A bounded-search iterated greedy algorithm for the distributed permutation flowshop scheduling problem

Abstract

As the interest of practitioners and researchers in scheduling in a multi-factory environment is growing, there is an increasing need to provide efficient algorithms for this type of decision problems, characterised by simultaneously addressing the assignment of jobs to different factories/workshops and their subsequent scheduling. Here we address the so-called distributed permutation flowshop scheduling problem, in which a set of jobs has to be scheduled over a number of identical factories, each one with its machines arranged as a flowshop. Several heuristics have been designed for this problem, although there is no direct comparison among them. In this paper, we propose a new heuristic which exploits the specific structure of the problem. The computational experience carried out on a well-known testbed shows that the proposed heuristic outperforms existing state-of-the-art heuristics, being able to obtain better upper bounds for more than one quarter of the problems in the testbed.

Full text

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1†1
1
∗
†

n
F m
F DF |prmu|Cmax
DF |prmu|Cmax
m > 2

n
F m
m
i j
pij f
πfnff
π π =[π1, . . . , πf, . . . , πF]Cf
i,j
j i f Cf
max =Cmax(πf)
f Cmax =Cmax(π)
πf[i]
f i fmax
Cfmax
max

j j = 2 · · · n
j
n
O(n3·m)k k ∈[1 · · · n]
k O(n2·m)O(n2·m)
k O(n·m)

mini(pi,l)∀pi,l >0i∈[1,· · · , m]l
l k
H|H| ≥ 1
k k
k ph,l
h∈H minh∈H(ph,l)minh∈H(ph,l )≥mini∈[1,··· ,m](pi,l )
l

n−1
m∑m
i=1 pi,l i∈[1,· · · , m]l

f
f

•
d
π1
•d
f
•
n2·m/F n2·m/F
n3·m/F 2
n
F n/F
L n2·m/F
•
T emperature T
T emperature =T·∑∀i∑∀jpi,j
n·m·10
πd
d π1
Cref
max

π:=
f= 1 F
πf:= πf
flag :=
flag
π:= π
flag :=
n/F ≤L
flag :=
flag
π:= π
flag :=
πb:= π
π1:= π
i= 1 d
π1:= π1πD
π2:= ConstructionF unction(π1, πD)
f= 1 F
πf
2:= πf
2
π3:= π2
n/F ≤L
π3:= π3
Cmax(π3)< Cmax (π)
π:= π3
Cmax(π3)< Cmax (πb)
πb:= π3
random ≤exp{−(Cmax(π3)−Cmax(π))/T emperature}
π:= π3
πb

flag :=
flag
flag :=
f= 1 F
Cref
max =Cmax(πf)
πf[i]i f
πf[i]πf
Cmax(πf)< Cref
max
flag :=
πd
Cref
max
f Cf
max +mini(pi,πd)< Cref
max Cref
max
πdf Cref
max

π, πD
d= 1 D
πD[d]πf=1
Cref
max
pmin := πD[d]i
f= 2 F
Cf
max := f
Cf
max +pmin < Cref
max
πD[d]f
C0
max
C0
max < Cref
max
Cref
max =C0
max
π:= πD[d]
π
T, d, L

π
h(f) = 1, f = 1 ···F
i(f) = 1, f = 1 · · · F
πb:= π π =[π1,· · · , πf,· · · , πF]
fmax C′
max
i(fmax)< nfmax
j:= h(fmax)nfmax
π0:= πfmax [j]πfmax
Cfmax
max := π0
pmin := πfmax [j]i
f= 1 F
Cf
max := f
Cf
max +pmin < C′
max
πfmax [j]f
π:= πfmax [j]
fmax Cmax
Cmax < C′
max
C′
max =Cmax
i(fmax) = 1
πb:= π
i(fmax)++
h(fmax)++
πb

π
h(f) = 1, f = 1 · · · F
i(f) = 1, f = 1 · · · F
πb:= π π =[π1,· · · , πf,· · · , πF]
fmax C′
max
i(fmax)< nfmax
Caux
max =C′
max
flag :=
j:= h(fmax)nfmax
f= 1 F
g= 1 nf
f̸=fmax
π0:= πfmax [j]πfmax
π1:= πf[g]πf
C1
max πfmax [j]π1
P osf
C0
max πf[g]π0
P osfmax
C1
max < Caux
max&C0
max < Caux
max
flag :=
BestP osfmax := P osfmax
BestP osf:= P osf
choseng:= g
chosenf:= f
Caux
max =max (C0
max, C 1
max)
flag
πfmax := πchosenf[choseng]fmax
BestP osfmax
πchosenf:= πfmax [j]chosenf
BestP osf
π πfmax πchosenf
fmax Cmax
Cmax < C′
max
C′
max =Cmax
i(fmax) = 1
πb:= π
i(fmax)++
h(fmax)++
πb

d
T
L
d, L T
•T∈[0.1,0.2,0.3,0.4,0.5]
•d∈[3,4,5,6,7]
•L∈[15,20,25,30]
RP D =Cmax −Base
Base ·100
Cmax Base
n, m F
n∈[20,50,100,200] m∈[5,10,15,20] F∈[2,3,4,5,6,7]
n·m·F·1.5
d
d= 5 L= 20 T= 0.4

•
•
•
n∈[20,50,100,200,500] m∈[5,10,15,20] F∈
[2,3,4,5,6,7]
n·m·F·0.5n·m·F·1n·m·F·2
RP D
RP Di=Cmax,i −Best
Best ·100
Cmax,i i Best
ARP D n ·m·F·0.5
n·m·F·1

n m
ARP D n m
n·m·F·0.5
f
ARP D f
n·m·F·0.5
n·m·F·2

n m
ARP D n m
n·m·F·1
f
ARP D f
n·m·F·1

n m
ARP D n m
n·m·F·2
f
ARP D f
n·m·F·2

t n ·m·F·0.5
t n ·m·F·1
t
ARP D
p
n m
t n ·m·F·2

f
mini(pi,l)i∈[1,· · · , m]
m
f
m

n m
n m
t
p
n·m·F·2

m
n