Properties of Fuzzy Soft Codes and Their Role in Decision-Making

Masresha Wassie Woldie; Jejaw Demamu Mebrat; Mihret Alamneh Taye

doi:10.12688/f1000research.161824.1

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Research Article

Properties of Fuzzy Soft Codes and Their Role in Decision-Making

[version 1; peer review: awaiting peer review]

Masresha Wassie Woldie ^1,2, Jejaw Demamu Mebrat³, Mihret Alamneh Taye¹

PUBLISHED 16 Dec 2025

Author details Author details

¹ Demartment of Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia
² Mathematics, DebreTabor University, Debre Tabor, Amhara, Ethiopia
³ Mathematics, Debark University Department of Mathematics, Bahir Dar, Amhara, Ethiopia

Masresha Wassie Woldie
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation

Jejaw Demamu Mebrat
Roles: Conceptualization, Methodology, Writing – Review & Editing

Mihret Alamneh Taye
Roles: Conceptualization, Supervision, Writing – Review & Editing

OPEN PEER REVIEW

REVIEWER STATUS AWAITING PEER REVIEW

Abstract

Background

Fuzzy soft sets are well-established for decision-making under uncertainty. However, integrating soft set theory with fuzzy codes remains unexplored, as fuzzy soft code sets have not addressed these concepts. This study presents novel fuzzy soft codes and discusses their application in diagnosing medical conditions and decision-making, bridging a critical gap in the literature.

Methods

Numerous scholars have examined fuzzy sets, fuzzy codes, and the theory of fuzzy soft sets and their applications over the years. This paper introduces the novel concept of fuzzy soft codes by combining fuzzy soft set theory with the notion of fuzzy codes, inspired by Ozkan’s 2002 definition of fuzzy codes.

Results

We define fuzzy soft codes as a parameterized family of fuzzy codes, detailing their matrix representation, set-theoretic operations, and algebraic properties. A new decision-making method using choice and score values from comparison tables is introduced. An example demonstrates its effectiveness for selecting optimal objects, proving its real-world potential in areas like medical diagnosis. This bridges a key theoretical gap and enables future extensions like intuitionistic fuzzy soft codes.

Conclusions

This study addresses a gap in the integration of soft set theory with the notion of fuzzy codes. The primary objective is to introduce the concept of fuzzy soft codes within the context of decision-making by merging fundamental ideas from soft set theory and fuzzy coding, while also examining several of their key properties. Additionally, a practical example is provided to demonstrate how this approach can be effectively utilized across a range of real world problem.

Keywords

Fuzz set, soft sets, fuzzy soft sets, decision making, fuzzy codes, fuzzy soft codes.

Corresponding author: Masresha Wassie Woldie

Competing interests: No competing interests were disclosed.

Grant information: The author(s) declared that no grants were involved in supporting this work.

Copyright: © 2025 Woldie MW et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

How to cite: Woldie MW, Mebrat JD and Taye MA. Properties of Fuzzy Soft Codes and Their Role in Decision-Making [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:1403 (https://doi.org/10.12688/f1000research.161824.1) First published: 16 Dec 2025, 14:1403 (https://doi.org/10.12688/f1000research.161824.1) Latest published: 16 Dec 2025, 14:1403 (https://doi.org/10.12688/f1000research.161824.1)

1. Introduction

Making the right decision is critical because we frequently face problems in our daily activities. Several scholars investigated various properties of fuzzy soft set theories and its application.

In 1999, Molodtsov¹ proposed mathematical method using Soft set theory for dealing with complexities that existing mathematical tools that cannot be managed and controlled. Molodtsov developed a general mathematical method to handle uncertainty in complex problems by combining classical and modern mathematical methods. Maji, Biswas, and Roy in (2003)² studied the theory of soft sets initiated by Molodtsov.

Mostafa, et al. (2020)³ reviewed soft sets and its application in coding theory. S. Acharjee and A. Oza (2022)⁴ established the idea of soft set equivalent as well as the appropriate operations and relationships for soft sets based on equivalence. In their 2007, Roy and Maji⁵ explored theoretical approaches to problem solving using fuzzy sets, along with associated operations. Later, in 2010, Majumdar and Samanta⁶ focused on the application of generalized fuzzy soft sets in decision-making problems, examining relations and attributes within this framework. Further, Maji, et al. (2023)² laid the groundwork by defining fundamental concepts such as complement, null set, absolute set, subset, superset, and equality of soft sets. Ali, M. et al. (2018)⁷ introduced soft linear algebraic codes, which use soft sets, are a new class of linear algebraic codes.

The positive aspect of these codes is how they can simultaneously send m-distinct notifications to m-receivers. A method for constructing, and decoding of these new categories of linear algebraic soft code has been discovered. Feng, F. et al. (2010)⁸ investigated an adjustable approach to fuzzy soft sets based on decision making problem. Ozkan, A. and Ozkan, E.M (2002)⁹ presented fuzzy codes and fuzzy linear codes based on the idea of fuzzy subsets, and the notion of relative weight rate of a code word. Amudhambigai,B.and Neeraja, A (2019)¹⁰ developed a new perspective on fuzzy codes, and investigated the arithmetic operations of fuzzy codes. Cagman, et al. (2011)¹¹ developed a fuzzy soft set theory and the properties that go along with it, as well as a fuzzy soft aggregation operator that enables the development of more effective decision-making techniques. Suebsan,P. and Kanchanasuk, S (2022)¹² studied the applications of fuzzy soft sets over semigroups based on the grey theory. Kong, Z. et al. (2011)¹³ developed an application of fuzzy soft set theory in day -to-day problems. Gogoi, K., Kr. Dutta, A., & Chutia, C. (2014)¹⁴ studied an application of fuzzy soft set and its theoretic approaches. Herawan, T. et al. (2010)¹⁵ presented Soft set theoretic approach for dimensionality reduction. Wajid Ali, Tenzeela Shaheen, H. A. (2023)¹⁶ developed a novel decision theoretic rough set model based on Intuitionistic fuzzy data and its application. Mohamed, et al. (2023)¹⁷ presented AOEHO: A new hybrid data replication method in fog computing for IoT application.

One can observe that, the above studies are based on the combination of fuzzy sets with soft sets, and fuzzy sets with the concepts of binary codes. Thus it is important to study soft sets in new approaches by defining the fuzzy soft codes on decision-making, which parametrized family of fuzzy codes. This motivated by the forementioned works, the present study attempts to introduce the theoretical concepts of fuzzy soft sets on fuzzy codes and we develop a novel combination of fuzzy soft set to fuzzy codes. Also, we indicate the applications of fuzzy soft codes on the decision-making problems.

In this study, Section 2 contained fundamental definitions and features of preliminary concepts. Whereas section 3, Fuzzy soft codes, and some of its properties are considered.

2. Methods

We explain some basic concepts which we utilize to conclude our findings in this section.

Definition 2.1.

q-array code is a collection of symbol patterns wherein each member is chosen from such a collection of q different elements $Fq = {a_{1}, a_{2}, a_{3}, \dots, a_{q}} .$ ⁹

Definition 2.2.

Let $F_{q}^{n}$ is denoted by the collection of all ordered n-tuples $a = a_{1}, a_{2}, a_{3}, \dots, a_{n}$ , and each $a_{i} \in F_{q}$ is denoted by distinct letters, where each elements of $F_{q}^{n}$ are words is defined by different letters, where each elements of $F_{q}^{n}$ are words.¹⁰

Definition 2.3.

Let a be any code word of C. Let $a_{1}, a_{2}, a_{3}, \dots, a_{k}$ be the set of positions of 1’s in a. Then the sum $a_{1} + a_{2} + a_{3} + \dots + a_{k}$ is termed the relative weight of a code word a and is represented by $f (a)$ and is defined as⁹

(1)

f (a) = a_{1} + a_{2} + a_{3} + \dots + a_{k}

For example, If $a = 10111$ is a $C$ code word in $F_{2}^{5},$ then $f (a) = 1 + 3 + 4 + 5 = 13 .$ But since $111 \dots 1$ is a word a in the vector space $F_{2}^{n}$ , then its relative weight is $1 + 2 + 3 + \dots + n .$ That is,⁹

(2)

1 + 2 + 3 + \dots + n

This is called the maximum relative weight of a code C in $F_{2}^{n}$ .

$f (a)$ denotes the relative weight rate of a code word a in $n^{th}$ dimensional vector space $F_{2}^{n}$ and can be defined

as f (a).

Definition 2.4.

Let $C$ be a code, and $a, b \in C$ . A mapping $f : C^{'} \to [0, 1]$ is said to be a fuzzy code if it satisfies the following conditions:⁹

i. $f (a + b) \geq min {f (a), f (b)},$ for all $a, b \in C;$
ii. $f (- a) = f (a), for all a \in C;$
iii. $f (ab) \leq max {f (a), f (b)}, for all a, b \in C .$

Definition 2.5.

Let the set of parameters be denoted by E. Let P(U) denote all subsets of the initial universal set U, and let A be a proper subset of E. A soft set over U is a pair (F, A) if and only if F is a mapping defined by:²

(3)

F : A \to P (U)

The soft set, therefore, is a parameterized family of all the U subsets.

Any F(e_i) from it, for any e ∈ A, can be seen as a set of e-approximates of the set (F, A), where F (e) can be random, some of them may be empty, and the others may have a non-empty set.

Where parameters are usually things’ qualities, characteristics, or properties.

Definition 2.6.

Consider the set of all fuzzy subsets in a universal set $U$ is denoted by $I^{U}$ and define $E$ to be a collection of parameters, and $A \subseteq E$ . Then a pair $(S, A)$ is known as a fuzzy soft set over $U$ , where $F$ is a mapping given by:⁶

(4)

F : A \to I^{U}

Definition 2.7.

If a fuzzy soft set $(F_{1}, A)$ is both subset and super subset of a fuzzy soft set $(F_{2}, B)$ then $(F_{1}, A)$ and $(F_{2}, B)$ are equal with the identity universe $U$ .⁶

Definition 2.8.

${(F, A)}^{∁}$ is denotes the complement of a fuzzy soft set $(F, A)$ over universal set U and is defined as ${(F, A)}^{∁} = (F^{∁}, \neg A),$ where $F^{∁}$ is a mapping given by:⁶

(5)

F^{∁} : \neg A \to I^{U}

And $F^{∁} (e)$ is a fuzzy complement of $F (\neg e)$ for all $e \in A .$

Results

3. Fuzzy soft codes in decision-making

In this subsection, we introduce fuzzy soft codes and its properties.

Consider the set of $k$ code of objects $U_{C} = {o_{c_{1}}, o_{c_{2}}, \dots, o_{c_{k}}} .$ This set of objects may be described by a set of parameters ${A_{1}, A_{2}, \dots, A_{i}}$ . One way to express the parameter space $E$ is as $A_{i} \subseteq E$ . Let $A_{i}$ be the $ith$ class of parameters, and let each parameter set $A_{i}$ represent a particular property set. It is assumed here that these property sets can be thought of as fuzzy sets.

Now that we have the above information, we can create a fuzzy soft code $(FSC, A_{i})$ to describe a set of objects with the parameter set $A_{i}$ .

Definition 3.1.

Let $P (U_{C})$ is the set of the power sets of $U_{C}$ . Assume $E$ is a collection of parameters and $A$ is subset $E$ . Then the order pair $(FSC, A)$ is said to be a fuzzy soft code over $K_{2}^{n}$ if and only if each $S (e)$ is a fuzzy code, where $FSC$ is a mapping given by:

FSC : A \to P (U_{C})

and

U_{C}

: C → [0, 1] is a fuzzy code,

C \subseteq K_{2}^{n}

, and defined as:

(6)

U_{C} = \frac{\sum_{i = 1}^{n} p_{i}}{\frac{n (n + 1)}{2}}

for all

a \in C, n \in Z +, p^{'} s

, are defined to be the places of

1^{'} s

in the code word. Equation (6) can be generalised as follows:

(7)

U_{C} (a) = {\begin{matrix} 1, & if \sum_{i = 1}^{n} p_{i} = \frac{n (n + 1)}{2} \\ 0 < m < 1, & if \sum_{i = 1}^{n} p_{i} < \frac{n (n + 1)}{2} \\ 0, & \sum_{i = 1}^{n} p_{i} = 0 . \end{matrix}

Example 3.2.

Consider

$C {0000, 0001, 0100, 0110, 0011, 1110, 1011, 1111}$ and given that a fuzzy soft code $(FSC, A)$ over the universal set $U_{C} (a)$ for any $a \in C$ , where $U_{C}$ denotes the relative weight rate of the code $C$ defined by a mapping:

U_{C} : C \to [0, 1], \forall C \subseteq K^{4} .

Hence, using Equation (6) the set of fuzzy code is as

U_{C} (a) = {0, 0.2, 0.4, 0.5, 0.6, 0.7, 0.8, 1} .

Assume the set of parameters given by $A = {e_{1}, e_{2}, e_{3}}$

Let us consider

\begin{matrix} FSC (e_{1}) = {U_{C} (0001) / 0.4, U_{C} (1110) / 0.6, U_{C} (0100) / 0.2, U_{C} (1111) / 1} \\ FSC (e_{2}) = {U_{C} (0000) / 0, U_{C} (0110) / 0.5, U_{C} (1111) / 1, U_{C} (0011) / 0.7} \\ FSC (e_{3}) = {U_{C} (0110) / 0.5, U_{C} (1011) / 0.8} \end{matrix}

That is,

\begin{matrix} FSC (e_{1}) = {0.4, 0.6, 0.2, 1} \\ FSC (e_{2}) = {0, 0.5, 1, 0.7} \\ FSC (e_{3}) = {0.5, 0.8} \end{matrix}

Now, the fuzzy soft code $(FSC, A)$ is characterised as:

(FSC, A) = {(e_{1}, {0.4, 0.2, 1}), (e_{2}, {0, 0.5, 1}), (e_{3}, {0.5})}

Definition 3.3.

Assume $({FSC}_{1}, A),$ and $({FSC}_{2}, B)$ were two fuzzy soft codes that share the common universe of a fuzzy soft code $U_{C} (a) .$ Then we concluded $({FSC}_{1}, A)$ is said to be a fuzzy soft code subset of $({FSC}_{2}, B)$ if and only if the conditions listed below holds:

i. $A \subset B$
ii. $\forall e \in A, {FSC}_{1} (e) \subset {FSC}_{2} (e)$

Definition 3.4.

(Fuzzy soft codes equality): Over the common universe $U_{C} (a),$ if a fuzz soft code $({FSC}_{1}, A)$ is both fuzzy soft code subset and super subset of a fuzz soft code of $({FSC}_{2}, B)$ , then $({FSC}_{1}, A),$ is equal to $({FSC}_{2}, B),$ i.e., ${FSC}_{1} (e)$ is equal to ${FSC}_{2} (e)$ , $\forall e$ .

Definition 3.5.

(Fuzzy soft code complement): The fuzzy soft code complement is represented by ${(FSC, A)}^{∁}$ and is described by ${(FSC, A)}^{∁}$ = $({FSC}^{∁}$ , ¬A), such that ${FSC}^{∁}$ is a mapping given by:

(8)

{FSC}^{∁} : \neg A \to U_{C} (a)

and

{FSC}^{∁} (e) = {1 - U_{C} (a)}

is a fuzzy soft code complement of

FSC (\neg e)

for all

e \in A .

Example 3.6.

Consider $(FSC, A)$ is a fuzzy soft code given by:

\begin{matrix} FSC (e_{1}) = {U_{C} (0000), UC (1101)} and \\ FSC (e_{2}) = {U_{C} (0100), UC (1100), UC (1111)} \end{matrix}

over the field

K_{2}^{4}

.

The relative weight rates of the code words over the field $K_{2}^{4}$ can be written as

U_{C} (0000) = 0, U_{C} (1101) = 0.7, U_{C} (0100) = 0.2, U_{C} (1100) = 0.3 and U_{C} (1111) = 1

Then

FSC (e_{1}) = {0, 0.7} and FSC (e_{2}) = {0.2, 0.3, 1}

The fuzzy soft code is a pair:

(FSC, A) = {(e_{1}, {0, 0.7}), (e_{2}, {0.2, 0.3, 1})} .

Now, the fuzzy soft code complements of $(FSC, A)$ is given as follows:

\begin{matrix} FSC (\neg e_{1}) = {1 - 0, 1 - 0.7} \\ = {1, 0.3} \end{matrix}

\begin{matrix} FSC (\neg e_{2}) = {1 - 0.2, 1 - 0.3, 1 - 1} \\ = {0.8, 0.7, 0} \end{matrix}

Therefore,

{(FSC, A)}^{∁} = {(\neg e_{1}, {1, 0.3}), (\neg e_{2}, {0.8, 0.7, 0})}

Definition 3.7.

Over the universal set $U_{C} (a),$ a fuzzy soft code $(FSC, A)$ is said to be type one fuzzy soft code, if the associated relative weight rate of $FSC (e_{i})$ is the same degree of memberships.

Example 3.8.

Suppose $(FSC, A)$ to be a fuzzy soft code over the universe $U_{C} (a)$ for some $a \in K_{2}^{5}$ .

Consider

\begin{matrix} FSC (e_{1}) = {U_{C} (00000), U_{C} (11111), U_{C} (10110), U_{C} (01001)} \\ FSC (e_{2}) = {U_{C} (00000), U_{C} (11111), U_{C} (11001), U_{C} (00110)} \\ FSC (e_{3}) = {U_{C} (00000), U_{C} (11111), U_{C} (00111), U_{C} (11001)} \end{matrix}

Then the associated relative weight rate of $FSC (e_{i})$ is given by:

\begin{matrix} FSC (e_{1}) = {U_{C} (00000), U_{C} (11111), U_{C} (10110), U_{C} (01001)} = {0, 1, \approx 0.53, \approx 0.467} \\ FSC (e_{2}) = {U_{C} (00000), U_{C} (11111), U_{C} (11001), U_{C} (00110)} = {0, 1, \approx 0.53, \approx 0.467} \\ FSC (e_{3}) = {U_{C} (00000), U_{C} (11111), U_{C} (00101), U_{C} (01001)} = {0, 1, 0.8, \approx 0.533} \end{matrix}

Now the fuzzy soft code is as follows:

(FSC, A) = {(e_{1}, {0, 1, \approx 0.53, \approx 0.467}), (e_{2}, {0, 1, \approx 0.533, \approx 0.467}), (e_{3}, {0, 1, \approx 0.533, \approx 0.467})}

From this fuzzy soft code, the relative weight rate of a code is identical.

Hence, $(FSC, A)$ is type $1$ fuzzy soft codes.

3.1 Fuzzy soft code matrix representation

The matrix form of fuzzy soft codes as well as their basic operations, and characteristics are described in this subsection.

Definition 3.1.1.

Let the set of parameters with respect to the universal set $U_{C} (a)$ denoted by $E$ and $A$ is subset of $E$ .

Consider $({FSC}_{A}, E)$ be a fuzzy soft code over the set $U_{C} (a)$ . Then a subset $M_{A}$ of $U_{C} \times E$ , uniquely described as $M_{A} = {(u, e) : e \in A, u \in {FSC}_{A} (e)}$ which is known as the fuzzy soft code’s relation form of $({FSC}_{A}, E) .$ Now, if $U_{C} (a) = {u_{c_{1}} (a), u_{c_{2}} (a), u_{c_{3}} (a), \dots, u_{c_{m}} (a)}$ and $E = {e_{1}, e_{2}, e_{3}, \dots, e_{n}},$ then $({FSC}_{A}, E)$ can be represented by the fuzzy soft code matrix form:

(9)

{[t_{ij}]}_{m \times n} = {[\begin{matrix} t_{11} & t_{12} \dots & t_{1 n} \\ ⋮ & ⋱ & ⋮ \\ t_{m 1} & t_{m 2} \dots & t_{mn} \end{matrix}]}_{m \times n}

Where $t_{ij} = (u_{c_{i}} (a), e_{i})$

As a result, any fuzzy soft code can be represented as matrix form.

Example 3.1.2.

Let $C = {0000, 1100, 1001, 0111, 0010}$ is a code in the vector space with dimension $4$ , and this fuzzy code $U_{C}$ is given by $U_{C} (a) = {u_{0000} / 0, u_{1100} / 0.3, u_{1001} / 0.5, u_{0111} / 0.8, u_{0010} / 0.3},$ for any $a \in C$ , is the universal set, and let a set of parameters: $E = {e_{1}, e_{2}, e_{3}, e_{4}} over U_{C} (a) \subseteq [0, 1] .$

Let $A = {e_{1}, e_{3}, e_{4}},$ such that

\begin{matrix} {FSC}_{A} (e_{1}) = {u_{3} / 0.5, u_{4} / 0.8}, \\ {FSC}_{A} (e_{3}) = {u_{2} / 0.3, u_{3} / 0.5, u_{4} / 0.8, u_{5} / 0.3} \\ {FSC}_{A} (e_{4}) = {u_{1} / 0, u_{3} / 0.5, u_{5} / 0.3} \end{matrix}

Then the fuzzy soft code $({FSC}_{A}, E)$ is given by

({FSC}_{A}, E) = {(e_{1}, {u_{3} / 0.5, u_{4} / 0.8}), (e_{3}, {u_{2} / 0.3, u_{3} / 0.5, u_{4} / 0.8, u_{5} / 0.3}, (e_{4}, {u_{1} / 0, u_{3} / 0.5, u_{5} / 0.3})} .

The relation form $R_{A}$ of $({FSC}_{A}, E)$ is given by

M_{A} = {(u_{3} / 0.5, e_{1}), (u_{4} / 0.8, e_{1}), (u_{2} / 0.3, e_{3}), (u_{3} / 0.5, e_{3}), (u_{4} / 0.8, e_{3}), (u_{5} / 0.3, e_{3}), (u_{1} / 0, e_{4}), (u_{3} / 0.5, e_{4}), (u_{5} / 0.3, e_{4})}

That is,

M_{A} = {(0.5, e_{1}), (0.8, e_{1}), (0.3, e_{3}), (0.5, e_{3}), (0.8, e_{3}), (0.3, e_{3}), (0, e_{4}), (0.5, e_{4}), (0.3, e_{4})},

Hence the matrix form of the fuzzy soft code $({FSC}_{A}, E)$ is as follows:

{[t_{ij}]}_{5 \times 4} = {[\begin{matrix} (u_{1} / 0, e_{1}) (u_{1} / 0, e_{2}) (u_{1} / 0, e_{3}) (u_{1} / 0, e_{4}) \\ (u_{2} / 0, e_{1}) (u_{2} / 0, e_{2}) (u_{2} / 0.3, e_{3}) (u_{2} / 0, e_{4}) \\ (u_{3} / 0.5, e_{1}) (u_{3} / 0, e_{2}) (u_{3} / 0.5, e_{3}) (u_{3} / 0.5, e_{4}) \\ (u_{4} / 0.8, e_{1}) (u_{4} / 0, e_{2}) (u_{4} / 0.8, e_{3}) (u_{4} / 0, e_{4}) \\ (u_{5} / 0, e_{1}) (u_{4} / 0, e_{2}) (u_{4} / 0.3, e_{3}) (u_{4} / 0.3, e_{4}) \end{matrix}]}_{5 \times 4}

This matrix can be simplified as the following ways

{[t_{ij}]}_{5 \times 4} = {[\begin{matrix} 0 0 0 0 \\ 0 0 0.3 0 \\ 0.5 0 0.5 0.5 \\ 0.8 0 08 0 \\ 0 0 0.3 0.3 \end{matrix}]}_{5 \times 4}

Definition 3.1.3.

Let $M_{1} = {[l_{ij}]}_{m \times n}, M_{2} = {[m_{ij}]}_{m \times n}$ be two fuzzy soft code matrix form. Then, a fuzz soft code matrix $N = {[t_{ij}]}_{m \times n}$ is said to be the

i. Union of $M_{1}$ and $M_{2}$ , defined $N = M_{1} ⊔ M_{2}$ , if $[t_{ij}] = max {[l_{ij}], [m_{ij}]}$ for all $i$ and $j;$
ii. Intersection of $M_{1}$ and $M_{2}$ , defined $N = M_{1} ⊓ M_{2}$ , if $[t_{ij}] = max {[l_{ij}], [m_{ij}]}$ for all $i$ and $j;$
iii. Complement of $M_{1}$ , described as $N = M^{∁}$ , if $[t_{ij}] = [1 - l_{ij}]$ for all $i$ and $j;$
iv. Difference of $M_{2}$ from $M_{1}$ , also called the relative complement of $M_{2}$ in $M_{1}$ , denoted $N = M_{1} - M_{2}$ or $N = M_{1} \ M_{2} if N = M_{1} ⊓ M_{2}^{RC}$ .

In view of the (ii) above, $M_{1}$ and $M_{2}$ are said to be disjoint if $M_{1} ⊓ M_{2}^{RC} = [0]$

Proposition 3.1.4.

Let $M_{1} = [l_{ij}], M_{2} = [m_{ij}]$ and $N = [tij]$ be an $m \times n$ rectangular fuzzy soft matrices. Then the following properties are true:

i. Idempotent: $M_{1} ⊔ M_{1} = M_{1}$ and $M_{1} ⊓ M_{1} = M 1$
ii. Identity: $M_{1} ⊔ [0] = M_{1}$ and $M_{1} ⊔ [I] = M_{1}$
iii. Domination: $M_{1} ⊓ [I] = [I]$ and $M_{1} ⊓ [0] = [0]$
iv. ${[0]}^{∁}$ = [I] and $I^{∁}$ = [0]
v. De’ Morgan’s law: ${(M_{1} ⊔ M_{2})}^{∁} = M^{∁} ⊓ M^{∁}$ and ${(M_{1} ⊓ M_{2})}^{∁} = M^{∁} ⊔ M^{∁}$
vi. ${(M^{C})}^{C} = M$
vii. Commutativity: $M_{1} ⊔ M_{2} = M_{2} ⊔ M_{1}$ and $M_{1} ⊓ M_{2} = M_{2} ⊓ M_{1}$
viii. Associativity: $M_{1} ⊔ (M_{2} ⊔ N) = (M_{1} ⊔ M_{2}) ⊔ N$ and $M_{1} ⊓ (M_{2} ⊓ N) = (M_{1} ⊓ M_{2}) ⊓ N$
ix. Distributivity: $M_{1} ⊔ (M_{2} ⊓ N) = (M_{1} ⊔ M_{2}) ⊓ (M_{1} ⊔ N)$ and $M_{1} ⊓ (M_{2} ⊔ N) = (M_{1} ⊓ M_{2}) ⊔ (M_{1} ⊓ N)$

Note that, $[I]$ and $[0]$ denotes the identity and zero fuzzy soft code matrix respectively.

Proof:

Let us prove property $(v), (vi)$ and $(ix)$

$(v),$ Let $M_{1} = [l_{ij}]$ and $M_{2} = [m_{ij}]$ for each $i$ and $j,$ then

\begin{matrix} {(M_{1} ⊔ M_{2})}^{C} = ([l_{ij}] ⊔ m_{ij}])^{∁} \\ = {[max {l_{ij}, m_{ij}}]}^{∁} \\ \begin{matrix} = [1 - max {l_{ij}, m_{ij}}] \\ = [min {1 - l_{ij}, 1 - m_{ij}}] \\ \begin{matrix} = {[l_{ij}]}^{C} ⊓ {[m_{ij}]}^{C} \\ = {M_{1}}^{∁} ⊓ {M_{2}}^{∁} \end{matrix} \end{matrix} \end{matrix}

To prove the other side, the researchers followed the same procedure. For any $i$ and $j,$

\begin{matrix} {(M^{C})}^{C} = {({[l_{ij}]}^{∁})}^{∁} \\ = {[1 - l_{ij}]}^{∁} = 1 - (1 - l_{ij}) = 1 - 1 + l_{ij} \\ = l_{ij} = M_{1} \end{matrix}

$(ix)$ Let $M_{1} = [l_{ij}]$ , $M_{2} = [m_{ij}]$ and $N = [t_{ij}]$ for any $i$ and $j,$ then

\begin{matrix} M_{1} ⊔ (M_{2} ⊓ N) = [l_{ij}] ⊔ ([m_{ij}] ⊓ [t_{ij}]) \\ = [l_{ij}] ⊔ [min {m_{ij}, t_{ij}}] \\ \begin{matrix} = [max {l_{ij}, min {m_{ij}, t_{ij}}] \\ = [min {max {l_{ij}, m_{ij}, max {l_{ij}, t_{ij}}] \\ \begin{matrix} = [min {[l_{ij}] ⊔ [m_{ij}], [l_{ij}] ⊔ [t_{ij}]}] \\ = ([l_{ij}] ⊔ [m_{ij}]) ⊓ ([l_{ij}] ⊔ [t_{ij}]) \\ = (M_{1} ⊔ M_{2}) ⊓ (M_{1} ⊔ N) \end{matrix} \end{matrix} \end{matrix}

Also, the other sides of the proof is the same pattern.

The observer decided that this form of representation was used. If $U_{C} (a) \in FSC (e_{j}),$ then

3.2 Choice and score values of an object

The value of an object that has been chosen $o_{i} / U_{C} (a) \in χ / U_{C} (a)$ is u_i which is given by

(10)

u_{i} = \sum_{j} o_{ij} / U_{C} (a)

Where u_i indicates the row sum of each degree of memberships of an $FSC$ table.

The comparison table is a rectangular table with entries $o_{ij}, i, j = 1, 2, \dots, n,$ and $p_{ij}$ is the parameter count for which the degree of membership values of $o_{i} / U_{C} \geq o_{j} / U_{C} .$

For all $i, j, 0 \leq p_{ij} \leq l$ and $p_{ij} = l .$

As a result, $p_{ij}$ denotes an integer measure, and $o_{i} / U_{C}$ characterizes $o_{j} / U_{C}$ in $p_{ij}$ number of parameters different from $l$ parameters.

$R_{i}$ denoted the add up of the rows of the object $o_{i} / U_{C}$ and is calculated as follows:

(11)

R_{i} = \sum_{i = 1}^{n} p_{ij}

Likewise, the add up of the columns of the object $o_{j} / U_{C}$ is denoted by $C_{j}$ and computed as:

(12)

C_{j} = \sum_{i = 1}^{n} p_{ij}

We denoted the total number of parameters by the integer number $u_{j}$ in which all the members of $U_{C}$ dominates

$o_{j} / U_{C}$ .

The object score values of $o_{i}$ are determined by S_i and described as

(13)

V_{i} = R_{i} - C_{j}

Note: The choice value and score value is to choose or identify the object from a set of offered materials based on a set of screening parameters $e_{i}$ .

3.3 Algorithm for selection of an object

Example 3.3.1

Let $O / S C = {o_{1} / U_{C} (a), o_{2} / U_{C} (a), o_{3} / U_{C} (a), o_{4} / U_{C} (a), o_{5} / U_{C} (a)}$ is a collection of fuzzy code objects with varying characteristics based on color and quality of the objects.

Consider the parameter set

\begin{matrix} E = {pink color, magneta color, violet color, purple color, \\ high quality, low quality, medium quality, very low quality} . \end{matrix}

Let $A$ and $B$ are parameter subsets of $E$ , such that

$A$ stands for the colors of the object and

$B$ stands for the qualities of the object. That is,

$A = {pink color, magneta color, violet color, purple color}$ and $B = {high quality, low quality, medium quality, very low quality}$

Assuming that the fuzzy soft code $({FSC}_{1}, A)$ describes ‘colorful objects,’ the fuzzy soft code $({FSC}_{2}, B)$ describes ‘sizeful’ objects. The challenge is to find an unknown object by the multi-observer interms of fuzzy soft codes $({FSC}_{1}, A)$ and $({FSC}_{2}, B)$ . This fuzzy soft code can be calculated as follows.

The fuzzy soft code $({FSC}_{1}, A)$ is as shown below:

\begin{matrix} ({FSC}_{1}, A) = {Objects having pink color = {o_{1} / U_{C} (0001), o_{2} / U_{C} (1001), o_{3} / U_{C} (0111), \\ o_{4} / U_{C} (1011), o_{5} / U_{C} (1111)}, \\ \begin{matrix} Objects having magenta color = {o_{1} / U_{C} (0101), o_{2} / U_{C} (1011), o_{3} / U_{C} (0110), \\ o_{4} / U_{C} (1011), o_{5} / U_{C} (1110)}, \\ \begin{matrix} Objects having violet color = {o_{1} / U_{C} (1001), o_{2} / U_{C} (1011), o_{3} / U_{C} (1101), \\ o_{4} / U_{C} (1011), o_{5} / U_{C} (1101)}, \\ \begin{matrix} Objects having purple color = {o_{1} / U_{C} (0001), o_{2} / U_{C} (1101), o_{3} / U_{C} (0111), \\ o_{4} / U_{C} (1011), o_{5} / U_{C} (1001)}} \end{matrix} \end{matrix} \end{matrix} \end{matrix}

The fuzzy soft code $({FSC}_{2}, B)$ is defined as follows:

\begin{matrix} ({FSC}_{2}, B) = {Objects having high quality = {o_{1} / UC (0101), o_{2} / UC (1100), o_{3} / UC (0111), \\ o_{4} / UC (1011), o_{5} / UC (1011)}, \\ \begin{matrix} Objects having medium quality = {o_{1} / UC (0001), o_{2} / UC (1001), o_{3} / UC (0111), \\ o_{4} / UC (1011), o_{5} / UC (1111)}, \\ \begin{matrix} Objects having low quality = {o_{1} / UC (0101), o_{2} / UC (1011), o_{3} / UC (0110), \\ o_{4} / UC (0011), o_{5} / UC (1110)}, \\ \begin{matrix} Objects having very low quality = {o_{1} / UC (0001), o_{2} / UC (1101), o_{3} / UC (0111), \\ o_{4} / UC (1011), o_{5} / UC (1001)}} \end{matrix} \end{matrix} \end{matrix} \end{matrix}

\begin{matrix} \Rightarrow ({FSC}_{1}, A) = {Objects having pink color = {o_{1} / 0.4, o_{2} / 0.5, o_{3} / 0.9, o_{4} / 0.8, o_{5} / 1}, \\ Objects having magenta color = {o_{1} / 0.6, o_{2} / 0.8, o 3 / 0.5, o_{4} / 0.8, o_{5} / 0.6}, \\ \begin{matrix} Objects having violet color = {o_{1} / 0.5, o_{2} / 0.8, o_{3} / 0.7, o_{4} / 0.8, o_{5} / 0.7}, \\ Objects having purple color = {o_{1} / 0.4, o_{2} / 0.7, o_{3} / 0.9, o_{4} / 0.8, o_{5} / 0.5}} \end{matrix} \end{matrix}

And

\begin{matrix} ({FSC}_{2}, B) = {Objects having high quality = {o_{1} / 0.6, o_{2} / 0.3, o_{3} / 0.9, o_{4} / 0.8, o_{5} / 0.8}, \\ Objects having medium quality = {o 1 / 0.4, o_{2} / 0.5, o_{3} / 0.9, o_{4} / 0.8, o_{5} / 1}, \\ \begin{matrix} Objects having low quality = {o_{1} / 0.6, o_{2} / 0.8, o_{3} / 0.5, o_{4} / 0.7, o_{5} / 0.6}, \\ Objects very low quality = {o_{1} / 0.4, o_{2} / 0.7, o_{3} / 0.9, o_{4} / 0.8, o_{5} / 0.5}} \end{matrix} \end{matrix}

Algorithm 1

The following algorithms were followed by the observer to selected the objects.

i. Compute fuzzy soft code (FSC) $(FSC, A);$
ii. Input Mr. $X^{’} s$ choice parameter set $A$ , which is a subset of $E$ ;
iii. Select one reduct FSC to say $(FSC, B)$ of $(FSC, A);$
iv. Find $k$ , such that $u_{k} = max u_{i}$
v. Then $u_{k}$ is chosen by the observer

Algorithm 2

i. Inter the fuzzy soft codes $({FSC}_{1}, A)$ and $({FSC}_{2}, B);$
ii. Fill in the observer’s observed parameter set $P$ ;
iii. Build the fuzzy soft code comparison table (V, E) from the fuzzy soft codes $({FSC}_{1}, A)$ and $({FSC}_{2}, B)$ ;
iv. Generate a comparison table for the fuzzy soft code $(V, E)$ and compute $R_{i}$ and $C_{j}$ for $o_{i} / U_{C}$ , $\forall i;$
v. Calculate the $o_{i} / U_{C}$ score values, $\forall i;$
vi. Determine $k$ in which $S_{k} = max V_{i}$ . The observer then selected $V_{k}$ .

Tables 1 and 2 shows the tabular representations of the fuzzy soft codes $({FSC}_{1}, A)$ and $({FSC}_{2}, B)$ , respectively: Let $({FSC}_{1}, A)$ and $({FSC}_{2}, B)$ , be any two fuzzy soft codes from the same universe $O / U_{C} (a) .$ Another fuzzy soft code is obtained after operating two fuzzy soft codes, for some specific parameters $A$ and $B$ .

Table 1. Comparison of fuzzy soft code of $({FSC}_{1}, A)$ .

$O / U_{C}$	Pink color = A₁	Magenta color = A₂	Violet color = A₃	Purple color = A₄	$u_{i} = \sum_{j} h_{ij} / R_{wr}$
$o_{1}$ / $U_{C}$ (a)	0.4	0.6	0.5	0.4	1.9
$o_{2}$ / $U_{C}$ (a)	0.5	0.8	0.8	0.7	2.8
$o_{3}$ / $U_{C}$ (a)	0.9	0.5	0.7	0.9	3.0
$o_{4}$ / $U_{C}$ (a)	0.8	0.8	0.8	0.8	3.2
$o_{5}$ / $U_{C}$ (a)	1	0.6	0.7	0.5	2.8

Table 2. Comparison of fuzzy soft codes of $({FSC}_{2}, B)$ .

$O / U_{C}$	High quality = B₁	Medium quality = B₂	Low quality = B₃	Very low quality = B₃	$u_{i} = \sum_{j} h_{ij} / R_{wr}$
$o_{1}$ / $U_{C}$ (a)	0.4	0.6	0.5	0.4	1.9
$o_{2}$ / $U_{C}$ (a)	0.5	0.8	0.8	0.7	2.8
$o_{3}$ / $U_{C}$ (a)	0.9	0.5	0.7	0.9	3.0
$o_{4}$ / $U_{C}$ (a)	0.8	0.8	0.8	0.8	3.2
$o_{5}$ / $U_{C}$ (a)	1	0.6	0.7	0.5	2.8

The resultant fuzzy soft code of $({FSC}_{1}, A)$ is the name given to the new fuzzy soft code $({FSC}_{2}, B) .$

Let’s assume the operation of the above two fuzzy soft codes by

$({FSC}_{1}, A) ⊔ ({FSC}_{2}, B),$ then we’ll have $4 \times 4 = 16$ number of parameters of the form $e_{ij}$ , were $e_{ij} = A_{i} ⊔ B_{j},$ for all $i, j = 1, 2, 3, 4 .$

If the parameters are activated in a fuzzy soft code, then $N =$ ${N_{1} = A_{1} ⊔ B_{3}, N_{2} = A_{2} ⊔ B_{4}, N_{3} = A_{3} ⊔ B_{3}, N_{4} = A_{4} ⊔ B_{1}, N_{5} = A_{2} ⊔ B_{3}, N_{6} = A_{2} ⊔ B_{4}},$ from the fuzzy soft code $({FSC}_{1}, A)$ and $({FSC}_{2}, B)$ let say $({FSC}_{3}, N)$ . So, after performing the table $({FSC}_{1}, A) ⊔ ({FSC}_{2}, B),$ the resultant fuzzy soft code $({FSC}_{3}, N)$ will look like as follows, for some parameters:

The resultant fuzzy soft code comparison table is shown below:

The maximum score from the above table is $V_{i} = 12$ , scored by the fuzzy code $o 3 / U_{C} (a)$ and the decision has been taken to use o3/SC(a).

Tables 3 and 4 shows the resultant fuzzy soft code of $({FSC}_{1}, A) ⊔ ({FSC}_{2}, B)$ and the resultant fuzzy soft code comparisons.

Table 3. The resultant fuzzy soft code of $({FSC}_{1}, A) ⊔ ({FSC}_{2}, B)$ .

$O / U_{C}$	$N_{1}$	$N_{2}$	$N_{3}$	$N_{4}$	$N_{5}$	$N_{6}$
$o_{1}$ / $U_{C}$ (a)	0.6	0.6	0.8	0.6	0.6	1.9
$o_{2}$ / $U_{C}$ (a)	0.8	0.8	0.8	0.7	0.8	2.8
$o_{3}$ / $U_{C}$ (a)	0.9	0.9	0.8	0.9	0.5	3.0
$o_{4}$ / $U_{C}$ (a)	0.8	0.8	0.7	0.8	0.8	3.2
$o_{5}$ / $U_{C}$ (a)	1	0.6	0.6	0.8	0.6	2.8

Table 4. The resultant fuzzy soft code comparison table.

$O / U_{C}$	$o_{1} / U_{C} (a)$	$o_{2} / U_{C} (a)$	$o_{3} / U_{C} (a)$	$o_{4} / U_{C} (a)$	$o_{5} / U_{C} (a)$	Row sum ( $R_{i}$ )	Column sum ( $C_{i}$ )	Scor value ( $V_{i} = R_{i} - C_{i}$ )
$o_{1}$ / $U_{C}$ (a)	6	2	2	2	4	16	25	−9
$o_{2}$ / $U_{C}$ (a)	5	6	2	4	4	21	21	0
$o_{3}$ / $U_{C}$ (a)	5	5	6	5	4	25	13	12
$o_{4}$ / $U_{C}$ (a)	5	5	1	6	5	22	17	5
$o_{5}$ / $U_{C}$ (a)	4	3	2	2	6	17	23	−6

4. Conclusion

In this study we analysed the theoretical approaches of fuzzy soft sets to fuzzy codes on decision-making problems. This new concept is introduce by combining fuzzy code and fuzzy soft set theory. This new concept is a powerful and eective extension of fuzzy soft sets and fuzzy codes which deals with parameterized values of the alternative. This paper is an extension model of fuzzy soft sets and a new mathematical tool that has great advantages in dealing with uncertain information and is proposed by combining fuzzy soft sets and fuzzy codes. In the future work, it can be address fuzzy soft cyclic code, Interval-valued fuzzy soft code, Generalised fuzzy soft code (GFSC), Generalised interval-valued fuzzy soft code (GIVFSC), Intuitionistic fuzzy soft code, and so on.

Ethics and consent

No ethical approval and consent was required for this study.

Data availability

No data associated in this research.

References

1. Molodtsov D: Soft Set Theory First Results.1999; 37: 19–31.
2. Maji PK, Biswas R, Roy AR: Soft set theory. Comput. Math. Appl. 2003; 45(4–5): 555–562. Publisher Full Text
3. Mostafa SM, Kareem FF, Jad HA: Brief Review of Soft Sets and Its Application in Coding Theory.2020; 33, 95–106. Reference Source
4. Acharjee S, Oza A: On correct structures and similarity measures of soft sets.n.d.; 1–17.
5. Roy AR, Maji PK: A fuzzy soft set theoretic approach to decision making problems.2007; 203: 412–418. Publisher Full Text
6. Majumdar P, Samanta SK: Generalised fuzzy soft sets. Comput. Math. Appl. 2010; 59(4): 1425–1432. Publisher Full Text
7. Ali M, Khan H, Son LH, et al.: New soft set based class of linear algebraic codes. Symmetry. 2018; 10(10). Publisher Full Text
8. Feng F, Jun YB, Liu X, et al.: An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math. 2010; 234(1): 10–20. Publisher Full Text
9. Ozkan AO, Mehmet Ozkan E: Different aproach to codind theory.pdf. Pak. J. Appl. Sci. 2002; 2(11): 1032–1033.
10. Amudhambigai B, Neeraja A: A new view on fuzzy codes and its application. Jordan J. Math. Stat. 2019; 12(4): 455–471.
11. Caǧman N, Enginoǧlu S, Citak F: Fuzzy soft set theory and its applications. Iran. J. Fuzzy Syst. 2011; 8(3): 137–147.
12. Suebsan P, Kanchanasuk S: Mathematics, I. J. Interdiscip. Math. 2022; 25(8): 2653–2670. Publisher Full Text
13. Kong Z, Wang L, Wu Z: Application of fuzzy soft set in decision making problems based on grey theory. J. Comput. Appl. Math. 2011; 236(6): 1521–1530. Publisher Full Text
14. Gogoi K, Kr. Dutta A , Chutia C: Application of Fuzzy Soft Set Theory in Day to Day Problems. Int. J. Comput. Appl. 2014; 85(7): 27–31. Publisher Full Text
15. Herawan T, Ghazali R, Deris MM: Soft set theoretic approach for dimensionality reduction. Int. J. Database Theory Appl. 2010; 3(2): 47–60.
16. Ali W, Shaheen T, Haq IU, et al.: A Novel Interval-Valued Decision Theoretic Rough Set Model with Intuitionistic Fuzzy Numbers Based on Power Aggregation Operators and Their Application in Medical Diagnosis. Mathematics. 2023; 11(19). Publisher Full Text
17. Mohamed AA, Abualigah L, Alburaikan A, et al.: AOEHO: A New Hybrid Data Replication Method in Fog Computing for IoT Application. Sensors. 2023; 23(4). Publisher Full Text

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Author details Author details

¹ Demartment of Mathematics, Bahir Dar University Department of Mathematics, Bahir Dar, Amhara, Ethiopia
² Mathematics, DebreTabor University, Debre Tabor, Amhara, Ethiopia
³ Mathematics, Debark University Department of Mathematics, Bahir Dar, Amhara, Ethiopia

Masresha Wassie Woldie
Roles: Conceptualization, Formal Analysis, Investigation, Methodology, Writing – Original Draft Preparation

Jejaw Demamu Mebrat
Roles: Conceptualization, Methodology, Writing – Review & Editing

Mihret Alamneh Taye
Roles: Conceptualization, Supervision, Writing – Review & Editing

Competing interests

No competing interests were disclosed.

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The author(s) declared that no grants were involved in supporting this work.

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https://doi.org/10.12688/f1000research.161824.1

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Woldie MW, Mebrat JD and Taye MA. Properties of Fuzzy Soft Codes and Their Role in Decision-Making [version 1; peer review: awaiting peer review]. F1000Research 2025, 14:1403 (https://doi.org/10.12688/f1000research.161824.1)

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[1] 1. Molodtsov D: Soft Set Theory First Results.1999; 37: 19–31.

[2] 2. Maji PK, Biswas R, Roy AR: Soft set theory. Comput. Math. Appl. 2003; 45(4–5): 555–562. Publisher Full Text

[3] 3. Mostafa SM, Kareem FF, Jad HA: Brief Review of Soft Sets and Its Application in Coding Theory.2020; 33, 95–106. Reference Source

[4] 4. Acharjee S, Oza A: On correct structures and similarity measures of soft sets.n.d.; 1–17.

[5] 5. Roy AR, Maji PK: A fuzzy soft set theoretic approach to decision making problems.2007; 203: 412–418. Publisher Full Text

[6] 6. Majumdar P, Samanta SK: Generalised fuzzy soft sets. Comput. Math. Appl. 2010; 59(4): 1425–1432. Publisher Full Text

[7] 7. Ali M, Khan H, Son LH, et al.: New soft set based class of linear algebraic codes. Symmetry. 2018; 10(10). Publisher Full Text

[8] 8. Feng F, Jun YB, Liu X, et al.: An adjustable approach to fuzzy soft set based decision making. J. Comput. Appl. Math. 2010; 234(1): 10–20. Publisher Full Text

[9] 9. Ozkan AO, Mehmet Ozkan E: Different aproach to codind theory.pdf. Pak. J. Appl. Sci. 2002; 2(11): 1032–1033.

[10] 10. Amudhambigai B, Neeraja A: A new view on fuzzy codes and its application. Jordan J. Math. Stat. 2019; 12(4): 455–471.

[11] 11. Caǧman N, Enginoǧlu S, Citak F: Fuzzy soft set theory and its applications. Iran. J. Fuzzy Syst. 2011; 8(3): 137–147.

[12] 12. Suebsan P, Kanchanasuk S: Mathematics, I. J. Interdiscip. Math. 2022; 25(8): 2653–2670. Publisher Full Text

[13] 13. Kong Z, Wang L, Wu Z: Application of fuzzy soft set in decision making problems based on grey theory. J. Comput. Appl. Math. 2011; 236(6): 1521–1530. Publisher Full Text

[14] 14. Gogoi K, Kr. Dutta A , Chutia C: Application of Fuzzy Soft Set Theory in Day to Day Problems. Int. J. Comput. Appl. 2014; 85(7): 27–31. Publisher Full Text

[15] 15. Herawan T, Ghazali R, Deris MM: Soft set theoretic approach for dimensionality reduction. Int. J. Database Theory Appl. 2010; 3(2): 47–60.

[16] 16. Ali W, Shaheen T, Haq IU, et al.: A Novel Interval-Valued Decision Theoretic Rough Set Model with Intuitionistic Fuzzy Numbers Based on Power Aggregation Operators and Their Application in Medical Diagnosis. Mathematics. 2023; 11(19). Publisher Full Text

[17] 17. Mohamed AA, Abualigah L, Alburaikan A, et al.: AOEHO: A New Hybrid Data Replication Method in Fog Computing for IoT Application. Sensors. 2023; 23(4). Publisher Full Text

Properties of Fuzzy Soft Codes and Their Role in Decision-Making

Abstract

Background

Methods

Results

Conclusions

Keywords

1. Introduction

2. Methods

Definition 2.1.

Definition 2.2.

Definition 2.3.

(1)

(2)

Definition 2.4.

Definition 2.5.

(3)

Definition 2.6.

(4)

Definition 2.7.

Definition 2.8.

(5)

3. Fuzzy soft codes in decision-making

Definition 3.1.

(6)

(7)

Example 3.2.

Definition 3.3.

Definition 3.4.

Definition 3.5.

(8)

Example 3.6.

Definition 3.7.

Example 3.8.

3.1 Fuzzy soft code matrix representation

Definition 3.1.1.

(9)

Example 3.1.2.

Definition 3.1.3.

Proposition 3.1.4.

Proof:

3.2 Choice and score values of an object

(10)

(11)

(12)

(13)

3.3 Algorithm for selection of an object

Example 3.3.1

Algorithm 1

Algorithm 2

Table 1. Comparison of fuzzy soft code of (FSC1,A) .

Table 2. Comparison of fuzzy soft codes of (FSC2,B) .

Table 3. The resultant fuzzy soft code of (FSC1,A)⊔(FSC2,B) .

Table 4. The resultant fuzzy soft code comparison table.

4. Conclusion

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Table 1. Comparison of fuzzy soft code of $({FSC}_{1}, A)$ .

Table 2. Comparison of fuzzy soft codes of $({FSC}_{2}, B)$ .

Table 3. The resultant fuzzy soft code of $({FSC}_{1}, A) ⊔ ({FSC}_{2}, B)$ .