Agri-Food Management and Sustainable Practices: A Fuzzy Clustering Application Using the Galois Lattice

Abstract

The objective of this study was to generate groups of agri-food producers with high affinity in relation to their sustainable waste management practices. The aim of conforming these groups is the development of synergies, knowledge management, and policy- and decision-making by diverse stakeholders. A survey was conducted among the most experienced farmers in the region of Nuevo Urecho, Michoacán, Mexico, and a total of eight variables relating to sustainable waste management practices, agricultural food loss, and the waste generated at each stage of the production process were examined. The retrieved data were treated using the maximum inverse correspondence algorithm and the Galois Lattice was applied to generate clusters of highly affine producers. The results indicate 163 possible elements that generate the power set, and 31 maximum inverse correspondences were obtained. At this point, it is possible to determine the maximum number of relationships, called affinities. In general, all 15 considered farmers shared the measure of revaluation of food waste and 90% of the farmers shared affinity in measures related to ecological care and the proper management of waste. A practical implication of this study is the conformation of highly affine clusters for both policy and strategic decision-making.

1. Introduction

In recent years, the agricultural exploitation of land has increased at exponential levels. The intensification of agricultural activity increases productivity and profitability while increasing environmental costs [1]. In order to minimize or eliminate the associated damages, numerous sustainable agriculture narratives have emerged, proposing solutions for the challenges facing the food system [2]. New paradigms that have emerged in recent decades focus on sustainable agriculture and new biorefineries that increase food, biofuels, and ecosystem security [3].

The ability of agriculture to meet the food demand imposed by the accelerating growth of the world’s population without devastating the environment poses a major challenge [4]. Sustainable agriculture based on active collaboration between farmers relates to important concepts that have a significant impact on the development of sustainable agriculture, striving for social and economic development, as well as supporting a reduction of the impact on the environment [5]. A concept linked to sustainable agriculture is agri-food management, which covers the use and handling of all foods that originate in the fields and those that are handled and treated by the companies that transform or distribute them. Mourad [6] refers to the choice of measures aimed at reducing, reusing, and recycling any product considering economic and environmental factors [7]. Agri-food management includes food waste, defined as food initially produced for human consumption but then discarded [8]. Various actors are involved in the food waste system based on several factors, such as the environmental impacts associated with the inefficient use of natural resources (i.e., water, energy, and land) and disposal to landfill, which causes pollution [7].

There is a pressing need to move towards a more advanced technological approach to revalorize waste food into higher-value products [9]. Current food transformation processes must face the food waste issue by developing valorization processes to reintroduce by-products into the economic cycle, thus contributing to the circular economy, generating social and economic value, and ensuring the permanence of agricultural and rural activities [8]. The trend in food waste valorization focuses on those activities whose main target consists of the development of cheaper or improved raw materials and substitutive ingredients for the production of high-value-added products [10].

Agricultural sustainability represents an innovative paradigm of agriculture in which external inputs are minimized and farm assets are highly valued [7]. Agroecological production is based on farmers’ direct management of resources, active participation in the agricultural knowledge and innovation system, and interaction with the surrounding ecological natural systems [11]. Agricultural sustainability intensification involves increasing the yield of existing agricultural land while reducing its environmental impact [6]. Therefore, agricultural sustainability seeks to optimize on-farm irrigation and fertilizer scheduling concerning optimal crop yields and environmental impacts, which collectively represent trade-offs between conflicting objectives [7]. The study of Pagliarino et al. [11] focused on the experience of farmers, scientists, government officials, and private company managers experimenting with agroecology in rice production. This study paper uses ethnographic methods such as direct observations and in-depth interviews to understand the role of participatory research in sustainable agriculture and what good participation entails.

The main motivation for the research was the generation of a Galois lattice, which shows various levels of groups of highly affine farmers and the specific characteristics that they share. The main contribution is to analyze and describe the Nuevo Urecho region as it possesses a high intensity of agriculture and sustainable practices. Please note that this region supplies agricultural products to a large part of Mexico [12]. In addition, once the level of synergies is established, through the graphical representation, we can know precisely the characteristics that are compatible between farmers, which can enhance public policies to enhance regional development.

This article is divided into six parts. First, an introduction is presented, followed by a preliminary section, the materials and methods, the results, the discussion, and finally the conclusions.

2. Preliminary Section

In this section, we present the Galois lattice and Galois theory.

The concept of the Galois lattice is beneficial for knowledge discovery since it straightforwardly illustrates different relationships between concepts and their subconcepts [13].

The Galois lattice is a graphic method of representing knowledge structures [14]. Galois lattices arose from group theory and geometry [15] and allow us to know exactly which groups are robust at determined characteristics with a significant level of confidence, thus allowing the creation of synergies at the allocation of resources [16].

Galois theory establishes a link between field theory and group theory. It enables us to simplify certain problems in field theory by translating them into problems in group theory. This translation aids in comprehending the problems more clearly and solving them more straightforwardly. Initially, Galois employed permutation groups to illustrate the interrelations among the different roots of a given polynomial equation [17].

Galois theory is founded on a remarkable relationship discovered between subgroups of the Galois group associated with a particular extension field E/F and the intermediate fields lying between E and F [16].

If G = Gal (E/F) is the Galois group of the extension E/F. If H is a subgroup of G, the fixed field of H is the set of elements fixed by every automorphism in H, that is:

$$F\left(H\right)=\left\{x\in E: \sigma \left(x\right)=x for every \sigma \in H\right\}$$

If K is an intermediate field, that is, $F\le K\le E,$ define:

$$G\left(K\right)=Gal\left(E/K\right)=\left\{\sigma \in G: \sigma \left(x\right)=x for every x \in K\right\}$$

In Galois theory, when we fix a subgroup K for the Galois group G(K), where G(K) represents the group of automorphisms of E that keep K fixed, the theory explores the connection between the fields that remain fixed under these automorphisms and the corresponding fixing groups. This concept of fixing groups and fixed fields forms the essence of Galois theory, as described by Edwards and Tardieu [17] and Artin [18].

Definitions of the theory according to [16].

Definition 1.

A lattice is a structure where elements are partially ordered, meaning there is a relationship between them, and any two elements always have a lowest common upper bound (LUB) and a greatest common lower bound (GLB). A complete lattice extends this concept by ensuring that every set within it possesses both a LUB and a GLB.

Definition 2.

In a given context, denoted as K, there are three components: O, F, and ζ. O represents a collection of objects, F stands for a set of attributes, and ζ is a function that maps pairs of objects and attributes to either 0 or 1.

Definition 3.

In the context K = (O,F,ζ), we can define two mappings from the power set of O (P(O)) to the power set of F (P(F)), and from the power set of F (P(F)) to the power set of O (P(O)). These mappings will be denoted by the same notation ζ.

$$\forall A\subset O,A^{\prime }=\left\{f \in \left.F\right|\forall o \in A,\zeta \left(o,f\right)=1\right\}$$

$$\forall B\subset F,B^{\prime }=\left\{o \in \left.O\right|\forall o \in B,\zeta \left(o,f\right)=1\right\},$$

$A^{\prime }$is called the dual of A, similarly $B^{\prime }$is called the dual of B.

Definition 4.

In the context K = (O,F,ζ), a pair C = (A,B) is termed a concept of K if and only if A’s complement with respect to F equals B, and B’s complement with respect to F equals A.

Definition 5.

The set A is referred to as the extent of the concept C, while B is termed its intent. This relationship is symbolized as A = extent(C) and B = intent(C). By establishing an ordering relationship based on the inclusion of intents, one can construct a Galois lattice or concept lattice.

Definition 6.

The complete lattice L(K) composed of concepts derived from the context K is termed the (general) Galois lattice or concept lattice.

The lattice (please see [19]) appears to be infinite and indirectly irreducible. However, despite its infinite nature, the variety it generates encompasses only a finite number of other varieties, contradicting a previous conjecture. If we envision this lattice as encircling a cylinder, elements with the same labels on the sides can be matched. As we progress from left to right, the central part of the diagram moves upwards, creating a helical pattern when wrapped around the cylinder. While the lattice extends infinitely (see [20]), we can narrow our focus to the variables relevant to our analysis and incorporate them into our model, as suggested by [19].

3. Materials and Methods

An empirical study was conducted in Nuevo Urecho, located in the Federal State of Michoacán, Mexico. This region represents an essential source of income, jobs, and development for the economy of the State due to its agricultural productivity. For this reason, the empirical study was conducted in a sample of farmers from this region. Information retrieval was performed through an in-depth personal survey of 15 expert farmers. These farmers are considered experts due to having 25 to 35 years of experience producing crops using sustainable techniques. Most of them have been engaged in agriculture for 30 years. The agricultural products harvested yearly from the farms under study include mango, soursop, cassava, guava, mamey, and lemon.

3.1. Retrieval of Information

The proposed questionnaire was designed based on the UN [21] Sustainable Development Goals and the sustainable theory (see, e.g., [22,23,24,25,26,27,28,29]). It was carried out personally with each farmer, and 100% valid responses were obtained. The survey had 36 questions divided into 8 variables under study, which were included based on the work of several authors: Sustainability (15 items) ([30,31,32,33,34,35,36,37,38]). 1. Social (zero hunger); 2. Economy (profits, workforce, sale of compost, and benefit of eco-friendly products); 3. Environmental (sustainable agriculture) and agri-food management (10 items) ([9,10,39,40,41,42,43]); 4. Waste (management and sustainable fertilizer) 5. Revaluation (co-products and by-products); Supply chain (11 items) ([44,45,46,47,48]); 6. Post-harvest handling and storage (food waste, just-in-time); 7. Processing and packaging (food waste and certifications); and 8. Distribution (food waste, food monitoring, and transportation).

The first group of questions was related to the sustainability of farmers in the social dimension. It included questions on cooperation to end malnutrition, capacity to adapt to climate change, contribution to maintaining ecosystems, and ensuring agricultural food production for society; the second group of questions focused on the environmental dimension, asking about sustainable agriculture, the use of ecological products, the rational use of natural resources, prevention practices, and the reduction and recycling of food waste and inorganic waste; the third group of questions was oriented to the economic dimension, mainly regarding the operating, transportation, and investment costs of implementing sustainable practices.

The fourth and fifth groups of questions corresponded to agri-food management, specifically focusing on the waste and valorization of agricultural products. Questions were asked about handling and treating waste, particular areas for its correct decomposition, and valorization using the waste to make jams, pulp, or organic compost.

The distribution chain was studied in the last groups of questions; the sixth group corresponds to post-harvest handling and storage, addressing the aspects of information on the shelf life of the product, the amount of product that is wasted at this stage and how it is used, and the just-in-time delivery system; the seventh group of questions is composed of processing and packaging, the amount of food wasted at this stage, the use of biodegradable packaging, and the seals/labels of compliance with quality standards in packaging; finally, the last group of questions focused on distribution; farmers were asked about the amount of food wasted at this stage, the food return system, the use of refrigeration chambers in transport vehicles, temperature-controlled systems, and systems for detecting gas emissions.

The indicators to be measured were established for the questionnaire. The instrument presented five response alternatives on a Likert-type scale for each item, with a numerical value for each alternative, where respondents could only choose one option. The response options were as follows:

(1)

Totally false

(2)

False

(3)

Neither true or false

(4)

True

(5)

Totally true

In order to estimate the validity of the questionnaire, an SEMPLS model was used to create a matrix with the data to be used when running the algorithm. The measured indicators were used as pure data to estimate each of the different constructs. The partial regressions of the models estimated with the SEMPLS algorithm and its iterative procedure include two stages: in the first, the scores of the constructs are estimated while in the second stage, the weights and loadings are estimated, as well as the coefficients of the structural model and the R2 value of the endogenous latent variables [49]. Once the validity of the questionnaire had been assessed using SEMPLS, the results shown in the following table were obtained (

Table 1. Results of internal consistency reliability test.

	Alpha Cronbach	Reliability
Supply chain	0.889	0.906
Agri-food management	0.896	0.903
Sustainability	0.807	0.913

).

With all these questions, the objective is to determine whether the farmers of Nuevo Urecho (please see details of the surveyed farmers in

Table 2. Survey information.

Study Factor	Factor Value
Location	Nuevo Urecho, Michoacán, México
Time	May 2023
Farmers	15
Number of responses	15
Experience in agrifoods	25 to 50 years
Products	mango, guanabana, yucca, guava, mamey, and lime

) comply with the aforementioned sustainable characteristics, in order to promote synergy between them or their linkage with entrepreneurs and investors interested in this branch of agriculture.

Table 3. Sustainable agri-food variables.

Letter	Variable
A	Social
B	Economy
C	Ecology
D	Waste
E	Recovery/valorization
F	Post-harvest handling and storage
G	Processing and packaging
H	Distribution

shows the variables used in the study. These variables were retrieved from the literature review.

Table 4. Codification of farmers.

Letter	Codification
a	Farmer 1
b	Farmer 2
c	Farmer 3
d	Farmer 4
e	Farmer 5
f	Farmer 6
g	Farmer 7
h	Farmer 8
i	Farmer 9
j	Farmer 10
k	Farmer 11
l	Farmer 12
m	Farmer 13
n	Farmer 14
o	Farmer 15

codifies the 15 farmers using alphabetical letters.

Table 5. Linguistic ratings.

Totally True	T.C.	1
True	C	0.75
Neither true or false	NCF	0.50
False	F	0.25
Totally false	T.F.	0

presents the set of five linguistic scales used to measure each farmer’s competitive variable; the rating is set from T.C. (totally true) to T.F. (totally false).

In this part, the variables for the research development were detailed and the questionnaire was explained, as well as the parts that it is composed of and what is to be measured with each stage of the questionnaire. The farmers who responded to the surveys were also presented. From the responses, one can observe that 35% of the farmers proposed that, for the sustainability of waste, they are neither True nor False; for the majority of the responses, 29% were True concerning the sustainability that they apply in their processes for food waste, with 19% being Totally True, 10% Totally False, and 7% False.

Groups of Farmers

We applied the maximum inverse correspondence algorithm [50] to obtain general results. The aim was to generate groups of farmers who share common characteristics based on their responses to the applied questionnaire. In [51], we found a series of steps to apply the maximum inverse correspondence algorithm. We briefly mention the steps as follows:

Step 1. Each response of the interviewed farmer is registered as follows:

$$\begin{gathered} \begin{array}{c}i\\ ~ \end{array}=\begin{array}{c}A\\ \boxed{{{\mu }_A}^{\left(i\right)}}\end{array}\begin{array}{c}B\\ \boxed{{{\mu }_B}^{\left(i\right)}}\end{array}\cdots \begin{array}{c}N\\ \boxed{{{\mu }_N}^{\left(i\right)}}\end{array} \\ \begin{array}{c}i=a,b,c,\dots ,m\\ \hspace{1em}\hspace{1em}{{\mu }_j}^{\left(i\right)}\in \left[0,1\right] , j=A,B,\dots ,N \end{array} \end{gathered}$$

Thus constructing the following fuzzy matrix,

$$\left[\begin{array}{c}R\\ ~ \end{array}\right]=\begin{array}{cccc}a& \begin{array}{c}A\\ \boxed{{{\mu }_A}^{\left(a\right)}}\end{array}\begin{array}{c}B\\ \boxed{{{\mu }_B}^{\left(a\right)}}\end{array}& \cdots & \begin{array}{c}N\\ \boxed{{{\mu }_N}^{\left(a\right)}}\end{array}\\ b& \boxed{{{\mu }_A}^{\left(b\right)}}\boxed{{{\mu }_A}^{\left(b\right)}}& \cdots & \boxed{{{\mu }_N}^{\left(b\right)}}\\ & \dots & \dots & \dots \\ m& \boxed{{{\mu }_A}^{\left(m\right)}}\boxed{{{\mu }_A}^{\left(m\right)}}& \dots & \boxed{{{\mu }_N}^{\left(m\right)}}\end{array}$$

Step 2. Obtain a homogeneity matrix. Here, homogeneity indicates the desired level of each variable to be included in the model. For each sustainable agri-food variable, we evaluate and transform it by:

$$\begin{gathered} {{\mu }_j}^{\left(i\right)}\ge {\theta }_{j , }{{\beta }_j}^{\left(i\right)}=1 \\ {{\mu }_j}^{\left(i\right)}<{\theta }_{j , }{{\beta }_j}^{\left(i\right)}=0 \\ \begin{array}{c}i=a,b,\dots ,m\\ j=A,B,\dots ,N\end{array} \end{gathered}$$

Step 3. Choose the subset that presents the fewest elements in the homogeneity matrix.

Step 4. Create a power set (i.e., all the possible combinations of the subset with the fewest elements).

Step 5. For each entry in the power set, establish the connection to the right set, that is, the subset’s corresponding elements that have not been chosen for having a more significant number of elements.

Step 6. From every non-void subset of the connection to the right, include the corresponding subset with the more significant number of elements of the power set.

Step 7. Generate a visual representation of the resulting relationships. Here, the algorithm yields homogeneous groups of affine elements. These relationships create a Galois Lattice.

4. Results

Table 6. Results of the linguistic scale.

	A	B	C	D	E	F	G	H
A	0.50	0.50	0.75	0.50	0.75	1	0.50	0
B	0.75	0.25	0.75	0.50	0.50	0.50	0.50	0
C	0.50	1	0.75	0.25	1	0.50	0.50	0
D	0.75	1	0.75	1	1	0.75	0.50	0.50
E	0.50	1	0.50	0.50	1	0.50	0.75	0.25
F	0.75	0.75	0.50	0.50	1	0.50	0.50	0
G	0.75	0.75	0.50	0.50	1	0.50	0.50	0
H	0.75	1	0.75	0.50	1	0.75	0.75	0
I	0.75	0.75	0.75	0.25	0.75	0.75	0.50	0
J	0.50	0.75	0.75	0.25	1	0.50	0.50	0
K	0.50	1	1	0.25	1	0.50	0.50	0
L	0.75	1	0.75	0.25	1	0.75	0.50	0.25
M	0.75	1	1	0.75	1	0.75	0.50	0
N	0.75	1	0.50	0.75	1	0.50	0.50	0
O	0.75	0.50	0.75	0.50	0.75	0.50	0.50	0

shows the evaluation of each farmer according to their suitability with respect to each of the variables indicated.

Table 7. Evaluation matrix.

	A	B	C	D	E	F	G	H
a	0.6	0.6	0.8	0.6	0.8	1.0	0.6	0.2
b	0.8	0.4	0.8	0.6	0.6	0.6	0.6	0.2
c	0.6	1.0	0.8	0.4	1.0	0.6	0.6	0.2
d	0.8	1.0	0.8	1.0	1.0	0.8	0.6	0.6
e	0.6	1.0	0.6	0.6	1.0	0.6	0.8	0.4
f	0.8	0.8	0.6	0.6	1.0	0.6	0.6	0.2
g	0.8	0.8	0.6	0.6	1.0	0.6	0.6	0.2
h	0.8	1.0	0.8	0.6	1.0	0.8	0.8	0.2
i	0.8	0.8	0.8	0.4	0.8	0.8	0.6	0.2
j	0.6	0.8	0.8	0.4	1.0	0.6	0.6	0.2
k	0.6	1.0	1.0	0.4	1.0	0.6	0.6	0.2
l	0.8	1.0	0.8	0.4	1.0	0.8	0.6	0.4
m	0.8	1.0	1.0	0.8	1.0	0.8	0.6	0.2
n	0.8	1.0	0.6	0.8	1.0	0.6	0.6	0.2
ñ	0.8	0.6	0.8	0.6	0.8	0.6	0.6	0.2

presents the results for each farmer concerning the variables studied in the linguistic scale.

The desired level of homogeneity (i.e., the level at which farmers must comply according to specific characteristics, qualities, and peculiarities for each studied variable) is set as:

Please note that for our study, vector θ follows regional agri-food sustainable practices. This vector may vary depending on the specific conditions for each case, and it is retrieved directly from the final policy markers or study designers. A sensibility analysis, including various homogeneity vectors, is also recommended (

Table 8. Homogeneity matrix.

	A	B	C	D	E	F	G	H
A	0	0	1	1	1	1	0	0
B	1	0	1	1	0	0	0	0
C	0	1	1	0	1	0	0	0
D	1	1	1	1	1	1	0	1
E	0	1	0	1	1	0	1	1
F	1	1	0	1	1	0	0	0
G	1	1	0	1	1	0	0	0
H	1	1	1	1	1	1	1	0
I	1	1	1	0	1	1	0	0
J	0	1	1	0	1	0	0	0
K	0	1	1	0	1	0	0	0
L	1	1	1	0	1	1	0	1
M	1	1	1	1	1	1	0	0
N	1	1	0	1	1	0	0	0
Ñ	1	0	1	1	1	0	0	0

).

For our case, we define θ as:

θA = 0.8, θB = 0.8, θC = 0.8, θD = 0.6, θE = 0.8, θF = 0.8, θG = 0.8, θH = 0.4

Maximum inverse correspondence algorithm.

We used the theory of affinities to group the most affine farmers. Specifically, the maximum inverse matching algorithm was applied [50].

Studies and applications of this algorithm in economic and business sectors have led to relevant results when dealing with uncertain conditions [51]. The steps followed are described below:

Step 1. From the homogeneity matrix, the set with the fewest elements is chosen. In our case, this is: {A, B, C, D, E, F, G, H}

Step 2. The power set is created, which represents all the possible combinations of the set with fewer elements; in our case, this is:

{θ, A, B, C, D, E, F, G, H, AB, AC, AD, AE, AF, AG, AH, BC, BD, BE, BF, BG, BH, CD, CE, CF, CG, CH, DE, DF, DG, DH, EF, EG, EH, FG, FH, GH, ABC, ABD, ABE, ABF, ABG, ABH, ACD, ACE, ACF, ACG, ACH, ADE, ADF, ADG, ADH, AEF, AEG, AEH, AFG, AFH, AGH, BCD, BCE, BCF, BCG, BCH, BDE, BDF, BDG, BDH, BEF, BEG, BEH, BFG, BFH, BGH, CDE, CDF, CDG, CDH, CEF, CEG, CEH, CFG, CFH, CGH, DEF, DEG, DEH, DFG, DFH, DGH, EFG, EFH, EGH, FGH, ABCD, ABCE, ABCF, ABCG, ABCH, ABDE, ABDF, ABDG, ABDH, ABEF, ABEG, ABEH, ABFG, ABFH, ABGH, BCDE, BCDF, BCDG, BCDH, BCEF, BCEG, BCEH, BCFG, BCFH, BCGH, CDEF, CDEG, CDEH, CDFG, CDFH, CDGH, DEFG, DEFH, DEGH, EFGH, ABCDE, ABCDF, ABCDG, ABCDH, ABCEF, ABCEG, ABCEH, ABCFG, ABCFH, ABCGH, BCDEF, BCDEG, BCDEH, BCDFG, BCDFH, BCDGH, CDEFG, CDEFH, CDEGH, DEFGH, ABCDEF, ABCDFG, ABCDGH, ABCEFG, ABCEGH, ABCFGH, BCDEFG, BCDEGH, BCDFGH, CDEFGH, ABCDEFG, ABCDEGH, ABCDFGH, BCDEFGH, ABCDEFGH}.

Step 3. For each element of the power set, include the corresponding elements of the conjunct that have not been chosen for having a more significant number of elements. In our case, this is:

Table 9. Connection to the right. Source: Self-elaborated.

A	bdfghilmno	ABH	dl	DEF	Adhm	DEFG	h
B	cdefghijklmn	ACD	bdhmo	DEG	Eh	DEFH	d
C	abcdhijklmo	ACE	bdhilmo	DEH	De	DEGH	e
D	abdefghmno	ACF	dhilm	DFG	H	EFGH	-
E	acdefghijlkmno	ACG	h	DFH	D	ABCDE	dhm
F	adhilm	ACH	dl	DGH	-	ABCDF	dhm
G	eh	ADE	dfghmno	EFG	H	ABCDG	h
H	del	ADF	dhm	EFH	Dl	ABCDH	d
AB	dfghilmn	ADG	h	EGH	E	ABCEF	dhilm
AC	bdhilmn	ADH	d	FGH	-	ABCEG	h
AD	bdfghmno	AEF	dhilm	ABCD	Dhm	ABCEH	dl
AE	dfghilmno	AEG	h	ABCE	Dhilm	ABCFG	h
AF	dhilm	AEH	dl	ABCF	Dhilm	ABCFH	dl
AG	h	AFG	h	ABCG	H	ABCGH	-
AH	dl	AFH	dl	ABCH	Dl	BCDEF	dhilm
BC	cdhijklm	AGH	-	ABDE	Dfghmn	BCDEG	h
BD	defghmn	BCD	dhm	ABDF	Dhm	BCDEH	h
BE	cdefghijklmn	BCE	cdhijklm	ABDG	H	BCDFG	h
BF	dhilm	BCF	dhilm	ABDH	D	BCDFH	dhilm
BG	eh	BCG	h	ABEF	Dhilm	BCDGH	-
BH	del	BCH	dl	ABEG	H	CDEFG	h
CD	abdhmo	BDE	defghmn	ABEH	Dl	CDEFH	d
CE	acdhijklmo	BDF	dhm	ABFG	H	CDEGH	-
CF	adhilm	BDG	eh	ABFH	Dl	DEFGH	-
CG	h	BDH	de	ABGH	-	ABCDEF	dhm
CH	dl	BEF	dhilm	BCDE	Dm	ABCDFG	h
DE	adefghmno	BEG	eh	BCDF	Dhm	ABCDGH	-
DF	adhm	BEH	del	BCDG	H	ABCEFG	h
DG	eh	BFG	h	BCDH	D	ABCEGH	-
DH	del	BFH	dl	BCEF	Dhilm	ABCFGH	-
EF	adhilm	BGH	e	BCEG	H	BCDEFG	-
EG	eh	CDE	adhmo	BCEH	Dl	BCDEGH	-
EH	del	CDF	adhm	BCFG	H	BCDFGH	-
FG	h	CDG	h	BCFH	Dl	CDEFGH	-
FH	dl	CDH	d	BCGH	-	ABCDEFG	-
GH	e	CEF	adhilm	CDEF	Adhm	ABCDEGH	-
ABC	dhilm	CEG	h	CDEG	H	ABCDFGH	-
ABD	dfghmn	CEH	dl	CDEH	D	BCDEFGH	-
ABE	dfghilmn	CFG	h	CDFG	H	ABCDEFGH	-
ABF	dhilm	CFH	dl	CDFH	D
ABG	h	CGH	-	CDGH	-

presents the set of elements that have not been chosen because there is a power set that contains them; in other words, there is a power set with a larger number of elements.

Step 4. From every non-void conjunct of the connection to the right, we choose the corresponding conjunct of the power set that possesses a greater number of elements. In our case (

Table 10. Maximum inverse correspondence matrix.

A	bdfghilmno	BE	cdefghijklmn	ADE	dfghmno	CDEF	adhm
C	abcdhijklmo	CD	abdhmo	BCE	cdhijklm	ABCDEF	dhm
D	abdefghmno	CE	acdhijklmo	BEH	Del	ABCEF	dhilm
E	acdefghijklmno	DE	adefghmno	CDE	adhmo	ABCEH	dl
AC	bdhilmn	ABD	dfghmn	CEF	adhilm	ABCFH	dl
AD	bdfghmno	ABE	dfghilmn	DEG	Eh	BCDEF	dhilm
AE	dfghilmno	ACD	bdhmo	DEH	De	BCDFH	dhilm
BD	defghmn	ACE	bdhilmo	ABDE	dfghmn

):

At this point, we have found the maximum number of relations, named affinities. The algorithm creates the most considerable number of groups below the chosen homogeneity level through a precise method.

Step 5. The relationships between both conjuncts create a Galois lattice. Such a representation allows for a systematic demonstration of the homogeneous groups and a proposed structure of the elements. Figure 1 represents the proposed Galois lattice for our case:

Figure 1 shows the Galois lattice, which highlights that most farmers are affine in the variables measured individually; according to the combination of variables carried out, fewer farmers share these affinities. A total of 93.3% of farmers in the Nuevo Urecho area shared the characteristic of recovery/valorization, showing their concern and interest in giving new use to food waste. In this way, for stakeholders interested in the production of co-products or by-products, all of the farmers in this area can be considered to be suitable.

A total of 73.3% of the farmers were sympathetic to the ecological aspect of sustainability, while 66.6% of the farmers shared an affinity for the social characteristic of sustainability and care and interest regarding food waste. Eight possible combinations were found with affinity in two variables, most corresponding to waste and recovery/valorization. For a combination of three variables, 11 possible combinations were found, where the highest affinity was presented by the aspects of sustainability, social aspects, economic aspects, and recovery/valorization, with 53.3% of farmers presenting an affinity. Finally, the lowest affinity found was waste, recovery/valorization, and processing and packaging, with 13.3% affinity in farmers.

With five common characteristics, five categories were found, in which 33.3% of the farmers were similar in three categories, 20% in one category, and 13.3% in two possible combinations. In the combination of four related characteristics, only two groups were found, with 40% of farmers sharing the sustainable, social, and economic aspects and the characteristics of waste and recovery-/-valorization, while 26.6% of farmers were related in terms of ecology, waste, recovery-/-valorization, and post-harvest handling and storage. Finally, the maximum combination found, with six related characteristics including all sustainable aspects (i.e., social, economic, ecological, waste recovery-/-valorization, post-harvest handling, and storage), covered 20% of the farmers.

5. Discussion

Given that the adoption of best practices in terms of sustainability requires the adoption of techniques, culture, and information, it may be essential to highlight the value of grouping individual farmers that are affine. Through finding like-minded stakeholders, critical mass can be achieved quicker to influence consumers and institutions. One aspect that could hold a strong linkage could be eco-labeling, which informs consumers about the products they buy and how they were obtained or made. In the case of food products, agri-food management practices related to sustainability issues are communicated and shared via labels [36]. However, in Mexico and many other emerging markets, eco-label use is primarily driven by market leaders and multinational corporations, not small farm owners [52]. By identifying these affine linkages, cooperation can be initiated. Trade organizations and other forms of collaborative partnerships can be used to influence norms, trade policies, market awareness, and other important drivers toward more responsible production and consumption practices.

Individual businesses may have a set pattern of attributes that match competitors or collaborators but may also find that they are missing some potentially important attributes; that is, the information may also hold some benchmarking value to individuals who may not know what they are missing in terms of market standards. Furthermore, using unbiased grouping strategies such as the Galois Lattice can also help to determine patterns that would otherwise be overlooked.

Finally, it could be essential to understand whether the matching characteristics that define the sets of food producers are a local phenomenon or have a special resolution. It may be the case that different practices and groupings may occur for other regions with different agricultural practices, land, and cultivars, which may require different policy and management practices. This is to say that there may exist a relationship between the region under study and the results obtained through the questionnaire.

6. Conclusions

The objective of this study was to generate groups of farmers with high affinity regarding the sustainable management of waste for the creation of synergies, linkages, and stakeholder decision-making. For this purpose, a model based on farmer clusters was proposed, which compares the exposed variables inherent to their sustainability values, allowing for the grouping of farmers with similar affinities.

The model was based on the theory of affinities, specifically the algorithm of maximum inverse correspondence [51], to group the most affine farmers while, for the purpose of graphic representation, the Galois group theory was used. Through employing this process, the farmers were grouped with a certain level of significance. Ordered visualization of the groupings was conducted, allowing for the observation that the greater the number of variables studied, the smaller the number of farmers with these characteristics. In the other direction, the smaller the number of variables studied, the greater the number of farmers with affinity.

Based on the results, it can be stated with certainty that most of the farmers in Nuevo Urecho are very interested in carrying out activities in favor of the reduction of food waste, the recovery-/-valorization of food, and the social and ecological aspects of sustainability; other characteristics that they shared, with a smaller proportion, include post-harvest handling and storage, the economic aspect of sustainability, and processing and packaging, while the trait with the lowest affinity was distribution. One of the advantages of the Galois lattice is the visualization, at structural levels, of the composition of all shared characteristics among related farmers. Through the Galois lattice, farmers with similar affinities are presented, which may allow them to create work teams and synergies to achieve their objectives more quickly and efficiently. It also allows them to identify the variables that they are not considering within their fields, providing them with business and growth opportunities.

This contributes to the world’s sustainable development objectives, which have come under great pressure in recent years in order to meet global food consumption and waste management needs. The present work intends to shed light on this academic field by offering a model based on groupings with theoretical variables and subjective characteristics according to farmers’ perceptions, thus facilitating decision-making and guiding decision-makers by means of the results obtained, in such a manner that they can create synergies, linkages, and shareholder decision-making. Some of the limits of the study are that it does not contemplate the variable of green marketing and does not include the entire State of Michoacán. For future lines of investigation, it would be convenient to replicate the study with farmers from other regions of the country, in order to expand the results, compare them, and reach more precise conclusions.