Robust Design for Profit Maximization under Uncertainty of Consumer Choice Model Parameters Using the Delta Method

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 design decision variables. This process is described for the multinomial logit model, and a case study demonstrates implementation of the method to find the optimal design characteristics of a midsize consumer automobile.