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Foods and Raw Materials, 2020, vol. 8, no. 1 .„..,,,.. .„._
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DOI: http://doi.org/10.21603/2308-4057-2020-1-12-19 Available online at http://jfrm.ru/en/
Multi-criteria food products identification by fuzzy logic methods
Lev A. Oganesyants , Vladislav K. Semipyatniy* , Aram G. Galstyan , Ramil R. Vafin , Sergey A. Khurshudyan , Anastasia E. Ryabova
All-Russian Scientific Research Institute of Brewing, Non-Alcoholic and Wine Industry, Moscow, Russia
* e-mail: [email protected]
Received May 17, 2019; Accepted in revised form July 11, 2019; Published February 25, 2020
Abstract: The paper deals with the theory of fuzzy sets as applied to food industry products. The fuzzy indicator function is shown as a criterion for determining the properties of the product. We compared the approach of fuzzy and probabilistic classifiers, their fundamental differences and areas of applicability. As an example, a linear fuzzy classifier of the product according to one-dimensional criterion was given and an algorithm for its origination as well as approximation is considered, the latter being sufficient for the food industry for the most common case with one truth interval where the indicator function takes the form of a trapezoid. The results section contains exhaustive, reproducible, sequentially stated examples of fuzzy logic methods application for properties authentication and group affiliation of food products. Exemplified by measurements of the criterion with an error, we gave recommendations for determining the boundaries of interval identification for foods of mixed composition. Harrington's desirability function is considered as a suitable indicator function of determining deterioration rate of a food product over time. Applying the fuzzy logic framework, identification areas of a product for the safety index by the time interval in which the counterparty selling this product should send it for processing, hedging their possible risks connected with the expiry date expand. In the example of multi-criteria evaluation of a food product consumer attractiveness, Harrington's desirability function, acting as a quality function, was combined with Weibull probability density function, accounting for the product's taste properties. The convex combination of these two criteria was assumed to be the decision-making function of the seller, by which identification areas of the food product are established.
Keywords: Fuzzy logic, Harrington's desirability function, identification criteria of food products, identification areas
Please cite this article in press as: Oganesyants LA, Semipyatniy VK, Galstyan AG, Vafin RR, Khurshudyan SA, Ryabova AE. Multi-criteria food products identification by fuzzy logic methods. Foods and Raw Materials. 2020;8(1):12-19. DOI: http://doi. org/10.21603/2308-4057-2020-1-12-19.
INTRODUCTION
In the food industry, the task of identification - that is, determining the attribution of a food product to a particular class in terms of condition, quality and taste characteristics - stands alone. For the solution of this task there exist: a set of criteria both measurable and expert; typical characteristics that product clusters must meet; and stratifying borderline values [1-5].
At the same time, all the obtained relations are empirical. Besides, as discriminatory criteria are construed, product clusters often intersect according to some measured parameters, so it makes sense to introduce a characteristic of attribution [6]. The latter would be a unit ("the sample certainly belongs to this product cluster") in cluster centers and would decrease at the borders ("the sample belongs to some extent to one cluster and to some extent to the neighboring one"). This would allow making product identification mere transparent and applicable to real food applications [), 8].
The method of fuzzy sets theory application to the problems of the food industry, proposed in this paper, will create lax regulatory restrictions on the composition, quality and sanitary characteristics of the product, taking into account the varied errors of methods and measurements. The purpose of this research was to provide food industry experts with a tool that allows building a robust multiparameter identification criteria based on empirical product data.
STUDY OBJECTS AND METHODS
The concept of fuzzy sets as applied to the food industry. In order to define fuzzy set A for elements rf R", enter the indicator membership function':
XiS*) n [0,l],x n R" (1)
: Hereafter: the indicator function and the membership function are interchangeable concepts
Copyright © 2020, Oganesyants et al. This is an op en access erticle distssbutsd enderthe te rms of the Creative Commons Attri)ution 4.0 International License (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, evsn eommercially, provided ttie original work is ermpeelysited and etates its 1)ense.
Fuzzy logic
¿5, §
1.0 0.8 0.6 0.4 0.2 0.0
Boolean logic
6 8
Figure 1 Fuzzy and Boolean approaches to the definition of a set consisting of an element {4}
Concurrently, the set in the classical sense of, defined in this way, is a special case of ^4*, a fuzzyset:
^.(x)e{0,3},xe
(2)
Thus, fuzzy logic extends theBoolean onewith two values {0,1} to the continuum of values in the interval of [0,1]. The difference between the approachesisshown in Fig. 1. Mostoften,the value X4OO isinterpretedas a subjective assessment of x as attributedto A,forexample X4 O) = 0.9 means that x is 90%of A [9].
The example of interpretation contains the word "subjective", which presupposes the possibility of each subject having an opinion concerning the relationship of each specific set attribution on the basis of their own indicator function. For food industry, this means the need of using a consensus membership function for each criterion; the function based on a particular food industry experts' consolidatedopinion, aswell as confirmedexperimentaldata[10].
The subjectivity of assessment also implies the existence of a method of translating psycholinguistic conclusions about the considered attribution to the digitaldomainof the indicator function.
The concept of a linguistic variable includes the object under study, as well as a set of natural language phrases (linguisticlexemes)thatthevariable can take in a fuzzy sense. The method of establishingthe relevance is individually selected for each industry and case of study. Common sense is one of the primary factors, since the number of linguistic lexemes used by experts and intended for digital transformation is extremely diverse, for example: "true", "false", "almost false", "almost true", "unknown", "possible", "sometimes", "maybe",etc.(Fig.2).
The main prerequisite for the use of fuzzy logic as applied to the food industry is the inability to build clear relations and criteria that link the qual"ty and performance of products and are not subject to multiparameter, unamenable to expression, factors of influence and measurement errors[11].
In a way, the definition of a fuzzy sel via the indicator function contains neither lack of focus nor
ambiguity, so it is possible to use the fuzzy logic framework for setting standards and identification methoda in the food iudustry.
Basic operations with fuzzy sets. Let us examine in mooe detai1 the passible fuzzy sets manipulations and highlight the most common operations in terms of the food industry (ia fuzzy logic it is impossible to identify a fintte aet ol basit Ounctions, thoougt which ull the others could be expressed; besides, operations on sets become "Mutrsd") U2-1"].
Canstder "he sets A, Bun —u. Tha rulotionof inclusion of the A set B into:
¿cß^x^xOrS^CxXVxe
(3)
The most practical option for constructing fuzzy negation  is:
-,0° = 1 -X4OO
(4)
There is an unlimited number of simpla ftirzy negations; besides, this methad is convenient foe constructing linguistic expert models, for example, thenegationfor "uakoown" (/¿(CO a m5, witl also bo "unknown".
The expansion o( conjunction (operation "AND") loir fuzzysets iscalled the t-nonm (01 rriengularnorm), and the expansion of dtsjunctian (operation "nR") ii oa(Od thes-norm.Inpractice, mostcommonly used are: Thelogicalproduct of .,405 ant( sutn .Alls:
X^bM = mmCr^xUiOx)) J4iB to = maxC^M^Cx)) (5)
Thealgebraicproductof .4tB andsum yl+B
False True
Almost false
Unknown
Almost true
Figure 2 An exampleofrelationbetweenthelinguistic "attribution" variable andtheintervalsoftheindicatorfunction
x^bM mto4 0j0 -/BRO
X4+b to 8 rZ to + Xb to o- X4 to ■ "Tb to (6)
The presented pairs 08 l and s- norms are called dual, since when usingthe abovenegation, de Morgan's laws are implemested in a fuzzy form, which makes their application praciimalgy convenient iu calculations.
Failure of the law of complementarity in the general case must be noted as an important fealure of fuzzr logic. Denoting t=norm as &, s-norm as |s we have:
^&rto ^ 0
n^GO < 1 (7)
The postulate (es Bookau algebra "some criterion and ies negotion are simultaeerutty unjurt" elates -he introduction oS intermediate variants. In particular, that ofthelexeme "ut^nrwn", since it and it- negation nre aseumed to be si^^Datt^gu-S;^ end equally Mr. This fart demwnetrates the coexngeenl) of ihe uroferty and its negation.
With muiti-crieetis ieemtifieation of -Food produ^s it is oftan ndcfseary So assign weoght numbers )or each individiaal criterion whi-e obtaoning ilie eggregoit indicator quality funetiof. uo do tins, cunve x iniegtation with A coefficient: (denoted os Sjeaaf1 "t used:
*(xaj+Ato = ^S*) + (1 m >a)tfs (.jj:) iS)
This formulm is eatity gennrolized for the easa o-criteria. auppoaing (h8reare fezey seds te2, ... , tem, wliere jijCH" thoir convex integration will have the form:
m
zd+o + ¿e^mm"-)
¿ = S
(a \ A
iS 7 (9r
Zm
A/ m 1
l = 1
Construcding indieator functions,, it is use1ul to control thesmoothness anrl ¿peel o° the transition of one linguistic concept to anofher. To do this, we use a power function that dffims jfa as fofiows:
)^eCOmipCif(mi(>o (lot
If a < h, the function reduces the requirements for membership to tlue set i4a with iespect to A, at a m h, thefunctionclarifiesit.
Linear fuzzy classification. Fro> tlri standpoine of the nrobabilihy tltewry -he indicztor function can be interpreted ns conCitiowcl prodabilify
e^to as Pi* n >4 I -5- = a (11)
that is, the probability of mem>eruhip to the set of a random variabla X, provided that it was implemented
Fat content, %
—•— Fat-free —— Low-fat Medium-fat —a— High-fat
Figure 3 Fuzzy classification of drinking milk by fat content
by x value. It should be noted that this is the basic dffferenee between the approaches: fuzzy logic operates by the deg=ee of membership to a particular set. While probability thoory (and "probabilistic" logic) indicates the probability of occurrence of mutually exclusive ev e nds.
As an example, consider the fuzzy classification of drinOing milk by fat candent .Fig 3f" A"cording to this classification, milk with a fat content of 3.75% is both 0.5 mediumqfat and 0.5 Wgh-fat. We consciously give no percentages here, becwuse O is not a matter of probability (otherwise, in a batch of milk with the samefert contentof 3.75%, halfof She bottles wouee be rer+anized ns "medium-fat", und the otlter half - as "high-fat", which makes no sense). Fuzzy sets exist in superposition with each ethew, this being the(r main ndvan+age im food idsntiPoation. (Sontiuuing the ewampie on the same classifier, ..8 milk of average fat content isactually ihe same as 0.2 of exfra fwt cootent, and this 8as a direut interpretation since two lineuistic postulates describing different degrees of one measurable criterion are associated.At the swme time, it sho+ld (o noted thft combiningprobabilistic and fuzzy methods has its own scope; besides, pro ability distributions can be used as indicator functions, as will be shown below.
In the example with milk fat linear functions are used to determine the degree of membership, being the most practically applicab{e for the food ind=itry due to th8 simplicity of construction and linguistic explanation of the result [15, 16]. In order to construct a linear characteristic function forsome criterion A on the domain^ n R there are three steps to follow: (1) Determinotron cO tCe intervaUr th jO, i = S ... n, w^eer; rwOh = r that is, belonging to such intervals is characterized bythelexeme "certainty teer"; (i) Determtnation of the intervals (n?, j-), y = 1 .., m, wwhee rett that is, their linguistic characteristics is "certainly No";
n State Standard 31450-2013. Drinking milk. Specifications. Moscow: Standartinform; 2014. 9 p.
1.0
0.8
0.6
0.4
0.2
Fat content, %
■ Fat-free
—a— Low-fat
Medium-fat
Fat content, % —High-fat
Figure 4 Approximationoffuzzyclassificationofdrmkmg milkbyfat content
(3) Combinaeion of "=1(0 iottrvals into oos "is) with tho length of k = or + ri find tori tham iee ascending oroler of the leftbordea. (incem the resulting list the intervals with the characteristics "certainty Yea" and "certainly No" win alternalte, gV remains oney to connect the boundarins lay Vnear funclion.
(a) goo OOies soquenrie (n;, y.O; (*;+h yg-i) tlie funclion wil° look like:
s yt
= ~ r _ _-ttn0'X 6 fright. (12)
'yel /i '"•^-t-i Ju
(It) =oo otie seqoonce 0x-,n'V (xi+t,—:0i) -he Uuvction wili look like:
so y/
hoiOO v--=77^-;::-rrv^ [yiUiet] (1t)
xiet 3t xiet
Of oruuee, in paction the most oommou case is that
with one trotV internal, and the funntioe italics hhe ieorm
of a trapezoid, os soen in thu grajili af milkclassification.
For one truth Internal the covvement app+txima-
tio n it: ^
(14)
1 n (="-"0/
where c isthe center of the interval, ) isits rungs, p is the smoothing fit. In the context of the example, rhe indicatorfunctionsfordniny pro ductswilltakethe form: ills.
1 +
lo.75j 1 + 1-ÜT75;-; 1 +11575; ^iTiTJ
for fat-free, lowfat,mediumfat and high fu products, respectively. The patterns of these funutions, as wett ae comparisonofthe two approachesare shown inFig. 0.
RESULTS AND DISCUSSION
To determine thevalueoftheindicatorfunction jfoOO at a particular point, it is sometimes necessary Oo resort to nested fuzzy sets. This happens, for example, when the values of the linguistic variables of the expert group differ for the samecriterionata point.Whena indicator function of a set is realized not by a specific number, but
byanother "ngicator function i) is cal"ed a second orOer fuzzy set. In practice, it is very difficult to ure sucle items, and they are absolutely unsuitable fot entabSssUing teg-rh relations between contractors o° the food Industry, in particular, producers and conoumeesi In Olus coer. insieod of the nested indicator function at a point. its integral valueisconsidered, for example, the consensus of 0X0611 e or the probability valua, if the "unction was reprcsenied bytheprobability density[17-l9b
As an exam=le, ^ons^id^ier a gritorion of a food product, whiah accordine to regulator. ""ooumsnts should fall into the interval [-1,1] (we nonsider ii as a fuzzy set I)with a meuyuoing deogce arrorof ± 0.5 The error was deliberate^ taten as com/ffable to the length of the interval for a mora visual demonslraiion of the behnvioi oU the mdicntor function ai the bohncltiry.
euppove Ohtt the rieaturement error e ~ ¿v(0, o-2) is a normally distributed random vtriable with zero expectntion and dcspeesion, whose value can = e determined from the instrument errot. In we assume thai 95.6% (which corresponds to ttie pt^obaliv^^ith ot a normally ditriliuted random variahle tailing witltin the range h relative to tie m^an value) of mnasurements of x fallwithin the range x ±0.5 (t+ie assumption cvn tie strengthtined or weakened depending on the conUitivno and the nature of the error). it meats; l^leat:
1
0 .5 = Iff, a = — 16
(16)
To construct the indicator function of ihe nriierion, let us ask:"what probability does theproduct setiefy tie criterion with if its measurement showed the result of?". Obviously,asecond ordtr luzzy snt emergos: foe na(h u there isan error probability density that can serve (after some manipulations) asa nested indicate! fynction. As previously stated, it is mtire ctinvenietit to aseume t°e ijoO(or;il vatuw as Ohe rulue a): ilie onini, l=ot is, fi"oni n proaabllisltc ri^wtiht of vtew, to coloulate the condttioval probahihty P(;«s- e 11 x), whvoo g = x -o e is the real valve of the indieator. In Ohis case, Xi ^^ao-O is the possMe reat valuo x n £ on the interval [-t,(] probability density
-3-2-10 1 2 3
x
-Criterion interval -----Specified interval
Figure 5 Indicator functionsofthe interval criterion in strict and fuzzy approaches
integral (i.e., the probability of x + £ falling into the specifi ed interval):
co-b fi x + £ < ee = p(+ + £ < n) - P(I +
(( £ < -b) = d(£ < b y x) y P(£ < -b - x) (17)
Th) last expression is nothini1 but lhe difference bn)ween the disti^^bution functitns i>£) ( of thn random varitble e. The formula is:
x/00 g ^b - x) - ^-b -b
f*) ag^ + erf^VgOg (18)
erf x is the errorfunction, mcludedfor convenience of calculations in matty packajjes of m^a^tl^ematicnl data processing, in particular, MS Excel.
In the classical aparoach to identification, any measurement that falls witdin tbe interva< [u),= c 0.5 will be recognized as correspondingto the criterion (to simplify the example, the questions o= cdd^m- and outlier measurements are omitted here), while already at the values -1 and 1 the level o) aelonging to th) criterion in the fuzzy approach will be equal to only 0.5 (Fig. 5), and when approaching the boundaries of a largeinterval-1.5 and 1.5, there is no chancefor the criterion.Moreover, toprovide the characteristic "most likely the product has a criterion" (function value 0.8), the measurement value must fall within the range [-0.79, 0.79].
This approach should be taken into account specifically at the boundaries of the interval identification. For example, when establishing a boundary for foods of mixed composition with milk fat content the following definition is proposed: if milk fat content exceeds 51% of the total fat phase, the product is called milk-based. If it makes less than 50%
- milk-containing, respectively, with a measurement error of ± 0.5%. In this case1 product s containing milk fat inthe range oh [50.25%, 50.75%] wid cot belong to any specified class with a sufficiently high level of confidence.
Despite measurement errors, the boundary of identification clmsaes should be set withouh taking them into account. Regardless of the nature iexcept for the assumption of distribuiihn S2mmetiy) and the type of error at the point of the boundary, the indicator function of both daises wi.l be equa. to 0.5. TMs is a logical assumption to refer the product to a particular class if the measurement gave a boundary indicator. In the aboveexample,this b oundiiy wiU be the point 50%.
However, if indkator fnnctions of two identification classes, beieg adjncent linguistic characteristics of the same criterion, take the sa me value of 0.5 at a point, it makessensesetting a beunetary Cetween these classes at thispoint.Forthe multidimeesio nal case, the boundary willbe represented by a hyperplane, but in practice the dimension eechealing two ts rarefy considmred.
Harrington's Ounctioc as ae oxnmpae ot 'nOicntca function. One of (he cppliid tools ic the qualitaaice assessment of the developed food industry identification methodsisHarrington's de sirabil ity tunciion [20].
TWe iiea of Hnrrinmton's function is to transform the velues of We criteria into a dimensionless (esraMity seale that allows comparing and combining the chirecleriities of ernduets of different nature. It establishes (ompliance between experts' psycho-linguistic assessmetrts aed ea'ural iadiaators of criteria. In addition, it has all tlia necessary practical properties of the indicator function, wlùch allow s using it actively in fuzzy logic applicalioni.
Generally,Harrington's function is of the form of:
cî(x) m e6-"FW,X d ca cwf
where is a Xunmtian thai establishes a relation between the values of the ex^rimental variable and the d'mewsioxkis scale [21]. In practice, it is almnit always linear, being accountable for the shift and steepness of Harrington's function curve in accordance with apjdicetion needt. h is so as to correspond to ehe well-established mapping of the function value ieteavals 2o the lmgui(tic variable of desirability: "very good" - 0.8, 1; "good" - 0.63, 0.8; "satisfactory" - 0.37, 0.63; "bad" -0.2, 0.37; "verybad"-h0.37.
If there are n criteria with correseonding desirability functions ¿¡(x^the contolidafed esiimgteisexpressedas a weighted geometric mean:
d(xs.....xJm^ndixOrC Î=lWl (20f
The useful property of the function is insensitivity within the range of 0 to a values (estimates "very bad" and "very good", respectivelyf. It can be used in the construction of criteria linked to the product's shelf life.
Figure 6 Product qualityindicator function, classically
In food products - complex biological systems -quality deterioration is often subject to the exponential law, and one of the key Cactors is the change m microbiological parameters. So, Harrington's function can be considered as x^Ct) indicaSor function of Q fuzzy set - i.e., the products correspondlng to public hfalth reyulitions. He re storage time t > 0 isused as a product characteristic.
Sipyesing a product has 4 days' shelf-life. Let us first consider its vgidity iediuaior fuectien without the use of fuzzy logic (Fig. 6).
Identification areas a "the product meets the standards and is ready for consumption" and III "the product must be disposed pf" are shown, respectively. Within strict logic, at pgint {4}, it is expected that the function has a gap of the first kind. As an applicable rule, the function teflects rathur Cinderella's carriage qualitative characteristics before and after midnight than those of the actual food product.
Since shulf life urually has a margin of 20-25%, consider the following indic ator function:
X<2(t) = C
(21)
The function is a fuzzy negation of Harrington's function, but this is natural when smaller values are assumed to have a larger desirability value. Its graph is shown in Fig. 7.
In addition to the identification areas I and "H, whose linguistic characteristics remain the same, there appears area II - "theproduct is safe for use, but already undergoing degrading qualitative changes". In this sense, products from identification area II are no longer suitable for end-users and must be sent for extending shelf life processing (sterilization, canning, etc.). At the same time, it should be noted that, for example, when canning, a fuzzy logic device must alsa be used to establish the final shelf life of the produca from raw materials within the boundaries of identification area H.
As it was mentioned above, changaf in microbiological parameters have a direct impact on the quality of the product. Logically the phases
of microbiological cultures' development can be compared with Harrington's function identification areas; in particular, area I corresponds to the lag phase, area II - to acceleration and exponential growth of microorganisms phase, and area III - to deceleration and stationarity phase. At the same time, substrate and other biotechnological characteristics of change in the population of microorganisms are calculated for each specific product, which may lead to diversities in the general matches given.
Combination of the product's quality characteristics. The construction of indicator functions is inextricably linked with decision-making systems. In the case of one criterion (for example, safety, as described above), the fuzzy logic apparatus gives no clear advantage over a strict approach. In the end, all the contractors of the food industry (consumers, manufacturers, law enforcement agencies, etc.) make a binary decision whether a particular product sample complieswitha criterion [22, 23]. Due to the fact that the criterion is unique (for example, expiration date) they identify the above decision with the function of the ultimate goal ("buying" vs. "not buying", "recalling" vs. "not recalling", "fining" vs. "not fining").
In the example with the fety function, three clear identification areas can be introduced. For them, for instance, the seller will have a system of specific actions (I - "selling", II - "reselling for recycling", III - "recycling").
However, even when the second criterion in the decision-making system is engaged, it is much harder to establish the precise boundaries of identification classes.
Consider the instance with the safety criterion with an additional indicator "consumer quality1 - a characteristic that demonstrates the taste en" overd° satisfaction from the consumption of the produce -added. In the fuzzy logic the unction of this indicator decreases faster than the safety function. For example, for baking and confectionery products, taste profiles degrade much earlier than the products become unfit for use. The taste of "fresh bread' fs o- great vahie to the consumer and has its impact fe their purchase prefere^es, but it is not unique or decisive, as shelf life es al to takee iafo aaceuait.
To construrt an exampln o0 ihe consumer quahty functio> Xr(t), let us usr tUe profubility theory apparatus, afsumifg thot fresh (t = o) producthassome taste profile lost on expiry [24]. In practice, it makes sense to put an experiment to determine the distribution histogram oy )he moment of fresh taste degradation. However, for the peaposes of exemplificutian, it will be simulated with the help of Wfibull distribution, used in survival analyais, givit" a good -pproximation in the study of products' storagn stabiltty [25]. The density of tUis yif!ribuliof Av(k,2)(t) has the form:
AvtMutO = ^tfc-1e-(t/A)ft,t > a (22)
Figure7 Productqualityindicator function based on Harrington's desirability function
As a quality ihnctigg we take the survival function Sw(fc, a) (i) fer the giveia a^stribution. It is equa( to the probabihty that the vahie of the raedom earisble under study will exeeed "n this fese, lhe frobabdity that the taste has not yet Ceen lost by i For ihe Weibull distribution, it has a convanicnherpprsdion:
^^ifceAilt) rp et-li/A)fe,t > a (23)
Suppling titat fo r a produa) w wt;h the safetyfunction describ ed by formula (23) the ave)age taste profile is lnii on she sta)nd yay, an approximaten of distribution parameters wi)h x = e, h = e can be derived.
The problem of food products' seller ii to establish the time when the product should afready be sold at the residual price (the time of entering area III) taking into account the safety and cunrumer quality criteria.
If the weight of safety indicator is set at 0.6, and the weight of consumer quality indic2tor is set at 0.4, respectively, a convex criteria combination (8) will take the form:
^y^lt) = 0.6(e e ee5dtc w 0.4 e-tt/2)a (24)
Solving the equation (0.63 being Harrington's
-xeW - safety criterion -----xr(t) - consumer quality
..........XiQ+Tf*® - combined qualityfunction
Figure 8 Convex combination of two consumer functions
function upper exponent for "satisfactory"):
*(Q+rvft6tO = 0.f3 (25a
we obtein t = 4. This means that after four days the product must be sold in traditional or alternative ways. Guided by rate expiry date only, the seller would get the value of 5, thus having no time left for operational maneuvers. The type of function graphs and their convex combination is shown in Fig. 8.
CONCLUSION
Thus, the apparatus of fuzzy logic allows building multi-criteria decision-making systems in the food industry. They help effectively make decisions about products' quality and safety and, in the case of violations and arbitral bodies' involvement, differentiate the administrative impact on the contractors of the food industry.
CONFLICT OF INTEREST
The authors declare that there is no conflict of interest related to this article.
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ORCID IDs
Lev A. Oganesyants https://orcid.org/0000-0001-8195-4292 Vladislav K. Semipyatniy https://orcid.org/0000-0003-1241-0026 Aram G. Galstyan https://orcid.org/0000-0002-0786-2055 RamilR. Vafin https://orcid.org/0000-0003-0914-0053 Sergey A. Khurshudyan https://orcid.org/0000-0001-7735-7356 Anastasia E. Ryabova https://orcid.org/0000-0002-5712-2020
