%.(-%(% +, %()* %’& % ! “# $

!

! “#$ :”#$ % -%# “#$ & ( “+, ()* “‘ .# “#$ % & 6 7+8 9 (0 #: ;< ,! (+)= .0 / /#0 1+, 234 5 + @>? 6 % 7+8 AB)9 .# >? <% & % (0 +8 ( + /#$ ,! -+ ! (+)= 2C ! .2C ;< &% -+ # % +:3 ( C .# (? /# /#$ ! <% & (+)= (3 E F +:3 (GE 9 -+ &% .5 ( “+C #3+, % D ##* # 9+:3 (+)= .# #9+H #3+, (C 92)! D # +:3
&% (5 + J, ID8 (3G K< .#0 IH “#$ # % +:3 F9 0
.2C /# (G L M (< % +*+ K% C -+

“+C #3+, H M (< “#$ % & :
; .$+( $1 = $1 $ *)I+ )
$+ !$% )I+ ) 0H5 0- 1F *$1 D = & $+( #$% 0H5 1( ( 0 8 1( #$% ; .[J] + B< $1 = ; #8 ( .$+( $1 & /< = 0$1( & * ; !$+ *)I+ ) 0H5 3 )I+ ) 1F #$% ) .$+( #$% & ‘( ) 0( D K/
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$; $KF : * & 0 ” I& – *'( L !$+ $; #<& 0+ M2 ( *'( 0C; ( 7 91: 0<( .[P&O] ( N” ; *$+( 1F #$% ; *$/+( / *A & G- Q; 2 9- *R 0<( ) .< )I+ ) *!$// U S;T 0<( ) #/ S * 0$/(
( EX 6&$/D #$% ) /W7 .[V]$+ @/+
.$/+( ?9 *)I+ ) 1F #$% ; ( D $( *'( ; 3 0;$ /2 I& ( 0I& ,ZVP G2 .$$+ $; $( ‘( 07C; Y

Orthogonal
Distributor‘s Pallet Loading
Bin-Packing Problem

.

!” – ( .[,] $+( #$% & ‘( ) *$+ # .$/+( 0$1( 2 0$1( & *0$1( 3 $/ ) & ) 04 5 6& ‘( *0$1( 3 ) 78 ‘( ( !+ *0$1( 2 ) 0( / 7 91: #$% & 0$1( /< = /< $ *0$1( & ‘( ; 3 . 3 ‘( ?@ 7 *A&B ‘( 0C; 3 0 ( < .[>] $+(
/< 65 0C; *$( ‘( 6& )( 9; 9;
6E- /< = 0C; #8 ( D = & $/< & 1F #$% ; G1 *0$1( & ‘( ; .< 1F #$% ; 3 .$+( 0$1( & 0H5 3 0$1(

// //
. “#!

‘ $() * #+, $%&
mjasgharpour@yahoo.com
-0 12 #+,
hatefima@yahoo.com
Cutting & Packing Problems
Cutting Problem
Guillotine
0;$ )(F 0$( D ( *G$ @
.[,>]S2 . 2( # ;$ /% & 6& G( 07 + 0 l 5< F [7 / ‘& 3 Y$F .[,J]+ $; 0$1( & 0C; *$”& A 92 ‘( 0C; 1N #4 0( g( $” 3 *[C;
G$ [l” n7 m( 0;$ g( $” G .<
.$+ $7@ 6& G- 02 I 0$1( 3 07C; Y *Y& Y$F .< D ; 0;$ *Y2 Y$F & + 92 !& 07 + $1 U: $( g& *G$ 07 S&$ ً 0$1( 15 l “_ & $+( 6& A (( C; ( $+( 1F A U: (( *< I 7 + A&B S: E S&$ ً .S2 65 $1( (( 8 G- G 1l D ( C; $/+( G- [7 1F 1F S2 p 0 0;$ R; & $/ #9 G1 $/8 U/ D ( S2 I *C; 1( 7 !$ / n45 ( /W7 .$+( C; 1F ); ( 5- .$/ $+ – ( G$ [l” *1F 2T 0< !$ & $/+( bD $( 1( Y *0;$ 070;$ ( N 4 *N” ; .S l 12& ); ( 5- .$+( N 04 ( 0g( [l” & < D #$% & ‘( ) !” Yl 07 Y # #I * -( g & E- ( ( l/ R; .$+( I = ; 3 !$+
0I( 07 ‘& ( )” #$% & ‘( ) N .S2 !$] @T

. m$1 /W7 & W A ،L G- ( )I+ ) F&
O i 1F i = 1,2,.., m S2 L )I+ ) 1F
S9 ; .$+( Vi ‘ & Wi A Li G- 0
1F ‘ l 0- *6& 65 1F #$%
.< /]( *6& !$+ !$% C 9 ( .$ S(_ ( S: 1F S2 ( A5 G- ( 0 8 G- $+ !$% 6& 0- $( 1F ( S: ( S2 ( A5 /W7 .< F *6&
9- 65 ; .< 0: 6& */ ( 1F 7 *S(_ pf] 2 / V / O / R D ( ! 2/+ .[qr]<
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#8 – $+ & ?2 #2 $; #/ 07 SF ( : ( *'( 0C; 3 / *[C; ” 7
‘( ; 3 I& 8 .[P&O]+ $; * 2
.$ /< 2 07SF 0/9 ( *3 5 ( D $( ‘& ]T ; ; 3 *1F 0N 07S&$ ( \2/ 0C; ( * 1F 7 0( #8 – & + ;5 0$1( 3
; *Y& 5 .< 1 6& G- 02 0$1( 3 F \ ( #8 S2 0C ]T ; U( \N .< $ [7 / 0$1( 3 07C; 5< 5 !$+ )D” /( *Y& 5 ]T ; ; ^$7
3 ]T ; ; )” 0( & .$/+( 3 92 I& .$ !B2 T 04 ( ‘& *0$1( & S5< F ) ?2 7$1( 0< & <
.[`]S]< $ #$% & ‘( ) )” 0( #8 #( _ ( /<= ‘( ; G1 #$% ; 3 0;$ 1F +a7 0< : S2 [ )D *$+( +a7 % *< G G$ 07 S&$ C% C; #8 #( $] G1 *A&B #$% 0C; 3 1F
*#$% 0( *N 04 ( !” G” 7 ( .$+(
( *: 070;$ \ = & S2 !$+ 0&$1 07
.[d&,]$/+( bD $ 07c 3 . 0H5 0 !2 G$ ,ZdP G2 e% +< Kf ( * 0H5 1F 7 *0& G$ ( A&B 1F & 0( .[Z]+ @/+ 1F /T !$+ !B2 S&$ 02% *$/+( +$ +a7 [7 ) 3 ( &$ *G$ ) S: 0& .S2 i@ ); ( 0& .&45 ^$7 U( ( *0h<g
G$ .$/+( bD $( 1( Y *0;$ F 0( .[Z&,k]$ ED #8 j & $ F L $ !B2 0/( c % *[7 ( S9 A&B 1F & 0<
Y& 1F S2 e% *T *g( *G& 1F .S2 !$+
$ < % 0/( 07c I + \: *< F
.$+ BD ( & C( 3
0H5 1F S1F 4 0( I; *+ 1 * 1F /T e% +< Kf 1 c % !B2 0l( *1F +a7 0< : 0< $ 0- ( + !B2 0/( c & *S: I *S;” 7 + S;” % GK” m( *65 c & g1 M2 ( /W7 .[,,]( < % :

7 Overlap
!
m
MAX U =∑V i ×ti (,)
i =1
St. (I)
n
∑tij × = ×DjLi ti ∀ =i1,2,…,m (>)
j=1 n
∑tij × ≤W iW ∀ =i1,2,…,n (J)
i =1
∑n D j ≤ L (O) j =1 n−1
ti1 + +tin ∑

tij −ti j, +1

≤ 2 (P) j =1
D j ≥ 0 ,t tiij Binary (V)

7 Dj & 7 ti j 0< $ m( (>) 07S&$ *G$
.$+ 1F G- M2 (
$// G $( 7 #$% 0g( $” *(J) 07S&$
G 7$( A l 0g( $” 4 (O) S&$ & 1F 3 7$( $/ H (P) S&$ .$/
* 1F 3 04: ( $/+( [7 7 < F 8 .C D

(I) .
7 .$+( #8 02 @ (I) G$ ) ] ?@ & !$WT ( G$ (P) s 0&$ *+ L”E *&$ 02 02 @ 0( ‘E .$+( @ = *G$ ) S: .+ G$ / = #$+ [l” m( 9; .S2 (P) s 0&$ ^R” *Y$F ;& @T #8 ( @ 0: &8 $7@ :& ( EI] #2 $; I& M2 ( *) Y& Y$F .$+ $7@
[2 3 #8 – < – ;$ * &
$; 0$1( 3 0C; 3 *I ” & 7 *0 ” *(7 $() 0$1( 3( 07C; *0 ” I& .[P&O]+
3 ?2 0$1( 3 07C; .$/7 pf] * D G$
.$+ $/7@ $; _ G$ _ G$ ( *G$ (J) s 0&$ *) G ( + (,k) (y) G$ D ( D G$ & $+ )/ & S2 $( j -1 0&” (II) G$ $/ A5 #/ .$+ $7@ 0( .[/ $; j $( *0 ” I& – S2 F 0 ( @8 $( *$+( ^$7 U( 9( H $$: $( *^$7 U( 2 x j \N (II) G$ ( t( zI a2 .$C( L (II) G$ zI a2 0 ( G” .$+( B/

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.
1F S1F $$: s1 $( *$$: ‘& ( 0;$ 0(
.< 6& ( .$C( L $+( 0$1( & 0C; 3 , )I+ 0 l ( 0$1( & 0C; *0 ‘( t@ $
3 0C; 3 [I” *$( 7 0- ( + [ 7$(
.$+( 0$1( &B 07A 0 7$( *$/ L”E 7 G- .S2 6& G- (( 8 A U: 0- ( *$/7
\ ( .$+( 6& A (( & !( 0& 4 7$(
:S2 s1 )(F 07 l

j = 1 ,2,., n .$ F Y j $( 1F l :STR j
i = 1,2 ,.,m . F 8 Y i 1F 7$( l : Bi
:SB< # , )I+ Gu 0(

.STRD3 ={2,5,6,7} & B7 ={D2,D3,D4}

.

*7$( $1 < $; S2 F C; 0( + :
.$/7 pf] B& STR07 l & 7$( A G$ 9- .+ + (I)G$ *C; 0$( s1 M2 (
= & $+ $7@ 3 0& ti < F 6& i 1F <
.+ BD D
& $+ $7@ 3 (( ti j $+( k $( i 1F < /W7
.S2 Dj ،j $( A N .< BD D = .$+( @= (I) G$
HG ،(I) K 2 (Q) 5 1/ A ,A
.4$! RS/, ،(III) K () 5
m
MAX Z = +θ π∑ i ×Pi (T)
i =1
St. (III)
m
∑Pi × ≤W iW ()
i =1
∀ ∈ti(0,1) ()
8V W8 8 Pi * 4 i BU * CK (III) K8 D8 E * .$ ! FG HG X $
R8N SN $ &4 O (II)K x j ‘4 JY
. ! 0O P
(II) K 2 (III) K A !+4M 2 9 (II) K C4$ ;< A C$ ! /$4 (IV)HG (II) I K .$ 0 ZG A V 2 ![\’ 0 ZG DN /J .$!
P RN ,M9 $ V ? Z< 14M A .,A! ]N[ > R\$
m
MAX U =∑Vi ×ti ()
i =1
St. (IV)
n
.∑tij × = ×DjLi ti ∀ =i1,2,…,m (^_)
j=1 n
∑D j ≤ L (>) j =1
∀ ≥D j0 0 ≤∀ ≤ti1 (>>)

M A X U = ∑m V i × t i ()
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=1
n
St.

tDLt i1,2,… m
=1
∑n D j ≤ L ()
j =1
D j ≥ 0, ti Binary ()
. m+1
%&’ π .$ ! “# m+1
:*! DP t = (πθ π ππ πt , ) ; t = ( 1, 2,…., m ) ()
-,/ 0* 1$2 3’4 π +,
58 -,/ 0* 62 74 θ $ ! () 5
? @84 (>) 8 8 D j !8,9 ;8< 8,= . ()
.$! Pj = (t1j ,t2 j ,…..,t mj ,1) (>)

;
:

;



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