Reliable Infrastructure Location Design under
Interdependent Disruptions
Xiaopeng Li, Ph.D.
Department of Civil and Environmental Engineering,
Mississippi State University
Joint work with
Yanfeng Ouyang, University of Illinois at Urbana-Champaign
Fan Peng, CSX Transportation
The 20th International Symposium on Transportation and Traffic Theory
Noordwijk, Netherlands, July 17, 2013
2
Outline
•
Background
 Infrastructure network design
 Facility disruptions
•
Mathematical Model
 Formulation challenges
 Modeling approach
•
Numerical Examples
 Solution quality
 Case studies
3
Logistics Infrastructure Network
 Facilities are to be built to serve spatially distributed customers
 Trade-off
 one-time facility investment
 day-to-day transportation costs
 Optimal locations of facilities?
Customer
…
Facility
Transp.
cost
Facility
cost
3
4
Infrastructure Facility Disruptions
 Facilities may be disrupted
due to
 Natural disasters
 Power outages
 Strikes…
 Adverse impacts
 Excessive operational cost
 Reduced service quality
 Deteriorate customer
satisfaction…
 Effects on facility planning
 Suboptimal system design
 Erroneous budget estimation
4
5
Impacts of Facility Disruptions
 Excessive operations cost (including travel & penalty)
 Visit the closest functioning facility within a reachable distance
 If all facilities within the penalty distance fail, the customer will receive a
penalty cost
 Reliable design?
Facility cost
Reachable Distance
Operations Cost
6
Literature Review
 Traditional models
 Deterministic models (Daskin, 1995; Drezner, 1995)
 Demand uncertainty (Daskin, 1982, 1983; Ball and Lin, 1993; Revelle and
Hogan, 1989; Batta et al., 1989)
 Continuum approximation (Newell 1973; Daganzo and Newell, 1986;
Langevin et al.,1996; Ouyang and Daganzo, 2006)
 Reliable models
 I.i.d. failures (Snyder and Daskin, 2005; Chen et al., 2011; An et al.,2012)
 Site-dependent (yet independent) failures (Cui et al., 2010;)
 Special correlated failures (Li and Ouyang 2010, Liberatore et al. 2012)
 Most reliable location studies assume disruptions are independent
6
7
Disruption Correlation
 Many systems exhibit positively correlated disruptions
 Shared disaster hazards
 Shared supply resources
Power Plant
Factories
Northeast Blackout (2003)
Hurricane Sandy (2012)
7
8
Prominent Example: Fukushima Nuclear Leak
Earthquake
→ Power supply failure
→ Reactors meltdown
Power supply
for cooling systems
Reactors
(Sources: ibtimes.com; www.pmf.kg.ac.rs/radijacionafizika)
9
Research Questions
 How to model interdependent disruptions in a simple way?
 How to design reliable facility network under correlated
disruptions?
 minimize system cost in the normal scenario
 hedge against high costs across all interdependent disruption scenarios
Initial
investment
Operations
cost
normal scenario
Operations
cost
correlated
disruption scenarios
10
Outline
•
Background
 Infrastructure network design
 Facility disruptions
•
Mathematical Model
 Formulation challenges
 Modeling approach
•
Numerical Examples
 Solution quality
 Case studies
11
Probabilistic Facility Disruptions
 A facility is either disrupted or functioning
 Disruption probability = long-term fraction of time when the
facility is in the disrupted state
 Facility state combination specifies a scenario
Facility 1
Facility 2
Facility 3
Normal
scenario
Scenario
1
Disrupted state
Normal
scenario
Scenario
2
Scenario
3
Normal
scenario
Functioning state
time
12
Modeling Challenges
 Deterministic facility location problem is NP-hard
 Even for given location design, # of failure scenarios
increases exponential with # of facilities
 Difficult to consolidate scenarios under correlation
…
Scenario 2
…
Scenario N+1
…
Scenario 2N
…
…
Scenario 1
…
Disrupted
Functioning
13
Correlation Representation:
Supporting Structure
 Each supporting station is disrupted independently with an
identical probability (i.i.d. disruptions)
 A service facility is operational if and only if at least one of its
supporting stations is functioning
Supporting Stations:
…
Service Facilities:
…
14
Supporting Structure Properties
 Proposition: Site-dependent facility disruptions(Cui et al.,
2010) can be represented by a properly constructed
supporting structure
 Idea: # of stations connected to a facility determines
disruption probability
…
…
15
Supporting Structure Properties
 Proposition: General positively-correlated facility disruptions
can be represented by a properly constructed supporting
structure.
 Structure construction formula:
N'
p (u N ' ) 

i0


QL 


 L  N \ N ', L  N \ N '  i

A
B
i 1
, N ' N
C
16
System Performance - Expected Cost
 Supporting stations K:
k: cons. cost ck
(i.i.d. failure probability p)
j: cons. cost fj
 Service facilities J:
i: demand – li; penalty pi
 Customers I:
 All scenarios S = {s}; each scenario s occurs at probability Ps
 In s, i is assigned to js ; js ∈ J (functioning facility), or js = 0, di0 := pi (penalty)
 Expected total system cost:
𝑘∈𝐾 𝑐𝑘
Construction cost
+
𝑗∈𝐽 𝑓𝑗
+
𝑠∈𝑆 𝑃𝑠
𝑖∈𝐼 𝜆𝑖
𝑑𝑖𝑗𝑠
Expected operations cost
17
Expected System Cost Evaluation
 Consolidated cost formula
 i I  r 1 l i (1  p ) p
R
r 1
m in {p i , d ij ,  j  J k } 
 i I l i p i p
 Scenario consolidation principles
 Separate each individual customer
 Rank infrastructure units according to a customer’s visiting sequence
R
18
Reliable Facility Location Model
m in
X ,Y ,Z

fjX j 
j J
cY
k
kK
R
k

l
i I
r 1
i
p
r 1

(1  p )  

 kK
subject to


d ij Z ijkr  p i Z i 0 r
j J k



Expected system cost
Z ijkr  Z i 0 r  1,  i  I , r  1, 2,
,R
Assignment feasibility
k  K j J k
Z ijkr  X j ,  i  I , j  J , k  K j , r  1, 2,
Facility existence
,R
R
Z
ijkr
 Yk ,  i  I , k  K
Station existence
j J k r 1
Z ijkr , Z i 0 r  {0,1},  i  I , j  J , k  K j , r  1, 2,
, R;
X j , Yk ,  {0,1},  j  J , k  K .
Compact Linear Integer Program
Integrality
19
Outline
•
Background
 Infrastructure network design
 Facility disruptions
•
Mathematical Model
 Formulation challenges
 Modeling approach
•
Numerical Examples
 Solution quality
 Case studies
20
Hypothetical Example
 Supporting stations are given
 Identical network setting except for # of shared stations
 Identical facility disruption probabilities
 Case 1: Correlated disruptions
 Neighboring facilities share stations
…
 Case 2: Independent disruptions (not sharing stations)
 Each facility is supported by an isolated station
…
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Comparison Result
Case 1: Correlated disruptions
Facility disruption
probability
0
0.3
0.6
0.9
Facility
construction
Transportation Penalty
cost
cost
cost
1800
3000
0
1800
3264
4
3000
3722
653
3000
6849 56485
Facility locations
j3 j6 j9
j3 j6 j9
j2 j4 j6 j8 j10
j2 j4 j6 j8 j10
Total
cost
4800
5068
7375
66335
Case 2: Independent disruptions
Facility disruption
probability
0
0.3
0.6
0.9
Facility locations
j3 j6 j9
j3 j5 j8 j9
j2 j3 j5 j6 j8 j10
j2 j3 j4 j5 j6 j7 j8 j9 j10
Facility
Transportation
construction cost
cost
1800
3000
2400
2656
3600
2586
5400
6680
Penalty
cost
0
0.01
18
10467
Total
cost
4800
5055
6204
22547
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Case Study
 Candidate stations: 65 nuclear power plants
 Candidate facilities and customers: 48 state capital cities & D.C.
Data sources: US major city demographic data from Daskin, 1995
eGRID http://www.epa.gov/cleanenergy/energy-resources/egrid/index.html
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Optimal Deployment
Supporting station:
Service facility:
24
Summary
 Supporting station structure
 Site-dependent disruptions
 Positively correlated disruptions
 Scenario consolidation
 Exponential scenarios → polynomial measure
 Integer programming design model
 Solved efficiently with state-of-the-art solvers
 Future research
 More general correlation patterns (negative correlations)
 Application to real-world case studies
 Algorithm improvement
25
Acknowledgment
 U.S. National Science Foundation
 CMMI #1234936
 CMMI #1234085
 EFRI-RESIN #0835982
 CMMI #0748067
Thank You!
Xiaopeng Li
xli@cee.msstate.edu