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k

=

k

′

/

j

k

/ 1

k

′

/

j

k

′

<

0

#

j

#

k

′

!0 N ! ! # A # > # !

! A! ! !! F ! ! A! ! A !

-$#

O

(

n

2 )

5+!6 8

! ! A$ ! ! ," P % '# #

5 ! ! !! J K5 $

A # 5 !% ! J

= 1

K

A! 5 ! P V $ A! !!

! ! ! !! A! ! 5

! P ! > # $ # ! > $5

! # ! $ ! ! A ! !

! ! %

! ! A!!!!! !

A$ ! ! ! !! !

(52)

0

50

100

150

200

250

300

0

20

40

60

80

100

120

140

160 Solution

Jobs

GRASP with local search 1

GRASP with local search 2

GRASP with local search 3

VNS

6 H/ * # A! (&" $ !

A!-

65> #,%"%( $+(+

'!A!#- !

A! V # + !! 5

T 7 H # ! ! - T5 H 734

P2 #*+ $

&`,-* ! AF !A! A

" H T + % # ! !"#$ $

!

6 PO/ ! V #*+"

& , "!

*+" P ," P/ N "A !!

*+" 6 ," 6/ N "A V !

*+" H ," H/ '# N V !

(53)

0 1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

0

20

40

60

80

100

120

140

160 Solution

Jobs

GRASP with local search 1

GRASP with local search 2

GRASP with local search 3

VNS

6 T/ *# A!! (&" $!

A! -

0 1000

2000

3000

4000

5000

6000

7000

8000

0

20

40

60

80

100

120

140

160 Solution

Jobs

GRASP with local search 1

GRASP with local search 2

GRASP with local search 3

VNS

6 B/ * # A! ! (&" $ !

(54)

$*9*)

!-5I5A!P75PB56756B5H75O75P77 PB7>5 )!

$H7A! $ ! )!

V ! !

# # $ ! ! %- P75777 1 5 !

!

! ! # A >5 ! X

$ # (&" # ! ! !(&" ! A! *+"

! >5 A ! V # *+" !

# ! ! f 6 H ! ! ! % #

! (&"!5 6 PO!A ! V #*+"

0

20

40

60

80

100

120

140

0

20

40

60

80

100

120

140

160 Time [sec]

Jobs

GRASP with local search 1

GRASP with local search 2

GRASP with local search 3

VNS

6 O/ Y ℄ # A! ! (&"

$! A! -

6 B 56 H 6 T!#!!#*+"

!(&"5 !$#/ ! 5

A!!5!

'!!# 5!(&"!

A! ! O7>

! J6 BK!(&"!#

(55)

!##!5 A! ! P77>!(&"

! !A!- J6 T K5

- 5 ! (&" #5 A! O7

>

+ # ! 5 ! # ! ! A ! (&"

+A6 O 5!(&"!A

# *+"5$#

6 8! +% # ! # A!PB7 >

R 5 !! (&" ! ! *+" P H5

QBU# ! ) A! A !% # ! (&"

! *+" 65 ! (&" ! $ A!

6 P8!A ! +# ! 5 ! ! #

!%#JPP7K!! 5#PP67!

%$ J6PH7K ! R ! ! !

5$#

2

4

6

8

10

12

14

16

18

20

22

0

1

2

3

4

5 APD

Algorithm

Confidence Intervals for 150 job instances

GRASP with local search 1

GRASP with local search 2

GRASP with local search 3

VNS

6 8/ +% #PB7> *+"P5*+"6 *+"H

! *+" ! ! P56 H5 $ J

(56)

6 P8/ + # !(&" ! )! # V

#>

65A ("- $#%7+ % $ $+$%.

'!! A A! !! $ X F

! A!

! *+" ! $ X @-5 A$ $

V ! ! A!*+"A

!! (&" # ! ! A

R # - ! (&" ! A!

!#*+"5!A! V! +$4 !

5A !!(&" $ #A!O7>

5$# ! ! #! V!

(57)

F$A!!5!!*+"

!&)3 % ! % #! V A

! ! ! # $4!

# &)3%! '+ -2 !A$4!

! A *+" - ! A *+" &)3

" ! (&" ! # # !

A! 5 #!AFA $#

(58)

854 "

'!!5A V 5 +2! 2

!

! > # AF A! ! # ! -

` AX

! !

' ! ! ! AF !

A! !5 5 ! !

> ! '# > $

! )!>! ! %!5 ! !!A

>! 'F! > V $!

! R!!%!>5! #!

- !-!!! #!->J

% K V # ! # > ! ! )! >

! J- A! ! > ! K A! J$K

!!AA! - 5!A$!>#

! > A!

856 $(+

! #A A$ ! 5 # $4 A #

!5A % ! A !! !

(59)

' #>

# #!

im

#> $!

ii

′

m

# !5 #> > -

i

#>

ν

i

$ X #>

ii

′

m

A ! # > ! #

(

i, i

′

)

g

ii

′

_m

=

p

_im

+

s

_ii

′

_m

C

m

&# !

$ "X$

+*$*)

! % $

JPQO7 K5#! >!! +5 ! $

! # % !

A

• ti

/ # >

i

M

• α

im

/

1

#>

i

!

m,

0

!AM

• β

ii

′

_m

/

1

#>

i

′

!5

i

#

i

′

_,

0

!AM

• ρ

i

/ #>

i

M

• Z

/ F J- K

Z

m

F# !

m

Z

t

F

, % %

E

m

!# # >

(

i, i

′

)

!A $!

!

m

!', A

min

Z

+

N

i

(

ρ

i

· ν

i

)

,-(.

M

m

α

im

= 1

,

∀

i

∈

N

,-*.

Z

≥

t

i

+

p

im

−

(1

−

α

im

)

· G,

∀

i

∈

N, m

∈

M

,--.

(60)

(1

−

α

im

)

· G

+ (1

−

α

i

′

m

)

· G

+ (1

−

β

ii

′

m

)

· G

+

t

i

′

≥

t

i

+

p

im

+

s

ii

′

m

,

∀

(

i, i

′

)

∈

E

m

, m

∈

M

,-0.

(1

−

α

im

)

· G

+ (1

−

α

i

′

m

)

· G

+

β

ii

′

m

· G

+

t

i

≥

t

i

′

+

p

i

′

m

+

s

i

′

im

,

∀

(

i, i

′

)

∈

E

m

, m

∈

M

,-1.

α

im

∈ {

0 ,

1 }

,

∀

i

∈

N, m

∈

M

β

ii

′

m

∈ {

0 ,

1 }

,

∀

(

i, i

′

)

∈

E

m

, i

′

∈

N, m

∈

M

t

i

, ρ

i

≥

0 ,

∀

i

∈

N

Z

≥

0

,-0. ,-1. 2 3

4" ,(51+.

3

46

3

β

2 44 3

) 6

,-/.

ρ

i

2 3 7 2

4 6 8 2 4

2 6 4 4 3

4

3 6 2 6

4 ,--.

9 2 2 :

; ,-*. & 4 4

+*$*$ "

49 ,(505. 3

2 6 4

2

3

• t

(o)

m

8

o

th

2

m

• α

(o)

im

8

1

2

i

4

m

o

th

0

<

• β

ii

(o)

′

m

8

⎧

⎪

⎨

⎪

⎩

1

2

i

′

4

m

o

th

₍

_o

_{+ 1)}

th

4

0

(61)

• Z

8

96 4 47

9 =2 3

min

Z

+

N

i

(

ρ

i

· ν

i

)

,->.

M

m

|N

|

o=1

α

(o)

im

= 1

,

∀

i

∈

N

,-?.

N

i

α

(o)

im

≤

1 ,

∀

m

∈

M, o

= 1

, . . . ,

|

N

|

,-5.

N

i

α

(o)

im

≤

N

i

α

(o−1)

im

,

∀

m

∈

M, o

= 2

, . . . ,

|

N

|

,-(+.

d

i

+

ρ

i

≥

t

(o)

m

+

p

im

−

1 −

α

(o)

im

· G,

∀

m

∈

M,

∀

i

∈

N, o

= 1

, . . . ,

|

N

|

,-((.

Z

≥

t

(|N

m

|)

+

N

i

α

(|N|)

im

· p

im

,

∀

m

∈

M

,-(*.

t

(1)

m

= 0

,

∀

m

∈

M

,-(-.

β

ii

(o−1)

′

m

≥

1 −

2 −

α

(o−1)

im

−

α

(o)

i

′

m

· G,

∀

m

∈

M,

∀

i, i

′

∈

N, o

= 2

, . . . ,

|

N

|

,-(/.

t

(o)

m

≥

t

(o−1)

m

+

N

i

α

(o−1)

im

· p

im

+

N

i

N

−{i}

i

′

β

(o−1)

ii

′

m

· s

ii

′

m

,

∀

m

∈

M, o

= 2

, . . . ,

|

N

|

,-(0.

α

(o)

im

, β

(o)

ii

′

m

∈ {

0 ,

1 }

,

∀

i

∈

N, i

′

_∈

_{N, m}

_∈

_{M, o}

_{= 1}

_{, . . . ,}

_|

_N

_|

ρ

i

≥

0 ,

∀

i

∈

N

t

(o)

m

≥

0 ,

∀

m

∈

M, o

= 2

, . . . ,

|

N

|

Z

≥

0

2

47 42

32 ,->.6 7

4 @2:4 ,-?. 2

,-5.

32

o

−

1

2

o

≥

2

,-(+. 3 2

o

th

_m

(62)

;2= 4

ρ

i

,-((. 3 ; 2

4 ;

,-(*.

3 42 6 ;7,-(-.

3 2

i

2

i

′

o

−

1 o

m

2

i

′

m

2

i

o

−

1

2

i

′

o

,-(/.

3 2 ;

2

,-(0. ,-(/.

,

α

(o−1)

im

=

α

(o)

i

′

m

= 0

. ,-(/.

β

(o−1)

ii

′

m

≥

1 −

2

· G

β

(o−1)

ii

′

m

0

1

7 ,-(0.

β

(o−1)

ii

′

m

;

0

858 .$ %. % (-".#

0 / + 2!?2

/ "

node

/ 20

node

U B

←

node

M

/

n

∈

03

node

/

n

< U B

n

M

U B

=

M

f irstN ode

M

U B

M

)A) 6 4 )A) 8

' 6

,.

@

4 4

: 3

,. , 7 .

(63)

-( 7

0/ F# ! 2?2 !4 !J*+"K

bestSol

M45 M

localSol

M45 M

! "

iterations

M#

1

M

bestSol

M

/ ! "

"

iterations

jobs

←

1 J

N

KM

/

i

←

1 , iterations

jobsRand

←

$

jobs

M

jobsRand

M

%M

/

localSol < bestSol

bestSol

←

localSol

M

+*+*) 2 - %3"

' 7 4

6 6 3 8

3

3 4 , *++/.

424,.'

4 44 6 4

4

f

(

x

)

∼

1 /x

f

(

x

)

4

x

th

2

: & 44 2 34

: 2

%

,. 4 3

4 ; & 8 6

@4 4

(64)

+*+*$

3 3 *

6 2 ;

6 32;

36 2 4 @'' 32 6

6 : @

2 3

--3 3

B 62

2 @2 6

3

-/ &-(

3 6 !)

&

i

!),

i

. C) 3 :

0/ !!

/ #

"

i

/ 22

&

1 ,

1

M

job

←

jobs

[

i

]

M

/

m

←

'( ) "

job, m

M

/

presentSol

< bestSol

#

i

+ 1

M

*

job, m

M

+*+*+

3 4 )A)!)

4

4 !) ,

LB

k

.

,

LB

t

. 3

LB

k

(65)

()M1

()M2

()M1

()M2

(1)M1

()M2

(1,3,5)M1

(2,4)M2

(1)M1

()M2

(1,3,5)M1

(2,4)M2

(1,5)M1:3

(2,4)M2

(1)M1:3>5

(2,4)M2

()M1:3>5>1

(2,4)M2

()M1:3>5>1

(2)M2:4

()M1:3>5>1

()M2:4>2

First Branching.

Each time a job is assigned to a machine, a new node in the

enumeration tree is created.

Second Branching.

The process decides the processing order on each

machine, at a time. On each machine, the first job

to be processed is chosen, then the second, until

there are no more jobs assigned to that machine.

At node Z a feasible solution is completed. All the

jobs are sequenced.

1,

A

B

Z

At node A, no jobs are assigned.

At node B, job one is assigned to machine M1

At node Z, all jobs are assigned to a

machine, but not sequence is considered.

Z

1

(66)

•

& 2

2

n

u

n

, 2 . 3

LB

kt

•

%: 2

m

n

′

n

′

4

m

, 2

m

. 3

LB

ka1

m

LB

k

46 @; ,-(1-(?.

N

u

2

N

a

m

2

m

S

a

m

2

N

a

m

LB

k

_{= max}

LB

kt

|

M

|

,

M

max

m

LB

m

ka1

,-(1.

LB

kt

=

N

u

i

M

min

m

(

p

im

) +

M

m

N

a

m

i

p

im

+

n

· u

,-(>.

LB

m

ka1

=

N

_m

a

i

(

p

im

) +

S

_m

a

{ii

′

m}

(

s

ii

′

m

)

,-(?.

' 2 @; ,-(5-*+.

LB

k

3 ,

4 .

n

2

n

3

LB

k

N

as

m

2

m

6

N

au

m

2

m

@;

0/ " ! !

/ &

i, m

"

/ 0

bestSol

←

presentSol

M

/ 36

m

&

1 , m

+ 1

M

/

job

←

'(

m

"

job, m, i

M /

presentSol

< bestSol

&

i

+ 1

, m

M