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

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. Zobel, C.W. (2014). Quantitatively (2003). A framework to quantitatively assess 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

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? 4

Resource allocation model 𝑆 ∗ 𝛾, 𝑌, 𝑈 = 1 − 𝛾𝑌𝑈 𝑈 ∗ Factor as a function of resource allocation decision maximize 𝑆 ∗ 𝛾 𝑨 𝛾 , 𝑌 𝑨 𝑌 , 𝑈 𝑨 𝑈 minimize 𝛾 𝑨 𝛾 ∗ 𝑌 𝑨 𝑌 ∗ 𝑈 𝑨 𝑈 subject to 𝑨 𝛾 + 𝑨 𝑌 + 𝑨 𝑈 ≤ 𝑎 Budget 𝑨 𝛾 , 𝑨 𝑌 , 𝑨 𝑈 ≥ 0 5

Allocation functions • 𝛾 𝑨 𝛾 , 𝑌 𝑨 𝑌 , and 𝑈 𝑨 𝑈 describe ability to allocate resources to reduce each factor of resilience • Requirements – Factor should decrease as more resources are 𝑒𝛾 𝑒𝑌 𝑒𝑈 allocated: 𝑒𝑨 𝛾 , 𝑒𝑨 𝑌 , and 𝑒𝑨 𝑈 are less than 0 – Constant returns or marginal decreasing improvements as more resources are allocated: 𝑒 2 𝛾 𝑒 2 𝑌 𝑒 2 𝑈 2 , 2 , and 2 are greater than or equal to 0 𝑒𝑨 𝛾 𝑒𝑨 𝑌 𝑒𝑨 𝑈 6

Four allocation functions 1. Linear 2. Exponential 3. Quadratic 4. Logarithmic 7

Linear allocation function − 𝑏 𝛾 𝑨 𝛾 𝛾 𝑨 𝛾 = 𝛾 − 𝑏 𝑌 𝑨 𝑌 𝑌 𝑨 𝑌 = 𝑌 − 𝑏 𝑈 𝑨 𝑈 𝑈 𝑨 𝑈 = 𝑈 ≥ 𝑏 𝛾 𝑎 , 𝑌 ≥ 𝑏 𝑌 𝑎 , and 𝑈 ≥ 𝑏 𝑈 𝑎 • Assume 𝛾 • Decision maker should only allocate resources to reduce one of the three resilience factors based on 𝑏 𝛾 𝑏 𝑌 𝑏 𝑈 max , , 𝛾 𝑌 𝑈 • Focuses resources on the factor whose initial parameter is already small and where effectiveness is large 8

Exponential allocation function exp −𝑏 𝛾 𝑨 𝛾 𝛾 𝑨 𝛾 = 𝛾 exp −𝑏 𝑌 𝑨 𝑌 𝑌 𝑨 𝑌 = 𝑌 exp −𝑏 𝑈 𝑨 𝑈 𝑈 𝑨 𝑈 = 𝑈 • Decision maker should only allocate resources to reduce one of the three resilience factors based on max 𝑏 𝛾 , 𝑏 𝑌 , 𝑏 𝑈 • Decision depends only the effectiveness and not the initial values 9

Quadratic allocation function − 𝑐 𝛾 𝑨 𝛾 + 𝑏 𝛾 𝑨 𝛾 2 𝛾 𝑨 𝛾 = 𝛾 − 𝑐 𝑌 𝑨 𝑌 + 𝑏 𝑌 𝑨 𝑌 2 𝑌 𝑨 𝑌 = 𝑌 − 𝑐 𝑈 𝑨 𝑈 + 𝑏 𝑈 𝑨 𝑈 2 𝑈 𝑨 𝑈 = 𝑈 𝑐 𝛾 2𝑏 𝛾 , 𝑨 𝑌 ≤ 𝑐 𝑌 2𝑏 𝑌 , 𝑨 𝑈 ≤ 𝑐 𝑈 Assume 𝑨 𝛾 ≤ 2𝑏 𝑈 so that functions are always decreasing 10

Optimal to allocate to three factors 9.2 2.7 14.1 11

Logarithmic allocation functions − 𝑏 𝛾 log 1 + 𝑐 𝛾 𝑨 𝛾 𝛾 = 𝛾 − 𝑏 𝑌 log 1 + 𝑐 𝑌 𝑨 𝑌 𝑌 = 𝑌 − 𝑏 𝑈 log 1 + 𝑐 𝑈 𝑨 𝑈 𝑈 = 𝑈 12

Optimal to allocate to three factors 8.5 4.0 13.4 13

Simulation ~𝑉(0,1) , 𝑌 ~𝑉(0,1) , 𝑈 ~𝑉(0,30) • 𝛾 • Effectiveness parameters: 𝑏 𝛾 , 𝑐 𝛾 , 𝑏 𝑌 , 𝑐 𝑌 , 𝑏 𝑈 , 𝑐 𝑈 – Uniform distribution – Allocation functions are not negative – Other requirements are met Percent of simulations Quadratic Logarithmic allocation allocation Sufficient conditions 0.4 51 met Optimal to allocate 1.3 91 to all 3 factors 14

Uncertainty with independence , 𝑌 , 𝑈 , 𝑏 𝛾 , 𝑐 𝛾 , 𝑏 𝑌 , 𝑐 𝑌 , 𝑏 𝑈 , 𝑐 𝑈 have known • Assume 𝛾 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 15

Exponential allocation, uncertainty Always a convex optimization problem 16

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 , 𝑌 , and 𝑈 – Allocation may be influenced by 𝛾 • Quadratic and logarithmic: no special properties 17

Uncertainty without probabilities • Each parameter is bounded above and below, ≤ 𝛾 ≤ 𝛾 and 𝑏 𝛾 ≤ 𝑏 𝛾 ≤ 𝑏 𝛾 i.e. 𝛾 • Maxi-min approach maximize min 𝑆 ∗ 𝛾 𝑨 𝛾 , 𝑌 𝑨 𝑌 , 𝑈 𝑨 𝑈 • Same rules as the case with certainty but choose worst-case parameters to determine and 𝑏 𝛾 allocation, i.e. 𝛾 18

Summary Allocation Certainty Uncertainty Uncertainty Uncertainty function with with with no independence dependence probabilities Linear Reduce 1 Reduce 1 Reduce 1 or factor factor 3 factors Exponential Reduce 1 Reduce 1, 2, or Same as case factor 3 factors with certainty but use Quadratic May reduce 3 May reduce 3 Reduce 1, 2, worst-case factors but factors but not or 3 factors parameters not likely likely Logarithmic Often reduce Often reduce 3 3 factors factors Email: camacken@nps.edu 19