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Unformatted text preview: Solutions 1. Exercise 11.4.1 from the course notes. Solution: (a) The ﬁrst row is 5 1 5 l
Z+<1+ﬁ)$4+<0+1—2‘)$5+(0+E)$6=14+Z the cut is Emiri +3 >1
124 12“”5 12x5—'4' The second row is $+o+11 + 1+11 + 1+59 *1 3
3 60 $4 12 $5 60 m6" +4 with cut
11 + 11 . + 59 > 3
609‘"4 12“5 60336 — 4 The third row is 1. 1 9 1
1122+(0+I6)€E4+(0+§)$5+(—1+ﬁ)$5—1+§ with cut The fourth row is l 5 l 1
m1+<0+ﬁ)m4+(0+ﬁ)m5+<O+ﬁ)16_1+1 with cut
1 + 5 + 1 >1
12CB4 12m5 12$6 ‘ 4 (b) For the ﬁrst out we have 5 1 5 + ,_ 1
12“ 12"”5 12‘36 3— 4’ the leaving variable is 3. Compute mm 1712 112 512 _1
1 125’121’125 "1’ so 1:5 (or $5) enters. For the second out we have ~—1—1—x———llm—§—9os+s——§
604 12° 606 ‘ 4 the leaving variable is 3. Compute ~ 11% i1? 3% i
mm 1211’1211’1259 “11’ the entering variable is :35. For the third out we have 1_ 1‘ 9~+ _1
10954 215 10%6 8— 2 the leaving variable is 3. Compute .1710 12 510 _1
mm 121’121’129 _6’ the entering variable is $5. For the last out we have
1 5 1 1
3954—5965 ‘ 123966”— “:1 1712112 512 _1
(a? E? a?) — 3’
again the entering variable is 935.
(c) From the initail ILP we have m4 = 10—m1 ~5m3, m5 = 1—371 *3321(173 and m6 = —6m1+5m2,
substituting into the cuts we get
i.3a:1— 2.732 + 2233 S 4,
ii. 7301 — 4mg 3 2,
iii. 6931 ~ 4932 S 1 and
iv: $1 5 1. 2. Exercise 11.4.2 from the course notes. Solution:
The ﬁrst tableau for the relaxed LP is: z + x1 — 32:2 ' : 0
331 — (1:2 + $3 = 2
2:31 + 4mg I + .734 = 15 Entering variable: x2 Leaving variable: m4 Pivoting gives the optimal tableau: Z + (5/2)$1  (3/4)m4 = 45/4
(3/2):I:1 + :03 —— (1/4)a;4 = 23/4
(1/2):E1 + 1122  (ll/40374 = 15/4 Gomory cuts from either constraint are identical (1/2cc1 + 1/4234 2 3/4). Adding this cut to
the tableau gives: » Z + (5/2)IL‘1 + (3/4)$4 = 45/4 (3/2)x1 + $3 + . (1/4)a:4 = 23/4 (1/2)x1 + x2 + (1/4)m4 = 15/4 — (1/2)m1 — (1/4):l?4 + C135 = —3/4 Applying the dual simplex with leaving variable m5 and entering variable 5134 gives: Z ‘l' 1'1 —— 35135 = 9
£31 + {£3  935 = 5
$2  (135 = 3
2.1)] + (134 — 45E5 = 3 An optimal solution to the original problem is [0, 3]T. 3. Exercise 11.4.3 from the course notes. Solution: (80 (b) The second equation of the tableau is 1“ 1' _3
$2+§L3~§l4§. We split each of the coefﬁcients for $3 and x4 and the right—handside value into their
integer part and fractional part. 932 + $013 + (—1 + $134 = (1+ é).
We further reorganize this equation to get:
1 1 1
§$3+5$4—§ =1"$2+$4.
Since we want all the variables to be integers, 1 — 272 + x4 must be intege1. Thus,
2:33 + %m4 — — must be integer as well. We also need 1all the variables to be nonnegative,
thus1 m—3—I—2 —ac4__ > 0 which implies2 :v—3—l—2 l:14 — — 1> ——.Since1—x3+%m4 ~ 1 1s an integer
if it is larger than — ,it must be greater2 than 01 equal to 0 Thus the resulting Gomory
cut is:
1 1 l
5:123 + 511:4 2 E. In the basic feasible solution for B = {1,2}, 333 and 1'4 are 0, thus the current solution
violates this Gomory cut. The constraint %m3+%m4—% 2 —% is satisﬁed by all nonnegative
solutions but we also know that the left— hand— side must be integer (from above). So
integer solutions that satisfy 21r3+2 —a"4 — — 12 —— will satisfy 2m3+§1m4 — — > 0 although
fractional solutions may not Add slack variable $5 to the Gomory cut and make $5 a basic variable, i.e., we add the
following equation to our tableau: 1 1 1
"“5133 ‘ 5334 + $5 : "E.
The augmented tableau looks as follows:
2 + $3 + —§ 3:4 2 12
m1  £133 + x4 = 2
$2 — $123 — i334 = g
 59103 — 5564 + 335 = —§ This new tableau corresponds to a basic primal infeasible solution and a dual feasible
basis. Dual simplex method will choose x5 as the leaving variable and :L4 as the entering variable since 3334:— — 111in{_—‘1——/2,—_T/L§}. Pivoting 011 T4 and 235 gives us the following
tableau:
z + —%:C3 + 335 = 3
{1:1 1 + 2.15 = 1
£172 + $3 — {135 = 2
{113 + $4  21125 = 1 We have a primal feasible and dual feasible basis, so B = {1,2,4} is optimal for the
new LP relaxation. Since the corresponding basic feasible solution, m* =' (1,2,0,1)T, 1s
integral, (5* is also the optimal solution for the original ILP. ...
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 Fall '10
 GUENIN

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