A function   L:[0, [ —> [0, 1]    is called reference function if it meets the conditions

• L(0) = 1   and
• L is not ascending in [0, [

Example: L(u) = Max {0, 1–u} with  > 0 or
   L(u) = exp(-u )   with  > 0 (exponential function).

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A = (a; o; ; ), and

µ A (x) =		L((a – x) / ) in case x  a
		in case a < x  o
		R((x – o) / ) in case o < x

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A = (a; ; ), and

µ A (x) =		L((a – x) / ) in case x  a,  > 0
		R((x – a) / ) in case a < x,  > 0

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µ _B (a) = 1 = L(0) is culmination of the fuzzy number, , are called span.
i=1,…,m; j=1…,n.

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Fuzzy numbers A = (a; ; ) and B = (b; ; ) of the same LR type are linked by

• extended addition

(a; ; ) (b; ; ) = (a + b;  + ;  + )

• extended multiplication
  (a; ; ) (b; ; ) = (ab; a  + b –  ; a + b + )

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Restriction to the -niveau in order to limit the supporting set after the required additions / multiplications.

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named after Slowinsky

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Optimization model KO1 (No-Fuzzy) going by the way of Kacprzyk and Orlovski:

z(x) =    c_j x_j     —>   Max

in compliance with the subconditions

 a_ij x_j  b_i i = 1,...,m

x_j  0, j = 1,...,n.

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z(x) =    c_j x_j     —>   Max

in compliance with the subconditions

 a_ij x_j  b_i i = 1,...,m

x_j   0, j = 1,...,n.

back to 3.1