linopt::Transparent::simplex
--
finish the current phase of the 2-phase simplex algorithmlinopt::Transparent::simplex(
tableau)
finishes the current phase of the 2-phase simplex algorithm using the
given tableau tableau
.
linopt::Transparent::simplex(tableau)
tableau |
- | a simplex tableau of domain type
linopt::Transparent |
a simplex tableau of domain type linopt::Transparent
or
the empty set if there was no feasible solution found.
linopt::Transparent
, linopt::Transparent::autostep
,
linopt::Transparent::convert
,
linopt::Transparent::dual_prices
,
linopt::Transparent::phaseI_tableau
,
linopt::Transparent::result
,
linopt::Transparent::suggest
,
linopt::Transparent::userstep
linopt::Transparent::simplex
runs the current phase of
the 2-phase simplex algorithm until the end, i.e. if phase I was
explicitly started (see linopt::Transparent::phaseI_tableau
)
the first phase will lead the optimal tableau. Sometimes it can be
necessary to eliminate some slack variables of phase one by using
linopt::Transparent::clean_basis
.(linopt::Transparent)::simplex
returns the last optimal
tableau or the empty set if there was no feasible solution found.We apply linopt::Transparent::simplex
to an
ordinary simplex tableau of a linear program and we get the optimal
tableau:
>> k := [[x + y >= 2], x, NonNegative]: t := linopt::Transparent(k); t := linopt::Transparent::simplex(t)
+- -+ | "linopt", "restr", slk[1], x, y | | | | "obj", 0, 0, 1, 0 | | | | slk[1], -2, 1, -1, -1 | +- -+ +- -+ | "linopt", "restr", slk[1], x, y | | | | "obj", 0, 0, 1, 0 | | | | y, 2, -1, 1, 1 | +- -+
Let us proof the obtained result:
>> linopt::Transparent::suggest(t)
OPTIMAL
>> delete k, t:
If the first phase of the simplex algorithm was started
explicitly,
linopt::Transparent::simplex
returns only the optimal
tableau of the first phase:
>> k := [[x + y >= 2], x, NonNegative]: t := linopt::Transparent(k): t := linopt::Transparent::phaseI_tableau(t); t := linopt::Transparent::simplex(t)
+- -+ | "linopt", "restr", slk[2], slk[1], x, y | | | | "obj", -2, 0, 1, -1, -1 | | | | slk[2], 2, 1, -1, 1, 1 | +- -+ +- -+ | "linopt", "restr", slk[2], slk[1], x, y | | | | "obj", 0, 1, 0, 0, 0 | | | | x, 2, 1, -1, 1, 1 | +- -+
The next step of the simplex algorithm is computed:
>> linopt::Transparent::suggest(t)
"linopt::Transparent::phaseII_tableau"
With linopt::Transparent::autostep
we execute the
first step of the second phase of the simplex algorithm. One can see
that the simplex algorithm is not finished yet:
>> t := linopt::Transparent::autostep(t): linopt::Transparent::suggest(t);
x, y
If we then apply
linopt::Transparent::simplex
again we get the optimal
solution. Here we don't had to use linopt::Transparent::clean_basis
, before
using linopt::Transparent::autostep
, because there are no
artificial variables in the basis computed by the first
linopt::Transparent::simplex
call above:
>> t := linopt::Transparent::simplex(t); linopt::Transparent::suggest(t)
+- -+ | "linopt", "restr", slk[1], x, y | | | | "obj", 0, 0, 1, 0 | | | | y, 2, -1, 1, 1 | +- -+ OPTIMAL
>> delete k, t:
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Nemhauser, George L; Wolsey, Laurence A: Integer and Combinatorial Optimization. New York, Wiley, 1988.
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