Symmetry, Volume 10, Issue 9 (September 2018) – 66 articles

Hill’s quadratic orthotropic yield criterion is used for revealing the effect of plastic anisotropy on the distribution of stresses and strains within rotating annular polar orthotropic disks of constant thickness under plane stress. The associated flow rule is adopted for connecting the stresses and strain rates. Assuming that unloading is purely elastic, the distribution of residual stresses and strains is determined as well. The solution for strain rates reduces to one nonlinear ordinary differential equation and two linear ordinary differential equations, even though the boundary value problem involves two independent variables. The aforementioned differential equations can be solved one by one. This significantly simplifies the numerical treatment of the general boundary value problem and increases the accuracy of its solution. In particular, comparison with a finite difference solution is made. It is shown that the finite difference solution is not accurate enough for some applications. Full article

The uncertainty and concurrence of randomness are considered when many practical problems are dealt with. To describe the aleatory uncertainty and imprecision in a neutrosophic environment and prevent the obliteration of more data, the concept of the probabilistic single-valued (interval) neutrosophic hesitant fuzzy set is introduced. By definition, we know that the probabilistic single-valued neutrosophic hesitant fuzzy set (PSVNHFS) is a special case of the probabilistic interval neutrosophic hesitant fuzzy set (PINHFS). PSVNHFSs can satisfy all the properties of PINHFSs. An example is given to illustrate that PINHFS compared to PSVNHFS is more general. Then, PINHFS is the main research object. The basic operational relations of PINHFS are studied, and the comparison method of probabilistic interval neutrosophic hesitant fuzzy numbers (PINHFNs) is proposed. Then, the probabilistic interval neutrosophic hesitant fuzzy weighted averaging (PINHFWA) and the probability interval neutrosophic hesitant fuzzy weighted geometric (PINHFWG) operators are presented. Some basic properties are investigated. Next, based on the PINHFWA and PINHFWG operators, a decision-making method under a probabilistic interval neutrosophic hesitant fuzzy circumstance is established. Finally, we apply this method to the issue of investment options. The validity and application of the new approach is demonstrated. Full article

Rough sets provide a useful tool for data preprocessing during data mining. However, many algorithms related to some problems in rough sets, such as attribute reduction, are greedy ones. Matroids propose a good platform for greedy algorithms. Therefore, it is important to study the combination between rough sets and matroids. In this paper, we investigate rough sets and matroids through their operators, and provide a matroidal method for attribute reduction in information systems. Firstly, we generalize four operators of rough sets to four operators of matroids through the interior, closure, exterior and boundary axioms, respectively. Thus, there are four matroids induced by these four operators of rough sets. Then, we find that these four matroids are the same one, which implies the relationship about operators between rough sets and matroids. Secondly, a relationship about operations between matroids and rough sets is presented according to the induced matroid. Finally, the girth function of matroids is used to compute attribute reduction in information systems. Full article

As a generalization of single value neutrosophic rough sets, the concept of multi-granulation neutrosophic rough sets was proposed by Bo et al., and some basic properties of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators were studied. However, they did not do a comprehensive study on the algebraic structure of the pessimistic (optimistic) multigranulation neutrosophic rough approximation operators. In the present paper, we will provide the lattice structure of the pessimistic multigranulation neutrosophic rough approximation operators. In particular, in the one-dimensional case, for special neutrosophic relations, the completely lattice isomorphic relationship between upper neutrosophic rough approximation operators and lower neutrosophic rough approximation operators is proved. Full article

The development of lateral asymmetries in athletes could have an influence on performance or injuries. The aim of this study was to determine the within-day reliability of the symmetry tests and the performance tests, and explore the relationship between them. Eighteen male volleyball players (18.1 ± 2.1 years) participated in this study. Seven lateral symmetry assessments were used, namely: lateral symmetry through tensiomyography (LS), active knee extension (AKE), Y-balance test (YBT), muscular electrical activity in attack jump (MEA-AJ), single-leg squat jump (SLSJ), triple hop test for distance (THTD), and bilateral maximum repetition in leg press (1RMSL); and three volleyball performance tests, namely: the T-test, counter-movement jump (CMJ), and attack jump (AJ). Three in-day measurements were taken from each volleyball player after the recovery was completed. The reliability was calculated through the intraclass correlation coefficient and the coefficient of variation, and the relationship was calculated through Pearson’s bivariate correlation coefficient (p < 0.05). The results indicate that AKE, YBT, and LS are the symmetry tests with increased reproducibility. THTD correlates positively with the AKE test and 1RMSL test, and a greater symmetry in the YBT correlates with a greater performance in the CMJ and AJ performance tests. In conclusion, AKE, LS, and YBT are the best tests to determine, with reliability, the asymmetries in volleyball players, and a greater symmetry in the YBT seems to influence the height of bilateral vertical jump. Full article

We construct families of ground state representations of the $\mathrm{U}\left(1\right)$ -current net and of the Virasoro nets ${\mathrm{Vir}}_c$ with central charge $c\ge 1$ . We show that these representations are not covariant with respect to the original dilations, and those on the $\mathrm{U}\left(1\right)$ -current net are not solitonic. Furthermore, by going to the dual net with respect to the ground state representations of ${\mathrm{Vir}}_c$ , one obtains possibly new family of Möbius covariant nets on $S^1$ . Full article

The detection of human emotions has applicability in various domains such as assisted living, health monitoring, domestic appliance control, crowd behavior tracking real time, and emotional security. The paper proposes a new system for emotion classification based on a generative adversarial network (GAN) classifier. The generative adversarial networks have been widely used for generating realistic images, but the classification capabilities have been vaguely exploited. One of the main advantages is that by using the generator, we can extend our testing dataset and add more variety to each of the seven emotion classes we try to identify. Thus, the novelty of our study consists in increasing the number of classes from N to 2N (in the learning phase) by considering real and fake emotions. Facial key points are obtained from real and generated facial images, and vectors connecting them with the facial center of gravity are used by the discriminator to classify the image as one of the 14 classes of interest (real and fake for seven emotions). As another contribution, real images from different emotional classes are used in the generation process unlike the classical GAN approach which generates images from simple noise arrays. By using the proposed method, our system can classify emotions in facial images regardless of gender, race, ethnicity, age and face rotation. An accuracy of 75.2% was obtained on 7000 real images (14,000, also considering the generated images) from multiple combined facial datasets. Full article

Let $k$ be an algebraically closed field of characteristic zero, and $R/I$ and $S/J$ be algebras over $k$ . ${\Omega }_1\left(R/I\right)$ and ${\Omega }_1\left(S/J\right)$ denote universal module of first order derivation over $k.$ The main result of this paper asserts that the first nonzero Fitting ideal ${\Omega }_1\left(R/I{\otimes }_{k}S/J\right)$ is an invertible ideal, if the first nonzero Fitting ideals ${\Omega }_1\left(R/I\right)$ and ${\Omega }_1\left(S/J\right)$ are invertible ideals. Then using this result, we conclude that the projective dimension of ${\Omega }_1\left(R/I{\otimes }_{k}S/J\right)$ is less than or equal to one. Full article

The modeling and control of the multi-rope parallel suspension lifting system (MPSLS) are investigated in the presence of different and spatial distributed tensions; unknown boundary disturbances; and multiple constraints, including time varying geometric constraint, input saturation, and output constraint. To describe the system dynamics more accurately, the MPSLS is modelled by a set of partial differential equations and ordinary differential equations (PDEs-ODEs) with multiple constraints, which is a nonhomogeneous and coupled PDEs-ODEs, and makes its control more difficult. Adaptive boundary control is a recommended method for position regulation and vibration degradation of the MPSLS, where adaptation laws and a boundary disturbance observer are formulated to handle system uncertainties. The system stability is rigorously proved by using Lyapunov’s direct method, and the position and vibration eventually diminish to a bounded neighborhood of origin. The original PDEs-ODEs are solved by finite difference method, and the multiple constraints problem is processed simultaneously. Finally, the performance of the proposed control is demonstrated by both the results of ADAMS simulation and numerical calculation. Full article

We study a new exactly solvable model of coupling of the Dark Energy and Dark Matter, in the framework of which the kernel of non-gravitational interaction is presented by the integral Volterra-type operator well-known in the classical theory of fading memory. Exact solutions of this isotropic homogeneous cosmological model were classified with respect to the sign of the discriminant of the cubic characteristic polynomial associated with the key equation of the model. Energy-density scalars of the Dark Energy and Dark Matter, the Hubble function and acceleration parameter are presented explicitly; the scale factor is found in quadratures. Asymptotic analysis of the exact solutions has shown that the Big Rip, Little Rip, Pseudo Rip regimes can be realized with the specific choice of guiding parameters of the model. We show that the Coincidence problem can be solved if we consider the memory effect associated with the interactions in the Dark Sector of the universe. Full article

Chromatographic interaction between infiltrating solutions of racemic mixtures of enantiomers and enantiomorphic minerals with chiral excess has been proposed as a scenario for the emergence of biomolecular homochirality. Enantiomer separation is supposed to be produced by different partition coefficients of both enantiomers with regard to crystal faces or walls of capillary tubes in the enantiomorphic mineral. Besides quartz, nepheline is the only common magmatic mineral with enantiomorphic symmetry. It crystallizes from SiO₂-undersaturated melts with low viscosity and is a promising candidate for chiral enrichment by autocatalytic secondary nucleation. Under liquidus conditions, the dynamic viscosity of silicate melts is mainly a function of polymerization. Melts with low concentrations of SiO₂ (<55 wt%) and rather high concentrations of Na₂O (>7 wt%) are only slightly polymerized and hence are characterized by low viscosities. Such melts can ascend, intrude or extrude by turbulent flow. Fourteen volcanic and subvolcanic samples from alkaline provinces in Africa and Sweden were chemically analyzed. Polished thin sections containing fresh nepheline phenocrysts were etched with 1% hydrofluoric acid at 20 °C for 15 to 25 min. Nepheline crystals suitable for a statistical evaluation of their etch figures were found in four samples. Crystals with chiral etch figures are mainly not twinned. Their chiral proportions in grain percentages of single crystals are close to parity in three samples. Only one sample shows a slight chiral excess (41.67% L-type vs. 58.33% D-type) but at a low level of significance (15 vs. 21 crystals, respectively). Full article

This paper investigates the boundary value in the heat conduction problem by a variational iteration method. Applying the iteration method, a sequence of convergent functions is constructed, the limit approximates the exact solution of the heat conduction equation in a few iterations using only the initial condition. This method does not require discretization of the variables. Numerical results show that this method is quite simple and straightforward for models that are currently under research. Full article

Without the design for inherent security, the Border Gateway Protocol (BGP) is vulnerable to prefix/subprefix hijacks and other attacks. Though many BGP security approaches have been proposed to prevent or detect such attacks, the unsatisfactory cost-effectiveness frustrates their deployment. In fact, the currently deployed BGP security infrastructure leaves the chance for potential centralized authority misconfiguration and abuse. It actually becomes the critical yield point that demands the logging and auditing of misbehaviors and attacks in BGP security deployments. We propose a blockchain-based Internet number resource authority and trustworthy management solution, named BGPcoin, to facilitate the transparency of BGP security. BGPcoin provides a reliable origin advertisement source for origin authentication by dispensing resource allocations and revocations compliantly against IP prefix hijacking. We perform and audit resource assignments on the tamper-resistant Ethereum blockchain by means of a set of smart contracts, which also interact as one to provide the trustworthy origin route examination for BGP. Compared with RPKI, BGPcoin yields significant benefits in securing origin advertisement and building a dependable infrastructure for the object repository. We demonstrate it through an Ethereum prototype implementation, and we deploy it and do experiment on a locally-simulated network and an official Ethereum test network respectively. The extensive experiment and evaluation demonstrate the incentives to deploy BGPcoin, and the enhanced security provided by BGPcoin is technically and economically feasible. Full article

The generalization of Farey graphs and extended Farey graphs all originate from Farey graphs. They are simultaneously scale-free and small-world. A labeling of the vertices for them are proposed here. All of the shortest paths between any two vertices in these two graphs can be determined only on their labels. The number of shortest paths between any two vertices is the product of two Fibonacci numbers; it is increasing almost linearly with the order or size of the graphs. However, the label-based routing algorithm runs in logarithmic time O(logn). Our efficient routing protocol for Farey-type models should help contribute toward the understanding of several physical dynamic processes. Full article

Soft set theory is a mathematical tool for handling uncertainty. This paper investigates the limits of the interval type of soft sets ( $it$ -soft sets). The notion of $it$ -soft sets is first introduced. Then, the limits of $it$ -soft sets are proposed and their properties obtained. Next, point-wise continuity of $it$ -soft sets and continuous $it$ -soft sets is discussed. Finally, an application for rough sets is given. Full article

Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields ${\mathbb{Z}}_p$. Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field. Full article

Hypergraph theory is the most developed tool for demonstrating various practical problems in different domains of science and technology. Sometimes, information in a network model is uncertain and vague in nature. In this paper, our main focus is to apply the powerful methodology of fuzziness to generalize the notion of competition hypergraphs and fuzzy competition graphs. We introduce various new concepts, including fuzzy column hypergraphs, fuzzy row hypergraphs, fuzzy competition hypergraphs, fuzzy k-competition hypergraphs and fuzzy neighbourhood hypergraphs, strong hyperedges, kth strength of competition and symmetric properties. We design certain algorithms for constructing different types of fuzzy competition hypergraphs. We also present applications of fuzzy competition hypergraphs in decision support systems, including predator–prey relations in ecological niche, social networks and business marketing. Full article

Parts manufacturers use sudden death testing to reduce the testing time of experiments. The sudden death testing plan in the literature can only be applied when all observations of failure time/parameters are crisp. In practice however, it is noted that not all measurements of continuous variables are precise. Therefore, the existing sudden death test plan can be applied if failure data/or parameters are imprecise, incomplete, and fuzzy. The classical statistics have the special case of neutrosophic statistics when there are no fuzzy observations/parameters. The neutrosophic fuzzy statistics can be applied for the testing of manufacturing parts when observations are imprecise, incomplete and fuzzy. In this paper, we will design an original neutrosophic fuzzy sudden death testing plan for the inspection/testing of the electronic product or parts manufacturing. We will assume that the lifetime of the product follows the neutrosophic fuzzy Weibull distribution. The neutrosophic fuzzy operating function will be given and used to determine the neutrosophic fuzzy plan parameters through a neutrosophic fuzzy optimization problem. The results of the proposed neutrosophic fuzzy death testing plan will be implemented with the aid of an example. Full article

Sound-event classification has emerged as an important field of research in recent years. In particular, investigations using sound data are being conducted in various industrial fields. However, sound-event classification tasks have become more difficult and challenging with the increase in noise levels. In this study, we propose a noise-robust system for the classification of sound data. In this method, we first convert one-dimensional sound signals into two-dimensional gray-level images using normalization, and then extract the texture images by means of the dominant neighborhood structure (DNS) technique. Finally, we experimentally validate the noise-robust approach by using four classifiers (convolutional neural network (CNN), support vector machine (SVM), k-nearest neighbors(k-NN), and C4.5). The experimental results showed superior classification performance in noisy conditions compared with other methods. The F1 score exceeds 98.80% in railway data, and 96.57% in livestock data. Besides, the proposed method can be implemented in a cost-efficient manner (for instance, use of a low-cost microphone) while maintaining high level of accuracy in noisy environments. This approach can be used either as a standalone solution or as a supplement to the known methods to obtain a more accurate solution. Full article

Type-2 intuitionistic fuzzy set (T2IFS) is a powerful and important extension of the classical fuzzy set, intuitionistic fuzzy set to measure the vagueness and uncertainty. In a practical decision-making process, there always occurs an inter-relationship among the multi-input arguments. To deal with this point, the motivation of the present paper is to develop some new interval type-2 (IT2) intuitionistic fuzzy aggregation operators which can consider the multi interaction between the input argument. To achieve it, we define a symmetric triangular interval T2IFS (TIT2IFS), its operations, Hamy mean (HM) operator to aggregate the preference of the symmetric TIT2IFS and then shows its applicability through a multi-criteria decision making (MCDM). Several enviable properties and particular cases together with following different parameter values of this operator are calculated in detail. At last a numerical illustration is to given to exemplify the practicability of the proposed technique and a comparative analysis is analyzed in detail. Full article

The eye blink rate, a major human physiological response, directly affects ocular diseases, such as keratitis and dry eye syndrome. It has been shown that the eye blink rate in normal eyes has a certain frequency for individuals, from 6–30 times/min. It was suggested in a previous study that the eye blink rate can be decreased during the viewing of high-intensity and realistic content. Therefore, in this paper, we examine the change of the eye blink rate during the HMD (head-mounted display) viewing of VR (virtual reality) contents; accordingly, we propose an algorithm to measure the eye blink rate as well as compare and analyze this rate in three different environments (natural, monitor, and HMD). We confirmed that IPD (interpupillary distance) and phoria affected the eye blink rate in each environment. In this experiment, 21 subjects (28.38 ± 6.87 years) were selected, and a paired t-test was performed for changes in the eye blink rate over 1 min for each environment. The IPD and phoria effects on the eye blink rate were confirmed using the Spearman’s correlation coefficient. In this experiment, the eye blink rate was decreased in the monitor and HMD environments compared with the natural environment, while that in the HMD environment was decreased compared with the monitor environment. The results of the correlation analysis of far IPD and the eye blink rate show no statistical significance or correlation. The correlation analysis of near IPD and the eye blink rate showed a strong positive correlation of the eye blink rate in the monitor environment. The correlation analysis of distance phoria and the eye blink rate showed a strong negative correlation of the eye blink rate in the HMD environment. The correlation analysis of near-field phoria and the eye blink rate showed a strong negative correlation of the eye blink rate in the HMD environment. It is expected that the results of this study will be used as a VR-viewing recommendation. Full article

This article performs the cryptanalysis of an image encryption algorithm using an S-box generated by chaos. The algorithm has the advantages of simple structure, high encryption efficiency, and good encryption performance. However, an attentive investigation reveals that it has some undiscovered security flaws. The image cryptosystem is totally breakable under proposed chosen-plaintext attack, and only two chosen plain-images are required. An array equivalent to the S-box is constructed by an elaborately designed chosen-plaintext image, and the cipher-image is deciphered without having to know the S-box itself. Both mathematical deduction and experimental results validate the feasibility of the attacking scheme. Furthermore, an improved encryption scheme is proposed, in which a feedback mechanism is introduced, a bidirectional diffusion scheme is designed, and values of the ciphertext are associated with more parameters in each diffusion process. Testing results and security analysis verify that the improved cryptographic system can achieve a higher security level and has a better performance than some of the latest encryption algorithms. Full article

We study and examine the rotational hypersurface and its Gauss map in Euclidean four-space ${\mathbb{E}}^4$ . We calculate the Gauss map, the mean curvature and the Gaussian curvature of the rotational hypersurface and obtain some results. Then, we introduce the third Laplace–Beltrami operator. Moreover, we calculate the third Laplace–Beltrami operator of the rotational hypersurface in ${\mathbb{E}}^4.$ We also draw some figures of the rotational hypersurface. Full article

To protect the security of vector maps, we propose a novel reversible watermarking scheme for vector maps based on a multilevel histogram modification. First, a difference histogram is constructed using the correlations of adjacent coordinates, and the histogram is divided into continuous regions and discontinuous regions by combining the characteristics of vector map data. Second, the histogram bins that require modification are determined in the continuous regions through the optimal peak value, and the peak values are chosen from the flanking discontinuous regions in both directions; the watermarks are embedded by adopting the multilevel histogram modification strategy. The watermark extraction process is the reverse of the embedding process, and after completing the watermark extraction, the carrier data can be recovered losslessly. The experimental results show that the proposed algorithm has good invisibility and is completely reversible. Compared with similar algorithms reported previously, it achieves higher watermark embedding capacity under the same embedding distortion with lower complexity, thereby having a higher application value. Full article

Recently, we have proposed models of topological field theory including gravity in Mod. Phys. Lett. A 2016, 31, 1650213 and Phys. Rev. D 2017, 96, 024009, in order to solve the problem of the cosmological constant. The Lagrangian densities of the models are BRS (Becchi-Rouet-Stora) exact and therefore the models can be regarded as topological theories. In the models, the coupling constants, including the cosmological constant, look as if they run with the scale of the universe and its behavior is very similar to the renormalization group. Motivated by these models, we propose new models with an the infrared fixed point, which may correspond to the late time universe, and an ultraviolet fixed point, which may correspond to the early universe. In particular, we construct a model with the solutions corresponding to the de Sitter space-time both in the ultraviolet and the infrared fixed points. Full article

The goal of this paper is to define the $(p,q)$ -analogue of tangent numbers and polynomials by generalizing the tangent numbers and polynomials and Carlitz-type q-tangent numbers and polynomials. We get some explicit formulas and properties in conjunction with $(p,q)$ -analogue of tangent numbers and polynomials. We give some new symmetric identities for $(p,q)$ -analogue of tangent polynomials by using $(p,q)$ -tangent zeta function. Finally, we investigate the distribution and symmetry of the zero of $(p,q)$ -analogue of tangent polynomials with numerical methods. Full article

The toxic transformation products of hydrazines are of great concern. These products’ properties combined with their formation mechanisms are needed to assess their potential environmental and human impacts. In this study, the gas-phase reaction of hydrazine (N₂H₄), monomethyldrazine (MMH) and unsymmetrical dimethyhydrazine (UDMH) with O₃ have been studied at varying reactant ratios, both in the presence and absence of a radical trap. Gas chromatography-mass spectroscopy (GC-MS) has been implied to follow reactant consumption and product formation. Apart from the reported products detected by Fourier transform infrared spectroscopy (FT-IR), the newly found compounds (hydrazones, formamides, dimethylamine, 1,1,4,4-tetramethyl-1,2-tetrazene,dimethylamino-acetonitrile, N₂, H₂O, et al.) are identified by GC-MS. The relative yields of the organic products vary considerably at different O₃/MMH or UDMH ratios. UDMH and MMH are confirmed as high potential precursors of N-nitrosodimethylamine (NDMA). The presence of hydroxyl radicals (HO·) hinders NDMA formation in MMH-O₃ system. Meanwhile, it increases NDMA formation in UDMH-O₃ system. The suggested reaction mechanisms which account for the observed products are discussed. Full article

In this paper, a new multi-criteria problem solving method—the Full Consistency Method (FUCOM)—is proposed. The model implies the definition of two groups of constraints that need to satisfy the optimal values of weight coefficients. The first group of constraints is the condition that the relations of the weight coefficients of criteria should be equal to the comparative priorities of the criteria. The second group of constraints is defined on the basis of the conditions of mathematical transitivity. After defining the constraints and solving the model, in addition to optimal weight values, a deviation from full consistency (DFC) is obtained. The degree of DFC is the deviation value of the obtained weight coefficients from the estimated comparative priorities of the criteria. In addition, DFC is also the reliability confirmation of the obtained weights of criteria. In order to illustrate the proposed model and evaluate its performance, FUCOM was tested on several numerical examples from the literature. The model validation was performed by comparing it with the other subjective models (the Best Worst Method (BWM) and Analytic Hierarchy Process (AHP)), based on the pairwise comparisons of the criteria and the validation of the results by using DFC. The results show that FUCOM provides better results than the BWM and AHP methods, when the relation between consistency and the required number of the comparisons of the criteria are taken into consideration. The main advantages of FUCOM in relation to the existing multi-criteria decision-making (MCDM) methods are as follows: (1) a significantly smaller number of pairwise comparisons (only n − 1), (2) a consistent pairwise comparison of criteria, and (3) the calculation of the reliable values of criteria weight coefficients, which contribute to rational judgment. Full article

Decision making is the key component of people’s daily life, from choosing a mobile phone to engaging in a war. To model the real world more accurately, probabilistic linguistic term sets (PLTSs) were proposed to manage a situation in which several possible linguistic terms along their corresponding probabilities are considered at the same time. Previously, in linguistic term sets, the probabilities of all linguistic term sets are considered to be equal which is unrealistic. In the process of decision making, due to the vagueness and complexity of real life, an expert usually hesitates and unable to express its opinion in a single term, thus making it difficult to reach a final agreement. To handle real life scenarios of a more complex nature, only membership linguistic decision making is unfruitful; thus, some mechanism is needed to express non-membership linguistic term set to deal with imprecise and uncertain information in more efficient manner. In this article, a novel notion called probabilistic hesitant intuitionistic linguistic term set (PHILTS) is designed, which is composed of membership PLTSs and non-membership PLTSs describing the opinions of decision makers (DMs). In the theme of PHILTS, the probabilities of membership linguistic terms and non-membership linguistic terms are considered to be independent. Then, basic operations, some governing operational laws, the aggregation operators, normalization process and comparison method are studied for PHILTSs. Thereafter, two practical decision making models: aggregation based model and the extended TOPSIS model for PHILTS are designed to classify the alternatives from the best to worst, as an application of PHILTS to multi-attribute group decision making. In the end, a practical problem of real life about the selection of the best alternative is solved to illustrate the applicability and effectiveness of our proposed set and models. Full article

As air pollution becomes increasingly severe, express trains play a more important role in shifting road freight and reducing carbon emissions. Thus, the design of railway express shipment service networks has become a key issue, which needs to be addressed urgently both in theory and practice. The railway express shipment service network design problem (RESSNDP) not only involves the selection of train services and determination of service frequency, but it is also associated with shipment routing, which can be viewed as a service network design problem (SNDP) with railway characteristics. This paper proposes a non-linear integer programming model (INLP) which aims at finding a service network and shipment routing plan with minimum cost while satisfying the transportation time constraints of shipments, carrying capacity constraints of train services, flow conservation constraint and logical constraints among decision variables. In addition, a linearization technique was adopted to transform our model into a linear one to obtain a global optimal solution. To evaluate the effectiveness and efficiency of our approach, a small trial problem was solved by the state-of-the-art mathematical programming solver Gurobi 7.5.2. Full article