Solution methods for the two-machine blocking flow shop scheduling problem with position-based learning effect

This paper addresses the two-machine blocking flow shop scheduling problem with a learning effect, aiming to minimize the makespan. Under certain conditions, we show that the problem can be reduced to a polynomially solvable case and further reduced to an equivalent single-machine scheduling problem. We derive dominance rules and theoretical lower bounds. For small instances, we develop a mixed-integer programming (MIP) formulation and a constraint programming model. In addition, a warm-start strategy is designed and effectively used to initialize the MIP. For large instances involving up to 300 jobs, we design a beam search algorithm enhanced with a variable neighborhood descent improvement phase. Extensive computational experiments are reported, including comparisons with a well-established metaheuristic from the literature.