N-PSI - Neutrosophic PSI

1. 1. Build the SVN decision matrix; complement cost cells <F, 1−I, T>; shifted score s'_ij = (3 + T − 2I − F)/4 in [0, 1].

   $$s_{ij}^{\prime }=\frac{3+T_{ij}-2I_{ij}-F_{ij}}{4}$$

2. 2. Sum normalisation N_ij = s'_ij / Σ_i s'_ij for every column (cost columns were complemented in Step 1). Column sums equal 1, so PV_j < 1 and the PSI deviation stays positive.

   $$N_{ij}=\frac{s_{ij}^{\prime }}{{\sum }_{k=1}^m{s}_{kj}^{\prime }}$$

3. 3. Mean preference value per criterion.

   $${\bar{N}}_j=\frac{1}{m}\sum _{i=1}^m{N}_{ij}$$

4. 4. Preference variation value PV_j: sum of squared deviations from the mean.

   $$P{V}_j=\sum _{i=1}^m(N_{ij}-{\bar{N}}_j{)}^2$$

5. 5. Preference deviation φ_j = 1 − PV_j and PSI weight DPV_j = φ_j / Σ_k φ_k (Maniya & Bhatt 2010, Eqs. 4-5): less spread means more weight, as in crisp PSI.

   $${\phi }_j=1-P{V}_j,DP{V}_j=\frac{{\phi }_j}{{\sum }_{k=1}^n{\phi }_k}$$

6. 6. Preference selection index Ψ_i = Σ_j N_ij · DPV_j.

   $${\Psi }_i=\sum _{j=1}^n{N}_{ij}\cdot DP{V}_j$$

7. 7. Rank alternatives in descending order of Ψ_i.

   $$\text{rank descending by }{\Psi }_i$$

8. 8. PSI is weight-free: DPV_j replaces external weights; supplied weights are echoed with a warning and not used.

   $$\text{weight-free; }DP{V}_j\text{ replaces }{w}_j$$