Exactly solving packing problems with fragmentation

Mathematical models and decomposition algorithms for cutting and packing problems

“…• bm-BPPIF (8), a set of 540 instances that was proposed by Casazza and Ceselli [57] and is a copy of benchmark (7). Parameter B is removed and parameter F is set to 0.5 F * where F * is the optimal solution of the corresponding fm-BPPIF instance.…”

“…Now an item is allowed to be fractionally packed into different bins, as long as the sum of the fractions is equal to the weight of the item. Several variants of the BPPIF have been studied in the literature, see, e.g., the recent work by Casazza and Ceselli [57].…”

“…If LB and the solutions given by the model do not match after K = 3, we solved (4.33)-(4.37). Table 4.5 provides the results obtained by the branch-and-price algorithm of Casazza and Ceselli [57], the adaptation of arc-flow for the BPPIF, and the adaptation of reflect with a time limit of 3600 seconds. The results from the literature were copied from the original papers, and the time was adjusted to incorporate unsolved instances in the average time.…”

Enhanced Pseudo-polynomial Formulations for Bin Packing and Cutting Stock Problems

“…The bin packing problem with item fragmentation (BPPIF) is the BPP generalization in which items are allowed to be fractionally packed into different bins. Several BPPIF variants have been studied in the literature, see, e.g., Casazza and Ceselli (2016). In this section, we show how to solve the two main problem variants: (i) minimize the number of bins used for the packing while the total number of fragmentations is at most F (bm-BPPIF); and (ii) minimize the number of fragmentations while the number of bins is at most B (fm-BPPIF).…”

“…For each bin type t, we define an arbitrarily large availability b t , a capacity c t = tc, and a cost p t = t. Under this construction, using a bin of type t in the VSBPP corresponds to using t bins of capacity c in the BPPIF. This derives from the fact that a fractional packing in t bins can always be obtained by using no more than t − 1 fragmentations (see, e.g., Casazza and Ceselli 2016). Thus, an optimal bm-BPPIF solution can be obtained by solving a modified F V S RE that includes the additional constraint…”