SF-CODAS - Spherical extension of CODAS

1. 1. Structure the decision problem: alternatives, criteria, and K decision makers with linguistic SF importance.

   $$(alternatives,criteria,DMs)$$

2. 2. Each DM provides SF criterion weights and an SF decision matrix using linguistic terms. Compute DM weights ŵ_s.

   $${\hat{w}}_{s}fromlinguisticDMimportance$$

3. 3. Aggregate SF criterion weights via SWAM. Defuzzify with Ch.12 score function and normalise to obtain crisp w̄_sj with Σ w̄_sj = 1.

   $$Score(\alpha )=(\mu -\pi /2{)}^2-(\nu -\pi /2{)}^2[Ch.12score]{\bar{w}}_{s}j=Score({\tilde{w}}_j);{\bar{w}}_{s}j\leftarrow {\bar{w}}_{s}j/\Sigma {\bar{w}}_{s}j$$

4. 4. Aggregate the K SF decision matrices into a collective SF decision matrix via SWAM using DM weights ŵ_s.

   $${\tilde{x}}_{i}j=SWA{M}_{\hat{w}}({\tilde{x}}^{(}1{)}_{i}j,\dots ,{\tilde{x}}^{(}K{)}_{i}j)$$

5. 5. Construct the weighted SF decision matrix by multiplying each x̃_ij with w̄_sj using SF scalar multiplication.

   $${\tilde{x}}_i{j}^w={\bar{w}}_{s}j\cdot {\tilde{x}}_{i}j$$

6. 6. Determine the SF Negative-Ideal Solution (NIS) per criterion. CODAS uses ONLY the NIS (no PIS). For benefit criteria: NIS = min by score; for cost criteria: NIS = max by score.

   $$NI{S}_j=argmi{n}_{i}Score({\tilde{x}}_i{j}^w)forbenefitcriteriaNI{S}_j=argma{x}_{i}Score({\tilde{x}}_i{j}^w)forcostcriteria$$

7. 7. Compute the SF normalized Euclidean distance D_E,i and SF normalized Spherical distance D_S,i of each alternative from the NIS.

   $$D_E,i=sqrt(1/(2n)\cdot {\Sigma }_j(({\mu }_{i}j-{\mu }_{N}IS,j{)}^2+({\nu }_{i}j-{\nu }_{N}IS,j{)}^2+({\pi }_{i}j-{\pi }_{N}IS,j{)}^2))D_S,i=(2/(n\pi ))\cdot {\Sigma }_{j}arccos({\mu }_{i}j\cdot {\mu }_{N}IS,j+{\nu }_{i}j\cdot {\nu }_{N}IS,j+{\pi }_{i}j\cdot {\pi }_{N}IS,j)$$

8. 8. Construct the relative assessment matrix RA with entries s_ik = (D_E,i − D_E,k) + Φ(D_E,i − D_E,k) · (D_S,i − D_S,k), where Φ is a threshold function with τ = 0.02 default: Φ(x) = 1 if |x| ≥ τ, else Φ(x) = 0.

   $$s_{i}k=(D_E,i-D_E,k)+\Phi (D_E,i-D_E,k)\cdot (D_S,i-D_S,k)\Phi (x)=1if|x|\ge \tau ,0otherwise;\tau =0.02(default)$$

9. 9. Compute assessment score AS_i = Σ_k s_ik for each alternative.

   $$A{S}_i={\Sigma }_{k=\mathrm{1..}m}{s}_{i}k$$

10. 10. Rank alternatives in DESCENDING order of AS_i. The highest-AS_i alternative is the best ranked.

    $$rank:descendingA{S}_i$$

11. 11. Sensitivity analysis: vary τ ∈ {0.01, 0.02, 0.05} and verify stability of the top alternative.

    $$vary\tau ;checkrankstability$$