Fuzzy modeling of biometric variables development of tomato crop under irrigation and water salinity effects

. Tomato is a demanding crop in terms of handling, mainly because irrigation has a strong influence on fruit production and quality. Salinity changes the absorption, transport, assimilation, and distribution of nutrients in the plant. In general, such effects are analyzed using statistical tests. However, fuzzy models allow simulations between points that are not verified in agricultural experimentation. Currently, systems with artificial intelligence have excelled in the field of applied sciences, particularly fuzzy systems applied to mathematical modeling. The objective of this research was to use fuzzy modeling to analyze the biometric variables during the development of hybrid tomatoes under two different conditions: the first concerning different water tensions in the soil and the second concerning different salinity doses in irrigation. To this end, two models were developed based on an experiment carried out at São Paulo State University (UNESP), School of Agriculture, Botucatu, São Paulo State, Brazil. Both models sought to estimate the values of biometric variables of the tomato crop. Thus, two models were developed: Model 1 regarded water tensions and days after sowing (DAS), while Model 2 featured salinity and DAS. Fuzzy models provided results that verified the effects of irrigation and salinity layers. Two Fuzzy Rule-Based Systems (FRBS), an input processor with two variables, a set of linguistic rules defined from statistical procedures with percentiles, the Mamdani fuzzy inference method, and the center of gravity method to defuzzification were elaborated for this purpose. The range between −25 and −10 kPa (for Model 1) and between 0.08 and 3 dS m − 1 (for Model 2) provided the development within the ideal parameters for the complete development of the plant cycle. The use of fuzzy logic has shown effectiveness in evaluating the development of tomato crops, thus showing potential for use in agricultural sciences. Moreover, the created fuzzy models showed the same characteristics of the experiment, allowing their use as an automatic technique to estimate ideal parameters for the complete development of the plant cycle. The development of applications (software) that provide the results generated by the artificial intelligence models of the present study is the aim of future research.

The quality of water used for irrigation is a key factor for plants to express their maximum development and productive potential (Gomaa et al., 2021;Rosa et al., 2020;Ullah, Santiago-Arenas, Ferdous, Attia, & Datta, 2019;Wang & Xing, 2017).On the other hand, the use of water with moderate salinity levels has Acta Scientiarum.Agronomy, v. 46, e63515, 2024 become a common practice, both due to the decrease in the availability of high-quality water and as a strategy to improve production (García-Caparrós & Lao, 2018;Nachshon, 2018;Yasuor, Yermiyahu, & Ben-Gal, 2020).
In the presently developed experiments, the effects are analyzed using statistical tests that aim to assist in interpreting the results and, thus, verify the real implications caused by the different adopted treatments.In this way, mathematical models seek to provide more accurate answers.Fuzzy models allow simulations between points that are not verified in agricultural experimentation to be more accurate than regression models.
According to Benini and Rinaldi (2015), fuzzy logic is the basis for developing methods and algorithms for modeling and process control, allowing the reduction of design and implementation complexity, and becoming the solution to identification problems hitherto intractable by classical techniques.
Generally, fuzzy modeling is used in several areas to better understand the observed phenomenon or find optimal points.As an example, Oliveira, Viais Neto, Maeda, and Gabriel Filho (2020) assessed the electricity consumption of a higher education institution to classify it over the months of the year for a better understanding of the phenomenon.Viais Neto et al. (2018) used fuzzy modeling to study the effects of different doses of polymers applied to the substrate with irrigation levels on the development of cherry tomato seedlings up to the transplanting stage, aiming to weave optimal points relative to seedling management.Also, Castro, Saad, and Gabriel Filho (2022) used artificial intelligence techniques applied to the optimization of micro-irrigation systems by the Zimmermann-Werner method for the resolution of fuzzy linear programming problems.
Several applications involving fuzzy logic can be found in agricultural sciences, such as crop yield improvement and fertilizer use efficiency (Prabakaran, Vaithiyanathan, & Ganesan, 2018), in which rules generated by experts have been used.It is also found in agrometeorological models for estimating yield (Luydmila, Mikhail, Imran, Tamara, & Anatoliy, 2017), in which the degree of agronomic suitability of the cultivation site of a given crop is expressed as a number between 0 and 1.
Particularly, the use of fuzzy models aimed at mathematical modeling of agronomic experiments has proved to be highly necessary not only for analyzing the effects of variation in response variables but also for the creation of future applications/software to support researchers and farmers.
Therefore, this research aimed to study the effects of biometric variables during the development of the hybrid tomato (Lycopersicum esculentum) under two different conditions using fuzzy modeling: the first condition was about different soil water tensions and the second one was about different salinity doses in irrigation.

Description of the experiment
The experimental data used for the fuzzy modeling of this study were analyzed statistically (Silva Junior, 2018) and employed in the elaboration of the other fuzzy modeling used by Viais Neto et al. (2019a and b).The experiment was carried out between June and October 2011 in a protected environment installed in an area located in the Department of Rural Engineering of the São Paulo State University (UNESP), School of Agriculture, city of Botucatu, São Paulo State, Brazil, with an average altitude of 786 m, latitude 22°51′03″ South, and longitude 48°25′37″ West.
The used pots had a capacity of 15 dm 3 and were filled with the soil taken from the 0-0.20-m depth layer, classified as a medium-textured Rhodic Ferralsol.The soil was taken, sieved, and air-dried until it reached a water content of 4%.The method proposed by Klar (1998) was used to determine soil water content.
Liming and planting and topdressing fertilization were carried out based on the soil analysis, following the recommendation suggested by Trani and Raij (1996).The tomato variety used was the cultivar Katia.Seedlings were prepared by the Department of Horticulture and transplanted into pots 45 days after sowing.Sodium chloride (NaCl) was used for water salinization.For this purpose, a 2 M solution (2 times the molecular weight of NaCl) was prepared to have a liter of solution.A conductivity meter was used to obtain the doses of 0.08, 3, and 5 dS m −1 of water salinity by diluting the respective proportions of 0, 31.07, and 53.96 mL of the solution in a liter of supply water.
A soil sample was collected in the 0-0.20-m layer and sent to the Laboratory of Soil Plant Water Atmosphere Relationship in the Department of Rural Engineering to determine the water content in the soil as a function of soil potentials (Ψ).The soil water retention curve points, adjusted by the Soil Water Retention Curve program (Dourado Neto, Lier, Botrel, & Libardi, 1990), allowed the determination of the soil water content as a function of soil potentials using the model proposed by van Genucinen (1980).Table 1 shows the equation parameters.The following soil characteristic curve was generated after calculating the parameters: Matric requirements (Ψ) of −60, −30, and −10 kPa was established to cause water stress and irrigation was determined based on the mass of the pot.Soil water content was inferred from the established Van Genuchten equation, and the amount of daily water to reach the established levels was calculated.
According to Silva Junior ( 2018), plant height corresponded to the distance between the soil base and the plant apex; the stem diameter was determined in the basal region of the plant, close to the soil; the leaf area was calculated using an estimate using dry leaf phytomass; the calculation of both green and dry phytomass corresponded to the sum of the masses of leaflets, petioles, clutches, and stem; and the percentage of deficient fruits was calculated for the fruits that presented apical rot.
The evaluations were destructive seeking to meet the four analyses throughout the cycle (75, 90, 105, and 120 days after sowing) and 27 plants were used to assess the effects of irrigation and water salinity on the tomato crop.Thus, 108 plants (4 × 27) were required for the complete study.

Fuzzy modeling
Two fuzzy models were performed to evaluate the development characteristics of the experiment analyzed in this study, which used the management of different soil water tensions and salinity doses in irrigation throughout the tomato cycle.
The first modeling (Model 1) aimed to estimate the values of biometric variables of the tomato crop over the days after sowing (DAS) versus different soil water tensions (Irrigation) (Figure 1a).Model 2 also estimated the biometric variables over DAS, but now testing the effect of different doses of irrigation salinity (Salinity) (Figure 1b).
Analogously to the methodology proposed by Viais Neto et al. (2019a) for the elaboration of both Fuzzy Rule-Based Systems (FRBS), an input processor, a set of linguistic rules, a fuzzy inference method, and an output processor were defined so that a real number was generated in the end as an output.

Fuzzy sets of input variables
The variable DAS is an input variable for both models, defined using the evolution periods of the experiment (75, 90, 105, and 120 days), with four fuzzy sets named Very Low (VL), Low (L), Medium (M), and High (H).Specifically in Model 1, the other input variable is Irrigation, in which the fuzzy sets were defined according to Viais Neto et al. (2019a).In Model 2, the other variable is Salinity, in which the fuzzy sets were defined according to Viais Neto et al. (2019b) (Figure 2).

Fuzzy sets of output variables
Several biometric variables were analyzed according to the representative factors of both Models 1 and 2: Irrigation x DAS and Salinity x DAS (Table 2).Among the analyzed biometric variables, those that were considered for the models showed interactions between such factors and/or significant simultaneous differences between factors.The only variable that did not meet these criteria was Plant Height (Heig) and therefore not used in the models.Legend: Factor A -Irrigation (−kPa) for Model 1 and Salinity (dS m −1 ) for Model 2; Factor B -DAS (days); (*) Significant by the F-test with a 5% probability of error.
In Table 2, the variable Heig of Model 1 shows the absence of differences between the levels of analysis factors (Irrigation and DAS) and also the interaction between them.The variables Diam, GP, and DP had the same behavior, showing differences in the levels of each factor and an absence of interaction between analysis factors.Finally, there is an interaction between the analysis factors of the variable LA.In general, the coefficients of variation ranged from 8 to 18%.
The variable Hight of Model 2 shows differences between the levels of the factor DAS but has no differences between the factor Salinity as well as no differences of interaction between analysis factors (Salinity and DAS).The other variables show an interaction between analysis factors.In general, the coefficients of variation remained between 9 and 12%.
The methodology used to create the membership functions of the fuzzy sets of output variables in the modeling performed in this study was similar to that used by Viais Neto et al. (2019a) and is defined in Table 3, using percentiles varying at 12.5%.Importantly, one of the two disjunct intervals of the support, whose points have no degree of membership 1, was defined with an amplitude equal to 1 for the membership functions L1 and H3.
Similar to those studies, the highest degrees of relevance of each treatment median were calculated after the elaboration of the fuzzy output sets, associating the input variables with the output variables.
The association of combinations of fuzzy sets of the input variables with a fuzzy set of each output variable of both Models 1 and 2 was performed similarly to the model established by Viais Neto et al. (2019a and b).

Inference, defuzzification methods, and software
The inference method used in this study was the Mamdani (Mamdani, & Assilian, 1975).The center of gravity method was adopted for defuzzification.The value of the linguistic output variable inferred by the fuzzy rules is translated into real value.
The development of FRBS of the present study required the software Matlab ® , licensed for the São Paulo State University (UNESP), School of Sciences and Engineering, Tupã campus, São Paulo State, Brazil.Specifically, the Fuzzy Logic Toolbox of the software was used to prepare three-dimensional graphs and contour maps.

Statistical analysis
The biometric variables were analyzed statistically using ANOVA (analysis of variance) in two ways: considering the factors irrigation and DAS (3 × 4) and later the factors salinity and DAS (3 × 4), thus consisting of 12 treatments in each analysis.
Each treatment had three repetitions.There was no need for a comparison between treatment means, as the purpose of the analysis was to identify variables with significant differences between their treatments.Thus, variables that did not present significant differences between their treatments were not mathematically modeled.
All statistical analyses were performed using the R software 4.2.0 (R Core Team, 2022), adopting a value of p < 0.05 as statistically significant.

Results and discussion
Both models were prepared, after calculating the percentiles of the measured data, with the definition of the membership functions of the fuzzy sets of the output variables stem diameter, leaf area, green phytomass, and dry phytomass (Figure 3).Once the membership functions were defined, the highest pertinence degree was taken for each biometric variable analyzed, referring to the type of treatment applied during the tomato development.Thus, the situation that fits the variable under analysis could be assessed by comparing it with the other treatments, thus rating the rules base for the Models (Table 4).Table 4. Rule-base of fuzzy systems relating all combinations of fuzzy sets of the input variables with those of the output variables (Diam -Stem Diameter, LA -Leaf Area, GP -Green Phytomass, and DP -Dry Phytomass).Input 1 is defined as "Irrigation" for Model 1 and "Salinity" for Model 2.

Input variable Output variable
Input The power of analysis coming from the present mathematical modeling stands out, allowing a simultaneous comparison of all contour maps of the analyzed response variables.
Additionally, the way of obtaining the rules base contributes to performing the present method of creating Fuzzy Rule-Based Systems.Most fuzzy systems that use the Mamdani inference method are elaborated using expert interviews to create their rules base.The rules base is also an important result since the understanding of the present system with a table (rules base) becomes easier for many rural producers when faced with the interpretation of three-dimensional surfaces.
The Mamdani method was employed in this present study but without the use of interviews to create the rules base.It was generated from a specific methodology using percentiles and the association of fuzzy sets.
Therefore, Table 4 could be elaborated, which is an extremely important result for the practical use of the models developed in this paper.
The graphical results of the models are presented in the form of surfaces (Figure 4) and contour maps (Figure 5) for all biometric variables.
The Stem Diameter of Model 1 (Figure 5a) had the highest value at a soil water tension of −10 kPa and 105 DAS.Soil water tensions between −25 and −10 kPa led to a gradual increase in diameter until the end of the tomato cycle.In contrast, this development only occurred after 90 days for tensions between −30 and −60 kPa.Biomass allocation in stem diameter shows linear increases with increasing water depths, being associated with a reduction in phytomass production activity, as energy must be redirected to fruit production (Ullah et al., 2021;Zhou et al., 2019).In Model 2, Stem Diameter (Figure 5b) reaches its maximum at a salinity of 0.08 dS m −1 and 115 DAS.The minimum value was inferred at 90 DAS for a salinity of 5 dS m −1 .Salinity caused a reduction in stem diameter, as salts accumulate in this case, which affects the osmotic adjustment (Abdeldym, El-Mogy, Abdellateaf, & Atia, 2020;Feng et al., 2019).
The Leaf Area of Model 1 (Figure 5a) has a maximum value at a soil water tension of −10 kPa and 90 DAS.The leaf area size was similar at 120 DAS, regardless of the soil water tension.Water deficit decreases the number of leaves per plant, leaf area, and leaf longevity due to reduced soil water potential.The most visual effect of water deficit is the reduction in leaf area.It leads to reduced water loss and lower energy expenditure to control stoma opening and thus minimize impacts (Khapte et al., 2019;Ullah et al., 2021).For Model 2 (Figure 5b), Leaf Area had the highest value at 90 DAS and salinity of 0.08 dS m −1 .In general, the behavior of this variable is very similar throughout the tomato cycle according to the treatments, regardless of the salinity dose in irrigation.Morphological and anatomical changes in plants are common under salt stress conditions, which reflect in the reduction of transpiration as an alternative to maintain the low absorption of saline water; one of these adaptations is the reduction in leaf area (Pérez-Labrada et al., 2019;Sassine et al., 2020).
The Green Phytomass in Model 1 (Figure 5a) obtained a maximum at soil water tension of −10 kPa in the last evaluation period.There is no development of green phytomass between −60 to −30 kPa and 75 to 105 DAS.Similar effects have been observed in studies with the application of irrigation depths so that the reduction in green phytomass accumulation is linear as a function of the reduction in the applied depth (Hatamleh et al., 2022;Salgado et al., 2021).The highest development for Model 2 (Figure 5b) occurred throughout the cycle for the treatment with a salinity of 0.08 dS m −1 .The opposite of this situation occurred at 105 DAS for a salinity of 5 dS m −1 .Finally, the Dry Phytomass behavior in Model 1 (Figure 5a) was similar to that of green phytomass.A similar phenomenon occurred in Model 2 (Figure 5b).However, salinity doses in Model 2 influenced the dry phytomass more intensely for values above 3 dS m −1 .In general, the inhibition in the growth and production of phytomass by plants is a response to nutritional imbalance and toxicity, which reflect in-breath loss, root expansion, water absorption, and CO 2 fixation (Alzahib et al., 2021;Martínez-Andújar et al., 2021;Sousa et al., 2022).The decrease in aerial phytomass production by plants irrigated with saline water is almost always the result of early senescence caused by the toxic effects of excess salts in the water, which limits leaf area expansion and, therefore, reduces dry matter yield (Huang, Zhang, Zhai, Lu, & Zhu, 2019;Okon, 2019;Zörb, Geilfus, & Dietz, 2019).
The normalization of references to the regions of each contour map of the biometric variables (Figure 5) was performed by defining as Region A the site with the highest values of the biometric variable (warmer colors or red), while Region B will be the one with the lowest values (cooler colors or blue).
Thus, Region A in Model 1 is partially or fully inserted between the irrigation levels 60 and −30 kPa for all variables.In contrast, Region B is fully inserted between the irrigation levels −25 and −10 kPa.Thus, water deficit negatively influences all the variables analyzed throughout the tomato cycle.For Model 2, Region A represents the lowest values inferred for each variable, while Region B represents the highest values.The variables most affected by the increase in salinity doses in irrigation are stem diameter, green phytomass, and dry phytomass, as Region A of these variables is located between salinity doses in irrigation of 4 and 5 dS m −1 .In addition, Region B for all variables is partially or fully inserted between salinity doses in irrigation of 0.08 and 3 dS m −1 , which is in line with Ayres and Westcot (1976), who stated that tomato is considered moderately sensitive to salts, with the threshold for the crop at 2.5 dS m −1 .

Conclusion
All the biometric variables of the tomato crop analyzed in Model 1 showed a better development for high soil water tensions.Water deficit negatively influenced the variables analyzed throughout the tomato cycle.In addition, a certain homogeneity was observed in the effects caused by water stresses in the soil on the development of the variables stem diameter and green and dry mass.In Model 2, high salinity doses in irrigation severely affected the variables stem diameter, green phytomass, and dry phytomass throughout the tomato cycle.In addition, low amounts of salinity in irrigation provided the best development for all analyzed biometric variables.The use of fuzzy logic helped in modeling the behavior of the tomato crop under different management conditions whether due to water deficit or salinity, thus supporting the decision-making regarding the most appropriate management method.Thus, it proved to be an important tool to assist in the decision-making process of producers.It can be seen in the present mathematical model, which classified different conditions according to the development time, salinity, and irrigation conditions offered to the tomato crop.These different conditions proved to be favorable for the crop in certain cases but revealed situations not so favorable in other cases.Sometimes, these situations are characteristic of the planting site and therefore cannot be controlled.However, the model represents, even in such a scenario, an excellent decision-making support system, as the rural producer will be able to predict the biometric variables that tomato plants will have naturally before planting.The producer can carry out the most appropriate possible management for the crop regarding controllable factors, such as Irrigation, which is present in one of the models in this study.However, there are situations in which the amount of water is limited, and, in this case, the optimal point shown in the models cannot be obtained by the rural producer.Still in this case, the results of this study are useful, as the predictability of the behavior of biometric variables helps in the decisionmaking by the rural producer.Another conclusion is that the created fuzzy models showed the same characteristics of the experiment, allowing their use as an automatic technique to estimate the ideal parameters for the complete development of the plant cycle.The development of applications (software) that provide the results generated by the artificial intelligence models of the present study is the aim of future research.

Figure 1 .
Figure 1.Tomato fuzzy systems with four output variables (biometric variables): stem diameter, leaf area, green phytomass, and dry phytomass.In both systems, days after sowing (DAS) is one of the input variables.In (a) Model 1, the other input variable is soil water tension.In (b) Model 2, the second input variable is salinity doses in irrigation.

Figure 2 .
Figure 2. Triangular relevance functions of the Very Low (VL), Low (L), Medium (M), and High (H) fuzzy sets for the input variables DAS, Irrigation, and Salinity.

Figure 3 .
Figure 3. Membership functions of the fuzzy sets of the output variables of Stem diameter, Leaf area, Green phytomass, and Dry phytomass (a) for Model 1 and (b) Model 2.

Table 1 .
Parameters of the Van Genuchten equation related to the soil water retention curve.

Table 2 .
Mean Squares (MS) and Coefficients of Variation (CV) of the Analysis of Variance (ANOVA) of the biometric variables to be the output variables of Models 1 and 2, labeled Plant Height (Heig), Stem Diameter (Diam), Leaf Area (LA), Green Phytomass (GP), and Dry Phytomass (DP), under a 3 × 4 Factorial Scheme.