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In**format**ion, Noise and Lies:

The Evolving Discovery of Misin**format**ion

In Rail Transportation

Tenth International Conference on Stochastic Programming

**University** of Arizona

October, 2004

© 2003 Warren B. Powell, **Princeton** **University**

Warren Powell

**CASTLE** **Lab**oratory

**Princeton** **University**

http://www.castlelab.princeton.edu

© 2004 Warren B. Powell

Slide 1

© 2004 Warren B. Powell

Slide 2

Norfolk Southern

© 2004 Warren B. Powell

Slide 3

Customers

Option 1: Send directly to customers

Regional depots

Option 1: Send directly to customers

Option 2: Send to regional depots

Classification yards

Option 1: Send directly to customers

Option 2: Send to regional depots

Option 3: Send to classification yards

© 2004 Warren B. Powell

Slide 7

The importance of uncertainty

Available resource:

© 2004 Warren B. Powell

Possible customer

demand

Possible customer

demand

Slide 8

The importance of uncertainty

Why do stochastic models work?

$100

» The newsvendor effect:

0

Revenue from an

additional boxcar.

5

© 2004 Warren B. Powell

Point

forecast

Demand

Cost of an

additional

car.

Slide 9

The importance of uncertainty

Why do stochastic models work?

» The newsvendor effect:

0

Revenue from an

additional boxcar.

4

7

8

© 2004 Warren B. Powell

Demand

Cost of an

additional

car.

Slide 10

The car cycle

Empty movement

Load car

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Loaded movement

Unload car

Slide 11

The planning process

“This week” “Next week”

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 12

The planning process

Phone calls Activities

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 13

The planning process

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 14

A myopic model

Cars

© 2004 Warren B. Powell

Orders

Slide 15

If a customer releases a car in the

beginning of the week, you have

to send it somewhere right away.

We can’t hold cars, wait for an

order, and then send them.

The planning process

?

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 17

Forecasting

Forecast Actual

Actual vs. predicted car demands

© 2004 Warren B. Powell

Slide 18

Forecasting

Actual vs. predicted car supplies

© 2004 Warren B. Powell

Slide 19

Two-stage problems

Two-stage resource allocation under uncertainty

© 2004 Warren B. Powell

Slide 20

Two-stage problems

If we model different regions as separable….

© 2004 Warren B. Powell

Slide 21

Optimization frameworks

We obtain piecewise linear recourse functions for each

regions.

© 2004 Warren B. Powell

Slide 22

Optimization frameworks

The function is piecewise linear on the integers.

Profits

We approximate the value of cars

in the future using a separable

approximation.

0 1 2 3 4 5

Number of vehicles at a location

© 2004 Warren B. Powell

Slide 23

Optimization frameworks

Using standard network trans**format**ion:

Each link captures the marginal

reward of an additional car.

© 2004 Warren B. Powell

Slide 24

Two-stage problems

© 2004 Warren B. Powell

Slide 25

Two-stage problems

© 2004 Warren B. Powell

Slide 26

Literature review

Piecewise linear bounds:

1987: Wallace, S.W.,” A Piecewise Linear Upper Bound on the Network

Recourse Function”

1988: Birge, J.R. and S. W. Wallace, “A Separable Piecewise Linear

Upper Bound for Stochastic Linear Programs”

Static piecewise linear approximations:

1987: Powell, W.B., “An Operational Planning Model for the Dynamic

Vehicle Allocation Problem with Uncertain Demands”

1988: Powell, W.B., “A Comparative Review of Alternative Algorithms

for the Dynamic Vehicle Allocation Problem”

© 2004 Warren B. Powell

Slide 27

Literature review

Static piecewise linear approximations (cont’d)

1990: Frantzeskakis, L.F. ,W. B. Powell, “A Successive Linear

Approximation Procedure for Stochastic Dynamic Vehicle Allocation

Problems”

1992, Powell, W.B. and L.F. Frantzeskakis, “Restricted Recourse

Strategies for Stochastic, Dynamic Networks”

1994: Powell, W.B. and R. Cheung, “Stochastic Programs over Trees with

Random Arc Capacities”

1996: Cheung, R. and W.B. Powell, “An Algorithm for Multistage

Dynamic Networks with Random Arc Capacities, with an Application

to Dynamic Fleet Management”

© 2004 Warren B. Powell

Slide 28

Literature review

Adaptive linear approximations

1997: Powell, W.B. and T. Carvalho, “Dynamic Control of

Multicommodity Fleet Management Problems”

1998: Powell, W.B. and T. Carvalho, “Dynamic Control of Logistics

Queueing Network for Large-Scale Fleet Management”

2000: Powell, W.B. and T. Carvalho, “ A Multiplier Adjustment Method

for Dynamic Resource Allocation Problems

© 2004 Warren B. Powell

Slide 29

Literature review

Adaptive nonlinear/piecewise linear approximations

2000: Cheung, R. and W.B. Powell, “SHAPE: A Stochastic Hybrid

Approximation Procedure for Two-Stage Stochastic Programs”

2001: Godfrey, G. and W. B. Powell, “An Adaptive, Distribution-Free

Approximation for the Newsvendor Problem with Censored Demands,

with Applications to Inventory and Distribution Problems”

2002: Godfrey, G. and W.B. Powell, “An Adaptive, Dynamic Programming

Algorithm for Stochastic Resource Allocation Problems I and II”

2003: Topaloglu, H. and W.B. Powell,“An Algorithm for Approximating

Piecewise Linear Concave Functions from Sample Gradients”

(to appear): Topaloglu, H. and W.B. Powell, “Dynamic Programming

Approximations for Stochastic, Time-Staged Integer Multicommodity

Flow Problems”

(to appear): Powell, W. B., A. Ruszczynski and H. Topaloglu, “Learning

Algorithms for Separable Approximations of Discrete Stochastic

Optimization Problems”

© 2004 Warren B. Powell

Slide 30

Two-stage problems

© 2004 Warren B. Powell

Slide 31

Two-stage problems

We estimate the functions by sampling from our distributions.

Marginal value:

n

vˆ ( ω )

1

n

vˆ ( ω )

2

n

vˆ ( ω )

3

n

vˆ ( ω )

4

n

vˆ ( ω )

5

n

R1 →

n

R2 →

n

R3 →

n

R4 →

n

R5 →

© 2004 Warren B. Powell

n

D ( ω )

1

n

D ( ω )

2

n

D3 ( ω )

n

D ( ω )

C

Slide 32

Two-stage problems

Profits

v −

v +

⎫

⎬

⎭

© 2004 Warren B. Powell

Duals

Number of vehicles

Slide 33

Two-stage problems

Left and right derivatives are used to build up a nonlinear

approximation of the subproblem.

V ˆ k

it (R1t )

k

R1t © 2004 Warren B. Powell

R 1t

Slide 34

Two-stage problems

Left and right derivatives are used to build up a nonlinear

approximation of the subproblem.

Vi ( Ri)

Left derivative

v −

k

R1t Right derivative

v +

© 2004 Warren B. Powell

R 1t

Slide 35

Two-stage problems

Each iteration adds new segments, as well as refining old ones.

ˆ k + 1

it ( it )

V R

© 2004 Warren B. Powell

v −

k+1

R1t v +

R 1t

Slide 36

Two-stage problems

Approximate value function

Functional Value, f(s) = ln(1+s)

2.5

2.0

1.5

1.0

0.5

0.0

0 1 2 3 4 5 6 7 8 9 10

Number Variable of Value, resources s

© 2004 Warren B. Powell

Slide 37

Exact

1 Iter

2 Iter

5 Iter

10 Iter

15 Iter

20 Iter

Two-stage problems

Two-stage stochastic programming.

© 2004 Warren B. Powell

Slide 38

Two-stage problems

Exact solutions using Benders:

“L-Shaped” decomposition

(Van Slyke and Wets)

Stochastic decomposition

(Higle and Sen)

CUPPS

(Chen and Powell)

© 2004 Warren B. Powell

V0

V0

V0

x0

x0

x0

Slide 39

Two-stage problems

120,000.00

100,000.00

80,000.00

60,000.00

40,000.00

20,000.00

0.00

Variations on Bender’s decomposition

25 50 100 250 500 1000 2500 5000

Iterations

© 2004 Warren B. Powell

Slide 40

Two-stage problems

120,000.00

100,000.00

80,000.00

60,000.00

40,000.00

20,000.00

0.00

Separable approximation

Variations on Bender’s decomposition

25 50 100 250 500 1000 2500 5000

Iterations

© 2004 Warren B. Powell

Slide 41

Two-stage problems

120,000.00

100,000.00

80,000.00

60,000.00

40,000.00

20,000.00

0.00

Separable approximation

Variations on Bender’s decomposition

25 50 100 250 500 1000 2500 5000

Iterations

© 2004 Warren B. Powell

Deterministic

approximation

Slide 42

We have different box car

types, and some customers are

flexible about which types we

send them.

© 2004 Warren B. Powell

Slide 43

Two-stage problems

Two-stage resource allocation under uncertainty

© 2004 Warren B. Powell

Slide 44

Two-stage problems

We normally think of sending cars to locations…

The set of locations

© 2004 Warren B. Powell

Slide 45

Two-stage problems

© 2004 Warren B. Powell

The set of car types.

Slide 46

Two-stage problems

© 2004 Warren B. Powell

The set of locations

Slide 47

Two-stage problems

© 2004 Warren B. Powell

Geographical substitution

Slide 48

Two-stage problems

© 2004 Warren B. Powell

Functional substitution

Slide 49

Two-stage problems

… actually, we “create” cars with different attributes.

The set of attributes

© 2004 Warren B. Powell

Slide 50

Multiattribute resources

A

Assets can have a number of attributes:

≈

a =

[ Location]

200

⎡ Location ⎤

⎢

Boxcar type

⎥

⎣ ⎦

2000

© 2004 Warren B. Powell

Slide 51

A car arriving on Thursday can

serve an order that was

available on Tuesday. We

should penalize this, but we

can do it.

© 2004 Warren B. Powell

Slide 52

Empty movement

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Loaded movement

Slide 53

Empty movement

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Loaded movement

Slide 54

Multiperiod travel times

Available:

t t+1 t+2 t+3

© 2004 Warren B. Powell

Slide 55

Multiperiod travel times

Available:

t t+1 t+2 t+3

© 2004 Warren B. Powell

Slide 56

Multiperiod travel times

© 2004 Warren B. Powell

The set of locations

Slide 57

Multiperiod travel times

© 2004 Warren B. Powell

Available now

Available t+1

Available t+2

Available t+2

Slide 58

Multiattribute resources

A

If the longest trip is seven days…

≈

a =

[ Location]

⎡ Location ⎤⎡Location

⎤

⎢

Boxcar type

⎥⎢

⎣ ⎦ Boxcar type

⎥

⎢ ⎥

⎢⎣ Time to dest. ⎥⎦

200 2000

14,000

© 2004 Warren B. Powell

Slide 59

But the travel times are not

that long! We can load and

unload cars multiple times

within two weeks.

Empty movement

Load car

Loaded movement

Unload car

M T W Th F Sa Su M T W Th F Sa Su

?

M T W Th F Sa Su M T W Th F Sa Su

Multistage problems

t

© 2004 Warren B. Powell

Slide 63

Multistage problems

© 2004 Warren B. Powell

Slide 64

Multistage problems

© 2004 Warren B. Powell

Slide 65

© 2004 Warren B. Powell

Slide 66

© 2004 Warren B. Powell

Slide 67

© 2004 Warren B. Powell

Slide 68

© 2004 Warren B. Powell

Slide 69

Multiattribute resources

A

For multiperiod problems, we have to capture the

time period:

≈

a =

[ Location]

⎡ Location ⎤⎡Location

⎤

⎢

Boxcar type

⎥⎢

⎣ ⎦ Boxcar type

⎥

⎢ ⎥

⎢⎣ Time to dest. ⎥⎦

200 2000

14,000

© 2004 Warren B. Powell

Slide 70

Multiattribute resources

A

For multiperiod problems, we have to capture the

time period:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥

⎣ ⎦

⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥

⎢ ⎥

⎣Time to dest. ⎦

4,000 40,000 280,000

© 2004 Warren B. Powell

Slide 71

That’s great! Let’s see how it

works on our past history!

History!!?? Watch

out! History is

deterministic!

A single commodity flow problem

A pure (deterministic) network:

© 2004 Warren B. Powell

Slide 73

Percentage of Network Optimal Solution

Percent of LP optimal

100%

95%

90%

85%

80%

75%

70%

0 20 40 60 80 100 120 140 160 180 200

Number of Iterations Number of CAVE of iterations Training on Expectation

Multicommodity flow

© 2004 Warren B. Powell

Slide 75

% of Objective Upperbound

100

95

90

85

80

1

4

7

10

13

16

19

The optimal solution

(continuous)

22

25

28

31

34

37

40

43

46

Using separable piecewise linear

approximations (integer)

49

52

55

Iteration No.

58

© 2004 Warren B. Powell

61

64

67

70

73

76

79

82

Agg_PWLinear_1

Agg_PWLinear_2

Agg_PWLinear_3

DisAgg_Linear

DisAgg_PWLinear

Decomp_Location

85

88

91

94

Slide 76

97

100

Wait a minute! When the

shipper books an order, he

doesn’t tell us where it is

going!

Multiattribute resources

Locations

1

2

3

4

5

In**format**ion at time t

Actionable time

t t+1 t+2

?

?

© 2004 Warren B. Powell

t+3 t+4 t+5

?

Slide 78

Multiattribute resources

2

3

Locations 1

4

5

In**format**ion at time t+1

t

Actionable time

t+1 t+2

?

© 2004 Warren B. Powell

t+3 t+4 t+5

?

Slide 79

Multiattribute resources

Locations

1

2

3

4

5

In**format**ion at time t+2

t

Actionable time

t+1 t+2

© 2004 Warren B. Powell

t+3 t+4 t+5

Slide 80

Multiattribute resources

Locations

1

2

3

4

5

In**format**ion at time t+2

t

Actionable time

t+1 t+2

© 2004 Warren B. Powell

t+3 t+4 t+5

Slide 81

You are assuming that the

travel times between points

takes an integer number of

days. This is just not realistic!

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 83

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 84

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 85

M T W Th F Sa Su M T W Th F Sa Su

© 2004 Warren B. Powell

Slide 86

Multistage problems

We had assumed that the time discretization of the

in**format**ion process matched the physical process.

© 2004 Warren B. Powell

Slide 87

Multistage problems

If we want finer time steps, we had to run more

iterations:

© 2004 Warren B. Powell

Slide 88

Multiattribute resources

A

With one day time steps:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥

⎣ ⎦

⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥

⎢ ⎥

⎣Time to dest. ⎦

4,000 40,000 280,000

© 2004 Warren B. Powell

Slide 89

Multiattribute resources

A

With four hour time steps for in**format**ional and

physical processes:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥

⎣ ⎦

⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥

⎢ ⎥

⎣Time to dest. ⎦

24,000 240,000 40,080,000

© 2004 Warren B. Powell

Slide 90

Multiattribute resources

A

With four hour time steps for just physical

processes:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥

⎣ ⎦

⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥

⎢ ⎥

⎣Time to dest. ⎦

4,000 40,000 1,680,000

© 2004 Warren B. Powell

Slide 91

But the cars break down… we

have to model the repair status

so we capture the right number

of usable cars.

Repair status

Acceptable

Minor

repair

© 2004 Warren B. Powell

Major

repair

Slide 93

The flow of in**format**ion

In practice, there are a number of parallel in**format**ion

processes taking place:

Order is made

Empty transit time to shipper becomes known.

Customer inspects car and accepts or rejects.

Customer loads car (we learn the release time)

Destination of order becomes known

© 2004 Warren B. Powell

Time

Slide 94

Multiattribute resources

A

When we add in repair status:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥

⎣ ⎦

⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥

⎢ ⎥

⎣Time to dest. ⎦

4,000 40,000 1,680,000

© 2004 Warren B. Powell

Slide 95

Multiattribute resources

A

When we add in repair status:

≈

a = ⎡ Time ⎤⎡

Time ⎤⎡Time

⎤ ⎡ Time ⎤

⎢

Location

⎥⎢

Location

⎥⎢

⎢ ⎥

Location

⎥ ⎢

⎣ ⎦

Location

⎥

⎢ ⎥ ⎢ ⎥

⎢⎣Boxcar type⎥⎦⎢Boxcar

type ⎥ ⎢ Boxcar type ⎥

⎢ ⎥ ⎢ ⎥

⎣Time to dest. ⎦ ⎢Time to dest. ⎥

⎢

⎣Repair status⎥

⎦

4,000 40,000 1,680,000 5,040,000

© 2004 Warren B. Powell

Slide 96

Two-stage problems

We need to compute duals for all the attributes…

Marginal value:

n

vˆ ( ω )

1

n

vˆ ( ω )

2

n

vˆ ( ω )

3

n

vˆ ( ω )

4

n

vˆ ( ω )

5

n

R1 →

n

R2 →

n

R3 →

n

R4 →

n

R5 →

© 2004 Warren B. Powell

n

D ( ω )

1

n

D ( ω )

2

n

D3 ( ω )

n

D ( ω )

C

Slide 97

Two-stage problems

… even the ones with zero flow.

Marginal value:

n

vˆ ( ω )

1

n

vˆ ( ω )

2

n

vˆ ( ω )

3

n

vˆ ( ω )

4

n

vˆ ( ω )

5

R > 0 →

n

1

R > 0 →

n

2

R = 0 →

n

3

R = 0 →

n

4

R = 0 →

n

5

© 2004 Warren B. Powell

n

D ( ω )

1

n

D ( ω )

2

n

D3 ( ω )

n

D ( ω )

C

Slide 98

Optimization frameworks

⎫

⎪

⎪

⎬

⎪

⎪

⎪⎭

© 2004 Warren B. Powell

Attribute space is now

too large to enumerate.

Slide 99

Optimization frameworks

© 2004 Warren B. Powell

The biggest computational

expense is simply

generating the links out of

the nodes.

Slide 100

Aggregation

TX

v PA =

?

© 2004 Warren B. Powell

vNE

NE region

PA

Slide 101

vˆ( ω)

ˆ( )

v ω

vˆ( ω)

vˆ( ω)

v

=

vˆ + vˆ + vˆ

1

a

1 2 3

3

ˆ( )

v ω

vˆ( ω)

vˆ( ω)

Percent of Optimal

Aggregation

100.00%

98.00%

96.00%

94.00%

92.00%

90.00%

88.00%

86.00%

84.00%

82.00%

80.00%

1 x 1 grid

1 71 141 211 281 351 421 491 561 631 701 771 841 911 981

Number of Iterations

© 2004 Warren B. Powell

2x2 Grid

5x5 Grid

10x10 Grid

20x20 Grid

1x1 Grid

Slide 107

Percent of Optimal

Aggregation

100.00%

98.00%

96.00%

94.00%

92.00%

90.00%

88.00%

86.00%

84.00%

82.00%

80.00%

2 x 2 grid

1 71 141 211 281 351 421 491 561 631 701 771 841 911 981

Number of Iterations

© 2004 Warren B. Powell

5x5 Grid

10x10 Grid

20x20 Grid

1x1 Grid

2x2 Grid

Slide 108

Percent of O ptim al

Aggregation

100.00%

98.00%

96.00%

94.00%

92.00%

90.00%

88.00%

86.00%

84.00%

82.00%

80.00%

5 x 5 grid

1 72 143 214 285 356 427 498 569 640 711 782 853 924 995

Number of Iterations

© 2004 Warren B. Powell

10x10 Grid

20x20 Grid

1x1 Grid

2x2 Grid

5x5 Grid

Slide 109

Percent of O p tim al

Aggregation

100.00%

98.00%

96.00%

94.00%

92.00%

90.00%

88.00%

86.00%

84.00%

82.00%

80.00%

10 x 10 grid

1 74 147 220 293 366 439 512 585 658 731 804 877 950

Number of Iterations

© 2004 Warren B. Powell

20x20 Grid

1x1 Grid

2x2 Grid

5x5 Grid

10x10 Grid

Slide 110

Multiattribute resources

A

We can define a family of aggregation functions:

≈

3

G ( a)

a = ⎡ Time ⎤ ⎡ Time ⎤ ⎡ Time ⎤ ⎡ Time ⎤

⎢

Location

⎥ ⎢

Location

⎥ ⎢

Location

⎥ ⎢

Location

⎥

⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦

⎢ Boxcar type ⎥ ⎢ Boxcar type ⎥ ⎢⎣ Boxcar type⎥⎦

⎢ ⎥ ⎢ ⎥

⎢Time to dest. ⎥ ⎣Time to dest. ⎦

⎢

⎣Repair status⎥

⎦

5,040,000

2

G ( a)

1,680,000

1

G ( a)

40,000

© 2004 Warren B. Powell

0

G ( a)

4,000

Slide 112

Hierarchical aggregation

We can then use all levels of aggregation at the

same time:

a ( g) a

( g)

a

g

v = ∑ w v

Weight on gth level of aggregation

Estimate at gth level of aggregation

© 2004 Warren B. Powell

Slide 113

Hierarchical aggregation

Optimal weights change as the algorithm progresses:

Weights

Weights

0.35

0.3

0.25

0.2

0.15

0.1

0.05

Weight on most disaggregate level

0

0 200 400 600

Iteration

Iterations

800 1000 1200

© 2004 Warren B. Powell

Aggregation level

Weight on most aggregate levels

1

3

2

4

5

6

7

Slide 114

Objective function

Hierarchical aggregation

1900000

1850000

1800000

1750000

1700000

1650000

1600000

1550000

1500000

1450000

1400000

Disaggregate

0 100 200 300 400 500 600 700 800 900 1000

Iterations

© 2004 Warren B. Powell

Aggregate

Slide 115

Objective function

Hierarchical aggregation

1900000

1850000

1800000

1750000

1700000

1650000

1600000

1550000

1500000

1450000

1400000

Weighted Combination

Disaggregate

0 100 200 300 400 500 600 700 800 900 1000

Iterations

© 2004 Warren B. Powell

Aggregate

Slide 116

Norfolk Southern

© 2004 Warren B. Powell

Slide 117

Flows from history

© 2004 Warren B. Powell

Slide 118

Flows from history

Flows from the model

© 2004 Warren B. Powell

Slide 119

But we just don’t move our cars

that way! If you try to send cars

from Buffalo through the

Cleveland yard, it comes in on the

wrong tracks and can’t make the

connection.

Expert knowledge

Usual modeling approach: bottom up engineering

“Behavior”

x

*

Subject to

= arg min cx

:

“Physics”

Ax

= b,

x

≥

© 2004 Warren B. Powell

0

Objectives

Slide 121

Expert knowledge

Bottom up/top down modeling:

Specify costs,

driver availability,

work rules, routing

preferences, load avail.

Engineering

Patterns

Specify the behaviors

you want at a general

level.

© 2004 Warren B. Powell

Slide 122

Expert knowledge

Pattern matching

*

x = cx+ H x

arg min θ ( , ρ)

The “happiness” function –

measures the degree to which

model behavior agrees with a

knowledgeable expert.

Hx ( , ρ) = || Gx ( ) −ρ||

where Gx ( ) is an aggregation function

© 2004 Warren B. Powell

Slide 123

Historical patterns

© 2004 Warren B. Powell

Slide 124

Expert knowledge

Our expert:

© 2004 Warren B. Powell

Slide 125

Flows from history

Flows from the model

© 2004 Warren B. Powell

Slide 126

Flows from history

Flows from the model

© 2004 Warren B. Powell

Slide 127

You can’t trust what the

customer asks for. They don’t

believe we are going to give

them what they need, so they

ask for more than they need.

ars

600

500

400

300

200

100

0

12/31/2001

Weekly Cars ordered Car Order/Supply/Load and the cars actually at loaded CJ229

Cars ordered in advance

1/14/2002

1/28/2002

2/11/2002

2/25/2002

3/11/2002

3/25/2002

4/8/2002

Week Ending

© 2004 Warren B. Powell

4/22/2002

Cars loaded

Booked Orders Cars Supplied Cars Loaded

5/6/2002

5/20/2002

6/3/2002

6/17/2002

Slide 129

C

© Warren B. Powell 2004

© 2004 Warren B. Powell

Slide 130

© 2004 Warren B. Powell

Slide 131

Dollars

1800000

1600000

1400000

1200000

1000000

800000

600000

400000

200000

0

1

5

9

13

17

Profits

21

25

29

33

37

41

45

49

53

57

Iterations

61

65

69

73

77

81

85

89

93

97

Dollars

1800000

1600000

1400000

1200000

1000000

800000

600000

400000

200000

0

1

5

9

13

17

Profits

Late service penalties

21

25

29

Total revenue

Empty repositioning costs

33

37

41

45

49

53

57

Iterations

61

65

69

73

77

81

85

89

93

97

60%

50%

40%

30%

20%

10%

0%

Empty Ratio miles of Empty as Miles a percent to Total Miles of total Traveled miles

History

“Optimized” without adaptive learning

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

60%

50%

40%

30%

20%

10%

0%

Empty Ratio miles of Empty as Miles a percent to Total Miles of total Traveled miles

History

“Optimized” without adaptive learning

“Optimized” with adaptive learning

1 5 9 13 17 21 25 29 33 37 41 45 49 53 57 61 65 69 73 77 81 85 89 93 97

© 1999 Warren B. Powell, **Princeton** **University**

© 2004 Warren B. Powell

Slide 136

© 1999 Warren B. Powell, **Princeton** **University**