Outline
The role of information What is information? Different types of information Controlling information
September, 1999 © 1999 Warren B. Powell Slide 2
How do we optimize systems like this?
The role of information
September, 1999 © 1999 Warren B. Powell Slide 3
The role of information
Consider how we normally solve big optimization problems:
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 4
The role of information
Normally, we would formulate a big optimization problem:
0
:subject to
min
11
0
kt
tk
kt
kt
kt
kt
kt
kt
k
T
t
ktt
x
x
ux
bxBxA
xc
September, 1999 © 1999 Warren B. Powell Slide 5
The role of information
Normally, we would formulate a big optimization problem:
0
:subject to
min
11
0
kt
tk
kt
kt
kt
kt
kt
kt
k
T
t
ktt
x
x
ux
bxBxA
xc
)1,0(ktxInteger!
September, 1999 © 1999 Warren B. Powell Slide 6
qD
qD
qD
qD
qDqD
qD
qD
qD
qD qD
qD
The role of information
September, 1999 © 1999 Warren B. Powell Slide 7
The role of information
qD
Control class (move plane)
Decision horizon (e.g. 8 hours)
Att
ribu
te s
ubs
pacePlanner A
q=
September, 1999 © 1999 Warren B. Powell Slide 8
The role of information
qD
3qD
The forward reachable set. qM
2qD1qD
September, 1999 © 1999 Warren B. Powell Slide 9
qD
qD
qD
qD
qDqD
qD
qD
qD
qD qD
qD
“Harry”
The role of information
September, 1999 © 1999 Warren B. Powell Slide 10
The role of information
The information wall:
Data errors
“The wall”
Info
rmat
ion
cos
t
September, 1999 © 1999 Warren B. Powell Slide 11
The role of information
How do we optimize … Harry?
September, 1999 © 1999 Warren B. Powell Slide 12
The role of information
How do we make decision makers smarter?
You have to raise their “IQ”!
You mean hiresmarter people?
No, we have togive them more
information
What’s that?
Outline
The role of information What is information? Different types of information Controlling information
September, 1999 © 1999 Warren B. Powell Slide 14
What is information?
What is data?
… knowledge?
… information?
Advertisement from Business Week
September, 1999 © 1999 Warren B. Powell Slide 15
What is information?
Data:
001100111110011100001010011100001000010010111101010000111010111110011100001010011100001000010010111101010000111010011011110101000011101001100111110011100001010011100001000010010
1111010111110011100001010000111010011001001110000100001001001011111001110011011001001110001010000111010110001000010010
00110011111001110000101001110000100001001011110101000011101001100111110011100001010011100001000010010111101010000111010111110011100001010011100001000010010111101010000111010011011110101000011101001100111110011100001010011100001000010010
1111010111110011100001010000111010011001001110000100001001001011111001110011011001001110001010000111010110001000010010
00110011111001110000101001110000100001001011110101000011101
Bits and bytes
September, 1999 © 1999 Warren B. Powell Slide 16
What is information?
Knowledge» Knowledge comes in two forms:
• Exogeneously derived data• Relationships which allow us to use knowledge to make
inferences about data elements that are not yet known to our system.
» Can we have knowledge about something that we do not know perfectly?
September, 1999 © 1999 Warren B. Powell Slide 17
What is information?
Example» Commodities prices
» Assume that we derive the functional relationship:
Old prices 0
Current price 0
Future prices 0t
t
P t
t
5 4 3 2 1 0 1
UnknownKnowledge
, , , , , ,P P P P P P P
1 1 2 2 3 3 4 4 5 5t t t t t t t t t t t tP a P a P a P a P a P
September, 1999 © 1999 Warren B. Powell Slide 18
What is information?
Without the “knowledge” of this functional relationship, our “knowledge” of the future price would be captured by:
With the knowledge of the functional relationship:
Lik
elih
ood
Price
Price
Lik
elih
ood
September, 1999 © 1999 Warren B. Powell Slide 19
What is information?
The field of information theory captures the information content about a piece of data using entropy:
Let:
( ) Probability of outcome
Vector of probabilities
Entropy( ) ( ) ln ( )
ˆIf we have perfect knowledge, then for some outcome , we have
ˆ( ) 1, and
ˆ ˆEntropy( ) - ( ) ln ( ) 1 ln(1) 0.
So
x
p x x
p x p x
x
p x
p x p x
p
p
p
, Entropy( )>0 is a measure of the imprecision with which we know something.p
September, 1999 © 1999 Warren B. Powell Slide 20
What is information?
Information classes:
, ,
,
,
,
Set of information classes
,
Set of information classes with static attributes
Set of information classes with dynamic attributes
Set of resource classes
Inform
I
I s I d
I s
I d
R I d
(c)
C
C C
C
C
C C
E ation elements in class c IC
September, 1999 © 1999 Warren B. Powell Slide 21
What is information?
What is information?
e1
e4
e7e8
e5
e3e6
e1
e2 e2a =
hours
status
type
location
E
September, 1999 © 1999 Warren B. Powell Slide 22
What is information?
The “knowledge base”» This is what we know at time t:
» When we have multiple agents, we have to represent what each agent knows:
|t e tK a e E
September, 1999 © 1999 Warren B. Powell Slide 23
What is information?
e1
e4
e7e8
e5
e3e6
e1
e2
1Ee1
e4
e7e8
e5
e3e6
e1
e2
2E
e1
e4
e7e8
e5
e3e6
e1
e2
3Ee1
e4
e7e8
e5
e3e6
e1
e2
4E
We have multiple sets of information:
September, 1999 © 1999 Warren B. Powell Slide 24
What is information?
The “knowledge base”
Let:
The information elements that decision maker
has at time .
The knowledge base of decision maker is then:
|
qt
qt e qt
q
t
q
K a e
E
E
September, 1999 © 1999 Warren B. Powell Slide 26
What is information?
t txExogenous
inputEndogenous
input (decisions)
September, 1999 © 1999 Warren B. Powell Slide 27
What is information?
Systems evolve through a cycle of exogenous and endogenous information
Time
1 2 3 4 5 6
1x 2x 3x 4x 5x 6x
September, 1999 © 1999 Warren B. Powell Slide 28
What is information?
Both kinds of information evolve over time:
forecast"A " ,...,...,, 21 t
plan"A " ,...,...,, 21 txxxx
A plan is a forecast of a decision.
September, 1999 © 1999 Warren B. Powell Slide 29
What is information?
We can’t always predict the future...
be the set of possible events in the future.
is the new information that will become available in the future.
= ( )1
23 4 5
6 7
September, 1999 © 1999 Warren B. Powell Slide 30
What is information?
P.2) System dynamics:» Evolution due to exogenous information processes
Time
1 2 3 4 5 6 7
tF = The “history” of the information process
"filtration a" :assumption Standard 1tt FF
… But this assumes that we never forget anything!
September, 1999 © 1999 Warren B. Powell Slide 31
What is information? Better notation:
."preservingn informatio" is say that then we, :If
:where
),(
:function update""an using updated is This
tat time base" knowledge"Our
1
Kt
t
ttK
t
t
U
U
t
t
FK
FK
KK
K
Our current knowledge base
New information
Our new knowledge base
September, 1999 © 1999 Warren B. Powell Slide 32
What is information?
When exogenous information arrives, the update is usually pretty simple:
t
'
'
'
fuel
status
location
fuel
emaintenanc
status
type
location
ea
tK
'
'
'
fuel
status
location
emaintenanc
type
1tK
So how does a decision xt change the system?
September, 1999 © 1999 Warren B. Powell Slide 33
What is information?
Our decision function usually looks like:
0xq bq
Aq xq
min cq xqarg)( qq Ix
qI
.subproblemour of IQ"" theis say that can We qI
September, 1999 © 1999 Warren B. Powell Slide 34
What is information?
The modify function:
action) complete torequired (time timeDwell
ons.contributior costs of (Vector)
resource. modified of vector Attribute '
:
n)informatio (exogenous at time base Knowledge
control) s(endogenou dimplemente beingDecision
modified. being resource of vctor Attribute
:
),,'(),,(
c
a
Outputs
tK
d
a
Inputs
caKdaM
t
t
ttt
September, 1999 © 1999 Warren B. Powell Slide 35
What is information?
Discuss: decision functions
A decision function is a mapping from information to decisions:
:
The modify function (or transfer function) is a mapping from
decisions to information:
:
q q q
q
x I x
M x I
q
September, 1999 © 1999 Warren B. Powell Slide 36
What is information?
The information cycle:
( , , ) ( ', , )M a x K a c The modify function uses decisions to create information ...
( )q q qx I x … The decision function turns information into decisions.
Outline
The role of information What is information? Different types of information Controlling information
September, 1999 © 1999 Warren B. Powell Slide 38
Different types of information
What types of information are there?
September, 1999 © 1999 Warren B. Powell Slide 39
Different types of information
There are four classes of information:
» What we “know”:
» Forecasts of exogenous information:
» Future plans:
» “Values”: the impact of our decisions on other problem components.
qK
qW
px
qMV ��������������
September, 1999 © 1999 Warren B. Powell Slide 40
Different types of information
How do we increase ?
» We start with
qIqq KI
We canexpand the
scope...
September, 1999 © 1999 Warren B. Powell Slide 41
Different types of information
Using knowledge:» Decision functions that only use knowledge are called
myopic.
» Over the last two decades, we have seen a dramatic increase in our ability to collect, transmit and store “information” (knowledge).
September, 1999 © 1999 Warren B. Powell Slide 42
Different types of information
How do we increase ?
» We start with
qIqq KI
We canexpand the
scope...
Or we can expand the
time horizon, but future
events are not inqK
September, 1999 © 1999 Warren B. Powell Slide 43
Different types of information
How do we increase ?
» We can add forecasting:
» What do we forecast?• Resources
– Arrivals (e.g. customer demands)– Departures (cancellations, equipment breakdowns, people
quitting).• Process parameters (the Modify function)
– Prices/costs– Times– Engineering parameters
qI
qqq ,KI
September, 1999 © 1999 Warren B. Powell Slide 44
Different types of information
Types of forecasts:
Number of units required
Lik
elih
ood
of
occu
rren
ce
50 10
100% Point
Number of units required
Lik
elih
ood
of
occu
rren
ce50 10
100% Distributional
September, 1999 © 1999 Warren B. Powell Slide 45
T0 1TimeE
volu
tio
n o
f S
imu
lati
on
TS
0 E0 E
0 E
E0
E0
E0
0
Different types of information
September, 1999 © 1999 Warren B. Powell Slide 46
Network
80%
82%
84%
86%
88%
90%
92%
94%
96%
98%
100%
0 10 20 30 40 50 60 70 80 90
Horizon Length (4 hour periods)
Per
cent
age
of O
ptim
al P
oste
rior
Bou
ndDifferent types of information
Planning horizon
P
erce
nt
of p
oste
rior
bou
nd
Deterministic, rolling horizon
We need more information!Posterior boundThat’s not very
good
September, 1999 © 1999 Warren B. Powell Slide 47
Different types of information
How do we increase ?
» We can do planning:
• A plan is a forecast of a decision.
qI
, , pq q q qx I K
September, 1999 © 1999 Warren B. Powell Slide 48
Different types of information
Types of planning:
» Plans
» Patterns
» Policies
September, 1999 © 1999 Warren B. Powell Slide 49
Different types of information The second form of head knowledge is patterns -
standard actions given the state of the system.
Movements of sleeper teams for a trucking company:
September, 1999 © 1999 Warren B. Powell Slide 50
Different types of information
Concept: pattern matching» Old modeling approach: Bottom up modeling
0, :Subject to
minarg*
xbAx
cxx
Objectives
“Physics”
“Behavior”
To get the right “behavior” we have to specify the right costs and the right constraints.
If you don’t like the behavior, you have to fix the data!
September, 1999 © 1999 Warren B. Powell Slide 52
Different types of information
Concept: pattern matching» New modeling approach: Top down, bottom up
modeling
||)(||minarg* pxxGcxx
Cost function
“Behavior”
Aggregationfunction
Pattern databasefrom history
Scaling parameter
The difference between the model solution and historical patterns.
September, 1999 © 1999 Warren B. Powell Slide 53
Different types of information
Historical:
The flows are not the same, but they have the same pattern.
Forecasted
September, 1999 © 1999 Warren B. Powell Slide 54
Different types of information
So far, we have three types of decision functions:» Myopic
» With forecasts:
» With forecasts and plans:
» Now we have to bring in our impact on others.
0 0
0 0 0 0
min
subject to: , 0
c x
A x b x
0 01
1 1
min
subject to: , 0
T
t tt
t t t t t t
c x c x
A x B x b x
0 01
1 1
min ( )
subject to: , 0
Tp
t tt
t t t t t t
c x c x G x x
A x B x b x
September, 1999 © 1999 Warren B. Powell Slide 55
Different types of information
Consider how we normally solve big optimization problems:
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 56
Different types of information
Real problems are decomposed over space . . .
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 57
Different types of information
… and time.
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 58
Different types of information
We use approximations of subproblems to model interactions:
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 59
Different types of information
… and then approximate the problem we just solved...
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 60
Different types of information
… so other people can understand how their decisions impact us!
Monday Tuesday Wednesday
September, 1999 © 1999 Warren B. Powell Slide 61
Different types of information
Using value function approximations, we find ourselves solving problems that look like:
' ' ''
ˆminq
q x q q qq qq qqq M
V c x V R x
Forward reachable set
September, 1999 © 1999 Warren B. Powell Slide 65
R0
R1
R2
D0D2
D0
D2
t = 0 t = 1 t = 2
D0
D1
R0
R1
R2
R0
R1
R2
R0
R1
R2
Different types of information
September, 1999 © 1999 Warren B. Powell Slide 66
R0
R1
R2
D0D2
D0
D2
t = 0 t = 1 t = 2
D0
D1
R0
R1
R2
R0
R1
R2
R0
R1
R2
Different types of information
September, 1999 © 1999 Warren B. Powell Slide 67
R0
R1
R2
D0D2
D0
D2
t = 0 t = 1 t = 2
D0
D1
R0
R1
R2
R0
R1
R2
R0
R1
R2
Different types of information
September, 1999 © 1999 Warren B. Powell Slide 68
80%
82%
84%
86%
88%
90%
92%
94%
96%
98%
100%
0 10 20 30 40 50 60 70 80 90
Horizon Length (4 hour periods)
Per
cent
age
of O
ptim
al P
oste
rior
Bou
ndDifferent types of information
Planning horizon
P
erce
nt
of p
oste
rior
bou
nd
Deterministic, rolling horizon
Adaptive dynamic programming
Can we do better?Posterior bound
The value of addingV’s to the information set.
September, 1999 © 1999 Warren B. Powell Slide 69
Different types of information
80
85
90
95
1001 4 7
10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 61 64 67 70 73 76 79 82 85 88 91 94 97
100
Iteration No.
% o
f O
bje
ctiv
e U
pp
erb
ou
nd
Agg_PWLinear_1
Agg_PWLinear_2
Agg_PWLinear_3
DisAgg_Linear
DisAgg_PWLinear
Decomp_Location
LP relaxation
Integer solution obtained using value function approximation
September, 1999 © 1999 Warren B. Powell Slide 70
Different types of information There are four classes of information:
» Knowledge
» Forecast of exogenous events
» Plans (forecast of future decisions)
» Values (impact of decisions on other subproblems)
qK
q
pqx
qMV
qM
(data and relationships)
September, 1999 © 1999 Warren B. Powell Slide 71
Different types of information
The information set shapes the decision function:» Myopic decision rules
» Rolling horizon procedures
» Proximal point algorithms
» Dynamic programming
q qI K
,q q qI K
, , pq q q qI K x
, , ,q
pq q q qI K x V
M ,
qq qI K V
Mor
Outline
The role of information What is information? Different types of information Controlling information
September, 1999 © 1999 Warren B. Powell Slide 73
Controlling information
How do I control an operation?
PricesGoalsIncentives
DataCommunicationsForecastingOptimization
LocomotivesTrainsCrews
September, 1999 © 1999 Warren B. Powell Slide 74
Controlling information
The information optimization problem:
qeq
tqq
teq
e
ea
qe
ea
tq
q
tq,
EI
E
E
,
:isset n informatioour given So
' to from flows"n informatio" ofVector
Otherwise0
tat time ' subproblem sent to is element If1
element of attributes Data
at time subproblemfor elementsn Informatio
,',
,'
September, 1999 © 1999 Warren B. Powell Slide 75
Controlling information
The information flow problem:
e1
e4
e7e8
e5
e3e6
e1
e2
e1
e4
e7e8
e5
e3e6
e1
e2
q
q’
qE
'qE'eq
September, 1999 © 1999 Warren B. Powell Slide 77
Controlling information
The information cost functions:
q
qqxq
X xcF qI
:resources moving ofcost The
n vector informatio ofcost The
:ninformatio providing ofcost in the add tohave weNow
IF
September, 1999 © 1999 Warren B. Powell Slide 78
Controlling information
The information optimization problem is now:
IX FF max
Subject to system dynamics.
Cost of moving information
Cost of moving resources
September, 1999 © 1999 Warren B. Powell Slide 79
Controlling information In many ways, the economics of moving information is very similar to
moving flow:» The function may be linear:
e
eq
qIqc
Phone calls, communication links
September, 1999 © 1999 Warren B. Powell Slide 80
Controlling information In many ways, the economics of moving information is very similar to
moving flow:» It may have a fixed charge:
e
eq
qIqc
Cost of constructing databases, screens,communication links
September, 1999 © 1999 Warren B. Powell Slide 81
Controlling information In many ways, the economics of moving information is very similar to
moving flow:» It may be convex:
e
eq
qIqc
September, 1999 © 1999 Warren B. Powell Slide 82
Controlling information In many ways, the economics of moving information is very similar to
moving flow:» … or concave:
e
eq
qIqc
September, 1999 © 1999 Warren B. Powell Slide 83
Controlling information In many ways, the economics of moving information is very similar to moving flow:
» It may be separable:
» or highly nonseparable. There are joint economies of production, just as in discrete parts manufacturing.
q
qIq
I cF
September, 1999 © 1999 Warren B. Powell Slide 84
Controlling information
But there is one way in which the flow of information is very different from the flow of resources...