 
The amount of water diverted from the source to the conveyance system.

The net volume of crop water requirement is computed by the method developed
as:
where:
EP_{k} 
= 
Potential evaporation. 
KC_{k} 
= 
Crop coefficient 
Xk 
= 
Cropped area of crop k. 
ILLUSTRATIVE APPLICATION
Three sequences of 26year (1978  2003) seasonal flow and crop waterrequirement
records and related evaporation and effective rainfall data (the Nong Wei Irrigation
Project in the Northeast region of Thailand) during dry season were considered
for illustrating the application of the proposed approach.

Fig. 2: 
Locations of the Nong Wei irrigation project 
Figure 2 presents the location of the Nong Wei Irrigation
Project. It indicates that the Ubolratana Dam released the water to the project
directly. However, in the dry season the available water is insufficient for
cultivating the entire area of the irrigation project. Therefore, irrigation
efficiency is important factor in this irrigation project. The overall area
of irrigation project is 259,400 Rai (1 Rai = 1,600 m^{2}).
Table 1 shows the available inflow, requested irrigationarea,
farmer participation in water resource management and the irrigation efficiency
during dry season for 26 years.
Table 1: 
Historical data of an available inflow, requested irrigationarea,
farmer participation in water resource management and actual irrigation
efficiency 

Note: The overall area of irrigation project = 259,400 Rai
(1 Rai = 1,600 m^{2}),* = Yearly data for validation, ** = Irrigation
efficiency exceeds 100% 
Table 2: 
Example of fuzzy rule bases for estimating irrigation efficiency
(the numbers of membership function = 3 groups) 

The farmer participation were calculated via the relationship between the requested
irrigationarea and the overall area of irrigation project. They indicate that
the maximum and the minimum seasonal inflow are 521 MCM and 51 MCM, respectively.
The seasonal requested irrigationareas are varying during 10,100 and 230,100
Rai (1 Rai = 1,600 m^{2}). The least and the highest farmer participation
are 3.89 and 88.70%, respectively. The actual seasonal irrigation efficiency
in 1983, 1988, 1996 and 1997 are greater than 100%, so the data of these years
are not accepted for computing.
Table 2 shows an example of fuzzy rule bases using AND
and OR operators. The numbers of membership function of each variable is 3 groups
(less, medium and high). However, the numbers of membership function will be
searched from 1 to 5 groups that cover the preliminary cluster of the historical
data. Further more, the 5 types of membership function were used to search for
finding the optimal condition providing the highest coefficient of determination.
RESULTS AND DISCUSSION
The optimal condition of all types of membership functions using the genetic
algorithm for calibration that provides the highest coefficient of determination
as shown in Table 3. Coefficients of determination of Gaussian
type with the number 443 is the highest values 0.9948. The least value of
coefficients of determination is 0.8859 of the sigmoid type. These results present
that the optimal conditions of all types provide the high coefficient of determination.
Therefore, these types are suitable for estimating irrigation efficiency given
the optimal condition of calibration. The functions were further validated using
actual irrigation efficiencies which were not considered (1979, 1986, 1987,
1998 and 2001) for constructing model.
Table 4 shows the estimated irrigation efficiency of each
type of membership function and their average errors.
Table 3: 


Note: Inf. = Inflow, Far. = Farmer participation and Irr.
= Irrigation efficiency 
Table 4: 
Estimated irrigation efficiencies of each type of membership
function and the average error of each type 

The results show that the average error of Gaussian type is quite small (4.22%),
as compared with those of the other types whereas; the average error of trapezoidal
type is the highest of 6.53%. However, these values are not significantly different.
They indicate that the optimal conditions of fuzzy model give the estimated
irrigation efficiency close to the actual irrigation efficiency for all types
of membership function. Moreover, it concludes that the farmer participation
in water resource management can be calculated via the proportion of seasonal
requiredarea and the overall land area of the irrigation project.
CONCLUSIONS
This paper developed the fuzzy sets model for finding the irrigation efficiency of limited water resource area. The calibration process of the fuzzy model was used the genetic algorithm technique to search the optimal condition of membership function. The developed model was applied to determine the irrigation efficiency of the Nong Wei Irrigation Project (in the Northeast region of Thailand). Results showed that the fuzzy model which used in this study can be used to estimate the irrigation efficiencies, given the total available water resources and the farmer participation in water resource management. The optimal conditions of these types; trapezoidal, generalized bell, sigmoid, triangular and Gaussian membership function provided the estimated irrigation efficiency close to the actual irrigation efficiency similarly. Furthermore, the results indicated that the farmer participation in water resource management can be calculated via the proportion of seasonal requiredarea and the whole area of the irrigation project.
ACKNOWLEDGMENT
The authors would like to express their appreciation to the Faculty of Engineering, Mahasarakham University for financial support.