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Chapter 6

Integer Programming: Branch-and-Bound Algorithm

FIRST A

PP

ROXLMATION

i\n Integer program

is

a linear program with the odd requirement that

ull v a n a ~ e s

be integers

~

Chnptor I. Thcrdor

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CHAP. 6) tNHGER I'ROGRAMMlt-:G:

BRAN

CHANO.BOUND ALGORITHM

t25

approximation

is slllll

nonintegrnl, then the iillegtl" proyafil whicb

g a v ~ rise

to

Lllat first a)lproxlmetion

lx't'Om

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126 INTEGER PROG

RAMMING

: BRANCH-AND-BOUND AI GOIUTHM

(Cit"P. 6

COMPUTATIONAL CONSIDERATIONS

Ont alway' branches from thot progrm

whi

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128 INTFOF. R PROORAM

MI

'

: ,, : ~ 9

~

.Sl

\ >

tth

' \:

MnM U\

C

nd ont-2

II""

n\ldn

C>lhtt

than

111OIIIuon

to Prl)(lr= J

the ~ ~ unc ' ; I. ; ll. , ; 161 1.

( 4999. lt.Jth : - usooo.

1.5

1 ~

t.hr

cri"'f'\ tnohcd n rouodtn& 1he i int ~ p p r o ~ t m a t t O n '

to

the O n ~ J n a l pRIJI.lm tn

Problcmt6.= and 6 4

to

tnlcgcn

and

lhcn l.t\.1111

t ~

am ' '

the uptunal

a.>nei

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C

HAP

. 6)

IKTEGER P

RO

G

RAMMI

NG: JlRAt-ICHANDBOUND ALGORITHM

129

6.6

T h ~ fi rs uppro,imatlorn in Probkm

6.2 was

= 2.25

.xi

- LS

.

We wi

(b 10 round

10

the

do,sest

i o ~ e ~ r pOint in tlrr.fNsiblt

rl1)i01t

.

Now.

or rhe: rour integer points surrounding tbe first approximation. only

one, Cl. I). 15 round O I ~

the

feasib

le

region. Thus

we

lakt r - 2. " ) - I, with a o r c ~ p o n d

m g

: = 10,

the proposed C\palmzd sdu1roo The

tnac

optimal

$01 u1

km w

as

found ali ;

=

~

th

UJ the rounded

$0

lu

1ion

dC"vii IU."J

rvm

lhr true S\)h

llit>n

by more lhD T16 perct'Jll.

Thlppcd (rom f11ry I to rctOJ1CI 1

500

boCI

from

f>

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14/16

CHAP

7)

INTF.GF.

R PROGRAMMIN():

CU T

A

LGOR

.I

THMS

137

after lhrtenerations.The fiat approximation to r o ~ r a m (I)

is

lhus x: I.H. x; a 7,x; a 5with :. -:;a: 279.

x

.

x.

~ .

I 0 0 - 0.3

0.005

0 - 1.6 0

us

x,

0 0

I

0.2

-

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15/16

13k IN1 CtoCR PR

CKJ

RAMMtNO: CUT AI.

GO

RI11fM S

[C

H.AI'.

v:uiil.ble> ; l _ f ~ rcquirccJ lo be nounc:pthc and lntC gfal il rouo,...& wt '.li lr;u;i. " 'lc- no ubot.\tC tun.ahiC' U1UJol be

1n"de

trcnuer

thtth oc eqma l to L Thb ht tur.:o lmpl.es

that

the sum or aU lhc acmb.sic

V

.n lbln

u1w:t

be

LWde gr-eater th

:in

or equal to t. I ( hii condnioo rs usod as the nev. c : t . I ~ L m 10 beadjoin'*s

to

the r i . g ~ n . a

l o t c ~ r

progrum.

< ~ ~ ~

ha'e

the t ~ l g . o r i t b m llrst ~ l c d . y l'>ao1.jg.

7.7 Use

the

cut

aJgoriLhm developed In

Pr

oblen\ 7.6

10

solve

n u x t m t l t

-: - ,

1

+ 4.v.

11bject1

:

h , +

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CHAI'

7)

INTEGER PROGRAM MING: CUT

AL