Multiple criteria group decision-making with generalized interval-valued fuzzy numbers based on signed distances and incomplete weights

Decision-making information provided by decision makers is often imprecise or uncertain, due to lack of data, time pressure, or the decision makers’ limited attention and information-processing capabilities. Interval-valued fuzzy sets are associated with greater imprecision and more ambiguity than are ordinary fuzzy sets. For these reasons, this paper presents a signed distance-based method for handling fuzzy multiple-criteria group decision-making problems in which individual assessments are provided as generalized interval-valued trapezoidal fuzzy numbers, and the information about criterion weights are not precisely but partially known. First, concerning the relative importance of decision makers and the group consensus of fuzzy opinions, all individual decision opinions were aggregated into group opinions using a hybrid average with weighted averaging and signed distance-based ordered weighted averaging operations. Next, considering a decision situation with incomplete weight information of criteria, an integrated programming model was developed to estimate criterion weights and to order the priorities of various alternatives based on signed distances. In addition, several deviation variables were introduced to mitigate the effect of inconsistent evaluations on the importance of criteria. Finally, the feasibility of the proposed method is illustrated by a numerical example of a multi-criteria supplier selection problem. Furthermore, a comparative analysis with other methods was conducted to validate the effectiveness and applicability of the proposed methodology.     

Statistically based multiwavelet denoising

In this work, we consider a statistically based multiwavelet thresholding method which acts on the empirical wavelet coefficients in groups, rather than individually, in order to obtain an edge-preserving image denoising technique. Our strategy allows us to exploit the dependencies between neighboring coefficients to make a simultaneous thresholding decision, so that estimation accuracy is increased.

By interpreting the multiwavelet analysis in a statistical context, we propose a new weighted multiwavelet matrix thresholding rule, based on the statistical modeling of empirical coefficients. This allows the thresholding decision to be adapted to the local structure of the underlying image, hence producing edge-preserving denoising. Extensive numerical results are presented showing the performance of our denoising procedure.     

Group decision making with simplified neutrosophic ordered weighted distance operator

Simplified neutrosophic set is a convenient tool proposed for dealing with complex problems; it is effective in providing more data for decision‐making process. In this study, we develop a simplified neutrosophic ordered weighted distance operator which combines the neutrosophic distance measures and the ordered weighted average distance in the same formulation. It is a new handy aggregation operator that considers the situations where the input data are represented in simplified neutrosophic numbers, and it also contains diverse distance aggregation operators. Parameterized families of simplified neutrosophic ordered weighted distance operator are handled. Moreover, we establish a new neutrosophic group decision‐making method based on the simplified neutrosophic ordered weighted distance operator, which has 2 extended approaches for determining the weights of decision makers and decision attributes in decision‐making process, respectively. Finally, an illustrative example demonstrates the application of the proposed method. The effectiveness and advantages of the proposed method are shown by the comparative analysis with existing relative methods.     

Probability distribution based weights for weighted arithmetic aggregation operators

One key point in the multiple attribute decision making is to determine the associated weights. In this paper, we first briefly review some main methods for determining the weights by using distribution functions. Then, motivated by the idea of data distribution, we develop some novel methods for obtaining the weights associated with the weighted arithmetic aggregation operators. The methods can relieve the influence of biased data on the decision results by weighting these data with small values based on the corresponding probability of data. Furthermore, some commonly used probability distribution methods are used to solve the problems in different conditions. Finally, four practical examples are provided to illustrate the weighting method.     

直觉模糊多属性的意见集中排序法

研究了属性权重信息不完全确定,属性值为直觉模糊集的多属性决策问题。首先根据直觉模糊数的得分函数和精确函数对决策矩阵中的评价值比较大小,进而按属性集中的每个属性对方案排成线性序;然后通过计算赋权模糊优先矩阵确定方案的优属度,建立规划模型确定属性的权重;再利用加权算术算子对方案集结,得到专家对方案的排序,从而得到一种新的意见集中排序的决策方法。数值实例说明该方法的有效性和实用性,可为解决直觉模糊多属性决策提供新方法     

An improved method for ranking alternatives in multiple criteria decision analysis

Scoring rules are an important disputable subject in data envelopment analysis (DEA). Various organizations use voting systems whose main object is to rank alternatives. In these methods, the ranks of alternatives are obtained by their associated weights. The method for determining the ranks of alternatives by their weights is an important issue. This problem has been the subject at hand of some authors. We suggest a three-stage method for the ranking of alternatives. In the first stage, the rank position of each alternative is computed based on the best and worst weights in the optimistic and pessimistic cases, respectively. The vector of weights obtained in the first stage is not a singleton. Hence, to deal with this problem, a secondary goal is used in the second stage. In the third stage of our method, the ranks of the alternatives approach the optimistic or pessimistic case. It is mentionable that the model proposed in the third stage is a multi-criteria decision making (MCDM) model and there are several methods for solving it; we use the weighted sum method in this paper. The model is solved by mixed integer programming. Also, we obtain an interval for the rank of each alternative. We present two models on the basis of the average of ranks in the optimistic and pessimistic cases. The aim of these models is to compute the rank by common weights.     

Weighting method for bi-level linear fractional programming problems

In this paper we consider the solution of a bi-level linear fractional programming problem (BLLFPP) by weighting method. A non-dominated solution set is obtained by this method. In this article decision makers (DMs) provide their preference bounds to the decision variables that is the upper and lower bounds to the decision variables they control. We convert the hierarchical system into scalar optimization problem (SOP) by finding proper weights using the analytic hierarchy process (AHP) so that objective functions of both levels can be combined into one objective function. Here the relative weights represent the relative importance of the objective functions.     

Weighted Euclidean centers

This article addresses the derivation of a Euclidean center, which is defined as the point in decision space from which one can inscribe the largest sphere contained by the constraints. We extend this concept by introducing its weighted version, which we term the weighted Euclidean center. We show that by assigning weights to the different decision variables we can traverse the entire decision space. In addition, we show that the concept of a weighted Euclidean center and that of the achievement scalarizing function introduced by Wierzbicki are intimately related.     

A new proposed model of restricted data envelopment analysis by correlation coefficients

The concept of efficiency in data envelopment analysis (DEA) is defined as weighted sum of outputs/weighted sum of inputs. In order to calculate the maximum efficiency score, each decision making unit (DMU)’s inputs and outputs are assigned to different weights. Hence, the classical DEA allows the weight flexibility. Therefore, even if they are important, the inputs or outputs of some DMUs can be assigned zero (0) weights. Thus, these inputs or outputs are neglected in the evaluation. Also, some DMUs may be defined as efficient even if they are inefficient. This situation leads to unrealistic results. Also to eliminate the problem of weight flexibility, weight restrictions are made in DEA. In our study, we proposed a new model which has not been published in the literature. We describe it as the restricted data envelopment analysis ((ARIII(COR))) model with correlation coefficients. The aim for developing this new model, is to take into account the relations between variables using correlation coefficients. Also, these relations were added as constraints to the CCR and BCC models. For this purpose, the correlation coefficients were used in the restrictions of input–output each one alone and their combination together. Inputs and outputs are related to the degree of correlation between each other in the production. Previous studies did not take into account the relationship between inputs/outputs variables. So, only with expert opinions or an objective method, weight restrictions have been made. In our study, the weights for input and output variables were determined, according to the correlations between input and output variables. The proposed new method is different from other methods in the literature, because the efficiency scores were calculated at the level of correlations between the input and/or output variables.     

运筹学中若干线性目标规划和线性规划的人工智能-代数解法

运筹学中的线性目标规划和线性规划问题一直分别采用修正单纯形法和单纯形法求解.当变量稍多时计算还是有些繁琐、费时,最近作者通过研究发现,可应用人工智能-代数方法求得这两类问题的解,而且具有相当广泛的适用性.若干例题说明,本法的结果和传统方法的结果由于传统算法在计算中发生的错误,除少数例外大都是一致的.本文的一个 重要目的是希望和广大读者一起研究该方法是否具有通用性.     

基于新型区间直觉三角模糊熵的多属性决策方法

在进行区间直觉模糊多属性决策时,有时属性权重是未知的,针对这一问题,提出一种新型区间直觉三角模糊熵的决策方法.首先,给出该新型区间直觉三角模糊熵定义和相关定理,应用该区间直觉三角模糊熵确定属性的权重.然后,基于逼近理想解排序法(TOPSIS)的思想,采用改进的加权欧几里得距离,进行区间直觉模糊群决策,并给出决策步骤.最后,将该方法应用在供应链选择的群决策问题中,通过算例实验验证了该方法的有效性与可行性.     

考虑决策者风险偏好的区间直觉模糊多属性群决策方法

提出了一种考虑决策者风险偏好且属性权重信息不完全的区间直觉模糊数多属性群决策方法。同时考虑相似度和接近度,确定每一属性的决策者权重。为了考虑决策者风险偏好对决策结果的影响和避免区间直觉模糊矩阵的渐进性,引入了决策者风险偏好系数,将集结后的综合决策矩阵转换成区间数矩阵。然后,为了客观地求出属性权重信息不完全环境下属性的权重,构建了基于区间直觉模糊交叉熵的属性权重目标规划模型,该模型不仅考虑了评价值的偏差,也强调了评价值自身的可信度。最后,通过研发项目选择问题的实例分析说明了所提方法的合理性和优越性。     

基于模糊熵的直觉模糊多属性群决策方法

针对专家权重未知、专家判断信息以直觉模糊集给出的多属性群决策问题,提出了一种新的决策方法.通过定义直觉模糊集的模糊熵计算专家判断信息的模糊程度,进而确定每位专家的权重.然后定义直觉模糊集的模糊交叉熵确定备选方案距理想方案和负理想方案的距离,再根据加权算术算子集结专家的判断信息,得到方案的排序.最后,通过一个实例分析验证了方法的有效性.     

Compatible weighting method with rank order centroid: Maximum entropy ordered weighted averaging approach

In a situation where imprecise attribute weights such as a rank order are captured, various approximate weighting methods have been proposed to aid multiattribute decision analysis. Among others, it is well known that the rank order centroid (ROC) weights result in the highest performance in terms of the identification of the best alternative under the ranked attribute weights. In this paper, we aim to reinterpret the meaning of the ROC weights and to develop a compatible weighting method that is based on other well-established academic disciplines. The ordered weighted averaging (OWA) method is a nonlinear aggregation method in that the weights are associated with the objects reordered according to their magnitudes in the aggregation process. Some interesting semantics can be attached to the approximate weights in view of the measure developed in the OWA method. Furthermore, the weights generated by the maximum entropy method show equally compatible performance with the ROC weights under some condition, which is demonstrated by theoretical and simulation analysis.     

Determining attribute weights to improve solution reliability and its application to selecting leading industries

In multiple attribute decision analysis, many methods have been proposed to determine attribute weights. However, solution reliability is rarely considered in those methods. This paper develops an objective method in the context of the evidential reasoning approach to determine attribute weights which achieve high solution reliability. Firstly, the minimal satisfaction indicator of each alternative on each attribute is constructed using the performance data of each alternative. Secondly, the concept of superior intensity of an alternative is introduced and constructed using the minimal satisfaction of each alternative. Thirdly, the concept of solution reliability on each attribute is defined as the ordered weighted averaging (OWA) of the superior intensity of each alternative. Fourthly, to calculate the solution reliability on each attribute, the methods for determining the weights of the OWA operator are developed based on the minimax disparity method. Then, each attribute weight is calculated by letting it be proportional to the solution reliability on that attribute. A problem of selecting leading industries is investigated to demonstrate the applicability and validity of the proposed method. Finally, the proposed method is compared with other four methods using the problem, which demonstrates the high solution reliability of the proposed method.     

A complex multi-attribute large-group PLS decision-making method in the interval-valued intuitionistic fuzzy environment

In the complex multi-attribute large-group decision-making (CMALGDM) problems in interval-valued intuitionistic fuzzy (IVIF) environment, attributes of the alternatives are often stratified and correlated. This paper proposes a decision-making method for these problems based on partial least squares (PLS) path modelling, which not only fully exploits the decision information of decision makers (DMs), but also effectively addresses the relativity problem in the decision attributes and objectively assigned weights to the primary decision attributes (i.e., “latent variables for decision making”). The method can be outlined in three steps. First, a two-stage method is proposed to transform the interval-valued intuitionistic fuzzy number (IVIFN) samples into single-valued samples. In this step, an improved C-OWA operator is first given to transform the IVIFN samples into intuitionistic fuzzy number (IFN) samples, which makes the preference information of the DMs more objectively aggregated. Then a proposed membership-based method is applied to reduce the information loss and transform the IFN samples into single-valued samples. Second, the estimated values and weights of the “latent variables for decision-making” are obtained by means of the PLS path modelling algorithm. Finally, a multi-alternative sorting method is devised in accordance with the estimated values and weights. An example is provided to illustrate the proposed technique and evaluate its feasibility and validity.     

A QFD-based fuzzy MCDM approach for supplier selection

Supplier selection is a highly important multi-criteria group decision making problem, which requires a trade-off between multiple criteria exhibiting vagueness and imprecision with the involvement of a group of experts. In this paper, a fuzzy multi-criteria group decision making approach that makes use of the quality function deployment (QFD) concept is developed for supplier selection process. The proposed methodology initially identifies the features that the purchased product should possess in order to satisfy the company’s needs, and then it seeks to establish the relevant supplier assessment criteria. Moreover, the proposed algorithm enables to consider the impacts of inner dependence among supplier assessment criteria. The upper and the lower bounds of the weights of supplier assessment criteria and ratings of suppliers are computed by using the fuzzy weighted average (FWA) method. The FWA method allows for the fusion of imprecise and subjective information expressed as linguistic variables or fuzzy numbers. The method produces less imprecise and more realistic overall desirability levels, and thus it rectifies the problem of loss of information. A fuzzy number ranking method that is based on area measurement is used to obtain the final ranking of suppliers. The computational procedure of the proposed framework is illustrated through a supplier selection problem reported in an earlier study.     

Liberating the weights

A partial answer to why quasi-Monte Carlo (QMC) algorithms work well for multivariate integration was given in Sloan and Woźniakowski (J. Complexity 14 (1998) 1–33) by introducing weighted spaces. In these spaces the importance of successive coordinate directions is quantified by a sequence of weights. However, to be able to make use of weighted spaces for a particular application one has to make a choice of the weights.In this work, we take a more general view of the weights by allowing them to depend arbitrarily not only on the coordinates but also on the number of variables. Liberating the weights in this way allows us to give a recommendation for how to choose the weights in practice. This recommendation results from choosing the weights so as to minimize the error bound. We also consider how best to choose the underlying weighted Sobolev space within which to carry out the analysis.We revisit also lower bounds on the worst-case error, which change in many minor ways now, since the weights are allowed to depend on the number of variables, and we do not assume that the weights are uniformly bounded as has been assumed in previous papers. Necessary and sufficient conditions for QMC tractability and strong QMC tractability are obtained for the weighted Sobolev spaces with general weights.In the final section, we show that the analysis of variance decomposition of functions from one of the Sobolev spaces is equivalent to the decomposition of functions with respect to an orthogonal decomposition of this space.     

基于改进的粗糙集条件信息熵和灰关联投影法的工程项目投标决策研究

投标决策是建筑企业面临的一大难题.在分析建设工程投标风险因素的基础上,提出一种基于改进的粗糙集条件信息熵和灰关联投影法的投标决策方法.文中投标风险因素的客观权重经两次修正,更具全面性和合理性;然后结合依靠专家经验确定的主观权重得到综合权重;最后将综合权重应用于灰关联投影法进行投标决策.通过应用实例,验证决策方法的可操作性和合理性,为承包商实际的投标工作提供一定的借鉴和参考.     

A primal-dual interior-point method for linear programming based on a weighted barrier function

One motivation for the standard primal-dual direction used in interior-point methods is that it can be obtained by solving a least-squares problem. In this paper, we propose a primal-dual interior-point method derived through a modified least-squares problem. The direction used is equivalent to the Newton direction for a weighted barrier function method with the weights determined by the current primal-dual iterate. We demonstrate that the Newton direction for the usual, unweighted barrier function method can be derived through a weighted modified least-squares problem. The algorithm requires a polynomial number of iterations. It enjoys quadratic convergence if the optimal vertex is nondegenerate.The research of the second author was supported in part by ONR Grants N00014-90-J-1714 and N00014-94-1-0391.