The gummy candies were prepared following a formulation and process provided for academic purposes by a confectionery company in Mineral de la Reforma, Hidalgo, Mexico. Due to the nature of the project, citing the details of the process or the quantities of ingredients and food additives used was not permitted, except in the final stage related to the addition of acidifying agents. At this stage, for a mixture of water, refined sucrose, and corn syrup previously cooked at a specific temperature and combined with hydrated gelatin wholly dissolved, a liquor prepared with artificial orange flavoring, orange artificial coloring, water, and 15 g of an acid blend (citric, fumaric, and malic acids) was added, adjusted according to the simplex-lattice mixture design shown in Table 1. Finally, the mixture of all ingredients and food additives was homogenized and dosed into dropshaped silicone molds. The gummy candies were left to solidify and rest for a specified time, then de-molded, coated with refined sugar, and left to rest for an additional period. Finally, they were stored in airtight glass containers until used for sensory analysis.

TABLE 1. Experimental design and mass fraction of the three acids used in the formulation of orange-flavored gummy candies according to the simplex-lattice mixture design.

TABLE 1 TABLE 1 Experimental design and mass fraction of the three acids used in the formulation of orange-flavored gummy candies according to the simplex-lattice mixture design

	Uncoded values (fractions of the acids)			Coded values (g)
Index	x1	x2	x3	Citric acid	Malic acid	Fumaric acid
M1	1	0	0	15	0	0
M2	0.5	0.5	0	7.5	7.5	0
M3	0	1	0	0	15	0
M4	0.5	0	0.5	7.5	0	7.5
M5	0	0.5	0.5	0	7.5	7.5
M6	0	0	1	0	0	15
M7	0.75	0.25	0	11.25	3.75	0
M8	0.75	0	0.25	11.25	0	3.75
M9	0.5	0.25	0.25	7.5	3.75	3.75
M10	0.25	0.75	0	3.75	11.25	0
M11	0.25	0.5	0.25	3.75	7.5	3.75
M12	0.25	0.25	0.5	3.75	3.75	7.5
M13	0	0.75	0.25	0	11.25	3.75
M14	0.25	0	0.75	3.75	0	11.25
M15	0	0.25	0.75	0	3.75	11.25

The sensory analysis was conducted as described in previous research, with modifications [23,24,25]. A group of 30 trained panelists, all students of the Bachelor’s Degree in Food Chemistry at the Universidad Autónoma del Estado de Hidalgo, Mexico, conducted the sensory evaluation to identify the analyzed attributes (sourness and flavor). The samples were individually presented to evaluators on cardboard plates labeled with three-digit characters. The panelists provided scores from 1 to 5 for each attribute analyzed as follows: Sourness: 1 = very slightly intense, 2 = slightly intense, 3 = adequate, 4 = intense, and 5 = very intense. Flavor: 1 = dislike, 2 = neither like nor dislike, 3 = like slightly, 4 = like moderately, and 5 = like very much.

The number of scores for each point on the hedonic scale was multiplied by the corresponding coefficient, and the cumulative sum for each evaluated formulation was calculated to use the value as a response variable. Equation 1 represents this:

(1) y = ∑ i = 1 5 n i c i

where y is the response value (sourness, flavor, or combined level) n i is the number of scores for point i on the scale, and c i is the coefficient for point i on the scale.

All scripts used to produce the results of the analyses presented in this document were implemented using the R programming language (v.4.1.2, “Bird Hippie”) and the integrated development environment RStudio® (v.2024.04.2) on a computer running the elementary OS 7.1 Horus operating system (based on Ubuntu 22.04.3 LTS, Linux 6.5.0-44-generic). The libraries used included ‘mixexp’, ‘Ternary’, ‘viridisLite’, ‘PlotTools’, ‘stats’, and ‘dplyr’. All scripts are also provided as executable R files with their respective documentation, datasets, and model summaries. Everything has been uploaded to GitLab so that interested parties can access them for analysis and implementation

(https://gitlab.com/FoodChem-DataSci-Lab/orange-flavored-gummy-candies ).

First, an R file named “simplex_lattice_design.R” was created, and the script used to define and configure the mixture design was developed. This design was established according to the procedures described in previous research [26,27,28]. In the script, the mixexp library was used to set up a simplex-lattice mixture design with three factors: citric acid (x₁), malic acid (x₂), and fumaric acid (x₃), as well as two additional levels at zero using the SLD function (simplex_lattice_desing_1 <- SLD(fac = 3, lev = 2)). Subsequently, the design was enriched with interior points using the Fillv function (simplex_lattice_ design_2 <- Fillv(3, simplex_lattice_design_1)), ensuring complete coverage of the experimental space, as this configuration generated 15 different mixtures [14,15,29].

The factors were coded by multiplying the values by 15 to obtain the amounts of the acids in grams (simplex_lattice_desing_2$citric_acid <- 15 * simplex_lattice_ desing_2$x1, etc.). To visualize the combinations, the ‘TernaryPlot’ library was used to set up the ternary plot, and the AddToTernary function was used to add the points and experiment numbers, thereby generating an index of experiments within the plot (AddToTernary(graphics::points, data_points, pch = 21, cex = 4, bg = “white”)). This plot can be seen in Figure 1. Finally, the design was exported to a CSV file (write. csv(simplex_lattice_design_2, file = “simplex_lattice_design_ORANGE.csv”, row.names = FALSE)), facilitating the subsequent analysis of experimental data and the optimization of the gummy candies’ sourness and flavor levels. The resulting experiment matrix can be seen in Table 1.

FIGURE 1 FIGURE 1 Arrangement of mixture points using the simplex-lattice design method in a ternary plot. FIGURE 1 Arrangement of mixture points using the simplex-lattice design method in a ternary plot.

FIGURE 1: Arrangement of mixture points using the simplex-lattice design method in a ternary plot.

The selection of the simplex-lattice mixture design was driven by its specific suitability for experiments where the response variables are functions of the proportions of the components in a mixture rather than their absolute quantities. The elements’ proportions (factors) are involved in mixture designs, and the mix’s total amount is usually fixed. This means that the quantity of each component is proportional to the total mixture, and unlike in factorial designs, the proportions of the elements cannot vary independently since they are constrained by the requirement that their sum must be constant (1 or 100%). This interdependency is a critical characteristic that distinguishes mixture designs from other approaches like Central Composite Design or Box-Behnken Design, where factors can vary independently. The analysis in mixture designs is conducted through a response surface, which allows for the identification of the optimal mixture [30,31]. This approach is particularly appropriate for optimizing formulations in food science, where the balance of ingredients directly influences the product’s sensory attributes. Previous studies have demonstrated the effectiveness of this design in similar contexts, supporting its application in this work [24,28,29,30,31]. The advantages of using a simplexlattice mixture design include a comprehensive exploration of the experimental space with fewer trials and the ability to capture complex interactions between components [32], thereby providing a robust framework for optimizing the sensory properties of the orange-flavored gummy candies.

The file “combined_model.R” was created to develop a combined model integrating the sourness and flavor levels in orange-flavored gummy candies. The goal was to determine the optimal combination of citric acid (x1), malic acid (x2), and fumaric acid (x3) to maximize both responses. The following describes each stage of the process, including the code used and an explanation of its purpose and functionality. This script has been uploaded to GitLab as an executable R file in the project directory.

Initially, necessary libraries such as ‘mixexp’, ‘Ternary’, ‘viridisLite’, ‘PlotTools’, ‘stats’, and ‘dplyr’ were loaded. The datasets containing experimental results for sourness and flavor were imported using read.csv(“sourness.csv”) and read.csv(“flavor.csv”), respectively.

Separate linear regression models were developed for sourness and flavor levels using the lm function, considering main effects and interactions among acids (sourness_model <- ... and flavor_model <- ...), excluding the intercept [28,33]. A combined dataset was created by merging the sourness and flavor datasets and calculating a combined response variable (combined_dataset <- …). The combined response variable y_combined is calculated as the average of y_sourness and y_flavor, giving equal weight to both sensory properties. This combined measure provided a holistic view of the product’s sensory profile.

The combined model was then developed using the lm function (combined_model_ lm <- lm(y_combined ~ x1 + x2 + x3 + x1:x2 + x1:x3 + x2:x3 + x1:x2:x3 -1, data = combined_dataset)), excluding the intercept, and the summary was saved to a text file (“summary_combined_model.txt”). A function, combined_model, was defined to predict responses based on the combined model (combined_model <- function(a, b, c) ... ). It uses three arguments, a, b, and c, representing the proportions of acids to create a new data frame. Then, the predict function was used to generate predictions from the combined_model_lm model with this new data set, returning the predicted result [26].

For visualization, the EffPlot function was used to create an effects plot, saved as a PNG file. A ternary plot was generated to illustrate the response surface, including contour lines representing the combined response (TernaryPlot(alab = “Citric acid (x1)”, ... ) [14,15]. The optimal combination of x1, x2, and x3, yielding the highest combined response, was identified using a grid search approach (predict_best_combined_mixture()), with results saved to a text file (“optimal_combined_level_results.txt”). The optimal point was also highlighted in the ternary plot (AddToTernary(graphics::points, matrix(unlist(optimal_values), ncol = 3), pch = 19, col = “red”, cex = 2)). This analysis comprehensively evaluated the optimal mixture to balance sourness and flavor in the orange-flavored gummy candies.

The sourness level model demonstrated high accuracy, as indicated by the Multiple R-squared value of 0.9926 and an adjusted R-squared of 0.9926 (Table 2). These values suggested that the model explained approximately 99.26% of the variance in the sourness response, highlighting the model’s predictive solid capability. The coefficients for the primary factors, citric acid (x₁), malic acid (x₂), and fumaric acid (x₃), were all statistically significant, with p-values below 0.05. The coefficient estimates were 80.759 for x₁, 89.474 for x₂, and 90.545 for x₃, indicating that these acids contributed significantly to the sourness level. However, the interaction terms, including x₁:x₂, x₁:x₃, x₂:x₃ and x₁:x₂:x₃, were not statistically significant, suggesting that the combined effects of these acids did not significantly influence the sourness beyond their individual contributions. The model’s residual standard error was 11.140, reflecting the average deviation of the observed data points from the fitted model, and the F-statistic of 153.4 indicated a highly significant model overall, with a p-value of 6.919×10⁻⁸.

Table 2. Model coefficients and statistical summary for sourness level, flavor level, and combined levels models.

Table 2 Table 2 Model coefficients and statistical summary for sourness level, flavor level, and combined levels models. Table 2 Model coefficients and statistical summary for sourness level, flavor level, and combined levels models.

Model	Coefficients	Standard Error	R2	R2adjusted	Residual Standard Error	p-value	F-statistic
Sourness level	x1: 80.759 x2: 89.474 x3: 90.545 x1:x2: 42.836 x1:x3: 26.550 x2:x3: 40.836 x1:x2:x3: 40.296	x1: 9.905 x2: 9.905 x3: 9.905 x1:x2: 46.054 x1:x3: 46.054 x2:x3: 46.054 x1:x2:x3: 286.911	0.9926	0.9926	11.140	6.919×10−8	153.4
Flavor level	x1: 96.045 x2: 113.545 x3: 107.688 x1:x2: 3.122 x1:x3: 43.693 x2:x3: −48.593 x1:x2:x3: 216.296	x1: 9.702 x2: 9.702 x3: 9.702 x1:x2: 45.113 x1:x3: 45.113 x2:x3: 45.113 x1:x2:x3: 281.047	0.9945	0.9945	10.910	2.119×10−8	206.7
Combined level	x1: 88.402 x2: 101.509 x3: 99.116 x1:x2: 22.979 x1:x3: 35.122 x2:x3: −3.878 x1:x2:x3: 128.296	x1: 6.112 x2: 6.112 x3: 6.112 x1:x2: 28.417 x1:x3: 28.417 x2:x3: 28.417 x1:x2:x3: 177.036	0.9975	0.9975	6.872	8.845×10−10	459.5

The flavor model also exhibited strong performance, with a Multiple R-squared value of 0.9945 and an adjusted R-squared of 0.9945, indicating that 99.45% of the variance in the flavor response was explained by the model. The coefficient estimates for citric acid (x₁), malic acid (x₂), and fumaric acid (x₃) were 96.045, 113.545, and 107.688, respectively. These values demonstrated that malic acid had the highest influence on flavor, followed by fumaric and citric acids. The standard errors for these coefficients were relatively small, suggesting precise estimates. Unlike the sourness model, the flavor model showed a significant interaction term for x₁:x₃, with a coefficient estimate of 43.693, indicating a notable combined effect of citric and fumaric acids on flavor. The model’s residual standard error was 10.910, slightly lower than the sourness model’s, indicating a better fit. The F-statistic was 206.7, with a p-value of 2.119×10⁻⁸, confirming the overall statistical significance of the model.

The combined model, which integrated sourness and flavor responses, achieved the highest accuracy among the three models. It had a Multiple R-squared value of 0.9975 and an adjusted R-squared of 0.9975, suggesting that the model explained 99.75% of the variance in the combined response. The coefficient estimates for citric acid (x₁), malic acid (x₂), and fumaric acid (x₃) were 88.402, 101.509, and 99.116, respectively, indicating that malic acid was the dominant contributor to the combined response, followed closely by fumaric and citric acids. The interaction term x₁:x₂ was also notable, with a coefficient estimate of 128.296, suggesting a significant three-way interaction effect on the combined response. The residual standard error of 6.872 was the lowest among the models, indicating an exact fit. The F-statistic was 459.5, with a p-value of 8.845×10⁻¹⁰, underscoring the solid overall significance of the combined model.