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

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

Transportation disruptions

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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.

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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

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Static model

minimize 𝑔

𝑗 𝐫 𝑜 𝑗=1

𝑨

𝑘 ≥ 0, 𝑨0 ≥ 0

𝑕𝑗 𝐫

𝑜 𝑗=1

subject to 𝑟𝑘 = 𝑟 𝑘exp −𝑙𝑘𝑨

𝑘 − 𝑙0𝑨0

𝑨0 + 𝑨

𝑘 𝑛 𝑘=1

≤ 𝑎

Cost for firm 𝑗 Delay for firm i Vector (length 𝑛) of inoperability for each transportation infrastructure, mode, route Number of firms Initial inoperability for transportation infrastructure 𝑘 Effectiveness of allocating to transportation j Allocation to transportation 𝑘 Total budget Allocation to general transportation Effectiveness of general allocation

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Illustrative example

Inoperability (𝒓𝒌) 𝒍𝒌 Water 0.3 1 Railroad 0.3 1 Highway 0.3 1 Water Water Railroad Railroad Highway Highway Origin Destination Transfer point

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Illustrative example

𝑔

𝑗 𝐫 = 𝑑𝑗,𝑥𝑟𝑥 + 𝑑𝑗,𝑠𝑟𝑠 + 𝑑𝑗,ℎ𝑟ℎ

Linear cost function for firm 𝑗 𝑑𝑗,𝑥 𝑑𝑗,𝑠 𝑑𝑗,ℎ Firm 1 3 1 1 Firm 2 3 2 Firm 3 5 𝑕𝑗 𝐫 = 𝑒𝑗,𝑥𝑟𝑥 + 𝑒𝑗,𝑠𝑟𝑠 + 𝑒𝑗,ℎ𝑟ℎ Linear delay function for firm 𝑗 𝑒𝑗,𝑥 𝒆𝑗,𝑠 𝒆𝑗,ℎ Firm 1 1 2 1 Firm 2 3 1 Firm 3 1 Water Railroad Highway

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs
• 5. Impact of allocation with respect to time

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs
• 5. Impact of allocation with respect to time

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Pareto front

Create Pareto front for cost and delay

minimize α 𝑔

𝑗 𝐫 𝑜 𝑗=1

+ 1 − α 𝑕𝑗 𝐫

𝑜 𝑗=1

0 ≤ 𝛽 ≤ 1

Cost for firm 𝑗 Delay for firm 𝑗 Vector (length 𝑛) of inoperability for each transportation infrastructure, mode, route Tradeoff parameter between cost and delay

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1.8 2 2.2 2.4 2.6 2.8 3 3.2 3.4 1 1.2 1.4 1.6 1.8 2 Cost (money) Delay (time)

Pareto front for different budgets

Budget = 2 Budget = 1.5 Budget = 1

minimize α 𝑔

𝑗 𝐫 𝑜 𝑗=1

+ 1 − α 𝑕𝑗 𝐫

𝑜 𝑗=1

Budget = 1.25 Budget = 1.75

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs
• 5. Impact of allocation with respect to time

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1 2 3 1 2 3 4 5 6 7 8 9 10

Optimal allocation for each transportation mode

Objective function (=0.8) Budget Water Railroad Highway

If 𝒜𝟏 = 𝟏

𝑨

𝑘 ∗ = 1

𝑙𝑘 log 𝑟 𝑘𝑙𝑘𝑦𝑘 𝜇∗ 𝑦𝑘 = 𝛽 𝜖𝑔

𝑗

𝜖𝑟𝑘

𝑜 𝑗=1

+ 1 − 𝛽 𝜖𝑕𝑗 𝜖𝑟𝑘

𝑜 𝑗=1

Change in objective function per change in inoperability Lagrange multiplier for budget constraint Optimal allocation to transportation 𝑘

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Optimal allocation for railroad and highway

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0.2 0.4 0.6 0.8 1 k2 k3

Allocate more to highway Allocate more to railroad Equal allocation

Comparing allocation to railroad and highway

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs
• 5. Impact of allocation with respect to time

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When is 𝒜𝟏 > 𝟏?

𝑨0 > 0 if and only if 𝑙0 ≥ 𝑙0

∗ 𝑙0

∗ =

𝜇∗ 𝑟 𝑘exp −𝑙𝑘𝑨

𝑘 ∗ 𝑦𝑘 𝑛 𝑘=1

Lagrange multiplier for budget constraint Optimal allocation to transportation 𝑘 Change in objective function per change in inoperability Initial inoperability for transportation 𝑘 1 2 3 4 5 0.35 0.4 0.45 0.5 Budget k0

*

Effectiveness of general allocation

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs
• 5. Impact of allocation with respect to time

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Impact of 𝒍𝟏 on optimal allocation

For larger budgets, allocate higher proportion of budget to 𝑨0

0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.6 0.2 0.4 0.6 0.8 1 k0 Proportion of budget allocated to z0

Proportion of budget to general allocation

Budget = 1 Budget = 2 Budget = 3 Budget = 4

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If 𝒜𝟏 > 𝟏

Calculate 𝑨

𝑘 ∗ as a function of 𝑨0

𝑟 𝑘 𝑨0 = 𝑟 𝑘exp −𝑙0𝑨0

Inoperability for transportation 𝑘 as a function of 𝑨0

1 2 3 1 2 3 4 5 Objective function (=0.8) Budget

Optimal allocation when k0=0.4

Highway General

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Alternative interpretation for 𝑨0

• 𝑨0 could represent preparedness activities in

advance of the disruption

• Initial inoperability decreases as 𝑨0 increases
• But model assumes that planners can prepare

for transportation disruption with certainty

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Tradeoff between 𝒍𝟏 and budget

k0 Budget

Contour plot of objective function

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 1 2 3 4 5

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Modeling philosophy

• 1. Pareto front
• 2. Solution if 𝑨0 = 0
• 3. Conditions when 𝑨0 = 0
• 4. Tradeoffs with budget
• 5. Impact of allocation with respect to time

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Discrete time dynamic model

minimize 𝐾 = α 𝑔

𝑗 𝐫 𝑢 𝑜 𝑗=1

+ 1 − α 𝑕𝑗 𝐫 𝑢

𝑜 𝑗=1 𝑢𝑔 𝑢=0

𝑨

𝑘 𝑢 ≥ 0, 𝑨0 𝑢 ≥ 0

subject to 𝑟𝑘 𝑢 + 1 = 𝑟𝑘 𝑢 exp −𝑙𝑘 𝑢 𝑨

𝑘 𝑢 − 𝑙0 𝑢 𝑨0 𝑢

𝑨0 𝑢 + 𝑨

𝑘 𝑢 𝑛 𝑘=1 𝑢𝑔 𝑢=0

≤ 𝑎

𝐫 0 = 𝐫 Cost for firm 𝑗 Delay for firm i Inoperability as a function of time Number of firms Initial inoperability for transportation infrastructure Allocation to transportation 𝑘 at time 𝑢 Allocation to general transportation at time 𝑢 Effectiveness of allocation at time 𝑢 Inoperability for transportation infrastructure j at time 𝑢 + 1 Final time period Budget constraint

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Dynamic models and effect of time

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Dynamic models and effect of time

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Dynamic models and effect of time

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Discrete time dynamic model

minimize 𝐾 = α 𝑔

𝑗 𝐫 𝑢 𝑜 𝑗=1

+ 1 − α 𝑕𝑗 𝐫 𝑢

𝑜 𝑗=1 𝑢𝑔 𝑢=0

𝑨

𝑘 𝑢 ≥ 0, 𝑨0 𝑢 ≥ 0

subject to 𝑟𝑘 𝑢 + 1 = 𝑟𝑘 𝑢 exp −𝑢𝑙𝑘𝑨

𝑘 𝑢 − 𝑢𝑙0𝑨0 𝑢

𝑨0 𝑢 + 𝑨

𝑘 𝑢 𝑛 𝑘=1 𝑢𝑔 𝑢=0

≤ 𝑎 𝐫 0 = 𝐫

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Results

1 2 3 4 5 6 7 8 9 10 5 10 2 4 6 Optimal allocation when k0 = 0 1 2 3 4 5 6 7 8 9 10 5 10 2 4 6 Optimal allocation when k0 = 0.4 Number of periods since disruption Budget Allocation per period

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Comparing allocations when budget = 2

1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 Optimal allocation when k0= 0 1 2 3 4 5 6 7 8 9 10 0.2 0.4 0.6 Number of periods since disruption Allocation per period Optimal allocation when k0= 0.4 Water Railroad Highway General

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Conclusions

• Static model

– Optimal allocation as function of initial inoperability, effectiveness of allocation, and importance of transportation infrastructure to firms – Solution approach based on effectiveness of allocation to 𝑨0 – Different tradeoffs in model illustrated with numerical example

• Dynamic model

– If effectiveness of resources is constant or decreases with time, optimal to allocate immediately – If effectiveness of resources increases with time, may be desirable to wait to allocate

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This work was supported by

• The National Science Foundation, Division of

Civil, Mechanical, and Manufacturing Innovation, under award 0927299 Email: cmackenzie@ou.edu

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End of Presentation

contact: cmackenzie@ou.edu