Sudip Kumar Sahana - Advances in computational intelligence: proceedings of international

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Part I
Fuzzy Logic

Springer Science+Business Media Singapore 2017

Sudip Kumar Sahana and Sujan Kumar Saha (eds.) Advances in Computational Intelligence Advances in Intelligent Systems and Computing 10.1007/978-981-10-2525-9_1

A New Approach to Interval-Valued Fuzzy Soft Sets and Its Application in Decision-Making

B. K. Tripathy 1

(1)

SCOPE, VIT University, Vellore, Tamilnadu, India

B. K. Tripathy (Corresponding author)

Email:

T. R. Sooraj

Email:

R. K. Mohanty

Email:

Abstract

Soft set (SS) theory was introduced by Molodtsov to handle uncertainty. It uses a family of subsets associated with each parameter. Hybrid models have been found to be more useful than the individual components. Earlier interval-valued fuzzy set (IVFS) was introduced as an extension of fuzzy set (FS) by Zadeh. Yang introduced the concept of IVFSS by combining and soft set models. Here, we define IVFSS through the membership function approach to define soft set by Tripathy et al. very recently. Several concepts, such as complement of an IVFSS, null IVFSS, absolute IVFSS, intersection, and union of two IVFSSs, are redefined. To illustrate the application of IVFSSs, a decision-making (DM) algorithm using this notion is proposed and illustrated through an example.

Keywords

SS FS FSS IVFS IVFSS DM

Introduction

Fuzzy set introduced by Zadeh [].

It is well known that IVFSS introduced by Yang [] while keeping the authenticity intact.

Definitions and Notions

By P ( U ) and I ( U ), we denote the power set and the fuzzy power set of U, respectively.

Definition 2.1

( Soft Set ) A pair ( F, E ) is called as a soft set over the universal set U, where

(2.1)

The pair ( U , E ), which is a combination of a universal set U and a parameter set E, is called a soft universe.

Definition 2.2

We denote a FSS over ( U, E ) by ( F, E ), where

(2.2)

Let I ([0, 1]) denote the set of all closed subintervals of [0, 1].

Definition 2.3 (IVFS)

An IVFS X on a universe U is a mapping, such that

(2.3)

Moreover, . Here, and represent as the lower and upper degrees of membership of x to X.

Interval-Valued FSS

In this section, we follow the membership function approach introduced in [).

Definition 3.1

Given , we define

(3.1)

For any two IVFSS ( F , E ) and ( G , E ), we define the following operations.

Definition 3.2

The union of ( F, E ) and ( G, E ) is the IVFSS ( H, E ), and and , we have

(3.2)

where and denotes the lower and upper membership value of the IVFSS.

Definition 3.3

The intersection of ( F, E ) and ( G, E ) is the IVFSS ( H, E ), and and , we have

(3.3)

Definition 3.4

( F, E ) is said to be interval valued fuzzy soft subset of ( G, E ), . Then, , ,

(3.4)

Definition 3.5

We say that ( F, E ) is equal to ( G, E ) written as ( F, E ) = ( G, E ) if ,

(3.5)

Definition 3.6

For any two IVFSSs ( F, E ) and ( G, E ) over a common soft universe ( U, E ), we define the complement ( H, E ) of ( G, E ) in ( F, E ) as and .

(3.6)

Definition 3.7

The complement of an IVFSS over a soft universe ( U, E ) can be derived from the above Definition by taking ( F, E ) as U and ( G, E ) as ( F, E ). We denote it by

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