A fusion of decision-making method and neutrosophic linguistic considering multiplicative inverse matrix for coastal erosion problem

Abstract

The recent boom of decision-making under uncertain information has attracted many researchers to the field of integrating various types of sets with decision-making methods. In this paper, a combined decision-making trial and evaluation laboratory (DEMATEL) with single-valued neutrosophic sets is proposed to solve the decision problem. This new model combines the advantages of multiplicative inverse of decision matrix in DEMATEL and neutrosophic numbers in linguistic variables, which can find the interrelationship among factors of decision problem. Differently from the typical multiplicative inverse of DEMATEL, which directly used inverse of matrix using real numbers, this method introduces the concept of inverse of matrix using the proposed left–right neutrosophic numbers. This step will enhance the validity of multiplicative inverse of decision matrix in the DEMATEL with neutrosophic numbers. The proposed neutrosophic DEMATEL is also be compared with the DEMATEL and fuzzy DEMATEL. This paper includes a case study that demonstrates the applicability of the neutrosophic DEMATEL in establishing the relationship among influential factors of coastal erosion. Extensive empirical studies using 12 factors of coastal erosion were presented to study the benefits of the proposed method. The result unveils the cause-and-effect relationships among the factors, where seven factors are identified as cause factors and five factors are grouped as effect factors. It is discovered that the factor ‘imbalance sediment supply’ gives a significant influence to coastal erosion. It is also shown that the degree of importance of the factors is almost consistent with the other two methods despite differences in type of numbers used in defining linguistic variables.

Appendix A

Theorem

Consider the nonnegative SVNS matrix $\tilde{A} = \left( {\left\langle {\tilde{T},\tilde{I},\tilde{F}} \right\rangle } \right)_{n \times n}$ and let $\tilde{A}$ be the element of the class of all nonnegative matrices with nonnegative inverses. Moreover, suppose that $\tilde{F}_{ij} = 0$ if $\tilde{T}_{ij}$ is zero; then,

$$ \left( {\tilde{T}^{ - 1} ,\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{I}\tilde{T}^{ - 1} } \right),\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{F}\tilde{T}^{ - 1} } \right)} \right), $$

is the left $\varepsilon$-inverse of $\tilde{A}$, where $\varepsilon > 0$ is chosen in the following interval,

$$ \left[ {\hbox{max} \left\{ {\mathop {\hbox{max} }\limits_{ij} \left\{ {\frac{{\tilde{F}_{ij} }}{{\tilde{T}_{ij} }}} \right\},\quad \mathop {\hbox{max} }\limits_{ij} \left\{ {\frac{{\tilde{I}_{ij} }}{{\tilde{T}_{ij} }}} \right\}} \right\},\quad 1 + \mathop {\hbox{max} }\limits_{ij} \left\{ {\frac{{\tilde{I}_{ij} }}{{\tilde{T}_{ij} }}} \right\} \, } \right],\quad {\text{where }}\tilde{T}_{ij} \ne 0 $$

This theorem can be proved using the concept of inverse matrix, $\tilde{A} \times \tilde{B} = I$, where $\tilde{B}$ is the inverse of $\tilde{A}$.

Proof

We must prove

$$ \left( {\tilde{T},\tilde{I},\tilde{F}} \right) \otimes \left( {\tilde{T}^{ - 1} ,\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{I}\tilde{T}^{ - 1} } \right),\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{F}\tilde{T}^{ - 1} } \right)} \right) = \left( {I,\varepsilon I,\varepsilon I} \right) $$

Based on the above assumption on $\varepsilon$, the SVNS matrix

$$ \tilde{B} = \left( {\tilde{T}^{ - 1} ,\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{I}\tilde{T}^{ - 1} } \right),\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{F}\tilde{T}^{ - 1} } \right)} \right) $$

is positive. Thus, it can be rewritten as

$$ \left( {\tilde{T}\tilde{T}^{ - 1} ,\tilde{T}\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{I}\tilde{T}^{ - 1} } \right) + \tilde{I}\tilde{T}^{ - 1} ,\tilde{T}\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{F}\tilde{T}^{ - 1} } \right) + \tilde{F}\tilde{T}^{ - 1} } \right) = \left( {I,\varepsilon I,\varepsilon I} \right) $$

Or equivalently

$$ \left\{ {\begin{array}{*{20}l} {\tilde{T}\tilde{T}^{ - 1} = I} \hfill \\ {\tilde{T}\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{I}\tilde{T}^{ - 1} } \right) + \tilde{I}\tilde{T}^{ - 1} = \varepsilon I} \hfill \\ {\tilde{T}\tilde{T}^{ - 1} \left( {\varepsilon I - \tilde{F}\tilde{T}^{ - 1} } \right) + \tilde{F}\tilde{T}^{ - 1} = \varepsilon I} \hfill \\ \end{array} } \right. $$

which is a valid system. □