SF-MARCOS - Spherical extension of MARCOS

1. 1. Construct SF decision matrix K_ij from DM linguistic evaluations of m alternatives × n criteria.

   $$K_{i}j=(({\mu }_{i}j,{\nu }_{i}j,{\pi }_{i}j){)}_{m\times n}(Eq25)$$

2. 2. Convert linguistic variables to spherical fuzzy numbers using Table 1 9-point scale (ALI..AMI).

   $$LV\mapsto SFN(Table1)$$

3. 3. Determine ideal A_id (max-μ for benefit / min-ν for cost) and anti-ideal A_ai over alternatives.

   $$A_{i}d=(ma{x}_i{x}_{i}j:j\in B_{m}ax,mi{n}_i{x}_{i}j:j\in C_{m}in)(Eq27);A_{a}i=(mi{n}_i{x}_{i}j:j\in B_{m}ax,ma{x}_i{x}_{i}j:j\in C_{m}in)(Eq26)$$

4. 4. Form weighted SF decision matrix v_ij = w_j ⊗ x_ij using SF multiplication.

   $$v_{i}j=(\surd ({\mu }_w^2\cdot {\mu }_x^2+{\nu }_w^2\cdot {\nu }_x^2-{\nu }_w^2\cdot {\nu }_x^2)form...)(Eqs28-29)$$

5. 5. Compute utility degrees K^+_i (to ideal) and K^-_i (to anti-ideal) via arccos spherical distance.

   $$K_i^{+}=(2/(n\cdot \pi ))\cdot {\Sigma }_{j}arccos({\mu }_v\cdot {\mu }_{i}d+{\nu }_v\cdot {\nu }_{i}d+{\pi }_v\cdot {\pi }_{i}d)(Eq30);K_i^{-}=(2/(n\cdot \pi ))\cdot {\Sigma }_{j}arccos({\mu }_v\cdot {\mu }_{a}i+{\nu }_v\cdot {\nu }_{a}i+{\pi }_v\cdot {\pi }_{a}i)(Eq31)$$

6. 6. Compute utility functions f(K^+_i), f(K^-_i) and final score F_i.

   $$f(K_i^{+})=K_i^{-}/(K_i^{+}+K_i^{-})(Eq33);f(K_i^{-})=K_i^{+}/(K_i^{+}+K_i^{-})(Eq34);F_i=(K_i^{+}+K_i^{-})/(1+(1-f(K^{+}))/f(K^{+})+(1-f(K^{-}))/f(K^{-}))(Eq32)$$

7. 7. Rank alternatives by F_i in descending order; highest F_i is most preferred.

   $$rank:argsor{t}_{d}esc(F_i)$$