Algorithms for Marketing-Mix Optimization

TL;DR

This paper develops efficient algorithms for optimizing marketing mix by balancing quality, cost, and pricing to maximize profit in saturated markets, with solutions for different product quality dimensions.

Contribution

It introduces new algorithms for marketing-mix optimization, including an exact $O(n ext{log} n)$ solution for linear products and approximation algorithms for multi-quality products.

Findings

Exact $O(n ext{log} n)$ algorithm for $d=1$ case.

Approximation algorithms for constant $d$ with complexity $O(n( ext{log} n)^{d+1})$.

Complexity bound for arrangements of homothetic simplices in $ extbf{R}^d$.

Abstract

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.