!
1
* 2
! (hkarimi@mail.kntu.ac.ir)
! (setak@kntu.ac.ir)
" # $% #
&
' ( $)
*+#
, - +" &. # / , 0 1 , 2'
2 /
3 4 5 6#
- #' 7 8 9
. 5 , 2 " 1 / ; #
7 9 $% # 2 ; 0 "
" 5< 2 " ; # = 1 > ? .
= #' @A, , B = " A & "
C D <
' " 4 E " F ) " &) &@G = #' )# 5< 2 7 3 H I 2
.
" , "
; # ' J / K L # = - @ 8 M ' L " N/ 1 2'
# / # @/ 1 7 "
O #
7 / P 2
. -" ?9, * " ) < Q $% # &G 2 "
1 7 . '
#! " R
* P ($@M ' =
.
: - +" $) & " # ' " ' -" ?9, * 2'
2
1 -
"
# ! " # $ %
& % "' (
)
* + ( - . / 0 " + 1 % 2 ' .
4 5' 6 7 8 9 4 ' - :; ' "
. 5 :;
$ 7 < + 9 4 = >
? & @ ' 3
]4
1 [ : 0 F G ' ( '
7 51
]5
2 [
* ; 4 7 /
6
] 3 [ /
* ' KL 7 ]
4 [
+ M /
* ' KL 7
] 5 [
$' 1 /
* LPQ 8
]9
6 [
! 9 / ]10
7 [ 4 % /
* ' KL 11
] 8 [
–S
T -E -+# 2 1 2 9
* U V
; % # - W :
3Dantzig
4Ramser
5Bodin
6Golden
7Desrochers
8Laporte
9Osman
10 Fisher
11 Eksioglu
8 ' .
% 8 7 + 6 T U /< V 7 '
" ' $ W 8 6 T + 9 4 = X;
"
:- " ' .
" ' % ; . " 7 T ' /8 *' 5' 6 "
.
* + 7 LY$
7 + Z8 "' 8Y ' % F LQ [ - 5' 6 * ! " # $ \ "
Q
< % " +
*]
8 K!
. ' ' \ * + [ - * & 7 LY$ % Q /2 P "
%
: 0 5 9 _$ K 9 $ / `] @
"
. 2 / ; 7 L a b% "
% ' % [ - "
Y @ 7 ' . ' - / % .
' 0 * L( = Q @ 8c$ d . " $ \ " K ' .
' /2 P "
8 ; . % "
$ \ 2 + / " [ - "
\ ' G "
% ) 2 P "
] 9 [ .(
K
" ' " "+ 9 7 \4 e ' /8 : - = Q @ X0 " K ' +
% 8 4 % < V * $
1
* ' KL ] 10 [
% ' . f ' + 2 "
X "+
f ' /* L( = Q @ 8c$ d "+
"& '
+ /f ' 7 ' % 8 ; L . " 9 _$ g + $ 7 LY$ a
\ )
2 P "
] 11 [ .(
/' c ' K ' ' * $
+ 5' 6 '
"
% ' 2
. ' + "
. a 7 . a /"
9 4 e ' " 1 .
' . 7 7 ; L . g + $ ' ' 8 h ' $ Qb( i /8; . 7
. '
/8; . 7 ' /g- + * L(
8; . 8G 8 L% "+ 9 ; L . "& '
.
% $1 + T
< ' ' "' c ' K ' ' + T ' 7 0 $ X $ / 4
' "&
% ) L
] 12 [ (
' ' ' ' T
8; . L$ j / ' $ Qb(
' >
) L ] 13 [ .(
8; . 7 '
+ F LQ % 8 + "
\ "
% .
5 \$ + K % 8 7 " K ' ' c ' K ' kX "
X + / K ' 7 ' ' "+
\
% . X + U 7 ' % 8 l 0 /' "+
' c 7 $ k% / T 7KL 8; . 7 $
l ] '
. k% /' ` 0 a ' % : 7 ' * L( = Q + ' # 7 $
> 4 k L_$ "
.
X + / * l; [ /< V 7 ' 8 \ ' "+
.
+ + "
X 8 : - = Q 8; . ' $ \ d . ' "
8 5 ' ' "+
.
* /2 P * Q
* ' KL 3
] 14 [ U 7 + X '
$ % ' "+
% \ ! 8T' 7 $ .
.
4 L
* ' KL ]
15 [ X 4 ' 9 " K ' 7 + / .
m ' ] ] 13 [ /
+ 8; . 7 9 4 e ' ' n _Y$
& " "
X 5' 6 n _Y$ "
2 ' "+
% . "+
. '
< V X / "
8 ' # 7 ! "+ L% h ' "+
. F G $ / d + U 7
] ; ' % ' (
* ' KL 5
] 16 [ ' .
T % " 8; . 7 $ U - % + \ K ' / c
+ 8 l "
. K ' '
k% T 2 L . " ' % : - " K ' 7 j /8 L% T 2 L . " ' F LQ 7KL 8; . 7 $ / ; L . & c g - )
@ / + a 9 $ 8 Q * /l ' " 0 @ / L
0 ( 8 ' % / .
% 8 G K ' /l ] 7 ! "+ L% K ' /g- '
$ 8; . 7 ' *
\
L .]
16 [ + 5' 6 % 8 : - = Q @ " ' /o 7 \ * + / # . < V ' "
.
% $ :;
+ 5' 6 '
0 $ / " "
' 0 / "
.
1Kagaya
2 Interval
3Yaman
4Montemanni
5Aissi
* L
* ' KL L @ $ 8; . 7 ' 4 ' 9 / ' k FbG- % ' ( ]
15 [ ' - :; '
8 9 4 .
% 7 ; +
' < V 7 ' "
X o /8 9 4 ' - \ "' o / ` Y "+
T ` 0 .
X ` % ' % ' ( o 7 j ; % 4 " p9 ' ' "+
] 1
17 [
8 *
.
' % 7 "
` 0 7 $ + ' q c a /
T X .
6 7 ; ' T % 8 5'
2 /= 8L - ' ' q c "+
T .
X f ' +' &>$ "+
' = 8L - ' 2
. +' &>$ k ' ; d ' =' XM 8L -
> = > 8 T T ' ] 8L - ' " 4
.
2 -
%
&
8X0 q 4
{ }
V A ' G= , e ' '. Q L> % o V
* L% Q L> A 8 0 "
. '
o Q L>
/ 0 4
* ! 8
. q 4 7 " ' e '
T .
* L% "
* + a a )
&
( + 5' 6 % ' 0 \ "
[ ]
t ,ij tijij /
ij t
t < L 0 .
X + q "+
8 L ' $ M 7 9 / 7 ' )
# $ * & h
i) (d 8 ' :$ 7 2 / ' ]
( %
*] r ! k% " ' 8
* + 7 $ )
&
( \ . 9 4 e ' "' s :$ h q 7 .
j ; % % "' K ' 0 $ )
1997 (
% . : 1 ( "' K ' /[
2 (
q c K '
' . 8 ` Y = K ' # . < V ' .
' e ' @ ' " ! l $ *] "' + .
2 " '
* \ ' :$ / "+
.
s :$
1 .
" ' '
* + + L - a /R )
&
( 5' 6 \
[ ]
ij ijR
ij t t
t ,
` Y .
s :$
2 . ` 0 a " ' q c a ' ' f
"
7 X & ` 0 7 & 7 5 \$ /R
' 7 ' ` 0 .
` 0 a " ' q c * &
' ' f
"
5' 6 R ) , (R f ' 7 ' ` 0 7 X RD
5' 6 ) (R * ! B .
s :$
3 . k% " ' % k 4 ' ` 0 ' ` 0 a L$ 7 ' q c * & 7 ! /7 $
0 " '
. q g $ * L - * s ' :$ 0 $ )
8 9 t + \ = Q & / \ * + &
(/
T + p-
.
p- . ` 0 " q c * & 7 ! > f
* + % )
&
( 2 " \
* ' "
2 ' T ' 7 ! /` 0 k% / ' ' p. ` 0 ' %
' T ' 7 $ .
5 Gu . 0 @ $ > " ' f `
F / .
q c % 8 u 2 . " ' '
"
R L L% / F +
8 .
{ } { } ={ } { } { } { }
=
f R B j i
F ij R
B f j i
F ij f
R B j i
R ij R
B f j i
R ij R
B j i
R ij f
j i
R
ij t t t t t
t f
R RD
\ ) ( , ) (
\ ,
\ ) ( , ) (
\ , ) ( , ,
) , (
s :$ h /F
{ }
i,j f \B(R)t tijF ijR
{ }
i j B R ft
tijR ijF , 1( )\ 6
v .
1Kouvelis
2Bender’s decomposition
{ } { } { } { } { } { } { } { }
{ } { } ( , )
) (
) , (
) ( , ,
) ( , ,
\ ) ( , ) ( , ,
\ ) ( , ) ( , ,
f F RD t
t
t t
t t
t t
t t
f R RD
F B j i
F ij f
j i
F ij
R B j i
F ij f
j i
F ij f
R B j i
F ij R
B f j i
F ij f
j i
F ij f
R B j i
F ij R
B f j i
F ij f
j i
F ij
=
= +
=
p- 0 $
"
5' GQ 5' 6 q g $ / )
1 ( T .
) 1
( )
(i j i
ij R All ij
i j i y ij R S ij
R All
x max t x min t y
min
Q L>
8 ' L$ S
All Z 8; . L$ k
8 "
.
p- 0 $ .
1 5' GQ ) 2 ( 5' GQ + ) 1 ( 8 ] .
) 2
( )
(( )
(
+)
i j i
ij ij ij ij All ij
i j i y ij All ij
x t x min t t t x y
min
5' GQ ' ) 2 ( w 0 $ % 0 /
"
xij
> ' ' a / T >
% . 7 X ' 2 .
X 4 ' 7 ` 0 X i
/ 2 _$ j
=1 yij
5' 6 7 x ' Z
=0 yij
T .
+] w a /q g $ * % T "
s :$
5 ' GQ w 7 0 $ / )
3 ( ) 4 ( T 8
] .
) 3 (
i j i ij ij
All
x t x
min
) 4
( )
(( )
(
t t t x y)
y Alli j i
ij ij ij ij ij+
%W - 5' GQ ' % 8
) 4 (/
yij
$
* ! ' ` 0 a < L < X 8 u *] /5' GQ 7 '
8 . 5' GQ 0 $ )
3 ( ) 4 ( 8; . L$ ` 0 " p9 % 8 9 % X $ / KL "
7 All) ( >
.
"
p9 + ' % 7 2 a b% "
\ .
5 ' GQ ' " p9 7 )
5 (/
) 6 (/
) 7 (/
) 8 (/
) 9 (/
) 10 ( ) 11 (
8 * !
.
) 5 } (
0 {
\
1 i
x
i j
ij =
) 6 } (
0 {
\
1 j
x
j i
ij =
) 7 } (
0 {
\ )
(d C x0 i
C
ui + i i
) 8 j (
i j i
x d d C Cx C d u
uj i + j + ij +( j i) ji \{0}, \{0},
) 9 } (
0 {
\ i C
u di i
) 10 } (
0 {
\
0 i
ui
) 11
{ }
i j i j (xij 0,1 , ,
0 4
8 4 * L
.
i w $ @ $ % 8 "' * & u
. i L .
5 ' GQ
) 5 ( ) 6 (/
a 4 + 0 T " ' :$ % U # 7 .
5' GQ
"
) 7 (/
) 8 ( ) 9 (
" /
\ ' $ + qy.
) MTZ q :
( L0 M ' " ' % @ $ "
* ' KL 1
] 18 [
9 : .
5' GQ ) 10 ( ) 11 (
* ! ' ` 0 " p9 o " w k .
2 0 $ "+
) 5 ' GQ : )
3 (
$ ) 11 ( ( 5' GQ @ $ . " 6 K! / )
4 (
>
.
' / 51 . L$ /g- '
. . ' j L
% . ; G- - F G * + ' ' 2 7 * $ .
f ' + K + U 7 8 "
\ +' &>$ f ' /2 ' T h ' % 8 &>$ f ' /
T .
3 -
&) *+ # ,-. / )
* L 2 / ! / G- 8L - " X ' % "' ( "+
8 c + " + :$ " ' )
4 ( 8 .
+ K
+' @ $ % 8 +' f ' /U # 7 . " K ' ]
19 [
8 d '
. 51 ' 7 " ' %
j ; % ]
17 [
* ' KL L / ]
15 [
8 4 '
. L% :$ X $ % 8 7 +' f ' 6 +
8 c
"
) 4 (
k X ` 0
.
8 c 7 9 4 e ' * 2 /+' f ' @ $ . "
"
) 4 ( . .
1 7 ' % 8 %W =+
. )
"+ ] (
" /` 0 * c + " 4 0 ; / )
12 ( T 9 # .
) 12
i j i (
ij ijx t
iter 2 . 9 : / ' L * Q
` 0 % "
' All L
iter .
` 0 : All $ "
K$
iter ' .
= 4 / * l; 0 $ 8 + 5' 6 k ' ; "
:
= 4
\6 1 .
=
{ }
iter= Alliter
.
2 = 4 . +] 2 )
3 (
$ ) 11 ( 8 c " 0 % 5' 6 7 / % . ' )
4 (
8 c +
) 13 (
\
% .
) 13
( )
(( )
( )
iteri j i
ij ij ij ij
ij t t x y y All
t +
6 >
` 0 % 4
> "
. $ . ' 2 iter
% .
` 0 ] 8 "
xiter iter / q g $ * &
LBiter
8 +' f ' ' 7 . * L % .
= = 4 .
L. a b%
. / 6 + ] 8 ` 0 0 $ .
'
* + ' /8; . 7 )
&
( ' + o 7 \ )
14 ( 8
' - \ ' a b% ' % ]
4 .
) 14
( )
ij ij ijiter (ij
ij t t t x
t = +
+' f ' Q 9 % / 7 .
iter /8 ] T 8 y
. . " / 7 0 $
5' GQ /+' f ' "1 )
15 (
>
.
) 15
( )
(( )
(
+)
=
i j i
iter ij iter ij ij ij ij
i j i
iter ij ij
iter t x t t t x y
UB
1Miller
= = 4 .
{ } 4
iter
iter UB LB
,..., 2 ,
1min
s- k ' ; / ' Z
x 7 15' 6 +
=iter iter
iter iter
iter All y
All = 2 = 4
.
4 - .1 2 3 4 5 ' - < +] ' 8 d ' % "+' &>$ f ' k ' ; /8L - 7 ' 4
.
@ $ k ' ; 7
MATLAB 7.8 GAMS 23.5
8 % G % $ 5' 6 .
CPLEX+
% . * Q
= ' & 9 '
GAMS23.5
8 ` Y
. + ' @ $ 0 L$
2.53 GHz e9 .
2.99GB 8 = >
. +
% > ] + / G 0 7 " 5 '
' 9 @G$ "
@ $ % 4
* ' KL L
] 15 [
\ / >
8 4
. 7 h' ] '
http://www.idsia.ch/~roberto/RTSPinst.zip 0
. 7 G$ "
\6 o / e ' "
8 z 9 o * L .
" # $ " ' d
10 4 ) 7 9 4 e ' (
9 4 ' - < +] ' %
o " l $ $ 0
$ 9 0 / 91 / 13 / 92 / 64 / 10 / 28 / 55 / 96 97 8 .
" ' d " 7 L 17
4
) 7 9 4 e ' (
o " l $ $ 0
$ 16 0
/ 43 / 92 / 80 / 96 / 66 / 4 / 85 / 94 / 68 / 76 / 75 / 40 / 66 / 18 / 71
4 8 . 7 9 _$ { c6 Q a % \ : $ + / 7 ; $ "
0
$ 100 .
8 9 t
"
2 0 ' 1
8 ]
.
( k ' ; + \ * 51 . L$ ' @ $ . | 9' .
+ 4'& 0 $ /k
) . 10 4 ( 8 e9 . + + }' T 8 G * T ' - / \ ' @ $ 2 / .
. o + ~ ; 7 L /
8 G * f' &4 ' - k ' ; + \ * .
& M *]
nY! % "
@ ' 0 $ 8 )
4 ( " 2 + /
"
10 + T 4 F L
k . ; $ * + /
F G T + .
0 - T ' @ $ 2 / \ +' &>$ k ' ; + % )$
17
4 ( 8 .
" ' d $ G c ~ X $ 8L - 7 ' 10
17 8 ' ] 4 .
"
20 4 / ! + }' T
8 \ ' ' e9 . .
d ; 7 L
•'&
$ Q L> ' ' - < +] ' / 9 : "
8 9 . 2 0 ' ] 8 ~ 1
8 ]
. * ' = /2 0 7 2 * %
.
* ! = * 4
8 9 t 8 \ '
. q g $ ' )
3 ( 8 * ! = * ' / .
k> =' XM *
* ' *] . * + +' &>$ k ' ; " ' K$ :$ l $ $
% .
"#
$ 1 :
" '() * +, $" - .
- ! ) 0 ( 2) "3) 4"5 6 ) "
7 8
9
355 / 1 4
284 R-10-100.1 200
520 / 1 4
216 400
096 / 1 3
224 R-10-100.2 200
006 / 1 3
194 400
032 / 1 3
224 R-10-100.3 200
898 / 0 3
194 400
592 / 1 3
255 R-10-100.4 200
366 / 1 4
211 400
371 / 1 4
281 R-10-100.5 200
039 / 1 3
191 400
133 / 2 4
2525 R-10-1000.1 200
050 / 1 3
1298 400
635 / 0 2
1597 R-10-1000.2 200
937 / 0 3
1460 400
249 / 1 3
1828 R-10-1000.3 200
326 / 1 4
1457 400
842 / 1 5
2614 R-10-1000.4 200
989 / 0 3
1816 400
328 / 2 4
2618 R-10-1000.5 200
312 / 2 4
1867 400
577 / 60 2
1487 gr17.rm.0.25.1 480
328 / 59 2
1401 800
318 / 59 2
1545 gr17.rm.0.25.2 480
697 / 58 2
1464 800
524 / 61 2
1608 gr17.rm.0.25.3 480
830 / 51 2
1381 800
234 / 59 2
1540 gr17.rm.0.25.4 480
186 / 54 2
1389 800
617 / 60 2
1443 gr17.rm.0.25.5 480
831 / 56 2
1372 800
2 0 ' ' ] ~ )
1 (
* ! 8 ' ` 0 +' &>$ k ' ; T T * + ' %
.
@ ' :$ /7 •L )
4 ( d " 2 :9 X ` 0 ' % k% T / .
' ( 7
97 / 2 .
FbP
" ' % d "
17 / 4 2
' ) 4 ( 8 ' ] = &;
.
5 -
.7 3 89 : #- ;
' + "
2 8 : - = Q 8; . ' "
* + "
)
&
(
< V 7 ' \
8 '
. 2 a "+
4V h @ Y "& '
8 / "
. . * • ;
/2 8 d ' +' &>$ f ' .
0 = Q ; " ' + T /5 ' @G$ "
0 "
4 > b6 7 " 2 /*] + j
. .
* ! ] 8 ~ ' +' k ' ; %
8 u ' . * + < % .
$] 5 :; * Q k ' ; /* $
. T L " [ - "
7 •L / % . ( 7 " ' f T k ' ; + * $
" " X ! +' &>$
d ' .
X d L $ G % $ "+
* K 5 LQ ; $
X .
=>
[1] Dantzig, G. B., & Ramser, J. H., "The Truck Dispatching Problem", Management Science, Vol.6, pp.80-91, 1959
[2] Bodin, L., "A Taxonomic Structure for Vehicle Routing and Scheduling Problems", Computers & Urban Society, Vol.1, pp.11-29, 1975.
[3] Bodin, L., & Golden, B., "Classification in Vehicle Routing and Scheduling",Networks,Vol.11, pp.97-108, 1981.
[4] Bodin, L., Golden, B., Assad, A. A., & Ball, M, "Routing and Scheduling of Vehicles and Crews", Computers
& Operation Research, Vol.10, pp.63-211, 1983.
[5] Desrochers, M., Lenstra, J. K., & Savelsbergh, M. W., "A Classification Scheme for Vehicle Routing and Scheduling Problems",European Journal of Operational Research, Vol.46, pp.322-332, 1990.
[6] Laporte, G., & Osman, I. H., "Routing problems: A bibliography", Annals of Operations Research, Vol.61, pp.227-262, 1996.
[7] Fisher, M., Vehicle Routing. In Ball. M. O., "Handbooks in OR & MS (Chapter 1) ",Elsevier Science, Vol.8, 1995.
[8] Eksioglu, B., Vural, A. V., & Reisman, A., "The vehicle routing problem: A taxonomic review",Computers
& Industrial Engineering, Vol.57, pp.1472–1483, 2009.
[9] Yu, G., & Yang, J., "On the Robust Shortest Path Problem",Computers & and Operations Research, Vol.25, pp.457-468, 1998.
[10] Kagaya, S., Kikuchi, S., & Donnelly, R. A., "Use of A Fuzzy Theory Technique for Grouping of Trips in The Vehicle Routing and Scheduling Problem", European Journal of Operational Research, Vol.76, pp.143-158, 1994.
[11] Yaohuang, G., Binglei, X., & Qiang, G., "Overview of Stochastic Vehicle Routing Problem", Journal of Southwest Jiaotong University, Vol.10, ppp.113–121, 2002.
[12] Alumar, S. A., Stefan, N., & Saldanha-da-Gama, F., "Hub Location under Uncertainty", Transportation Research Part B, Vol.46, pp.529–543, 2012.
[13] Pereira, J., & Averbakh, I., "Exact and Heuristic Algorithm sor The Interval Data Robust Assignment Problem",Computers & Operations Research, Vol.38, pp.1153–1163, 2011.
[14] Yaman, H., Karasan, O. E., & Pinar, M. C., "The Robust Spanning Tree Problem with Interval Data", Operations Research Letters, Vol.29, pp.31–40 ,2001.
[15] Montemanni, R., Barta, J., Mastrolilli, M., & Gambar, L., "The Robust Traveling Salesman Problem with Interval Data", Transportation Science, Vol.41, pp.366–381, 2007.
[16] Aissi, H., Bazgan, C., & Vanderpoote, D., "Min–Max and Min–Max Regret Versions of Combinatorial Optimization Problems: A Survey", European Journal of Operational Research, Vol.197, pp.427–438, 2009.
[17] Kouvelis, P., & Yu, G., "Robust Discrete Optimization and its applications", Kluwer Academic Publishers, 1997.
[18] Miller, C., Tucker, A., & Zemlin, R., "Integer Programming Formulation of Travelling Salesman Problems",
Journal of the Association for Computing Machinery, Vol.7, pp.326-329, 1960.
[19] Benders, J. F., "Partitioning Procedures for Solving Mixed Integer Variables Programming Problmes", Numerische Mathematik, Vol.4, pp.238-252, 1962.