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3.2 Less-Than-Or-Equal-Relation

Optimization model LH2

On the basis of the optimization model LH1 we receive the optimizatin model LH2.

z(x) =    c_j x_j     —> Max

on the condition

x  X  := {x = (x₁,…,x_n) | µ_i (x)  ,     [0, 1]     i = 1, ..., m;   x_j  0}

The affilation functions   µ_i (x)   are

• for     a_ijx_j  b_i    totally fulfilled, i.e.    µ_i (x) = 1.

• for     a_ijx_j  b_i + _i    totally hurt, i.e.    µ_i (x) = 0.
  (_i  is the tolerance quantity which is set by the decision-maker.)

• for     a_ijx_j   ]b_i; b_i + _i[    monoton descending:
  The more products (raw materials etc.) are needed over the set limit   b_i  , the less satisfied is the decision-maker.

Then the optimization model LH2 is described as

z(x) = c_jx_j    —>   Max

on the condition

 a_ijx_j  b_i + (1 – )_i i = 1, …, m    and      [0, 1]

x_j  0,     j = 1, …, n.

The decision-maker sets according to the demand a suitable   .
Now you can calculate the optimal result with common algorithms.

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