An Exact Algorithm for Constrained Two-Dimensional Two-Staged Cutting Problems

Abstract

The constrained two-dimensional cutting (C_TDC) problem consists of determining a cutting pattern of a set of n small rectangular piece types on a rectangular stock plate S with length L and width W, to maximize the sum of the profits of the pieces to be cut. Each piece type i, i=1,…,n, is characterized by a length li, a width wi, a profit (or weight) ci, and an upper demand value bi. The upper demand value is the maximum number of pieces of type i that can be cut on S. In this paper, we study the two-staged C_TDC problem, noted C_2TDC. It is a classical variant of the C_TDC where each piece is produced, in the final cutting pattern, by at most two cuts. We solve the C_2TDC problem using an exact algorithm that is mainly based on a bottom-up strategy. We introduce new lower and upper bounds and propose new strategies that eliminate several duplicate patterns. We evaluate the performance of the proposed exact algorithm on problem instances extracted from the literature and compare it to the performance of an existing exact algorithm.

ALMA, A Logistic Mobile Application Based on Internet of Things

“…Reference is made to [21] and [22] for details on the two-stage two-dimensional method based on strip generation and beam search. A maximum of twenty processors were allocated to this particular problem.…”

Approximate and exact algorithms for the double-constrained two-dimensional guillotine cutting stock problem

“…is the maximum profit that can be achieved by filling region P = S \ R. This bound is the tightest of bounds obtained by solving-as in [22,24,26]-an UTDC, a single bounded knapsack (SBK), and a relaxed bounded knapsack (RSBK).…”

“…Herein, we find an initial feasible solution to DCTDC using an approximate two-stage procedure inspired from Hifi and M'Hallah [24]. The procedure solves CTDC, a DCTDC relaxation obtained by ignoring the lower demand constraints.…”

A worst case analysis of a dynamic programming-based heuristic algorithm for 2D unconstrained guillotine cutting

“…Several different exact algorithms have been suggested for this problem, including those described in Christofides and Whitlock (1977), Wang (1983), Viswanathan and Bagchi (1988), Vasko (1989), Dowsland et al (1992), Christofides and Hadjiconstantinou (1995) and Hifi and M'Hallah (2005). Approximate algorithms for this problem have also been proposed by Morabito and Arenales (1996), Cung et al (2000), Belov and Scheithauer (2006), Hifi and M'Hallah (2006), Gongalves (2007) and Cui (2008).…”