Modeling of recipes of special purpose bakery products

Ainash Rustemova; Nurudin Kydyraliev; Tatyana Kirillova; Madina Sadygova; Nurgul Batyrbayeva

doi:10.1051/bioconf/20202700017

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Volume 27 (2020)

BIO Web Conf., 27 (2020) 00017

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Issue		BIO Web Conf. Volume 27, 2020 International Scientific-Practical Conference “Agriculture and Food Security: Technology, Innovation, Markets, Human Resources” (FIES 2020)


Article Number		00017
Number of page(s)		6
DOI		https://doi.org/10.1051/bioconf/20202700017
Published online		25 November 2020

BIO Web of Conferences 27, 00017 (2020)

Modeling of recipes of special purpose bakery products

Ainash Rustemova¹^*, Nurudin Kydyraliev², Tatyana Kirillova³, Madina Sadygova³ and Nurgul Batyrbayeva¹

¹ Аlmaty Technological University, 050012 Almaty, Kazakhstan
² Kyrgyz-Turkish Manas University, 720038 Bishkek, Kyrgyzstan
³ Saratov State Agrarian University named after N.I. Vavilov, 410012 Saratov, Russia

^* Corresponding author: aist_2707@mail.ru

Abstract

In this article the results of calculations of experimental-statistical methods are given, using the optimized contents of components in “Puff buns” and “Grain bars” recipes. The proposed products are balanced in the content of nutrients (proteins, fats, carbohydrates), contain a wide range of vitamins and essential amino acids, which allows expanding the consumer food market and more efficiently using vegetable resources. In order to obtain the percentage ratio of the components in the flour products composition, the complete factor experiment – a central compositional rotatable planning experiment – was applied. According to the results of mathematical modeling for “Puff buns”, the optimal percentage ratio of rice bioflour and pumpkin was chosen as 20:20, for “Grain bar” the optimal dosage of mashed bananas and Jerusalem artichoke is 28:12 respectively.

© The Authors, published by EDP Sciences, 2020

This is an Open Access article distributed under the terms of the Creative Commons Attribution License 4.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1 Introduction

Food products design is a process of creation rational recipes that can provide a high level of adequacy of the complex of properties of food products to the consumers’ requirements and to the standardized values of nutrients and energy content [1-7]. According to the modern views, the term design of products includes the development of models that describes steps of products creation using the given quality. It also represents mathematical dependence of changing one or several key parameters on which base they are being developed, including the optimization of choice and the ratio of components [8-11].

Purpose of the study – ingredients’ optimization in bakery products’ recipe – “Puff buns”, “Grain bars” which by quantitative content and quality composition will match the formula of balanced diet, as much as possible, meet biomedical requirements and own high consumer properties.

2 Materials and Research Methods

The optimization of the prescribed composition of “Puff buns”, “Cereal bars” was performed by experimental statistical methods. To construct the mathematical models of function dependency, the response of input factors, we used central compositional rotable planning of the experiment (CCRP).

The regression equation with two variables in central compositional rotable planning is presented as the second order equation: $Y = b_{o} + b_{1} X_{1} + b_{2} X_{2} + b_{12} X_{1} X_{2} + b_{11} X_{1}^{2} + b_{22} X_{2}^{2}$ $Y = b_{o} + b_{1} X_{1} + b_{2} X_{2} + b_{12} X_{1} X_{2} + b_{11} X_{1}^{2} + b_{22} X_{2}^{2}$ (1)

The number of experiments using CCRP in the two-factor experiment (n = 2) equals N=13. To the full factor experiment 2² let us add 5 experiments in centre of the plan and four “star” points with coordinates (+a; 0); (- a;0); (0;+a); and (0; -a).

Using CCRP for the regression equations coefficients (1) and relevant marks of dispersions, we use formulas from the tutorial [12].

3 Results and Discussion

The grafic interpritation of the regression equation is response surface. Analysis of two-dimensional sections of response surface allows preliminarily determining areas of factor space, where we achieved an optimal value of output parameters. Figures 1-16 show a graphic picture of regression models as paraboloids and a two-dimensional section of paraboloids.

The comparison of estimated values of the Student criterion with the table at a significance level of α = 0,05 and the number of degrees of freedom N(n₀ -1)=16 (t_m=1.7459) allows choosing significant regression coefficients.

Table values of the Fisher criterion at a significance level of α = 0,05 and the number of degrees of freedom of numerator ${\ddot{f}}_{1} = 3$ $\ddot{f}_{1} = {3}$ and denominator ${\ddot{f}}_{2} = 4$ $\ddot{f}_{2} = {4}$ and equal F_T = 6.59. The comparison of the Fisher criterion estimated value and table shows that regression equations are adequate to experimental data.

Built regression models may be used for recipe optimization of «Cereal bars», «Puff buns».

For the ratio of components determination in the «Cereal bars» recipe, let`s build a mathematical model of optimization by four criteria: $\begin{matrix} 6, 00 + 0, 45 X_{2} + 0, 39 X_{1}^{2} + 0, 24 X_{2}^{2} \to m i a x; \\ 6, 03 + 0, 4 X_{1} + 0, 34 X_{2} - 0, 28 X_{1}^{2} - 0, 29 X_{2}^{2} \to max; \\ 28, 18 + 0, 9 X_{1} + 0, 52 X_{1}^{2} + 1, 479 X_{2}^{2} \to max x; \\ 94, 35 - 3, 55 X_{1} + 0, 39 X_{1} X_{2} - 12, 342 X_{1}^{2} - 10, 63 X_{2}^{2} \to max \\ while limiting X_{1}^{2} + X_{2}^{2} \leq 2; \end{matrix}$ $\begin{gathered} {6},00 + 0,{45}X_{{2}} + 0,{39}X_{1}^{{2}} + 0,{24}X_{2}^{{2}} \to miax; \hfill \\ {6},0{3} + 0,{4}X_{{1}} + 0,{34}X_{{2}} - 0,{28}X_{1}^{{2}} - 0,{29}X_{2}^{{2}} \to {\text{max}}; \hfill \\ {28},{18} + 0,{9}X_{{1}} + 0,{52}X_{1}^{{2}} + {1},{479}X_{2}^{{2}} \to {\text{max}}\,x; \hfill \\ {94},{35} - {3},{55}X_{{1}} + 0,{39}X_{{1}} X_{{2}} - {12},{342}X_{1}^{{2}} - {1}0,{63}X_{2}^{{2}} \to {\text{max}} \hfill \\ \text{while limiting } X_{1}^{{2}} { + }X_{2}^{{2}} \le {2}; \end{gathered}$

here X₁, X₂ – coded values of factors, related with natural values xi by ratios: $X_{1} = \frac{x_{1} - 10, 5}{5, 5}; X_{2} = \frac{x_{2} - 26}{6} .$ $X_{1} = \frac{{x_{1} - 10,5}}{5,5};\,\,\,\,\,\,\,\,\,\,X_{2} = \frac{{x_{2} - 26}}{6}.$

For determination of the components ratio in the recipe of «Puff buns» let`s build a mathematical model of optimization by four criteria: $\begin{matrix} 2, 40 + 0, 12 X_{1} - 0, 11 X_{1}^{2} - 0, 10 X_{2}^{2} \to max; \\ 3, 20 + 0, 27 X_{2} - 0, 11 X_{1}^{2} - 0, 12 X_{2}^{2} \to max; \\ 88, 10 + 1, 36 X_{1} + 2, 29 X_{1} - 2, 41 X_{1}^{2} - 0, 55 X_{2}^{2} \to max x; \\ 89, 84_{1} + 3, 63 X_{1} + 5, 23 X_{1} - 8, 49 X_{1}^{2} - 6, 79 X_{2}^{2} \to max \\ while limiting X_{1}^{2} + X_{2}^{2} \leq 2; \end{matrix}$ $\begin{gathered} 2,40 + 0,12X_{1} - 0,11X_{1}^{2} - 0,10X_{2}^{2} \to \max ; \hfill \\ 3,20 + 0,27X_{2} - 0,11X_{1}^{2} - 0,12X_{2}^{2} \to \max ; \hfill \\ 88,10 + 1,36X_{1} + 2,29X_{1} - 2,41X_{1}^{2} - 0,55X_{2}^{2} \to \max x; \hfill \\ 89,84_{1} + 3,63X_{1} + 5,23X_{1} - 8,49X_{1}^{2} - 6,79X_{2}^{2} \to \max \hfill \\ \text{while limiting } X_{1}^{{2}} { + }X_{2}^{{2}} \le {2}; \end{gathered}$

here X₁, X₂ – coded values of factors, related with natural values xi by ratios: $X_{1} = \frac{x_{1} - 15}{5}; X_{2} = \frac{x_{2} - 15}{5} .$ $X_{1} = \frac{{x_{1} - 15}}{5};\,\,\,\,\,\,\,\,\,\,X_{2} = \frac{{x_{2} - 15}}{5}.$

The task of multi-criteria optimization by the target programming method transformation into a mono-criteria task of minimization of the number of deviations with some indicator p: $G = {(\sum_{k = 1}^{k} {w_{k} | \frac{f_{k} (x, y, z) - {\bar{f}}_{k}}{{\bar{f}}_{k}} |}^{p})}^{\frac{1}{p}} \to min,$ $G = \left( {\sum\limits_{k = 1}^{k} {w_{k} \left| {\frac{{f_{k} \left( {x,y,z} \right) - \overline{f}_{k} }}{{\overline{f}_{k} }}} \right|}^{p} } \right)^{\frac{1}{p}} \to \min ,$ (2)

where w_k- some weight coefficients that characterize the importance of one or other criterion, ${\bar{f}}_{1}, {\bar{f}}_{2}, . . ., {\bar{f}}_{K}$ $\overline{f}_{{1}} ,\overline{f}_{{2}} ,...,\overline{f}_{K}$ – values of target functions on the optimal plan for each criteria, p – parameter, k - number of target functions.

At p=2 and $w_{k} = 1$ $w_{k} = 1$ we get the following task of minimization of the amount and limitations: $G = {(\sum_{k = 1}^{4} {w_{k} | \frac{f_{k} (x, y, z) - {\bar{f}}_{k}}{{\bar{f}}_{k}} |}^{2})}^{\frac{1}{2}} \to min X_{1}^{2} + X_{2}^{2} \leq 2;$ $G = \left( {\sum\limits_{k = 1}^{4} {w_{k} \left| {\frac{{f_{k} \left( {x,y,z} \right) - \overline{f}_{k} }}{{\overline{f}_{k} }}} \right|}^{2} } \right)^{\frac{1}{2}} \to \min \,X_{1}^{2} + X_{2}^{2} \le 2;$

where ${\bar{f}}_{1}$ $\overline{f}_{{1}}$ – maximum of the first criterion, ${\bar{f}}_{2}$ $\overline{f}_{{2}}$ - maximum of the second criterion. ${\bar{f}}_{3}$ $\overline{f}_{{3}}$ – maximum of the third criterion, ${\bar{f}}_{4}$ $\overline{f}_{{4}}$ - maximum of the fourth criterion.

According to the results of the calculations, experimental and statistical methods optimized the contents of the components in the formulation of the developed products, which are confirmed by the results of trial laboratory baking.

The optimal solution of the single criterion task is point $x_{1} = 12, x_{2} = 28$ $x_{{1}} = {12 },x_{{2}} = {28}$ .

The optimal dose of banana and Jerusalem artichoke puree is 28% and 12% respectively.

The optimal solution of the single criterion task is point $x_{1} = 20, x_{2} = 20$ $x_{{1}} = { 2}0 \, ,x_{{2}} = { 2}0$ .

The optimal dose of rice flour and pumpkin puree is 20% and 20% respectively.

Table 1.

Planning matrix and results of the experiment «Grain bars»

Table 2.

Dependences of response function of input factors «Grain bars»

Fig. 1.

General form of response surface Y₁

Fig. 2.

Two-dimensional section of response Y₁

Fig. 3.

General form of response surface Y₂

Fig. 4.

Two-dimensional section of response surface Y₂

Fig. 5.

General form of response surface Y₃

Fig. 6.

Two-dimensional section of response surface Y₃

Fig. 7.

General form of response surface Y₄

Fig. 8.

Two-dimensional section of response surface Y₄

Table 3.

Planning matrix and results of the experiment «Puff buns»

Table 4.

Results of coefficients significant check of models and the adequacy of the regression equation «Puff buns»

Table 5.

Dependences of the response function of input factors «Puff buns»

Fig. 9.

General form of response surface Y₁

Fig. 10.

Two-dimensional section of response surface Y₁

Fig. 11.

General form of response surface Y₂

Fig. 12.

Two-dimensional section of response surface Y₂

Fig. 13.

General form of response surface Y₃

Fig. 14.

Two-dimensional section of response surface

Fig. 15.

General form of response surface Y₄

Fig. 16.

Two-dimensional section of response surface Y₄

4 Conclusion

Thus, the presented results of mathematical modeling of recipes “Grain bars” and “Puff buns”, based on the use of raw materials with different chemical composition and functional and technological properties, make it possible to obtain the ratio of recipe components, providing high organoleptic and physicochemical indicators of specialized products.

References

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All Tables

Table 1.

Planning matrix and results of the experiment «Grain bars»

In the text

Table 2.

Dependences of response function of input factors «Grain bars»

In the text

Table 3.

Planning matrix and results of the experiment «Puff buns»

In the text

Table 4.

Results of coefficients significant check of models and the adequacy of the regression equation «Puff buns»

In the text

Table 5.

Dependences of the response function of input factors «Puff buns»

In the text

All Figures

	Fig. 1. General form of response surface Y₁
In the text

	Fig. 2. Two-dimensional section of response Y₁
In the text

	Fig. 3. General form of response surface Y₂
In the text

	Fig. 4. Two-dimensional section of response surface Y₂
In the text

	Fig. 5. General form of response surface Y₃
In the text

	Fig. 6. Two-dimensional section of response surface Y₃
In the text

	Fig. 7. General form of response surface Y₄
In the text

	Fig. 8. Two-dimensional section of response surface Y₄
In the text

	Fig. 9. General form of response surface Y₁
In the text

	Fig. 10. Two-dimensional section of response surface Y₁
In the text

	Fig. 11. General form of response surface Y₂
In the text

	Fig. 12. Two-dimensional section of response surface Y₂
In the text

	Fig. 13. General form of response surface Y₃
In the text

	Fig. 14. Two-dimensional section of response surface
In the text

	Fig. 15. General form of response surface Y₄
In the text

	Fig. 16. Two-dimensional section of response surface Y₄
In the text

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