Allocating resources to enhance resilience Cameron MacKenzie, - - PowerPoint PPT Presentation

Allocating resources to enhance resilience

Cameron MacKenzie, Assistant Professor,

Defense Resources Management Institute, Naval Postgraduate School

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Disaster resilience

• Disaster resilience is the

ability to (Bruneau et al. 2003)

– Reduce the chances of a shock – Absorb a shock if it occurs – Recover quickly after it occurs

• Nonlinear disaster recovery

(Zobel 2014)

Bruneau, M., Chang, S.E., Eguchi, R.T., Lee, G.C., O’Rourke, T.D., Reinhorn, A.M., Shinozuka, M., Tierney, K., Wallace, W.A., & von Winterfeldt, D. (2003). A framework to quantitatively assess and enhance the seismic resilience of

• communities. Earthquake Spectra, 19(4), 733-

752. Zobel, C.W. (2014). Quantitatively representing nonlinear disaster

• recovery. To appear in Decision

Sciences.

Quantifying disaster resilience

𝑆∗ 𝛾, 𝑌, 𝑈 = 1 − 𝛾𝑌𝑈 𝑈∗

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𝑌 𝑈 𝑈∗ 𝛾

Research questions

• 1. How should a decision maker allocate

resources among the three factors in order to maximize resilience?

• 2. What are possible functions that determine

effectiveness of allocating resources?

• 3. When is it optimal to allocate resources to

reduce all three factors?

• 4. Does the optimal decision change when

there is uncertainty?

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Resource allocation model

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maximize 𝑆∗ 𝛾 𝑨𝛾 , 𝑌 𝑨𝑌 , 𝑈 𝑨𝑈 subject to 𝑨𝛾 + 𝑨𝑌 + 𝑨𝑈 ≤ 𝑎 𝑨𝛾, 𝑨𝑌, 𝑨𝑈 ≥ 0 𝑆∗ 𝛾, 𝑌, 𝑈 = 1 − 𝛾𝑌𝑈 𝑈∗

Factor as a function of resource allocation decision Budget

minimize 𝛾 𝑨𝛾 ∗ 𝑌 𝑨𝑌 ∗ 𝑈 𝑨𝑈

Allocation functions

• 𝛾 𝑨𝛾 , 𝑌 𝑨𝑌 , and 𝑈 𝑨𝑈 describe ability to

allocate resources to reduce each factor of resilience

• Requirements

– Factor should decrease as more resources are allocated:

𝑒𝛾 𝑒𝑨𝛾, 𝑒𝑌 𝑒𝑨𝑌, and 𝑒𝑈 𝑒𝑨𝑈 are less than 0

– Constant returns or marginal decreasing improvements as more resources are allocated:

𝑒2𝛾 𝑒𝑨𝛾

2,

𝑒2𝑌 𝑒𝑨𝑌

2, and

𝑒2𝑈 𝑒𝑨𝑈

2 are greater than or equal to 0

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Four allocation functions

• 1. Linear
• 2. Exponential
• 3. Quadratic
• 4. Logarithmic

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Linear allocation function

𝛾 𝑨𝛾 = 𝛾 − 𝑏𝛾𝑨𝛾 𝑌 𝑨𝑌 = 𝑌 − 𝑏𝑌𝑨𝑌 𝑈 𝑨𝑈 = 𝑈 − 𝑏𝑈𝑨𝑈

• Assume 𝛾

≥ 𝑏𝛾𝑎, 𝑌 ≥ 𝑏𝑌𝑎, and 𝑈 ≥ 𝑏𝑈𝑎

• Decision maker should only allocate resources to

reduce one of the three resilience factors based on max

𝑏𝛾 𝛾 , 𝑏𝑌 𝑌 , 𝑏𝑈 𝑈

• Focuses resources on the factor whose initial

parameter is already small and where effectiveness is large

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Exponential allocation function 𝛾 𝑨𝛾 = 𝛾 exp −𝑏𝛾𝑨𝛾 𝑌 𝑨𝑌 = 𝑌 exp −𝑏𝑌𝑨𝑌 𝑈 𝑨𝑈 = 𝑈 exp −𝑏𝑈𝑨𝑈

• Decision maker should only allocate resources

to reduce one of the three resilience factors based on max 𝑏𝛾, 𝑏𝑌, 𝑏𝑈

• Decision depends only the effectiveness and

not the initial values

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Quadratic allocation function 𝛾 𝑨𝛾 = 𝛾 − 𝑐𝛾𝑨𝛾 + 𝑏𝛾𝑨𝛾

2

𝑌 𝑨𝑌 = 𝑌 − 𝑐𝑌𝑨𝑌 + 𝑏𝑌𝑨𝑌

2

𝑈 𝑨𝑈 = 𝑈 − 𝑐𝑈𝑨𝑈 + 𝑏𝑈𝑨𝑈

2

Assume 𝑨𝛾 ≤

𝑐𝛾 2𝑏𝛾, 𝑨𝑌 ≤ 𝑐𝑌 2𝑏𝑌, 𝑨𝑈 ≤ 𝑐𝑈 2𝑏𝑈 so that

functions are always decreasing

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Optimal to allocate to three factors

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9.2 14.1 2.7

Logarithmic allocation functions 𝛾 = 𝛾 − 𝑏𝛾 log 1 + 𝑐𝛾𝑨𝛾 𝑌 = 𝑌 − 𝑏𝑌 log 1 + 𝑐𝑌𝑨𝑌 𝑈 = 𝑈 − 𝑏𝑈 log 1 + 𝑐𝑈𝑨𝑈

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Optimal to allocate to three factors

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8.5 13.4 4.0

Simulation

• 𝛾

~𝑉(0,1), 𝑌 ~𝑉(0,1), 𝑈 ~𝑉(0,30)

• Effectiveness parameters: 𝑏𝛾, 𝑐𝛾, 𝑏𝑌, 𝑐𝑌, 𝑏𝑈, 𝑐𝑈

– Uniform distribution – Allocation functions are not negative – Other requirements are met

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Quadratic allocation Logarithmic allocation Sufficient conditions met 0.4 51 Optimal to allocate to all 3 factors 1.3 91 Percent of simulations

Uncertainty with independence

• Assume 𝛾

, 𝑌 , 𝑈 , 𝑏𝛾, 𝑐𝛾, 𝑏𝑌, 𝑐𝑌, 𝑏𝑈, 𝑐𝑈 have known distributions

• Assume independence
• Maximize expected resilience 𝐹 𝑆∗ 𝛾, 𝑌, 𝑈

= 1 −

𝐹 𝛾 𝐹 𝑌 𝐹 𝑈 𝑈∗

• Linear and quadratic allocation functions (same as

with certainty)

• Logarithmic allocation function: more likely to

allocate to reduce all three factors than with certainty

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Exponential allocation, uncertainty Always a convex optimization problem

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Uncertainty with dependence

• Assume dependence among uncertain

parameters

• Linear: allocate to reduce one or all three

factors

• Exponential

– Convex optimization problem – May allocate to reduce one, two, or three factors – Allocation may be influenced by 𝛾 , 𝑌 , and 𝑈

• Quadratic and logarithmic: no special

properties

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Uncertainty without probabilities

• Each parameter is bounded above and below,

i.e. 𝛾 ≤ 𝛾 ≤ 𝛾 and 𝑏𝛾 ≤ 𝑏𝛾 ≤ 𝑏𝛾

• Maxi-min approach

maximize min 𝑆∗ 𝛾 𝑨𝛾 , 𝑌 𝑨𝑌 , 𝑈 𝑨𝑈

• Same rules as the case with certainty but

choose worst-case parameters to determine allocation, i.e. 𝛾 and 𝑏𝛾

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Summary

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Allocation function Certainty Uncertainty with independence Uncertainty with dependence Uncertainty with no probabilities Linear Reduce 1 factor Reduce 1 factor Reduce 1 or 3 factors Same as case with certainty but use worst-case parameters Exponential Reduce 1 factor Reduce 1, 2, or 3 factors Reduce 1, 2,

• r 3 factors

Quadratic May reduce 3 factors but not likely May reduce 3 factors but not likely Logarithmic Often reduce 3 factors Often reduce 3 factors Email: camacken@nps.edu