INTRODUCTION
In designing multicomponentI food products, of great
importance is an opportunity to model characteristics
of the finished product and predict its quality, as well as
functional and technological properties [1, 2].
Designing multicomponent products is based on the
principle of food combinatorics. This process involves
creating new formulations through a careful selection
of raw materials, ingredients, as well as dietary and
biologically active additives. Such combinations make
the product balanced and ensure the required sensory
and physicochemical properties, as well as nutritional,
biological, and energy values [3, 4].
The information base created by many years of
I Multicomponent products are a combination of various types
of raw materials, ingredients, food additives, etc.
Russian scientific efforts is highly instrumental in
improving food formulations through the use of design
criteria and concepts.
This article offers a review of some theoretical and
practical results achieved by the Russian science of “food
combinatorics” from its foundation to the present day.
RESULTS AND DISCUSSION
A.M. Brazhnikov and I.A. Rogov were the first
Soviet scientists who formulated the principles for
mathematical design of multicomponent foods with a
required set of consumer properties [5, 6].
Back then, food design meant developing models
to govern all stages of creating a product of required
quality. At the same time, it prioritized a need to express
quality 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 International
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Foods and Raw Materials, 2020, vol. 8, no. 1
E-ISSN 2310-9599
ISSN 2308-4057
3
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
Figure 1 Changes in limiting amino acid scores in binary
compositions depending on the X1/X2 ratio [5]. The C2m – C1m
line (red) is the changing score of the limiting m-th amino
acid; the C2n – C1n line (blue) is the changing score of the
limiting n-th amino acid; CF is the ‘ideal’ protein score; X*
is the optimal ratio of components corresponding to C*, the
maximum score of the composition
A.M. Brazhnikov et al. classified food products
into three groups to develop the analytical method [5].
Group I included those products (porridge, curd cheese,
paste) whose components were interchangeable, both in
terms of their relation to each other and their position
in the general system. The relationships between the
components were not taken into account. To describe the
properties of those products, the authors used the general
principles of thermodynamics.
Group II covered those products (minced meats,
sausages, bread, butter, vegetable pastes, etc.) whose
components could interact with each other in various
ways without having a fixed position in the system.
Their distinctive feature was that the physicochemical
interaction of their components during processing
could have highly significant effects on the quality of
the finished product. The principle of superposition
could not be applied to Group II in the same way
as it could be applied to Group I. Thus, the authors
concluded that designing Group II products required
a greater awareness of the product characteristics and
a quantitative expression of relationships between the
components.
Finally, Group III included products (cakes, ready-toeat
foods, etc.) with interchangeable components and a
rigidly fixed structure.
Thus, the authors set out the initial provisions of the
analytical approach to designing meat products [5]. This
approach was further developed by creating methods to
determine specific quality indicators.
In 1980–1990, the most developed methods were
those for designing binary systemsII. It was difficult
to achieve a specific amino acid profile in the protein
systems of three or more components. In 1980,
V.A. Shaternikov proposed the first analytical
approach to designing food products with a binary
composition [7].
The mass fraction of any j-th amino acid in the
binary 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 amino
acid in the first and second components, g/100 g protein;
X1 and X2 are mass fractions of the first and second
type proteins in the binary system (X1 + X2 = 1).
The scores of the m-th and n-th essential amino
acids (used to optimize the binary composition) were
calculated as:
n
1 1n 2 2n
n
m
1 1m 2 2m
m F
C X A X A
F
C X A X A
+
=
+
= ;;
n
1 1n 2 2n
n
m
1 1m 2 2m
m F
C X A X A
F
C X A X A
+
=
+
= ; (2)
where X1, X2 are mass fractions of the first and second
type 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 the
first and second type proteins, g/100 g protein;
Fm and Fn are mass fractions of the m-th and n-th
essential amino acids in the reference protein, g/100 g
protein.
Below are proposed solutions for three typical
situations.
First situation. If both proteins have a limited
content of the same essential amino acid (given Cm = Cn),
the composition protein score is a constant value equal
to Cm = Cn, regardless of X1 and X2.
Second situation. If the first protein has a limited
content of the m-th essential amino acid, while the
second protein has it in excess (compared to the
reference protein), the optimal ratio of X1 and X2 is
determined by solving a system of linear equations:
( ) 


− ⋅ + =
= −
2 1m 2 2m m
1 2
1 X A X A F
X 1 X
(3)
Another condition is needed for system (3) to
determine 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 limited
content of the m-th amino acid and an excessive content
of the n-th amino acid ( 1n n 1m m A > F ;; A < F1n n 1m m A > F ; A < F ), while
the second component has a limited content of the
n-th amino acid and an excessive content of the m-th
amino acid ( 2m m 2n n A > F ;; A < F2m m 2n n A > F ; A < F ), the optimal ratio
between X1 and X2 in the binary system is determined
by the graphical method (Fig. 1). This method allows a
quick determination of the required values that ensure
the maximum score of the limiting amino acid in the
composition.
In 1983, this approach was approved by the USSR
Ministry of Health within Guidelines No. 2688-83 for
using milk and soy proteins in meat production.
4
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
In 1981, N.A. Mikhailov (whose research supervisors
were I.A. Rogov, Doctor of Technical Sciences and
V.G. Vysotsky, Doctor of Medicine) developed some
basic analytical principles of designing combined
products based on modelling the biological value of
protein. He used those principles to create combined
paste, as well as a number of cereals and diabetic
protein-wheat bread with an increased biological
value [8–12].
In addition, N.A. Mikhailov proposed a
comprehensive statistical model of protein biological
value to determine the optimal composition of
ingredients in combined products or correct the initial
ratio of ingredients to ensure a specific biological value
after heat treatment.
N.N. Lipatov (Jr.) proposed a completely different
classification of food products that is still used today for
designing functional products [13–16]. In particular, it
includes three generations of industrial foods:
‒ products that have sensory characteristics similar to
traditional ones, with raw materials partially replaced
with hydrated components equivalent in protein content;
‒ multicomponent products with a nutrient ratio close
to a statistically sound standard that take into account
the metabolism in specific population groups united by
nationality, age or other characteristics; and
‒ products with a specially selected combination
of components that can ensure their targeted use as
functional products by certain population groups.
In addition, N.N. Lipatov developed six basic principles
for formulating balanced foods and diets [13–16],
namely:
‒ compliance with a rationally balanced formulation;
‒ compliance of an amino acid composition of proteincontaining
ingredients with a statistically sound
reference protein;
‒ a possibility of changing the fatty acid composition by
adding fat-containing ingredients;
‒ the nearest approximation to a desirable ratio of
saturated, monounsaturated, and polyunsaturated fatty
acids in any combination of fat-containing ingredients;
‒ taking into account the composition of other dishes
and foods in the diet; and
‒ a balanced multicomponent composition for a single
or daily ration in terms of energy value, macro- and
micronutrients, and ballast agents.
These principles are still used as a foundation for
research in the field of food combinatorics.
N.N. Lipatov et al. conducted several studies to
develop methods for creating foods with a specified
nutritional value [13–16]. In doing so, they assumed that
the mechanical processing of raw materials to ensure the
required level of dispersion or structural and mechanical
characteristics did not violate the principle of
superposition with respect to their biologically valuable
nutrients. They used this hypothesis in making logical
constructions about deterministic formalized approaches
to measuring the quantity of individual ingredients. As
a result, the authors made valid and reliable conclusions
about formulating products with a specified nutritional
value and formalized the qualitative and quantitative
conceptions about the rationality of using essential
amino acids in the technology of adequate exotrophy.
Formalization takes into account the mutual balance of
essential amino acids.
The scientists formulated the main principle and
criterion for the rational use of essential amino acids in
new types of foods. The principle gives preference to
such combinations of n-protein-containing components
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i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n i
i
p
i
k
j
n
i
i p
i
X p
X p 


 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
  
  
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min

 

 

m
ki ij j
a b x
2
is the total mass fraction of assimilated
essential amino acids that can be used by the body for
anabolic purposes without further degradation;
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n i
k
j
n
i
X 


 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
    
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
is the actual sum of assimilated essential amino
acids.
The authors transformed the above formula
introducing pi as a mass fraction of digestible protein
in the i-th component (%) and aij as a mass fraction
of the ј-th essential amino acid in the protein of the
i-th component (g/100 g). Criterion (4) for finding
a preferable ratio of the mass fractions 
 

 
     
     
     
     
 
 
p      
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
1
min
1
1 1
1
1
1 1 
    

     








 

 


 
p
i
k
j
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
  min 1,2,3
2
1
1
0  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
1 1 1             

  


 


 

m
m
m
L
m
m
i L
m
i L
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X a X Y X p a Y A
  of these
components in the designed product with regard to
rational use of the k essential amino acids can be
presented as:
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 



 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1
1 1 1 1 1
    
     
         
         
 



 


 
 


 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
     
;   min
min


 

 


 
 
p
i
p
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n i
p
i
k
j
n
i
p
i
X X 


 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
   
         
         
 



 


 
 



 

 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 





 
n
m
P z zi bijx 2
( ) 0 min
;
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

    


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n i
p
i
k
j
n
i
p
i
X p
X 


 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
    
     
         
         
 



 


 
 



 


 

n
i m
m
i L
i
m
i L
i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X X p a X Y X p a Y X X p a
A
    
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

   


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1
     
   
         
       
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i ij
L
l
i j
X X p X Y X p Y X X p
X Y X p a Y X X p a
A
  
 
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
     
; min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n i
k
j
n
i
X X 


 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1

    
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
(5)
where
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


            









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
     
  

 

 
m
n
i i i ij
m
i
m
i i i ij
m
i
L
i i i i ij
X X p a X Y X p a Y X X p a
  
is the mass fraction of j-th
essential amino acid in the protein of the designed
product with the fixed j, g/100 g protein;
Arj is the reference mass fraction of the j-th essential
amino acid, g/100 g protein.
5
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
N.N. Lipatov (Jr.) developed the following methodological
approaches to designing foods with the
required set of nutritional indicators.
The first stage involves modelling the amino acid
composition of protein in the designed product and
selecting 
 
     
     
     
     
 
 
 
 
  
  
     
     
     
     
     
;   min
min


 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     









 

 



p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
 min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
V V
n
k
m
j
j
m
j
kj j
k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
  
  
 
 

  


  


 
n
i
m
j
z zi bijx j
1
2
1
) 0 min
 




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
k i
b x
a b x
A A
1
2
1
0 1 min; 1,2
 min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
V V
n
k
m
j
j
m
j
kj j
k
that provide the minimum functional
values (4).
The second stage involves modelling the fatty
acid composition, given that the mass fractions of
components L(p)
i X containing protein, as well as fat, are
constant and predetermined by the first stage. Based on
the modelling results, mass fractions L
i X are selected that
together with L(p)
i X provide the required approximation to
the physiologically determined ratio of saturated, monoand
polyunsaturated fatty acids.
The third stage involves calculating the energy
value Qp of the designed product, taking into account
only those c(p L)
i X , which are sources of protein and/or
fat. The result is then compared with the required Q.
If the estimated energy value is less than Q, the product
is supplemented with additional technologically
permissible carbohydrate-containing components in
quantities that ensure the required Q. If Qp is greater
than Q, L
i X are recalculated. If necessary, L
i X with
excessively high Li values can be replaced with those
with lower Li values.
Using the Mitchell-Block principle, N.N. Lipatov
developed a number of indicators, namely: the utilization
coefficient for essential amino acids; the utilization
coefficient for the amino acid composition of the
product, g/100 g protein; the ratio of amino acids as
a balance of essential amino acids in relation to the
physiologically determined norm (standard); and the
indicator of excess in the content of essential amino
acids as the total amount of essential amino acids that
are not used for anabolic purposes [17, 18].
Thus, we can conclude that the main studies of
N.N. Lipatov were devoted to the trophological,
mathematical, informational, and algorithmic aspects
of food design. He supervised the creation of ordinary
and specialised products for baby and gerodietetic
nutrition. Finally, he established a scientific school to
improve the quality of foods considered as objects of a
single exotrophic chain of production, consumption, and
assimilation of nutrients by the human body.
A.B. Lisitsyn combined the mathematical methods
of I.A. Rogov and N.N. Lipatov for calculating the
amino acid composition and total protein digestibility in
multicomponent mixtures [19–21].
Protein digestibility is one of the most important
indicators of the product’s biological value, along with
its amino acid balance. A.B. Lisitsyn understood the
need to take into account individual protein digestibility
of all components when estimating the product’s
biological value and study their effect on the amino
acid composition of total protein. The mathematical
interpretation of his concept can be presented as follows:
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
  min
1
1
     

     





p
i
j
n rj
i
i
p
i
C X
X p
n i
i
p
i
X p


1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
(6)
Σ Σ Σ
= = + = +
= = ≤
m
i L
i
n
i m
i
m
i
Xi X Y X
1 1 1
1; 1;
where Aj is the content of j-th amino acid, g/100 g
protein;
i is the mass fraction of the i-th component in the
mixture, unit fraction;
πi is the dimensionless characteristic (coefficient) of
protein digestibility of the i-th component;
pi is the mass fraction of protein in the i-th
component, % or unit fraction;
aij is the mass fraction of j-th amino acid in the
protein of the i-th component, g/100 g protein;
n is the total number of ingredients in the
formulation;
(n-m) is the number of replacement ingredients
during modelling;
L is the number of ingredients that are not replaced
during modelling;
(m-L) is the number of ingredients varying (replaced)
during modelling;
Y is the total amount of varying ingredients in the
formulation.
Thus, A.B. Lisitsyn substantiated the principles of
designing meat products with a given biological value,
taking into account individual protein digestibility and
the amino acid composition of every ingredient in the
formulation. His mathematical formulas allow us to
devise the amino acid composition of multicomponent
systems, taking into account individual protein
digestibility of every component.
Yu.A. Ivashkin combined the structural and the
parametric optimization approaches in his works.
Structural optimization is the determination of optimal
structural parameters of the formulation. Parametric
optimization involves calculating optimal deviations
from the norm. Yu.A. Ivashkin et al. suggested using
structural and parametric optimization for every
criterion with pairwise comparison and quality
assessment of the resulting product using an independent
quality functional and desirability scales [22, 23]. The
multicriteria optimization of the combined product
(nutritional and biological values) consists in building its
model according to the specified adequacy and quality
parameters, depending on the composition of initial
components.
For this, a parametric model of the product is
devised, taking into account:
6
Lisitsyn 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 (key
ingredients, fiber, biologically active additives, enzymes,
etc.); and
– structural relationships of biological value indicators
(amino and fatty acid compositions) according to various
compliance criteria.
Consideration is also given to what makes a balanced
diet for a certain population group.
An objective function is the minimum deviation
from the given structural group of nutritional and
biological indicators [22, 23], namely the criteria below.
(1) The optimization criterion for chemical elements
that determine the nutritional value P(z) of the designed
product:
 
;   min
min
1
min
1
1
1
1 1 
     

   
     

   






 



 
p
i
j
rj
n i
i
i
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i
L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
(7)
where
 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  



  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
is the reference content of the i-th element of
nutritional value;
bij is the specific content of the i-th element of
chemical composition in the j-th component of the
designed product;
xj is the mass fraction of the j-th component.
(2) The criterion of the minimum deviation from the
given structural indicators of biological value Pi(A), for
example, the monostructure of essential amino acids
(i = 1) and fatty acids (i = 2):
    
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  

 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj i k
(8)
where Ak
0 is the reference content of the k-th
monostructural indicator of biological value;
aki is the specific content of the k-homonostructural
ingredient in the i-th element of chemical composition;
bij is the specific content of the i-th element of
chemical composition in the j-th component of the
designed product;
xj is the mass fraction of the j-th component.
(3) The criterion of the minimum deviation from the
given structure Pi(V) of the vitamin composition (i = 1),
minerals (i = 2), and carbohydrates (i = 3):
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
1 1 
     

     


     

     









 

 


 
p
i
k
j
rj
p
n i
i
i
p
i
k
j
n
i
i ij
p
i
k
j
n rj
i
i
p
i
k
j
n
i
i ij
p
i
C X
C X A
X p
X p a
A
X p
X p a
n j
i
i
p
i
k
j
n
i
i ij
p
i
A
X p
X p a




 
1
1 1
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k
,
1 1 1 1 1
1 1 1 1 1
     
     
         
         
 



 


 
 



 


 

n
i m
i i i
m
i L
i
m
i L
i i i
m
i L
i
L
l
i i i
m
i L
i
m
i L
n
i m
i i i ij
m
i L
i
m
i L
i i i ij
m
i L
i
L
l
i i i i ij
j
X X p X Y X p Y X X p
X X p a X Y X p a Y X X p a
A
  
  
 
 

  


  


 
n
i
m
j
P z zi bijx j
1
2
1
( ) 0 min
  




  
     


     


 
n
k
m
j
ij j
m
j
ki ij j
i k i
b x
a b x
P A A
1
2
1
0 1 min; 1,2
  min 1,2,3
2
1
1
0 1  
     


     


 




 i
x
b x
P V V
n
k
m
j
j
m
j
kj j
i k ;
     
;   min
min


 

 


 
 
p
i
p
i
p
i
r
p
i C X
A X A X
A X A  

 
;   min
min
1
min
1
1 1
1
1
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where bkj is the specific content of the k-th ingredient in
the j-th element of chemical composition;
xj is the mass fraction of the j-th component.
Yu.A. Ivashkin proposed to use simulation modelling
to solve the problem of structural and parametric
optimization of a multicomponent product in various
combinations of linear and non-linear criteria and
restrictions. It involves “playing out” all possible
combinations of initial ingredients with subsequent
verification of restrictions and calculation of criteria.
Noteworthily, any problem of the NPIII class can
be solved by simulation modelling. The complexity
depends on the number of ingredients in the formulation.
If the space of problem solutions is very large, this
method may take longer than a “reasonable” time to
produce results.
A.E. Krasnov et al. used new information
technologies to produce foods of given quality. In
particular, they created sausage meat formulations
under varying conditions of uncertainty. Their study
showed how to transform the problem of stochastic
programming with uncertain target criteria into
the linear programming problem with stochastic
constraints [24].
Considerable attention is now paid to modelling
interactions between food components based on the
laws of equilibrium statistical thermodynamics. For the
first time, a polynomial dependence was found between
the properties of food mixtures and the mass fractions
of their ingredients. Its relationship with macroscopic
thermodynamic parameters of the mixtures was also
shown.
Scientific modelling of multi-component food
products with a specific set of nutritional and energy
indicators is still relevant worldwide. An ever wider
circle of researchers are engaged in various aspects of
improving food technology.
E.I. Muratova et al. proposed an object-oriented
approach to designing multicomponent food products
(for example, confectionery). A distinctive feature of this
approach is presenting a formulation as a hierarchical
structure (the Saati method) [25, 26].
Each vertex of this structure is an object (raw
materials – semi-finished product – finished product).
Each level is a certain stage of food technology that
can have its own number of vertices located lower
in the hierarchy. The algorithm for calculating a
multicomponent product begins with the lowest level in
the 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 the
second is the number of a component in the formulated
mixture. When several semi-finished products are used
at the same level, their first index becomes a composite
and is indicated as a list (i, j), where i is the level number
and j is the serial number of the semi-finished product at
the i-th level. This composite index is used lower in the
hierarchy (shown by the dashed arrow).
The algorithm for calculating the multiphase
formulation begins with the lowest level in the longest
III In the theory of algorithms, the NP (non-deterministic polynomial)
class refers to a multitude of decision problems whose solutions can
be verified on a Turing machine within a certain input polynomial
time, if there is some additional information (the so-called solution
certificate) [39].
7
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
Figure 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 to
Fig. 2, the calculation of the formulation begins with
the semi-finished product FP (2, 1), since the path to its
components is the longest in the hierarchy. The initial
data for calculating the lowest level include the loading
of all types of raw materials and semi-finished products,
loss of dry matte, and a given amount of finished
products equal to 1 t.
According to the authors, the main advantage of this
approach is the object-oriented representation. It allows
for inheriting properties and methods while adding new
calculation formulas that take into account new raw
materials, production features, as well as technical and
economic indicators of the processes.
O.N. Musina and P.A. Lisin proposed a methodology
for system modelling of multicomponent food products
[27–29]. They defined system modelling as a strategy
for studying and creating biosystems, particularly
food products, their formulations, and production
technologies.
The basic principle of system modelling is the
decomposition of a complex biosystem into simpler
subsystems. This is a principle of the system hierarchy.
In this case, the mathematical model of the system
is based on the block principle: the general model is
divided into blocks which can have relatively simple
mathematical descriptions. All subsystems interact
with each other and constitute a common unified
mathematical model.
Figure 3 shows a visual interpretation of the basic
principles of system modelling of multicomponent
products.
System modelling principles allow for the
decomposition of the production system at the stage
of formulating composite mixtures using linear
models. In such models, mathematical dependencies
(equalities or inequalities) are linear with respect to
all variables in the model. Problems of this kind are
used to select the optimal option from a set of possible
formulations according to a given criterion. In 1939,
the Russian mathematician L. Kantorovich and the
American scientist G. Danzig began to develop what
was later called “the simplex metho”. It became a
universal method of linear programming used in solving
optimization problems.
A.A. Borisenko proposed a methodology for
optimizing multicomponent food mixtures using
universal mathematical methods. His methodology
allows for the development of foods with a given nutrient
composition [30, 31]. Taking into account certain
restrictions and permissible deviations of nutrient
mass fractions from the reference amounts, the author
proposed to use the Lagrange function and the system
FP
level 1
RM 1,1 RM 1,2
RM
(2,1),1
level 2
level 3
RM 3,1 RM 3,k
RM
(2,2),1
RM
(2,2),2
SFP 1,1 SFP 1,2
SFP
(2,1)
Liebig’s principle
Minimization
Biosystems operate optimally when
the body receives minimum (reference)
amounts of every nutrient
For example, a food product can be
considered as a combination of vitamins,
minerals, fatty acids, and other systems
or as a combination of chemical elements
Every system is complex so multiple
models are required to understand
how it works, each describing
only one of its aspects
Multiplicity of food system
descriptions
Product properties
Functionality
Manifestation of certain properties
(functions) during interaction
with the external environment
The designed product is considered as a whole,
and its ingredients, as subsystems
Integrity
 The biosystem is considered as a whole
 The biosystem’s integrity means that each
ingredient in a multicomponent product contributes
to its quality
Compliance
Components are integral parts of the product,
they are structural elements that make it a whole
and without which it cannot exist
The level of nutrients’ compliance
with reference values
Basic principles of system modelling
of multicomponent foods
Hierarchy
 Analysis of the system elements (ingredients)
and their relations in the product structure
 The system’s functioning is determined by
the properties of the product structure, rather than
the properties of its individual elements (ingredients)
Structuredness
Manifestation of certain properties (functions) during
interaction with the external environment
8
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
of equations in the form of conditions of the Kuhn-
Tucker theorem for convex programming. Solving these
problems produces a vector of component mass fractions
to ensure the most balanced nutrient composition.
The author concluded that the most balanced
formulation cannot always guarantee the highest quality
of the finished product. Therefore, in most cases, there
is a need for a fairly wide range of formulated options.
To achieve that, he proposed to optimize formulations
in two stages. The first stage of modelling a formulation
involved determining all possible quantitative ratios
of the ingredients. The second stage was a qualitative
assessment and selection of several most optimal
variants. The author used Harrington’s desirability
function as a general criterion for quality assessment.
A.Yu. Prosekov developed the principles of forming
dispersed food systems and designing functional
products from modern perspectives [32–34].
T.V. Sanina and Yu.S. Serbulov proposed a
differentiated approach to a comprehensive assessment
of highly nutritional bakery products. The authors
believe that consumers should select key quality
indicators for foods with increased nutritional value to
make their assessment objective. In addition, quality
assessment should check if the product satisfies certain
needs consistent with its purpose [35].
A.A. Zaporozhsky et al. formulated new gerodietetic
products with specified qualitative characteristics
based on natural raw materials. For this, they used a
methodological approach and the principles of modern
nutrition, qualimetry, food combinatorics, and neural
network approximation of theoretical (estimated) and
experimental data [36, 37].
T.Yu. Reznichenko et al. substantiated an integrated
technological approach to the development of functional
foods enriched with biologically active substances and
dietary fibre. They studied the factors that determine the
quality of specialized products and critical control points
that identify their functional character at the stages
of production and distribution. They also developed a
range of consumer properties that included functional
indicators in addition to sensory and physicochemical
characteristics. Finally, the authors developed an
algorithm to examine a functional cereal breakfast
bar [38, 39].
V.M. Kiselev and E.G. Pershina looked at the
production and consumption of functional foods as
a multi-factor system subjected to comprehensive
assessment. They used the methods of food
combinatorics, parity of needs, and the vital concept,
taking into account modern requirements of nutrition.
With this approach, the authors studied a possibility of
evolutionary development of functional food design
based on food combinatorics. They identified consumer
preferences for functional foods and systematized them
in a model of consumer value [40].
O.N. Krasulya et al. considered the design of
multicomponent foods based on the functional and
technological properties (FTP) of their main raw
materials and ingredients. They also took into account
the kinetics of biochemical and colloidal processes, as
well as analytical and empirical relations characterizing
the main patterns of heterogeneous disperse systems
with varying physicochemical factors [42, 43].
In the age of digital (information) technologies,
the design of multicomponent food formulations
can be improved by using linear, experimental and
statistical programming methods, or an object-oriented
approach. M.S. Koneva et al. proposed using neural
network technologies [44]. The relationship between
sensory criteria and the quantitative composition of
the formulation was identified by neural network and
regression analysis of the ranking score of sensory
characteristics. The model parameters were obtained
with Statistica software. The convolution of the
balancing index and sensory evaluation was proposed as
a multiplicative desirability function. MathCAD scripts
were used to optimize the composition of antianemic
smoothie for pregnant women.
N.A. Berezina et al. developed a program in Object
Pascal for designing gerodietetic bread compositions
[45]. The technological adequacy of the flour mix,
which ensured a stable quality of the final product, was
modelled by introducing the flour technological indicator
(“falling number”) calculated using the Perten formula.
The mathematical foundations of solving singlecriterion
optimization problems are quite well studied
today. However, various areas of engineering, research
and management have multicriteria problems in which
several criteria need to be simultaneously optimized.
M.A. Nikitina and I.M. Chernukha proposed using the
Pareto method for multicriteria optimization [46].
The informational aspects of modelling and
evaluating the nutritional adequacy of raw materials and
finished products are very important in improving the
quality and technology of specialized multicomponent
food products.
CONCLUSION
The analysis of literature on the principles and
methods of designing balanced foods showed that
the initial stage in this process involved formalizing
qualitative and quantitative assumptions about
the rational use of essential amino acids in the
adequate exotrophy technology. N.N. Lipatov’s
contribution to designing balanced formulations in
Russia cannot be underestimated. His principles of
creating multicomponent foods and balanced diets
are still relevant today. Further development of food
combinatorics was related to informational and
algorithmic aspects of food design.
The conceptual approaches to computer-aided
food design proposed by N.N. Lipatov (Jr.) are used to
model functional products with specified qualitative
characteristics. Based on the optimal choice of raw
materials and ratios of ingredients, they result in
formulations whose nutrients (essential amino acids,
9
Lisitsyn A.B. et al. Foods and Raw Materials, 2020, vol. 8, no. 1, pp. 2–11
unsaturated fatty acids, macro- and microelements, and
vitamins) are consistent with the medical and biological
requirements in terms of quantity and quality.
The computer systems and software products
actively used in Russia to automate technological
calculations for food and diet formulations include
Etalon, Generic 2.0, Food & Life, CheesePro 1.0,
ShkoOptiPit, and others. They are based on the
databases of foods and raw materials, scientific research
and industrial experience, as well as mathematical
methods of modelling and designing food covered
in the works of I.A. Rogov, A.M. Brazhnikov, N.N.
Lipatov (Jr.), and other scientists. With the help of
those systems, new types of products were developed
by Moscow State University of Applied Biotechnology,
Gorbatov All-Russia Meat Research Institute, Research
Institute of Baby Food, All-Russia Research Institute
of Dairy Industry, and other institutes. These products
had 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) are
based on calculating the individual’s daily energy intake
and their need for basic nutrients.
Designing foods in the digital age, we need to take
into account not only nutritional and biological values,
but also medical, technological, economic, social, and
other factors. Computer technologies allow us to address
problems with numerous parameters, alternatives,
and criteria, as well as restrictions and conditions.
By processing and formalizing data, they help us find
optimal solutions based on complex optimization models
and objective assessment of options.
A need for “digital nutritiology”, a new scientific field,
was highlighted in Decree of the Presidium of the Russian
Academy of Sciences No. 178 dated November 27,
2018 “On the Current Problems of Optimizing the
Population of Russia: Role of Science” (paragraph 11).
This new direction is supposed to translate into the
language of numbers our physiological needs for energy,
nutrients, biologically active substances, and balanced
diets, on the one hand, and the chemical composition of
foods and general diets, on the other.
CONTRIBUTION
The authors were equally involved in writing the
manuscript and are equally responsible for plagiarism.
CONFLICT OF INTEREST
The authors state that there is no conflict of interest.

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