Unformatted text preview: TA B L E 37
Tableau Indicatinu Inlaasihility tor Bum (Intrasihla) Basic
1‘ I1 52 3' a! a, a, , rhs Variable
l 1—2M 0 0 M 0 3—4M 30+6M z=6M+30
0 j 0 1 o 0 —% % s1 = 5
0 —2 0 0 —1 1 —3 6 a2 = 6
0 1 l 0 0 0 1 10 x2 = 10 Note that when the Big M method is used, it is difﬁcult to determine how large M should
be. Generally, M is chosen to be at least 100 times larger than the largest coefﬁcient in the
original objective function. The introduction of such large numbers into the problem can
cause roundoff errors and other computational diﬂiculties. For this reason, most computer
codes solve LPs by using the twophase simplex method (described in Section 4.13). PROBLEMS Group A
Use the Big M method to solve the following LPs: x min 2— — 3x1
1minz=4x1+4x2+x3 s..t 2361+ X2>6
s.t. x1+x2+ X352 3'x'Ili—Zx"2=4
2x1+X2 S3 xlaxZZO
2xl+x2+3x3—>—3 5 minz=x1+x2
xl,x2,x3 Z 0 s.t. 2xl + x2 + x3 = 4
2 minz=2x1+3x2 x1+x2+2x3=2
s.t. le + x2 2 4 x1, x2, x3 2 0
xl—xZZl 6 minz=x1+x2
x1, x2 2 0 s.t. x1 + XZ= 2
xmaxz=3x1+x2 le+2x2=4
i s.t. x1 + 1'2 2 3 x1! x2 2 0
2X1 + 1'2 5 4
XI + 1'2 =
351,352 2 0 4.13 The TwoPhase Simplex Memoir 178 When a basic feasible solution is not readily available, the twophase simplex method may
be used as an alternative to the Big M method. In the twophase simplex method, we add ar
tiﬁcial variables to the same constraints as we did in the Big M method. Then we ﬁnd a bfs
to the original LP by solving the Phase I LP. In the Phase I LP, the objective ﬁmction IS to
minimize the sum of all artiﬁcial variables. At the completion of Phase I, we reintroduce the
original LP’ s objective function and determine the optimal solution to the original LP. The following steps describe the twophase simplex method. Note that steps 1—3 for
the two—phase simplex are identical to steps 1—3 for the Big M method. TThis section covers topics that may be omitted with no loss of continuity. c H mm 4 Thu Simplex Aluuritllm and lloal Pmuramminu ...
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 Spring '09
 VLADIMIRLBOGINSKI

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