A point process approach to inventory models (Q2756674)

The modern theory of point processes is used to study optimal solutions for single-item inventory. Demand for goods is assumed to occur according to a compound Poisson process and production occurs continuously and deterministically between times of demand, such that the inventory evolves according to a Markov process in continuous time. The author considers a situation where demand occurs at random times \(T_1< T_2 < \ldots\); \(\;N(t) = \sum_{n=1}^\infty {\mathbf 1}(T_u \leq t)\) denotes the associated counting measure. The amount of goods a customer demands at time \(T_n\) is \(Z_n\) and is assumed to be a non-negative random variable. The process \(X=\{X(t)\}_{t\geq 0}\), where \(X(t) = \sum_{n=1}^{N(t)} Z_n\), measures the total demand for goods over the period \([0, t].\) The author assumes that \(X\) is a compound Poisson process, that is \(N(t)\) is some Poisson process with continuous intensity \(\lambda (t)\) and the sequence \((T_n)_n\) is independent of \((Z_n)_n\), where \(Z_n\)'s are i.i.d. A general Markov process \(I =\{I(t)\}_{t\geq 0}\) for describing the inventory is of the form NEWLINE\[NEWLINE I(t) = I_0 +\int_0^t b(s,I(s)) ds -X(t), NEWLINE\]NEWLINE where \(I_0\) is the initial level, and \(b\) is a continuous function playing the role of the production rate. Let \(\tau_n = \inf\{ T_k/ T_{k-1}>\tau_{n-1}, I(T_k)< 0\}.\) The main theorem is the following one: The intensity process of \((\tau_n)_n\) exists and is given by NEWLINE\[NEWLINE\lambda^{\ast}(t) = \lambda(t) [{\overline G}(I(t)){\mathbf 1} (I(t)\geq 0)+{\mathbf 1}(T(t)<0)], \quad \overline G(x) =1 -G(x)NEWLINE\]NEWLINE where \(G(x)\) is the common distribution function of \(Z_i.\) Let NEWLINE\[NEWLINEh(t, x) =c(t, x) +\mu\lambda [{\overline G} {\mathbf 1}(x\geq 0) + {\mathbf 1}(x<0)], \quad J(I) = e\int_0^T h(s, I(s)) ds.NEWLINE\]NEWLINE Evaluation of the expected cost over time \(J(I)\) is based on the previous theorem. The problem is to find the process \(I\) such that \(J(I)\) is minimized. The author considers some examples.