A MILP model and two heuristics for the Bin Packing Problem with Conflicts and Item Fragmentation

Bin Packing Problem with Conflicts and Item Fragmentation (BPPC-IF) is a variant of the classical bin packing problem in which fragments of the same item can be packed in different bins (item fragmentation), and some pairs of items cannot be packed in the same bin (item conflicts). To solve the problem, we propose a new mixed-integer linear programming (MILP) model and two heuristics. The MILP model represents BPPC-IF as a generalized transportation problem in which sources correspond to items, and sinks are defined for all maximal independent sets in the graph representing conflicts between items. The proposed heuristics, called Sequential Packing Heuristics 1 and 2 (SPH1 and SPH2, respectively) are modifications of the previously proposed Sequential Maximum Degree Packing Heuristic (SMDPH) for BPPC-IF, which fills bins by solving a series of MILP subproblems. Both new heuristics use improved criteria for selecting items to pack. Additionally, in SPH1, bins are filled heuristically without solving any MILP subproblems. Computational experiments carried out on a large set of benchmark problem instances from the literature show that (i) SPH1 obtains good quality solutions and is orders of magnitude faster than the MILP-based heuristics, (ii) compound heuristic SPH1+SPH2 finds on average better solutions than any other heuristic and is on average faster than any other MILP-based heuristic, and (iii) the proposed MILP model greatly outperforms the best previously proposed MILP model.