Ordonnancement d’ateliers de traitements de surfaces pour ...
Intro Ordonnancement
Transcript of 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
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t1, . . . , tn ti < tj
i < j
O(n log n)
2
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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 = ∞
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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,
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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
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p
D
D
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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
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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]
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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
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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
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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
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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
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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
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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
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