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Unformatted text preview: Assignment 4
Due: Wednesday November 16 at the BEGINNING of class 1. Given the following linear programs, write down the corresponding dual linear program: (a) maximize 2501 + x2 —— $3 + 3x4 + x5
subject to $1 + 2m + $4 + x5
~$1+ $2 — 21133 + 2$5
$1—$2+x3+x4—x5 $1,$2,$3,$4,$5 llll
094100.: IV II (b) maximize x1 — 331:2 + 2:133 + 21r4 + m5
subject to $1 + 21172 — 2903 + :54 $1 " $3 + $4 '— $5 —:r1+x2+333+2$4—m5 331 $2, $5 $3, $4 /\VI/\ II IV
moocohum
CD
(D (c) minimize —3x1 + $2 + 2333 subject to 2301 + x2 — $3 31132 + 7$3 —a:1 +132 + x3 x1 I 332 + $3 $1 free
0
0 II VI/\ 
Ipr—IN
H /\ IV 2. For the following, you are given a feasible solution 33* for the linear program and a fea—
sible solution if for the corresponding dual. Show if they are both optimal solutions
or not using complementary slackness conditions. (a) maximize 5:121 — 6532 + 8373 + 4534 + 235
subject to 23:1 — :52 + $3 —I— 904
m1+3x2—x3+x4+x5
2961+ 3$3 + x4 + 335
$101,332, $3,334,135 x* = (0,1,0,2,4)T, y* = (3, —1,2)T IV II  
coop—a (b) maximize m1 — x2 —I— 5x3 + $4 — 5x5
subject to 2331 ~ :02 + x3 + 964
3:1 + 2$3 + 2935
—$1+$2+3$3 —$4+$5
$1, $2, $3, $4, $5
113* = (7,11,0,0,0)T, y* = (5, —4, 4)T IV II II II'
<3»wa 3. Consider the graph G = (V, E) where V = {5,(1, t}, E = {5a, at, 325}, and the length of
the edges are em = 1, cat = 1 and est = 3. a S 3 z: The integer program to ﬁnd the shortest st—path has duality gap zero. Find the short—
est st—path by solving a linear program using the Simplex Method. Verify that the
complementary slackness conditions hold. Use your solution to ﬁnd a set of rivers that can cross the graph Where the Widths of
the rivers is no more than the length of any edge it crosses and the total Widths of the rivers is maximized. 4. For the following linear program, maximize :51 + 032 + 333 + m4
subject to 3:1 + 332 + x5 =
331 + $3 + $6 = x3 + $4 + $7
$2 + $4 + (L‘s
371, $2, $3, $4, 555, $6,377, $8 I!
Oi—ll—‘i—‘H IV ll use complementary slackness conditions to determine if the given feasible solution 110*
is optimal or not.
(Note: This is in fact an LP for a maximum matching problem in SEF.) 17* = (0,1,1,0,0,0,0,0)T
(Hint: There will be many possible y’s. Try one with simple values of 0 or 1.)
53* I = (0, 0, 1,0, 1,0,0, 1)T (a) (b) ...
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This note was uploaded on 01/26/2012 for the course ECON 401 taught by Professor Burbidge,john during the Fall '08 term at Waterloo.
 Fall '08
 Burbidge,John

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