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2.2 Less-Than-Or-Equal-Relation 

Optimization model KO2:

The fuzzifized version of the optimization model KO1 is the optimization model KO2 (Fuzzy):

z(x) =    e_j x_j    K

in compliance with the restrictions

 a_ij x_j    b_i,    j = 1,…,m

x_j  0,   j = 1,…,n.

This can be read as follows:

The target function    z(x) =    e_j x_j    should be "really less than or equal" to a value K (capacity limit), and the restrictions     a_ij x_j    should be "really less or equal" to the right sides   b_i.

   Or, in other words: The inequations should be met as sharp as possible.
We formulate the expression "really less or equal", written , in the following way:

Let   H := (h_kj)   be the matrix which you receive by adding to the matrix   A = (a_ij)    the row vector   (e_j)   as first row    (i = 1,…,m; j = 1,…,n; k = 1,…,m+1).

We define the function

zk((Hx)k) :=		1		for   (Hx)k  wk
		(Hx)k-wk 1 -    k		for   wk  <   (Hx)k  wk +  k
		0		for   (Hx)k  >  wk +  k

and   w_k   ist the vector of the initial rights sides   b_i   , completed by the value   K   , i.e.

w^T = (w₁, …, w_m+1)^T = (K, b₁, …, b_m)^T

where _k are the permitted tolerances for crossings of the restrictions.

Should all restrictions of optimization model   KO2    be met, a new target function results from this, the fuzzy decision

µ_D(x) =  z_k((Hx)_k).

The optimal result x ^* =   (x₁^*,...,x_n^* ) is in demand, so that

 z_k((Hx)_k)    —>   Max_x

or in other words:

 (w_k /  _k    –    (Hx)_k /  _k) =:  (w_{_k} – (Hx)_{_k})    —>   Max _{x = (x1,…,xn)}

This is equivalent to:

in the way that an optimal result of this also is an optimal result for opotimization model   KO2   .

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