Lagrangian approach to minimize makespan of non-identical parallel batch processing machines

- Proposes a Lagrangian Relaxation of the non-identical parallel batch processing machine problem.
- Compares to Particle Swarm Optimization (PSO), Random Keys Genetic Algorithm (RKGA), and CPLEX.
- Conducts experiments for two and four machine problems of various number of jobs.
- Shows Lagrangian Relaxation identifies new, improved solutions for several benchmark instances.

This research is motivated by the testing operations conducted at an electronics manufacturing facility. Printed Circuit Boards (PCBs) of varying size are assembled on multiple assembly lines. The PCBs from different assembly lines are later grouped to form batches that are scheduled for testing on non-identical Environmental Stress Screening (ESS) chambers. The ESS chambers can process multiple PCBs simultaneously as long as the total size of all PCBs in the batch does not exceed the chamber's capacity. The testing time of the batch depends on the composition of the batch. The chambers are referred to as Batch Processing Machines (BPMs) and PCBs are jobs in this paper. Scheduling non-identical BPMs to minimize the makespan objective is known to be NP-hard. Consequently, the Particle Swarm Optimization (PSO) and Random Keys Genetic Algorithm (RKGA) approaches were proposed in the literature. In this research, a Lagrangian Relaxation (LR) approach is proposed. The solution from the LR approach is compared to the solution from PSO, RKGA, and a commercial solver. An experimental study is conducted on benchmark instances and the LR approach identified a new improved solution for several problem instances.