© The Authors, 2023, Published by the Universidad del Zulia*Corresponding author: sergiovladimirores@gmail.com
Keywords:
Response surface methodology
Yield optimization
Corn
Zea mays L.
Application of the response surface methodology for yield optimization in maize (Zea mays L.)
Aplicación de la metodología de supercie de respuesta para la optimización del rendimiento en
maíz (Zea mays L.)
Aplicação da metodologia de superfície de resposta para otimização de produtividade em milho
(Zea mays L.)
Rev. Fac. Agron. (LUZ). 2023, 40(4): e234035
ISSN 2477-9407
DOI: https://doi.org/10.47280/RevFacAgron(LUZ).v40.n4.04
Crop production
Associate editor: Dr. Jorge Vilchez-Perozo
University of Zulia, Faculty of Agronomy
Bolivarian Republic of Venezuela
Abstract
The objective of this study was based on the application of the response
surface methodology (RSM) for yield optimization in maize (Zea mays L.).
The hybrid INIA SQ-1 was used, and the Response Surface Methodology was
used using the Box-Behnken design (DBB), with which the following factors
were evaluated: plant density, nitrogen (N) dose and phosphorus (P) dose
at three levels each; for the optimization of the response variables: “yield”
(kg.ha
-1
) and the “number of grains per square meter” (g.m
2
). The response
surface method provided a statistically validated predictive model, which
through adjustments was adapted to an established optimization process. For
the variable “yield”, a maximum response was found with the application of
150 kg.ha
-1
of N and 90 kg.ha
-1
of P. In relation to the number of grains per
square meter (g.m
2
), the optimum was obtained using 75,000 plants.ha
-1
and
an applied dose of 150 kg.ha
-1
.
1
Facultad de Ciencias de la Salud, Universidad Autónoma de
Chile, Chile.
2
Facultad
de
Ciencias
Humanas,
Universidad
Bernardo
O´Higgins, Santiago-Chile.
3
Departamento
de
Ingeniería
Civil,
Facultad
de
Ingeniería,
Universidad
Católica
de
la
Santísima
Concepción,
Concepción, Chile.
4
Universidad Central de Venezuela. Venezuela.
5
Instituto
de
Geografía,
Ponticia
Universidad
Católica
de
Valparaíso, Chile.
Received: 09-10-2023
Accepted: 22-11-2023
Published: 29-11-2023
Román Montaña
1
Ángel
Roco-Videla
2
Nelson Maureira-Carsalade
3
Ana Nieves
4
Sergio Flores
5*
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Rev. Fac. Agron. (LUZ). 2023, 40(4): e234035. October-December. ISSN 2477-9407.2-6 |
to increase the yield per hectare of the crop and in turn improve the
nal quality of the grain, and thus satisfy the requirements of national
consumption that contributes to the sustainability of the corn circuit,
of great strategic importance for the Venezuelan diet.
The reduction in crop yield can be attributed to several factors,
mainly the use of varieties with low productive potential, water
deciency, low fertility of cultivated soils, inadequate planting time,
planting density and inadequate control of insects and harmful plants
(Masood et al., 2011). However, fertilization and plant density are
some essential technological components within modern agriculture,
considering nitrogen fertilization as one of the key aspects in corn crop
management because, among other aspects, it plays a fundamental
role in plant growth and yield and, above all, in increasing the number
of grains (Moreno et al., 2017; Vargas et al., 2021). Similarly, the
application of phosphorus contributes to the formation of nucleic
acids, cellular respiration, and metabolic activity which, when
applied together with nitrogen, inuence grain yield, forage quality,
plant height and number of leaves per plant (Medina et al., 2018;
Romero et al., 2022). Thus, the interest in growing corn using reduced
spacing has grown in recent years in dierent producing regions,
mainly among producers who work with planting densities higher
than 50,000 plants.ha
-1
, as reported by Rodríguez et al. (2021). In
this sense, planting density has been one of the factors that producers
frequently modify to increase grain yield, although they do not
always establish the correct density, increasing competition for light,
water, and nutrients, causing a decrease in root volume, number of
ears, grain quantity and quality per plant (Shari et al., 2014; Bouras
et al., 2021). Indeed, there is a challenge to increase crop productivity
and eciency in the use of fertilizers, leading to the search for new
ways that lead to a more sustainable agriculture, being necessary the
application of tools to evaluate dierent scenarios and to obtain an
optimal performance (Yaguas, 2017; Torralbo et al., 2023).
Therefore, the objective of this study was based on the
application of the response surface methodology for yield
optimization in corn (Ganugi et al., 2022), evaluating the factors
plant density (plants.ha
-1
), nitrogen (N) dose and phosphorus (P)
dose; for the response variables “yield” (kg.ha
-1
) and “number of
grains per square meter” (g.m
2
).
Materials and methods
Study area
The present investigation was carried out during the rainy period
(May-October) of 2020, in the farm “El Angel”, specically in the
rural area of Mariara, Diego Ibarra Municipality, Carabobo State,
Venezuela, at the coordinates 10°17´16´´ North and longitude
67°43´10´´ West, at an altitude of 442 masl. This area is characterized
by average annual rainfall and temperature of 800 msl. And 28 °C
respectively. The climate of the area corresponds to dry tropical,
according to the Köppen climate classication. The material used
corresponded to the INIAsq-1 hybrid.
Experimental units were used, made up of plots of 3.6 x 4.0 m,
with four rows of 4 m long and 0.9 m apart, the distance between
plants was 0.20 m. The size of the useful plot was 10 m
2
Variables
Corn yield (kg.ha
-1
)
Ten squared meters were harvested manually in each experimental
unit; the harvested ears were threshed manually and grain moisture at
=
(
1
,
2
)
+ 1
= (
1
,
2
, … . ,
)
1
Resumen
El
objetivo
de
este
estudio
se
basó
en
la
aplicación
de
la
metodología de supercie de respuesta (MSS) para la optimización
del
rendimiento
en
maíz
(Zea
mays
L.).
Se
utilizó
el
híbrido
INIA SQ-1 y la Metodología de Supercie de Respuesta mediante el
diseño Box-Behnken (DBB), con el cual se evaluaron los siguientes
factores:
densidad
de
plantas,
dosis
de
nitrógeno
(N)
y
dosis
de
fósforo
(P)
en
tres
niveles
cada
una;
para
la
optimización
de
las
variables
de
respuesta:
“rendimiento”
(kg.ha
-1
)
y
“número
de
granos
por
metro
cuadrado” (g.m
2
). El método de supercie de respuesta proporcionó
un modelo predictivo validado estadísticamente, que mediante ajustes
se adaptó a un proceso de optimización establecido. Para la variable
“rendimiento”, se encontró una respuesta máxima con la aplicación
de 150 kg.ha
-1
de N y 90 kg.ha
-1
de P. En relación con el número de
granos
por
metro
cuadrado
(g.m
2
),
el
óptimo
se
obtuvo
utilizando
75.000 plantas.ha
-1
y una dosis aplicada de 150 kg.ha
-1
.
Palabras clave:
metodología de supercie de respuesta, optimización
de rendimiento, maiz, Zea mays
L.
Resumo
O objetivo deste estudo baseou-se na aplicação da metodologia
de superfície de resposta (RSM) para a otimização da produtividade
do
milho
(Zea
mays
L.).
Foi
utilizado
o
híbrido
INIA
SQ-1
e
a
Metodologia
de
Superfície
de
Resposta
foi
utilizada
por
meio
do
delineamento
Box-Behnken
(DBB),
com
o
qualforam
avaliados
os
seguintes fatores: densidade de plantas, dose de nitrogênio (N) e dose
de fósforo (P) em trêsníveis cada; para a otimização das variáveis de
resposta: “produtividade” (kg.ha
-1
) e o “número de grãos por metro
quadrado” (g.m
2
). O método de superfície de resposta forneceu um
modelo preditivo validado estatisticamente, que, por meio de ajustes,
foi
adaptado
a
um
processo
de
otimização
estabelecido.
Para
a
variável “rendimento”, foi encontrada uma resposta
máxima
com a
aplicação de 150 kg.ha
-1
de N e 90 kg.ha
-1
de P. Em relaçãoao
número
de
grãos por metro quadrado (g.m
2
), o
ótimo
foi obtido com 75.000
plantas.ha
-1
e uma dose aplicada de kg.ha
-1
.
Palavras-chave:
metodologia de superfície de resposta, otimização
de rendimento, milho,
Zea mays
L.
Introduction
Corn is an important crop in the Venezuelan vegetable agricultural
sector, considered one of the most important strategic and priority crops
of the nation, due to its importance in the daily diet of Venezuelans. In
addition, corn is a source of employment, due to the large number of
people who grow it throughout almost the entire national geography,
being
concentrated
mainly
in
the
states
of
Portuguesa,
Barinas,
Cojedes, Guárico, Apure and Yaracuy; 80 % of the production is used
for the manufacture of precooked our and the rest for the processing
of corn our and animal consumption. Hence, precooked corn our
represents the rst source of calories and the third source of protein
in the Venezuelan diet, generating a great demand for corn at levels
that
are
not
covered
by
domestic
production,
according
to
FAO-
ESS
(2021)
(Erenstein
et
al.,
2022).
Therefore,
it
is
convenient
to
evaluate
the
factors
aecting
the
agricultural
practices
of
the
crop,
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Montaña et al. Rev. Fac. Agron. (LUZ). 2023 40(4): e234035
3-6 |
physiological maturity was determined with a moisture meter. Yield
was expressed at 12 % grain moisture.
Number of grains per square meter (g.m
2
)
The number of grains per m
2
was calculated as the quotient
between yield (on a dry basis) and individual grain weight. The latter
variable was determined by averaging two samples of 200 grains
each and dried in a forced-air oven for 10 days.
Statistical methods
Response Surface Methodology was used to optimize yield
(kg.ha
-1
) and number of grains per square meter (g.m
2
) in corn.
Response Surface Methodology is a collection of mathematical
and statistical techniques used to develop, improve, and optimize
processes. It is also applicable in the design, development, and
formulation of new products, as well as the improvement of
existing product designs. One of the most widespread applications
of these techniques is to model and analyze problems in which a
response of interest (there may be more than one) is inuenced by
several quantitative factors, being the objective to optimize this
response bydetermining the optimal values of the factors involved
(Montgomery, 2004). The relationship will be given by:
(1)
Which is assumed to be continuous, in E where represents the
noise or error observed in the response, whose distribution is assumed
to be normal with zero mean. The variables in the equation (2) are
natural variables since they are expressed in the natural units of
measurement. However, it is common to transform them into coded
variables The variables, without dimensions, with zero mean and the
same standard deviation. Thus, the actual value expected to be taken
by the response variable implies a relationship that can be represented
by a hypersurface called response surface.
(2)
The form of the true response function is unknown, so we must
approximate it and the choice of appropriate factors is therefore
important. The success of applying the MSR technique also lies in the
fact that the response can be tted to a polynomial of rst or second
degree.
Box-Behnken Design
We used the Box-Behnken design (DBB), designed by Box and
Behnken (1960), often used to rene models after important factors
have been determined using screening designs or factorial designs,
especially if curvature of the response surface is suspected. The
mathematical model t to a DBB is determined by the following
equation:
(3)
Where is the independent term, y y are the coecients of the i-s
main eects and their quadratic eect respectively, is the interaction
coecient between the i-s and the j-s factor and y is the random error.
This equation, having quadratic terms, provides a response surface
with some curvature, thus better approximating the real model than in
the case of the k-factor design.
Factors evaluated
In the present work, the factors plant density (plants.ha
-1
), n dose
and p dose were evaluated. Table 1 shows the levels used for each
of these factors with their respective coding. The details ofthe Box-
Behnken design are presented in table 2, where each of the levels used
for each factor are shown in table 1.
Table 1. Treatment levels and coded values of the factors
evaluated.
FACTORS LEVELS
-1 0 1
Density of plants (plants.ha
-1
) 50000 75000 100000
Nitrogen dosage (kg.ha
-1
) 100 150 200
Phosphorus dosage (kg.ha
-1
) 60 90 120
Table 2. Box-Behnken design matrix for response surface analysis,
with natural and coded factors.
Randomization
Order
CODED FACTORS NATURAL FACTORS
Density Nitrogen Phosphorus Density Nitrogen Phosphorus
8 -1 -1 0 50 100 90
7 -1 1 0 50 200 90
11 1 -1 0 100 100 90
12 1 1 0 100 200 90
1 -1 0 -1 50 150 60
4 -1 0 1 50 150 120
14 1 0 -1 100 150 60
2 1 0 1 100 150 120
13 0 -1 -1 75 100 60
3 0 -1 1 75 100 120
15 0 1 -1 75 200 60
9 0 1 1 75 200 120
6 0 0 0 75 150 90
5 0 0 0 75 150 90
10 0 0 0 75 150 90
The use of the results obtained here is not directly applicable in
the eld. The way the treatments were assigned to the experimental
units was an unrestricted randomization, where each treatment,
including the control, had the same probability of being assigned to
any of the available experimental units. Table 2 shows the order of
randomization.
Results and discussion
Corn yield (kg.ha
-1
)
Table 3 shows the results of the analysis of variance for the
variable “yield” (kg.ha
-1
), showing that the model studied is signicant
(p<0.05), which indicates that at least one of the factors evaluated
has a signicant inuence on the measured response. The non-
signicance of the lack of t (p>0.05) allows inferring that the model
ts the data adequately. Likewise, it is observed that the linear eect
of the density and nitrogen factors, in addition to the quadratic eect
of the density factor and all interactions, didnot present signicant
dierences; therefore, their eects are considered negligible and are
eliminated to increase the model t.
=
(
1
,
2
)
+ 1
= (
1
,
2
, … . ,
)
1
=
0
+
=1
+
2
=1
+
<=2
+
1
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Rev. Fac. Agron. (LUZ). 2023, 40(4): e234035. October-December. ISSN 2477-9407.4-6 |
Table 3. Analysis of variance for yield variable (kg.ha
-1
).
df
Sum of
squares
Mean
squares
Fc p
Model 9 500867 55652 5.65 0.0355*
Density 1 14113 14113 1.43 0.2854
Nitrogen 1 7080 7080 0.72 0.4349
Phosphorus 1 46512 46512 4.72 0.0818
Density*density 1 3142 3142 0.32 0.5961
Nitrogen*nitrogen 1 312985 312985 31.75 0.0024*
Phosphorus* phosphorus 1 66682 66682 6.76 0.0483*
Interactions 3 50323 16774.33 1.70 0.2491
Error 5 49289 9857.8
Lack of adjustment 3 25724 8574.67 0.73 0.6321
Pure error 2 23565 11782.5
Total 14 540126
Adjust= 74.45 %; * p<0.05
Table 4 shows the analysis of variance excluding the non-
signicant eects from the previous table, showing how the model t
goes from 74.45 % (table 3) to 90.28 %, which indicates that 90.28 %
of the corn yield is explained by the new model.
(4)
Table 4. Analysis of variance for yield variable (kg.ha
-1
).
df
Sum of
Squares
Middle
Management
Fc p
Model 3 500867 166956 14.82 0.0028*
Phosphorus 1 46512 46512 4.13 0.0667
Nitrogen*Nitrogen 1 312985 312985 27.78 0.0002*
Phosphorus*Phosphorus 1 66682 66682 5.92 0.0332*
Error 11 123947 11267.91
Total 14 540126
Adjust= 90,28 %; * p<0.05
Figure 1 shows the maximum curvature reached by the maize
yield variable; when studying the surface, it is established that this
optimum point is achieved when using a medium level of N doses
(150 kg.ha
-1
), and a medium level of the phosphorus factor, that is,
an application of 90 kg.ha
-1
of P. Similar results were obtained by
barrios et al. (2018), who obtained higher yields (9763 kg.ha
-1
) with
doses of 150 kg.ha
-1
of N, determining that higher doses of N did
notsignicantly inuence yield, thus allowing a more ecient use of
N at a lower dose. Barrios et al. (2016) also reported that excess n
accumulates in leaf tissues, saturating the N absorption capacity of the
crop, resulting in a poor root system, soft tissue, weak plants, delayed
production, and lower yields. Gómez et al. (2016) indicated that 200
kg.ha
-1
of n were required to obtain maximum grain yield; however,
with the application of 90 kg.ha
-1
of P, they obtained higher values in
plant and ear height, ear health and grain yield. However, there are
studies (Guo et al., 2016) indicating the case of the loess plateau of
China where a potential yield of 98 to 108 % was achieved; varying
the optimum dose of N from 207 to 222 kg.ha
-1
with a ratio of 65 to
80 kg of grain/kg of N applied. In relation to planting density, it was
showed that the eect was not signicant, allowing to establish that
the three density levels evaluated had no eect on the yield variable.
These results agree with those reported by Tadeo et al. (2020), who
determined that plant density is not a factor that inuences grain
yield; however, they recommend the use of a population of 65,000
plants.ha
-1
to avoid spending more seed. Similarly, Youngerman et
al. (2018) indicated that yield at low density did not dier from the
standard, suggesting no signicant changes.
Figure 1. Response surface for the response variable yield (kg.ha
-1
).
Table 5 shows the analysis of variance for the variable “grains
per square meter”, nding signicance (p<0.05) for the proposed
model, which indicates that at least one of the selected factors has a
signicant inuence on the evaluated variable, this eect being either
linear or quadratic. The non-signicance of the lack of adjustment
allows us to establish that the model proposed achieves a good
adjustment for the variable studied. Likewise, the eect of linear
density, quadratic phosphorus and the dierent interactions that form
the factors studied, did notpresent signicant eects; therefore, these
eects were eliminated from the model to improve its t.
Table 6 shows that the signicant eects are nitrogen, both
linearly and quadratically, and plant density quadratically, which
indicates that these factors allow nding an optimal response for the
measured variable. The improvement in the model t, from 89.79 %
to 93.5 %, is noteworthy.
Figure 2 shows the maximum curvature reached for the number
of grains per square meter (g.m
2
), achieving this optimum value with
a planting density of 75,000 plants.ha
-1
, and an N dosage of 150
kg.ha
-1
. This is in agreement with what was reported by Tadeo et al.
(2012), where the stocking density of 70,000 plants.ha
-1
presented
numerically higher yields for a given hybrid, compared to other
densities. In case the number of plants per area goes beyond the
optimum, there would be detrimental consequences for ear ontogeny,
resulting in sterility and a decrease in the number of grains per square
meter (g.m
2
), according to Jun et al (2017). In other study (Martinez et
al., 2017) the components ear length, mean ear diameter and mean ear
diameter were sensitive to the increase in plant population per hectare,
recommending a density of 65,000 plants.ha
-1
. With respect to the
dose of N applied, it has been previously showed that the number of
grains per square meter (g.m
2
) had a positive and signicant response
to the intermediate dose of N (150 kg.ha
-1
of N), favoring the size and
number of grains per ear, important components in determining the
level of yield in corn (Barrios et al., 2018).
= 9104,63 + 6,879 0,0241
2
+ 0,003524
2
1
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Montaña et al. Rev. Fac. Agron. (LUZ). 2023 40(4): e234035
5-6 |
The response surface method provided a statistically validated
predictive model, which through adjustments was adapted to
an established optimization process. For the variable “yield”, a
maximum response was found with the application of 150 kg.ha
-1
of
N and 90 kg.ha
-1
of P. In relation to the number of grains per square
meter (g.m
2
), the optimum was obtained using 75,000 plants.ha
-1
and
an applied dose of 150 kg.ha
-1
.
Conclusions
The present study demonstrates that the response surface
methodology is a valuable tool to optimize maize yield in Venezuela.
The results indicate that planting density and the amount of
nitrogen are key factors aecting maize yield, and that the optimal
planting density to maximize the number of grains per square meter
was 75,000 plants per hectare. In addition, it was found that the
improvement in the t of the model, from 89.79 % to 93.5 %, is
signicant and demonstrates the eectiveness of the response surface
methodology in optimizing maize yield. These results are important
for corn production in Venezuela since corn is an essential crop for
the economy and diet of the population.
Increasing the yield per hectare can improve the economy of
Venezuela and guarantee the food security of the population. In
addition, improving the quality of corn production can reduce the
need to import corn from other countries, which can have a positive
impact on the trade balance not only locally, but also regionally.
Cited literature
Barrios, M., and Basso, C. (2018). Eect of nitrogen fertilization on yield
components and grain nutritional quality of six maize hybrids. Bioagro,
30(1), 39-48. http://ve.scielo.org/pdf/ba/v30n1/art04.pdf
Bouras, H., Bouaziz, A., Choukr-Allah, R., Hirich, A., Devkota, K. & Bouazzama,
B. (2021). Phosphorus fertilization enhances productivity of forage corn
(Zea mays L.) Irrigated with Saline Water. Plants (Basel, Switzerland),
10(12), 2608. https://doi.org/10.3390/plants10122608
Box, G. and Behnken, D. (1960). Some new three-level designs for the study of
quantitative variables. Technometrics, 30 (1960), pp. 1-40.https://doi.org/
10.1080/00401706.1960.10489912
Erenstein, O., Jaleta, M., Sonder, K., Mottaleb, K., & Prasanna, B. M. (2022).
Global maize production, consumption and trade: trends and R&D
implications. Food Security, 14, 1295–1319. https://doi.org/10.1007/
s12571-022-01288-7
FAO-ESS. (2021). Estadísticas de Cultivos: Conceptos, Deniciones y
Clasicaciones. División de Estadística de la FAO (ESS), Roma.
https://www.fao.org/economic/the-statistics-division-ess/methodology/
methodology-systems/crops-statistics-concepts-definitions-and-
classications/en/.
Ganugi, P., Fiorini, A., Rocchetti, G., Bonini, P., Tabaglio, V., & Lucini, L. (2022).
A response surface methodology approach to improve nitrogen use
eciency in maize by an optimal mycorrhiza-to-Bacillus co-inoculation
rate. Frontiers in Plant Science, 13, 956391. https://doi.org/10.3389/
fpls.2022.956391
Gómez, N., Palemón, F., Reyes, G., Hernández, C., Cantú, M., Juárez, P.
&Ascencio, A. (2016). Rendimiento de grano y características fenotípicas
de maíz: efecto de ambiente y dosis de fertilización. Revista Mexicana De
Ciencias Agrícolas, 7(8), 1801–1813. https://doi.org/10.29312/remexca.
v7i8.93
Guo, J., Wang, Y., Fan, T.; Chen, X. & Cui, Z. (2016). Designing corn management
strategies for high yield and high nitrogen use eciency. Agronomy
Journal 108(2):922-929. https://doi.org/10.2134/agronj2015.0435
Jun, X., Rui-Zhi, X., Wang-Feng, Z., Ke-Ru, W., Peng, H., Bo, M., Ling, G. &
Shao-Kun, L. (2017). Research progress on reduced lodging of high-yield
and -density maize. Journal of Integrative Agriculture, 16(12):2717-
2725. https://doi.org/10.1016/S2095-3119(17)61785-4
Martínez U., Raúl A., Kolln, O. & de Castro, G. (2017). Evaluación de la densidad
de plantas, componentes fenológicos de producción y rendimiento de
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23-30. https://dx.doi.org/10.4067/S0718-34292017005000401
Table 5. Analysis of variance for the variable “number of grains
per square meter” (g.m
2
).
df
Sum of
squares
Mean
squares
Fc p
Model 9 215407 23934 4.95 0.0468
Density 1 8522 9522 1.97 0.2854
Nitrogen 1 32381 32381 6.69 0.0491*
Phosphorus 1 5555 6555 1.35 0.0818
Density*density 1 34416 34416 7.11 0.0445*
Nitrogen*nitrogen 1 116768 116768 24.13 0.0044*
Phosphorus*phosphorus 1 2725 2725 0.56 0.4879
Interactions 3 15040 5013 1.04 0.2491
Error 5 24200 4840
Lack of adjustment 3 18080 6027 1.97 0.6321
Pure error 2 6121 3060
TOTAL 14 239607
Adjustment: 89.79 %; * p< 0.05
Table 6. Adjusted analysis of variance for the variable “number
of grains per square meter” (g.m
2
).
df
Sum of
Squares
Middle
Squares Fc p
Modelo 3 183656 61219 12.02 0.0008*
Nitrógeno 1 32381 32381 6.36 0.0283*
Densidad*-
densidad
1 34416 34416 6.76 0.0247*
Nitrógeno*-
nitrógeno
1 116768 116768 22.92 0.0001*
Error 11 56042 5095
Total 14 239607
Adjustment: 93.5 %; *signicant dierence at 5 %.
Figure 2. Response surface for the response variable “grain”
(g.m
2
).
This scientic publication in digital format is a continuation of the Printed Review: Legal Deposit pp 196802ZU42, ISSN 0378-7818.
Montaña et al. Rev. Fac. Agron. (LUZ). 2023 40(4): e234035
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