Modeling Of Transport Problems Research Articles

On the use of spherical harmonic approximations in nonclassical particle transport problems

Presented here is the application of the spherical harmonic expansion to the nonclassical transport equation to derive a system of equations for the nonclassical flux moments, namely nonclassical spherical harmonic approximations (NSHA). The nonclassical transport equation is a recently developed mathematical model that allows the modeling of transport problems wherein the particle flux does not undergo exponential attenuation. We employ a Spectral Approach technique to depict the nonclassical flux by expressing it as a truncated series of Laguerre polynomials with respect to the free-path variable s. This yields a system of equations for the nonclassical flux moments structured similarly to the classical PN equations. We show that the NSHA is reduced to the classical PN equation in a special case. Numerical results for slab-geometry test problems are given to verify the derivation.

Interval transportation problem: feasibility, optimality and the worst optimal value

We consider the model of a transportation problem with the objective of finding a minimum-cost transportation plan for shipping a given commodity from a set of supply centers to the customers. Since the exact values of supply and demand and the exact transportation costs are not always available for real-world problems, we adopt the approach of interval programming to represent such uncertainty, resulting in the model of an interval transportation problem. The interval model assumes that lower and upper bounds on the data are given and the values can be independently perturbed within these bounds. In this paper, we provide an overview of conditions for checking basic properties of the interval transportation problems commonly studied in interval programming, such as weak and strong feasibility or optimality. We derive a condition for testing weak optimality of a solution in polynomial time by finding a suitable scenario of the problem. Further, we formulate a similar condition for testing strong optimality of a solution for transportation problems with interval supply and demand (and exact costs). Moreover, we also survey the results on computing the best and the worst optimal value. We build on an exact method for solving the NP-hard problem of computing the worst (finite) optimal value of the interval transportation problem based on a decomposition of the optimal solution set by complementary slackness. Finally, we conduct computational experiments to show that the method can be competitive with the state-of-the-art heuristic algorithms.

Geometric Mean Method Combined With Ant Colony Optimization Algorithm to Solve Multi-Objective Transportation Problems in Fuzzy Environments

The transportation problem (TP) is a well-known subject in the field of optimization and a very prevalent challenge for businesspeople. The goal is to reduce the total transportation cost, of delivering resources from sources to destinations. The literature demonstrates that various approaches have been designed with a single goal in mind, although TPs are not always developed with a bi-goal in mind. Solving transportation difficulties with several objectives is a common task. In this study, a new method for addressing multi-criteria TP using geometric means, along with a novel approach of the Ant Colony Optimization algorithm (ACO) for solving multi-objective TP in a fuzzy environment. Fuzzy numbers have been used to solve real-world problems in various domains, including operations research and optimization. The ACO Algorithm has long been recognized as a viable alternative strategy for solving optimization problems. The purpose of this study is to provide a unique approach for organizing fuzzy numbers as well as enhancements to the ACO algorithm for solving the Multi-Objective TP model. Our method, such as Geometric Mean Ant Colony Optimization Algorithm (GMACOA), outperforms other methods in terms of objective values. Numerical examples are provided to demonstrate the method in comparison to various current methods.

Development of Two-Stage Transportation Problem Model with Fixed Cost for Opening the Distribution Centers

The Two-Stage Transportation Problem (TSTP) is a model of product transportation in the supply chain. The transportation starts from the factory to customers through a distribution center (DC), which considers the fixed cost of opening a DC, the transportation cost per unit from a certain factory to a certain DC, and the transportation cost per unit from a certain DC to certain customers. This study develops a model by allowing direct delivery from the factory to the customer. From the numerical example given, the proposed model that allows direct delivery from the factory to a customer could result in a total distribution cost that is minimized to the initial model. Both models were compared with calculation using Lingo 19.0. The model developed is expected to provide options and consideration for the management to determine the right distribution and logistics strategy for their products.

A new ranking technique on heptagonal fuzzy numbers to solve fuzzy transportation problem

In this work, a general fuzzy transportation problem model is discussed and proposed a new ranking procedure to the heptagonal fuzzy numbers using which convert any fuzzy valued transportation problem to a crisp valued transportation problem which then could be solved using the MODI method. The proposed method serves as an efficient method in ranking heptagonal fuzzy numbers which is illustrated through a numerical example. The purpose of this paper is to give a new ranking index to rank fuzzy numbers effectively. Fuzzy numbers cannot be easily ordered as ordinary real numbers. Several proposals have addressed this problem, each with some drawbacks and limitations. This paper renounces to the idea of finding a universal ordering method for fuzzy numbers.

Evaluation of transport policy packages in the excess commuting framework: The case of Xiamen, China

Transport policies are often carried out in package form, but, presently, transport policy package (TPP) remains a theoretical concept whose practicality has rarely been investigated. Additionally, jobs-housing balance has become a major urban planning and public policy issue that has received considerable scholarly attention. This study probes into evaluating TPPs using the minimum commute, a proxy for jobs-housing balance and an important statistic in the excess commuting framework. Using a brief transportation problem model of linear programming, the minimum commute statistics in scenarios with different TPPs are calculated by repeatedly solving the proposed mathematical model. Moreover, this study considers multimodal commuting for better reflecting realistic situations. Drawing on an empirical study in Xiamen, China, we show that the mathematical model provides indicators for urban form evaluation.

A reliable transportation problem

In today’s market, managers are being pressurized to deliver the product in a reliable and cost effective manner. The aim of the present work is to model a transportation problem where decision maker has a flexibility to set his own reliability level when he minimizes the overall transportation cost. To achieve this goal, the overall reliability of transportation system is calculated and a mathematical model of transportation problem is formulated. A numerical experiment is also conducted to illustrate the proposed model.

Algebraic approximation of sub-grid scales for the variational multiscale modeling of transport problems

Variational Multiscale (VMS) Finite Element Methods (FEMs) are robust for the development of general formulations for the solution of multiphysics and multiscale transport problems. To obtain a tractable and computationally efficient model, VMS methods often rely on a residual-based algebraic approximation of the sub-grid scales (small or unresolved features of the solution field not captured by the discretization) using a so-called intrinsic time scales matrix, which depends on the problem’s overall differential operator and represents the main model parameter. A novel technique for approximating the intrinsic time scales matrix for generic transport problems in a relatively inexpensive manner (e.g., does not rely on eigenvalue computations) is presented. The method is denoted Transport-Equivalent Scaling (TES) and is based on the monolithic casting of the transport problem as a system of transient–advective–diffusive–reactive (TADR) equations and a subsequent scaling of the coefficient matrices such to preserve each type of transport flux. An algebraic VMS formulation incorporating the TES method is complemented with a discontinuity-capturing (DC) approach and implemented within a FEM solver for the solution of TADR problems. The solution of the global discrete system is accomplished using a generalized-alpha time-stepper together with a globalized inexact Newton–Krylov nonlinear solver. The effectiveness of the TES formulation is verified with the simulation of benchmark incompressible, compressible, and magnetohydrodynamic flow problems. The results demonstrate that the TES method seamlessly handles incompressible–compressible flows in a unified manner (e.g., without assessing the compressibility of the flow). The convergence process using the TES approach and a more standard approximation for the intrinsic time scales, as well as the effect of the DC approach, are also investigated. Analysis of the intrinsic time scales for a one-dimensional incompressible flow model reveals the similitudes and differences between the TES formulation and other conventional methods.

One of the methods for optimizing a part of supply chain by using transportation problem model

Supply chain is coordinated system of people, resources, information, etc., involved in the movement of materials from a state of raw materials to final customer, and to ensure the integration of all participants and the exchange of information along the entire supply chain.. One of potential problem of SC that is the subject of optimization and with whose improvement is achieved a series of effects - economic, environmental, etc. is transportation problem. The choice of method for optimization of this problem has a significant impact on the quality of the solutions, which are shown in this paper on the example of the supply chain of fruit producing company. The paper applied two algorithms to optimize part of the supply chain: exact and heuristic. The result of their use is the reduction of the transport operation, as well as insight into the overall way that the transport part of the supply chain function, and the possibilities for its improvement.

PoreFlow: A complex pore-network model for simulation of reactive transport in variably saturated porous media

This study introduces PoreFlow, a pore-network modeling tool capable of simulating fluid flow and multi-component reactive and adsorptive transport under saturated and variably saturated conditions. PoreFlow includes a variety of modules, such as: pore network generator, drainage simulator, calculation of pressure and velocity distributions, and modeling of reactive solute transport accounting for advection and diffusion. The pore space is represented using a multi-directional pore-network capable of capturing the random structure of a given porous media with user-defined directional connectivities for anisotropic pore structures. The chemical reactions can occur within the liquid phase, as well as between the liquid and solid phases which may result in an evolution of porosity and permeability. Under variably saturated conditions the area of interfaces changes with degree of the fluid saturation.PoreFlow uses complex formulations for more accurate modeling of transport problems in presence of the nonwetting phase. This is done by refining the discretization within drained pores. An implicit numerical scheme is used to solve the governing equations, and an efficient substitution method is applied to considerably minimize computational times. Several examples are provided, under saturated and variably saturated conditions, to demonstrate the model applicability in hydrogeology problems and petroleum fields. We show that PoreFlow is a powerful tool for upscaling of flow and transport in porous media, utilizing different pore scale information such as various interfaces, phase distributions and local fluxes and concentrations to determine macro scale properties such as average saturation, relative permeability, solute dispersivity, adsorption coefficients, effective diffusion and tortuosity. Such information can be used as constitutive relations within continuum scale governing equations to model physical and chemical processes more accurately at the larger scales.

The numerical modelling of advective transport in the presence of fluid pressure transients

AbstractConventional modelling of transport problems for porous media usually assumes that the Darcy flow velocities are steady. In certain practical situations, the flow velocity can exhibit time‐dependency, either due to the transient character of the flow process or time dependency in the boundary conditions associated with potential flow. In this paper, we consider certain one‐ and three‐dimensional problems of the advective transport of a chemical species in a fluid‐saturated porous region. In particular, the advective flow velocity is governed by the piezo‐conduction equation that takes into account the compressibilities of the pore fluid and the porous skeleton. Time‐ and/or mesh‐refining adaptive schemes used in the computational modelling are developed on the basis of a Fourier analysis, which can lead to accurate and optimal solutions for the advective transport problem with time‐ and space‐dependent advective flow velocity distributions. Copyright © 2006 John Wiley & Sons, Ltd.

Error bound comparisons for aggregation/disaggregation techniques applied to the transportation problem

A priori and a posteriori error bounds for a transportation problem model at different levels of aggregation were statistically compared. An experimental design was used to (1) examine the size and significance of correlation between all pairs of the a priori error bound, a posteriori error bound and actual error and (2) quantify the size and significance of the difference of a posteriori error bound from actual error. Two different methods for calculating a posteriori error bounds were utilized. Results are for aggregating customers in a transportation model using one aggregation strategy and varying the level of aggregation on a set of randomly generated problems. Results show significant correlation between a posteriori error and actual error. A priori error is not significantly correlated with actual error. These preliminary results indicate that calculating the a posteriori error bound to select the appropriate aggregation level is a helpful strategy since the a posteriori bound varies in the same way that the actual error varies. In addition, one method of calculating the a posteriori bound is determined to be significantly tighter than the other method of calculating the a posteriori error bound. Scope and purpose Aggregation/disaggregation techniques are used to reduce a large model to a smaller model. Error bounds for the objective function of an aggregated linear programming model quantify the loss of information due to aggregation. There are two types of error bounds, a priori and a posteriori, and several methods to calculate the a posteriori bound. This is an exploratory study to determine if these error bounds vary consistently when different levels of aggregation are applied to the customers of the same transportation problem model and to statistically compare the tightness of these bounds. Results show that under the conditions of this study there is significant correlation between the actual aggregation error and a posteriori error. This implies that aggregation levels that produce large a posteriori error bounds are associated with large actual error and aggregation levels that produce small a posteriori error bounds are associated with small actual error. Results also show that of the two methods used to calculate the a posteriori bound, one consistently determines a tighter bound.