Node-based Formulation Research Articles

A location-routing problem for local supply chains

This study addresses a local Supply Chain problem by proposing the Collection and Delivery Location Open Routing Problem (CDLORP), a variant of the classical location-routing problem that considers both collection and distribution routes to decide where to locate facilities. Collection routes are responsible for picking up the required raw material from local suppliers, while distribution routes deliver the final product to local customers. This study is motivated by a public initiative that desires to seize the by-product generated by the agro-industrial local sector to produce and provide healthy and nutritional snacks to students in the public elementary school system in Saltillo, Coahuila, Mexico. A node-based formulation and a flow-based formulation are proposed to solve instances of the problem. The formulations are valid for the symmetric and asymmetric cases of the problem. Four families of valid inequalities from the literature are adapted for the problem and are used to implement two branch-and-cut algorithms based on the proposed formulations. Computational experiments using 42 benchmark instances of different sizes (25 to 260 vertices) are performed to assess the efficacy of the proposed formulations and algorithms. The best results are obtained by the branch-and-cut algorithm using the node-based formulation. Finally, a case study is also addressed and optimally solved with the branch-and-cut algorithm using the node-based formulation. The results obtained for the benchmark instances and the case study show that the proposed approach can solve real-life problems to develop local supply chains for new products.

Exploring the numerical performance of node-based smoothed finite elements in coupled hydro-mechanical problems

Node-based smoothed finite element (NS-FEM) formulations for the coupled hydro-mechanical problem are now becoming popular in soil mechanics, but there has been limited comparative study of their performance. In this work, a node-based formulation is developed for the coupled hydro-mechanical problem (u−pw formulation) using low-order, equal-order interpolants and an implicit time marching scheme for the global problem. The numerical properties of the resulting formulation are then compared to different implementations of classical finite elements. The use of nodal smoothing does not render by itself the solution stable at the undrained limit, on the contrary, the range of undrained instability is increased. Stabilizing terms previously employed in FEM are equally successful in stabilizing NS-FEM solutions, but the computational cost of the stabilized solution is much higher for NS-FEM. It is also shown that NS-FEM does not present by itself any particular advantage in terms of mesh dependency over classic FEM: when strain localization is present the solution obtained is strongly dependent on mesh size.

Low energy fold paths in multistable origami structures

Origami design concepts are finding pervasive utility in engineering applications due to their ability to map complex shape transformations into a series of folding actions. The interplay between stretching, folding, and facet bending modes in origami structures also generates a complex energy landscape of multistable states to leverage for engineering applications. However, identifying rigid and deformable folding paths in this high-dimensional and non-convex energy landscape remains a challenge. To help address this challenge, we first introduce a global, constraint-based approach to modeling origami that uses a redundant kinematic description of the facets and nodes, and treats the kinematic compatibility between these redundant descriptors as a constraint. This approach allows for complex facet shapes without increasing the dimensionality of the system, as would be necessary in truss-based and other node-based formulations in order to stiffen the facet. Secondly, we adopt the nudged elastic band method, that is widely used in computational chemistry, to identify minimum energy folding paths. This strategy addresses, from a global perspective, the difficulty of piecing together sequences of local folding steps in order to connect two different points in configuration space. We implement this path finding approach on both the kinematic constraint formulation and a truss-based model, and compare their behaviors on a series of folding and multistable origami examples.

Unconstrained binary models of the travelling salesman problem variants for quantum optimization

Quantum computing is offering a novel perspective for solving combinatorial optimization problems. To fully explore the possibilities offered by quantum computers, the problems need to be formulated as unconstrained binary models, taking into account limitation and advantages of quantum devices. In this work, we provide a detailed analysis of the travelling salesman problem with time windows (TSPTW) in the context of solving it on a quantum computer. We introduce quadratic unconstrained binary optimization and higher-order binary optimization formulations of this problem. We demonstrate the advantages of edge-based and node-based formulations of the TSPTW problem. Additionally, we investigate the experimental realization of the presented methods on a quantum annealing device. The provided results pave the path for utilizing quantum computer for a variety of real-world tasks which can be cast in the form of travelling salesman problem with time windows.

Unified Formulations for Combined-Cycle Units

In this letter, we describe and develop edge-based and node-based formulations for combined-cycle units (CCUs) with different levels of accuracy in respect of the physical constraints of each turbine, from no requirement to partially and fully respecting all physical constraints, including min-up/-down time, capacity, and ramping constraints. We develop corresponding models for independent system operators (ISOs) and vertically integrated utilities (VIUs), respectively. These models demonstrate complexity in terms of the size of decision variables and show the tradeoff between accuracy and computational complexity. To the best of our knowledge, this letter provides one of the first models that can capture both configuration transition restrictions and all the physical constraints of each turbine within CCUs, for both ISOs and VIUs.

The location-routing problem with simultaneous pickup and delivery: Formulations and a heuristic approach

In this paper, we consider a variant of the Location-Routing Problem (LRP), namely the LRP with simultaneous pickup and delivery (LRPSPD). The LRPSPD seeks to minimize total cost by simultaneously locating the depots and designing the vehicle routes that satisfy pickup and delivery demand of each customer at the same time. We propose two polynomial-size mixed integer linear programming formulations for the problem and a family of valid inequalities to strengthen the formulations. While the first formulation is a node-based formulation, the second one is a flow-based formulation. Furthermore, we propose a two-phase heuristic approach based on simulated annealing, tp_SA, to solve the large-size LRPSPD and two initialization heuristics to generate an initial solution for the tp_SA. We then empirically evaluate the strengths of the proposed formulations with respect to their ability to find optimal solutions or strong lower bounds, and investigate the performance of the proposed heuristic approach. Computational results show that the flow-based formulation performs better than the node-based formulation in terms of the solution quality and the computation time on small-size problems. However, the node-based formulation can yield competitive lower bounds in a reasonable amount of time on medium-size problems. Meantime, the proposed heuristic approach is computationally efficient in finding good quality solutions for the LRPSPD.

Edge Elements and Current Diffusion

Current flow in the neighborhood of sharp conducting edges and corners is encountered in many applications of computational electromagnetics. Some examples include resonant cavities and waveguides of rectangular shape, cracks in metal components in nondestructive testing, and the edges and corners of the rail–armature contact in an electromagnetic launcher. Although the number of analytical solutions to such problems is small, it is generally believed that such solutions often involve singularities, high gradients, or discontinuities in one or more field components near the edge or corner. It has been demonstrated that the so-called edge elements can provide much more accurate predictions of global quantities (like resonant frequencies and scattering parameters) in the case of full-wave electromagnetics when field discontinuities or singularities are present. They have also been shown to better represent field discontinuities at material interfaces in magnetostatics and reentrant corners in low-frequency electromagnetics. This feature of edge elements is due to their ability to allow necessary jumps in field components at geometric or material discontinuities, as opposed to nodal elements, with the Coulomb gauge enforced, which can overconstrain the fields in certain situations. The main question of interest in this paper is whether edge elements give a similar advantage in the representation of current diffusion around a sharp edge. Both edge- and node-based formulations are considered, and the results are discussed.

An Information-Theoretic View of Network Management

We present an information-theoretic framework for network management for recovery from nonergodic link failures. Building on recent work in the field of network coding, we describe the input-output relations of network nodes in terms of network codes. This very general concept of network behavior as a code provides a way to quantify essential management information as that needed to switch among different codes (behaviors) for different failure scenarios. We compare two types of recovery schemes, receiver-based and network-wide, and consider two formulations for quantifying network management. The first is a centralized formulation where network behavior is described by an overall code determining the behavior of every node, and the management requirement is taken as the logarithm of the number of such codes that the network may switch among. For this formulation, we give bounds, many of which are tight, on management requirements for various network connection problems in terms of basic parameters such as the number of source processes and the number of links in a minimum source-receiver cut. Our results include a lower bound for arbitrary connections and an upper bound for multitransmitter multicast connections, for linear receiver-based and network-wide recovery from all single link failures. The second is a node-based formulation where the management requirement is taken as the sum over all nodes of the logarithm of the number of different behaviors for each node. We show that the minimum node-based requirement for failures of links adjacent to a single receiver is achieved with receiver-based schemes.

Performance comparison of finite-element approaches for electromagnetic waveguides

The accuracy of finite-element-method techniques used for waveguide modal analysis has usually been assessed by testing the precision of the propagation constant. As a consequence, no useful criteria have been proposed for checking the spatial distribution of the evaluated unknown field. To overcome this lack, two error figures have been introduced and applied to different finite-element-method formulations. In particular, the following approaches have been compared: the one based on the transverse magnetic field, those based on the so-called edge elements, and a new one presented by the authors. The new approach, obtained by simply operating on the matrices of the original node-based formulation, directly solves for the propagation constant at a given frequency, preserves the matrix sparsity, and directly evaluates all of the unknown magnetic field components. Results demonstrate the applicability of the proposed approach and the usefulness of the introduced figures for a deep waveguide analysis.