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(nlog n) time algorithm is given for the case, d=1, of products having a single quality, 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.
