Consistency test and weight generation for additive interval fuzzy preference relations

Some simple yet pragmatic methods of consistency test are developed to check whether an interval fuzzy preference relation is consistent. Based on the definition of additive consistent fuzzy preference relations proposed by Tanino (Fuzzy Sets Syst 12:117–131, 1984), a study is carried out to examine the correspondence between the element and weight vector of a fuzzy preference relation. Then, a revised approach is proposed to obtain priority weights from a fuzzy preference relation. A revised definition is put forward for additive consistent interval fuzzy preference relations. Subsequently, linear programming models are established to generate interval priority weights for additive interval fuzzy preference relations. A practical procedure is proposed to solve group decision problems with additive interval fuzzy preference relations. Theoretic analysis and numerical examples demonstrate that the proposed methods are more accurate than those in Xu and Chen (Eur J Oper Res 184:266–280, 2008b)

Goal programming approaches to deriving interval fuzzy preference relations

This article investigates the consistency of interval fuzzy preference relations based on interval arithmetic, and new definitions are introduced for additive consistent, multiplicative consistent and weakly transitive interval fuzzy preference relations. Transformation functions are put forward to convert normalized interval weights into consistent interval fuzzy preference relations. By analyzing the relationship between interval weights and consistent interval fuzzy preference relations, goal-programming-based models are developed for deriving interval weights from interval fuzzy preference relations for both individual and group decision-making situations. The proposed models are illustrated by a numerical example and an international exchange doctoral student selection problem

Rating alternatives from pairwise comparisons by solving tropical optimization problems

We consider problems of rating alternatives based on their pairwise comparison under various assumptions, including constraints on the final scores of alternatives. The problems are formulated in the framework of tropical mathematics to approximate pairwise comparison matrices by reciprocal matrices of unit rank, and written in a common form for both multiplicative and additive comparison scales. To solve the unconstrained and constrained approximation problems, we apply recent results in tropical optimization, which provide new complete direct solutions given in a compact vector form. These solutions extend known results and involve less computational effort. As an illustration, numerical examples of rating alternatives are presented.Comment: 16 pages. arXiv admin note: substantial text overlap with arXiv:1503.0400

Incomplete interval fuzzy preference relations and their applications

This paper investigates incomplete interval fuzzy preference relations. A characterization, which is proposed by Herrera-Viedma et al. (2004), of the additive consistency property of the fuzzy preference relations is extended to a more general case. This property is further generalized to interval fuzzy preference relations (IFPRs) based on additive transitivity. Subsequently, we examine how to characterize IFPR. Using these new characterizations, we propose a method to construct an additive consistent IFPR from a set of n − 1 preference data and an estimation algorithm for acceptable incomplete IFPRs with more known elements. Numerical examples are provided to illustrate the effectiveness and practicality of the solution process

On the normalization of a priority vector associated with a reciprocal relation.

In this paper we show that the widely used normalization constraint SUM(i=1,n) wi = 1 does not apply to the priority vectors associated with reciprocal relations, whenever additive transitivity is involved. We show that misleading applications of this type of normalization may lead to unsatisfactory results and we give some examples from the literature. Then, we propose an alternative normalization procedure which is compatible with additive transitivity and leads to better results.reciprocal relation; fuzzy preference relation; priority vector; normalization

Approaches to improving consistency of interval fuzzy preference relations

This article introduces a consistency index for measuring the consistency level of an interval fuzzy preference relation (IFPR). An approach is then proposed to construct an additive consistent IFPR from a given inconsistent IFPR. By using a weighted averaging method combining the original IFPR and the constructed consistent IFPR, a formula is put forward to repair an inconsistent IFPR to generate an IFPR with acceptable consistency. An iterative algorithm is subsequently developed to rectify an inconsistent IFPR and derive one with acceptable consistency and weak transitivity. The proposed approaches can not only improve consistency of IFPRs but also preserve the initial interval uncertainty information as much as possible. Numerical examples are presented to illustrate how to apply the proposed approaches

An overview on managing additive consistency of reciprocal preference relations for consistency-driven decision making and Fusion: Taxonomy and future directions

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The reciprocal preference relation (RPR) is a powerful tool to represent decision makers’ preferences in decision making problems. In recent years, various types of RPRs have been reported and investigated, some of them being the ‘classical’ RPRs, interval-valued RPRs and hesitant RPRs. Additive consistency is one of the most commonly used property to measure the consistency of RPRs, with many methods developed to manage additive consistency of RPRs. To provide a clear perspective on additive consistency issues of RPRs, this paper reviews the consistency measurements of the different types of RPRs. Then, consistency-driven decision making and information fusion methods are also reviewed and classified into four main types: consistency improving methods; consistency-based methods to manage incomplete RPRs; consistency control in consensus decision making methods; and consistency-driven linguistic decision making methods. Finally, with respect to insights gained from prior researches, further directions for the research are proposed

Nanostructured Layered Copper Hydroxy Dodecyl Sulfate: A Potential Fire Retardant for Poly(vinyl Ester) (PVE)

Composites of poly(vinyl ester) (PVE) with copper hydroxy dodecyl sulfate (CHDS) were prepared by thermal curing. The efficiency of the additive, CHDS, in reducing flammability is demonstrated via cone calorimetry and thermogravimetric analysis (TGA). The addition of 1-10% by mass of the CHDS additive resulted in significant increments in char formation (~4-11%) from thermogravimetric analysis (TGA). Incorporation of the CHDS into the polymer matrix at these low concentrations leads to substantial reductions in the total heat release (~20-30%) but no significant change in the peak heat release rate. The composite materials generally ignite more quickly, however, the flame extinguishes faster for the composites relative to the virgin polymer. X-ray diffraction (XRD) and infrared spectroscopic analyses of the residues collected at various stages during thermal decomposition of the composities, suggest the participation of copper-containing species in promoting enhanced thermal stability of PVE

Toward a systems understanding of plant–microbe interactions

Plants are closely associated with microorganisms including pathogens and mutualists that influence plant fitness. Molecular genetic approaches have uncovered a number of signaling components from both plants and microbes and their mode of actions. However, signaling pathways are highly interconnected and influenced by diverse sets of environmental factors. Therefore, it is important to have systems views in order to understand the true nature of plant–microbe interactions. Indeed, systems biology approaches have revealed previously overlooked or misinterpreted properties of the plant immune signaling network. Experimental reconstruction of biological networks using exhaustive combinatorial perturbations is particularly powerful to elucidate network structure and properties and relationships among network components. Recent advances in metagenomics of microbial communities associated with plants further point to the importance of systems approaches and open a research area of microbial community reconstruction. In this review, we highlight the importance of a systems understanding of plant–microbe interactions, with a special emphasis on reconstruction strategies