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Unformatted text preview: are nonpositive, and
the objective coefkient is () for
maximization [or (+) for minimization]
for that variable
then, the problem is unbounded in the
direction of that variable
l l ernate Optimal Problem Special Cases
Unbounded uftxr20~~+$04 Iteration 0 Sk .4yt.I*50 0 2~+4~s 100 @
Basic Xl ?I 5 x4 Soln I 2 1 0 0 0 x3 1 1 1 0 10 4 1 0 0 1 40 nr . . Special Cases Alternate Optimal Aiternate Optimal
Iteration 0 If in an optimal tableau
The coeffkient of any nonbasic
variable is zero in the zrow
then, the probkm has alternate optimal
solutions with equioptimal objective
values Basic l z l x1 I
I x2 x4 0 4 2 saln x3 010 I xJ 1 2 10 5 4 1 1 0 4 . . EM602 / QM710 (NJ) Lecture 5
Page 57 1 Alternate Optimal (iterationa 1 L 2) ILh 0 2 0 1 1 1 0 1 2 see problem 2l 7 1 1 Infeasible Problem X 24 3 3 I[q
~~~ I 01101 Special Cases
Infeasible
Iteration8 0 & 1 Infeasible
Basic if, in an optimal tableau,
An artifictai variable is basic and has
value > zero, and
The objective function value contains
M if minimizing [or (A) if maximizing]
then the probtem is infeasible
Note: if the artiftciai variable is basic but
with zero value, the problem is
feasible, but had one or more
redundant constraints L l x1 ~2 q ~q R, SoIn
u 0 0 0 01 0 112 2 0 M 2+4M 0 4SY 1
0 0
1 1
4 0
1 2
4 33M_24M ;I:: 4l 1 1215 l L l+Skl g2 2 Ri 5 .. Homework Assignment
Due in 1 we&k
l Note: Textbook problem # 341 l l The remainder of the slides are
presented for your reference only
They were not discussed in class and
are thus not included in the scope of
the course EM602 / QM710 (NJ) Lecture 5
Page 58 Dual Problem Degenerate For every LP problem (called the Primal)
we can derive a related symmetrkal LP
(called the Dual)
l The solution of the Primal LP
automatically gives the optlmal
solution to the Dual
l The solution of the Dual LP
automatkally gives the optimal
solution to the Primal
l The decision variables and objective
for the primal are XJ and r; and for the
dual they are y and w If In successive iterations of the tableau
l The objective function value stays the
same
l the nonbasic variable’s zrow entry is nonzero
l the bask variable’s zrow entries are zero
then the solution at this point is degenerate
Note: Degeneracy may be temporary
(degenerate point is suboptimal) or
permanent (degenerate at optlmallty) Dual Problem Dual Problem l l l For every prlmal constraint there is a
dual variable
For every primal variable there is a
dual constraint
The co...
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 Spring '94
 DonaldC.Johnson

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