Sample Test Problems Research Articles

Solving Three-Dimensional Path Planning Problem Using a Visibility-Based Graphical Representation of the Design Space

Abstract Planning the shortest collision-free path among scattered obstacles is an NP-complete problem. As reviewed in this paper, a variety of deterministic as well as heuristic methods have been developed to address different instances of the problem. The focus of the deterministic methods is primarily on the optimality of the final solution and has been applied exclusively to regular shapes such as spheres or cubes. Nevertheless, due to the problem's intrinsic complexities (especially in 3D), researchers mainly resort to heuristics that offer acceptable (yet possibly suboptimal) solutions with reasonable resources. Therefore, less attention has been given to further the state-of-the-art in deterministic methods, which for 3D problems primarily focuses on approximating the solution. However, with the advancements in high-performance computing, we believe it is time to focus on solution quality. As such, this study aims to further the efforts in deterministic optimization methods for 3D path planning by overcoming some challenges of the existing methods and improving the optimality of the solution. The proposed approach is rooted in visibility-based planning methods where the obstacle-free space is modeled as a connectivity graph to be searched for the shortest path. The advantage of the proposed method in constructing the representative graph is that it does not make approximations to identify the graph nodes, unlike the existing methods. Nor does it limit the objects’ geometries to specific shapes such as blocks or spheres. The capability of the method in finding the shortest collision-free paths in environments cluttered with convex polyhedra is demonstrated using sample test problems.

The Parameters Affecting the Mathematical Programming Model of the Appropriate Transportation System

Due to the importance of public transportation, which has become today an important urban problem with special conditions, this study was conducted aimed at determining the parameters affecting the mathematical programming model of the appropriate transportation system. The study was performed by modeling method, and the problem was solved in three different sizes (small, medium and large) to show the efficiency of the model and the proposed solution methods. Some common test problems in the literature were selected and the required characteristics of the problem under study were defined for them as needed, since there are no sample test problems in the literature to examine the performance of the developed algorithms. The results indicated that some of the most important parameters affecting the designed mathematical model were movement risks, fatigue and drowsiness while driving, demands at the source node, demands at the destination node, number of vehicles across the network, the time interval between requests, and the time interval between two consecutive requests. The mentioned parameters were so effective that little changes in parameters such as demands at the source node, demands at the destination node, and the number of vehicles across the network significantly increased time and decreased profitability. the company has a significant effect on the debt cost rate and capital cost rate and does not have a significant effect on stock prices. Finally, the share price shows little sensitivity to the company's financial structure.

A mathematical model to design dynamic cellular manufacturing systems in multiple plants with production planning and location–allocation decisions

In this research, a mixed-integer nonlinear programming model is formulated to design cellular manufacturing systems (CMSs) in multiple plants in a dynamic condition. The proposed mathematical model integrates significant manufacturing characteristics in designing CMSs along with the main strategies of production planning (PP) and location–allocation (LA) problems. In addition, there are some novel characteristics that make the designed model remarkable regarding the literature including: (1) multi-plant location; (2) allocation of plants capacity to fulfill multiple markets demand; and (3) the integration of dynamic cell formation, PP and LA decisions. The objective function terms are to optimize the sale revenue and total costs of machine operating, machine overhead, inter-cell material handling, inventory holding, outsourcing, machine installation/uninstallation, products transportation, dispersing machines, establishing plants, and forming cells. A sample test problem is solved by CPLEX solver to show the achievement obtained by the characteristics incorporated into the model. Since the proposed model is NP-hard, two meta-heuristics algorithms as grey wolf optimization and genetic algorithm are developed. To verify the computational effectiveness of the employed meta-heuristics in comparison with that of CPLEX solver, its performance is tested using a number of test problems.

A new methodology for solving bi-criterion fractional stochastic programming

<p style='text-indent:20px;'>Solving a bi-criterion fractional stochastic programming using an existing multi criteria decision making tool demands sufficient efforts and it is time consuming. There are many cases in financial situations that a nonlinear fractional programming, generated as a result of studying fractional stochastic programming, must be solved. Often management is not in needs of an optimal solution for the problem but rather an approximate solution can give him/her a good starting for the decision making or running a new model to find an intermediate or final solution. To this end, this author introduces a new linear approximation technique for solving a fractional stochastic programming (CCP) problem. After introducing the problem, the equivalent deterministic form of the fractional nonlinear programming problem is developed. To solve the problem, a fuzzy goal programming model of the equivalent deterministic form of the fractional stochastic programming is provided and then, the process of defuzzification and linearization of the problem is presented. A sample test problem is solved for presentation purposes. There are some limitations to the proposed approach: (1) solution obtains from this type of modeling is an approximate solution and, (2) preparation of approximation model of the problem may take some times for the beginners.</p>

A semi-implicit, energy- and charge-conserving particle-in-cell algorithm for the relativistic Vlasov-Maxwell equations

Conventional explicit electromagnetic particle-in-cell (PIC) algorithms do not conserve discrete energy exactly. Time-centered fully implicit PIC algorithms can conserve discrete energy exactly, but may introduce large dispersion errors in the light-wave modes. This can lead to intolerable simulation errors where accurate light propagation is needed (e.g. in laser-plasma interactions). In this study, we selectively combine the leap-frog and Crank-Nicolson methods to produce an exactly energy- and charge-conserving relativistic electromagnetic PIC algorithm. Specifically, we employ the leap-frog method for Maxwell's equations, and the Crank-Nicolson method for the particle equations. The semi-implicit formulation still features a timestep CFL, but facilitates exact global energy conservation, exact local charge conservation, and preserves the dispersion properties of the leap-frog method for the light wave. The algorithm employs a new particle pusher designed to maximize efficiency and minimize wall-clock-time impact vs. the explicit alternative. It has been implemented in a code named iVPIC, based on the Los Alamos National Laboratory VPIC code (https://github.com/losalamos/vpic). We present numerical results that demonstrate the properties of the scheme with sample test problems: relativistic two-stream instability, Weibel instability, and laser-plasma instabilities.

A framework to exploit the structure of and solve set packing problems with a semi-block-angular structure

Real-world, large-scale problems often have coefficient matrices with special structures that can be obtained by reordering their rows and columns. One fundamental problem in the field of combinatorial optimization is the Set Packing Problem (SPP), which takes on added significance in view of its far-reaching applications in routing and scheduling problems, scheduling surgical procedures, the winner determination problem, forestry, production management, etc. The coefficient matrix of this problem has low density (the number of 1’s is much less than the number of 0’s in the matrix); therefore, by reordering the matrix, one can aggregate its 1’s in order to exploit special structures like semi-block-angular ones.In this paper, a decomposition technique based on constraint partitioning is proposed for exploiting the semi-block-angular structures of SPPs and solving the original problem through solving the sub-problems of the obtained structure. In addition, a theorem is developed to relate the optimal values of the original problem to its sub-problems.The results of implementing the proposed algorithm on some sample test problems demonstrate the ability of the algorithm to exploit semi-block-angular structures and achieve optimal solutions for SPPs. In addition, the application of the proposed algorithm to solve SPPs without a semi-block-angular structure is also investigated on some test problems of OR-library.

Layout and configuration design for a satellite with variable mass using hybrid optimization method

This article deals with configuration design and layout optimization of a GEO communication satellite. Layout design optimization of satellite objects are performed based on two-stage algorithm. Finite Circle Method (FCM) is used to express geometric constraints mathematically, and provide a smooth design space. For mathematical expression of performance constraints such as electromagnetic and thermal requirements, the distance constraints between objects have been developed. Then a hybrid method is proposed to optimize layout design which is a combination of simulated annealing optimization and quasi-Newton methods. This approach provides global and deterministic optimal solutions. Developed hybrid method validation is performed on a sample test problem. Finally, the proposed algorithm has been implemented for configuration and optimal layout design of a communication satellite. The results show that considered constraints of communication satellite are satisfied.

A two-parameter continuation algorithm for vortex pinning in rotating Bose–Einstein condensates

We describe an efficient two-parameter continuation algorithm combined with spectral collocation methods for computing the ground state and central vortex state solutions of rotating Bose–Einstein condensates in optical lattices, where the first kind and second kind Chebyshev polynomials are used as the basis functions for the trial function space. By treating the chemical potential and angular velocity as the continuation parameters simultaneously under the additional constraint of normalization condition, the proposed algorithm can effectively compute numerical solutions for a rich variety of physical phenomena observed in physical experiments with very little cost. Comparisons with various numerical methods on some sample test problems are reported.

Fast multiresolution methods for density functional theory in nuclear physics

We describe a fast real-analysis based O(N) algorithm based on multiresolution analysis and low separation rank approximation of functions and operators for solving the Schrödinger and Lippman-Schwinger equations in 3-D with spin-orbit potential to high precision for bound states. Each of the operators and wavefunctions has its own structure of refinement to achieve and guarantee the desired finite precision. To our knowledge, this is the first time such adaptive methods have been used in computational physics, even in 1-D. Accurate solutions for each of the wavefunctions are obtained for a sample test problem. Spin orbit potentials commonly occur in the simulations of semiconductors, quantum chemistry, molecular electronics and nuclear physics. We compare our results with those obtained by direct diagonalization using the Hermite basis and the spline basis with an example from nuclear structure theory.

SUPERCONVERGENCE OF FEMS AND NUMERICAL CONTINUATION FOR PARAMETER-DEPENDENT PROBLEMS WITH FOLDS

We study finite element approximations for positive solutions of semilinear elliptic eigenvalue problems with folds, and exploit the superconvergence of finite element methods (FEM). In order to apply the superconvergence of FEM for Poisson's equation in [Chen & Huang, 1995; Huang et al., 2004, 2006; Lin & Yan, 1996] to parameter-dependent problems with folds, this paper provides the framework of analysis, accompanied with the proof of the strong monotonicity of the nonlinear form. It is worthy to point out that the superconvergence of the nonlinear problem in this paper is different from that in [Chen & Huang, 1995]. A continuation algorithm is described to trace solution curves of semilinear elliptic eigenvalue problems, where the Adini elements are exploited to discretize the PDEs. Numerical results on some sample test problems with folds and bifurcations are reported.

Model-based data assessment for terrestrial carbon processes: implications for sampling strategy in FACE experiments

The value of different types of data in the estimation of different carbon transfer parameters is investigated. A carbon accounting model is used with different observation operators to generate data. The effectiveness of the inversion is assessed by observing relative errors of estimators and likelihood ratios. It is demonstrated that for an observation operator that relative errors vary widely with the sample test problems. An effective strategy to test types of data is to test the effectiveness of corresponding observation operators on an ensemble of sample problems for which parameters are selected from the space of admissible parameters. The selection is carried out under the assumption that the test parameters themselves are random variables uniformly distributed over the space of admissible parameters.

The structure of constitutive equations for finite deformation of elastic–plastic materials involving strain‐induced anisotropy with applications

AbstractConstitutive relations based on finite‐deformation‐valid analysis require the formulation of coupling effects that were not needed in earlier developments which included small deformation restrictions. However, the feasibility to introduce such coupling effects mathematically requires that the new terms introduced must be in accord with physically based concepts of the phenomena involved. An example came to light in which objectivity of a constitutive relation, considered to be generalized for finite deformation applications involving strain‐induced plastic anisotropy, had been achieved by use of the Jaumann derivative in place of the material derivative for the growth of a tensor variable which expressed the anisotropic characteristics. The spin terms thus added to the infinitesimal displacement analogue were not in accord with a physically based assessment of the underlying micro‐mechanical phenomena and in fact led to the prediction of anomalous stress response. This paper comprises comments on the subsequent development of the theory, including a tensor equation for the evolution of the anisotropy. By solving a sample test problem with different initial conditions both the numerical implementation of the constitutive relation and the response according to the proposed growth law of the anisotropy are described and discussed.

Elastic-plastic analysis with NASTRAN user elements

Described herein are two user-developed elements that have been implemented in the elastioplastic capability of NASTRAN. The first element is a thin-walled beam. The second is a flat, triangular shell element. The elements are shown to give good agreement with plastic limit analysis for five sample test problems. Application is made to static crush of auto frames. Both beam and shell models are investigated and good comparison is obtained with test data.