SF-TODIM - Spherical extension of TODIM

1. 1. normalization

   $${\tilde{a}}_{ij}^c=({\nu }_{ij},{\mu }_{ij},{\pi }_{ij})\text{for cost criteria}$$

2. 2. ranking_computation

   $${\bar{w}}_{jr}=\frac{w_j}{w_r},w_r=\underset{j}{\max }{w}_j$$

3. 3. ranking_computation

   $${\varphi }_j(A_i,A_k)=\{\begin{array}{ll}{\bar{w}}_{jr}\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\bar{w}}_{jr}}{{\bar{w}}_{jr}}}& \text{gain}\\ -\frac{1}{\theta }\sqrt{\frac{d({\tilde{a}}_{ij},{\tilde{a}}_{kj}){\sum }_j{\bar{w}}_{jr}}{{\bar{w}}_{jr}}}& \text{loss}\\ 0& \text{tie}\end{array}$$

4. 4. ranking_computation

   $$\delta (A_i,A_k)=\sum _{j=1}^n{\varphi }_j(A_i,A_k)$$

5. 5. ranking_computation

   $${\xi }_i=\frac{{\sum }_k\delta (A_i,A_k)-{\min }_i{\sum }_k\delta (A_i,A_k)}{{\max }_i{\sum }_k\delta (A_i,A_k)-{\min }_i{\sum }_k\delta (A_i,A_k)},\text{rank descending}$$