Virtual enterprise partner selection integrating LINMAP and TOPSIS

Virtual enterprise (VE) has become an ever-increasing trend in today’s highly competitive markets. A more scientific decision-making process for selecting partner is very important during the formation phase of VE. Partner selection is formulated as a type of fuzzy hybrid multi-criteria group decision-making problems with fuzzy truth degrees of alternatives’ comparisons represented as trapezoidal fuzzy numbers (TrFNs). Integrating Technique for Order Preference by Similarity to Ideal Solution (TOPSIS) and Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP), we use the relative closeness degrees to define fuzzy consistency and inconsistency indices. The decision makers’ weight vector is derived by using the relative entropy. Criteria weights are estimated through constructing a new fuzzy linear programming model with TrFNs, which is solved by the developed fast and efficient method. Collective ranking matrix of alternatives is generated through constructing multi-objective assignment model. Example analysis demonstrates the effectiveness of the proposed method.     

Axiomatic properties of inconsistency indices for pairwise comparisons

Pairwise comparisons are a well-known method for the representation of the subjective preferences of a decision maker. Evaluating their inconsistency has been a widely studied and discussed topic and several indices have been proposed in the literature to perform this task. As an acceptable level of consistency is closely related to the reliability of preferences, a suitable choice of an inconsistency index is a crucial phase in decision-making processes. The use of different methods for measuring consistency must be carefully evaluated, as it can affect the decision outcome in practical applications. In this paper, we present five axioms aimed at characterizing inconsistency indices. In addition, we prove that some of the indices proposed in the literature satisfy these axioms, whereas others do not, and therefore, in our view, they may fail to correctly evaluate inconsistency.     

Analysis of pairwise comparison matrices: an empirical research

Pairwise comparison (PC) matrices are used in multi-attribute decision problems (MADM) in order to express the preferences of the decision maker. Our research focused on testing various characteristics of PC matrices. In a controlled experiment with university students (N=227) we have obtained 454 PC matrices. The cases have been divided into 18 subgroups according to the key factors to be analyzed. Our team conducted experiments with matrices of different size given from different types of MADM problems. Additionally, the matrix elements have been obtained by different questioning procedures differing in the order of the questions. Results are organized to answer five research questions. Three of them are directly connected to the inconsistency of a PC matrix. Various types of inconsistency indices have been applied. We have found that the type of the problem and the size of the matrix had impact on the inconsistency of the PC matrix. However, we have not found any impact of the questioning order. Incomplete PC matrices played an important role in our research. The decision makers behavioral consistency was as well analyzed in case of incomplete matrices using indicators measuring the deviation from the final order of alternatives and from the final score vector.     

Inconsistency indices for pairwise comparisons and the Pareto dominance principle

We consider the problem of quantifying inconsistency in pairwise comparisons and valued-preferences. A wide range of indices have been proposed in the literature to perform this task, and two sets of conditions have been introduced to validate such indices. We summarize some criticisms from the literature and we add more evidence to show that neither of the two systems is adequate in its current formulation. Thanks to the widely accepted concept of weak Pareto dominance, we formulate a new property. We argue that a simple regularity condition and this new property can overcome the shortcomings of the two axiomatic systems, and represents a significantly simpler framework. Finally, we claim that, if we had resorted to strict Pareto dominance, we would have needed just one axiom.     

The unprovability of small inconsistency

Summary We show that a consistent, finitely axiomatized, sequential theory cannot prove its own inconsistency on every definable cut. A corollary is that there are at least three degrees of global interpretability of theories equivalent modulo local interpretability to a consistent, finitely axiomatized, sequential theory U.     

A formal system for fuzzy reasoning

A formal system for fuzzy reasoning is described which is capable of dealing rationally with evidence which may be inconsistent and/or involve degrees of belief. The basic idea is that the meaning of each formal sentence should be given by a certain commitment or bet associated with it. Each item of evidence is first expressed in the form of such a (hypothetical) bet, which is then written as a formal sentence in a language related to ?ukasiewicz logic. The sentences may be weighted to express the relative reliability of the various informants. A sentence is considered to “follow” from the evidence if the bet it represents can be offered by a speaker without fear of loss, on the assumption that the bets representing various items of evidence have been offered to him. A detailed account, illustrated by concrete examples, is given of the procedures by which an arbitrary sentence in common language can be translated into a formal sentence. The treatment of inconsistency, degrees of belief, and weights is illustrated by a practical example which is solved in full. It is shown that in most practical cases the computations involved in the process of formal reasoning reduce to a problem in linear programming. In the last section the relation between this system and the procedures advocated by Zadeh is examined. It is shown that, subject to certain modifications in formulas, there is general agreement in the region of overlap.     

Measuring and repairing inconsistency in probabilistic knowledge bases

In this paper we present a family of measures aimed at determining the amount of inconsistency in probabilistic knowledge bases. Our approach to measuring inconsistency is graded in the sense that we consider minimal adjustments in the degrees of certainty (i.e., probabilities in this paper) of the statements necessary to make the knowledge base consistent. The computation of the family of measures we present here, in as much as it yields an adjustment in the probability of each statement that restores consistency, provides the modeler with possible repairs of the knowledge base. The case example that motivates our work and on which we test our approach is the knowledge base of CADIAG-2, a well-known medical expert system.     

Extension of the LINMAP for multiattribute decision making under Atanassov’s intuitionistic fuzzy environment

The aim of this article is further extending the linear programming techniques for multidimensional analysis of preference (LINMAP) to develop a new methodology for solving multiattribute decision making (MADM) problems under Atanassov’s intuitionistic fuzzy (IF) environments. The LINMAP only can deal with MADM problems in crisp environments. However, fuzziness is inherent in decision data and decision making processes. In this methodology, Atanassov’s IF sets are used to describe fuzziness in decision information and decision making processes by means of an Atanassov’s IF decision matrix. A Euclidean distance is proposed to measure the difference between Atanassov’s IF sets. Consistency and inconsistency indices are defined on the basis of preferences between alternatives given by the decision maker. Each alternative is assessed on the basis of its distance to an Atanassov’s IF positive ideal solution (IFPIS) which is unknown a prior. The Atanassov’s IFPIS and the weights of attributes are then estimated using a new linear programming model based upon the consistency and inconsistency indices defined. Finally, the distance of each alternative to the Atanassov’s IFPIS can be calculated to determine the ranking order of all alternatives. A numerical example is examined to demonstrate the implementation process of this methodology. Also it has been proved that the methodology proposed in this article can deal with MADM problems under not only Atanassov’s IF environments but also both fuzzy and crisp environments.     

Approaches to measuring inconsistency for stratified knowledge bases

A number of proposals have been proposed for measuring inconsistency for knowledge bases. However, it is rarely investigated how to incorporate preference information into inconsistency measures. This paper presents two approaches to measuring inconsistency for stratified knowledge bases. The first approach, termed the multi-section inconsistency measure (MSIM for short), provides a framework for characterizing the inconsistency at each stratum of a stratified knowledge base. Two instances of MSIM are defined: the naive MSIM and the stratum-centric MSIM. The second approach, termed the preference-based approach, aims to articulate the inconsistency in a stratified knowledge base from a global perspective. This approach allows us to define measures by taking into account the number of formulas involved in inconsistencies as well as the preference levels of these formulas. A set of desirable properties are introduced for inconsistency measures of stratified knowledge bases and studied with respect to the inconsistency measures introduced in the paper. Computational complexity results for these measures are presented. In addition, a simple but explanatory example is given to illustrate the application of the proposed approaches to requirements engineering.     

Correcting Jaccard and other similarity indices for chance agreement in cluster analysis

Correcting a similarity index for chance agreement requires computing its expectation under fixed marginal totals of a matching counts matrix. For some indices, such as Jaccard, Rogers and Tanimoto, Sokal and Sneath, and Gower and Legendre the expectations cannot be easily found. We show how such similarity indices can be expressed as functions of other indices and expectations found by approximations such that approximate correction is possible. A second approach is based on Taylor series expansion. A simulation study illustrates the effectiveness of the resulting correction of similarity indices using structured and unstructured data generated from bivariate normal distributions.     

Kronecker indices of Lie algebras and invariants degrees estimate

The concept of Kronecker indices of a Lie algebra as integer characteristics naturally connected to its structure tensor is introduced. A lower bound for the degrees of polynomial invariants of the co-adjoint action in terms of Kronecker indices is proved.     

A Syntax-based approach to measuring the degree of inconsistency for belief bases

Measuring the degree of inconsistency of a belief base is an important issue in many real-world applications. It has been increasingly recognized that deriving syntax sensitive inconsistency measures for a belief base from its minimal inconsistent subsets is a natural way forward. Most of the current proposals along this line do not take the impact of the size of each minimal inconsistent subset into account. However, as illustrated by the well-known Lottery Paradox, as the size of a minimal inconsistent subset increases, the degree of its inconsistency decreases. Another lack in current studies in this area is about the role of free formulas of a belief base in measuring the degree of inconsistency. This has not yet been characterized well. Adding free formulas to a belief base can enlarge the set of consistent subsets of that base. However, consistent subsets of a belief base also have an impact on the syntax sensitive normalized measures of the degree of inconsistency, the reason for this is that each consistent subset can be considered as a distinctive plausible perspective reflected by that belief base, whilst each minimal inconsistent subset projects a distinctive view of the inconsistency. To address these two issues, we propose a normalized framework for measuring the degree of inconsistency of a belief base which unifies the impact of both consistent subsets and minimal inconsistent subsets. We also show that this normalized framework satisfies all the properties deemed necessary by common consent to characterize an intuitively satisfactory measure of the degree of inconsistency for belief bases. Finally, we use a simple but explanatory example in requirements engineering to illustrate the application of the normalized framework.     

Stream-based inconsistency measurement

Inconsistency measures have been proposed to assess the severity of inconsistencies in knowledge bases of classical logic in a quantitative way. In general, computing the value of inconsistency is a computationally hard task as it is based on the satisfiability problem which is itself NP-complete. In this work, we address the problem of measuring inconsistency in knowledge bases that are accessed in a stream of propositional formulæ. That is, the formulæ of a knowledge base cannot be accessed directly but only once through processing of the stream. This work is a first step towards practicable inconsistency measurement for applications such as Linked Open Data, where huge amounts of information is distributed across the web and a direct assessment of the quality or inconsistency of this information is infeasible due to its size. Here we discuss the problem of stream-based inconsistency measurement on classical logic, in order to make use of existing measures for classical logic. However, it turns out that inconsistency measures defined on the notion of minimal inconsistent subsets are usually not apt to be used in the streaming scenario. In order to address this issue, we adapt measures defined on paraconsistent logics and also present a novel inconsistency measure based on the notion of a hitting set. We conduct an extensive empirical analysis on the behavior of these different inconsistency measures in the streaming scenario, in terms of runtime, accuracy, and scalability. We conclude that for two of these measures, the stream-based variant of the new inconsistency measure and the stream-based variant of the contension inconsistency measure, large-scale inconsistency measurement in streaming scenarios is feasible.     

FE‐formulation for plate and beam elements to simulate onset and propagation of discrete cracks in solid structures

The aim of this paper is to present some numerical aspects related to the modeling both the formation and the propagation of discrete cracks in solid structures. The presented formulation corresponds to the concept of embedded discontinuities [1], and will be applied to a plate and to a beam element. The failure of solid structures is often triggered by a highly localized pattern of inelastic deformation in the form of narrow bands. Characteristic examples are shear bands in metals and soils, or localized bands of cracking in brittle materials, like concrete or rocks. A well known difficulty associated with classical (local, rate‐independent) continuum theories with strain softening attributes is that numerical solutions are found to lack invariance with respect to the choice of spatial discretization. For quasi‐static boundary problems, this mathematical inconsistency causes the loss of ellipticity for the governing equations (material instability). To regularize this inconsistency, several strategies have been applied. In the presented formulation, additional degrees of freedom are considered. Within the concept of embedded discontinuities, the regular displacements are enriched by discontinuities. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

关于判断矩阵一致性检验与调整的一个注记

首先分析了判断矩阵不一致形成的原因,认为一个判断矩阵中的不一致是由强矛盾判断,弱矛盾判断,标度离散性,标度有限性共同作用的结果,并通过两个例子指出现有一致性检验与调整方法中存在的问题,最后在已有研究基础上给出了判断矩阵一致性调整的新步骤.     

基于吴方法的多值模型检验

大型复杂系统的开发过程中不可避免的涉及到非确定或不一致信息的处理,而多值模型检验作为经典模型检验的一种扩展,是处理和分析包含此类信息模型的一种有效手段.提出了一种系统化的多值逻辑(涵盖经典逻辑)的代数表示方法,使用吴方法的基本思想和框架实现复杂系统形式验证中基于多值逻辑的模型检验的代数化,建立了通过吴方法实现多值模型检验技术的整体框架.这种代数化的多值模型检验方法可以作为现有方法的有力补充.     

基于基本回路修正的AHP一致性调整方法研究

为解决AHP一致性问题,提出一种基于基本回路修正的调整方法,能够同时解决数值不一致和逻辑不一致问题,同时保证对原始信息的修改量最小。数值不一致和逻辑不一致均由决策者的不准确判断引起,其中数值不一致可以通过降低一致性比率(CR)值进行改善,而逻辑不一致只有将判断矩阵中所有三阶回路去除才能得到解决。因此,通过对n阶判断矩阵进行基本矩阵分解,得到C³_n个3阶的基本矩阵,其中存在三阶回路的称为基本回路,从而将判断矩阵的一致性修正问题转化为基本回路的一致性修正问题。通过对基本回路的一致性比较,提出了两种确定最不一致元素的方法,即CR和最大法和优化法,并设计了优化模型对最不一致元素进行修正。最后,通过算例分析验证了本文方法的可行性,与已有方法的对比结论证明了本文方法更为有效。     

基于深度学习的直觉模糊集隶属度确定方法

直觉模糊集隶属度、非隶属度和犹豫度的确定方法是直觉模糊集理论与应用研究中一个十分重要的问题,其直接影响着相关方法的可扩展性及应用结果。然而,现有方法存在主观性强、标准难以统一等问题,并且大多基于模拟数据进行实验,难以应用至实际数据。针对上述问题以及大规模非结构化数据,提出一种基于深度学习的直觉模糊集隶属度、非隶属度和犹豫度确定方法。新方法克服了传统方法的技术和思维局限,拓展了直觉模糊集相关问题的研究思路,为其实际应用提供了更多可能。     

From inconsistency handling to non-canonical requirements management: A logical perspective

As a class of defects in software requirements specification, inconsistency has been widely studied in both requirements engineering and software engineering. It has been increasingly recognized that maintaining consistency alone often results in some other types of non-canonical requirements, including incompleteness of a requirements specification, vague requirements statements, and redundant requirements statements. It is therefore desirable for inconsistency handling to take into account the related non-canonical requirements in requirements engineering. To address this issue, we propose an intuitive generalization of logical techniques for handling inconsistency to those that are suitable for managing non-canonical requirements, which deals with incompleteness and redundancy, in addition to inconsistency. We first argue that measuring non-canonical requirements plays a crucial role in handling them effectively. We then present a measure-driven logic framework for managing non-canonical requirements. The framework consists of five main parts, identifying non-canonical requirements, measuring them, generating candidate proposals for handling them, choosing commonly acceptable proposals, and revising them according to the chosen proposals. This generalization can be considered as an attempt to handle non-canonical requirements along with logic-based inconsistency handling in requirements engineering.     

On constraint qualifications in nonlinear programming

In this paper we give conditions for deriving the inconsistency of an inequality system of positively homogeneous functions starting from the inconsistency of another one. When the impossibility of the starting system represents a necessary optimality condition for an inequality constrained extremum problem and the positively homogeneous functions involved have suitable properties of convexity, such conditions collapse into the well known constraint qualifications.