Algorithm of Sequential Improving the Size Coefficient for Solving the Problem of Partitioning the Multiple Connected Orthogonal Polygon

Abstract

The problem of geometrical partitioning the multiple connected orthogonal polygon is considered in the given paper. The problem refers to the NP-hard class of problems because it is necessary to fulfill exhaustive search for assured finding the optimal solution. It stipulates the interest for developing efficient heuristic methods for solving the above problem. Some multiply-connected orthogonal polygon is supposed to be parted into a set of rectangles avoiding their intersecting and their crossing the polygon borders. The goal function represents certain minimization of summarized length of boundary junctions in the process of partitioning. The mathematical model of the problem is offered. The algorithm based on improving the size coefficient, characterizing the degree of rectangle elongation, has been developed. The algorithm consists of two procedures applied sequentially: the first one is for generating the primary polygon partitioning, and the second is for primary partitioning transformation taking into account the adjacent elements and the compound adjacent elements. The computing experiment has been carried out and the results are shown. Keywords—problem of geometrical partitioning, multiple connected orthogonal polygon, size coefficient, primary partition, composite united, minimization of joints length partition. I. PREFACE The partition problems refer to the class of discrete optimization problems. Actually, there are many problems based on the similar models, for example, scheduling cuttingpacking problems, nesting and many others. The solving of such problems is aimed at resource saving. The problems of orthogonal packing into a quadrant [7,8], on sheets [8,10] and into a semi-endless strip [9,14,15,17] are of special importance. No less important is determining of the lower and upper boundaries in cutting-packing problems [8]. Besides, there are classes of regular and non-regular packing problems [11]. All the above mentioned problems are considered for 2D and 3D space [12, 13, 16]. The researchers in this field have been developing new approaches to solving different specific problems that arise in practice. The partitioning problems have been known and are being researched to present day [1, 2]. In these problems it is required to divide some object into parts to reach definite aim. The partitioning problems are considered to be problems of rational use of resources, therefore the solving of the above problems results in real economic effect for business. The efficient solution to such problems is urgent from both theoretical and practical points of view. The partitioning problem can be considered as a separate problem as well as a sub-problem of some technological process [3, 4]. Technological processes in different applied branches of industry are known to use the stage of cutting or the stage of element placement taking into account their geometrical properties [3]. The process is very important for resource saving but it is difficult for decision making. The classification of the basic models for geometrical placement problems is given in papers [2, 5]. The process of creating a card for geometrical partitioning the multiple connected orthogonal polygon into rectangular domains is considered in the given paper. II. THE PROBLEM OF GEOMETRICAL PARTITIONING THE MULTIPLE CONNECTED ORTHOGONAL POLYGON Let us consider in greater detail the problem of geometrical partitioning the multiple connected orthogonal polygon (MOP), where it is required to divide some domain into minimum number of elements (rectangles). At that, the total length of boundary junctions of the elements can be considered as an important criterion of partitioning. When solving such problems, the existence of the so-called prohibited zones in the area to be divided is of great importance because the search for the algorithm depends on it. The proposed mathematical model of the problem of geometrical partitioning MOP is a modification of the subproblem to the complex problem of geometrical covering and orthogonal cutting [3, 6]. 7th Scientific Conference on Information Technologies for Intelligent Decision Making Support (ITIDS 2019) Copyright © 2019, the Authors. Published by Atlantis Press. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/). Advances in Intelligent Systems Research, volume 166