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


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.


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


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


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


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


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


Modeling philosophy

1. Pareto front



4. Tradeoffs



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


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


Modeling philosophy

1. Pareto front



4. Tradeoffs



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


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


Modeling philosophy

1. Pareto front



4. Tradeoffs



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


Modeling philosophy

1. Pareto front



4. Tradeoffs



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


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


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


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


Modeling philosophy

1. Pareto front



4. Tradeoffs with budget



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


Dynamic models and effect of time


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


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


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


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


This work was supported by

• The National Science Foundation, Division of Civil, Mechanical, and Manufacturing Innovation, under award 0927299

Email: [email protected]


End of Presentation

contact: [email protected]

optimal resource allocation for recovery from … 111205...d optimal allocation when k 0 = 0.4 water...

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