Recovery from Multimodal Transportation Disruptions Cameron - PowerPoint PPT Presentation

Optimal Resource Allocation for Recovery from Multimodal Transportation Disruptions Cameron MacKenzie Kash Barker, PhD Society for Risk Analysis Annual Meeting December 5, 2011

Transportation disruptions MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 2

Previous studies • Disrupt one or more transportation modes • Determine alternate transportation routes for each firm • Calculate additional transportation cost or economic impact J. K. Kim, H. Ham, and D. E. Boyce (2002), “Economic impacts of transportation network changes: Implementation of a combined transportation network and input- output model,” Papers in Regional Science 81: 223-246. J. Sohn, T. J. Kim, G. J. D. Hewings, J. S. Lee, and S.- G. Jang (2003), “Retrofit priority of transport network links under an earthquake,” Journal of Urban Planning & Development 129: 195-210. P. Gordon, J. E. Moore II, H. W. Richardson, M. Shinozuka, D. An, and S. Cho (2004), “Earthquake disaster mitigation for urban transportation systems: An integrated methodology that builds on the Kobe and Northridge experiences,” in Y. Okuyama and S. E. Chang, eds., Modeling Spatial and Economic Impacts of Disasters , 205-232. MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 3

Research goals • Develop optimal resource allocation model to repair disrupted transportation infrastructure – Given firms’ alternate transportation routes – Given additional costs or delays experienced by firms • Calculate optimal allocation as function of parameters (e.g., initial inoperability, effectiveness of allocation, additional costs and delays) • Explore whether decision changes over time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 4

Number of firms Static model Cost for firm 𝑗 𝑜 minimize 𝑔 𝑗 𝐫 Vector (length 𝑛 ) of 𝑗=1 inoperability for each 𝑜 transportation infrastructure, 𝑕 𝑗 𝐫 mode, route Initial inoperability for Delay for firm i 𝑗=1 transportation infrastructure 𝑘 Effectiveness of general allocation subject to 𝑟 𝑘 = 𝑟 𝑘 exp −𝑙 𝑘 𝑨 𝑘 − 𝑙 0 𝑨 0 Effectiveness of allocating to transportation j Allocation to Allocation to transportation 𝑘 general 𝑛 transportation 𝑨 0 + 𝑨 ≤ 𝑎 𝑘 Total budget 𝑘=1 𝑨 𝑘 ≥ 0 , 𝑨 0 ≥ 0 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 5

Illustrative example Water Water Destination Origin Railroad Railroad Transfer Highway Highway point Inoperability ( 𝒓 𝒌 ) 𝒍 𝒌 Water 0.3 1 Railroad 0.3 1 Highway 0.3 1 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 6

Railroad Water Highway Illustrative example 𝑔 𝑗 𝐫 = 𝑑 𝑗,𝑥 𝑟 𝑥 + 𝑑 𝑗,𝑠 𝑟 𝑠 + 𝑑 𝑗,ℎ 𝑟 ℎ Linear cost function for firm 𝑗 𝑑 𝑗,𝑥 𝑑 𝑗,𝑠 𝑑 𝑗,ℎ Firm 1 3 1 1 Firm 2 0 3 2 Firm 3 0 0 5 𝑕 𝑗 𝐫 = 𝑒 𝑗,𝑥 𝑟 𝑥 + 𝑒 𝑗,𝑠 𝑟 𝑠 + 𝑒 𝑗,ℎ 𝑟 ℎ Linear delay function for firm 𝑗 𝑒 𝑗,𝑥 𝒆 𝑗,𝑠 𝒆 𝑗,ℎ Firm 1 1 2 1 Firm 2 0 3 1 Firm 3 0 0 1 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 7

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 8

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 9

Pareto front Create Pareto front for cost and delay Tradeoff parameter between cost and delay 𝑜 𝑜 minimize α 𝑔 𝑗 𝐫 + 1 − α 𝑕 𝑗 𝐫 0 ≤ 𝛽 ≤ 1 𝑗=1 𝑗=1 Cost for firm 𝑗 Delay for firm 𝑗 Vector (length 𝑛 ) of inoperability for each transportation infrastructure, mode, route MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 10

Pareto front for different budgets 𝑜 𝑜 minimize α 𝑔 𝑗 𝐫 + 1 − α 𝑕 𝑗 𝐫 𝑗=1 𝑗=1 2 Budget = 1 1.8 Budget = 1.25 Delay (time) 1.6 Budget = 1.5 1.4 Budget = 1.75 Budget = 2 1.2 1 1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 Cost (money) MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 11

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 12

If 𝒜 𝟏 = 𝟏 𝑜 𝑜 𝑦 𝑘 = 𝛽 𝜖𝑔 + 1 − 𝛽 𝜖𝑕 𝑗 Change in objective 𝑗 function per change 𝜖𝑟 𝑘 𝜖𝑟 𝑘 𝑗=1 𝑗=1 in inoperability log 𝑟 𝑘 𝑙 𝑘 𝑦 𝑘 ∗ = 1 Lagrange multiplier Optimal allocation to 𝑨 for budget constraint transportation 𝑘 𝑘 𝜇 ∗ 𝑙 𝑘 Objective function (  =0.8) Optimal allocation for each transportation mode Water Railroad 3 Highway 2 1 0 1 2 3 4 5 6 7 8 9 10 Budget MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 13

Optimal allocation for railroad and highway Comparing allocation to railroad and highway 1 Allocate more to highway 0.8 0.6 Equal k 3 allocation 0.4 0.2 Allocate more to railroad 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k 2 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 14

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 15

When is 𝒜 𝟏 > 𝟏? ∗ 𝑨 0 > 0 if and only if 𝑙 0 ≥ 𝑙 0 Lagrange multiplier Effectiveness of general allocation for budget constraint 0.5 𝜇 ∗ ∗ = 0.45 𝑙 0 ∗ 𝑦 𝑘 𝑛 𝑟 𝑘 exp −𝑙 𝑘 𝑨 𝑘=1 𝑘 * k 0 0.4 0.35 Optimal allocation to transportation 𝑘 1 2 3 4 5 Change in objective Budget Initial inoperability function per change for transportation 𝑘 in inoperability MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 16

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 17

Impact of 𝒍 𝟏 on optimal allocation For larger budgets, allocate higher proportion of budget to 𝑨 0 Proportion of budget to general allocation 1 Budget = 1 0.8 Proportion of budget Budget = 2 allocated to z 0 Budget = 3 0.6 Budget = 4 0.4 0.2 0 0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 k 0 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 18

If 𝒜 𝟏 > 𝟏 Inoperability for 𝑟 𝑘 𝑨 0 = 𝑟 𝑘 exp −𝑙 0 𝑨 0 transportation 𝑘 as a function of 𝑨 0 ∗ as a function of 𝑨 0 Calculate 𝑨 𝑘 Optimal allocation when k 0 =0.4 Objective function (  =0.8) Highway 3 General 2 1 0 1 2 3 4 5 Budget MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 19

Alternative interpretation for 𝑨 0 • 𝑨 0 could represent preparedness activities in advance of the disruption • Initial inoperability decreases as 𝑨 0 increases • But model assumes that planners can prepare for transportation disruption with certainty MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 20

Tradeoff between 𝒍 𝟏 and budget Contour plot of objective function 5 4 3 Budget 2 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 k 0 MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 21

Modeling philosophy 1. Pareto front 2. Solution if 𝑨 0 = 0 3. Conditions when 𝑨 0 = 0 4. Tradeoffs with budget 5. Impact of allocation with respect to time MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 22

Discrete time dynamic model Number of firms Delay for firm i Cost for firm 𝑗 Final time period 𝑢 𝑔 𝑜 𝑜 minimize 𝐾 = α 𝑔 𝑗 𝐫 𝑢 + 1 − α 𝑕 𝑗 𝐫 𝑢 𝑢=0 𝑗=1 𝑗=1 Inoperability as a function of time Inoperability for transportation infrastructure j at time 𝑢 + 1 Effectiveness of allocation at time 𝑢 subject to 𝑟 𝑘 𝑢 + 1 = 𝑟 𝑘 𝑢 exp −𝑙 𝑘 𝑢 𝑨 𝑘 𝑢 − 𝑙 0 𝑢 𝑨 0 𝑢 Allocation to transportation 𝑘 at time 𝑢 Allocation to 𝑢 𝑔 𝑛 general transportation 𝑨 0 𝑢 + 𝑨 𝑘 𝑢 ≤ 𝑎 Budget constraint at time 𝑢 𝑢=0 𝑘=1 𝑨 𝑘 𝑢 ≥ 0 , 𝑨 0 𝑢 ≥ 0 Initial inoperability for 𝐫 0 = 𝐫 transportation infrastructure MacKenzie and Barker, Optimal Resource Allocation for Multimodal Transportation Disruptions 23