A fix-and-relax matheuristic method for lot-sizing and scheduling problem in a dairy industry

The lot-sizing and scheduling problem (LSP) focuses on determining the production quantities and the sequence of production activities, simultaneously. This article addresses the LSP in a flexible flow shop environment within the real-world dairy industry in Iran. We consider perishability, which is a common issue in the food industry, as well as cleaning in place (CIP) of the machines. The problem can be described as multi-product, multi-level, multi-period, with sequence-dependent setup and CIP time. To maximise profits, we present a mixed-integer linear programming (MILP) model that subtracts total revenue from the total cost, including tardiness, inventory holding, sequence-dependent setup, CIP, and disposal of perished products costs. Due to the NP-hardness of this problem, we propose a fix-and-relax (F&R) matheuristic method to solve it. The numerical experiments on randomly generated instances with up to 40 products on the proposed model show that while the exact MILP model becomes computationally intractable for large-scale instances, the proposed F&R method is able to generate high-quality feasible solutions within reasonable computation times.