Optimal Resource Allocation for Recovery from … 111205...d Optimal allocation when k 0 = 0.4 Water...
Transcript of Optimal Resource Allocation for Recovery from … 111205...d Optimal allocation when k 0 = 0.4 Water...
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 𝑓𝑖 𝐪
𝑛
𝑖=1
𝑧𝑗 ≥ 0, 𝑧0 ≥ 0
𝑔𝑖 𝐪
𝑛
𝑖=1
subject to 𝑞𝑗 = 𝑞 𝑗exp −𝑘𝑗𝑧𝑗 − 𝑘0𝑧0
𝑧0 + 𝑧𝑗
𝑚
𝑗=1
≤ 𝑍
Cost for firm 𝑖
Delay for firm i
Vector (length 𝑚) of inoperability for each transportation infrastructure, mode, route
Number of firms
Initial inoperability for transportation infrastructure 𝑗
Effectiveness of allocating to transportation j Allocation to transportation 𝑗
Total budget
Allocation to general transportation
Effectiveness of general allocation
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Illustrative example
Inoperability (𝒒𝒋) 𝒌𝒋
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 𝑓𝑖 𝐪 = 𝑐𝑖,𝑤𝑞𝑤 + 𝑐𝑖,𝑟𝑞𝑟 + 𝑐𝑖,ℎ𝑞ℎ Linear cost function for firm 𝑖
𝑐𝑖,𝑤 𝑐𝑖,𝑟 𝑐𝑖,ℎ
Firm 1 3 1 1
Firm 2 0 3 2
Firm 3 0 0 5
𝑔𝑖 𝐪 = 𝑑𝑖,𝑤𝑞𝑤 + 𝑑𝑖,𝑟𝑞𝑟 + 𝑑𝑖,ℎ𝑞ℎ Linear delay function for firm 𝑖
𝑑𝑖,𝑤 𝒅𝑖,𝑟 𝒅𝑖,ℎ
Firm 1 1 2 1
Firm 2 0 3 1
Firm 3 0 0 1
Water Railroad Highway
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Modeling philosophy
1. Pareto front
2. Solution if 𝑧0 = 0
3. Conditions when 𝑧0 = 0
4. Tradeoffs
5. Impact of allocation with respect to time
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Modeling philosophy
1. Pareto front
2. Solution if 𝑧0 = 0
3. Conditions when 𝑧0 = 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 𝑓𝑖 𝐪
𝑛
𝑖=1
+ 1 − α 𝑔𝑖 𝐪
𝑛
𝑖=1
0 ≤ 𝛼 ≤ 1
Cost for firm 𝑖 Delay for firm 𝑖 Vector (length 𝑚) 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 3.41
1.2
1.4
1.6
1.8
2
Cost (money)
Del
ay (
tim
e)Pareto front for different budgets
Budget = 2
Budget = 1.5
Budget = 1
minimize α 𝑓𝑖 𝐪
𝑛
𝑖=1
+ 1 − α 𝑔𝑖 𝐪
𝑛
𝑖=1
Budget = 1.25
Budget = 1.75
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Modeling philosophy
1. Pareto front
2. Solution if 𝑧0 = 0
3. Conditions when 𝑧0 = 0
4. Tradeoffs
5. Impact of allocation with respect to time
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1
2
3
0 1 2 3 4 5 6 7 8 9 10
Optimal allocation for each transportation mode
Ob
ject
ive
fun
ctio
n (
=0.8
)
Budget
Water
Railroad
Highway
If 𝒛𝟎 = 𝟎
𝑧𝑗∗ =
1
𝑘𝑗log
𝑞 𝑗𝑘𝑗𝑥𝑗
𝜆∗
𝑥𝑗 = 𝛼 𝜕𝑓𝑖𝜕𝑞𝑗
𝑛
𝑖=1
+ 1 − 𝛼 𝜕𝑔𝑖𝜕𝑞𝑗
𝑛
𝑖=1
Change in objective function per change in inoperability
Lagrange multiplier for budget constraint
Optimal allocation to transportation 𝑗
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Optimal allocation for railroad and highway
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
k2
k 3
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 𝑧0 = 0
3. Conditions when 𝑧0 = 0
4. Tradeoffs
5. Impact of allocation with respect to time
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When is 𝒛𝟎 > 𝟎?
𝑧0 > 0 if and only if 𝑘0 ≥ 𝑘0∗
𝑘0∗ =
𝜆∗
𝑞 𝑗exp −𝑘𝑗𝑧𝑗∗ 𝑥𝑗
𝑚𝑗=1
Lagrange multiplier for budget constraint
Optimal allocation to transportation 𝑗
Change in objective function per change in inoperability
Initial inoperability for transportation 𝑗
1 2 3 4 5
0.35
0.4
0.45
0.5
Budget
k 0*
Effectiveness of general allocation
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Modeling philosophy
1. Pareto front
2. Solution if 𝑧0 = 0
3. Conditions when 𝑧0 = 0
4. Tradeoffs
5. Impact of allocation with respect to time
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Impact of 𝒌𝟎 on optimal allocation
For larger budgets, allocate higher proportion of budget to 𝑧0
0.25 0.3 0.35 0.4 0.45 0.5 0.55 0.60
0.2
0.4
0.6
0.8
1
k0
Pro
po
rtio
n o
f b
ud
get
allo
cate
d t
o z
0
Proportion of budget to general allocation
Budget = 1
Budget = 2
Budget = 3
Budget = 4
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If 𝒛𝟎 > 𝟎
Calculate 𝑧𝑗∗ as a function of 𝑧0
𝑞 𝑗 𝑧0 = 𝑞 𝑗exp −𝑘0𝑧0 Inoperability for transportation 𝑗 as a function of 𝑧0
1
2
3
0 1 2 3 4 5Ob
ject
ive
fun
ctio
n (
=0.8
)
Budget
Optimal allocation when k0=0.4
Highway
General
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Alternative interpretation for 𝑧0
• 𝑧0 could represent preparedness activities in advance of the disruption
• Initial inoperability decreases as 𝑧0 increases
• But model assumes that planners can prepare for transportation disruption with certainty
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Tradeoff between 𝒌𝟎 and budget
k0
Bu
dge
t
Contour plot of objective function
0 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 𝑧0 = 0
3. Conditions when 𝑧0 = 0
4. Tradeoffs with budget
5. Impact of allocation with respect to time
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Discrete time dynamic model
minimize 𝐽 = α 𝑓𝑖 𝐪 𝑡
𝑛
𝑖=1
+ 1 − α 𝑔𝑖 𝐪 𝑡
𝑛
𝑖=1
𝑡𝑓
𝑡=0
𝑧𝑗 𝑡 ≥ 0, 𝑧0 𝑡 ≥ 0
subject to 𝑞𝑗 𝑡 + 1 = 𝑞𝑗 𝑡 exp −𝑘𝑗 𝑡 𝑧𝑗 𝑡 − 𝑘0 𝑡 𝑧0 𝑡
𝑧0 𝑡 + 𝑧𝑗 𝑡
𝑚
𝑗=1
𝑡𝑓
𝑡=0
≤ 𝑍
𝐪 0 = 𝐪
Cost for firm 𝑖 Delay for firm i
Inoperability as a function of time
Number of firms
Initial inoperability for transportation infrastructure
Allocation to transportation 𝑗 at time 𝑡 Allocation to general transportation at time 𝑡
Effectiveness of allocation at time 𝑡
Inoperability for transportation infrastructure j at time 𝑡 + 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 𝐽 = α 𝑓𝑖 𝐪 𝑡
𝑛
𝑖=1
+ 1 − α 𝑔𝑖 𝐪 𝑡
𝑛
𝑖=1
𝑡𝑓
𝑡=0
𝑧𝑗 𝑡 ≥ 0, 𝑧0 𝑡 ≥ 0
subject to 𝑞𝑗 𝑡 + 1 = 𝑞𝑗 𝑡 exp −𝑡𝑘𝑗𝑧𝑗 𝑡 − 𝑡𝑘0𝑧0 𝑡
𝑧0 𝑡 + 𝑧𝑗 𝑡
𝑚
𝑗=1
𝑡𝑓
𝑡=0
≤ 𝑍
𝐪 0 = 𝐪
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Results
1 2 3 4 5 6 7 8 9 1005
100
2
4
6
Optimal allocation when k0 = 0
1 2 3 4 5 6 7 8 9 1005
100
2
4
6
Optimal allocation when k0 = 0.4
Number of periods since disruptionBudget
Allo
cati
on
per
per
iod
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Comparing allocations when budget = 2
1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
Optimal allocation when k0= 0
1 2 3 4 5 6 7 8 9 100
0.2
0.4
0.6
Number of periods since disruption
Allo
cati
on
per
per
iod
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 𝑧0
– 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: [email protected]
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End of Presentation
contact: [email protected]