qR-TOPSIS - q-Rung Orthopair extension of TOPSIS

1. 1. Validate native ratings and complement cost criteria.

   $${\mu }_{ij},{\nu }_{ij}\in [0,1],{\mu }_{ij}^q+{\nu }_{ij}^q\le 1,1\le q<\infty ;R_{ij}=\{\begin{array}{ll}({\nu }_{ij},{\mu }_{ij}),& j\text{ cost}\\ ({\mu }_{ij},{\nu }_{ij}),& j\text{ benefit}\end{array}$$

2. 2. Apply scalar criterion weights.

   $$0\le w_j\le 1,\sum _j{w}_j=1;V_{ij}=\left((1-(1-{\mu }_{ij}^{Rq}{)}^{w_j}{)}^{1/q},({\nu }_{ij}^R{)}^{w_j}\right),V_{ij}=(0,1)\text{ if }{w}_j=0$$

3. 3. Construct positive and negative ideals.

   $$A_j^{+}=(\underset{i}{\max }{\mu }_{ij}^v,\underset{i}{\min }{\nu }_{ij}^v),A_j^{-}=(\underset{i}{\min }{\mu }_{ij}^v,\underset{i}{\max }{\nu }_{ij}^v)\text{after cost-swap (benefit)}$$

4. 4. Calculate weighted q-ROF Euclidean distances.

   $${\pi }_{ij}^{Vq}=1-({\mu }_{ij}^V{)}^q-({\nu }_{ij}^V{)}^q;d_i^{\pm }=\sqrt{\sum _j\frac{w_j}{2}\left[{\left(({\mu }_{ij}^V{)}^q-({\mu }_j^{\pm }{)}^q\right)}^2+{\left(({\nu }_{ij}^V{)}^q-({\nu }_j^{\pm }{)}^q\right)}^2+{\left(({\pi }_{ij}^V{)}^q-({\pi }_j^{\pm }{)}^q\right)}^2\right]}$$

5. 5. Calculate relative closeness and equal-score ranks.

   $$d_i^{+}+d_i^{-}>0;C{C}_i=\frac{d_i^{-}}{d_i^{+}+d_i^{-}},r_i=1+\#\{k:C{C}_k>C{C}_i\}$$