Post on 07-Jun-2020
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
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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
Dynamic Fleet Management
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Problem Statement
Dynamic Fleet Management
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
Dynamic Fleet Management
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
Dynamic Fleet Management
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
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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
Dynamic Fleet Management
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
Method Objec,ve/Focus Researcher
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
Dynamic Fleet Management
Background
• ApproximaLon Methods in the Context of Fleet Management
1. limit the size of the problem we can consider
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Method Objec,ve/Focus Researcher
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
Dynamic Fleet Management
Background
• Online Vehicle RouLng Problem
1. limit the size of the problem we can consider
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Method Objec,ve/Focus Researcher
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
Dynamic Fleet Management
Model Development
• Network:
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{ }:
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
Dynamic Fleet Management
Model Development
• Trucks and Tasks:
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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
Dynamic Fleet Management
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=
Dynamic Fleet Management
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Truck Inventories
Ψ
t 1t +ReposiLoning Task Performance
Model Development
• ObjecLve FuncLon:
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,
,
: ($ / )
: ($ / )
: ($ / )
:
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
Dynamic Fleet Management
(1)
(2)
(3)
(4)
SoluLon Approach
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: 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
=∑
Dynamic Fleet Management
(5) (6)
(2)-‐(4) and (6)
(7) Bellman Eq.
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Initialization
Forward Simulation
Solve the network sub-problem (2)-(4),(6)- (7)
CAVE Update
Store
Determine
SoluLon Approach
• Fipng Concave FuncLonal ApproximaLons (CAVE*)
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( )sπ −
( )sπ +
0v
1v
2v3 0v =ˆ ( )it itV S
itS
Update Interval
1 2 3u s s u s uε ε− +− +
Dynamic Fleet Management
* 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
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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 $1
• Tasks become available over Lme on routes
Dynamic Fleet Management
• The algorithm is coded in C++ and run on a desktop computer with 2.67 GHz CPU and 3 GB memory • CPLEX is called to solve the forward simulaLon. • Number of routes in the last iteraLon = 14
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Numerical Results
Dynamic Fleet Management
0 500
1000 1500 2000 2500
0 1 2 3 4 5 6 7
Tota
l Ben
efit
($)
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
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Dynamic Fleet Management
Thanks Leila Hajibabai, Ph.D.
Washington State University leila.hajibabai@wsu.edu