Fuzzy WPM (Kahraman-Birgun-Yenen 2008) - Triangular Fuzzy Weighted Product

Overview

Fuzzy WPM applies the crisp WP formula componentwise on TFN (l, m, u) ratings and weights, then defuzzifies via centroid. Note this is multiplicative: unit-mismatched criteria can be compared without normalisation, but each rating must be strictly positive.

Output

utility (product), higher is better

Data

Fuzzy (TFN), TFN per cell (strictly positive); TFN weights

Weights

Needs a weight source

Size

2+ alternatives, 3-10 criteria works best

Used for

Fuzzy MCDM where criteria are dimensionally heterogeneous, Industrial system selection without explicit normalization, Engineering / manufacturing system evaluation

How it works

1. 1

   Construct decision matrix X_tilde = [x_tilde_ij] and weight vector w_tilde, all TFNs (l, m, u).

   $${\tilde{x}}_{ij}=(l_{ij},m_{ij},u_{ij});{\tilde{w}}_j=(l_j,m_j,u_j)$$

   Kahraman 2008 Ch.7 Sec.3.2 p.205

2. 2

   Compute fuzzy weighted product per alternative componentwise per Eq.(38): each TFN parameter is raised to its matching weight TFN parameter, then multiplied across criteria.

   $$\tilde{V}(A_i)=\left(\prod _{j=1}^n{l}_{ij}^{l_j},\prod _{j=1}^n{m}_{ij}^{m_j},\prod _{j=1}^n{u}_{ij}^{u_j}\right)$$

   Kahraman 2008 Ch.7 Eq.(38), p.205

3. 3

   Defuzzify each V_tilde(A_i) via TFN centroid (l + m + u) / 3.

   $$C_i=(l_i+m_i+u_i)/3$$

   Kahraman 2008 Ch.7 p.206 (book uses fuzzy comparison via Fig.6; centroid is standard equivalent)

4. 4

   Rank alternatives by defuzzified centroid C_i in descending order.

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

   Kahraman 2008 Ch.7 p.206 (FMS_1 selected over FMS_2)

Fits when / Look elsewhere when

Assumptions to verify

• All ratings strictly positive
• Partial compensation acceptable: a very weak rating dominates the product via low exponent

Edge cases and pitfalls

• Ratings must be > 0 (multiplicative method); zero or negative values cause undefined exponents.

• Eq.(38) raises the l-component to the weight's l, m to m, u to u; do not cross-multiply parameters.

• Fuzzy WPM output is on a different scale than Fuzzy SAW; do not compare scores across methods.

Works with

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