Difference sequence-based distance measure for intuitionistic fuzzy sets and its application in decision making process

Abstract

Distance measures are among the most studied phenomena for deciding the degree of belongingness between the two distinct objects. This paper presents a new distance measure among intuitionistic fuzzy sets (IFSs) using the generalized difference sequence spaces within p-summable intuitionistic fuzzy bounded variation (IFBV). The IFBV is a procedure used to approximate the arc-length of an intuitionistic fuzzy-valued function (IFVF) over the IFS, i.e. distance function of all data points of IFS distributed over the geometrical structure of IFBV with power p. The topological uniqueness in the proposed distance measure made it distinguishable, independent from the reflective relationship, and free from the reflective symmetry. All distance measure characteristics are satisfied, and various situations showing the inclusive relations within distance metrics are drawn. Moreover, the proposed distance measure is applied for multi-attribute decision-making approaches, namely the TOPSIS (Technique for Order Preference by Similarity to Ideal Solution) and GRA (Grey Relational Analysis) techniques. An improved Intuitionistic Fuzzy TOPSIS (IFTOPSIS) and Trapezoidal Intuitionistic Fuzzy GRA (TrIFGRA) techniques are developed to demonstrate the superiority of proposed distance measure with several established ones. With suitable examples, we thoroughly illustrate the functioning of the IFTOPSIS and TrIFGRA decision-making procedures. Extensive comparisons with other popular distance measures are performed to check the effectiveness of our proposed distance measure.

Appendices

Appendix 1

The distances for IFN used in Table 7 for the IFTOPSIS comparison process:

Atanassov (1999):

$$\begin{aligned} D_{H}(A,B)= & {} \bigg ( \frac{1}{2}\sum _{s=1}^{m} \big [\big |w_{A}(x_s)-w_B(x_s)\big | \\&+\big |u_{A}(x_s)-u_{B}(x_s)\big | \big ]\bigg ) \end{aligned}$$

Szmidt and Kacprzyk (2000):

$$\begin{aligned} D_{NE}(A,B)= & {} \bigg ( \frac{1}{4m}\sum _{s=1}^{m} \big [\big (w_{A}(x_s)-w_B(x_s)\big )^2\\&+\big (u_{A}(x_s)-u_{B}(x_s)\big )^2 \big ]\bigg )^{\frac{1}{2}} \end{aligned}$$

Hung and Yang (2004):

$$\begin{aligned} D_{HD}(A,B)= & {} \sum _{s=1}^{m} max \bigg (\big |w_{A}(x_s)-w_B(x_s)\big |,\\&\big |u_{A}(x_s)-u_{B}(x_s)\big | \bigg ) \end{aligned}$$

Solanki et al. (2016):

$$\begin{aligned}&\!\!\!C_{M}(A,B)\\&\!\!\!\quad =1 - \frac{1}{m}{\sum \limits _{s=1}^{m}} \frac{ \big |w_{A}(x_s)-w_B(x_s)\big | +\big |u_{A}(x_s)-u_{B}(x_s)\big |}{1+\big |w_{A}(x_s)-w_B(x_s)\big | +\big |u_{A}(x_s)-u_{B}(x_s)\big |} \end{aligned}$$

Ashraf et al. (2019):

$$\begin{aligned} D_{bv_p}(A,B)= & {} \frac{1}{\root p \of {2m}}\Bigg (|w_{A}(x_1)-w_{B}(x_1)|^p \\&+|u_{A}(x_1)-u_{B}(x_1)|^p\\&+\sum _{s=2}^{m}(|\varDelta w_{A}(x_s)-\varDelta w_{B}(x_s)|^p\\&+|\varDelta u_{A}(x_s)-\varDelta u_{B}(x_s)|^p)\Bigg )^{\frac{1}{p}} \end{aligned}$$

Ashraf et al. (2019):

$$\begin{aligned} D_{bv_p}^h(A,B)&= \frac{1}{\root p \of {2m}} \Bigg ( |w_{A}(x_i)-w_{B}(x_i)|^p\\&+|u_{A}(x_i)-u_{B}(x_i)|^p\\&+\sum \limits _{s\in \rho (s)}(|w_{A}(x_{s})-w_{B}(x_{s})|^p\\&+ |u_{A}(x_{s})-u_{B}(x_{s})|^p ) \\&+\sum \limits _{s\in \tau (s)-i} ( |\varDelta w_{A}(x_{s})-\varDelta w_{B}(x_{s})|^p\\&+ |\varDelta u_{A}(x_{s})-\varDelta u_{B}(x_{s})|^p )\Bigg )^\frac{1}{p} \end{aligned}$$

Ashraf et al. (2019) :

$$\begin{aligned} D_{G_p}(A,B)= & {} \frac{1}{\root p \of {2m}}\Bigg (|w_{A}(x_i)-w_{B}(x_i)|^p\\&+|u_{A}(x_i)-u_{B}(x_i)|^p \\&+\sum \limits _{s\in \tau (s)-i} \big ( |\varDelta _t^rw_{A}(x_{s-t})-\varDelta _t^rw_{B}(x_{s-t})|^p \\&+|\varDelta _t^ru_{A}(x_{s-t})-\varDelta _t^ru_{B}(x_{s-t})|^p\big )\Bigg )^\frac{1}{p} \end{aligned}$$

Appendix 2

The distances for TrIFN used in Table 10 for the TrIFGRA comparison process:

Let $\tilde{a}=\bigg (\big (\big [a_1,b_1,c_1,d_1\big ];w_{\tilde{a_1}},u_{\tilde{a_1}}), \big (\big [a_2,b_2,c_2,d_2\big ]; w_{\tilde{a_2}},u_{\tilde{a_2}})\cdots ,$

$\big (\big [a_m,b_m,c_m,d_m\big ];w_{\tilde{a_m}},u_{\tilde{a_m}})\bigg )$ and

$\tilde{b}=\bigg (\big (\big [f_1,g_1,h_1,l_1\big ];w_{\tilde{b_1}},u_{\tilde{b_1}}), \big (\big [f_2,g_2,h_2,l_2\big ];w_{\tilde{b_2}},u_{\tilde{b_2}}) \cdots ,$

$\big (\big [f_m,g_m,c_m,d_m\big ];w_{\tilde{b_m}},u_{\tilde{b_m}})\bigg )$ be two intuitionistic trapezoidal fuzzy sets (TrIFSs). Then, Wan (2013):

$$\begin{aligned} d_{E}(\tilde{a},\tilde{b})= & {} \bigg [\frac{1}{2}\sum _{s=1}^{m} \bigg (\big (\tilde{a_{s}}-\tilde{f_{s}}\big )^2+\big (\tilde{b_{s}}-\tilde{g_{s}}\big )^2\\&+\big (\tilde{c_{s}}-\tilde{h_{s}}\big )^2+\big (\tilde{d_{s}}-\tilde{l_{s}}\big )^2\\&+\text{ max }\big \{\big (w_{\tilde{a_{s}}}-w_{\tilde{b_{s}}}\big )^2,\big (u_{\tilde{a_{s}}}-u_{\tilde{b_{s}}}\big )^2 \big \}\bigg )\bigg ]^{\frac{1}{2}} \end{aligned}$$

Wan (2013)::

$$\begin{aligned} d_{H}(\tilde{a},\tilde{b})= & {} \bigg [\frac{1}{2}\sum _{s=1}^{m}\bigg ( \big |\tilde{a_{s}}-\tilde{f_{s}}\big |+\big |\tilde{b_{s}}-\tilde{g_{s}}\big |+\big |\tilde{c_{s}}-\tilde{h_{s}}\big |\\&+\big |\tilde{d_{s}}-\tilde{l_{s}}\big |+\text{ max }\big \{\big |w_{\tilde{a_{s}}}-w_{\tilde{b_{s}}}\big |,\big |u_{\tilde{a_{s}}}-u_{\tilde{b_{s}}}\big | \big \}\bigg )\bigg ] \end{aligned}$$

Zhang et al. (2013):

$$\begin{aligned} d_{NH}(\tilde{a},\tilde{b})= & {} \frac{1}{2m}\bigg [\sum _{s=1}^{m}\bigg ( \big |\big (1+w_{\tilde{a_{s}}}-u_{\tilde{a_{s}}})a_{s}\\&-\big (1+w_{\tilde{b_{s}}}-u_{\tilde{b_{s}}})f_{s}\big |\\&+\big |\big (1+w_{\tilde{a_{s}}}-u_{\tilde{a_{s}}})b_{s}-\big (1+w_{\tilde{b_{s}}}-u_{\tilde{b_{s}}})g_{s}\big | \\&+ \big |\big (1+w_{\tilde{a_{s}}}-u_{\tilde{a_{s}}})c_{s}-\big (1+w_{\tilde{b_{s}}}-u_{\tilde{b_{s}}})g_{s}\big |\\&+ \big |\big (1+w_{\tilde{a_{s}}}-u_{\tilde{a_{s}}})d_{s}-\big (1+w_{\tilde{b_{s}}}-u_{\tilde{b_{s}}})l_{s} \bigg )\bigg ] \end{aligned}$$

Ashraf et al. (2019):

$$\begin{aligned} d_{bv_p}(\tilde{a},\tilde{b})= & {} \frac{1}{\root p \of {2m}}\bigg [\big (\big |a_1-f_1\big |^p+\big |b_1-g_1\big |^p+\big |c_1-h_1\big |^p\\&+\big |d_1-l_1\big |^p+\big |w_{\tilde{a_1}}-w_{\tilde{b_1}}\big |^p+\big |u_{\tilde{a_1}}-u_{\tilde{b_1}}\big |^p\big ) \\&+\sum _{s=2}^{m}\bigg (\big |\varDelta \tilde{a_{s}}-\varDelta \tilde{f_{s}}\big |^p+|\varDelta \tilde{b_{s}}-\varDelta \tilde{g_{s}}\big |^p\\&+|\varDelta \tilde{c_{s}}-\varDelta \tilde{h_{s}}\big |^p+|\varDelta \tilde{d_{s}}-\varDelta \tilde{l_{s}}\big |^p\\&+\big |\varDelta w_{\tilde{a_s}}-\varDelta w_{\tilde{b_s}}\big |^p +\big |\varDelta u_{\tilde{a_s}}-\varDelta u_{\tilde{b_s}}\big |^p\bigg )\Bigg ]^{\frac{1}{p}} \end{aligned}$$

Ashraf et al. (2019):

$$\begin{aligned} d_{bv_p}^h(\tilde{a},\tilde{b})= & {} \frac{1}{\root p \of {2m}}\bigg [\big (\big |a_i-f_i\big |^p+\big |b_i-g_i\big |^p\\&+\big |c_i-h_i\big |^p+\big |d_i-l_i\big |^p\\&+\big |w_{\tilde{a_i}}-w_{\tilde{b_i}}\big |^p+\big |u_{\tilde{a_i}}-u_{\tilde{b_i}}\big |^p\big ) \\&+ \sum _{s\in \rho (s)}\big (\big |a_s-f_s\big |^p+\big |b_s-g_s\big |^p+\big |c_s-h_s\big |^p\\&+\big |d_s-l_s\big |^p+\big |w_{\tilde{a_s}}-w_{\tilde{b_s}}\big |^p+\big |u_{\tilde{a_s}}-u_{\tilde{b_s}}\big |^p\big ) \\&+ \sum _{s\in \tau (s)-i} \bigg ( \big |\varDelta \tilde{a_{s}}-\varDelta \tilde{f_{s}}\big |^p+|\varDelta \tilde{b_{s}}-\varDelta \tilde{g_{s}}\big |^p\\&+|\varDelta \tilde{c_{s}}-\varDelta \tilde{h_{s}}\big |^p+|\varDelta \tilde{d_{s}}-\varDelta \tilde{l_{s}}\big |^p \\&+\big |\varDelta w_{\tilde{a_s}}-\varDelta w_{\tilde{b_s}}\big |^p+\big |\varDelta u_{\tilde{a_s}}-\varDelta u_{\tilde{b_s}}\big |^p\bigg )\Bigg ]^{\frac{1}{p}} \end{aligned}$$

Ashraf et al. (2019):

$$\begin{aligned} d_{G_p}(\tilde{a},\tilde{b})= & {} \frac{1}{\root p \of {2m}}\bigg [\big (\big |a_i-f_i\big |^p+\big |b_i-g_i\big |^p\\&+\big |c_i-h_i\big |^p+\big |d_i-l_i\big |^p\\&+\big |w_{\tilde{a_i}}-w_{\tilde{b_i}}\big |^p+\big |u_{\tilde{a_i}}-u_{\tilde{b_i}}\big |^p\big ) \\&+\sum \limits _{s\in \tau (s)-i}\bigg (\big |\varDelta _t^r \tilde{a}_{s-t}-\varDelta _t^r \tilde{f}_{s-t}\big |^p\\&+|\varDelta _t^r \tilde{b}_{s-t}-\varDelta _t^r \tilde{g}_{s-t}\big |^p+|\varDelta _t^r \tilde{c}_{s-t}-\varDelta _t^r \tilde{h}_{s-t}\big |^p \\&+|\varDelta _t^r \tilde{d}_{s-t}-\varDelta _t^r \tilde{l}_{s-t}\big |^p +\big |\varDelta _t^r w_{\tilde{a}_{s-t}}-\varDelta _t^r w_{\tilde{b}_{s-t}}\big |^p\\&+\big |\varDelta _t^r u_{\tilde{a}_{s-t}}-\varDelta _t^r u_{\tilde{b}_{s-t}}\big |^p\bigg )\Bigg ]^{\frac{1}{p}} \end{aligned}$$