A heuristic and an exact method for the gate matrix connection cost minimization problem

Abstract

In many applications, a sequencing of patterns (electronic circuit nodes, cutting patterns, product orders etc.) has to be found in order to optimize some given objective function, giving rise to the so-called Open Stack Problems. We focus on a problem related to the optimization of Gate Matrix Layouts: electronic circuits are obtained by connecting gates and one seeks a gate layout permutation that minimizes connection costs under restrictions on the circuit area. In the literature, the connection costs and the circuit area are also known as Time of Open Stacks and Maximum Number of Open Stacks, respectively. We propose a genetic algorithm providing heuristic solutions and a branch-and-cut algorithm based on a new linear integer programming formulation that represents, to the best of our knowledge, the first exact method proposed in the literature. The algorithms have been tested on real instances and on data sets from the literature. The computational results give evidence that the proposed methods provide solutions that improve the ones found by the approaches presented in the literature

Optimal patchings for consecutive ones matrices

“…Actually, some of the examples mentioned in [27] are of this type, like the ones in archaeology and film-making. We give here a short description of some other typical applications and variants (see, e.g., [10] for additional details).…”

“…The MOS Problem then seeks for such an X that minimizes the maximum number of 1s in each column, while the TOS Problem reduces to an MLC1PP Problem. The MOS problem has been investigated in the literature, see, for instance, Baptiste [2], de la Banda and Stuckey [11], and [10].…”

“…Then, a feasible solution to the GMCCP is a positive patching X of M and the number of required tracks for each net is the number of 1s in the corresponding column of X . Therefore, it is not difficult to see that GMCCP reduces to an MLC1PP Problem with the further request to have a bounded value for the maximum number of 1s in each column, see [10].…”

Mathematical models for the minimization of open stacks problem

“…The MOSP belongs to this class of problems. Other examples of pattern sequencing problems include the minimization of order spread problem (MORP), which minimizes the total sum of the lifespan of the stacks (Foerster and Wäscher, 1998; De Giovanni et al., 2013b); the minimization of tool switches problem (MTSP), which minimizes the number of tool switches in a machine of a flexible manufacturing system (Tang and Denardo, 1988; Silva et al., 2020); and, the minimization of discontinuities problem, which is also known as the consecutive blocks minimization problem (Soares et al., 2020). These three classes of problems are usually solved sequentially, despite some integrated approaches in the literature (Armbruster, 2002; Rinaldi and Franz, 2007; Yanasse and Lamosa, 2007).…”

“…To the best of our knowledge, there is no approach in the literature comparing the relative computational performance of these formulations. However, the analysis of these formulations and their constraints is relevant, since limiting the maximum number of simultaneously open stacks is also considered as a constraint in some approaches (Belov and Scheithauer, 2007; Matsumoto et al., 2011; De Giovanni et al., 2013b).…”