A new MILP model and fast heuristics for the variable-sized bin packing problem with time windows

Abstract

Variable-Sized Bin Packing Problem with Time Windows (VSBPPTW) is a variant of the classical bin packing problem in which several types of bins with different sizes and costs are available, and the total cost of bins used to pack all items is minimized. Additionally, a time window is defined for each item, and the set of items packed into each bin must be time-compatible, i.e., the items in a bin must have at least one common time point within their time windows. We propose a new mixed-integer linear programming (MILP) model for VSBPPTW that (i) ensures time-compatibility by filling each bin with items from one maximal time-compatible set only and (ii) uses a compressed network flow formulation to represent the packing of items into bins under each maximal time-compatible set. New fast heuristics for VSBPPTW are also proposed to warm-start the MILP solver. Computational experiments are carried out on two sets of benchmark problem instances from the literature. We show that the new fast heuristics obtain on average much better solutions than the previously proposed fast heuristic, and the new MILP model greatly outperforms the previously proposed MILP model. While the previously proposed MILP model solves to optimality only 13% of the problem instances in the first set and none in the second, the new MILP model solves to optimality over 70% problem instances in both sets.