Programação de Horários em Instituições Educacionais ... - Decom

XXXIX SBPO [2827]

×
×
XXXIX SBPO [2828]

×
XXXIX SBPO [2829]

XXXIX SBPO [2830]

XXXIX SBPO [2831]

XXXIX SBPO [2832]

XXXIX SBPO [2833]

p
f1(x), f2(x), . . . , fp(x) wk ∈ R +
k
f(x)
f(x) =
p
wk fk(x)
k=1
wk
×
XXXIX SBPO [2834]

P p ∈ P 1, . . . , |P |
T t ∈ T 1, . . . , |T |
D d ∈ D
1, . . . , |D|
H h ∈ H
1, . . . , |H|
˜ R |P |×|T | ˜rpt
p t
˜ P |P |×|D|×|H| ˜ppdh = 1
p d h ˜ppdh = 0
1 p t d h
0
xptdh =
xptdh ∀p ∈ P, t ∈ T, d ∈ D, h ∈ H
d∈D h∈H
xptdh = ˜rpt ∀p ∈ P, t ∈ T
xptdh ≤ 1 ∀t ∈ T, d ∈ D, h ∈ H
p∈P
t∈T
xptdh ≤ ˜ppdh ∀p ∈ P, d ∈ D, h ∈ H
xptdh ∈ {0, 1} ∀p ∈ P, t ∈ T, d ∈ D, h ∈ H
d∈D
h∈H ˜ppdh < |D|.|H|
XXXIX SBPO [2835]

×
×
XXXIX SBPO [2836]

×
×
P p ∈ P 1, . . . , |P |
T t ∈ T 1, . . . , |T |
D d ∈ D
1, . . . , |D|
H h ∈ H
1, . . . , |H|
˜ R |P |×|T | ˜rpt
p t
˜ P |P |×|D|×|H| ˜ppdh = 1
p d h ˜ppdh = 0
˜ M |P |×|T | ˜mpt
p t
˜mpt ∈ {1, 2} ∀p ∈ P, t ∈ T
˜ G |P |×|T | ˜gpt
p
t
XXXIX SBPO [2837]

W ′
|P | w ′
p
p
W ′′
|P | w ′′
p
p
W ′′′
|P | w ′′′
p
p
Gpd ⊂ H
p d
˜pptdh.˜pptdh+1 = 1∀h ∈ Gpd
F1
xptdh p
t d h (xptdh = 1) (xptdh = 0)
vpd bpd gpt
XXXIX SBPO [2838]

X X
p∈P d∈D
w ′
p.bpd + X
X
p∈P d∈D
p .vpd + X X
w ′′
p∈P t∈T
w ′′′
p .gpt
X X
xptdh = ˜rpt ∀p ∈ P, t ∈ T
d∈D h∈H
X
xptdh ≤ 1 ∀t ∈ T, d ∈ D, h ∈ H
p∈P
X
xptdh ≤ ˜ppdh ∀p ∈ P, d ∈ D, h ∈ H
t∈T
X
xptdh ≤ ˜mpt ∀p ∈ P, t ∈ T, d ∈ D
h∈H
vpd ≥ X
xptdh ∀p ∈ P, d ∈ D, h ∈ H
t∈T
apd ≤ (|H| + 1) − (|H| + 1 − h) X
t∈T
xptdh ∀p ∈ P, d ∈ D, h ∈ H
apd ≥ h. X
xptdh ∀p ∈ P, d ∈ D, h ∈ H
t∈T
bpd ≥ apd − apd + vpd − X X
xptdh ∀p ∈ P, d ∈ D
t∈T h∈H
yptdh ≤ xptdh ∀p ∈ P, t ∈ T, d ∈ D, h ∈ b Gpd
yptdh ≤ xptdh+1 ∀p ∈ P, t ∈ T, d ∈ D, h ∈ b Gpd
h+2 X
h ′ =h
≤ 1 ∀p ∈ P, t ∈ T, d ∈ D, h ∈ b Gpd|h < |H| − 1
gpt ≥ ˜gpt − X
X
yptdh ∀p ∈ P, t ∈ T
d∈D h∈ b xptdh ∈ {0, 1}
Gpt
∀p ∈ P, t ∈ T, d ∈ D, h ∈ H
yptdh ∈ {0, 1} ∀p ∈ P, t ∈ T, d ∈ D, h ∈ b vpd ∈ {0, 1}
apd ∈ {0, . . . , |H|}
apd ∈ {0, . . . , |H|}
bpd ∈ {0, . . . , |H| − 2}
Gpd
∀p ∈ P, d ∈ D
∀p ∈ P, d ∈ D
∀p ∈ P, d ∈ D
∀p ∈ P, d ∈ D
× vpd
p d
bpd
apd apd
p
t d h yptdh
yptdh xptdh
yptdh
gpt
XXXIX SBPO [2839]

p
t
n
r
f
s
XXXIX SBPO [2840]

n
XXXIX SBPO [2841]

xit i t
XXXIX SBPO [2842]

N
Ei i ∈ 1, . . . , N
B N B = E1, . . . , EN
D
T
M
CN×N cij
i j
tk
k
N−1
F1(i) =
proximidade(ti, tj) =
N
i=1 j=i+1
N−1
i=1 F1(i)
M
N
j=i+1
cij proximidade(ti, tj)
2 5 /2 |ti−tj| |ti − tj| ≤ 5
0
cij λ(ti, tj) = 0
λ(ti, tj) =
1 ti = tj
0
XXXIX SBPO [2843]

XXXIX SBPO [2844]

XXXIX SBPO [2845]

p1(i) i p1(i) =
j∈N(i) d(j)
N(i) i d(j)
j p1(i)
p2(i) =
j∈N(i) p1(j) pk+1(i) =
j∈N(i) pk(j)
i
p1
pk(i) k > 1
XXXIX SBPO [2846]

i
i
cj j
j
r Cr
Cr i
j r Cr
cj cj i
N (s)
s s
XXXIX SBPO [2847]

i rij
j
uij
Q
Q
XXXIX SBPO [2848]

Qm×p
m p p =
|D| |P |
i Q i
qik ∈ {−1, 0, 1, 2, . . . , n} i k
f(Q) = ω × f1(Q) + δ × f2(Q) + ρ × f3(Q)
ω δ ρ
f
ω > δ ≫ ρ
f1(Q)
k lk
k sk
k
f1(Q) =
p
(lk + sk)
k=1
Q
ei
i
f2(Q) =
m
i=1
ei
ω ′
i ω ′′
f3(Q) =
m
i=1
(ω ′
i × bi + ω ′′
i × vi + ω ′′′
i × gi)
bi
vi
gi
i ω ′′′
i
i
gi = max
0, geminadas(Q requerido
i
) − geminadas(Q corrente
i )
XXXIX SBPO [2849]

f2
f3
Q (f1(Q) =
f2(Q) = 0) j Gj = (Vj, Aj) Vj
j Aj
Aj = (k, ¯ k) : j k
¯ k ¯ k }
(k, ¯ k) ∈ Gj ∆fi(k, ¯ k)
i k ¯ k
f3
∆fi(k, ¯ k) = fi( ¯ k) − fi(k)
f(.) = (ρ × f3)(.)
Q1
i (i = P 1, P 2, P 3, P 4)
k (k = H1, H2, H3, H4, H5) qik
i k
fi
ρ = 1 ω ′
i
= 1 ω′′ i
= ω′′′
i
f(Q1) = fP 1 + fP 2 + fP 3 + fP 4 = 1 + 1 + 0 + 0 = 2
= 0 ∀i
Q1
fi
GA
(H1, H5) −1 P 1
H1 H5
1 (∆fP 1(H1, H5) = fP 1(H5) − fP 1(H1) = 0 − 1 = −1)
XXXIX SBPO [2850]

P1
1
-1
P2
P4 P3
P3
P4
-1
H1
H2
H1
H2
-1
H5
H3
GA
-1
1
0
H5
H3
G ′ A
Gj
{(H1, H5), (H5, H3), (H3, H1)}
−1 (= −1 + (−1) + 1)
Q ′ 1
G ′ A
GA Q ′ 1
f(Q ′ 1) = fP 1 + fP 2 + fP 3 + fP 4 = 0 + 0 + 1 + 0 = 1
1
P1
P2
0
1
H4
H4
Q ′ 1
fi
P2
P2
XXXIX SBPO [2851]

P1
P2
P1
P4
P3
H1
H2
H3
H4
H5
-1
0
H6
0
H7
H8
H9
H10
dia 1 dia 2
G ′ A
Q2
f
ρ = 1 ω ′
i
= 1 ω′′ i
P1
P3
P4
P4
P2
= 2 ω′′′
i
0
= 0 ∀i
f1(Q2) = f2(Q2) = 0
f(Q2) = fP 1 + fP 2 + fP 3 + fP 4 = 5 + 4 + 4 + 2 = 15
= −1 + 0 + 0 + 0
Q ′ 2
f(Q ′ 2) = fP 1 + fP 2 + fP 3 + fP 4 =
6 + 4 + 4 + 2 = 16
XXXIX SBPO [2852]

P1
P2
P3
P4
P5
H1
H2
H3
H4
H5
0
-2
H6
dia 1 dia 2
1
P1
-1
H11
dia 3
H6 H11
(k, k) ∈ Gj
L
Gj L
L
j Gj
Gj
j
P1
XXXIX SBPO [2853]

Intraturmas(Q, f(.))
j ← 1 ;
← ;
( j ≤ n )
( Gj
)
Gj j ;
← ;
( = j = n )
j ← 1 ;
← ;
j ← j + 1 ;
Intraturmas
j ¯j
j k ¯ k
cj(k, ¯ k)
i k ¯ k
ī
¯ k k
¯j
ī
k ¯ k
c¯j( ¯ k, k) ¯ k k G¯j
cj(k, ¯ k) + c¯j( ¯ k, k) < 0 ī
j ¯j ¯ k ¯j j k
XXXIX SBPO [2854]

i
i
k
k
0
c j (k,k)

Interturmas(Q, f(.))
j ← 1 ;
← ;
( j ≤ n )
( (k, ¯ k) ∈ Gj )
ī ¯ k ;
¯j ¯j = j ī k ;
cj(k, ¯ k) j
k ¯ k ;
c¯j( ¯ k, k) ¯j
¯ k k
( cj(k, ¯ k) + c¯j( ¯ k, k) < 0
)
Gj j ;
G¯j ¯j ;
;
← ;
( = j = n )
j ← 1;
← ;
j ← j + 1;
Interturmas
Q2
fi
Q ′ 2
fi
XXXIX SBPO [2856]

II(Q, f(.))
Q f1(Q) = 0
Q ∗
(f2(Q) = 0)
Q ← Intraturmas(Q, f2)
(f2(Q) = 0) Q ← Interturmas(Q, f2)
(f2(Q) = 0)
Q ← Intraturmas(Q, f3)
Q ← Interturmas(Q, f3)
Q ∗ ← Q
Q ∗
II
XXXIX SBPO [2857]

s 0
s ∗
s ← s0
s∗ ← s0
( )
N T
s ′ ∈ N (s)
s ← s 0
s
T
f(s) < f(s∗ )
s ∗ ← s
T
XXXIX SBPO [2858]

T1 T2
|T1| < |T2|
Q |P |×|D|×|H| qpdh
p d h
qpdh ∈ {1, . . . , |T |} qpdh = 0
×
XXXIX SBPO [2859]

\ · · · |D| × |H|
· · ·
× × · · ·
× × · · ·
· · ·
× · · ·
ω
d h c
t
max{c − 1, 0} × ω ω
× δ
Q
N (Q)
m
p ∈ P
(d1, h1) (d2, h2) qpd1h1 = qpd2h2 d1 × |H| + h1 < d2 × |H| + h2
N (Q) O(|P | · (|D| · |H|) 2 · EV )
EV
p (d1, h1) (d2, h2)
⊕
m
validadeT abu(m)
validadeT abu(m)
validT abu ϕ ∈ [0, 1]
validT abu = ∞
XXXIX SBPO [2860]

×
´ M |P |×|T |
´mpt p
t ´tpt
´mpt
´tpt =
max{ ´mpt|p ∈ P, t ∈ T }
´ppa1a2
p d1, h1 d2, h2 a1 = qpd1h1
a2 = qpd2h2 f(Q)
´ppa1a2 =
⎧
⎨
⎩
´tpa1 × f(Q) a1 = 0 a2 = 0
´tpa2 × f(Q) a1 = 0 a2 = 0
(´tpa1 + ´tpa2)/2 × f(Q) a1 = 0 a2 = 0
` M |P |×|T |×|D|×|H|
`mptdh
p t d h
`rptdh
`rptdh =
`mptdh
max{ `mptdh|p ∈ P, t ∈ T, d ∈ D, h ∈ H}
XXXIX SBPO [2861]

`pptdh
p t d h
`pptdh = `rptdh × f(Q)
ativDiv
itDiv
ativDiv
XXXIX SBPO [2862]

Q, ativDiv, itDiv, validTabu, ϕ
Q ∗
Q ∗ ← Q ListaTabu ← ∅
semMelhora ← 0 it ← 0
∆ ← ∞ it + +
melhorMov ←
m Q ⊕ m ∈ N (Q)
penalidade ← 0
semMelhora ativDiv < itDiv it ≥ ativDiv
penalidade m
∆ ′
← f(Q ⊕ m) − f(Q)
((∆ ′
+ penalidade < ∆) (m /∈ ListaTabu)) (f(Q ⊕ m) <
f(Q ∗ ) ∆ ′ < ∆)
melhorMov ← m
∆ ← ∆ ′
f(Q ⊕ m) ≥ f(Q ∗
) ∆ ← ∆ + penalidade
Q ← Q ⊕ melhorMov
validadeTabu(melhorMov) ←
random(⌊validTabu − ϕ × validTabu⌋, ⌈validTabu + ϕ × validTabu⌉) + it
ListaTabu
f(Q) < f(Q ∗ )
Q ∗ ← Q semMelhora ← 0
semMelhora + +
XXXIX SBPO [2863]

s 0
s ∗
t 0
t ← t0 s ← s0
s∗ ← s0
( )
s ′
∈ N (s)
∆f = f(s ′
) − f(s)
r [0, 1]
(∆f < 0) (r < e−∆f/t
)
s ← s ′
f(s) < f(s∗
)
s∗ ← s
t
XXXIX SBPO [2864]

P
s∗
s ∗ ← arg min
s∈P
f(s)
( )
i ← 1 nrF ilhos
s, s ′ ∈ P
s ′′′ ← (s ′ , s ′′ )
P ← P ∪ {s ′′′ }
f(s ′′′ ) < f(s∗ )
s∗ ← s ′′′
P
s ∗ ← arg min
s∈P
f(s)
P
XXXIX SBPO [2865]

F1
XXXIX SBPO [2866]

F1
h ′ , . . . , h ′′
XXXIX SBPO [2867]

p d h ′′ − h ′ + 1
h ′′
h=h ′ ˜ppdh = h ′′ − h ′ + 1 {h ′ , . . . , h ′′ − 1} ⊆ Gpd
h ′′ −1
yptdh ≤ vpd
|{h ′ , . . . , h ′′ }|
t∈T h=h ′
t
d h ′ , . . . , h ′′
p∈P h∈ b Gpd∩{h ′ ,...,h ′′ −1}
yptdh ≤
2
{h ′ , . . . , h ′′ }
2
p d t
h
h1∈ b Gpd\{h,h−1}
yptdh1 + xptdh ≤ vpd
d∈D
vpd ≥ max
t∈T ˜rpt
|H|
˜rpt
, max
t∈T ˜mpt
P
∀p ∈ P
t∈T rpt
|H|
d
p d
h∈H ˜ppdh < |H| ˙ppk
k p
k k
k=1 ˙ppk ≥
t∈T ˜rpt
|
k
˙ppk ≥
k=1
t∈T
vpd ≥ k ∀p ∈ P, k ∈ D
d∈D
k−1
˜rpt ∧ (k − 1 = 0 ∨
k=1
˙ppk <
t∈T
˜rpt)
XXXIX SBPO [2868]

max
p k k
k=1 ˙ppk ≥
t∈T ˜rpt
d ′ k−1
k=1 ˙ppk +
h∈H ˜ppd ′ h <
t∈T ˜rpt
|
k
d ′′ =1
˙ppd
vpd ≥
d∈D
′′ ≥ ˜rpt ∧ (k − 1 = 0 ∨
t∈T
k−1
k ′′ =1
k−1
d ′′ =1
˙pk ′′ +
h∈H
t∈T
(k + 1)xptd ′ h
∀p ∈ P, d ′ ∈ D
˙ppd
′′ < ˜rpt),
t∈T
˜ppd ′ h <
t∈T
˜rpt)
t
{ ˙pt1, . . . , ˙pt5} {5, 5, 4, 2, 2}
t
xptdh
˜mptvpd ≥
xptdh ∀p ∈ P, t ∈ T, d ∈ D
h∈H
˘ bpdh ˘ bpdh
h h + 2
h + 1 p d
Bpd
XXXIX SBPO [2869]

p d
h ∈ Bpd ˜ppdh = ˜ppdh+2 = 1
˘bpdh ≥ X
(xptdh + xptdh+2 − xptdh+1) − 1 ∀p ∈ P, d ∈ D, h ∈ b Bpd
t∈T
bpd ≥ X
˘ bpdh ∀p ∈ P, d ∈ D
h∈ b Bpd
F2
˘ Pp
p
˘xpjth = 1 p j
h t ˘xpjth = 0
j p cpj
˘gpjt p
t j λpjd
j d p λpjd = 1
λpjd = 0 d ′
p
(p ∈ P, j ∈ ˘ Pp)
˘ Ppd
˘ Pp F2
XXXIX SBPO [2870]

p∈P j∈ ˘ Ppd
j∈ ˘ Ppd
d∈D h∈D
d∈D j∈ ˘ Ppd
j∈ ˘ Ppd
λpjd.cpj +
w ′′′
p .gpt
p∈P d∈D j∈ ˘ Ppd
p∈P t∈T
λpjd = 1 ∀p ∈ P, d ∈ D
λpjd.˘xpjth ≤ 1 ∀t ∈ T, d ∈ D, h ∈ H
λpjd.˘xpjth = ˜rpt ∀p ∈ P, t ∈ T
λpjd.˘gpjt + gpt ≥ ˜gpt ∀p ∈ P, t ∈ T
λpjd ∈ {0, 1} ∀p ∈ P, d ∈ D, j ∈ ˘ Ppd
gpt ∈ IN ∀p ∈ P, t ∈ T
F2
F2 F2r
F2r
λ
µ ν π κ
p d ˘xth
(t ∈ T, h ∈ H) ˘xth = 1 p
t d h ˘xth = 0
p d Ppd
F2
XXXIX SBPO [2871]

Ppd =
⎧
⎪⎨
cpd = ˘cpd − ˘ dpd
X
˘xth ≤ min{ ˜mpt, ˜rpt} ∀t ∈ T
h∈H
˘cpd = ˘b.w ′
p + ˘v.w ′′
˘dpd =
p
µpd + X X
νtdh.˘xth +
t∈T h∈H
X X
.πpt.˘xth +
t∈T h∈H
X X
a ≤
t∈T
X
(+h − |H|).˘xth + |H| ∀h ∈ H
a ≥
t∈T X
h.˘xth ∀h ∈ H
˘b =
t∈T X X
1 + a − a − ˘xth
X
t∈T h∈H
˘xth ≤ ˜ppdh.˘v h ∈ H
t∈T
h∈ b ˘yth.κpt
Gpd ⎪⎩
˘yth ≤ ˘xth ∀t ∈ T, h ∈ b ˘yth ≤
Gpd
˘xth+1 ∀t ∈ T, h ∈ b ˘yth ≥
Gpd
˘xth + ˘xth+1 − 1 ∀t ∈ T, h ∈ b ˘xth ∈ {0, 1}
˘yth ∈ {0, 1}
Gpd
∀t ∈ T, h ∈ H
∀t ∈ T, H ∈ b ˘v ∈ {0, 1}
a ∈ IN
a ∈ IN
˘b ∈ IN
Gpd
Ppd ∀p ∈ P, d ∈ D
|P | × |D|
F2r ∄ cpd < 0 ∀p ∈ P, d ∈ D
Ppd
|H| |H|
˜mpt
F2
F1
F2
F ′ 2 ăpdh p
h d ăpdh = 1 ăpdh = 0
ăptdh p h
t d ăptdh = 1 ăptdh = 0 ˘ Ppdh
˘ Ppd ˘ Ppdh
h ˘ Pptdh ˘ Ppd
XXXIX SBPO [2872]

h p t
λ
λ
h∈{0,...,|H|}
j∈ ˘ Ppdh
ăpdh = 1 ∀p ∈ P, d ∈ D
λpjd = ăpdh ∀p ∈ P, d ∈ D, h ∈ H
ăpdh ∈ {0, 1} ∀p ∈ P, d ∈ D, h ∈ H
ăptdh = 1 ∀p ∈ P, t ∈ T, d ∈ D
h∈{0,...,|H|}
j∈ ˘ Pptdh
λpjd = ăptdh ∀p ∈ P, t ∈ T, d ∈ D, h ∈ H
ăptdh ∈ {0, 1} ∀p ∈ P, t ∈ T, d ∈ D, h ∈ H
F ′ 2 P ′ pd
p d Ppd
λ P ′ pd
Ppd ăh ∈ {0, 1} ∀h ∈ {0, . . . , |H|}
h p d ăh =
1, ăh = 0 ăth ∀t ∈ T, h ∈ {0, . . . , |H|}
h p
t d ăth = 1, ăth = 0
h.ăh =
h∈H
h∈{0,...,H}
h∈H t∈T
˘xth
ăh = 1
ăh ∈ {0, 1} ∀h ∈ {0, . . . , |H|}
h.ăth =
h∈H
h∈{0,...,H}
h∈H
˘xth ∀t ∈ T
ăth = 1 ∀t ∈ T
ăth ∈ {0, 1} ∀t ∈ T, h ∈ {0, . . . , |H|}
P ′ pd ˘ dpd
˘ dpd P ′ pd ι ϑ
XXXIX SBPO [2873]

˘dpd = µpd +
νtdh.˘xth +
πpt.˘xth+
t∈T h∈ b Gpd
t∈T h∈H
t∈T h∈H
˘yth.κpt +
ăh.ιpdh +
h∈H
t∈T h∈H
ăth.ϑptdh
F ′ 2
F ′ 2
h.ăpdh =
h∈H d∈D
h∈H d∈D
t∈T
˜rpt ∀p ∈ P
h.ăptdh = ˜rpt ∀p ∈ P, t ∈ T
p ˙ C
(d1, h1), . . . , (d ˙c, h ˙c), di ∈ D∀i ∈ {1, . . . , ˙c}, hi ∈ H∀i ∈ {1, . . . , ˙c}
i∈{1,..., ˙c}
hi.ăpdihi >
F ′ 2
i∈{1,..., ˙c}
t∈T
˜rpt
ăpdihi ≤ ˙c − 1
|H|
ă
XXXIX SBPO [2874]

F ′ 2
PL PL
Si
ăpdh ∀p ∈ P, d ∈ D, h ∈
{0, . . . , h} Spt
ăptdh ∀p ∈ P, t ∈ T, d ∈ D, h ∈ {0, . . . , h}
F ′ 2
cpd >= 0∀p ∈ P, d ∈ D
Ppd
F ′ 2 Ppd
x
XXXIX SBPO [2875]

λ f(λ)
PL
F ′ 2 λ
PL
novasColunas ← 1
novosCortes ← 0
(novosCortes + novasColunas > 0)
novasColunas ← 0
p ∈ P
d ∈ D|
h∈H ˜ppdh
> 0
P ′
pd
(cpd < 0)
PL
novasColunas ← novasColunas + 1
novosCortes ← 0
p ∈ P
Sp
PL
novosCortes ← novosCortes + 1
PL λ ă
(novasColunas = 0) ((λ))
XXXIX SBPO [2876]

XXXIX SBPO [2877]

XXXIX SBPO [2878]

XXXIX SBPO [2879]

XXXIX SBPO [2880]

XXXIX SBPO [2881]

XXXIX SBPO [2882]