The introduction is divided into two main sections: interval-valued intuitionistic fuzzy (IVIF) graph representations [1] and alpha-beta decomposition. In the mathematical field of lattice theory, every pair of elements in a substantially ordered set has a unique upper and lower bound [2]. Lattice theory has significant applications in domains such as artificial intelligence, computer science, and decision-making, demonstrating considerable potential for the grid idea. Intuitionistic fuzzy (IF) graphs, an extension of fuzzy graphs, form the foundation of the IVIF graphs [3–5]. IVIF graphs incorporate interval-valued intuitionistic fuzzy sets (IVIFSs), which represent membership degrees (MD) and non-membership degrees (NMD) using intervals rather than single values [6, 7]. This approach effectively captures uncertainty and ambiguity in data representation. The alpha-beta decomposition technique divides a lattice into two distinct components: the alpha component and the beta component. The alpha component consists of elements with a single immediate successor, representing the lower portion of the lattice. These elements are directly connected to the minimal element of the lattice. Conversely, the beta component comprises elements with a single immediate predecessor, representing the upper portion of the lattice and including those directly associated with the maximal element [8].

The alpha-beta decomposition systematically organizes the lattice structure, making it particularly useful for concept analysis, knowledge representation, and decision-support systems. By integrating the alpha-beta decomposition with intuitionistic fuzzy graph representation in C language, researchers can visualize and analyze uncertain lattice systems more effectively, reducing statistical uncertainty, improving intuition, and enhancing fuzzy logic capabilities [9, 10]. This combined approach provides a deeper understanding of lattice structures and broadens their applications. In mathematical terms, a lattice is a discrete group of elements in n-dimensional Euclidean space (ℜ_n) that repeats periodically. It can be visualized as an infinite grid or mesh, with points in the lattice represented as integer linear combinations of linearly independent basis vectors. Lattices are used in many mathematical fields, including algebra, wide variety theory, geometry, and physics.

Lattice structures exhibit several key properties:

Periodicity: If the lattice contains a point P, it also contains any point Q obtained by adding or subtracting a linear combination of basis vectors from P. In other words, the lattice oscillates back and forth.

Invariance under translation: Any point derived from translating P by a lattice vector is also within the lattice if a point P is in it, indicating the absence of a preferred origin.

Symmetry: Lattices often display symmetry with respect to rotations, assessments, or other transformations.

Fuzzy lattices [11] extend classical lattice theory to simplify inaccurate or misleading statistics. In fuzzy lattices, every element is a fuzzy set, with interactions governed by MD rather than strict ordering. This makes connectional representation more flexible, accommodating uncertain and imprecise information. Lattice operations are generalized to accommodate fuzzy degrees, with the join operation yielding the least upper bound and the meet operation yielding the greatest lower bound. In set theory, “decomposition” is defined as dividing a set into exhaustive and mutually exclusive subsets [12]. Fuzzy decomposition divides fuzzy sets into overlapping subgroups (often called additives or divisions) that reflect distinct qualities or functions. With MD 0–1, furnished sets provide factor departments with more options and nuance. Fuzzy decomposition enables the division of items into multiple subgroups instead of crisp units.

Alpha-beta fuzzy decomposition splits a complex fuzzy set into two components: beta (β) and alpha (α). The alpha component, which is critical to fuzzy sets, represents elements with high confidence or comfort in the set, often associated with stronger membership. In contrast, the beta component, a non-critical element of fuzzy sets, represents elements with lower confidence or weaker membership. This decomposition facilitates pattern recognition, fuzzy logic management, and decision-making.

Formal concept analysis (FCA) was first introduced by Wille [13] as a method for analyzing binary data matrices using principles from applied modern algebra. The information matrix ( ) represents the relationships ( ) between objects ( ) and a chosen attribute set ( ). FCA employs mathematical techniques to uncover hidden patterns, referred to as formal concepts, within the data matrix [14, 15]. These discovered concepts, when organized within the concept lattice, can facilitate various information processing tasks across domains [16, 17]. Fuzzy logic has been incorporated into FCA [18] to handle datasets containing fuzzy attributes [19]. Using an interval-valued fuzzy context [20, 21] and graphical visualization techniques [22, 23], this approach allows for the representation of bipolarity [24, 25] in heterogeneous data. When datasets exhibit similar ambiguous qualities, composition methods can establish links between attributes. Alternatively, analysis can be conducted using set-based approximations or different possibility analysis frameworks [26, 27]. To address attribute ambiguity and uncertainty, recent research employs two-way fuzzy spaces, a two-way decision framework, and their associated partial ordering [28–31]. These models can also be applied through differential equations and have found utility in ensuring the sustainability of supply chain management [32–34]. Given the critical role of the supply chain in business operations, the integration of fuzzy logic into differential equations provides a novel method for managing uncertainties, with applications extending to data analysis [35–37].

This study developed a framework to better understand two-way formal concepts and their hierarchical representation within the concept lattice. The framework enables a thorough examination of deficiencies in attributes related to truth membership functions (TMFs), falsehood membership functions (FMFs), and indeterminacy membership degrees (IMDs) [38, 39]. Previous research found that single-valued fuzzy membership methods were inadequate for addressing issues of partial ignorance, inconsistent representations, and unpredictability in datasets. Various approaches have been proposed to address these challenges, including interval-valued fuzzy formal concept analysis, interval-valued fuzzy graphs for concept lattice representation, and interval-valued fuzzy lattices [29, 40, 41]. However, despite these advancements, accurately capturing uncertainty in data remains a limitation. For example, in a dataset where 40 out of 100 voters support a petition, 30 oppose it, and 20 are undecided, IVIFSs struggle to adequately represent this distribution of uncertainty. Significant research has focused on exploring the properties of IVIFSs and their networks, as well as their potential applications in knowledge processing. One notable method for addressing partial ignorance in knowledge representation involves the separate use of interval-valued TMFs and FMFs to capture uncertainty more effectively.

According to Djouadi and Prade [42], using interval-valued fuzzy sets alone is insufficient to fully capture the partial uncertainty inherent in the provided fuzzy characteristics. To address this limitation, we propose employing interval-valued fuzzy sets as a key approach. These sets include the interval-valued TMF [℧^LI(xi), ℧^UI(xi)] and the interval-valued FMF [Ω^LI(xi),Ω^UI(xi)] which coexist within intuitionistic fuzzy sets I in a three-way decision space. These functions are defined over the domain of objects . Recent applications of these features span various fields, including relational databases, graph modeling [34], multi-decision-making processes [43], and FCA. Interval-valued fuzzy properties, which account for partial ignorance in data, provide valuable insights for deeper investigations. Examples include IVIF graphs, logic, and two-way fuzzy concept lattices [44, 45] By leveraging interval-valued fuzzy properties, significant hidden patterns in data can be identified and represented in formats that balance specificity and generality, enhancing comprehension. However, when users select specific core concepts at their desired granularity for truth and falsity membership values, the computational cost increases. Examples of computationally intensive approaches include bipolar fuzzy contexts, two-way fuzzy contexts, fuzzy concept lattices, formal concepts [46–48], interval-valued contexts, and triarchic reductions of formal contexts. To mitigate this issue, we propose an alternative strategy: identifying key conceptual themes through (α, β)-cuts, which independently define the TMF (α-cut) and FMF (β-cut). The goal of reduced interval-valued intuitionistic concept generation is to streamline decision-making and minimize computational overhead.

Existing methods using fuzzy sets, particularly IVIFSs, have been applied to address data ambiguity. However, they often fail to produce models that accurately represent the intricate relationships within complex datasets. This study seeks to demonstrate how IVIF graphs can enhance the accuracy and clarity of hidden pattern detection in complex datasets by introducing interval-valued intuitive properties in conceptual networks. By combining fuzzy sets and graphs with (α, β)-cuts for distinct truth and falsehood values, our proposed method offers a more nuanced understanding and representation of data patterns. These advancements over existing approaches are validated through practical examples, highlighting their benefits and potential for improved decision-making in fields such as data mining and knowledge discovery. To achieve these objectives, the research focuses on the following:

• · Proposing an approach based on the characteristics of IVIFSs to efficiently express partial uncertainty in data with fuzzy attributes.

• · Demonstrating how hidden patterns can be identified within an IVIF framework using the properties of fuzzy graphs and lattices.

• · Introducing (α, β)-cuts as a method to analyze the iIVIF environment, specifically targeting truth (α-cut) and uncertainty (β-cut) values, with an example illustrating the approach.

• · Using correlation techniques and operator results to empirically evaluate the proposed method.

In this section, we present the features of interval-valued fuzzy sets and their graphical sequence diagrams, which form the foundation for IVIFFCs. Using the proposed method, we construct and analyze bipolar gap-valued concepts, enabling a comprehensive examination of interval-valued fuzzy data.

4.1 IVIF Graph

Definition 4.1. Let xi be an element of the set , representing a collection of factors or objects. In this context, I within can be characterized using an interval-valued TMF denoted as ℧_I (ξ) and an interval-valued FMF denoted as Ω_I (ξ). For each specific point xi in , the values ℧_I (ξ) and Ω_I (ξ) are subsets of the interval [0, 1]. The representation of IVIFSs is as follows:

where ℧_I(§), Ω_I(§) ⊆ [0, 1].

Definition 4.2. Let I₃ represent the intersection of two interval-valued fuzzy sets, I₁ and I₂, within a universal set . The TMF and FMF of I₃ are computed as follows:

This operation identifies the infimum of any two given bipolar IVIFFCs.

Definition 4.3. Let I₃ represent the union of two interval-valued fuzzy sets I₁ and I₂, within a given universal set ,. The TMF and FMF of I₃ are computed as follows:

This operation identifies the supremum of any two given bipolar IVIFFCs.

Definition 4.4. Let G = (⋎,E) be an IVIF graph where the vertices (⋎) are described by an interval-valued TMF [℧Ξ⋎L',℧Ξ⋎I'] and an interval-valued FMF[ΩΞ⋎L',ΩΞ⋎I'], such as

Similarly, edges (E) can be characterized by IVIF relationships

for all ⋎ × ⋎ ∈ E such that

The IVIF graph is complete if and only if

Here, (℧_E (Ξ_⋎Ξ_⋌), Ω_E (Ξ_⋎,Ξ_⋌)) = (0, 0) ∀(vi, vi) ∈ (⋎ × ⋎/E).

The following section introduces a technique that examines the latent pattern within the IVIF context by leveraging the characteristics of the concept lattice across various granulations, as well as a technique that generates IVIF concepts using the previously defined IVIFSs and accounting for its concept lattice, partial sorting principle, and graph.

4.2 Suggested Process for Developing IVIF Concepts

We propose a method to identify formal concepts in IVIF data, leveraging intuitionistic logic, IVIF graphs, interval-valued lattices, and Galois connections. Assume an IVIF context , where , andℜ˜={(§⋎,⋎⋌),[℧ℜ˜ (§⋎,⋎⋌)L',℧ℜ˜ (§⋎,⋎⋌)I'],[Ωℜ˜ (§⋎,⋎⋌)L',Ωℜ˜ (§⋎,⋎⋌)I']},

Using this context, the concept of an IVIF can be defined as follows:

Definition 4.5. Suppose an IVIFS of attributesB={⋎⋌,[℧B(⋎⋌)L',℧B(⋎⋌)I'],[ΩB(⋎⋌)L',ΩB(⋎⋌)I']:∈[0,1]2:∀⋎⋌∈N}.

Be represented by an interval-valued TMF [℧B(⋎⋌)L',℧B(⋎⋌)I'] and an interval-valued FMF[℧B(⋎⋌)L',℧B(⋎⋌)I'], independently. The exterior component set for a chosen IVIF attribute set is determined as follows:

These components are distinguished by their respective interval-valued TMF [℧B(⋎⋌)L',℧B(⋎⋌)I'] and interval-valued FMF[℧A(⋎⋌)L',℧A(⋎⋌)I'], separately.

Formal concepts are defined as IVIFSs for objects and attributes, denoted as (A, B), which exhibit closure under a Galois connection: A^↑ = B and B^↓ = A. This relationship identifies objects with a maximum interval-valued TMF and a minimum FMF, integrating information from a shared set of attributes. A component-wise approach simplifies this integration within the interval-valued fuzzy space [0, 1]². The method ensures that no additional objects or attributes can increase the IVIF membership values (IVIFMVs) of the identified sets. To exclusively identify the set of IVIFSs (A,B) as formal concepts, A is regarded as the ’extent’ and B as the ’intent’. These formal concepts can be represented through complete lattices and as nodes in IVIF graphs. The following algorithm outlines the process:

• Step 1: Consider an IVIF context, denoted as , where , and ℜ̃ represents the IVIF relationship.

  [℧ℜ˜ (§⋎,⋎⋌)L',℧ℜ˜ (§⋎,⋎⋌)I'],[Ωℜ˜ (§⋎,⋎⋌)L',Ωℜ˜ (§⋎,⋎⋌)I'].

• Step 2: To uncover hidden patterns in the IVIF environment K, identify all subsets of attributes that satisfy the IVIF conditions 2^m, characterized by a TMF of [1, 1] and an FMF of [0, 0].

• Step 3: Adjust the membership values for generating IVIF concepts based on user preferences. The selected membership values ensure maximal alignment with the chosen subsets S_⋌, where ⋌ is less than or equal to 2^m.

• Step 4: Identify the covered items for each subset (S_⋌) by applying the lower operator (↓), denoted as BS⋌↓=AS⋌.

• Step 5: Compute the TMF for the resulting set of objects:

  ℧AS⋌L'=min⋌∈℧BS⋌L'[μ℧L'ℜ˜(§⋎,⋎⋌)] ,℧AS⋌I'=min⋌∈℧BS⋌I' [μ℧I'ℜ˜(§⋎,⋎⋌)],ΩAS⋌L'=max⋌∈ΩBS⋌I' [μΩL'ℜ˜(§⋎,⋎⋌)] ,ΩAS⋌I'=max⋌∈ΩBS⋌I' [μΩI'ℜ˜(§⋎,⋎⋌)].

  Here, ℜ̃ denotes the IVIF relationship between the attribute sets. The computed membership value represents the highest degree of membership necessary to satisfy the constraint and identify the minimal desired property.

• Step 6: Determine the covering attribute set for the object set to reveal latent patterns using the upper operator (↑) of the Galois connection on the assembled object sets, denoted as AS⋌↑=BS⋌.

• Step 7: Calculate the membership score for the attributes covering the object set

  ℧BS⋌L'=min⋎∈℧AS⋌L'[μ℧L'ℜ˜(§⋎,⋎⋌)] ,℧BS⋌I'=min⋎∈℧AS⋌I' [μ℧I'ℜ˜(§⋎,⋎⋌)],ΩBS⋌L'=max⋎∈ΩAS⋌L' [μΩL'ℜ˜(§⋎,⋎⋌)] ,ΩBS⋌I'=max⋎∈ΩAS⋌I' [μΩI'ℜ˜(§⋎,⋎⋌)].

• Step 8: Define the formal fuzzy concept as the pair (A_S⋌,B_S⋌).

• Step 9: Construct the IVIF concept lattice to represent the hierarchical relationships among the concepts.

• Step 10: Apply the proposed strategy using the identified object sets, as demonstrated in Algorithm 1.

Complexity: The pseudocode for the proposed algorithm generates IVIF concepts in a context with ‘n’ objects, ‘m’ attributes, and an IVIF relation ℜ̃. The time complexity for Step 1 is 2^m, while identifying covering objects (defined by TMF and FMF) requires 2m * n operations. Therefore, the overall time complexity is O(2^m * 2m * n).

Example 4.1. Consider a company that aims to allocate funds to the following entities:

• 1. A car manufacturer §₁

• 2. A food company §₂.

• 3. The computer company §₃

• 4. A company specializing in the production of arms §₄

The decision-making process is guided by the following criteria ( ):

• 1. Risk analysis ⋎₁

• 2. Growth analysis ⋎₂

• 3. Environmental impact analysis ⋎₃.

The determination parameter can be identified through TMF and FMF independently. The organization has the ability to gather data utilizing these two-way decision defined and present them in a tabular matrix structure. In this structure, the enumerated organization can be viewed as a set of entities, denoted as = (§₁ = car company, §₂ = food company, §₃ = computer company, and §₄ = arms company), while the decision parameters can be seen as a set of features, denoted as = (⋎₁ = risk analysis, ⋎₂ = growth analysis, ⋎₃ = environmental impact analysis). Let’s assume that the impact of decision attribute risk analysis, denoted as ⋎₁, on the making of a car company is 0.43 in terms of total market value (TMV) and 0.34 in terms of fair market value (FMV). This relationship can be expressed as ℜ̃; (§₁,⋎₁) = (0.43, 0.34) using interval type-2 fuzzy sets in the tabular matrix, as illustrated in Table 1. As, for the establishment of other companies represented by objects (§₂, §₃, §₄), the corresponding decision parameters are detailed in Tables 2, 3, and 4, respectively. The company is now interested in uncovering concealed patterns within the IVIF context, as presented in Table 5. Analyzing investment preferences based on specified decision parameters stands as a significant challenge for the company. To address this concern, the company seeks patterns, specifically formal concepts derived from the data presented in Table 5, utilizing the given attribute set. To meet this requirement, Algorithm 1 is proposed. This algorithm facilitates the generation of IVIF concepts from the context illustrated in Table 5. , where , and ℜ̃ represents IVIF relationship among them i.e.,

The following describe each of the produced subsets of attributes displayed in Table 5.

From Step 3 in Algorithm 1, determine the value of the acceptance membership = ([1, 1], [0, 0]) for each subsets.

• 1. The subsequent intuitionistic fuzzy set can serve as a representation for this.

  {[(0.00,0.00),(1.00,1.00)]/⋎1,[(0.00,0.00),(1.00,1.00)]/⋎2,[(0.00,0.00),(1.00,1.00)]/⋎3},

• 2. {[(1.00, 1.00), (0.00, 0.00)]/⋎₁},

• 3. {[ (1.00, 1.00), (0.00, 0.00) ] / ⋎₂},

• 4. {[(1.00, 1.00), (0.00, 0.00)] / ⋎₃},

• 5. {[(1.00, 1.00), (0.00, 0.00)]⋎₁, [(1.00, 1.00), (0.00, 0.00)]⋎₂},

• 6. {[(1.00, 1.00), (0.00, 0.00)]⋎₂, [(1.00, 1.00), (0.00, 0.00)]⋎₃},

• 7. {[(1.00, 1.00), (0.00, 0.00)]⋎₁, [(1.00, 1.00), (0.00, 0.00)]⋎₃},

• 8. {[(1.00, 1.00), (0.00, 0.00)]⋎₁, [(1.00, 1.00), (0.00, 0.00)]⋎₂, }.

• 9. {[(1.00, 1.00), (0.00, 0.00)]⋎₃}.

  Step 2: Let’s select the initial subset of characteristics.

  {[(0.00,0.00),(1.00,1.00)]/⋎1,[(0.00,0.00),(1.00,1.00)]/⋎2,[(0.00,0.00),(1.00,1.00)]/⋎3}.

  Next, use the to determine its covered object set, down operator (↓) as given below:

  {[(0.00,0.00),(1.00,1.00)]/⋎1,[(0.00,0.00),(1.00,1.00)]/⋎2,[(0.00,0.00),(1.00,1.00)]/⋎3}↓={[(0.78,0.90),(0.43,0.56)]/§1,[(0.65,0.78),(0.87,0.90)]/§2,[(0.56,0.65),(0.78,0.90)]/§3,[(0.78,0.87),(0.87,0.90)]/§4}.

Apply the UP (upper) operator (↑) on this assembled set of objects to identify the additional covering attribute set. {[(0.78, 0.90), (0.43, 0.56)]/§₁, [(0.65, 0.78), (0.87, 0.90)]/§₂, (0.56, 0.65), (0.78, 0.90)]/§₃, [(0.78, 0.87), (0.87, 0.90)]/§₄}^↑ = {[(0.34, 0.56), (0.34, 0.43)]/⋎₁, [(0.43, 0.65), (0.34, 0.43)]/⋎₂, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}. It encompasses the subsequent IVIF principles

• 1. Extent: {[(0.78, 0.90), (0.43, 0.56)]/§₁, [(0.65, 0.78), (0.87, 0.90)]/§₂, [(0.56, 0.65), (0.78, 0.90)]/§₃, [(0.78, 0.87), (0.87, 0.90)]/§₄}.

  Intent: {[(0.34, 0.56), (0.34, 0.43)]/⋎₁, [(0.43, 0.65), (0.34, 0.43)]/⋎₂, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}.

  Step 3: Likewise, employing alternative subsets presented in Step 1 allows for the creation of additional IVIF concepts.

• 2. Extent: {[(0.43, 0.56), (0.34, 0.43)]/§₁, [(0.65, 0.78), (0.21, 0.34)]/§₂,[(0.34, 0.65), (0.34, 0.43)]/§₃, [(0.78, 0.87), (0.12, 0.21)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₁, [(0.43, 0.65), (0.34, 0.43)]/⋎₂, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}.

• 3. Extent: {[(0.43, 0.56), (0.21, 0.43)]/§₁, [(0.65, 0.78), (0.21, 0.34)]/§₂, [(0.56, 0.65), (0.34, 0.43)]/§₃, [(0.65, 0.78), (0.12, 0.34)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₂, [(0.43, 0.56), (0.34, 0.43)]/⋎₁, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}.

• 4. Extent: {[(0.78, 0.90), (0.43, 0.56)]/§₁, [(0.34, 0.56), (0.87, 0.90)]/§₂, [(0.34, 0.65), (0.78, 0.90)]/§₃, [(0.65, 0.78), (0.87, 0.90)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₃, [(0.34, 0.56), (0.34, 0.43)]/⋎₁, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}.

• 5. Extent: {[(0.43, 0.56), (0.34, 0.43)]/§₁, [(0.65, 0.78), (0.21, 0.34)]/§₂, [(0.34, 0.65), (0.34, 0.43)]/§₃, [(0.65, 0.78), (0.12, 0.34)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₁, [(1.00, 1.00), (0.00, 0.00)]/⋎₂, [(0.34, 0.56), (0.87, 0.90)]/⋎₃}.

• 6. Extent: {[(0.43, 0.65), (0.43, 0.56)]/§₁, [(0.34, 0.56), (0.87, 0.90)]/§₂, [(0.43, 0.56), (0.78, 0.90)]/§₃, [(0.65, 0.78), (0.87, 0.90)]/§₄}.

  Intent: {[(0.34, 0.56), (0.34, 0.43)]/⋎₁, [(1.00, 1.00), (0.00, 0.00)]/⋎₂, [(1.00, 1.00), (0.00, 0.00)]/⋎₃}.

• 7. Extent: {[(0.43, 0.56), (0.43, 0.56)]/§₁, [(0.34, 0.56), (0.87, 0.90)]/§₂, [(0.34, 0.65), (0.78, 0.90)]/§₃, [(0.65, 0.78), (0.87, 0.90)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₁, [(0.43, 0.65), (0.34, 0.43)]/⋎₂, [(1.00, 1.00), (0.00, 0.00)]/⋎₃}.

• 8. Extent: {[(0.43, 0.56), (0.43, 0.56)]/§₁, [(0.34, 0.56), (0.87, 0.90)]/§₂, [(0.34, 0.65), (0.78, 0.90)]/§₃, [(0.65, 0.78), (0.87, 0.90)]/§₄}.

  Intent: {[(1.00, 1.00), (0.00, 0.00)]/⋎₁, [(1.00, 1.00), (0.00, 0.00)]/⋎₂, [(1.00, 1.00), (0.00, 0.00)]/⋎₃}.

• The suggestion number 1 indicates that the company is inclined to favor investments guided by the attribute [(0.43, 0.65), (0.3, 0.4)]/⋎₂, specifically in terms of growth analysis. This is evident through its maximum TMV at (0.43, 0.65) and minimum FMV at (0.34, 0.43), showcasing superior performance calculated to alternative attributes.

• The suggestion number 2 signifies that the object [(0.78, 0.87), (0.12, 0.21)]/§₄ attains the maximum TMV at (0.78, 0.87) and the minimum FMV at (0.12, 0.21) compared to other objects in terms of embracing quality ⋎₁ (i.e., risk analysis). Similarly, the analysis can be extended to other objects, yielding the sequence §₄, §₂, §₃, §₁ for the company’s investment order based on risk analysis (⋎₁). This analytical conclusion from the suggested approach agrees rather well with the predictions.

• · The suggestion number 3 signifies that the entity [(0.65, 0.78), (0.12, 0.34)]/§₄ attains maximum TMV with coordinates (0.65, 0.78) and minimum FMV with coordinates (0.12, 0.34) in comparison to other entities for the acceptance of attribute ⋎₂ (i.e., growth analysis). Likewise, a similar analysis can be conducted for other entities, resulting in the sequence §₄, §₂, §₃, §₁ for investing money in companies based on growth analysis (⋎₂). This analysis, derived from the proposed method, aligns well with the data.

• · The suggestion number 4 signifies that the object [(0.78, 0.90), (0.43, 0.56))]/§₁ attains maximum TMV at (0.78, 0.90) and minimum FMV at (0.43, 0.56) when compared to other objects for the assessment of attribute ⋎₃ (i.e., environmental impact analysis). Similarly, an evaluation of other objects yields the sequence §₁, §₄, §₂, §₃ for investment prioritization based on environmental impact analysis (⋎₃). This analysis, resulting from the proposed methodology, aligns well with expectations.

• · The suggestion number 5 signifies that the object [(0.65, 0.78), (0.12, 0.34)]/§₄ exhibits the highest TMV at (0.65, 0.78) and the lowest FMV at (0.12, 0.34) in comparison to other objects, concerning the acceptance of attributes ⋎₁ and ⋎₂ (i.e., risk and growth analysis). Similarly, one can analyze other objects, resulting in the following investment order: §₄, §₂, §₃, §₁ for the company, based on the analysis of risk and growth (⋎₁,⋎₂). This analytical approach, as derived from the proposed method, aligns well with the overall agreement.

• · The suggestion number 6 signifies that the object [(0.65, 0.78), (0.87, 0.90)]/§₄ exhibits the highest TMV at (0.65, 0.78) and the lowest FMV at (0.87, 0.90) compared to other objects for the acceptance of attributes ⋎₂ and ⋎₃ (specifically, growth and environmental impact analysis). Likewise, a similar analysis can be applied to other objects, establishing the sequence §₄, §₁, §₂, §₃ for investment decisions based on growth and environmental impact analysis. This analysis, stemming from the proposed method, aligns well with expectations.

• · The suggestion number 7 implies that the object [(0.65, 0.78), (0.87, 0.90)]/§₄ exhibits the highest TMV at (0.65, 0.78) and the lowest FMV at (0.87, 0.90) compared to other objects, specifically concerning the acceptance of attributes ⋎₁ and ⋎₃ (i.e., risk and environmental impact analysis). In a similar vein, an examination of other objects establishes the sequence §₄, §₁, §₂, §₃ for company investment decisions based on risk and environmental impact analysis. The outcomes of this analysis, derived from the proposed method, align well with each other.

• · The suggestion number 8 is centered on the acceptance of attributes ⋎₁, ⋎₂, ⋎₃ the object [(0.65, 0.78), (0.87, 0.90)]/§₄ has TMV (0.65, 0.78) and minimal FMV (0.87, 0.90) When contrasted with other items, a similar approach can be applied to assess additional objects, resulting in the sequence §₄, §₁, §₂, §₃ for investing money in the company based on individual attributes. The analysis produced through the suggested method aligns well with expectations.

The lattice derived from the aforementioned concepts, known as the IVIF concept lattice, is illustrated in Figure 2. This representation conveys the following information. The analysis obtained from the suggested approach closely mirrors the correlation method and aggregation operator, but with a more in-depth examination. Achieving this involves the proposed method generating concepts exponentially through slight variations in membership values. Consequently, identifying key concepts based on user-specified parameters becomes a challenging endeavor in this scenario. To address this issue, an alternative approach is suggested in Algorithm 2 for the dissection of the interval-valued neutrosophic context at a granulation defined by the user as (α, β). The following section provides an illustration of the decomposition of the IVIF context.

4.3 A New Approach for the (α, β)-Decomposition of the IF Context

The proposed approach presented in Algorithm 1 yields a considerable number of IVIF concepts, characterized by minimal variation in TMF and FMF. This abundance of concepts may hinder knowledge processing tasks, particularly when users seek essential concepts tailored to specific requirements within a defined granularity. Granular computing, a mathematical tool, offers a means to refine expansive contexts into smaller information granules. The chosen level of granulation facilitates efficient processing of large contexts by modularizing complex problems into well-defined subproblems at minimal computational cost. Recently, the principles of granular computing have found application in formal contexts, formal fuzzy contexts, interval-valued contexts, bipolar fuzzy contexts, two-polar fuzzy contexts, and multi-scaled concept lattices, aiding in the identification of crucial concepts at user-defined granulation levels. This paper focuses on decomposing an IVIF context by considering its TMV and FMV separately, utilizing a user-defined (α, β)-cut.

• Step 1: Let’s consider an IF context denoted as , where , and ℜ̃ signifies the IF relationship among them.

  (℧ℜ˜ (§,⋎),Ωℜ˜ (§,⋎)).

• Step 2: Now, specify the granulation parameters for TMV and FMV, denoted as (α, β).

• Step 3: The provided IF context can be broken down at the selected granularity (α, β) using TMV and FMV separately, as outlined below:

  If and only if the Transaction Message Validator (TMV) is part of the identical decomposed context:

  Kα℧={℧ℜ˜ (§,⋎)∣μ℧ℜ˜ (§,⋎)≥α}.

  In this context, let §_⋎ denote an object and ⋎_⋌ an attribute. Express them as the maximum acceptance of the TMV, specifically as 1.00 for the selected α-cut, or alternatively, as 0.00. The FMV is part of the identical decomposed context if:

  KβΩ={Ωℜ˜ (§,⋎)∣μΩℜ˜ (§,⋎)≥β}.

  In this context, where §_⋎ represents an object and ⋎_⋌ signifies an attribute, they are expressed as the maximum acceptance of the FMV. Specifically, denoted as 0.00 for the selected β-cut, otherwise marked as 1.00. Therefore, a binary context has been established for K_α,β through a defined granulation for the TMV and FMV.

• Step 4: The equality provided below pertains to the decomposed contexts, denoted as K_α,β, within a user-defined granulation.

  K=∪α,β Kα,β,∀α,β.

  In this context, α represents a specified granulation on TMV, and β pertains to a defined granulation as well.

• Step 5: The disintegrated context also adheres to the subset conditions, namely, K_{α1, β1} ⊆ Kα₂, β₂ when α₁ ≤ α₂, β₁ ≤ β₂. This implies that the quantity of concepts and the dimensions of the IF concept lattice can be regulated by employing the specified granulation for the TMV and FMV.

• Step 6: The method described above can be applied to determine both lower and upper limits for the TMV and FMV in the context of the provided IVIF relations.

• Step 7: At the end, formulate the decomposed “IF” context for tasks related to knowledge processing. The pseudocode illustrating this proposed algorithm is presented in Table 7.

Complexity:Algorithm 2 presents the pseudocode outlining the decomposition process for a provided intuitionistic fuzzy context. Here, let the count of objects (| |) be denoted as n, and the count of attributes (| |) as m within the specified IVIF context. The suggested approach decomposes the context using a user-defined (α, β)-cut, with a time complexity of O(m³) or O(n²) in the case of a IVIF context. For the decomposition of IVIFSs, the cut is specified individually for both the lower and upper bounds associated with the given attribute O(m) or object set O(n). Consequently, the overall time complexity of the suggested approach amounts to O(n *m³) or O(n² *m). This methodology effectively minimizes the computational burden for handling the provided IVIF context in comparison to the approach detailed in Table 5. The following section provides a demonstration of both proposed methods through an illustrative example.

In the following paragraphs, we illustrate both the proposed algorithms with an example. The first algorithm presents hierarchical order visualization in addition to IVIF concept demonstrations. The second approach, on the other hand, presents an assortment of these ideas at a customized (α, β)-cut, which refines the knowledge. To illustrate these methods and conduct a comparative analysis of the results obtained, we employ an example.

5.1 IVIF Concept Lattice

In recent times, the exploration of the three-way decision space and its approximation has significantly influenced the trajectory of knowledge processing tasks, particularly in the in-depth examination of partial ordering, concept lattice, and their applications across various research domains. A key focal point for researchers engaged in knowledge processing tasks is the generation of formal concepts from a provided IVIF context. This paper makes a concerted effort to scrutinize data utilizing IVIF attributes, leveraging the characteristics of IVIFSs, the IVIF graph, and its associated concept lattice. The aim is to offer a more precise visualization of partial ignorance, inconsistency, and incompleteness within a given dataset compared to alternative approaches. To realize this objective, the paper introduces a method outlined in Table 5 for identifying all IVIF concepts within a given context.

5.2 The (α, β)-Decomposition of IVIF Context

Lately, granular computing has attracted a lot of interest from scholars who want to learn more about the two-way decision environment. This method is a technique for contextual processing that considers the problem’s structure, functioning, and closeness. Numerous formal fuzzy contexts, such as formal fuzzy ideas, fuzzy concept lattice, interval-valued fuzzy information, bipolar fuzzy information, two-polar fuzzy information, and multi-scaled concept lattice, have been added to its list of uses. These characteristics are utilized to pinpoint fundamental ideas at user-specified granularities. This work focuses on breaking down the IF context through the use of granular computing features to accurately identify important patterns hidden within the given IVIF context. Algorithm 2 presents a strategy to accomplish this goal. A single-valued intuitionistic fuzzy (SVIF) environment is modified in order to compare the analysis produced by the suggested method with the intended methodology in order to demonstrate the suggested way. Below is an example of it.

Example 5.1. Let’s consider an SVIF context for examining the allocation of funds among companies = ( §₁ =automobile company, §2 =food company, §3 =computer company, and §₄ =defense company ) based on a single-valued TMF and single-valued FMF using the decision parameters = (⋎₁ = risk analysis, ⋎₂ = growth analysis, ⋎₃ = environmental impact analysis). The formal representation of this dataset in the context of IF context properties is illustrated in Table 6.

The challenge facing the company lies in uncovering concealed patterns within the provided SVIF context presented in Table 6, with the objective of investing funds in the establishment of new ventures. To realize this objective, capital is allocated to companies exhibiting a maximum single-valued TMV exceeding 0.56 and a minimal single-valued FMV close to 0.00 based on the specified decision parameters. Assuming the company adopts a two-polar granulation (0.65, 0.12) to assess investment preferences, the context outlined in Table 6 can be disentangled through the proposed method detailed in Algorithm 2 for the defined (0.65, 0.12)-cut, as outlined below:

• 1. ℧_{ℜ̃(§1, ⋎1)}< 0.65, Ω_{ℜ̃(§1, ⋎1)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 2. ℧_{ℜ̃(§1, ⋎2)}< 0.65, Ω_{ℜ̃(§1, ⋎2)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 3. ℧_{ℜ̃(§1, ⋎3)}< 0.65, Ω_{ℜ̃(§1, ⋎3)}> 0.12.

  Therefore, the decomposition offers a non-acceptance region within the interval (0, 1).

• 4. ℧_{ℜ̃(§2, ⋎1)}< 0.65, Ω_{ℜ̃(§2, ⋎1)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 5. ℧_{ℜ̃(§2, ⋎2)}< 0.65, Ω_{ℜ̃(§2, ⋎2)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 6. ℧_{ℜ̃(§2, ⋎3)}< 0.65, Ω_{ℜ̃(§2, ⋎3)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 7. ℧_{ℜ̃(§3, ⋎1)}< 0.65, Ω_{ℜ̃(§3, ⋎1)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 8. ℧_{ℜ̃(§3, ⋎2)}< 0.65, Ω_{ℜ̃(§3, ⋎2)}> 0.12.

  Therefore, the decomposition offers a non-acceptance region within the interval (0, 1).

• 9. ℧_{ℜ̃(§3, ⋎3)}< 0.65, Ω_{ℜ̃(§3, ⋎3)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 10. ℧_{ℜ̃(§4, ⋎1)}> 0.65, Ω_{ℜ̃(§4, ⋎1)}< 0.12.

  Therefore, the decomposition establishes an acceptance region, denoted as (1, 0), for both the TMV and FMV.

• 11. ℧_{ℜ̃(§4, ⋎2)}< 0.65, Ω_{ℜ̃(§4, ⋎2)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

• 12. ℧_{ℜ̃(§4, ⋎3)}< 0.65, Ω_{ℜ̃(§4, ⋎3)}> 0.12.

  Therefore, the decomposition establishes a non-acceptance region within the interval (0, 1).

The decomposition computed above is presented in Table 7 in the form of a contextualized framework for processing knowledge. In this representation, (1, 0) and (0, 1) signify acceptance and non-acceptance values, respectively, for the specified decision parameters. Let’s consider the granulation with two poles at (0.65, 0.12). Under these conditions, the optimal choice is limited to object §₄, specifically an arms company. Therefore, investing money in an arms company emerges as a highly favorable option.

The analysis resulting from the suggested approach bears similarity to the hybrid vector similarity method and a three-way fuzzy concept lattice. However, the proposed method achieves this by requiring computational time of O(m³) or O(n²). Consequently, it offers a more customizable granulation, allowing users to refine knowledge according to their specific requirements. To illustrate, if a user wishes to assess preferences based on a (0.65, 0.21)-cut for the context presented in Table 7, the newly decomposed context will be displayed as depicted in Table 8.

• 1. ℧_{ℜ̃(§1, ⋎1)}< 0.65, Ω_{ℜ̃(§1, ⋎1)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 2. ℧_{ℜ̃(§1, ⋎2)}< 0.65, Ω_{ℜ̃(§1, ⋎2)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 3. ℧_{ℜ̃(§1, ⋎3)}< 0.65, Ω_{ℜ̃(§1, ⋎3)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 4. ℧_{ℜ̃(§2, ⋎1)}> 0.65, Ω_{ℜ̃(§2, ⋎1)}< 0.21.

  Acceptance region (1, 0)∀ the TMV and FMV is thus provided by the decomposition.

  ℧ℜ˜ (§2,⋎2)>0.65,Ωℜ˜ (§2,⋎2)<0.21.

  Acceptance region (1, 0)∀ the TMV and FMV is thus provided by the decomposition.

• 5. ℧_{ℜ̃(§2, ⋎3)}< 0.65, Ω_{ℜ̃(§2, ⋎3)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 6. ℧_{ℜ̃(§3, ⋎1)}< 0.65, Ω_{ℜ̃(§3, ⋎1)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 7. ℧_{ℜ̃(§3, ⋎2)}< 0.65, Ω_{ℜ̃(§3, ⋎2)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 8. ℧_{ℜ̃(§3, ⋎3)}< 0.65, Ω_{ℜ̃(§3, ⋎3)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 9. ℧_{ℜ̃(§4, ⋎1)}> 0.65, Ω_{ℜ̃(§4, ⋎1)}< 0.21.

  Acceptance region (1, 0)∀ the TMV and FMV is thus provided by the decomposition.

• 10. ℧_{ℜ̃(§4, ⋎2)}> 0.65, Ω_{ℜ̃(§4, ⋎2)}< 0.21.

  Acceptance region (1, 0)∀ the TMV and FMV is thus provided by the decomposition.

• 11. ℧_{ℜ̃(§4, ⋎3)}< 0.65, Ω_{ℜ̃(§4, ⋎3)}> 0.21.

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

The disassembled context reveals that selecting object §₄ (arms company) and §₂ (food company) can be determined using a (0.65, 0.21)-cut. Consequently, investing money in the arms and food companies would be the most preferable choice. This analysis aligns with both the vector similarity method and the three-way concept lattice. The application of the proposed method to the IVIF context is illustrated below: Table 5 serves as an illustration of the IVIF context decomposition utilizing the proposed method, as depicted in Algorithm 2. The issue pertaining to Table 5 lies in the company’s objective to identify crucial patterns for investment based on specified decision parameters. In a broader sense, to uncover these patterns, the company requires a maximal interval-valued TMV range of [0.65, 0.78], along with a minimal interval-valued FMV range of [0.12, 0.21], to validate the acceptance of the ideal conditions. The decomposed context, guided by this selected granulation, is presented in Table 9.

• 1. ℧_{ℜ̃(§1, ⋎1)}< (0.65, 0.78), Ω_{ℜ̃(§1, ⋎1)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 2. ℧_{ℜ̃(§1, ⋎2)}< (0.65, 0.78), Ω_{ℜ̃(§1, ⋎2)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 3. ℧_{ℜ̃(§1, ⋎3)}> (0.65, 0.78), Ω_{ℜ̃(§1, ⋎3)}> (0.12, 0.21).

  Acceptance region (1, 1)∀ the TMV and FMV is thus provided by the decomposition.

  ℧ℜ˜ (§2,⋎1)>(0.65,0.78),Ωℜ˜ (§2,⋎1)>(0.12,0.21).

  Acceptance region (1, 1)∀ the TMV and FMV is thus provided by the decomposition.

• 4. ℧_{ℜ̃(§2, ⋎2)}> 0.65, Ω_{ℜ̃(§2, ⋎2)}> (0.12, 0.21).

  Acceptance region (1, 1)∀ the TMV and FMV is thus provided by the decomposition.

• 5. ℧_{ℜ̃(§2, ⋎3)}< (0.65, 0.78), Ω_{ℜ̃(§2, ⋎3)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 6. ℧_{ℜ̃(§3, ⋎1)}< (0.65, 0.78), Ω_{ℜ̃(§3, ⋎1)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 7. ℧_{ℜ̃(§3, ⋎2)}< (0.65, 0.78), Ω_{ℜ̃(§3, ⋎2)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 8. ℧_{ℜ̃(§3, ⋎3)}< (0.65, 0.78), Ω_{ℜ̃(§3, ⋎3)}> (0.12, 0.21).

  Therefore, non-acceptance zone (0, 1) is provided by the decomposition.

• 9. ℧_{ℜ̃(§4, ⋎1)}> (0.65, 0.78), Ω_{ℜ̃(§4, ⋎1)}< (0.12, 0.21).

  Therefore, for any TMV and FMV, the decomposition yields an acceptability region of (1, 0).

• 10. ℧_{ℜ̃(§4, ⋎2)}> (0.65, 0.78), Ω_{ℜ̃(§4, ⋎2)}> (0.12, 0.21).

  As a result, the breakdown offers an acceptance region (1, 1) the TMV and FMV.

• 11. ℧_{ℜ̃(§4, ⋎3)}> (0.65, 0.78), Ω_{ℜ̃(§4, ⋎3)}> (0.12, 0.21).

  Recognition region (1, 1) is thus provided by the decomposition for all TMV and FMV.

Table 9 illustrates that the object §₄ exhibits a threshold measure value of 1 for each attribute, while object §₂ attains the maximum TMV for two specific attributes, namely ⋎₁ and ⋎₂. Consequently, the arms company (i.e., §₄) emerges as the primary preference for investment, followed by the food company (i.e., §₂) as the secondary option. These findings align well with the correlation method, aggregation operator, and the associated concept lattice depicted in Figure 1. Moreover, for a more detailed analysis of preferences, the company can explore variations in granulation, aiming for increased refinement within either O(n*m³) or O(n² *m) computational time. The methods proposed in this paper are anticipated to be highly beneficial for researchers engaged in the realm of interval-valued multi-decision-making processes. Anticipates scholars studying interval-valued multi-decision-making processes will find great benefit from the techniques this paper offers.

The analysis and processing of large amounts of technological data are integral to modern decision-making. To address this need, we have developed techniques and tools designed to compute accurate information and enhance efficiency. Utilizing graph and lattice methods to condense large datasets into single values is resource-intensive. However, the concept lattice in fuzzy sets, integrated with interval-valued fuzzy sets, provides a powerful framework for scenarios where each item has a range of possible values defined by MD and NMD. The lattice function differs in two key aspects: periodicity and symmetry at the origin. Consequently, current concept lattice rules have been adapted to the IVIF network environment. The advantages and contributions of these techniques can be summarized as follows:

• · We introduced the term lattice environment of an IVIF network and established its fundamental properties. This foundational step enabled the creation of a comprehensive framework for analyzing and assessing IVIF contexts.

• · This study thoroughly examines the primary features of IVIF contexts, including the technical intricacies of their scoring systems, precise computations, and the formulation of ordering principles tailored specifically for IVIF contexts using concept lattices. This detailed evaluation provides a robust foundation for understanding the core concepts and methodologies underpinning IVIF environments.

• · We developed mathematical methods optimized for processing complex value data. These operators facilitate the identification of numerous optimal values, each with distinct characteristics, ensuring robust computational efficiency.

• · Our approach is meticulously designed to exploit the unique attributes of IVIF contexts in conjunction with concept lattices, enabling intelligent and efficient decisionmaking processes that effectively address the complexities of interval-valued data.

• · The methodology focuses on uncovering IVIF concepts and decomposing intuitionistic fuzzy contexts based on user-defined (α, β)–cuts for truth (α-cut) and falsity (β-cut) membership values. The results of these methods align closely with those established correlation and aggregation techniques

• · Concept lattices hold significant potential for extending the applicability of IVIF contexts to various industries and domains, including healthcare, robotics, information retrieval systems, intelligent systems, artificial intelligence, ecological science, and logistics management. While this study does not address the interdependencies between these applications, future research will explore these relationships to further enhance the utility of IVIF-based methodologies.

No potential conflict of interest relevant to this article was reported.

Table 1. (§₁) involves IVIF relationship shown in Figure 1.

Table 2. IVIF relation for §₂, shown in Figure 1.

Table 3. An IVIF relations for §₃.

Table 4. An IVIF relationship by §₄.

Table 5. IVIF representation for Tables 1 to 4.

Table 6. A three-way fuzzy context representation using intuitionistic fuzzy sets.

Table 7. A decomposition shown in Table 6.

Table 8. An analysis of the background presented in Table 6.

Table 9. Context breakdown of Table 6 (0.12, 0.21).

Algorithm 1. Proposed algorithm to develop the IVIF concept.

Input:	An IVIF envirnment , whereas , and ℜ̃ express IVIF relation.
Output:	Set of IVIFFCs
Extent:	{§⋎,[℧A(§⋎)L',℧A(§⋎)I'],[ΩA(§⋎)L',ΩA(§⋎)I']},
Intent:	{⋎⋌,[℧B(yj)L',℧B(⋎⋌)I'],[ΩB(yj)L',ΩB(⋎⋌)I']}, where ⋎ ≤ n and ⋌ ≤ m.
1	Determine the subsets S⋌ of given IV IF attribues 2m.
2	for ⋌ = 1 to 2m.
3	Determine the value of the acceptance membership = ([1.00, 1.00], [0.00, 0.00]) for each subsets
4	Use the scrolling operator to locate the encompassing object set (↓):{⋎⋌,[℧BS⋌L'(⋎⋌),℧BS⋌I'(⋎⋌)],[ΩBS⋌L'(⋎⋌),ΩBS⋌I'(⋎⋌)]}↓.
5	The down operator (↓) provides the following object set: {§⋎,[℧A(§⋎)L',℧A(§⋎)I'],[ΩA(§⋎)L',ΩA(§⋎)I']}
6	The calculation of the membership value for the acquired set of objects can be determined in the following manner:℧AS⋌L'=min⋌∈℧BS⋌L'[μ℧L'ℜ˜(§⋎,⋎⋌)] ,℧AS⋌I'=min⋌∈℧BS⋌L' [μ℧I'ℜ˜(§⋎,⋎⋌)],ΩAS⋌L'=max⋌∈ΩBS⋌L' [μΩL'ℜ˜(§⋎,⋎⋌)] ,ΩAS⋌I'=max⋌∈ΩBS⋌I' [μΩI'ℜ˜(§⋎,⋎⋌)].
7	Determine the attribute set that covers the specified object set using the upper operator, denoted as ↑ (UP). {§⋎,[℧A(§⋎)L',℧A(§⋎)I'],[ΩA(§⋎)L',ΩA(§⋎)I']}↑.
8	Now, calculate the membership value for the obtained attribute set using the following procedure:℧BS⋌L'=max⋎∈℧AS⋌L' [μ℧L'ℜ˜(§⋎,⋎⋌)] ,℧BS⋌I'=min⋎∈℧AS⋌I' [μ℧I'ℜ˜(§⋎,⋎⋌)],ΩBS⋌L'=max⋎∈ΩAS⋌L' [μΩL'ℜ˜(§⋎,⋎⋌)] ,ΩBS⋌I'=max⋎∈ΩAS⋌I' [μΩI'ℜ˜(§⋎,⋎⋌)].
9	The formal concepts (AS⋌, BS⋌) are derived from the selected subset.
10	end for
11	In a similar way, more concepts can be developed with the other subsets.
12	Build the IVIF concept lattice.

Algorithm 2. Suggested algorithm for breaking down the IF environment into α and β.

Input:	An IF environment where , and (ℝ̃ = (℧ℜ̃(§, ⋎), Ωℜ̃(§, ⋎))
Output:	The set of decomposed environment Kα, β
1	Let us assume an IF environment .
2	Define the granulation for the TMV and FMV i.e., (α, β).
3	Now the decomposed the given environment Kα, β as follows:
4	if {℧ℜ̃(§, ⋎)|μ℧ℜ̃(§,⋎) ≥ α}then represent 1.00 at the place of TMV,else 0.00 at the place of TMV.
5	if {Ωℜ̃(§, ⋎)|μΩℜ̃(§,⋎) ≥ β}then represent 0.00 at the place of FMV,else 1.00 at the place of FMV.
6	The decomposed environment follows the equality:K = ∪α, βKα, β.
7	The decomposed binary environment write for the user-defined granulation.
8	Derive the knowledge from the decomposed environment Kα, β.