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Unformatted text preview: CO 350 NAME: M—
Section #: _l___ I.D. #: ﬂ.— CO 350 Linear Optimization Midterm Examination  Spring 2004
June 17, 2001, 7—9 p.m. Instructors: . Tuncel (87 ion 1), C. B. Chua (Section 2) Instructions: 1. There are 14 pages including this cover page. Make sure that you have a complete copy.
Answer all questions on these pages. Use backs of pages if necessary. 2. Calculators are not permitted. 3. Some questions require you to prove or disprove certain statements. In doing so, you may
refer to any result from the lecture notes except the particular one you are trying to prove or
disprove. NW. QM CO 350 NAME:
1. [15 points]
[5] (a) Convert the following problem to standard inequality form:
maximize 10:61 — 5362
subject to
$1 “l" 3552 _>_ “—8
2381 ~ 7x2 = 12
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[5] (b) Convert the following problem to standard equality form:
minimize 93:1 — 611:2
subject to
$1 — 4x2 2 —20
3331 + 5.162 S 1
931 a $2 2 0 d?) “A” ”3X: 14qu 5+ —?<J +tm2< g. 20.
3% {MS’XZ é} QCI/IX2 7/0 €557) 1/”.ch a» 3% ~é Saw “V ’k +1 I'm +k3 =20
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[5] (c) Prove algebraically that the following linear programming problem is unbounded.
maximize z = 5x1 + x2 — _:v3 — 7504 + :65
subject to
131 + 2323 — 2374 + 305 = 1 *8
332 + 1133 — 9.734 v 1135 = 4 X‘
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° 5 ' CO 350 NAME: 4
2. [20 points]
4 1 3 5 10
A:= 6 5’ 1 ,b:= 10 ,c:= 20
2 8 8 2 30
describe the data for an LP problem in standard inequality form.
[5] (3.) Introduce slack variables $4, :35, 3:6 2 0 and transform the given problem to an LP problem
in standard equality form.
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[15] (b) Using the simplex method with the smallest subscript rule, solve the LP problem you
obtained in part (a). { 6 £1 ,. I W, Z q.
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Let describe the data for an LP problem in standard equality form. [10] (a) Introduce one artiﬁcial variable x5 2 0 and write down an auxiliary LP problem whose kiwi QWK ‘ CO 350 NAME: 7 [10] (b) Using the simplex method With the largest coefﬁcient rule, solve the auxiliary LP problem
you obtained in part (a). If the original LP problem is feasible then use the ﬁnal tableau
from Phase—I to make a feasible simplex tableau for the original LP problem. M» “all base: (5:: {M53 l (1 £0 3:2) Y: /€0v°5 Ma) @0395“ 8" (, L13 422505.: W QM CO 350 NAME: 8
4. [15 points]
[9] (a) Consider the following tableau. Assume that the initial basis was B = {5,6,7}. with
AB = I. For each nonbasic variable which is eligible to enter the basis, ﬁnd the corresponding
leaving variable according to the Lexicographic Rule; do not pivot. Show er. 64km whats: 4Q (67:00) “(s (323876)).
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“3) WZZ léavﬁs (lie £35595, \/ . N [M 09/ 0i CO 350 NAME: [6] (b) For each of the following statements, determine whether it is true or false (and mark T or F) Proof is not required. Each wrong answer will be assigned (1) (instead of zero);
however, negative total scores for part (b) will be treated as zero. g ’ (i) Every optimal solution of a linear programming problem is an extreme point of its
feasible region. l: ‘ (ii) A linear programming problem is unbounded if and only if its feasible region is un—
bounded. T (iii) The feasible region of a linear programming problem is convex.
I: (iv) Every convex set has at least one extreme point.
(2 (v) Every subset of a convex set is convex. (ﬁvi) The intersection of any two convex sets in R" is convex. a) Fake, , 6f) [:45er (In) (ls/“Vt, ~ Q0) Faéx , (u) 9% ~ «
6/1) 77%., ‘ (O / M“  co 350 NAME: 10 5. [10 points] [3] (a) Deﬁne the terms convex set, extreme point and basis. x 5043
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L3. Q%((0V"5 <€W LW 4/3673”). WK WM 11 [7] (b) Let F be the set of x satisfying
11:1 — 484984543 + $4 = 3
9/ 2x2 + 1048576x3 = 2
$1 7 $2 7 {E3 3 134 Z 0 (i) Find two extreme points of F. Justify your answer. “0 ~ OL x3 «m, J 3 (ML/g 0(: ‘1 fwsw
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éérﬂw‘“ SQKJCS’ M 5 s C éa‘xiék S0 ems fc’rJ'NP/Tcy F‘v’v‘r’i“‘5 ,/ (ii) Find a point of F that is not an extreme point. Justify your answer. m 564* F: B Yﬂ‘eéozc/«lolﬁ Nafm oﬁ~ovi LP
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ﬁﬂ?’ aﬂ @XM’V‘L. fbM/E, t . co 350 NAME: &1 §(6?( 0 8f 6L 12 6. [10 points]
Prove that the set {3: : Ax = b, :1: 2 0} is unbounded if and only if there exists 0 such that the linear programming problem T maximize c cc
subject to
A7: = b
a: _>_ 0
is unbounded.
2:25" gm gm. W, m W / (L‘ JU/Ménzl ’EZ/Ké g 3% :7/ng2/ﬂﬂ.
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Suppose that in an LP problem we replace the unrestricted (free) variable :31 by ul — 121,
where U1 and 111 are nonnegative variables, to create a new LP problem. Prove that if
the point (ul,vl,$2, x3, . . . ,mn) is an extreme point of the feasible region of the new LP
problem, then we cannot have both ul > 0 and 211 > 0. 69w MW‘X ’ 061“ Cowblravts A~~ wows} an “an al?"‘ 0* 35 A128 :2 b j ’X“ 20. @dea ZéwM/jjj?
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M (MGM w’ 96M)  CO 350 NAME: 6% Qf’ék 14 8. (BONUS) [10 points] Suppose that the matrix [A l D] is m by n with rank m. Deﬁne [ cc y* ] to be a basic feasible solution of the system (Ax + Dy = b, y 2 0) if there exist u* and v* such that
u* 56* = u* — v" and [ 21* ] is a basic feasible solution of the system
y* [A l~—A][:]+Dy=b, [:]20,y20. Z Prove that if [ a: 11* ] is an extreme point of the set {[3}: Ax+Dy= b, we} then it is a basic feasible solution of the system( (Am + Dy: b, y >0
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