The complexity of two supply chain scheduling problems

We consider an assembling manufacture system where several suppliers provide component parts to a manufacturer, who assembles products from all the supplied component parts. The delivery time of any product is the maximum of the suppliers' delivery time of the parts needed for that product. We assume that the final stage is non-bottleneck. Chen and Hall (2007) show that the suppliers' problem of minimizing the total weighted number of tardy parts is binary (i.e., weakly) NP-hard and left open whether it is unary (i.e., strongly) NP-hard. In this paper we resolve this open problem by showing that it is unary NP-hard. Moreover we show that the suppliers' problem of minimizing the total late work of parts is also unary NP-hard. (C) 2013 Elsevier B.V. All rights reserved.