R&D Portfolio Optimization One Stage R&D Portfolio Optimization with an Application to Solid Oxide Fuel Cells Lauren Hannah 1 , Warren Powell 1 , Jeffrey Stewart 2 1 Princeton University, Department of Operations Research and Financial Engineering 2 Lawrence Livermore National Laboratory
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R&D Portfolio Optimization One Stage R&D Portfolio Optimization with an Application to Solid Oxide Fuel Cells Lauren Hannah 1, Warren Powell 1, Jeffrey.
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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
R&D Portfolio Optimization
The R&D Portfolio Problem
• A government or large corporation wants to spend resources on research. What is the best allocation?
• Cost is a function of technologies at time 1, C(ρ1).
• The problem becomes…
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R&D Portfolio Optimization
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):
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R&D Portfolio Optimization
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:
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R&D Portfolio Optimization
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:
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R&D Portfolio Optimization
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
R&D Portfolio Optimization
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.
R&D Portfolio Optimization
Stochastic Gradient Portfolio Optimization
• Idea: linearly approximate by
• Iteratively estimate marginal value i at iteration n by • Choose portfolio xn+1 by solving
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R&D Portfolio Optimization
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.
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R&D Portfolio Optimization
Stochastic Gradient Portfolio Optimization• Get value for original portfolio• Get perturbed technology change, , from perturbed
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)
R&D Portfolio Optimization
Results
Marginal values vary with portfolio make-up.
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gina
l cos
t of
a p
roje
ct
R&D Portfolio Optimization
Results
Value of selected portfolio for as a function of time for single run. SGPO gravitates to a “good” value quickly..
R&D Portfolio Optimization
Results
Empirical density function of portfolio selected at algorithm termination in terms of cost per kWh.
Fra
ctio
n of
pro
ject
s
R&D Portfolio Optimization
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.