Linear one-dimensional cutting-packing problems: numerical experiments with the sequential value correction method (SVC) and a modified branch-and-bound method (MBB)

Abstract

Two algorithms for the one-dimensional cutting problem, namely, a modified branch-and-bound method (exact method) and a heuristic sequential value correction method are suggested. In order to obtain a reliable assessment of the efficiency of the algorithms, hard instances of the problem were considered and from the computational experiment it seems that the efficiency of the heuristic method appears to be superior to that of the exact one, taking into account the computing time of the latter. A detailed description of the two methods is given along with suggestions for their improvements.

Pricing-based primal and dual bounds for selected packing problems

“…in which the formula defines each new pseudo-price value to be the weighted sum of its previous value and the material consumption norm within the last obtained pattern. Some years later the SVC scheme was employed again by Mukhacheva et al [97] in one-dimensional CSP, and by Verkhoturov and Sergeyeva [131] for the two-dimensional CSP with irregular shapes. The latter dealt the issue of the item placement by defining the pseudo-prices as functions of the angle of rotation and such angles are chosen in order to maximize the pseudo-price values.…”

Mathematical models and decomposition algorithms for cutting and packing problems

“…In the early noughties Mukhacheva et al [222] proposed a pattern oriented branchand-bound algorithm for both the BPP and the CSP, while Korf [179,180] proposed a "bin completion" algorithm (later improved on by Schreiber and Korf [252]) in which decision nodes are produced by assigning a feasible set to a bin. However, starting from the late nineties, branch-and-price (see Section 2.6) proved to be very effective, and became the most popular choice for the exact solution of the BPP.…”

One-dimensional heuristics adapted for two-dimensional rectangular strip packing

“…Here we adapt the method sequential value correction (SVC), already well-known e.g. for 1D stock cutting (Mukhacheva et al, 2000, Belov andScheithauer, 2003).…”

“…A similar algorithm from (Mukhacheva and Valeeva, 2000) fills every hole by solving a subset-sum problem with item widths; then, items are ordered by non-increasing heights. Inspired by Sub and the 1D-CSP heuristic SVC (Mukhacheva et al, 2000, Belov andScheithauer, 2003), we propose the following modification SubKP of Sub. Namely, to fill the empty spaces solving a one-dimensional knapsack problem.…”

“…A thorough investigation of SVC for 1D cutting has been done in (Mukhacheva et al, 2000, Belov andScheithauer, 2003). The idea is to fill every bin by solving a knapsack problem while assigning some heuristic "pseudo-values" to the items.…”