INTRODUCTIONIn designing multicomponentI food products, of greatimportance is an opportunity to model characteristicsof the finished product and predict its quality, as well asfunctional and technological properties [1, 2].Designing multicomponent products is based on theprinciple of food combinatorics. This process involvescreating new formulations through a careful selectionof raw materials, ingredients, as well as dietary andbiologically active additives. Such combinations makethe product balanced and ensure the required sensoryand physicochemical properties, as well as nutritional,biological, and energy values [3, 4].The information base created by many years ofI Multicomponent products are a combination of various typesof raw materials, ingredients, food additives, etc.Russian scientific efforts is highly instrumental inimproving food formulations through the use of designcriteria and concepts.This article offers a review of some theoretical andpractical results achieved by the Russian science of “foodcombinatorics” from its foundation to the present day.RESULTS AND DISCUSSIONA.M. Brazhnikov and I.A. Rogov were the firstSoviet scientists who formulated the principles formathematical design of multicomponent foods with arequired set of consumer properties [5, 6].Back then, food design meant developing modelsto govern all stages of creating a product of requiredquality. At the same time, it prioritized a need to expressquality in quantitative terms.Copyright © 2020, Lisitsyn et al. This is an open access article distributed under the terms of the Creative Commons Attribution 4.0 InternationalLicense (http://creativecommons.org/licenses/by/4.0/), allowing third parties to copy and redistribute the material in any medium or format and to remix,transform, and build upon the material for any purpose, even commercially, provided the original work is properly cited and states its license.Foods and Raw Materials, 2020, vol. 8, no. 1E-ISSN 2310-9599ISSN 2308-40573Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11Figure 1 Changes in limiting amino acid scores in binarycompositions depending on the X1/X2 ratio [5]. The C2m – C1mline (red) is the changing score of the limiting m-th aminoacid; the C2n – C1n line (blue) is the changing score of thelimiting n-th amino acid; CF is the ‘ideal’ protein score; X*is the optimal ratio of components corresponding to C*, themaximum score of the compositionA.M. Brazhnikov et al. classified food productsinto three groups to develop the analytical method [5].Group I included those products (porridge, curd cheese,paste) whose components were interchangeable, both interms of their relation to each other and their positionin the general system. The relationships between thecomponents were not taken into account. To describe theproperties of those products, the authors used the generalprinciples of thermodynamics.Group II covered those products (minced meats,sausages, bread, butter, vegetable pastes, etc.) whosecomponents could interact with each other in variousways without having a fixed position in the system.Their distinctive feature was that the physicochemicalinteraction of their components during processingcould have highly significant effects on the quality ofthe finished product. The principle of superpositioncould not be applied to Group II in the same wayas it could be applied to Group I. Thus, the authorsconcluded that designing Group II products requireda greater awareness of the product characteristics anda quantitative expression of relationships between thecomponents.Finally, Group III included products (cakes, ready-toeatfoods, etc.) with interchangeable components and arigidly fixed structure.Thus, the authors set out the initial provisions of theanalytical approach to designing meat products [5]. Thisapproach was further developed by creating methods todetermine specific quality indicators.In 1980–1990, the most developed methods werethose for designing binary systemsII. It was difficultto achieve a specific amino acid profile in the proteinsystems of three or more components. In 1980,V.A. Shaternikov proposed the first analyticalapproach to designing food products with a binarycomposition [7].The mass fraction of any j-th amino acid in thebinary composition was calculated as:j 1 1j 2 2 j A = X A + X A (1)where Aj is the content of the j-th essential amino acid,g/100 g protein;A1j and A2j are the contents of the essential aminoacid in the first and second components, g/100 g protein;X1 and X2 are mass fractions of the first and secondtype proteins in the binary system (X1 + X2 = 1).The scores of the m-th and n-th essential aminoacids (used to optimize the binary composition) werecalculated as:n1 1n 2 2nnm1 1m 2 2mm FC X A X AFC X A X A+=+= ;;n1 1n 2 2nnm1 1m 2 2mm FC X A X AFC X A X A+=+= ; (2)where X1, X2 are mass fractions of the first and secondtype proteins in the binary system (X1 + X2 = 1);II Binary systems are protein systems made of two components.A1j and A2j are mass fractions of the j-th amino acid(including the n-th and m-th essential amino acids) in thefirst and second type proteins, g/100 g protein;Fm and Fn are mass fractions of the m-th and n-thessential amino acids in the reference protein, g/100 gprotein.Below are proposed solutions for three typicalsituations.First situation. If both proteins have a limitedcontent of the same essential amino acid (given Cm = Cn),the composition protein score is a constant value equalto Cm = Cn, regardless of X1 and X2.Second situation. If the first protein has a limitedcontent of the m-th essential amino acid, while thesecond protein has it in excess (compared to thereference protein), the optimal ratio of X1 and X2 isdetermined by solving a system of linear equations:( ) − ⋅ + == −2 1m 2 2m m1 21 X A X A FX 1 X(3)Another condition is needed for system (3) todetermine the optimal ratio of X1 and X2, namely:1n n A ≥ F and 2n n A ≥ F .Third situation. If the first component has a limitedcontent of the m-th amino acid and an excessive contentof the n-th amino acid ( 1n n 1m m A > F ;; A < F1n n 1m m A > F ; A < F ), whilethe second component has a limited content of then-th amino acid and an excessive content of the m-thamino acid ( 2m m 2n n A > F ;; A < F2m m 2n n A > F ; A < F ), the optimal ratiobetween X1 and X2 in the binary system is determinedby the graphical method (Fig. 1). This method allows aquick determination of the required values that ensurethe maximum score of the limiting amino acid in thecomposition.In 1983, this approach was approved by the USSRMinistry of Health within Guidelines No. 2688-83 forusing milk and soy proteins in meat production.4Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11In 1981, N.A. Mikhailov (whose research supervisorswere I.A. Rogov, Doctor of Technical Sciences andV.G. Vysotsky, Doctor of Medicine) developed somebasic analytical principles of designing combinedproducts based on modelling the biological value ofprotein. He used those principles to create combinedpaste, as well as a number of cereals and diabeticprotein-wheat bread with an increased biologicalvalue [8–12].In addition, N.A. Mikhailov proposed acomprehensive statistical model of protein biologicalvalue to determine the optimal composition ofingredients in combined products or correct the initialratio of ingredients to ensure a specific biological valueafter heat treatment.N.N. Lipatov (Jr.) proposed a completely differentclassification of food products that is still used today fordesigning functional products [13–16]. In particular, itincludes three generations of industrial foods:‒ products that have sensory characteristics similar totraditional ones, with raw materials partially replacedwith hydrated components equivalent in protein content;‒ multicomponent products with a nutrient ratio closeto a statistically sound standard that take into accountthe metabolism in specific population groups united bynationality, age or other characteristics; and‒ products with a specially selected combinationof components that can ensure their targeted use asfunctional products by certain population groups.In addition, N.N. Lipatov developed six basic principlesfor formulating balanced foods and diets [13–16],namely:‒ compliance with a rationally balanced formulation;‒ compliance of an amino acid composition of proteincontainingingredients with a statistically soundreference protein;‒ a possibility of changing the fatty acid composition byadding fat-containing ingredients;‒ the nearest approximation to a desirable ratio ofsaturated, monounsaturated, and polyunsaturated fattyacids in any combination of fat-containing ingredients;‒ taking into account the composition of other dishesand foods in the diet; and‒ a balanced multicomponent composition for a singleor daily ration in terms of energy value, macro- andmicronutrients, and ballast agents.These principles are still used as a foundation forresearch in the field of food combinatorics.N.N. Lipatov et al. conducted several studies todevelop methods for creating foods with a specifiednutritional value [13–16]. In doing so, they assumed thatthe mechanical processing of raw materials to ensure therequired level of dispersion or structural and mechanicalcharacteristics did not violate the principle ofsuperposition with respect to their biologically valuablenutrients. They used this hypothesis in making logicalconstructions about deterministic formalized approachesto measuring the quantity of individual ingredients. Asa result, the authors made valid and reliable conclusionsabout formulating products with a specified nutritionalvalue and formalized the qualitative and quantitativeconceptions about the rationality of using essentialamino acids in the technology of adequate exotrophy.Formalization takes into account the mutual balance ofessential amino acids.The scientists formulated the main principle andcriterion for the rational use of essential amino acids innew types of foods. The principle gives preference tosuch combinations of n-protein-containing componentswithmass fractions                                                                  ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p a  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji kin which the maximumproportion of assimilable essential amino acids in theprotein, given equal provision of the body with anabolicmaterial, can be used for anabolic purposes withoutdegradation in the biosynthesis of nonessential aminoacids, let alone biological oxidation in compensating forthe energy expenditure of the body [13].Taking into account a possibility of C ≥1 min (whereCmin is the minimum score of essential amino acids inthe protein of the designed product in relation to thereference protein) or a possibility of C <1 min , the selectioncriterion                                                        ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an iipikjnii ijpiX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2can be symbolically presented as     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p a  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k1 1 1 1 11 1 1 1                                   ni mmi Limi Li i imi LiLli i imi Limi Lni mmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X X X p a X Y X p a Y X A           nimjP z zi bijx j121( ) 0 min             nkmmjki ij ji k ib xa b xP A A120 1 min; 1,2(4)  where  ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an iipikjnii piX pX p  11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                 ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min  mki ij ja b x2is the total mass fraction of assimilatedessential amino acids that can be used by the body foranabolic purposes without further degradation;     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an ikjniX  11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                 ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 minis the actual sum of assimilated essential aminoacids.The authors transformed the above formulaintroducing pi as a mass fraction of digestible proteinin the i-th component (%) and aij as a mass fractionof the ј-th essential amino acid in the protein of thei-th component (g/100 g). Criterion (4) for findinga preferable ratio of the mass fractions                         p      ;   minmin    pipipirpi C XA X A XA X A   ;   min1min11 1111 1             pikjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p a  min 1,2,32110              ixb xP V Vnkmjjmjkj ji k1 1 1                 mmmLmmi Lmi Lmi Li i i ijmi LiLli i i i ijjX X a X Y X p a Y A  of thesecomponents in the designed product with regard torational use of the k essential amino acids can bepresented as:     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 11 1 1 1 1                                 ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2     ;   minmin    pippirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an ipikjnipiX X  11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA      nmP z zi bijx 2( ) 0 min;     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                       pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an ipikjnipiX pX  11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                 ni mmi Limi Li imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X X p a X Y X p a Y X X p aA         ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                      pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1                              ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i ijLli jX X p X Y X p Y X X pX Y X p a Y X X p aA          nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2     ; minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an ikjniX X  11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                 ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA    (5)where     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                          pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k         mni i i ijmimi i i ijmiLi i i i ijX X p a X Y X p a Y X X p a  is the mass fraction of j-thessential amino acid in the protein of the designedproduct with the fixed j, g/100 g protein;Arj is the reference mass fraction of the j-th essentialamino acid, g/100 g protein.5Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11N.N. Lipatov (Jr.) developed the following methodologicalapproaches to designing foods with therequired set of nutritional indicators.The first stage involves modelling the amino acidcomposition of protein in the designed product andselecting                                                       ;   minmin   pipipirpi C XA X A XX A   ;   minmin1min11 1111 1             pikjrjpn iiipikjnii ijpikjn rjiipinii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1 min 1,2,32110 1              ixb xV Vnkmjjmjkj jk,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijX X p X Y X p Y X X pX X p a X Y X p a Y X X p a           nimjz zi bijx j121) 0 min              nkmjij jmjki ij jk ib xa b xA A1210 1 min; 1,2 min 1,2,32110 1              ixb xV Vnkmjjmjkj jkthat provide the minimum functionalvalues (4).The second stage involves modelling the fattyacid composition, given that the mass fractions ofcomponents L(p)i X containing protein, as well as fat, areconstant and predetermined by the first stage. Based onthe modelling results, mass fractions Li X are selected thattogether with L(p)i X provide the required approximation tothe physiologically determined ratio of saturated, monoandpolyunsaturated fatty acids.The third stage involves calculating the energyvalue Qp of the designed product, taking into accountonly those c(p L)i X , which are sources of protein and/orfat. The result is then compared with the required Q.If the estimated energy value is less than Q, the productis supplemented with additional technologicallypermissible carbohydrate-containing components inquantities that ensure the required Q. If Qp is greaterthan Q, Li X are recalculated. If necessary, Li X withexcessively high Li values can be replaced with thosewith lower Li values.Using the Mitchell-Block principle, N.N. Lipatovdeveloped a number of indicators, namely: the utilizationcoefficient for essential amino acids; the utilizationcoefficient for the amino acid composition of theproduct, g/100 g protein; the ratio of amino acids asa balance of essential amino acids in relation to thephysiologically determined norm (standard); and theindicator of excess in the content of essential aminoacids as the total amount of essential amino acids thatare not used for anabolic purposes [17, 18].Thus, we can conclude that the main studies ofN.N. Lipatov were devoted to the trophological,mathematical, informational, and algorithmic aspectsof food design. He supervised the creation of ordinaryand specialised products for baby and gerodieteticnutrition. Finally, he established a scientific school toimprove the quality of foods considered as objects of asingle exotrophic chain of production, consumption, andassimilation of nutrients by the human body.A.B. Lisitsyn combined the mathematical methodsof I.A. Rogov and N.N. Lipatov for calculating theamino acid composition and total protein digestibility inmulticomponent mixtures [19–21].Protein digestibility is one of the most importantindicators of the product’s biological value, along withits amino acid balance. A.B. Lisitsyn understood theneed to take into account individual protein digestibilityof all components when estimating the product’sbiological value and study their effect on the aminoacid composition of total protein. The mathematicalinterpretation of his concept can be presented as follows:  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k  min11          pijn rjiipiC XX pn iipiX p1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k(6)Σ Σ Σ= = + = += = ≤mi Lini mimiXi X Y X1 1 11; 1;where Aj is the content of j-th amino acid, g/100 gprotein;i is the mass fraction of the i-th component in themixture, unit fraction;πi is the dimensionless characteristic (coefficient) ofprotein digestibility of the i-th component;pi is the mass fraction of protein in the i-thcomponent, % or unit fraction;aij is the mass fraction of j-th amino acid in theprotein of the i-th component, g/100 g protein;n is the total number of ingredients in theformulation;(n-m) is the number of replacement ingredientsduring modelling;L is the number of ingredients that are not replacedduring modelling;(m-L) is the number of ingredients varying (replaced)during modelling;Y is the total amount of varying ingredients in theformulation.Thus, A.B. Lisitsyn substantiated the principles ofdesigning meat products with a given biological value,taking into account individual protein digestibility andthe amino acid composition of every ingredient in theformulation. His mathematical formulas allow us todevise the amino acid composition of multicomponentsystems, taking into account individual proteindigestibility of every component.Yu.A. Ivashkin combined the structural and theparametric optimization approaches in his works.Structural optimization is the determination of optimalstructural parameters of the formulation. Parametricoptimization involves calculating optimal deviationsfrom the norm. Yu.A. Ivashkin et al. suggested usingstructural and parametric optimization for everycriterion with pairwise comparison and qualityassessment of the resulting product using an independentquality functional and desirability scales [22, 23]. Themulticriteria optimization of the combined product(nutritional and biological values) consists in building itsmodel according to the specified adequacy and qualityparameters, depending on the composition of initialcomponents.For this, a parametric model of the product isdevised, taking into account:6Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11– the required chemical composition (protein, fat,carbohydrates, etc.);– mass fractions of the main components (keyingredients, fiber, biologically active additives, enzymes,etc.); and– structural relationships of biological value indicators(amino and fatty acid compositions) according to variouscompliance criteria.Consideration is also given to what makes a balanceddiet for a certain population group.An objective function is the minimum deviationfrom the given structural group of nutritional andbiological indicators [22, 23], namely the criteria below.(1) The optimization criterion for chemical elementsthat determine the nutritional value P(z) of the designedproduct: ;   minmin1min1111 1                   pijrjn iiipikjn rjiipikjnii ijpiC XC X AX pAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmiLiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k(7)where ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji kis the reference content of the i-th element ofnutritional value;bij is the specific content of the i-th element ofchemical composition in the j-th component of thedesigned product;xj is the mass fraction of the j-th component.(2) The criterion of the minimum deviation from thegiven structural indicators of biological value Pi(A), forexample, the monostructure of essential amino acids(i = 1) and fatty acids (i = 2):    ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA          nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj i k(8)where Ak0 is the reference content of the k-thmonostructural indicator of biological value;aki is the specific content of the k-homonostructuralingredient in the i-th element of chemical composition;bij is the specific content of the i-th element ofchemical composition in the j-th component of thedesigned product;xj is the mass fraction of the j-th component.(3) The criterion of the minimum deviation from thegiven structure Pi(V) of the vitamin composition (i = 1),minerals (i = 2), and carbohydrates (i = 3):     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipikjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                  ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k ;     ;   minmin    pipipirpi C XA X A XA X A   ;   minmin1min11 1111 1                        pikjrjpn iiipkjnii ijpikjn rjiipikjnii ijpiC XC X AX pX p aAX pX p an jiipikjnii ijpiAX pX p a 11 1  min 1,2,32110 1             ixb xP V Vnkmjjmjkj ji k,1 1 1 1 11 1 1 1 1                                 ni mi i imi Limi Li i imi LiLli i imi Limi Lni mi i i ijmi Limi Li i i ijmi LiLli i i i ijjX X p X Y X p Y X X pX X p a X Y X p a Y X X p aA           nimjP z zi bijx j121( ) 0 min               nkmjij jmjki ij ji k ib xa b xP A A1210 1 min; 1,2  min 1,2,32110 1              ixb xP V Vnkmjjmjkj ji k ( 9)where bkj is the specific content of the k-th ingredient inthe j-th element of chemical composition;xj is the mass fraction of the j-th component.Yu.A. Ivashkin proposed to use simulation modellingto solve the problem of structural and parametricoptimization of a multicomponent product in variouscombinations of linear and non-linear criteria andrestrictions. It involves “playing out” all possiblecombinations of initial ingredients with subsequentverification of restrictions and calculation of criteria.Noteworthily, any problem of the NPIII class canbe solved by simulation modelling. The complexitydepends on the number of ingredients in the formulation.If the space of problem solutions is very large, thismethod may take longer than a “reasonable” time toproduce results.A.E. Krasnov et al. used new informationtechnologies to produce foods of given quality. Inparticular, they created sausage meat formulationsunder varying conditions of uncertainty. Their studyshowed how to transform the problem of stochasticprogramming with uncertain target criteria intothe linear programming problem with stochasticconstraints [24].Considerable attention is now paid to modellinginteractions between food components based on thelaws of equilibrium statistical thermodynamics. For thefirst time, a polynomial dependence was found betweenthe properties of food mixtures and the mass fractionsof their ingredients. Its relationship with macroscopicthermodynamic parameters of the mixtures was alsoshown.Scientific modelling of multi-component foodproducts with a specific set of nutritional and energyindicators is still relevant worldwide. An ever widercircle of researchers are engaged in various aspects ofimproving food technology.E.I. Muratova et al. proposed an object-orientedapproach to designing multicomponent food products(for example, confectionery). A distinctive feature of thisapproach is presenting a formulation as a hierarchicalstructure (the Saati method) [25, 26].Each vertex of this structure is an object (rawmaterials – semi-finished product – finished product).Each level is a certain stage of food technology thatcan have its own number of vertices located lowerin the hierarchy. The algorithm for calculating amulticomponent product begins with the lowest level inthe longest branch of the hierarchical structure (Fig. 2).Figure 2 shows a three-level hierarchy of formulation,where the first index is the level number and thesecond is the number of a component in the formulatedmixture. When several semi-finished products are usedat the same level, their first index becomes a compositeand is indicated as a list (i, j), where i is the level numberand j is the serial number of the semi-finished product atthe i-th level. This composite index is used lower in thehierarchy (shown by the dashed arrow).The algorithm for calculating the multiphaseformulation begins with the lowest level in the longestIII In the theory of algorithms, the NP (non-deterministic polynomial)class refers to a multitude of decision problems whose solutions canbe verified on a Turing machine within a certain input polynomialtime, if there is some additional information (the so-called solutioncertificate) [39].7Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11Figure 2 Hierarchical structure of the product formulation:FP – finished product; RM – raw materials;SFP – semi-finished product [23]Figure 3 Basic principles of system modelling of multicomponent products [27]branch of the hierarchical structure. According toFig. 2, the calculation of the formulation begins withthe semi-finished product FP (2, 1), since the path to itscomponents is the longest in the hierarchy. The initialdata for calculating the lowest level include the loadingof all types of raw materials and semi-finished products,loss of dry matte, and a given amount of finishedproducts equal to 1 t.According to the authors, the main advantage of thisapproach is the object-oriented representation. It allowsfor inheriting properties and methods while adding newcalculation formulas that take into account new rawmaterials, production features, as well as technical andeconomic indicators of the processes.O.N. Musina and P.A. Lisin proposed a methodologyfor system modelling of multicomponent food products[27–29]. They defined system modelling as a strategyfor studying and creating biosystems, particularlyfood products, their formulations, and productiontechnologies.The basic principle of system modelling is thedecomposition of a complex biosystem into simplersubsystems. This is a principle of the system hierarchy.In this case, the mathematical model of the systemis based on the block principle: the general model isdivided into blocks which can have relatively simplemathematical descriptions. All subsystems interactwith each other and constitute a common unifiedmathematical model.Figure 3 shows a visual interpretation of the basicprinciples of system modelling of multicomponentproducts.System modelling principles allow for thedecomposition of the production system at the stageof formulating composite mixtures using linearmodels. In such models, mathematical dependencies(equalities or inequalities) are linear with respect toall variables in the model. Problems of this kind areused to select the optimal option from a set of possibleformulations according to a given criterion. In 1939,the Russian mathematician L. Kantorovich and theAmerican scientist G. Danzig began to develop whatwas later called “the simplex metho”. It became auniversal method of linear programming used in solvingoptimization problems.A.A. Borisenko proposed a methodology foroptimizing multicomponent food mixtures usinguniversal mathematical methods. His methodologyallows for the development of foods with a given nutrientcomposition [30, 31]. Taking into account certainrestrictions and permissible deviations of nutrientmass fractions from the reference amounts, the authorproposed to use the Lagrange function and the systemFPlevel 1RM 1,1 RM 1,2RM(2,1),1level 2level 3RM 3,1 RM 3,kRM(2,2),1RM(2,2),2SFP 1,1 SFP 1,2SFP(2,1)Liebig’s principleMinimizationBiosystems operate optimally whenthe body receives minimum (reference)amounts of every nutrientFor example, a food product can beconsidered as a combination of vitamins,minerals, fatty acids, and other systemsor as a combination of chemical elementsEvery system is complex so multiplemodels are required to understandhow it works, each describingonly one of its aspectsMultiplicity of food systemdescriptionsProduct propertiesFunctionalityManifestation of certain properties(functions) during interactionwith the external environmentThe designed product is considered as a whole,and its ingredients, as subsystemsIntegrity The biosystem is considered as a whole The biosystem’s integrity means that eachingredient in a multicomponent product contributesto its qualityComplianceComponents are integral parts of the product,they are structural elements that make it a wholeand without which it cannot existThe level of nutrients’ compliancewith reference valuesBasic principles of system modellingof multicomponent foodsHierarchy Analysis of the system elements (ingredients)and their relations in the product structure The system’s functioning is determined bythe properties of the product structure, rather thanthe properties of its individual elements (ingredients)StructurednessManifestation of certain properties (functions) duringinteraction with the external environment8Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11of equations in the form of conditions of the Kuhn-Tucker theorem for convex programming. Solving theseproblems produces a vector of component mass fractionsto ensure the most balanced nutrient composition.The author concluded that the most balancedformulation cannot always guarantee the highest qualityof the finished product. Therefore, in most cases, thereis a need for a fairly wide range of formulated options.To achieve that, he proposed to optimize formulationsin two stages. The first stage of modelling a formulationinvolved determining all possible quantitative ratiosof the ingredients. The second stage was a qualitativeassessment and selection of several most optimalvariants. The author used Harrington’s desirabilityfunction as a general criterion for quality assessment.A.Yu. Prosekov developed the principles of formingdispersed food systems and designing functionalproducts from modern perspectives [32–34].T.V. Sanina and Yu.S. Serbulov proposed adifferentiated approach to a comprehensive assessmentof highly nutritional bakery products. The authorsbelieve that consumers should select key qualityindicators for foods with increased nutritional value tomake their assessment objective. In addition, qualityassessment should check if the product satisfies certainneeds consistent with its purpose [35].A.A. Zaporozhsky et al. formulated new gerodieteticproducts with specified qualitative characteristicsbased on natural raw materials. For this, they used amethodological approach and the principles of modernnutrition, qualimetry, food combinatorics, and neuralnetwork approximation of theoretical (estimated) andexperimental data [36, 37].T.Yu. Reznichenko et al. substantiated an integratedtechnological approach to the development of functionalfoods enriched with biologically active substances anddietary fibre. They studied the factors that determine thequality of specialized products and critical control pointsthat identify their functional character at the stagesof production and distribution. They also developed arange of consumer properties that included functionalindicators in addition to sensory and physicochemicalcharacteristics. Finally, the authors developed analgorithm to examine a functional cereal breakfastbar [38, 39].V.M. Kiselev and E.G. Pershina looked at theproduction and consumption of functional foods asa multi-factor system subjected to comprehensiveassessment. They used the methods of foodcombinatorics, parity of needs, and the vital concept,taking into account modern requirements of nutrition.With this approach, the authors studied a possibility ofevolutionary development of functional food designbased on food combinatorics. They identified consumerpreferences for functional foods and systematized themin a model of consumer value [40].O.N. Krasulya et al. considered the design ofmulticomponent foods based on the functional andtechnological properties (FTP) of their main rawmaterials and ingredients. They also took into accountthe kinetics of biochemical and colloidal processes, aswell as analytical and empirical relations characterizingthe main patterns of heterogeneous disperse systemswith varying physicochemical factors [42, 43].In the age of digital (information) technologies,the design of multicomponent food formulationscan be improved by using linear, experimental andstatistical programming methods, or an object-orientedapproach. M.S. Koneva et al. proposed using neuralnetwork technologies [44]. The relationship betweensensory criteria and the quantitative composition ofthe formulation was identified by neural network andregression analysis of the ranking score of sensorycharacteristics. The model parameters were obtainedwith Statistica software. The convolution of thebalancing index and sensory evaluation was proposed asa multiplicative desirability function. MathCAD scriptswere used to optimize the composition of antianemicsmoothie for pregnant women.N.A. Berezina et al. developed a program in ObjectPascal for designing gerodietetic bread compositions[45]. The technological adequacy of the flour mix,which ensured a stable quality of the final product, wasmodelled by introducing the flour technological indicator(“falling number”) calculated using the Perten formula.The mathematical foundations of solving singlecriterionoptimization problems are quite well studiedtoday. However, various areas of engineering, researchand management have multicriteria problems in whichseveral criteria need to be simultaneously optimized.M.A. Nikitina and I.M. Chernukha proposed using thePareto method for multicriteria optimization [46].The informational aspects of modelling andevaluating the nutritional adequacy of raw materials andfinished products are very important in improving thequality and technology of specialized multicomponentfood products.CONCLUSIONThe analysis of literature on the principles andmethods of designing balanced foods showed thatthe initial stage in this process involved formalizingqualitative and quantitative assumptions aboutthe rational use of essential amino acids in theadequate exotrophy technology. N.N. Lipatov’scontribution to designing balanced formulations inRussia cannot be underestimated. His principles ofcreating multicomponent foods and balanced dietsare still relevant today. Further development of foodcombinatorics was related to informational andalgorithmic aspects of food design.The conceptual approaches to computer-aidedfood design proposed by N.N. Lipatov (Jr.) are used tomodel functional products with specified qualitativecharacteristics. Based on the optimal choice of rawmaterials and ratios of ingredients, they result informulations whose nutrients (essential amino acids,9Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11unsaturated fatty acids, macro- and microelements, andvitamins) are consistent with the medical and biologicalrequirements in terms of quantity and quality.The computer systems and software productsactively used in Russia to automate technologicalcalculations for food and diet formulations includeEtalon, Generic 2.0, Food & Life, CheesePro 1.0,ShkoOptiPit, and others. They are based on thedatabases of foods and raw materials, scientific researchand industrial experience, as well as mathematicalmethods of modelling and designing food coveredin the works of I.A. Rogov, A.M. Brazhnikov, N.N.Lipatov (Jr.), and other scientists. With the help ofthose systems, new types of products were developedby Moscow State University of Applied Biotechnology,Gorbatov All-Russia Meat Research Institute, ResearchInstitute of Baby Food, All-Russia Research Instituteof Dairy Industry, and other institutes. These productshad an improved composition of chemical elements,amino and fatty acids, as well as better energy values,quality indicators, etc. The experimental and theoretical(mathematical) data were 98% reliable.The foreign software solutions (DietPlan, Nutri-Survey, NutriBase, NUT, MyFitnesspal, and 8fit) arebased on calculating the individual’s daily energy intakeand their need for basic nutrients.Designing foods in the digital age, we need to takeinto account not only nutritional and biological values,but also medical, technological, economic, social, andother factors. Computer technologies allow us to addressproblems with numerous parameters, alternatives,and criteria, as well as restrictions and conditions.By processing and formalizing data, they help us findoptimal solutions based on complex optimization modelsand objective assessment of options.A need for “digital nutritiology”, a new scientific field,was highlighted in Decree of the Presidium of the RussianAcademy of Sciences No. 178 dated November 27,2018 “On the Current Problems of Optimizing thePopulation of Russia: Role of Science” (paragraph 11).This new direction is supposed to translate into thelanguage of numbers our physiological needs for energy,nutrients, biologically active substances, and balanceddiets, on the one hand, and the chemical composition offoods and general diets, on the other.CONTRIBUTIONThe authors were equally involved in writing themanuscript and are equally responsible for plagiarism.CONFLICT OF INTERESTThe authors state that there is no conflict of interest.