Crisp Weights Research Articles

Approaches to group decision making with linguistic preference relations based on multiplicative consistency

Order consistency and multiplicative consistency for LPRs are introduced.A consistency index is defined.Two linear optimization models are established to generate the weight vector.Two GDM methods are investigated, and they are proved to be convergent.Several examples are provided to illustrate the behavior of the proposed methods. A key step in group decision making (GDM) with linguistic preference relations (LPRs) is to derive the priority weight vector of the alternatives. However, the lack of consistency in GDM can lead to inconsistent conclusions. In this paper, two new GDM methods are developed to improve the multiplicative consistency of LPRs until they are acceptable, and the priority weight vector of the alternatives is derived from adjusted LPRs. First, the new concepts of order consistency and multiplicative consistency for LPRs are introduced. Then, a consistency index is defined to measure whether a LPR is of acceptable multiplicative consistency. Two linear optimization models are established to generate the normalized crisp weight vector for both individual and group LPRs with the principle of minimizing the deviation values. In addition, two GDM methods are investigated to improve LPRs with unacceptable multiplicative consistency until the adjusted LPRs are acceptable multiplicative consistent, and they can help the decision makers (DMs) to obtain the reasonable and reliable decision making results. Finally, several numerical examples are provided, and comparative analyses with existing approaches are performed to demonstrate that the proposed methods are both valid and practical to deal with GDM problems.

The utility based non-linear fuzzy AHP optimization model for network selection in heterogeneous wireless networks

Next generation wireless networks will integrate various heterogeneous technologies like WLAN, WiMax and cellular technologies etc., to support multimedia services with higher bandwidth and guaranteed quality of service (QoS). In order to keep the mobile user always connected to the best wireless network in terms of QoS parameters and user preferences, an optimal network selection technique in heterogeneous networks is required. This paper proposes a novel fuzzy-Analytic Hierarchy Process (AHP) based network selection in heterogeneous wireless networks. Triangular fuzzy numbers are used to represent the elements in the comparison matrices for voice, video and best effort applications. Deriving crisp weights from these fuzzy comparison matrices is a challenging task. When extent analysis method is applied, irrational zero weights are obtained for some attributes. Due to this, many important criteria are not considered in the decision making process. To overcome this problem, a new non-linear fuzzy optimization model for deriving crisp weights from fuzzy comparison matrices for network selection is presented. The weights obtained from this model are more consistent than the existing optimization models. Also, parameterized utility functions are used to model the different Quality of Service (QoS) attributes (bandwidth, delay, jitter, bit error rate) and user preferences (cost) for three different types of applications. Finally, scores are calculated exclusively for each network by three MADM (Multiple Attribute Decision Making) methods Simple Additive Weighting (SAW), TOPSIS (The Technique for Order of Preference by Similarity to Ideal Solution) and MEW (Multiplicative Exponential Weighting). Results show that the MEW method gives more appropriate scores with utility functions than the SAW and TOPSIS methods.

Deriving crisp and consistent priorities for fuzzy AHP-based multicriteria systems using non-linear constrained optimization

Fuzzy optimization models are used to derive crisp weights (priority vectors) for the fuzzy analytic hierarchy process (AHP) based multicriteria decision making systems. These optimization models deal with the imprecise judgements of decision makers by formulating the optimization problem as the system of constrained non linear equations. Firstly, a Genetic Algorithm based heuristic solution for this optimization problem is implemented in this paper. It has been found that the crisp weights derived from this solution for fuzzy-AHP system, sometimes lead to less consistent or inconsistent solutions. To deal with this problem, we have proposed a consistency based constraint for the optimization models. A decision maker can set the consistency threshold value and if the solution exists for that threshold value then crisp weights can be derived, otherwise it can be concluded that the fuzzy comparison matrix for AHP is not consistent for the given threshold. Three examples are considered to demonstrate the effectiveness of the proposed method. Results with the proposed constraint based fuzzy optimization model are more consistent than the existing optimization models.

A constrained non-linear optimization model for fuzzy pairwise comparison matrices using teaching learning based optimization

Non-linear optimization models have been recently proposed to derive crisp weights from fuzzy pairwise comparison matrices. In this paper, a TLBO (Teaching Learning Based Optimization) based solution is presented for solving an optimization model as a system of non-linear equations to derive crisp weights from fuzzy pairwise comparison matrices in AHP (Analytic Hierarchy Process). This fuzzy-AHP method is named as TLBO-1. It has been found that TLBO-1 can lead to inconsistent or less consistent weights. To solve the problem of inconsistent weights, a new constrained non-linear optimization model is proposed in this paper. This model is based on the min-max approach for fuzzy pairwise comparison ratios of weights. TLBO is again used to solve this optimization model, and crisp weights are derived. This fuzzy AHP method is named as TLBO-2. The effectiveness of the proposed model is illustrated by three examples. For each example, the consistency of the derived crisp weights is compared with other optimization models. The results show that the TLBO-2 method can derive more consistent weights for the fuzzy AHP based Multi-Criteria Decision Making (MCDM) systems as compared to the other optimization models.

A Fuzzy Analytic Hierarchy Process (FAHP) Approach for Optimal Selection of Low-carbon Energy Technologies

In process systems engineering, one may encounter design problems that require choosing one among the predefined set of alternatives in a complex decision making environment. In most cases, the selection process involves multiple conflicting criteria that can be either qualitative or quantitative in nature. The use of multiple attribute decision making (MADM) technique thus has been proven effective in providing a transparent, systematic and rational approach to problem solving. In this study, a variant of the widely used MADM technique known as the Analytic Hierarchy Process (AHP) has been developed and applied for the selection of low-carbon technologies. This approach enables the selection of optimal alternative by incorporating not only the judgment of domain experts but also their degree of confidence through the use of fuzzy numbers for the pairwise comparison ratios in the AHP decision framework. The proposed fuzzy AHP then determines a set of crisp weights that maximizes the degree of consistency of all judgments taken together. The said approach can also derive crisp weights from an incomplete fuzzy pairwise comparative judgment matrix (PCJM). To illustrate the technique, a case study involving comparison of electricity storage technologies for renewable energy systems is considered. This selection problem is important due to the intermittent nature of many forms of renewable energy, which thus require cost-effective auxiliary energy storage subsystems for successful deployment.

The incorporation of the Taguchi and the VIKOR methods to optimize multi-response problems in intuitionistic fuzzy environments

The Taguchi method is a powerful method of solving quality problems in various fields of engineering. However, this method was developed to optimize single-response processes. In many multi-response optimization problems, the important response is determined subjectively, based on knowledge or experience. However, using only exact numbers to represent this importance is problematic, because there is uncertainty and vagueness. The concept of intuitionistic fuzzy sets (IFSs) is a powerful method for characterization, using a membership function and a non-membership function. This paper proposes an efficient VIKOR method that optimizes multi-response problems in intuitionistic fuzzy environments. The importance weights of various responses are evaluated in terms of IFSs. In the proposed method, the similarity measure between IFSs is used to determine the crisp weights of the responses. This scheme eliminates the need for complicated intuitionistic fuzzy arithmetic operations and increases efficiency in solving multi-response optimization problems in intuitionistic fuzzy environments. Two case studies: plasma-enhanced chemical vapor deposition and a double-sided surface mount technology electronic assembly operation are used to demonstrate the effectiveness of the proposed method.

A fuzzy goal programming based procedure for Machine Tool Selection

The evaluation and selection of the appropriate machine tools is one of the most critical decisions in the design and development of an efficient production environment and the use of fuzzy set theory allows incorporating qualitative and partially known information into the decision model. This paper describes a fuzzy goal programming method for evaluation and selection of machine tool alternatives for a manufacturing company in Iran. Our approach applied triangular numbers into traditional goal programming method, and we investigated deriving fuzzy weights of criteria from the pair-wise comparison matrix with fuzzy elements. To our knowledge, no previous work investigated such problem with fuzzy goal programming. In almost fuzzy decision making problems other fuzzy methods like fuzzy AHP, fuzzy TOPSIS and etc. are used. And most of them obtained the crisp weights from fuzzy pair-wise decision matrix. Moreover, in this method we obtain the fuzzy weights by solving a simple linear programming. As a result of the study, we find that the proposed method is practical for ranking machine tool alternatives with respect to multiple conflicting criteria.

A novel meta-heuristic based method for deriving priorities from fuzzy pairwise comparison judgments

This paper proposes a new method to derive the priority vector from fuzzy pairwise comparison matrices. Unlike several known methods, the proposed method derives crisp weights from consistent and inconsistent fuzzy comparison matrices. Therefore, the crisp weights obviate the need of additional aggregation and ranking procedures. To derive the priority vector, a Modified Fuzzy Logarithmic Least Square Model (MFLLSM) is proposed. In order to solve the MFLLSM, a framework based on genetic algorithm is proposed. In the proposed framework, a heuristic algorithm of population initialization, a heuristic algorithm for simulating fuzzy numbers and a heuristic algorithm of fitness evaluation are proposed.The solution of the prioritization problem requires finding priorities such that their ratio approximately satisfies the initial judgments. Computational results reveal the superiority of the proposed method in comparison with five well known methods of literature from the viewpoint of satisfaction of initial judgments by the obtained priority vector. It is shown by ten different examples that the deviation of the priorities ratio from initial judgments in the proposed method is less than five existing methods of literature. In addition, unlike several methods of literature, the proposed method considers fuzzy judgments represented by both triangular and trapezoidal fuzzy numbers. Furthermore, the proposed method for the first time considers judgments represented by triangular shaped fuzzy numbers and trapezoidal shaped fuzzy numbers which are discussed in the paper.

A Fuzzy SQFE approach in supplier’s performance monitoring

SQFE (Suivi Qualite Par le Fournisseur Exterieur) approach will be used to measure and improve the quality of supplier products. It has been introduced by three automotive corporations: Peugeot, Citroen and Renault. As a useful procedure not only in monitoring the supplier performance, but also in verifying the quality of any kind of products, SQFE has not been considered in researches appropriately. In this method, the level of products’ quality is determined based on the several categories of demerits which are identified in advance. There is a drawback in the classical SQFE. It assumes that the different degrees of severity related to the same demerit category must be presented as a crisp weight and equally treated. But, operators, gauges and environmental conditions as the parts of the measurement system, lead to the inevitable uncertainty. Therefore, in this research, the newly developed model, based on fuzzy concepts provides an estimate which accurately reflects the reality than those obtained using the procedures currently presented in standard. According to this scheme, due to taking the severity of different types of demerits into consideration, linguistic weights are applied to solve the weight assignment problem of classical SQFE, on the other hand sample observations are converted to triangular fuzzy numbers. After introducing the classical SQFE, developed fuzzy SQFE procedure is described in three steps. Moreover, a real empirical study is demonstrated for evaluation of effectiveness of the proposed model in practice.

Estrategias para el entrenamiento de redes neuronales de números difusos

El propósito de este artículo es presentar estrategias generales de entrenamiento para redes neuronales de números difusos utilizadas en el aprendizaje de sistemas a partir de información lingüística. Se exponen brevemente las principales tendencias en el entrenamiento de este tipo de sistemas y con base en ellas se proponen nuevas estrategias. La primera de ellas se basa en la retropropagación del error cuadrático medio en todos los a-cortes para pesos crisp. La segunda hace uso de un algoritmo genético con codificación real para redes con pesos crisp. La tercera consiste en la retropropagación del error en el valor promedio y la ambigüedad en todos los a-cortes para pesos difusos. Por último, se presenta una basada en la retropropagación de una medida difusa del error para redes con pesos difusos. Se realiza una etapa experimental en la que se implementan los algoritmos desarrollados junto con algunos de los más representativos reportados en el estado del arte, permitiendo identificar para qué conjuntos de datos particulares resulta útil cada una de las estrategias. Finalmente, se aplican dichas estrategias para la implementación de un sistema de evaluación de impacto ambiental en vertederos.

Automatic procedures for group decision making with intuitionistic fuzzy preference relations

The intuitionistic fuzzy preference relation (IFPR), whose elements are intuitionistic fuzzy values, is a powerful structure in expressing the comprehensive preference information of a person. In this paper, after reviewing different kinds of consistency for IFPR, we construct a mathematical model to derive the underlying crisp weighting vector from a multiplicative consistent IFPR. Such method is very easy to be understood. As to multiplicative inconsistent IFPR, we claim that there are two ways to repair it, which are interactive method and automatic method. When the expert does not want to interact with the analyst, or if the analyst cannot find the initial expert to re-evaluate and alter his/her preferences, or if consistency must be urgently obtained, the proposed automatic algorithm is efficient and useful. Finally, we propose a procedure for group decision making with IFPRs based on the multiplicative consistency of IFPRs. It is proven that if all individual IFPRs are multiplicative consistent, their overall IFPR fused by IFWG operator is also multiplicative consistent. Some numerical examples are given to illustrate the validity and applicability of the proposed methods.

Interval weight generation approaches for reciprocal relations

In this paper, we propose methods to derive interval weight vectors from reciprocal relations for reflecting the inconsistency when decision makers provide preferences over alternatives (or criteria). Several goal programming models are established to minimize the inconsistency based on multiplicative and additive consistency, respectively. Especially, if we obtain a crisp weight vector from a reciprocal relation, then it is consistent. Then, we extend the proposed methods to incomplete reciprocal relations and interval reciprocal relations and develop the corresponding models to derive interval weight vectors. Several examples are also given to compare the developed methods with the existing ones.

A bi‐level neural‐based fuzzy classification approach for credit scoring problems

The credit scoring is a risk evaluation task considered as a critical decision for financial institutions in order to avoid wrong decision that may result in huge amount of losses. Classification models are one of the most widely used groups of data mining approaches that greatly help decision makers and managers to reduce their credit risk of granting credits to customers instead of intuitive experience or portfolio management. Accuracy is one of the most important criteria in order to choose a credit‐scoring model; and hence, the researches directed at improving upon the effectiveness of credit scoring models have never been stopped. In this article, a hybrid binary classification model, namely FMLP, is proposed for credit scoring, based on the basic concepts of fuzzy logic and artificial neural networks (ANNs). In the proposed model, instead of crisp weights and biases, used in traditional multilayer perceptrons (MLPs), fuzzy numbers are used in order to better model of the uncertainties and complexities in financial data sets. Empirical results of three well‐known benchmark credit data sets indicate that hybrid proposed model outperforms its component and also other those classification models such as support vector machines (SVMs), K‐nearest neighbor (KNN), quadratic discriminant analysis (QDA), and linear discriminant analysis (LDA). Therefore, it can be concluded that the proposed model can be an appropriate alternative tool for financial binary classification problems, especially in high uncertainty conditions. © 2013 Wiley Periodicals, Inc. Complexity 18: 46–57, 2013

An Algorithm for Extracting Intuitionistic Fuzzy Shortest Path in a Graph

We consider an intuitionistic fuzzy shortest path problem (IFSPP) in a directed graph where the weights of the links are intuitionistic fuzzy numbers. We develop a method to search for an intuitionistic fuzzy shortest path from a source node to a destination node. We coin the concept of classical Dijkstra's algorithm which is applicable to graphs with crisp weights and then extend this concept to graphs where the weights of the arcs are intuitionistic fuzzy numbers. It is claimed that the method may play a major role in many application areas of computer science, communication network, transportation systems, and so forth. in particular to those networks for which the link weights (costs) are ill defined.

A Fuzzy Group Prioritization Method for Deriving Weights and its Software Implementation

Several Multi-Criteria Decision Making (MCDM) methods involve pairwise comparisons to obtain the preferences of decision makers (DMs). This paper proposes a fuzzy group prioritization method for deriving group priorities/weights from fuzzy pairwise comparison matrices. The proposed method extends the Fuzzy Preferences Programming Method (FPP) by considering the different importance weights of multiple DMs. The elements of the group pairwise comparison matrices are presented as fuzzy numbers rather than exact numerical values, in order to model the uncertainty and imprecision in the DMs' judgments. Unlike the known fuzzy prioritization techniques, the proposed method is able to derive crisp weights from incomplete and fuzzy set of comparison judgments and does not require additional aggregation procedures. A prototype of a decision tool is developed to assist DMs to implement the proposed method for solving fuzzy group prioritization problems in MATLAB. Detailed numerical examples are used to illustrate the proposed approach.

Entropy based fuzzy classification of images on quality assessment

Referenced image quality assessment methods require huge memory and time involvement, therefore not suitable to use in real time environment. On the other hand development of an automated system to assessing quality of images without reference to the original image is difficult due to uncertainty in relations between features and quality of images. The paper aims at developing a fuzzy based no-reference image quality assessment system by utilizing human perception and entropy of images. The proposed approach selects important features to reduce complexity of the system and based on entropy of feature vector the images are partitioned into different clusters. To assign soft class labels to different images, continuous weights are estimated using entropy of mean opinion score (MOS) unlike the previous works where crisp weights were used. Finally, fuzzy relational classifier (FRC) has been built using MOS based weight matrix and fuzzy partition matrix to establish correlation between features and class labels. Quality of the distorted/decompressed test images are predicted using the proposed fuzzy system, showing satisfactory results with the existing no-reference techniques.

Fuzzy analytical hierarchy process using preference ratio: A case study for selecting management short course in a business school

: Received 1 January 2010 Received in revised form 10 May 2010 Accepted 18 May 2010 Available online 18 May 2010 Analytic Hierarchy Process (AHP) is one of the most popular approaches in the area of multiple attribute decision making (MADM). However, it is not practical any more if input information are fuzzy. In this paper, we propose a new method for fuzzy AHP which is especially useful to make decisions for multiple attribute problems. The method is developed by applying preference ratio concept which makes it practical since it assigns crisp weights and crisp scores to different alternatives. Two algorithms are proposed in this paper: The first one defines crisp and normalized weight by pairwise comparison with fuzzy data while the second one calculates fuzzy consistency ratio. The proposed method is applied to prioritize different short courses in a management school.

An information mining method for deriving weights from an interval comparison matrix

This paper provides a simple and practical approach for obtaining the weight information from an interval comparison matrix. First, a series of weight vectors are acquired by three linear programming models. Next, the crisp comparison matrix is gained by the convex combination of the vertices. The interval comparison matrix is approximately replaced by the crisp comparison matrix. Then, a method of the deviation degree is used to get the crisp weight vector. Finally, a numerical example and the comparison shows the validity and practicality of our developed method. More potential weight information is mined.

An eigenvector method for generating normalized interval and fuzzy weights

Crisp comparison matrices produce crisp weight estimates. It is logical for an interval or fuzzy comparison matrix to give an interval or fuzzy weight estimate. In this paper, an eigenvector method (EM) is proposed to generate interval or fuzzy weight estimate from an interval or fuzzy comparison matrix, which differs from Csutora and Buckley’s Lambda-Max method in several aspects. First, the proposed EM produces a normalized interval or fuzzy eigenvector weight estimate through the solution of a linear programming model, while the Lambda-Max method gives a series of nonnormalized interval eigenvector weight estimate through an iterative solution process. Next, the EM directly solves the principal right eigenvector of an interval or fuzzy comparison matrix, while the Lambda-Max method needs transforming a fuzzy comparison matrix into a series of interval comparison matrices using α- level sets and the extension principle and therefore needs the solution of a series of eigenvalue problems. Finally, the Lambda-Max method needs the help of the principal right eigenvector of a crisp comparison matrix to determine the final interval weights, while the EM has no such requirement. It is also found that not all interval or fuzzy comparison matrices can generate normalized interval or fuzzy eigenvector weights. Situations where the EM is inapplicable are analyzed and the aggregation of local interval or fuzzy weights into global interval or fuzzy weights is also discussed. Three numerical examples including a hierarchical structure decision-making problem are examined using the EM to demonstrate its applications.

Numerical solution of fuzzy polynomials by fuzzy neural network

In this paper, an architecture of fuzzy neural networks is proposed to find a real root of fuzzy polynomials (if exists) by introducing a learning algorithm. We proposed a learning algorithm from the cost function for adjusting of crisp weights. Finally, we illustrated our approach by numerical examples.