DARPA ULTRALOG Final Report - Industrial and Manufacturing ...

Manuscript for IEEE Transactions on Automatic Control 14
D i (t) as a quantitative representation of the component’s position. D i (t) is the required time gap
between the system’s and the component’s completion times at time t. Each component needs to
complete at less than or equal to T-D i (t) to keep the completion time T. Components without
successors have depth equal to 0 but components with successors have positive depths. A
component a can keep its depth if its predecessors’ depth is D a (t) plus its total service time for
the last arriving tasks in the worst case. So, a component i’s depth to keep the depths of its all
successors is the maximal of the required depths from its successors represented as:
D i( t ) = max[ Da( t ) + f a( va
) / MRAa
( t )] , (10)
a∈i
∑
b∈a
in which i denotes successors of component i and a predecessors of component a.
Though it is possible to refine the naïve decision model by incorporating the depths as
variables, the model complexity increases because each component’s constraint will be
intertwined with the decision variables of all connected components. So, we simply estimate
components’ depths through the decisions used at the last control point. At each control point
each successor informs its predecessors of required depths and each predecessor chooses the
maximal one as its depth. As a result, we can consider the depth as constant rather than variable
so that the refined model has no increase in complexity. We call the refined model in (11) as
stable decision model. If we don’t consider the depth, i.e., D i (t)=0, the model becomes equivalent
to the naïve decision model. Also, the stable decision model becomes an exact CPM/PERT
formulation as a special case found in project management literature when one consider D i (t) as
variable.