Extension of VIKOR method for decision making problem based on hesitant fuzzy set

The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are all based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. The VIKOR method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum of an “individual regret” for the “opponent”, which is an effective tool in multi-criteria decision making, particularly in a situation where the decision maker is not able, or does not know to express his/her preference at the beginning of system design. The TOPSIS method determines a solution with the shortest distance to the ideal solution and the greatest distance from the negative-ideal solution, but it does not consider the relative importance of these distances. And, the hesitant fuzzy set is a very useful tool to deal with uncertainty, which can be accurately and perfectly described in terms of the opinions of decision makers. In this paper, we develop the E-VIKOR method and TOPSIS method to solve the MCDM problems with hesitant fuzzy set information. Firstly, the hesitant fuzzy set information and corresponding concepts are described, and the basic essential of the VIKOR method is introduced. Then, the problem on multiple attribute decision marking is described, and the principles and steps of the proposed E-VIKOR method and TOPSIS method are presented. Finally, a numerical example illustrates an application of the E-VIKOR method, and the result by the TOPSIS method is compared.     

Extended VIKOR method in comparison with outranking methods

The VIKOR method was developed to solve MCDM problems with conflicting and noncommensurable (different units) criteria, assuming that compromising is acceptable for conflict resolution, the decision maker wants a solution that is the closest to the ideal, and the alternatives are evaluated according to all established criteria. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria, and on proposing compromise solution (one or more). The VIKOR method is extended with a stability analysis determining the weight stability intervals and with trade-offs analysis. The extended VIKOR method is compared with three multicriteria decision making methods: TOPSIS, PROMETHEE, and ELECTRE. A numerical example illustrates an application of the VIKOR method, and the results by all four considered methods are compared.     

A fuzzy closeness approach to fuzzy multi-attribute decision making

The aim of this paper is to develop a new fuzzy closeness (FC) methodology for multi-attribute decision making (MADM) in fuzzy environments, which is an important research field in decision science and operations research. The TOPSIS method based on an aggregating function representing “closeness to the ideal solution” is one of the well-known MADM methods. However, while the highest ranked alternative by the TOPSIS method is the best in terms of its ranking index, this does not mean that it is always the closest to the ideal solution. Furthermore, the TOPSIS method presumes crisp data while fuzziness is inherent in decision data and decision making processes, so that fuzzy ratings using linguistic variables are better suited for assessing decision alternatives. In this paper, a new FC method for MADM under fuzzy environments is developed by introducing a multi-attribute ranking index based on the particular measure of closeness to the ideal solution, which is developed from the fuzzy weighted Minkowski distance used as an aggregating function in a compromise programming method. The FC method of compromise ranking determines a compromise solution, providing a maximum “group utility” for the “majority” and a minimum individual regret for the “opponent”. A real example of a personnel selection problem is examined to demonstrate the implementation process of the method proposed in this paper.     

基于时间度的动态直觉模糊多属性妥协决策

针对当前动态直觉模糊多属性决策方法存在的不足,提出一种基于时间度的动态直觉模糊妥协决策方法。引入时间度准则,基于逼近理想解法融合主客观两类赋权法,获得兼顾主观偏好和样本客观信息的时序权重,克服现有时序权重主观赋值的随意性,同时运用直觉模糊熵(IFE)确定不同时序状态下各属性权重;根据动态直觉模糊加权几何算子(DIFWG)集结不同时序直觉模糊决策矩阵,构造动态直觉模糊综合决策矩阵,并利用VIKOR法,提供兼顾群体效用最大化与个体后悔最小化的各方案妥协折中排序,得到与理想解最近的妥协方案;以分布式创新企业合作伙伴选择为例,验证该方法在实际决策过程中的可行性和有效性。     

A new methodology for fuzzy multi-attribute group decision making with multi-granularity and non-homogeneous information

The aim of this paper is to develop a new methodology for solving fuzzy multi-attribute group decision making problems with non-homogeneous information, including multi-granular linguistic term sets, fuzzy numbers, interval values and real numbers. In this methodology, different distances are defined to measure differences between alternatives and the ideal solution as well as the negative ideal solution. A relative closeness method is developed by introducing the multi-attribute ranking index based on the particular measure of closeness to the IS. The proposed method determines a compromise solution for the group, providing a maximum “group utility” for the “majority” and a minimum of an individual regret for the “opponent”. The implementation process, effectiveness and feasibility of the method proposed in this paper are illustrated with a real example of the missile weapon system design project selection.     

Extension of VIKOR method for decision making problem with interval numbers

The VIKOR method was developed for multi-criteria optimization of complex systems. It determines the compromise ranking list and the compromise solution obtained with the initial (given) weights. This method focuses on ranking and selecting from a set of alternatives in the presence of conflicting criteria. It introduces the multi-criteria ranking index based on the particular measure of “closeness” to the “ideal” solution. The aim of this paper is to extend the VIKOR method for decision making problems with interval number. The extended VIKOR method’s ranking is obtained through comparison of interval numbers and for doing the comparisons between intervals, we introduce α as optimism level of decision maker. Finally, a numerical example illustrates and clarifies the main results developed in this paper.     

Extensions of TOPSIS for large scale multi-objective non-linear programming problems with block angular structure

This paper focuses on multi-objective large-scale non-linear programming (MOLSNLP) problems with block angular structure. We extend the technique for order preference by similarity ideal solution (TOPSIS) to solve them. Compromise (TOPSIS) control minimizes the measure of distance, provided that the closest solution should have the shortest distance from the positive ideal solution (PIS) as well as the longest distance from the negative ideal solution (NIS). As the measure of “closeness” L_P-metric is used. Thus, we reduce a q-dimensional objective space to a two-dimensional space by a first-order compromise procedure. The concept of a membership function of fuzzy set theory is used to represent the satisfaction level for both criteria. Moreover, we derive a single objective large-scale non-linear programming (LSNLP) problem using the max–min operator for the second-order compromise operation. Finally, a numerical illustrative example is given to clarify the main results developed in this paper.     

线性无量纲化方法对比及反向指标正向化方法

线性无量纲化方法的对比及反向指标的正向化方法都是综合评价的重要研究内容。从指标差异信息的角度,以TOPSIS、基于街区距离的TOPSIS和线性加权综合法为例,基于理论推导和实证分析对比了常用的线性无量纲化方法,并提出了两种反向指标正向化方法。研究发现,对于线性加权综合法和TOPSIS,不同线性无量纲化方法下同一指标归一化极差的不同是导致排序结果存在差异的关键因素;本文提出的反向指标正向化方法,不仅可以保证正向化前后TOPSIS、基于街区距离的TOPSIS的评价值不变,也可以实现反向指标正向化后线性加权综合法与基于街区距离的TOPSIS在排序目的上的等效性。最后,本文提出了线性无量纲化方法和反向指标正向化方法的应用建议。     

基于VIKOR算法的区域经济发展状况评价

区域经济发展状况评价是多属性方案决策,为防止个别较差指标的消极影响被其他指标中和,提高决策的合理性,采用简化的VIKOR算法完成决策矩阵的规范化处理,在此基础上对区域经济发展状况进行排序.并通过算例说明VIKOR算法的有效性和可行性.     

TOPSIS for bi-level MODM problems

TOPSIS (technique for order preference by similarity to ideal solution) is a multiple criteria method to identify solutions from a finite set of alternatives based upon simultaneous minimization of distance from an ideal point and maximization of distance from a nadir point. This paper proposes a fuzzy TOPSIS algorithm to solve bi-level multi-objective decision-making (BL-MODM) problems, and in which the objective function at each level are non-linear functions which are to be maximized. The proposed model for getting the satisfactory solution of the BL-MODM problems includes the membership functions for the upper level decision variables vector with possible tolerances, the membership function of the distance function from the positive ideal solution (PIS) and the membership function of the distance function from the negative ideal solution (NIS). A numerical illustrative example is given to clarify the proposed TOPSIS approach of this paper.     

基于有序梯形模糊灰色关联TOPSIS的多属性决策方法

研究了有序梯形模糊数来表示不确定语言环境下的灰色关联TOPSIS多属性决策问题。首先应用有序梯形模糊数标度方案属性偏好信息,在传统梯形模糊数基础上增加了一个方向属性,使得决策信息的表示更加细腻;提出了有序梯形模糊环境下多属性决策灰色关联TOPSIS综合优选算法,引入了距离和灰色关联度相结合的综合贴近度公式,实现最优方案与理想方案的位置与曲线形状的一致性;最后通过制造系统内流动控制实例说明了所提出有序梯形模糊灰色关联TOPSIS方法的可行性和有效性。     

An extreme-distance approach to multiple criteria ranking

A distance approach based on extreme points, or predefined ideal and anti-ideal points, is proposed to improve on the TOPSIS (Technique for Order Performance [or Ordered Preference] by Similarity to Ideal Solution) method of multiple criteria ranking. Two case studies demonstrate how the analysis procedure works, and provide a basis for comparison of the proposed method to the original TOPSIS and similar methods. In applications, the new method produces results that are generally consistent with the original technique, but offers new features such as a clear interpretation of extreme points, more flexibility in setting extreme points, no normalization distortion, and the ability to handle non-monotonic criteria.     

区间数型模糊VIKOR方法

针对备选方案的属性值和各属性权重为区间数的多属性决策问题,讨论了区间数型模糊V IKOR方法。该方法以最接近理想解为基本思想,在决策过程中采用线性规范方法,利用心态指标对区间数进行排序,在可接受优势和决策过程的稳定条件下对备选方案进行选择,得到折衷解,实现了群体效用最大化,个体遗憾最小化。最后,在最大群体效用权重为0.5的情况下,用实例说明了该方法的有效性和可行性,结果显示不同的心态,最后的选择是不同的。     

三角直觉模糊数型VIKOR方法

针对备选方案的属性值为三角直觉模糊数且权重为实数的多属性决策问题,研究了三角直觉模糊数型VIKOR方法。首先,本文提出了一种基于偏好指标的三角直觉模糊数排序方法;其次,根据VIKOR方法的基本思想,提出了求解三角直觉模糊数型VIKOR方法的步骤,并在可接受优势和决策过程的稳定条件下对备选方案进行排序,得到折衷解;最后,在最大群体效用权重为0.5的情况下,用第三方物流服务商选择为例说明了该方法的有效性和可行性。     

An extension of TOPSIS for group decision making

An extension of TOPSIS (technique for order performance by similarity to ideal solution), a multi-attribute decision making (MADM) technique, to a group decision environment is investigated. TOPSIS is a practical and useful technique for ranking and selection of a number of externally determined alternatives through distance measures. To get a broad view of the techniques used, we provide a few options for the operations, such as normalization, distance measures and mean operators, at each of the corresponding steps of TOPSIS. In addition, the preferences of more than one decision maker are internally aggregated into the TOPSIS procedure. Unlike in previous developments, our group preferences are aggregated within the procedure. The proposed model is indeed a unified process and it will be readily applicable to many real-world decision making situations without increasing the computational burden. In the final part, the effects of external aggregation and internal aggregation of group preferences for TOPSIS with different computational combinations are compared using examples. The results have demonstrated our model to be both robust and efficient.     

Weight determination for consistently ranking alternatives in multiple criteria decision analysis

One of the most difficult tasks in multiple criteria decision analysis (MCDA) is determining the weights of individual criteria so that all alternatives can be compared based on the aggregate performance of all criteria. This problem can be transformed into the compromise programming of seeking alternatives with a shorter distance to the ideal or a longer distance to the anti-ideal despite the rankings based on the two distance measures possibly not being the same. In order to obtain consistent rankings, this paper proposes a measure of relative distance, which involves the calculation of the relative position of an alternative between the anti-ideal and the ideal for ranking. In this case, minimizing the distance to the ideal is equivalent to maximizing the distance to the anti-ideal, so the rankings obtained from the two criteria are the same. An example is used to discuss the advantages and disadvantages of the proposed method, and the results are compared with those obtained from the TOPSIS method.     

基于一类新的Picture模糊距离的VIKOR多属性决策方法

针对现有Picture模糊距离的不足。本文构建了一种带有参数的Picture模糊距离,该参数能够反映决策者的态度偏好。其次,将新距离拓展到多准则妥协解排序法(VIKOR)中,并利用新距离计算各备选方案的群体效益值和个体遗憾值进而获得决策结果。最后,通过算例验证所提决策方法的有效性和优点,并对参数进行灵敏度分析。     

The interval-valued fuzzy TOPSIS method and experimental analysis

The purpose of this paper is to extend the TOPSIS method based on interval-valued fuzzy sets in decision analysis. Hwang and Yoon developed the technique for order preference by similarity to ideal solution (TOPSIS) in 1981. TOPSIS has been widely used to rank the preference order of alternatives and determine the optimal choice. Considering the fact that it is difficult to precisely attach the numerical measures to the relative importance of the attributes and to the impacts of the alternatives on these attributes in some cases, therefore, the TOPSIS method has been extended for interval-valued fuzzy data in this paper. In addition, a comprehensive experimental analysis to observe the interval-valued fuzzy TOPSIS results yielded by different distance measures is presented. A comparative analysis of interval-valued fuzzy TOPSIS rankings from each distance measure is illustrated with discussions on consistency rates, contradiction rates, and average Spearman correlation coefficients. Finally, a second-order regression model is provided to highlight the effects of the number of alternatives, the number of attributes, and distance measures on average Spearmen correlation coefficients.     

一种基于前景理论和改进TOPSIS的模糊随机多准则决策方法及其应用

针对准则值为区间灰数直觉模糊数、准则权系数部分已知以及自然状态出现概率为灰数的多准则决策问题,提出一种结合前景理论和改进TOPSIS的决策方法。该方法首先定义了灰色直觉模糊数的前景价值函数和概率权重函数,并利用前景理论构建出前景决策矩阵;接着从两个方面对传统TOPSIS决策方法进行改进:(1)过定义方案间综合差异的概念,采用离差最大化思想,建立平均综合差异最大化规划模型,给出了一种兼顾主客观权重信息确定准则权系数的新方法;(2)用灰关联替换备选方案与正负理想方案的距离,据此刻画了各方案与正负理想方案的贴近度。进而利用改进TOPSIS决策方法中的综合贴近度对方案进行了排序。最后通过实例验证了该方法的有效性。     

A ratio model for discriminant analysis using linear programming

The innovative hybrid model for discriminant analysis via linear programming, was introduced along with the β normalization, which was subsequently replaced by a new normalization. The primary weakness of the β normalization, and the reason for replacing it with the new normalization, is that it distorts solutions and consequently does not always find the best solution. It is shown here, that unfortunately, both normalizations are affected by distance distortion. In addition, whether a model finds the best solution is highly dependent on the criterion or criteria by which “best solution” is defined. A ratio criteria and associated ratio model are proposed, which avoid the problems associated with distance distortion. The nonlinear ratio may be linearized in much the same way that Data Envelopment Analysis is linearized.