KSL-03-08
## Incremental Computation of Resource-Envelopes in Producer-Consumer Models

**Reference: **
Kumar, T. K. S. Incremental Computation of Resource-Envelopes in Producer-Consumer Models. The Ninth International Conference on Principles and Practice of Constraint Programming (CP 2003), September, 2003.

**Abstract:** Interleaved planning and scheduling employs the idea of extending partial
plans by regularly heeding to the scheduling constraints during search.
One of the techniques used to analyze scheduling and resource consumption
constraints is to compute the so-called {\it resource-envelopes}. These
envelopes can then be used to derive effective heuristics to guide the
search for good plans and/or dispatch given plans optimally. The key to
the success of this approach however, is in being able to recompute the
envelopes incrementally as and when partial commitments are made. The
resource-envelope problem in producer-consumer models is as follows: A
directed graph $\mathcal{G}=\langle \mathcal{X}, \mathcal{E} \rangle$ has
$\mathcal{X}=\{X_0, X_1 \ldots X_n\}$ as the set of nodes corresponding to
events ($X_0$ is the ``beginning of the world'' node and is assumed to be
set to $0$) and $\mathcal{E}$ as the set of directed edges between them. A
directed edge $e=\langle X_i, X_j \rangle$ in $\mathcal{E}$ is annotated
with the simple temporal information $[LB(e), UB(e)]$ indicating that a
consistent schedule must have $X_j$ scheduled between $LB(e)$ and $UB(e)$
seconds after $X_i$ is scheduled ($LB(e) \le UB(e)$). Some nodes (events)
correspond physically to production or consumption of resources and are
annotated with a real number $r(X_i)$ indicating their levels of
production or consumption of a given resource. Given a consistent schedule
$s$ for all the events, the total production (consumption) by time $t$ is
given by $P_s(t)$ ($C_s(t)$). The goal is to build the envelope functions
$g(t) = max_{\{s \hspace{0.05cm} \mbox{is a consistent schedule}\}}
(P_s(t) - C_s(t))$ and $h(t) = min_{\{s \hspace{0.05cm} \mbox{is a
consistent schedule}\}} (P_s(t) - C_s(t))$. In this paper, we provide
efficient incremental algorithms for the computation of $g(t)$ and $h(t)$,
along with flexible consistent schedules that actually achieve them for
any given time instant $t$.

Full paper available as pdf, pdf.

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