General model of tool path problem for the CNC sheet cutting machines

The formalization of the tool path problem for the CNC sheet metal/material cutting equipment is considered. General model of tool path problem for laser/plasma/gas/water-jet machines is offered. Model uses the term “the basic cutting segment” proposed by author. The existing classification of the tool path problem is expanded. In the paper also the discretization of offered general statement of optimization problem is described. Results of computing experiments for some instances are given. © 2019, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved.Russian Foundation for Basic Research, RFBR: № 17-08 -01385The work was supported by the Russian Foundation for Basic Research ( grant № 17-08 -01385) and by Act 211 Government of the Russian Federation, contract № 02.A03.21.0006

The routing problems with optimization of the starting point: dynamic programming

The extreme routing problem focused on engineering applications in mechanical engineering is considered. We mean the well-known task of tool controlling in the CNC sheet cutting machines. A mathematical model is presented which includes a system of megalopolises (nonempty finite sets) and cost functions depending on the list of tasks. Megalopolises are constructed on the basis of discretization of equidistant curves of part contours. The dependence on the list of tasks is connected with reasons associated with the dynamic constraints that arise in the process of task completion. Among all restrictions, the conditions of precedence are distinguished (earlier cutting of the inner contours and more earlier cutting of large parts). Rational consideration of the precedence conditions allows one to reduce the complexity of calculations when widely understood dynamic programming (DP) is used in the implementation that develops R. Bellman's scheme. This approach makes it possible to solve the problem of optimizing complexes, which include the initial state (starting point), the method of numbering megalopolises in the order of their visits, and the specific trajectory of the process. For a problem complicated by the dependence of the terminal function on the initial state, a decomposition algorithm is used, which allows, in a substantial part of the procedure, the application of a single (for all initial states) DP scheme. The optimal algorithm based on DP is implemented as a program for PC; a computational experiment is conducted

О задаче последовательного обхода мегаполисов с условиями предшествования и функциями стоимости с зависимостью от списка заданий

A constrained routing problem with complicated cost functions is studied. The construction of the cost functions can be difficult, and therefore the stages of this construction are elements of the solution of the problem. This situation arises, in particular, in studying the engineering problem of dismantling radiation hazardous elements, where, in the framework of a problem statement traditional for discrete optimization, it takes an unacceptably long time to construct a cost matrix whose entries characterize the radiation doses received by performers at the stage of displacement and dismantling. It is assumed that, at the stage of the computational implementation of the resulting optimal algorithm, the corresponding “parts” of the matrix may be not fed to the computer’s memory but calculated as needed. Possible applications of the developed methods may be related to the problem of dismantling a decommissioned generator unit of an NPP. © Krasovskii Institute of Mathematics and Mechanics.Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00573) and is a part of the research carried out at the Ural Mathematical Center

On the Problem of Sequential Traversal of Megalopolises with Precedence Conditions and Cost Functions Depending on a List of Tasks

A constrained routing problem with complicated cost functions is studied. The construction of the cost functions can be difficult,and therefore the stages of this construction are elements of the solution of the problem. This situation arises, in particular, in studyingthe engineering problem of dismantling radiation hazardous elements, where, in the framework of a problem statement traditional fordiscrete optimization, it takes an unacceptably long time to construct a cost matrix whose entries characterize the radiation dosesreceived by performers at the stage of displacement and dismantling. It is assumed that, at the stage of the computational implementationof the resulting optimal algorithm, the corresponding “parts” of the matrix may be not fed to the computer’s memory but calculated as needed.Possible applications of the developed methods may be related to the problem of dismantling a decommissioned generator unit of a nuclear power plant. © 2021, Pleiades Publishing, Ltd.Russian Foundation for Basic Research, РФФИ: 19-01-00573This work was supported by the Russian Foundation for Basic Research (project no. 19-01-00573) and within the research carried out at the Ural Mathematical Center

The routing problems with optimization of the starting point: Dynamic programming

The extreme routing problem focused on engineering applications in mechanical engineering is considered. We mean the well-known task of tool controlling in the CNC sheet cutting machines. A mathematical model is presented which includes a system of megalopolises (nonempty finite sets) and cost functions depending on the list of tasks. Megalopolises are constructed on the basis of discretization of equidistant curves of part contours. The dependence on the list of tasks is connected with reasons associated with the dynamic constraints that arise in the process of task completion. Among all restrictions, the conditions of precedence are distinguished (earlier cutting of the inner contours and more earlier cutting of large parts). Rational consideration of the precedence conditions allows one to reduce the complexity of calculations when widely understood dynamic programming (DP) is used in the implementation that develops R. Bellman’s scheme. This approach makes it possible to solve the problem of optimizing complexes, which include the initial state (starting point), the method of numbering megalopolises in the order of their visits, and the specific trajectory of the process. For a problem complicated by the dependence of the terminal function on the initial state, a decomposition algorithm is used, which allows, in a substantial part of the procedure, the application of a single (for all initial states) DP scheme. The optimal algorithm based on DP is implemented as a program for PC; a computational experiment is conducted. © 2019 A.G. Chentsov, P.A. Chentsov.Russian Foundation for Basic Research, RFBR: 17–08–01385Funding. This research was supported by the Russian Foundation for Basic Research (projects no. 17–08–01385)

The Dependence of Actual Laser Cutting Speed on CNC Sheet Equipment on Number of NC Program Commands for Metal Grades 1.0114 and AWAIMg3

One of the complex optimization problems arising in technical applications is the cutting path problem for CNC sheet metal cutting machines. This problem is to optimize the tool path for the CNC technological equipment when cutting a fixed set of parts allocated on sheet metal. The time of cutting process is one of objective functions for the optimization problem. This objective function depends on six parameters of cutting process: 1) cutting speed; 2) lengths of the cutting tool path; 3) speed of idling motion of tool; 4) length of idling motion; 5) number of piercings; 6) time of one piercing. When programming NC program, speed of cutting is defined by the user of the CAM (Computer-Aided-Manufacturing) system and considered as a constant. However, this is actually not the case. Consequently, the problem of exact calculation of objective function is arisen. In order to solve this problem, the correction coefficients for the cutting speed value defined by the user must been calculated. Obviously, these coefficients will vary for different thicknesses and grades of metal. The paper presents the results of investigates, which made it possible to find the functional dependences of the cutting speed for the CNC laser cutting machine (CO2) on the number of NC program commands for the following metal grades: 1.0114 (thickness Δ = 1-10 mm) and AWAIMg3 (Δ = 1-5 mm).The statistical materials of experiments were processed by using software "Mathcad". The received results show that the actual average value of cutting speed is monotonically decreasing function depended on number of NC program commands. In addition, the calculation of the actual cutting speed provides not only an accurate calculation of the value of the objective function, but also, as a result, the correct finding of the optimal tool path. © Published under licence by IOP Publishing Ltd.Russian Foundation for Basic Research, RFBR: 19-01-00573The work was supported by the Russian Foundation for Basic Research (grant № 19-01-00573) and by Act 211 Government of the Russian Federation, contract №02.A03.21.0006

Routing in cnc cutting machines: Engineering constraints

The paper is devoted to the problems of cutting path optimization with engineering specifics. There are two severe constraints of thermal cutting: thermal expansion and sheet stiffness constraints. Our solution uses the dependence of the cost functions on the previous part of the route and on the direction of movement of the tool along the contour. © 2020, Budapest Tech Polytechnical Institution. All rights reserved.This work was supported by Russian Foundation for Basic Research, grant 20-08-00873

Об одной задаче последовательного обхода множеств

The problem of sequential traversal of megapolises with precedence conditions is investigated; this problem is oriented to mechanical engineering — CNC metal cutting machines. There is the following setting singularity: the terminal component of additive criterion contains the dependence on the starting point. This singularity leads to the fact that the natural solution procedure based on dynamic programming must be applied individually for every starting point. The investigation goal consists in the construction of an optimizing algorithm for determining a complex including a route (a variant of megapolis numbering), a trajectory, and a starting point. The proposed algorithm realizes an idea of directed enumeration of starting points. This algorithm is realized as a program for PC; computations for model examples are made. © 2021 Udmurt State University. All rights reserved.The study was funded by the Russian Foundation for Basic Research (project No. 20–08– 00873)

Optimization Models of Tool Path Problem for CNC Sheet Metal Cutting Machines

The problem of tool path optimization for CNC sheet metal cutting equipment is considered. Sheet metal cutting equipment includes laser/plasma/gas/water-jet machines and some others. Users of CAD/CAM systems develop numerical control programs for the cutting equipment after nesting of parts onto the sheet. The control programs contain information about tool path. The tool path is a routing of cutter head used for cutting of sheet material. Classification and the correspondent mathematical models of tool path problem are considered. The tasks of cost/time minimization for various types of cutting techniques are formalized. Mathematical formalization of technological constraints for these tasks is also described. Unlike the known analogs this formalization allows to consider constraints of thermal cutting. In some cases the optimization tasks can be interpreted as discrete optimization problem (generalized travel salesman problem with additional constraints, GTSP). In paper also the developed exact algorithm and some heuristic algorithms of tool path optimization based on described models is reported. Results of computing experiments for some instances are given © 2016The work was supported by Act 211 Government of the Russian Federation, contract № 02.A03.21.0006