Lemon-Flavored Gummy Candies: Sourness, Flavor and Overall Acceptance Optimization Using Lattice-Simplex Mixture Design Implemented with Python Programming Language ^†

Abstract

Acidulants enhance the tartness and flavor of gummy candies. This study employed Python, a free, open-source tool, to perform a simplex–lattice mixture design, optimizing the sourness, flavor, and overall acceptance of lemon-flavored gummies. Citric, malic, and fumaric acids were evaluated, and a combined model measured their sensory attributes. The results showed high R² values and significant effects for individual and interaction terms. The optimal mixture (5.85 g citric, 4.8 g malic, 4.35 g fumaric acids) achieved a combined sensory score of 100.11. This study highlights Python’s utility in product development, providing an accessible resource for researchers and food developers.

1. Introduction

Gummy candies are confectionery products primarily made with saturated refined sucrose and corn syrup solutions, though polyols are sometimes used as a substitute for sucrose. These ingredients are mixed and heated to a cooking temperature between 106 and 120 °C, after which prehydrated gelatin is added. The mixture is then combined with colorants, flavorings, and acidulants. The resulting mixture is poured into molds, commonly made of starch, to shape the gummies. Once unmolded, they are coated with a mix of refined sucrose and acidulants, chocolate, powdered chili, or glazing agents such as beeswax, carnauba wax, or palm oil [1,2,3].

Acidulants are added to gummies and candies to enhance flavors by intensifying certain desired notes, making them stronger and longer lasting [4]. They are also added to impart a tart and tangy taste. Citric acid is the most commonly used food acid because it causes minimal degradation of food components. Other food acids, such as malic acid, create different flavor profiles compared to citric acid [5].

Mixture design is a statistical tool used in the food industry and research to combine different ingredients to develop or modify products and achieve specific and optimal characteristics. The simplex–lattice mixture design (SLMD) has been successfully applied to optimize food formulations [6].

Python is ideal for analyzing and optimizing food formulations due to its free, open-source accessibility. It helps bridge the digital divide between research institutions and food companies of various sizes and economic capacities that cannot afford proprietary software. Python is a powerful and flexible tool with libraries like ‘NumPy’, ‘SciPy’, ‘Pandas’, ‘Scikit-learn’, and ‘Matplotlib’ that facilitate complex analyses and optimization. Its reliability is supported by an active developer community that provides ongoing support and updates. Additionally, Python integrates seamlessly with other tools and platforms, enhancing collaboration and efficiency in research and development projects.

While previous studies have utilized mixture design techniques to optimize food products, this study stands out by integrating computational modeling with sensory evaluation through Python programming. Unlike traditional methods that rely on proprietary statistical software, this research demonstrates the potential of open-source tools in food product development. The novelty lies in the comprehensive application of Python for experimental design and data analysis, which showcases it as a cost-effective, flexible, and reproducible approach. This approach opens new possibilities for innovative applications in food science.

Based on all the above, the objective of this contribution was to develop a script in the Python programming language using the simplex–lattice mixture design to optimize the pleasantness of some lemon-flavored gelatin gummies to provide a practical case study that demonstrates how these tools have the potential to improve the sensory properties of products in the confectionery industry.

2. Materials and Methods

2.1. Gummy Candies Production

For academic purposes, the gummy candies were produced by a confectionery company located in Mineral de la Reforma, Hidalgo, Mexico. The company requested anonymity to protect its trade practices. Due to the nature of the project, the specific quantities of ingredients and food additives used (except for acidulants) and the cooking times and temperatures were not disclosed. However, the process is described in general terms for context:

A candy mixture was prepared by heating sucrose, 42 DE glucose syrup, and water to a specific temperature. After cooling, prehydrated pork gelatin was added and stirred continuously until a uniform mixture was achieved. Subsequently, a liquor made with lemon essence, green colorant, water, and 15 g of an acid blend (citric, fumaric, and malic acids) was incorporated and adjusted based on the simplex–lattice mixture design in

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

x1	x2	x2	Mix Index	Citric Acid (g) *	Malic Acid (g) *	Fumaric Acid (g) *
0	0	1	1	0	0	15
0	0.25	0.75	2	0	3.75	11.25
0	0.5	0.5	3	0	7.5	7.5
0	0.75	0.25	4	0	11.25	3.75
0	1	0	5	0	15	0
0.25	0	0.75	6	3.75	0	11.25
0.25	0.25	0.5	7	3.75	3.75	7.5
0.25	0.5	0.25	8	3.75	7.5	3.75
0.25	0.75	0	9	3.75	11.25	0
0.5	0	0.5	10	7.5	0	7.5
0.5	0.25	0.25	11	7.5	3.75	3.75
0.5	0.5	0	12	7.5	7.5	0
0.75	0	0.25	13	11.25	0	3.75
0.75	0.25	0	14	11.25	3.75	0
1	0	0	15	15	0	0

* The mass fractions of citric acid, malic acid, and fumaric acid were adjusted according to the simplex–lattice mixture design to optimize the sourness, flavor, and overall acceptance of the lemon-flavored gummy candies.

. This amount of acid ensured a concentration of 10 g per kg of product, within the recommended range of 2 to 10 g per kg for gummy candies, depending on the flavoring used.

The mixture of all ingredients and food additives was homogenized and poured into molds. The gummies were allowed to solidify and rest for a specific period, and were then de-molded, humidified, coated with refined sugar, and left to rest for an additional period. Finally, they were stored in airtight glass containers until they were used in the sensory analysis.

2.2. Sensory Analysis

The sensory analysis was conducted as described in previous studies, with some modifications [7,8]. The evaluation was performed by 30 trained panelists, all students from the Food Chemistry undergraduate program at the Autonomous Universidad Autónoma del Estado de Hidalgo, Mexico. The panelists were trained to assess the analyzed attributes (sourness, flavor, and overall acceptance). Samples were presented individually to the evaluators on cardboard plates labeled with three-digit codes. The panelists assigned scores from 1 to 5 for each attribute 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.
• Overall acceptance: 1 = dislike; 2 = neither like nor dislike; 3 = like slightly; 4 = like moderately; and 5 = like very much.

2.3. Python Scripts

All scripts used to produce the analysis results presented in this paper were implemented using the Anaconda suite (base: Python 3.12.2) and Visual Studio Code (v.1.91.1) as the integrated development environment 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).

All scripts are provided as executable Jupyter Notebooks, along with datasets and model summaries, and are available from this GitLab repository: https://gitlab.com/FoodChem-DataSci-Lab/lemon-flavored-gummy-candies (accessed on 1 December 2024). These resources have been made publicly accessible to allow researchers, students, food developers, and other interested individuals to download, analyze, and implement the materials in their own projects.

2.3.1. Experimental Design

A Python script was developed and is documented in the “simplex-lattice.ipynb” file. The code was used to create simplex–lattice mixture designs to analyze the impact of different component ratios on responses (sourness, flavor, and overall acceptance levels), using the statistical framework reported in previous studies [8,9,10]. The levels for the factors ‘x₁’, ‘x₂’, and ‘x₃’ (citric, malic, and fumaric acids) were set at [0, 0.25, 0.5, 0.75, 1] (‘levels = [0, 0.25, 0.5, 0.75, 1]’). Combinations of these levels that summed to one were generated (‘np.isclose(x₁ + x₂ + x₃, 1)’) and stored in a pandas data frame (‘df = pd.DataFrame(combinations, columns = factors)’). Each combination received a unique index (‘df[‘mix_index’] = range(1, len(df) + 1)’), and the amounts of acid (in g) were calculated by multiplying the factor values by 15.

A ternary plot was created using the ‘mpltern’ library to visualize these combinations (‘ax.scatter(df[‘x₁’], df[‘x₂’], df[‘x₃’], facecolors = ‘black’, edgecolors = ‘black’, s = 20)’). Points were labeled with their mixture index for clarity (‘ax.text(row[‘x₁’], row[‘x₂’], row[‘x₃’], str(int(row[‘mix_index’])), fontsize = 10, ha = ‘center’, va = ‘bottom’, color = ‘blue’)’). The data were saved in a CSV file (‘df.to_csv(‘simplex-lattice-mixture-design.csv’, index = False)’) and as a high-resolution PNG image (‘plt.savefig(‘ternary-diagram-mixture-design.png’, dpi = 300, bbox_inches = ‘tight’)’) with both elements shown in Table 1 and Figure 1, respectively. This process aided in optimizing formulations and provided a clear visual representation for further analysis.

2.3.2. Python Script for Analyzing the Results

The following script was adapted to analyze sourness, flavor, and overall acceptance levels. However, the explanation focused on the analysis of the flavor level. The script for modeling flavor was saved in a file named “flavor.ipynb”. Initially, the dataset containing the mixture components (‘x₁’, ‘x₂’, ‘x₃’) and the response (‘y’) was loaded using pandas (‘data = pd.read_csv(‘flavor.csv’)’). Interaction terms, including two-way and three-way interactions (‘X[‘x₁:x₂’] = X[‘x₁’] × X[‘x₂’]’, etc.), were added. A linear regression model without an intercept was fitted using the ‘statsmodels’ library to estimate the effects of the components (‘model = sm.OLS(y, X, hasconst = False).fit()’). The model’s coefficients were saved, and a function was defined to predict the flavor level based on these coefficients (‘def flavor_level(x, coefficients):…’). A grid of possible mixtures was created, and predictions were made to identify the optimal mixture (‘optimal_mixture = None, max_level = -np.inf’).

A ternary plot was generated using the ‘mpltern’ library to visualize the predicted responses across different mixtures, and the plot was saved as a high-resolution PNG image (‘plt.savefig(“Ternary_plot_flavor.png”, dpi = 300)’). The second part of the code calculated the centroid of the design (‘centroid_x₁ = data[‘x₁’].mean()’, etc.), included quadratic terms, and used an adjusted model to create an effect plot showing the impact of deviations from the centroid on the response (‘plt.savefig(“ModelEff_plot_flavor.png”, dpi = 300)’). This visualization illustrated the influence of each component on the predicted flavor level.

2.3.3. Combined Response Analysis for Sourness, Flavor, and Overall Acceptance

A Python script was developed and saved in a file named “combined-model.ipynb”. The script was designed to combine and analyze datasets for sourness, flavor, and overall acceptance, aiming to find an optimal mixture of components. The datasets were loaded using the ‘pandas’ library (‘sourness_dataset = pd.read_csv(‘sourness.csv’)’, etc.), and interaction terms were added to capture the combined effects of the components (‘df[‘x₁:x₂’] = df[‘x₁’] × df[‘x₂’]’, etc.). Separate regression models were fitted for each dataset using the ‘statsmodels’ library (‘model = sm.OLS(y, X, hasconst = False).fit()’), capturing the influence of the factors and their interactions.

A combined dataset was then created, including all three response variables and a new variable representing the average combined response (‘combined_dataset[‘y_combined’] = (combined_dataset[‘y_sourness’] + combined_dataset[‘y_flavor’] + combined_dataset[‘y_overall_acceptance’])/3’). This dataset was also enriched with interaction terms, and a combined model was fitted to predict the average response (‘combined_model = sm.OLS(y_combined, X_combined, hasconst = False).fit()’). The model’s coefficients were saved, and a function was created to predict responses based on component ratios (‘def combined_model_predict(x₁, x₂, x₃):…’). The optimal mixture was identified by evaluating predictions across a grid of possible mixtures, and the best combination was recorded.

Visualizations included a ternary plot of the predicted responses (‘ax.tricontour(grid[:, 0], grid[:, 1], grid[:, 2], predicciones, levels = sorted(levels))’) and an effect plot showing the impact of deviations from the centroid on the response (‘ax.plot(x_vals, y_x₁, label = ‘x₁’, color = ‘black’)’). These plots were saved as high-resolution images for further analysis. The script provided a comprehensive method for evaluating the combined effects of sourness, flavor, and overall acceptance in mixtures, facilitating optimization and visualization of the best formulation.

3. Results and Discussion

The comparative analysis of the polynomial models for sourness, flavor, overall acceptance, and combined attributes in lemon-flavored gummy candies is shown in

Table 2. Comparative analysis of polynomial models for sourness, flavor, overall acceptance, and combined attributes in lemon-flavored gummy candies.

Response	Polynomial Equation	R2	R2-Adjusted	F-Statistic	AIC *	BIC *
Sourness level	y = 79.53x1 + 92.96x2 + 71.74x3 + 0.02x1x2 + 34.02x1x3 − 8.26x2x3 + 295.70x1x2x3	0.995	0.990	208.5	111.798	116.754
Flavor level	y = 83.70x1 + 87.20x2 + 92.77x3 − 19.27x1x2 + 3.02x1x3 − 26.41x2x3 + 145.78x1x2x3	0.997	0.994	385	102.974	107.930
Overall Acceptance level	y = 109.47x1 + 117.75x2 + 103.39x3 + 18.77x1x2 + 27.63x1x3 − 21.51x2x3 + 81.19x1x2x3	0.993	0.987	163.2	123.807	128.763
Combined level	y = 90.90x1 + 99.30x2 + 89.30x3 − 0.16x1x2 + 21.56x1x3 − 18.73x2x3 + 174.22x1x2x3	0.998	0.997	624	98.484	103.440

* AIC = Akaike Information Criterion; BIC = Bayesian Information Criterion.

. Each model provides a unique perspective on how different acids (x₁, x₂, x₃) and their interactions influence the sensory characteristics and acceptability of the products. The models show high levels of fit, as indicated by the R² values, which are all above 0.99. This suggests that the chosen variables and interactions effectively capture the variance in the observed data.

In the sourness level model, the coefficients for the main effects of x₁ (79.53), x₂ (92.96), and x₃ (71.74) are all significant and positive, indicating strong individual contributions of citric, malic, and lactic acids to perceived sourness. The interaction terms, particularly the three-way interaction x₁:x₂:x₃ (295.70), also show a positive contribution, although it is not statistically significant (p = 0.22). This suggests a potential synergistic effect that aligns with the sensory impact of these acids on sourness [5].

For the flavor level model, the main effects of x₁ (83.70), x₂ (87.20), and x₃ (92.77) are highly significant, demonstrating that each acid independently enhances flavor perception. However, the interaction terms, including x₁:x₂ (−19.27) and x₁:x₃ (3.02), have smaller coefficients and are not significant, indicating minimal synergistic effects. The three-way interaction term x₁:x₂:x₃ (145.78) shows a large coefficient but lacks statistical significance (p = 0.40). This pattern implies that each acid primarily influences distinct flavor notes, contributing to additive rather than synergistic effects.

In the overall acceptance model, the coefficients for x₁ (109.47) and x₂ (117.75) are substantial and significant, reflecting their strong positive impact on consumer preference. The coefficient for x₃ (103.39) is slightly lower but still significant, highlighting its role in acceptance. The interaction terms, such as x₁:x₂ (18.77) and x₁:x₂:x₃ (81.19), show lower coefficients and lack significance, indicating that combined effects among the acids are less influential on overall acceptance than individual contributions.

Finally, the combined response model incorporates data from the three individual models (sourness, flavor, and overall acceptance) to identify the optimal mix of x₁, x₂, and x₃. The model’s high R² value and significant individual effect coefficients confirm the critical roles of citric, malic, and lactic acids in shaping the sensory profile. The three-way interaction term (positive in all models but not significant) suggests a consistent, albeit modest, combined effect across sensory attributes. This reinforces the nuanced interplay between these acids in creating a balanced sensory experience.

Figure 2 shows the contour and effect (Piepel direction) plots, which illustrate the relationships among the factors citric acid (x₁), malic acid (x₂), and fumaric acid (x₃) concerning sourness level (Figure 2a), flavor level (Figure 2b), overall acceptance level (Figure 2c), and a combined metric of these attributes (Figure 2d). Figure 2d aggregates the sourness, flavor, and overall acceptance levels results, aiming to find the best mix of x₁, x₂, and x₃ to maximize the overall sensory qualities of the gummies. Together, these plots provide a comprehensive analysis for optimizing product formulation to achieve a balanced sensory profile.

The combined model analysis identified the optimal mixture of acids to achieve the highest overall sensory qualities of the lemon-flavored gummy candies. The best combination was determined to be 5.85 g of citric acid, 4.8 g of malic acid, and 4.35 g of fumaric acid. This formulation resulted in a combined score (y) of 100.11 for sourness, flavor, and overall acceptance. The combined model analysis identified the optimal mixture of acidulants to achieve the highest overall sensory qualities of the lemon-flavored gummy candies. The ideal combination was 5.85 g of citric acid, 4.8 g of malic acid, and 4.35 g of fumaric acid. This formulation yielded a theoretical combined score (y) of 100.11 for sourness, flavor, and overall acceptance, indicating that this specific ratio could achieve the desired balance of sourness, flavor intensity, and consumer acceptance. Confirmatory experiments are recommended for future research to validate these models’ effectiveness. This study’s primary aim was to develop and implement the necessary Python scripts to generate simplex–lattice mixture designs and subsequently analyze the results, providing an alternative to R and proprietary software. This approach offers a valuable tool for researchers, students, food developers, and others interested in the field.

These models highlighted the importance of understanding both the individual and interactive effects of acidulants in lemon-flavored gummy candies, as they can significantly influence sensory attributes and consumer acceptance. This information is valuable for formulating and optimizing similar products, helping to tailor products to specific taste profiles and consumer preferences.

4. Conclusions

This study successfully utilized Python to optimize the flavor profile of lemon-flavored gummy candies using a simplex–lattice mixture design. The main findings demonstrated the optimal combination of acidulants, highlighting the effectiveness of the combined model in fine-tuning sensory attributes. Python proved advantageous due to its accessibility and flexibility, offering a cost-effective alternative to proprietary software. This approach provided an open-source tool for researchers, students, and food developers and supports more inclusive innovations in product development. Future confirmatory experiments are recommended to validate the robustness of these models. The findings underscore the importance of understanding the individual and interactive effects of acidulants in confectionery products, paving the way for tailored product optimization. This research is of great interest to both academia and industry, providing valuable insights and practical tools for enhancing food product development.

Future research could expand the application of this methodology to a wider range of confectionery products, including jellies, marshmallows, and hard candies, to optimize sensory attributes beyond acidity and flavor. Moreover, integrating advanced computational tools, such as machine learning algorithms, could enhance predictive modeling capabilities, enabling more precise formulation optimization. Exploring alternative mixture designs implemented in Python and applying multi-objective optimization approaches would also be valuable for addressing complex formulation challenges that require balancing multiple sensory or functional attributes. Python’s versatility in managing large datasets and performing complex analyses further presents opportunities for real-time quality control and automated formulation adjustments in industrial settings. These potential research directions highlight the importance of computational tools in advancing food science and driving innovation in product development.