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Special section: New Problem Formulations for Design Under Uncertainty

Robust Design for Profit Maximization With Aversion to Downside Risk From Parametric Uncertainty in Consumer Choice Models

[+] Author and Article Information
Camilo B. Resende

 Mechanical Engineering, Carnegie Mellon University, Pittsburgh 15217, PAcamilo.cbr@gmail.com

C. Grace Heckmann

 Mechanical Engineering, Carnegie Mellon University, Pittsburgh 15217, PAchristine.grace.heckmann@gmail.com

Jeremy J. Michalek1

Mechanical Engineering, Engineering and Public Policy,  Carnegie Mellon University, Pittsburgh 15217, PAjmichalek@cmu.edu

“Econometric models that are based explicitly on the consumer’s maximization problem and whose parameters are parameters of the consumers’ utility functions or of their constraints are referred to as structural models.” [52]

Though it is possible to conduct sensitivity analysis on the parameters defining the firm’s utility function, misspecification of functional form remains, and interpretation of parametric sensitivity is generally cumbersome.

Results vary depending on which respondents are drawn.

1

Corresponding author.

J. Mech. Des 134(10), 100901 (Sep 28, 2012) (12 pages) doi:10.1115/1.4007533 History: Received January 12, 2012; Revised July 30, 2012; Published September 21, 2012; Online September 28, 2012

In new product design, risk averse firms must consider downside risk in addition to expected profitability, since some designs are associated with greater market uncertainty than others. We propose an approach to robust optimal product design for profit maximization by introducing an α-profit metric to manage expected profitability vs. downside risk due to uncertainty in market share predictions. Our goal is to maximize profit at a firm-specified level of risk tolerance. Specifically, we find the design that maximizes the α-profit: the value that the firm has a (1 − α) chance of exceeding, given the distribution of possible outcomes. The parameter α ∈ (0,1) is set by the firm to reflect sensitivity to downside risk (or upside gain), and parametric study of α reveals the sensitivity of optimal design choices to firm risk preference. We account here only for uncertainty of choice model parameter estimates due to finite data sampling when the choice model is assumed to be correctly specified (no misspecification error). We apply the delta method to estimate the mapping from uncertainty in discrete choice model parameters to uncertainty of profit outcomes and identify the estimated α-profit as a closed-form function of decision variables for the multinomial logit model. An example demonstrates implementation of the method to find the optimal design characteristics of a dial-readout scale using conjoint data.

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Copyright © 2012 by American Society of Mechanical Engineers
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Figures

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Figure 6

Comparison of simulated and approximated g function

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Figure 7

(a) and (b) Comparison of simulated and approximated market share

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Figure 8

(a)–(f) Utility levels

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Figure 1

α-profit shown for (a) probability density function of profit and (b) cumulative distribution function of profit

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Figure 2

Expected profit vs. downside risk: the expected profit for design 1 is higher than the expected profit for design 2 (π¯1>π¯2); however, design 2 has a higher profit at the α-level than design 1 (π2α>π1α)

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Figure 3

Illustration of probability distribution functions and their approximations using the delta method

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Figure 4

CDF of profit distribution illustrating that different designs are preferred for α = 0.10 versus α = 0.90 and maximum expected profit for coefficients estimated using n = 250 data points

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Figure 5

CDF of profit distributions illustrating that α = 0.10 and α = 0.90 designs converge to the expected value design as data increase using n = 9200 data points

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