Structure of Class
1
Formula,tionn: Lcc. I
Ctcomct,ry: Lcc. 2-4
Simplex l\lcthoi:l: Lcc. 5-8
Duality Thcory: Lc,:. 9-11
Sensitivity .-inalysin: Lei:. 12
Robust Optimization: Lcc. 1 3
Large ncalc ol:,timizat,ion: Lcc. 14-15
Nctmork F l o ~ i s :Lcc. 16-17
The Ellipsi:,ii:l methi:,,$: Lcc. 18-19
Interior 1:)oint mcthoi:ls: Lcc. 20-21
.
2
Scmii:lcfinitc opt,imizatii:,n: Lcc. 22
~ b..
il ~ c~t cOpt,imizatii:,n:
,
Lcc. 24-2;
Requirements
Homcmorkn: 30%)
Mii:ltcrm Exam: 30%)
Final Exam: 40% Iml:,ortant tic brakcr: cont,ribut,ionst,o ,class
Lnc of CPLEX fi:n si:,lving ol:,timiza,tion problems
3
Lecture Outline
History of Optimizatio~l
Whcrc LOPS Arisc?
Examplcs of Fi:,rmulatii:,ns
4
History of Opti~rlization
Fermat, 1638: Newton, 1670 mi11 f(r)
x : scalar
Euler, 1755 Lagrange, 1707 nlin ~ ( s .I. .,. r,")
s.t,.
(Jk(1.1,.
. . : r,")= 0
k = 1 , .. . , 77L
Euler, Lagrange Prol:,lcms in in fin it,^ dimcnsii:,ns, iralirulus of variat,ions.
5
5.1
6
6.1
Nonlinear Optirrlizatiovl
The general problem
What is Linear Optimization?
Forlnt~lat,ioli
+
+
mininlizc
31.1
1.2
sul,jcct ti:,
1.1
21.2
21.1
+ 1.2
1.1
~ninimizc
sul,jcct ti:,
2 0.1.2
22
23
>0
c'z
2h
z20
Az
7
History of LO
7.1
Tlie pre-algorit,hmic period
Fonrier, 1826 Mct,hoil for solving synbcm of linear inc,:lualit,ics.
de la Val1i.e Poussirl simplcx-like mcthoi:l fin ol,jcct,ivc functii:,n wit,h al:,ii:,lute values.
vo11 Nenlnann, 1028 game t,hcory, iluality.
Farkas, Minkowski: Caratl~i.odory,1870-1930 Fi:,un,Sa,tionn
7.2
Tlie lnoderll period
George Dantzig. 1047 51ml,lcx mcthoil
1950s Applicat,ions.
1960s Large Siralc Opt,imizat,ion.
1970s Complexity thci:,ry.
1979 The cllipsi:,ii:l algi:,rit,hm.
1980s Interior pi,int algi:,rit,hmn.
1990s Scnlidcfinit,c all<\conic ~ ~ t i ~ n i r a t i o n .
2000s Ri:,bust Ol:,timizat,ion.
Where do LOPS Arise?
8
.
8.1
Wide Applicabilit,y
Transportat,ion
Air traffic ci:,nt,ri:,l,Crew schci:luling,
>Iovcmcnt i:,f Truck Loails
Trar~sportatiollProblem
9
..
.
Dat,a
9.1
171
.si supply
9.2
9.2.1
rij
plants.
ti,?
11
~snrchouscs
of i t h plant, i = 1. . . m
i:lcrnnn,S of j t h >iarchousc, j = I . . . 11
Decision Variables
Fur~~l~~latiuri
= numl:,cr
i:,f
units t,o send i i j
10
Sorting through LO
11
Invest ~ r ~ e vurlder
~ t taxation
..
You have purchnsci:l
.5i
sharcs
Cl~rrrcntprice of stock i is pi
i:,f
stock i at pricc
yi.
i = 1, . . .
.
11
You cxpcct that the l:,ricc i:,f stock i i:,nc scar fii:,m now will bc ri
You ]:,a- a cal:,ital-gains t,ax a t thc rat? of 30% ,311 any capital gains at t,hc
t,imc i:,f the sale.
You want
ti:,
raise C ami:,unt of crash aft,cr taxes.
You pay 1%)
in transaction costs
Example: You sell 1.000 shares a t $50 per sharc; you ha,~:cbought them
a t $30 per sharc: Vet cash is:
SO x 1.000
-
0.30 x (SO - 30) x 1,000
Five invcstmcnt choices .-i. B. C , D.E
.-i. C , and
D arc
a,~:ailal:,lcin 1993.
B is availablc in 1994.
Cash carns 6% per year.
S1.OOO.OOO in 1993.
12.1
C;asli Flowper Dollar Ilivest,ed
.
Decision Variables
12.2.1
.I, . . . E: amount invcst,cd in Y; millions
C'~I,../I~:
ami:,unt invcstcd in cash in pcrii:,d t , t = 1.2 , 3
max 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
+
+
Manufacturing
13
13.1
Data
11
l:,roilucts, m raw nlatcrials
a,~:ailal:,lcunits of material i .
hi:
aij:
13.2
13.2.1
rj =
<
# units of nlatcrial i proi:luct j needs in ori:lcr
Formulat,ion
Decision variables
amount i:,f pri:,,Suct j pri:,iSucc,S.
n
max C qis,i
j=l
ti:,
lbc proi:lucc,S.
Capacity Exparlsiovi
14
14.1
Data and C;olistraiilt,s
Dt: fc,rccast,ccl clcmani:l fi:n electricit,:: a t ::car t
Et: cxisti~lgcal:,acity (in oil) nvailablc nt t
c,:
ci:,st ti:, l:,roclucc 11\I\\'
lit:
cost
ti:,
using coal capncity
proi:lucc 1MTV using nuclcnr c n p a c i t , ~
No morc than 20% nu,rlca,r
Coal plants last 20 ycnri
Nuclear plants Inst 15 :7cars
14.2
Decision Variables
r,: ami:,unt of
yt: nmount i:,f
ci:,al cal:,acity brought on lint in ycar t.
capncit,:: I:,rought i:,n line in ycnr t.
u:,: tot,nl ci:,al cal:,acity in ycar
zt: t,otal
15
15.1
t.
cnpacit,:: in :;car t .
Scheduling
Decision variables
Hospital ~ i a n t ,ti:,
s mnkc >icckl:: night,shift fi:,r its llurscs
D , iicmancl fi:n nurses, j = 1. . . T
Evcry llursc works 5 clays in a ri:,m
Goal: hire mininl~lmnnrnbcr of nurses
Decision Variables
nurscs startiqg t,hcir week i:,n cia- j
rj: #
16
Revenue Managerrlerlt
16.1 The indust,ry
Deregulation in 1978
-
-
Clarricrs only allowc~iti:, fly ircrt,ain ri:,utcs. Hcnirc airlincs such as
iYi:,rt,h~scst.Eastern, Southwest, ctc.
Fares ilct,cr~nincilI:,::
Clivil Acrona,utics Boaril (CAB) lbascil on mileage
and othcr ci:,st,s (C.4B no li:,ngcr exists)
Post D e r c g l ~ l a t i ~ n
anyone ,ran fly, anywhcrc
farcs ~ i c t c r m i n c ~
I:,yi carrier (and thc markct)
17
.
Revenue Managerrlerlt
Huge sunk anil fixcil cost,s
Vcry li:,m variablc ci:,st,s pcr passenger ($lO/passcngcr i:,r lcss)
Strong cci:,nomically ci:,mpctitivc cnvironmcnt
Ncar-pcrfcct infi:,rmat,ion and ncgligil:,lc ci:,st of infi,rmatii:,n
Highly pcrishal:,lc invcnt,ory
Result: l\lult~il~lc
farcs
SLII)E26
Revenue Managerrlerlt
18
18.1
.
.
.
Data
11
origins.
11
ilcstinatii:,ns
I hub
2 irlasscs (for simplicity), (2-class. Ti-class
R c ~ c n u c sT,! ZJ
Ca1:)acitics:
1.j.;.
i = I .. . . T I ;
Expected iicnlancls:
18.2
LO Forillulatioil
..
Decision Variables
18.2.1
D:)
Q,,:
# o f Q-class cu~tomersme accept from i t o j
1;,: # of Y-class cu~tomers accept from i t o j
maximize
19
C,0.i. .I. = I , . . . I L
zr.,::c2%,
+v:,Y;,
Revenue Managerrlerlt
F\'c c s t i m a t , ~t,hat RVI hns gcncrat,cil $1.4 billion in in,rrcmcntal rcvcnuc fi,r
Anlcri,ran Airlines in the last three ycnrs ali:,nc. This is not a i:,nc-time benefit.
F\'c c x p r t RM t o gcncrat,c at lcnst $500 millii:,n nnnually for the fi:,rcsccnl:,lc
future. .-is me continue t o invest in the cnhanccmcnt of DIV.-i?rlO me cxl:,cct t,o
cnpt,urc nn even lnrgcr rcvcnuc premium.
20
20.1
Messages
How t,o formulate?
1. Define your ilccision variables clearly.
2. ITrritc ci:,nstraints ani:l ol:,jcctivc funct,ion.
What is a good LO for~ril~latiorl?
A fi,rmulation with
numl:,cr of variaI:,lcs anil const,raint,s,anil t,hc mat,rix
A is sparse.
a
21.1
22
.
.
23
The general proble~ll
Convex furlctiorls
f :.S+R
For all sl.s 2E ,Y
+ ( l h ) s ? )5 Xf(sl)+(lPA)f(s2)
f(As1
,f (z) ci:,ncavc if
f (z)ci:,nvcx.
-
0 x 1 the power of LO
min
.5.t.
f (z)= maxi,
Az
3b
dn z
+ ci,
24
0x1the power of LO
min
n.t.
Idea: 1x1 = max{s, s ]
C
t,j
Az
l.cjl
2b
C
mi11
qiqi
5.t. Ax 2 b
r.i 5 z.?
" j
5 zj
Message: >Iinimizing Picirc~sisclincnr convex functii:,n ,ran lbc moi:lcllcil I:,y LO
MIT OpenCourseWare
http://ocw.mit.edu
6.251J / 15.081J Introduction to Mathematical Programming
Fall 2009
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