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.     

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。     

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.     

三角模糊数去模糊化对VIKOR妥协解的影响研究

针对准则值和准则权重均为三角模糊数的多准则决策问题,研究了不同三角模糊数去模糊化方法适用的数学运算规则,应用VIKOR方法进行三角模糊数去模糊化的必要环节和前提条件,分析了去模糊化对群体效用值和个体遗憾值以及对妥协解的影响机制,给出了一种拓展的VIKOR方法的决策步骤,最后运用算例说明了方法的实施过程和有效性。     

Compromise solution by MCDM methods: A comparative analysis of VIKOR and TOPSIS

The multiple criteria decision making (MCDM) methods VIKOR and TOPSIS are based on an aggregating function representing “closeness to the ideal”, which originated in the compromise programming method. In VIKOR linear normalization and in TOPSIS vector normalization is used to eliminate the units of criterion functions. 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”. 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. A comparative analysis of these two methods is illustrated with a numerical example, showing their similarity and some differences.     

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.     

On the algebraic solution of fuzzy linear systems based on interval theory

In this paper, fuzzy linear systems involving a crisp square matrix and a fuzzy right-hand side vector are considered. A new approach to solve such systems based on interval theory and the new concept “interval inclusion linear system” is proposed. Also, new necessary and sufficient conditions are derived for obtaining the unique algebraic solution. Numerical examples are given to illustrate the efficiency of the proposed method.     

基于泛性模糊数的VIKOR方法研究

建立了一种泛性模糊数可比较的度量,对决策信息通常为泛性模糊数的决策问题进行加工和扩展,提出了基于泛性模糊数不确定信息的VIKOR决策方法,实现了属性为泛性模糊数的多属性群决策及信息融合的目的.     

A VIKOR-based method for hesitant fuzzy multi-criteria decision making

Since it was firstly introduced by Torra and Narukawa (The 18th IEEE International Conference on Fuzzy Systems, Jeju Island, Korea, 2009, pp. 1378–1382), the hesitant fuzzy set has attracted more and more attention due to its powerfulness and efficiency in representing uncertainty and vagueness. This paper extends the classical VIKOR (vlsekriterijumska optimizacija i kompromisno resenje in serbian) method to accommodate hesitant fuzzy circumstances. Motivated by the hesitant normalized Manhattan distance, we develop the hesitant normalized Manhattan $L_p$ —metric, the hesitant fuzzy group utility measure, the hesitant fuzzy individual regret measure, and the hesitant fuzzy compromise measure. Based on these new measures, we propose a hesitant fuzzy VIKOR method, and a practical example is provided to show that our method is very effective in solving multi-criteria decision making problems with hesitant preference information.     

一种基于证据推理的信息不完全的多准则决策方法

针对权系数信息不完全、准则值不确定且不完全的多准则决策问题,提出了一种基于证据推理的方法.该方法通过证据推理算法构造方案的目标函数,结合不完全信息的权系数建立非线性规划模型,使用遗传算法求解模型得到效用值的区间数,从而得到整个方案集的排序.最后以实例表明该方法的有效性和可行性.     

Minimum variance capacity identification

In the framework of multi-criteria decision making whose aggregation process is based on the Choquet integral, we present a maximum entropy like method enabling to determine, if it exists, the “least specific” capacity compatible with the initial preferences of the decision maker. The proposed approach consists in solving a strictly convex quadratic program whose objective function is equivalently either the opposite of a generalized entropy measure or the variance of the capacity. The application of the proposed approach is illustrated on two examples.     

Multi-criteria classification – A new scheme for application of dominance-based decision rules

We are considering the problem of multi-criteria classification. In this problem, a set of “if … then …” decision rules is used as a preference model to classify objects evaluated by a set of criteria and regular attributes. Given a sample of classification examples, called learning data set, the rules are induced from dominance-based rough approximations of preference-ordered decision classes, according to the Variable Consistency Dominance-based Rough Set Approach (VC-DRSA). The main question to be answered in this paper is how to classify an object using decision rules in situation where it is covered by (i) no rule, (ii) exactly one rule, (iii) several rules. The proposed classification scheme can be applied to both, learning data set (to restore the classification known from examples) and testing data set (to predict classification of new objects). A hypothetical example from the area of telecommunications is used for illustration of the proposed classification method and for a comparison with some previous proposals.     

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.     

Multi-objective geometric programming problem with ∊-constraint method

In multi-objective geometric programming problem there are more than one objective functions. There is no single optimal solution which simultaneously optimizes all the objective functions. Under these conditions the decision makers always search for the most “preferred” solution, in contrast to the optimal solution. A few mathematical programming methods namely fuzzy programming, goal programming and weighting methods have been applied in the recent past to find the compromise solution. In this paper ?$?$-constraint method has been applied to find the non-inferior solution. A brief solution procedure of ?$?$-constraint method has been presented to find the non-inferior solution of the multi-objective programming problems. Further, the multi-objective programming problems is solved by the fuzzy programming technique to find the optimal compromise solution. Finally, two numerical examples are solved by both the methods and compared with their obtained solutions.     

Behavioral multi-criteria decision analysis: the TODIM method with criteria interactions

In this paper a multi-criteria decision aiding model is developed through the use of the Choquet integral. The proposed model is an extension of the TODIM method, which is based on nonlinear Cumulative Prospect Theory. The paper starts by reviewing the first steps of behavioral decision theory. A presentation of the TODIM method follows. The basic concepts of the Choquet integral as related to multi-criteria decision aiding are reviewed. It is also shown how the measures of dominance of the TODIM method can be rewritten through the application of the Choquet integral. From the ordering of decision criteria the fuzzy measures of criteria interactions are computed, which leads to the ranking of alternatives. A case study on the forecasting of property values for rent in a Brazilian city illustrates the proposed model. Results obtained from the use of the Choquet integral are then compared against a previously made usage of the TODIM method. It is concluded that significant advantages exist derived from the use of the Choquet integral. The paper closes with recommendations for future research.     

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.     

An efficient and simple model for multiple criteria supplier selection problem

Simply looking for vendors offering the lowest prices is not “efficient sourcing” any more. Selection of suppliers is a multiple criteria decision. We propose a weighted linear program for the multi-criteria supplier selection problem. In addition to mathematical formulation, this paper studies a transformation technique which enables our proposed model to be solved without an optimizer. The model for multi-criteria supplier selection problem can be easily implemented with a spreadsheet package. The model can be widely applied to practical situations and does not require the user with any optimization background.     

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.     

Induced aggregation operators in the VIKOR method and its application in material selection

In this paper, a hybrid decision making approach integrating induced aggregation operators into VIKOR is proposed for tackling multicriteria problems with conflicting and noncommensurable (different units) criteria. For doing so, we develop a new distance aggregation operator called the induced ordered weighted averaging standardized distance (IOWASD) operator. It is an aggregation operator that provides a wide range of standardized distance measures between the maximum and the minimum by using the induced OWA (IOWA) operator. The main advantage of the IOWA-based VIKOR (IOWA-VIKOR) is that it is able to reflect the complex attitudinal character of the decision maker by using order inducing variables and provide much more complete information for decision making. We also studied some of the IOWASD’s main properties and different particular cases and further generalized it by using the induced generalized OWA (IGOWA) operator. Finally, we apply the integrated IOWA-VIKOR method in a multi-criteria decision making problem regarding the selection of materials and the results are compared for different types of standardized distance aggregation operators.