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Unformatted text preview: 1. (30 points total) Consider the linear prograﬁnﬁng problem maximize 53:1 + 0n + 0333 + 1.0% at In + 1322 + 133 + 13:4 3 4 ‘— 111'31 + 2332 + 3173 + 31'4 S 1 8:31 — 6592 + 7333 + 2374 S 2
l 9131 + 10232 — 11.33 + 45174 S $1201$22Q$32013420 (a) (5 points) Write down the linear programming dual of (Let yi be the dual
variable associated with the ith Iess—than—or—equal—to constraint of M'xrx LI" \‘l‘i K 13‘” 2‘ .a:€”¥ﬁ“'gt%kf ..
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A‘Qirtwﬁg 2 (c) (5 points) Explain why the vectors as": (4, —1,0,0)T and y = (0,0,D,1) are n__ot
,Cornplementary for this pair of problems (P) and ‘rs 5‘ ‘r I
a% XE «KBLfEFb easel QK\+[oxs_ull><3+LlKég=tu>
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a K17“: We “as lawsgy raises
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(d) (15 points) You need to check whether the feasible solution for (P) given by
331=l,:32=l,1;3=D,:t4=0 is optimal. Use your answers to (a) and (b) to determine this; do not use the simplex
method. State any results that you are using. On . (“ﬁery 1c. ﬁnewﬂ_ E amstQLwMﬁfj
as ﬂﬁaiﬂia I 2(th Eb} “ﬁvkaﬁb 2. (30 points total) We are solving a linear programming problem of the form maximize c2:
subject to Ar S b
:L' 2 0, by the revised simplex method with explicit basis inverse The matrix A is 4 X 5. The
objective function coefﬁcients of the ﬁve decision variables are: CI = 40102 : 5163 z 50,64 = —20,cs = 32. The columns of coefﬁcients Aj of the ﬁve decision variableskand the right—handside vector b are:
A1 A2 A3 A4 14.5 b 1 0 2 —1 2 50 '
1 2 l O 0 20
IO 1 9 0 0 205
O 0 1.5 l —1 5 Slack variables 2:5,m7,3:3,$g were added, and after two pivots of the revised simplex
method the vector of ordered basic indices is B = (5, 1, 8, 9) and 0.5 —0.5 0 0 15
11_ 0 1 0 0 —+ 20
AB — 0 —10 1 0 b‘ 5 0.5 —0.5 0 1 20 Perform one iteration of the revised simplex method using the explicit A; Verify that
your calculation of y is correct by checking that 53 = 0 for each of the basic indices j. Use the maximum Ej rule for column selection. Make sure to show how you updated 5
and A31. B : {3,13%a\ a gag: (arena) {:33 3; (3111.10) 0) s3 3. (12 points total) We are solving a large sparse linear programming maximization problem
by the revised simplex method. For the next calculation we require the inner product yAg. (Vile have calculated 3; already.) ' x (a) (3 pants) WhiCh of the 3 major Steps 0f thé Simplex iteration are we in? Explain in ‘i
a few words. _ I
J ' ‘ ‘m mﬁu»
Q eluww ﬂick/b71993, 3A3 15 cbWai claim
' C3 =2 513‘" 5 it: 4 (b) (9 points)
The arrays VAL( ), ROI/W ) and FIRSTNZ( ) begin with 15 16 7 8 9 10 11 21 13 41 32 16\
18 52 81 110 112 6
24
17 3 4 5
62 31 4.5
15 16
15 21 i: 1 2
VAL(i): ID —3
ROWU): 2 4 8
FIRSTNZ(I'): 1 7 12 1 13
125 143 149\915 968 Explain 'as speciﬁcally as possible how yA3 would be computed. How many arith 
metic operations are performed in this calculation? I 4. 128 points total) Consider a. general linear programing problem (P) of the form: max ca,"
(P) 3.75. AI 5 b
x 2 D where A is an m x n matrix. (a) (3 points) Write down the duel linear programming problem (b) (4 points) State in full detail the Weak Duality Theorem as it applies tO' such pairs (P) and (D)_
A , , mm A: ’ Wong
\ l? ’Eiu M: l)? gwﬂufu 1) WWW; ‘\ {‘3 9;]; v . Q3 ﬂ; 8 (c) (9 points) Prove the Weak Duality Theorem as it applies to such pairs (P) and “ M: ii? \ $5 ﬁght: @2ng
k Dr {1le ﬂ” 5‘19 r‘\
i?” 'm “$3 1513‘“ my) “LEA; Fax:
A »~ LC;
’\”:>»%9¢w $3 :5“ P (d) (5 points) Suppose that our (P) and (D) are both feasible. State precisely the Strong Duality result for (P) and  I
mm'tw x . \r\ M W _
can a».
Cl?) it” Q” A @ﬂic‘ilx 9 (e) (7 points) We discussed how the simplex method can be used to provide a construc ative proof of the strong duality result above. In a few lines give an overview of that
argument. \ C t) :93 ab am; )wfﬁrg j
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