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

𝑅∗ 𝛽, 𝑋, 𝑇 = 1 −𝛽𝑋𝑇

𝑇∗

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𝑋

𝑇

𝑇∗

𝛽

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 𝑅∗ 𝛽 𝑧𝛽 , 𝑋 𝑧𝑋 , 𝑇 𝑧𝑇

subject to 𝑧𝛽 + 𝑧𝑋 + 𝑧𝑇 ≤ 𝑍

𝑧𝛽 , 𝑧𝑋, 𝑧𝑇 ≥ 0

𝑅∗ 𝛽, 𝑋, 𝑇 = 1 −𝛽𝑋𝑇

𝑇∗

Factor as a function of resource allocation decision

Budget

minimize 𝛽 𝑧𝛽 ∗ 𝑋 𝑧𝑋 ∗ 𝑇 𝑧𝑇

Allocation functions

• 𝛽 𝑧𝛽 , 𝑋 𝑧𝑋 , and 𝑇 𝑧𝑇 describe ability to

allocate resources to reduce each factor of resilience

• Requirements

– Factor should decrease as more resources are

allocated:𝑑𝛽

𝑑𝑧𝛽,

𝑑𝑋

𝑑𝑧𝑋, and

𝑑𝑇

𝑑𝑧𝑇 are less than 0

– Constant returns or marginal decreasing improvements as more resources are allocated: 𝑑2𝛽

𝑑𝑧𝛽2 ,

𝑑2𝑋

𝑑𝑧𝑋2 , and

𝑑2𝑇

𝑑𝑧𝑇2 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

𝛽 𝑧𝛽 = 𝛽 − 𝑎𝛽𝑧𝛽

𝑋 𝑧𝑋 = 𝑋 − 𝑎𝑋𝑧𝑋 𝑇 𝑧𝑇 = 𝑇 − 𝑎𝑇𝑧𝑇

• Assume 𝛽 ≥ 𝑎𝛽𝑍, 𝑋 ≥ 𝑎𝑋𝑍, and 𝑇 ≥ 𝑎𝑇𝑍

• Decision maker should only allocate resources to reduce one of the three resilience factors based on

max𝑎𝛽

𝛽 ,𝑎𝑋

𝑋 ,𝑎𝑇

𝑇

• Focuses resources on the factor whose initial parameter is already small and where effectiveness is large

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Exponential allocation function

𝛽 𝑧𝛽 = 𝛽 exp −𝑎𝛽𝑧𝛽

𝑋 𝑧𝑋 = 𝑋 exp −𝑎𝑋𝑧𝑋 𝑇 𝑧𝑇 = 𝑇 exp −𝑎𝑇𝑧𝑇

• Decision maker should only allocate resources to reduce one of the three resilience factors

based on max 𝑎𝛽 , 𝑎𝑋, 𝑎𝑇

• Decision depends only the effectiveness and not the initial values

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Quadratic allocation function

𝛽 𝑧𝛽 = 𝛽 − 𝑏𝛽𝑧𝛽 + 𝑎𝛽𝑧𝛽2

𝑋 𝑧𝑋 = 𝑋 − 𝑏𝑋𝑧𝑋 + 𝑎𝑋𝑧𝑋2

𝑇 𝑧𝑇 = 𝑇 − 𝑏𝑇𝑧𝑇 + 𝑎𝑇𝑧𝑇2

Assume 𝑧𝛽 ≤𝑏𝛽

2𝑎𝛽, 𝑧𝑋 ≤

𝑏𝑋

2𝑎𝑋, 𝑧𝑇 ≤

𝑏𝑇

2𝑎𝑇 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

𝛽 = 𝛽 − 𝑎𝛽 log 1 + 𝑏𝛽𝑧𝛽

𝑋 = 𝑋 − 𝑎𝑋 log 1 + 𝑏𝑋𝑧𝑋 𝑇 = 𝑇 − 𝑎𝑇 log 1 + 𝑏𝑇𝑧𝑇

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Optimal to allocate to three factors

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8.5 13.4 4.0

Simulation

• 𝛽 ~𝑈(0,1), 𝑋 ~𝑈(0,1), 𝑇 ~𝑈(0,30)

• Effectiveness parameters: 𝑎𝛽, 𝑏𝛽, 𝑎𝑋, 𝑏𝑋, 𝑎𝑇, 𝑏𝑇

– 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 𝛽 , 𝑋 , 𝑇 , 𝑎𝛽, 𝑏𝛽, 𝑎𝑋, 𝑏𝑋, 𝑎𝑇, 𝑏𝑇 have known

distributions

• Assume independence

• Maximize expected resilience 𝐸 𝑅∗ 𝛽, 𝑋, 𝑇 = 1 −𝐸 𝛽 𝐸 𝑋 𝐸 𝑇

𝑇∗

• 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 𝛽 , 𝑋 , and 𝑇

• Quadratic and logarithmic: no special properties

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Uncertainty without probabilities

• Each parameter is bounded above and below,

i.e. 𝛽 ≤ 𝛽 ≤ 𝛽 and 𝑎𝛽 ≤ 𝑎𝛽 ≤ 𝑎𝛽

• Maxi-min approach

maximize min 𝑅∗ 𝛽 𝑧𝛽 , 𝑋 𝑧𝑋 , 𝑇 𝑧𝑇

• Same rules as the case with certainty but choose worst-case parameters to determine

allocation, i.e. 𝛽 and 𝑎𝛽

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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, or 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