Machine scheduling with soft precedence constraints

This paper introduces a new concept, soft precedence constraint (SPC), in machine scheduling problems. Similar to the conventional precedence constraint, SPC specifies some partial order over the jobs; however, an SPC can be violated, but with a certain penalty or cost. The scheduling problem is to balance the tradeoff between the SPC violation penalty and other criteria relative to job completion times. We focus on studying a special case where SPC is defined by a bipartite network. This case is motivated by the berth allocation problem at a transshipment port, where the SPC models any missed container connections from feeder vessels to ocean-going vessels. We discuss the complexity of the problems for different scenarios and develop approximation algorithms.