Data Structures and Route Reduction Procedures in the Problem of Distribution Transport Flows in a Communication Network

The paper discusses the methodology of mathematical modeling of the two-stage problem of optimization of the backbone hierarchical communication network with multicommodity discrete flows and parameters. The methodology is based on the sequential solution of the problem of optimizing the network structure and the problem of distribution and routing of discrete correspondence flows. As a rule, such networks consist of a decentralized backbone network and fragmented networks in the internal service areas of the backbone nodes. There are four types of network nodes and three levels of its hierarchy. In a multicommodity network, each node can exchange correspondence (products, goods, cargo, messages) with other nodes. Correspondence is characterized by a source node, a drain node and a value, which for transport networks is given by the number of packaged goods, cargo in a package of a unified size, and for data transmission networks – by the number of bytes, kilobytes, etc. In the transport backbone network, all correspondence is first sorted by destination addresses, packed in transport blocks (containers), and then transported in vehicles along the transport highways. In data networks, correspondence is also sorted by destination addresses (multiplexed), packaged in virtual transport blocks, and then transmitted over trunk communication channels. The size (capacity, volume) of the transport blockt is set by the parameter, and is determined by the number of units of correspondence that fit into it. Mathematical models of problems of optimization of network structure, distribution and routing of flows, and an example of numerical modeling of solving problems on a transport network containing 120 nodes and 300 unoriented arcs are presented. Experimental studies have shown high computational efficiency of the proposed algorithms and programs, and they can be recommended for the practical solution of problems of optimizing the processes of processing and transporting flows in communication networks of large dimensions.

The article discusses methodological approaches to the construction of multicommodity hierarchical communication networks and identifies the main tasks of processing and distribution of discrete correspondence flows, which allow to create favorable conditions for reducing material, financial and labor costs in transport systems with further mechanization and automation of production. Multicommodity communication networks are characterized by the presence of a multitude of sources and drains of correspondence flows (products or requirements). Correspondence is understood as a pair of different network nodes, between which there is a directed (addressed) discrete flow of elements (for example, indivisible loads of uniform size, bits or symbols) of a given size. In a multicommodity network, all correspondence flows are subject to a one-time transfer from sources to drains. In general, a certain set of types (categories) of correspondence can be defined on the network, differing in weight, dimensions and other characteristics, but having common sources and drains. The hierarchical structure of the network and the principles of sorting, distribution and routing of flows are given.

This article is the second part of the work, which proposes a methodology for mathematical modeling of the step-by-step development of nodes and transport routes in a hierarchical network with multicommodity discrete correspondence flows. As a rule, such networks consist of a decentralized trunk network and networks in the internal service areas of trunk nodes. In a multicommodity network, each node can exchange correspondence (products, goods, cargo, messages) with other nodes. In the trunk network, all correspondence is transmitted via communication channels or transported in vehicles in transport blocks of a given size (capacity, volume). In this part of the work, using the example of transport networks, it is experimentally shown that the step-by-step solution of the problems of optimizing the structure of the trunk network and the distribution and routing of flows allows obtaining initial data for building dynamic deterministic and stochastic models of their development. It is also shown how these tasks can be used for operational redistribution of flows in case of equipment failures in nodes and on communication lines (exceeding the throughput capacities of nodes and communication channels, the carrying capacity of vehicles, etc.). Keywords: multicommodity hierarchical networks, discrete flows, combinatorial optimization problems, mathematical models, computer modeling.

Introduction. The article discusses mathematical models of problems of constructing circular routes of vehicles in a multicommodity hierarchical network. As a rule, such networks consist of a decentralized backbone network and networks in the internal service areas of the backbone nodes (internal networks). In multicommodity networks, each node can exchange products (goods, cargo) with other nodes. In contrast to the distribution problems of a homogeneous interchangeable product, in multicommodity problems the flows of products are not interchangeable, the flow of each product must be delivered from a specific primary object to a specific customer. It is assumed that the multi-level structure of the transport network is defined and the geographical location of the main hubs and its internal service areas with a set of nodes for the delivery and collection of goods (customers) are known. Therefore, the problems of determining the main routes of vehicles and constructing circular routes of internal vehicles are considered independently of each other. The types of costs of real transport processes, which should be taken into account in the formation of the objective function of routing problems, are discussed and mathematical models of problems for constructing circular delivery routes with a heterogeneous fleet of vehicles are proposed. The possibility of solving the formulated problems with the help of well-known packages of mixed and integer linear programming is noted. Purpose. The aim of the article is to formulate new mathematical models of the problem of constructing circular routes of vehicles in the internal networks of servicing the main nodes, which take into account the real costs of transport processes and the geographical features of internal networks. The technique. The research methodology is based on the system analysis of many modern models, methods and algorithms for solving the problems of constructing circular routes for customer service in the internal zones of the main nodes of the hierarchical network. Results. On the basis of the review and analysis of known mathematical models, several new variants of mathematical formulation of problems of designing routes of vehicles for the transportation of discrete cargo in the internal zones of the central nodes of the network have been developed. To solve the problems, precise, heuristic and metaheuristic methods and algorithms can be used, implemented in many commercial and non-commercial packages of mixed and integer programming programs, for example, IBM ILOG CPLEX, GAMS, AIMMS, Gurobi Optimizer, ABACUS, COIN-OR, GLPK, lp_solve. Many of them are available for free on the NEOS server (https://neos-server.org/neos/). Scientific novelty and practical significance. The novelty of the work lies in the formulation of mathematical models of the problem of constructing circular routes of vehicles, which take into account the real costs of transport processes and geographical features of internal networks. The materials of the article form the methodological basis for the development of modern mathematical support for solving the problems of long-term, current and operational planning and management in the internal zones of the trunk nodes of the global hierarchical network. Keywords: problems of combinatorial optimization, mathematical models of circular routes of vehicles.

The article discusses methodological approaches to the construction of multicom-modity hierarchical communication networks and identifies the main tasks of pro-cessing and distribution of discrete correspondence flows, which allow to create favorable conditions for reducing material, financial and labor costs in transport systems with further mechanization and automation of production. The number of levels of hierarchy in the network, as a rule, is determined by the administrative division of the territory, the subordination of territorial administration bodies, the adopted technology for processing and distributing the flows of goods and information. Three levels of hierarchy are identified in the network — trunk, zonal and internal, and four types of nodes. Nodes of the first, second and third types, located on the trunks of the transport network or data transmission network and connecting them sections of vehicle routes or communication channels, constitute the trunk network. All trunk nodes have service areas that form zonal levels of the trunk network. Nodes of the fourth type are located in the internal service area of any main node and together with it form an internal network. Multicommodity communication networks are characterized by the presence of multiple sources and drains of correspondence flows (products or requirements). Correspondence is understood as a pair of different network nodes, between which there is a directed discrete flow of elements of a given value. In the backbone network, all correspondence is transmitted through communication channels or transported in vehicles in transport blocks of a given size (capacity, volume). The size of a transport block is measured by the number of units of correspondence that fit into it. All trunk nodes are sorting centers, in which correspondence is first sorted by destination addresses, and then packed into transport blocks. This article some aspects of the creation and implementation of spatial data infrastructure, information system and special mathematical support for transport networks and data transmission networks are presented. A conceptual approach to building an information-analytical decision support system for managing the processes of handling and distribution of discrete flows in hierarchical communication networks is considered. The issues of creating a toolkit and information platform (portal) for automating decision-making processes in the tasks of operational management, current planning and long-term development for trunk transport networks and data transmission backbone networks are discussed.

The article is devoted to the study of the optimization problem for the hierarchical structure of a multicommodity communication network with discrete flows and parameters when its density changes (the ratio of the number of network arcs to the maximum possible number of arcs for a given number of nodes in the network). The network has three levels of hierarchy — a backbone, a zonal and a internal and four types of nodes — backbone nodes of the first, second and third types, forming the backbone and zonal levels of the network, and nodes of the fourth type, which subordinate to each backbone node and form the internal levels of the network. The types of nodes different from one another in functionality. The main task of the study is to establish how the structure of the backbone network changes (the number and location of backbone nodes of the first, second and third types), the scheme of processing and distribution of flows and technical and economic indicators of the network's functioning for various degrees of its density. The principles of organizing the sorting and distribution of flows in a three-level network and its mathematical model are given. A mathematical model of the problem of optimizing the structure of the backbone network and the schemes of sorting and distribution of flows is formulated. The algorithms for solving the problem are based on the discrete analogue of the local descent method proposed earlier by the authors, when the neighborhoods of the metric space of possible solutions are chosen from heuristic considerations, taking into account the peculiarities of the problem being solved. Computer modeling of the problem on a homogeneous network containing 100 nodes with a change in the degree of nodes from 2 to 99 is carried out. The modeling was carried out on the example of a road transport network for the transportation of cargo using a computer program, which is part of the instrumental software of the Information and Analytical Decision Support System (IA DSS), which is being developed at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine. An experimental study of the solution to the problem showed that the best technical, economic and operational indicators of its functioning are achieved with a degree nodes from 9 to 14, when high network connectivity is ensured, and is significantly reduced the average and maximum time of delivery of cargos to recipients. The proposed computer technology for solving problem with changing network density allows you to interactively modeling various options of a network , changing the topology, hierarchical structure, flows, parameters and constraints of the model and from the family of the resived results choose the best option, taking into account the selected a goal function and the accepted constraints; calculate preliminary technical and economic indicators of the network's functioning, estimate the cost of additional resources and plan the amount of investment required for the modernization and construction of its structural elements, which ultimately makes it possible to increase the efficiency functioning of the network by optimizing use of its resources and reducing the operating costs for the processing and transportation of flows.

The article considers the importance of transport and communication as key sectors for the development of the economy of Ukraine, as well as the need to create applied mathematical software for the optimization of transport processes and the development of information and analytical systems. Special attention is paid to the development and study of multi-product communication networks with discrete flows of small consignments, which have a significant impact on market transformations in Ukraine, especially in the field of transportation of small consignments. The study focuses on modern and future communication networks, which, as a rule, have a multi-level structure determined by the administrative division of the territory, management hierarchy and technologies for the processing and distribution of goods and information. The authors of the article emphasize the need to design multi-level communication networks, which would take into account the complexity of the system, as well as the impossibility of complete centralization of information processing and management in one link. The article also focuses on the analysis of existing research and publications in this field, paying attention to the contribution of foreign and domestic scientists to the development of models and methods of network resource management. Most research focuses on deterministic models and models based on the mathematical foundations of mass service theories and random Markov processes. The authors indicate the inadequacy of the existing mathematical tools for modeling and analyzing the functioning of multi-product networks with discrete flows, which requires the development of new mathematical models and methods.The purpose of the research is the development and analysis of effective approaches to the design of multi-tracks pipeline transport networks with a special focus on discrete trunk data flows. The authors note that this includes the development of new approaches and methods for effective management and optimization of resource allocation, as well as modeling the specifics of discrete flows in the context of trunk transport systems. An important part of the research is the creation of a hierarchical system structure for automated resource management and flow distribution in large-scale networks. The authors emphasize the need to develop a methodology that integrates already existing achievements in this field and allows to effectively solve practical problems related to long-term development, ongoing planning and operational management in large-scale networks. Such a methodology should include models of different levels of aggregation and be suitable for all levels of the network’s hierarchical structure. The study also includes consideration of mathematical models and algorithms for modeling and analyzing the functioning of multi-product networks with discrete flows.

In the work the improvement of a method for balanced queue management on the interface of a telecommunication network router is proposed. The novelty of the proposed method is to provide a consistent solution to the problems of aggregation and distribution of packet flows among queues (Congestion Management tasks), as well as problems of balancing the allocation of the interface band-width among the class queues (Resource Allocation Tasks) in accordance with the principles of the Traffic Engineering Queues concept. Unlike the known approaches, the proposed solution has a positive effect on the computational complexity of the calculations, which is connected, first of all, with the decrease in the size of optimization problems and their linear nature. This was conditioned by the fact that the consistency of the Congestion Management and Resource Allocation processes was ensured on the basis of their agreed solution, since the decision on aggregation and distribution of flows among queues was not connected with the results of the solutions of the Resource Allocation problem. The study of Congestion Management and Resource Allocation processes using the proposed method has confirmed its efficiency and adequacy of the obtained solutions. The aggregation and distribution of packet flows in corresponding queues were carried out in accordance with the values of their classes. While the optimal balanced allocation of the interface bandwidth among the formed class queues was carried out in accordance with the principles of the concept of Traffic Engi-neering Queues - the queue with a higher value of the class had a lower utilization factor.

The article is devoted to the study of the problem of optimizing the hierarchical structure of a multicommodity communication network with discrete flows when changing its important parameters, such as the capacity of network arcs in transport blocks and the size of the transport block for transportation of the discrete small-lot cargo or of the data transmission in a digital communication network. The network has three levels of hierarchy – a backbone, a zonal and an internal and four types of nodes - backbone nodes of the first, second and third types, forming the backbone and zonal levels of the network, and nodes of the fourth type, which subordinate to each backbone node and forming the internal levels of the network. Types of nodes differ from each other in terms of functionality. The main task of the study is to establish how the structure of the backbone network changes (the number and location of backbone nodes of the first, second, and third types), the flow processing and distribution scheme, and the technical and economic indicators of the network's functioning for different values of its parameters. The principles of organization of sorting and distribution of flows in a three-level network and its mathematical model are given. A mathematical model of the optimization problem of the backbone network structure and flow sorting and distribution scheme is formulated. Algorithms for solving the problem are based on the discrete analogue of the local descent method proposed by the authors earlier, when the neighborhoods of the metric space of possible solutions are chosen based on heuristic considerations, taking into account the specifics of the problem being solved. Computer modeling of the problem on a network containing 10 nodes at the specified change in the both parameters is carried out. The modeling was carried out on the example of the transport network of cargo transportation using a computer program that is part of the instrumental software of the Information and Analytical Decision Support System (IA DSS), which is being developed at the Institute of Telecommunications and Global Information Space of the National Academy of Sciences of Ukraine. An experimental study of solution the problem showed that the structure of the network is weakly dependent on the change in the carrying capacity of the arcs and the size of the transport block. The proposed computer technology for solving problem when the network parameters are changed allows you to interactively modeling various options of a network, changing the topology, hierarchical structure, flows, parameters and constraints of the model and from the set of the obtained solutions to choose a best option, taking into account the selected a goal function and the accepted constraints; calculate preliminary technical and economic indicators of the network's functioning, estimate the cost of additional resources and plan the amount of investment required for the modernization and construction of its structural elements, which ultimately makes it possible to increase the efficiency functioning of the network by optimizing use of its resources and reducing the operating costs for the processing and transportation of flows.

Purpose. The purpose of the article is a critical analysis of the existing methods of freight car flows distribution on the railway network and the determination of the directions of these methods adaptation to the operating conditions of the Ukrainian railway transport. Methodology. The research was carried out on the basis of the processing of literary sources in the scientometric databases Scopus and Google Scholar. Results. The distribution of car flows on the railway network is one of the stages of developing a plan for the trains formation. The methods of solving this problem, which are currently used in the practical work of railway transport, were developed during the time of the planned state economy. They are based on the solution of a single-criterion optimization problem aimed at finding such a distribution of car flows on the network that minimizes the cost of transportation for the railway. Reforming the railway transport market with the aim of introducing competition requires the complication of methods of solving the problem of distribution of car flows to take into account the conflicting goals of independent participants in the transport process. The conducted studies showed that the main method of modeling the structure of the railway transport network and the volumes of car flows in it is the theory of graphs; determination of the carrying capacity of elements of the railway network is carried out by the methods of the theory of railway operation, simulation modeling and correlation analysis; determination of the optimal distribution of bandwidth is carried out by comparing competing options, as well as by linear and non-linear programming methods, adaptive control methods using neural networks; the behavior of participants in the transportation process during the allocation of bandwidth is investigated by game theory methods. It has been established that in modern conditions, the task of determining the criterion of equal access to congested sections on the branched railway network and the task of distributing car flows on the network in conditions of fluctuating car flows during transportation by regular trains and trains to order require additional research. Originality. The scientific novelty of the work consists in the critical analysis and generalization of modern methods used to solve the problem of distribution freight car flows distribution on railway networks. Practical value. The practical significance of the results lies in determining the inconsistency of the methods that are practically used on the railways of Ukraine for the distribution of car flows, with the modern conditions of operation of railway transport. Scientific methods have been established that allow to improve approaches to solving the problem of rational distribution of car flows, taking into account the interests of various participants in the transportation process.

The article is devoted to the study of the subproblem of distribution and merging of correspondence flows in separate zones of the backbone network, which arises when solving the general problem of optimizing the hierarchical structure of a multicommodity communication network with discrete flows and parameters. In a multicommodity network, each node can exchange correspondence (products, goods, cargo, messages) with other nodes. Correspondence is characterized by a source node, a drain node and a value, which for transport networks is given by the number of packaged goods in a package of a unified size, and for data transmission networks – by the number of bytes, kilobytes, etc. In the backbone network, all correspondence is transported in vehicles in transport units of a given size (capacity, volume) or transmitted via communication channels. The size of a transport block is measured by the number of units of correspondence that fit into it (for example, 40 packaged goods, 100 gigabytes). All trunk nodes are sorting centers in which correspondence is first sorted by destination addresses (nodes) and then packed as consolidated correspondence into transport blocks. Since the size of individual correspondence is much smaller than the size of the transport block, they can be combined (packed) with correspondence with other destination addresses several times and in different nodes during sorting. There are three levels of hierarchy in the network – backbone, zonal and internal and four types of nodes – trunk nodes of the first, second and third types, forming the backbone and zonal levels of the network and nodes of the fourth type, which are subordinate to each trunk node and form internal levels of the network. Node types differ from each other in functionality. The main task of the study is to develop a mathematical model and algorithms for solving the subproblem of optimizing the distribution and merging (sorting) of correspondence flows at the zonal levels of the network. It is shown that it can be formulated as a linear programming problem with a block structure of constraints and the Danzig-Wolf decomposition method and other methods of integer programming can be used to solve it. To solve the problem on real networks, approximate algorithms based on the construction of the shortest paths are proposed.

The paper proposes a new approach to solving the problem of distribution of information flows on the communication network based on the method of constraints. A multi-pole multi-product graph is used as a mathematical model of the communication network structure. The problem of distribution of information flows is solved in three stages. At the first stage, the structural analysis of the network is carried out using the mathematical apparatus of Boolean algebra. As a result, we get a lot of ways to distribute information flows. At the second stage, the structural reliability of the found set of information flow distribution paths is calculated using the orthogonalization algorithm. At the final stage, using the method of constraints, a compromise version of the distribution of information flows of corresponding pairs of nodes is found.

The vehicle routing problem is a problem of combinatorial optimization and operational research, which has been largely studied for many years, as well as its variants. With the increased distribution of product and service flows, many logistics applications increasingly requires customers to be grouped into clusters that facilitate delivery or any processing upstream or downstream of customers. In this article, we propose an approach using the known problem of generalized vehicle routing (GVRP) which is an extension of the problem of vehicle routing (VRP) and the problem of split delivery vehicle routing (SDVRP). The GVRP handles a set of clients grouped into separate groups, while the SDVRP allowed for split deliveries. The constraint that each customer must be visited only once is then removed. In this article, using a simple example and combining the two previous methods, we give an illustration of the problem of multisplit delivery. This example presents the new concept of problems of clustered vehicle routings that overlap with multisplit deliveries. This can be considered as an extension of previous work. We also give a mathematical formulation of this type of problem. A two-level variant will be presented, and we propose an algorithm inspired by those proposed in the references.

Contributors

Understanding the dynamics of groundwater–surface water interaction is needed to evaluate and simulate water and solute transport in catchments. However, direct measurements of the contributions of different flow routes from specific surfaces within a catchment toward the surface water are rarely available. For this study, we physically separated the tile drain discharge toward a 43.5 m ditch transect from the groundwater–plus–overland flow routes. Direct groundwater flow and ephemeral overland flow were jointly captured in three sheet pile in‐stream reservoirs, while the effluent from three tile drain outlets was collected in vessels. Our flux measurements showed that, in response to a rainfall event, the tile drain contribution to the total ditch discharge decreased from 80% to 28%. We used these flow route measurements to calibrate a field‐scale integrated water transport model. The HydroGeoSphere code was used because it simultaneously solves the flow regimes in the variably saturated domain, the tile drain domain, and the surface flow domain. This simultaneous solution is needed for a correct representation of the mutual interactions between groundwater flow, tile drain flow, and ditch water flow. Our model produced a flow distribution between the flow paths which deviated only 2% from the measured flow distribution. A sensitivity analysis showed that model parameters related to tile drain entrance resistance and to the resistance to water flow through the surface water system controlled the water flow route distribution but with little effect on groundwater levels. This indicates that a model calibration based on groundwater levels alone does not necessarily produce a correct representation of the flow route contributions.