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.     

Stackelberg solutions for fuzzy random two-level linear programming through level sets and fractile criterion optimization

This paper considers Stackelberg solutions for two-level linear programming problems under fuzzy random environments. To deal with the formulated fuzzy random two-level linear programming problem, an α-stochastic two-level linear programming problem is defined through the introduction of α-level sets of fuzzy random variables. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced and the α-stochastic two-level linear programming problem is transformed into the problem to maximize the satisfaction degree for each fuzzy goal. Through fractile criterion optimization in stochastic programming, the transformed stochastic two-level programming problem can be reduced to a deterministic two-level programming problem. An extended concept of Stackelberg solution is introduced and a numerical example is provided to illustrate the proposed method.     

Multi-person multi-attribute decision making models under intuitionistic fuzzy environment

Intuitionistic fuzzy numbers, each of which is characterized by the degree of membership and the degree of non-membership of an element, are a very useful means to depict the decision information in the process of decision making. In this article, we investigate the group decision making problems in which all the information provided by the decision makers is expressed as intuitionistic fuzzy decision matrices where each of the elements is characterized by intuitionistic fuzzy number, and the information about attribute weights is partially known, which may be constructed by various forms. We first use the intuitionistic fuzzy hybrid geometric (IFHG) operator to aggregate all individual intuitionistic fuzzy decision matrices provided by the decision makers into the collective intuitionistic fuzzy decision matrix, then we utilize the score function to calculate the score of each attribute value and construct the score matrix of the collective intuitionistic fuzzy decision matrix. Based on the score matrix and the given attribute weight information, we establish some optimization models to determine the weights of attributes. Furthermore, we utilize the obtained attribute weights and the intuitionistic fuzzy weighted geometric (IFWG) operator to fuse the intuitionistic fuzzy information in the collective intuitionistic fuzzy decision matrix to get the overall intuitionistic fuzzy values of alternatives by which the ranking of all the given alternatives can be found. Finally, we give an illustrative example.     

A method of fuzzy preference relation to authorize the worldwide interoperability for microwave access license in Taiwan

This study proposes a preference relation based evaluation framework to help the National Communication Commission (NCC) in Taiwan authorize a worldwide interoperability for microwave access (WiMAX) license under a fuzzy environment where the uncertainty, subjectivity and vagueness are dealt with linguistic variables parameterized by triangular fuzzy numbers. This study applies the fuzzy multi-criteria decision making approach to determine the importance weights of evaluation criteria and consolidate the performance ratings of possible alternatives. Aggregated the evaluators’ opinions toward the criteria and alternatives, the fuzzy preference relation approach is utilized to obtain the non-dominated degree of each alternative for the decision makers to make a final decision. Simultaneously, an empirical case involving sixteen quantitative and fifteen qualitative evaluation criteria, thirteen telecommunication applicants assessed by twelve specialists from various fields of telecommunication industry in Taiwan is solicited to demonstrate the proposed approach.     

An integrated model-based interactive approach to FMAGDM with incomplete preference information

Incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations are very useful to express decision makers’ incomplete preferences over attributes or alternatives in the process of decision making under fuzzy environments. The aim of this paper is to investigate fuzzy multiple attribute group decision making problems where the attribute values are represented in intuitionistic fuzzy numbers and the information on attribute weights is provided by decision makers by means of one or some of the different preference structures, including weak ranking, strict ranking, difference ranking, multiple ranking, interval numbers, incomplete fuzzy preference relations, incomplete multiplicative preference relations, and incomplete linguistic preference relations. We transform all individual intuitionistic fuzzy decision matrices into the interval decision matrices and construct their expected decision matrices, and then aggregate all these expected decision matrices into a collective one. We establish an integrated model by unifying the collective decision matrix and all the given different structures of incomplete weight preference information, and develop an integrated model-based approach to interacting with the decision makers so as to adjust all the inconsistent incomplete fuzzy preference relations, inconsistent incomplete linguistic preference relations and inconsistent incomplete multiplicative preference relations into the ones with acceptable consistency. The developed approach can derive the attribute weights and the ranking of the alternatives directly from the integrated model, and thus it has the following prominent characteristics: (1) it does not need to construct the complete fuzzy preference relations, complete linguistic preference relations and complete multiplicative preference relations from the incomplete fuzzy preference relations, incomplete linguistic preference relations and incomplete multiplicative preference relations, respectively; (2) it does not need to unify the different structures of incomplete preferences, and thus can simplify the calculation and avoid distorting the given preference information; and (3) it can sufficiently reflect and adjust the subjective desirability of decision makers in the process of interaction. A practical example is also provided to illustrate the developed approach.     

Mental representations of important real-life decisions

Two studies investigated how decision makers characterize alternatives in important real-life decisions, which they themselves had made (to leave a partner, to choose an education and to choose a home). First, the participants indicated a very high degree of involvement in the decisions studied and about half of the participants gave maximum involvement ratings for the partner decision. Second, the results showed that concepts that are essential in most decision theories, such as, consequence, probability and value were important characteristics of the decisions. Third, emotion, positive and negative affect were also important characteristics. Fourth, value and emotion were uncorrelated. Fifth, the patterns of characteristics of decisions made in the past did not differ markedly from the characteristics given to future decisions. Principal component analyses were performed on the ratings of applicability of the different characteristics across participants for each decision situation. Three factors were extracted. There was one factor for positive affect/emotions and another factor for negative affect/emotions verified in oblique solutions. Thus, different scales are needed to represent emotion/affect components (and not bipolar scales) in real-life important decisions. The third factor represented the way in which a decision was represented (moving pictures dialogue etc.). An analysis restricted to the participants who rated 100% involvement showed an additional fourth factor with “what others would think”, “similar situations”, “values” and “money” as the most prominent characteristics. This points to the importance of controlling for participant involvement in studies of human decision making to enable generalizations to real-life decisions.     

Exploring the critical pillars and causal relations within the NRI: An innovative approach

The Global Information Technology Report released by the World Economic Forum (WEF) has employed networked readiness index (NRI) to measure the global competitiveness of a country’s information and communication technologies (ICT) diffusion. The final NRI overall scores were measured by an arithmetic mean aggregation of the composite pillars scores, which implicitly assumed that all the pillars have constant weights. The Report did not explore the critical pillars and causal relations for better decision making. To add values to this Report, the objective of this paper is to propose an innovative approach by using data mining techniques and partial least squares path modeling to scrutinize the critical pillars within the NRI and to further explore the causal relations amongst them. An empirical analysis based on the latest Report (2009-2010) is carried out. The results show that “business usage,” “business readiness,” and “market environment” are the three root drivers—critical pillars to manipulate the NRI overall scores; whereas “government readiness,” which is further mostly affected by the “government usage,” is the foremost enabler to the NRI overall scores. Based on the results, policy makers are suggested to allocate limited resources with priority to the three root drivers and one foremost enabler to frog-leap the global competitiveness of national ICT diffusion.     

Comparison of first aggregation and last aggregation in fuzzy group TOPSIS

The Technique for Order Preference by Similarity to the Ideal Solution (TOPSIS), one of the major multi attribute decision making (MADM) techniques, ranks the alternatives according to their distances from the ideal and the negative ideal solution. In real evaluation and decision making problems, it is vital to involve several people and experts from different functional areas in decision making process. Also under many conditions, crisp data are inadequate to model real-life situations, since human judgments including preferences are often vague and cannot estimate his preference with an exact numerical value. Therefore aggregation of fuzzy concept, group decision making and TOPSIS methods that we denote “fuzzy group TOPSIS” is more practical than original TOPSIS.     

Stackelberg solutions for fuzzy random two-level linear programming through probability maximization with possibility

This paper considers Stackelberg solutions for decision making problems in hierarchical organizations under fuzzy random environments. Taking into account vagueness of judgments of decision makers, fuzzy goals are introduced into the formulated fuzzy random two-level linear programming problems. On the basis of the possibility and necessity measures that each objective function fulfills the corresponding fuzzy goal, together with the introduction of probability maximization criterion in stochastic programming, we propose new two-level fuzzy random decision making models which maximize the probabilities that the degrees of possibility and necessity are greater than or equal to certain values. Through the proposed models, it is shown that the original two-level linear programming problems with fuzzy random variables can be transformed into deterministic two-level linear fractional programming problems. For the transformed problems, extended concepts of Stackelberg solutions are defined and computational methods are also presented. A numerical example is provided to illustrate the proposed methods.     

Strategic and operational decisions in restaurant revenue management

The paper addresses restaurant revenue management from both a strategic and an operational point of view. Strategic decisions in restaurants are mainly related to defining the most profitable combination of tables that will constitute the restaurant. We propose new formulations of the so-called “Tables Mix Problem” by taking into account several features of the real setting. We compare the proposed models in a computational study showing that restaurants, with the capacity of managing tables as renewable resources and of combining different-sized tables, can improve expected revenue performances. Operational decisions are mainly concerned with the more profitable assignment of tables to customers. Indeed, the “Parties Mix Problem” consists of deciding on accepting or denying a booking request from different groups of customers, with the aim of maximizing the total expected revenue. A dynamic formulation of the “Parties Mix Problem” is presented together with a linear programming approximation, whose solutions can be used to define capacity control policies based on booking limits and bid prices. Computational results compare the proposed policies and show that they lead to higher revenues than the traditional strategies used to support decision makers.     

A new multiple attribute group decision making method in intuitionistic fuzzy setting

The multiple attribute group decision making (MAGDM) problem with intuitionistic fuzzy information investigated in this paper is very useful for solving complicated decision problems under uncertain circumstances. Since experts have their own characteristics, they are familiar with some of the attributes, but not others, the weights of the decision makers to different attributes should be different. We derive the weights of the decision makers by aggregating the individual intuitionistic fuzzy decision matrices into a collective intuitionistic fuzzy decision matrix. The expert has a big weight if his evaluation value is close to the mean value and has a small weight if his evaluation value is far from the mean value. For the incomplete attribute weight information, we establish some optimization models to determine the attribute weights. Furthermore, we develop several algorithms for ranking alternatives under different situations, and then extend the developed models and algorithms to the MAGDM problem with interval-valued intuitionistic fuzzy information. Numerical results finally illustrate the practicality and efficiency of our new algorithms.     

Fuzzy linear programming models for NPD using a four-phase QFD activity process based on the means-end chain concept

Quality function deployment (QFD) is a customer-driven approach in processing new product development (NPD) to maximize customer satisfaction. Determining the fulfillment levels of the “hows”, including design requirements (DRs), part characteristics (PCs), process parameters (PPs) and production requirements (PRs), is an important decision problem during the four-phase QFD activity process for new product development. Unlike previous studies, which have only focused on determining DRs, this paper considers the close link between the four phases using the means-end chain (MEC) concept to build up a set of fuzzy linear programming models to determine the contribution levels of each “how” for customer satisfaction. In addition, to tackle the risk problem in NPD processes, this paper incorporates risk analysis, which is treated as the constraint in the models, into the QFD process. To deal with the vague nature of product development processes, fuzzy approaches are used for both QFD and risk analysis. A numerical example is used to demonstrate the applicability of the proposed model.     

Shapley mappings and the cumulative value for n-person games with fuzzy coalitions

In this paper we prove existence and uniqueness of the so-called Shapley mapping, which is a solution concept for a class of n-person games with fuzzy coalitions whose elements are defined by the specific structure of their characteristic functions. The Shapley mapping, when it exists, associates to each fuzzy coalition in the game an allocation of the coalitional worth satisfying the efficiency, the symmetry, and the null-player conditions. It determines a “cumulative value” that is the “sum” of all coalitional allocations for whose computation we provide an explicit formula.     

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。     

Consistency analysis of triangular fuzzy reciprocal preference relations

In order to simulate the uncertainty associated with impression or vagueness, a decision maker may give her/his judgments by means of triangular fuzzy reciprocal preference relations in the process of decision making. The study of their consistency becomes a very important aspect to avoid a misleading solution. Based on the reciprocity property, this paper proposes a new definition of consistent triangular fuzzy reciprocal preference relations. The new definition is different from that reduced by consistent fuzzy reciprocal preference relations proposed by Buckley (1985). The properties of consistent triangular fuzzy reciprocal preference relations in the light of the new definition are studied in detail. In addition, the shortcomings of the proof procedure of the proposition given by Wang and Chen (2008) are pointed out. And the proposition is reproved by using the new definition of consistent triangular fuzzy reciprocal preference relations. Finally, using the (n − 1) restricted comparison ratios, a method for obtaining consistent triangular fuzzy reciprocal preference relations is proposed, and an algorithm is shown to make a consistent decision ranking. Numerical results are further calculated to illustrate the new definition and the obtained algorithm.     

区间直觉模糊集的模糊熵群决策方法

针对专家权重未知、专家判断信息以区间直觉模糊集给出的多属性群决策问题,提出了一种新的模糊熵决策方法。通过定义区间直觉模糊集的模糊熵判断专家信息的模糊程度,进而确定每位专家的权重;然后计算备选方案距理想方案和负理想方案的模糊交叉熵距离,得到每个专家对方案的排序;再分别利用加权算术算子和加权几何算子集结专家的排序结果,得到专家群体对方案的排序。实例分析验证了方法的有效性。     

From decision theory to decision aiding methodology

The paper presents the author’s partial and personal historical reconstruction of how decision theory is evolving to a decision aiding methodology. The presentation shows mainly how “alternative” approaches to classic decision theory evolved. In the paper it is claimed that all such decision “theories” share a common methodological feature, which is the use of formal and abstract languages as well as of a model of rationality. Different decision aiding approaches can thus be defined, depending on the origin of the model of rationality used in the decision aiding process. The concept of decision aiding process is then introduced and analysed. The paper’s ultimate claim is that all such decision aiding approaches can be seen as part of a decision aiding methodology.     

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.     

Meta-data envelopment analysis: Finding a direction towards marginal profit maximization

This paper discusses a new meta-DEA approach to solve the problem of choosing direction vectors when estimating the directional distance function. The proposed model emphasizes finding the “direction” for productivity improvement rather than estimating the “score” of efficiency; focusing on “planning” over “evaluation”. In fact, the direction towards marginal profit maximization implies a step-by-step improvement and “wait-and-see” decision process, which is more consistent with the practical decision-making process. An empirical study of U.S. coal-fired power plants operating in 2011 validates the proposed model. The results show that the efficiency measure using the proposed direction is consistent with all other indices with the exception of the direction towards the profit-maximized benchmark. We conclude that the marginal profit maximization is a useful guide for determining direction in the directional distance function.     

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.