Intelligent decision support in the optimization of irrigation systems in agriculture

. The issues of determining the optimal values of the regulatory parameters of irrigation systems engaged in the cultivation of agricultural crops are considered. Following the requirements of a market economy, the main emphasis is placed on taking into account two types of criteria: maximizing the yield of agricultural crops and minimizing monetary costs. The proposed method for solving the multi-criteria optimization problem is based on the combination of the minimax criterion and the medium-step convolution, which makes it possible to scalarize the vector optimality criterion with access to smooth optimization methods. Concerning the case of priority uncertainty according to particular optimality criteria, an intelligent algorithm is proposed based on the approximation of the preference function of the decision-maker by the fuzzy Mamdani model. The multi-criteria optimization of the irrigation system used for growing cotton results differ favorably from the average values. The one hectare yield in the republic-increased by 2%, monetary costs - reduced by 4.5%. It could be concluded that the developed methodology makes it possible to bypass the computational difficulties that arise when solving problems of multi-criteria optimization of irrigation systems engaged in the cultivation of agricultural crops and to obtain real results in conditions of certainty and uncertainty goals.


Introduction
One of the effective ways to solve the problems of rational use of water and land resources in water and agriculture, increasing crop yields is using system analysis, mathematical modeling, and optimization in planning, preparing, and managing the technological processes of irrigation systems.
A feature of setting the optimization problem in a market economy about irrigation systems of agriculture is multi-criteria, since along with technological (yield), it involves the use of economic optimality criteria (cash costs for irrigation, mineral fertilizers, etc.).The vector optimality criterion gives rise to two problems in practice [1][2][3][4][5].
First, the existing vector optimality criteria scalarization methods often lead to nonsmooth functions.Conventional numerical optimization methods under these conditions turn out to be ineffective due to the emerging «jamming» effect.Overcoming this problem is possible based on smoothing the scalar optimality criterion and is in the plane of solving ill-posed problems by regularization methods [6][7][8].The practical implementation of these methods encounters several difficulties due to the need for additional functional analysis and relatively high computational costs.
Secondly, most vector optimality criteria scalarization methods require setting priorities (weight coefficients) for each particular optimality criteria.In practice, setting priorities is not a trivial task and leads to uncertainty.The solution to the problem of «uncertainty of priorities» is possible based on the introduction of a preference function (PF) of a decision maker (DM), followed by its approximation by fuzzy or neural network approximation methods [9][10][11][12].
Thus, further progress in solving problems of multi-criteria optimization in the preparation and control of technological processes of irrigation systems of agriculture provides for the creation of effective methods and algorithms that combine traditional numerical optimization methods, regularization methods for ill-posed problems, and intelligent decision support.

Multicriteria optimization of irrigation systems used for cultivation of agricultural crops under conditions of certainty.
The problem statement for optimizing the irrigation system has the form In (1) 1 y is the yield (2) Applying the combined convolution method, we transform problem (1) as follows [2,15].
Multiplying both parts of the criterion constraint y 1 ≥t 1task in expression (2) by -1, we obtain constraints of the form -y 1 ≤t1 task and a particular criterion 1 where δ i is the estimate of the scattering of the i-th output parameter, which is set based on practical considerations or is determined using the method of statistical tests; i D is weight coefficients that determine the relative importance of particular criteria ; 1, 2 i i y .
Applying the maximin convolution, we obtain a scalar optimality criterion 1,2 min max, ( ) ( ) where D is a set in which direct restrictions on the variable parameters with the help of an appropriate substitution, for example, x j = x jmax + (x jmin -x jmax ) * sin 2 ( j xc ), converted to functional; ; 1, j j n x c -new independent variables.
Let's smooth the criterion (4) using the exponential function and the power-mean convolution i i x exp z x I { , J is parameter introduced to control convergence in the vicinity of the optimum point.
The final optimization problem (1) will take the following form In (6), F(x) is the modified optimality criterion.
As applied to additive regression, for the smoothness of the modified criterion, it is necessary that the partial derivatives be continuous [15].When the condition of smoothness of the modified criterion is satisfied, as applied to problem (6), the simplest ' ( , ) ; 1, f x a i n smooth optimization algorithms can be applied in practice.

Intelligent Decision Support in the Problem of Multicriteria Optimization of Irrigation Systems under Uncertainty of Priorities.
When solving problem ( 6), the values may not be known in advance, which leads to the , 1,2 i i D uncertainty of priorities.In this case, the general statement of the problem of multi-criteria optimization of irrigation systems is formulated as follows.A vector function is given, whose components are particular optimality criteria and defined on the set of alternatives of the vector of variable parameters For each fixed vector, the combined convolution method reduces the solution of problem (6) to the solution of a single criterion optimization problem of the form: If the solution of problem ( 7) is unique for each this means that each of the admissible vectors $ corresponds to a single vector .

R ] $ o
Then the problem of multi-criteriative optimization is reduced to the choice of We will assume that ] it is a linguistic variable that takes a certain number of finite ] «Average» corresponds to 0 3 z = , the value ] «Well» corresponds to 0 4 z = , and the value ] «Very Well» corresponds to 0 5 z = .
This, the problem of multi-criteria optimization is reduced to finding a vector * D $ $ , that provides the maximum of the discrete function ( ) those, to the approximation of the PF DM.
The general scheme for solving such a problem is iterative and has several stages [1,13].
At the first stage, n vectors A 1 , A 2 ,..., A m are randomly generated.The order of the following actions is as follows.
A one-criteria problem is solved: ] is constructed.Further, the procedure continues according to the scheme of the second stage until the DM decides to stop the calculations.At each iteration, a «rollback» is allowed to change the previously introduced estimates of its PF DM.

Results of practical application
The above optimization technique has been applied to the irrigation system used for cotton cultivation.
The software implementation of optimization algorithms was carried out in the MATLAB 2015 environment on a computer with an Intel(R) Core (TM) i5-9400 CPU @ 2.90 GHz and 8.00 GB of RAM.
The vector of input parameters of the technological process included: irrigation rates per complex hectare x 1 (thousand m 3 /hectare), cash costs x 2 (thousand sum/hectare), costs of nitrogen fertilizers x 3 (Ton/hectare), costs of phosphate fertilizers x 4 (Ton/hectare), labor costs x 5 (person-days/hectare).
On the output parameters y 1 (yield) and y 2 (monetary costs), the input parameters 1 5 x x y of the technological process of cotton irrigation, restrictions were imposed, constituting a set of permissible solutions , ( ) a x y a x a x a x a x a x (11) where a 1 = 420.06;a 2 = 6.49; a 3 =0.00000002; a 4 =0.075; a 5 =0.075; a 6 =0.01505.
The optimization problem under conditions of certainty was to maximize parameter y 1 and minimize parameter y 2 .A set of variable parameters make up parameters x 1 , x 3, and x 4 .
The scattering estimates of the output parameter values were selected as follows: 1, 2 i i a = . The results of the optimization are summarized in Table 1.Bold indicates a situation in which the functional limits on the output parameters are violated.Optimization in the face of uncertainty of priorities was carried out as follows.
The formation of function values ] was carried out based on the rules given in Table 2.When solving the problem (6), the method of coordinate descent was used, and the method of the golden ratio was used to solve the problem (10).The intermediate points of the function were determined using a cubic spline.The approximation of the function ] was carried out using a fuzzy Mamdani model, which was implemented using the Fuzzy Logic Toolbox MATLAB 2015 extension [16,17].
The semi-complete fuzzy output system (see Fig. 1) has two inputs (weigh1t1, weigh1t2), a Mamdani fuzzy output mechanism, and one output (function).The input variables are 1 2 , a a the weighting coefficients of the particular criteria of optimality y 1 and y 2 ; the output variable corresponds to the function ] .
The input and output variables correspond to the coziness of the membership function, which were given as a symmetrical Gaussian function.
The input variables correspond to three types of membership functions: small, middle, big, which correspond to a small, medium, and large value of the weighting coefficients α 1 and α 2 .The output variable corresponds to five types of accessory functions, which have been assigned names -VB ( Very bad ), B ( Bad ), A ( Average ), W (Well ), VW ( Very Well ).

Fig. 1. Fuzzy Output System
The set of rules that specify the relationship between the input and output variable is of the form 1 2 ; 1 2 ; Fine-tuning the fuzzy output model at each step of the problem solution (10) was implemented using the fmincon function of the extension Optimization Toolbox MATLAB 2015 [18][19][20].
In Fig. 2, an illustration of a fuzzy model is given Mamdani, obtained after tuning in the next step of solving the problem (10).3 and illustrated in Fig. 3.
In Table 3 The maximum time for one iteration when solving the problem (10) was 4.6 s.The total time for solving the problem (9) was 30 seconds.  1.The proposed method of solving the problem of optimizing the values of parameters of irrigation systems makes it possible to bypass the computational difficulties associated with the complexity and incorrectness of the problem.
2. The described algorithm of intellectual decision support makes it possible to solve the problem of multicriteria optimization of irrigation system parameters in conditions of uncertainty of priorities according to particular criteria of optimality.
3. The software implementation of the developed methodology and algorithm for optimizing the parameters of irrigation systems is very effective.It can be widely used in practice to solve the problems of multicriteria optimization of various irrigation systems in agriculture.

Preference function graph
introduce estimates of the degree of fulfillment of constraints for each of the output parameters of the form y x a .Based on this, you can build some PF DM ( ) ] $ , defined on the set D $ : :

.
The static optimization problem in proposition (6) was solved by the method of coordinate descent for different sets of values ;

Fig. 2 .
Fig. 2. Fuzzy Mamdani model after fine-tuning: a-editor window of the functions belonging to the variable weight 1; b-window of the editor of the functions belonging to the variable weight 2; c-the function editor window of the function membership variable; d-output Rule Editor window; e -output rule viewer window; f-solution Viewer windowWhen solving the problem(10), the number of "overclocking" solutions n was chosen equal to six: A 1 , A 2 , ..., A 6 .Moreover, the extreme values A 1 , A 6 were chosen at the boundaries of the area of change in the weighting coefficients 1 D and 2 D , and the average values A 2 , A 3 , ..., A 5 were randomly generated.The results of solving the multi-criteria optimization problem are shown in Table3and illustrated in Fig.3.In Table3, the «overclocking» iterations are highlighted in gray.The optimal value

Fig. 3 .
Fig. 3. Graph of the face preference function, the decision maker

Table 1 .
Results of software implementation of the optimization model.

Table 2 .
Function value generation rules.
. At the same time, the values of the partial optimality criteria were: f 1 =38.560centner/hectare and f 2 =79023.222thousand sum/hectare.The optimal values of the variable parameters obtained during optimization

Table 3 .
Optimization of problem solution results.