Consider a standard additive model consisting of rules of the form of If…
2015
Consider a standard additive model consisting of rules of the form of
If \(𝑥\) is \(𝐴_𝑖\) AND \(𝑦\) is \(𝐵_𝑖\) THEN \(𝑧\) is \(𝐶_𝑖\).
Given crisp inputs \(𝑥=𝑥_0,𝑦=𝑦_0\) the output of the model is :
- A.
\(z=\Sigma_i \mu_{A_i} (x_0) \mu_{B_i} (y_0) \mu_{C_i} (z)\) - B.
\(z=\Sigma_i \mu_{A_i}(x_0) \mu_{B_i} (y_0)\) - C.
\(z=\text{centroid } (\Sigma_i \mu_{A_i} (x_0) \mu_{B_i} (y_0) \mu_{C_i} (z))\) - D.
\(z=\text{centroid } (\Sigma_i \mu_{A_i} (x_0) \mu_{B_i} (y_0)\)
Attempted by 43 students.
Show answer & explanation
Correct answer: C
Correct output (crisp value):
Step-by-step procedure:
Compute each rule's firing strength: for each rule i, w_i = μ_Ai(x0) × μ_Bi(y0).
Form the aggregated output membership function by summing the consequents weighted by the firing strengths: μ_C'(z) = Σ_i [w_i × μ_Ci(z)].
Defuzzify the aggregated fuzzy set to obtain a single crisp value using the centroid (center of gravity): z = (∫ z · μ_C'(z) dz) / (∫ μ_C'(z) dz).
Remarks:
The expression Σ_i μ_Ai(x0)·μ_Bi(y0)·μ_Ci(z) represents the aggregated fuzzy membership function μ_C'(z), not a crisp z by itself. A defuzzification step is required to get a numeric output.
If each consequent μ_Ci(z) is a singleton located at z_i, the centroid reduces to the weighted average: z = (Σ_i w_i·z_i) / (Σ_i w_i).