An optimized multi-attribute decision-making approach to construction supply chain management by using complex picture fuzzy soft set

(Zadeh, 1965) If ${\hat{M}}_{\hat{T}}\left(\hat{s}\right)$ is the truth membership grade of $\hat{s}\in \hat{\mathbb{S}}$ then FS ${\hat{A}}_F$ is defined as ${\hat{A}}_F=\left\{\left(\hat{s},{\hat{M}}_{\hat{T}}\left(\hat{s}\right)\right):\hat{s}\in \hat{\mathbb{S}}\right\}$ where ${\hat{M}}_{\hat{T}}:\hat{\mathbb{S}}\to {\mathbb{I}}^{\left[0,1\right]}$ is membership mapping provided that $0\le {\hat{M}}_{\hat{T}}\left(\hat{s}\right)\le 1$. The family of all fuzzy subsets is represented by Σ_FS.

(Atanassov, 1986) If ${\hat{M}}_{\hat{T}}\left(\hat{s}\right)$ and ${\hat{M}}_{\hat{F}}\left(\hat{s}\right)$ are truth and false membership grades of $\hat{s}\in \hat{\mathbb{S}}$ then IFS ${\hat{A}}_{IF}$ is defined as ${\hat{A}}_{IF}=\left\{\left(\hat{s},〈{\hat{M}}_{\hat{T}}\left(\hat{s}\right),{\hat{M}}_{\hat{F}}\left(\hat{s}\right)〉\right):\hat{s}\in \hat{\mathbb{S}}\right\}$ where ${\hat{M}}_{\hat{T}},{\hat{M}}_{\hat{F}}:\hat{\mathbb{S}}\to {\mathbb{I}}^{\left[0,1\right]}$ are membership mappings provided that $0\le {\hat{M}}_{\hat{T}}\left(\hat{s}\right)+{\hat{M}}_{\hat{F}}\left(\hat{s}\right)\le 1$ with hesitancy grade ${\hat{M}}_{\hat{H}}\left(\hat{s}\right)=1-\left({\hat{M}}_{\hat{T}}\left(\hat{s}\right)+{\hat{M}}_{\hat{F}}\left(\hat{s}\right)\right)$ within 𝕀^[0,1]. The family of all IF-subsets is represented by Σ_IFS.

(Cuong & Kreinovich, 2013) If ${\hat{M}}_{\hat{P}}\left(\hat{s}\right)$, ${\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right)$ and ${\hat{M}}_{\hat{N}}\left(\hat{s}\right)$ are positive, neutral and negative membership grades of $\hat{s}\in \hat{\mathbb{S}}$ then pFS $\hat{\Omega }$ is defined as $\hat{\Omega }=\left\{\left(\hat{s},〈{\hat{M}}_{\hat{P}}\left(\hat{s}\right),{\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right),{\hat{M}}_{\hat{N}}\left(\hat{s}\right)〉\right):\hat{s}\in \hat{\mathbb{S}}\right\}$ where ${\hat{M}}_{\hat{P}},{\hat{M}}_{\widehat{Ne}},{\hat{M}}_{\hat{N}}:\hat{\mathbb{S}}\to {\mathbb{I}}^{\left[0,1\right]}$ are membership mappings provided that $0\le {\hat{M}}_{\hat{P}}\left(\hat{s}\right)+{\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right)+{\hat{M}}_{\hat{N}}\left(\hat{s}\right)\le 1$ with refusal membership grade ${\hat{M}}_{\hat{R}}\left(\hat{s}\right)=1-\left({\hat{M}}_{\hat{P}}\left(\hat{s}\right)+{\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right)+{\hat{M}}_{\hat{N}}\left(\hat{s}\right)\right)$ within 𝕀^[0,1].

(Akram, Bashir & Garg, 2020a; Qu et al., 2022) If ${\hat{M}}_{\hat{P}}\left(\hat{s}\right)$, ${\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right)$ and ${\hat{M}}_{\hat{N}}\left(\hat{s}\right)$ are positive, neutral and negative membership grades of $\hat{s}\in \hat{\mathbb{S}}$ then cpFS $\hat{\Theta }$ is defined as $\hat{\Theta }=\left\{\left(\hat{s},〈{\hat{M}}_{\hat{P}}\left(\hat{s}\right),{\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right),{\hat{M}}_{\hat{N}}\left(\hat{s}\right)〉\right):\hat{s}\in \hat{\mathbb{S}}\right\}$ where ${\hat{M}}_{\hat{P}},{\hat{M}}_{\widehat{Ne}},{\hat{M}}_{\hat{N}}:\hat{\mathbb{S}}\to {\mathbb{C}}^{\left[0,1\right]}$ are complex fuzzy membership mappings such that ${\hat{M}}_{\hat{P}}\left(\hat{s}\right)={\alpha }_{\hat{P}}\left(\hat{s}\right){exp}^{j{\beta }_{\hat{P}}\left(\hat{s}\right)}$, ${\hat{M}}_{\widehat{Ne}}\left(\hat{s}\right)={\alpha }_{\widehat{Ne}}\left(\hat{s}\right){exp}^{j{\beta }_{\widehat{Ne}}\left(\hat{s}\right)}$ and ${\hat{M}}_{\hat{N}}\left(\hat{s}\right)={\alpha }_{\hat{N}}\left(\hat{s}\right){exp}^{j{\beta }_{\hat{N}}\left(\hat{s}\right)}$ provided that $0\le {\alpha }_{\hat{P}}\left(\hat{s}\right)+{\alpha }_{\widehat{Ne}}\left(\hat{s}\right)+{\alpha }_{\hat{N}}\left(\hat{s}\right)\le 1$ and $0\le {\beta }_{\hat{P}}\left(\hat{s}\right)+{\beta }_{\widehat{Ne}}\left(\hat{s}\right)+{\beta }_{\hat{N}}\left(\hat{s}\right)\le 2\pi$. The ${\alpha }_{\hat{P}}\left(\hat{s}\right),{\alpha }_{\widehat{Ne}}\left(\hat{s}\right)$, ${\alpha }_{\hat{N}}\left(\hat{s}\right)$ are called the amplitude terms and ${\beta }_{\hat{P}}\left(\hat{s}\right),{\beta }_{\widehat{Ne}}\left(\hat{s}\right)$, and ${\beta }_{\hat{N}}\left(\hat{s}\right)$ are known as phase terms. The refusal membership grade ${\hat{M}}_{\hat{R}}\left(\hat{s}\right)=\left[1-{\alpha }_{\hat{P}}\left(\hat{s}\right)-{\alpha }_{\widehat{Ne}}\left(\hat{s}\right)-{\alpha }_{\hat{N}}\left(\hat{s}\right)\right]ex{p}^{j\left[2\pi -{\beta }_{\hat{P}}\left(\hat{s}\right)-{\beta }_{\widehat{Ne}}\left(\hat{s}\right)-{\beta }_{\hat{N}}\left(\hat{s}\right)\right]}$ within ℂ^[0,1]. The family of all cpFSs over $\hat{\mathbb{S}}$ is symbolized as ${\Delta }_{cpFS}\left(\hat{\mathbb{S}}\right)$.

(Molodtsov, 1999) If $\hat{\Xi }$ is consisting of attributes then an SST $\hat{\Psi }$ is defined as $\hat{\Psi }=\left\{\left(\hat{a},\hat{\zeta }\left(\hat{a}\right)\right):\hat{a}\in \hat{\Xi }\right\}$ where $\hat{\zeta }:\hat{\Xi }\to 2^{\hat{\mathbb{S}}}$ is an approximate mapping with $\hat{\zeta }\left(\hat{a}\right)$ as $\hat{a}$-approximate element in $\hat{\mathbb{S}}$.

(Maji, Biswas & Roy, 2001a) If $\hat{\Xi }$ is consisting of attributes then FSS ${\hat{\Psi }}_F$ is defined as ${\hat{\Psi }}_F=\left\{\left(\hat{a},{\hat{\zeta }}_F\left(\hat{a}\right)\right):\hat{a}\in \hat{\Xi }\right\}$ where ${\hat{\zeta }}_F:\hat{\Xi }\to {\Sigma }_{FS}$ is an approximate mapping with ${\hat{\zeta }}_F\left(\hat{a}\right)$ as $\hat{a}$-approximate element in $\hat{\mathbb{S}}$.

(Maji, Biswas & Roy, 2001b) If $\hat{\Xi }$ is consisting of attributes then an IFSS ${\hat{\Psi }}_{IF}$ is defined as ${\hat{\Psi }}_{IF}=\left\{\left(\hat{a},{\hat{\zeta }}_{IF}\left(\hat{a}\right)\right):\hat{a}\in \hat{\Xi }\right\}$ where ${\hat{\zeta }}_{IF}:\hat{\Xi }\to {\Sigma }_{IF}$ is an approximate mapping with ${\hat{\zeta }}_{IF}\left(\hat{a}\right)$ as $\hat{a}$-approximate element in $\hat{\mathbb{S}}$.

If $\hat{\Lambda }$ is the subset of a set $\hat{\Xi }$ consisting of attributes and ${\hat{M}}_{\hat{P}}\left(\hat{a}\right)$, ${\hat{M}}_{\widehat{Ne}}\left(\hat{a}\right)$, ${\hat{M}}_{\hat{N}}\left(\hat{a}\right)$ are positive, neutral and negative membership grades of $\hat{a}\in \hat{\Lambda }$ then the cpFSS $\hat{\Pi }$ is defined as where ${\hat{M}}_{\hat{P}},{\hat{M}}_{\widehat{Ne}},{\hat{M}}_{\hat{N}}:\hat{\Lambda }\to {\Delta }_{cpFS}\left(\hat{\mathbb{S}}\right)$ are complex fuzzy approximate mappings such that ${\hat{M}}_{\hat{P}}\left(\hat{a}\right)={\alpha }_{\hat{P}}\left(\hat{a}\right){exp}^{j{\beta }_{\hat{P}}\left(\hat{a}\right)}$, ${\hat{M}}_{\widehat{Ne}}\left(\hat{a}\right)={\alpha }_{\widehat{Ne}}\left(\hat{a}\right){exp}^{j{\beta }_{\widehat{Ne}}\left(\hat{a}\right)}$ and ${\hat{M}}_{\hat{N}}\left(\hat{a}\right)={\alpha }_{\hat{N}}\left(\hat{a}\right){exp}^{j{\beta }_{\hat{N}}\left(\hat{a}\right)}$ provided that $0\le {\alpha }_{\hat{P}}\left(\hat{a}\right)+{\alpha }_{\widehat{Ne}}\left(\hat{a}\right)+{\alpha }_{\hat{N}}\left(\hat{a}\right)\le 1$ and $0\le {\beta }_{\hat{P}}\left(\hat{a}\right)+{\beta }_{\widehat{Ne}}\left(\hat{a}\right)+{\beta }_{\hat{N}}\left(\hat{a}\right)\le 2\pi$. The ${\alpha }_{\hat{P}}\left(\hat{a}\right),{\alpha }_{\widehat{Ne}}\left(\hat{a}\right)$, ${\alpha }_{\hat{N}}\left(\hat{a}\right)$ are called the amplitude terms and ${\beta }_{\hat{P}}\left(\hat{a}\right),{\beta }_{\widehat{Ne}}\left(\hat{a}\right)$, and ${\beta }_{\hat{N}}\left(\hat{a}\right)$ are known as phase terms. The refusal membership grade ${\displaystyle {\hat{M}}_{\hat{R}}\left(\hat{a}\right)=\left[1-{\alpha }_{\hat{P}}\left(\hat{a}\right)-{\alpha }_{\widehat{Ne}}\left(\hat{a}\right)-{\alpha }_{\hat{N}}\left(\hat{a}\right)\right]ex{p}^{j\left[2\pi -{\beta }_{\hat{P}}\left(\hat{a}\right)-{\beta }_{\widehat{Ne}}\left(\hat{a}\right)-{\beta }_{\hat{N}}\left(\hat{a}\right)\right]}}$

within ℂ^[0,1]. For convenience, $〈{\alpha }_{\hat{P}}\left(\hat{a}\right){exp}^{j{\beta }_{\hat{P}}\left(\hat{a}\right)},{\alpha }_{\widehat{Ne}}\left(\hat{a}\right){exp}^{j{\beta }_{\widehat{Ne}}\left(\hat{a}\right)},{\alpha }_{\hat{N}}\left(\hat{a}\right){exp}^{j{\beta }_{\hat{N}}\left(\hat{a}\right)}〉$ is called a complex picture fuzzy soft number (cpFSN). The family of all cpFSSs over $\hat{\mathbb{S}}$ is symbolized as ${\Delta }_{cpFSS}\left(\hat{\mathbb{S}}\right)$.

Let $\hat{\mathbb{S}}=\left\{{\hat{s}}_1,{\hat{s}}_2,{\hat{s}}_3,{\hat{s}}_4\right\}$ be an initial space of objects and $\hat{\Xi }=\left\{{\hat{a}}_1,{\hat{a}}_2,{\hat{a}}_3,{\hat{a}}_4,{\hat{a}}_5,{\hat{a}}_6\right\}$ be a set of attributes with $\hat{\Lambda }=\left\{{\hat{a}}_1,{\hat{a}}_2,{\hat{a}}_5,{\hat{a}}_6\right\}\subseteq \hat{\Xi }$ then approximate elements of cpFSS $\hat{\Pi }$ are computed as ${\displaystyle \hat{\delta }\left({\hat{a}}_1\right)=\left\{\begin{array}{c}\left({\hat{s}}_1,〈0.11ex{p}^{j2\pi \left(0.22\right)},0.25ex{p}^{j2\pi \left(0.31\right)},0.42ex{p}^{j2\pi \left(0.15\right)}〉\right),\hfill \\ \left({\hat{s}}_2,〈0.23ex{p}^{j2\pi \left(0.18\right)},0.32ex{p}^{j2\pi \left(0.21\right)},0.51ex{p}^{j2\pi \left(0.22\right)}〉\right),\hfill \\ \left({\hat{s}}_3,〈0.33ex{p}^{j2\pi \left(0.25\right)},0.35ex{p}^{j2\pi \left(0.30\right)},0.20ex{p}^{j2\pi \left(0.45\right)}〉\right),\hfill \\ \left({\hat{s}}_4,〈〉0.15ex{p}^{j2\pi \left(0.30\right)},0.21ex{p}^{j2\pi \left(0.45\right)},0.33ex{p}^{j2\pi \left(0.60\right)}〈〉\right)\hfill \end{array}\right\},}$ ${\displaystyle \hat{\delta }\left({\hat{a}}_2\right)=\left\{\begin{array}{c}\left({\hat{s}}_1,〈0.12ex{p}^{j2\pi \left(0.24\right)},0.26ex{p}^{j2\pi \left(0.30\right)},0.35ex{p}^{j2\pi \left(0.40\right)}〉\right),\hfill \\ \left({\hat{s}}_2,〈0.22ex{p}^{j2\pi \left(0.31\right)},0.25ex{p}^{j2\pi \left(0.34\right)},0.28ex{p}^{j2\pi \left(0.37\right)}〉\right),\hfill \\ \left({\hat{s}}_3,〈0.23ex{p}^{j2\pi \left(0.32\right)},0.26ex{p}^{j2\pi \left(0.35\right)},0.29ex{p}^{j2\pi \left(0.38\right)}〉\right),\hfill \\ \left({\hat{s}}_4,〈0.24ex{p}^{j2\pi \left(0.33\right)},0.27ex{p}^{j2\pi \left(0.36\right)},0.30ex{p}^{j2\pi \left(0.39\right)}〉\right)\hfill \end{array}\right\},}$ ${\displaystyle \hat{\delta }\left({\hat{a}}_5\right)=\left\{\begin{array}{c}\left({\hat{s}}_1,〈0.11ex{p}^{j2\pi \left(0.41\right)},0.12ex{p}^{j2\pi \left(0.31\right)},0.40ex{p}^{j2\pi \left(0.51\right)}〉\right),\hfill \\ \left({\hat{s}}_2,〈0.21ex{p}^{j2\pi \left(0.30\right)},0.24ex{p}^{j2\pi \left(0.33\right)},0.27ex{p}^{j2\pi \left(0.36\right)}〉\right),\hfill \\ \left({\hat{s}}_3,〈0.22ex{p}^{j2\pi \left(0.31\right)},0.25ex{p}^{j2\pi \left(0.34\right)},0.28ex{p}^{j2\pi \left(0.37\right)}〉\right),\hfill \\ \left({\hat{s}}_4,〈0.23ex{p}^{j2\pi \left(0.32\right)},0.26ex{p}^{j2\pi \left(0.35\right)},0.29ex{p}^{j2\pi \left(0.38\right)}〉\right)\hfill \end{array}\right\},}$ ${\displaystyle \hat{\delta }\left({\hat{a}}_6\right)=\left\{\begin{array}{c}\left({\hat{s}}_1,〈0.13ex{p}^{j2\pi \left(0.44\right)},0.14ex{p}^{j2\pi \left(0.55\right)},0.44ex{p}^{j2\pi \left(0.66\right)}〉\right),\hfill \\ \left({\hat{s}}_2,〈0.25ex{p}^{j2\pi \left(0.18\right)},0.28ex{p}^{j2\pi \left(0.21\right)},0.31ex{p}^{j2\pi \left(0.24\right)}〉\right),\hfill \\ \left({\hat{s}}_3,〈0.26ex{p}^{j2\pi \left(0.19\right)},0.29ex{p}^{j2\pi \left(0.22\right)},0.32ex{p}^{j2\pi \left(0.25\right)}〉\right),\hfill \\ \left({\hat{s}}_4,〈0.27ex{p}^{j2\pi \left(0.20\right)},0.30ex{p}^{j2\pi \left(0.23\right)},0.33ex{p}^{j2\pi \left(0.26\right)}〉\right)\hfill \end{array}\right\}.}$ The cpFSS $\hat{\Pi }$ is constructed as $\hat{\Pi }=\left\{\left({\hat{a}}_1,\hat{\delta }\left({\hat{a}}_1\right)\right),\left({\hat{a}}_2,\hat{\delta }\left({\hat{a}}_2\right)\right),\left({\hat{a}}_5,\hat{\delta }\left({\hat{a}}_5\right)\right),\left({\hat{a}}_6,\hat{\delta }\left({\hat{a}}_6\right)\right)\right\}$. It can be represented in matrix notation as $\hat{\Pi }=$ ${\displaystyle \left(\begin{array}{cccc}\hfill 〈0.11^{0.22},0.25^{0.31},0.42^{0.15}〉\hfill & \hfill 〈0.23^{0.18},0.32^{0.21},0.51^{0.22}〉\hfill & \hfill 〈0.33^{0.25},0.35^{0.30},0.20^{0.45}〉\hfill & \hfill 〈0.15^{0.30},0.21^{0.45},0.33^{0.60}〉\hfill \\ \hfill 〈0.12^{0.24},0.26^{0.30},0.35^{0.40}〉\hfill & \hfill 〈0.22^{0.31},0.25^{0.34},0.28^{0.37}〉\hfill & \hfill 〈0.23^{0.32},0.26^{0.35},0.29^{0.38}〉\hfill & \hfill 〈0.24^{0.33},0.27^{0.36},0.30^{0.39}〉\hfill \\ \hfill 〈0.11^{0.41},0.12^{0.31},0.40^{0.51}〉\hfill & \hfill 〈0.21^{0.30},0.24^{0.33},0.27^{0.36}〉\hfill & \hfill 〈0.22^{0.31},0.25^{0.34},0.28^{0.37}〉\hfill & \hfill 〈0.23^{0.32},0.26^{0.35},0.29^{0.38}〉\hfill \\ \hfill 〈0.13^{0.44},0.14^{0.55},0.44^{0.66}〉\hfill & \hfill 〈0.25^{0.18},0.28^{0.21},0.31^{0.24}〉\hfill & \hfill 〈0.26^{0.19},0.29^{0.22},0.32^{0.25}〉\hfill & \hfill 〈0.27^{0.20},0.30^{0.23},0.33^{0.26}〉\hfill \end{array}\right).}$

For ${\hat{\Pi }}_1$, ${\hat{\Pi }}_2$ ∈ ${\Delta }_{cpFSS}\left(\hat{\mathbb{S}}\right)$ then following set theoretic operations are valid:

1. ${\hat{\Pi }}_1\subseteq {\hat{\Pi }}_2$ if ${\alpha }_{\hat{P}}^1\left(\hat{a}\right)\le {\alpha }_{\hat{P}}^2\left(\hat{a}\right)$, ${\alpha }_{\widehat{Ne}}^1\left(\hat{a}\right)\ge {\alpha }_{\widehat{Ne}}^2\left(\hat{a}\right)$, ${\alpha }_{\hat{N}}^1\left(\hat{a}\right)\ge {\alpha }_{\hat{N}}^2\left(\hat{a}\right)$ and ${\beta }_{\hat{P}}^1\left(\hat{a}\right)\le {\beta }_{\hat{P}}^2\left(\hat{a}\right)$, ${\beta }_{\widehat{Ne}}^1\left(\hat{a}\right)\ge {\beta }_{\widehat{Ne}}^2\left(\hat{a}\right)$, ${\beta }_{\hat{N}}^1\left(\hat{a}\right)\ge {\beta }_{\hat{N}}^2\left(\hat{a}\right)$.

2. The ${\hat{\Pi }}_1\cap {\hat{\Pi }}_2$ is another cpFSS ${\hat{\Pi }}_3$ with ${\alpha }_{\hat{P}}^3\left(\hat{a}\right)=min\left\{{\alpha }_{\hat{P}}^1\left(\hat{a}\right),{\alpha }_{\hat{P}}^2\left(\hat{a}\right)\right\}$, ${\alpha }_{\widehat{Ne}}^3\left(\hat{a}\right)=max\left\{{\alpha }_{\widehat{Ne}}^1\left(\hat{a}\right),{\alpha }_{\widehat{Ne}}^2\left(\hat{a}\right)\right\}$, ${\alpha }_{\hat{N}}^3\left(\hat{a}\right)=max\left\{{\alpha }_{\hat{N}}^1\left(\hat{a}\right),{\alpha }_{\hat{N}}^2\left(\hat{a}\right)\right\}$ and ${\beta }_{\hat{P}}^3\left(\hat{a}\right)=min\left\{{\beta }_{\hat{P}}^1\left(\hat{a}\right),{\beta }_{\hat{P}}^2\left(\hat{a}\right)\right\}$, ${\beta }_{\widehat{Ne}}^3\left(\hat{a}\right)=max\left\{{\beta }_{\widehat{Ne}}^1\left(\hat{a}\right),{\beta }_{\widehat{Ne}}^2\left(\hat{a}\right)\right\}$, ${\beta }_{\hat{N}}^3\left(\hat{a}\right)=max\left\{{\beta }_{\hat{N}}^1\left(\hat{a}\right),{\beta }_{\hat{N}}^2\left(\hat{a}\right)\right\}$.

3. The ${\hat{\Pi }}_1\cup {\hat{\Pi }}_2$ is another cpFSS ${\hat{\Pi }}_4$ with ${\alpha }_{\hat{P}}^4\left(\hat{a}\right)=max\left\{{\alpha }_{\hat{P}}^1\left(\hat{a}\right),{\alpha }_{\hat{P}}^2\left(\hat{a}\right)\right\}$, ${\alpha }_{\widehat{Ne}}^4\left(\hat{a}\right)=min\left\{{\alpha }_{\widehat{Ne}}^1\left(\hat{a}\right),{\alpha }_{\widehat{Ne}}^2\left(\hat{a}\right)\right\}$, ${\alpha }_{\hat{N}}^4\left(\hat{a}\right)=min\left\{{\alpha }_{\hat{N}}^1\left(\hat{a}\right),{\alpha }_{\hat{N}}^2\left(\hat{a}\right)\right\}$ and ${\beta }_{\hat{P}}^4\left(\hat{a}\right)=max\left\{{\beta }_{\hat{P}}^1\left(\hat{a}\right),{\beta }_{\hat{P}}^2\left(\hat{a}\right)\right\}$, ${\beta }_{\widehat{Ne}}^4\left(\hat{a}\right)=min\left\{{\beta }_{\widehat{Ne}}^1\left(\hat{a}\right),{\beta }_{\widehat{Ne}}^2\left(\hat{a}\right)\right\}$, ${\beta }_{\hat{N}}^4\left(\hat{a}\right)=min\left\{{\beta }_{\hat{N}}^1\left(\hat{a}\right),{\beta }_{\hat{N}}^2\left(\hat{a}\right)\right\}$.

4. The ${\hat{\Pi }}_1^c$ is again cpFSS with ${\alpha }_{\hat{P}}^c\left(\hat{a}\right)={\alpha }_{\hat{N}}^1\left(\hat{a}\right)$, ${\alpha }_{\widehat{Ne}}^c\left(\hat{a}\right)={\alpha }_{\widehat{Ne}}^1\left(\hat{a}\right)$, ${\alpha }_{\hat{N}}^c\left(\hat{a}\right)={\alpha }_{\hat{P}}^1\left(\hat{a}\right)$ and ${\beta }_{\hat{P}}^c\left(\hat{a}\right)={\beta }_{\hat{N}}^1\left(\hat{a}\right)$, ${\beta }_{\widehat{Ne}}^c\left(\hat{a}\right)={\beta }_{\widehat{Ne}}^1\left(\hat{a}\right)$, ${\beta }_{\hat{N}}^c\left(\hat{a}\right)={\beta }_{\hat{P}}^1\left(\hat{a}\right)$.

Reassuming the data from Example 3.2, the following two cpFSSs are constructed with matrix representations as given below