In multi-attribute models in marketing, a consumer’s preference for a brand in a product class is expressed as a weighted sum of the brand’s attribute values. However, marketing is abundant with examples where two brands may have approximately the same attribute values but enjoy very different market shares, e.g., Coke and Pepsi may have the same values for “sweetness,” “carbonation,” “calories” and “price” but have quite different market shares; two political candidates may take approximately the same position on relevant political issues but enjoy different levels of voter support. Defining the “brand-specific effect” to be the component of overall preference not explained by the attributes used in the multi-attribute model, the aim is to empirically estimate the brand-specific effects for the different brands. It turns out that the estimation problems have a close relationship to some minimum cost network flow models in Operations Research. An empirical application of the proposed approach in the context of consumers’ choices of different primary health care physicians in a rural area reveals that the brand-specific component substantially improves the validity of the multi-attribute model.