A multi-stage decision-making process for multiple attributes analysis under an interval-valued fuzzy environment

Abstract

This paper presents a multi-stage decision-making process for multiple attributes analysis under an interval-valued fuzzy environment. The proposed method does not demand weights of conflicting attributes through the decision-making process under uncertainty. The performance ratings of potential alternatives with respect to selected conflicting attributes are first described by linguistic terms and then are represented as interval-valued triangular fuzzy numbers. Second, an outranking matrix is proposed to denote the frequency with which one potential alternative dominates all the other alternatives according to each selected attribute. Consequently, the outranking matrix is triangularized in order to provide an implicit preordering or provisional order of potential alternatives under uncertainty. Finally, the tentative order of alternatives undergoes different operations of the screening and balancing that requires sequential application of a balancing principle to the advantages–disadvantages table. It hybridizes the conflicting attributes with the pair-wise comparisons of the potential alternatives for the multiple attributes analysis. Furthermore, an application example is used for the decision-making in a selection problem to illustrate the feasibility and applicability of proposed method in an interval-valued fuzzy environment.

Appendix

Consider two interval-valued fuzzy numbers ${N_x} = \left[ {N_x^{ - };N_x^{ + }} \right]$ and ${M_y} = \left[ {M_y^{ - };M_y^{ + }} \right]$, we have [16, 17, 22]:

Definition A.1

If ∈ (+, −, ×, ÷), then $N \cdot M\left( {x \cdot y} \right) = \left[ {N_x^{ - } \cdot M_y^{ - };N_x^{ + } \cdot M_y^{ + }} \right]$, for a positive non-fuzzy number (v), $v \cdot M\left( {x \cdot y} \right) = \left[ {v \cdot M_y^{ - };v \cdot M_y^{ + }} \right]$.

Definition A.2

Let $\widetilde{N}$ and $\tilde{M}$ be two interval-valued fuzzy numbers. $\widetilde{N}$ and $\tilde{M}$ can then be represented as follows:

$$ \widetilde{N} = \left[ {\left( {{N_1},N_1^{\prime }} \right);{N_2};\left( {N_3^{\prime },{N_3}} \right)} \right],\widetilde{M} = \left[ {\left( {{M_1},M_1^{\prime }} \right);{M_2};\left( {M_3^{\prime },{M_3}} \right)} \right]. $$

Let

$$ h\left( {\widetilde{N}} \right) = \frac{{{N_1} + N_1^{\prime } + 2{N_2} + N_3^{\prime } + {N_3}}}{6}, $$

and

$$ h\left( {\widetilde{M}} \right) = \frac{{{M_1} + M_1^{\prime } + 2{M_2} + M_3^{\prime } + {M_3}}}{6}, $$

We say $\widetilde{N} > \widetilde{M}$ if $h\left( {\widetilde{N}} \right) > h\left( {\widetilde{M}} \right)$.

Definition A.3

Normalizing interval-valued fuzzy numbers. Consider ${\widetilde{x}_{{ij}}} = \left[ {\left( {{a_{{ij}}},a_{{ij}}^{\prime }} \right);{b_{{ij}}};\left( {c_{{ij}}^{\prime },{c_{{ij}}}} \right)} \right]$, the normalized performance rating can be calculated as:

$$ \begin{array}{*{20}l} {{\widetilde{n}_{{ij}} = {\left[ {{\left( {\frac{{a_{{ij}} }} {{c^{ + }_{i} }},\frac{{a^{\prime }_{{ij}} }} {{c^{ + }_{i} }}} \right)};\frac{{b_{{ij}} }} {{c^{ + }_{i} }};{\left( {\frac{{c^{\prime }_{{ij}} }} {{c^{ + }_{i} }},\frac{{c_{{ij}} }} {{c^{ + }_{i} }}} \right)}} \right]},} \hfill} & {{j = 1, \ldots ,m,\quad i \in \Omega _{b} } \hfill} \\ {{\widetilde{n}_{{ij}} = {\left[ {{\left( {\frac{{a^{ - }_{j} }} {{c^{\prime }_{i} }},\frac{{a^{ - }_{j} }} {{c_{i} }}} \right)};\frac{{a^{ - }_{j} }} {{b_{i} }};{\left( {\frac{{a^{ - }_{j} }} {{\prime _{i} }},\frac{{a^{ - }_{j} }} {{a_{i} }}} \right)}} \right]},} \hfill} & {{j = 1, \ldots ,m,\quad i \in \Omega _{c} } \hfill} \\ \end{array} $$

where

$$ \begin{array}{*{20}c} {{c^{ + }_{i} = \begin{array}{*{20}c} {{{\text{Max}}c_{{ij}} ,}} \\ {j} \\ \end{array} \,\,\,\,\,\,\,i \in \Omega _{b} }} \\ {{a^{ - }_{i} = \begin{array}{*{20}c} {{{\text{Min}}\,a_{{ij}} ,}} \\ {j} \\ \end{array} \,\,\,\,\,\,\,i \in \Omega _{c} }} \\ \end{array} $$

where b is associated with benefit attributes, and c is associated with cost attributes.