Formula,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 ~ , b l~c tc 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

of

Opti~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

z 2 0

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

h

Tlie 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<\ ~ ~ t i ~ n i r a t i o n .

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 . Air

9 Trar~sportatioll

Dat,a

171 11 ~snrchouscs

.si supply i 1 m

ti,? i:lcrnnn,S j t h >iarchousc, j = 11

9.2.1 Fur~~l~~ la t iur i

r i j = numl:,cr i:,f t,o send i i j

Sorting

11 Invest ~ r ~ e v ~ t urlder

You have purchnsci:l .5i sharcs i:,f i pricc y i . i 11

Cl~rrrcnt price i pi

10

Problem

9.1 . plants. . of i th plant, = . . . . of I . . .

9.2 Decision Variables

units

through LO

taxation . stock at = 1, . . . . . of stock is

You 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 a t $50 bought a t sharc: cash is:

SO x 1.000 0.30 x (SO 30) x 1,000

that stock

a a t of 30% capital gains at

choices E

arc in 1993.

B is in 1994.

Cash

S1.OOO.OOOin 1993.

12.1 Dollar

Variables

.I, invcst,cd Y;

C'~I,../I~: ami:,unt invcstcd pcrii:,d = 1.

1.0GCu.~h3 1.00B 1 . i 5 D + 1.40E s.t . . l+ C:+D+ C'udhl 5 1

CIA.S/L~ B 5 0.3.1+ l . l C + l.OGC'u.~hl Cil.sI~3 l.OE < l . O l + 0.3A 1.06C'ash2

11 l:,roilucts, m raw nlatcrials

h i : a,~:ailal:,lc i .

a i j : # 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

max

Exparlsiovi

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

l i t : ti:, proi:lucc 1MTV using nuclcnr cnpaci t ,~

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

~ ian t , 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

a t 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 a

hire 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 Dercg l~ la t i~n

anyone ,ran anywhcrc

farcs ~ic tcrminc~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 . . T I ; C 0.i. .I = I , . I L

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 c s t ima t ,~ t,hat RVI hns gcncrat,cil in,rrcmcntal rcvcnuc fi,r Anlcri,ran the three ycnrs ali:,nc. i:,nc-time benefit. F\'c c x p r t 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

t o at $500 future. continue t o 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 ( l h ) s ? ) 5 Xf(sl)+ (lPA)f(s2)

,f ci:,ncavc - (z) ci:,nvcx.

0 x 1 the LO

(z) dn z ci,

.5.t. 3 b

20 Messages

20.1 How formulate?

2.

What is a good LO A of A is

21.1 The general

22 Convex

f

. (z) if f

power of 23

. For all

+

min f = maxi, + Az

0x1 the

C t,j l.cjl n . t . Az 2 b

1x1 = max{s, s ] mi11 C qiqi

Ax 2 b

r.i 5 z.? " j 5 z j

>Iinimizing Picirc~sisc lincnr convex functii:,n ,ran lbc moi:lcllcil I:,y

24 power of LO

min

Idea:

5.t.

Message: LO

MIT OpenCourseWarehttp://ocw.mit.edu

For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.

6.251J / 15.081J Introduction to Mathematical ProgrammingFall 2009