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Second Bank
First CP-III
First Bank
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Second CP-III
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Used for Component Placement
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Second Bank
First Bank
First CP-III
Second Bank
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Primary Family (Permanent Placement)
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First Bank
Second Bank
Second CP-III
First Bank
Second Bank
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AB
A
AC
A
Used for Component Placement
Used for Offline Setup
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As mentioned in the main article, a family in our manufacturing environment is a
group of products (boards) that can be built with a single setup on the component
placement machines. The greedy board heuristic is one way of assigning products
to families for printed circuit assembly. The only data required for the greedy board
algorithm is the list of components and the expected volume for each board. Each
family is created by the repetitive addition of products, one at a time, until slot
availability is exhausted.
In the following example assume there are eight component slots available per
family and that the following boards must be assigned to families.
Board
Expected Volume (vi)
Components
Alpha
Tango
Delta
Echo
Beta
Lambda
Gamma
1400
132
2668
1100
668
1332
900
A, F, K, M
C, K
H, D, R, F, K
R, J, S, K
G, F, T, L
H, D, F, K
A, J, E, K
Board
New Parts
(si)
Expected Volume
(vi)
Greedy Ratio
(Gi = si/vi)
Alpha
Tango
Echo
Beta
Lambda
Gamma
2
1
2
3
0
3
1400
132
1100
668
1332
900
0.0014
0.0076
0.0018
0.0045
0.0000
0.0033
The Lambda board is the one with the lowest greedy ratio because its components
are a complete subset of the components already in the family. Since adding
Lambda to the family does not require the addition of any components to the
family, the greedy ratios given above still apply for the selection of the next board.
The Alpha board has the next lowest ratio and it adds two new components (A and
M) to the family. This brings the total number of components in the family to
seven—one slot left.
After adding the Alpha board to the family, the new part-to-volume ratios for the
remaining unassigned boards become:
The board with the lowest greedy ratio is the first one added to the current family
being created.
Board
New Components
(si)
Expected Volume
(vi)
Greedy Ratio
(Gi = si/vi)
Board
New Parts
(si)
Expected Volume
(vi)
Greedy Ratio
(Gi = si/vi)
Alpha
Tango
Delta*
Echo
Beta
Lambda
Gamma
4
2
5
4
4
4
4
1400
132
2668
1100
668
1332
900
0.0029
0.0152
0.0019
0.0036
0.0060
0.0030
0.0040
Tango
Echo
Beta
Gamma
1
2
3
3
132
1100
668
900
0.0076
0.0018
0.0045
0.0033
Delta is the board with the lowest greedy ratio so it becomes the first member of
the family. It has the highest product volume added per component slot used.
Delta adds five components, leaving three slots to fill this family.
Now Echo has the lowest ratio. However, the Echo board has two components,
and since we already have seven components, adding the Echo components to
the family would exceed our limit of eight components per family. Therefore, Tango
is the only board that will fit even though it has the lowest theoretical contribution.
Adding the Tango board fills up the family allotment. Finally, the components in the
family include H, C, D, A, R, F, K, and M.
The next family is defined by following the above procedure for the remaining
boards: Echo, Beta, and Gamma.
With the components H, D, R, F, and K already in the family, for the next board the
ratios are computed as follows:
,) *3003;-2+ 7)'8-327 463:-() % &6-)* 3:)6:-); 3* 731) 3*
8,) &%7-' '32')487 3* *9>>= 7)8 8,)36= %440-'%&0) 83 8,) 834B
-'7 (-7'977)( -2 8,-7 4%4)6 36 136) %&398 *9>>= 7)8 8,)36=
7)) 6)*)6)2') Definition. 3 &)+-2 396 (-7'977-32 3* *9>>= 7)87 ;) ()*-2)
8,) 92-:)67) 3* (-7'3967) ! ?<<…<-@ %2( 0)8 <- ()B
238) 8,) ()+6)) 3* 1)1&)67,-4 *36 *9>>= 7)8 32 92-:)67)
! *36 )0)1)28 <- ,) ()+6)) 3* 1)1&)67,-4 *92'8-32 *36
*9>>= 7)8 -7 < Ů #
$ ;,)6) 6)46)7)287 8,) ;)%/)78
1)1&)67,-4 -2 % 7)8 %2( 6)46)7)287 8,) 78632+)78 1)1&)6B
7,-4 -2 % 7)8
ǒ< -Ǔ
ǒ< Ǔ
20-/) 8,) '0%77-'%0 =)7 %2( 23 36 232*9>>= 7)87
9>>= 7)8 < )AAA ) < *9>>= 7)87 %003; 136) :%6=-2+ 36 4%68-%0 ()+6))7 3* 1)1&)6B
7,-4 *36 8,)-6 -2(-:-(9%0 )0)1)287 7)) -+ 32')489%00=
;,)6) 8,) ,36->328%0 &%6 -7 238 % 5938-)28 &98 % ()0-1-8)6
320= % *); 2%896%0 4,)231)2% '390( &) %77-+2)( % '6-74
Examples. ,) *3003;-2+ )<%140)7 7,3; (-**)6)28 8=4)7 3*
1)1&)67,-4 :%09) 3* )-8,)6 =)7 36 23 ;-8,398 %2= (39&8
*9>>= 7)87
2 8,) 38,)6 ,%2( 1378 3* 8,) 6)%0B;360(7 3&.)'87 ):)287
• 91&)6 %7 % *9>>= 7)8
0-2+9-78-' )<46)77-327 36 %2= %&786%'8 59%0-8-)7 ;) )<4)6-)2')
2-:)67) ?
@
-2 396 ):)6=(%= 0-*) 8)2( 83 &) 136) 79-8)( *36 % *9>>-)6 7)8
9>>= 7)8 1)1&)67,-4 9>>= 7)87 %003; 8,)-6 )0)1)287 83 &)032+ 83
9>>= 7)8 1-+,8 &) ()7'6-&)( 0-2+9-78-'%00= %7 A.978 %&398
1908-40) 7)87 6)+%6(0)77 3* 8,) 6)0%8-327,-4 %132+ 8,) 7)87
&)'%97) ,%7 8,) ,-+,)78 ()+6)) 3* 1)1&)67,-4 -2
2 74-8) 3* 8,)-6 8)2()2'= 83 7))1 -146)'-7) *9>>= 7)87 %6)
*9>>= 7)8 92%1&-+93970= ()*-2)( %032+ ;-8, 8,)-6 %773'-%8)( 34)6%B
• )*-2-2+ 4)340) -2 8)617 3* 8,)-6 46)*)6)2') *36 ')68%-2
8-327 %2( 4634)68-)7 ,) *9>>= 7)87 97)( -2 8,) *9>>= *%1-0=
%0'3,30-' &):)6%+)7
%77-+21)28 %2( 1%',-2) &%0%2'-2+ ,)96-78-' )<-78 -2 92-B
2-:)67) " ?&))6 ;-2) 74-6-87@ ?= … =@
:)67)7 3* (-7'3967) 8,%8 %6) *-2-8) %2( '3928%&0)
92) );0)88B%'/%6( 3962%0
1
L
L(w)
Degree of Membership
1
0
0
45
(72.41)
55
(88.5)
65
(104.59)
mi/h
(km/h)
Speed
0600
1200
1800
2400
Time (Hours)
A fuzzy set representation of the term lunch hour.
(a)
Operations. Operations such as union, intersection, and comĆ
plement are defined in terms of their membership functions.
For fuzzy sets A and B on Universe X we have the following
calculations:
• Union: AƔB(x) = A(x) Ɵ B(x)
Degree of Membership
1
or ôx i : AƔBǒx iǓ + Maxǒ Aǒx iǓ, Bǒx iǓǓ
(see Fig. 7a).
1
0
50
(80.45)
60
(96.54)
mi/h
(km/h)
A B
B
A B (x)
A
Speed
(b)
Bǒy iǓ
yi
= 0.1/beer + 0.3/wine + 0.0/spirits
might describe somebody who doesn't drink much but
prefers wine and dislikes spirits
Fuzzy set C = 0.8/beer + 0.2/wine + 0.1/spirits might
describe a beer lover
Fuzzy set D = 0.0/beer + 0.0/wine + 0.0/spirits might deĆ
scribe a person who doesn't indulge in alcoholic beverages
Fuzzy set E = 0.8/beer + 0.7/wine + 0.8/spirits might
describe a heavy drinker.
• Defining a person in terms of their cultural heritage:
Universe Z = {Zirconia, Opalinia, Topazia} and F(zi) repreĆ
sents a degree of cultural heritage from the three provinces
in some imaginary gemĆproducing country.
Fuzzy set F = 0.3/Zirconian + 0.5/Opalinian + 0.1/Topazian
might describe someone who was born in Western Opalina,
attended a university in Zirconia, and married a Topazian
living in Diamond City, Zirconia.
• Lunch hour:
Universe W = Day (continuous time of 24 hours)
The fuzzy set L (1100 to 1300) might represent the term
lunch hour" as shown in Fig. 6.
1
A
B
A B (x)
Fuzzy set B =
x
(a)
A B
x
(b)
1
A
A (x)
Crispy and fuzzy representation of the notion of average
driving speed. (a) In the fuzzy representation the membership class
average driving speed varies from zero for t45 mi/h or u65 mi/h
to 100% at 55 mi/h. (b) In a crisp representation membership in the
average driving speed set is 100% in the range from 50 to 60 mi/h
only.
A
x
(c)
Fuzzy set operations. (a) Union. (b) Intersection.
(c) Complement.
June 1994 HewlettĆPackard Journal
1
1
AƒA = X
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x
1
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A A (x)
Ɣ!
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A A (x)
25 ô; , m Ɠǒ; ,Ǔ + ,1ǒm ǒ; ,Ǔ m ǒ; ,ǓǓ
6(( ,* %
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A A (x)
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x
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$1' 7+$7 ,6 $ )8==< 5(/$7,21 21 81,9(56( " # 2 ),1' 7+(
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)8==< 6(7 21 81,9(56( " 7+(1 7+( )8==< 5(/$7,21 Ě 0$36 81,9(56( ! 72 81,9(56( " ,( ,6 $ 5(/$7,21 21 81,@
9(56( ! " +( 6<0%2/ Ě '(127(6 $ &20326,7,21 23(5$@
7,21 :+,&+ &20387(6 7+( 675(1*7+ 2) 7+( 5(/$7,21 %(7:((1
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81( (:/(77@$&.$5' 2851$/
ƪ ǒ
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• 80@52'8&7 m ǒ7 ,-Ǔ +
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Y
B
AƒB
AƒB
AƒB
AĚBƒAĚY
A
X
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30. &26"5*0/ Fuzzy Relation Membership Value
1.0
0.9
1 .$0..0/"-*5: .70-6.& .4-05
Commonality
0.8
8)&3&
0.7
#: &26"5*0/ $0..1 1 "/%
$0..1 1 0.6
0.5
0.4
Volume
0.3
0.2
0.1
Slots
0
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
Slots Membership Value
6;;: 3&-"5*0/ .&.#&34)*1 7"-6&4 '03 063 &91&3*.&/5"'6/$5*0/"- "1130"$) 50 '".*-: "44*(/.&/5
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1 $0..0/"-*5: 70-6.& 4-05
• %&."/%1 "%%*5*0/"- 4-0541 "/%
1 $0..0/"-*5: 70-6.& 4-05
)*4 "1130"$) 563/&% 065 50 #& 500 $0.1-&9 "/% $6.#&3<
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01&3"503 5)& '6;;: $0.104*5*0/ 4* 3* Ě 1* '03 5)*4 &9".1-&
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5 *4 *.1035"/5 50 /05*$& 5)"5 */ (&/&3"- /0 3&453*$5*0/4 "3&
*.104&% 0/ '6;;: .&.#&34)*1 7"-6&4 #65 4*/$& 8& 64&% 5)&
6.<30%6$5 '6;;: $0.104*5*0/ 01&3"503 5)&/ 5)& 46.."5*0/
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5)& "44*(/.&/5 "-(03*5). 50 4&-&$5 " 130%6$5 50 "44*(/ 50 "
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4&-&$5*7*5: .&"463& "/% *4 5)&3&'03& "44*(/&% 50 5)& '".*-:
#&*/( '03.&% %63*/( 5)*4 *5&3"5*0/
'5&3 5)& '6;;: '".*-: "44*(/.&/5 "-(03*5). "44*(/4 5)&
130%6$54 50 5)&*3 $033&410/%*/( '".*-*&4 5)& '6;;: ."$)*/&
#"-"/$&3 53*&4 50 "44*(/ &"$) '".*-:4 $0.10/&/54 50 5)&
1-"$&.&/5 ."$)*/&4 )& 13*."3: 0#+&$5*7& *4 50 )"7& &"$)
4*%& 0' 5)& "44&.#-&% 13*/5&% $*3$6*5 "44&.#-: 64& 5)& 580
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4*%& '30. 5)& */)&3&/5 $0/453"*/54 */530%6$&% #: '".*-*&4
."/6'"$563*/( 3&"-*5: #3*/(4 " '&8 41&$*"- $"4&4 0' "-3&"%:
13&%&5&3.*/&% ."$)*/& "44*(/.&/54 "/% $0/453"*/54
Physical Process Constraints. */$& 5)& 0#+&$5*7& *4 50 )"7& "4
.6$) 4&561 4-05 300. "4 1044*#-& $&35"*/ 1):4*$"- 130$&44
3&-"5&% -*.*5"5*0/4 "3*4& 03 &9".1-& $0/453"*/54 0/ 5)& 7&3:
-"45 4-05 "7"*-"#-& 0/ 5)& #"/, %0 /05 "--08 " 580<4-05<8*%&
'&&%&3 50 #& .06/5&% 0/ 5)& -"45 4-05 ' 8& )"7& 0/& 4-05
45*-- "7"*-"#-& 0/ &"$) ."$)*/& "/% 8& )"7& 50 1-"$& " 580<
4-05<8*%& '&&%&3 8& /&&% 50 .07& " 0/&<4-05<8*%& '&&%&3
'30. 0/& ."$)*/& 50 .",& 300. '03 5)& 580<4-05<8*%&
'&&%&3 */"--: " $0.10/&/5 8)04& 1"$,"(& *4 )*()&3 5)"/
.. .645 #& 1-"$&% #: 5)& 4&$0/% ."$)*/& 4*/$& 5)&
$0.10/&/5 )&*()5 .*()5 */5&3'&3& 8*5) 5)& 1-"$*/( /0;;-& 0/
" %&/4&-: 1016-"5&% 13*/5&% $*3$6*5 "44&.#-:
6/& &8-&55<"$,"3% 063/"-
Primary Family Products. / =1. 9;27=.- ,2;,>2= *<<.6+5B 6.6G
+.;< 27 =1. 9;26*;B /*625B ,8>5- +. +*5*7,.- @2=18>= ,87<2-G
.;*=287 /8; =1. ;.6*27270 9;27=.- ,2;,>2= *<<.6+52.< =1*= ><.
* 98;=287 8/ =1. 9;26*;B /*625B +*5*7,270 ,8>5- 9;8+*+5B +.
*,12.?.- *= =1. .A9.7<. 8/ =1. ;.6*27270 9;8->,=< 26+*5G
*7,. &1>< 2= 2< ,;>,2*5 =1*= +*5*7,270 /8; 9;26*;B /*625B
9;8->,=< =*4. 27=8 ,87<2-.;*=287 =1. ;.6*27270 9;8->,=<
Nonprimary Family Products. 7 =1. ,*<. 8/ 7879;26*;B /*625B
9;8->,=< =1. 9;8+5.6 2< 3><= =1. 8998<2=. 8/ =1. 9;8+5.6
.7,8>7=.;.- /8; 9;26*;B /*625B 9;8->,=< 8; 7879;26*;B
/*625B 9;8->,=< +*5*7,270 1*< =8 27,8;98;*=. =1. ,86987.7=
*<<2076.7=< *5;.*-B ,8662==.- +B =1. 9;26*;B /*625B
+*5*7,270 9;8,.->;.
Placement Time Estimation. &1. =;>. 95*,.6.7= =26. /8; *
9;27=.- ,2;,>2= *<<.6+5B 2< * />7,=287 8/ =1. 95*,.6.7= <.G
:>.7,. @12,1 27,5>-.< =1. 95*,.6.7= =*+5. 68?.6.7= =1.
95*,270 1.*- ;8=*=287 <9..- *7- =1. /..-.; +*74 68?.6.7=
&1. 875B 27/8;6*=287 @. 1*?. *?*25*+5. 2< =1. 95*,270 1.*;8=*=287 <9..- *7- .?.7 =1*= 2< *7 *99;8A26*=287 &1. 6*A2G
6>6 *558@*+5. <9..- /8; =1. 95*,270 1.*- ;8=*=287 2< -.=.;G
627.- /;86 * ,86987.7=< 985*;2=B 9;.<.7=*=287 9*,4*0.
=B9. <2C. *7- 92,4>9 78CC5. <2C. >;=1.;68;. =1. 95*,270
1.*- 1*< =@8G78CC5. <=*=287< =1*= *;. *55 27/5>.7,.- +B
=1. 1.*-< <9..- <.5.,=287 (. *99;8A26*=.- =1. 95*,.6.7=
<9..- +B 8+=*27270 =1. <9..- 8/ =1. 1.*- ;8=*=287
Products with Inherent Imbalance. 7 ,.;=*27 ,*<.< 875B =1.
->952,*=287 8/ * ,86987.7=< *?*25*+252=B *6870 =1. "G
95*,.6.7= 6*,127.< @8>5- 5.*- =8 088- +*5*7,. 8; .A*6G
95. 2= 2< 98<<2+5. =1*= * 9;27=.- ,2;,>2= *<<.6+5B< <2-. ;.G
:>2;.< * 95*,.6.7=G27=.7<2?. ,86987.7= =1*= 0;.*=5B .A,..-<
=1. =8=*5 95*,.6.7= 8/ =1. ;.6*27270 ,86987.7=< !75B *7
*?*25*+252=B 8/ =1*= ,86987.7= 27 +8=1 8/ =1. "G <.=>9<
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Fuzzy Machine Balancing
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