Chapte  INTEGE PROGRAMMIN G Rober Bosc h Oberlin College Oberlin OH USA Michae Tric k Carnegie Mellon University Pittsburgh PA USA

Chapte INTEGE PROGRAMMIN G Rober Bosc h Oberlin College Oberlin OH USA Michae Tric k Carnegie Mellon University Pittsburgh PA USA - Description

INTRODUCTIO N Ove th las 2 years th combinatio o faste computers mor reliabl e data an improve algorithm ha resulte i th nearroutin solutio o man y intege program o practica interest Intege programmin model ar use d a wid variet o apphcations includ ID: 25014 Download Pdf

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Chapte INTEGE PROGRAMMIN G Rober Bosc h Oberlin College Oberlin OH USA Michae Tric k Carnegie Mellon University Pittsburgh PA USA

INTRODUCTIO N Ove th las 2 years th combinatio o faste computers mor reliabl e data an improve algorithm ha resulte i th nearroutin solutio o man y intege program o practica interest Intege programmin model ar use d a wid variet o apphcations includ

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Chapte INTEGE PROGRAMMIN G Rober Bosc h Oberlin College Oberlin OH USA Michae Tric k Carnegie Mellon University Pittsburgh PA USA




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Chapte 3 INTEGE PROGRAMMIN G Rober Bosc h Oberlin College Oberlin OH, USA Michae Tric k Carnegie Mellon University Pittsburgh PA, USA 3. INTRODUCTIO N Ove th las 2 years th combinatio o faste computers mor reliabl e data an improve algorithm ha resulte i th near-routin solutio o man y intege program o practica interest Intege programmin model ar use d a wid variet o apphcations includin scheduling resourc assignment , planning suppl chai design auctio design an many man others I thi s tutorial w outlin som o th majo theme involve i creatin an solvin g intege programmin models

. Th foundatio o muc o analytica decisio makin i linea program ming I a linea program ther ar variables, constraints, an a objective function. Th variables o decisions tak o numerica values Constraint ar e use t limi th value t a feasibl region Thes constraint mus b linea r th decisio variables Th objectiv functio the define whic particu la assignmen o feasibl value t th variable i optimal i i th on tha t maximize (o minimizes dependin o th typ o th objective th objec tiv function Th objectiv functio mus als b linea i th variables Se e Chapte 2 fo mor detail abou Linea Programming . Linea

program ca mode man problem o practica interest an mode m linea programmin optimizatio code ca find optima solution t problem s wit hundred o thousand o constraint an variables I i thi combina tio o modelin strengt an solvabiHt tha make Hnea programmin s o important .
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BOSCH AND TRICK Intege programmin add additiona constraint t linea programming A n intege progra begin wit a linea program an add th requiremen tha t som o al o th variable tak o intege values Thi seemingl innocu ou chang greatl increase th numbe o problem tha ca b modeled , bu als make th model mor difficul t

solve I fact on frustratin as pec o intege programmin i tha tw seemingl simila formulation fo th e sam proble ca lea t radicall differen computationa experience on e formulatio ma quickl lea t optima solutions whil th othe ma tak e excessivel lon tim t solve . Ther ar man key t successfull developin an solvin intege pro grammin models W conside th followin aspects : b creativ i formulations , find intege programmin formulation wit a stron relaxation , avoi symmetry , conside formulation wit man constraints , conside formulation wit man variables , modif branch-and-boun searc parameters . fix

ideas w wil introduc a particula intege programmin model , an sho ho th mai intege programmin algorithm branch-and-bound , operate o tha model W wil the us thi mode t illustrat th ke idea s successfu intege programming . 3.1. Facilit Locatio n conside a faciht locatio problem A chemica compan own fou r factorie tha manufactur a certai chemica i ra form Th compan woul d lik t ge i th busines o refinin th chemical I i intereste i buildin g refinin facilities an i ha identifie thre possibl sites Tabl 3. contain s variabl costs fixed costs an weekl capacitie fo th thre possibl refinin g facilit

sites an weekl productio amount fo eac factory Th variabl e cost ar i dollar pe wee an includ transportatio costs Th fixed cost s ar i dollar pe year Th productio amount an capacitie ar i ton pe r week . Th decisio make wh face thi proble mus answe tw ver differen t type o questions question tha requir numerica answer (fo example , ho man ton o chemica shoul factor / sen t th site- refinin facilit y eac week? an question tha requir yes-n answer (fo example shoul d th site- facilit b constructed?) Whil w ca easil mode th first typ e questio b usin continuou decisio variable (b lettin Xij equa

th e
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INTEGER PROGRAMMING 7 1 Table 3.1. Facilit locatio problem . Variabl cos t Fixe cos t Capacit y factor 1 factor 2 factor 3 factor 4 1 5 5 0 5 50 00 0 150 0 Sit e 2 0 5 5 5 50000 0 150 0 3 5 0 5 5 50000 0 150 0 Productio n 100 0 100 0 50 0 50 0 numbe o ton o chemica sen fro factor / t sit j eac week) w cannot thi wit th second W nee t us intege variables I w le yj equa 1 th site- refinin faciht i constructe an 0 i i i not w quickl arriv e a I formulatio o th problem : minimiz 5 25xi + 5 20xi + 5 15xi 3 5 . 15x2 + 5 25x2 + 5 20x2 3 5 20x3 + 5 15x3 + 5 25x3 3 5 25x4 + 5 15x4

+ 5 15x4 3 500000^ + 50 000^ +50 000>' 3 subjec t ^1 + x\2 + Xi = 100 0 X2\ +X2 +JC2 = 100 0 •^3 +-^32 -^3 = 50 0 X4 + X4 + X4 = 50 0 •^1 +-^2 +-^3 +M\ < 1500)^ 1 x\2 + X2 + X3 + X4 < 1500_y 2 •^1 + -^2 + -^3 + -^4 < 1500^ 3 Xij > 0 fo al / an j >^y€{0 1 fo al 7 Th objectiv i t minimiz th yearl cost th su o th variabl cost s (whic ar measure i dollar pe week an th fixed cost (whic ar mea sure i dollar pe year) Th first se o constraint ensure tha eac factory' s weekl chemica productio i sen somewher fo refining Sinc factor 1 produce 100 ton o chemica pe week factor 1 mus shi a tota o 100 0 ton

o chemica t th variou refinin facilitie eac week Th secon se t constraint guarantee tw things (1 i a faciht i open i wil operat e o belo it capacity an (2 i a facilit i no open i wil no operat e all I th site- facilit i ope (yi = 1 the th factorie ca sen i u p 1500^ = 150 1 = 150 ton o chemica pe week I i i no ope n
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BOSCH AND TRICK (_y = 0) the th factorie ca sen i u t 1500}; = 150 0 = 0 ton pe r week . Thi introductor exampl demonstrate th nee fo intege variables I t als show tha wit intege variables on ca mode simpl logica require ment (i a facilit i open i ca refin u t a

certai amoun o chemical ; not i canno d an refinin a all) I turn ou tha wit intege variables , on ca mode a whol hos o logica requirements On ca als mode fixed costs sequencin an schedulin requirements an man othe proble as pects . 3.1. Solvin th Facilit Locatio I P Give a intege progra (IP) ther i a associate Hnea progra (LR ) calle th linear relaxation. I i forme b droppin (relaxing th integralit y restrictions Sinc (LR i les constraine tha (IP) th followin ar imme diate : (IP i a minimizatio problem th optima objectiv valu o (LR i s les tha o equa t th optima objectiv valu o (IP) . (IP i a

maximizatio problem th optima objectiv valu o (LR i s greate tha o equa t th optima objectiv valu o (IP) , (LR i infeasible the s i (IP) . al th variable i a optima solutio o (LR ar integer-valued the n tha solutio i optima fo (IP too . I th objectiv functio coefficient ar integer-valued the fo mini mizatio problems th optima objectiv valu o (IP i greate tha n equa t th ceilin o th optima objectiv valu o (LR). Fo max imizatio problems th optima objectiv valu o (IP i les tha o r equa t th floor o th optima objectiv valu o (LR). summary solvin (LR ca b quit useful i provide a boun o th e optima

valu o (IP) an ma (i w ar lucky giv a optima solutio t o (IP) . Fo th remainde o thi section w wil le (IP stan fo th Facilit Lo catio intege progra an (LR fo it linea programmin relaxation Whe n
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INTEGER PROGRAMMING 3 solv (LR) w obtai n Objectiv e •^1 X.\2 Xi 3 X2\ X2 -^2 3 •^3 -^3 -^3 3 X4 X^2 -^4 3 yi 3^ 3 334000 0 100 0 100 50 50 22 2 33 3 Thi solutio ha factor 1 sen al 100 ton o it chemica t sit 3 factor y sen al 100 ton o it chemica t sit 1 factor 3 sen al 50 ton t sit e an factor 4 sen al 50 ton t sit 2 I construct two-third o a refinin g facilit a eac site Althoug i

cost onl 334 00 dollar pe year i canno t implemented al thre o it intege variable tak o fractiona values . i temptin t tr t produc a feasibl solutio b rounding Here i w e roun y\, yi, an ^ fro 2/ t 1 w ge luck (thi i certainl no alway th e case! an ge a intege feasibl solution Althoug w ca stat tha thi i a goo solution—it objectiv valu o 384000 i withi 15 o th objectiv e valu o (LR an henc withi 15 o optimal—w canno b sur tha i i s optimal . ho ca w find a optima solutio t (IP) Examinin th optima l solutio t (LR) w se tha >'] yi, an _y ar fractional W wan t forc y\, yi, an y2, t b intege

valued W star b branching o _yi creatin tw ne w intege programmin problems I one w ad th constrain y\ = 0. I th e other w wil ad th constrain ^' = 1 Not tha an optima solutio t (IP ) mus b feasibl fo on o th tw subproblems . Afte w solv th hnea programmin relaxation o th tw subproblems , ca displa wha w kno i a tree a show i Figur 3.1 . Not tha th optima solutio t th lef subproblem' L relaxatio i in tege valued I i therefor a optima solutio t th lef subproblem Sinc e ther i n poin i doin anythin mor wit th lef subproblem w mar i t wit a "x an focu ou attentio o th righ subproblem . Bot y2 an

y^ ar fractiona i th optima solutio t th righ subprob lem' L relaxation W wan t forc bot variable t b intege valued Al thoug w coul branc o eithe variable w wil branc o ^2 Tha is w e wil creat tw mor subproblems on wit y2 = 0 an th othe wit j = 1 Afte w solv th L relaxations w ca updat ou tree a i Figur 3.2 . Not tha w ca immediatel " out th lef subproblem th optima l solutio t it L relaxatio i intege valued I addition b employin a bounding argument w ca als x ou th righ subproblem Th argumen t goe lik this Sinc th objectiv valu o it L relaxatio (3636666| i s greate tha th objectiv valu o ou

newl foun intege feasibl solutio n
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4 BOSCH AND TRICK j]= 0 334000 0 -100 0 100 50 50 2 2 3 3 >'1= 1 373000 0 100 0 50 50 0 50 50 1 347000 0 100 50 50 0 \ 100 0 2 1 Figure 3.1. Intermediat branc an boun tree . (3470000) th optima valu o th right subproble mus b highe tha n (wors than th objectiv valu o ou newl foun intege feasibl solution . ther i n poin i expendin an mor effor o th right subproblem . Sinc ther ar n activ subproblem (subproblem tha requir branching) , ar done W hav foun a optima solutio t (IP) Th optima solutio n ha factorie 2 an 3 us th site- refinin facilit

an factorie 1 an 4 us th e site- facility Th site- an site- facihtie ar constructed Th site- facilit y not Th optima solutio cost 347000 dollar pe year 37000 dollar s pe yea les tha th solutio obtaine b roundin th solutio t (LR) . Thi metho i calle branch and bound, an i th mos commo metho d fo finding solution t intege programmin formulations . 3.1. Difficultie wit Intege Program s Whil w wer abl t ge th optima solutio t th exampl intege pro gra relativel quickly i i no alway th cas tha branc an boun quickl y solve intege programs I particular i i possibl tha th bounding aspect s branc an

boun ar no invoked an th branc an boun algorith ca n the generat a hug numbe o subproblems I th wors case a proble m wit n binar variable (variable tha hav t tak o th valu 0 o 1 ca n hav 2 subproblems Thi exponentia growt i inheren i an algorith fo r intege programming unles P = N (se Chapte 1 fo mor details) du t o th rang o problem tha ca b formulate withi intege programming .
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INTEGER PROGRAMMING 5 Ji= 0 334000 0 -100 0 100 50 50 2 2 3 3 y,=l 373000 0 100 0 50 50 0 50 50 1 ^2= 0 347000 0 100 0 100 50 50 ^ 2 ' 3 3 }^2= 1 347000 0 100 0 50 0 1 100 0 50 0 1 363 666 f 100 0 50

50 0 I 50 50 1 Figure 3,2. Fina branc an boun tree . Despit th possibilit o extrem computatio time ther ar a numbe r technique tha hav bee develope t increas th likelihoo o finding optima solution quickly Afte w discus creativit i formulations w wil l discus som o thes techniques . 3. B CREATIV I FORMULATION S first, i ma see tha intege programmin doe no offe muc ove lin ea programming bot requir linea objective an constraints an bot hav e numerica variables Ca requirin som o th variable t tak o intege r value significantl expan th capabiHt o th models Absolutely intege r programmin model g fa

beyon th powe o Hnea programmin models . Th ke i th creativ us o integralit t mode a wid rang o commo n
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BOSCH AND TRICK structure i models Her w outlin som o th majo use o intege vari ables . 3.2. Intege Quantitie s Th mos obviou us o intege variable i whe a intege quantit i s required Fo instance i a productio mode involvin televisio sets a inte gra numbe o televisio set migh b required Or i a personne assignmen t problem a intege numbe o worker migh b assigne t a shift . Thi us o intege variable i th mos obvious an th mos over-used . Fo man applications th adde ''accuracy i

requirin intege variable i s fa outweighe b th greate difficult i finding th optima solution Fo r instance i th productio example i th numbe o television produce i s th hundred (sa th fractiona optima solutio i 202.7 the havin a pla n wit th rounde of valu (20 i thi example i likel t b appropriat i n practice Th uncertaint o th dat almos certainl mean tha n productio n pla i accurat t fou figures! Similarly i th personne assignmen proble m fo a larg enterpris ove a year an th linea programmin mode suggest s 154. peopl ar required i i probabl no worthwhil t invok a intege r programmin mode i

orde t handl th fractiona parts . However ther ar time whe intege quantitie ar required A productio n syste tha ca produc eithe tw o thre aircraf carrier an a personne l assignmen proble fo smal team o five o si peopl ar examples I n thes cases th additio o th integralit constrain ca mea th differenc e betwee usefu model an irrelevan models . 3.2. Binar Decision s Perhap th mos use typ o intege variabl i th binary variable: a n intege variabl restricte t tak o th value 0 o 1 W wil se a numbe o f use o thes variables Ou first exampl i i modelin binar decisions . Man practica decision ca b see a

''yes o ''no decisions Shoul w e construc a chemica refinin facifit i sit j (a i th introduction) Shoul d inves i projec B Shoul w star producin ne produc Y Fo man y thes decisions a binar intege programmin mode i appropriate I n suc a model eac decisio i modele wit a binar variable settin th e variabl equa t 1 correspond t makin th "yes decision whil settin i t 0 correspond t goin wit th "no decision Constraint ar the forme d correspon t th effect o th decision . a example suppos w nee t choos amon project A B C an d Eac projec ha a capita requiremen ($ milHon $2. million $ milhon , an $

millio respectively an a expecte retur (say $ million $ million ,
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INTEGER PROGRAMMING 11 $1 million an $1 million) I w hav $ millio t invest whic project s shoul w tak o i orde t maximiz ou expecte return ? ca formulat thi proble wit binar variable XA A:B XQ, an X D representin th decisio t tak o th correspondin project Th effec o f takin o a projec i t us u som o th fund w hav availabl t invest . Therefore w hav a constraint : JC + 2.5.^ + 4x + 5X < 7 Ou objectiv i t maximiz th expecte profit : Maximiz 3x + 6x + 13^ + 15x 4 thi case binar variable le u mak th yes-n decisio o

whethe t o inves i eac fund wit a constrain ensurin tha ou overal decision ar e consisten wit ou budget Withou intege variables th solutio t ou mode l woul hav fractiona part o projects whic ma no b i keepin wit th e need o th model . 3.2. Fixe Charg Requirement s man productio applications th cos o producin x o a ite i s roughl linea excep fo th specia cas o producin n items I tha case , ther ar additiona saving sinc n equipmen o othe item nee b pro cure fo th production Thi lead t o. fixed charge structure Th cos fo r producin x o a ite i s 0 if = 0 C + C2X, i X > 0 fo constant c\ C2 Thi typ

o cos structur i impossibl t embe i a linea program Wit h intege programming however w ca introduc a ne binar variabl y. Th e valu _ = 1 i interprete a havin non-zer production whil > = 0 mean s production Th objectiv functio fo thes variable the become s c\y + C2X whic i appropriatel linea i th variables I i necessary however t ad d constraint tha lin th x an y variables Otherwise th solutio migh b e = Q an x = 10 whic w d no want I ther i a uppe boun M o n ho larg x ca b (perhap derive fro othe constraints) the th constrain t < My correctl link th tw variables I _ = 0 the x mus equa 0 i > =

1 the x ca tak o an value Technically i i possibl t hav th value x = 0 an d
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BOSCH AND TRICK = \ wit thi formulation bu a lon a thi i modelin a fixed cos (rathe r tha a fixed profit) thi wil no b a optima (cos minimizing solution . Thi us o "M value i commo i intege programming an th resul i s calle a "Big- model" Big- model ar ofte difficul t solve fo reason s wil see . sa thi fixed-charge modelin approac i ou initia facilit locatio n example There th y variable corresponde t openin a refinin facilit y (incurrin a fixed cost) Th x variable correspon t assignin a factor y th

refinin facility an ther wa a uppe boun o th volum o ra w materia a refiner coul handle . 3.2. Logica Constraint s Binar variable ca als b use t mode complicate logica constraints , capabilit no availabl i linea programming I a facilit locatio proble m wit binar variable y\, yi, y^, y^, an ys correspondin t th decision t o ope warehouse a location 1 2 3 4 an 5 respectively complicate rela tionship betwee th warehouse ca b modele wit linea function o th e variables Her ar a fe examples : A mos on o location 1 an 2 ca b opened y\+ yi <\. Locatio 3 ca onl b opene i locatio 1 i _y < >'i . Locatio

4 canno b opene i location 2 o 3 ar suc tha y^ + yi S^ or>' + >' < 1 I locatio 1 i open eithe location 2 o 5 mus b y2 + y5>y\- Muc mor complicate logica constraint ca b formulate wit th addi tio o ne binar variables Conside a constrain o th form ?>x\ +Ax2 < 1 0 Ax\ - 2x > 12 A written thi i no a linea constraint However i w e le M b th larges eithe \'ix\ +4jC2 o |4x - 2x2 ca be the w ca defin e ne binar variabl z whic i 1 onl i th first constrain i satisfie an 0 onl i th secon constrain i satisfied The w ge th constraint s 3x - 4x < 1 - ( - 10)( - z) 4x -f 2x > 1 - ( - \2)z Whe z = 1 w obtai n

3x 4-4x < 1 0 4x + 2x > -M Whe z = 0 w obtai n 3x -f 4x < M 4x -f-2x > 1 2
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INTEGER PROGRAMMING 7 9 Thi correctl model th origina nonlinea constraint . w ca see logica requirement ofte lea t Big-M-typ formulations . 3.2. Sequencin Problem s Man problem i sequencin an schedulin requir th modelin o th e orde i whic item appea i th sequence Fo instance suppos w hav a mode i whic ther ar items wher eac ite / ha a processin tim o a machin pt I th machin ca onl handl on ite a a tim an w le ti a (continuous variabl representin th star tim o ite / o th machine , the w ca ensur tha

item / an j ar no o th machin a th sam tim e wit th constraint s tj > ti + Pi I tj > ti ti > tj + pj I tj < ti Thi ca b handle wit a ne binar variabl j/ whic i 1 i ti < tj an 0 otherwise Thi give th constraint s tj >ti+pi-Mil -y) ti > tj + Pj - My fo sufficientl larg M I j i 1 the th secon constrain i automaticall y satisfie (s onl th first i relevant whil th revers happen fo y = 0. 3. FIN FORMULATION WIT STRON G RELAXATION S th previou sectio mad clear intege programmin formulation ca n use fo man problem o practica interest I fact fo man problems , ther ar man alternativ intege programmin

formulations Findin a ''good " formulatio i ke t th successfu us o intege programming Th defini tio o a goo formulatio i primaril computational a goo formulatio i s on fo whic branc an boun (o anothe intege programmin algorithm ) wil find an prov th optima solutio quickly Despit thi empirica aspec t th definition ther ar som guideline t hel i th searc fo goo for mulations Th ke t succes i t find formulation whos linea relaxatio n no to differen fro th underlyin intege program . sa i ou first exampl tha solvin hnea relaxation wa ke t th e basi intege programmin algorithm I th solutio t th

initia hnea relax atio i integer the n branchin nee b don an intege programmin i s harde tha linea programming Unfortunately finding formulation wit h thi propert i ver har t do Bu som formulation ca b bette tha othe r formulation i thi regard .
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BOSCH AND TRICK Le u modif ou facilit locatio proble b requirin tha ever factor y assigne t exactl on refiner (incidentally th optima solutio t ou r origina formulatio happene t mee thi requirement) Now instea o hav in Xij b th ton sen fro factor / t refiner j w defin X/ t b 1 i f factor / i service b refiner j Ou formulatio become s

Minimiz 100 5 25xi + 100 5 20xi + 100 5 \5xu 100 5 15x2 + 100 5 25x2 + 100 5 20x2 3 50 5 20x3 + 50 5 15x3 + 50 5 25x3 3 50 5 25x4 + 50 5 15x4 + 50 5 15x4 3 500000^ + 500000); + 500000}; 3 Subjectt xii+xi + xi = l Xl\ + X2 + ^2 = 1 •^3 +-^32+-^3 = 1 X4 + X4 + X4 = 1 1000x1 + 1000x2 +500x3 +500x4 < 1500>' i 1000x1 + 1000x2 + 500x3 + 500x4 < \5my2 1000X1 + 1000X2 + 500X3 + 500X4 < 1500^ 3 Xij €{0,1 fo al / an j >',€{0 1 fo ally . Le u cal thi formulatio th base formulation. Thi i a correc formulatio n ou problem Ther ar alternativ formulations however Suppos w ad d th bas formulatio th se o

constraint s Xij < yj fo al / an j Cal th resultin formulatio th expanded formulation. Not tha i to i s appropriat formulatio fo ou problem A th simples level i appear s tha w hav simpl mad th formulatio larger ther ar mor constraint s o th linea program solve withi branch-and-boun wil likel tak longe t o solve I ther an advantag t th expande formulation ? Th ke i t loo a non-intege solution t linea relaxation o th tw o formulations w kno th tw formulation hav th sam intege solution s (sinc the ar formulation o th sam problem) bu the ca diffe i non intege solutions Conside th solutio xi = 1,X2

= l,-^3 = l,-^4 = >' = 2/3 y2 = 2/3 _y = 2/3 Thi solutio i feasibl t th Hnea relax atio o th bas formulatio bu i no feasibl t th linea relaxatio o th e expande formulation I th branch-and-boun algorith work o th bas e formulation i ma hav t conside thi solution wit th expande formula tion thi solutio ca neve b examined I ther ar fewe fractiona solution s explor (technically fractiona extrem poin solutions) branc an boun d wil typicall terminat mor quickly .
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INTEGER PROGRAMMING 8 1 Sinc w hav adde constraint t ge th expande formulation ther i s non-intege solutio t th hnea

relaxatio o th expande formulatio n tha i no als feasibl fo th linea relaxatio o th bas formulation W e sa tha th expande formulatio i tighter tha th bas formulation . general tighte formulation ar t b preferre fo intege programmin g formulation eve i th resultin formulation ar larger O course ther e ar exceptions i th siz o th formulatio i muc larger th gai fro m th tighte formulatio ma no b sufficien t offse th increase linea pro grammin times Suc case ar definitel th exception however almos t invariably tighte formulation ar bette formulations Fo thi particula in stance th Expande

Formulatio happen t provid a intege solutio with ou branching . Ther ha bee a tremendou amoun o wor don o finding tighte formu lation fo differen intege programmin models Fo man type o problems , classe o constraint (o cuts) t b adde ar known Thes constraint ca n adde i on o tw ways the ca b include i th origina formulatio n the ca b adde a neede t remov fractiona values Th latte cas e lead t a branch and cut approach whic i th subjec o Sectio 3.6 . cu relativ t a formulatio ha t satisf tw properties first, ever fea sibl intege solutio mus als satisf th cut second som fractiona solutio n tha i

feasibl t th linea relaxatio o th formulatio mus no satisf th e cut Fo instance conside th singl constrain t 3;c + 5JC + 8;c + 10x < 1 6 wher th Xi ar binar variables The th constrain x^ -\- X4 < 1 i a cu t (ever intege solutio satisfie i and fo instanc x = (0 0 .5 1 doe not ) bu X] + ^ + -^ + ^ < 4 i no a cu (n fractiona solution removed no i s -^ + ^ + ^ 2 (whic incorrectl remove ; = (1 1 1 0) . Give a formulation finding cut t ad t i t strengthe th formulatio n no a routin task I ca tak dee understanding an a bi o luck t find improvin constraints . On generall usefu approac i calle th

Chvata (o Gomory-Chvatal ) procedure Her i ho th procedur work fo ''< constraint wher al th e variable ar non-negativ integers : Tak on o mor constraints multipl eac b a non-negativ constan t (th constan ca b differen fo differen constraints) Ad th resultin g constraint int a singl constraint . Roun dow eac coefficien o th left-han sid o th constraint . Roun dow th right-han sid o th constraint .
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BOSCH AND TRICK Th resul i a constrain tha doe no cu of an feasibl intege solutions . ma b a cu i th effec o roundin dow th right-han sid o th con strain i mor tha th effec o roundin

dow th coefficients . Thi i bes see throug a example Takin th constrain above le u s tak th tw constraint s 2>xx + 5;c + 8;c + 10;c < 1 x^<\ w multipl eac constrain b 1/ an ad the w obtai n 3/9JC + 5/9x + 9/9JC + \0/9x^ < 17/ 9 Now roun dow th left-han coefficient (thi i vali sinc th x variable s ar non-negativ an i i a "< constraint) : ^3+^ < 17/ 9 Finally roun dow th right-han sid (thi i vah sinc th x variable ar e integer t obtai n + ^ : 1 whic turn ou t b a cut Notic tha th thre step hav differin effect s feasibihty Th first step sinc i i jus takin a linea combinatio o con straints neithe

add no remove feasibl values th secon ste weaken th e constraint an ma ad additiona fractiona values th thir ste strengthen s th constraint ideall removin fractiona values . Thi approac i particularl usefu whe th constant ar chose s tha n o roundin dow i don i th secon step Fo instance conside th followin g se o constraint (wher th xt ar binar variables) : •^ + -^ : 1 ^ + -^ : 1 x\ -^ X3 < I Thes type o constraint ofte appea i formulation wher ther ar list s mutuall exclusiv variables Here w ca multipl eac constrain b 1/ 2 an ad the t obtai n + ^ + ^ < 3/ 2 Ther i n roundin dow o th left-han

side s w ca mov o t round in dow th right-han sid t obtai n +^ +-^ < 1 which fo instance cut of th solutio x = (1/2 1/2 1/2) . case wher n roundin dow i neede o th left-han sid bu ther e roundin dow o th right-han side th resul ha t b a cu (relativ e th include constraints) Conversely i n roundin dow i don o th e right-han side th resul canno b a cut .
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INTEGER PROGRAMMING 8 3 th formulatio section w mentione tha "Big-M formulation ofte n lea t poo formulations Thi i becaus th linea relaxatio o suc a for mulatio ofte allow fo man fractiona values Fo instance conside th e

constrain (al variable ar binary ) +A; + X < 1000} ; Suc constraint ofte occu i facilit locatio an relate problems Thi s constrain correctl model a requiremen tha th x variable ca b 1 onl y y i als 1 bu doe s i a ver wea way Eve i th x value o th e linea relaxatio ar integer y ca tak o a ver smal valu (instea o th e require 1) Here eve for = (1,1,1) y nee onl b 3/100 t mak th e constrain feasible Thi typicall lead t ver ba branch-and-boun trees : th linea relaxatio give littl guidanc a t th "true value o th variables . Th followin constrain woul b better : •^ + ^ + ^ < 'iy whic force _ t tak o

large values Thi i th concep o makin th M Big- a smal a possible Bette stil woul b th thre constraint s < ^ X2 X2 whic forc y t b intege a soo a th x value are . Findin improve formulation i a ke concep t th successfu us o in tege programming Suc formulation typicall revolv aroun th strengt h th linea relaxation doe th relaxatio well-represen th underlyin inte ge program Findin classe o cut ca improv formulations Findin suc h classe ca b difficult bu withou goo formulations intege programmin g model ar unlikel t b successfu excep fo ver smal instances . 3. AVOI SYMMETR Y Symmetr ofte cause

intege programmin model t fail Branch-and boun ca becom a extremel inefficien algorith whe th mode bein g solve display man symmetries . Conside agai ou facilit locatio model Suppos instea o havin jus t on refiner a a site w wer permitte t hav u t thre refinerie a a site W coul modif ou mode b havin variable yj, Zj an Wj fo eac h sit (representin th thre refineries) I thi formulation th cos an othe r coefficient fo yj ar th sam a fo Zj an Wj Th formulatio i straight forward bu branc an boun doe ver poorl o th result . Th reaso fo thi i symmetry fo ever solutio i th branch-and-boun d tre wit a

give y, z, an w, ther i a equivalen solutio wit z takin o y' s values w takin o z' an _ takin o w. Thi greatl increase th numbe r
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BOSCH AND TRICK solution tha th branch-and-boun algorith mus conside i orde t find an prov th optimalit o a solution . i ver importan t remov a man symmetrie i a formulatio a pos sible Dependin o th proble an th symmetry thi remova ca b don e addin constraints fixing variables o modifyin th formulation . Fo ou faciht locatio problem th easies thin t d i t ad th con straint s yj ^ Zj > Wj fo al j Now a a refiner site Zj ca b non-zer onl i yj i

non-zero an Wj i s non-zer onl i bot yj an Zj are Thi partiall break th symmetr o thi s formulation thoug othe symmetrie (particularl i th x variables remain . Thi formulatio ca b modifie i anothe wa b redefinin th variables . Instea o usin binar variables le yj b th numbe o refinerie pu i n locatio j Thi remove al o th symmetrie a th cos o a weake linea r relaxatio (sinc som o th strengthening w hav explore requir binar y variables) . Finally t illustrat th us o variabl fixing, conside th proble o col orin a grap wit K colors w ar give a grap wit nod se V an edg e se E an wis t determin i w

ca assig a valu v{i) t eac nod / suc h tha v(i) e {I,,.., K} and vii) # v(j) fo al (/ ; e E. ca formulat thi proble a a intege programmin b definin a binar variabl xtk t b 1 i / i give colo k an 0 otherwise Thi lead t o th constraint s Xik = 1 fo al / (ever nod get a color ) k Xik + Xjk = 1 fo al k, (i, j) e E (n adjacen ge th same ) Xik {0 1 fo al / k Th grap colorin proble i equivalen t determinin i th abov se o f constraint i feasible Thi ca b don b usin branch-and-boun wit a n arbitrar objectiv value . Unfortunately thi formulatio i highl symmetric Fo an colorin o f graph ther i a

equivalen colorin tha arise b permutin th colorin g (tha is permutin th se {1,... A: i thi formulation) Thi make branc h an boun ver ineffectiv fo thi formulation Not als tha th formulatio n ver weak sinc settin xik = 1// fo al i,k is a feasibl solutio t th e linea relaxatio n matte wha E is . ca strengthe thi formulatio b breakin th symmetr throug vari abl fixing. Conside a cliqu (se o mutuall adjacen vertices o th graph . Eac membe o th cliqu ha t ge a differen color W ca brea th e symmetr b finding a larg (ideall maximu sized cHqu i th grap an d
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INTEGER PROGRAMMING 8 5

settin th color o th cHqu arbitrarily bu fixed S i th cliqu ha siz e kc, w woul assig th color 1,... ^ t member o th cliqu (addin i n constraint forcin th correspondin x value t b 1) Thi greatl reduce s th symmetry sinc no onl permutation amon th color kc + \,..., K ar valid Thi als remove th xik = l/k solutio fro consideration . 3. CONSIDE FORMULATION WIT MAN Y CONSTRAINT S Give th importanc o th strengt o th linea relaxation th searc fo r improve formulation ofte lead t set o constraint tha ar to larg t o includ i th formulation Fo example conside a singl constrain wit non negativ

coefficients : a]X] + a2X2 + fl3^3 H h n-^ < b wher th x ar binar variables Conside a subse S o th variable suc h tha Ylies ^' ^ b- T^ constrain t i€S vali (i i no violate b an feasibl intege solution an cut of frac tiona solution a lon a S i minimal Thes constraint ar calle cover constraints. W woul the lik t includ thi se o constraint i ou formu lation . Unfortunately th numbe o suc constraint ca b ver large I general , i exponentia i n, makin i impractica t includ th constraint i th e formulation Bu th relaxatio i muc tighte wit th constraints . handl thi problem w ca choos t

generat onl thos constraint s tha ar needed I ou searc fo a optima intege solution man o th e constraint ar no needed I w ca generat th constraint a w nee them , ca ge th strengt o th improve relaxatio withou th hug numbe o f constraints . Suppos ou instanc i s Maximiz 9x 4 14^: + lOxj + 32^ 4 Subjec t 3x] - 5x + 8x - IOX < 1 6 Xi €{0,1 } Th optima solutio t th linea relaxatio i x = (1,0.6 0 1 wit objec tiv 49.4 No conside th se 5 = (xj X2 X4) Th constrain t .^ + X + X < 2
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BOSCH AND TRICK a cu tha x violates I w ad tha constrain t ou problem w ge a tighte formulation Solvin

thi mode give solutio ^ = (1 0 0.375 1 an d objectiv 48.5 Th constrain t + ^ : 1 a vaH cove constrain tha cut of thi solution Addin thi constrain t an solvin give solutio x = (0 1 0 1 wit objectiv 46 Thi i th op tima solutio t th origina intege program whic w hav foun onl b y generatin cove inequalities . thi case th cove inequalitie wer eas t see bu thi proces ca n formalized A reasonabl heuristi fo identifyin violate cove inequali tie woul b t orde th variable b decreasin a/x the ad th variable s th cove S unti X]/G ^i ^ ^' ^^^ heuristi i no guarantee t find violate cove inequalitie (fo that

a knapsac optimizatio proble ca b e formulate an solved bu eve thi simpl heuristi ca creat muc stronge r formulation withou addin to man constraints . Thi ide i formalize i th branch-and-cut approac t intege program ming I thi approach a formulatio ha tw parts th explicit constraints (denote Ax < b) an th implicit constraints {A'x < b'). Denot th ob jectiv functio a Maximiz ex. Her w wil assum tha al x ar integra l variables bu thi ca b easil generalized . Step L Solv th linea progra Maximiz ex subjec io Ax t ge optima l relaxatio solutio x* . Step 2. I X integer the stop x i optimal . Step 3.

Tr t find a constrain a'x < b' fro th implici constraint suc tha t a'x^ > b. I found ad a'x < b to th Ax < b constrain se an g t ste 1 . Otherwise d branch-and-boun o th curren formulation . orde t creat a branch-and-cu model ther ar tw aspects th defini tio o th implici constraints an th definitio o th approac i Ste 3 t o find violate inequahties Th proble i Ste 3 i referre t a th separation problem an i a th hear o th approach Fo man set o constraints n o goo separatio algorith i known Note however tha th separatio prob le migh b solve heuristically i ma mis opportunitie fo separatio an d

therefor invok branch-and-boun to often Eve i thi case i ofte hap pen tha th improve formulation ar sufficientl tigh t greatl decreas e th tim neede fo branch-and-bound . Thi basi algorith ca b improve b carryin ou cu generatio withi n th branc an boun tree I ma b tha b fixing variables differen con straint becom violate an thos ca b adde t th subproblems .
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INTEGER PROGRAMMING 8 7 3. CONSIDE FORMULATION WIT MAN Y VARIABLE S Jus a improve formulation ca resul fro addin man constraints , addin man variable ca lea t ver goo formulations Le u begi wit h ou grap colorin example

Recal tha w ar give a grap wit vertice s an edge E an wan t assig a valu v{i) t eac nod / suc tha t v{i) ^ v{j) fo al (/ y e E. Ou objectiv i t us th minimu num be o differen value (before w ha a fixed numbe o color t use i thi s sectio w wil us th optimizatio versio rathe tha th feasibilit versio n thi problem) . Previously w describe a mode usin binar variable xi^ denotin whethe r nod / get colo k o note A a alternativ model le u concentrat o th e se o node tha get th sam color Suc a se mus b a independent set ( a se o mutuall non-adjacen nodes o th graph Suppos w fisted al inde penden set o

th graph 5i ^2,... 5^ The w ca defin binar variable s J25... J wit th interpretatio tha yj = 1 mean tha independen se t Sj i par o th coloring an yj = 0 mean tha independen se Sj i no par t th coloring No ou formulatio become s Minimiz Tj^ j j Subjec t V j = 1 fo al / e V j'JeSj yj e {0 Ijforal j {l,...,m } Th constrain state tha ever nod mus b i som independen se o th e coloring . Thi formulatio i a muc bette formulatio tha ou xi^ formulation Thi s formulatio doe no hav th symmetr problem o th previou formulatio n an result i a muc tighte linea relaxation Unfortunately th formulatio n

impractica fo mos graph becaus th numbe o independen set i ex ponentia i th numbe o nodes leadin t a impossibl larg formulation . Jus a w coul handl a exponentia numbe o constraint b generatin g the a needed w ca als handl a exponentia numbe o variable b y variable generation: th creatio o variable onl a the ar needed I orde r understan ho t d this w wil hav t understan som ke concept s fro linea programming .
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BOSCH AND TRICK Conside a linea program wher th variable ar indexe b j an th e constraint indexe b / : Maximiz /^^CjXj j Subjec t y^<3!/7-^/ < bi fo al / j Xj > 0 fo al

j Whe thi linea progra i solved th resul i th optima solutio x*. I n addition however ther i a valu calle th dual value, denote jr, associate d wit eac constraint Thi valu give th margina chang i th objectiv e valu a th right-han sid fo th correspondin constrain i changed S i f th right-han sid o constrain / change t Z? + A the th objectiv wil l chang b 7r, (ther ar som technica detail ignore her involvin ho w larg A ca b fo thi t b a vali calculation sinc w ar onl concerne d wit margina calculations w ca ignor thes details) . Now suppos ther i a ne variabl Xn+\, no include i th origina for

mulation Suppos i coul b adde t th formulatio wit correspondin g objectiv coefficien c„+ an coefficient a/,„+i Woul addin th variabl e th formulatio resul i a improve formulation Th answe i certainl y "no i th cas whe n i thi case th valu gaine fro th objectiv i insufficien t offse th e cos charge marginall b th effec o th constraints W nee Cn+\ J2i <^i,n+i^i > 0 i orde t possibl improv o ou solution . Thi lead t th ide o variabl generation Suppos yo hav a formula tio wit a hug numbe o variables Rathe tha solv thi hug formulation , begi wit a smalle numbe o variables Solv th linea relaxatio an

ge t dua value n. Usin TT, determin i ther i on (o more variable whos in clusio migh improv th solution I not the th linea relaxatio i solved . Otherwise ad on o mor suc variable t th formulatio an repeat . Onc th linea relaxatio i solved i th solutio i integer the i i op timal Otherwise branc an boun i invoked wit th variabl generatio n continuin i th subproblems . Ke t thi approac i th algorith fo generatin th variables Fo a hug numbe o variable i i no enoug t chec al o them tha woul d to tim consuming Instead som sor o optimizatio proble mus b e define whos solutio i a improvin variable W

illustrat thi fo ou r grap colorin problem .
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INTEGER PROGRAMMING 8 9 Suppos w begi wit a limite se o independen set an solv ou re laxatio ove them Thi lead t a dua valu 7ti fo eac node Fo an othe r independen se 5 i X]/ ^ > 1 the S correspond t a improvin vari able W ca writ thi proble usin binar variable zt correspondin t o whethe / i i S o not : Maximiz Y^^/^ / Subjec t Zi + Z < 1 fo al (/ y e E Zi e {0,1 fo all / Thi proble i calle th maximum weighted independent set (MWIS prob lem and whil th proble i formall hard effectiv method hav bee n foun fo solvin i fo problem o

reasonabl size . Thi give a variabl generatio approac t grap coloring begi wit a smal numbe o independen sets the solv th MWI problem addin i n independen set unti n independen se improve th curren solution I th e variable ar integer the w hav th optima coloring Otherwis w nee t o branch . Branchin i thi approac need specia care W nee t branc i suc a wa tha ou subproble i no affecte b ou branching Here i w simpl y branc o th yj variable (s hav on branc wit yj = 1 an anothe wit h yj = 0) w en u no bein abl t us th MWI mode a a subproblem . th cas wher yj = 0 w nee t find a improvin set excep

tha Sj doe no coun a improving Thi mean w nee t find th secon mos t improvin set A mor branchin goe on w ma nee t find th thir mos t improving th fourt mos improving an s on T handl this specialize d branchin routine ar neede (involvin identifyin node that o on sid o f th branch mus b th sam colo and o th othe sid o th branch canno t th sam color) . Variabl generatio togethe wit appropriat branchin rule an variabl e generatio a th subproblem i a metho know a branch and price. Thi s approac ha bee ver successfu i attackin a variet o ver difficul prob lem ove th las fe years . summarize model

wit a hug numbe o variable ca provid ver y tigh formulations T handl suc models i i necessar t hav a variabl e generatio routin t find improvin variables an i ma b necessar t mod if th branchin metho i orde t kee th subproblem consisten wit tha t routine Unlik constrain generatio approaches heuristi variabl generatio n routine ar no enoug t ensur optimality a som poin i i necessar t o prov conclusivel tha th righ variable ar included Furthermore thes e variabl generatio routine mus b applie a eac nod i th branch-and boun tre i tha nod i t b crosse ou fro furthe analysis .
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BOSCH AND TRICK 3. MODIF BRANCH-AND-BOUN PARAMETER S Intege program ar solve wit compute programs Ther ar a numbe r compute program availabl t solv intege programs Thes rang fro m basi spreadsheet-oriente system t open-sourc researc code t sophis ticate commercia applications T a greate o lesse extent eac o thes e code offer parameter an choice tha ca hav a significan affec o th e solvabilit o intege programmin models Fo mos o thes parameters th e onl wa t determin th bes choic fo a particula mode i experimenta tion an choic tha i uniforml dominate b anothe choic woul no b e include i th

software . Her ar som common ke choice an parameters alon wit som com ment o each . 3.7. Descriptio o Proble m Th first issu t b handle i t determin ho t describ th intege r progra t th optimizatio routine(s) Intege program ca b describe d spreadsheets compute programs matri descriptors an higher-leve lan guages Eac ha advantage an disadvantage wit regard t suc issue a s ease-of-use solutio power flexibility an s on Fo instance implementin g branch-and-pric approac i difficul i th underlyin solve i a spread shee program Usin ''callabl hbraries tha giv acces t th underlyin g optimizatio routine

ca b ver powerful bu ca b time-consumin t de velop . Overall th interfac t th softwar wil b define b th software I i s generall usefu t b abl t acces th softwar i multipl way (callabl e libraries hig leve languages comman lin interfaces i orde t hav ful l flexibihty i solving . 3.7. Linea Programmin Solve r Intege programmin relie heavil o th underlyin linea programmin g solver Thousand o ten o thousand o linea program migh b solve i n th cours o branch-and-bound Clearl a faste linea programmin cod ca n resul i faste intege programmin solutions Som possibihtie tha migh t offere ar prima

simplex dua simplex o variou interio poin methods . Th choic o solve depend o th proble siz an structur (fo instance , interio poin method ar ofte bes fo ver large block-structure models ) an ca diffe fo th initia linea relaxatio (whe th solutio mus b foun d ''fro scratch" an subproble linea relaxation (whe th algorith ca n us previou solution a a startin basis) Th choic o algorith ca als b e affecte b whethe constrain and/o variabl generatio ar bein used .
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INTEGER PROGRAMMING 9 1 3.7 Choic o Branchin Variabl e ou descriptio o branch-and-bound w allowe branchin o an frac

tiona variable Whe ther ar multipl fractiona variables th choic o f variabl ca hav a bi effec o th computatio time A a genera guideline , mor 'Important variable shoul b branche o first I a facilit locatio n problem th decision o openin a facilit ar generall mor importan tha n th assignmen o a custome t tha facility s thos woul b bette choice s fo branchin whe a choic mus b made . 3.7. Choic o Subproble t Solv e Onc multipl subproblem hav bee generated i i necessar t choos e whic subproble t solv next Typica choice ar depth-firs search breadth firs search o best-boun search Depth-firs searc

continue fixin variable s fo a singl proble unti integraht o infeasibilit results Thi ca lea d quickl t a intege solution bu th solutio migh no b ver good Best boun searc work wit subproblem whos linea relaxatio i a larg (fo r maximization a possible wit th ide tha subproblem wit goo linea r relaxation ma hav goo intege solutions . 3.7. Directio o Branchin g Whe a subproble an a branchin variabl hav bee chosen ther ar e multipl subproblem create correspondin t th value th variabl ca tak e on Th orderin o th value ca affec ho quickl goo solution ca b e found Som choice her ar a fixe orderin o

th us o estimate o th e resultin linea relaxatio value Wit fixe ordering i i generall goo t o firs tr th mor restrictiv o th choice (i ther i a difference) . 3.7. Tolerance s i importan t not tha whil intege programmin problem ar pri maril combinatorial th branch-and-boun approac use numerica linea r programmin algorithms Thes method requir a numbe o parameter giv in allowabl tolerances Fo instance i ;c = 0.99 shoul Xj b treate a s th valu 1 o shoul th algorith branc o Xj ? Whil i i temptin t giv e overl bi value (t allo fo faste convergence o smal value (t b ''mor e accurate") eithe extrem ca

lea t problems Whil fo man problems th e defaul value fro a qualit cod ar sufficient thes value ca b th sourc e difficultie fo som problems .
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BOSCH AND TRICK 3. TRICK O TH TRAD E Afte readin thi tutorial al o whic i abou ''trick o th trade" i i s eas t thro one' hand u an giv u o intege programming Ther ar e man choices s man pitfalls an s muc chanc tha th combinatoria l explosio wil mak solvin problem impossible Despit thi complexity , intege programmin i use routinel t solv problem o practica interest . Ther ar a fe ke step t mak you intege programmin implementatio n well .

Us state-of-the-ar software I i temptin t us softwar becaus i i s easy o available o cheap Fo intege programming however no hav in th mos curren softwar embeddin th lates technique ca doo m you projec t failure No al suc softwar i commercial Th COIN projec i a open-sourc effor t creat high-quaht optimizatio n codes . Us a modehn language A modelin language suc a OPL Mosel , AMPL o othe languag ca greatl reduc developmen time an al low fo eas experimentatio o alternatives Callabl Hbrarie ca giv e mor powe t th user bu shoul b reserve fo ''fina implementa tions" onc th mode an solutio approache

ar known . I a intege programmin mode doe no solv i a reasonabl amoun t time loo a th formulatio first, no th solutio parameters Th e defaul setting o curren softwar ar generall prett good Th prob le wit mos intege programmin formulation i th formulation no t th choic o branchin rule fo example . Solv som smal instance an loo a th solution t th Hnea re- laxations Ofte constraint t ad t improv a formulatio ar quit e obviou fro a fe smal examples . Decid whethe yo nee "optimal solutions I yo ar consistentl y gettin withi 0. % o optimal withou provin optimality perhap yo u shoul declar succes an

g wit th solution yo have rathe tha n tryin t hun dow tha final gap . Tr radicall differen formulations Often ther i anothe formulatio n wit completel differen variables objective an constraint tha wil l hav a muc differen computationa experience . 3. CONCLUSION S Intege programmin model represen a powerfu approac t solvin har d problems Th bound generate fro linea relaxation ar ofte sufficien t
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INTEGER PROGRAMMING 9 3 greatl cu dow o th searc tre fo thes problems Ke t successfu l intege programmin i th creatio o goo formulations A goo formulatio n on wher th linea relaxatio

closel resemble th underlyin intege pro gram Improve formulation ca b develope i a numbe o ways includin g finding formulation wit tigh relaxations avoidin symmetry an creatin g an solvin formulation tha hav a exponentia numbe o variable o con straints I i throug th judiciou combinatio o thes approaches combine d wit fas intege programmin compute code tha th practica us o intege r programmin ha greatl expande i th las 2 years . SOURCE O ADDITIONA INFORMATIO N Intege programmin ha existe fo mor tha 5 year an ha develope d hug literature Thi bibliograph therefor make n effor t b compre hensive

bu rathe provide initia pointer fo furthe investigation . Genera Intege Programmin Ther ar a numbe o excellen recen mono graph o intege programming Th classi i Nemhause an Wolse y (1988) A boo updatin muc o th materia i Wolse (1998) Schri jve (1998 i a outstandin referenc book coverin th theoretica l underpinning o intege programming . Intege Programmin Formulation Ther ar relativel fe book o for mulatin problems A exceptio i William (1999) I addition mos t operation researc textbook offe example an exercise o formu lations thoug man o th example ar no o realisti size Som e choice ar Winsto

(1997) Tah (2002) an HilHe an Lieberma n (2002) . Branc an Boun Branc an boun trace bac t th 1960 an th wor k Lan an Doi (1960) Mos basi textbook (se above giv a n outlin o th metho (a th leve give i thi tutorial) . Branc an Cu Th cuttin plan approac date bac t th lat 1950 an d th wor o Gomor (1958) whos cuttin plane ar applicabl t an y intege program Juenge e al (1995 provide a surve o th us o f cuttin plan algorithm fo specialize proble classes . a computationa technique th wor o Crowde e al (1983 showe d ho cut coul greatl improv basi branch-and-bound . Fo a exampl o th succes o suc

approache fo solvin extremel y larg optimizatio problems se Applegat e al (1998) . Branc an Pric Bamhar e al (1998 i a excellen surve o thi ap proach .
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BOSCH AND TRICK Implementation Ther ar a numbe o ver goo implementation tha al lo th optimizatio o realisti intege programs Som o thes ar e commercial lik th CPLE implementatio o ILOG Inc (CPLEX , 2004) Bixb e al (1999 give a detaile descriptio o th advance s tha thi softwar ha made . Anothe commercia produc i Xpress-M fro Dash wit th text boo b Guere e al (2002 providin a ver nic se o example an d applications . COIN-O (2004

provide a open-sourc initiativ fo optimization . Othe approache ar describe b Ralph an Ladany (1999 an b y Cordieretal (1999) . Reference s Applegate D. Bixby R. Chvatal V an Cook W. 1998 O th solutio o f travelin salesma problems in Proc. Int. Congress of Mathematicians, Doc. Math. J. DMV, Vol 645 . Bamhart C Johnson E L. Nemhauser G L. Savelsbergh M W P an d Vance P H. 1998 Branch-and-price colum generatio fo hug intege r programs Oper. Res. 46:316 . Bixby R E. Fenelon M. Gu Z. Rothberg E an Wunderling R. 1999 , MIP: Theory and Practice Closing the Gap, Proc. 19th IF IP TC7 Conf. on System

Modelling, Kluwer Dordrecht pp 19-50 . Commo Optimizatio INterfac fo Operation Researc (COIN) 2004 a t http://www.coin-or.or g Cordier C Marchand H. Laundy R an Wolsey L A. 1999 bc-opt a branch-and-cu cod fo mixe intege programs Math. Program. 86:335 . Crowder H. Johnson E L an Padberg M W. 1983 Solvin larg scal e zero-on linea programmin problems Oper Res. 31:803-834 . Gomory R E. 1958 Outlin o a algorith fo intege solution t linea r programs Bulletin AMS 64:275-278 . Gueret C Prins C an Sevaux M. 2002 Applications of Optimization with Xpress-MP, S Heipcke transl. Das Optimization Blisworth

UK . Hillier F S an Lieberman G J. 2002 Introduction to Operations Research, McGraw-Hill Ne York . ILO CPLE 9. Referenc Manual 2004 ILOG . Juenger M. Reinelt G an Thienel S. 1995 Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization, DI AC Se rie i Discret Mathematic an Theoretica Compute Science Vol Ill , America Mathematica Society Providence RI . Land A H an Doig A G. 1960 A Automati Metho fo Solvin Discret e Programmin Problems Econometrica 28:83-97 .
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INTEGER PROGRAMMING 9 5 Nemhauser G L an Wolsey L A. 1998 Integer and Combinatorial Opti-

mization, Wiley Ne York . Ralphs T K an Ladanyi L. 1999 SYMPHONY: A Parallel Framework for Branch and Cut, Whit paper Ric University . Schrijver A. 1998 Theory of Linear and Integer Programming, Wiley Ne w York . Taha H A. 2002 Operations Research: An Introduction, Prentice-Hall Ne w York . WilHams H R 1999 Model Building in Mathematical Programming, Wiley , Ne York . Winston W, 1997 Operations Research: Applications and Algorithms, Thom son Ne York . Wolsey L A. 1998 Integer Programming, Wiley Ne York . XPRESS-M Extende Modelin an Optimisatio Subroutin Library Ref erenc Manual 2004 Das

Optimization BHsworth UK .