Relating Domain Concepts Intensionally by Ordering Connections - Formal Methods and Hybrid Real-Time Systems - page 205

6.1 Intrinsics

The intrinsics of a project plan is that it is a directed acyclic graph (a DAG )
of works to be done. The ordering of the DAG specifies that some works can be
done in parallel, whereas other works need to be done in sequence due to certain
dependencies. Such work dependencies may lie deeply in the nature of the works
and may not be possible to formalize explicitly. Certainly, project planning can
be much more than this. However, in this paper, we shall abstract from notions
like time, deadlines, milestones, follow-up specifications and other administrative
aspects. Such notions we consider additional and not intrinsic as we certainly
can have project plans without them.
We model a project plan (pp:PP) as a map from nodes (g: Γ ) to sets of nodes.
Nodes correspond to works, and the edges define a partial ordering of works.
The execution order follows the direction of the graph.
From a node in a project plan, we can observe the work to be performed; i.e.
the operation type (obs_On_ Γ ) and the object aspects (obs_Xs_ Γ ). We can
also observe the kinds of resources to be used (obs_Rn_ Γ ). Here, we need two
definitions.

Definition 9 (Resource Usage). By a resource usage (re:Rn m Q) ,weun-

derstand the resources which are consumed by an operation.

Definition 10 (Relevant Resource Usage). By a relevant resource usage ,
we understand a resource usage which concerns a given object aspect.

From a node in a project plan, we can observe the resource usage of the work
to be done (obs_Rm_ Γ ). Also, from a node and an object aspect of a project
plan, we can observe the relevant resource usage concerning the object aspect
(obs_rel_res). The relevant resource usage for a node in a project plan is given
by adding together (i.e. applying join ; see Sect. 5.2) pair-wise on the resource
usage for each work concerning each object aspect involved in the operation.
A project plan is well-formed if and only if it satisfies the criteria of being a
directed, acyclic graph.
We define a partial ordering of project plans based on the idea that a project
plan is a sub-plan of another project plan if and only if its graph is a sub-
graph of the other project plan. Furthermore, for nodes existing in both graphs,
the resource usage of the sub-graph must be a sub-map (see Sect. 5.1). Thus,
specialization of project plans is based on the principle that restricting to a sub-
graph implies narrowing the scope of the total work. That is, the products or
services being the results of executing the project plan, are limited.
As for budgets, we assume that the same nodes may exist in different project
plans so project plans can be compared.
We define two operations meet and join for combining project plans. meet
takes two project plans and gives the combination which is a specialization of
both argument project plans. meet gives the project plan for which each node
is in both argument project plans. For each such node, the set of object aspects
is the intersection set and the resource usage maps the common resource types