A weighted centroids approach based trapezoidal interval type-2 fuzzy TOPSIS method for evaluating agricultural risk management tools

Abstract

This study proposes a trapezoidal interval type-2 fuzzy TOPSIS method to evaluate agricultural risk management tools. The weighted centroids based on AHP is used in the proposed method to defuzzify the fuzzy weighted ratings of each alternative versus each criterion. By this way, this method considers all centroid values of the upper and lower membership functions and uses the weight as a parameter to consider the relative importance of controid values of the upper and lower membership functions. Additionally, the left and the right upper membership function and lower membership function of the multiplication of two positive trapezoidal IT2FNs can be developed based on the α-cuts. The proposed method is applied to help farmers compare agricultural risk management tools and select the most suitable one. This is the first study using fuzzy decision-making methods to help farmers evaluate agricultural risk management tools. To show the proposed method’s feasibility, a numerical example is displayed. Finally, some comparisons are made to show the advantages of the proposed method, and an experiment is conducted to show the robustness of the proposed method.

Appendices

Appendix A

Theorem

Let $q$ and $w$ be two trapezoidal IT2FNs, $q,w \in R^{ + } , \, and$${\text{ q}} = ((a_{1}^{{\text{U}}} ,a_{2}^{{\text{U}}} ,a_{3}^{{\text{U}}} ,a_{4}^{{\text{U}}} ;H_{q}^{{\text{U}}} ), \, (a_{1}^{{\text{L}}} ,a_{2}^{{\text{L}}} ,a_{3}^{{\text{L}}} ,a_{4}^{{\text{L}}} ;H_{q}^{{\text{L}}} ))$, $w = ((b_{1}^{{\text{U}}} ,b_{2}^{{\text{U}}} ,b_{3}^{{\text{U}}} ,b_{4}^{{\text{U}}} ;H_{w}^{{\text{U}}} ),(b_{1}^{{\text{L}}} ,b_{2}^{{\text{L}}} ,b_{3}^{{\text{L}}} ,b_{4}^{{\text{L}}} ;H_{w}^{{\text{L}}} ))$. The multiplication of $q$ and $w$ is defined by the upper membership function $\overline{f}_{q \otimes w}^{{}} (x)$, including left upper membership function $\overline{f}_{q \otimes w}^{L} (x)$ and the right upper membership function $\overline{f}_{q \otimes w}^{R} (x)$, and lower membership functions $\underline {f}_{q \otimes w} (x)$, including left lower membership function $\underline {f}_{q \otimes w}^{L} (x)$ and the right lower membership function $\underline {f}_{q \otimes w}^{R} (x)$, as follows.

$$ \overline{f}_{q \otimes w} (x) = \left\{ \begin{gathered} \overline{f}_{q \otimes w}^{L} (x) = \frac{{ - F_{1} { + }\sqrt {F_{1}^{2} - 4E_{1} (G_{1} - x)} }}{{2E_{1} }}, \, G_{1} \le x < G_{2} , \hfill \\ H_{q \otimes w}^{{\text{U}}} , \, G_{2} \le x \le G_{3} , \hfill \\ \overline{f}_{q \otimes w}^{R} (x) = \frac{{ - F_{2} - \sqrt {F_{2}^{2} - 4E_{2} (G_{4} - x)} }}{{2E_{2} }},G_{3} < x \le G_{4} ; \hfill \\ \end{gathered} \right. $$

and

$$ \underline {f}_{q \otimes w} (x) = \left\{ \begin{gathered} \underline {f}_{q \otimes w}^{L} (x) = \frac{{ - F_{3} + \sqrt {F_{3}^{2} - 4E_{3} (Q_{1} - x)} }}{{2E_{3} }}, \, Q_{1} \le x < Q_{2} , \hfill \\ H_{q \otimes w}^{{\text{L}}} , \, Q_{2} \le x \le Q_{3} , \hfill \\ \underline {f}_{q \otimes w}^{R} (x) = \frac{{ - F_{4} - \sqrt {F_{4}^{2} - 4E_{4} (Q_{4} - x)} }}{{2E_{4} }},Q_{3} < x \le Q_{4} . \hfill \\ \end{gathered} \right. $$

where $E_{1} = (a_{2}^{{\text{U}}} - a_{1}^{{\text{U}}} )(b_{2}^{{\text{U}}} - b_{1}^{{\text{U}}} )$, $F_{1} = a_{1}^{{\text{U}}} ((b_{2}^{{\text{U}}} - b_{1}^{{\text{U}}} ) + b_{1}^{{\text{U}}} (a_{2}^{{\text{U}}} - a_{1}^{{\text{U}}} )$, $E_{2} = (a_{3}^{{\text{U}}} - a_{4}^{{\text{U}}} {)}(b_{3}^{{\text{U}}} - b_{4}^{{\text{U}}} )$,$F_{2} = a_{4}^{{\text{U}}} (b_{3}^{{\text{U}}} - b_{4}^{{\text{U}}} ) + b_{4}^{{\text{U}}} (a_{3}^{{\text{U}}} - a_{4}^{{\text{U}}} {)}$, $G_{1} = a_{1}^{{\text{U}}} b_{1}^{{\text{U}}}$, $G_{2} = a_{2}^{{\text{U}}} b_{2}^{{\text{U}}}$, $G_{3} = a_{3}^{{\text{U}}} b_{3}^{{\text{U}}}$, $G_{4} = a_{4}^{{\text{U}}} b_{4}^{{\text{U}}}$, $H_{q \otimes w}^{{\text{U}}} = \min (H_{q}^{{\text{U}}} ,H_{w}^{{\text{U}}} )$; $E_{3} = (a_{2}^{{\text{L}}} - a_{1}^{{\text{L}}} )(b_{2}^{{\text{L}}} - b_{1}^{{\text{L}}} )$, $F_{3} = a_{1}^{{\text{L}}} (b_{2}^{{\text{L}}} - b_{1}^{{\text{L}}} ) + b_{1}^{{\text{L}}} (a_{2}^{{\text{L}}} - a_{1}^{{\text{L}}} )$, $E_{4} = (a_{3}^{{\text{L}}} - a_{4}^{{\text{L}}} )(b_{3}^{{\text{L}}} - b_{4}^{{\text{L}}} )$,$F_{4} = a_{4}^{{\text{L}}} (b_{3}^{{\text{L}}} - b_{4}^{{\text{L}}} ) + b_{4}^{{\text{L}}} (a_{3}^{{\text{L}}} - a_{4}^{{\text{L}}} )$, $Q_{1} = a_{1}^{{\text{L}}} b_{1}^{{\text{L}}}$, $Q_{2} = a_{2}^{{\text{L}}} b_{2}^{{\text{L}}}$, $Q_{3} = a_{3}^{{\text{L}}} b_{3}^{{\text{L}}}$, $Q_{4} = a_{4}^{{\text{L}}} b_{4}^{{\text{L}}}$, $H_{q \otimes w}^{{\text{L}}} = \min (H_{q}^{{\text{L}}} ,H_{w}^{{\text{L}}} )$.

Proof

By using Eq. (3) and Eqs. (14)–(22) based on $\alpha {\text{ - cuts}}$ and interval arithemetic operations of fuzzy numbers, the upper membership function $\overline{f}_{q \otimes w}^{{}} (x)$, including $\overline{f}_{{v_{ij} }}^{L} (x)$ and $\overline{f}_{{v_{ij} }}^{R} (x)$, and lower membership functions $\underline {f}_{q \otimes w} (x)$, including $\underline {f}_{{v_{ij} }}^{L} (x)$ and $\underline {f}_{{v_{ij} }}^{R} (x)$, can be developed as shown in the theorem; in which $\overline{f}_{q \otimes w}^{L} (x) = \frac{{ - F_{1} - \sqrt {F_{1}^{2} - 4E_{1} (G_{1} - x)} }}{{2E_{1} }}$ and $\underline {f}_{q \otimes w}^{L} (x) = \frac{{ - F_{3} - \sqrt {F_{3}^{2} - 4E_{3} (Q_{1} - x)} }}{{2E_{3} }}$ are not allowed because negative grades are not allowed.where $E_{1} = (a_{2}^{{\text{U}}} - a_{1}^{{\text{U}}} )(b_{2}^{{\text{U}}} - b_{1}^{{\text{U}}} )$, $F_{1} = a_{1}^{{\text{U}}} ((b_{2}^{{\text{U}}} - b_{1}^{{\text{U}}} ) + b_{1}^{{\text{U}}} (a_{2}^{{\text{U}}} - a_{1}^{{\text{U}}} )$, $E_{2} = (a_{3}^{{\text{U}}} - a_{4}^{{\text{U}}} {)}(b_{3}^{{\text{U}}} - b_{4}^{{\text{U}}} )$,$F_{2} = a_{4}^{{\text{U}}} (b_{3}^{{\text{U}}} - b_{4}^{{\text{U}}} ) + b_{4}^{{\text{U}}} (a_{3}^{{\text{U}}} - a_{4}^{{\text{U}}} {)}$, $G_{1} = a_{1}^{{\text{U}}} b_{1}^{{\text{U}}}$, $G_{2} = a_{2}^{{\text{U}}} b_{2}^{{\text{U}}}$, $G_{3} = a_{3}^{{\text{U}}} b_{3}^{{\text{U}}}$, $G_{4} = a_{4}^{{\text{U}}} b_{4}^{{\text{U}}}$, $H_{q \otimes w}^{{\text{U}}} = \min (H_{q}^{{\text{U}}} ,H_{w}^{{\text{U}}} )$; $E_{3} = (a_{2}^{{\text{L}}} - a_{1}^{{\text{L}}} )(b_{2}^{{\text{L}}} - b_{1}^{{\text{L}}} )$, $F_{3} = a_{1}^{{\text{L}}} (b_{2}^{{\text{L}}} - b_{1}^{{\text{L}}} ) + b_{1}^{{\text{L}}} (a_{2}^{{\text{L}}} - a_{1}^{{\text{L}}} )$, $E_{4} = (a_{3}^{{\text{L}}} - a_{4}^{{\text{L}}} )(b_{3}^{{\text{L}}} - b_{4}^{{\text{L}}} )$,$F_{4} = a_{4}^{{\text{L}}} (b_{3}^{{\text{L}}} - b_{4}^{{\text{L}}} ) + b_{4}^{{\text{L}}} (a_{3}^{{\text{L}}} - a_{4}^{{\text{L}}} )$, $Q_{1} = a_{1}^{{\text{L}}} b_{1}^{{\text{L}}}$, $Q_{2} = a_{2}^{{\text{L}}} b_{2}^{{\text{L}}}$, $Q_{3} = a_{3}^{{\text{L}}} b_{3}^{{\text{L}}}$, $Q_{4} = a_{4}^{{\text{L}}} b_{4}^{{\text{L}}}$, $H_{q \otimes w}^{{\text{L}}} = \min (H_{q}^{{\text{L}}} ,H_{w}^{{\text{L}}} )$.

Moreover, $\overline{f}_{q \otimes w}^{R} (x) = \frac{{ - F_{2} + \sqrt {F_{2}^{2} - 4E_{2} (G_{4} - x)} }}{{2E_{2} }}$ and $\underline {f}_{q \otimes w}^{R} (x) = \frac{{ - F_{4} + \sqrt {F_{4}^{2} - 4E_{4} (Q_{4} - x)} }}{{2E_{4} }}$ are not allowed because their grades are not zero when applying $x = G_{4}$ and $x = Q_{4}$, respectively.□

Appendix B

Corollary 1

$\overline{f}_{q \otimes w}^{L} (x) = 0$ if $x = G_{1}$ and $\underline {f}_{q \otimes w}^{L} (x) = 0$ if $x = Q_{1}$.

Proof

Apply $x = G_{1}$ to $\overline{f}_{q \otimes w}^{L} (x) = \frac{{ - F_{1} { + }\sqrt {F_{1}^{2} - 4E_{1} (G_{1} - x)} }}{{2E_{1} }}$ to obtain.

$$ \overline{f}_{q \otimes w}^{L} (x) = \frac{{ - F_{1} { + }\sqrt {F_{1}^{2} - 4E_{1} (G_{1} - G_{1} )} }}{{2E_{1} }} = \frac{{ - F_{1} { + }F_{1} }}{{2E_{1} }} = 0 $$

Similarly, $\underline {f}_{q \otimes w}^{L} (x)$ = 0 if $x = Q_{1}$.□

Corollary 2

$\overline{f}_{q \otimes w}^{R} (x) = 0$ if $x = G_{4}$ and $\underline {f}_{q \otimes w}^{R} (x) = 0$ if $x = Q_{4}$.

Proof

Apply $x = G_{4}$ to $\overline{f}_{q \otimes w}^{R} (x) = \frac{{ - F_{2} - \sqrt {F_{2}^{2} - 4E_{2} (G_{4} - x)} }}{{2E_{2} }}$ to obtain.

$$ \overline{f}_{q \otimes w}^{R} (x) = \frac{{ - F_{2} - \sqrt {F_{2}^{2} - 4E_{2} (G_{4} - G_{4} )} }}{{2E_{2} }} = \frac{{ - F_{2} - \left( { - F_{2} } \right)}}{{2E_{2} }} = 0 $$

Similarly, $\underline {f}_{q \otimes w}^{R} (x) = 0$ if $x = Q_{4}$. □