The representation of partially-concurrent open shop problems

The partially-concurrent open shop scheduling problem is introduced. The standard open shop scheduling problem is generalized by allowing some operations to be processed concurrently. A schedule for the partially-concurrent problem is represented by a digraph. We show that the scheduling problem is equivalent to a problem of orienting a given undirected graph, called a conflict graph. The schedule digraph is then modeled by a matrix, generalizing the rank matrix representation. The problem is shown to be NP-hard. The representation can be used to generalize previously discussed standard open shop issues. It is demonstrated by generalizing the theoretical concept of reducibility and also by using standard open shop heuristic solutions to the partially-concurrent scenario. The presented problem is directly motivated from a real-life timetabling project of assigning technicians to airplanes in an airplane garage.