r&d portfolio optimization one stage r&d portfolio optimization with an application to solid...

R&D Portfolio Optimization

One Stage R&D Portfolio Optimization with an Application

to Solid Oxide Fuel Cells

Lauren Hannah1, Warren Powell1, Jeffrey Stewart2

1Princeton University, Department of Operations Research and Financial Engineering

2Lawrence Livermore National Laboratory


The R&D Portfolio Problem

• A government or large corporation wants to spend resources on research. What is the best allocation?

• Examples:– Wind technologies– Storage technologies– Solid oxide fuel cells– Carbon capture and storage projects


The R&D Portfolio Problem• The setup:

– Choose a set of projects to fund– Research occurs on those projects….which then

changes the state of the world– We must live with that new state

• The constraints:– Budget– 0/1 project funding

• The goal:– Maximize the expectation of a utility function based on

the state of the world after research occurs



• Challenges:– Projects are not (necessarily) independent

• Outcomes may be dependent• Project areas may overlap• Projects may be focused at different levels (“wind

technologies” vs. “bipolar plate coating”)

– Utility function is not a sum of project values– Problem is fundamentally stochastic knapsack

optimization


Solid Oxide Fuel Cells

• Ceramic (solid oxide) electrolyte• Used for stationary power production; high running temps

allow use of CH4 and other non-H2 gases• Research areas: anode, cathode, electrolyte, bipolar

plates, seal and pressure vessel


• Project dependence: what is a project?– May be specific component (ie anode or

cathode), or it may be an entire area (ie “wind technologies”)

– Can affect multiple underlying technologies, such as surface area and production cost of anode

• This allows sophisticated project interactions

– Outcomes occur for technologies, not overall projects

• Lets dependent outcomes be easily modeled



SOFC Components and Technologies

Component Technology Parameter Technology Parameter Technology Parameter

Anode Surface Area Power Density

Production Cost

Cathode Surface Area Power Density

Production Cost

Electrolyte Reaction Stability

Degradation Production Cost

Bipolar Plates

Temperature Stability

Conductivity Production Cost

Seal Temperature Stability

Chemical Stability

Production Cost

Pressure Vessel

Design ---- ---- Production Cost

SAA PD

A COSTA

SAC PD

C COSTC

RSE DEG

E COSTE

TSB CON

B COSTB

TSS CS

S COSTS

DESP COST

P


Projects and Technologies

Technology changes are additive and denoted by ̂


R&D Portfolios, Mathematically

• M projects• Decision x in {0,1}M

• There is a fixed budget, b

• Each project has cost ci, so Σcixi ≤ b

• Cost is a function of technologies at time 1, C(ρ1).

• The problem becomes…

.,...,2,1 },1,0{

, ..

)ˆ,,( min

1

01

Mix

bxcts

xCΕ

i

M

iii

x


SOFC Cost Function

• Find power output for a fuel cell, assume 1,000cm2 footprint:

• Find the capital cost of the fuel cell:

For each possible combination of components I = i1 (anode index), i2 (cathode index), i3 (electrolyte), i4 (bipolar plates), i5 (seal), and i6 (pressure vessel), find cost per kWh, that is (production cost) / (lifetime x kW output):

},min{000,1 )()()()()(2

11114

SAiC

PDiC

SAiA

PDiA

CONiB

kWI cmC

COSTiP

COSTiS

COSTiB

COSTiE

COSTiC

COSTiA

COSTIC )()()()()()( 654321

)(000,1


SOFC Cost Function, Continued• Find the lifetime for the fuel cell (minimum of component

lifetimes):

• Get an unpenalized cost per kWh:

• Calculate penalties for not meeting temperature and design specifications. First, calculate minimum operating temperature:

},,min{ )()()( 533

CSiS

DEGiE

RSiE

LIFEIC

LIFEI

kWI

COSTIkWhC

I CC

CC

/

},min{ )()( 54

TSiS

TSiB

TEMPIC


SOFC Cost Function, Continued• Create the penalty term:

• Add penalty to cost per kWh:

• The cost for the state of technologies is the cost of the best fuel cell:

22)( }0,~max{}0,~max{

6

TEMPTEMPI

TEMPDESiP

DESDESPENI CaaC

PENI

kWhCII CCC /

IAI

CxC

min)),(( 1


Mathematical Challenges

• The number of possible portfolios grows combinatorially– 10 projects out of 30 = ~10 million portfolios– 20 projects out of 60 = ~ 4.2 x 1015 portfolios

• Cost function may not be convex or separable

• Expectation of cost function is hard to compute given a portfolio


Previous Approaches

• R&D literature:– Simplifies problem to use math programming– Does not often address uncertainty or

complex project interactions

• Stochastic Combinatorial Optimization:– Not used for R&D problems– Uses metaheuristics such as branch and

bound, simulated annealing, nested partitions, ant colony optimization, etc.

– Performance uncertain (doubtful?) for R&D.


Stochastic Gradient Portfolio Optimization

• Idea: linearly approximate by

• Iteratively estimate marginal value i at iteration n by • Choose portfolio xn+1 by solving

)ˆ,,( 01 xCE

M

i

nii

n vxxV1

)(

niv

}.1,0{

, ..

max

1

1

i

M

iii

M

i

nii

x

x

bxcts

vx


Stochastic Gradient Portfolio Optimization

• To determine ith stochastic gradient, , create new portfolio , perturbed around ith project

• If project i is in the old portfolio, take it out. If it is not, add it in.

1ˆ niv

inx ,ˆ

. },1,0{

,1

, subject to

maxargˆ

M

1j

1

,

ijy

xy

byc

yvx

j

nii

jj

M

jj

nj

y

in


Stochastic Gradient Portfolio Optimization• Get value for original portfolio• Get perturbed technology change, , from perturbed

portfolio• Update technology parameters for by

• Obtain value for ,

• Smooth into previous estimate, ,

inx ,ˆinx ,ˆ

1ˆ nvin ,1ˆ

inin ,10

,11 ˆ

inx ,ˆ

.0 if )(ˆ

,1 if ˆ)(ˆ

,11

1

1,111

ni

inn

ni

ninni

xCv

xvCv

niv

nin

nin

ni vvv )1(ˆ 11


Comparisons

• SGPO– Stochastic gradient portfolio optimization

• EPI-MC– Evolutionary Policy Iteration (Chang, Lee, Fu, Markus,

2005)– Modified to avoid assumption that expectation can be

computed exactly.– Provably convergent by using increasing number of

samples every iteration to estimate expectation.

• SA– Simulated annealing (Gutjahr and Pflug, 1996)


Results

Marginal values vary with portfolio make-up.

Mar

gina

l cos

t of

a p

roje

ct


Results

Value of selected portfolio for as a function of time for single run. SGPO gravitates to a “good” value quickly..


Results

Empirical density function of portfolio selected at algorithm termination in terms of cost per kWh.

Fra

ctio

n of

pro

ject

s


Results

Statistics for terminal portfolio, based on problem class and run time. The “x choose y” problems give all SOFC projects equal costs, the knapsack problems do not.