15.093J/2.098J Optimization Methods
Professor: Dimitris Bertsimas
1 Structure of Class
Linear Optimization (LO): Lec. 1-9
Slide 1
Network Flows: Lec. 10-11
Discrete Optimization: Lec. 12-15
Dynamic Optimization: Lec. 16
Nonlinear Optimization (NLO): Lec. 17-24
2 Requirements
Slide 2
Homeworks: 30%
Midterm Exam: 30%
Final Exam: 40%
Class Participation: important tie breaker
Use of commercial software for solving optimization problems
3 Lecture Outline
Slide 3
History of Optimization
Where LOPs Arise?
Examples of Formulations
4 History of Optimization
Fermat, 1638 Newton, 1670
min f(x) x: scalar
df(x) = 0
dx
1
Slide 4
Euler, 1755
min f(x1 : : : xn)
rf(x) = 0
Lagrange, 1797
min f(x1 : : : xn)
s.t. gk (x1 : : : xn) = 0 k = 1 : : : m
Euler, Lagrange Problems in innite dimensions, calculus of variations.
5 Nonlinear Optimization
5.1 The general problem
Slide 5
min f(x1 : : : xn)
s:t: g1(x1 : : : xn) 0
...
gm (x1 : : : xn) 0:
6 What is Linear Optimization?
6.1 Formulation
minimize 3x1 + x2
subject to x1 + 2x2 2
2x1 + x2 3
x1 0 x2 0
3 x = x1 b = 2 1
x2
3
minimize c x
subject to Ax b
Slide 6
c=
A = 12 12
0
x0
7 History of LO
7.1 The pre-algorithmic period
Fourier, 1826 Method for solving system of linear inequalities.
de la Vallee Poussin simplex-like method for objective function with abso
lute values.
2
Slide 7
Kantorovich, Koopmans, 1930s Formulations and solution method.
von Neumann, 1928 game theory, duality.
Farkas, Minkowski, Caratheodory, 1870-1930 Foundations.
7.2 The modern period
Slide 8
George Dantzig, 1947 Simplex method.
1950s Applications.
1960s Large Scale Optimization.
1970s Complexity theory.
Khachyan, 1979 The ellipsoid algorithm.
Karmakar, 1984 Interior point algorithms.
8 Where do LOPs Arise?
8.1 Wide Applicability
Slide 9
Transportation
Air trac control, Crew scheduling, . . .
Movement of Truck Loads
Telecommunications
Manufacturing
Medicine
Engineering
Typesetting (TEX, LATEX)
9 Transportation Problem
9.1 Data
m plants, n warehouses
si supply of ith plant, i = 1 : : :m
dj demand of jth warehouse, j = 1 : : :n
cij : cost of transportation i ! j
3
Slide 10
9.2 Decision Variables
9.2.1 Formulation
xij = number of units to send i ! j
m n
XX
min
cij xij
i=1 j =1
m
X
s:t:
xij = dj
i=1
n
X
xij = si
j =1
xij 0
Slide 11
j = 1 : : :n
i = 1 : : :m
10 Sorting through LO
Slide 12
Given n numbers c1 c2 : : : cn
The order statistic c(1) c(2) : : : c(n): c(1) c(2) : : : c(n) P
Use LO to nd ki=1 c(i) .
n
X
min
ci xi
i=1
n
X
s:t:
xi = k
i=1
0 xi 1 i = 1 : : : n
11 Investment under taxation
You have purchased si shares of stock i at price qi, i = 1 : : : n.
Current price of stock i is pi
You expect that the price of stock i one year from now will be ri.
You pay a capital-gains tax at the rate of 30% on any capital gains at the
time of the sale.
You want to raise C amount of cash after taxes.
You pay 1% in transaction costs.
Example: You sell 1,000 shares at $50 per share you have bought them
at $30 per share Net cash is:
50 1 000 ; 0:30 (50 ; 30) 1 000;
0:01 50 1 000 = $43 500:
4
Slide 13
11.1 Formulation
n
X
max
i=1
n
Slide 14
ri (si ; xi )
X
n
n
X
X
pi xi ; 0:30 (pi ; qi)xi ; 0:01 pi xi C
i=1
i=1
i=1
0 xi si
s:t:
12 Investment Problem
Slide 15
Five investment choices A, B, C, D, E.
A, C, and D are available in 1993.
B is available in 1994.
E is available in 1995.
Cash earns 6% per year.
$1,000,000 in 1993.
12.1 Cash Flowper Dollar Invested
Year:
A
B
C
D
E
1993
1994
1995
1996
-1.00
+0.30
+1.00
0
0
-1.00
+0.30
+1.00
-1.00
+1.10
0
0
-1.00
0
0
+1.75
0
0
-1.00
+1.40
$500,000
NONE
$500,000
NONE
$750,000
LIMIT
Slide 16
12.2 Formulation
12.2.1 Decision Variables
A : : :E: amount invested in $ millions
Casht : amount invested in cash in period t, t = 1 2 3
max 1:06Cash3 + 1:00B + 1:75D + 1:40E
s:t: A + C + D + Cash1 1
Cash2 + B 0:3A + 1:1C + 1:06Cash1
Cash3 + 1:0E 1:0A + 0:3B + 1:06Cash2
A 0:5 C 0:5 E 0:75
A : : : E 0
5
Slide 17
Solution: A = 0:5M , B = 0, C = 0, D = 0:5M , E = 0:659M , Cash1 = 0,
Cash2 = :15M , Cash3 = 0 Objective: 1:7976M
13 Manufacturing
13.1 Data
Slide 18
n products, m raw materials
cj : prot of product j
bi : available units of material i.
aij : # units of material i product j needs in order to be produced.
13.2 Formulation
13.2.1 Decision variables
Slide 19
xj = amount of product j produced.
n
X
max
cj xj
j =1
s:t: a11x1 + + a1nxn b1
..
.
am1 x1 + + amn xn bm
xj 0 j = 1 : : :n
14 Capacity Expansion
14.1 Data and Constraints
Dt : forecasted demand for electricity at year t
Et : existing capacity (in oil) available at t
ct : cost to produce 1MW using coal capacity
nt : cost to produce 1MW using nuclear capacity
No more than 20% nuclear
Coal plants last 20 years
Nuclear plants last 15 years
6
Slide 20
14.2 Decision Variables
Slide 21
14.3 Formulation
Slide 22
xt : amount of coal capacity brought on line in year t.
yt : amount of nuclear capacity brought on line in year t.
wt : total coal capacity in year t.
zt : total nuclear capacity in year t.
min
T
P
ct xt + ntyt
t=1
Pt
wt =
xs s=max(0t;19)
Pt
zt =
ys s=max(0t;14)
wt + zt + Et Dt
zt 0:2(wt + zt + Et)
xt yt wt zt 0:
s:t:
t = 1 : : :T
t = 1 : : :T
15 Scheduling
15.1 Decision variables
Slide 23
Hospital wants to make weekly nightshift for its nurses
Dj demand for nurses, j = 1 : : : 7
Every nurse works 5 days in a row
Goal: hire minimum number of nurses
Decision Variables
xj : # nurses starting their week on day j
15.2 Formulation
P7 x
min
j
j =1
s:t: x1+
x1+
x1+
x1+
x1+
xj 0
Slide 24
x2
x2 +
x2 +
x2 +
x2 +
x3
x3 +
x3 +
x3 +
x3 +
x4 + x5 + x6 +
x5 + x6 +
x6 +
x4 +
x4 + x5
x4 + x5 + x6
x4 + x5 + x6 +
7
x7
x7
x7
x7
x7 d1
d2
d3
d4
d5
d6
d7
16 Revenue Management
16.1 The industry
Deregulation in 1978
Prior to Deregulation
{ Carriers only allowed to y certain routes. Hence airlines such as
Northwest, Eastern, Southwest, etc.
{ Fares determined by Civil Aeronautics Board (CAB) based on mileage
and other costs (CAB no longer exists)
Slide 25
Post Deregulation
anyone can y, anywhere
fares determined by carrier (and the market)
Slide 26
17 Revenue Management
17.1 Economics
Huge sunk and xed costs
Very low variable costs per passenger ($10/passenger or less)
Strong economically competitive environment
Near-perfect information and negligible cost of information
Highly perishable inventory
Result: Multiple fares
Slide 27
18 Revenue Management
18.1 Data
n origins, n destinations
1 hub
2 classes (for simplicity), Q-class, Y-class
Revenues rijQ rijY
Capacities: Ci0, i = 1 : : :n C0j , j = 1 : : :n
Expected demands: DijQ , DijY
8
Slide 28
18.2 LO Formulation
18.2.1 Decision Variables
Slide 29
: # of Q-class customers we accept from i to j
: # of Y-class customers we accept from i to j
�
Qij
�
Yij
maximize
subject to
X
+r
Q
Y
ij Yij
rij Qij
X
ij
n
�0
(Q + Y ) � C 0
ij
ij
i
X
j
n
�0
(Q + Y ) � C0
ij
i
0�Q
ij
ij
Q
� Dij
j
0�Y
ij
Y
� Dij
19 Revenue Management
19.1 Importance
Robert Crandall, former CEO of American Airlines:
Slide 30
We estimate that RM has generated $1.4 billion in incremental revenue for
American Airlines in the last three years alone. This is not a one-time benet.
We expect RM to generate at least $500 million annually for the foreseeable
future. As we continue to invest in the enhancement of DINAMO we expect to
capture an even larger revenue premium.
20 Messages
20.1 How to formulate?
1. Dene your decision variables clearly.
2. Write constraints and objective function.
3. No systematic method available.
What is a good LO formulation?
A formulation with a small number of variables and constraints, and the matrix
A is sparse.
9
Slide 31
21 Nonlinear Optimization
21.1 The general problem
Slide 32
min f(x1 : : : xn)
s:t: g1(x1 : : : xn) 0
...
gm (x1 : : : xn) 0:
22 Convex functions
f : S ;! R
For all x1 x2 2 S
f(x1 + (1 ; )x2 ) f(x1 ) + (1 ; )f(x2 )
Slide 33
f(x) concave if ;f(x) convex.
23 On the power of LO
23.1 LO formulation
min f(x) = maxk dk x + ck
s:t: Ax b
min z
s:t: Ax b
dk x + ck z 8 k
0
Slide 34
0
24 On the power of LO
24.1 Problems with
j:j
min
s:t:
Idea: jxj = maxfx ;xg
min
s:t:
P c jx j
j j
Ax b
Pc z
j j
Ax b
xj zj
;xj zj
Message: Minimizing Piecewise linear convex function can be modelled by LO
10
Slide 35
MIT OpenCourseWare
http://ocw.mit.edu
15.093J / 6.255J Optimization Methods
Fall 2009
For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.
-
1