Outline The role of information What is information? Different types of information Controlling...

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



Consider how we normally solve big optimization problems:

Monday Tuesday Wednesday



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



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!


qD

qD

qD

qD

qDqD

qD

qD

qD

qD qD

qD




qD

Control class (move plane)

Decision horizon (e.g. 8 hours)

Att

ribu

te s

ubs

pacePlanner A

q=



qD

3qD

The forward reachable set. qM

2qD1qD


qD

qD

qD

qD

qDqD

qD

qD

qD

qD qD

qD

“Harry”




The information wall:

Data errors

“The wall”

Info

rmat

ion

cos

t



How do we optimize … Harry?



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



What is information?

What is data?

… knowledge?

… information?

Advertisement from Business Week



Data:

001100111110011100001010011100001000010010111101010000111010111110011100001010011100001000010010111101010000111010011011110101000011101001100111110011100001010011100001000010010

1111010111110011100001010000111010011001001110000100001001001011111001110011011001001110001010000111010110001000010010

00110011111001110000101001110000100001001011110101000011101001100111110011100001010011100001000010010111101010000111010111110011100001010011100001000010010111101010000111010011011110101000011101001100111110011100001010011100001000010010

1111010111110011100001010000111010011001001110000100001001001011111001110011011001001110001010000111010110001000010010

00110011111001110000101001110000100001001011110101000011101

Bits and bytes



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?



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



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



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



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




e1

e4

e7e8

e5

e3e6

e1

e2 e2a =

hours

status

type

location

E



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



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:



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



What is a…

decision?



t txExogenous

inputEndogenous

input (decisions)



Systems evolve through a cycle of exogenous and endogenous information

Time

1 2 3 4 5 6

1x 2x 3x 4x 5x 6x



Both kinds of information evolve over time:

forecast"A " ,...,...,, 21 t

plan"A " ,...,...,, 21 txxxx

A plan is a forecast of a decision.



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



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!


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



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?



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



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



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



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



Different types of information

What types of information are there?



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 ��



How do we increase ?

» We start with

qIqq KI

We canexpand the

scope...



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).




» We start with

qIqq KI

We canexpand the

scope...

Or we can expand the

time horizon, but future

events are not inqK




» 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



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


T0 1TimeE

volu

tio

n o

f S

imu

lati

on

TS

0 E0 E

0 E

E0

E0

E0

0



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




» We can do planning:

• A plan is a forecast of a decision.

qI

, , pq q q qx I K



Types of planning:

» Plans

» Patterns

» Policies


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:



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!

Yellow Freight System



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.



Historical:

The flows are not the same, but they have the same pattern.

Forecasted



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



Consider how we normally solve big optimization problems:




Real problems are decomposed over space . . .




… and time.




We use approximations of subproblems to model interactions:




… and then approximate the problem we just solved...




… so other people can understand how their decisions impact us!




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

qD

qD

qD

qD

qDqD

qD

qD

qD

qDqD

qD


R0

R1

R2

D0D2

D0

D2

t = 0 t = 1 t = 2

D0

D1

R0

R1

R2

R0

R1

R2

R0

R1

R2



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.



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


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)



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



Controlling information

How do I control an operation?

PricesGoalsIncentives

DataCommunicationsForecastingOptimization

LocomotivesTrainsCrews



The information optimization problem:

qeq

tqq

teq

e

ea

qq

qe

ea

tq

qq

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

,',

,'



The information flow problem:

e1

e4

e7e8

e5

e3e6

e1

e2

e1

e4

e7e8

e5

e3e6

e1

e2

q

q’

qE

'qE'eq



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



The information optimization problem is now:

IX FF max

Subject to system dynamics.

Cost of moving information

Cost of moving resources


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



moving flow:» It may have a fixed charge:

e

eq

qIqc

Cost of constructing databases, screens,communication links



moving flow:» It may be convex:

e

eq

qIqc



moving flow:» … or concave:

e

eq

qIqc


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



But there is one way in which the flow of information is very different from the flow of resources...

Outline The role of information What is information? Different types of information Controlling...

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