Single-machine batch scheduling minimizing weighted flow times and delivery costs with job release times

This paper addresses scheduling a set of weighted jobs on a single machine in presence of release date for delivery in batches to customers or to other machines for further processing. The problem is a natural extension of minimizing the sum of weighted flow times by considering the possibility of delivering jobs in batches and introducing batch delivery costs. The classical problem is NP-hard and then the extended version of the problem is NP-hard. The objective function is that of minimizing the sum of weighted flow times and delivery costs. The extended problem arises in a real supply chain network by cooperation between two layers of chain. Structural properties of the problem are investigated and used to devise a branch-and-bound solution scheme. Computational experiments show the efficiency of suggested algorithm for solving instances up to 40 jobs.