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International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 378-386

Published online December 25, 2024

https://doi.org/10.5391/IJFIS.2024.24.4.378

© The Korean Institute of Intelligent Systems

Generalized Interval-Valued Fuzzy Soft Code and Its Properties on Decision Makings

Masresha Wassie Woldie1, Jejaw Demamu Mebrat2, and Mihret Alamneh Taye1

1Department of Mathematics, Bahar Dar University, Bahir Dar, Ethiopia
2Department of Mathematics, Debark University, Gondar, Ethiopia

Correspondence to :
Masresha Wassie Woldie (masreshawassie28@gmail.com)

Received: March 18, 2024; Revised: November 23, 2024; Accepted: December 10, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

An interval-valued fuzzy soft set theory is a strong instrument that can offer the uncertain data processing capacity in an imprecise environment. In this study, a generalized interval-valued fuzzy soft code (GIVFSC), which combines an interval-valued fuzzy soft set and a fuzzy code, is proposed. The GIVFSC framework supports its operation and decision-making processes. Various features of the GIVFSC are examined to improve the combination of interval-valued fuzzy soft sets and fuzzy codes.

Keywords: Fuzzy decoding, Fuzzy soft code, Interval-valued fuzzy soft set, Generalized interval-valued fuzzy soft set, Decision-Making, Generalized interval-valued fuzzy soft code

Molodtsov [1] established soft set theories and models as new mathematical tools for handling uncertainties that cannot be controlled by existing mathematical tools. Amudhambigai and Neeraja [2] studied the fuzzy codes and its application. Hong and Qin [3] discussed algebraic structures of fuzzy soft sets. Roy and Maji [4] presented a fuzzy soft theoretical approach to decision-making problems and shared several results on the application of fuzzy soft sets to these problems. Lal [5] applied fuzzy algebra to coding theory. Smarandache [6] initiated soft sets using a rough set of codes to create soft codes (soft linear codes). Ali et al. [7] developed and designed novel type of linear algebraic codes known as “soft linear algebraic codes” by employing soft sets. Alkhazaleh [8] introduced the concept of effective fuzzy soft sets, explored their operations, and investigated several of their properties. Kong et al. [9] examined the use of soft sets with fuzzy properties in grey theory-based decision-making problems. Gogoi et al. [10] designed a fuzzy softset theory applied to regular problems. Prade [11] proposed an interval-valued fuzzy set. Bustince [12] explored interval-valued fuzzy sets in soft computing. Majumdar and Samanta [13] introduced generalized fuzzy soft sets. Yang et al. [14] introduced the concept of interval-valued soft sets with fuzzy values by merging the notions of soft set models with interval-valued fuzzy sets. Chetia and Das [15] developed an application of interval-valued fuzzy software for medical diagnosis. Alkhazaleh and Salleh [16] introduced the concept of the generalized interval-valued fuzzy soft set (GIVFSs). A generalized Z-fuzzy soft covering-based rough matrix was defined by Sivaprakasam and Angamuthu [17], and its various algebraic properties were investigated. They used generalized Z-fuzzy soft covering-based rough matrices to offer a unique multi-attribute group decision-making problem model. Das and Granados [18] proposed a method to address group decision-making problems using fuzzy parameterized intuitionistic multi-fuzzy N-soft sets. They introduced an induced fuzzy parameterized hesitant N-soft set as an extension of multi-fuzzy N-soft sets, tailored for group decision-making problems. Mukherjee [19] introduced the concept of the intuitionistic fuzzy soft rough sets. Jiang et al. [20] proposed the interval-valued intuitionistic fuzzy soft set theory. Hayat and Mahmood [21] introduced an insight into a bipolar soft set in the union of two isomorphic hemirings. They characterized type-2 soft graphs as the underlying subgroups of simple graphs. Hayat et. al. [22] discussed the aggregation concepts of the selection of design parameter values by merging the acceptable and satisfactory level requirements of customers. Yang, et al. [23] developed the basic Dombi operational laws for spherical fuzzy soft numbers.

1.1 Motivation

Alkhazaleh and Salleh [16] introduced the concept of a GIVFSs and its application to their characteristics. In [2] and [24], the authors defined fuzzy codes by applying the concepts of classical fuzzy sets and binary codes.

Based on these concepts, we introduce the concept of a generalized interval-valued fuzzy soft code (GIVFSC). The parameterization of fuzzy sets is related to the degree of the GIVFSs concept. However, in our study, for the GIVFSC, the degree was attached to the parameterization of the fuzzy code. This study aims to develop new ideas by integrating the definition of a GIVFSs and fuzzy codes, from which a novel fuzzy soft set model can be derived based on the GIVFSC and its operations. The novelty of this study is that it develops a combination of interval-valued fuzzy soft sets and fuzzy codes to generate an interval-valued fuzzy soft code. GIVFSCs are strong tools for processing uncertain data and are utilized in numerous different applications, such as ranking alternatives, processing uncertain data, and solving multigroup decision-making problems.

Section 2 presents the fundamental definitions and properties of the preliminary concepts. Section 3 presents the GIVFSC. In Section 4, the basic operations of GIVFSC are presented. In Section 5, the operations of GIVFSC and its applications are discussed.

In this section, we discuss some basic concepts used to interpret our findings.

Definition 1 [1]. Let E be a set of parameters and U the initial universe set. If and only if F maps E into the set of all subsets of set U, pair (F,E) is referred to as a soft set over U.

In particular, the soft set is a parametrized family of the set U. Every set F(e), eE, from this family may be considered as the set of e approximate elements of the soft set.

Definition 2 [4]. Let {A1, A2, … , Ai} ⊆ E be a set of parameters that can be used to characterize a set of k objects, U = {u1, u2, … , uk}. Let each parameter set Ai represent the ith class of parameters and each property set be represented by its elements. Consider the set of all the fuzzy sets of U as P(U). The fuzzy soft sets over U are pairs (Fi,Ai), where Fi is the mapping indicated by

Fi:AiP(U).

Definition 3 [24]. Let C be the code, and the codewords with different lengths in C be {Cr1, Cr2, … , Crm}. The membership value of C is μC(x), which is described as follows in terms of the relative weight of each codeword:

If C = {Cr1, Cr2, … , Crm}, then μC(x) = {J(Cr1), J(Cr2), … , J(Crm)}, where J(Cri) denotes the relative weight of the codeword Cri, for i = 1, 2, …, m.

Definition 4 [7]. Let F = Z2 symbolize a field of dimension two. Let P(W) be every subset within set W and W = Fm be a vector space of size m over field F. If for each d in D; DV , f(d) is a linear algebraic code of W, where V denotes the collection of parameters, then (f,D) is considered as an algebraic soft code over F.

Definition 5 [13]. The universal set of elements is U = {x1, x2, … , xm}, and the universal set of parameters is E = {e1, e2, … , en}. We refer to pair (U, E) as a soft universe. Considering that a fuzzy subset of E is μ such that μ : EI = [0, 1], let F : EIU, where IU is the collection of all fuzzy subsets of U. Consider the function Fμ : EIU × I, which is defined as follows:

Fμ = (F(e), μ(e)), where F(e) ∈ IU. A generalized fuzzy soft set over the soft universe (U,E) is thus referred to as Fμ.

Definition 6 [16]. Considering a universal set of parameters, E = {e1, e2, … , en}, and a universal set of elements, U = {x1, x2, … , xm}. A soft universe is applied to pair (U,E). Define as the set of all interval-valued fuzzy subsets on U, and let μ be a fuzzy subset of E, that is, μ : EI = [0, 1]. Define function μ : E (U) × I.

A GIVFSs over the soft universe (U, E) is thus referred to as μ = ( (e), μ(e)).

In this section, we introduce the GIVFSC, which combines a GIVFSs with a fuzzy code.

Definition 7. Let E = {e1, e2, … , em} be the standard set of parameters and K = {c1, c2, … , cn} be the set of codewords or the universal set of elements in F2n. We refer to pair (K,E) as a soft code universe. Let A be a mapping A : C′ → [0, 1], CF2n, and F : EIK and μ be a fuzzy code of E, that is, μ : EA(x), defined as

A(x)=2ipik(k+1),

where IC denotes the set of all interval-valued fuzzy subsets on K and pi denotes the location of 1s. Consider the function Fμ : EIK × A(x) defined by

Fμ(x)=(F(e),μ(x)).

Subsequently, Fμ is called a GIVFSC over the soft universe (K,E), for each parameter ei.

Over the soft universe (K,E), Fμ is referred to as a GIVF-SCs.

For every ei parameter, Fμ(ei) = (F(ei)(c), μ(ei)) describes the degree to which such belongingness is possible, and it is symbolized by μ(ei), as well as the extent to which the components of C in F(ei) belong. We can define Fμ(ei) as follows:

Fμ(ei)=({c1F(ei)(c1),c2F(ei)(c2),,cnF(ei)(cn)},μ(ei)).

Example 1. Let K = {c1, c2, c3} be a collection of universes in F25, E = {e1, e2, e3} be a set of parameters, and C′ = {1000, 0110, 0111} be a set of codewords in F24. Let μ : EA(x), where A is a mapping: A : C′ → [0, 1], CF2n, μ(e1) = 0.1, μ(e2) = 0.5, and μ(e9) = 0.9 defined as (1), A(1000) = 0.1, A(0110) = 0.5, and A(0111) = 0.9, define a function Fμ : EIK × A(x) as follows:

Fμ(e1)=({c1[0.1,0.6],c2[0.1,0.8],c3[0.1,0.8]},0.1),Fμ(e2)=({c1[0.5,0.7],c2[0.5,0.6],c3[0.5,0.5]},0.5),Fμ(e3)=({c1[0.9,0.9],c2[0.9,1],c3[0.9,0.9]},0.9).

Then, Fμ is GIVFSCS over (K,E).

In matrix notation,

Fμ=[[0.1,0.6]   [0.1,0.8]   [0.1,0.8],0.1[0.5,0.7]   [0.5,0.6]   [0.5,0.5],0.5[0.9,0.9]   [0.9,1]   [0.9,0.9],0.9].

Definition 8. Let Fμ and Gδ be two GIVFSCs over (K,E). Fμ, expressed as FμGδ, is a GIVFSC subset of Gδ, if

  • μ(e) ⊆ δ(e), ∀ eE,

  • F(e) ⊆ G(e), ∀ eE.

Example 2. Let E = {e1, e2, e3} be a set of parameters, where e1 = cheap, e2 = expensive, and e3 = more expensive, and K = {c1, c2, c3} be the codes of the three cars in F27. Let C1 = {0101, 1001, 0011} and C2 = {0011, 1011, 0111} be the codewords in F24. Let μ : EA(x), for each xC1, and A a function expressed as A : C1 → [0, 1] and described as follows in (1): μ(e1) = 0.5, μ(e2) = 0.7, μ(e3) = 0.6, and A(0101) = 0.6, A(1001) = 0.5, A(0011) = 0.7.

Let Fμ be a GIVFSC over (K, E) defined as follows:

Fμ(e1)=({c1[0.3,0.5],c2[0.5,0.6],c3[0.5,0.8]},0.5),Fμ(e2)=({c1[0.4,0.7],c2[0.1,0.7],c3[0.6,0.7]},0.7),Fμ(e3)=({c1[0.1,0.6],c2[0,0.6],c3[0.3,0.6]},0.6).

Consider another GIVFSC over (K, E) with the following definition for Gσ:

σ : EA(x), for any xC2, and A is a function expressed as A : C2 → [0, 1], A(0011) = 0.7, A(1011) = 0.8, A(0111) = 0.9, and σ(e1) = 0.7, σ(e2) = 0.8, σ(e3) = 0.9.

Gσ(e1)=({c1[0.4,0.7],c2[0.7,0.7],c3[0.7,0.9]},0.7),Gσ(e2)=({c1[0.6,0.8],c2[0.4,0.8],c3[0.8,0.8]},0.8),Gσ(e3)=({c1[0.3,0.9],c2[0.2,0.9],c3[0.5,0.9]},0.9).

Hence, Fμ is a GIVFSC subset of Gσ.

Definition 9. We write Fμ = Gσ if Fμ is a GIVFSC subset of Gσ and Gσ is a GIVFSC subset of Fμ. Two GIVFSCs Fμ and Gσ over (K, E) are considered equal if the following conditions are satisfied

  • (i) μ(e) = σ(e), ∀eE,

  • (ii) F(e) = G(e), ∀eE.

Definition 10. The GIVFSC is identified by ∅︀μ, and is characterized as follows: ∅︀μ : EIK× A(x) such that

μ(e)=(F(e)(x),μ(e)).

Here, F(e) = [0, 0] = [0] and μ(e) = 0 for all eE.

Definition 11. The GIVFSC is represented by the symbol 1μ if 1μ : EIK × A(x) and defined as follows:

  • (iii) 1μ(e) = (F(e)(x), μ(e)), (5)

  • (iv) Here, F(e) = [1, 1] = [1] and μ(e) = 1 for all eE.

Definition 12. Considering a GIVFSCS over (K,E) as Fμ. The complement of Fμ is thus represented as FμC; it is characterized as FμC = 1 − Fμ = Gπ, such that for any e in E, π(e) = C (μ(e)) and G(e) = C (F(e)) with c being a complement notation.

Example 3. Consider a GIVFSCs Fμ over (K,E) in Example 1,

Fμ=[[0.1,0.6]   [0.1,0.8]   [0.1,0.8],0.1[0.5,0.7]   [0.5,0.6]   [0.5,0.5],0.5[0.9,0.9]   [0.9,1]   [0.9,0.9],0.9].Then FμC=[[0.4,0.9]   [0.2,0.9]   [0.2,0.9],0.9[0.3,0.5]   [0.4,0.5]   [0.5,0.5],0.5[0.1,0.1]   [0,0.1]   [0.1,0.1],0.1].

Proposition 1. Consider a GIVFSCs over (K,E) as Fμ. The following conclusions were drawn.

(FμC)C=Fμ.

Proof. As FμC = 1 − Fμ, then from the definition, we have

(FμC)C=(1-Fμ)C=1-(1-Fμ)=1-1+Fμ=Fμ.

This is complete the proof.

Definition 13 (Union of two GIVFSCs). FμGπ is the union of two GIVFSCs (Fμ,A) and (Gπ,B), which generates a GIVFSCs (Hσ,C), where C = AB and Hσ : EIK × A(x) is defined by

Hσ(e)=(H(e),σ(e)),

such that H(e) = F(e)∪ G(e) and σ(e) = s(μ(e), π(e)), where s denotes an s-norm and H(e) = [sup(μF(e), μG(e)), sup(μ+F(e), μ+G(e))].

Definition 14 (Intersection of two GIVFSCs). FμGπ is the intersection of two GIVFSCs (Fμ,A) and (Gπ,B), which generates a GIVFSCs (Hσ,C), where C = AB and Hδ : EIU × A(x) is defined by

Hσ(e)=(H(e),δ(e)),

such that H(e) = F(e)∩G(e) and σ(e) = s(μ(e), π(e)), where s is an s-norm and H(e) = [inf(μF(e), μG(e)), inf (μ+F(e), μ+G(e))].

Example 4. Consider the GIVFSCs Fμ and Gπ presented in Example 2, where FμGπ = Hσ is expressed as follows:

Hσ(e1)=({c1[inf[0.3,0.4],inf[0.5,0.7]],c2[inf[0.5,0.7],inf[0.6,0.7]],c3[inf[0.5,0.7],inf[0.8,0.9]]},min(0.5,0.7)),Hσ(e2)=({c1[inf[0.4,0.6],inf[0.7,0.8]],c2[inf[0.1,0.2],inf[0.7,0.8]],c3[inf[0.1,0.5],inf[0.6,0.7]]},min(0.8,0.9)),Hσ(e3)=({c1[inf[0.1,0.3],inf[0.6,0.9]],c2[inf[0,0.2],inf[0.6,0.9]],c3[inf[0.3,0.5],inf[0.6,0.9]]},min(0.6,0.9)),Hσ(e1)=({c1[0.3,0.5],c2[0.5,0.6],c3[0.5,0.8]},0.5),Hσ(e2)=({c1[0.4,0.7],c2[0.1,0.2],c3[0.1,0.6]},0.7),Hσ(e3)=({c1[0.1,0.6],c2[0,0.6],c3[0.3,0.6]},0.6).

Proposition 2. Consider any three GIVFSCs Fμ, Gπ, and Hσ. The following results are valid.

  • (i) FμGπ = GπFμ, if A × B = B × A.

  • (ii) FμGπ = GπFμ, if A × B = B × A.

  • (iii) Fμ ∪(GπHσ) = (FμGπ)∪Hσ, if A× B = B× A.

  • (iv) Fμ ∩(GπHσ) = (FμGπ)∩Hσ, if A× B = B× A .

  • (v) Fμ ∪1μ = 1μ.

  • (vi) Fμ ∪∅︀μ = Fμ.

  • (vii) Fμ ∩1μ = Fμ.

  • (viii) Fμ ∩∅︀μ = ∅︀μ.

Proof. (iii) Assume GπHσ = Jβ, then we have Fμ ∪ (GπHσ) = FμJβ = Bθ.

From the definition, we have, Bθ = (B(e)(c), θ(e)) such that B(e) = F(e)∪J(e) and θ(e) = t(μ(e), β(e)).

However, J(e) = G(e)∪H(e) and β(e) = t(π(e), δ(e)).

Now, B(e) = F(e)∪(G(e)∪H(e)) and

θ(e)=t(μ(e),t(π(e),σ(e))).

B(e) = (F(e)∪G(e))∪H(e) (because the union of intervalvalued fuzzy sets is associative) and θ (e) = t(t(μ(e), π(e)), σ(e)) (because t-norm is associative).

Then, Bθ = (FμGπ)∪Hσ.

Hence, Fμ ∪(GπHσ) = (FμGπ)∪Hσ.

The remaining proofs follow directly from the definition.

Proposition 3. Let Fμ and Gπ be any two GIVFSCs. The following results are valid.

  • (i) (FμGπ)C = FμcGπC.

  • (ii) (FμGπ)C = FμcGπC.

Proof. (i) FμCGπC

=(1-Fμ,1-μ(e))(1-Gπ,1-π(e))=((1-Fμ1-Gπ),(1-μ(e)1-π(e)))=(1-(FμGπ),1-(μ(e)π(e)))=(FμGπ)C.

The proof of (ii) follows from (i).

Proposition 4. Consider any two GIVFSCs, Fμ and Gπ. The following conclusions are drawn.

  • (i) Fμ ∪(GπHσ) = (FμGπ)∩(FμHσ).

  • (ii) Fμ ∩(GπHσ) = (FμGπ)∪(FμHσ).

Proof. (i) For all xE,

λF(x)(G(x)H(x))(x)=[sup(λ-F(x)(x),λ-G(x)H(x)(x)),sup(λ+F(x)(x),λ+G(x)H(x)(x))]=[sup(λ-F(x)(x),inf(λ-G(x)(x),λ-H(x)(x))),sup(λ+F(x)(x),inf(λ+G(x)(x),λ+H(x)(x)))]=[inf(sup(λ-F(x)(x),λ-G(x)(x)),sup(λ-F(x)(x),λ-H(x)(x)))inf(sup(λ+F(x)(x),λ+G(x)(x)),sup(λ+F(x)(x),λ+H(x)(x))]=λ(F(x)G(x))(F(x)H(x)(x),

and also,

γμ(x)(π(x)σ(x))(x)=max{γμ(x)(x),γπ(x)σ(x)(x)}=max{γμ(x)(x),min(γπ(x)(x),γσ(x)(x))}=min{max(γμ(x)(x),γπ(x)(x)),max(γμ(x)(x),γσ(x)(x))}=min{γμ(x)π(x)(x),γμ(x)σ(x)(x)}=γ(μ(x)π(x))(μ(x)σ(x))(x).

(ii) Similar to Proof (i).

The concepts of AND and OR operations on GIVFSCs are presented in this section, along with an example of their application to a decision-making problem.

Definition 15. Assume that a GIVFSC over (K,E) is (Fμ,A). Then, the complement of (Fμ,A) is represented by (Fμc, ¬A) and can be described as Fμc = Gπ, where σ(e) = μc(e) = 1 − μ(e) and G(e) = Fc(e), ∀eE.

Definition 16. Suppose there exist two GIVFSCs over (K,E) : (Fμ,A) and (Gπ,B). Then, (Fμ,A)AND(Gπ,B) is represented as (Fμ,A)∧(Gπ,B) and is defined as follows:

(Fμ,A)(Gπ,B)=(Hδ,A×B),

where Hδ(α, β) = (H(α, β), σ(α, β)) for all (α, β) ∈ A × B, such that H(α, β) = F(α)∩G(β) and δ(α, β) = t(μ(α), π(β)), where t is a t-norm.

Example 5. Assume that there are three machines K = {c1, c2, c3} in F25, C = {0100, 0110, 0101} and consider the collection of parameters such that the length of their strings according to a certain task is described by E1 = {e1, e2, e3} = {011, 10110, 0111010} and E2 = {e4, e5, e6} = {0101, 11110, 0111011}. We assume that a business wants to buy a single machine of this type based on only two parameters.

Let two GIVFSCs (Fμ,A) and (Gπ,B) be defined as follows: μ(e1) = 0.83, μ(e2) = 0.53, and μ(e3) = 0.54, π(e4) = 0.6, π(e5) = 0.67 and π(e6) = 0.82,

Fμ(e1)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.83),Fμ(e2)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Fμ(e3)=({c1[0.5,0.6],c2[0.8,0.9],c3[0.7,0.8]},0.54),

and

Gπ(e4)=({c1[0.3,0.4],c2[0.2,0.3],c3[0.4,0.5]},0.6),Gπ(e5)=({c1[0.4,0.5],c2[0.6,0.7],c3[0.8,0.9]},0.67),Gπ(e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.82).

Determine the “∧” between the two GIVFSCs over (K, E) as follows: (Fμ,A)∧(Gπ,B) = (Hδ,A × B), where

Hσ(e1,e4)=({c1[0.1,0.2],c2[0.2,0.3],c3[0.3,0.4]},0.6),Hσ(e1,e5)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.67),Hσ(e1,e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.82),Hσ(e2,e4)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Hσ(e2,e5)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Hσ(e2,e6)=({c1[0,0.1],c2[0.2,0.3],c3[0.3,0.4]},0.53),Hσ(e3,e4)=({c1[0.3,0.4],c2[0.2,0.3],c3[0.4,0.5]},0.54),Hσ(e3,e5)=({c1[0.4,0.5],c2[0.6,0.7],v3[0.7,0.8]},0.54),Hσ(e3,e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.54).

Now, for each gP, first, we determine the statistical grade RgP(ci) to identify the optimal machine.

RgP(ci)=iu((Ci--μH(gi)-(c))+(Ci+-μH(gi)+(c)).

The outcome is displayed in Tables 1 and 2.

Let P = {g1 = (e1, e4), g2 = (e1, e5), … , g9 = (e3, e6)}.

We now record the best possible scores in numbers, denoted by parentheses, in each row, with the exception of the final row, which represents the machine’s grade of belongingness in relation to each set of parameters (refer to Table 2 for details). Currently, all of these numerical grades’ items combined with the matching value of δ is used to calculate each machine’s result. The desired machine yields the best results. Because both parameters are identical, we do not consider the machine’s numerical grades against the pair (ei, ej), i = 1, 2, 3, j = 4, 5, 6:

Result(c1)=0,Result(c2)=(1.2*0.67)+(1.2*0.82)+(0.2*0.53)+(1.2*0.54)=2.542,Result(c3)=(0.6*0.6)+(1.2*0.53)+(1.2*0.53)+(0.8*0.53)+(0.6*0.54)+(1*0.54)=2.92.

Therefore, machine c3, which yielded the best outcome was selected by the company.

Definition 17. Considering two GIVFSCs (Fμ,A) and (Gπ, B), (Fμ,A)∨(Gπ,B) is defined as follows:

(Fμ,A)(Gπ,B)=(Hσ,A×B),

where Hσ(α, β) = (H(α, β), σ(α, β)) for all (α, β) ∈ A× B, such that H(α, β) = F(α)∪G(β) and δ(α, β) = s(μ(α), π(β)), where s is an s norm.

Proposition 5. Assume any two GIVFSCs, let (Fμ,A) and (Gπ,B). The following outcomes are valid.

((Fμ,A)(Gπ,B))C=(Fμ,A)c(Gπ,B)C,((Fμ,A)(Gπ,B))C=(Fμ,A)C(Gπ,B)C.

Proof. (i) Considering two GIVFSCs (Fμ,A) and (Gπ,B), we have

(Fμ,A)C(Gπ,B)C=(FμC,¬A)(GπC,¬B)=(Hσ,¬A׬B)

Here,

Hσ(¬ɛ,¬ɛ)=FμC(¬ɛ)GπC(¬ɛ)=(Hσ,¬(A×B)).

Hence, we have

(FμC,A)C(GπC,B)C=(Hσ,¬(A×B)).

Assume that

(FμC,A)(GπC,B)=(Hσ,A×B).

Then we have

((FμC,A)(GπC,B))C=(Hσ,A×B)C=(HσC,¬(A×B)),(ɛ,)A×B,HσC(¬ɛ,¬)=(HσC(ɛ,))C=(F(ɛ)G())C=(F(ɛ))C(G())C=F¯C(¬ɛ)G¯C(¬ɛ)=(HσC,¬(A×B)).

Therefore, (Fμ,A)∧(Gπ,B) = (Hσ,¬(A× B)).

Consequently, based on our discussion, we have a result.

((Fμ,A)(Gπ,B))C=(Fμ,A)C(Gπ,B)C.

The same method is applied to demonstrate the second.

To illustrate the significance of our work in the decision-making process, we compared the GIVFSs first described by Alkhazaleh and Salleh [6]. Both GIVFSCs and GIVFSs are used to solve real-world issues and identify the best options based on candidate grades. While our model GIVFSCs yield the parameterized set as the set of fuzzy codes—that is, after fuzzifying the universal set, we parameterized it—GIVFSs yield the parameterized set as the universal set/objects.

In this study, we present a GIVFSC and investigate some of its properties using an interval-valued fuzzy soft set and fuzzy code. The complement, intersection, union, AND, and OR operations were defined using the GIVFSC. Decisions regarding the study were made. The limitation of this study is the integration of a generalized interval-valued fuzzy set and fuzzy codes, because this is a new formulation. Future researchers in this discipline will find this study as an introduction to related fields. Further studies could investigate generalized interval-valued fuzzy soft cyclic codes, generalized interval-valued fuzzy soft linear codes, and other relevant areas.

Table. 1.

Table 1. Statistical grades of GIVFSCs.

Hσc1c2c3σ
(e1, e4)[0.1, 0.2][0.2, 0.3][0.3, 0.4]0.6
(e1, e5)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.67
(e1, e6)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.82
(e2, e4)[0, 0.1][0.2, 0.3][0.4, 0.5]0.53
(e2, e5)[0, 0.1][0.2, 0.3][0.4, 0.5]0.53
(e2, e6)[0, 0.1][0.2, 0.3][0.3, 0.4]0.53
(e3, e4)[0.3, 0.4][0.2, 0.3][0.4, 0.5]0.54
(e3, e5)[0.4, 0.5][0.5, 0.6][0.7, 0.8]0.54
(e3, e6)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.54

Table. 2.

Table 2. Numeral rating RgP(ci).

Hσx1x2x3σ
g1−0.60(0.6) 0.6
g2−1.2(1.2)00.67
g3−1.2(1.2)00.82
g4−1.20(1.2)0.53
g5−1.20(1.2)0.53
g6−1(0.2)(0.8)0.53
g70−0.6(0.6)0.54
g8−0.8−0.2(1)0.54
g9−1.2(1.2)00.54

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MasreshaWassieWoldie received an M.Sc. degree in 2015 from Bahir Dar University, Ethiopia. His research interests are fuzzy systems, fuzzy soft code and related topics, and fuzzy soft algebras.

E-mail: masreshawassie28@gmail.com

Jejaw Demamu Mebrat received a Ph.D. degree in 2012 from National Institute of TechnologyWarangal,Warangal, India. His research interests are fuzzy systems, fuzzy soft code, coding theory, fuzzy decoding and related topics, and fuzzy soft algebras.

E-mail: jejaw@yahoo.com

Mihret Alamneh Taye received Ph.D. degree in 2012 from India. His research interests are skew Heyting almost distributive lattices, dual skew Heyting almost distributive lattices, d-fuzzy and injective fuzzy ideals in distributive lattices, and fuzzy soft codes.

E-mail: mihretmahlet@yahoo.com

Article

Original Article

International Journal of Fuzzy Logic and Intelligent Systems 2024; 24(4): 378-386

Published online December 25, 2024 https://doi.org/10.5391/IJFIS.2024.24.4.378

Copyright © The Korean Institute of Intelligent Systems.

Generalized Interval-Valued Fuzzy Soft Code and Its Properties on Decision Makings

Masresha Wassie Woldie1, Jejaw Demamu Mebrat2, and Mihret Alamneh Taye1

1Department of Mathematics, Bahar Dar University, Bahir Dar, Ethiopia
2Department of Mathematics, Debark University, Gondar, Ethiopia

Correspondence to:Masresha Wassie Woldie (masreshawassie28@gmail.com)

Received: March 18, 2024; Revised: November 23, 2024; Accepted: December 10, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License (http://creativecommons.org/licenses/by-nc/3.0/) which permits unrestricted noncommercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An interval-valued fuzzy soft set theory is a strong instrument that can offer the uncertain data processing capacity in an imprecise environment. In this study, a generalized interval-valued fuzzy soft code (GIVFSC), which combines an interval-valued fuzzy soft set and a fuzzy code, is proposed. The GIVFSC framework supports its operation and decision-making processes. Various features of the GIVFSC are examined to improve the combination of interval-valued fuzzy soft sets and fuzzy codes.

Keywords: Fuzzy decoding, Fuzzy soft code, Interval-valued fuzzy soft set, Generalized interval-valued fuzzy soft set, Decision-Making, Generalized interval-valued fuzzy soft code

1. Introduction

Molodtsov [1] established soft set theories and models as new mathematical tools for handling uncertainties that cannot be controlled by existing mathematical tools. Amudhambigai and Neeraja [2] studied the fuzzy codes and its application. Hong and Qin [3] discussed algebraic structures of fuzzy soft sets. Roy and Maji [4] presented a fuzzy soft theoretical approach to decision-making problems and shared several results on the application of fuzzy soft sets to these problems. Lal [5] applied fuzzy algebra to coding theory. Smarandache [6] initiated soft sets using a rough set of codes to create soft codes (soft linear codes). Ali et al. [7] developed and designed novel type of linear algebraic codes known as “soft linear algebraic codes” by employing soft sets. Alkhazaleh [8] introduced the concept of effective fuzzy soft sets, explored their operations, and investigated several of their properties. Kong et al. [9] examined the use of soft sets with fuzzy properties in grey theory-based decision-making problems. Gogoi et al. [10] designed a fuzzy softset theory applied to regular problems. Prade [11] proposed an interval-valued fuzzy set. Bustince [12] explored interval-valued fuzzy sets in soft computing. Majumdar and Samanta [13] introduced generalized fuzzy soft sets. Yang et al. [14] introduced the concept of interval-valued soft sets with fuzzy values by merging the notions of soft set models with interval-valued fuzzy sets. Chetia and Das [15] developed an application of interval-valued fuzzy software for medical diagnosis. Alkhazaleh and Salleh [16] introduced the concept of the generalized interval-valued fuzzy soft set (GIVFSs). A generalized Z-fuzzy soft covering-based rough matrix was defined by Sivaprakasam and Angamuthu [17], and its various algebraic properties were investigated. They used generalized Z-fuzzy soft covering-based rough matrices to offer a unique multi-attribute group decision-making problem model. Das and Granados [18] proposed a method to address group decision-making problems using fuzzy parameterized intuitionistic multi-fuzzy N-soft sets. They introduced an induced fuzzy parameterized hesitant N-soft set as an extension of multi-fuzzy N-soft sets, tailored for group decision-making problems. Mukherjee [19] introduced the concept of the intuitionistic fuzzy soft rough sets. Jiang et al. [20] proposed the interval-valued intuitionistic fuzzy soft set theory. Hayat and Mahmood [21] introduced an insight into a bipolar soft set in the union of two isomorphic hemirings. They characterized type-2 soft graphs as the underlying subgroups of simple graphs. Hayat et. al. [22] discussed the aggregation concepts of the selection of design parameter values by merging the acceptable and satisfactory level requirements of customers. Yang, et al. [23] developed the basic Dombi operational laws for spherical fuzzy soft numbers.

1.1 Motivation

Alkhazaleh and Salleh [16] introduced the concept of a GIVFSs and its application to their characteristics. In [2] and [24], the authors defined fuzzy codes by applying the concepts of classical fuzzy sets and binary codes.

Based on these concepts, we introduce the concept of a generalized interval-valued fuzzy soft code (GIVFSC). The parameterization of fuzzy sets is related to the degree of the GIVFSs concept. However, in our study, for the GIVFSC, the degree was attached to the parameterization of the fuzzy code. This study aims to develop new ideas by integrating the definition of a GIVFSs and fuzzy codes, from which a novel fuzzy soft set model can be derived based on the GIVFSC and its operations. The novelty of this study is that it develops a combination of interval-valued fuzzy soft sets and fuzzy codes to generate an interval-valued fuzzy soft code. GIVFSCs are strong tools for processing uncertain data and are utilized in numerous different applications, such as ranking alternatives, processing uncertain data, and solving multigroup decision-making problems.

Section 2 presents the fundamental definitions and properties of the preliminary concepts. Section 3 presents the GIVFSC. In Section 4, the basic operations of GIVFSC are presented. In Section 5, the operations of GIVFSC and its applications are discussed.

2. Preliminary

In this section, we discuss some basic concepts used to interpret our findings.

Definition 1 [1]. Let E be a set of parameters and U the initial universe set. If and only if F maps E into the set of all subsets of set U, pair (F,E) is referred to as a soft set over U.

In particular, the soft set is a parametrized family of the set U. Every set F(e), eE, from this family may be considered as the set of e approximate elements of the soft set.

Definition 2 [4]. Let {A1, A2, … , Ai} ⊆ E be a set of parameters that can be used to characterize a set of k objects, U = {u1, u2, … , uk}. Let each parameter set Ai represent the ith class of parameters and each property set be represented by its elements. Consider the set of all the fuzzy sets of U as P(U). The fuzzy soft sets over U are pairs (Fi,Ai), where Fi is the mapping indicated by

Fi:AiP(U).

Definition 3 [24]. Let C be the code, and the codewords with different lengths in C be {Cr1, Cr2, … , Crm}. The membership value of C is μC(x), which is described as follows in terms of the relative weight of each codeword:

If C = {Cr1, Cr2, … , Crm}, then μC(x) = {J(Cr1), J(Cr2), … , J(Crm)}, where J(Cri) denotes the relative weight of the codeword Cri, for i = 1, 2, …, m.

Definition 4 [7]. Let F = Z2 symbolize a field of dimension two. Let P(W) be every subset within set W and W = Fm be a vector space of size m over field F. If for each d in D; DV , f(d) is a linear algebraic code of W, where V denotes the collection of parameters, then (f,D) is considered as an algebraic soft code over F.

Definition 5 [13]. The universal set of elements is U = {x1, x2, … , xm}, and the universal set of parameters is E = {e1, e2, … , en}. We refer to pair (U, E) as a soft universe. Considering that a fuzzy subset of E is μ such that μ : EI = [0, 1], let F : EIU, where IU is the collection of all fuzzy subsets of U. Consider the function Fμ : EIU × I, which is defined as follows:

Fμ = (F(e), μ(e)), where F(e) ∈ IU. A generalized fuzzy soft set over the soft universe (U,E) is thus referred to as Fμ.

Definition 6 [16]. Considering a universal set of parameters, E = {e1, e2, … , en}, and a universal set of elements, U = {x1, x2, … , xm}. A soft universe is applied to pair (U,E). Define as the set of all interval-valued fuzzy subsets on U, and let μ be a fuzzy subset of E, that is, μ : EI = [0, 1]. Define function μ : E (U) × I.

A GIVFSs over the soft universe (U, E) is thus referred to as μ = ( (e), μ(e)).

3. Generalized Interval-Valued Fuzzy Soft Code

In this section, we introduce the GIVFSC, which combines a GIVFSs with a fuzzy code.

Definition 7. Let E = {e1, e2, … , em} be the standard set of parameters and K = {c1, c2, … , cn} be the set of codewords or the universal set of elements in F2n. We refer to pair (K,E) as a soft code universe. Let A be a mapping A : C′ → [0, 1], CF2n, and F : EIK and μ be a fuzzy code of E, that is, μ : EA(x), defined as

A(x)=2ipik(k+1),

where IC denotes the set of all interval-valued fuzzy subsets on K and pi denotes the location of 1s. Consider the function Fμ : EIK × A(x) defined by

Fμ(x)=(F(e),μ(x)).

Subsequently, Fμ is called a GIVFSC over the soft universe (K,E), for each parameter ei.

Over the soft universe (K,E), Fμ is referred to as a GIVF-SCs.

For every ei parameter, Fμ(ei) = (F(ei)(c), μ(ei)) describes the degree to which such belongingness is possible, and it is symbolized by μ(ei), as well as the extent to which the components of C in F(ei) belong. We can define Fμ(ei) as follows:

Fμ(ei)=({c1F(ei)(c1),c2F(ei)(c2),,cnF(ei)(cn)},μ(ei)).

Example 1. Let K = {c1, c2, c3} be a collection of universes in F25, E = {e1, e2, e3} be a set of parameters, and C′ = {1000, 0110, 0111} be a set of codewords in F24. Let μ : EA(x), where A is a mapping: A : C′ → [0, 1], CF2n, μ(e1) = 0.1, μ(e2) = 0.5, and μ(e9) = 0.9 defined as (1), A(1000) = 0.1, A(0110) = 0.5, and A(0111) = 0.9, define a function Fμ : EIK × A(x) as follows:

Fμ(e1)=({c1[0.1,0.6],c2[0.1,0.8],c3[0.1,0.8]},0.1),Fμ(e2)=({c1[0.5,0.7],c2[0.5,0.6],c3[0.5,0.5]},0.5),Fμ(e3)=({c1[0.9,0.9],c2[0.9,1],c3[0.9,0.9]},0.9).

Then, Fμ is GIVFSCS over (K,E).

In matrix notation,

Fμ=[[0.1,0.6]   [0.1,0.8]   [0.1,0.8],0.1[0.5,0.7]   [0.5,0.6]   [0.5,0.5],0.5[0.9,0.9]   [0.9,1]   [0.9,0.9],0.9].

Definition 8. Let Fμ and Gδ be two GIVFSCs over (K,E). Fμ, expressed as FμGδ, is a GIVFSC subset of Gδ, if

  • μ(e) ⊆ δ(e), ∀ eE,

  • F(e) ⊆ G(e), ∀ eE.

Example 2. Let E = {e1, e2, e3} be a set of parameters, where e1 = cheap, e2 = expensive, and e3 = more expensive, and K = {c1, c2, c3} be the codes of the three cars in F27. Let C1 = {0101, 1001, 0011} and C2 = {0011, 1011, 0111} be the codewords in F24. Let μ : EA(x), for each xC1, and A a function expressed as A : C1 → [0, 1] and described as follows in (1): μ(e1) = 0.5, μ(e2) = 0.7, μ(e3) = 0.6, and A(0101) = 0.6, A(1001) = 0.5, A(0011) = 0.7.

Let Fμ be a GIVFSC over (K, E) defined as follows:

Fμ(e1)=({c1[0.3,0.5],c2[0.5,0.6],c3[0.5,0.8]},0.5),Fμ(e2)=({c1[0.4,0.7],c2[0.1,0.7],c3[0.6,0.7]},0.7),Fμ(e3)=({c1[0.1,0.6],c2[0,0.6],c3[0.3,0.6]},0.6).

Consider another GIVFSC over (K, E) with the following definition for Gσ:

σ : EA(x), for any xC2, and A is a function expressed as A : C2 → [0, 1], A(0011) = 0.7, A(1011) = 0.8, A(0111) = 0.9, and σ(e1) = 0.7, σ(e2) = 0.8, σ(e3) = 0.9.

Gσ(e1)=({c1[0.4,0.7],c2[0.7,0.7],c3[0.7,0.9]},0.7),Gσ(e2)=({c1[0.6,0.8],c2[0.4,0.8],c3[0.8,0.8]},0.8),Gσ(e3)=({c1[0.3,0.9],c2[0.2,0.9],c3[0.5,0.9]},0.9).

Hence, Fμ is a GIVFSC subset of Gσ.

Definition 9. We write Fμ = Gσ if Fμ is a GIVFSC subset of Gσ and Gσ is a GIVFSC subset of Fμ. Two GIVFSCs Fμ and Gσ over (K, E) are considered equal if the following conditions are satisfied

  • (i) μ(e) = σ(e), ∀eE,

  • (ii) F(e) = G(e), ∀eE.

Definition 10. The GIVFSC is identified by ∅︀μ, and is characterized as follows: ∅︀μ : EIK× A(x) such that

μ(e)=(F(e)(x),μ(e)).

Here, F(e) = [0, 0] = [0] and μ(e) = 0 for all eE.

Definition 11. The GIVFSC is represented by the symbol 1μ if 1μ : EIK × A(x) and defined as follows:

  • (iii) 1μ(e) = (F(e)(x), μ(e)), (5)

  • (iv) Here, F(e) = [1, 1] = [1] and μ(e) = 1 for all eE.

4. Basic Operations on GIVFSC

Definition 12. Considering a GIVFSCS over (K,E) as Fμ. The complement of Fμ is thus represented as FμC; it is characterized as FμC = 1 − Fμ = Gπ, such that for any e in E, π(e) = C (μ(e)) and G(e) = C (F(e)) with c being a complement notation.

Example 3. Consider a GIVFSCs Fμ over (K,E) in Example 1,

Fμ=[[0.1,0.6]   [0.1,0.8]   [0.1,0.8],0.1[0.5,0.7]   [0.5,0.6]   [0.5,0.5],0.5[0.9,0.9]   [0.9,1]   [0.9,0.9],0.9].Then FμC=[[0.4,0.9]   [0.2,0.9]   [0.2,0.9],0.9[0.3,0.5]   [0.4,0.5]   [0.5,0.5],0.5[0.1,0.1]   [0,0.1]   [0.1,0.1],0.1].

Proposition 1. Consider a GIVFSCs over (K,E) as Fμ. The following conclusions were drawn.

(FμC)C=Fμ.

Proof. As FμC = 1 − Fμ, then from the definition, we have

(FμC)C=(1-Fμ)C=1-(1-Fμ)=1-1+Fμ=Fμ.

This is complete the proof.

Definition 13 (Union of two GIVFSCs). FμGπ is the union of two GIVFSCs (Fμ,A) and (Gπ,B), which generates a GIVFSCs (Hσ,C), where C = AB and Hσ : EIK × A(x) is defined by

Hσ(e)=(H(e),σ(e)),

such that H(e) = F(e)∪ G(e) and σ(e) = s(μ(e), π(e)), where s denotes an s-norm and H(e) = [sup(μF(e), μG(e)), sup(μ+F(e), μ+G(e))].

Definition 14 (Intersection of two GIVFSCs). FμGπ is the intersection of two GIVFSCs (Fμ,A) and (Gπ,B), which generates a GIVFSCs (Hσ,C), where C = AB and Hδ : EIU × A(x) is defined by

Hσ(e)=(H(e),δ(e)),

such that H(e) = F(e)∩G(e) and σ(e) = s(μ(e), π(e)), where s is an s-norm and H(e) = [inf(μF(e), μG(e)), inf (μ+F(e), μ+G(e))].

Example 4. Consider the GIVFSCs Fμ and Gπ presented in Example 2, where FμGπ = Hσ is expressed as follows:

Hσ(e1)=({c1[inf[0.3,0.4],inf[0.5,0.7]],c2[inf[0.5,0.7],inf[0.6,0.7]],c3[inf[0.5,0.7],inf[0.8,0.9]]},min(0.5,0.7)),Hσ(e2)=({c1[inf[0.4,0.6],inf[0.7,0.8]],c2[inf[0.1,0.2],inf[0.7,0.8]],c3[inf[0.1,0.5],inf[0.6,0.7]]},min(0.8,0.9)),Hσ(e3)=({c1[inf[0.1,0.3],inf[0.6,0.9]],c2[inf[0,0.2],inf[0.6,0.9]],c3[inf[0.3,0.5],inf[0.6,0.9]]},min(0.6,0.9)),Hσ(e1)=({c1[0.3,0.5],c2[0.5,0.6],c3[0.5,0.8]},0.5),Hσ(e2)=({c1[0.4,0.7],c2[0.1,0.2],c3[0.1,0.6]},0.7),Hσ(e3)=({c1[0.1,0.6],c2[0,0.6],c3[0.3,0.6]},0.6).

Proposition 2. Consider any three GIVFSCs Fμ, Gπ, and Hσ. The following results are valid.

  • (i) FμGπ = GπFμ, if A × B = B × A.

  • (ii) FμGπ = GπFμ, if A × B = B × A.

  • (iii) Fμ ∪(GπHσ) = (FμGπ)∪Hσ, if A× B = B× A.

  • (iv) Fμ ∩(GπHσ) = (FμGπ)∩Hσ, if A× B = B× A .

  • (v) Fμ ∪1μ = 1μ.

  • (vi) Fμ ∪∅︀μ = Fμ.

  • (vii) Fμ ∩1μ = Fμ.

  • (viii) Fμ ∩∅︀μ = ∅︀μ.

Proof. (iii) Assume GπHσ = Jβ, then we have Fμ ∪ (GπHσ) = FμJβ = Bθ.

From the definition, we have, Bθ = (B(e)(c), θ(e)) such that B(e) = F(e)∪J(e) and θ(e) = t(μ(e), β(e)).

However, J(e) = G(e)∪H(e) and β(e) = t(π(e), δ(e)).

Now, B(e) = F(e)∪(G(e)∪H(e)) and

θ(e)=t(μ(e),t(π(e),σ(e))).

B(e) = (F(e)∪G(e))∪H(e) (because the union of intervalvalued fuzzy sets is associative) and θ (e) = t(t(μ(e), π(e)), σ(e)) (because t-norm is associative).

Then, Bθ = (FμGπ)∪Hσ.

Hence, Fμ ∪(GπHσ) = (FμGπ)∪Hσ.

The remaining proofs follow directly from the definition.

Proposition 3. Let Fμ and Gπ be any two GIVFSCs. The following results are valid.

  • (i) (FμGπ)C = FμcGπC.

  • (ii) (FμGπ)C = FμcGπC.

Proof. (i) FμCGπC

=(1-Fμ,1-μ(e))(1-Gπ,1-π(e))=((1-Fμ1-Gπ),(1-μ(e)1-π(e)))=(1-(FμGπ),1-(μ(e)π(e)))=(FμGπ)C.

The proof of (ii) follows from (i).

Proposition 4. Consider any two GIVFSCs, Fμ and Gπ. The following conclusions are drawn.

  • (i) Fμ ∪(GπHσ) = (FμGπ)∩(FμHσ).

  • (ii) Fμ ∩(GπHσ) = (FμGπ)∪(FμHσ).

Proof. (i) For all xE,

λF(x)(G(x)H(x))(x)=[sup(λ-F(x)(x),λ-G(x)H(x)(x)),sup(λ+F(x)(x),λ+G(x)H(x)(x))]=[sup(λ-F(x)(x),inf(λ-G(x)(x),λ-H(x)(x))),sup(λ+F(x)(x),inf(λ+G(x)(x),λ+H(x)(x)))]=[inf(sup(λ-F(x)(x),λ-G(x)(x)),sup(λ-F(x)(x),λ-H(x)(x)))inf(sup(λ+F(x)(x),λ+G(x)(x)),sup(λ+F(x)(x),λ+H(x)(x))]=λ(F(x)G(x))(F(x)H(x)(x),

and also,

γμ(x)(π(x)σ(x))(x)=max{γμ(x)(x),γπ(x)σ(x)(x)}=max{γμ(x)(x),min(γπ(x)(x),γσ(x)(x))}=min{max(γμ(x)(x),γπ(x)(x)),max(γμ(x)(x),γσ(x)(x))}=min{γμ(x)π(x)(x),γμ(x)σ(x)(x)}=γ(μ(x)π(x))(μ(x)σ(x))(x).

(ii) Similar to Proof (i).

5. AND (∧) and OR (∨) Operations on GIVFSC with Applications

The concepts of AND and OR operations on GIVFSCs are presented in this section, along with an example of their application to a decision-making problem.

Definition 15. Assume that a GIVFSC over (K,E) is (Fμ,A). Then, the complement of (Fμ,A) is represented by (Fμc, ¬A) and can be described as Fμc = Gπ, where σ(e) = μc(e) = 1 − μ(e) and G(e) = Fc(e), ∀eE.

Definition 16. Suppose there exist two GIVFSCs over (K,E) : (Fμ,A) and (Gπ,B). Then, (Fμ,A)AND(Gπ,B) is represented as (Fμ,A)∧(Gπ,B) and is defined as follows:

(Fμ,A)(Gπ,B)=(Hδ,A×B),

where Hδ(α, β) = (H(α, β), σ(α, β)) for all (α, β) ∈ A × B, such that H(α, β) = F(α)∩G(β) and δ(α, β) = t(μ(α), π(β)), where t is a t-norm.

Example 5. Assume that there are three machines K = {c1, c2, c3} in F25, C = {0100, 0110, 0101} and consider the collection of parameters such that the length of their strings according to a certain task is described by E1 = {e1, e2, e3} = {011, 10110, 0111010} and E2 = {e4, e5, e6} = {0101, 11110, 0111011}. We assume that a business wants to buy a single machine of this type based on only two parameters.

Let two GIVFSCs (Fμ,A) and (Gπ,B) be defined as follows: μ(e1) = 0.83, μ(e2) = 0.53, and μ(e3) = 0.54, π(e4) = 0.6, π(e5) = 0.67 and π(e6) = 0.82,

Fμ(e1)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.83),Fμ(e2)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Fμ(e3)=({c1[0.5,0.6],c2[0.8,0.9],c3[0.7,0.8]},0.54),

and

Gπ(e4)=({c1[0.3,0.4],c2[0.2,0.3],c3[0.4,0.5]},0.6),Gπ(e5)=({c1[0.4,0.5],c2[0.6,0.7],c3[0.8,0.9]},0.67),Gπ(e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.82).

Determine the “∧” between the two GIVFSCs over (K, E) as follows: (Fμ,A)∧(Gπ,B) = (Hδ,A × B), where

Hσ(e1,e4)=({c1[0.1,0.2],c2[0.2,0.3],c3[0.3,0.4]},0.6),Hσ(e1,e5)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.67),Hσ(e1,e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.82),Hσ(e2,e4)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Hσ(e2,e5)=({c1[0,0.1],c2[0.2,0.3],c3[0.4,0.5]},0.53),Hσ(e2,e6)=({c1[0,0.1],c2[0.2,0.3],c3[0.3,0.4]},0.53),Hσ(e3,e4)=({c1[0.3,0.4],c2[0.2,0.3],c3[0.4,0.5]},0.54),Hσ(e3,e5)=({c1[0.4,0.5],c2[0.6,0.7],v3[0.7,0.8]},0.54),Hσ(e3,e6)=({c1[0.1,0.2],c2[0.5,0.6],c3[0.3,0.4]},0.54).

Now, for each gP, first, we determine the statistical grade RgP(ci) to identify the optimal machine.

RgP(ci)=iu((Ci--μH(gi)-(c))+(Ci+-μH(gi)+(c)).

The outcome is displayed in Tables 1 and 2.

Let P = {g1 = (e1, e4), g2 = (e1, e5), … , g9 = (e3, e6)}.

We now record the best possible scores in numbers, denoted by parentheses, in each row, with the exception of the final row, which represents the machine’s grade of belongingness in relation to each set of parameters (refer to Table 2 for details). Currently, all of these numerical grades’ items combined with the matching value of δ is used to calculate each machine’s result. The desired machine yields the best results. Because both parameters are identical, we do not consider the machine’s numerical grades against the pair (ei, ej), i = 1, 2, 3, j = 4, 5, 6:

Result(c1)=0,Result(c2)=(1.2*0.67)+(1.2*0.82)+(0.2*0.53)+(1.2*0.54)=2.542,Result(c3)=(0.6*0.6)+(1.2*0.53)+(1.2*0.53)+(0.8*0.53)+(0.6*0.54)+(1*0.54)=2.92.

Therefore, machine c3, which yielded the best outcome was selected by the company.

Definition 17. Considering two GIVFSCs (Fμ,A) and (Gπ, B), (Fμ,A)∨(Gπ,B) is defined as follows:

(Fμ,A)(Gπ,B)=(Hσ,A×B),

where Hσ(α, β) = (H(α, β), σ(α, β)) for all (α, β) ∈ A× B, such that H(α, β) = F(α)∪G(β) and δ(α, β) = s(μ(α), π(β)), where s is an s norm.

Proposition 5. Assume any two GIVFSCs, let (Fμ,A) and (Gπ,B). The following outcomes are valid.

((Fμ,A)(Gπ,B))C=(Fμ,A)c(Gπ,B)C,((Fμ,A)(Gπ,B))C=(Fμ,A)C(Gπ,B)C.

Proof. (i) Considering two GIVFSCs (Fμ,A) and (Gπ,B), we have

(Fμ,A)C(Gπ,B)C=(FμC,¬A)(GπC,¬B)=(Hσ,¬A׬B)

Here,

Hσ(¬ɛ,¬ɛ)=FμC(¬ɛ)GπC(¬ɛ)=(Hσ,¬(A×B)).

Hence, we have

(FμC,A)C(GπC,B)C=(Hσ,¬(A×B)).

Assume that

(FμC,A)(GπC,B)=(Hσ,A×B).

Then we have

((FμC,A)(GπC,B))C=(Hσ,A×B)C=(HσC,¬(A×B)),(ɛ,)A×B,HσC(¬ɛ,¬)=(HσC(ɛ,))C=(F(ɛ)G())C=(F(ɛ))C(G())C=F¯C(¬ɛ)G¯C(¬ɛ)=(HσC,¬(A×B)).

Therefore, (Fμ,A)∧(Gπ,B) = (Hσ,¬(A× B)).

Consequently, based on our discussion, we have a result.

((Fμ,A)(Gπ,B))C=(Fμ,A)C(Gπ,B)C.

The same method is applied to demonstrate the second.

6. Comparative Analysis

To illustrate the significance of our work in the decision-making process, we compared the GIVFSs first described by Alkhazaleh and Salleh [6]. Both GIVFSCs and GIVFSs are used to solve real-world issues and identify the best options based on candidate grades. While our model GIVFSCs yield the parameterized set as the set of fuzzy codes—that is, after fuzzifying the universal set, we parameterized it—GIVFSs yield the parameterized set as the universal set/objects.

7. Conclusion

In this study, we present a GIVFSC and investigate some of its properties using an interval-valued fuzzy soft set and fuzzy code. The complement, intersection, union, AND, and OR operations were defined using the GIVFSC. Decisions regarding the study were made. The limitation of this study is the integration of a generalized interval-valued fuzzy set and fuzzy codes, because this is a new formulation. Future researchers in this discipline will find this study as an introduction to related fields. Further studies could investigate generalized interval-valued fuzzy soft cyclic codes, generalized interval-valued fuzzy soft linear codes, and other relevant areas.

Conflict of Interest

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

Conflict of Interest:

The author(s) did not declare any conflicts of interest or shared interests.

Table 1 . Statistical grades of GIVFSCs.

Hσc1c2c3σ
(e1, e4)[0.1, 0.2][0.2, 0.3][0.3, 0.4]0.6
(e1, e5)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.67
(e1, e6)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.82
(e2, e4)[0, 0.1][0.2, 0.3][0.4, 0.5]0.53
(e2, e5)[0, 0.1][0.2, 0.3][0.4, 0.5]0.53
(e2, e6)[0, 0.1][0.2, 0.3][0.3, 0.4]0.53
(e3, e4)[0.3, 0.4][0.2, 0.3][0.4, 0.5]0.54
(e3, e5)[0.4, 0.5][0.5, 0.6][0.7, 0.8]0.54
(e3, e6)[0.1, 0.2][0.5, 0.6][0.3, 0.4]0.54

Table 2 . Numeral rating RgP(ci).

Hσx1x2x3σ
g1−0.60(0.6) 0.6
g2−1.2(1.2)00.67
g3−1.2(1.2)00.82
g4−1.20(1.2)0.53
g5−1.20(1.2)0.53
g6−1(0.2)(0.8)0.53
g70−0.6(0.6)0.54
g8−0.8−0.2(1)0.54
g9−1.2(1.2)00.54

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