Talk at Odysseus 2012, Mykonos

1. Robust
Empty
Reposi.oning
in
Large-­‐Scale
Freight
Consolida.on
Networks
Alan
Erera1,
Antonio
Carbajal1,
Mar.n
Savelsbergh2
1
School
of
Industrial
and
Systems
Engineering,
Georgia
Tech
2
University
of
Newcastle,
Australia
Odysseus
2012

2. What
to
remember
1. Robust
models
for
empty
mobile
resource
management
pragma.c
and
effec.ve
2. Empty
resource
hubs
useful
for
very
large-­‐
scale
reposi.oning
networks
3. Rolling-­‐horizon
deployments
of
two-­‐stage
robust
op.miza.on
models
should
u.lize:
– short
robust
horizons
– rolling
robust
constraints

3. Consolida.on
networks
Customer
Satellite terminal
Hub terminal
origin
destination

4. Consolida.on
networks
Net weekly surplus -3 Satellite terminal
of empty trailers Hub terminal
+5
-5
-1
+2
+4 +1
+1 -1 +1
+1
+3 +4
+3
-4
-3
0
+3 +2
+1 -3
0 -6 +1
-2 -1

5. Dynamic
trailer
reposi.oning
• Large-­‐scale
terminal
network
– 250+
satellites
and
hubs
• Dynamics
– Several
decision
epochs
daily
– Today’s
projected
demand
for
trailers
accurate
– Tomorrow’s
(and
beyond)
significantly
uncertain
• Goal
– Best
empty
reposi.oning
plan
each
epoch

6. Modeling
approaches
• Determinis.c
rolling-­‐horizon
network
flow
LP
– Assume
that
trailer
demands
tomorrow
(and
beyond)
behave
as
expected
• Stochas.c
models
– Minimize
expected
costs
given
probabilis.c
model
of
demand
– Powell
(87),
Frantzeskakis
and
Powell
(90),
Cheung
and
Powell
(96),
Godfrey
and
Powell
(02a,
02b)
– Crainic
(93),
Di
Francesco,
et
al.
(09)

7. Modeling
approaches
• Robust
op.miza.on
models
– Bertsimas
and
Sim
(03),
Atamturk
and
Zhang
(07)
– Morales
(06),
Erera
et.
al.
(09)
• Two-­‐stage
model
• Explicit
focus
on
future
feasibility
• Minimize
cost
of
planned
movements
such
that
a
feasible
set
of
recovery
movements
exists
for
each
non-­‐
extreme
scenario

8. Two-­‐stage
robust
reposi.oning
First stage decisions
t=0 t=1 t=2 t=3 t=4 t=5
A
B
C
D
E

9. Two-­‐stage
robust
reposi.oning
First stage net supply bi
t=0 t=1 t=2 t=3 t=4 t=5
A
Initial trailers
B
+2
Known and expected future loaded moves
C
+6 -2 -1
D
E
+2 +1

10. Two-­‐stage
robust
reposi.oning
Second stage “recovery” decisions
t=0 t=1 t=2 t=3 t=4 t=5
A
B
C
D
E

11. Two-­‐stage
robust
reposi.oning
Second stage uncertain demand
t=0 t=1 t=2 t=3 t=4 t=5
A
B
Intervals on future loaded moves [a , a ]
C
[0, 2]
D
E

12. Two-­‐stage
robust
reposi.oning
• First
stage
network
flow
min c a xa
a
xa − x a = bi ∀i∈N
a∈δ + (i) a∈δ − (i)
xa ≥ 0 and integer ∀ a ∈ A
• Second
stage
“recovery
flow”
for
each
scenario
wa (ω) − wa (ω) = bi (ω) − bi ∀i∈N
a∈δ + (i) a∈δ − (i)
xa + wa (ω) ≥ 0 ∀a∈A

13. Two-­‐stage
robust
reposi.oning
• Key
result:
Existence
of
Recovery
Flow
xa ≥ ν(U ) ∀ U is inbound-closed
a∈δ + (U )∩I
t=0 t=1 t=2 t=3 t=4 t=5
A
B
C

14. Two-­‐stage
robust
reposi.oning
• Inbound
closed
set
– node
set
with
no
incoming
recovery
transporta.on
arcs
t=0 t=1 t=2 t=3 t=4 t=5
A
B
C

15. Two-­‐stage
robust
reposi.oning
• Inbound
closed
set
– this
example
not
inbound-­‐closed
t=0 t=1 t=2 t=3 t=4 t=5
A
B
C

16. Two-­‐stage
robust
reposi.oning
• Worst-­‐case
vulnerability
of
inbound-­‐closed
set
xa ≥ ν(U ) ∀ U is inbound-closed
a∈δ + (U )∩I
ν(U ) = a − a + bi
a∈δ + (U ) a∈δ − (U ) i∈U
t=0 t=1 t=2 t=3 t=4 t=5
A
[0, 2]
B
[1, 5]
C

17. Challenges
(1) Smart
recovery
network
– Low-­‐cost
moves
(since
costs
not
modeled)
– Opera.onally
simple
(2) Appropriate
use
of
two-­‐stage
model
– Controlling
conserva.sm
pragma.cally
– Special
considera.ons
for
rolling
horizon
implementa.on
– Solvable
(but
very
large
scale)
MIPs

18. Smart
recovery
network
Empty hubs
Region 1
Region 2
Region 3

19. Controlling
conserva.sm
• Exclude
extreme
scenarios
– Narrow
the
width
of
intervals
[a , a ]
– Limit
to
k
the
number
of
uncertain
quan..es
that
may
simultaneously
take
on
an
extreme
quan.ty
• Challenges
for
large
.me-­‐expanded
networks
– Very
large
numbers
of
inbound-­‐closed
sets
and
associated
robust
constraints:
O(τ nS )
– Difficult
to
judge
in
advance
which
robust
constraints
will
be
ac.ve

20. Two-­‐stage
robust
reposi.oning
• Bounded
vulnerability
of
inbound-­‐closed
set
 
 
max (a − a )za + (a − a )za | za = k
z  
a∈δ + (U ) a∈δ − (U )
t=0 t=1 t=2 t=3 t=4 t=5
A
[0, 2]
B
[1, 5]
C

21. Appropriate
use
of
two-­‐stage
model
Terminal limit
known future
A
B
C
D
– inbound-­‐closed
sets
with
L+1
terminals
or
fewer

22. Appropriate
use
of
two-­‐stage
model
Robust horizon
known future
A
B
C
D
robust horizon
– inbound-­‐closed
sets
include
no
nodes
aher
RH

23. Appropriate
use
of
two-­‐stage
model
Rolling-horizon robust constraints
known future
A
B
C
D
robust horizon
– add
constraints
now
for
future
horizon
rolls
• assume
that
demand
intervals
do
not
change

24. Appropriate
use
of
two-­‐stage
model
Rolling-horizon robust constraints
known future
A
B
C
D
robust horizon
– add
constraints
now
for
future
horizon
rolls
• assume
that
demand
intervals
do
not
change

25. Tes.ng
the
ideas
• Givens
– Historical
data
from
a
na.onal
consolida.on
trucking
carrier
– Loaded
moves
involve
264
terminals
– Reposi.oning
moves
(truck
and
rail)
– 10
empty
hubs
– At
most
4
daily
dispatch
.mes
per
terminal
– Wide
forecast
intervals
on
loaded
demands
(+/-­‐
50%
of
actual)

26. Tes.ng
the
ideas
• Horizons
– 14
weeks
of
data
– Planning
horizon
of
7
days
for
each
model
• Network
size
for
7-­‐day
planning
horizon
– 5,000
.me-­‐space
nodes
– 300,000
arcs
• Primarily
reposi.oning
arcs
• Limited
connec.ons

27. Tes.ng
the
ideas
• Simulate
– Assume
today’s
loaded
demands
known
– Solve
model,
implement
today’s
decisions
• Assume
trailer
deficits
covered
by
an
outsourced
trailer
– Draw
realiza.on
of
tomorrow’s
demands
– Repeat

28. Figure 18: Unmet demands on a given day - Scenario 2
Results
Figure 19: Cumulative unmet demands - Scenario 2

29. Figure 20: Execution costs on a given day - Scenario 2
Results
Figure 21: Cumulative execution costs - Scenario 2

30. Short
planning
horizons
Figure 22: Cumulative unmet demands with different planning horizons - Scenario 1

31. Next
steps
• Refinements
– More
reasonable
model
of
true
uncertainty
in
demand
– Understand
sources
of
cost
escala.on,
including
if
and
where
excessive
conserva.sm
introduced
• Empty
hub
selec.on
• Fleet
size
versus
reposi.oning
cost
for
robust
plans

32. What
to
remember
1. Robust
models
for
empty
mobile
resource
management
pragma.c
and
effec.ve
2. Empty
resource
hubs
useful
for
very
large-­‐
scale
reposi.oning
networks
3. Rolling-­‐horizon
deployments
of
two-­‐stage
robust
op.miza.on
models
should
u.lize:
– short
robust
horizons
– rolling
robust
constraints