## 3 Methodology
### Preliminaries  
#### 3.1.1 Spherical Fuzzy Sets  
Spherical fuzzy sets (SFS), an extension of fuzzy sets, have recently been introduced by [PERSON] et al. [23]. The decision-uncertain maker's judgment is stated in each level of membership, non-membership, and hesitance, as defined below. **Definition 1**. : _The SFS \(\widetilde{A}\) of the universe of \(L\) is described in Equation (1) [23]:_  
\[\widetilde{A}=\left\{1,\theta_{\widetilde{A}}(1),\mu_{\widetilde{A}}(1),\pi_{ \widetilde{A}}(1)|1\in L\right\} \tag{1}\]  
_where \(\theta_{\widetilde{A}},\mu_{\widetilde{A}},\pi_{\widetilde{A}}(1):L\to[0,1]\) and \(0\leq\theta_{\widetilde{A}}^{2}(l)+\mu_{\widetilde{A}}^{2}(l)+\pi_{ \widetilde{A}}^{2}(l)\leq 1,\forall l\in L\)_  
_The numbers \(\theta_{\widetilde{A}}(l)\), \(\mu_{\widetilde{A}}(l)\), \(\pi_{\widetilde{A}}(l)\) are the levels of membership, non-membership, and hesitance of \(1\) to \(\widetilde{A}\)._  
**Definition 2**. : _The SFS of two values \(\widetilde{A}=\left(\theta_{\widetilde{A}},\mu_{\widetilde{A}},\pi_{ \widetilde{A}}\right)\) and \(\widetilde{B}=\left(\theta_{\widetilde{B}},\mu_{\widetilde{B}},\pi_{ \widetilde{B}}\right)\) of the universe of \(L_{1}\) and \(L_{2}\) are illustrated based on some calculations demonstrated by the following Equations (2)-(5) [23]:_  
_Addition_  
\[\widetilde{A}\oplus\widetilde{B}=\left(\sqrt{\theta_{\widetilde{A}}^{2}+ \theta_{\widetilde{B}}^{2}-\theta_{\widetilde{A}}^{2}\theta_{\widetilde{B}}^{ 2}},\mu_{\widetilde{A}}\mu_{\widetilde{B}},\sqrt{\left(1-\theta_{\widetilde{ B}}^{2}\right)\pi_{\widetilde{A}}^{2}+\left(1-\theta_{\widetilde{A}}^{2} \right)\pi_{\widetilde{B}}^{2}-\pi_{\widetilde{A}}^{2}\pi_{\widetilde{B}}^{2}}\right) \tag{2}\]  
Figure 1: The proposed approach. Multiplication_  
\[\widetilde{A}\otimes\widetilde{B}=\left(\vartheta_{\widetilde{A}}\vartheta_{ \widetilde{B}^{\prime}}\sqrt{\mu_{\widetilde{A}}^{2}+\mu_{\widetilde{B}}^{2}- \mu_{\widetilde{A}}^{2}\mu_{\widetilde{B}^{\prime}}^{2}}\sqrt{\left(1-\mu_{ \widetilde{B}}^{2}\right)\pi_{\widetilde{A}}^{2}+\left(1-\mu_{\widetilde{A}}^ {2}\right)\pi_{\widetilde{B}}^{2}-\pi_{\widetilde{A}}^{2}\pi_{\widetilde{B}}^{2 }}\right) \tag{3}\]  
Multiplication by a scalar \(\left(\lambda>0\right)\)  
\[\lambda\widetilde{A}=\left(\sqrt{1-\left(1-\vartheta_{\widetilde{A}}^{2} \right)^{\lambda}},\mu_{\widetilde{A}^{\prime}}^{\lambda}\sqrt{\left(1- \vartheta_{\widetilde{A}}^{2}\right)^{\lambda}-\left(1-\vartheta_{\widetilde {A}}^{2}-\pi_{\widetilde{A}}^{2}\right)^{\lambda}}\right) \tag{4}\]  
Power of \(\widetilde{A}\) (\(\lambda>0\))  
\[\widetilde{A}^{\lambda}=\left(\vartheta_{\widetilde{A}^{\prime}}^{\lambda} \sqrt{1-\left(1-\mu_{\widetilde{A}}^{2}\right)^{\lambda}},\sqrt{\left(1-\mu_ {\widetilde{A}}^{2}\right)^{\lambda}-\left(1-\mu_{\widetilde{A}}^{2}-\pi_{ \widetilde{A}}^{2}\right)^{\lambda}}\right) \tag{5}\]  
**Definition 3**. : _Spherical weighted geometric mean (SWG) and spherical weighted arithmetic mean (SWA) are described through the weight vector \(\omega=\left(\omega_{1},\omega_{2},\ldots,\omega_{n}\right)\), where \(0\leq\omega_{i}\leq 1\) and \(\sum_{i=1}^{n}\omega_{i}=1\) by the following Equations (6) and (7) [23]:_  
\[