Faaliyetler İş Araştırma ve İktisat Integer Linear Programming Chapter 11 integer Linear Programming • Types,of Integer Linear Progranurung Models • Graphical and Computer Solutions for an AU- J Integer Linear Program • Applications Involving 0-1 Variables ""_ .~ Modeling Flexibility hovided by 0-1 Variables C 2006 Thomson South-Western. An R,j" ghis ~rved. . Slide 1 , .. ~, ' :. .-- . • Consider the follow-lOg aU~in"teger linear program: . Max '- 3x(¥-2l:/" '. . ·s·t ~ -:3~n·-- i Xl·--'-;-? : :/ 1 ~_ ji;:.-~ !- : ·;,"'::'f!;.~tr ;..i!;.. --/ }:_;~ .. .. ;~; 'Xl'i~~:~d '~fegei . . -.. : .. ', -. . ..E"amplei~:i~.tegerL.P . • l.P Relaxation . . -Xl + x 2 ~ \'--+~---"3,. +" :%2 ~ 9 . Slide3 (25. 1.5) 1 7 x, 02006 ThODlSOn South·Western.. Slide 5 .~ '~ .' Types of Integel" Programming Models • An LP in which aIlllbevariables are restricted to be integers is called .. ;all-integer linear program (ILP). • The LP that resu~hom_dTopping the intege r requirements is caIIIIif1Jltl.p Relaxation of the ILP. • If only a subset of lIE variables are restricted to be integers, the problra is called a mixed-integer linear program (MILP). • Binary variables _mriables whose values are restri€-ted-to-be-O- OI'_ 1..-lLalL\[ariables are restricted to ~-OT 1, the pr·o •• 7." ......... ·j~cdd~.~a"=ll~o~'~ ·b"m~ · ~a~ryyjin!ilit e~g~e ~' linear program: 02006 ThOU\S9n Sou~ AU Rights Reserved. Slide 2 Exaaple: All-Integer LP • LP Relaxation Solving the~eIl"! ·as a linear progra....-n ignoring .· the integer COJ'"lSb;lintsI the optimal solution-to the' linear program cjifts fractional values for both Xl and x 2 . From the g:r .. ium the next slide, we see that I l'ie: ·- .' , i' op~al S91utionlD~ linear program is: . . ·1 .II =-on x 2 ;; 1.5, Z = 10.5 ~ 2006 TholnSOn 5 "'t Exomple: All-Integer L.P . . .. . ... -• ·.Ro~·lding Up . . - - If ,we' • OD tOe fractional solution (Xl = 2.5, . x;;' 1.5) t: ') ie lax\$p problem we, get ~ - j -a:J'd~T.FnI--.-2he gra~e next slide, w e . : .. '., see that this paiim 1ie~ outside the feasTffie region, J making _ this _ . ~n infeasible. ~ ~ ~ Slide 6 1 '. Example: All-Integer LP • Rounde~ Up Solution " 5 4 \----/'--- 3r l + Xl .s 9 3k_~~-- Max 3r t '" 21'2 2 ______ n..p lnfeasible (3, 2) o LP Optimal (2,5, 1.5) 1 /' x1 .+3x2 S 7 1 2 3 4 5 6 7 " Cl2006 Tho~n'So\lth-Westem. All Rights ~rved . Slidt 7 . ' Q:>0p'1~.~e .~u .meration of Feasible JLP Solutions .~ -'-:li:u:~re ~t:~'-ejght feasible integer solu'ti6~ to this 'ptobleni: " . '. < . '. '. • . ' . . '" . ". : '. • ". • • \ ·~~\-_i\1.'· -~·- ·b2~ ' ~6 ' . . ·..-. ·.:'·.·· · 2·> 1->·0 · 3 '.: : ,: : ,"';:'-;,i.";3~;';i:"':": d C . :6 .. .... ::(\0~.~(;~ ;,{~~~t . . : .J1 - O~timal ~Jution ' .. '. ~::·.T.;Z .1-.~~_,~.1.;.(.,:2 .r>~~· . - . .' .. ,~ ,,', " ' , > , •. · .. ::E:)x~mple:All-lnteger LP . _Pa .~.J:f;p.'~.(isl,eet SJom"ir'gProblem. Data , 1 , lHSCcefficlents COllSlra nt X1 I >:2 #2 , -"' '., I , CbJ.COEffic,enlsl 3 2 - --- Cl2006 ThOInSon South-Western. All Rights Reserved. Slide 11 Example: A ll-Integer LP - ' Rounding Down ....•. '.' = By rounding the optimal solution down to Xl "" 2, x 2 =], we see lbal this solution indeed is an integer s6fuG-on within the feasibl e region, and substituting in the objective function, it gives z "" 8. We have found a feasible aU-in~eger solution, but have we found ~ OPTIMAL all-integer solution? The answel"is NO! The op.t:imaLsolution is XI - 3 ·amr.:rf:U-gtvift&z = 9, as evi~enced in the next two slides. - ._. -::~- . 0 2006 Thomson ~e;u.'m. AU rughts Reserved. Example: All-lntegerLP " 5 '.: 1 . Example: .All~lnteger, LP . . Slide 8 ; : ,,:! .. ", :'~ . ...., ', . '.' . Stich:· ID . -. ".:- .. . ... B Ie! D 141 ~l ~,~'scsetC4'SDS~ w }\ .:;:'S'SCS9+Cs·eOS9 ...0 2006 lboUlSOn South-~ All Rights Reserv~. Slide 12 , , 2 - Exam pIe: All-Integer LP • Partial Spreadsheet Sho'\oVi.ng Optimal Solution 9 I '<:- . 3 , . : "(:; , 15 ~_ #3: . ~ -< ·3 . . ~.~-~ _ 0 2006 Thorruon South-We\$lem. AU Righ!5 ~rved. Metropolitan Mia:o~~y~~(}Q(~ planning to expand its oper.tious into other~ :'-', . i lectronic"iippliances: Th~ toin~a'hy: .; "EJ ~as identified sey _~n n?~._~r~d~c~ .~~S":4 ' .. It ca n carry. Relevant inJorm'atidn : . . about each line. (o llo ws oriili~il~'( ~[jde. : . - :, ·::;;';;?:f?::~';~ .',; . V" .. ... .. ,. ' ; , ' . '. -~ . ~ . .. - ". ' - " . ' ~- ~' -: .'- . - ' - . . -. ", .- Slide J3 Slid.e 1:; Example: Metrop~li~ii.IVlici-Qwav~ r® Metropolitanhas..d.ed.de<1;:Nit,they shQ uld not ~- ' - stock proJectipn TV~ ttnJ,eS§ :t1teYc.-s~cx;.k el!:her -,-,- . ~_ =JY/'lCRs. _ o. ~ colc, + .102(14.000)x, -+- .105(lSOOO)xs + .141(2000)r6 + .132(32oo0)x, C 2006 Thomson So\Jth-Western. All Righl\$ ~l"IIed Slide 19 _~ • . r :( • Definet!l~'C;ci~tiaiilts (con~ued) ~)- Do :ril;~J'\$I~~ Qot)l:yc:Rs and OVD playeJs: . . '. ~ '. ~ ~ ~ ~ ~ ~ '.., . . - . '. -. x, + xs-,S 1 . ~) .. S~.DI}xii~.ep:g~,tn~8: if...' .hey slock colo~ TV's: . r~ -:i~.~O ~. ~ . - . : . ~) : .~ti~~~ ,~i)~!l~~ ·~~~~~ -lines: .. ~c:i'~i~f~fltt:;:6+ x, ~ 3 ....... .. ... ... · :., ~;<~~:<~'~~-G:-i;~~f+' 'c:~; .~\::'~~- .,., .{'.,'.-{'.,'-'( '.{., I; 02006 Thomson South-Western. ~ rughts Re5erved. Slide 23 Example:: Metropolitan Microwaves • Define the ~ints I) Money: 6h7 1~--+2DxJ + 14x, + ]Sx s + 2r6 + 32x, ~ 45 2) Space: 125x 1 +15lIl:K:" +200]'3 +40x 4 +40x s +20.:r:6 + l OOx 7 .:s. 420. 3) Stock projmtion lVs only if . -.-:!~k TV fV'CRs or color TV s: Xl + Xl>'" or Xl + X2 - X3 ~ 0 Q 2006 Thomson s.......eslem_ . AU Rights Reserved Slide 20 Exaroplr; Metropolitan MicrO~aYeS ® • Partial Spr~1 Showing Problem 0ata ::,' ... -- Ccnstraims ; 1 3 #1 : #2 :'S 150 ObiruncC~f -.15 1060 :E~ilInple: Metropolitan Micrc)w,)'v,;, o 2OQ(; Thomson So~em. All Rights Re!.erved. Slide·2:4 4 Example: Metropolitan Microwaves • Solver Options Dialog Box r C 2006 Thomson So,:th,Western, All Right\$ R~~rv~ , Slide 25 Example: Metropoij~l\lh~'~owaves ~ ~ ' .... ~ " - ' . .. :.,.-." ., '.: '.: ';" :: -:":-:, :--:".: " " ... ~! --, " ., ' 1 ' -.. ~'. '. ,',. '. Example: Tiha' ~t.;u9P.!rg · Tina~s Tailoring has fiye' idle'ta-il~d a~d f~u~ " ,' :::';:-.1 ~.-:~-;-cUsto~ garments to Ul.i!.ke--:-inti ~'tl~t~d 'fime..(~ . ~ ~ h?)~) it : wonJd·take eadRail~r 'to-make e'ach,g . arment ~ ,-: -C. i§~ .in --the next sli~ 'X' iidhe tatile·iitdkates 1 an unac~eptai5re tailor-garment ~~~i~*:nti ' . , Tailor - 1 - Garmen' 1- , J ~g.g-OWTl 1~ 21 Clown'costume 11 14 X 12 I Admiral' s uniform 12 8 11 X . " Bullfighter's outfit X 20 20 -18 ~ - C> 2006 Tho=O" So"