This document summarizes a presentation on dynamic vehicle dispatch strategies for same-day delivery systems. It discusses how last-mile home delivery is costly due to poor economies of scale, and same-day delivery adds further challenges. Dynamic dispatch strategies that determine dispatch times and route assignments dynamically may provide better performance than fixed dispatch waves. Simple rollout policies that adapt pre-computed optimal solutions can produce good results for the dynamic dispatch problem even with stochastic order arrival times.

1. Dynamic
Dispatch
Waves
for
Same-­‐Day
Delivery
Mathias
Klapp,
Alejandro
Toriello,
Alan
Erera
School
of
Industrial
and
Systems
Engineering
Georgia
Tech
UC-­‐Berkeley
ITS
Friday
Seminar
February
20,
2015

2. What
to
remember
1. Last-­‐mile
home
delivery
logis=cs
costly
due
to
poor
scale
economies,
and
same
day
delivery
adds
to
challenge
2.
Dynamic
vehicle
dispatch
strategies
for
SDD
systems
may
provide
significant
value
over
fixed
wave
strategies
3. Simple
rollout
policies
produce
high
quality
dynamic
solu=ons
in
idealized
seJng

3. This
talk
is
not
about…

4. Last-­‐mile
home
delivery
• Weak
scale
economies
– Ton-­‐miles
/
operator-­‐hour
compara=vely
low
– Small
vehicles,
opera=ng
cost
inefficient
Distribution
center
Consumer delivery locations

5. Home
delivery
e-­‐commerce

6. Home
delivery
e-­‐commerce

7. Same-­‐day
home
delivery

8. Same-­‐day
home
delivery

9. Pick/pack/load
and
vehicle
dispatch
• Both
benefit
from
order
batching
– Pick
density
for
warehouse
opera=ons
– Stop
density
for
vehicle
rou=ng
opera=ons

10. Pick/pack/load
and
vehicle
dispatch
• Both
benefit
from
order
batching
– Pick
density
for
warehouse
opera=ons
– Stop
density
for
vehicle
rou=ng
opera=ons
Distribution
center
dense = shorter travel
time per delivery

11. Pick/pack/load
and
vehicle
dispatch
• Both
benefit
from
order
batching
– Pick
density
for
warehouse
opera=ons
– Stop
density
for
vehicle
rou=ng
opera=ons
Distribution
center
sparse = longer travel
time per delivery

12. Next-­‐day
vs.
same-­‐day
yesterday today time
orders arrive
Next-day Local Distribution System
order pick, pack, and load
vehicles for delivery dispatched

13. Next-­‐day
vs.
same-­‐day
yesterday today time
orders arrive
Same-day Local Distribution System
order pick, pack, and load
vehicles for delivery dispatched

14. Pick/pack/load
batching
economies?
yesterday today time
orders arrive
Same-day Local Distribution System
order pick, pack, and load
vehicles for delivery dispatched
many orders arrive after
first picks must be made

15. Dispatch
batching
economies?
yesterday today time
orders arrive
Same-day Local Distribution System
order pick, pack, and load
vehicles for delivery dispatched
some vehicles should
be dispatched before all
orders are ready

16. Vehicle
dispatch
challenges
• Each
vehicle
dispatched
mul=ple
=mes
during
opera=ng
day
(10-­‐12
opera=ng
hours)
– When
to
dispatch
vehicles?
• Tradeoffs
between
wai=ng
to
dispatch,
dispatching
long
routes,
dispatching
short
routes
– When
to
wait
to
accumulate
stop
density?
– Which
orders
to
serve
with
each
vehicle
dispatch?

17. Dynamic
Dispatch
Waves
Problem
• Determine
dispatch
epochs
dynamically
• Explore
tradeoffs
ini=ally
with
single
vehicle
and
simplified
geography
time
wait
dispatch 1 dispatch 2 dispatch 3

18. Simplified
geography:
stops
on
line
di
• Order
loca=on
– round-­‐trip
travel
=me
from
DC
• No
stop
=me
• Dispatch
– serves
all
ready
orders
di
di
{j : dj  di}

19. Order
ready
Mme
process
di
time
Ready is picked, packed
for loading (no duration)
T 0⌧i
Orders served and
vehicle back to DC
by time 0

20. Order
ready
Mme
process
di
time
Ready orders for first
dispatch of day
T 0⌧i

21. Order
ready
Mme
process
di
time
Orders that come
available later in
operating day,
and unknown when
planning at time T
T 0⌧i

22. Dynamic
Dispatch
Waves
Problem
• Each
=me
vehicle
at
distribu=on
center,
decide:
– Whether
to
dispatch
vehicle,
or
wait
– If
dispatched,
which
unserved
ready
orders
to
include
in
the
route
• Given
set
of
poten.al
orders
– Round-­‐trip
dispatch
=me
– Stochas=c
=me
(or
wave)
when
order
ready
– Penalty
if
order
remains
unserved
• Operate
to
minimize
total
cost
of
all
dispatches
plus
total
unserved
order
penal=es
N = {1, ..., n}
di
⌧i
i

23. Dynamic
programming
formulaMon
for
DDWP
on
the
line
• State:
– Number
of
remaining
waves,
– Ready
and
unserved
orders,
– Poten=al
orders
not
yet
ready,
• Ac=ons:
wait
one
wave,
or
serve
– Cost:
– Possible
dispatch
ac=ons:
– Must
return
by
0:
• At
end
horizon,
pay
penal=es
for
unserved
orders
(t, R, P)
t
R
P
S ✓ R
|R|
x = maxi2S di
x  t

24. Dynamic
programming
formulaMon
for
DDWP
on
the
line

25. Bellman
recursion
for
DDWP

26. DeterminisMc
DDWP
on
line
• Request
ready
=mes
known
in
advance,
but
requests
cannot
be
served
before
ready
=me
• Proper=es
of
op=mal
solu=on
– Dispatch
lengths
x
strictly
decreasing
– No
wai=ng
a]er
first
dispatch

27. DeterminisMc
DDWP
on
line
• Request
ready
=mes
known
in
advance,
but
requests
cannot
be
served
before
ready
=me
• Proper=es
of
op=mal
solu=on
– Dispatch
lengths
x
strictly
decreasing
– No
wai=ng
a]er
first
dispatch

28. DeterminisMc
DDWP
on
line
• New
DP
state:
remaining
waves
t,
length
d
of
prior
dispatch
• Recursion
O(n2
T)

29. Using
determinisMc
DDWP
• Es=ma=ng
an
a
posteriori
cost
lower
bound
– Average
cost
for
sample
of
order
realiza.on
days
– Any
dynamic
policy
for
stochas=c
DDWP
can
have
no
lower
expected
cost
• Building
a
priori
policy
solu=ons
to
the
stochas=c
DDWP

30. A
priori
soluMon
• Before
first
dispatch
(i.e.,
at
wave
T),
find
complete
set
of
vehicle
dispatches:
• Theorem
{(xk
, tk
)}
Optimal a priori solution is solution to
deterministic DDWP where each order i
replicated for each wave t 2 {T, ..., 1} with
known ready time t and penalty i Pr(⌧i = t)

31. Dynamic
policies
1. Implement
a
priori
solu=on,
but
adjust
during
opera=ons
– Shorten,
delay,
and
cancel
some
dispatches
2. Rollout
using
a
priori
solu=ons
– Execute
first
decision
in
adjusted
a
priori
solu=on
– Build
new
a
priori
plan
any
=me
vehicle
at
distribu=on
center,
using
new
informa=on

32. Experiment
1
• Request
ready
=me
process
– Condi=onal
arrival
likelihoods
each
wave
• Request
loca=ons
and
penal=es
– Loca=on
discrete
uniform
up
to
a
maximum
– Penal=es
discrete
uniform
on
quarters
of
•
20
random
instances
for
class
– r
measures
=me
flexibility
✓i
T
`
`
✓
n, `, r =
T
`
◆

33. Experiment
1:
Results
AvgGaptoaposteriorilowerbound

34. Experiment
1:
Results
AvgGaptoaposteriorilowerbound

35. Experiment
1:
Results,
r
=
2
AvgGaptoaposteriorilowerbound

36. Dynamic
policies
via
ALP
• Dual
LP
reformula=on
of
Bellman’s
equa=on
– massive
LP:
exponen=al
variables,
constraints
max ERT
[CT (RT , N RT )]
C0(R, P) 
X
i2R
i
Ct(R, P)  EF t
1
[Ct 1(R [ Ft
1, P Ft
1)]
Ct(R, P)  d + EF t
d
[Ct d(Rd [ Ft
d, P Ft
d)]

37. Dynamic
policies
via
ALP
• ALP
restric=on
provides
lower
bound,
and
poten=ally
useful
approxima=on
of
C
• Restrict
C
:
• “cost
of
unserved
known
requests”,
“cost
of
unserved
poten=al
requests”,
“value
of
remaining
waves”
Ct(R, P) ⇡
X
i2R
at
i +
X
j2P
bt
j
tX
k=1
vk

38. Dynamic
policies
via
ALP
• Proposi=ons
– Using
this
restric=on
in
dual
LP,
the
ALP
lower
bound
LP
requires
variables
and
constraints
– (*)
For
determinis=c
problems,
the
ALP
lower
bound
is
=ght,
equal
to
op=mal
cost
• Hybrid
ALP-­‐A
priori
rollout
policy
– Use
a
priori
rollout
first,
then
switch
to
ALP
rollout
later
in
opera=ng
period
O(nT) O(n2
T)

39. Experiment
1:
Results
AvgGaptoaposteriorilowerbound

40. Experiment
2
• New
ready
=me
process
– p:
Likelihood
ready
by
T
– q:
Likelihood
of
no
request
– Remaining
likelihood
discrete
uniform:
• Request
loca=ons
and
penal=es
as
before
• 20
random
instances
(n = 20, ` = 10, r = 3)
µi
µi + vµi v

41. Experiment
2:
Results
vs.
q
-­‐
p
AvgGaptoaposteriorilowerbound

42. Experiment
2:
Results
vs.
v
AvgGaptoaposteriorilowerbound

43. ObservaMons:
1
• Dynamic
soluMons
valuable
– Dispatching
scheme
from
A
priori-­‐rollout
approach
usually
provides
significant
savings
over
instance-­‐specific
A
priori
solu=ons
– Schemes
with
fixed
dispatch
waves
could
be
no
befer
in
this
seJng
time
wave I wave II wave III

44. ObservaMons:
1
• Fixed-­‐but-­‐flexible
dispatch
waves
(?)
– Fixed
planning
waves
useful
to
DC
pick/pack/load,
and
for
customer
order
management
– Design
and
performance
of
a
fixed-­‐but-­‐flexible
dispatch
wave
system?
time
wave I wave II wave III

45. ObservaMons:
2
• LocaMons
on
line
creates
maximum
batching
benefit
– Compounded
by
assump=on
of
no
fixed
stop
=me
required
per
delivery
– Incen=ve
to
wait
and
batch
may
be
too
strong
– Inves=ga=ng
problems
with
fixed
stop
=mes
and
two-­‐dimensional
delivery
loca=ons

46. ObservaMons:
2
T 0⌧i
latest wave and
dispatch duration

47. Other
extensions
• MulMple
vehicles
per
delivery
zone
– How
to
coordinate
dispatch
waves
for
two
vehicles
serving
a
single
zone?
Other
configura=ons?
• Customer
order
management
– Reject/not
offer
same
day
delivery
op=on
dynamically
as
orders
are
received

48. What
to
remember
1. Last-­‐mile
home
delivery
logis=cs
costly
due
to
poor
scale
economies,
and
same
day
delivery
adds
to
challenge
2.
Dynamic
vehicle
dispatch
strategies
for
SDD
systems
may
provide
significant
value
over
fixed
wave
strategies
3. Simple
rollout
policies
produce
high
quality
dynamic
solu=ons
in
idealized
seJng