The Use of the Permutation Algorithm for Suboptimising the Position of Used Nozzles on the Field Sprayer Boom

Abstract

The worn-out nozzles of field sprayers cause agricultural treatment to be uneven and therefore ineffective. Spray nozzles are consumable elements of the field sprayer that are subject to inspection and in the event of their excessive wear should be replaced with new ones to ensure the proper execution of agricultural treatment. The aim of the study is to propose, using operational research methods, an expert methodology allowing further operation of worn-out and often expensive sprayer nozzles, including standard, universal, anti-drift, or ejector nozzles. The previous attempts, performed with the use of the random computer optimisation method, did not guarantee a global solution in the entire population of all possible permutations without repetitions of 24 worn-out nozzles (for a field boom with a width of 12 m) or even estimating approximation to this solution. The process of measuring the wear of nozzles, the simulation of the entire virtual field boom, and the permutation algorithm proposed here allow you to specify a suboptimal solution of an NP-hard problem separately for each sprayer, i.e., to indicate in a very short time such a permutation out of 24! ≈ 6.20448 × 10⁺²³ permutations of nozzles with variable degrees of wear, which is close to the optimal permutation of used nozzles on the field sprayer boom, in terms of the coefficient of variation. The use of expert methodology allows for reducing the operating costs of sprayers by using a relatively cheap automated expert service instead of the costly purchase of a set of new nozzles for field sprayers. Many areas of application of this methodology have been indicated.

1. Introduction

Spraying liquid is currently a technically and technologically important process that has found application both on conventional and ecological farms as well as in industries producing food, pharmaceutical drugs, in steelworks, power plants, rolling stock, coatings, and in other sectors of the economy [1]. A flat fan spray nozzle serves the purpose of spraying onto a surface or an object moving in a transverse direction with respect to the direction of the jet, a typical example being the nozzles in a car washing tunnel. Farmers have always sought to facilitate their day-to-day labour. Already in the 19th century, when chemical preparations had begun to be used in agriculture, one of the devices which allowed crop cultivation, gardening, and fruit growing to flourish was the sprayer. Currently, it is argued that the maintenance of rich agricultural landscape, low consumption of plant protection agents, and efficiently functioning control systems are a good foundation for the systematic and sustainable development of rural environments [2].

The effectiveness of each agrochemical treatment results primarily from the cultivation conducted and soil conditions, on top of which is the purposefulness of the treatment affecting the soil, plant, or agrophage, from the preparation and equipment employed, as well as the precision of its performance. The precision of the performance is primarily the outcome of the working pressure and speed of passage that determine the quantity of the preparation dose applied. Each of these factors determines the effectiveness of field works. Hence the heterogeneity and predictability of the distribution of sprayed liquid are the principal features of sprayers. The quality of agricultural spraying depends, to a large extent, on the uniformity of the sprayed liquid dispersal over the sprayed surface. Presently, in field cultivation, flat fan nozzles, which wear out during their operation, are used the most frequently [3,4]. A worn-out nozzle causes the quantity of the outflowing liquid to increase, thus disrupting the uniformity of liquid distribution. The current Directive of the European Parliament and Council 2009/128EC on the sustainable use of pesticides in agriculture contains guidelines concerning, among others, tests of spraying nozzles in agricultural sprayers [5]; the parameter being subject to tests in field sprayers regarding the uniformity of the working liquid distribution over the sprayed surface. The ISO standard regarding the examination of used agricultural sprayers stipulates that the coefficient of variation (CV) cannot exceed 10% [6]. Specialist devices with a grooved table are used for measuring the uniformity of spray distribution [2,6,7]. The degree of nozzle wear impacts the obtained measurement of the coefficient CV [8]. During periodic approval inspections of agricultural sprayers, the value of this coefficient determines whether a sprayer is approved for use. The technical condition of the nozzles installed on the boom of the sprayer also affects the quality of spraying and the application of liquid [9]. This uniform spraying of plants mainly determines the effectiveness and quality of agricultural spraying, reduces the quantity of pesticides used in agricultural production, and affects the safety of the environment.

The development of nozzle design and the positioning on the sprayer boom significantly affect the uniformity of the sprayed liquid falling onto the sprayed surface. Flat fan nozzles were initially made of metal, but their durability was not satisfactory. A rapid development of new designs has been observed since the emergence of the possibility of manufacturing nozzles from plastic.

The current standards for ground-level spraying systems in field cultivation, used for spraying the entire surface, assume that 24 single flat fan nozzles are to be placed on the linear boom of a field sprayer with an operational width of 12 m with 0.5 m spacing (Figure 1). The distance between the nozzles does not change.

Every agricultural sprayer should undergo obligatory technical condition inspections once every three years, except for new machines that are inspected before a five-year period expiry date. So far, in practice it is usually assumed that when some nozzles of a field sprayer are worn out, it is necessary to replace the set of flat fan sprayers in the sprayer concerned [8,10,11]. This inevitably results in the purchase of an expensive set of new flat fan sprayers.

However, already in the work [6] it is indicated that the homogeneity of the sprayed liquid distribution and the application of pesticides is conditioned not only by the improvement of the manufacturers of sprayer components in meeting the operating standards of these entire devices but by maintaining the parameter uniformity of the newly produced flat fan nozzles. Parafiniuk and Tarasińska note that the uniformity of the distribution of chemical plant protection products, as required by relevant regulations, can also be obtained by permutating the worn-out flat fan sprayers on the beam of field sprayers [12]. The methods of the monitoring search for the extremum of the objective function are characterised by great simplicity and naturalness, because they consist in finding the value of the function at each point of the set of feasible solutions. However, indicating such a permutation of 24 flat fan nozzles, which is close to the optimal permutation regarding the variation coefficient out of all 24! = 6.20448 × 10⁺²³ permutations of flat fan nozzles with a varying wear degree, turns out to be an NP-difficult computational problem. (The problem becomes even more computationally complicated when we consider the possibility of rotating each of the nozzles on the boom by 180°. Then, as many as 2⁽²⁴⁻¹⁾·24! = 5.2047 × 10⁺³⁰ different nozzle-positioning results may be obtained.) Illustratively, this means that in a reasonable amount of time this problem cannot be ultimately and optimally solved even with the use of good equipment. Therefore, the authors of the cited work suggested a search for the best satisfactory solution out of only 10,000 randomly drawn permutations related to the ordering of a group of only 20 flat fan nozzles. Unfortunately, this means drawing conclusions about the distribution of the coefficient of variation based on a random sample constituting only a 1.61174 × 10⁻²⁰ part of the population of all permutations (respectively, a 1.92134 × 10⁻²⁷ part of the population of all various possible sprayer settings) and searching for the best global minimum of the examined parameter only in such an insignificant part of all permutations (respectively, all possible nozzle settings). The solution obtained in this way is optimal only for the group of 10,000 permutations taken into consideration, but it is impossible to apply this result to the actual optimal solution among all permutations (respectively, all possible settings of the nozzles). Moreover, this method guarantees neither approximation to the requirements of the standard nor approximation to the global optimum. Thus, even with its multiple application to selected flat fan sprayers, one cannot achieve any resolution as to whether it is worth looking even further by selecting blindly another group of random, e.g., 10,000, permutations. The purpose of the present work is to propose a way out of this often endless sequence of computations performed for a blind sampling of 10,000 permutations. It seems that, for example, one should observe fragments of a single twenty-four-element permutation and then intelligently modify only those fragments that need to be changed or which will further decrease the variation coefficient of the transverse spray distribution obtained for the entire boom of a field sprayer. This is the motivation to employ a heuristic permutation algorithm to sub-optimise the described problem.

The process of measuring the wear of nozzles, the simulation of the entire virtual field boom, and the permutation algorithm proposed here allow you to specify a suboptimal solution of an NP-hard problem separately for each sprayer, i.e., to indicate in a very short time such a permutation out of 24! = 6.20448 × 10⁺²³ permutations of nozzles with a variable degree of wear, which is close to the optimal permutation of used nozzles on the field sprayer boom in terms of the coefficient of variation (CV). The use of the expert methodology allows for reducing the operating costs of sprayers by using a relatively cheap automated expert service instead of the costly purchase of a set of new nozzles for field sprayers.

The remaining part of the article is organised as follows: Section 2 presents a brief description of the nature-inspired methodology of the genetic algorithm. In Section 3, four groups of experimental data sets are presented. Section 4 presents the mathematical model. Section 5 describes one of the algorithms implemented here and others that are possible. Section 6 covers suboptimal algorithm performance and conclusions pertaining to experimental data sets. Section 7 contains a detailed discussion. Finally, the possibility of application and further adaptation of the methodology, conclusions, and future efforts are presented in Section 8.

2. Materials and Methods

The use of any algorithm that generates all possible permutations of an N-element set to search for the optimal permutation, i.e., sprayer settings, seems to be pointless for small N numbers. This is the effect of a very rapid increase in the value of the N power function. This NP-difficult computational problem for permutation may be associated with sequence modification in the chain of a linear DNA molecule, which is the chemical carrier of genetic information. This information is constituted by appropriate sequencing of deoxyribonucleotides in DNA chains. All possible settings of the flat fan nozzles are coded as individuals, also called chromosomes. Chromosomes are made up of genes, each of which codes for a specific distribution of the flow rate from a respective flat fan sprayer. For integer-encoded genotypes, permutations are used to model a mutation that introduces random changes to the genotype. Thus, in the problem considered here, the following were used to encode the chromosomes: (a) binary values encoding a 180° rotation of the sprayer and (b) integer number representation of the permutations of the 24 flat fan nozzles on the sprayer boom. The total integer number representation in successive genes denotes the numbers of the flat fan nozzles listed in the order of their installation on this boom. (In the example shown in

Table 1. Suboptimal chromosomes encoding the sequencing and method of mounting slot nozzles on the boom of a field sprayer.

A Group of Nozzles	A Set of Selected Binary Values	The Number of the Next Algorithm Run	A Suboptimal Chromosome Encoding the Sequencing and Method of Mounting Slot Nozzles on the Boom of a Field Sprayer																								The Value of the Coefficient of Variation [%] for Tables with Grooves of a Width
																											0.05 m		0.10 m
																											Initial	Final	Initial	Final
A	{0}	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	25.0%	16.5%	20.7%	13.3%
			23	19	22	15	2	16	20	7	17	24	5	13	8	10	14	1	21	12	6	18	9	4	11	3
	{0;1}	1	0	0	0	0	0	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	25.0%	16.7%	20.7%	13.0%
			3	4	5	13	21	19	2	8	6	9	22	10	7	20	1	12	15	16	14	17	24	18	11	23
B	{0}	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	21.9%	13.7%	16.3%	11.1%
			11	8	14	10	22	15	24	18	4	16	20	19	3	23	13	17	2	5	6	9	7	12	21	1
	{0}	2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	13.7%	13.6%	11.1%	11.1%
			11	18	14	20	17	22	15	4	16	19	23	12	5	24	7	6	9	13	3	8	10	2	21	1
	{0;1}	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	21.9%	13.8%	16.3%	11.6%
			1	16	18	8	4	10	17	20	2	5	14	6	9	13	3	22	15	24	19	23	7	12	11	21
	{0;1}	2	0	0	0	0	0	0	0	0	0	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	13.8%	13.7%	11.6%	11.7%
			11	16	19	13	7	20	2	22	5	24	10	17	15	8	4	18	14	6	9	3	12	23	21	1
C	{0}	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	10.3%	8.4%	9.0%	7.5%
			18	1	4	6	8	12	22	7	20	13	5	16	2	19	14	15	24	21	9	11	17	23	3	10
	{0;1}	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	10.3%	8.8%	9.0%	8.1%
			20	1	24	17	13	11	12	22	6	5	16	15	19	14	10	2	7	8	4	9	21	3	23	18
	{0;1}	2	0	0	0	0	0	0	0	0	1	0	1	1	0	0	0	1	0	0	0	0	0	0	0	0	8.8%	8.8%	8.1%	8.1%
			7	21	15	24	10	22	5	11	12	9	1	2	19	13	4	14	16	20	8	17	6	23	18	3
	{0;1}	3	0	0	0	0	0	1	0	0	1	0	0	0	0	0	0	1	0	0	1	0	0	0	0	0	8.8%	8.7%	8.1%	8.0%
			18	21	15	24	10	22	5	11	12	9	1	2	19	13	4	14	16	20	8	17	6	23	7	3
D	{0}	1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	10.0%	7.8%	8.6%	6.5%
			15	24	4	11	1	17	12	9	23	19	13	2	16	20	7	10	3	22	6	14	5	21	8	18
	{0;1}	1	0	0	0	0	0	0	0	0	0	0	0	1	1	0	0	0	0	0	0	0	0	0	0	0	10.0%	8.1%	8.6%	6.9%
			15	10	13	12	2	6	4	1	7	9	3	19	14	11	17	20	23	22	16	24	5	21	18	8

, there are four groups of 24 flat fan nozzles randomly numbered from 1 to 24 in each group.) The representation of the way these flat fan nozzles are mounted on the boom is a two-row chromosome:

$$\left[\begin{array}{ccc}\begin{array}{c}0\\ 3\end{array}& \left[\begin{array}{c}0\\ 4\end{array}\right]& \begin{array}{cccccccccccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 1& 1& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 5& 13& 21& 19& 2& 8& 6& 9& 22& 10& 7& 20& 1& 12& 15& 16& 14& 17& 24& 18& 11& 23\end{array}\end{array}\right]$$

wherein each column is a single gene each time separately encoding a specific flow rate distribution of a randomly numbered flat fan nozzle by means of another 50 real numbers. The numbers 0 and 1 in the first row of the above chromosome perform binary encoding of the effect of the sprayer’s rotation by 180° on the boom, i.e., how this distribution should be considered, whether from left to right or vice versa. Therefore, for example, for the group A of exhausted nozzles, the gene with the number $\left[\begin{array}{c}0\\ 4\end{array}\right]$, highlighted in the above chromosome, encodes the sequence from left to right in a 50-element vector of real numbers: ${a^{\prime }}_4=[0 0 0 0 0 0 0 0 0 0 0.04 0.12 0 0.9 2 0.44 0.08 4.76 9.82 13.18 8.96 8.07 18.34$ $14.72 20.56 17.76 46.91 25.47 13.57 8.21 8 8.29 5.65 5.94 2.04 1.2 4.06 3.71 0.08 0 0.36 0 0$ $0 0 0 0 0 0 0]$ flow rates of the flat fan sprayer No. 4 measured for 10 s on a 2.5 m table in each of the 0.05 m wide grooves (cf. Figure 2).

Additionally, by turning the sprayer No. 4 by 180°, the sequencing is reversed in the 50-element vector of real numbers denoting the flow rate of the flat fan nozzle No. 4. It is encoded by the gene $\left[\begin{array}{c}1\\ 4\end{array}\right]$.

Moreover, depending on the spray angles of the flat fan sprayers, some non-zero extreme components of the above-mentioned vectors of adjacent spray nozzles aggregate as the liquid streams exiting adjacent flat fan nozzles partially feed some adjacent spray areas as shown in Figure 1. The nozzle outlets are placed at a distance of 0.5 m regardless of the spraying angles; therefore, when measuring on a 2.5 m table with 0.05 m wide grooves for results in the form of vectors ${a^{\prime }}_k$ with a size of 1 × 50, the sum of the vector values for successive genes requires them to be shifted by 10 positions each, which together creates a vector 1 × 280 of fluid flow rate from the entire chromosome. Additionally, since the extreme sections beyond both ends of the boom, including the 0.5 m sections under the boom, are not included in the calculations, it means that the optimized spray uniformity index is determined disregarding two 1 × 35 extreme sub-vector components of the vector of 1 × 280 liquid flow rate from the entire chromosome. Therefore, the order of the flat fan nozzles on the boom and their rotation by 180° influences the value of the liquid fall irregularity index, which is the coefficient of variation (CV) in the transverse distribution of the sprayed liquid determined for the entire boom, including 210 central elements of the vector of 1 × 280 liquid flow rate from the entire chromosome. The measurements corresponding to a table with 0.1 m wide grooves required only summing up two adjacent values of this vector.

The use of one of the genetic algorithms, i.e., a kind of heuristic search of the space of permissible solutions to the problem to find the best solutions, seems to be appropriate for solving the optimisation problem. The main idea behind genetic algorithms is based on the biological process, whereby parents pass on genetic information to the next generation. Therefore, the elementary genetic algorithm is built from the following three operations: reproduction, crossing, mutation.

3. Experimental Data

For four randomly numbered groups containing 24 flat fan nozzles each, with an opening angle of 110°, their flow rate was measured for 10 s at a distance of 0.5 m from the nozzle on a 2.5 m table of the measuring device equipped with grooves 0.05 m wide (Figure 2). On this table, measurements can be made for slotted nozzles XR 110 02 and MM RS 110 04 from a distance of 0.5 m at a spray angle $\alpha \le 2\cdot \arctan \left(2.5\right)\approx 136.4°$. The groups A and B were worn-out old nozzles, while groups C and D were new ones, but not of the highest quality.

Filling 50 measuring cups in Figure 2 forms the bell lines as in Figure 2, Figure 3 and Figure 4. The measurement results of 50 cups are in the form of 50-element vectors ${a^{\prime }}_k$ of the distribution of nozzle flow rates. Examples of the values of these vectors ${a^{\prime }}_k$ for several worn-out flat fan nozzles and for several new ones in groups A and C are presented in Figure 3 and Figure 4.

The simulation of the entire virtual field boom, illustrating the work of the sprayer with the analysis of spray uniformity, has been carried out [6]. Below, there is a presentation of the mathematical model depicting the distribution of a stream observed over a section of 14 m and exiting from the entire boom of a virtual field sprayer containing 24 nozzles undergoing measurement here.

4. Mathematical Model for Any Group with the Spray Angle α ≤ 136.4°

4.1. Designations and Formulas

Independently, in each group A, B, C, and D under consideration for each $k\in K$ for a gene are assigned (a) a binary number $x_k$ and (b) $y_k$-other of the numbers from 1 to 24. Moreover, for each $k\in K$ for the genes $\left[\begin{array}{c}0\\ y_k\end{array}\right]$ and $\left[\begin{array}{c}1\\ y_k\end{array}\right]$ are assigned other of the two 50-element real number vectors ${a^{\prime }}_k=\left[\begin{array}{ccccc}{a}_{1;k}& a_{2;k}& \dots & a_{49;k}& a_{50;k}\end{array}\right]$ or $\left[\begin{array}{ccccc}{a}_{50;k}& a_{49;k}& \dots & a_{2;k}& a_{1;k}\end{array}\right]$ of the distribution of a flat fan sprayer flow rate for one of the two ways of mounting the sprayer. In each group A, B, C, and D the chromosome encoding the random initial sequencing of mounting the flat fan sprayers on the boom of the field sprayer with settings compatible with the manner of setting each sprayer in the measuring device from Figure 2 has the form:

$$\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]=\left[\begin{array}{cccccccccccccccccccccccc}0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 1& 2& 3& 4& 5& 6& 7& 8& 9& 10& 11& 12& 13& 14& 15& 16& 17& 18& 19& 20& 21& 22& 23& 24\end{array}\right]$$

For each group, by placing these 24 two-element column vectors $\left[\begin{array}{c}{x}_k\\ y_k\end{array}\right]$ (binary number $x_k$ and all different integer number $y_k\in K$) in a random sequence, we obtain a chromosome $\left[\begin{array}{c}x\\ y\end{array}\right]=\left[\begin{array}{ccc}\left[\begin{array}{c}{x}_1\\ y_1\end{array}\right]& \cdots & \left[\begin{array}{c}{x}_{24}\\ y_{24}\end{array}\right]\end{array}\right]$, which corresponds to a certain value $CV\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)$ of the coefficient of variation for the transverse distribution of the sprayed liquid determined for the entire boom.

For $J=\left\{\begin{array}{cccc}1,& 2,& \dots & 50\end{array}\right\}$ and $K=\left\{\begin{array}{cccc}1,& 2,& \dots & 24\end{array}\right\}$, let 24 columns $a_k$ of the matrix $A=\left[\begin{array}{ccc}{a}_1& \dots & a_{24}\end{array}\right]=\left[a_{j;k}\right]$, containing 24 flow rate distributions for 24 flat fan nozzles for $\left(j;k\right)\in J\times K$. All these columns contain 50 real numbers derived from the measurement on the table as shown in Figure 2, Figure 3 and Figure 4.

For flat fan nozzles with the spraying angle $\alpha \le 136.4°$, maximum five streams overlap. Therefore, an additional set of $S=\left\{\begin{array}{ccccc}0,& 1,& 2,& 3,& 4\end{array}\right\}$ auxiliary indices was adopted. Moreover, let PK denote the permutation group of the set K.

Let $K_0=\left\{\begin{array}{cccc}-3,& -2,& \dots ,& 28\end{array}\right\}$ $L=\left\{\begin{array}{cccc}1,& 2,& \dots & 280\end{array}\right\}$. Let the function $p:L\times S\to J$ where $\left(i,\hspace{0.17em}s\right)\in L\times S$ and

$$p\left(i,s\right)=\{\begin{array}{l}10\cdot \left(s+1\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}mod(i,10)=0\\ 10\cdot s+mod(i,10)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\mathrm{if}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}mod(i,10)\ne 0\end{array}$$

(1)

is used to indicate the row number of the matrix A. The mod (number, divisor) function returns the remainder of two numbers after division. For example, $mod\left(125;10\right)=5$. Let the function $q:L\times S\to K_0$ where

$$q\left(i,\hspace{0.17em}s\right)=ceil\left(\raisebox{1ex}{$i$}\!\left/ \!\raisebox{-1ex}{$10$}\right.\right)-s$$

(2)

indicates the number of the decision variable responsible for the nozzle number of $\left(i,\hspace{0.17em}s\right)\in L\times S$. The ceiling function maps x to the least integer greater than or equal to x, denoted ceil(x). We assume that $a_{j;k}=0\hspace{0.17em}\hspace{0.17em}\mathrm{for}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}k\notin K$.

Then the function $f:L\times {\left\{\begin{array}{cc}0;& 1\end{array}\right\}}^{24}\times PK\to R$ is such that

$$f\left(i,\left[\begin{array}{c}x\\ y\end{array}\right]\right)={\displaystyle \sum _{s=0}^4{a}_{\left[51-2\cdot p\left(i,s\right)\right]\cdot x_{q\left(i,s\right)}+p\left(i,s\right);\hspace{0.17em}{y}_{q\left(i,s\right)}}}$$

(3)

for $i\in L$, $x_k\in \left\{0;1\right\}$, $y_k\in K$ expresses the distribution of the stream observed over the section $280\times 0.05=14$ m, exiting from the entire boom of the virtual field sprayer containing 24 nozzles. For fixed values of decision variables $x=\left[\begin{array}{ccc}{x}_1,& \dots ,& x_{24}\end{array}\right]$, this $y=\left[\begin{array}{ccc}{y}_1,& \dots ,& y_{24}\end{array}\right]$ distribution is expressed as 280 values $f\left(i,\left[\begin{array}{c}x\\ y\end{array}\right]\right)$ for $i\in L$ corresponding measurements on a 14 m virtual table with grooves being 0.05 m wide.

Then, for any determined 24 flat fan nozzles with the spraying angle $\alpha =110°$, the design task is to find such optimal arguments x, y, which minimises the liquid fall unevenness indicator calculated according to the dependency:

$$CV\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\frac{\sqrt{\frac{1}{210}{\displaystyle \sum _{i=36}^{245}{\left(f\left(i,\left[\begin{array}{c}x\\ y\end{array}\right]\right)-f_a\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)\right)}^2}}}{f_a\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)}\cdot 100\%$$

(4)

where

$CV\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)$—(coefficient of variation)—liquid fall unevenness indicator dependent on the values of function $f\left(i,\left[\begin{array}{c}x\\ y\end{array}\right]\right)$,

$i=36$—beginning of liquid output measurement on a 12 m field sprayer boom,

$i=245$—end of liquid output measurement on a 12 m field sprayer boom,

$f_a\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)$—arithmetic mean of the liquid output from the field sprayer boom within the area between $i=36$ and $i=245$ calculated according to formula:

$$f_a\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)=\frac{1}{210}{\displaystyle \sum _{i=36}^{245}f\left(i,\left[\begin{array}{c}x\\ y\end{array}\right]\right)}$$

(5)

4.2. Decision-Making Model

Hence, the permutational decision-making model takes the form of:

• Two 24-element vectors of decision variables:

The $x=\left[\begin{array}{ccc}{x}_1,& \dots ,& x_{24}\end{array}\right]$ binary vector expresses the method of mounting any nozzle mounted in the k-th position on the field sprayer boom for $k\in K$:

$$x_k=\{\begin{array}{l}0, \mathrm{if} \mathrm{the} k-\mathrm{th} \mathrm{nozzle} \mathrm{is} \mathrm{in} \mathrm{the} \mathrm{standard} \mathrm{position}\hfill \\ 1, \mathrm{if} \mathrm{the} k-\mathrm{th} \mathrm{nozzle} \mathrm{is} \mathrm{set} \mathrm{at} \mathrm{an} \mathrm{angle} \mathrm{of} 180°\hfill \end{array}$$

(6)

The vector $y=\left[\begin{array}{ccc}{y}_1,& \dots ,& y_{24}\end{array}\right]$ contains the numbers of the $y_k$ nozzle mounted in the k-th position on the field sprayer boom for $k\in K$.

• Objective function (4):

$$CV\left(\left[\begin{array}{c}x\\ y\end{array}\right]\right)\to \min$$

(7)

• The limiting conditions shall be equivalent to the boundary conditions:

$$x\in {\left\{\begin{array}{cc}0;& 1\end{array}\right\}}^{24}, y\in PK$$

(8)

5. Algorithms

Genetic algorithms are good at finding approximations of the extremes of functions that cannot be computed analytically, for example, problems of discrete and combinatorial optimisation. One of the main advantages of genetic algorithms is the ability to break out of local extremes by acting on the population with the mutation operator. Avoiding this phenomenon is also the result of starting the search from the entire population of possible solutions, not from a single point as in traditional deterministic gradient optimisation methods. A review of genetic algorithms can be found in [13]. Examples of popularised computational tools that find application here include the MATLAB Genetic Algorithm Solver [14] or the Solver—product from Frontline Systems Inc., [15] that supports the Excel spreadsheet.

The latter found a simple application in the experiment presented below, due to the previously used method of collecting data on worn-out nozzles.

5.1. The Microsoft Office Excel Solver

The array form of the INDEX function in Excel returns the value of an element from the matrix A of 24 distributions of flow rates for 24 flat fan nozzles, based on the row and column numbers you specify in (4). In the Solver Parameters dialog box, under Subject to the Constraints, you just need to click Add. If you click bin for x, binary appears in the Constraint box. If you click dif for y, alldifferent appears in the Constraint box. This condition generates the group PK of the permutation of the set K. The alldifferent constraint for y is a specialized constraint that forces every decision variable in the vector y to assume a value different from the value of every other decision variable in that vector. (For example, a constraint such as A1:A5 = alldifferent, where A1:A5 are 5 decision variable cells, requires that these cells must be integers in the range 1 to 5, with each variable different from all the others at the solution. Hence, A1:A5 will contain a permutation of integers, such as 1, 2, 3, 4, 5 or 1, 3, 5, 2, 4.) An alldifferent constraint can be used to model problems involving ordering or sequencing of choices, such as the Traveling Salesman Problem [16].

The Microsoft Office Excel Solver tool uses several algorithms to find optimal solutions. However, its two algorithms using gradient-based optimisation methods are not effective in addressing the problems of the discrete search space. Therefore, heuristic methods are most often applied where algorithms allowing one to find accurate solutions quickly enough are not known. The Evolutionary Solving Method for non-smooth optimisation uses a variety of genetic algorithm and local search methods, implemented by Frontline Systems, Inc., Incline Village-Crystal Bay, NV, USA [17].

In the Solver Options dialog box, the following standard options were selected on the All Methods tab [18]:

 Constraint Precision: 0.000001

 Use Automatic Scaling: Yes

 Integer Optimality (%): 1

 Solving Limits

   Max Time (Seconds): Missing

   Iterations: Missing

 Evolutionary and Integer Constraints:

   Max Subproblems: Missing

   Max Feasible Solutions: Missing

For the Evolutionary solving method, standard option values were selected for the calculations [19]:

 Convergence: 0.0001

 Mutation Rate: 0.075

 Population Size: 100

 Random Seed: 0

 Maximum Time without improvement: 30

 Require Bounds on Variables: Yes

5.2. Other Implementations of Genetic Algorithms

There are libraries of ready-made modules available on the Internet, containing implementations of genetic algorithms in Java, C#, C++, Python [16,17,18], or in the R environment [19,20,21,22,23]. For samples of alldifferent constraints in Java, C#, C++, Python, MATLAB Application Programming Interface of CP Optimizer, etc., see [24,25,26,27,28].

5.3. Other Metaheuristics

One of the popular classifications of metaheuristics is based on the inspiration drawn from evolutionary algorithms, swarm intelligence algorithms, physics-based methods, and human-based methods [29,30]. The theoretical studies published in the literature can be classified into three sections: modifying the current algorithms, hybridising various algorithms, and proposing novel algorithms. The reason why researchers do not use a single algorithm is because there is no optimisation algorithm to solve all optimisation problems according to the No Free Lunch theorem [31].

It should be noted that the algorithms useful for permutation should accept the alldifferent condition described above. For example, in the MATLAB language they are available for Arithmetic Optimization Algorithm (AOA) and Dwarf Mongoose Optimization Algorithm source codes, because of which there exists a possibility that using the alldifferent condition is indispensable here [32,33]. Not being able to easily apply the alldifferent condition renders every implementation of any metaheuristic inefficient or useless.

The efficiencies of newly emerging metaheuristic algorithms are sometimes compared against one another, using a wide range of benchmark and test functions as well as benchmark problems in various fields of engineering [34,35]. However, this is only performed for one specified level of control parameters governing these algorithms. This implies that the range of conclusions resulting from comparisons published in the literature is very limited. In the case of selecting the algorithm optimising the permutation of the spraying beam nozzles on an agricultural sprayer, the problem is complicated by the multiplicity of manufacturers and types of nozzles as well as their polyvalent characteristics, depending on the multiplicity of spray angles and the quality of the material from which the nozzles are made. This signifies the random character of the spray distribution patterns subject to specific permutation optimisation. Moreover, as described above, the multidimensional objective function is itself non-linear. Therefore, in the optimisation problem analysed here, future comparative research of the algorithms requires the collection of possibly the largest database of spray distribution patterns for both new and used nozzle sets. Only the analysis of a large base of spray distributions patterns for nozzles produced by various manufacturers can be used to select a useful implementation group for metaheuristic optimisation algorithms, together with selecting the best levels of the parameters controlling these algorithms. This requires further research on the issues described here through the cooperation of computer scientists and environmental engineers.

6. Results

The bold values $CV\left(\left[\begin{array}{c}{x}^{*}\\ y^{*}\end{array}\right]\right)$ obtained as a result of sub-optimisation for the four studied groups are presented in Table 1 for the 2.5 m table of the measuring device equipped with grooves 0.05 m (or 0.10 m) wide. The underlined initial values of the coefficient of variation $CV\left(\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]\right)$ [%] listed in Table 1 correspond to initial sequencing $\left[\begin{array}{c}{x}_0\\ y_0\end{array}\right]$, i.e., those that occurred before the first run of the algorithm. There, one can notice drops in the optimised parameter by several percentage points as a result of applying sub-optimal solutions already obtained in the analysis from 4100 up to 14,000 sub-problems during 35 to 70 s of the genetic algorithm iteration with its standard input parameters. Characteristics of the used computer system: Processor Intel(R) Pentium(R) CPU B960 @ 2.20 GHz, Cores: 2, Logical CPUs: 2; 4.00 GB RAM; 64-bit Microsoft Windows 10 Pro; Microsoft Excel 2013—Microsoft Corporation, Bellevue, WA, USA. It is surprising that considering the possibility of rotating the nozzles (the top line may contain values of one) most often shortened the computation time but did not necessarily lead to lower values of the coefficient of variation.

Table 2. The value of the coefficient of variation [%] for groups A–D on tables with 0.1 m wide grooves, depending on the sequencing and method of mounting flat fan nozzles on the boom of a field sprayer.

Groups of Nozzles		Value of the Coefficient of Variation [%]			Spread [Percentage Points]	Quotient [%]: of the Maximum Value and the Minimum Value	The Quotient [%]: of the Spread and the Minimum Value
		Initial	Maximum	Minimum
Old	A	20.7%	25.4%	13.0%	12.4	195.7%	95.7%
	B	16.3%	20.6%	11.1%	9.4	184.7%	84.7%
New	C	9.0%	11.5%	7.5%	4.0	152.5%	52.5%
	D	8.6%	10.4%	6.5%	3.9	160.2%	60.2%

expands the scope of proposals that are already noticeable in Table 1. It contains, among others, the results of two different directions of sub-optimisation using the permutation algorithm. Table 2 shows that although the random initial setting of new nozzles occurring in the tested experiment led to the initial value of the coefficient of variation lower than 10%, which permits groups C and D to be used, the same new nozzles in a different configuration are disqualified due to them exceeding the 10% barrier. This is expressed, among others, by as much as four percentage points of the spread, i.e., the difference between the maximum and minimum value of the coefficient of variation. It also means that the uncontrolled way of installing new nozzles may increase the value of the countered spray irregularity parameter by 50–60%. In the group of older nozzles, the importance of selecting their optimal permutation on the sprayer boom becomes more obvious, due to the two-fold and three-fold increase in the spread of the coefficient of variation in groups A and B compared with groups C and D. Furthermore, the last column of Table 2 indicates that as a result of operation, the nozzles wear out unevenly and asymmetrically and the negligence in controlling the permutation of the nozzles on the boom may increase the value of the countered spray irregularity parameter by as much as 85–95%, i.e., making it almost double. These comments indicate the importance and validity of the research undertaken here on the permutation of nozzles.

7. Discussion

Obtaining optimal performance parameters of the spraying nozzles used in agricultural sprayers is conditioned by many factors of operational nature. These factors have a significant impact on the effectiveness of the pesticides used [36]. In the construction of an agricultural sprayer, it is the spraying nozzle that determines the quality of the obtained sprayed liquid stream. In the countries where mandatory sprayer inspection has been adopted, various methods are used to determine the degree of wear of sprayer nozzles. Methods of measuring the outflow rate and methods of measuring the fall of the sprayed liquid are used [37]. A method of testing single nozzles was also proposed, and the results obtained from single nozzles are used to build a virtual boom of a field sprayer [38,39]. The results of the transverse distribution of the sprayed liquid obtained from single nozzles can be combined in any way to obtain a satisfactory distribution coefficient for the entire field sprayer boom. Currently, there are many computing applications for simulating the assumed operating parameters [40,41]. The quality of spray nozzle operation has a significant impact on spraying efficiency as well as the spectrum of droplets formed. Worn-out nozzles may produce droplets of different size classes than their equivalents in new nozzles [42]. The size of the produced droplets has a significant impact on their susceptibility to drift, which has a significant impact on the safety of the environment and the area surrounding the sites of plant protection treatments [43,44,45]. This parameter may also determine the effectiveness of treatment performance [46,47]. According to the authors’ knowledge, the first paper that attempts to optimise the sequence and orientation of the nozzles used on the sprayer beam dates to 2005 [48]. The authors of this paper employ sparse matrices, i.e., such matrices that typically contain only zeros except for block submatrices, which have only ones on the diagonals. Thereby, they attempt to describe the phenomenon of the permutation of nozzles on a linear beam with the possibility of changing their orientation by applying instruments of linear algebra. Unfortunately, this paper demonstrates the authors’ limited experience and mathematical imagination. The paper contains a multitude of factual errors, leading to a departure from the problem formulated in the paper. Hence, according to Figure 1 of the paper quoted here, because there are five submatrices which are not of full column rank the permutation of these submatrices of matrix A, which is presented there, does not correspond to a permutation of the nozzles but to a replacement by other nozzles that the authors will not have at their disposal. The situation is similar with the attempt at changing orientation. On the other hand, according to Figure 4 of the paper quoted here, there are five submatrices, all of which are already of full column rank, and these present five types of nozzles, three identical ones for each type. Such a situation does not exist in the real world. In turn, finding solutions to a system of equations with a number of equations greater than the number of unknowns normally leads to contradictions, unless they are linearly interdependent. The expected homogeneity of spraying, on the other hand, i.e., the identicality of the elements of vector B, may unavoidably lead to a contradiction. Even if one could solve such a system of linear equations, its solution has nothing to do with permutations or reorientation, but with the theoretical determination of the distribution of five types of nozzles that, in an assumed order, would lead to a continuous uniform distribution. Again, such a situation corresponds neither to the problem formulated here nor to the necessities of life. Similarly, the measurement of only a few nozzle permutations does not in any way correspond to optimalisation of the nozzle sequence on the sprayer beam. Unfortunately, in addition to the formulation of the problem, an enormous number of factual errors alone makes any further modelling considerations of the paper quoted here irrelevant. Not even the randomisation of an apparently huge group of 10,000 permutations proposed in [6] provides any insight into the optimisation problem. It is only genetic algorithm methods that render a suboptimal solution more trustworthy. The proposed method of juxtaposing nozzles on the sprayer boom to obtain satisfactory effects of the uniform distribution of the sprayed liquid is conducive to satisfying the requirements necessary to obtain a technical approval for a field sprayer [49,50]. Extremely short times required to determine individual nozzle output distributions, for example, using the equipment from Figure 2 equipped with a storage tank for 50 nozzles, and the short time of operation of the genetic algorithm are conducive to the implementation of the method proposed here. Furthermore, the increasingly widespread availability of advanced software contributes to such modernisation of the sprayer diagnostics process.

8. Conclusions

A flat fan spray nozzle serves the purpose of spraying onto a surface or an object moving in a transverse direction with respect to the direction of the jet, a typical example being the nozzles in a car washing tunnel. The reduction of the unevenness of the transverse sprayed liquid dosage is of interest to agricultural and horticultural bodies, to railway maintenance services, in production lines of foamed foam, in the paper, food, or pharmaceutical industries, in the spraying of rubber surfaces and asphalt, in road defrosting, in air humidification, in sprinkling and continuous painting lines, in fire protection systems, in the creation of a uniform flame by jet engines, etc. Moreover, the use of atomised scrubbing liquids to capture particulate matter and/or gaseous pollutants in liquid droplets, e.g., in wet scrubbers [1]. This means that the solution to the problem presented here may be easily adapted to the situation in which nozzles are placed in a configuration different from the one on the spraying beam of a field sprayer. It may apply, for example, to nozzles placed in a circle or inside a workspace, such as in a jet engine or in a device used for producing food or pharmaceuticals, for example, in the process of spray drying or in the processes of agglomeration, coating, or encapsulation [51,52].

The value of the coefficient of variation for the sprayed liquid falling from the entire boom sprayer does not have to result only from the random sequence of mounting individual nozzles on the boom but may also be the result of planning the sequence of mounting nozzles on the boom.

The retrofitting of a data collection and processing system with the above-described permutation algorithm in automatic devices for the assessment of the technical condition of agricultural sprayers will allow for modelling the quality of the field boom operation in laboratory conditions. This applies both to the correction of the uniformity parameter in the equipment already operated and the improvement of the coefficient of variation in the newly manufactured equipment, allowing a certain predetermined tolerance for deviations from the standard for each nozzle.

Further work is required to collect a large database of spray distribution patterns to compare the possible metaheuristic implementations and to select the best parameters for controlling them.

The comprehensive use of the permutation algorithm may significantly improve the condition of plant protection equipment and, by increasing the uniformity of spraying, reduce the number of preparations used in agricultural treatments.

The algorithm also allows for selecting out of a given number of N ≥ 24 nozzles, even ones with decalibration, n ≤ N nozzles that best meet the standards of coverage uniformity (n ≤ 24). The application of, among others, the above-mentioned permutation algorithm, not only for field sprayers, allows one to select nozzles for replacement and thus reduce the operating costs of sprayers. This means using a relatively cheap automated expert service instead of the costly purchase of a set of new nozzles.