Report

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 ' ) i0 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: ∈ Construction cost + ∈ + ∈ ∈ 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 kK R k l i I r 1 i p r 1 (1 p ) kK 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 … 21 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 22 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 23 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 [email protected]