Multi-Objective Optimization Methods for Optimal Funding Allocations to Mitigate Chemical and Biological Attacks

Roshan Rammohan, Ryan SchnalzerMahmoud Reda Taha, Tim Ross and Frank GilfeatherUniversity of New Mexico

Ram PrasadNew Mexico State University

OutlineOutlineIntroduction

MIDST: Exploration Mode

MIDST: Optimization mode

Alterative Optimization Methods

Case study

Conclusions

IntroductionIntroduction

What is the optimal budget $B and its distribution to N investment units in order to reduce the consequences of S number of CB events?

MIDST: Problem Statement

IntroductionIntroduction

$ $ $ $ $ $ $ $

$

Consequences “No Investment”

Total $ Budget

θ θ θ θ θ θ θ θ

Investment Units “N”

Cex

CB event Class

ExpectedConsequences

All possible individual CB events

“Effectiveness”

IntroductionIntroduction

MIDST: Exploration ModeMIDST: Exploration Mode

Data Cards

Interpolate

Investment unit Effectiveness

$X1

$X2

$XmFuse Abilities Overall

Effectiveness

LnL2L1

I1 InI2

Expected Consequences

Expectation

Soliciting Information: Data CardsSoliciting Information: Data Cards

Soliciting Information: Data CardsSoliciting Information: Data Cards

Establishing effectivity functionEstablishing effectivity functionPolynomial or spline interpolationMultivariate interpolation (See Prasad et al. tommorow!)

$X Funding

e (Effectivity)

Establishing effectivity functionEstablishing effectivity functionUsing this method we establish the matrix of effectivity

⎪⎪⎪

⎭

⎪⎪⎪

⎬

⎫

⎪⎪⎪

⎩

⎪⎪⎪

⎨

⎧

=

N,Si,S2,S1,S

N,mi,m2,m1,m

N,2i,22,21,2

N,1i,12,11,1

eeee

eeeeeeeeeeee

e

K

MMMMM

KK

KK

KK

For N: investment units and S: CB events

Fusing EffectivitiesFusing EffectivitiesConsidering the interaction between IUs on the final

consequences we have to fuse these effectivities

Many fusion operators exist. Example 2D fusion:

Very conservative Very optimistic

Expected ConsequencesExpected ConsequencesThe fusion operation results in

{ }fNS

fNm

fN3

fN2

fN1

fN eee,e,ee KK=For S: CB events

The expected consequence for each CB event can be computed as

( ) 0m

km

km Ce1C −=

For each CB eventConsidering the likelihoods of the CB events we can

compute the overall expected consequences as

∑==

S

1m

kmm

k CLCVector of consequences at $k investment

MIDST: Optimization ModeMIDST: Optimization Mode

Data Card

Interpolate

Investment unit Effectiveness

$X1

$X2

$Xm

Fuse Abilities Overall Effectiveness

Optimize and Rank Order

LnL2L1

I1 InI2

Expected Consequences

Expectation

Optimal Solution

Y

X

C

If we have a bimodal surface

Minimum

consequences

What does optimization mean?What does optimization mean?

We need to identify “x” that results in minimum “C”

Our optimization challenges are

- The surface of our function is not bimodal

-There might be many local minima

-There is more than one objective and they are not necessary achievable all together

- Computing time, space and accuracy resolution

- Practical interests

- Derivative based optimization- Gradient descent method- Levenberg Marquadrt- Many other

- Non-derivative based optimization- Genetic algorithms- Simulated annealing- Many other

MethodsMethods- To address the risk associated with the previously listed concerns/challenges, a group of optimization methods was examined

Genetic Algorithms (GA) mimics laws of Natural Evolution which emphasizes “survival of the fittest”.

In GA a “population” that contains different possible solutions to the problem is created.

DerivativeDerivative--free optimizationfree optimization

Genetic Algorithms Genetic Algorithms

1001011001100010101001001001100101111101

. . .

. . .

. . .

. . .

Initialgeneration

1001011001100010101001001001110101111001

. . .

. . .

. . .

. . .

Nextgeneration

The process is repeated until evolution happens“a solution is found!”

Selection

Crossover

Mutation

Elitism

MultiMulti--Objective OptimizationObjective Optimization- It is practical to assume that the decision maker might have priorities on the different objectives casualties/mission disruption and time to recover.

-In this case, usually there exist more than one optimal solution to the problem (Named Pareto solution)

- Based on the preferences, these solutions can be rank ordered.

MultiMulti--Objective OptimizationObjective Optimization

- Three major issues differentiate between single and multi- objective optimizations

- Multiple (three) goals instead of one

- Dealing with multiple search spaces not one

- Artificial fixes affect results

- We are looking for a set of Pareto-optimal solutions

MultiMulti--Objective OptimizationObjective Optimization

Pareto Optimal Solutions

Domain of All Feasible Solutions

Casualties (Objective 1)

Mis

sion

Dis

rupt

ion

(Obj

ectiv

e 2)

- Global criteria method- Require target values for the functions- Can incorporate weights for preferences

- Hierarchical optimization method- Optimize the top priority function- Specify constraints to prevent deteriorating the optimized function

- Multi-Objective Genetic Optimization (MOGA)- Non-dominated Sorting Genetic Algorithm

MultiMulti--Objective Optimization MethodsObjective Optimization Methods

MultiMulti--Objective OptimizationObjective OptimizationHierarchical Method

- Rank order the objective functions

-The j-1 function is used as constraint in optimizing the jth function.

)x(f100

11)x(f 1j

1jj

1j−

−− ⋅⎟⎟⎠

⎞⎜⎜⎝

⎛ −±≤Σ

- is a lexicographic increment %

− Ηοw much error is allowed in losing optimal solution for (j-1) given more optimization in (j)

jΣ

MultiMulti--Objective OptimizationObjective OptimizationGlobal Criterion

- The threshold vector is defined by

w can also be implemented to represent preferences as weights

[ ]0k

03

02

01

0i ff,f,ff K=

∑=

⎟⎟⎠

⎞⎜⎜⎝

⎛ −=

k

i

P

i

iii f

xffwxf1

0

0 )()(

P is integer 1 or 2

MultiMulti--Objective OptimizationObjective OptimizationNon-dominated Sorting Genetic Algorithm (NSGA)

- While similar to GA, NSGA sorts the population according to non-domination principles.

- Population is classified into a number of mutually exclusive classes

- Highest fitness is assigned to class that are closest to the Pareto-optimal front

- The use of non-dominated sorting allows diversity to solutions and thus guarantees reaching the Pareto-front.

-NSGA also includes elitism principles which allows it to find higher number of Pareto-solutions.

Selection

CrossoverMutation

Parent Set (Pi)MIDST

SimulationMulti-Objective

Fitness Evaluation

Parent Subset (Pi)

Offspring Population Set (Oi)

MID

ST

Sim

ula

tion

Pi

Oi

Rejection

Cro

wdi

ng S

ortin

g

Par

ent

Set

(Pi+

1 )

Non

-Dom

inat

ed

Sort

ing

S1i

S2i

S4i

S3i

S5i

S6i

NSGANSGA--IIII

Merits and shortcomingsMerits and shortcomings- Derivative based

- If the space is continuum, it converges very fast and an optimal solution is guaranteed- If too many local minima exist, the algorithm might be trapped and cannot find global minima

- Non-derivative based- If the space is non-continuum, GA will be able to find the solution- Whether local minima exist or not, it will converge.- GA is better equipped with some aiding optimization technique to narrow search domain

Reduction in Mission Disruption

Reduction on Recovery Time

Reduction in Casualties

Case studyCase study- For a given group of data cards and inputs we identified

Case studyCase study- For a given group of data cards and inputs we identified

0

50

100

150

200

250

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26

Investment Units

$M o

ver 1

0 ye

ars

Base Optimal

Case studyCase study- At the optimal level, we can identify the funding portfolio

Portfolio for Base Funding C1 = 21, C2 = 21. C3 = 42

Portfolio for Optimal Funding C1 = 11, C2 = 12. C3 = 12

-We demonstrated the possible use of multi-objective genetic optimization for allocation of funding for investment units to reduce consequences of CB events

- Classical gradient based versus gradient free optimization techniques have been examined in search for Pareto solutions

- The presented work is part of MIDST: A robust mathematical framework that can be used to help decision makers for funding allocations considering multiple objectives and priorities

Research is currently on-going to integrate fuzzy rank ordering module as part of the optimization process.

ConclusionsConclusions

This research is funded by

Defense Threat Reduction Agency (DTRA) Strategic Partnership Program.

The authors gratefully acknowledge this funding.

AcknowledgmentAcknowledgment

Questions

DerivativeDerivative--based optimizationbased optimization

gGoldnew ηθθ +=

Gradient descent method- Assumes continuous and differentiable function

-g is the derivative of the objective function

- G is a positive definite matrix

- η

is the step size

( ) ( ) ( ) ( ) T

n21

E....EEE)(g ⎥⎦

⎤⎢⎣

⎡∂∂

∂∂

∂∂

=∇=θθ

θθ

θθθθ

DerivativeDerivative--based optimizationbased optimization

( ) gIH 1oldnew

−+−= ληθθ

Levenberg-Marquardt (LM) method

- A modified version of classical Newton’s method. It also assumes continuous and differentiable function

- g is the gradient, I is the identity matrix, λ is some nonnegative value and H is the Hessian matrix

− η is the step size as defined before

( ) ( ) ( ) ( ) T

2n

2

22

2

21

22 E....EEE)(H ⎥

⎦

⎤⎢⎣

⎡

∂∂

∂∂

∂∂

=∇=θθ

θθ

θθθθ

Levenberg-Marquardt

Newton

Steepest Descent

incre

asing

decr

easi

ng

Func

tion

cont

our

Tangent

λ

DerivativeDerivative--based optimizationbased optimization