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 composition

Fuzzy Max–Product composition is defined as:

fuzzy composition

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:

relation matrix R
Relation R
relation matrix S
Relation S

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,

relation matrix T
Composition of relation R and S

Max-Min Composition:

Max-min composition is defined as,

fuzzy composition

From the given relations R and S,

ÎŧT(x1, z1) = max ( min( ÎŧR(x1, y1), ÎŧS(y1, z1)), min( ÎŧR(x1, y2), ÎŧS(y2, z1)) )

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

ÎŧT(x1, z2) = max ( min( ÎŧR(x1, y1), ÎŧS(y1, z2)), min( ÎŧR(x1, y2), ÎŧS(y2, z2)) )

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

ÎŧT(x1, z3) = max ( min( ÎŧR(x1, y1), ÎŧS(y1, z3)), min( ÎŧR(x1, y2), ÎŧS(y2, z3)) )

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

ÎŧT(x2, z1) = max ( min( ÎŧR(x2, y1), ÎŧS(y1, z1)), min( ÎŧR(x2, y2), ÎŧS(y2, z1)) )

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

ÎŧT(x2, z2) = max ( min( ÎŧR(x2, y1), ÎŧS(y1, z2)), min( ÎŧR(x2, y2), ÎŧS(y2, z2)) )

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

ÎŧT(x2, z3) = max ( min( ÎŧR(x2, y1), ÎŧS(y1, z3)), min( ÎŧR(x2, y2), ÎŧS(y2, z3)) )

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

max min composition
Max-Min composition of Relations

Max-Product Composition:

fuzzy composition

ÎŧT(x1, z1) = max ( (ÎŧR(x1, y1) × ÎŧS(y1, z1)), ( ÎŧR(x1, y2) × ÎŧS(y2, z1)) )

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

ÎŧT(x1, z2) = max ( ( ÎŧR(x1, y1) × ÎŧS(y1, z2)), ( ÎŧR(x1, y2) × ÎŧS(y2, z2)) )

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

ÎŧT(x1, z3) = max ( ( ÎŧR(x1, y1) × ÎŧS(y1, z3)), ( ÎŧR(x1, y2) × ÎŧS(y2, z3)) )

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

ÎŧT(x2, z1) = max ( ( ÎŧR(x2, y1) × ÎŧS(y1, z1)), ( ÎŧR(x2, y2) × ÎŧS(y2, z1)) )

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

ÎŧT(x2, z2) = max ( ( ÎŧR(x2, y1) × ÎŧS(y1, z2)), ( ÎŧR(x2, y2) × ÎŧS(y2, z2)) )

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

ÎŧT(x2, z3) = max ( ( ÎŧR(x2, y1) × ÎŧS(y1, z3)), ( ÎŧR(x2, y2) × ÎŧS(y2, z3)) )

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

max product composition
Max-Product composition

Watch on YouTube:

fuzzy composition

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,

relation r

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

relation s

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 !

14 Responses

  1. Komal Goyani says:

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  2. Nazeera Sheth says:

    Very systematic and easy way to learn!!!!

  3. Himani says:

    Thanks to this blog of fuzzy composition. Its too much helpful and specifically to ME students. Explained in minutely detailed fashion.

  4. Bhavani M says:

    Easy to learn

  5. Gaurav says:

    Is this the answer
    .8 .8 .8 .9
    .8 .8 .8 .9
    .8 .8 .8 .9
    .8 .8 .8 .9

  6. Om says:

    I think last row will be (.8 .8 .8 .9)

  7. prashant says:

    fine website

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