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Ranking and comparing grey numbers represent a very important decision-making procedure in any given grey environment. The purpose of this paper is to study the existing approaches of ordering interval grey numbers in the context of decision making by surveying existing definitions.

Different methods developed for comparing grey numbers are presented along with their disadvantages and advantages in terms of comparison outcomes. Practical examples are employed to show the importance and necessity of using appropriate methods to compare grey numbers.

Most the available methods are not suitable for pointing out which number is larger when the nuclei of the grey numbers of concern are the same. Also, these available methods are also considered in terms of partial order and total order. Kernel and degree of greyness of grey numbers method is more advantageous than other methods and almost eliminates the shortcomings of other methods.

Different methods for ranking grey numbers are presented where each of them has disadvantages and advantages. By using different methods, grey interval numbers are compared and the results show that some methods cannot make grey number comparisons in some situations. The authors intend to find a method that can compare grey numbers in any situation. The findings of this research can prevent errors that may occur based on inaccurate comparisons of grey numbers in decision making. There are various research studies on the comparison of grey numbers, but there is no research on the comparison of these methods and their disadvantages, advantages or their total or partial order.

The purpose of this paper is to survey and express the advantages and disadvantages of the existing approaches for solving grey linear programming in decision-making problems.

After presenting the concepts of grey systems and grey numbers, this paper surveys existing approaches for solving grey linear programming problems and applications. Also, methods and approaches for solving grey linear programming are classified, and its advantages and disadvantages are expressed.

The progress of grey programming has been expressed from past to present. The main methods for solving the grey linear programming problem can be categorized as Best-Worst model, Confidence degree, Whitening parameters, Prediction model, Positioned solution, Genetic algorithm, Covered solution, Multi-objective, Simplex and dual theory methods. This survey investigates the developments of various solving grey programming methods and its applications.

Different methods for solving grey linear programming problems are presented, where each of them has disadvantages and advantages in providing results of grey linear programming problems. This study attempted to review papers published during 35 years (1985–2020) about grey linear programming solving and applications. The review also helps clarify the important advantages, disadvantages and distinctions between different approaches and algorithms such as weakness of solving linear programming with grey numbers in constraints, inappropriate results with the lower bound is greater than upper bound, out of feasible region solutions and so on.

In plane elasticity, a general expression for a mutual work difference integral (MWDI) derived from two stress fields is introduced. Once two physical stress fields are known beforehand, the relevant MWDI can be evaluated exactly from the coefficients in the complex potentials. A biaxial tension model for evaluating defect energy is introduced. A particular MWDI from two fields, one is for the damaged medium under remote biaxial tension and other is for an infinite perfect plate under the same remote biaxial tension, can be defined as a suitable measure of stiffness for the damaged medium, which is called the defect energy ( E (a) ). The suggested model can deal with the cracks, holes, and elastic inclusions in a unique way. The model can also evaluate the defect energies for different damages exactly without dependence on the orientation of damages. Physically, the higher is the defect energy achieved, the more are the involved damages in the medium. The defect energy may be negative, which means a more rigid inclusion is included in the medium. For 3D‐elasticity, a triaxial tension model is introduced for evaluating the defect energy for the damaged medium. For some particular cases, for example, the dissimilar elastic spherical inclusion, or the elliptic flat crack, the relevant defect energies are evaluated.

For extending the common definitions and concepts of grey system theory to the optimization subject, a dual problem is proposed for the primal grey linear programming problem.

The authors discuss the solution concepts of primal and dual of grey linear programming problems without converting them to classical linear programming problems. A numerical example is provided to illustrate the theory developed.

By using arithmetic operations between interval grey numbers, the authors prove the complementary slackness theorem for grey linear programming problem and the associated dual problem.

Complementary slackness theorem for grey linear programming is first presented and proven. After that, a dual simplex method in grey environment is introduced and then some useful concepts are presented.

In this paper, elastic analysis for an edge‐cracked plate of functionally graded materials (FGMs) is carried out. The cracked plate is subject to a longitudinal tension. The property of FGMs is assumed to be exponential function form in both x‐ and y‐directions. The finite element method is suggested to solve the boundary value problem. An indirect method, the energy release method, is developed to evaluate the stress intensity factors (SIFs) at the crack tip. Under the applied loading, the amount of the release energy from the crack length “a” to “ a + Δa ” can be evaluated from relevant displacements at the top face of plate. The SIFs can be obtained from a relation between the energy release rate and SIF. The obtained result shows that the property of FGMs has a significant influence to the value of SIF at crack tip. Numerical results are given which are useful for engineer to assess the safety of the cracked plate of FGMs.

Three-way decision (3WD) and probabilistic rough sets (PRSs) are theoretical tools capable of simulating humans' multi-level and multi-perspective thinking modes in the field of decision-making. They are proposed to assist decision-makers in better managing incomplete or imprecise information under conditions of uncertainty or fuzziness. However, it is easy to cause decision losses and the personal thresholds of decision-makers cannot be taken into account. To solve this problem, this paper combines picture fuzzy (PF) multi-granularity (MG) with 3WD and establishes the notion of PF MG 3WD.

An effective incomplete model based on PF MG 3WD is designed in this paper. First, the form of PF MG incomplete information systems (IISs) is established to reasonably record the uncertain information. On this basis, the PF conditional probability is established by using PF similarity relations, and the concept of adjustable PF MG PRSs is proposed by using the PF conditional probability to fuse data. Then, a comprehensive PF multi-attribute group decision-making (MAGDM) scheme is formed by the adjustable PF MG PRSs and the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) method. Finally, an actual breast cancer data set is used to reveal the validity of the constructed method.

The experimental results confirm the effectiveness of PF MG 3WD in predicting breast cancer. Compared with existing models, PF MG 3WD has better robustness and generalization performance. This is mainly due to the incomplete PF MG 3WD proposed in this paper, which effectively reduces the influence of unreasonable outliers and threshold settings.

The model employs the VIKOR method for optimal granularity selections, which takes into account both group utility maximization and individual regret minimization, while incorporating decision-makers' subjective preferences as well. This ensures that the experiment maintains higher exclusion stability and reliability, enhancing the robustness of the decision results.

This paper aims to propose the new incremental and parallel training algorithm of proximal support vector machines (Inc-Par-PSVM) tailored on the edge device (i.e. the Jetson Nano) to handle the large-scale ImageNet challenging problem.

The Inc-Par-PSVM trains in the incremental and parallel manner ensemble binary PSVM classifiers used for the One-Versus-All multiclass strategy on the Jetson Nano. The binary PSVM model is the average in bagged binary PSVM models built in undersampling training data block.

The empirical test results on the ImageNet data set show that the Inc-Par-PSVM algorithm with the Jetson Nano (Quad-core ARM A57 @ 1.43 GHz, 128-core NVIDIA Maxwell architecture-based graphics processing unit, 4 GB RAM) is faster and more accurate than the state-of-the-art linear SVM algorithm run on a PC [Intel(R) Core i7-4790 CPU, 3.6 GHz, 4 cores, 32 GB RAM].

The new incremental and parallel PSVM algorithm tailored on the Jetson Nano is able to efficiently handle the large-scale ImageNet challenge with 1.2 million images and 1,000 classes.

The purpose of this paper is to present a new methodology, used for dynamic reliability analysis of a gear transmission system (GTS) of wind turbine (WT), which could be used for assembly decision-making of the parts with errors to improve the GTS’s performance.

This paper involves the dynamic and dynamic reliability analysis of a GTS. The history curves of dynamic responses of the parts are obtained with the developed gear-bearing coupling dynamic model considering the random errors, failure dependency and random load. Then, the surrogate models of the mean and standard deviation of responses are presented by statistics, rain flow counting method and corrected-partial least squares regression response surface method. Further, a novel dynamic reliability model based on the maximum extreme theory, a theory of sequential statistics, equivalent principles and the inverse transform theory of random variable sampling, is developed to overcome the limitations of traditional methods.

The dynamic reliability of GTS considering the different impact factors are evaluated. The proposed reliability methodology not only overcomes the limitations associated with traditional approaches but also provides good guidance to assembly the parts in a GTS to its best performance.

Instead of constant errors, this paper considers the randomness of the impact factors to develop the dynamic reliability model. Further, instead of the limitation of the normal distribution of the random parameters in the traditional method, the proposed methodology can deal with the problems with non-normal distribution parameters, which is more suitable for the real engineering problems.

The purpose of this paper is to detect the degenerate scale of a 2D bending plate analytically and numerically.

To avoid the time-consuming scheme, the influence matrix of the boundary element method (BEM) is reformulated to an eigenproblem of the 4 by 4 matrix by using the scaling transform instead of the direct-searching scheme to find degenerate scales. Analytical degenerate scales are derived from the boundary integral equation (BIE) by using the degenerate kernel only for the circular case. Numerical results of the direct-searching scheme and the eigen system for the arbitrary shape are also considered.

Results using three methods, namely, analytical derivation, the direct-searching scheme and the 4 by 4 eigen system, are also given for the circular case and arbitrary shapes. Finally, addition of a constant for the kernel function makes original eigenvalues (2 real roots and 2 complex roots) of the 4 by 4 matrix to be all real. This indicates that a degenerate scale depends on the kernel function.

The analytical derivation for the degenerate scale of a 2D bending plate in the BIE is first studied by using the degenerate kernel. Through the reformed eigenproblem of a 4 by 4 matrix, the numerical solution for the plate of an arbitrary shape can be used in the plate analysis using the BEM.