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1.3 Less-Than-Or-Equal-Relation 

General

A_i (x) = A_i1 x₁  …  A_in x_n  B_i

Restriction with fuzzy interval A _ij = (a_ij ; o_ij ; _ij ; _ij) and fuzzy number B_i = (b_i; 0; _i) .

For         this is equivalent to:

     1.      a_ij – _ijL^-1 ()) x_j   b_i + _i L^-1 (), ], 1]   and

     2.      o_ij + _ij x_j   b_i + _i

Example of use

= 0.1, = 0.9
z(x, y) = 4x + 7y    —>   Max

with subconditions

(A)   I :       1.95x + 1.45y 20.8
(A)   I :       2.5x + 3y 28

(B)   II :       2.9x + 4.9y 49.2
(B)   II :       5.5x + 7y 60

(C)   III :       2.45y 18.7
(C)   III :       3.4y 25

x, y 0.

I    =>    x (20.8 – 1.45y) / 1.95    =>    x 5.2
I    =>    x 2.38

II    =>    x 4.55
I    =>    x 1.55
=>    x 1.55

III    =>    y 18.7 / 2.45 = 7. 63
III    =>    y 25 / 3.4 = 7.35
=>    y 7.35

This leads to the best possible result:

(x, y) = (1.55; 7.35)   and so   z(1.55; 7.35) = 57.65

Graph 6: Less-Than-Or-Equal-Relation 

For the niveau    = 1   you get with

o_ij x_j b_i    the following result:

I1:	2 • 1.55 + 2 • 7.35 = 17.8 < 20;	µB1 (17.8) = 1
II1:	4 • 1.55 + 6 • 7.35 = 50.3 > 48;	µB2 (50.3) = 0.83
III1:	3 • 7.35 = 22.05 > 18;	µB3 (22.05) = 0.48

Here are given borders crossed then. The decision-maker must consider carefully if he can tolerate it to this extent.
[µ_B1 (I₁)    to    µ_B3 (III₁)]

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