arbor, such that the total amount of waste generated is minimized while the ..... of measure which is concerned with length only; a board which is ten feet ... to 15 ). A random sample of boards is drawn and the usable width of each .... CA theoretical lower bound for each instance of the problem is obtained if we ... Page 25 ...
! "
#$ %&
' ! ( ! $ ( $ ( $!!$ ( $ ! ( ! ( ) ! * + ! (
( !! ! ( ' ), ! ) ( ! ( + ) ' ! " ! & ! -.
$! ! ( "/& + ) ( "0& + $ ! (
1 "2& ! "3& ! ' 4 5 " ( & + $ ! (
! ( ((6 ! ( )
!5 ! ( !$ % ( ! (! ,$ ! ( ! ( !5 !5 !$ (!! ' 5 ! ! ( !! (5 ( ( ! ! !( 7 ( ! , ( !! (5 ( ! ( !! * ! 5 5 $ + !$ ! ! ( 5 $ , (
!"""
#
$# %
# # #
# &
! " #$" % & ' ( ! # ' " $ ' )$" ! " $ !*" $ + , - ! - - . + / # ,' +/ $ " 0 '1 *"2 ' 3$ , - ' * ' ' )$" ! - ' ' "" ! # ! 4 5$$ 4 - ! 3$ , '" #
- 6$ " $ $$ " ) 7 8 ! ' $ ' " ' + / # - " ! $$ "$ ! - 6$ " ' ) , ( ! 6$ - ' ) 8 #"
! "" - " ! ' " "" 8 - " $ + + ! $$ '" ' " ' $ '"
!" #! $ % & " " !" #$
"! '(( )* " + !
,-(
*
* .
. * !*
-( / & " !* $
% #
%
0*! 1*
2 * ( *
" " 2*!*
2!" 2
!( & " . "
-( & " . " 3 &
!
%'
0*! !( & " . " $ . " ! . 0( & " . "
, (!! !
4! !
5 *
( !
%)
0*! !( & " . " 3 0( & " . " 6
-" ( $
0 % & " (* $
* 7 * $
2 ! 7!"
4! (
7 0(
-
3
5 *
* +
&)
0*! 8* 7 * & . 3 " 2 3 7" 6 )
,
*&
2* & " !" -! *
1** !" -
**
. "("! !"*
. "("! *!* !9 (
* & " * (!*: !* * ((/ -( & " ! ( (!* 6 2* ( * 4! " ( *
!"* " &* ! !* " !"* " ! !* " !"* " / ! !* " !"* " ! !* " 0! * * 0! * * -( & " "* ! & ;< = " * *
3
3 -( & " "* ! & ;2 4.2 & ;2 4.2 & ;2 " -;0 >' " 3
/
.
/
. (! * (! " " " &** *&!* * " !* & * " " ?(
" @ ( & A " " ( " !* " B* " * *&!* . !! " * ( !* !" " ( ! * " 9 "
" / (!! ( "& "!"
( ! * " " " " " ! & " C & * !*D " & " ?D " ( & !* !" @ " "& !
2 ( * !* # D * !" /!" B*
"* " " *
D *
( & * & !" /!" "* D
(& * " * & ?(& @ & (!* !* # *!"
1" EF ( "! & !"* *!" ,( * " !!( & ! * & ( ((( (!
& ! * !!
!"* & " " " * " !, (* B* " " GD " * & " ( * " " (!
D * *
" * & " " B* ?(!
@ 2*!" * & D " " "D * (
(*!
* #! " !" *! "
!
*
/!" * " (& !* # " " / ! ! " ( ( " !( (
.
# +
1" EF ! " ( &
: C" " & !" !
% " ( (* "
! & " " !!( " / "", " " (( & " " " " ( " " ( #* " * !9 ? " / ! " ( "* !* "*" " *" ( @ " H I " " ! " * " & *!* D
" ?(
" "&@ " !9 & !** !* !" "!"
(*! " * & " !* ? 1* @ " (* ?@ !* ( ! & ?@ !"* & " " " ? @ & *!* !9 " *
Board
Saw Blades Rotating Shaft
Cutting Channels
1* : 4,( !* "
$$$
$$$
$$$
$$$
$$
: . "("! !"* & * "!" ! * /(
" B* . *
& * "!" !! " " J "!" & * &
& " . D " " "D *
" & B* & & * ?& !" "!9 @D "*
" " " ?0 &! H &I * & *J "D !
!* !9 * & " "!9 D * " H &I & B* & & * @ " !"* & !* # (!
(! & . "(, "! !"* D "!" ! & ! " " D ( " * & ! " " (!
! *! "! D ! " !"* B*, & * & & * & & - " * & * !!( " * " &
&* : < ? @ ? @ " & !* !9 * * & & '!*
* ! A *" " ?(!
&
$
$
$$
$$$
,
,
,
,
,
,
$
6$$
: . "("! *!* !9 ( @ . ( & " * " & !"
" ( " ! &! & !" ?* *
3
@ " B* & !" *!* " " ( ? &@ *
" & & &B*! * " * " * ( & " / & "
! ?0D ( * * *
*!
* " ( & " !" .
*!"D " (
% (!
D " * &B*! * & " * * & " ! " @ . "("! * (
* " !* (D *!* (*
& !
& " J ( " (& "D " ( * & " " & ! " (! &B*! * & " ( * !
" * & (*! " !* & !* # " 9 !( " ( K (*! * & ?@ * D ? @ ( !* " & ! "!" & " " " !"* ? ( @ " " & " *
! "
!" !* B* " " & " ?(!
3 @ K
* " " * & !* " " & ! " " * & " " " ! " ! * (!! "!" B* " &*
" !* & !" & ! " " & " ( ?& @ * " *
" " & " H I 0 E F ? * B*@ " " B* & !" " " (,
*! ! /! ! ( ?L,$L (*! " " ! !!( @=
.%
# "0
- " * ! D 1" EF " C4,( 2 . 2!"* = ?42.2 @ &
: ? @ C1 ?@ & " " " ! (, B* ? !"* @D ? @ &B*! * & ! ? @D ?@ "& "D ?@ "D & ? @ ?D @ " ! ( B* & ! * ? &@ * "*" !" D *!" " " " " & & %= 0 " D " #! & " ( % " (*! " !* # * #! " ! " " (*! & !" " " *!" " ! ( " " D
" * & * !" " #! (!D " ( & !" "> " ((( 0 " !"D &!* " * (! * & "&D " C4,( 2 . 2!"* " . = ?@ &
: ?@ C4 ( D * /! = 0 (!* D &!* D "!" & *
.&
1
" &
!"!% " ! * " !", * :
* & ! " " " !"*
" " ! " " ?!" @D <
B* & & " " D <
* & ! ! " " *!* !9
" " ! ! " ?!" @D <
; & B* & " " ( D <
" ? &@ &
! " ( " D
" &
!"(
% # " !"( (
;! 0( !* ?* @ & 4,
2.2 > !* " * & !(* /( !( " (!* (!
%.
" ! ! ( (!* " & * " ( & " C!* *= ( * & ! (* 0& *!" ( * &*D ! " C!* *=D " (!
( " D " !"
" & * " ! & ! (* & * !
" C & =D ?@ . ! * !( " " !/ & % (
" 1* D
" ?@ " * & " #! &*! * 1 & ! ( D * * " ( " (! " " *!* GD " ?D !
( @ * * * *
&
( " " *
Start with a feasible solution S
Is there a solution S’ ∈ N(S) with f(S’) < f(S) ?
No
STOP
Yes S ← S’
1* : . ! ! ( (!* !" " " & " !* *D " * !" &
* & " * (! " "( " " * *
C *"= 0 ( * " " !/ & (!! ( D &*, ! : ?@ * ( *D ? @ " " *!* D ?@ !" 0 " / !D " * &
%. %..
+ +
0 * * ( & D B*! & !* !" " "& ( * ?D @ . * * ( !(* " /(! ! (
" ! !
%..
1
" ( * * " ! & " " *!* 1 D A " " *!* :
" "& " " &
: ?@ < ?.
" ! & ! !* !"
D & !* ( D " ( @ " % & " "& " " & ? @ " " * & !* !" ?! " & " " ! 0 " ! D " * & ! "
" "
* @ . * !9 & " " " *!* " J * !" & " * " * " " *!*D " " "D &
: ?@ < ?.
" ! & ! !*
!" D !" " " & " !" " " "D C(= " * & ! @ ! " & !" " " & !" " " "D " * D D ! " " & " "& 0 "
! D (& C = 9 &*
, " " (
! & ( : C & " & " "&D !* !" & " " * " " " "& D
C" & " "D " (! "
& " " " !" * &*
, " ?"
(
( " " (! " "& 9 !!* " !" @ " % & " * " " & # ? @ : & " " " ( ( (( " * "!" (D & " * * ! 0 " ! D " % & " " " & "
" ? @ ?!
" " " " *!* @
%..%
2!" " ( * * * " 0 " ,( D & !" !!(
!*D &*" !" "
" !* * 0 " ( ! D " " "D " " "
!"D " * " " !
! 1 ,
( ( * * D ! " / * *
#* & !" " " " ( ! ." * " " " !/ & !" " "!" " " "
!" " !" ! D * /!,
("! 0 * ( D " !"
& " !*
!" " !* " " !" K" , ( * &*D " !"
& " !" & " *
!* " & " 1
D " ! !" " & &*
, "
%.%
" !
" * ( & " ! ;0
" 1* D * " "& * " " " " " *!, * " " !
* *& (! " " " !
$ % &' )*& @ $ ( &' ?)*@) "!" ! D (! D 1*
Start with arbor R0 ∈ Ω
No
Is R0 better than all RU ∈ NU(R0) ?
Find a better RU ∈ NU(R0), R0 ← RU
Yes
No
Is R0 better than all RS ∈ NS(R0) ?
Find a better RS ∈ NS(R0), R0 ← RS
Yes STOP
1* : -;0 >2 " !" & " !( * " ( D (& ! !" " A " " *!* " !" ( " " & " ! !" " " (! " "D ! !" " (! " " " " (& "
Start with arbor R0 ∈ Ω
Is R0 better than all RS ∈ NS(R0) ?
No
Find a better RS ∈ NS(R0), R0 ← RS
No
Find a better RU ∈ NU(R0), R0 ← RU
Yes
Is R0 better than all RU ∈ NU(R0) ?
Yes STOP
1* : -;0 >' "
%.&
!
( !( & " &* * & (!! ! " # 0 " !D " !"* D " ! * * ( D " <
7* ( * & !" & " &* * (& & * " A !" * " * & * " "
"
" ! & " " * " " * & " ! ( ! * " ! * & " /(! & "
!
( * &* " "
0 ! & " " * * &* )* " " ! * & )* " " & " * 3
*
*&
)*&
)*
* & *
$
6
7
$33
333
3
2
6
7/*
3 66
33
7* ? &*@
6
6
: -( & " ! ( (!* 1*"D " " ! & " ( ,
"&D ! !( " " "* ! ( " &
, !"( " )* * !(* /(
6
& ! " !"( ( & !* & " * " &
(( !
&.
" !
2* . ?@ ) "!" ! ! !" 0 " (( *!! &*
(% ( !!* &
*!"
B*! !"* D 1 *
D (! D !* D & ! /( E F " & ! !" " " 9 " & ! D " " (*: " * * " ! ) D " * & " ! &*! ! GD *!" & * !! !
( * . * & ((!" " * "!" (
! " ( : " " ! ( * A * D " !( / & " " " ! ! D " !( & " !" '&* D & " " ( &! E F
$ $
( " ( ( ! (
!! " * " T*! & * ! ( D !
" " ! " " & "
!* * !!( !! " &
!!(! !D " ?@ " * & " #! &*! * :
U < ? @ ?@ $ ?.. @ < U 2 $
! "
* " ?$ @ " U
?@ ( D
! ( ( " ( U " ( " H *("
& % U
!!(I ! " (* D
*("
" "" ( & !!(! " & / (* . (!
( D " (*
9( ""D
! *
" !" !* EF " " (!
( & ( (, : ! D $ % % !
. ! ( ( VVVVVVVVVVVVVVVVVVVVVVVV ?@ 4 * 4 (*
2$
( ?@ ( ! ?@
; U < ? @ ?@
0& U $ " ?"
@
0& U 2 $ 4 * D $D " ?$ @ 0& $ %
" ?*("
@
? @ ' . < ?!@
?*! (*@
' " 3
" ((/ " ( * VVVVVVVVVVVVVVVVVVVVVVVV 0 " (D " " !* (*
" " (* *! !! " &
*! &*! : (!
*.
0 " " !"( D "* ! ((!" !* & !, ( : ! ( (!* ?;0 @D * ?2.@D ! " ?4. @ 2(!! ( & " "* ! ! & * #! " !"( !( " (&! & " "* ! & * B* !(* B* (!! ! & " ( " !* ! D )* (& " " (( * & " ( ! " * 2D ! !( " (&! & " )* * !(* /( 0 " & " !"(D B* * & *!D "
!" #$
* " !(* /( ( " " " & /( ! D " !*
*.
: 8 !
!
" " #! % " & & (*! " !* # 2D " 9 B* * & " D " (! ! ( " GD !* ! D " # !! " ( " D " (*! " " ! ( D ! (*! " " " ! /! " ! ( "&D # D " (! ! ( " D " ( B* * "! * " # ! ! (! : ?@ " D ! D *(( * " D ? @ $L * # J D "!
D " ! (*!
2D " B* & & (!! ! & " /(! ! ( " ! !(* !( " ( *I *D & ( * " ! 9 " D & " ( * 9D " * & " # ! * !"9 D !( " " & " "! *
*.%
"
" ! & " " * !( " (&! & "* ! (!* " " & /! " " " & " "* ! / (D ! !! ! % " ! " "* * !! * 1 " D ! & " ( " " !*
!* "
3
* " " " " !"! ! " ! .
D (! ! & " ( ! & ?@ & B*
" " " ! ( & ? !"* @D ? @ &B*! * & ! * D ?@ "& "D ?@ " 0 " !(* /( D A ! * * * " * " ! " " , * ! ! ! " & 3 D "
! * ! ! ! " &
" !"* D " " "D ( D D D !" & " ! A !"* " " * & ! " " ! ( !" !"* " " * & ! " " " !"* & A * & " "& " ! : 1 "
"D " * & 3 ! G!D & 3 < A
! & " ( * " !(* /(
*.&
8
" !( & " "* !
& # !* " ( ! & " !( (& " !* # " B* & " " " ! / * & !(* 1*"D * " ( " ?(!
D !* # (& @ "&D * ( * & !" & " " "* ! (& & $ * D (! D & !" " * & " * !" "* ! ?& *
6
" " @ " !" & " &* ( " ( * " "* ! & A (
" ( & " * !" "* ! D " !" "* ! !( " * & " * (!* /!* & "
! " ! ( ( * " * " "* ! " !( " ( * & " ! 1*"D " (! & " * &* ?& @ !" & " "* ! &
& " " " $ * ( " D !! * * " B* & " * & " & "!" " ( * 9 1
D & !" D " * * * ( * & )* !( " * * & (!9 "!" !* * " * & " (!9 !
?4,( . 2 @ !! ( & " ! * (!* & ( (( 1" EF 0 " !( D " )* * & * D 6D 3D $ !( " "* ! & < < D
" ! * *
" !" & " "
! * " ! ( " !"*
" "*"
" ! !" & " ( " ( * & " & "!" !( " * (!* " "* " / "
" " /(! & & ! (
" &* ?& @ * ( * & !" "* ! " * ?! " &
& " D !* * " " *@ " &
" " /(! & & ! ( $
" &* ?& @ * ( * & !" "* ! " D D $ * D (! " / ( " & " ! ( D " &
/
" " (! & " &*
* ( * & !" "* ! " " $ * D D D !( )* & <
< D " ! * *
"
" " !" & " ( " * & " * & * " & & * " / " " /(! & & " (! " !* # D (! D & " ( (& ( " & " D " " "D ( " /(! & & " (!D (! D & " * ( * & )* *
*.(
K " " &
* * /( " " : " B* " "* ! & !( * & ?@ $ & ! ! & )*D ? @ ! & D ?@ * & ? " & D " * " " $ * (@ .!
" &* ( ! D " * *
)* " * * ?& @ " " "* ! & $ * & " 1 " & "!" " & " " "* ! &* * "
)*D " *I *
*+ & " *I * )* 1 " & "!" " ( * 9D )* &* " ( * * 1 " & "!" " ( * 9D " (!, & " ! ( ( * " "! *( ,+D " (! "" ,-+ ?" , (! & " ( * ( ( D ! !" ! D )* " ( * " " $ * @ 1 " & "!" " ( * 9D " * )* " " $ * " (! "!" ,+ " "! ! 1
& " D " * * )* " " * ,+ & " * " " / 6 * .!
" &* ( D " * *
?& @ .+ & " )* 3 . "*" " * & " * "" ? " * & " " " !"* ! @D " /(! & & " (! & " *
" " & " * )*
2!"* 1 " " ?!" @ .
)
? &@
$
$$
$
3$
$
$$
: !"* " &* ! !* "
2!"* 1 " " ?!" @ .
)
? &@
$$
$$
$$
$$
$
$
$$
$
$
$
: !"* " ! !* "
2!"* 1 " " ?!" @ .
)
? &@
$
3
6
3$
$3
$
3
$
$
: !"* " / ! !* "
2!"* 1 " " ?!" @ .
)
? &@
$
$
$
$
$
$
$
: !"* " ! !* "
) " ; &
) " ; &
?!" @
!9
?!" @
!9
$
33
33
33
33
$
$
33
33
3
3
3
3
: 0! * *
) " ; &
) " ; &
?!" @
!9
?!" @
!9
$
$
$
$
3$
6
3
$
$3
$
6
$
$
$
$3
$
$
$
$
$
3
$
3
$
3
$
3
$
$
$
$
$
$
$
: 0! * *
/
.
)
.
)
.
)
.
)
0
6
,
$
,
,
,
-;0 >2
6
$
$
4.
6
$6
$
2.
,
,
,
,
,
,
,
,
-;0 >2
6
$
$
4.
6
$
$
2.
6
,
$
,
,
,
-;0 >2
6
$
$
4.
6
$
$
2.
6
,
$
,
,
,
-;0 >2
6
$
$
$
4.
6
$
$
2.
6
$
$
6$ 6$ 6$6 6 6 6$6
6
6
1 *
1 2
1 3
1 *4
5 *4 5 & ! -;0 >2
36 36 36 36 6$ 6$ 6$6 6$6
4.
36 36 36 36 6$ 366 6$6 6$6
2.
36 36 36 36 6$ 6$$ 6$3 6$6
0 *4 -;0 >2
3
3 3
6
6
4.
3
3 3
$
6
2.
3
3
3 3
$
$
6
: -( & " "* ! & ;< = " * * 3
/
.
)
.
)
.
)
.
)
0
,
,
,
,
,
,
,
,
-;0 >2
33
6$
6
4.
33
6$
6
2.
,
,
,
,
,
,
,
,
-;0 >2
33
6$
36
4.
33
6$
6$
2.
,
,
,
,
,
,
,
,
-;0 >2
33
6$
36
4.
33
6$
6$
6
2.
,
,
,
,
,
,
,
,
-;0 >2
33
3
36
4.
33
6$
6$
3
2.
33
,
,
6$
,
,
,
6
6
1 *
1 2
1 3
1 *4
5 *4 5 & !
6$ 6$ 6$6 6 6 6$6
-;0 >2
36 363 36 36 6$ 6$ 6 6
4.
36 363 36 36 6 6$ 6 6
2.
36 363
,
,
6
,
,
,
0 *4 -;0 >2
3
$6
3
4.
3
$6
3 3
$
$
2.
3
$6
,
,
,
,
,
3
3: -( & " "* ! & ;2
3
3
$3
36
4.
3
3
$
6
2.
,
,
,
,
,
,
,
,
-;0 >2
3
3
$3
3
4.
3
3
$
6$
3
2.
,
,
,
,
,
,
,
,
-;0 >2
3
3
$3
3
4.
3
$
6$
3
2.
,
,
,
,
,
,
,
,
-;0 >2
3
$
$3
3
4.
3
$
6$
3
2.
,
3
,
,
,
,
,
1 *
1 2
1 3
1 *4
5 *4 5 & !
6 6 6 6 6 6 6 6 $
-;0 >2
36 6$ 36 36 6 6$3 6 6
4.
36 6$ 36 36 6 6$6 6 6
2.
,
6$
,
,
6
,
,
,
-;0 >2
3
$
6
4.
3
$
6
2.
,
$3
,
6
,
,
,
0 *4
$: -( & " "* ! & ;2 4.2 & ;< = " ! * *
/ 678/%
.
)
.
)
.
)
.
)
&
6
$
3
L $L
$L
L
L
$L
$L
$L
3
3
$3
36
L $L
L
L
L
L
L
$&'9%
L
: -( & -;0 >2 4.2 & ;