Fuzzy SAW (Bonissone 1982) - L-R trapezoidal Simple Additive Weighting

Overview

Fuzzy SAW (Bonissone 1982 L-R variant) aggregates weighted L-R 4-tuple contributions per alternative, then defuzzifies via trapezoidal centroid. Output is the descending ranking of centroids.

Output

utility, higher is better

Data

Fuzzy LR Tr FN, L-R 4-tuples per cell + per criterion weight

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy MCDM with linguistic ratings, Industrial system selection under qualitative criteria, Engineering design evaluation

How it works

1. 1

   Construct decision matrix X_tilde = [x_tilde_ij] and weight vector w_tilde, all L-R 4-tuples.

   $${\tilde{x}}_{ij}=(a_{ij},b_{ij},{\alpha }_{ij},{\beta }_{ij});{\tilde{w}}_j=(a_j,b_j,{\alpha }_j,{\beta }_j)$$

   Kahraman 2008 Ch.7 Eq.(7)-(8), p.192; Bonissone 1982 p.331

2. 2

   Compute weighted contribution w_tilde_j (x) x_tilde_ij using Bonissone L-R product, Eq.(29).

   $$\tilde{M}\otimes \tilde{N}=(ac,bd,a\gamma +c\alpha -\alpha \gamma ,b\delta +d\beta +\beta \delta )$$

   Kahraman 2008 Ch.7 Eq.(29), p.200; Bonissone 1982

3. 3

   Aggregate weighted contributions per alternative via Bonissone L-R sum, Eq.(27).

   $${\tilde{U}}_i=\underset{j=1}{\overset{n}{⨁}}({\tilde{w}}_j\otimes {\tilde{x}}_{ij});(a+c,b+d,\alpha +\gamma ,\beta +\delta )$$

   Kahraman 2008 Ch.7 Eq.(27), (31), p.200; Bonissone 1982

4. 4

   Defuzzify each U_tilde_i via L-R trapezoidal centroid (left triangle + plateau + right triangle).

   $$C_i=\frac{({\alpha }_i/2)(a_i-{\alpha }_i/3)+(b_i-a_i)(a_i+b_i)/2+({\beta }_i/2)(b_i+{\beta }_i/3)}{(b_i-a_i)+({\alpha }_i+{\beta }_i)/2}$$

   Bonissone 1982 (canonical L-R centroid); standard trapezoidal area centroid

5. 5

   Rank alternatives by defuzzified centroid C_i in descending order.

   $$\text{rank}(A_i)={\text{argsort}}_{\downarrow }(C_i)$$

   Kahraman 2008 Ch.7 p.201 (U_2 >- U_1 >- U_3)

Fits when / Look elsewhere when

Assumptions to verify

• Decision matrix entries are valid L-R 4-tuples (a <= b, alpha >= 0, beta >= 0)
• Compensation assumption holds: a strong score on one criterion can offset a weak score on another

Edge cases and pitfalls

• Bonissone product Eq.(29) is non-commutative in alpha/gamma; preserve operand order (weight (x) rating).

• Book p.201 contains minor typos in U_1.beta and U_3.alpha; canonical algorithm gives 0.60 and 0.23 respectively.

Works with

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