Dynamic Dispatch Waves for Same-day Delivery

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

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

This talk is not about…

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

Home delivery e-‐commerce

Home delivery e-‐commerce

Same-‐day home delivery

Same-‐day home delivery

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



Distribution center

sparse = longer travel time per delivery

Next-‐day vs. same-‐day

yesterday today time

orders arrive

Next-day Local Distribution System

order pick, pack, and load

vehicles for delivery dispatched

Next-‐day vs. same-‐day


orders arrive

Same-day Local Distribution System



Pick/pack/load batching economies?


orders arrive




many orders arrive after first picks must be made

Dispatch batching economies?


orders arrive



vehicles for delivery dispatched some vehicles should be dispatched before all orders are ready

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?

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

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}

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


di

time

Ready orders for first dispatch of day

T 0⌧i


di

time

Orders that come available later in operating day, and unknown when planning at time T

T 0⌧i

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

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 )

tR

P

S ✓ R

|R|x = maxi2S di

x t

Dynamic programming formulaMon for DDWP on the line

Bellman recursion for DDWP

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

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

DeterminisMc DDWP on line •  New DP state: remaining waves t, length d of prior dispatch

•  Recursion

O(n2T )

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

A priori soluMon

•  Before first dispatch (i.e., at wave T), find complete set of vehicle dispatches:

•  Theorem {(xk

, t

k)}

Optimal a priori solution is solution todeterministic DDWP where each order ireplicated for each wave t 2 {T, ..., 1} withknown ready time t and penalty �i Pr(⌧i = t)

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

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

✓iT

``

✓n, `, r =

T

`

◆

Experiment 1: Results Av

g G

ap to

a p

oste

riori

low

er b

ound


g G

ap to

a p

oste

riori

low

er b

ound

Experiment 1: Results, r = 2 Av

g G

ap to

a p

oste

riori

low

er b

ound

Dynamic policies via ALP

•  Dual LP reformula=on of Bellman’s equa=on – massive LP: exponen=al variables, constraints

maxERT [CT (RT , N \RT )]

C0(R,P ) X

i2R

�i

Ct(R,P ) EF t1[Ct�1(R [ F t

1 , P \ F t1)]

Ct(R,P ) d+ EF td[Ct�d(Rd [ F t

d, P \ F td)]


•  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

ati +X

j2P

btj �tX

k=1

vk


•  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(n2T )


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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)

µiµi + vµi � v

Experiment 2: Results vs. q -‐ p Av

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Experiment 2: Results vs. v Av

g G

ap to

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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

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

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

ObservaMons: 2

T 0⌧i

latest wave and dispatch duration

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

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