Fuzzy Composition: Max-min and Max-product

Composition of fuzzy relation is defined over two fuzzy relations.

Fuzzy composition

Fuzzy composition can be defined just as it is for crisp (binary) relations. Suppose R is a fuzzy relation on X × Y, S is a fuzzy relation on Y × Z, and T is a fuzzy relation on X × Z; then,

Fuzzy Max–Min composition is defined as:

Fuzzy Max–Product composition is defined as:

Let us try to understand it with the help of an example. The fuzzy max-min approach is identical to that of the crisp max-min composition.

Example:

X = {x₁, x₂}, Y = {y₁, y₂},  and Z = {z₁, z₂, z₃}. Consider the following fuzzy relations:

Find the resulting relation, T which relates elements of universe X to elements of universe Z, i.e., defined on Cartesian space X × Z

• Using Max-Min composition and
• Using Max-Product composition

Solution:

So ultimately, we have to find the elements of the matrix,

Max-Min Composition:

Max-min composition is defined as,

From the given relations R and S,

μ_T(x₁, z₁) = max ( min( μ_R(x₁, y₁), μ_S(y₁, z₁)), min( μ_R(x₁, y₂), μ_S(y₂, z₁)) )

= max(min(0.7, 0.8), min(0.6, 0.1)) = max(0.7, 0.1) = 0.7

μ_T(x₁, z₂) = max ( min( μ_R(x₁, y₁), μ_S(y₁, z₂)), min( μ_R(x₁, y₂), μ_S(y₂, z₂)) )

= max(min(0.7, 0.5), min(0.6, 0.6)) = max(0.5, 0.6) = 0.6

μ_T(x₁, z₃) = max ( min( μ_R(x₁, y₁), μ_S(y₁, z₃)), min( μ_R(x₁, y₂), μ_S(y₂, z₃)) )

= max(min(0.7, 0.4), min(0.6, 0.7)) = max(0.4, 0.6) = 0.6

μ_T(x₂, z₁) = max ( min( μ_R(x₂, y₁), μ_S(y₁, z₁)), min( μ_R(x₂, y₂), μ_S(y₂, z₁)) )

= max(min(0.8, 0.8), min(0.3, 0.1)) = max(0.8, 0.1) = 0.8

μ_T(x₂, z₂) = max ( min( μ_R(x₂, y₁), μ_S(y₁, z₂)), min( μ_R(x₂, y₂), μ_S(y₂, z₂)) )

= max(min(0.8, 0.5), min(0.3, 0.6)) = max(0.5, 0.3) = 0.5

μ_T(x₂, z₃) = max ( min( μ_R(x₂, y₁), μ_S(y₁, z₃)), min( μ_R(x₂, y₂), μ_S(y₂, z₃)) )

= max(min(0.8, 0.4), min(0.3, 0.7)) = max(0.4, 0.3) = 0.4

Max-Product Composition:

μ_T(x₁, z₁) = max ( (μ_R(x₁, y₁) × μ_S(y₁, z₁)), ( μ_R(x₁, y₂) × μ_S(y₂, z₁)) )

= max((0.7 × 0.8), (0.6 × 0.1)) = max(0.56, 0.06) = 0.56

μ_T(x₁, z₂) = max ( ( μ_R(x₁, y₁) × μ_S(y₁, z₂)), ( μ_R(x₁, y₂) × μ_S(y₂, z₂)) )

= max( (0.7 × 0.5), (0.6 × 0.6)) = max(0.35, 0.36) = 0.36

μ_T(x₁, z₃) = max ( ( μ_R(x₁, y₁) × μ_S(y₁, z₃)), ( μ_R(x₁, y₂) × μ_S(y₂, z₃)) )

= max((0.7 × 0.4), (0.6 × 0.7)) = max(0.28, 0.42) = 0.42

μ_T(x₂, z₁) = max ( ( μ_R(x₂, y₁) × μ_S(y₁, z₁)), ( μ_R(x₂, y₂) × μ_S(y₂, z₁)) )

= max((0.8 × 0.8), min(0.3 × 0.1)) = max(0.64, 0.03) = 0.64

μ_T(x₂, z₂) = max ( ( μ_R(x₂, y₁) × μ_S(y₁, z₂)), ( μ_R(x₂, y₂) × μ_S(y₂, z₂)) )

= max((0.8 × 0.5), (0.3 × 0.6)) = max(0.4, 0.18) = 0.40

μ_T(x₂, z₃) = max ( ( μ_R(x₂, y₁) × μ_S(y₁, z₃)), ( μ_R(x₂, y₂) × μ_S(y₂, z₃)) )

= max((0.8 × 0.4), (0.3 × 0.7)) = max(0.32, 0.21) = 0.32

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Test Your Knowledge:

Let two sets P = {P₁, P₂, P₃, P₄} and D = {D₁, D₂, D₃, D₄} represent a set of variety of paddy plants and a set of plant diseases. In addition to these, also consider another set S = {S₁, S₂, S₃, S₄} to be the common symptoms of the diseases. Let, R be a relation on P x D, representing which plant is susceptible to which diseases, then R can be stated as,

Also, consider T to be another relation on D x S, which is given by

Find the association of plants with the different symptoms of the disease using max-min composition.

Please post your answer / query / feedback in comment section below !