Symmetry Groups and New Exact Solutions to （2＋1）-Dimensional Variable Coefficient Canonical Generalized KP Equation

In this paper, the modified CK＇s direct method to find symmetry groups of nonlinear partial differential equation is extended to （2＋1）-dimensional variable coeffficient canonical generalized KP （VCCGKP） equation. As a result, symmetry groups, Lie point symmetry group and Lie symmetry for the VCCGKP equation are obtained. In fact, the Lie point symmetry group coincides with that obtained by the standard Lie group approach. Applying the given Lie symmetry, we obtain five types of similarity reductions and a lot of new exact solutions, including hyperbolic function solutions, triangular periodic solutions, Jacobi elliptic function solutions and rational solutions, for the VCCGKP equation.

Lie Symmetry Groups of(2+1)-Dimensional BKP Equation and Its New Solutions

A modified direct method is developed to find finite symmetry groups of nonlinear mathematical physicssystems.Applying the modified direct method to the well-known (2+1)-dimensional BKP equation we get its symmetry.Furthermore,the exact solutions of (2+1)-dimensional BKP equation are obtained through symmetry analysis.

Symmetry Groups and New Exact Solutions of(2+1)-Dimensional Dispersive Long-Wave Equations

Using the modified CK＇s direct method, we derive a symmetry group theorem of （2＋1）-dimensional dispersive long-wave equations. Based upon the theorem, Lie point symmetry groups and new exact solutions of （2＋1）- dimensional dispersive long-wave equations are obtained.

A New Understanding on the Problem That the Quintic Equation Has No Radical Solutions

It is proved in this paper that Abel’s and Galois’s proofs that the quintic equations have no radical solutions are invalid. Due to Abel’s and Galois’s work about two hundred years ago, it was generally accepted that general quintic equations had no radical solutions. However, Tang Jianer et al. recently prove that there are radical solutions for some quintic equations with special forms. The theories of Abel and Galois cannot explain these results. On the other hand, Gauss et al. proved the fundamental theorem of algebra. The theorem declared that there were n solutions for the n degree equations, including the radical and non-radical solutions. The theories of Abel and Galois contradicted with the fundamental theorem of algebra. Due to the reasons above, the proofs of Abel and Galois should be re-examined and re-evaluated. The author carefully analyzed the Abel’s original paper and found some serious mistakes. In order to prove that the general solution of algebraic equation he proposed was effective for the cubic equation, Abel took the known solutions of cubic equation as a premise to calculate the parameters of his equation. Therefore, Abel’s proof is a logical circular argument and invalid. Besides, Abel confused the variables with the coefficients (constants) of algebraic equations. An expansion with 14 terms was written as 7 terms, 7 terms were missing. We prefer to consider Galois’s theory as a hypothesis rather than a proof. Based on that permutation group S5 had no true normal subgroup, Galois concluded that the quintic equations had no radical solutions, but these two problems had no inevitable logic connection actually. In order to prove the effectiveness of radical extension group of automorphism mapping for the cubic and quartic equations, in the Galois’s theory, some algebraic relations among the roots of equations were used to replace the root itself. This violated the original definition of automorphism mapping group, led to the confusion of concepts and arbitrariness. For the general cubic and quartic algebraic equations, the actual solving processes do not satisfy the tower structure of Galois’s solvable group. The resolvents of cubic and quartic equations are proved to have no symmetries of Galois’s soluble group actually. It is invalid to use the solvable group theory to judge whether the high degree equation has a radical solution. The conclusion of this paper is that there is only the Sn symmetry for the n degree algebraic equations. The symmetry of Galois’s solvable group does not exist. Mathematicians should get rid of the constraints of Abel and Galois’s theories, keep looking for the radical solutions of high degree equations.

基于广义正交三角模糊数的WSM-TOPSIS群决策方法

结合广义正交模糊集和三角直觉模糊数,给出广义正交三角模糊数的定义和运算法则,在此基础上,对WSM和TOPSIS方法进行拓展,提出一种多属性群决策方法WSM-TOPSIS。考虑到决策者和属性的权重,该方法使用WSM对决策者提出的决策矩阵进行第一次集结,通过TOPSIS计算各方案的相对贴近度得到方案的优劣排序。最后通过实例和对比分析验证了该方法的有效性和实用性。

TOPSIS with Belief Structure for Group Belief Multiple Criteria Decision Making

The technique for order performance by similarity to ideal solution （TOPSIS） is one of the major techniques in dealing with multiple criteria decision making （MCDM） problems, and the belief structure （BS） model has been used successfully for uncertain MCDM with incompleteness, impreciseness or ignorance. In this paper, the TOPSIS method with BS model is proposed to solve group belief MCDM problems. Firstly, the group belief MCDM problem is structured as a belief decision matrix in which the judgments of each decision maker are described as BS models, and then the evidential reasoning approach is used for aggregating the multiple decision makers＇ judgments. Subsequently, the positive and negative ideal belief solutions are defined with the principle of TOPSIS. To measure the separation from ideal solutions, the concept and algorithm of belief distance measure are defined, which can be used for comparing the difference between BS models. Finally, the relative closeness and ranking index are calculated for ranking the alternatives. A numerical example is given to illustrate the proposed method.

基于改进D-S证据和融合权集的安全风险评估

针对信息化系统安全风险评估过程中安全风险因素指标的重要性难以赋权的问题,本文以建筑工地施工现场为应用场景,提出一种基于改进的D-S证据理论与融合权集结合的安全风险评估模型.首先,在充分研究建筑工地安全风险评估流程和要素的基础上,建立建筑工地安全评价体系;其次,采用基于权值分配和矩阵分析的D-S合成算法改进AHP法和基于数据的熵权法计算评价体系中指标层中各指标的主、客观权重;然后,运用改进的D-S证据融合算法进行多源证据的合成,获取指标权重,避免单一赋权的片面性,得到最优综合权重;最后,根据TOPSIS评价算法计算建筑工地综合评价指数.分析表明,基于改进D-S证据理论-融合权集的安全风险评估模型,能有效评估建筑工地施工现场的安全性,降低评估结果的不确定性,提高风险评估结果可信度.

基于直觉模糊环境的海上风机选型

选择合适的海上风力发电机机型对海上风电场的长期高效运行起着至关重要的作用。针对决策者在风机选型决策过程中存在不确定性和主观偏好等问题,提出一种改进的多准则决策(multi-criteria decision making, MCDM)框架及方法:在权重求解算法中将群体决策和直觉模糊数与层次分析法相结合,提出群体直觉模糊层次分析法(group intuitionistic fuzzy analytic hierarchy process, GIAHP)计算指标权重;在备选方案排序算法中将余弦距离引入接近理想点法(technique for order preference by similarity to an ideal solution, TOPSIS),提出多距离TOPSIS确定备选方案排序。最后以山东省海上风电场风机选型为例,并通过敏感性分析验证框架及方法的鲁棒性。该框架及方法为中国海上风电场风机选型提供理论依据,可确保海上风电场长期稳定运行。

基于TOPSIS的直觉模糊多属性群决策方法

针对直觉模糊信息环境下的多属性群决策问题,对专家给出的直觉模糊集决策矩阵进行规范化处理,利用直觉模糊集的相关理论,提出基于直觉模糊熵的属性权重确定方法;为了便于度量任意两个专家之间决策信息的相似度,将各专家给出的直觉模糊集决策矩阵转化决策向量,提出基于直觉模糊相似度的专家权重确定方法;利用逼近于理想解的排序方法(TOPSIS)对专家群的决策信息进行综合集成,给出备选方案优劣排序。结合算例验证方法的有效性。

群决策的交互式TOPSIS方法及其应用

建立了一种多轮交互逐步逼近满意解的多属性群体决策的综合方法,对决策方案采用扩展的 TOPSIS(technique for order preference by similarity to ideal solution)法进行定性评价排序,以克服一类难以建立解析模型,分析求解困难的多属性群决策问题;设计了群体满意度等软指标来分析计算,判断群体一致性的达成,最终得出群体满意的方案排序.通过应用实例,说明了该方法的有效性和可行性,初步提出了可作为建立一类实际的群决策支持系统的方法和技术框架.

基于TOPSIS的出行方案评价指标权重的确定方法

旅客出行偏好选择问题是旅客出行行为分析理论的一个重要部分,而旅客对出行方案的选择是基于对各种方案的评价指标的综合分析.但旅客对各评价指标的侧重程度,即属性权重的计算,目前常用方法都有一定的局限性.本文以旅客出行偏好选择为背景,借助多指标评价方法TOPSIS的评价准则,以实际的大规模旅客调查数据为基础,分析、确定旅客对出行方案各属性指标的权重,为旅客出行行为分析提供重要、客观、可靠的依据.经过实例验证,证明了该方法的有效性,并将计算结果代入多项Logit模型,进一步验证了计算结果的合理性.此外,该方法计算简便、易于实现,对于各种交通方式下的旅客出行偏好选择问题具有普遍适用性.

基于TOPSIS法的群决策方法

根据群决策的特点，按照集议决策的习惯程序，提出了一种充分发挥集议决策优势的群决策方法。通过组织者有效的组织，专家群体经过充分的讨论和广泛的交流，使各专家行为有效集结。最后由运筹专业人员利用ＴＯＰＳＩＳ法对行为集结的最后结果（决策矩阵）进行数学集结，较好地做到了决策在定性和定量上的综合集成。

耦合Burgers方程的对称群和新精确解

利用推广的CK直接方法,求出了耦合Burgers方程的一般对称群,建立了方程新旧解之间的关系,进一步利用对称求得了耦合Burgers方程的不变量和若干约化,通过约化方程求得耦合Burgers方程大量新的精确解,包括有理函数解、三角函数解和双曲函数解.

考虑QoS偏好的多属性群决策服务选择策略

面对多个功能相同或相似的服务,服务的Qo S是服务选择中重要的考虑因素.将Qo S属性分为精确数型、区间数型和三角模糊数型.在此基础上,利用TOPSIS(technique for order preference by similarity to an ideal solution)条件下的多属性群决策模型给出了服务选择过程,该过程考虑了多个决策者在决策过程中所占的权重,以及多个决策者不同的Qo S偏好权重.通过一个实例验证了该方法的有效性.

一种基于TOPSIS的多属性决策方法研究

文章给出了一种基于TOPSIS的多属性决策方法,并通过实例检验了该方法的有效性。

TOPSIS法在基于疾病诊断相关组(DRGs)的医院绩效评价中的应用

目的探讨TOPSIS法在基于DRGs的医院绩效评价中的效果。方法收集2019年1~12月浙江省某三级甲等综合医院的出院患者病案首页信息,通过上海联众DRGs绩效考核系统运算,得出DRGs相关指标信息,运用TOPSIS评价法对院内各出院科室医疗服务进行综合绩效评价。结果DRGs组数为708组,占分组的85.7%;CMI为0.98,平均住院天数7.39 d,平均住院费用17862.52元;排名前5位的分别为重症医学科、心胸外科、急诊医学科、神经外科、感染病科,排名后5位的分别为介入医学科、眼科、整形外科、生殖内分泌科、产科。结论TOPSIS法能够集中地反应出各出院科室的优劣,更客观准确地反映不同类型科室的医疗服务、诊治范围、诊疗技术难度、服务效率等综合情况,较好地解决了不同科室间的医疗评价差别,为医院绩效管理、学科发展提供参考方向,值得应用推广。

基于组合赋权的非介入式终端辨识能力灰色TOPSIS评价

非介入式负荷辨识终端克服了传统侵入式终端成本高、安装复杂和维护不便的问题,得以广泛应用。但目前仅通过电量指标对非介入式终端负荷辨识能力进行评价,评价体系单一。为此,文中选取多维指标综合衡量终端对电器负荷的辨识能力,并据此构建了非介入式终端分层多组评价模型;利用熵权-层次分析法(AHP)组合评价技术得到指标的组合权值;在此基础上,利用灰色逼近理想解排序(TOPSIS)评价对电器组合案例的电器辨识能力进行分类、灰色排序,从而客观地得到终端整体辨识能力水平及不同案例下终端辨识能力好坏。最后,在非介入式终端实证平台选取4种典型电器,验证了文中方案可解决评估指标数据不完全的问题,结合实际工程的终端辨识能力校验,提高了非侵入负荷辨识终端辨识能力评价的科学性和有效性。

概率犹豫模糊环境下基于前景理论和TOPSIS法的多属性群决策模型

在研究多属性群决策问题的领域中,概率犹豫模糊术语集(hesitant probabilistic fuzzy set,HPFS)作为犹豫模糊集的一种扩展,正广受关注。针对目前在概率犹豫模糊语言环境下,考虑用主客观结合的方式来求解权重以及对方案排序的过程中存在的问题,提出了一种基于前景理论和逼近理想解排序法(technique for order preference by similarity to an ideal solution,TOPSIS)相结合的多属性群决策模型。首先根据已知的主观决策者权重,经过一致性调节运算得到决策者的综合权重;其次利用熵值法构建了属性权重的求解模型;在充分考虑决策者心理行为的前提下,求解出正、负理想解矩阵,并且基于TOPSIS方法实现多个备选方案之间的优劣排序;最后,通过实例验证了该模型的可行性和有效性。

基于TOPSIS和矩阵组合的县域永久基本农田储备区划定时序研究

永久基本农田储备区划定是巩固永久基本农田划定成果、确保国家粮食安全的重大举措,合理规划储备区范围及划定时序可以保障永久基本农田在被占用后能够快速得到补划。本文以河北省黄骅市为研究区,从耕地的地力质量、空间质量、工程质量和生态环境质量4个方面选取评价指标,运用逼近理想解排序法和热点分析法进行耕地质量综合评价及空间集聚性分析,综合二者结果采用矩阵组合法确定永久基本农田储备区的分级划定。结果表明:1)黄骅市非永久基本农田的耕地地力质量、空间质量、工程质量和生态环境质量差异显著,整体土地质量较好;部分地区受土壤盐渍化影响较大,工程质量较差;灌溉保证率低下是限制耕地工程质量的主要因素;综合质量以高值区和较高值区为主,分别占非永久基本农田耕地图斑面积和数量的44.22%和31.08%;空间分布上表现为西部永久基本农田周边的耕地综合质量较高,东部距海岸较近处耕地质量较低。2)研究将黄骅市永久基本农田储备区划分为优先划入区、适宜划入区和整治调控区3类。优先划入区的耕地综合质量高、集聚特征明显,面积占比为30.75%;适宜划入区的耕地综合质量或空间聚集性较差,需投入一定的人力物力进行质量建设;整治调控区面积最大,占比为41.19%,其整治和改良应致力于提升农业生态的长期安全。研究在考虑基本农田储备区划定的质量目标和生态环境目标的基础上增加空间利用规模的限制,可为黄骅市及同处于环渤海带的滨海平原区高质量完成永久基本农田储备区划定工作提供参考。

Synthetic security assessment for incomplete interval-valued information system

In order to understand the security conditions of the incomplete interval-valued information system （IllS） and acquire the corresponding solution of security problems, this paper proposes a multi-attribute group decision- making （MAGDM） security assessment method based on the technique for order performance by similarity to ideal solution （TOPSIS）. For IllS with preference information, combining with dominance-based rough set approach （DRSA）, the effect of incomplete interval-valued information on decision results is discussed. For the imprecise judgment matrices, the security attribute weight can be obtained using Gibbs sampling. A numerical example shows that the proposed method can acquire some valuable knowledge hidden in the incomplete interval-valued information. The effectiveness of the proposed method in the synthetic security assessment for IIIS is verified.