3.5 Hierarchical controls for other multilevel models

Sethi and Zhang (1992b), and Sethi and Zhang (1995a) extend the model in Section 3.1 to incorporate promotional or advertising decisions that influence the product demands. Zhou and Sethi (1994) demonstrate how workforce and production decisions can be decomposed hierarchically in a stochastic version of the classical HMMS model Holt, Modigliani, Muth, and Simon (1960). Manufacturing systems involving preventive maintenance are studied by Boukas and Haurie (1990), Boukas (1991), Boukas, Zhang, and Zhou (1993), and Boukas, Zhu and Zhang (1994). The maintenance activity involves lubrication, routine adjustment, etc., which reduce the machine failure rates. The objective in these systems is to choose the rate of maintenance and the rate of production in order to minimize the total discounted cost of surplus, production, and maintenance.

In this section, we shall only discuss the model developed in Sethi and Zhang (1995a), we consider the case when both capacity and demand are finite state Markov processes constructed from generators that depend on the production and promotional decisions, respectively. In order to specify their marketing-production problem, let $m(\varepsilon,t)\in {\cal M}$ as in Section 3.1 and${\mbox{\boldmath$z$ }}(\delta,t)\in\{{\mbox{\boldmath$z$ }}^0,{\mbox{\boldmath$z$ }}^1,\cdots,{\mbox{\boldmath$z$ }}^d\}$, for a given $\delta$, denote the capacity process and the demand process, respectively.

Definition 3.7   We say that a control$({\mbox{\boldmath$u$ }}^{\varepsilon}(\cdot),w(\cdot))=\{({\mbox{\boldmath$u$ }}^{\varepsilon}(t),w(t)):\; t\geq 0\}$ is admissible, if

(i)

$({\mbox{\boldmath$u$ }}^{\varepsilon}(\cdot),w(\cdot))$ is right-continuous having left-hand limit (RCLL);

(ii)

$({\mbox{\boldmath$u$ }}^{\varepsilon}(\cdot),w(\cdot))$ is$\sigma\{(m(\varepsilon,s),{\mbox{\boldmath$z$ }}(\delta,s)):\; 0 \leq s\leq t\}-\mbox{adapted}$, and satisfies ${\mbox{\boldmath$u$ }}^{\varepsilon}(t)\geq 0$,${\mbox{\boldmath$r$ }}\cdot {\mbox{\boldmath$u$ }}^{\varepsilon}(t)\leq m(\varepsilon,t)$ and$0\leq w(t)\leq 1$ for all $t\geq 0$.

We use ${\cal A}^{\varepsilon, \delta}({\mbox{\boldmath$x$ }},m,{\mbox{\boldmath$z$ }})$ to denote the set of all admissible controls. Then our control problem can be written as follows:

$$\begin{array}{l}{\cal P}^{\varepsilon, \delta}:\left\{\begin{arr......boldmath$u$ }}^{\varepsilon}(\cdot),w(\cdot))\end{array}\right.\\\end{array}$$ (3.31)

where by $m(\varepsilon,t)\sim \varepsilon^{-1}Q^m({\mbox{\boldmath$u$ }}^{\varepsilon}(t))$, we mean that the Markov process$m(\varepsilon,t)$ has the generator $\varepsilon^{-1}Q^m({\mbox{\boldmath$u$ }}^{\varepsilon}(t))$.
We use ${\cal A}^{0, \delta}$ to denote the admissible control space $$\begin{eqnarray*}{\cal A}^{0, \delta}=&&\{( U(t),w(t))=({\mbox{\boldmath$u$ }}...... }}(\delta,s): \ 0 \leq s\leqt\} \ \mbox{adapted and RCLL}\}.\end{eqnarray*}$$

$$\begin{array}{l}{\cal P}^{0, \delta}:\left\{\begin{array}{ll}\m......},{\mbox{\boldmath$z$ }}, U(\cdot) ,w(\cdot))\end{array}\right.\\\end{array}$$ (3.32)

Let $(U(\cdot),w(\cdot))\in {\cal A}^{0, \delta}$ denote an optimal open-loop control. We construct $$\begin{eqnarray*}{\mbox{\boldmath$u$ }}^{\varepsilon,\delta}(t)=\sum_{i=0}^m{\......varepsilon,t)=i\}}\mbox{ and }w^{\varepsilon,\delta}(t)=w(t).\end{eqnarray*}$$ Then $({\mbox{\boldmath$u$ }}^{\varepsilon,\delta}(t),w^{\varepsilon,\delta}(t))\in{\cal A}^{\varepsilon, \delta}$, and it is asymptotically optimal, i.e., $$\begin{eqnarray*}\lim_{\delta \to 0} \big \vert J^{\varepsilon,\delta}({\mbo......{\mbox{\boldmath$x$ }},m,{\mbox{\boldmath$z$ }})\big\vert=0.\end{eqnarray*}$$ Similarly, let $(U({\mbox{\boldmath$x$ }},{\mbox{\boldmath$z$ }}),w({\mbox{\boldmath$x$ }},{\mbox{\boldmath$z$ }}))\in{\calA}^{\varepsilon, \delta}$ denote an optimal feedback control for${\cal P}^{\delta}$. Suppose that $(U({\mbox{\boldmath$x$ }},{\mbox{\boldmath$z$ }}),w({\mbox{\boldmath$x$ }},{\mbox{\boldmath$z$ }}))$ is locally Lipschitz for each${\mbox{\boldmath$z$ }}$. Let $$\begin{eqnarray*}{\mbox{\boldmath$u$ }}^{\varepsilon,\delta}(t)=\sum_{i=0}^m......\mbox{\boldmath$z$ }}(\delta,t)) I_{\{m(\varepsilon,t)=i\}} \\end{eqnarray*}$$ and $$\begin{eqnarray*}w^{\varepsilon,\delta}(t)=w({\mbox{\boldmath$x$ }}^{\varepsilon, \delta}(t),{\mbox{\boldmath$z$ }}(\delta,t)).\end{eqnarray*}$$ Then the feedback control$({\mbox{\boldmath$u$ }}^{\varepsilon,\delta}(\cdot),w^{\varepsilon,\delta}(\cdot))$ is asymptotically optimal for ${\cal P}^{\varepsilon, \delta}$, i.e., $$\begin{eqnarray*}\lim_{\delta \to 0} \big \vert J^{\varepsilon,\delta}({\mbo......{\mbox{\boldmath$x$ }},m,{\mbox{\boldmath$z$ }})\big\vert=0.\end{eqnarray*}$$ We have described only the hierarchy that arises from a large$\delta$ and a small $\varepsilon$. In this case, promotional decisions are obtained under the assumption that the available production capacity is equal to the average capacity. Subsequently, production decisions taking into account the stochastic nature of the capacity can be constructed. Other possible hierarchies result when both $\delta$ and $\varepsilon$ are small or when $\varepsilon$ is large and $\delta$ is small. The reader is referred to Sethi and Zhang (1995a) for details on these.