INTRODUCTION
Water shortage and growing water demand, particularly in arid and semiarid
areas, is a worldwide issue. So considering efficient management of limited
water resources in irrigation reservoir operation policy is necessary to increase
crop productivity. Also, an irrigation reservoir operation policy should reflect
the economic value of stored versus released water. So in making decision for
reservoir operation both reservoir level and farm level should be considered
(Reddy and Kumar, 2007).
For optimal allocation of irrigation water, different models were developed
based on the basic classification of optimization techniques consists of Linear
Programming (LP), Dynamic Programming (DP) and NonLinear Programming (NLP).
Each of these techniques has been applied in a deterministic and stochastic
environment for planning purposes as well as realtime operation. So each method
has a certain distinguishing feature that separates it from the others. An excellent
review of the topic is given by Labadie (2004). A comparative
study on the applicability and computational difficulties of these models is
presented by Mujumdar and Narulkar (1992) and Azamathulla
et al. (2008).
Borhani and Eftekhar (2005) presented and compared
various types of stochastic dynamic programming models and also deterministic
dynamic programming for multipurpose dez reservoir dam located in southwest
of Iran.
Vedula and Mujumdar (1992) used a twostage Dynamic
Programming (DP)SDP to obtain a steady state optimal reservoir operating policy
for irrigation of multiple crops. Vedula and Kumar (1996)
developed an improved model using a twostage Linear Programming (LP)SDP approach
considering the soil moisture balance independently for each crop and actual
evapotranspiration in order to obtain cropwater allocation and the steadystate
optimal operating policy (Reddy and Kumar, 2007).
A significant contribution to the realtime reservoir approach was presented
by Mujumdar and Ramesh (1997) who addressed the issue
of short term realtime reservoir operation by forecasting the inflow for the
current period, a crop production state variable and a soil moisture state variable.
Their work was based on SDP but had all the limitations of SDP regarding the
curse of dimensionality (Azamathulla et al., 2008).
It is also noted that in a shortterm yearly reservoir operation model SDPbased
steady state are not used as they are useful for maximizing the longterm benefits
from an irrigation system. LP models have also limitation since various functional
relationships are assumed to be linear whereas this will not reflect the actual
situation in the field (Reddy and Kumar, 2007).
Among optimization methods stated above NonLinear Programming (NLP) has been
treated extensively in the literature of operation research that offers a more
general mathematical formulation of the reservoir problem (Simonovic,
1992). In this way, Ghahraman and Sepaskhah (2002)
developed a model using a twostage (NLP)SDP. In the first step they maximized
the total farm income in a season. In the second step for the convergent operating
policy over seasons for optimal expected farm income over a year.
Reddy and Kumar (2008) also proposed MultiObjective
Differential Evolution (MODE) approach for a multicrop irrigation reservoir
system. A nonlinear multiobjective optimization model to maximize total net
benefits by irrigating high economic value crops includes waterintensive crops
and a longer duration crop is formulated. They concluded that because of considerable
impact of the hydrologic conditions on net benefits and cropping pattern using
this model can be helpful for irrigation planning and reservoir policies to
select the best possible solution.
Kangrang and Compliew (2010) used an allocation LP
model to find an optimal crop pattern. They take into account heterogenetity
of water demand and yield of irrigation area for crop in their aim. They also
used a sensitivity analysis of irrigation efficiency in modified LP model. Results
showed that LP model is feasible for finding the optimal crop pattern.
A simulation model is usually characterized as a representation of a physical
system used to predict the response of the system under a given set of conditions.
The simulation model is not able to generate an optimal solution to a reservoir
problem directly. However, when making numerous runs of a model with alternative
decision policies it can detect an optimal or nearoptimal solution (Simonovic,
1992).
Simulation models as a prominent tool for reservoir systems planning and management
studies are more practible compared with optimization techniques. Simulation
models associated with reservoir operation are usually based on mass balance
equation and represent the hydrological behavior of reservoir systems using
inflows and other operating conditions. Some models however, represent economic
performance of the reservoir system. Simulation could be the starting point
in the planning of large scale systems but in view of the very large number
of options of configuration, capacity and operating policy, simulation without
preliminary screening through optimization would be very time consuming (Rani
and Moreira, 2010).
In the recent past, the Fuzzy logic techniques as a kind of simulation model
in the form of ifthen rules (i.e., human like reasoning in linguistic terms)
play a deserving role to simplify complicated nonlinear functions models with
simpler ones (Russel and Campbell, 1996; Shrestha
et al., 1996; Hasebe and Nagayama, 2002;
Panigrahi and Mujumdar, 2000; Dubovin
et al., 2002). Fuzzy logic are also suitable in models with multi
input and single output control variables which have been used jointly with
stochastic modeling (Mousavi et al., 2007). Methods
for deriving a rule base from observations in reservoir operation are presented
by Tilmant et al. (2002), Teegavarapu
and Simonovic (1999) and Dubovin et al. (2002),
with the difference that the expert knowledge for making Fuzzy Rule based (FR)
are either derived from implicit stochastic models or explicit ones (Mousavi
et al., 2005).
Comparison between Fuzzy inference System and other Artificial Intelligence
Models such as ANN and ANFIS in Water Resources Management are also studied
by Mpallas et al. (2011), Dastorani
et al. (2010) and Tareghian and Kashefipour (2007).
Due to complexity of optimization models and deviation of the their results
from the reality these models are used along with simulation models to permit
very detailed representation of complex physical, economic and social characteristics
of a reservoir system. The concepts inherent in the simulation approach are
easier to understand and make flexible operational rules in reservoir operations
(Depic and Simonovic, 2000). Moreover, the optimal releases
obtained by optimization models are not useful much in that form for operators
in the future well. Therefore, to solve this problem optimal time series, derived
by an optimization model in the first step, are reformed as some general operational
rules in the second step, which are easier to use for operations in the future
(Ponnambalam et al., 2001). Mousavi
et al. (2007) examined Fuzzy rule base in deriving operating rules
for reservoir operations optimization problem. The model was an implicit stochastic
optimization model for synthetically generated inflow scenarios having one year
of horizon each. This model used to decide how much release should be made from
the reservoir in each time period of a representative planning horizon minimizing
the sum of total deviations of releases from target releases. The optimal sets
obtained from the above model are then used in inferring operating rules using
Fuzzy Rule based (FR). Based on the results derived by Mousavi
et al. (2007), making decision for general operational rules at reservoir
level is possible, however, the point is the objective function of Mousavi
et al. (2007) cannot satisfy the decisions at the field level.
Rani and Moreira (2010) presented a review on different
approaches of simulation and optimization modeling in operating reservoir systems.
They studied different articles about application of simulation models such
as evolutionary computations, Fuzzy set theory and artificial neural networks,
classical optimization techniques and combined simulationoptimization modeling.
The outline of this survey can be helpful for future research to decide appropriate
methodology for application to their system.
Extensions of NLP to stochastic cases are rare due to intense computational
requirements and few applications are reported in literature. Also, it can be
noticed that in recent years interest has grown in using heuristic approaches
as an alternative to NLP, as they can easily handle both nonlinearity and uncertainty
(Rani and Moreira, 2010). However, none of the abovementioned
studies used NLP as optimization methods to derive the Fuzzy rules for optimal
operation of reservoir systems. Moreover, in spite of gaining popularity of
Fuzzy logic techniques, Fuzzy inference system also suffers from the problem
of a large number of rules. The concept of clustering to reduce Fuzzy rules
is extended in some studies. According to the study of Sivapragasam
et al. (2007), one effective way to reduce the rules is to use fewer
numbers of Fuzzy sets by dividing the training data into a number of clusters.
It is easy to represent the inputs by fewer Fuzzy sets by classifying the similar
inputs into a given class. By classifying the similar inputs into a given class,
it is easy to represent the inputs by fewer Fuzzy sets. Based on abovementioned
study to reduce the Fuzzy rules in highly condensed and meaningful rules we
consider single triangular membership function together with the clustering
methods with the difference that the structure of clustering in this study is
based on main cluster and subclusters by Kmean clustering algorithm.
The purpose of this study is to establish a synthesized model relied on objective
function by Ghahraman and Sepaskhah (2004) at the field
level and the Fuzzy model by Mousavi et al. (2007)
at the reservoir level. A Fuzzy model base reservoir operation model using a
nonlinear programming to obtain optimal sets to it is run for multiple crops.
The model has two steps. At the reservoir level the amount of water available
according to the objective function. At the field level, the model considers
monthly competition for water among multiple crops in each cropped area and
crop response to the water obtained by Fuzzy model. What is more, in each year
input data including monthly generated inflow series, water demand and storage
volume used in nonlinear programming. Monthly released volume as output data
are extracted from nonlinear programming. Then, the optimal values obtained
from a NonLinear Programming (NLP) model with input values are used in a Fuzzy
RuleBased (FRB) model to derive the operating policies (NLPFBR). Filling the
gap by comparing policies derived from the NLPFBR model and its equivalent classical
(crisp) NLP formulation has been considered in this study. To expand the Fuzzy
model, the description of clustering model is presented.
MATERIALS AND METHODS
Case study: Ilanjogh, a single purpose reservoir in the north of Khorasan province is chosen as a case study in 2007. This reservoir is supplied from Zangelanloo river.
Daregaz basin with 3129 km^{2} is located in the north of the Allahoakbar mountain. Zangelanloo river originated from the southeast Daregaz mountain is extended to Turkmenistan in the east of Daregaz. It is supposed that Ilanjogh dam will provide agricultural water needs of this region. water resources system plan is shown in Fig. 1.
Implementation of NonLinear Programming as an optimization tool: Objective function is set to maximize the Net Benefit of cultivation as follows:
where, A_{c} is the cultivated area of different crops including wheat,
barley and sorghum, f is the number of fields, B_{c} is the benefit
(price ha^{1}) and C_{c} is the cost (price ha^{1})
for each crop, Y_{a}/Y_{p} is relative yield.

Fig. 1: 
Scheme of the water resources system of llanjogh dam 
In this formula the only unknown variable is Y_{a} while the other
variables are either measured (e.g., A_{c}, B_{c}, C_{c})
or accounted for (e.g., Y_{a}).
The water product function is:
where, Y_{a} is the actual yield, Y_{m} is the maximum yield, i is a generic growth stage, N is the number of growth stages considered, K is the yield response factor at growth stages i and ET_{a} and ET_{m} are the actual and maximum evapotranspiration, respectively. Water which can be allocated as ET_{a} can be available from the reservoir, on which the following constraint is governed:
where, V_{t} and V_{t+1} are the state of storage volume of reservoir at the beginning and the end of month t, respectively, Q_{t} and RAIN_{t} are inflow to the reservoir and direct rainfall over the reservoir area during month t, respectively, REL_{t} and OVF_{t} are release and overflow volume from the reservoir and EVP_{t} is the evaporation lost from the reservoir during month t. The amount of water available during the month t (i.e., V_{t}) is logically bounded by dead storage (Vdead) and maximum capacity of the reservoir (V_{max}):
The conveyance efficiency (assumed 0.9) is used to convert the released water from the reservoir to the amount of water paid for crop irrigation:

Fig. 2: 
Crop calendar used for command area of ilanjogh reservoir
system. G1 to G2 are the establishment, vegetative, flowering, yield formation
and ripening stage, respectively 
the following term should be considered for different crops:
The objective function coupled with the relevant constraints was managed through
a nonlinear optimization procedure from a single purpose reservoir operation
(Doorenbos et al., 1979; Ghahraman
and Sepaskhah, 1999). Here we did not consider the water balance in soil
and just approximated the ET_{a}/ET_{p} by the ratio of applied
water to its optimum amount. To manage the relative yield, Y_{a}/Y_{p},
we adopted the sensitivity of crops to water from literature (Ghahraman
and Sepaskhah, 2004) and changed them to the monthly scale. Figure
2 shows the 5 growth stages including establishment, vegetative, flowering,
yield formation and ripening considered in this study. The year is divided into
2 unequal seasons of dormant and active seasons. In dormant season there is
no summer crop in the field and all winter crops are dormant, yet all crops
may be active in the other season. The dormant season in the region begins around
the 22nd of November up to the 20th of February, so there is no release from
the reservoir in this period. Considering that Wheat and Barley are winter crops
and Sorghum is a summer crop, there is a competition between all 3 crops for
water, especially in the middle and late of spring, when wheat and barley are
going to be harvested and sorghum is in establishment phase. In this period
of time there is a precise irrigation planning demand. Competing between 2 winter
crops (wheat and barley) and summer crop (sorghum) in the time of wheat and
barley cultivation and harvesting the sorghum, is very critical for irrigation
management.
Compiling NLP by considering various cultivation areas and initial reservoir storage volume: Different scenarios are adopted to compile NLP because there is different volume of water in reservoir storage at the beginning of the year and also different combinations of WheatBarleySorghum cultivation area. Five cases are considered for active reservoir storage: empty (dead storage), full and 0.25, 0.5 and 0.75full storage. At present time, acreage of wheat, barley and sorghum are 478, 530 and 761 hectares, respectively.
To be more comprehensive, we defined two bands for wheat and barley acreages,
respectively i.e., A and B. Another band is also defined for sum of acreages
of wheat and barley (Table 1). Based on Table
1, different scenarios for crop acreages are considered that are presented
in Table 2.
Table 1: 
The constraints of barley and wheat acreage 

^{a}The area of wheat, ^{b} The area of barely,
^{c }Total area 
Table 2: 
Different area states suggested for selected crops. A, B
and C are defined in Table 1 

^{a}Total acreage at present time (ha) 
Each acreage scenario in combination with each 5 classes of reservoir volumes
is managed through maximization of objective function (Eq. 1)
via NLP.
Decision variables effects on objective function: It is obvious that objective function includes two decision variables which have direct effect on the net benefit: water yield production and cultivated areas. To justify the same trend of decision variable and objective function, the water Yield productions in different stages of growth and areas as direct factor in the amount of benefit is investigated. The released water should be taken precisely into account because it is a variable in the Yield production and affect indirectly on the net benefit.
In addition, more area has more benefit, so we want to get the point to reduce the effect of area. In order to know which crop and area has the most effect on the economic objective function; precise study in each crop in different stages of the growth is inevitable. Moreover, we want to get reasonable results based on yield response factors and the amount of sensitivity for each crop particularly in more competitive months in the year with more deficiency.
Shortterm data for training and testing: Three hundred samples of monthly
inflow series which are synthetically generated by fragment methods, are considered
as inputs in nonlinear programming (Hosseinpourtehrani
et al., 2009). At first monthly and annual stochastic rainfall sequences
simulation are considered to generate data with longterm dependence or preserve
historical records distribution function. Annual and monthly rainfalls for Dargaz
station with 15 years records are satisfactory simulated by stochastic models.
A firstorder Markov process is used for annual data and the fragment method
for monthly data. Then, 5 different initial storage volumes (empty, full and
0.25, 0.5 and 0.75full reservoir volume) and 4 different area states were
set corresponding to the beginning of the cropping season, October month.
To evaluate the amount of water, which should be released from the dam reservoir in monthly steps to supply agricultural water, the monthly releases of nonlinear programming are extracted as output. Thus, monthly inflows, demands and also initial storage volume to reservoir are considered as independent variables and monthly releases from reservoir extracted from NLP model are set as dependent output variables. The optimal sets obtained from the nonlinear methods, are used in inferring general operating rules using Fuzzy logic each with one year of planning horizon for 300 years. Among 1500 years of monthly inputoutput data taken by NLP model half of them are used for training while the second half are used for testing and comparing the economic performance of Fuzzy model versus NLP one.
Implementation of Fuzzy logic in optimal reservoir operation: FIS parameters are determined from the data obtained by NLP. FIS models are used to infer the relation between premise part (independent variables; storage and inflow) and the consequent part (dependent variable; release from the reservoir), by representing them as Fuzzy IFTHEN rules.
Implementation of clustering method to expand the Fuzzy model: For each inputoutput pair, one single rule is used in optimal Fuzzy systems. By increasing the inputoutput data, the number of Fuzzy rules increases as well which causes two problems; building Fuzzy rules is timeconsuming and systematical errors are likely to increase. Thus, clustering technique, which classifies the inputoutput data into identical clusters, may be an effective way to reduce the rules in Fuzzy inference system. Similar inputs are put into the same clusters. Therefore, each cluster is known as a representative of a specific rule and consequently the number of rules reduces to the number of clusters.
In this study, the training data are divided into different clusters by Kmeansbased
clustering technique and each cluster is characterized by a single rule. Kmeans
treats each observation of data as an object having a location in space. Kmeans
clustering can best be described as a partitioning method. That is, the function
Kmeans partitions the observations into K mutually exclusive clusters and returns
a vector of indices indicating to which of the K clusters it has assigned each
observation. This technique finds a partition in which objects within each cluster
are as close to each other as possible and as far from objects in other clusters
as possible (Shamkoueyan et al., 2009). Each
Fuzzy set in each cluster has a single membership function for given variables.
As stated before, the present study is based on shortterm period in which each
scenario includes one year. Consequently, 300 synthetically generated samples
of inflow time series corresponding to 5 different initial reservoir storage
volumes, leads to generation of 1500 inputoutput data in which 750 samples
are used for training clustering and building Fuzzy rules and the rest are used
in simulation for testing. All input variables in each cluster, i.e., river
inflow, reservoir storage, irrigation demand and reservoir release volumes,
are fuzzified with a single triangular membership function. Each membership
function parameterized by its vertices (l, left vertex; c, centre and r, right
vertex of the membership function) (Sivapragasam et al.,
2007), as shown in Fig. 3. Clustering is used separately
for each month of years. The Kmean clustering method is done in two steps:
in the first step, according to the initial storage volume at the beginning
of the growing season the storage volume in each month classified into 5 main
different clusters with specific volume of inflow, demand and release. In the
second step in every main cluster each variable (inflow, demand and release)
falls into three sub clusters in accordance with silhouette width criteria.
As a result, in each month there are 15 Fuzzy rules for making decision about
the amount of released water from reservoir. The clustering is schematically
shown in Fig. 4.
Assessment of the Fuzzy logic model: A part of NLP model data is used
as historical data in Fuzzy model to make the Fuzzy rules. To assess the Fuzzy
model, the trend of Fuzzy model output in testing phase should be similar to
the historical data of NLP.

Fig. 4: 
The schematic example of clustering used in this study. C_{1}
to C_{5} are the main clusters according to the initial storage
volumes from dead to maximum one, respectively 
Moreover, three performance measurements, i.e., R^{2} (determination
coefficient), MAE (Mean Absolute Error) and RMSE (Root Mean Square error) are
introduced in the following equations to assess the performance of Fuzzy model:
where, r_{0} and
are the value and mean value of the released water volume, respectively, extracted
from NLP as observation data series (historical data) and r_{1} and
are as same as the previous parameters but taken from Fuzzy model.
RESULTS
Non linear programming: For inferring operating rules in which the maximization of Net Benefit cultivated by crops in a yearly base is the objective function monthly storage, release and demand volume series were extracted from optimization model. Considering 5 different amount of it did the effect of initial reservoir storage at the beginning of water year by considering 5 different amount of it. In addition, the effect of 4 different cultivation area scenarios (Table 2) on monthly water release and demand was considered in the following.
Different initial storage volume: We adopted 4 different cultivation
area scenarios (Table 2) and 5 different Initial Reservoir
Storage Volume (IRSV) at the beginning of agricultural year (23 th October).
For each case the NLP yielded the optimum reservoir release to reach the highest
amount of Net Benefit. For each case of IRSV, we averaged the reservoir volumes
for all cultivation area scenarios (Fig. 5). The results showed
that for a specific condition for initial reservoir storage, the storage volume
changed at different months (from October to September). This change would be
in accordance with the amount of water needs.

Fig. 5: 
Illustration of different optimal storage (MCM) paths. 1 is
the beginning of the season growth (October) 

Fig. 6: 
Illustration of different optimal mean (a): Water released
(CMC), (b): Water requirement (MCM): in order of increasing V1 to V5 is
dead, 0.25, 0.5, 0.75 full storage and full 
Figure 5 supports that the trends of these variations are
independent of initial reservoir storage volume; meanwhile the trends are approximately
parallel. Adopting a higher value for the initial reservoir storage also yields
in higher reservoir storages at all months of the year. As shown in Fig.
5, reservoir storage increased up to months 7 and 8 (May and June) when
there is a maximum competition between three crops for water demand. Two reasons
can explain this trend: First of all the higher river inflow and secondly low
water requirement. The stored water should be allocated afterwards to satisfy
the irrigation requirements. Reservoir storage, then, decreases up to the end
of the year, due to higher water requirement and also due to low river flow.
Monthly water release from the reservoir corresponding to different initial reservoir storages is shown in Fig. 6a. According to Fig. 6a, the average maximum release of irrigation with 5 different initial storage volumes occurs in May when all crops need water. Thus, it is logical to save much water in reservoir at preceding months when they need less water to release. It can be inferred from Fig. 6a that the amount of water released in a special month is nearly independent of initial storage volume such that it has almost identical results in different initial storage volumes. The result repeated in Fig. 6b showed that the amount of water requirement is independent of initial storage volume.
Different cultivation area scenarios: In a suitable irrigation reservoir operation, the optimum cultivated area and multiple cropping patterns are the most important factors for increasing the yield production and net benefit. Thus it is necessary to be sure about the efficiency of the model sensitivity analysis in different states of area with multiple cropping patterns.
In addition, the amounts of required and released water depend on the cultivation
area. In fact, released water affects directly to the yield production, which
has a crucial role in the amount of the net benefit. Figure 7
showed that the changing trend of released and required water in the downstream
was almost the same over a year. Besides, the most deficits in different states
of area occur in March and April when wheat and barley were in the second and
third stages of the growth.

Fig. 7: 
Illustration of different optimal mean (a) released water,
(b) required water (MCM) path. A_{1} to A_{4} are 4 different
states area accordance with Table 2 
This trend was repeated in June when sorghum was in the second stage of the
growth and Wheat and Barley were respectively in the fourth and fifth stages
of the growth.
Fuzzy logic based model: As stated before, the results of NLP indicated that specific initial reservoir storage at the beginning of cropping season plays an effective role on the reservoir storage in other months and they were positively related to each other. Moreover, the clustering was more pronounced by this variable, as was compared with the other variables of reservoir release and agricultural demand. Using either of these last two variables, did not lead to a good fuzzification in each month. So we considered the membership function of reservoir storage volume in each month as main cluster, where each cluster had three subclusters including specified river inflow, demand and release. In this study, because storage volume in each month with specified initial storage volume at the beginning of the cropping season completely differed from the same month with other initial storage volume at the beginning of cropping season, we made the clustering based on reservoir storage as the best choice for classification. The Silhouette width criteria which tested the similarity of any member to the appropriate clusters, confirmed the results.
Membership functions for clusters: We used triangular membership functions
in this study (Fig. 8, 9). μ was the
degree of membership function and C represented the cluster, such that in C_{i},
j, i was the number of the main cluster and j was the number of subcluster
corresponding to its main cluster. For instance, C_{12} showed the second
subcluster corresponding to the first main cluster. The proximity of maximum,
minimum and mean initial storage volumes to the limitation of l, c and r in
all five main clusters, confirmed the accuracy of the limitation of membership
functions of initial storage volume. In this paper the membership function of
May (month number 8) as the most critical month in which three crops of wheat,
barley and sorghum needed water simultaneously was chosen as a sample and was
shown in Fig. 8 and 9. Figure
8 showed the initial reservoir storage membership functions as the main
clusters. According to Fig. 8, for instance in C_{1}
the most degree of membership function was equal to 20 MCM; while the least
degree of membership functions was 10 and 35 MCM.

Fig. 8: 
The membership functions of intial storage volume in five
main clusters (C_{1} to C_{5}) in May 

Fig. 9: 
The membership functions of (a) storage volume (b) reservoir
release (c) agricultural demand and (d) river inflow in each cluster in
May 
However, the degree of membership function for 15 MCM corresponded to the value
of 0.5. Figure 9a represented the second main cluster, i.e.,
the initial storage volume was 0.25full at the beginning of the cropping season.
The membership function of release, demand and inflow as the subcluster of
the second main cluster with initial storage volume of 0.25full at the beginning
of the cropping season were shown in Fig. 9b, c
and d.
Fuzzy rule base and making decision for reservoir release: As stated
previously, each of five main clusters had three subclusters. Each subcluster
had its own rule. Therefore, for each month, a total of 15 rules were made.
Table 3: 
Assessment of the fuzzy logic model in both training and
test stage 

^{a}Mean absolute error, ^{b}Root mean square
error 
For instance, based on one of the rules obtained for May, if all variables
of reservoir storage, inflow and demand were “high”, then the release
would be “high”, so the decision making was rather “easy”.
On the other hand, if the reservoir initial storage in May 2was “low”
and both inflow and demand were “medium”, then the release volume
would be “medium”. So, making decision in this condition was more
“complicated” than the former one.
Monthly reservoir releases of Fuzzy model were less than those of nonlinear programming (Table 3). However, both models followed similar pattern. Although the amount of released water in Fuzzy model was less than NonLinear Programming, particularly in months with more water demands (for example in May and June when there was a competition between all 3 crops for water, the percentage of water deficit to the percentage of annual mean water deficit were respectively 0.57 and 0.81 in training and 0.93 and 1.145 in the test stage and in summer when Sorghum was irrigated this ratio were respectively 3.42, 0.55 and 2.53 in training and 1.45, 0.97 and 3.37 in test stage), it could not be used as an index for economic performance of the Fuzzy model. Because in this study there was no direct relationship between release volume and Net Benefit in objective function as the economic purpose.
Figure 10 and the two regression lines showed that by increasing the release, the difference between Fuzzy and nonlinear programming becomes more. So, what we deduced was in months with more water demands, the difference of released water between two models was more. Since stated before, the period between Decembers up to the end of February was neglected in Fig. 10 because of the period of dormant season.
As Table 3 shows, the highest difference between NLP and Fuzzy models occurred in summer (where Sorghum was irrigated) such that the percentage of water deficit to the percentage of annual mean water deficit were respectively in the months of summer and then for May and June. It should be noted that the results obtained in the test stage were better than training one.
Fuzzy model in different stages of growth: To investigate the efficiency of Fuzzy model and also the Benefit taken by Fuzzy model and the effect of decision variable on net benefit, it was necessary to have insightful study on three crops separately. It should be noted that in Fuzzy model the amount of released water was estimated directly and then based on required and released water, the yield production and net benefit were gained from it. Since the amount of area was determined by NLP in the previous step, the efficiency of Fuzzy model was merely based on released water and yield production of each crop.

Fig. 10: 
Comparison of mean monthly release using NLP and Fuzzy model
with 1:1 line (Dotted line: x = y, Straight line: Data in both models) 
According to the Fig. 11ae, the amount
of monthly released water in all areas of NLP model followed almost the same
trend in comparison with Fuzzy model. Over the 12 months period, these Figures
had risen from October to May but since then the Figures had fallen gradually.
The same trend of Fuzzy and NLP models emphasized that in different states of
initial storage volume, average of released water of all states of area had
no restriction.
Comparing 3 crops in different stages of growth revealed that in the second half of May as the third stage of the growth of wheat and barley, the YAP_{3} played the effective role in total yield production and the net benefit. However, in this period as the first stage of the growth of sorghum, YAP_{1} is not so effective. We inferred that even with the maximum yield production of sorghum and its benefit; the highest benefit could not be achievable.
According to the Fig. 12a, the released water in the first stage of growth of wheat in Fuzzy model as an optimum model was more than NLP model but the increase of release in the first stage of their growth had no effect on increase or decrease of Yield production in all 4 states of area. While in the third and fourth stage of growth the less water release resulted into the less Yield production and the differences between two models led to more decrease of yield production and vice versa. In the second stage of growth that relates to the last 10 days of March, the whole April and the first half of May took different results. Decrease and increase of released water for wheat caused a decrease and increase in the yield production, respectively. But the effect of water decrease was much more than water increase.
The results taken from Barley Fig. 12b indicated that decreasing of released water had no effect on the efficiency of Fuzzy model versus NLP, however, the effect of decreasing of water release on dropping the Yield production in the fourth stage of growth compared with other stages was much more.

Fig. 11: 
Comparison of average released water trend in fuzzy model
versus NLP in 5 different ststes of area (a) without separating the areas
(b) A_{1} (c) A_{2} (d) A_{3} (e) A_{4}.
The areas explained in Table 2 
Figure 12c showed that any changes in release made no trends
in the yield production of Sorghum except in the first stage of growth that
the increase of water releases in contrast to water decrease was more sensible.
But in general, there were no definite results in increasing or decreasing of
water in different stages of growth.
Fuzzy model vs. NLP in terms of different cultivation area scenarios: According to the Fig. 13, the changing percentage of yield production of wheat and barley compared with sorghum in the first state of area led into less Net Benefit. It should be noted that although the allocated area of wheat was more than sorghum, it could not affect to the net benefit. That means the effect of yield production outweighs of the area.
In the third state of area in which the best results were taken for yield production (the yield production of wheat and barley and sorghum were much closer to the ideal points) the maximum of net benefit was gained.

Fig. 12: 
Changing percentage right: yield production, left: released
water in different stages of growth for 3 cultivated crop (a) Wheat (b)
Barley and (c) Sorghum 

Fig. 13: 
Changing percentage of yield production for 3 cultivated crops
under 4 different states of area 
DISCUSSION
In this study the irrigation reservoir operation was considered in two level of decision making including reservoir level and farm level in the North of Khorasan. In reservoir level at the beginning of growing season with 5 states of initial storage volume in a certain period of time two models called NonLinear Programming and Fuzzy models were run. In Fuzzy model an approach was used based on clustering to make fewer rules with great importance. Clustering of inputoutput data taken from NLP as input data to Fuzzy system with 15 clusters and 15 Fuzzy rules were investigated. Because of simplicity and calculation speed of clustering, in making Fuzzy rules with the large amount of input output sets this approach can be replaced with common ones in optimized Fuzzy systems.
Since the mathematical models like NonLinear Programming cannot present different output in all conditions and different states of input and absolute optimal outputs have differences to a great degree, Fuzzy model as a good replacement was used in making decision as general in future. Results showed that the changing trend of releasing of the reservoir in both models was the same but in general the amount of annual releasing in Fuzzy model was less than NLP. In both model in competitive month (May) releasing was the most and March had the least releasing as Wheat and Barley needed the least water.
In addition, the water deficit in comparison with the amount of cultivated crops acreage played more effective role on net benefit and in the year with water deficit the amount of water released in competitive months should be more considered. To increase the net benefit, we should pay more attention to each crop separately in competitive month (May and June). In the first half of May when only wheat and barley were cultivated, wheat had more effect on net benefit (whether it would be faced with water deficit or not). More consideration on results showed that allocating less water to wheat had significant effect on the yield production of the crop compared with others. Therefore, in the years with much more deficit wheat must take into consideration much more in comparison with other crops, particularly in the second stage of growth. Moreover, in this period for sorghum that passed the first stage of growth, the water deficit had no effects on sorghum’s yield production that showed the preference of water for wheat and barley in this period.
Similarly, Irrigation regimes on wheat productivity and water use efficiency
in arid conditions in AlBarrak (2006) study showed
that there was a direct relationship between irrigation regimes with grain yields/ha
that is the grain yields was significantly increased as the water increased.
According to Reddy and Kumar (2007), a reservoir operation
model was developed for irrigation of multiple crops. EMPSO technique was used
for optimal utilization of available water resources to maximize the relative
yield. The main difference of Reddy and Kumar’s study with the present
study is that the model performance is evaluated for two types of objective
functions, OF_{1} and OF_{2}. The first objective function maximizes
the total relative yield of multiple crops, without considering the economic
benefit. The second objective function (OF_{2}), in addition, considers
the value of equivalent benefit coefficient (B_{c}), i.e., the model
objective function integrates arearelated economic benefits with crop growth
sensitivity. In contrast, in the present study relative yield of multiple crops
along with arearelated economic benefits have been considered in one equation.
In both study wheat is a sensitive crop particularly throughout its growth periods.
That is mean in Reddy and Kumar’s study OF_{2} model gives maximum
preference for wheat throughout its growth periods and allocates deficits to
other crops.
Comparing the results from this study with Mousavi et
al. (2007) in which the objective function had been presented in other
forms indicated the allocation of released water in this study in two steps,
reservoir level and farm level, would be more obvious for operator. In study
of Mousavi et al. (2007) making decision for
releasing was based on objective function with minimum loss unit while in present
study addition to making decision for monthly released water, allocation of
water to the crops based on their growth sensitivity had been also considered.
It means that for example a unit loss in different month had no the same meaning
and depends on the crop and the stage of its growth may be different.
Although the objective function in this study was the same as Ghahraman
and Sepaskhah (2002), the NLP model was used to maximize the sum of all
crop net benefits as an intraseasonal model for allocation decision. Furthermore,
they used a SDP model for making decisions over the seasons of a year resulting
in the maximization of expected economic system performance. The results of
module I and the seasonal transition probabilities of river inflow and those
of rainfall form the inputs to this module. Considering initial reservoir storage
class drew different conclusion from the present study. That is mean; the amount
of water released in a special month is nearly independent of initial storage
volume such that it has almost identical results in different initial storage
volumes. In other words, initial reservoir storage class has no effect on net
benefit but results taken by Ghahraman and Sepaskhah (2002)
showed that net benefit is sensitive to some classes of initial reservoir storage
class.
In addition, the structure of thinking to eliminate the linguistic rules by
clustering the data into different groups is somehow similar to Sivapragasam
et al. (2007) what distinguishes present study from Sivapragasam
et al. (2007) is the kind of clustering method. Also, use of subcluster
along with main cluster is another difference in both studies. Since the choice
of clusters and how the structure of clustering depends on the accuracy desired,
it seems to be suitable considering the membership function of reservoir storage
volume in each month as main cluster, where each cluster had three subclusters
including specified river inflow, demand and release for the present case study
whereas in the other study all variables were classified in different main cluster.
Both study with somehow differences in terms of the structure of clustering
results in highly condensed and meaningful rules.
In this study the efficiency of Fuzzy model in contrast to NLP in all states of area was smaller. Since the results from NLP as a mathematical model were absolute, the results from Fuzzy model fewer than NLP were closer to the reality.
Another point is that the fewer released water by Fuzzy model versus NLP does
not mean the better efficiency of it and it must be analyzed by objective function
of NLP to know how the efficiency of the Fuzzy model is (Hosseinpourtehrani
et al., 2011).
ACKNOWLEDGMENTS
The authors are grateful for Dr. Davary for his helpful and constructive comments.