The Linguistic Picture Fuzzy Set and Its Application in Multi-Criteria Decision-Making: An Illustration to the TOPSIS and TODIM Methods Based on Entropy Weight

Donghai Liu,Yan Luo,Zaiming Liu

doi:10.3390/sym12071170

Abstract

The paper considers the multi-criteria decision-making problem based on linguistic picture fuzzy information. Firstly, we propose the concept of linguistic picture fuzzy set(LPFS), where the positive-membership, the neutral-membership and the negative-membership are represented by linguistic variables, and its operation rules are also discussed. The linguistic picture fuzzy weighted averaging (LPFWA) operator and linguistic picture fuzzy weighted geometric (LPFWG) operator are developed based on the proposed operation rules. Secondly, we propose the generalized weighted distance measure, the generalized weighted Hausdorff distance measure, and the generalized hybrid weighted distance measure between LPFSs and discuss their properties. Thirdly, we extend the technique for order of preference by similarity to the ideal solution (TOPSIS) method and the TODIM (an acronym in Portuguese of interactive and multi-criteria decision-making) method to the proposed distance measure, and the linguistic picture fuzzy entropy method is proposed to calculate the weights of the criteria. Finally, an illustrative example is given to verify the feasibility and effectiveness of the proposed methods, the comparative analysis with other existing methods and sensitivity analysis of the proposed methods are also discussed.

Highlights

In 1965, Zadeh [1] proposed the fuzzy set (FS) F = {( x, μ F ( x ))| x ∈ X }, where μ F ( x ) represents the membership degree of x ∈ X to the set F
We proposed the LPFS based on picture fuzzy set (PFS) and linguistic term set (LTS), where the positive-membership, the neutral-membership, and the negative-membership are represented by linguistic variables, and the LPFS can deal with the vague and imprecise information in qualitative environment
We define some distance measures between LPFSs, and the TOPSIS method and TODIM method are developed to the proposed distance measures based on the linguistic picture fuzzy entropy

Summary

Introduction

In 1965, Zadeh [1] proposed the fuzzy set (FS) F = {( x, μ F ( x ))| x ∈ X }, where μ F ( x ) represents the membership degree of x ∈ X to the set F. In order to express such information, Cuong [3] proposed the concept of picture fuzzy set (PFS) P = {( x, μ P ( x ), ηP ( x ), v P ( x ))| x ∈ X }, where μ P ( x ), ηP ( x ) and v P ( x ) represent the positive membership degree, the neutral membership degree, and the negative membership degree of x ∈ X to the set P, respectively.

Methods

Results

Conclusion