FCILOS (Fuzzy Criterion Impact LOSs)

1. 1. Validate the ordered positive TFN matrix and map every minimising criterion to a maximising one by fuzzy division.

   $${\bar{r}}_{ij}=\frac{{\min }_i{\tilde{r}}_{ij}}{{\tilde{r}}_{ij}}=\left(\frac{{\min }_i{r}_{ij}^L}{r_{ij}^U},\frac{{\min }_i{r}_{ij}^M}{r_{ij}^M},\frac{{\min }_i{r}_{ij}^U}{r_{ij}^L}\right)$$

2. 2. Normalise every criterion column by the fuzzy column sum (cross-bound division).

   $${\tilde{s}}_{ij}=\frac{{\tilde{r}}_{ij}}{{\sum }_{i=1}^n{\tilde{r}}_{ij}}=\left(\frac{r_{ij}^L}{{\sum }_i{r}_{ij}^U},\frac{r_{ij}^M}{{\sum }_i{r}_{ij}^M},\frac{r_{ij}^U}{{\sum }_i{r}_{ij}^L}\right)$$

3. 3. Find the column maximum at each of the L, M and U points and build the three square matrices of the rows that attain them.

   $${\tilde{x}}_j=\underset{i}{\max }{\tilde{x}}_{ij},{\tilde{a}}_{ij}={\tilde{x}}_{k_{j}j},{\tilde{a}}_{jj}={\tilde{x}}_j$$

4. 4. Form the fuzzy relative-loss matrices, build the three homogeneous systems and solve each for its one-dimensional null space.

   $${\tilde{p}}_{ij}=\frac{{\tilde{x}}_j-{\tilde{a}}_{ij}}{{\tilde{x}}_j},f_{jj}=-\sum _{i=1}^m{\tilde{p}}_{ij},F^L{q}^T=0,F^M{q}^T=0,F^U{q}^T=0$$

5. 5. Average the three point solutions into the crisp criterion weight simplex reported as the Average row of Table 5.

   $$w_j=\frac{(q_j^L+q_j^M+q_j^U)/3}{{\sum }_k(q_k^L+q_k^M+q_k^U)/3}$$