Efficient constructive and composite heuristics for the Permutation Flowshop to minimise total earliness and tardiness

Abstract

In this paper we address the problem of scheduling jobs in a permutation flowshop with a just-in-time objective, i.e. the minimization of the sum of total tardiness and total earliness. Since the problem is NP-hard, there are several approximate procedures available for the problem, although their performance largely depends on the due dates of the specific instance to be solved. After an in-depth analysis of the problem, different cases or sub-problems are identified and, by incorporating this knowledge, four heuristics are proposed: a fast constructive heuristic, and three different local search procedures that use the proposed constructive heuristic as initial solution.

Full text

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F m|prmu|Ej+Tj
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F m|prmu|Tj
ΠIO := (πI O
1, . . . πI O
n)
Π1Π1=
(π1
1) = (πIO
1)
k= 2 k=n πI O
k
Πk−1k

πIO
kΠk−1
lΠk= (πk−1
1, . . . πk−1
l−1, πI O
k, πk−1
l, . . . πk−1
k−1
F m|prmu|Tj
n
n2·m n·m O(n3m)O(n2m)
F m|prmu|Tj

n
m j
tij djΠ := (π1, . . . πn)Cij(Π)
j i Π
Cm,πn(Π) = Cmax(Π)
Cij(Π)
Cij(Π) = max{Ci−1,j(Π), Ci,j −1(Π)}+tij
jΠTj(Π) = max{Cmj(Π)−dj,0}
Ej(Π) = max{dj−Cmj(Π),0}
Tj(Π) = ∀jmax{Cmj(Π) −dj,0}Ej(Π) = ∀jmax{dj−Cmj(Π),0}
F m|prmu|Ej+Tj
IF m|prmu|Ej+TjW M
dj≥W M ∀jI
F m|prmu| − CjI
W M
djCmj(Π) dj≥W M ≥Cm,j (Π),∀j, Π
∀jmax{Cm,j(Π) −dj,0}+∀jmax{dj−Cm,j(Π),0}= 0 + ∀jdj−Cm,j (Π) =
∀jdj−∀jCm,j(Π) = const −∀jCm,j(Π)
IF m|prmu|Ej+Tjdj≤
m
i=1 tij ∀jI
F m|prmu|CjI
dj≤tj∀j Cm,j(Π) ∀Π
djtjj∀jmax{Cm,j(Π) −

dj,0}+∀jmax{dj−Cm,j(Π),0}=∀j(Cm,j(Π) −dj) + 0 = ∀jCm,j(Π) −∀jdj=
∀jCm,j(Π) + const
F m|prmu|Cj
F m|prmu| − Cj
F m|prmu|Ej+Tj
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•
•
F m|prmu|Ej+Tj
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ξujk (Π) Πk:= (π1, ..., πk)
kUkujk
j j ∈[1, n −k]N Tk
kUkUkξujk (Πk)
Πkk+ 1 Πk+1
F m|prmu|Cj
F m|prmu| − Cj
F m|prmu|Ej+Tj
•F m|prmu|Cj
F m|prmu|Cj
ξ1
ujk (Πk)
uik
j∈[1, n −k]
ξujk (Πk) = ξ1
ujk (Πk) = (n−k−2)
4·ITujk (Πk) + Cm,ujk (Πk)
ITj(Πk)
ITujk (Πk) =
m

i=2
m·max{Ci−1,ujk (Πk)−Ci,πk(Πk),0}
i−1 + k·(m−i+ 1)/(n−2)

k
a
a·100% NTk/(n−k)≥a)a
F m|prmu|Cja
•F m|prmu| − Cj
F m|prmu| − Cj
ξujk (Πk) = −ξ1
ujk (Πk)
n−k > 3k
Cm,ujk (Πk)< dujk ∀j N Ek=n−k N Ek
Uk(n−k)·c
b·(n−k)≤NEk< n −k
ξ2Eujk (Πk)−ξ1
ξujk (Πk) = ξ2
ujk (Πk) = −(n−k−2)
4·ITujk (Πk)−Cm,ujk (Πk) + Eujk (Πk)
b c
•F m|prmu|Ej+Tj

ξujk (Πk)
NTk/(n−k)≥aξ1
ujk (Πk)
NTk>3
Cm,ujk (Πk)< dujk ∀j
−ξ1
ujk (Πk)
n−k > 3
NEk=n−k
Cm,ujk (Πk)< dujk ∀j
ξ2
ujk (Πk)
n−k > 3
b·(n−k)≤NEk< n −k
ξ3
ujk (Πk)
ξujk (Πk) = ξ3
ujk (Πk) = Eujk (Πk)
ξujk (Πk)
ITujk (Πk)
O(m)
m
m(n+1)·n
2·m∼O(n2·m

ITj,0CTj,0Ej,0
π1:= Ej,0ITj,0
Π1= (π1)
k= 1 n−1
ITujk (Πk)Cm,ujk (Πk)N TkN EkEujk (Πk)∀j∈[1, n−
k]
NTk/(n−k)≥a&NTk>3
j= 1 k−n
ξujk (Πk) = ξ1
ujk (Πk) = (n−k−2)
4·ITujk (Πk) + Cm,ujk (Πk)
&n−k > 3 & N Ek=n−k
j= 1 k−n
ξujk (Πk) = −ξ1
ujk (Πk) = −(n−k−2)
4·ITujk (Πk)−Cm,ujk (Πk)
&n−k > 3 & b·(n−k)≤N Ek< n −k
j= 1 k−n
ξujk (Πk) = ξ2
ujk (Πk) = −(n−k−2)
4·ITujk (Πk)−Cm,ujk (Πk) + Eujk (Πk)
j= 1 k−n
ξujk (Πk) = ξ3
ujk (Πk) = Eujk (Πk)
α ξujk (Πk)k
ITujk (Πk)
Πk+1 αΠk

(Π,Ej+Tj) = ACH1()
(Π,Ej+Tj) = BLS(Π,Ej+Tj)
P1P2
P1 = min{PAS, Pedd}
P2 = max{PAS, Pedd}
Pedd πjPAS πj
EπjTπj
•
πjP1P2
•
πjP1P2n
•

BLS Π, OF
OFb=OF
j= 1 n
Π0:= πjΠ
P1P2
πjP1P2 Π0
Π := πjj∈[P1, P 2] Π0
OF ′
OF ′< OFb
OFb=OF ′
Πb:= Π
ΠbOFb
(Π,Ej+Tj) = ACH1()
(Π,Ej+Tj) = iBRLS(Π,Ej+Tj)

iBRLS Π, OF
OFb=OF
h= 1
i= 1
Πb:= Π
i <=n
j:= h n
Π0:= πjΠ
P1P2
πjP1P2 Π0
Π := πjj∈[P1, P 2] Π0
OF ′
OF ′< OFb
OFb=OF ′
i= 1
Πb:= Π
i+ +
h+ +
ΠbOFb

(Π,Ej+Tj) = ACH1()
(Π,Ej+Tj) = iLS(Π,Ej+Tj)
iLS Π, OF
OFb=OF
flag :=
flag =
flag :=
j= 1 n
Π0:= πjΠ
πjΠ0
Π := πjjΠ0
OF ′
OF ′< OFb
OFb=OF ′
Πb:= Π
flag :=
ΠbOFb
πj
F m|prmu|Ej+TjΠkk
l j ∈[1, k + 1]

Πkl
j j
a b c
• B1
n={50,150,250,350}
m={10,30,50}T={0.2,0.4,0.6}R={0.2,0.6,1.0}T R
P·(1 −T−R/2) P·(1 −T+R/2)

P
• B2
n={50,150,250,350}m={10,30,50}
T={0.2,0.4,0.6}R={0.2,0.6,1.0}
a b c
B1
•a={0.8,0.85,0.9,0.95,1}
•b={0.4,0.45,0.5,0.55,0.6}
•c={25,30,35,40,45,50,55}
RP D1 = OF −Base
Base ·100
OF Base
p a b c
c a b c
a= 0.90 b= 0.55 c= 30

•F m|prmu|Tj
•
•
F m|prmu|Cj
•
F m|prmu|Cj
•
F m|prmu|Cj
•
F m|prmu|Cmax

F m|prmu|Ej+Tj
•
•
•
ARP D
ACT
ACTi=∀jTi,j
J
ARP Di=∀jRP D2i,j
J
Ti,j i j
J ARP DiRP D2i,j
i
RP D2i,j =OFi,j −BestConstj
BestConstj
·100
OFi,j i j BestConstj

AC T ARP Ti
i
ARP Ti=∀jRP Ti,j
J+ 1
RP Ti,j i j
RP Ti,j =Ti,j −ACTj
ACTj
RP D RP T e
ARP D ARP T
A
B2
ARP D n m
n m AC T ARP T
ACT
ARP T
ARP Ds

A
•1
•2 2
•3 4
•4 6
•5 8
•6 10
•7 12
•8
p
A

RP D
ACT
ARP T
ACT ARP T

T
T
T
R
R
R
n
n
n
n
m
m
m
ARP D

ARP D ACT ACT
AARP D
A
n·m·t/2
t= 5,10,15,20,25,30
ARP D

ARP D ARP T ACT
i Hip α/(k−i+ 1)
8
10
12
2
4
6

t
ARP D I LS
ARP Ds
ARP D t = 10
ARP D t = 30
•1t= 5
•2t= 10
•3t= 15
•4t= 20
•5t= 25
•6t= 30
p

i Hip α/(k−i+ 1)
t= 5
t= 10
t= 15
t= 20
t= 25
t= 30
F m|prmu|Cj
F m|prmu|−Cj
ARP D