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List of Contents     Rommelfanger     2.1     Lai, Hwang     Notations + Definitions

2.2 Less-Than-Or-Equal-Relation Z-less-than-or-equal

Optimization model KO2:

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

 z(x) =   Sum j = 1,...,n ej xj   Z-less-than-or-equal K

in compliance with the restrictions

 Sum j = 1,...,n aij xj   Z-less-than-or-equal bi,    j = 1,…,m

xj more-than-or-equal 0,   j = 1,…,n.

This can be read as follows:

The target function    z(x) =   Sum j = 1,...,n ej xj    should be "really less than or equal" to a value K (capacity limit), and the restrictions    Sum j = 1,...,n aij xj    should be "really less or equal" to the right sides   bi.

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

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

Further let be   (Hx)k := Sum j = 1,...,n hkjxj.

We define the function

zk((Hx)k) :=  open brace 1   for   (Hx)k less-than-or-equal wk
1 -    line
 beta k
for   wk  <   (Hx)k  less-than-or-equal wk +  beta k
0 for   (Hx)k  >  wk +  beta k


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

 wT = (w1, …, wm+1)T = (K, b1, …, bm)T

where betak 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

 µD(x) Min 1 <= k <= m+1 zk((Hx)k).

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

 Min 1 <= k <= m+1 zk((Hx)k)    —>   Maxx

or in other words:

 Min 1 <= k <= m+1 (wk / beta k    –    (Hx)k / beta k) =: Min 1 <= k <= m+1 (w_k – (Hx)_k)    —>   Max x = (x1,…,xn)

This is equivalent to:

 lambda    —>   Max lambda element out [0,1]

  in compliance with

 lambda less-than-or-equal w_k – (Hx)_k     with       k = 1, ..., m+1

 xj  more-than-or-equal 0,       j = 1, …, n


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

List of Contents     Rommelfanger     2.1     Lai, Hwang     Notations + Definitions

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© 1999-2001 Maria Oelinger
cand. math.
Fuzzy mathematics
Last Update: 25.04.2001
Address: http://www.oelinger.de/maria/en/fuzzy/ko_z.htm