Single period model

This case is a generalisation of the newsvendor model (which has only one component and one product). The problem involves three stages and we give one formation of the problem below^[2]

1. components acquired
2. demand realized
3. components allocated, products produced

We use the following notation^[1]

In the final stage when demands are known the optimization problem faced is to

${\displaystyle {\begin{aligned}{\text{minimize }}G(\mathbf {y} ,\mathbf {d} )&=\mathbf {h} '\mathbf {x} +\mathbf {p} '\mathbf {w} \\{\text{subject to }}A\mathbf {z} +\mathbf {x} &=\mathbf {y} \\\mathbf {z} +\mathbf {w} &=\mathbf {d} \\\mathbf {w} ,\mathbf {x} ,\mathbf {z} &\geq 0,\end{aligned}}}$

and we can therefore write the optimization problem at the first stage as

${\displaystyle {\begin{aligned}{\text{minimize }}&c(\mathbf {y} -\mathbf {x} _{0})+\mathbb {E} _{\mathbf {d} }[G(\mathbf {y} ,\mathbf {d} )]\\{\text{subject to }}&\mathbf {y} \geq \mathbf {x} _{0},\end{aligned}}}$

with x₀ representing the starting inventory vector and c the cost function for acquiring the components.

Continuous time

In continuous time orders for products arrive according to a Poisson process and the time required to produce components are independent and identically distributed for each component. Two problems typically studied in this system are to minimize the expected backlog of orders subject to a constraint on the component inventory, and to minimize the expected component inventory subject to constraints on the rate at which orders must be completed.^[3]