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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 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 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

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R&D Portfolio Optimization 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

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R&D Portfolio Optimization The R&D Portfolio Problem 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

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R&D Portfolio Optimization Solid Oxide Fuel Cells Ceramic (solid oxide) electrolyte Used for stationary power production; high running temps allow use of CH 4 and other non-H 2 gases Research areas: anode, cathode, electrolyte, bipolar plates, seal and pressure vessel

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R&D Portfolio Optimization 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 The R&D Portfolio Problem

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R&D Portfolio Optimization SOFC Components and Technologies ComponentTechnologyParameterTechnologyParameterTechnologyParameter AnodeSurface AreaPower Density Production Cost CathodeSurface AreaPower Density Production Cost ElectrolyteReaction Stability DegradationProduction Cost Bipolar Plates Temperature Stability ConductivityProduction Cost SealTemperature Stability Chemical Stability Production Cost Pressure Vessel Design---- Production Cost

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R&D Portfolio Optimization Projects and Technologies Technology changes are additive and denoted by

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R&D Portfolio Optimization R&D Portfolios, Mathematically M projects Decision x in {0,1} M There is a fixed budget, b Each project has cost c i, so Σc i x i b Cost is a function of technologies at time 1, C(ρ 1 ). The problem becomes…

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R&D Portfolio Optimization SOFC Cost Function Find power output for a fuel cell, assume 1,000cm 2 footprint: Find the capital cost of the fuel cell: For each possible combination of components I = i 1 (anode index), i 2 (cathode index), i 3 (electrolyte), i 4 (bipolar plates), i 5 (seal), and i 6 (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 portfolios Cost function may not be convex or separable Expectation of cost function is hard to compute given a portfolio

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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.

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R&D Portfolio Optimization Stochastic Gradient Portfolio Optimization Idea: linearly approximate by Iteratively estimate marginal value i at iteration n by Choose portfolio x n+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 portfolio Update technology parameters for by Obtain value for, Smooth into previous estimate,,

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R&D Portfolio Optimization

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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)

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R&D Portfolio Optimization Results Marginal values vary with portfolio make-up. Marginal cost of a project

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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..

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R&D Portfolio Optimization Results Empirical density function of portfolio selected at algorithm termination in terms of cost per kWh. Fraction of projects

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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.

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