intro ordonnancement

7/26/2019 Intro Ordonnancement

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9

F: fondations

M: murs

E: électricité T: toit

I: peinture intérieure

X: peinture extérieureP: plomberie T: fenêtres

N: nains de jardin



t1, . . . , tn ti < tj

i < j

O(n log n)

2



d

a

b

d

f

h

i

a b c

e

g

d

a b c

fe

p = ∞

t P (t)

∅

P (∅) = −∞

p

P (t)

d := 1

p

p

d

td,k d k

d := d + 1

p = 1 p = ∞



td,1 = ∅

P (td,1)

t > d t

> d t ≤ t

P (t) ≤ P (td,1)

P (t) ≤ P (td,1)

d

a

b

P (t) := t

2

P (t) := t

3

P (t)

t P (t) := 0

1, 2,



d P (td,2) < P (t) t td,2

td,2 = ∅ td,1 < t

P (t) ≥ P (td,1) t < t

T t

td,2 T t P (t) ≥ P (td,1)

d0 := 0 d1, d2, . . .

B0, B1, B2, . . .

Bk :=

tdk+1,1, tdk+1,2, . . . , tdk+1,1

.

B0 B1

2

2

2

2

p

3

e

a b c

f

d



p

D

D



2

n p

1 2 O(n + m)

3

p

∞ O(n + m)

m

n

I

i

ti i i

C i i

C

C

:= (C i)

pi i

pi,j (i, j)

pik i k

ri i

di i

T i i T i := (0, C i − di)

U i U i = 1 0



wi i

aij tj − ti ≥ aij

<

4

a

a 2 (a)

a 1 a

1

2

p1, . . . , pn m

i [ri, di]

A,B,C,D

pA = pB = pD = 2 pC = 4 dA = dD = 5 dB = 3 dC = 4 rA, rB, rC = 0 rD = 2

5 [0, 1] [1, 2] [2, 3] [3, 4] [4, 5]



ri di

(di) − (ri)

p1, . . . , pn m

C max

m := 3 n := 5 p1 := 10 p2 := 8 p3 := 4

p4 := 14 p5 := 1

B :=

i pi, 1

m

ni=1

pi

.

B ≤ C max

B = C max

m := 3 n := 5 p1 := 10 p2 := 8 p3 := 4

p4 := 14 p5 := 1

B pi ≤ B

ni=1 pi ≤ mB

B :=

i pi,

1

m (n

i=1 pi)

.

t := 0 k := 1

i

t + pi ≤ B i k t t + pi

t := t + pi



i k t B k + 1

0 pi − (B − t) k := k + 1 t := pi − (B − t)

m := 4 ( p1, . . . , p7) := (5, 10, 2, 8, 3, 4, 20)

m− 1

p1, . . . , pn m

C max

i pi

C max i j

pij

pij 4 3 C max

xij i j



xij

m

j=1xij ≤ C max, i = 1, . . . , n

ni=1

xij ≤ C max, j = 1, . . . , m

mj=1

xij

pij= 1, i = 1, . . . , n

xij ≥ 0, i = 1, . . . , n j = 1, . . . , m

C max

xij

C

:= 9

C

xij

C

xij M

ci lj

ci := C max −m

j=1 xij i

lj := C max −ni=1 xij j

1

C

M

M

M =

lk=1

λkP k,

P k

k λk = C

k λk P k



i j ti tj

tj − ti ≥ aij

{1, 2, 3, 4, 5}

i pi

2 3

2 3

4 1 2

5 3

0 6 t6 − t0

I



1 2

3

2

−11

i j C ij i j

C ij

C ij

l(

i, j) := −∞

C ij

l(i, j) c

i j l(i, j)

i ri := l(0, i)

ri



ri

0 O(m) m

ti

ri

O(n3)

ri

f i := l(0, n + 1) − l(i, n + 1).

ri ≤ ti ≤ f i

f i − ri i

i f i − ri = 0

i

ti

tn+1−t0 ti

1 2

intro ordonnancement

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