Single-lot, lot-streaming problem for a 1 + m hybrid flow shop

In this paper, we consider an application of lot-streaming for processing a lot of multiple items in a hybrid flow shop (HFS) for the objective of minimizing makespan. The HFS that we consider consists of two stages with a single machine available for processing in Stage 1 and m identical parallel machines in Stage 2. We call this problem a 1 + m TSHFS-LSP (two-stage hybrid flow shop, lot streaming problem), and show it to be NP-hard in general, except for the case when the sublot sizes are treated to be continuous. The novelty of our work is in obtaining closed-form expressions for optimal continuous sublot sizes that can be solved in polynomial time, for a given number of sublots. A fast linear search algorithm is also developed for determining the optimal number of sublots for the case of continuous sublot sizes. For the case when the sublot sizes are discrete, we propose a branch-and-bound-based heuristic to determine both the number of sublots and sublot sizes and demonstrate its efficacy by comparing its performance against that of a direct solution of a mixed-integer formulation of the problem by CPLEX®.