Pluralization: A Mathematical Model

This is the first of a series of simple mathematical models to explore the concept of decentralization. The primary focus is on pluralization as one of three aspects of decentralization. We derive expressions for cost and for net benefit as a function of the number of identical, uniformly dispersed facilities capable of servicing requests from sources that are uniformly distributed along a long, thin, geographical strip. The optimal, or cost-minimizing, number of such pluralized facilities is computed. We also derive conditions in terms of the basic parameters, such as the number of requests per month, transport costs per unit time, etc., which favor pluralization over centralization. The models are also analyzed for the secondary benefits of redundancy for more dependable service and to smooth fluctuations in demand. We found that, on balance of all the factors, historical trends—particularly the trend toward more requests per unit time—favor pluralization.