Set Of Test Problems Research Articles

ЭМПИРИЧЕСКОЕ ИССЛЕДОВАНИЕ МОЩНОСТИ СТАТИСТИЧЕСКОГО КРИТЕРИЯ КОЛМОГОРОВА–СМИРНОВА В ЗАДАЧАХ ПРОВЕРКИ ГИПОТЕЗ О ЗАКОНЕ РАСПРЕДЕЛЕНИЯ

Statistical tests that use A.N. Kolmogorov one-sample statistics and N.V. Smirnov two-sample statistics have been widely used for solving applied problems for almost a hundred years. Those criteria are present in many textbooks and implemented in computer software for data analysis. The purpose of the work is to empirically estimate the power of the Kolmogorov–Smirnov criterion on a set of test problems to check hypotheses about the distribution law, as well as to investigate the properties of estimates of the power of the criterion for various types of resampling. To obtain various estimates of the power of the Kolmogorov–Smirnov criterion in solving test problems, the classical bootstrap, nonparametric bootstrap, parametric bootstrap, bootstrap with a random term, and resampling without returns were used. Based on the results of multiple solution of test problems, medians of p-values, medians of estimates of the power of the Kolmogorov–Smirnov criterion were calculated, and medians of biases of estimates on test problems and medians of all pairwise differences of various estimates of the power of the criterion were found. The distribution laws for the considered Kolmogorov–Smirnov power estimates, for the biases of the power estimates, and for all pairwise differences of the power estimates were investigated. For test problems where real data were considered, calculating power estimates without resampling is impossible, but unbiased power estimates can be predicted as the half-sum of two other power estimates, one obtained with a nonparametric bootstrap and the other obtained with resampling without returns. Methods for restructuring sample data and software for estimating the power of a statistical test developed within the study are universal and can be used for analyzing the properties of power estimates of other one-sample statistical tests, as well as for developing the classical methodology for checking statistical hypotheses.

Improved Lebesgue Indicator-Based Evolutionary Algorithm: Reducing Hypervolume Computations

One of the major limitations of evolutionary algorithms based on the Lebesgue measure for multi-objective optimization is the computational cost required to approximate the Pareto front of a problem. Nonetheless, the Pareto compliance property of the Lebesgue measure makes it one of the most investigated indicators in the design of indicator-based evolutionary algorithms (IBEAs). The main deficiency of IBEAs that use the Lebesgue measure is their computational cost which increases with the number of objectives of the problem. On this matter, the investigation presented in this paper introduces an evolutionary algorithm based on the Lebesgue measure to deal with box-constrained continuous multi-objective optimization problems. The proposed algorithm implicitly uses the regularity property of continuous multi-objective optimization problems that has suggested effectiveness when solving continuous problems with rough Pareto sets. On the other hand, the survival selection mechanism considers the local property of the Lebesgue measure, thus reducing the computational time in our algorithmic approach. The emerging indicator-based evolutionary algorithm is examined and compared versus three state-of-the-art multi-objective evolutionary algorithms based on the Lebesgue measure. In addition, we validate its performance on a set of artificial test problems with various characteristics, including multimodality, separability, and various Pareto front forms, incorporating concavity, convexity, and discontinuity. For a more exhaustive study, the proposed algorithm is evaluated in three real-world applications having four, five, and seven objective functions whose properties are unknown. We show the high competitiveness of our proposed approach, which, in many cases, improved the state-of-the-art indicator-based evolutionary algorithms on the multi-objective problems adopted in our investigation.

A maximum-margin multisphere approach for binary Multiple Instance Learning

We propose a heuristic approach for solving binary Multiple Instance Learning (MIL) problems, whose objective is to categorize bags of instances. Considering the case with two classes of instances, on the basis of the standard MIL assumption, a bag is classified positive if it contains at least a positive instance and negative if all its instances are negative. Inspired by a well-established MIL Support Vector Machine type approach, our technique is based on iteratively separating the bags by means of successive maximum-margin spheres. Such spheres, whose number is automatically determined, are generated by computing, for each of them, the optimal radius in correspondence to a prefixed center. Numerical results are presented on a set of benchmark test problems, showing the effectiveness of our approach.

The Rainbow Steiner Tree Problem

Given an undirected and edge-colored graph with non-negative edge lengths, the aim of the Rainbow Steiner Tree Problem (RSTP) is to find a minimum Steiner Tree that uses at most one edge for each color. In this paper, the RSTP is introduced, a mathematical model is proposed to formally represent the problem and its theoretical properties are investigated. Since the RSTP belongs to the NP-class, two heuristic methods are designed: a Lagrangian relaxation approach and a multistart algorithm.Extensive computational experiments are carried out on a significant set of test problems to empirically evaluate the performance of the proposed approaches. The computational results show that the two approaches are both effective and efficient compared to the ILOG CPLEX solver.

Unbalanced budget distribution for automatic algorithm configuration

Optimization algorithms often have several critical setting parameters and the improvement of the empirical performance of these algorithms depends on tuning them. Manually configuration of such parameters is a tedious task that results in unsatisfactory outputs. Therefore, several automatic algorithm configuration frameworks have been proposed to regulate the parameters of a given algorithm for a series of problem instances. Although the developed frameworks perform very well to deal with various problems, however, there is still a trade-off between the accuracy and budget requirements that need to be addressed. This work investigates the performance of unbalanced distribution of budget for different configurations to deal with the automatic algorithm configuration problem. Inspired by the bandit-based approaches, the main goal is to find a better configuration that substantially improves the performance of the target algorithm while using a smaller run time budget. In this work, non-dominated sorting genetic algorithm II is employed as a target algorithm using jMetalPy software platform and the multimodal multi-objective optimization (MMO) test suite of CEC’2020 is used as a set of test problems. We did a comprehensive comparison with other known methods including random search, Bayesian optimization, sequential model-based algorithm configuration (SMAC), iterated local search in parameter configuration space (ParamILS), iterated racing for automatic algorithm configuration (irace), and many-objective automatic algorithm configuration (MAC) methods. In order to characterize, validate and evaluate the performance of these methods, hypervolume (HV), generational distance, and epsilon indicator (\(I_{{\epsilon }^{+}}\)) are used as performance indicators. The experimental results interestingly proved the efficiency of the proposed approach for automatic algorithm configuration with a minimum time budget in comparison with other competitors.

Constrained global optimization of multivariate polynomials using polynomial B-spline form and B-spline consistency prune approach

In this paper, we propose basic and improved algorithms based on polynomial B-spline form for constrained global optimization of multivariate polynomial functions. The proposed algorithms are based on a branch-and-bound framework. In improved algorithm we introduce several new ingredients, such as B-spline box consistency and B-spline hull consistency algorithm to prune the search regions and make the search more efficient. The performance of the basic and improved algorithm is tested and compared on set of test problems. The results of the tests show the superiority of the improved algorithm over the basic algorithm in terms of the chosen performance metrics for 7 out-off 11 test problems. We compare optimal value of global minimum obtained using the proposed algorithms with CENSO, GloptiPoly and several state-of-the-art NLP solvers, on set of 11 test problems. The results of the tests show the superiority of the proposed algorithm and CENSO solver (open source solver for global optimization of B-spline constrained problem) in that it always captures the global minimum to the user-specified accuracy.

Supply Chain Optimization Considering Sustainability Aspects

Supply chain optimization concerns the improvement of the performance and efficiency of the manufacturing and distribution supply chain by making the best use of resources. In the context of supply chain optimization, scheduling has always been a challenging task for experts, especially when considering a distributed manufacturing system (DMS). The present study aims to tackle the supply chain scheduling problem in a DMS while considering two essential sustainability aspects, namely environmental and economic. The economic aspect is addressed by optimizing the total delivery time of order, transportation cost, and production cost while optimizing environmental pollution and the quality of products contribute to the environmental aspect. To cope with the problem, it is mathematically formulated as a mixed-integer linear programming (MILP) model. Due to the complexity of the problem, an improved genetic algorithm (GA) named GA-TOPKOR is proposed. The algorithm is a combination of GA and TOPKOR, which is one of the multi-criteria decision-making techniques. To assess the efficiency of GA-TOPKOR, it is applied to a real-life case study and a set of test problems. The solutions obtained by the algorithm are compared against the traditional GA and the optimum solutions obtained from the MILP model. The results of comparisons collectively show the efficiency of the GA-TOPKOR. Analysis of results also revealed that using the TOPKOR technique in the selection operator of GA significantly improves its performance.

A multi-objective elitist feedback teaching–learning-based optimization algorithm and its application

Students generally attempt to improve themselves with different manners in their spare time. Inspired by this fact, this paper presents a multi-objective elitist feedback teaching–learning-based optimization (MEFTO) algorithm for multi-objective optimization problems. To promote the exploration capacity and convergence of the basic teaching–learning-based optimization (TLBO), a feedback phase that simulates the spare-time learning phenomenon is introduced at the end of the learner phase. Students should compare their score with the average class score to select an appropriate means for further improvement. Poorly performing students can learn from the teacher directly for rapid improvement, whereas high-performing students prefer to motivate themselves for reinforcement learning. Non-dominated sorting is incorporated to permit this heuristic to solve problems with several objectives. Crowding distance calculation is adopted to maintain the diversity of the obtained solution set in a single run. Elitism strategy is also employed to provide a great improvement for the algorithm. The performance of the MEFTO algorithm is compared with seven other well-known algorithms by using a set of unconstrained benchmark test problems and three constrained engineering optimization problems. The qualitative and quantitative results indicate that the proposed algorithm can provide considerably competitive results and outperforms the other algorithms.

The vehicle routing problem with release dates and flexible time windows

In the context of the increasing demands for home delivery of fresh food, this article addresses the terminal distribution problem in an urban environment, which is called the vehicle routing problem with flexible time windows and order release dates (VRPFTWRD). This study develops a mathematical model and presents a set of valid inequalities based on the problem characteristics. According to the model and the inequalities, a branch-and-cut algorithm (B&C) is proposed for solving the problem. To verify the effectiveness of B&C, a set of test problems of different sizes is generated. The results show that the VRPFTWRD problem with up to 55 nodes can be solved to be optimality within one hour. Furthermore, B&C is used to solve vehicle routing problem (VRP) variants, and the results demonstrate that B&C can effectively solve VRP variants within a reasonable time.

Improved Q ‐learning algorithm for solving permutation flow shop scheduling problems

Generally, scheduling problems refer to allocation of available shared resources and the sorting of production tasks, in order to satisfy the specified performance target within a certain time. The fundamental scheduling problem is that all jobs need to be processed on the same route, which is called flow shop scheduling problems (FSSP). The goal of FSSP, proven as an NP-hard problem, is to find a job sequence that minimizes the makespan. In this paper, an improved Q-learning algorithm is proposed for solving the FSSP. Firstly, a problem model based on the basic Q-learning algorithm is constructed. The makespan is used as the feedback signal, and the process of environmental state change is defined as the process of job selection. Q-learning gives the expected utility of taking a given action in a given state. Afterwards, combined with the NEH heuristic, the algorithm efficiency is enhanced by changing the job inserting mode. In order to validate the proposed method, several simulation experiments are carried out on a set of test problems having different scales. The obtained optimization results of the proposed algorithm are compared to the standard Q-learning algorithm and a hybrid algorithm. The discussion and analysis show that the proposed algorithm performs better than the others in solving the permutation FSSP. As a future direction, in order to shorten the running time, further improvements will be studied to increase the performance of the proposed algorithm and make it applicable and efficient for solving multi-objective optimization problems.

Modelling and optimization of integrated distributed flow shop scheduling and distribution problems with time windows

Production and distribution are two essential activities in supply chain management. Currently, integrated production and distribution problems receive much attention because decision-makers devote to improving the operation efficiency of both stages and try to achieve an optimal solution. This work proposes an integrated distributed production and distribution problem with consideration of time windows, in which a set of jobs (i.e., customer orders) needs to be assigned among factories and the jobs are processed on flow shop environments at their associated factories. Then, the completed jobs are delivered by capacitated vehicles to customers in different regions while satisfying given time windows as much as possible. Accordingly, to optimally solve the proposed problem, a mixed integer programming model with minimizing total weighted earliness and tardiness has been established. For the optimization task, an enhanced brain storm optimization algorithm with some particular strategies is designed to handle the considered problem. To assess the performance of the proposed optimization method, several experiments by adopting a set of benchmark test problems are performed, and state-of-the-art optimizers are chosen for comparisons. The obtained optimization results exhibit that the designed algorithm significantly outperforms its rivals and can be considered as an excellent optimizer for solving the studied problem. Besides, compared with the CPLEX solver, the designed optimizer also performs much better for solving large-size problems.

Pseudo-feasible solutions in evolutionary bilevel optimization: Test problems and performance assessment

This work presents a study about a special class of infeasible solutions called here as pseudo-feasible solutions in bilevel optimization. This work is focused on determining how such solutions can affect the performance of an evolutionary algorithm. After its formal definition, and based on theoretical results, two conditions to detect and deal with them are proposed. Moreover, a novel and scalable set of test problems with characterized pseudo-feasible solutions is introduced. Furthermore, an algorithm designed to solve bilevel optimization problems (BOP) is adapted with the above mentioned conditions and tested in already known test problems and also in the new testbed so as to analyze its performance when compared with state-of-the-art evolutionary approaches for BOPs. The obtained results suggest that the presence of pseudo-feasible solutions can be considered as a source of difficulty in this type of optimization problems, since their presence may lead to incorrect comparisons among algorithms.

Design of Computational Models for Hydroturbine Units Based on a Nonparametric Regression Approach with Adaptation by Evolutionary Algorithms

This article deals with the problem of designing regression models for evaluating the parameters of the operation of complex technological equipment—hydroturbine units. A promising approach to the construction of regression models based on nonparametric Nadaraya–Watson kernel estimates is considered. A known problem in applying this approach is to determine the effective values of kernel-smoothing coefficients. Kernel-smoothing factors significantly impact the accuracy of the regression model, especially under conditions of variability of noise and parameters of samples in the input space of models. This fully corresponds to the characteristics of the problem of estimating the parameters of hydraulic turbines. We propose to use the evolutionary genetic algorithm with an addition in the form of a local-search stage to adjust the smoothing coefficients. This ensures the local convergence of the tuning procedure, which is important given the high sensitivity of the quality criterion of the nonparametric model. On a set of test problems, the results were obtained showing a reduction in the modeling error by 20% and 28% for the methods of adjusting the coefficients by the standard and hybrid genetic algorithms, respectively, in comparison with the case of an arbitrary choice of the values of such coefficients. For the task of estimating the parameters of the operation of a hydroturbine unit, a number of promising approaches to constructing regression models based on artificial neural networks, multidimensional adaptive splines, and an evolutionary method of genetic programming were included in the research. The proposed nonparametric approach with a hybrid smoothing coefficient tuning scheme was found to be most effective with a reduction in modeling error of about 5% compared with the best of the alternative approaches considered in the study, which, according to the results of numerical experiments, was the method of multivariate adaptive regression splines.

On new methods to construct lower bounds in simplicial branch and bound based on interval arithmetic

Branch and Bound (B&B) algorithms in Global Optimization are used to perform an exhaustive search over the feasible area. One choice is to use simplicial partition sets. Obtaining sharp and cheap bounds of the objective function over a simplex is very important in the construction of efficient Global Optimization B&B algorithms. Although enclosing a simplex in a box implies an overestimation, boxes are more natural when dealing with individual coordinate bounds, and bounding ranges with Interval Arithmetic (IA) is computationally cheap. This paper introduces several linear relaxations using gradient information and Affine Arithmetic and experimentally studies their efficiency compared to traditional lower bounds obtained by natural and centered IA forms and their adaption to simplices. A Global Optimization B&B algorithm with monotonicity test over a simplex is used to compare their efficiency over a set of low dimensional test problems with instances that either have a box constrained search region or where the feasible set is a simplex. Numerical results show that it is possible to obtain tight lower bounds over simplicial subsets.

Balancing assembly lines operating with heterogeneous workers under uncertainty in task processing times

PurposeThis paper considers the assembly line worker assignment and balancing problem of type-2 (ALWABP-2) with fuzzy task times. This problem is an extension of the (simple) SALBP-2 in which task times are worker-dependent and concurrently uncertain. Two criteria are simultaneously considered for minimization, namely, fuzzy cycle time and fuzzy smoothness index.Design/methodology/approachFirst, we show how fuzzy concepts can be used for managing uncertain task times. Then, we present a multiobjective genetic algorithm (MOGA) to solve the problem. MOGA is devoted to the search for Pareto-optimal solutions. For facilitating effective trade-off decision-making, two different MO approaches are implemented and tested within MOGA: a weighted-sum based approach and a Pareto-based approach.FindingsExperiments over a set of fuzzified test problems show the effect of these approaches on the performance of MOGA while verifying its efficiency in terms of both solution and time quality.Originality/valueTo the author’s knowledge, no previous published work in the literature has studied the biobjective assembly line worker assignment and balancing problem of type-2 (ALWABP-2) with fuzzy task times.

Bayesian Testing of Scientific Expectations under Multivariate Normal Linear Models

The multivariate normal linear model is one of the most widely employed models for statistical inference in applied research. Special cases include (multivariate) t testing, (M)AN(C)OVA, (multivariate) multiple regression, and repeated measures analysis. Statistical criteria for a model selection problem where models may have equality as well as order constraints on the model parameters based on scientific expectations are limited however. This paper presents a default Bayes factor for this inference problem using fractional Bayes methodology. Group specific fractions are used to properly control prior information. Furthermore the fractional prior is centered on the boundary of the constrained space to properly evaluate order-constrained models. The criterion enjoys various important properties under a broad set of testing problems. The methodology is readily usable via the R package ‘BFpack’. Applications from the social and medical sciences are provided to illustrate the methodology.

Dynamic multi-objective evolutionary algorithm with objective space prediction strategy

To solve dynamic multi-objective optimization problems, a dynamic multi-objective evolutionary algorithm (DMOEA) must be able to deal with the dynamics of the environment, and such modifications can lead to new optimal solutions over time. Various algorithms have been proposed that modify the way a change is handled. Among them, prediction-based methods are promising for solving this kind of problem. They provide guided direction for population evolution through a prediction mechanism that assists the DMOEA to respond quickly to new changes. Based on these strategies, we propose a dynamic non-dominated sorting differential evolution improvement with prediction in the objective space (DOSP-NSDE). The proposal uses the objective space prediction (OSP) strategy for both the static evolutionary process (between changes) and the change reaction mechanism to predict the new optimal front location. Experiments were performed on a real-world problem and four sets of test problems: FDA, dMOP, UDF, and DF. Comparison of DOSP-NSDE with several algorithms in the literature, considering three metrics, is presented, showing that the proposal is competitive with most problems.

Generating set search using simplex gradients for bound-constrained black-box optimization

The optimization problems arising in modern engineering practice are increasingly simulation-based, characterized by extreme types of nonsmoothness, the inaccessibility of derivatives, and high computational expense. While generating set searches (GSS) generally offer a satisfying level of robustness and converge to stationary points, the convergence rates may be slow. In order to accelerate the solution process without sacrificing robustness, we introduce (simplex) gradient-informed generating set search (GIGS) methods for solving bound-constrained minimization problems. These algorithms use simplex gradients, acquired over several iterations, as guidance for adapting the search stencil to the local topography of the objective function. GIGS is shown to inherit first-order convergence properties of GSS and to possess a natural tendency for avoiding saddle points. Numerical experiments are performed on an academic set of smooth, nonsmooth and noisy test problems, as well as a realistic engineering case study. The results demonstrate that including simplex gradient information enables computational cost savings over non-adaptive GSS methods.

Optimization by moving ridge functions: derivative-free optimization for computationally intensive functions

A novel derivative-free algorithm, called optimization by moving ridge functions (OMoRF), for unconstrained and bound-constrained optimization is presented. This algorithm couples trust region methodologies with output-based dimension reduction to accelerate convergence of model-based optimization strategies. The dimension-reducing subspace is updated as the trust region moves through the function domain, allowing OMoRF to be applied to functions with no known global low-dimensional structure. Furthermore, its low computational requirement allows it to make rapid progress when optimizing high-dimensional functions. Its performance is examined on a set of test problems of moderate to high dimension and a high-dimensional design optimization problem. The results show that OMoRF compares favourably with other common derivative-free optimization methods, even for functions in which no underlying global low-dimensional structure is known.

An Eulerian Vlasov-Fokker–Planck algorithm for spherical implosion simulations of inertial confinement fusion capsules

We present a numerical algorithm that enables a phase-space adaptive Eulerian Vlasov–Fokker–Planck (VFP) simulation of inertial confinement fusion (ICF) capsule implosions. The approach relies on extending a recent mass, momentum, and energy conserving phase-space moving-mesh adaptivity strategy to spherical geometry. In configuration space, we employ a mesh motion partial differential equation (MMPDE) strategy while, in velocity space, the mesh is expanded/contracted and shifted with the plasma’s evolving temperature and drift velocity. The mesh motion is dealt with by transforming the underlying VFP equations into a computational (logical) coordinate, with the resulting inertial terms carefully discretized to ensure conservation. To deal with the spatial and temporally varying dynamics in a spherically imploding system, we have developed a novel nonlinear stabilization strategy for MMPDE in the configuration space. The strategy relies on a nonlinear optimization procedure that optimizes between mesh quality and the volumetric rate change of the mesh to ensure both accuracy and stability of the solution. Implosions of ICF capsules are driven by several boundary conditions: (1) an elastic moving wall boundary; (2) a time-dependent Maxwellian Dirichlet boundary; and (3) a pressure-driven Lagrangian boundary. Of these, the pressure-driven Lagrangian boundary driver is new to our knowledge. The implementation of our strategy is verified through a set of test problems, including the Guderley and Van-Dyke implosion problems — the first-ever reported using a Vlasov–Fokker–Planck model.