An exact global optimization method for deriving weights from pairwise comparison matrices

Some multiple-criteria decision making methods rank actions by associating weights to the different criteria or actions, which are pairwise compared via a positive reciprocal matrix A. There is a vast literature on proposals of different mathematical-programming methods to infer weights from such matrix A. However, it is seldom observed that such optimization problems may be multimodal, thus the standard local-search resolution techniques suggested may be trapped in local optima, yielding a wrong ranking of alternatives. In this note we show that standard tools of global optimization based on interval analysis, lead to globally optimal weights in reasonable time.     

A lexicographical compromise method for multiple criteria group decision problems with imprecise information

This paper addresses multiple criteria group decision making problems where each group member offers imprecise information on his/her preferences about the criteria. In particular we study the inclusion of this partial information in the decision problem when the individuals’ preferences do not provide a vector of common criteria weights and a compromise preference vector of weights has to be determined as part of the decision process in order to evaluate a finite set of alternatives. We present a method where the compromise is defined by the lexicographical minimization of the maximum disagreement between the value assigned to the alternatives by the group members and the evaluation induced by the compromise weights.     

Evaluation of rankings with regard to the possible number of agreements and conflicts

An interesting problem in group decision analysis is how many different agreements can occur, or conversely disagreements may exist, between two or more different rankings of a set of alternatives. In this paper it is assumed that a reference ranking has been established for the set of alternatives. This reference ranking may represent the ranking of a high authority decision maker or be just a virtual ranking to be used in determining the discrepancy between pairs of rankings. Then, the problem examined here is to evaluate the number of possible rankings when the ranking method is the number of agreements with some reference ranking. The analysis presented here illustrates that this problem is not trivial and moreover, its simple context conceals complexity in its depth. The purpose of this paper is to provide an evaluation of the number of possible agreements in rankings given to a set of concepts, alternatives or ideas, by two or more decision makers. The number of possible agreements takes on the values 0, 1, 2,…, n − 2, or n when n concepts are compared. This paper develops a recursive closed form formula for calculating the frequencies for the various numbers of agreements.     

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.     

Modeling subjective evaluation for fuzzy group multicriteria decision making

This paper presents a new fuzzy multicriteria decision making (MCDM) approach for evaluating decision alternatives involving subjective judgements made by a group of decision makers. A pairwise comparison process is used to help individual decision makers make comparative judgements, and a linguistic rating method is used for making absolute judgements. A hierarchical weighting method is developed to assess the weights of a large number of evaluation criteria by pairwise comparisons. To reflect the inherent imprecision of subjective judgements, individual assessments are aggregated as a group assessment using triangular fuzzy numbers. To obtain a cardinal preference value for each decision alternative, a new fuzzy MCDM algorithm is developed by extending the concept of the degree of optimality to incorporate criteria weights in the distance measurement. An empirical study of aircraft selection is presented to illustrate the effectiveness of the approach.     

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.     

Approach to group decision making based on determining the weights of experts by using projection method

The aim of this paper is to present a new approach for determining weights of experts in the group decision making problems. Group decision making has become a very active research field over the last decade. Especially, the investigation to determine weights of experts for group decision making has attracted great interests from researchers recently and some approaches have been developed. In this paper, the weights of experts are determined in the group decision environment via projection method. First of all, the average decision of all individual decisions is defined as the ideal decision. After that, the weight of expert is determined by the projection of individual decision on the ideal decision. By using the weights of experts, all individual decisions are aggregate into a collective decision. Then an ideal solution of alternatives of the collective decision, expressed by a vector, is determined. Further, the preference order of alternatives are ranked in accordance with the projections of alternatives on the ideal solution. Comparisons with an extended TOPSIS method are also made. Finally, an example is provided to illustrate the developed approach.     

A method for stochastic multiple criteria decision making based on pairwise comparisons of alternatives with random evaluations

This paper proposes a method for solving stochastic multiple criteria decision making (MCDM) problems, where evaluations of alternatives on considered criteria are random variables with known probability density functions or probability mass functions. Probabilities on all possible results of pairwise comparisons of alternatives are first calculated using Probability Theory. Then, all possible results of pairwise comparisons are classified into superior, indifferent and inferior ones using a predefined identification rule. Consequently, the probabilities on all possible results of pairwise comparisons are partitioned into superior, indifferent and inferior probabilities. Furthermore, based on the derived probabilities, an algorithm is developed to rank the alternatives. Finally, a numerical example is used to illustrate the feasibility and validity of the proposed method.     

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.     

Developing a straightforward approach for group decision making based on determining weights of decision makers

Group decision making is an active area of research within multiple attribute decision making. This paper assumes that all the decision makers (DMs) are not equally qualified to contribute equitably to the decision process. The aim of this paper is to develop an approach to determine weights of DMs, in which the decision information on alternatives with respect to attributes, provided by each DM, is represented in the form of interval data. We define the average of all individual decisions as the positive ideal decision (PID), and the maximum separation from PID as the negative ideal decision, which are characterized by a matrix, respectively. The weight of each DM is determined according to the Euclidean distances between the individual decision and ideal decisions. By using the obtained weights of DMs, all individual decisions are aggregated into a collective decision. Then the alternatives is ranked based on the collective decision. Meanwhile, this paper also gives a humanized decision method by using an optimistic coefficient, which is used in adjusting the relative importance between profit and risk. Finally, we give an example to illustrate the developed approach.     

基于ELECTRE方法的模糊多属性群决策

结合模糊集理论知识,将经典的ELECTRE方法拓展到模糊环境下,用来解决模糊多属性群决策问题.决策问题中涉及到的属性评价值与属性权值是以三角模糊数的形式给出.首先集结各决策者的评价矩阵,构造模糊群决策矩阵,在规范化的基础上,分别对其进行模糊和谐性检验与模糊非和谐性检验,最后通过设定否决门槛值得到一致性比较矩阵,最终评价结果也是以三角模糊数的形式给出.该方法操作方便,计算简单.给出的实例验证了此方法的可行性和有效性.     

基于广义WOWA算子的Pythagorean模糊决策方法

针对Pythagorean模糊信息的决策问题,构建广义Pythagorean模糊信息加权有序加权平均(PF-GWOWA)算子。首先,提出PF-GWOWA算子,并证明Pythagorean模糊广义加权平均(PF-GWA)算子、Pythagorean模糊加权有序加权平均(PF-WOWA)算子与Pythagorean模糊加权平均(PF-WA)算子均为PF-GWOWA算子的特例;其次,根据GWOWA算子属性综合权重计算模型,利用PF-GWOWA算子对信息进行集结;最后,通过算例分析和传统方法对比,说明本文提出方法的合理性与有效性。     

Projection method for multiple criteria group decision making with incomplete weight information in linguistic setting

Multiple criteria group decision making (MCGDM) problems have become a very active research field over the last decade. Many practical problems are often characterized by MCGDM. The aim of this paper is to develop a new approach for MCGDM problems with incomplete weight information in linguistic setting based on the projection method. Firstly, to reflect the reality accurately, a method to determine the weights of decision makers in linguistic setting is proposed by calculating the degree of similarity between 2-tuple linguistic decision matrix given by each decision maker and the average 2-tuple linguistic decision matrix. By using the weights of decision makers, all individual 2-tuple linguistic decision matrices are aggregated into a collective one. Then, to determine the weight vector of criteria, we establish a non-linear optimization model based on the basic ideal of the projection method, i.e., the optimal alternative should have the largest projection on the 2-tuple linguistic positive ideal solution (TLPIS). Calculate the 2-tuple linguistic projection of each alternative on the TLPIS and rank all the alternatives according to the 2-tuple linguistic projection value. Finally, an illustrative example is given to demonstrate the calculation process of the proposed method, and the validity is verified by comparing the evaluation results of the proposed method with that of the technique for order preference by similarity to ideal solution (TOPSIS) method.     

Multi-attribute decision making: A simulation comparison of select methods

Several methods have been proposed for solving multi-attribute decision making problems (MADM). A major criticism of MADM is that different techniques may yield different results when applied to the same problem. The problem considered in this study consists of a decision matrix input of N criteria weights and ratings of L alternatives on each criterion. The comparative performance of some methods has been investigated in a few, mostly field, studies. In this simulation experiment we investigate the performance of eight methods: ELECTRE, TOPSIS, Multiplicative Exponential Weighting (MEW), Simple Additive Weighting (SAW), and four versions of AHP (original vs. geometric scale and right eigenvector vs. mean transformation solution). Simulation parameters are the number of alternatives, criteria and their distribution. The solutions are analyzed using twelve measures of similarity of performance. Similarities and differences in the behavior of these methods are investigated. Dissimilarities in weights produced by these methods become stronger in problems with few alternatives; however, the corresponding final rankings of the alternatives vary across methods more in problems with many alternatives. Although less significant, the distribution of criterion weights affects the methods differently. In general, all AHP versions behave similarly and closer to SAW than the other methods. ELECTRE is the least similar to SAW (except for closer matching the top-ranked alternative), followed by MEW. TOPSIS behaves closer to AHP and differently from ELECTRE and MEW, except for problems with few criteria. A similar rank-reversal experiment produced the following performance order of methods: SAW and MEW (best), followed by TOPSIS, AHPs and ELECTRE. It should be noted that the ELECTRE version used was adapted to the common MADM problem and therefore it did not take advantage of the method's capabilities in handling problems with ordinal or imprecise information.     

群体多目标决策联合有效解类的几何特性

群体多目标决策是群体决策和多目标决策的一个交叉研究领域,借助供选方案的有效数,文[1]引进了群体多目标决策问题的联合有效解类概念,并且建立了这些解类的K－T最优条件,本文研究这类解的几何特性,得到若干基本的必要条件一充分条件。     

On the extent analysis method for fuzzy AHP and its applications

In a paper by Chang [D.Y. Chang, Applications of the extent analysis method on fuzzy AHP, European Journal of Operational Research 95 (1996) 649–655], an extent analysis method on fuzzy AHP was proposed to obtain a crisp priority vector from a triangular fuzzy comparison matrix. It is found that the extent analysis method cannot estimate the true weights from a fuzzy comparison matrix and has led to quite a number of misapplications in the literature. In this paper, we show by examples that the priority vectors determined by the extent analysis method do not represent the relative importance of decision criteria or alternatives and that the misapplication of the extent analysis method to fuzzy AHP problems may lead to a wrong decision to be made and some useful decision information such as decision criteria and fuzzy comparison matrices not to be considered. We show these problems to avoid any possible misapplications in the future.     

An environmental input-output model with multiple criteria

It is often claimed that there is a trade-off between economic goals and the quality of the environment. For this reason, an environmental input-output optimization model with multiple objectives is formulated. The criteria are the minimization of factor costs to produce the Gross National Product and the minimization of net pollution for a given level of final demand. Using the LeChatelier-Samuelson principle, we analyze the changes in the production of the sectors and in the prices of the goods (described by the dual model) due to the change in the preferences of the decision makers. It can be shown that higher weights for the environmental objectives imply — in tendency —non-decreasing production of the sectors andnon-decreasing abatement activities. The changes of prices are ambiguous. The condition for increasing prices is given. To some degree, the opposite results can be achieved, if maximization of the value of final demand (or of private consumption) and minimization of net pollution under the constraints for primary input are taken as objective functions. In this case, increasing weights for environmental goals will leadin tendency tonon-increasing final demand and tonon-increasing net pollution. Under given conditions, higher environmental quality will be achieved bynon-increasing gross production and abatement activities.     

基于Pythagorean模糊熵的风险型决策方法——考虑后悔与失望规避

针对决策信息为Pythagorean模糊数,属性权重完全未知的风险型多属性决策问题,提出了一种基于Pythagorean模糊熵的考虑决策者后悔与失望规避心理行为的决策方法。首先,计算备选方案和理想点各属性的效用值,从而获得各备选方案的后悔-欣喜值、失望-愉悦值及感知效用值。其次,构建了一种Pythagorean模糊熵,并给出基于该Pythagorean模糊熵的属性权重确定方法,利用属性权重加权求和获得备选方案综合感知效用值,从而对方案进行排序。最后,通过算例说明方法的可行性和优点,并分析了后悔规避系数δ和失望规避系数τ对决策结果的影响。     

Multicriteria analysis in decision making under information uncertainty

This paper presents results of research related to multicriteria decision making under information uncertainty. The Bellman–Zadeh approach to decision making in a fuzzy environment is utilized for analyzing multicriteria optimization models (X,M models) under deterministic information. Its application conforms to the principle of guaranteed result and provides constructive lines in obtaining harmonious solutions on the basis of analyzing associated maxmin problems. This circumstance permits one to generalize the classic approach to considering the uncertainty of quantitative information (based on constructing and analyzing payoff matrices reflecting effects which can be obtained for different combinations of solution alternatives and the so-called states of nature) in monocriteria decision making to multicriteria problems. Considering that the uncertainty of information can produce considerable decision uncertainty regions, the resolving capacity of this generalization does not always permit one to obtain unique solutions. Taking this into account, a proposed general scheme of multicriteria decision making under information uncertainty also includes the construction and analysis of the so-called X,R models (which contain fuzzy preference relations as criteria of optimality) as a means for the subsequent contraction of the decision uncertainty regions. The paper results are of a universal character and are illustrated by a simple example.     

Ranking alternatives using fuzzy numbers

We investigate the problem of employing expert opinion to rank alternatives across a set of criteria. The experts use fuzzy numbers to express their preferences and we employ fuzzy arithmetic to compute an issue's fuzzy ranking. This leads to a partition of the alternatives into sets H₁, H₂,… where H₁ contains the highest ranked issues, H₂ has all the second highest ranked alternatives, etc. The total ranking process is shown to possess a number of important properties. An example is presented to illustrate the method.