Solution Of Large-scale Problems Research Articles

A variant of the NSGA II for the speed scaling distributed flow shop scheduling problem with total tardiness minimisation

In this study, an energy-efficient distributed flow shop scheduling (DFSS) problem with total tardiness minimisation and machine-sequence dependent setup times is addressed. A mixed integer linear programming (MILP) model is proposed for the problem. A variant of the NSGA II algorithm is suggested for the solution of large scale problems. The proposed algorithm is compared with the state-of-the-art NSGA II, SPEA II, and multiobjective iterated local search algorithm. The computational results show that the proposed algorithm is efficient and effective for the problem. This is the first study to propose a heuristic algorithm for the distributed flow shop scheduling problem with total tardiness minimisation, speed scaling and setups.

A framework for inherently interpretable optimization models

With dramatic improvements in optimization software, the solution of large-scale problems that seemed intractable decades ago are now a routine task. This puts even more real-world applications into the reach of optimizers. At the same time, solving optimization problems often turns out to be one of the smaller difficulties when putting solutions into practice. One major barrier is that the optimization software can be perceived as a black box, which may produce solutions of high quality, but can create completely different solutions when circumstances change leading to low acceptance of optimized solutions. Such issues of interpretability and explainability have seen significant attention in other areas, such as machine learning, but less so in optimization. In this paper we propose an optimization framework that inherently comes with an easily interpretable optimization rule, that explains under which circumstances certain solutions are chosen. Focusing on univariate decision trees to represent interpretable optimization rules, we propose integer programming formulations as well as a heuristic method that ensure applicability of our approach even for large-scale problems. By presenting several extensions to the univariate decision tree approach, we showcase the generality of the proposed framework. Computational experiments using random and real-world data of a road network indicate that the costs of inherent interpretability can be very small.

Energy-Efficient UAVs Coverage Path Planning Approach

Unmanned aerial vehicles (UAVs), commonly known as drones, have drawn significant consideration thanks to their agility, mobility, and flexibility features. They play a crucial role in modern reconnaissance, inspection, intelligence, and surveillance missions. Coverage path planning (CPP) which is one of the crucial aspects that determines an intelligent system’s quality seeks an optimal trajectory to fully cover the region of interest (ROI). However, the flight time of the UAV is limited due to a battery limitation and may not cover the whole region, especially in large region. Therefore, energy consumption is one of the most challenging issues that need to be optimized. In this paper, we propose an energy-efficient coverage path planning algorithm to solve the CPP problem. The objective is to generate a collision-free coverage path that minimizes the overall energy consumption and guarantees covering the whole region. To do so, the flight path is optimized and the number of turns is reduced to minimize the energy consumption. The proposed approach first decomposes the ROI into a set of cells depending on a UAV camera footprint. Then, the coverage path planning problem is formulated, where the exact solution is determined using the CPLEX solver. For small-scale problems, the CPLEX shows a better solution in a reasonable time. However, the CPLEX solver fails to generate the solution within a reasonable time for large-scale problems. Thus, to solve the model for large-scale problems, simulated annealing for CPP is developed. The results show that heuristic approaches yield a better solution for large-scale problems within a much shorter execution time than the CPLEX solver. Finally, we compare the simulated annealing against the greedy algorithm. The results show that simulated annealing outperforms the greedy algorithm in generating better solution quality.

Bilevel learning for large-scale flexible flow shop scheduling

Many industrial practitioners are facing the challenge of solving large-scale scheduling problems within a limited time. In this paper, we propose a novel bilevel scheduler based on constraint Markov Decision Process to solve large-scale flexible flow shop scheduling problems (FFSP). There are many intelligent algorithms proposed to solve FFSP, but they take quite long time to execute or are even not working for large-scale problems. Our scheduler is able to decide the sequence of a large number of jobs in a limited time with the objective to minimize makespans. The upper level is designed to explore an initial global sequence, whereas the lower level aims to look for partial sequence refinements. In the implementation, Double Deep Q Network (DDQN) is used in the upper level and Graph Pointer Network (GPN) lies within the lower level. The two levels are connected by a sliding-window sampling mechanism. Based on datasets from public benchmarks and real-world industrial scenarios with over 5000 jobs, experiments show that our bilevel scheduler significantly outperforms seven baseline algorithms, including three state-of-the-art heuristics, three deep learning based algorithms, and another bilevel model, in terms of makespans and computational time. In particular, it only takes less than 200 s to get solutions of large-scale problems with up to 5000 jobs and matches the performance of the state-of-the-art heuristics.

A fast tool for the parametric analysis of human body exposed to LF electromagnetic fields in biomedical applications

A numerical procedure for analyzing electromagnetic (EM) fields interactions with biological tissues is presented. The proposed approach aims at drastically reducing the computational burden required by the repeated solution of large scale problems involving the interaction of the human body with EM fields, such as in the study of the time evolution of EM fields, uncertainty quantification, and inverse problems. The proposed volume integral equation (VIE), focused on low frequency applications, is a system of integral equations in terms of current density and scalar potential in the biological tissues excited by EM fields and/or electrodes connected to the human body. The proposed formulation requires the voxelization of the human body and takes advantage of the regularity of such discretization by speeding-up the computational procedure. Moreover, it exploits recent advancements in the solution of VIE by means of iterative preconditioned solvers and ad hoc parametric Model Order Reduction techniques. The efficiency of the proposed tool is demonstrated by applying it to a couple of realistic model problems: the assessment of the peripheral nerve stimulation, performed in terms of evaluation of the induced electric field, due to the gradient coils of a magnetic resonance imaging scanner during a clinical examination and the assessment of the exposure to environmental fields at 50 Hz of live-line workers with uncertain properties of the biological tissues. Thanks to the proposed method, uncertainty quantification analyses and time domain simulations are possible even for large scale problems and they can be performed on standard computers and reasonable computation time. Sample implementation of the method is made publicly available at https://github.com/UniPD-DII-ETCOMP/BioMOR.

The GMRES solver for the interpolating meshless local Petrov–Galerkin method applied to heat conduction

PurposeThe work presents a novel implementation of the generalized minimum residual (GMRES) solver in conjunction with the interpolating meshless local Petrov–Galerkin (MLPG) method to solve steady-state heat conduction in 2-D as well as in 3-D domains.Design/methodology/approachThe restarted version of the GMRES solver (with and without preconditioner) is applied to solve an asymmetric system of equations, arising due to the interpolating MLPG formulation. Its performance is compared with the biconjugate gradient stabilized (BiCGSTAB) solver on the basis of computation time and convergence behaviour. Jacobi and successive over-relaxation (SOR) methods are used as the preconditioners in both the solvers.FindingsThe results show that the GMRES solver outperforms the BiCGSTAB solver in terms of smoothness of convergence behaviour, while performs slightly better than the BiCGSTAB method in terms of Central processing Unit (CPU) time.Originality/valueMLPG formulation leads to a non-symmetric system of algebraic equations. Iterative methods such as GMRES and BiCGSTAB methods are required for its solution for large-scale problems. This work presents the use of GMRES solver with the MLPG method for the very first time.

Study of non-intrusive model order reduction of neutron transport problems

Model order reduction is one important technique to speed up the solution of large-scale problems. Two model order reduction techniques, i.e., the proper orthogonal decomposition with interpolation and the Gaussian process regression, are investigated to compute the neutron flux and the effective multiplication factor. Two reduced order models are constructed and it is shown that a speedup of 3 ~ 4 orders of magnitude is achieved with guaranteed accuracy. The Gaussian process regression model can predict the mean value of keff and more importantly, the uncertainty of the predicted value can be analyzed as well.

Optimized cycle basis in volume integral formulations for large scale eddy-current problems

We present a Volume Integral formulation for the solution of large scale eddy-current problems coupled with low-rank approximation techniques. Two alternative approaches are introduced to map the problem unknowns into a subset of grid elements forming a base of global or mixed (global and local) cycles, respectively, and guarantee the well-posedness of the problem both in simply and multiply connected domains. The paper shows that the adoption of mixed cycles is computationally more efficient than global ones. In particular, integral formulations based on global cycles cannot be safely coupled with low-rank approximation techniques, which, however, are crucial to increase the size of the largest solvable problem, like the ones involving conducting structures in magnetic confinement fusion devices. The aim of this paper is to demonstrate how such bottleneck can be overcome by considering local and global cycles differently, on the basis of the cohomology theory. An improved, efficient, and robust algorithm for computing a base of global cycles is described in detail. In particular, the presented algorithm is able to almost minimize the cohomology basis length, i.e. the number of mesh edges forming such a basis, in order to allow an efficient solution of large scale problems. Furthermore, a novel and general method to handle global and local cycles together, in the context of low-rank approximated matrices, is shown to be efficient for the solution of large scale eddy-current problems in multiply connected domains. Along the manuscript, pseudo-codes are given, which clarify the proposed methods and help to implement them by Volume Integral Equation practitioners.

EdgeCFD: a parallel residual-based variational multiscale code for multiphysics

High fidelity multiphysics simulations are ubiquitous in science and engineering but still face many challenges to run efficiently in today's supercomputers. This work reports advanced technologies present in EdgeCFD, a hybrid parallel variational multiscale multiphysics finite element software capable of running industrial problems involving turbulent incompressible, compressible, free-surface flows, advection-diffusion of multiple scalar fields, fluid-object interaction using an Arbitrary Lagrangian-Eulerian formulation, and, more recently, used to supporting uncertainty quantification. We also show the strategies to support the solution of large scale problems, including in-situ visualisation and how EdgeCFD handles computational steering. Industrial applications in the oil and gas industry are shown to demonstrate EdgeCFD capabilities. We present the analysis of vortex-induced vibration in cylindrical risers with appendages to reduce drag and lift, and the simulation of successive discharges of turbidity currents, a critical process for understanding the formation of oil reservoirs.

Minimizing the average searching time for an object within a graph

This paper presents a new graph search problem for which a searcher wishes to find an object that may be found at a set of locations. The searcher doesn’t know the object’s exact location, but does know the a-prior probability of finding the object at each location. He wishes to build a searching path for reaching the object that starts from a given location and ends when reaching the object (or after searching the entire set with a false result). The objective is to find a searching path which will minimize the average searching time. We consider two scenarios for this problem: one when there is an unknown number of objects on the set and another when there is exactly one object on the set (the sum of probabilities is equal to 1). We show that this problem is NP-Hard, and supply a branch and bound algorithm for finding an optimal solution for large scale problems. We also study greedy approaches and other heuristics and compare the performance of these algorithms in various situations.

Capacitated inspection scheduling of multi-unit systems

Abstract We consider the inspection scheduling problem of multi-unit systems where the inspections of individual units are coupled via a capacity constraint. Although the optimal inspection policy of the majority of single-unit systems can be characterized by a threshold policy, finding an optimal policy for multi-unit systems is significantly harder. Therefore, the current state-of-the-practice uses a periodical inspection policy for all units. Instead, we propose using a dynamic programming (DP) approach to solve small-scale problem instances to optimality and use solutions optimized for a single-unit system in an approximation scheme to obtain near-optimal solutions for large-scale problems. Our results show that taking individual properties of the units to be inspected into account and incorporating the single-unit solutions within an approximate DP framework significantly decrease the inspection cost compared to a periodical inspection policy. The proposed methods can help resource-constrained regulatory agencies such as US Food and Drug Administration (FDA) to optimize their inspection activities.

Phaseless Near-Field Far-Field Transformation Utilizing Combinations of Probe Signals

Phaseless near-field far-field transformations based on nonconvex optimization suffer from local stationary points. In order to avoid the corresponding suboptimal solutions, the incorporation of combinations of probe signals into the cost functional is proposed and investigated. The nonconvex optimization based on the Wirtinger Calculus relies on forward and adjoint radiation operator evaluations via the fast irregular antenna field transformation algorithm (FIAFTA) and is performed by a memory-limited Quasi-Newton method. This allows the solution of large scale problems with utilization of the full flexibility of the FIAFTA in terms of measurement probe correction and arbitrary sample locations. Results for synthetic and measured near-field data reveal significant improvements when the knowledge of probe signal combinations is utilized, especially in applications with a large number of unknowns.

Flow shop scheduling with a batch processor and limited buffer

This paper addresses flow shop scheduling problem with a batch processor followed by a discrete processor. Incompatible job families and limited buffer size are considered, and the objective is to determine a schedule such that the total completion time is minimised. Flexible buffer service policy is designed, and a greedy heuristic together with the worst-case analysis is developed. We also propose a hybrid method involving a Differential Evolution algorithm. Moreover, two tight lower bounds are provided to measure the performances of the proposed algorithms. Numerical results demonstrate that the proposed algorithms are capable of providing high-quality solutions for large-scale problems within a reasonable computational time.

Geographical Constraint-Based Optimization Framework for C-RAN Design [Invited

The centralized radio access network (C-RAN) is promising network architecture for enabling collaboration among macro/small cells, where a baseband unit (BBU) is centralized in a central office (CO), while remote radio heads (RRHs) are located at each local site. Unlike the distributed RAN (D-RAN), the C-RAN consists of sets of inter-coordinated RRHs called “clusters,” which should be carefully designed so that their coverage areas do not overlap with each other for reducing interference. Moreover, C-RAN should be designed to satisfy the constraints of the latency requirement between BBUs and RRHs, which lead to length restrictions of their connecting optical fiber cables. Therefore, the C-RAN should be constructed by utilizing a massive infrastructure (i.e., COs, space and power supplies for BBUs, installed fiber, and so on), which creates a strong geographical constraint. In this paper, we propose an optimization framework for designing the C-RAN under geographical constraints. Two algorithms are introduced: optimal and suboptimal design algorithms. The latter is introduced to quickly obtain a feasible solution for large-scale problems. Based on extensive simulations and various scenarios, we verify the effectiveness of the proposed algorithms, while the suboptimal algorithm is useful in terms of the accuracy of the solution and the CPU time.

Data-sparse approximation on the computation of a weakly singular Fredholm equation: A stellar radiative transfer application

Data-sparse representation techniques are emerging on computing approximate solutions for large scale problems involving matrices with low numerical rank. This representation provides both low memory requirements and cheap computational costs.In this work we consider the numerical solution of a large dimensional problem resulting from a finite rank discretization of an integral radiative transfer equation in stellar atmospheres. The integral operator, defined through the first exponential-integral function, is of convolution type and weakly singular.Hierarchically semiseparable representation of the matrix operator with low-rank blocks is built and data-sparse matrix computations can be performed with almost linear complexity. This representation of the original fully populated matrix is an algebraic multilevel structure built from a specific hierarchy of partitions of the matrix indices.Numerical tests illustrate the benefits of this matrix technique compared to standard storage schemes, dense and sparse, in terms of computational cost as well as memory requirements.This approach is particularly useful when a fine discretization of the integral equation is required and the resulting linear system of equations is of large dimension and numerically difficult to solve.

A hybrid approach for cost-optimized lateral transshipment in a supply chain environment

Purpose – When making sourcing decisions, both cost optimization and customer demand fulfillment are equally important for firm competitiveness. The purpose of this paper is to develop a stochastic search technique, hybrid genetic algorithm (HGA), for cost-optimized decision making in wholesaler inventory management in a supply chain network of wholesalers, retailers and suppliers. Design/methodology/approach – This study develops a HGA by using a mixture of greedy-based and randomly generated solutions in the initial population and a local search method (hill climbing) applied to individuals selected for performing crossover before crossover is implemented and to the best individual in the population at the end of HGA as well as gene slice and integration. Findings – The application of the proposed HGA is illustrated by considering multiple scenarios and comparing with the other commonly adopted methods of standard genetic algorithm, simulated annealing and tabu search. The simulation results demonstrate the capability of the proposed approach in producing more effective solutions. Practical implications – The pragmatic importance of this method is for the inventory management of wholesaler operations and this can be scalable to address real contexts with multiple wholesalers and multiple suppliers with variable lead times. Originality/value – The proposed stochastic-based search techniques have the capability in producing good-quality optimal or suboptimal solutions for large-scale problems within a reasonable time using ordinary computing resources available in firms.

Gradient methods and conic least-squares problems

This paper presents two reformulations of the dual of the constrained least-squares problem over convex cones. In addition, it extends Nesterov's excessive gap method 1 [Excessive gap technique in nonsmooth convex minimization, SIAM J. Optim. 16(1) (2005), pp. 235–249 (electronic)] to more general problems. The conic least-squares problem is then solved by applying the resulting modified method, or Nesterov's smooth method [Smooth minimization of non-smooth functions, Math. Program. 103(1, Ser. A) (2005), pp. 127–152], or Nesterov's excessive gap method 2 [Excessive gap technique in nonsmooth convex minimization, SIAM J. Optim. 16(1) (2005), pp. 235–249 (electronic)], to the dual reformulations. Numerical experiments show that this approach obtains relatively accurate solutions for large-scale problems using less CPU time than interior-point method-based state-of-art software do and is more accurate than these software on problem instances possibly with dual degeneracy [F. Alizadeh, J.-P.A. Haeberly, and M.L. Overton, Complementarity and nondegeneracy in semidefinite programming, Math. Program. 77(2, Ser. B) (1997), pp. 111–128]. The approach extends easily to related conic convex quadratic programs.

Recent Theory and Applications on Numerical Algorithms and Special Functions

In recent years, there has been extensive research activity on both the theory and application of efficient numerical methods for approximating the solutions of differential equations. Numerical algorithms and special functions have been a major thrust of this research and have numerous applications in science and engineering. Special functions play a tremendous role in mathematical analysis and in the development of newnumerical techniques. In particular, they allow a broad refinement of spectral methods for solving several types of differential equations in both finite and infinite domains. This is useful, in that there is a strong need for efficient and easy-to-use scalable techniques for the numerical solution of large scale problems. This special issue was focused on the analysis of numerical methods and special functions, while also addressing the practical nature of the two fields. In addition to the strong focus on computational mathematics, scientific computing, and theoretical physics, a special emphasis was given to orthogonal polynomials, the theory of numericalmethods for solution of differential equations, and applications of these fields. A modified generalized Laguerre-Gauss collocation scheme was reported for a class of fractional neutral differential equations on the half-line. Some efficient spectral algorithms were proposed for numerical solution third-order boundary value problems using novel operational matrices of generalized Jacobi polynomials. A mesh-free method for solving nonhomogeneous time-dependent problems was presented. The homotopy perturbation technique was developed for solving the Hamilton-Jacobi-Bellman equation. A 2-point block method for nonstiff higher-order ordinary differential equations was investigated. New bilateral type generating function associated with I-function, certain class of generating functions for the incomplete hypergeometric functions, complex transmission eigenvalues in one dimension, and certain matrices of Bernoulli numbers were other topics of our special issue. Fast and stable implementation of Clenshaw-Curtis and Fejer-type quadrature rules was investigated. An efficient computation of effective ground range using an oblate earth model and the boundary value problem for bihypermonogenic functions in Clifford analysis were presented. The bivariate spectral homotopy analysismethod for nonlinear evolution equations and the fractional Nagumo equationwith nonlinear diffusion and convection were distinct topics of our special issue. A coupled system of fractional integrodifferential equations with m-point boundary conditions on the half-line and global regularity for the ∂b-equation on CR manifolds of arbitrary codimension were analyzed.

A New Version of Fast Convergent GMRES(m) Algorithm

In this paper, a kind of modified GMRES(m) algorithm is proposed and its convergence theory is well established. By properly varying the restart parameter m in the GMRES(m) algorithm, the computational efficiency and precision can be significantly improved. Because the parameter m can be very small, the storage and computations are greatly saved. In addition, the new algorithm is successfully used to solve a heat conduction problem and a Laplace equation problem. Numerical results show that it is not only highly effective and accurate, but also can avoid slow convergence or failure behaviour of the GMRES(m) algorithm. The superiorities will be much more remarkable for the solution of large scale problems. So it has extensive prospect in scientific computing and engineering applications.

Supply chain network design under uncertainty with new insights from contracts

In this paper, the classical problem of supply chain network design is reconsidered to emphasize the role of contracts in uncertain environments. The supply chain addressed consists of four layers: suppliers, manufacturers, warehouses, and customers acting within a single period. The single owner of the manufacturing plants signs a contract with each of the suppliers to satisfy demand from downstream. Available contracts consist of long-term and option contracts, and unmet demand is satisfied by purchasing from the spot market. In this supply chain, customer demand, supplier capacity, plants and warehouses, transportation costs, and spot prices are uncertain. Two models are proposed here: a risk-neutral two-stage stochastic model and a risk-averse model that considers risk measures. A solution strategy based on sample average approximation is then proposed to handle large scale problems. Extensive computational studies prove the important role of contracts in the design process, especially a portfolio of contracts. For instance, we show that long-term contract alone has similar impacts to having no contracts, and that option contract alone gives inferior results to a combination of option and long-term contracts. We also show that the proposed solution methodology is able to obtain good quality solutions for large scale problems.