Resource Allocation Problems

Why¹

Definition

We want to allocate a resource among $n$ entities to maximize some measure of “return” or “profit.”

Let $B \in \R $. Let $R_i: \R \to \R $ be a function which gives the return for allocating entity $i$ the amount $x_i$ of the resource. Let $\mathcal{X} = \Set*{x \in \R ^n}{x \geq 0, \sum_i x_i = B}$ Define $f: \R ^n \to \R $ by $f(x) = \sum_{i = 1}^{n} R_i(x_i)$. We call the optimization problem $(\mathcal{X} , f)$ a single-resource allocation problem. In this case we call $x \in \R ^{n}$ an allocation and we call $B$ the budget.

1. Future editions will include. For now this sheet serves as an example of a continuous optimization problem. ↩︎