Contents
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Terminology for Solutions
feasible solution :
A **solution for which all the constraints are satisfied.
feasible region :
The collection of all feasible solutions.
infeasible solution :
A solution for which at least one constraint is violated, for a problem to have no feasible solutions.
optimal solution :
A feasible solution that has the most favorable value of the objective function.
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To initialize the method, one must transform all inequality constraints into equalities and must know one feasible, nonnegative solution.
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before that we should make $b_i$ nonnagitive, if not, mutiply -1.
if min z , ok, or if max z, make it min.
a free viriable should represented by: $x_i = x^{'}_i - x^{''}_i$
$x_i>c$ can be represeneted by: $x^{'}_i = x_i -c >0$, so $x_i = x^{'}_i + c$
after doing these, we then add slack or surplus or artificial viviables.
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for
$$ a_{i1} x_1 + a_{i2} x_2 + …. + a_{in} x_n < b_i $$
make
$$ a_{i1} x_1 + a_{i2} x_2 + …. + a_{in} x_n + x_{n+1} = b_i $$
for
$$ a_{i1} x_1 + a_{i2} x_2 + …. + a_{in} x_n > b_i $$
make
$$ a_{i1} x_1 + a_{i2} x_2 + …. + a_{in} x_n - x_{n+1} = b_i $$