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I HC QUC GIA H NII HC KHOA HC T NHIN

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THI KT THC MN HCNM HC 2011-2012

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Mn thi: Ti u haM mn hc: 49ABCD S tn ch: 2 s: 1Dnh cho sinh vin kho: Lp K52A2 Ngnh hc: Ton - Tin ng dng

Thi gian lm bi 90 pht (khng k thi gian pht )

Cu 1. (2 im) Cho hai tp li X, Y v cc s thc ,. Chng minh rng cc tp sau ycng l tp li:

(i) X Y;(ii) X = {x|x X};

(iii) X + Y = {x + y|x X,y Y}.

Cu 2. (4 im) Cho bi ton quy hoch tuyn tnh

L(x) = x1 + 3x2 + x3 min

x1 + x2 + x3 5

x2 + 2x3 = 8

x1 + 2x2 + x3 10

xi 0(i = 1,2,3)

i) a v dng chnh tc;ii) Gii bi ton trn bng phng php n hnh;

Cu 3. (4 im) Trong v bo lt va qua c 4 im B1, B2, B3, B4 b ngp nng, cn tip tlng thc vi yu cu tng ng l b = (10, 15, 20, 20) theo n v tn. Nh nc b trlng thc cu tr ba kho A1, A2, A3 vi tr lng tng ng l a = (15, 20, 30). Cc phvn chuyn t Ai n Bj l cij nh trong ma trn sau y:

3 4 7 8

4 3 7 9

6 5 4 11

Hy lp k hoch vn chuyn ti u sao cho cc im cn cu tr nhn s lng lngthc v tng s tn /km l nh nht.

1. Thit lp bng vn ti vi phng n c s theo phng php gc ty bc.2. Gii bi ton bng phng php th v.

Ch : Cn b coi thi khng gii thch g thm.

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I HC QUC GIA H NII HC KHOA HC T NHIN

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P N V THANG IMMnn hc: Ti u ha

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M mn hc: 49ABCD S tn ch: 2 s: 1Dnh cho sinh vin kho: Lp K52A2 Ngnh hc: Ton - Tin ng dng

Li gii 1. [2 im]

(i) Cho x,y X Y v 0 1. Taphi chng minh x + (1 )y X Y. Tht vy,do x,y X Y nn x,y X. Do X l li nn x + (1 )y X; tng t ta cng cx + (1 )y Y. T suy ra x + (1 )y X Y.

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(ii) Gi s ta c x1, x2 X vi x1, x2 X v 0 1. V X l li nn x1 + (1 )x2 X. Ta xt (x1) + (1 )x2 = (x1 + (1 )x2) X.

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(iii) Chng minh tng t phn trn. 0.5Li gii 2. [4 im]

i) Dng chnh tc: a thm bin ph x4, x5 l bin ph

L(x) = x1 + 3x2 + x3 min

x1 + x2 + x3 x4 = 5

x2 + 2x3 = 8

x1 + 2x2 + x3 + x5 = 10

xi 0(i = 1,2,3,4,5)

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ii) Gii bng phng php n hnh: a thm x6, x7 l bin gi, bi ton tr thnh

L(x) = x1 + 3x2 + x3 + Mx6 + Mx7 min

x1 + x2 + x3 x4 + x6 = 5

x2 + 2x3 + x7 = 8

x1 + 2x2 + x3 + x5 = 10

xi 0(i = 1,2,3,4,5,6,7)

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J cJ xJ 1 2 3 4 5 6 7

1 3 1 0 0 M M6 M 5 1 1 1 -1 0 1 0

7 M 8 0 1 [2] 0 0 0 1

5 0 10 1 2 1 0 1 0 0

L(x) 0 -1 -3 -1 0 0 0 0

13 1 2 [3] -1 0 0 0

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Do cn tn ti gi tr ln hn 0 nn cha c phng n ti u ta cn tm bin a voCt c gi ln nh nht ng vi x3 vy bin a vo l : x3 Hng c gi tr nh nht ngvi ct l hng 2

J cJ xJ 1 2 3 4 5 6 71 3 1 0 0 M M

6 M 1 [1] 1/2 0 -1 0 1

3 1 4 0 1/2 1 0 0 0

5 0 6 1 3/2 0 0 1 0

L(x) 4 [-1] -5/2 0 0 0 0

1 1 1/2 0 -1 0 0

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Do cn tn ti gi tr ln hn 0 nn cha c phng n ti u ta cn tm bin a voCt c gi ln nh nht ng vi x1 vy bin a vo l : x1 Hng c gi tr nh nht ng

vi ct l hng 1

J cJ xJ 1 2 3 4 5 6 7

1 3 1 0 0 M M

1 1 1 1 1/2 0 -1 0

3 1 4 0 1/2 1 0 0

5 0 5 0 1 0 0 1

L(x) 5 0 -2 0 -1 0

0 0 0 0 0 0

Phng n ti u ca bi ton m rng l : x = (1,0,4,0,5,0,0)Gi tr hm mc tiu t c l : F(x) = 5

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Li gii 3. [4 im]

1. y l bi ton vn ti dng min. Ni cung cp hng a1 = 15, a2 = 20, a3 = 30,v ninhn hng b1 = 10, b2 = 15, b3 = 20, b4 = 20. D thy y l bi ton vn ti cn bngthu pht.2. Phng n c s theo phng php gc ty bc trong bng 1.

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Ai Bj B1 : 10 B2 : 15 B3 : 20 B4 : 20

3 4 7 8A1 : 15 10 5

4 3 7 9A2 : 20 10 10

6 5 4 11A3 : 30 10 20

Bng 1.

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2. Vng lp th nht:Bc 1. Phng n x khng thoi ha v |G| = m + n 1 = 7. Tm cc th v

u1 + v1 = 3

u1 + v2 = 4

u2 + v2 = 3

u2 + v3 = 7

u3 + v3 = 4

u3 + v4 = 11

Cho u1 = 0 suy ra v1 = 3, v2 = 4 v u2 = 1, v3 = 8, u3 = 8, v4 = 15.

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Tnh ij nh bng 2

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Ai Bj B1 : 10 B2 : 15 B3 : 20 B4 : 20

0 0 1 7 + u1 = 0A1 : 15 10 5

-2 0 + 0 5 u2 = 1A2 : 20 10 10

-7 -5 0 + 0 u3 = 4A3 : 30 10 20

v1 = 3 v2 = 4 v3 = 8 v4 = 15Bng 2

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vi phm du hiu ti u (2,3) l a vo. Ta c vng

V = {(1, 4)+

, (3, 4)

, (3, 3)+

, (2, 3)

, (2, 2)+

, (1, 2)

}

Lng iu chnh q = min{x34, x23, x12} = min{20, 10, 5} = 5. Tng ng vi (1,2) yl loi ra ta c bng mi bng cch tnh li

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Ai Bj B1 : 10 B2 : 15 B3 : 20 B4 : 20

0 -7 -6 0 + u1 = 0A1 : 15 10 5

5 + 0 0 5 u2 = 7A2 : 20 15 5

0 -5 0 + 0 u1 = 7

A3 : 30 15 15v1 = 0 v2 = -7 v3 = -7 v4 = -7

Bng 3.

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Tm cc th vu1 + v1 = 0

u2 + v2 = 0

u2 + v3 =0

u3 + v3 = 0

u1 + v4 = 7

u3 + v4 = 0

Cho u1 = 0 suy ra v4 = 7, v1 = 0, v3 = 7, v2 = 7, u2 = 7, u3 = 7.

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Tnh ij nh bng 3 vi phm du hiu ti u (2,1) l a vo. Ta c vng

V = {(2, 1)+, (1, 1), (1, 4)+, (3, 4), (3, 3)+, (2, 3)}

Lng iu chnh q = min{x11, x34, x23} = min{10, 15, 5} = 5. Tng ng vi (2,3) y

l loi ra ta c bng mi bng cch tnh liaaaaaa

Ai Bj B1 : 10 B2 : 15 B3 : 20 B4 : 20

0 -2 -6 0 u1 = 0A1 : 15 5 10

0 0 -5 0 u2 = 5A2 : 20 5 15 5

0 0 0 0 u1 = 0A3 : 30 20 10

v1 = 0 v2 = 5 v3 = 0 v4 = 0Bng 4

Vy phng n ti u phi tm l

X =

5 0 0 10

5 15 0 0

0 0 20 10

Chi ph ti u l F(x) = 350.

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H ni, ngy 15 thng 12 nm 2011NGI LM P N

(k v ghi r h tn)

PGS.TS. Nguyn Hu in

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