BranchAndBound

8/17/2019 BranchAndBound

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ME 6973 Analytical Techniques in Engineering II

Spring 2015

Introduction to Branch &

Bound


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!"#$%&'#$('!)*$(


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,-#./01

2)$34(1" 5&1 678

.#-4.491 ! : ; ? ! ?

3*@A1%5 5) ! = > ? ! ? ≤ B

C ! = > ! ? ≤ ?:

? ! = > ? ! ? ≥ < ! =D ! ? # >

Integer and Combinatorial Optimization, Nemhauser and Wolsey, Chapter: II.4


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E&1 )/54.#0 3)0*54)$ )F G7H 438

! : >IJK==L ! J $ I=MK==L ! B ; JJ?K==

! =

I=K==L ! J $ I?K==L ! B ; JMK== ! ? $ ICK==L ! J $ I=K==L ! B ; B:K==

! C $ INK==L ! J $ IMK==L ! B ;


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U) T1 %)$34(1" 5&1 5T) 673D 67= #$( 67?8

! : >IJK==L ! J $ I=MK==L ! B ; JJ?K==

! = I=K==L ! J $ I?K==L ! B ; JMK==

! ? $ ICK==L ! J $ I=K==L ! B ; B:K==

! C $ INK==L ! J $ IMK==L ! B ;


6/32

U)0S4$P 5&1 G7H )F 67=D T1 )@5#4$8

! : > I


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E&1 F4"35 (4%54)$#"Q F)" 5&1 G7H )F 67? 438

! : >IJK==L ! J $ I=MK==L ! B ; JJ?K==

! =

I=K==L !

J$

I?K==L !

B ; JMK== ! ? $ ICK==L ! J $ I=K==L ! B ; B:K==

! C $ INK==L ! J $ IMK==L ! B ;


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Y"). 67=D T1 %)$34(1" 67J #$( 67B8

! : > I


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U)0S4$P 5&1 G7H )F 67JD T1 )@5#4$8

! : > ? ! M $


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U)0S4$P 5&1 G7H )F 67BD T1 )@5#4$8

! : > ? ! B $ J !


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,-#./01

Z"#/&4%#00Q8

E&1 )/54.#0 3)0*54)$ 43 ! = ; BD ! ? ; :D T45& ! : ;

?N

x2 ≤ 3

x1 ≤ 3

x2 ≥ 4

x1 ≥ 4

IP

IP1

IP3 IP4

IP2

Fathomed by

infeasibility


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[\7G ,-#./01


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67var x1 integer >= 0, = 0, = 0, = 0, = 0, = 0, = 1;

c2: x1 + x2 + 3* x6 >= 1;

c3: x1 + x2 + x6 >= 1;

c4: x3 - 3* x4 >= 1;

c5: x3 - 2* x4 -5* x5 >= 1;

c6: x4 + 3* x5 -4 *x6 >= 1;c7: x2 + x5 + x6 >= 1;

solve;

display x1, x2,x3, x4, x5, x6;

end;


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G7H• "#$ &' () *+ ,) '*-

• "#$ &! () *+ ,) '*-

• "#$ &. () *+ ,) '*-

• "#$ &/ () *+ ,) '*-

• "#$ &0 () *+ ,) '*-

• "#$ &1 () *+ ,) '*-

• 2#&32345 6789 :&' : !; &! :*• %=8 -= > ? ] -? ̂ ; =V _ [051"$#54S10Q8 -= > -? ^; =

• %?8 -= > -? > J] -M ^; =V

• %J8 -= > -? > -M ^; =V

• %B8 -J ' J] -B ^; =V

• %C8 -J ' ?] -B 'C] -C ^; =V

• %M8 -B > J] -C 'B ]-M ^; =V

• % -C > -M ^; =V

• >6?"5-

• @3>A?#B &'+ &!+&.+ &/+ &0+ &1-

• A$3CDE FGAD32#? H#?I59 JEKCF+ :L&' = !; &! =*• 5C@-


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• 6$51P1" 3)0*54)$-=XS#0 ; :

-?XS#0 ; =-JXS#0 ; B

-BXS#0 ; =

-CXS#0 ; :

-MXS#0 ; :R/54.#0 `#0*18 'BX?:::::

• G7 "10#-#54)$-=XS#0 ; :XJJJJJJJJJJJJJJJ

-?XS#0 ; :XMMMMMMMMMMMMMM<

-JXS#0 ; ?XMMMMMMMMMMMMM<

-BXS#0 ; :

-CXS#0 ; :XJJJJJJJJJJJJJJJ

-MXS#0 ; :

R/54.#0 `#0*18 'JXJJJJJJ


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• #./08 .)(10 /:X#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• ? 451"#54)$3D )@A1%54S1 'JXJJJJJJJJJ

• -= ; :XJJJJJJ

•

-? ; :XMMMMM<• -J ; ?XMMMM<

• -B ; :

• -C ; :XJJJJJJ

• -M ; :

• R/54.#0 `#0*18 'JXJJJJJJ

• #./08


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• p1.ampl

• Infeasible!

• #./08 .)(10 /?X#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• J 451"#54)$3D )@A1%54S1 'JXJ


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• #./08 .)(10 /JX#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• ? 451"#54)$3D )@A1%54S1 'JXNC


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• #./08 .)(10 /BX#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• = 451"#54)$3D )@A1%54S1 'JX


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• #./08 .)(10 /CX#./0

• /"13)0S18 %)$35"#4$5 %C %#$$)5 &)0(8

•

@)(Q ^; = %#$$)5 @1 a; ':XJCV (4FF1"1$%1 ;=XJC

• -= ; :

•

-? ; :• -J ; :

• -B ; :

• -C ; :

• -M ; :

• R/54.#0 `#0*18 :X::::::

• 6$F1#34@01b


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• #./08 .)(10 /MX#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• ? 451"#54)$3D )@A1%54S1 'BX:MMMMMMM<

• -= ; =

• -? ; :

• -J ; C

• -B ; :XJJJJJJ

• -C ; :XMMMMM<

• -M ; :XJJJJJJ

• R/54.#0 `#0*18 'BX:MMMM<


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• #./08 .)(10 /


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• #./08 .)(10 /NX#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

•

= 451"#54)$3D )@A1%54S1 'BX


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• #./08 .)(10 /WX#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

• = 451"#54)$3D )@A1%54S1 'BX?

• -= ; :

• -? ; =

• -J ; B

• -B ; =

• -C ; :

• -M ; :

•

R/54.#0 `#0*18 'BX?:::::

0 ( 0 =: 0


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• #./08 .)(10 /=:X#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

•

: 451"#54)$3D )@A1%54S1 'JXNJJJJJJJJ• -= ; :

• -? ; =

• -J ; J• -B ; :

• -C ; :XJJJJJJ

• -M ; :

• R/54.#0 `#0*18 'JXNJJJJJ

0 ( 0 =? 0


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• #./08 .)(10 /=?X#./0

• \6ORU CXC=8 )/54.#0 3)0*54)$ F)*$(X

•

= 451"#54)$3D )@A1%54S1 'CX


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`41T F"). 5&1 (4%54)$#"Q

"#$%& '#()#*$%+,

-&%#./ Dictionary D’

#0 %&1%,0 #$23%0#4&35

Branching

Note: this section are taken fromDr. Sankaranarayanan’s slides.


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,-#./01 I6G7L

var x1 >= 0, = 0, = 0,


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Y4$#0 c4%54)$#"Q


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c4%54)$#"Q [F51" !"#$%&

;#()#*$%+, 43(*.30


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c4#P"#.

D%+3$) 6*'3

E"#$%& ;#()#*$%+,F

-?#&' 6*'3/

GH'' $31 +*15

GD+#.%& I$23%0#4&3 4B) ;B%&

"3%0#4&3

-*$0#'3+ 'B%&

(*.


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!d! Z1$1"#0 Y)". c4%54)$#"Q

• Z4S1 3/1%4#0 5"1#5.1$5 5) @)*$(3 )$ S#"4#@013X

• \)(4FQ 5&1 U4./01- #0P)"45&. 4$ 5T) T#Q38

–

c4%54)$#"413 $)T S1 3/1%4#0 T#Q3 5) 5"#%e@)*$(3X

– 74S)54$P 43 .)(4F41(X

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