FSWARA Method
Fuzzy Step-wise Weight Assessment Ratio Analysist
Video Demonstration
Step-by-Step Algorithm
Determine the fuzzy comparative importance
Arrange the criteria in descending order according to their expected importance. The most important criterion is placed first.
\(C_1\) is the most important criterion and \(C_n\) is the least important criterion.
Starting from the second criterion, experts evaluate the relative importance of criterion \(C_j\) compared with the immediately preceding criterion \(C_{j-1}\). Linguistic judgments are represented by triangular fuzzy numbers.
Aggregated average values
When several decision-makers participate, their fuzzy judgments are aggregated using the minimum, arithmetic mean, and maximum values:
\(K\) is the number of decision-makers and \(S_j=(S_{jl},S_{jm},S_{ju})\) is the aggregated fuzzy comparative importance of criterion \(C_j\).
| Criterion | Aggregated Imp. ($S_j$) |
|---|---|
| $C_1$ | - |
| $C_2$ | (0.2, 0.3, 0.4) |
| $C_3$ | (0.4, 0.5, 0.6) |
Calculate the fuzzy coefficient
Calculate the coefficient for each criterion. The first criterion is used as the reference and therefore has a coefficient equal to one.
For a triangular fuzzy number, \(S_j=(S_{jl},S_{jm},S_{ju})\) :
| Criterion | Fuzzy Coeff. ($k_j$) |
|---|---|
| $C_1$ | (1,1,1) |
| $C_2$ | (1.2, 1.3, 1.4) |
| $C_3$ | (1.4, 1.5, 1.6) |
Calculate the fuzzy recalculated weight
Calculate the recalculated fuzzy weight sequentially. The first criterion receives a fuzzy weight of one. Each subsequent weight is obtained by dividing the previous fuzzy weight by the current fuzzy coefficient.
The fuzzy weights decrease progressively according to the comparative importance of the criteria.
| Criterion | Fuzzy Recalc. Weight ($q_j$) |
|---|---|
| $C_1$ | (1, 1, 1) |
| $C_2$ | (0.714, 0.769, 0.833) |
| $C_3$ | (0.446, 0.513, 0.595) |
Calculate the fuzzy relative weight
Normalize the recalculated fuzzy weights by dividing each fuzzy recalculated weight by the sum of all recalculated fuzzy weights.
The result is the fuzzy relative weight of criterion \(C_j\) :
| Criterion | Fuzzy Relative Weight ($w_j$) |
|---|---|
| $C_1$ | (0.41, 0.44, 0.46) |
| $C_2$ | (0.29, 0.34, 0.39) |
| $C_3$ | (0.18, 0.22, 0.28) |
Defuzzify the fuzzy weights
Convert each triangular fuzzy relative weight into a single crisp value. A commonly used centroid method is applied.
\(w_{j}'\) is the non-fuzzy (crisp) weight of criterion \(C_j\).
| Criterion | Defuzzified Weight ($w'_j$) |
|---|---|
| $C_1$ | 0.437 |
| $C_2$ | 0.340 |
| $C_3$ | 0.227 |
Normalize the crisp weights
Normalize the defuzzified weights so that the sum of all final criterion weights equals one.
\(W_j\) represents the final normalized weight of criterion \(C_j\).
| Criterion | Final Normalized Weight ($W_j$) |
|---|---|
| $C_1$ | 0.435 |
| $C_2$ | 0.339 |
| $C_3$ | 0.226 |
| Sum | 1.000 |
FSWARA Algorithm Flowchart
Calculate the fuzzy coefficient
Calculate the fuzzy recalculated weight