%./-. +, %()* %’& % ! “# $

! ” #

! ” #$% $& :#$%
2 # % 0″ $1- .% ‘ ()*+, $ – ‘” #-$) .
! : ;1 % ‘ 304 54 ’67- $ % 1 $% -89 # &% # -> – ? @%, .! ‘ $”) # )% <$= ()*+, ? @%, #E% – )A B 0> #$% ) ‘4%- – ;)% ‘C %- D 7 $ $*% – : ) FG ! % 5%$ .” .B #$% . ) FG $1JL9 )ML . ‘ H .” IJK 6% #$% FG #$ A $ ) – (Random Task Assignment) RTA N% ?-$6 O- % #*=)A
) FG ‘0″ #$,$% – ‘$% ! .PB Q ” )A -B
. ‘ 1U ‘% % .B ; ! $R ! F – S ):) T> % F

. :
#<4 I! 6&’ ‘ 8 $ V! = 6 6+ 7+ 5+
.[X]5F 345+ ! + A-/ 8< Y 9 5+ O ‘ 8 J’ WK [\E J 9 J’ Z J’ 344 F 6+ ! ‘ 8 8 8 ] … ,C I! + L/ 34 8 + P =] !
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I 5 C>% 8 N& -&% W7 4 6&’
.34 =8 + ! R’ 05 O 6 8 ^9 + ‘ 8 $ Q ! + IF &’ + I + =8 + _R\ * \ EC $ 8 .89 IF ‘ + 8 8 J’ ,-/ P 7 05 + 5+ O 6 5E+ 3&C 54 Z + ‘ 7 05 655] `a4 97 5O 8 =5$ L/ 5$+ 05 4 ! 8+’
5F 8 ]+ 55] 05 F ! : + 57 C 4+
.34 _O2 6 848 ‘ #Q 8 =8A4 + L 3-+C + ] + + 6 ! =E 5M [5 V+ + !

.
* + ,-/ ‘ ! % & #$ ! ” 8 9 + 7 + ‘ 6 5+ ‘ 34 0 12 + 4 ‘ : ; < + =>% ‘ 5+ ?1 #C 5B ‘ D’ 5B 8 $ @AB … ! 3852 %>H 6 ++ .[G] 5F 345+ 8<-&% E 4 I! ‘ 3 #$ ! ” 44
IF -‘ L MM ‘ 5$+ ! JK + ‘ :
.58 #<N ‘ 5 $ O -P =7 8 + ! #Q 8 I %+ #<4 I! ‘ 0 T :! 5R% -+ 5+ 0 12 + ! 8 ‘ =8 + U/N ” ‘W = 8 5E ‘ 8 $ V! = 6+ 7 I % + = 8 5E ‘ :8 T 8 $ #C 5B
0 12 + ! 8 ‘ 34 U/N ” 8

// //
.! #$”! . %&’ ! ()*+, ha_mousa@yahoo.com
. 2-&3 $ 4+, $”! -01
zegordi@modares.ac.ir
+ <R$ = & + 0 12 6 + ! 8 ‘ ! 05 6 8 .34 =5$ =8 8 #<4 I! ! JK r 8 IF I8 + $8 I + ] + ‘ r .5$+ I! 5B + 0 K 5B ;B + E r 6N+ + + + = #’ s4 6 + .5$+ = + 9 ?1 ‘ = IF r Lt-PB + 34 + + = + * + 5B + 0 K 5B ;B+ r ‘ F ! #<4 I! 8 I+ 0 12 5B + I! 5B ;B + #<4 I! I! 0 12 5B + 0 K 5B ;B + = W 8 $ + ‘ ‘ 34 6 05 6 8 [5 .5F 345+ 5O 6&’ + ‘ 8 $ V! = 6+ 7 + .5+ 8 $ R42 Y 9 c + ‘ = .34 =5$ Q E\C <+ c 65M 05 6 #B c ! #PB L ] 3A’ R42 I! + 6u& + Amen R .[GGGj] 34 97 P E\ <+ + E\C c M ‘ #Q + 8 8NK 05 #B p IF 8<-&% 8 R+ + 8 + =8 8 Q ‘ $ 8< ‘ 8 8 IN Scholl ‘ 8 + =8’ I+ $ + ,C *$ v>P P + *$ IF 5$+ =R$ *$ 6 ! <
.[GX] 8 8 Q =5$ 4 =5$ Q 05 ! : + _O2 6 8 8NK 05
‘W 5 8 8 W .34 97 #<$ Amen Bukchin
:57 C 4+ 5O 8 O9 = 8 ‘ 34 7 + Bukchin 05 _\ ‘ 8 8 8+’ ! Bukchin 05 W .5+ O4 5 =8A4 8 * \ 8 + Z WK [\E + O9 = 8 3B 6 8 ‘ 34 w^ .87 C 8 .8 + 5 + ] ‘W 05 6 ++ 87 C ! =8A4 8 $ =8A4 48 =84 ! < P + &’ ‘ ‘ 8 $ x%+ = 8 05 6 ++ 8 $ 8! = 8 5E 9 ?1 = 34 + 54 Z + 6 ++ .8 + 5 / + ] Bukchin 8 8 8 ] 8 5E ‘ : + ‘ 8 $ y9 5 8 ‘ ! 8 5E : F ! ‘ I y % 8 34 3+f U/N ‘ ! : I!
! =8A4 I< 8 8 ?1 ! J+ = +
.8 & 9 = 8 zN <A + 7<+ {5 Amen 05 8 -&% + I! 5B + 0 K 5B ;B+ ‘ R .5$+ !4 =8K #+C 4F + 34 8+’ r 6N+ s4 + ‘ = r R42
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,-/ K =5$ =8A4 05 6 + * + #Q #B IN + .[b] 8 $ 39 #Q ,-/ 8E+ + 8 ] &% c ! 8NK ‘ I8 8 ! =5F 348+ L ] =5$ =8A4 RTA :+ +8 8 d 6 4+ .887 O c 6 L ] +
3A’ e2 ! 8NK ‘ 58 IN
.5& #&% fg L + #B I! e2 + L ]

.
8 [5 .5$+ 0 T ” 7 05 6 3A7 I + 5$ 8 5E #C 5B =5$ =8 8 #<4 I! h ! + ‘ 34 6 6 = 8 5E J’ + 05 6 6 ++ .5F 345+ =
Giij 04 8 Erel Sarin .58 J’ ‘ : I! ‘ 597 Z 8 B 8 0 T &B = I! 6 ++ =8 + &B V! 8 6 8 .[k] 5& ! #<4 I! ! 34 6<& 5$+ [P + =5$ l ! 0 12 5+ P
Giik 04 8 Malakooti .5+ 8 IF ‘ 7 5] =5$ =8A4 = 6+ 8 ] ! IF 8 ‘ 8′ Q 5 34 =8 + 8 ] ! =8A4 = 8 5E J’ [5
: 5C 8 8NK 05 8 Sarker & Pan Giin 04 8 .[m] =5$ Q 05 8 .[o] 58 & <+ > J’ +
! = ! =8A4 I< McMullen & Frazier 4
8 5E 9P ‘ 34 6 [5 =8 + 0 H 8
.[p] 8 $ #C 5B ! 8 8 5E 97 Z 8 0 T ” Bukchin 4 =5$ Q 05 8
.[n] 5F 345+ 0 12 5+ ‘ 34 =5$ + <R$ = & + 0 12 6 + ! 8 ‘ ‘ : + .34 =5$ =8 8 #<4 I! ! JK + ‘ 8 8 8 ] WK [\E ! A-/ T ‘ ! 8 5E 5 8 WK [\E I + ] 8 5E ‘ ! : + 6 ++ .58 : ! + ‘ : I! ‘ 8 8 8 ] L/ + 5 + 7 34 5+ .8 + 5 A T + &’ I! =8 8 : ‘ ! N+ 8 5E 6 8 .3$ 8 5 q+ 5 3&C 5$+ $ 8 ! F : 6E 5+ W .87 C O9 = 8 05 ‘ U1/ = 2 87 C M = 8 ‘ 8 $
#C 5B 5 T & 5$+ M = +
.8 $ =5$ 97 Z 8 0 T ” Amen 4 =5$ Q 05 8
.[i] 5F 345+ 0 12 5+ ‘ 34
) ” &’ ( ! ” ” # $ %

. ] + ‘ ! + ‘ 8 $ y9 ” 6 8
+ ‘ 8 $ ,E 8&48 r … IF I8 + $8 I +
.5$+ I! 5B + 0 K 5B ;B
r ! 4 s4 + = r I! ‘ 8 $ R42 = IF + 9 ?1 ‘ r R42 + W .5$+ ‘ : I! I& =8A4 8 39 ‘ : I! r Lt-PB = I! T & + =5F 348+ 5O 5 $ V&] + =5$ r 887 O = IF + 9 ?1 ‘ = r 6$ 8 + .5F 348+ = 6 + .8F 348+ #’ I V&] + =5$ L^ #<4 I! 8 = r Z 348+ 0 12 5B + 5 $ L^ + ‘ 34 I! 5B + 0 K 5B ;B + r) 5F
0 12 5B + I! 5B ;B+ ‘ #<4 I! 8 IF I8’
348+ 0 12 5B + 0 K 5B ;B+ 5O 5$+
.(5F R42 Y 9 P + ‘ ‘ 3$ 8 ] 5+ I+ 0 K 5B ;B+ =8 + W7 4 #$ 8 $
5B + 0 K 5B ;B+ < P 8 8 $ 5< ‘ 8 T 8 6 < + 6 ++ .5$+ 0 12 6 ! 3& I8 8 IN + 57 C [5 V+ 8
=A4 !4sO + c ! E !
.8 $ J’ 8 E4 ” 6 [5 V+ ‘ ! ‘ 8 $ 6E 5+ ” 6 #B + 6 ++ 8 8
T & 5+ ?1 = : 5’ +
” 6 3852 ^9 4 .8 $ #C 5B
.34 ! ” &% 3852 I&

“# .!
I+ + ‘ } K 5+ ^ 05 Q ! #RC 5 <+ ^ + ;C 8 05 [5 V+ 3852
05 8 =8A4 8 =8 8 K .5 $ 9E
:! 5R%
=8 8 IN h i + ‘ `5 ‘ ‘ 8 5E =n
:5 $
n … = G X h n … = G X i
:  i ‘ : + ! 8 I! = ti
5B + 0 K 5B ;B + :  i ‘ r = li
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Downloaded from ijiepm.iust.ac.ir at 16:26 IRST on Saturday November 4th 2017

I!
.34 3EC ! 8 5$+ = IF + 9 ?1 ‘ ‘ ,-/ ‘ = K + q| &E ! r ‘ 8 $ =8a4 5$+ $8 + I4F + s4 + = IF r 7 34 A F 8&48 Z 8 ( r 6N+ +) = IF 8 8 ] ‘ 6 $8 8&48 R42 + 22P R P 6 8 8 $ 97 r 6N+ ‘ ‘ 34 6<& ! .3$ 8 5 / 8 ] 6<& ‘) = IF 8 8 ] ‘ 4 + O 8 8 8 <M ‘ + 4 (5 8 &’ r =8 + I4F + 34 r 6N+ + 34 + 54 Z + 6 ++ .5$+ $ 8 + 9 ?1 ‘ r ! 4 ‘
8 I! 87 C = IF r R42 R =
.5$+ ‘ : I! 5 =8A4 05 5< + F _A- 05 8 6 8 Y 9 } 0&% + 8 + IF 8 ‘ 87 #<$ _O2 6 8 8NK ! 6 ! 34 =5$ ]
.34 =5F 345+ F ! : + =5$ l4 ‘W 05 8 ! <

.
-‘ IF 8 ‘ 5$+ 0 T ! 05 6 8 x2+ 8 ” + <R$ = & + 0 12 + ! 8 ‘ 6 8 .34 =5$ =8 8 #<4 I! ‘ : I! ! JK
87 C ] 8 8 ” :34 ! P + IF ^9 R42 c ‘
. ‘ ! : + ‘ 8 $ y9 ” 6 8 : + =8A4 8 T .5$+ ! + ! T 8 5E ‘ 8 $ y9 34 : -E ‘ ,-/ ‘ 6+ 6 ++ 34 &’ ‘ 8 5E ‘ ! 8 5E = & $ 8 8 ] zN 8 .5& =8A4 T ! ‘ 5 8 8 ] + 5’ =8A4 T ! ‘ ‘ I + < P 65M : + ! I 8 8 U1/ = 6 8 .8 8 J’ 5 =8′ =8A4 ‘ 8 ‘ 8 $ U/N 5+ R42 + ” T M IF + =5$ =8 8 ?1 ‘ + ] + =
T ! = 8 ‘ 34 5+) 34 !
.(34 ! 8 < D’ 5B Z 8 + 9 F T ! #’ 8 5E I IF ! `K
8 -‘ 5 T 5 697 .8 & R42 !
n
∑xij =1,∀ i (b)
j=1
∑n (jxhj − jxij)≥0, ∀i,h;Pih =1 (k)
j=1
xij = i j (m)

\+ 6 0 J/+ .58 IN 05 6 [5 V+ (G) \+ r K # 8 5O .34 + * + L^ IF 8 #<4 I! ‘ 5$+ = ‘ l/ ‘ 34 6 #8 + ‘ l/ 8 ε 8 ] .34 =5$ ε 5 M ‘ + 85% H 6& + 88 AP
.34 =5$ 9^ 5O .58 IN 5 Y 9 \+ :8 J/+ R42 k T ! ! 8 #’ 8 5E K # 8 X ij Aik 5O 6N+ = 8 ‘ P 65+ .5’ 6 < P 8 .34 AP + + #PB ‘ 8 $ 39
8 k T ‘ 34 6 {58 IN 5$+ + + 5O
.34 =5$ =8A4 ‘W = 8 5F 348+ k T ! ! 8 8 5E < ! `K 5B I8 R 08E e2 + W2 W18 ] .8 $ L^ IF 5 8 0 K 5B ;B+ <) I8 8 IN + (5$+ 0 12 5B + 0 K 5B ;B+ .8 <+ 8 + 3R 8 6 ! 3& 8 $ O F 5A &% + 5 7 R 8 IF I8’ L^ + ‘ 5F 345+ I! 5B 8 5 .39 0 12 5B + 5 I #<4 I! 45-C : Z !4 sO + c ! I 6u& <+ =E 5M [5 V+ + ‘ 8 ] c 4 } + * + #<N ; 65+ .8’ =8A4 5 I8 8 IN + 8 $ [H+ Y 9 T 8 5B
=8A4 ! E ! 8 + 3R F ! 3&
.5$ 5 T & ‘ 5’ I+ =8 + #<4 I! 3852 (X) \+ #<4 I! ! 5R = + 9 ?1 ‘ I!
.5$+ N+ ‘ ‘ 5’ I+ =8 + ‘ z $ :5% 3852 (b) \+
‘ I & 5+ ?1 = + O9 O9 5+
.8 8 U1/ = 5M 8 + ‘ \& .58 IN ! JK + 3852 (k) \+ :  h ‘ ! JK :  i ‘ < P 8 34 8 N \+ 6 ! 5+ ?1 IF + :  i ‘ ‘ =&$ 5+ 5$+
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Downloaded from ijiepm.iust.ac.ir at 16:26 IRST on Saturday November 4th 2017

*
,E ! P + ‘ h i ‘ 6+ ! JK \+ = Pih :8 $

pih =⎧⎨1 .5$+ :  h’ #19>+ !JK :  i ‘ 7
⎩0 1 ‚ 8

8 ” ‘ F ! .5$+ = `5 = j
W 3 U/N IF 8 = 8 5E =8 + 0 T ! x2+ :8 + 5 ‘ 8 5E + + y9 JK P + = 8 5E

j = G X … n
#<4 I! = C
=8 8 IN k + T `5 ‘ T 8 5E =r : 8 $

k = G X … r
T : 5’ + ‘ ‘ 58 IN 34 U1/ } = Aik
:8 $ ,E ! P + } 6 .8 $ : !

ik ⎧1 .8 $ : :  k + :  i ‘ 7
A =⎨
⎩0 1 ‚ 8
k T 5 = Ck
[5 V+ 8 + * + I! = W1
[5 V+ 8 + * + I! = W2
8€O €’ &1€ €} =5€$ ,E K + ] +
:! 5R% 887 Q 05 4 F +
39 IF 5O ” #B + =8 + 05 &1 } = X ij
U1/ 5+ = : 5′ + ‘ ‘ 58 IN 8 $
:8 $ ,E ! P + } 6 .5+

⎧1 .5+ ?1 :  j = + :  i ‘ 7
xij =⎨
⎩0 1 ‚ 8
😯 69 + ‘ 34 05 &1 } X ij } 6 ++
.5F 345+ L ] IF
F + ‘ } K 9E ” ,E + ] + :8 8 IN # P + I ^ 05 5$ =$

⎛n⎞
777240111020

2029213-38945 minz1∑n ⎜⎜ C∑i=1nxijtili ⎟⎟+W2∑r Ck ⎛⎜⎜∑n⎟⎞⎟ (G)
=W⎜⎟
j=1 ⎜ε+∑xijti ⎟k=1⎝ j=1⎠
⎝i=1⎠
Subject to

n
∑xijti ≤C,∀ j (X)
i=1

#<4 I! + + + ‘ -‘ : I! IF 8 ‘ 8 $ Q D 7 5] = 8 ‘ ! P 6 8 5$+
+ (p) \+ ! 57 C
.5F 348

.
6K q+ 85B R42 = 2 + N+ $F 3] T r ‘ : I! ‘ (G) 05] 8 58 IN = /8 0D + =8A4 8
+ + ! q+ 6K 85B R42 +) 5+ Z
:(3 ! JK

.
G X b k m o p n i
m b k m k m G k o
b k X m G k b X G
x y x y x y x y x

+ y x T 8 =8 + Gj + + #<4 I! < P 8 ; + F 5 ‘ 5$+ $ 8 8 ] ‘ : + I! 5$+ K 5B Gjj mj + +
] + P 6 8 5$+ m G + + ; + :5F 348+ ! P + 6K q+ 85B 05] 6 +

6K 5B = m × GjG + G × bjj = njm q+ 5B = m × Xmj + G × omj = Gijj

ƒ = /8 ! JK + <R$ + 0D 6 + 6 ++ ! &’ njm ! IF [5 V+ 5O ‘ 39 I & + ]
U1/ = 2 ‘ 8 $ y9 7 0D H+ .5$+ N+ Gijj
:5$+ ! P + = + ‘

( G b ) – ( X k ) – ( m o p ) – ( n i )

8 34 ! = M + P 6 8
=8 + Gmj Gmj Gjj mj + + ; + = M 6
.5$+ kmj 5 T & + ! 4 s4 + = 8 r L^ ! `K ‘ 5$+ G/k X/p k/oX X/mm + + + ; + + + #’ T & #<4 I! 8 F I8′ + I! LB + .5F 348+ GGX/p
1268732040128Downloaded from ijiepm.iust.ac.ir at 16:26 IRST on Saturday November 4th 2017



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