FUZZY-MEREC (fuzzy MEthod based on the Removal Effects of Criteria)

1. 1. Read the aggregated TFN decision matrix and its criterion directions.

   $${\tilde{\xi }}_{ij}=(f_{ij},g_{ij},h_{ij}),i=1(1)m;j=1(1)n$$

2. 2. Normalise benefit criteria by the largest upper endpoint and cost criteria by fuzzy division of the smallest lower endpoint.

   $${\tilde{r}}_{ij}=\left(\frac{f_{ij}}{h_j^{\max }},\frac{g_{ij}}{h_j^{\max }},\frac{h_{ij}}{h_j^{\max }}\right)\text{(benefit)};{\tilde{r}}_{ij}=\left(\frac{f_j^{\min }}{h_{ij}},\frac{f_j^{\min }}{g_{ij}},\frac{f_j^{\min }}{f_{ij}}\right)\text{(cost)}$$

3. 3. Defuzzify the normalised TFNs with the graded mean integration representation.

   $$Crisp(\tilde{a})=\frac{a_1+4a_2+a_3}{6}$$

4. 4. Compute the overall performance of every alternative with the enhanced logarithm function.

   $$S_i=\frac{1}{n}\sum _{j=1}^n\ln (1-r_{ij})$$

5. 5. Recompute the performance of every alternative with each criterion removed in turn.

   $$S_{ij}^{\prime }=\frac{1}{n}\sum _{k=1,k\ne j}^n\ln (1-r_{ik})$$

6. 6. Sum the absolute deviations caused by removing each criterion.

   $$d_j=\sum _{i=1}^m|S_{ij}^{\prime }-S_i|$$

7. 7. Normalise the removal effects to the criterion weight simplex.

   $$w_j=\frac{d_j}{{\sum }_{k=1}^n{d}_k}$$