Algorithms for Marketing-Mix Optimization

Algorithms for determining quality/cost/price tradeoffs in saturated markets are considered. A product is modeled by $d$ real-valued qualities whose sum determines the unit cost of producing the product. This leads to the following optimization problem: given a set of $n$ customers, each of whom has certain minimum quality requirements and a maximum price they are willing to pay, design a new product and select a price for that product in order to maximize the resulting profit. An $O(n\log n)$ time algorithm is given for the case, $d=1$, of linear products, and $O(n(\log n)^{d+1})$ time approximation algorithms are given for products with any constant number, $d$, of qualities. To achieve the latter result, an $O(nk^{d-1})$ bound on the complexity of an arrangement of homothetic simplices in $\R^d$ is given, where $k$ is the maximum number of simplices that all contain a single points.