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# pg+178 - TA B L E 37 Tableau Indicatinu Inlaasihility tor...

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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 two-phase 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'Il-i—Zx"2=4 2x1+X2 S3 xla-xZZO 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—xZZ-l 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 Two-Phase Simplex Memoir 178 When a basic feasible solution is not readily available, the two-phase simplex method may be used as an alternative to the Big M method. In the two-phase 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 two-phase 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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