Chapter 1 Applications of network optimization

Network optimization is being an increasingly important and fundamental issue in the fields such as engineering, computer science, operations research, transportation, telecommunication, decision support systems, manufacturing, and airline scheduling. In many applications, however, there are several criteria associated with traversing each edge of a network. For example, cost and flow measures are both important in the networks. As a result, there has been recent interest in solving Bicriteria Network Optimization Problem. The Bicriteria Network Optimization Problem is known a NP-hard. The efficient set of paths may be very large, possibly exponential in size. Thus the computational effort required to solve it can increase exponentially with the problem size in the worst case. In this paper, we propose a genetic algorithm (GA) approach used a priority-based chromosome for solving the bicriteria network optimization problem including maximum flow (MXF) model and minimum cost flow (MCF) model. The objective is to find the set of Pareto optimal solutions that give possible maximum flow with minimum cost. This paper also combines Adaptive Weight Approach (AWA) that utilizes some useful information from the current population to readjust weights for obtaining a search pressure toward a positive ideal point. Computer simulations show the several numerical experiments by using some difficult-to-solve network design problems, and show the effectiveness of the proposed method.

Contributors

In this paper, we study the problem of technical transient gas network optimization, which can be considered a minimum cost flow problem with a nonlinear objective function and additional nonlinear constraints on the network arcs. Applying an implicit box scheme to the isothermal Euler equation, we derive a mixed-integer nonlinear program. This is solved by means of a combination of (i) a novel mixed-integer linear programming approach based on piecewise linearization and (ii) a classical sequential quadratic program applied for given combinatorial constraints. Numerical experiments show that better approximations to the optimal control problem can be obtained by using solutions of the sequential quadratic programming algorithm to improve the mixed-integer linear program. Moreover, iteratively applying these two techniques improves the results even further.

The Traveling Salesman Goes Shopping: The Systematic Deviations of Grocery Paths from TSP-Optimality

The flow maximization problem being a leading problem in network optimization is widely studied with many applications. In this work, we find the maximum flow pulled out from the source and sent to the sink using the uncertainty theory. We develop a flow function to get maximum flow for the network having uncertain arc capacities and sufficient storage capacities at the intermediate vertices. The centiles-method is used to find the maximum flow value and an algorithm is proposed to get non-conservative maximum flow. The result is also illustrated by an example. The flow values are compared graphically to see the dominance of the non-conservative maximum flow over the conservative maximum flow.

Despite the success of metaheuristic algorithms in solving complex network optimization problems, they often struggle with adaptation, especially in dynamic or high-dimensional search spaces. Traditional approaches can become stuck in local optima, leading to inefficient exploration and suboptimal solutions. Most of the widely accepted advanced algorithms do well either on highly complex or smaller search spaces due to the lack of adaptation. While recent approaches have attempted to address adaptability through various mechanisms, they typically require problem-specific tuning or sacrifice generalization capability for performance, failing to maintain consistent efficiency across different optimization tasks and varying problem scales. To address these limitations, we present ELENA (Epigenetic learning through evolved neural adaptation), a new evolutionary framework that incorporates epigenetic mechanisms to enhance the adaptability of the core evolutionary approach. ELENA leverages compressed representation of learning parameters improved dynamically through epigenetic tags that serve as adaptive memory. Three epigenetic tags (mutation resistance, crossover affinity, and stability score) assist with guiding solution space search, facilitating a more intelligent hypothesis landscape exploration. To assess the framework’s performance, we conduct experiments on three critical network optimization problems: the Traveling salesman problem (TSP), the Vehicle routing problem (VRP), and the Maximum clique problem (MCP). Experiments indicate that ELENA achieves competitive results, often surpassing state-of-the-art methods on network optimization tasks.

Decomposition based approaches are recalled from primal and dual point of view. The possibility of building partially disaggregated reduced master problems is investigated. This extends the idea of aggregated-versus-disaggregated formulation to a gradual choice of alternative level of aggregation. Partial aggregation is applied to the linear multicommodity minimum cost flow problem. The possibility of having only partially aggregated bundles opens a wide range of alternatives with different trade-offs between the number of iterations and the required computation for solving it. This trade-off is explored for several sets of instances and the results are compared with the ones obtained by directly solving the natural node-arc formulation. An iterative solution process to the route assignment problem is proposed, based on the well-known Frank Wolfe algorithm. In order to provide a first feasible solution to the Frank Wolfe algorithm, a linear multicommodity min-cost flow problem is solved to optimality by using the decomposition techniques mentioned above. Solutions of this problem are useful for network orientation and design, especially in relation with public transportation systems as the Personal Rapid Transit. A single-commodity robust network design problem is addressed. In this, an undirected graph with edge costs is given together with a discrete set of balance matrices, representing different supply/demand scenarios. The goal is to determine the minimum cost installation of capacities on the edges such that the flow exchange is feasible for every scenario. A set of new instances that are computationally hard for the natural flow formulation are solved by means of a new heuristic algorithm. Finally, an efficient decomposition-based heuristic approach for a large scale stochastic unit commitment problem is presented. The addressed real-world stochastic problem employs at its core a deterministic unit commitment planning model developed by the California Independent System Operator (ISO).

In computer science, there are many algorithms that finds a minimum spanning tree for a connected weighted undirected fuzzy graph. The minimum length (or cost) spanning tree problem is one of the nicest and simplest problems in network optimization, and it has a wide variety of applications. The problem is tofind a minimum cost (or length) spanning tree in G. Applications include the design of various types of distribution networks in which the nodes represent cities, centers etc.; and edges represent communication links (fiber glass phone lines, data transmission lines, cable TV lines, etc.), high voltage power transmissionlines, natural gas or crude oil pipelines, water pipelines, highways, etc. The objective is to design a network that connects all the nodes using the minimum length of cable or pipe or other resource in this paper we find the solution to the problem is to minimize the amount of new telephone line connection using matrixalgorithm with fuzzy graph.

The maximum multicommodity flow problem is a natural generalization of the maximum flow problem to route multiple distinct flows. Obtaining a 1-ε approximation to the multicommodity flow problem on graphs is a well-studied problem. In this paper we present an adaptation of recent advances in single-commodity flow algorithms to this problem. As the underlying linear systems in the electrical problems of multicommodity flow problems are no longer Laplacians, our approach is tailored to generate specialized systems which can be preconditioned and solved efficiently using Laplacians. Given an undirected graph with m edges and k commodities, we give algorithms that find 1-ε approximate solutions to the maximum concurrent flow problem and maximum weighted multicommodity flow problem in time O(m4/3poly(k,ε-1)).

Previous article Algorithms for the Assignment and Transportation ProblemsJames MunkresJames Munkreshttps://doi.org/10.1137/0105003PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAbout[1] Merrill M. Flood, The traveling-salesman problem, Operations Res., 4 (1956), 61–75 MR0078639 CrossrefISIGoogle Scholar[2] L. R. Ford, Jr. and , D. R. Fulkerson, A simple algorithm for finding maximal network flows and an application to the Hitchcock problem, RAND Corporation, Santa Monica, 1955, P-743, Sept. 26 Google Scholar[3] H. W. Kuhn, The Hungarian method for the assignment problem, Naval Res. Logist. Quart., 2 (1955), 83–97 MR0075510 0143.41905 CrossrefGoogle Scholar Previous article FiguresRelatedReferencesCited byDetails Scalable Semidefinite ProgrammingAlp Yurtsever, Joel A. Tropp, Olivier Fercoq, Madeleine Udell, and Volkan Cevher8 February 2021 | SIAM Journal on Mathematics of Data Science, Vol. 3, No. 1AbstractPDF (2365 KB)A Scalable Approach to Compute Delay Margin of a Class of Neutral-Type Time Delay SystemsAdrián Ramírez, Dimitri Breda, and Rifat Sipahi1 March 2021 | SIAM Journal on Control and Optimization, Vol. 59, No. 2AbstractPDF (802 KB)Closing Gaps in Asymptotic Fair DivisionPasin Manurangsi and Warut Suksompong8 April 2021 | SIAM Journal on Discrete Mathematics, Vol. 35, No. 2AbstractPDF (657 KB)A Privacy-Preserving Method to Optimize Distributed Resource AllocationOlivier Beaude, Pascal Benchimol, Stéphane Gaubert, Paulin Jacquot, and Nadia Oudjane24 August 2020 | SIAM Journal on Optimization, Vol. 30, No. 3AbstractPDF (968 KB)Multilevel Artificial Neural Network Training for Spatially Correlated LearningC. B. Scott and Eric Mjolsness29 October 2019 | SIAM Journal on Scientific Computing, Vol. 41, No. 5AbstractPDF (1559 KB)Index Reduction via Unimodular TransformationsSatoru Iwata and Mizuyo Takamatsu3 July 2018 | SIAM Journal on Matrix Analysis and Applications, Vol. 39, No. 3AbstractPDF (458 KB)Max-Plus Algebraic Statistical Leverage ScoresJames Hook16 November 2017 | SIAM Journal on Matrix Analysis and Applications, Vol. 38, No. 4AbstractPDF (2992 KB)A Seamless Multilevel Ensemble Transform Particle FilterA. Gregory and C. J. Cotter28 November 2017 | SIAM Journal on Scientific Computing, Vol. 39, No. 6AbstractPDF (537 KB)Community Detection in Temporal Multilayer Networks, with an Application to Correlation NetworksMarya Bazzi, Mason A. Porter, Stacy Williams, Mark McDonald, Daniel J. Fenn, and Sam D. Howison6 January 2016 | Multiscale Modeling & Simulation, Vol. 14, No. 1AbstractPDF (1688 KB)Multilevel Ensemble Transform Particle Filtering3 May 2016 | SIAM Journal on Scientific Computing, Vol. 38, No. 3AbstractPDF (479 KB)Algebraic Multilevel Preconditioner for the Helmholtz Equation in Heterogeneous Media9 October 2009 | SIAM Journal on Scientific Computing, Vol. 31, No. 5AbstractPDF (3632 KB)Combinatorial Analysis of Singular Matrix Pencils26 January 2007 | SIAM Journal on Matrix Analysis and Applications, Vol. 29, No. 1AbstractPDF (187 KB)Incremental Processing Applied to Munkres’ Algorithm and Its Application in Steinberg’s Placement ProcedureH. W. Carter, M. A. Breuer, and Z. A. Syed3 August 2006 | SIAM Journal on Algebraic Discrete Methods, Vol. 6, No. 2AbstractPDF (1082 KB)On the Assignment Polytope18 July 2006 | SIAM Review, Vol. 16, No. 4AbstractPDF (1033 KB)Algorithms for Obtaining Shortest Paths Visiting Specified Nodes2 August 2006 | SIAM Review, Vol. 15, No. 2AbstractPDF (772 KB)A Review of the Placement and Quadratic Assignment Problems18 July 2006 | SIAM Review, Vol. 14, No. 2AbstractPDF (2163 KB)Optimal and Suboptimal Algorithms for the Quadratic Assignment Problem13 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 10, No. 2AbstractPDF (867 KB)An Out-of-Kilter Method for Minimal-Cost Flow Problems10 July 2006 | Journal of the Society for Industrial and Applied Mathematics, Vol. 9, No. 1AbstractPDF (957 KB)The Backboard Wiring Problem: A Placement Algorithm18 July 2006 | SIAM Review, Vol. 3, No. 1AbstractPDF (1020 KB) Volume 5, Issue 1| 1957Journal of the Society for Industrial and Applied Mathematics History Submitted:08 August 1956Published online:10 July 2006 InformationCopyright © 1957 Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0105003Article page range:pp. 32-38ISSN (print):0368-4245ISSN (online):2168-3484Publisher:Society for Industrial and Applied Mathematics

In Part II of this two-part paper, a sum-rate-maximizing power allocation\nwith minimum power consumption is found for multiple-input multiple-output\n(MIMO) decode-and-forward (DF) two-way relaying (TWR) in a network optimization\nscenario. In this scenario, the relay and the source nodes jointly optimize\ntheir power allocation strategies to achieve network optimality. Unlike the\nrelay optimization scenario considered in part I which features low complexity\nbut does not achieve network optimality, the network-level optimal power\nallocation can be achieved in the network optimization scenario at the cost of\nhigher complexity. The network optimization problem is considered in two cases\neach with several subcases. It is shown that the considered problem, which is\noriginally nonconvex, can be transferred into different convex problems for all\nbut two subcases. For the remaining two subcases, one for each case, it is\nproved that the optimal strategies for the source nodes and the relay must\nsatisfy certain properties. Based on these properties, an algorithm is proposed\nfor finding the optimal solution. The effect of asymmetry in the number of\nantennas, power limits, and channel statistics is also considered. Such\nasymmetry is shown to have a negative effect on both the achievable sum-rate\nand the power allocation efficiency in MIMO DF TWR. Simulation results\ndemonstrate the performance of the proposed algorithm and the effect of\nasymmetry in the system.\n

The multicommodity flow problem arises when several different commodities are transshipped from specific supply nodes to the corresponding demand nodes through the arcs of an underlying capacity network. The maximum flow over time problem concerns to maximize the sum of commodity flows in a given time horizon. It becomes the earliest arrival flow problem if it maximizes the flow at each time step. The earliest arrival transshipment problem is the one that satisfies specified supplies and demands. These flow over time problems are computationally hard. By reverting the orientation of lanes towards the demand nodes, the outbound lane capacities can be increased. We introduce a partial lane reversal approach in the class of multicommodity flow problems. Moreover, a polynomial-time algorithm for the maximum static flow problem and pseudopolynomial algorithms for the earliest arrival transshipment and maximum dynamic flow problems are presented. Also, an approximation solution to the latter problem is obtained in polynomial-time.

Graph theory has gained considerable popularity in many areas of application due to its versatility, including ease of modeling and visualization. In general, a graph can represent an arbitrary model of real-world objects and the relationships between them, which allows applying any known algorithms to it and subject it to other types of further analysis. The theoretical foundations of graphs are also widely used in the field of operations research. For example, the travelling salesman problem, finding the minimum spanning tree of a weighted graph, finding the shortest path between two vertices etc. The purpose of this work is to analyze the practical applications of graph theory, both in the process of software development and for software development in general, as well as in the subsequent maintenance and support of software systems. In addition to using the theoretical foundations of graphs in other branches of science, graph theory has also found wide application in solving many practical problems in different areas of human life, such as chemistry, sociology, logistics, biology, economics, technology and so on. Similarly, in software development, graph theory has taken its rightful place. Many kinds of problems are solved with its help, both in the process of software engineering and for the development of software engineering itself. The analysis of the graph theory application opportunities for software development and maintenance has shown that at most stages of the software development cycle, graphs are used only for modeling and visualization of various kinds of information, but there is almost no automated algorithmic analysis of the obtained models. This may indicate a promising direction for discovering additional opportunities of graph theory usage in the software development processes.

In rider demand responsive systems riders submit requests to demand transit services and the incoming requests and vehicles are matched by the system. This demand-responsive transport problem is viewed by existing research as a kind of spatial matching problem between vehicles and riders. However, existing schemes mainly focus on maximizing the number of matching pairs. It neglects other factors like the length of pickup trajectories and riders’ waiting time. And the matching is not revocable, which loses the chance for further optimizations. Moreover, there is still not much work on handling and matching the appointment-based requests that play a key role in the demand-responsive transport market. In this article, we propose an algorithm called BMCF ( <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Bipartite Minimal-Cost Flow</i> ) to solve the taxi-rider matching problem with appointment-based rider requests on a time-dependent road network. Unlike existing solutions that map the problem into the max flow problem, we transform the optimal matching to a <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">minimal cost flow problem</i> which aims to maximize a utility value and could be solved efficiently. Riders and vehicles are modeled as vertices in a bipartite graph, and factors like the length of pickup trajectories, riders’ waiting time, etc. are abstracted as utilities and denoted as weights of the edges which are taken into account during the matching process. To the best of our knowledge, the proposed scheme is the first to efficiently integrate and process both the real-time and appointment-based requests within the same framework, and the assignments between vehicles and requests could be dynamically adjusted and revoked to maximise the overall utility. Experimental results show that the proposed algorithm can effectively increase the appointment-based matching ratio and decrease the riders’ waiting time (> 10.4%) and vacant vehicles’ picking up time (> 60.9%) compared with existing schemes, while at the cost of acceptable increase on the running time.

In the past few decades, the architecture of optical networks has undergone significant evolution, from the earliest wavelength-division multiplexing (WDM) optical networks to elastic optical networks (EONs) and later to space-division multiplexing (SDM) EONs, to address the continuous growth of Internet traffic. By 2024, Pbps-level optical networks are expected, far exceeding the capacity limit of single-mode fibers. The massive SDM era is on the horizon. In this context, a newly designed architecture for optical networks called the spatial channel network (SCN) architecture, which achieves high cost efficiency by means of practical hierarchical optical cross-connects, has recently been proposed. However, the evolution of optical network architectures will simultaneously present challenges related to network optimization. For instance, with the evolution from WDM optical networks to EONs, the kernel network optimization problem was transformed from the routing and wavelength assignment (RWA) problem into the routing and spectrum assignment (RSA) problem due to the additionally introduced constraint of spectrum contiguity. Similarly, specially designed algorithms are also expected to be essential for addressing the network optimization problem in SCNs. In this paper, we define this new problem as the routing, spatial channel, and spectrum assignment (RSCSA) problem. We propose an integer linear programming (ILP) model and a heuristic algorithm to solve the RSCSA problem. We examine the performance of the proposed approaches via simulation experiments. The results show that both proposed approaches are effective in finding the optimal solutions or solutions close to the lower bounds. To the best of our knowledge, this is the first work to focus on the network optimization problem in SCNs.