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

β–Ά
allocating resources to enhance resilience
SMART_READER_LITE
LIVE PREVIEW

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

Jul 31, 2023 279 likes β€’514 views

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,

slide-1
SLIDE 1

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

slide-2
SLIDE 2

2

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.

slide-3
SLIDE 3

Quantifying disaster resilience

π‘†βˆ— 𝛾, π‘Œ, π‘ˆ = 1 βˆ’ π›Ύπ‘Œπ‘ˆ π‘ˆβˆ—

3

π‘Œ π‘ˆ π‘ˆβˆ— 𝛾

slide-4
SLIDE 4

Quantifying disaster resilience

π‘†βˆ— π‘Œ , π‘ˆ = 1 βˆ’ π‘Œ π‘ˆ π‘ˆβˆ—

4

π‘Œ π‘ˆ π‘ˆβˆ— π‘Œ

slide-5
SLIDE 5

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

slide-6
SLIDE 6

Resource allocation model

6

maximize π‘†βˆ— π‘Œ π‘¨π‘Œ

, π‘ˆ π‘¨π‘ˆ

subject to π‘¨π‘Œ

+ π‘¨π‘ˆ ≀ π‘Ž

π‘¨π‘Œ

, π‘¨π‘ˆ β‰₯ 0

π‘†βˆ— π‘Œ , π‘ˆ = 1 βˆ’ π‘Œ π‘ˆ π‘ˆβˆ—

Factor as a function of resource allocation decision Budget

minimize π‘Œ π‘¨π‘Œ

βˆ— π‘ˆ π‘¨π‘ˆ

slide-7
SLIDE 7

Allocation functions

  • π‘Œ

π‘¨π‘Œ

and π‘ˆ π‘¨π‘ˆ describe ability to allocate

resources to reduce each factor of resilience

  • Requirements
  • Factor should decrease as more resources are

allocated:

π‘’π‘Œ π‘’π‘¨π‘Œ

and

π‘’π‘ˆ π‘’π‘¨π‘ˆ are less than 0

  • Constant returns or marginal decreasing

improvements as more resources are allocated:

𝑒2π‘Œ π‘’π‘¨π‘Œ

2 and

𝑒2π‘ˆ π‘’π‘¨π‘ˆ

2 are greater than or equal to 0

7

slide-8
SLIDE 8

Four allocation functions

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

8

slide-9
SLIDE 9

Linear allocation function

π‘Œ π‘¨π‘Œ

= π‘Œ

βˆ’ π‘π‘Œ

π‘¨π‘Œ

π‘ˆ π‘¨π‘ˆ = π‘ˆ βˆ’ π‘π‘ˆπ‘¨π‘ˆ

  • Decision maker should only allocate resources to

reduce one resilience factor based on max

π‘π‘Œ π‘Œ , π‘π‘ˆ π‘ˆ

  • Focuses resources on the factor whose initial

parameter is already small and where effectiveness is large

9

slide-10
SLIDE 10

Exponential allocation function π‘Œ π‘¨π‘Œ

= π‘Œ

exp βˆ’π‘π‘Œ

π‘¨π‘Œ

π‘ˆ π‘¨π‘ˆ = π‘ˆ exp βˆ’π‘π‘ˆπ‘¨π‘ˆ

  • Decision maker should only allocate resources

to reduce one resilience factors based on max π‘π‘Œ

, π‘π‘ˆ

  • Decision depends only the effectiveness and

not the initial values

10

slide-11
SLIDE 11

Quadratic allocation function

π‘Œ π‘¨π‘Œ

= π‘Œ

βˆ’ π‘π‘Œ

π‘¨π‘Œ + π‘π‘Œ π‘¨π‘Œ 2

π‘ˆ π‘¨π‘ˆ = π‘ˆ βˆ’ π‘π‘ˆπ‘¨π‘ˆ + π‘π‘ˆπ‘¨π‘ˆ

2

  • Assume π‘¨π‘Œ

≀ π‘π‘Œ 2π‘π‘Œ

, π‘¨π‘ˆ ≀

π‘π‘ˆ 2π‘π‘ˆ so that functions are always

decreasing

11

slide-12
SLIDE 12

Logarithmic allocation functions π‘Œ π‘¨π‘Œ

= π‘Œ

βˆ’ π‘π‘Œ

log 1 + π‘π‘Œ π‘¨π‘Œ

π‘ˆ π‘¨π‘ˆ = π‘ˆ βˆ’ π‘π‘ˆ log 1 + π‘π‘ˆπ‘¨π‘ˆ

12

slide-13
SLIDE 13

Uncertainty with independence

  • Assume π‘Œ

, π‘ˆ , π‘π‘Œ, π‘π‘Œ, π‘π‘ˆ, π‘π‘ˆ have known distributions

  • Assume independence
  • Maximize expected resilience 𝐹 π‘†βˆ— π‘Œ

, π‘ˆ

  • Linear, quadratic, and logarithmic allocation

functions 𝐹 π‘†βˆ— π‘Œ , π‘ˆ = 1 βˆ’ 𝐹 π‘Œ π‘¨π‘Œ

+ 𝐹 π‘ˆ π‘¨π‘ˆ +

π‘ˆβˆ—

  • May be optimal to allocate to reduce both factors

13

slide-14
SLIDE 14
  • Always a convex optimization problem
  • 𝐹

π‘π‘ˆ βˆ’ π‘π‘Œ

exp π‘π‘ˆ βˆ’ π‘π‘Œ π‘¨π‘Œ βˆ’ π‘π‘ˆπ‘Ž

= 0 Exponential allocation, uncertainty

14

slide-15
SLIDE 15

Uncertainty without probabilities

  • Each parameter is bounded above and below,

i.e. π‘Œ ≀ π‘Œ ≀ π‘Œ and π‘π‘Œ

≀ π‘π‘Œ ≀ π‘π‘Œ

  • Maximin approach

maximize min π‘†βˆ— π‘Œ π‘¨π‘Œ

, π‘ˆ π‘¨π‘ˆ

  • Same rules as the case with certainty but

choose worst-case parameters to determine allocation, i.e. π‘Œ and π‘π‘Œ

15

slide-16
SLIDE 16

Uncertainty without probabilities

  • Each parameter is bounded above and below,

i.e. π‘Œ ≀ π‘Œ ≀ π‘Œ and π‘π‘Œ

≀ π‘π‘Œ ≀ π‘π‘Œ

  • Maximin approach

maximize min π‘†βˆ— π‘Œ π‘¨π‘Œ

, π‘ˆ π‘¨π‘ˆ

  • Same rules as the case with certainty but

choose worst-case parameters to determine allocation, i.e. π‘Œ and π‘π‘Œ

16

slide-17
SLIDE 17

Simulation

17

Matching allocation functions to data

Budget

slide-18
SLIDE 18

Simulation results

18

Allocation function Percen- tage Certainty Linear 37 Exponential 6 Quadratic 27 Logarithmic 6 Uncertainty with independence Linear 0.4 Exponential 2 Quadratic Logarithmic Allocation function Percen- tage Uncertainty with dependence Linear 2 Exponential 2 Quadratic 4 Logarithmic 2 Robust allocation Linear 0.4 Exponential 2 Quadratic 0.3 Logarithmic

Percentage of simulations where π‘†βˆ— π‘Œ , π‘ˆ = 1

slide-19
SLIDE 19

Simulation results

19

Allocation function Percen- tage Certainty Linear Exponential Quadratic 65 Logarithmic 56 Uncertainty with independence Linear 9 Exponential 26 Quadratic 85 Logarithmic 85 Allocation function Percen- tage Uncertainty with dependence Linear 18 Exponential 23 Quadratic 55 Logarithmic 82 Robust allocation Linear Exponential Quadratic 46 Logarithmic 55

Percentage of simulations where π‘¨π‘Œ

> 0 and

π‘¨π‘ˆ > 0 given π‘†βˆ— π‘Œ , π‘ˆ < 1

slide-20
SLIDE 20

Simulation results

20

Allocation function Diffe- rence Certainty Linear 10.0 Exponential 10.0 Quadratic 9.0 Logarithmic 8.4 Uncertainty with independence Linear 9.7 Exponential 8.6 Quadratic 4.1 Logarithmic 5.9 Allocation function Diffe- rence Uncertainty with dependence Linear 9.2 Exponential 8.7 Quadratic 7.2 Logarithmic 6.3 Robust allocation Linear 10.0 Exponential 10.0 Quadratic 7.9 Logarithmic 7.7

Average absolute difference between π‘¨π‘Œ

and π‘¨π‘ˆ

slide-21
SLIDE 21

Simulation results

21

Allocation function Resi- lience Certainty Linear 0.98 Exponential 0.98 Quadratic 0.98 Logarithmic 0.98 Uncertainty with independence Linear 0.96 Exponential 0.97 Quadratic 0.96 Logarithmic 0.96 Allocation function Resi- lience Uncertainty with dependence Linear 0.96 Exponential 0.97 Quadratic 0.97 Logarithmic 0.96 Robust allocation Linear 0.82 Exponential 0.89 Quadratic 0.84 Logarithmic 0.80

Average resilience

slide-22
SLIDE 22

Conclusions

  • Assumptions impact optimal allocation
  • Linear or exponential allocation function with certainty οƒ 

allocate entire budget to reduce one factor

  • Quadratic or logarithmic οƒ  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

22

Email: camacken@nps.edu MacKenzie, C.A., & Zobel, C.W. (2014). Allocating resources to enhance

  • resilience. Under review. https://faculty.nps.edu/camacken/