A state of the art literature review of VIKOR and its fuzzy extensions on applications

Multi criteria decision making (MCDM) is one of the research areas of operations research and management science which has widely studied by researchers and practitioners. It finds a compromise solution for evaluating and ranking alternatives from the best to the worst under conflicting criteria with respect to decision maker(s) preferences. In a compromise approach, the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR; that means multi-criteria optimization and compromise solution) continues to be applied satisfactorily across different application areas. This paper conducts a state-of-the-art literature review to categorize, analyze and interpret the current research on VIKOR applications. It also discusses the extensions of VIKOR applied in fuzzy environments. A total of 343 papers are classified into 13 different application areas and a number of sub-application areas. Furthermore, all papers are also categorized with respect to publication year, published journal, country of origin, application type (real case study vs empirical study), and version of fuzzy sets used. This comprehensive literature review provides an insight for researchers and practitioners on VIKOR applications in terms of showing current state and potential areas for future attempts to be focused in the future.     

Compromise ratio method for fuzzy multi-attribute group decision making

The aim of this paper is to develop a compromise ratio (CR) methodology for fuzzy multi-attribute group decision making (FMAGDM), which is an important part of decision support system. Owing to fuzziness being inherent in decision data and group decision making processes, the crisp values are inadequate to model real-life situations. In this paper, the weights of all attributes and the ratings of each alternative with respect to each attribute are described by linguistic terms which can be expressed in trapezoid fuzzy numbers. A fuzzy distance measure is developed to calculate difference between trapezoid fuzzy numbers. The compromise ratio method for FMAGDM is developed by introducing the ranking index based on the concept that the chosen alternative should be as close as possible to the ideal solution and as far away from the negative-ideal solution as possible simultaneously. The computation principle and procedure of the compromise ratio method are described in detail in this paper. Moreover the TOPSIS method which was developed for multi-attribute decision making (MADM) with crisp decision data is analyzed and extended to multi-attribute group decision making (MAGDM) under fuzzy environments. A comparative analysis of the compromise ratio method and the extended fuzzy TOPSIS method is illustrated with a numerical example, showing their similarity and some differences.     

A novel TODIM-VIKOR approach based on entropy and Jensen–Tsalli divergence measure for picture fuzzy sets in a decision-making problem

The picture fuzzy set (PFS) has grown huge attention in the research area of uncertain information from the last few years. Information measures have been widely studied in various fuzzy environments. Therefore, in this paper, we study the entropy and divergence measures under the picture fuzzy environment. First, the paper introduced a new entropy measure to measure the fuzziness degree associated with a PFS. An example is established to show the capabilities of the proposed entropy measure. Second, the paper defines a new Jensen–Tsalli divergence measure for PFS to evaluate the information of discrimination between two PFS. We also discuss several properties of entropy and divergence measures in detail. Then we present a new method, based on proposed entropy and divergence measure, to determine the objective weights of experts for multicriteria group decision making with picture fuzzy information. The final criteria weights are obtained by combining subjective and objective weights for more reliable weightage of evaluation criteria. By using this comprehensive weight-determination technique, the proposed method can effectively reduce the unreasonable impact of the extreme evaluation data on the evaluation results. Further, a new multi-criteria decision-making approach is developed based on the combining concepts of the TODIM and VIKOR method under the picture fuzzy environment. We used TODIM to obtain the overall dominance degree which considers the bounded rationality of decision makers and VIKOR is used to obtain the compromise ranking of alternatives. Lastly, an application of the proposed integrated model is demonstrated to verify the feasibility and usefulness and the outcomes of the proposed model are compared with the outcomes of the existing approaches to indicate its validity. This integrated method can effectively reduce the distortion of decision information and provide extraordinary evaluation results. The proposed approach is used in detecting the major issues due to which a company is facing such breakdowns.     

三角模糊数型多准则群决策的VIKOR扩展方法

运用VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR)方法研究模糊多准则群决策问题常常将其分成模糊信息集结和VIKOR方法求解两个阶段。个体评估信息集结方法不同,所得到的群体集结结果也不同,获得的妥协解可能会存在较大差异。鉴于此,基于含有三角模糊数的多准则群决策问题,分析现有两种主流群体信息集结方法存在的缺陷,基于个体评估值与群体评估值的距离最优和较高的相似度两个目标,设计群体信息集结优化模型,提出一种拓展的VIKOR方法。最后通过实例分析说明了所提出方法的有效性和可行性。     

基于动态P-模糊语言算子的VIKOR决策方法

针对直觉模糊信息解决动态多属性决策问题时存在的不足,将Pythagorean模糊语言信息引入到动态多属性决策问题,提出一种基于Pythagorean模糊语言信息集成算子的多准则妥协排序（VIKOR）决策方法。引入Pythagorean模糊语言得分函数、精确函数、距离计算公式等概念,提出动态 Pythagorean模糊语言加权平均（DPFLWA）算子,并研究DPFLWA算子的基本性质。最后,基于DPFLWA算子和VIKOR方法,构建一种动态 Pythagorean模糊语言多属性决策方法。通过第三方逆向物流服务商的选择实例,表明该方法的可行性和有效性。     

Pythagorean fuzzy TOPSIS for multicriteria group decision-making with unknown weight information through entropy measure

In this study, a new technique for order preference by similarity to ideal solution (TOPSIS)-based methodology is proposed to solve multicriteria group decision-making problems within Pythagorean fuzzy environment, where the information about weights of both the decision makers (DMs) and criteria are completely unknown. Initially, generalized distance measure for Pythagorean fuzzy sets (PFSs) is defined and used to initiate a new Pythagorean fuzzy entropy measure for computing weights of the criteria. In the decision-making process, at first, weights of DMs are computed using TOPSIS through the geometric distance model. Then, weights of the criteria are determined using the entropy weight model through the newly defined entropy measure for PFSs. Based on the evaluated criteria weights, TOPSIS is further applied to obtain the score value of alternatives corresponding to each decision matrix. Finally, the score values of the alternatives are aggregated with the calculated DMs’ weights to obtain the final ranking of the alternatives to avoid the loss of information, unlike other existing methods. Several numerical examples are considered, solved, and compared with the existing methods.     

A new distance measure for interval valued intuitionistic fuzzy sets and its application to group decision making problems with incomplete weights information

The aim of this study is to introduce a novel generalized distance measure for interval valued intuitionistic fuzzy sets and to illustrate the applicability of the proposed distance measure to group decision making problems. Firstly, a generalized distance measure is proposed along with proofs satisfying its axioms. Then, a comparison between the proposed distance measure and well-known distance measures is performed in terms of counter-intuitive cases. Subsequently, the extension of TOPSIS method, in which the proposed distance measure is used to calculate separation measures, to an interval valued intuitionistic fuzzy (IVIF) environment is demonstrated to solve multi-criteria group decision making (MCGDM) problems using optimal criteria weights determined with linear programming model based on the concept of maximizing relative closeness coefficient. Finally, two illustrative examples are provided for proof-of-concept purposes and to demonstrate benefits of using the proposed distance measure over the existing ones in IVIF TOPSIS method for MCGDM problems.     

A developed distance method for ranking generalized fuzzy numbers

Fuzzy logic is one of the effective tools to handle uncertainty and vagueness in engineering and mathematics. One major part of fuzzy logic is ranking fuzzy numbers. In many fuzzy program systems, ranking fuzzy numbers has a remarkable role in decision making and data analysis. Despite the fact that a variety of methods exists for ranking fuzzy numbers, no one can rank fuzzy numbers perfectly in all cases and situations. In this paper, a new method for ranking fuzzy numbers based on the left and right using distance method and α-cut has been presented. To achieve this, a fuzzy distance measure between two generalized fuzzy numbers is proposed. The new measure is expanded with the help of the fuzzy ambiguity measure. The calculation of this method is derived from generalized trapezoidal fuzzy numbers and distance method concepts. Furthermore, a comparison of generalized fuzzy numbers between the proposed method and other resembled methods is provided.     

Fuzzy LINMAP method for multiattribute decision making under fuzzy environments

The Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP) developed by Srinivasan and Shocker [V. Srinivasan, A.D. Shocker, Linear programming techniques for multidimensional analysis of preference, Psychometrika 38 (1973) 337–342] is one of the existing well-known methods for multiattribute decision making (MADM) problems. However, the LINMAP only can deal with MADM problems in crisp environments. Fuzziness is inherent in decision data and decision making processes, and linguistic variables are well suited to assessing an alternative on qualitative attributes using fuzzy ratings. The aim of this paper is further extending the LINMAP method to develop a new methodology for solving MADM problems under fuzzy environments. In this methodology, linguistic variables are used to capture fuzziness in decision information and decision making processes by means of a fuzzy decision matrix. A new vertex method is proposed to calculate the distance between trapezium fuzzy number scores. 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 a fuzzy positive ideal solution (FPIS) which is unknown. The FPIS 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 FPIS 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 paper can deal with MADM problems under not only fuzzy environments but also crisp environments.     

Transition to a legacy- and reuse-based software life cycle

Augmenting domain engineering with legacy software reengineering can help an organization move into the software reuse mode. We present a generalized model of the software life-cycle that explicitly recognizes the critical contribution of legacy software to the attainment of software production from reusable software components. The model defines information products of the software life-cycle, denoted as states, leaving the choice of processes for moving between states open to various methodologies. Then we describe alternative methodologies for transitioning from current practices to the new software life-cycle. This includes evaluation criteria for enabling an informed decision concerning the selection of a transition methodology. Finally, the key to making the alternative transition methodologies feasible is the supporting automated environments. Our experience at integrating automated environments helps us to illustrate the transitions to, and operation within, the proposed software life-cycle     

Extension of Fuzzy ELECTRE based on VIKOR method

Many decision problems in real-world deal with conflicting criteria, uncertainty and imprecise information. Some also allow a group of decision makers (DMs) to make their opinions independently. Multi-criteria decision making (MCDM) is a well known decision method that can make the quality of group multiple criteria decisions better by creating a more explicit, rational and efficient process. A group of MCDM models known as “outranking methods” have been used to rank a set of alternatives. ELECTRE I is an outranking method which is simple, but provides partial ranking. So we consider VIKOR and try to mitigate this problem with regard to relations between VIKOR and ELECTRE. The objective of this paper is to extend ELECTRE I method based on VIKOR to rank a set of alternatives versus a set of criteria to show the decision maker’s preferences.     

An integrated fuzzy MCDM based approach for robot selection considering objective and subjective criteria

Robots with vastly different capabilities and specifications are available for a wide range of applications. Selection of a robot for a specific application has become more complicated due to increase in the complexity, advanced features and facilities that are continuously being incorporated into the robots by different manufacturers. The aim of this paper is to present an integrated approach for the optimal selection of robots by considering both objective and subjective criteria. The approach utilizes Fuzzy Delphi Method (FDM), Fuzzy Analytical Hierarchical Process (FAHP), Fuzzy modified TOPSIS or Fuzzy VIKOR and Brown–Gibson model for robot selection. FDM is used to select the list of important objective and subjective criteria based on the decision makers’ opinion. Fuzzy AHP method is then used to find out the weight of each criterion (both objective and subjective). Fuzzy modified TOPSIS or Fuzzy VIKOR method is then used to rank the alternatives based on objective and subjective factors. The rankings obtained are used to calculate the robot selection index based on Brown–Gibson model. The proposed methodology is illustrated with a case study related to selection of robot for teaching purpose. It is found that the highest ranked alternative based on Fuzzy VIKOR is closest to the ideal solution.     

Pythagorean Uncertain Linguistic Variable Hamy Mean Operator and Its Application to Multi-attribute Group Decision Making

Pythagorean fuzzy set (PFS) can provide more flexibility than intuitionistic fuzzy set (IFS) for handling uncertain information, and PFS has been increasingly used in multi-attribute decision making problems. This paper proposes a new multi-attribute group decision making method based on Pythagorean uncertain linguistic variable Hamy mean (PULVHM) operator and VIKOR method. Firstly, we define operation rules and a new aggregation operator of Pythagorean uncertain linguistic variable (PULV) and explore some properties of the operator. Secondly, taking the decision makers' hesitation degree into account, a new score function is defined, and we further develop a new group decision making approach integrated with VIKOR method. Finally, an investment example is demonstrated to elaborate the validity of the proposed method. Sensibility analysis and comprehensive comparisons with another two methods are performed to show the stability and advantage of our method.      

基于扩展VIKOR的不确定语言多属性群决策方法

针对不确定语言多属性群决策问题,提出了一种基于扩展多准则妥协优化解(VIKOR)的决策方法。该方法首先借鉴二元语义表示模型定义了不确定语言短语的运算规则及其距离公式,并使用运算规则集结了专家给出的不确定语言评价信息;然后结合属性的效益型和成本型特征分别定义了其正负理想点,通过将VIKOR法扩展到不确定语言环境来获取方案的折衷排序结果;最后,通过算例分析表明了所提出方法的有效性和优越性。     

云模型与VIKOR集成的多属性群决策方法

针对以往定性概念量化过程中只考虑到模糊性这一不确定性因素的不足,采用云模型从模糊性和随机性两个方面实现语义评价变量的量化,并对专家组的语义评价信息进行集结。针对以往云差异性度量方法的不足,基于云模型构成的本质特点,从云滴分布的角度提出了云距离测度算法,进而提出了云相似度算法。考虑到决策指标间相互影响关系,将云模型与决策试验与实验评估法（Decision-Making Trial and Evaluation Laboratory,DEMATEL）相结合,采用云DEMATEL法对专家主观评价给出的指标初始重要度进行修正,进而计算得到指标权重。采用云距离算法计算备选方案与正、负理想解间的距离,并最终由云VIKOR求得备选方案的妥协解。最后以实例验证了所提方法的可行性和有效性。     

Hesitant interval-valued Pythagorean fuzzy VIKOR method

In this article, we investigate multiple attribute decision-making problems with hesitant interval-valued Pythagorean fuzzy information. First, the concepts of hesitant interval-valued Pythagorean fuzzy set are defined, and the operation laws, the score function, and accuracy function have been developed. Then several distance measures for hesitant interval-valued Pythagorean fuzzy values have been presented including the Hamming distance, Euclidean distance, and generalized distance, and so on. Based on the operational laws, a series of aggregation operators have been developed including the hesitant interval-valued Pythagorean fuzzy weighted averaging (HIVPFWA) operator, the hesitant interval-valued Pythagorean fuzzy geometric weighted averaging (HIVPFGWA) operator, the hesitant interval-valued Pythagorean fuzzy ordered weighed averaging (HIVPFOWA) operator, and hesitant interval-valued Pythagorean fuzzy ordered weighed geometric averaging (HIVPFOWGA) operator. By using the generalized mean operator, we also develop the generalized hesitant interval-valued Pythagorean fuzzy weighed averaging (GHIVPFWA) operator, the generalized hesitant interval-valued Pythagorean fuzzy weighed geometric averaging (GHIVPFWGA) operator, the generalized hesitant interval-valued Pythagorean fuzzy ordered weighted averaging (GHIVPFOWA) operator, and generalized hesitant interval-valued Pythagorean fuzzy ordered weighted geometric averaging (GHIVPFOWGA) operator operator. We further develop several hybrid aggregation operators including the hesitant interval-valued Pythagorean fuzzy hybrid averaging (HIVPFHA) operator and the generalized hesitant interval-valued Pythagorean fuzzy hybrid averaging (GHIVPFHA) operator. Based on the distance measures and the aggregation operators, we propose a hesitant interval-valued Pythagorean fuzzy VIKOR method to solve multiple attribute decision problems with multiple periods. Finally, an illustrative example for evaluating the metro project risk is given to demonstrate the feasibility and effectiveness of the proposed method.     

基于犹豫模糊语言多属性群决策的VIKOR扩展方法

针对犹豫模糊语言多属性群决策问题,提出了一种基于可能度分布的VIKOR方法。该方法首先将基于犹豫模糊语言的评价信息转化成可能度分布值,定义了新的距离公式,避免了传统犹豫模糊语言评价信息在计算过程中造成的信息扭曲。然后,设计了基于最大群体效用与最小个体遗憾两个目标的群体信息集结优化模型,并给出多属性群决策的VIKOR扩展方法。运用一个交通建设方案选择的案例分析验证了方法的有效性和优越性。     

Fuzzy compromise programming for Group decision making

A multicriteria technique named fuzzy compromise programming is combined with a methodology known as group decision making under fuzziness to come up with a new technique that supports decision making with multiple criteria and multiple participants (or experts). All criteria (qualitative and quantitative) are modeled by way of fuzzy sets, utilizing the fact that criteria values in most water resources problems are vague, imprecise and/or ill defined. The involvement of multiple experts in the decision process is achieved by incorporating each participant's perception of criteria weights, best and worst criteria values, relative degrees of risk acceptance, as well as other parameters into the problem. The proposed methodology is illustrated with a case study taken from the literature, combined with the input of four expert individuals with diverse backgrounds. After processing the input from the experts, a group compromise decision is formulated.     

Multiple criteria decision analysis under complex uncertainty: A Pearson-like correlation-based Pythagorean fuzzy compromise approach

The aim of this article is to develop a novel multiple criteria decision analysis (MCDA) method using a Pearson-like correlation-based Pythagorean fuzzy (PF) compromise approach under complex uncertainty based on PF sets and interval-valued Pythagorean fuzzy (IVPF) sets. Because of the complexity and ambiguity involved in real-life decision-making situations, this article utilizes the theory of Pythagorean fuzziness, which is characterized by flexible degrees of membership, nonmembership, and indeterminacy to describe uncertain information more comprehensively. PF and IVPF sets possess exceptional abilities to accurately reflect the uncertainty, fuzziness, and vagueness inherent in the decision information. However, manipulating PF and IVPF information is a complicated and difficult task for most decision makers. In this regard, this article extends the well-known and widely used concept of correlation coefficients to develop simple and effective compromise models for solving MCDA problems in PF and IVPF contexts. This article conducts an extended analysis of Pearson-like correlation coefficients for PF and IVPF sets separately and introduces new concepts of PF and IVPF correlation coefficients to furnish a solid basis for the proposed methodology. Furthermore, this article develops useful concepts of PF and IVPF correlation-based closeness coefficients to simultaneously measure the relative closeness to the positive-ideal PF/IVPF solutions and the relative remoteness from the negative-ideal PF/IVPF solutions. On the basis of the developed concepts, this article proposes a novel Pearson-like correlation-based PF/IVPF compromise approach to address uncertain MCDA problems involving PF/IVPF information and determine the ultimate priority orders among competing alternatives. Finally, this article provides an illustrative application about a financing decision of working capital management to verify the developed approach and demonstrate its feasibility and practicality.     

一种基于二元语义信息处理的多属性群决策方法

为了解决评价信息为语言信息的多属性群决策问题,结合VIKOR方法,提出一种基于二元语义信息处理的多属性群决策方法．该方法使用二元语义信息集结算子获得决策群组的决策信息,通过最大化群效用和最小化个体遗憾来获得决策者满意的折衷方案．该方法计算简单,便于理解,可有效避免信息的丢失和扭曲,并可克服理想解方法不能反映出各方案与正负理想解的接近程度的不足．最后,算例计算结果表明了该方法的有效性．