Variable Neighbourhood Search Solving Sub-problems of a Lagrangian Flexible Scheduling Problem

New technologies allow the production of goods to be geographically distributed across multiple job shops. When optimising schedules of production jobs in such networks, transportation times between job shops and machines can not be neglected but must be taken into account. We have researched a mathematical formulation and implementation for flexible job shop scheduling problems, minimising total weighted tardiness, and considering transportation times between machines. Based on a time-indexed problem formulation, we apply Lagrangian relaxation, and the scheduling problem is decomposed into independent job-level sub-problems. This results in multiple single job problems to be solved. For this problem, we describe a variable neighbourhood search algorithm, efficiently solving a single flexible job (sub-) problem with many timeslots. The Lagrangian dual problem is solved with a surrogate subgradient search method aggregating the partial solutions. The performance of surrogate subgradient search with VNS is compared with a combination of dynamic programming solving sub-problems, and a standard subgradient search for the overall problem. The algorithms are benchmarked with published problem instances for flexible job shop scheduling. Based on these instances we present novel problem instances for flexible job shop scheduling with transportation times between machines, and lower and upper bounds on total weighted tardiness are calculated for these instances.