On an iterative procedure for solving a routing problem with constraints

The generalized precedence constrained traveling salesman problem is considered in the case when travel costs depend explicitly on the list of tasks that have not been performed (by the time of the travel). The original routing problem with dependent variables is represented in terms of an equivalent extremal problem with independent variables. An iterative method based on this representation is proposed for solving the original problem. The algorithm based on this method is implemented as a computer program. © 2013 Pleiades Publishing, Ltd

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)

ON ROUTING PROBLEM WITH STARTING POINT OPTIMIZATION

One problem focused on engineering applications is considered. It is assumed that sequential visits to megacities have been implemented. After all visits have been made, it is required to return to the starting point (a more complex dependence on the starting point is also considered). But the last requirement is not strict: some weakening of the return condition is acceptable. Under these assumptions, it is required to optimize the choice of starting point, route, and specific trajectory. The well-known dynamic programming (DP) is used for the solution. But when using DP, significant difficulties arise associated with the dependence of the terminal component of the criterion on the starting point. Starting point enumeration is required. We consider the possibility of reducing the enumeration associated with applied variants of universal (relative to the starting point) dynamic programming. Of course, this approach requires some transformation of the problem

Methods of extremal routing and their application to the control of sheet cutting on CNC machines

An extremal routing problem with constraints and complicated cost functions is considered. The investigated setting is oriented to application in engineering problems connected with sheet cutting on the machines with CNC. Nonstandard variant of dynamic programming is used for construction of an optimal solution including the starting point, the route (index permutation), and a concrete trajectory of the process. This procedure is implemented in the form of standard programs for a PC and a supercomputer. © 2019 Author(s)

OPTIMIZING THE STARTING POINT IN A PRECEDENCE CONSTRAINED ROUTING PROBLEM WITH COMPLICATED TRAVEL COST FUNCTIONS

We study the optimization of the initial state, route (a permutation of indices), and track in an extremal problem connected with visiting a finite system of megalopolises subject to precedence constraints where the travel cost functions may depend on the set of (pending) tasks. This problem statement is xemplified by the task to dismantle a system of radiating elements in case of emergency, such as the Chernobyl or Fukushima nuclear disasters. We propose a solution based on a parallel algorithm, which was implemented on the Uran supercomputer. It consists of a two-stage procedure: stage one determines the value (extremum) function over the set of all possible initial states and finds its minimum and also the point where it is achieved. This point is viewed as a base of the optimal process, which is constructed at stage two. Thus, optimization of the starting point for the route through megalopolises, connected with conducting certain internal tasks there, is an important element of the solution. To this end, we employ the apparatus of the broadly understood dynamic programming with elements of parallel structure during the construction of Bellman function layers

Dynamic programming and questions of solvability of route bottleneck problem with resource constraints

The article deals with the problem of admissible routing for a system of cycles each of which contains exterior permutation and works connected with megalopolises (non-empty finite sets) visiting. In the initial setting, a resource constraint is given; this constraint should be fulfilled for every cycle under permutation. The solvability conditions in this problem are connected with the extremum of the auxiliary bottleneck routing problem without above-mentioned constraint, in which the apparatus of widely understood dynamic programming (DP) is used. A particular case of the setting is the known bottleneck courier problem which can be used (in particular) for routing a vehicle (airplane or helicopter) aiming to realize the given shipping system with a limited fuel reserve on each flight. An algorithm implemented on a personal computer is constructed. © 2022 Authors. All rights reserved.Ministry of Education and Science of the Russian Federation, Minobrnauka, (075–02–2022–874)Funding. This work was funded within the framework of research at the Ural Mathematical Center supported by the Ministry of Science and Higher Education of the Russian Federation (agreement number 075–02–2022–874)

On the question of the optimization of permutations in the problem with dynamic constraints

The “additive” problem of sequentially visiting megalopolises (nonempty finite sets) is considered; some tasks are executed as the megalopolises are visited. Permutations and operations are evaluated by cost functions that admit a dependence on the list of tasks. There are restrictions of different types, which include precedence conditions used in the “positive direction” (to reduce the complexity of calculations). In addition, this conception involves dynamical restrictions that are formed in the process of task execution. This conception is oriented to engineering applications associated with sheet cutting on CNC machines. An approach to constructing optimal solutions based on a nonstandard version of dynamic programming (DP) is investigated. This approach takes into account restrictions of different types, including dynamic constraints which naturally arise in sheet cutting applications (in particular, it takes into account heat tolerances related to diffusion of heat in the vicinities of tie-in points). In this regard, a combination of “direct” interdictions of movements and cutting and a system of penalties is allowed; in the latter case, cost functions with a dependence on the list of tasks arise. The variant of DP that is used allows one to optimize the selection of a starting point, the route, which is identified with a permutation of the indexes, and the trajectory corresponding to the above-mentioned route. An economical variant of DP is used at the stage of construction of the Bellman function. This conception allows avoiding calculation of the whole array of values of this function; instead, only the system of its layers is defined (in the case of using the precedence conditions, which are typical of the problem of sheet cutting, this conception results in a considerable reduction of calculation costs). An optimal DP-based algorithm is constructed and realized on PC, and the results of the computational experiment are presented. © 2019 Udmurt State University. All rights reserved.Russian Foundation for Basic Research, RFBR: 18–01– 00410Funding. The research was funded by the Russian Foundation for Basic Research (project no. 18–01– 00410)

Minimax routing problem with a system of priority tasks

For a minimax routing problem with precedence conditions and cost functions that allow dependence on the list of tasks, we study the statement for which some of the tasks are allocated as first-priority ones. Other tasks can be started only after the fulfillment of priority tasks. The tasks themselves are connected with visiting megacities and, in particular, individual cities (terms corresponding to works in the field of solving the traveling salesman problem). One needs to find the extremum of arising two-stage problem with a minimax criterion, as well as the optimal compositional solution. In the paper, the optimal algorithm implemented on a PC is substantiated and built; a computational experiment is carried out. Possible applications may be related to some problems of aviation logistics in which it is required to ensure the visit of one vehicle (airplane or helicopter) to a system of aerodromes under a limited fuel reserve at each stage of the flight task; refueling is expected at points of visit (it is also assumed that a set of priority tasks is allocated). © 2024 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved

On reachable sets for one-pulse controls under constraints of asymptotic character

We study asymptotic versions of reachable sets of linear systems for two intuitive formalizations of onepulse controls given constraints of asymptotic character. The results are presented for the simplest example of linear control systems, the double integrator, though they admit a straightforward extension to a generic linear system. We suppose that the coefficient at the control is a piecewise continuous function. To illustrate the developed theoretical framework for both formalizations, we demonstrate examples of linear control systems, the double integrator, though they admit a straightforward extension to a generic linear system. We suppose that the coefficient at the control is a piecewise continuous function. To illustrate the developed theoretical framework for both formalizations, we demonstrate examples of attraction sets, asymptotic versions of reachable sets