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

Abstract

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] INTRODUCTIONOver the last three decades, a significant portion of the new product development (NPD) literature has been dedicated to the integration of engineering design and marketing processes for differentiated markets. Simple models to determine the most profitable characteristics of a single new product [1-2] have progressed to account for issues such as product-line design and preference heterogeneity [3][4][5][6][7], competitor reactions [8-10], cost structure [11][12], distribution channels [9,13-16], choice-set-dependent preferences [17], and coordination with constrained engineering design decisions [18][19][20][21][22][23][24][25][26].As Hsu and Wilcox [27] argue, the trend towards estimating marketing models at lower levels of aggregation that are more structural 1 in consumer behavior representation has led to models with many parameters and consequently greater uncertainty of those parameters. However, despite the advances in NPD methods, the research has not given much consideration to the intrinsic parameter uncertainty of the demand models. Demand uncertainty directly affects the risk of introducing a new product into the market, and firms evaluate potential projects not only in terms of expected return, but also in terms of risk.The purpose of this work is threefold. First, we define a robust a-profit metric and propose a general framework to incorporate demand uncertainty arising from choice model parameter estimation into the design decision process such that it accounts for varying levels of loss tolerance. Second, we apply the delta method to approximate the a-profit function in closed form for multinomial logit (MNL) demand models to be used efficiently in numerical optimization routines. Finally, we show how ignoring demand uncertainty can lead to suboptimal decisions for risk averse firms.We do not intend to consider all the various sources of demand model uncertainty [28], and several questions will remain open. In particular, we assume the discrete choice model is correctly specified and ignore uncertainty due to model misspecification, and we assume that the model parameters do not change over time or from the context in which the data were collected to the context in which predictions will be made. Nevertheless, the proposed methodology can be useful, and it serves as a first step in addressing design for profit maximization...

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“…First-order analysis of uncertainties, which is also known as the Delta method, is a rather straightforward and useful technique for the approximation of such uncertainties. This method is widely used in many fields of engineering due mainly to its ease of application to a wide variety of problems [17][18][19]. Mays stated that Delta method application is quite popular in many fields of engineering and, as a result, he developed a risk-based solution for storm sewers' design [17].…”

“…Their research showed that the average TSI number and its variation for mentioned reservoir oscillated between Mesotrophic to Eutrophic category [18]. Furthermore, Resende et al applied Delta method to estimate the mapping, from uncertainty in discrete choice model parameters to uncertainty of profit outcomes, and they identified the estimated -profit as a closed form function of design decision variables in computer science [19].…”

Sustainable product family design and a platform strategy

“…The Gumbel distribution is used to approximate closely normal distribution and yields a closed-form solution that leads to a computational advantage [194,195]. In summary, the MNL is applied to this study because of its advantages such as simplicity, interpretability, and its capability of extending the choice probability into new product design development [194,196,199].…”