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1.Fuzzy optimization by Rommelfanger

1.1 Introduction

1.1.1 Basics

We look at optimazation models of the form

z(x) =  x = c₁x₁ + … + c_nx_n    —>   Max

in compliance with the fuzzy restrictions

A_i1x₁  …  A_inx_n  B_i        i = 1, …, m

   and

x₁, …, x_n  0.

To simplify matters it should be considered for all coefficients A_ij and the right sides B_i in the following, that the reference functions are
L(u) = R(u) = Max {0, 1 – u} .
In general you can use all kinds of reference functions, of course.

Aij = (aij; oij; ij; ij)	are fuzzy intervals and
Bi = (bi; 0; i)	are fuzzy numbers on the -niveau,   ]0, 1].

1.1.2 Example of use

Target function should be

under the subconditions x, y  0

(A)     (2; 2; 0.5; 0.5) ^0.1 x      (1.5; 2; 0.5; 1) ^0.1 y    (20; 0; 8) ^0.1

(B)     (3; 4; 1; 1.5) ^0.1 x      (5; 6; 1; 1) ^0.1 y    (48; 0; 12) ^0.1

(C)      (2.5; 3; 0.5; 0.4) ^0.1 y    (18; 0; 7) ^0.1

List of Contents     1.2     1.3     Kacprzyk, Orlovski     Lai, Hwang     Notations + Definitions

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