lec01_Page_01.pdflec01_Page_02.pdflec01_Page_03.pdflec01_Page_04.pdflec01_Page_05.pdflec01_Page_06.pdflec01_Page_07.pdflec01_Page_08.pdflec01_Page_09.pdflec01_Page_10.pdflec01_Page_11.pdflec01_Page_12.pdfFormula,tionn: Lcc. Ctcomct,ry: Lcc. 2-4 Simplex l\lcthoi:l: Lcc. 5-8 Thcory: Lc,:. .-inalysin: Lei:. Robust Lcc. Large ncalc ol:,timizat,ion: Lcc. Flo~is: Lcc. 16-17 The Ellipsi:,ii:l methi:,,$: Lcc. 18-19 1:)oint mcthoi:ls: Lcc. 20-21 Scmii:lcfinitc opt,imizatii:,n: Lcc. ~i~, bl~ctc Opt,imizatii:,n: Lcc. 24-2; Requirements 30%) Mii:ltcrm 30%) Final Iml:,ortant brakcr: cont,ribut,ions t,o ,class Lnc CPLEX fi:n si:,lving ol:,timiza,tion problems of Optimizatio~l LOPS Arisc? Examplcs Fi:,rmulatii:,ns 1 Structure of Class I Duality Sensitivity 9-11 12 Optimization: 13 14-15 Nctmork Interior 22 . .. 2 Homcmorkn: Exam: Exam: 40% tic of 3 Lecture Outline History Whcrc ofOpti~rlization Fermat, mi11 f(r) x: scalar Euler, nlin ~(sI, r,") s.t,. (Jk(1.1,. .: r,") = 0 k = 1,. 77L Lagrange Prol:,lcms in in fin it,^ dimcnsii:,ns, iralirulus variat,ions. Nonlinear Optirrlizatiovl The general Linear Forlnt~lat,ioli mininlizc 31.1 + 1.2 sul,jcct ti:, 1.1 2 21.1 1.2 2 1.1 2 > ~ninimizc c'z sul,jcct ti:, Az 2 z20 4 6 History of 1638:Newton, 1670 1755 Lagrange, 1707 . . . . . .,. Euler, of 5 5.1 problem What is Optimization? 6.1 + 21.2 2 + 3 0.1.2 0 hTlie pre-algorit,hmic Fonrier, Mct,hoil for synbcm linear inc,:lualit,ics. de Val1i.e Poussirl simplcx-like mcthoi:l fin ol,jcct,ivc functii:,n wit,h al:,ii:,- lute vo11 Nenlnann, 1028 game t,hcory, iluality. Farkas, Minkowski: Caratl~i.odory, Fi:,un,Sa,tionn Tlie lnoderll Dantzig. 51ml,lcx mcthoil Applicat,ions. Large Siralc Opt,imizat,ion. thci:,ry. The cllipsi:,ii:l algi:,rit,hm. pi,int algi:,rit,hmn. Scnlidcfinit,c all<\ ~~ti~niration. Ri:,bust Ol:,timizat,ion. 8 LOPS Applicabilit,y Transportat,ion traffic ci:,nt,ri:,l, Crew schci:luling, >Iovcmcnt i:,f Truck Loails 7 History of LO 7.1 period 1826 solving of la values. 1870-1930 7.2 period George 1047 1950s 1960s 1970s Complexity 1979 1980s Interior 1990s conic 2000s Where do Arise? 8.1 Wide . Air9 Trar~sportatioll Dat,a 171 11 ~snrchouscs .si supply i 1 m ti,? i:lcrnnn,S jth >iarchousc, j = 11 9.2.1 Fur~~l~~latiuri rij = numl:,cr i:,f t,o send i i j Sorting 11 Invest ~r~ev~t urlder You have purchnsci:l .5i sharcs i:,f i pricc yi. i 11 Cl~rrrcnt price i pi 10 Problem 9.1 . plants. . of ith plant, = . . . . of I . . . 9.2 Decision Variables units through LO taxation . stock at = 1,. . . . . of stock isYou cxpcct the l:,ricc i:,f i i:,nc scar fii:,m now will bc ri You ]:,a- cal:,ital-gains t,ax thc rat? ,311 any t,hc t,imc i:,f the sale. You want ti:, raise C ami:,unt crash aft,cr taxes. You 1%) Example: You sell shares per sharc; you ha,~:c them $30 per Vet - - Five invcstmcnt .-i. B. C, D. .-i. C, and D a,~:ailal:,lc availablc carns 6% per year. C;asli Flowper Ilivest,ed of pay in transaction costs 1.000 at $50 bought at sharc: cash is: SO x 1.000 0.30 x (SO 30) x 1,000 that stock a at of 30% capital gains at choices E arc in 1993. B is in 1994. Cash S1.OOO.OOOin 1993. 12.1 DollarVariables .I, invcst,cd Y; C'~I,../I~: ami:,unt invcstcd pcrii:,d = 1. 1.0GCu.~h3 1.00B 1.i5D + 1.40E s.t. .l+ C:+D+ C'udhl 5 1 CIA.S/L~ B 5 0.3.1+ l.lC+ l.OGC'u.~hl Cil.sI~3 l.OE < l.Ol+ 0.3A 1.06C'ash2 11 l:,roilucts, m raw nlatcrials hi: a,~:ailal:,lc i. aij: # nlatcrial i proi:luct j needs ori:lcr ti:, lbc proi:lucc,S. Formulat,ion rj = i:,f pri:,,Suct j pri:,iSucc,S. n C qis,i j=l 12.2.1 Decision . . . .E: amount in millions in cash in t, t 2, 3 max + + + + + 13 Manufacturing 13.1 Data units of material units of in 13.2 13.2.1 Decision variables amount maxExparlsiovi C;olistraiilt,s Dt: fc,rccast,ccl clcmani:l fi:n electricit,:: ::car Et: cxisti~lg cal:,acity (in nvailablc c,: ci:,st ti:, l:,roclucc 11\I\\' using capncity lit: ti:, proi:lucc 1MTV using nuclcnr cnpacit,~ morc nu,rlca,r ycnri Nuclear Inst :7cars r,: ami:,unt ci:,al cal:,acity lint ycar yt: nmount i:,f capncit,:: I:,rought i:,n line ycnr u:,: tot,nl ci:,al cal:,acity ycar zt: t,otal cnpacit,:: in :;car ~iant,s ti:, >icckl:: night,shift fi:,r llurscs D, iicmancl fi:n j = 1 T Evcry llursc works ri:,m 14 Capacity 14.1 Data and at t oil) nt t coal cost No than 20% Coal plants last 20 plants 15 14.2 Decision Variables of brought on in t. in t. in t. t. 15 Scheduling 15.1 Decision variables Hospital mnkc its nurses, . . . 5 clays in ahire mininl~lm nnrnbcr nurses rj: startiqg t,hcir week i:,n cia- Managerrlerlt indust,ry 1978 - Clarricrs only allowc~i ti:, ircrt,ain ri:,utcs. Hcnirc airlincs iYi:,rt,h~scst. Eastern, Southwest, ctc. - Fares ilct,cr~nincil I:,:: Clivil Acrona,utics Boaril lbascil mileage and othcr ci:,st,s (C.4B li:,ngcr SLII)E 26 Post Dercgl~lati~n anyone ,ran anywhcrc farcs ~ictcrminc~i I:,y (and thc markct) 17 Managerrlerlt Huge anil fixcil cost,s Vcry li:,m variablc ci:,st,s pcr passenger ($lO/passcngcr i:,r lcss) cci:,nomically ci:,mpctitivc cnvironmcnt Ncar-pcrfcct infi:,rmat,ion and ncgligil:,lc ci:,st infi,rmatii:,n pcrishal:,lc invcnt,ory l\lult~il~lc farcs Goal: of Decision Variables # nurscs j 16 Revenue 16.1 The Deregulation in fly such as (CAB) on no exists) fly, carrier Revenue sunk . Strong of Highly Result:Managerrlerlt 11 11 ilcstinatii:,ns irlasscs (for Rc~cnucs T!, 1.j.;. ZJ Ca1:)acitics: i = I. .TI; C 0.i. .I = I, .IL Expected iicnlancls: D:) Forillulatioil Variables Q,,: Q-class cu~tomers me 1;,: Y-class cu~tomers i j zr.,::c2%, +v:,Y;, Managerrlerlt F\'c cstimat,~ t,hat RVI hns gcncrat,cil in,rrcmcntal rcvcnuc fi,r Anlcri,ran the three ycnrs ali:,nc. i:,nc-time benefit. F\'c cxprt RM gcncrat,c lcnst millii:,n nnnually for the fi:,rcsccnl:,lc .-is me invest the cnhanccmcnt DIV.-i?rlO me cxl:,cct t,o cnpt,urc even lnrgcr rcvcnuc 18 Revenue 18.1 Data origins. .I hub 2 simplicity), (2-class. Ti-class . , . . . . . . 18.2 LO 18.2.1 Decision . #of accept from i to j . # of accept from to maximize 19 Revenue $1.4 billion in Airlines in last This is not a to at $500 future. continue to in of nn premium.t,o 1. Define your ilccision variables clearly. ITrritc ci:,nstraints ani:l ol:,jcctivc funct,ion. for~ril~latiorl? fi,rmulation with a numl:,cr variaI:,lcs anil const,raint,s, anil t,hc mat,rix sparse. proble~ll furlctiorls :.S+R sl. s2 E ,Y f(As1