Multi-criteria decision-making using interval-valued hesitant fuzzy QUALIFLEX methods based on a likelihood-based comparison approach

QUALIFLEX is a very efficient outranking method to handle multi-criteria decision-making (MCDM) involving cardinal and ordinal preference information. Based on a likelihood-based comparison approach, this paper develops two interval-valued hesitant fuzzy QUALIFLEX outranking methods to handle MCDM problems within the interval-valued hesitant fuzzy context. First, we define the likelihoods of interval-valued hesitant fuzzy preference relations that compare two interval-valued hesitant fuzzy elements (IVHFEs). Then, we propose the concepts of the concordance/discordance index, the weighted concordance/discordance index and the comprehensive concordance/discordance index. Moreover, an interval-valued hesitant fuzzy QUALIFLEX model is developed to solve MCDM problems where the evaluative ratings of the alternatives and the weights of the criteria take the form of IVHFEs. Additionally, this paper propounds another likelihood-based interval-valued hesitant fuzzy QUALIFLEX method to accommodate the IVHFEs’ evaluative ratings of alternatives and non-fuzzy criterion weights with incomplete information. Finally, a numerical example concerning the selection of green suppliers is provided to demonstrate the practicability of the proposed methods, and a comparison analysis is given to illustrate the advantages of the proposed methods.

     

An interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions and its application to multiple criteria decision analysis

This paper presents an interval-valued intuitionistic fuzzy permutation method with likelihood-based preference functions for managing multiple criteria decision analysis based on interval-valued intuitionistic fuzzy sets. First, certain likelihood-based preference functions are proposed using the likelihoods of interval-valued intuitionistic fuzzy preference relationships. Next, selected practical indices of concordance/discordance are established to evaluate all possible permutations of the alternatives. The optimal priority order of the alternatives is determined by comparing all comprehensive concordance/discordance values based on score functions. Furthermore, this paper considers various preference types and develops another interval-valued intuitionistic fuzzy permutation method using programming models to address multiple criteria decision-making problems with incomplete preference information. The feasibility and applicability of the proposed methods are illustrated in the problem of selecting a suitable bridge construction method. Moreover, certain comparative analyses are conducted to verify the advantages of the proposed methods compared with those of other decision-making methods. Finally, the practical effectiveness of the proposed methods is validated with a risk assessment problem in new product development.     

Multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets

In this paper, we present a new method for multicriteria fuzzy decision making based on interval-valued intuitionistic fuzzy sets, where interval-valued intuitionistic fuzzy values are used to represent evaluating values of the decision-maker with respect to alternatives. First, we propose a new method for ranking interval-valued intuitionistic fuzzy values. Based on the proposed fuzzy ranking method of interval-valued intuitionistic fuzzy values, we propose a new method for multicriteria fuzzy decision making. The proposed multicriteria fuzzy decision making method outperforms Ye’s method (2009) due to the fact that the proposed method can overcome the drawback of Ye’s method (2009), where the drawback of Ye’s method is that it can not distinguish the ranking order between alternatives in some situations. The proposed method provides us with a useful way for dealing with multicriteria fuzzy decision making problems based on interval-valued intuitionistic fuzzy sets.     

Ranking and screening multiple criteria alternatives with partialinformation and use of ordinal and cardinal strength of preferences

This paper consists of three parts: 1) some theories and an efficient algorithm for ranking and screening multicriteria alternatives when there exists partial information on the decision maker's preferences; 2) generation of partial information using variety of methods; and 3) the existence of ordinal and cardinal functions based on and strengths of preferences. We demonstrate that strengths of preference concept can be very effectively used to generate the partial information on preferences. We propose axioms for ordinal and cardinal (measurable) value functions. An algorithm is developed for ranking and screening alternatives when there exists partial information about the preferences and the ordering of alternatives. The proposed algorithm obtains the same information very efficiently while by solving one mathematical programming problem many alternatives can be ranked and screened. Several examples are discussed and results of some computational experiments are reported     

A simulated annealing-based permutation method and experimental analysis for multiple criteria decision analysis with interval type-2 fuzzy sets

The aim of this paper is to develop a simulated annealing-based permutation method for multiple criteria decision analysis within the environment of interval type-2 fuzzy sets. The outranking methodology constitutes one of the most fruitful approaches in multiple criteria decision making and has been applied in numerous real-world problems. The permutation method is a classical outranking model, which generalizes Jacquet–Lagreze's permutation method and is based on a pairwise criterion comparison of the alternatives. Because modeling of the uncertainty in the decision-making process becomes increasingly important, an extension to the interval type-2 fuzzy environment is a useful generalization of the permutation method and is appropriate for handling uncertain and imprecise information in practical decision-making situations. This paper produces a signed-distance-based comparison among the comprehensive rankings of alternatives for concordance and discordance analyses. An integrated nonlinear programming model is constructed for estimation of the criterion weights and the optimal ranking order of the alternatives under incomplete preference information. To enhance the implementation efficiency, a simulated annealing-based permutation method and its meta-heuristic algorithm are developed to produce a polynomial time solution for the total completion time problem. Furthermore, computational experiments with notably large amounts of simulation data are conducted to test the solution approach and validate the correctness of the approximate solution compared with the optimal all-permutation-based result.     

Integrated evidential reasoning approach in the presence of cardinal and ordinal preferences and its applications in software selection

A combination of cardinal and ordinal preferences in multiple-attribute decision making (MADM) demonstrates more reliability and flexibility compared with sole cardinal or ordinal preferences derived from a decision maker. This situation occurs particularly when the knowledge and experience of the decision maker, as well as the data regarding specific alternatives on certain attributes, are insufficient or incomplete. This paper proposes an integrated evidential reasoning (IER) approach to analyze uncertain MADM problems in the presence of cardinal and ordinal preferences. The decision maker provides complete or incomplete cardinal and ordinal preferences of each alternative on each attribute. Ordinal preferences are expressed as unknown distributed assessment vectors and integrated with cardinal preferences to form aggregated preferences of alternatives. Three optimization models considering cardinal and ordinal preferences are constructed to determine the minimum and maximum minimal satisfaction of alternatives, simultaneous maximum minimal satisfaction of alternatives, and simultaneous minimum minimal satisfaction of alternatives. The minimax regret rule, the maximax rule, and the maximin rule are employed respectively in the three models to generate three kinds of value functions of alternatives, which are aggregated to find solutions. The attribute weights in the three models can be precise or imprecise (i.e., characterized by six types of constraints). The IER approach is used to select the optimum software for product lifecycle management of a famous Chinese automobile manufacturing enterprise.     

A new multi-criteria weighting and ranking model for group decision-making analysis based on interval-valued hesitant fuzzy sets to selection problems

The multi-criteria group decision-making methods under fuzzy environments are developed to cope with imprecise and uncertain information for solving the complex group decision-making problems. A team of some professional experts for the assessment is established to judge candidates or alternatives among the chosen evaluation criteria. In this paper, a novel multi-criteria weighting and ranking model is introduced with interval-valued hesitant fuzzy setting, namely IVHF-MCWR, based on the group decision analysis. The interval-valued hesitant fuzzy set theory is a powerful tool to deal with uncertainty by considering some interval-values for an alternative under a set regarding assessment factors. In procedure of the proposed IVHF-MCWR model, weights of criteria as well as experts are considered to decrease the errors. In this regard, optimal criteria’ weights are computed by utilizing an extended maximizing deviation method based on IVHF-Hamming distance measure. In addition, experts’ judgments are taken into account for computing the criteria’ weights. Also, experts’ weights are determined based on proposed new IVHF technique for order performance by similarity to ideal solution method. Then, a new IVHF-index based on Hamming distance measure is introduced to compute the relative closeness coefficient for ranking the candidates or alternatives. Finally, two application examples about the location and supplier selection problems are considered to indicate the capability of the proposed IVHF-MCWR model. In addition, comparative analysis is reported to compare the proposed model and three fuzzy decision methods from the recent literature. Comparing these approaches and computational results shows that the IVHF-MCWR model works properly under uncertain conditions.     

An extended GRA method for MCDM with interval-valued triangular fuzzy assessments and unknown weights

Multiple criteria decision making (MCDM) is the process of ranking the feasible alternatives and selecting the best one by considering multiple criteria. Owing to the complexity, fuzziness and uncertainties of the objective things, the criterion values often take the form of linguistic variables, which can be expressed in interval-valued triangular fuzzy numbers. The purpose of this paper is to develop an extended grey relational analysis (GRA) method for solving MCDM problems with interval-valued triangular fuzzy numbers and unknown information on criterion weights. In order to determine the criterion weights, some optimization models based on the basic idea of traditional GRA method are established. Then, calculation steps of extended GRA method for MCDM are given. Finally, a numerical example is shown to verify the developed method and to demonstrate its practicality and feasibility.     

A possibility degree method for interval-valued intuitionistic fuzzy multi-attribute group decision making

The ranking of interval-valued intuitionistic fuzzy sets (IVIFSs) is very important for the interval-valued intuitionistic fuzzy decision making. From the probability viewpoint, the possibility degree of comparison between two interval-valued intuitionistic fuzzy numbers (IVIFNs) is defined by using the notion of 2-dimensional random vector, and a new method is then developed to rank IVIFNs. Hereby the ordered weighted average operator and hybrid weighted average operator for IVIFNs are defined based on the Karnik–Mendel algorithms and employed to solve multi-attribute group decision making problems with IVIFNs. The individual overall attribute values of alternatives are obtained by using the weighted average operator for IVIFNs. By using the hybrid weighted average operator for IVIFNs, we can obtain the collective overall attribute values of alternatives, which are used to rank the alternatives. A numerical example is examined to illustrate the effectiveness and flexibility of the proposed method in this paper.     

Multiple attribute group decision-making methods with unknown weights in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting

To better solve the corresponding multiple attribute group decision-making problem with unknown weights, multiple attribute group decision-making methods with completely unknown weights of decision-makers and incompletely known weights of attributes are proposed in intuitionistic fuzzy setting and interval-valued intuitionistic fuzzy setting. In the group decision-making method, two weight models are proposed based on the score function to determine the weights of both experts and attributes from the intuitionistic fuzzy decision matrices and the interval-valued intuitionistic fuzzy decision matrices. Then, overall evaluation formulas of weighted scores for each alternative are introduced in the intuitionistic fuzzy setting and the interval-valued intuitionistic fuzzy setting to obtain the ranking order of alternatives and the most desirable one(s). Finally, two numerical examples demonstrate the applicability and benefit of the proposed methods.     

Interval-valued intuitionistic fuzzy cosine similarity measures for multiple attribute decision-making

In this work, considering all information carried by the membership degrees, nonmembership degree, and hesitancy degree in interval-valued intuitionistic fuzzy sets (IVIFSs) as 3-D vector representations, we propose a cosine similarity measure and a weighted cosine similarity measure for IVIFSs based on the extension of the cosine similarity measure (angular coefficient) between intuitionistic fuzzy sets in 2-D vector space. Then, the weighted cosine similarity measure for IVIFSs is applied to multiple attribute decision-making problems under the interval-valued intuitionistic fuzzy environment. Through the similarity measure between the ideal alternative and each alternative, the ranking order of all alternatives can be determined and the best alternative can be easily identified as well. Finally, two illustrative examples are given to demonstrate the applications and efficiency of the proposed decision-making method.     

An interval-valued intuitionistic fuzzy multiattribute group decision making framework with incomplete preference over alternatives

This article proposes a framework to handle multiattribute group decision making problems with incomplete pairwise comparison preference over decision alternatives where qualitative and quantitative attribute values are furnished as linguistic variables and crisp numbers, respectively. Attribute assessments are then converted to interval-valued intuitionistic fuzzy numbers (IVIFNs) to characterize fuzziness and uncertainty in the evaluation process. Group consistency and inconsistency indices are introduced for incomplete pairwise comparison preference relations on alternatives provided by the decision-makers (DMs). By minimizing the group inconsistency index under certain constraints, an auxiliary linear programming model is developed to obtain unified attribute weights and an interval-valued intuitionistic fuzzy positive ideal solution (IVIFPIS). Attribute weights are subsequently employed to calculate distances between alternatives and the IVIFPIS for ranking alternatives. An illustrative example is provided to demonstrate the applicability and effectiveness of this method.     

Extension of the ELECTRE method based on interval-valued fuzzy sets

Decision-making is the process of finding the best option among the feasible alternatives. In classical multiple criteria decision-making (MCDM) methods, the ratings and the weights of the criteria are known precisely. However, if decision makers cannot reach an agreement on the method of defining linguistic variables based on the fuzzy sets, the interval-valued fuzzy set theory can provide a more accurate modeling. In this paper, the interval-valued fuzzy ELECTRE method is presented aiming at solving MCDM problems in which the weights of criteria are unequal, using interval-valued fuzzy set concepts. For the purpose of proving the validity of the proposed model, we present a numerical example and build a practical maintenance strategy selection problem.     

Some interval-valued intuitionistic fuzzy Schweizer–Sklar power aggregation operators and their application to supplier selection

Supplier selection is an important multiple attribute group decision-making (MAGDM) problem. How to choose a suitable supplier is an evaluation process with different alternatives of multiple attributes, and it also relates to the expression of the evaluation value. Considering Schweizer–Sklar t-conorm and t-norm (SSTT) can make the information aggregation process more flexible than others, and the power average (PA) operator can eliminate effects of unreasonable data from biased decision-makers. So, we extend SSTT to interval-valued intuitionistic fuzzy numbers (IVIFNs) and define Schweizer–Sklar operational rules of IVIFNs. Then, we combine the PA operator with Schweizer–Sklar operations, and propose the interval-valued intuitionistic fuzzy Schweizer–Sklar power average operator, the interval-valued intuitionistic fuzzy Schweizer–Sklar power weighted average (IVIFSSPWA) operator, the interval-valued intuitionistic fuzzy Schweizer–Sklar power geometric operator and the interval-valued intuitionistic fuzzy Schweizer–Sklar power weighted geometric (IVIFSSPWG) operator, respectively. Furthermore, we study some desirable characteristics of them and develop two methods on the basis of IVIFSSPWA and IVIFSSPWG operators. At the same time, we apply the two methods to deal with the MAGDM problems based on supplier selection. Finally, an illustrative example of supplier selection problem is given to testify the availability of the presented operators.     

Multi-criteria decision-making method based on interval-valued intuitionistic fuzzy sets

Out of several generalizations of fuzzy set theory for various objectives, the notions introduced by Atanassov, 1983, Atanassov and Gargov, 1989 in defining intuitionistic fuzzy sets and interval-valued intuitionistic fuzzy sets are interesting and very useful in modeling real life problems. Ranking of interval-valued intuitionistic fuzzy sets plays a vital role in decision-making, data analysis, artificial intelligence and socioeconomic system and it was studied in Xu, 2007c, Xu and Chen, 2007a, Ye, 2009. In this paper a new method for ranking interval-valued intuitionistic fuzzy sets has been introduced and studied. The method is illustrated by numerical examples and compared with other methods. And then a new method for handling multi-criteria fuzzy decision-making problems based on interval-valued intuitionistic fuzzy sets is presented in which criterion values for alternatives are interval-valued intuitionistic fuzzy sets. The method proposed here can provide a useful way to efficiently help the decision-maker to make his decision. An illustrative example is given to verify the developed approach and to demonstrate its practicality and effectiveness.     

Soft computing based on maximizing consensus and fuzzy TOPSIS approach to interval-valued intuitionistic fuzzy group decision making

Multi-attribute group decision making (MAGDM) is an important research topic in decision theory. In recent decades, many useful methods have been proposed to solve various MAGDM problems, but very few methods simultaneously take them into account from the perspectives of both the ranking and the magnitude of decision data, especially for the interval-valued intuitionistic fuzzy decision data. The purpose of this paper is to develop a soft computing technique based on maximizing consensus and fuzzy TOPSIS in order to solve interval-valued intuitionistic fuzzy MAGDM problems from such two aspects of decision data. To this end, we first define a consensus index from the perspective of the ranking of decision data, for measuring the degree of consensus between the individual and the group. Then, we establish an optimal model based on maximizing consensus to determine the weights of experts. Following the idea of TOPSIS, we calculate the closeness indices of the alternatives from the perspective of the magnitude of decision data. To identify the optimal alternatives and determine their optimum quantities, we further construct a multi-choice goal programming model based on the derived closeness indices. Finally, an example is given to verify the developed method and to make a comparative analysis.     

Soft computing based on new interval-valued fuzzy modified multi-criteria decision-making method

In this paper, a new interval-valued fuzzy modified TOPSIS (IVFM-TOPSIS) method is proposed that can reflect both subjective judgment and objective information in real life situations. This proposed method is based on concepts of the positive ideal and negative ideal solutions for solving multi-criteria decision-making (MCDM) problems in a fuzzy environment. The performance rating values and weights of criteria are linguistic variables expressed as triangular interval-valued fuzzy numbers. Furthermore, we appraise the performance of alternatives against both subjective and objective criteria with multi-judges for decision-making problems. Finally, for the purpose of proving the validity of the proposed method a numerical example is presented for a robot selection problem.     

基于IITFN输入的复杂系统关联MAGDM方法

区间直觉梯形模糊数（Interval-valued intuitionistic trapezoidal fuzzy number,IITFN）是刻画复杂系统不确定性的有效工具. 基于进一步完善的IITFN 运算规则,讨论其局部封闭性. 由此定义IITFN 几何Bonferroni 平均算子,并验证该算子的相关性质. 针对决策者及属性之间均存在关联作用且权重均未知的多属性群决策（Multi-attribute group decision making,MAGDM）问题,提出基于前景混合区间直觉梯形几何 Bonferroni （Prospect hybrid interval-valued intuitionistic trapezoidal fuzzy geometric Bonferroni,PHIITFGB）平均算子 的关联多属性群决策方法. 该方法首先通过依次定义IITFN 的前景效应、前景价值函数和前景价值,获取前景价值矩阵;其次,将前景价值矩阵转化为前景记分函数矩阵,并综合运用基于灰关联深度系数的客观属性权重极大 熵模型和基于2-可加模糊测度与Choquet 积分联合的决策者权重确定模型,获取决策者权重及属性权重;再次,利 用PHIITFGB 算子集结各决策者的方案评估信息,结合决策者权重即可获取相应于各方案的综合前景价值;最后,计算综合前景记分价值函数,基于IITFN 的序关系判别准则确定方案排序. 案例验证决策方法的有效性和可行性.     

用区间直觉模糊集方法对属性权重未知的群求解其多属性决策

本文首先提出群区间直觉模糊有序加权几何(groupinterval-valuedintuitionistic fuzzy orderedweighted geometric,GIVIFOWG)算子和群区间直觉模糊有序加权平均(group interval-valued intuitionistic fuzzy ordered weighted averaging,GIVIFOWA)算子.利用GIVIFOWG算子或GIVIFOWA算子聚集群的决策矩阵以获得方案在属性上的综合区间直觉模糊决策矩阵(collectiveinterval-valuedintuitionistic fuzzy decision-matrix,CIVIFDM).然后定义了一个考虑犹豫度的区间直觉模糊熵(interval-valuedintuitionistic fuzzyentropy,IVIFE);通过熵衡量每个属性所含的信息来求解属性权重.最后,提出基于可能度的接近理想解的区间排序法(interval technique for order preference by similarity to an ideal solution,ITOPSIS)和区间得分函数法.在ITOPSIS法中,依据区间距离公式计算候选方案和理想方案的属性加权区间距离,进而采用ITOPSIS准则对各方案进行排序;在区间得分函数法中,算出CIVIFDM中各方案的得分值以及精确值,然后利用区间得分准则对各方案进行排序.实验结果验证了决策方法的有效性和可行性.     

Simulation-based evaluation of defuzzification-based approaches to fuzzy multiattribute decision making

This paper presents a simulation-based study to evaluate the performance of 12 defuzzification-based approaches for solving the general fuzzy multiattribute decision-making (MADM) problem requiring cardinal ranking of decision alternatives. These approaches are generated based on six defuzzification methods in conjunction with the simple additive weighting (SAW) method and the technique for order preference by similarity to the ideal solution method. The consistency and effectiveness of these approaches are examined in terms of four new objective performance measures, which are based on five evaluation indexes. The simulation result shows that the approaches, which are capable of using all the available information on fuzzy numbers effectively in the defuzzification process, produce more consistent ranking outcomes. In particular, the SAW method with the degree of dominance defuzzification is proved to be the overall best performed approach, which is followed by the SAW method with the area center defuzzification. These findings are of practical significance in real-world settings where the selection of the defuzzification-based approaches is required in solving the general fuzzy MADM problems under specific decision contexts.