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Generalized Neutrosophic Soft Set

Said Broumi

arXiv:1305.2724v1cs.AI

TL;DR

The paper addresses problems involving vague, indeterminate, or inconsistent information by combining generalized neutrosophic sets with soft sets in a parameterized model. It introduces operations and demonstrates the model in a blouse-selection decision problem, where blouse b4 receives the maximum score of 19.

  • Problem

    The paper targets problems whose data are vague, imprecise, indeterminate, or inconsistent, for which existing fuzzy-set approaches may not give satisfactory results.

    Source: §1 · p. 1

  • Method

    The paper maps parameters to generalized neutrosophic sets, combining generalized neutrosophic and soft-set concepts in a generalized neutrosophic soft set.

    Source: Definition 3.1 · p. 3, §3 · p. 3

  • Results

    Blouse b4 receives the maximum score of 19 and is selected as the best decision, followed by b5.

    Source: §4.1 · p. 13

  • Takeaways & Limitations

    The paper defines operations on generalized neutrosophic soft sets and demonstrates their use in a decision-making problem.

    Source: §5 · p. 13

    For technical applications, the paper uses values in [0,1] because the broader neutrosophic range is difficult to apply in scientific and engineering problems.

    Source: Definition 2.1 (see [3]). Neutrosophic… · p. 2

Abstract

from arXiv · show

In this paper we present a new concept called generalized neutrosophic soft set. This concept incorporates the beneficial properties of both generalized neutrosophic set introduced by A.A. Salama [7]and soft set techniques proposed by Molodtsov [4]. We also study some properties of this concept. Some definitions and operations have been introduced on generalized neutrosophic soft set. Finally we present an application of generalized neuutrosophic soft set in decision making problem.

1. INTRODUCTION

The paper motivates a model for problems involving vague, indeterminate, or inconsistent information by combining generalized neutrosophic sets with soft sets. It introduces the model, its operations, and a decision-making application.

1. INTRODUCTION

  • Neutrosophic sets address imprecise, indeterminate, and inconsistent data, while soft sets offer a way to handle uncertainty without setting a membership function.The paper presents these as complementary motivations for its proposed model.Source: §1 · p. 1
  • The paper combines generalized neutrosophic sets with soft sets to introduce the generalized neutrosophic soft set.It attributes the former to Salama and the latter to Molodtsov.Source: §1 · p. 1
  • The paper introduces operations on the model and applies it to a decision-making problem.The paper outline places preliminaries in section 2, the model in section 3, the application in section 4, and conclusions in section 5.Source: §1 · pp. 1–2

2 .Preliminaries

The preliminaries review neutrosophic, generalized neutrosophic, and soft sets as foundations for the paper’s model. They describe truth, indeterminacy, and falsity degrees, alongside parameter-based representations of objects.

Definition 2.1 (see [3]). Neutrosophic Set

  • A neutrosophic set assigns each universe element truth, indeterminacy, and non-membership degrees; for technical applications, the paper uses [0,1] rather than ]−0,1+.The broader range is described as difficult to apply in scientific and engineering problems.Source: Definition 2.1 (see [3]). Neutrosophic… · p. 2

Definition 2.2. (see [10])

  • Neutrosophic-set containment requires truth membership in A not to exceed B’s, while indeterminacy and falsity in A must be at least B’s.Source: Definition 2.2. (see [10]) · p. 2

Definition 2.3(see [7]). Generalized Neutrosophic Set

  • A generalized neutrosophic set represents each element through membership, indeterminacy, and non-membership functions; an example interprets these as goodness, indeterminacy, and poorness.The example assigns these degrees to capability, trustworthiness, and price characteristics.Source: Definition 2.3(see [7]). Generalized… · pp. 2–3

2 .Preliminaries

  • A soft set over U maps selected parameters to subsets of U, allowing a buyer’s house attributes to identify groups such as expensive or beautiful houses.The example uses a parameter set and a mapping to house subsets.Source: Definition 2.3(see [7]). Generalized… · p. 3

3. Generalized Neutrosophic Soft Set

The section defines the generalized neutrosophic soft set as a parameterized family of generalized neutrosophic sets. It illustrates the representation with blouse attributes and defines subset relations between such sets.

Definition 3.1

  • A generalized neutrosophic soft set maps each selected parameter to a generalized neutrosophic set over the universe, making it a parameterized family rather than a single generalized neutrosophic set.Source: Definition 3.1 · p. 3, Example 3.3 · p. 5

Example 3.3

  • Blouse qualities such as bright, cheap, costly, and colorful serve as parameters, with each parameter associated with generalized neutrosophic assessments of the blouses.The example represents each assessment with truth-membership, indeterminacy-membership, and falsity-membership values.Source: Example 3.3 · pp. 3–4
  • A generalized neutrosophic soft set is a subset of another when its parameter set is included and each corresponding generalized neutrosophic set satisfies the stated membership inequalities.Truth membership must be no greater, while indeterminacy and falsity must be no smaller.Source: Example 3.3 · p. 5
  • An example comparing sets describing object sizes and surface textures concludes that one is a generalized neutrosophic soft subset of the other.Source: Example 3.3 · pp. 5–6

Definition 3.7

The section defines equality between generalized neutrosophic soft sets through mutual subset relations and introduces the NOT set of a parameter set. These definitions extend the paper’s set operations and parameter notation.

Definition 3.7

  • Two generalized neutrosophic soft sets are equal when each is a subset of the other.Source: Definition 3.7 · p. 6

Definition 3.8

  • The NOT set of a parameter set contains the negated form of each parameter, with the paper noting that the two operators differ.Source: Definition 3.8 · p. 6

Example 3.9

Example 3.9 specifies a parameter set consisting of four negated blouse attributes.

  • The parameters are “not bright,” “not cheap,” “not costly,” and “not colorful.”Source: Example 3.9 · p. 6

Definition 3.10

Definition 3.10 defines the complement of a generalized neutrosophic soft set through a mapping that assigns a generalized neutrosophic soft complement to each parameter.

  • The complement (F,A)c is (Fc,A), with Fc mapping each parameter to its generalized neutrosophic soft complement.Source: Definition 3.10 · p. 6

Example 3.11

Example 3.11 illustrates a complement as the blouses’ “not attractiveness” and states the zero-membership condition for a null generalized neutrosophic soft set.

  • In the illustration of Example 3.2, the complement (F,A)c describes the ‘not attractiveness of the blouses’; the listed entries include b4 with values 0.5, 0.8, and 0.4.Source: Example 3.11 · p. 6
  • A null generalized neutrosophic soft set has truth, indeterminacy, and falsity memberships all equal to zero for every object and parameter.Source: Example 3.11 · p. 6

Example 3.13

The worked example uses five blouses and Bright, Cheap, and Colorful parameters, then presents operations and algebraic properties for generalized neutrosophic soft sets. It includes union, intersection, AND, and OR constructions.

  • The null-set example assigns zero truth, indeterminacy, and falsity values to the cheap and colorful parameters for all five blouses.Source: Example 3.13 · p. 7
  • Union forms (K,C) with C=A∪B; for shared parameters, truth uses the maximum while indeterminacy and falsity use the minimum.Source: Example 3.13 · p. 7
  • Intersection requires A∩B≠0, sets C=A∩B, and combines truth and indeterminacy by minimum and falsity by maximum.Source: Example 3.13 · p. 8
  • The stated laws include intersection idempotence and commutativity, union commutativity and identity with Φ, and double-complement involution.Source: Example 3.13 · p. 8
  • The example verifies distributivity of union over intersection by showing equal membership triples on both sides.The displayed values match for b1, b2, and b3.Source: Example 3.13 · p. 9
  • AND forms a GNSS over A×B, with K(α,β) defined from the intersection of the two parameter-specific sets.Source: Example 3.13 · p. 9
  • OR is defined over A×B, and the propositions state that complements exchange AND and OR.Source: Example 3.13 · pp. 10–11

4.An Application of Generalized Neutrosophic Soft Set in a Decision Making Problem

The paper applies generalized neutrosophic soft sets to choose a blouse for Mrs. X from her selected criteria. It ranks objects using a comparison matrix and scores, selecting b4 with a maximum score of 19.

4.An Application of Generalized Neutrosophic Soft Set in a Decision Making Problem

4.1. Definition: Comparison matrix.

  • The comparison matrix uses objects as rows and parameters as columns; each entry counts parameters where one object’s value equals or exceeds another’s.Entries are calculated as cij = a + d - c, with a, d, and c counting such comparisons for T, I, and F values, respectively.Source: §4.1 · pp. 11–12
  • The algorithm tabulates the generalized neutrosophic soft set, computes the comparison matrix and object scores, then selects the object with the highest score; ties permit choosing any maximum-scoring object.The example uses Mrs. X’s choice parameters P = {Cheap, Colorful, Polystyreneing, costly, Bright}.Source: §4.1 · p. 12
  • Score 19 gives blouse b4 the maximum, making it Mrs. X’s best choice, followed by b5.Source: §4.1 · p. 13

5. CONCLUSIONS

The conclusion presents generalized neutrosophic soft sets as a hybrid of generalized neutrosophic sets and soft sets, with defined operations and a decision-making application.

  • The paper combines generalized neutrosophic sets proposed by A. A. Salama with soft sets introduced by Molodtsov in the generalized neutrosophic soft set.Source: §5 · p. 13
  • The authors state that they defined operations on generalized neutrosophic soft sets and presented a decision-making application.Source: §5 · p. 13
  • The authors express hope that their model opens a new direction for engineers, mathematicians, computer scientists, and others.Source: §5 · p. 13

Authors

Broumi Said is a university administrator whose research spans soft set, fuzzy, intuitionistic fuzzy, and neutrosophic theories, as well as control systems.

  • Broumi Said’s research concentrates on soft set theory, fuzzy theory, intuitionistic fuzzy theory, neutrosophic theory, and control of systems.Source: Authors · p. 14
  • He worked at the University of Hassan II-Mohammedia–Casablanca for five years.Source: Authors · p. 14
  • He received a master’s degree in industrial automatic from the University of Hassan II–Ain Chok.Source: Authors · p. 14
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