Dynamic&Snow&Plow&FleetManagement … · – Special&case:&if&the&Lme&window&is&Lght,&& 1, , 0, to...

Leila Hajibabai1 and Yanfeng Ouyang2

1Washington State University 2University of Illinois at Urbana-‐Champaign

January 12, 2014

Dynamic Snow Plow Fleet Management under Uncertain Demand and Service DisrupLon

MoLvaLon

•  Safety impact of snow and ice storms (USDOT, 2014) –  24% of annual weather-‐related vehicle

crashes on snowy or icy pavement –  Over 1,300 fatal crashes in vehicle

crashes on snowy or icy roads

2

•  Economic implicaLons –  $300-‐$700 M for 1-‐day shutdown

(American Highway Users Alliance, 2010)

–  20% of state DOT maintenance budgets for snow and ice control (USDOT, 2014)

–  Over $110,000 cost per shiV on average for snow removal operaLons (District of Columbia, 2010)

Source: USDOT (FHWA Home Page), Accessed on April 20, 2014

ContribuLons

•  Dynamic fleet scheduling in long-‐storm condiLons:

•  Uncertain demand •  Service disrupLon

•  Route planning •  Fleet assignment •  Resource replenishment

3

Dynamic Fleet Management

Snow Plow Route Planning

Strategic Infrastructure Decisions

•  Resource planning and allocaLon: •  Resource replenishment faciliLes •  Roadway capacity expansion

•  Decision support tool

•  Problem Statement

•  Background

•  Model Development

•  SoluLon Approach

•  Numerical Results

•  Summary

4

Outline


5

Problem Statement


Number of available trucks on available tasks over time

Task availability over time

Temporal Demand Distribution

Service Truck Management

Available tasks in the network

Spatial Demand Distribution

Demand Pattern

Service Disruption

Problem Statement

•  ConsideraLons:

–  Random availability of the tasks and trucks

•  New trucks and tasks become available via some random process

–  Truck reposiLoning

•  There is no available task •  Tasks at other route have higher priority while service is disrupted

–  Truck deadheading

6


𝑖=3

𝑖=1 𝑖=2 𝑖=

4

. . .

Lme pe

riod, 𝑡

Set of available tasks Depot

𝑖=3

𝑖=1 𝑖=2 𝑖=

4

0

1

2

Problem Statement

Automatic Vehicle Location (AVL) Source: www.mee-‐pya-‐Lte.com, Accessed on Feb 25, 2014


𝑖=3

𝑖=1 𝑖=2 𝑖=

4

Background

•  Dynamic Programming (DP) Methods

•  Not as sensiLve to large state spaces, but suffer from large acLon spaces

•  Require the ability to esLmate the value of the system being in a parLcular state

•  Convergence proofs are only available under very strong assumpLons

8

Method Objec,ve/Focus Researcher

TradiLonal DP Discrete state and acLon spaces Puterman 1994

Forward DP Monte Carlo Bertsekas and Tsitsiklis 1996, Sueon and Barto 1998


Background

•  StochasLc Linear Programming Methods (Infanger 1994, Kall &

Wallace 1994, Birge & Louveaux 1997, Powell 2002)

1.  limit the size of the problem we can consider

9


Two-‐stage stochasLc linear programs

Large-‐scale opLmizaLon problems subject to non-‐anLcipaLvity constraints

Dantzig 1955, Rockafellar & Wets 1991

Approximate the second-‐stage recourse funcLon (linearizaLon methods, staLc/dynamic sampling)

Ermoliev 1988, Ruszczynski 1980, Van Slyke & Wets 1969, Higle & Sen 1991

MulLstage problems Nested Benders algorithm Birge 1985

Sampling-‐based methods Higle & Sen 1991, Pereira & Pinto 1991, Chen & Powell 1999


Background

•  ApproximaLon Methods in the Context of Fleet Management


10


Methods for conLnuous flows

Designed for non-‐integer flows Jordan & Turnquist 1983, Powell 1986

ApproximaLons that naturally produce integer soluLons

Not designed for problems with Lme windows (difficult to apply)

Powell 1987, Frantzeskakis & Powell 1990, Cheung & Powell 1996, Powell & Carvalho 1998

Linear approximaLon with a mulLplier adjustment procedure (LAMA)

Works only on determinisLc problems Carvalho & Powell 2000

CAVE (Concave AdapLve Value EsLmaLon) algorithm

•  Construct a concave, separable, piecewise-‐linear approximaLon of value funcLon

•  More flexible and responsive than linear approximaLons

Godfrey & Powell 2001, 2002


Background

•  Online Vehicle RouLng Problem


11


Online rouLng algorithms w/o service flexibility and rejecLon opLons

Minimize the Lme to visit a set of locaLons that are revealed incrementally over Lme

Jaillet & Wagner 2007, Jaillet & Lu 2011, Yang

Rolling horizon technique using a mixed integer programming formulaLon for the offline version

Online fleet assignment and scheduling Minimize costs of empty travels, jobs’ delayed compleLon Lmes, and job rejecLons w/o Lme windows

Yang, Jaillet, & Mahmassani 1998, 2002, 2004

SimulaLon framework using real-‐Lme info about vehicle locaLons and demands

Dynamic dispatching, load acceptance, and pricing strategies

Regan, Mahmassani, & Jaillet 1996


Model Development

•  Network:

12

{ }:

0,1,..., 1 , :

TTΓ = −

Ψ

the number of time periods in the planning horizon

the times at which decisions are made

the set of physical routes in the network


Model Development

•  Trucks and Tasks:

13

ˆ ˆ ˆ: ; ( ):

; (

it t it i

it

t

i ti t

κ κ κ

κ

κ

∈Ψ=

=

the number of trucks that first become available on route at timethe number of trucks available on route at time before any new arrivalshave been added in )

: ˆ ˆ ˆ: ; ( )

:

it i

it

it t it i

it

i t

A i t A A

A

κ

κ∈Ψ

+

∈Ψ=

the total number of trucks that are available to be used on route at time

the set of tasks that first become available on route at time

the set of tasks avail ˆ ; ( )

: ,

it

t it i

it

i t AA A

A i t∈Ψ

+

=

able on route at time before the new arrivals in are addedto the system

the set of tasks available to be services on route at time including the newly popped-‐up

0

: ˆˆW ( , ),

(W ) :

dit

t t t

Tt t

A i t

A tκ

=

=

tasks

the set of deadheads required to reach the available tasks on route at time

represents the new info arriving in time period

stochastic info process, with realiza ˆˆ W ( ) ( ( ), ( ))

t t t tAω ω κ ω ω= =tion

For each and ,t i∈Γ ∈Ψ


Model Development

•  Decision Variables:

•  State Variables:

–  Special case: if the Lme window is Lght,

1, ,0,

to

ti

tij

i t

i j t

µ

η

∈Ψ ∈Γ⎧= ⎨⎩

= ∈Ψ ∈Ψ ∈Γ

if a truck travels on route at timeo. w.

number of trucks reposition from at time

{ },t t tS Aκ=


14

Truck Inventories

Ψ

t 1t +ReposiLoning Task Performance

Model Development

•  ObjecLve FuncLon:

15

,

,

: ($ / )

: ($ / )

: ($ / )

:

pij

ra itd da itti r

i d

c i j mile

c a A mile

c a A mile

i tλ

λ

+∈

∈

cost of repositioning from route to route

the benefit from plowing task

the cost of deadheading link

the number of tasks on route at time

: t i tthe number of deadheads required to reach the tasks on route at time

Total benefit gained from decisions at Lme t Total reposiLoning cost at Lme t


(1)

(2)

(3)

(4)

SoluLon Approach

16

: etA tthe set of tasks that expire in time period

System Dynamics

Dynamic Programming

Value FuncLon ApproximaLon

ˆ ˆ( ) ( )t t it iti

V Vκ κ∈Ψ

=∑


(5) (6)

(2)-‐(4) and (6)

(7) Bellman Eq.

17

Initialization

Forward Simulation

Solve the network sub-problem (2)-(4),(6)- (7)

CAVE Update

Store

Determine

SoluLon Approach

•  Fipng Concave FuncLonal ApproximaLons (CAVE*)

18

( )sπ −

( )sπ +

0v

1v

2v3 0v =ˆ ( )it itV S

itS

Update Interval

1 2 3u s s u s uε ε− +− +


* Concave AdapLve Value EsLmaLon

0 00, 0it itv u= =

{ }{ }

{ } { })

1

1

min : (1 )

min : (1 )

min , ,max ,it it

n nit it it it

n nit it it it

n nit it

n n N v v

n n N v v

UI u u

α απ

α απ

κ ε κ ε− +

− + −

+ − +

− +

= ∈ ≤ − +

= ∈ < − +

⎡= − +⎣

Create new break points

, ,(1 ) ,

, if

, o.w.

n nit new it it old

nit it it it

it it

v v

v u

v

απ α

π κ

π

−

+

= + −

⎧ = <⎪⎨

=⎪⎩wherewhere

Numerical Results

19

Problem Characteris,cs A;ribute Values

number of routes Lake County, IL opLmal truck routes

number of trucks equal to the number of routes, but subject to failure

planning horizon length, T 8 Lme periods

number of tasks over simulaLon from Lake County, IL task links*

Lme period length (fixed) 20 min

net task revenue per mile $10

reposiLoning cost per mile $1

deadhead cost per mile $1

•  Tasks become available over Lme on routes


•  The algorithm is coded in C++ and run on a desktop computer with 2.67 GHz CPU and 3 GB memory •  CPLEX is called to solve the forward simulaLon. •  Number of routes in the last iteraLon = 14

20

Numerical Results


0 500

1000 1500 2000 2500

0 1 2 3 4 5 6 7

Tota

l Ben

efit

($)

0 20 40 60 80

100 120 140 160 180 200

0 1 2 3 4 5 6 7

Tota

l No.

of T

asks

Pe

rfor

med

0

1

2

3

4

0 1 2 3 4 5 6 7

Tota

l No.

of

Rep

ositi

ons

Time Period

0

500

1000

1500

2000

2500

0 1 2 3 4 5 6 7

Dynamic Programming with CAVE Update Greedy Algorithm

Time Period

Tota

l B

enef

it ($

)

•  Comparison with alternaLve algorithms –  5.8% difference over the planning

horizon

Summary

•  Dynamic fleet management for snow control acLviLes under uncertainty (operaLons)

–  Approximate Dynamic Programming (ADP) including a forward simulaLon followed by an update

–  Case study based on LCDOT truck routes

–  Comparison with a greedy approach

21


Thanks Leila Hajibabai, Ph.D.

Washington State University [email protected]

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