Allocating resources to enhance resilience Cameron MacKenzie, Assistant Professor, Defense Resources Management Institute, Naval Postgraduate School Christopher Zobel , Professor of Business Information Technology, Pamplin College of Business, Virginia Tech
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 Zobel, C.W. (2014). Quantitatively representing nonlinear disaster and enhance the seismic resilience of recovery. To appear in Decision communities. Earthquake Spectra , 19(4), 733- Sciences . 752. 2
Quantifying disaster resilience 𝑆 ∗ 𝛾, 𝑌, 𝑈 = 1 − 𝛾𝑌𝑈 𝑈 ∗ 𝛾 𝑌 𝑈 𝑈 ∗ 3
Quantifying disaster resilience 𝑈 , 𝑈 = 1 − 𝑌 𝑆 ∗ 𝑌 𝑈 ∗ 𝑌 𝑌 𝑈 𝑈 ∗ 4
Research questions 1. How should a decision maker allocate resources between reducing loss and decreasing time in order to maximize resilience? 2. What are possible functions that determine effectiveness of allocating resources? 3. How should the allocation change based on the assumptions in the allocation functions? 4. Does the optimal decision change when there is uncertainty? 5
Resource allocation model 𝑈 , 𝑈 = 1 − 𝑌 𝑆 ∗ 𝑌 𝑈 ∗ Factor as a function of resource allocation decision 𝑨 𝑌 maximize 𝑆 ∗ 𝑌 , 𝑈 𝑨 𝑈 𝑨 𝑌 minimize 𝑌 ∗ 𝑈 𝑨 𝑈 subject to 𝑨 𝑌 + 𝑨 𝑈 ≤ 𝑎 Budget 𝑨 𝑌 , 𝑨 𝑈 ≥ 0 6
Allocation functions 𝑨 𝑌 • 𝑌 and 𝑈 𝑨 𝑈 describe ability to allocate resources to reduce each factor of resilience • Requirements Factor should decrease as more resources are 𝑒𝑌 𝑒𝑈 allocated: and 𝑒𝑨 𝑈 are less than 0 𝑒𝑨 𝑌 Constant returns or marginal decreasing improvements as more resources are allocated: 𝑒 2 𝑌 𝑒 2 𝑈 2 and 2 are greater than or equal to 0 𝑒𝑨 𝑌 𝑒𝑨 𝑈 7
Four allocation functions 1. Linear 2. Exponential 3. Quadratic 4. Logarithmic 8
Linear allocation function 𝑨 𝑌 − 𝑏 𝑌 𝑌 = 𝑌 𝑨 𝑌 − 𝑏 𝑈 𝑨 𝑈 𝑈 𝑨 𝑈 = 𝑈 • Decision maker should only allocate resources to 𝑏 𝑌 𝑏 𝑈 reduce one resilience factor based on max , 𝑌 𝑈 • Focuses resources on the factor whose initial parameter is already small and where effectiveness is large 9
Exponential allocation function 𝑨 𝑌 exp −𝑏 𝑌 𝑌 = 𝑌 𝑨 𝑌 exp −𝑏 𝑈 𝑨 𝑈 𝑈 𝑨 𝑈 = 𝑈 • Decision maker should only allocate resources to reduce one resilience factors based on max 𝑏 𝑌 , 𝑏 𝑈 • Decision depends only the effectiveness and not the initial values 10
Quadratic allocation function 𝑨 𝑌 − 𝑐 𝑌 2 𝑌 = 𝑌 𝑨 𝑌 + 𝑏 𝑌 𝑨 𝑌 − 𝑐 𝑈 𝑨 𝑈 + 𝑏 𝑈 𝑨 𝑈 2 𝑈 𝑨 𝑈 = 𝑈 𝑐 𝑌 𝑐 𝑈 • Assume 𝑨 𝑌 ≤ , 𝑨 𝑈 ≤ 2𝑏 𝑈 so that functions are always 2𝑏 𝑌 decreasing 11
Logarithmic allocation functions 𝑨 𝑌 − 𝑏 𝑌 𝑌 = 𝑌 log 1 + 𝑐 𝑌 𝑨 𝑌 − 𝑏 𝑈 log 1 + 𝑐 𝑈 𝑨 𝑈 𝑈 𝑨 𝑈 = 𝑈 12
Uncertainty with independence , 𝑈 , 𝑏 𝑌 , 𝑐 𝑌 , 𝑏 𝑈 , 𝑐 𝑈 have known • Assume 𝑌 distributions • Assume independence , 𝑈 • Maximize expected resilience 𝐹 𝑆 ∗ 𝑌 • Linear, quadratic, and logarithmic allocation functions + 𝐹 𝑈 𝑨 𝑈 + 𝑨 𝑌 = 1 − 𝐹 𝑌 , 𝑈 𝐹 𝑆 ∗ 𝑌 𝑈 ∗ May be optimal to allocate to reduce both factors 13
Exponential allocation, uncertainty • Always a convex optimization problem • 𝐹 𝑏 𝑈 − 𝑏 𝑌 exp 𝑏 𝑈 − 𝑏 𝑌 𝑨 𝑌 − 𝑏 𝑈 𝑎 = 0 14
Uncertainty without probabilities • Each parameter is bounded above and below, and 𝑏 𝑌 ≤ 𝑌 ≤ 𝑌 i.e. 𝑌 ≤ 𝑏 𝑌 ≤ 𝑏 𝑌 • Maximin approach 𝑨 𝑌 maximize min 𝑆 ∗ 𝑌 , 𝑈 𝑨 𝑈 • Same rules as the case with certainty but choose worst-case parameters to determine and 𝑏 𝑌 allocation, i.e. 𝑌 15
Uncertainty without probabilities • Each parameter is bounded above and below, and 𝑏 𝑌 ≤ 𝑌 ≤ 𝑌 i.e. 𝑌 ≤ 𝑏 𝑌 ≤ 𝑏 𝑌 • Maximin approach 𝑨 𝑌 maximize min 𝑆 ∗ 𝑌 , 𝑈 𝑨 𝑈 • Same rules as the case with certainty but choose worst-case parameters to determine and 𝑏 𝑌 allocation, i.e. 𝑌 16
Simulation Matching allocation functions to data Budget 17
Simulation results , 𝑈 = 1 Percentage of simulations where 𝑆 ∗ 𝑌 Allocation Percen- Allocation Percen- function tage function tage Linear 37 Linear 2 Uncertainty Exponential 6 Exponential 2 Certainty with Quadratic 27 Quadratic 4 dependence Logarithmic 6 Logarithmic 2 Linear 0.4 Linear 0.4 Uncertainty Exponential 2 Exponential 2 Robust with allocation Quadratic 0 Quadratic 0.3 independence Logarithmic 0 Logarithmic 0 18
Simulation results Percentage of simulations where 𝑨 𝑌 > 0 and , 𝑈 < 1 𝑨 𝑈 > 0 given 𝑆 ∗ 𝑌 Allocation Percen- Allocation Percen- function tage function tage Linear 0 Linear 18 Uncertainty Exponential 0 Exponential 23 Certainty with Quadratic 65 Quadratic 55 dependence Logarithmic 56 Logarithmic 82 Linear 9 Linear 0 Uncertainty Exponential 26 Exponential 0 Robust with allocation Quadratic 85 Quadratic 46 independence Logarithmic 85 Logarithmic 55 19
Simulation results Average absolute difference between 𝑨 𝑌 and 𝑨 𝑈 Allocation Diffe- Allocation Diffe- function rence function rence Linear 10.0 Linear 9.2 Uncertainty Exponential 10.0 Exponential 8.7 Certainty with Quadratic 9.0 Quadratic 7.2 dependence Logarithmic 8.4 Logarithmic 6.3 Linear 9.7 Linear 10.0 Uncertainty Exponential 8.6 Exponential 10.0 Robust with allocation Quadratic 4.1 Quadratic 7.9 independence Logarithmic 5.9 Logarithmic 7.7 20
Simulation results Average resilience Allocation Resi- Allocation Resi- function lience function lience Linear 0.98 Linear 0.96 Uncertainty Exponential 0.98 Exponential 0.97 Certainty with Quadratic 0.98 Quadratic 0.97 dependence Logarithmic 0.98 Logarithmic 0.96 Linear 0.96 Linear 0.82 Uncertainty Exponential 0.97 Exponential 0.89 Robust with allocation Quadratic 0.96 Quadratic 0.84 independence Logarithmic 0.96 Logarithmic 0.80 21
Conclusions • Assumptions impact optimal allocation Linear or exponential allocation function with certainty allocate entire budget to reduce one factor Quadratic or logarithmic may allocate to reduce both factors • Heuristics Focus resources on small initial value and large effectiveness Uncertainty: divide resources approximately equal manner if marginal benefits decrease rapidly • Future work Apply allocation model to specific projects Resources can improve both factors simultaneously MacKenzie, C.A., & Zobel, C.W. (2014). Allocating resources to enhance resilience. Under review. https://faculty.nps.edu/camacken/ Email: camacken@nps.edu 22
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