A Rejoinder to Forman on AHP,with Emphasis on the Requirements of Composite Ratio Scales*

In an earlier paper [11], the problems of rank reversals and invalid composite priorities in AHP were addressed by modifications to the AHP procedure. That solution was subsequently criticized [5]. In this paper, we rebut these criticisms, and we show how rank reversals in AHP can arise merely from the process of normalizing local priorities.     

Ranking irregularities when evaluating alternatives by using some ELECTRE methods

The ELECTRE II and III methods enjoy a wide acceptance in solving multi-criteria decision-making (MCDM) problems. Research results in this paper reveal that there are some compelling reasons to doubt the correctness of the proposed rankings when the ELECTRE II and III methods are used. In a typical test we first used these methods to determine the best alternative for a given MCDM problem. Next, we randomly replaced a non-optimal alternative by a worse one and repeated the calculations without changing any of the other data. Our computational tests revealed that sometimes the ELECTRE II and III methods might change the indication of the best alternative. We treat such phenomena as rank reversals. Although such ranking irregularities are well known for the additive variants of the AHP method, it is the very first time that they are reported to occur when the ELECTRE methods are used. These two methods are also evaluated in terms of two other ranking tests and they failed them as well. Two real-life cases are described to demonstrate the occurrence of rank reversals with the ELECTRE II and III methods. Based on the three test criteria presented in this paper, some computational experiments on randomly generated decision problems were executed to test the performance of the ELECTRE II and III methods and an examination of some real-life case studies are also discussed. The results of these examinations show that the rates of the three types of ranking irregularities were rather significant in both the simulated decision problems and the real-life cases studied in this paper.     

PAHAP: A Pairwise Aggregated Hierarchical Analysis of Ratio-Scale Preferences

In this paper, we present a Pairwise Aggregated Hierarchical Analysis of Ratio-Scale Preferences (PAHAP), a new method for solving discrete alternative multicriteria decision problems. Following the Analytic Hierarchy Process (AHP), PAHAP uses pairwise preference judgments to assess the relative attractiveness of the alternatives. By first aggregating the pairwise judgment ratios of the alternatives across all criteria, and then synthesizing based on these aggregate measures, PAHAP determines overall ratio scale priorities and rankings of the alternatives which are not subject to rank reversal, provided that certain weak consistency requirements are satisfied. Hence, PAHAP can serve as a useful alternative to the original AHP if rank reversal is undesirable, for instance when the system is open and criterion scarcity does not affect the relative attractiveness of the alternatives. Moreover, the single matrix of pairwise aggregated ratings constructed in PAHAP provides useful insights into the decision maker's preference structure. PAHAP requires the same preference information as the original AHP (or, altematively, the same information as the Referenced AHP, if the criteria are compared based on average (total) value of the alternatives). As it is easier to implement and interpret than previously proposed variants of the conventional AHP which prevent rank reversal, PAHAP also appears attractive from a practitioner's viewpoint.     

Uses and Misuses of Multicriteria Decision Analysis (MCDA) in Environmental Decision Making

We focus on a class of multicriteria methods that are commonly used in environmental decision making—those that employ the weighted linear average algorithm (and this includes the popular analytic hierarchy process (AHP)). While we do not doubt the potential benefits of using formal decision methods of this type, we draw attention to the consequences of not using them well. In particular, we highlight a property of these methods that should not be overlooked when they are applied in environmental and wider decision-making contexts: the final decision or ranking of options is dependent on the choice of performance scoring scales for the criteria when the criteria weights are held constant. We compare this "sensitivity" to a well-known criticism of the AHP, and we go on to describe the more general lesson when it comes to using weighted linear average methods—a lesson concerning the relationship between criteria weights and performance scoring scales.     

Stochastic Judgments in the AHP: The Measurement of Rank Reversal Probabilities

This paper presents a methodology for analyzing Analytic Hierarchy Process (AHP) rankings if the pairwise preference judgments are uncertain (stochastic). If the relative preference statements are represented by judgment intervals, rather than single values, then the rankings resulting from a traditional (deterministic) AHP analysis based on single judgment values may be reversed, and therefore incorrect. In the presence of stochastic judgments, the traditional AHP rankings may be stable or unstable, depending on the nature of the uncertainty. We develop multivariate statistical techniques to obtain both point estimates and confidence intervals of the rank reversal probabilities, and show how simulation experiments can be used as an effective and accurate tool for analyzing the stability of the preference rankings under uncertainty. If the rank reversal probability is low, then the rankings are stable and the decision maker can be confident that the AHP ranking is correct. However, if the likelihood of rank reversal is high, then the decision maker should interpret the AHP rankings cautiously, as there is a subtantial probability that these rankings are incorrect. High rank reversal probabilities indicate a need for exploring alternative problem formulations and methods of analysis. The information about the extent to which the ranking of the alternatives is sensitive to the stochastic nature of the pairwise judgments should be valuable information into the decision-making process, much like variability and confidence intervals are crucial tools for statistical inference. We provide simulation experiments and numerical examples to evaluate our method. Our analysis of rank reversal due to stochastic judgments is not related to previous research on rank reversal that focuses on mathematical properties inherent to the AHP methodology, for instance, the occurrence of rank reversal if a new alternative is added or an existing one is deleted.     

Developing a supplier selection system through integrating fuzzy AHP and fuzzy DEA: a case study on an auto lighting system company in Taiwan

Supplier selection plays a very important role in supply chain management. This study intends to develop a novel performance evaluation method, which integrates both fuzzy analytical hierarchy process (AHP) method and fuzzy data envelopment analysis (DEA) for assisting organisations to make the supplier selection decision. Fuzzy AHP method is first applied to find the indicators’ weights through expert questionnaire survey. Then, these weights are integrated with fuzzy DEA. We use α -cut set and extension principle of fuzzy set theory to simplify the fuzzy DEA as a pair of traditional DEA model. Finally, fuzzy ranking using maximising and minimising set method is able to rank the evaluation samples. A case study on an internationally well-known auto lighting OEM company shows that the proposed method is very suitable for practical applications.     

目标导向层次分析方法

对含有抽象属性的多属性层次结构而言,层次分析法即AHP（包括DIS-AHP、ABS-AHP、IDE-AHP和SUP-AHP四种具体方法）会因比率比较基准缺失、权重内涵模糊不清或方案评价不保序而缺乏科学理性。为发展AHP,基于摆幅置权（SW）判断模式和多属性决策属性价值公度方法,首先给出了能为层次结构抽象属性上的SW判断提供支持的规约性多属性决策属性价值公度方法,然后由此并结合多属性价值理论给出了能够克服现有层次分析法内在缺陷的目标导向层次分析方法即ToAHP。相对于AHP,ToAHP在判断模式与权重内涵、方法建构的理论基础和相关假设检验、方案评价保序与其内在数理依据上具有明显的相对科学合理性。应用分析表明：在输入信息可比的条件下,ToAHP明显优于AHP的四种分析方法之中最具可信性的SUP-AHP方法。     

Strict Uncertainty: A Criterion for Moderately Pessimistic Decision Makers*

This paper proposes a decision rule to rank actions under strict uncertainty, the available information being limited to the states of nature, the set of alternative rows, and the consequence of choosing every row if a given state occurs. This rule is suitable to moderately pessimistic individuals and social groups, these agents being neither maximax nor maximin decision makers but people who assume that the best outcome from the action will not occur. For these decision makers the paper shows the existence of a consistent weight system in which one and only one weight is attached to each state of the world under plausible conditions of domination. Most of the traditional axioms are satisfied by the proposed ranking approach. In the frame of disappointment (measured by ranges of column dispersion), the meaning of some controversial postulates used in the literature is explained. The proposed criterion is a departure from Laplace's (1825) rule and from the remaining standard criteria. Only in the special case of equal column dispersion do both Laplace's rule and the proposed weights lead to the same solution.     

Adoption Curves and Social Interactions

This paper considers the observational implications of social influences on adoption decisions for an environment of perfect foresight adopters. We argue that social influences can produce two observable effects: (1) discontinuities in unconditional adoption curves and (2) pattern reversals in conditional adoption curves, in which earlier adoption is found for one group of actors versus another when fundamentals suggest the reverse ordering should occur; in turn the presence of either of these features can, under weak assumptions, be interpreted as evidence of social influences. As such, these properties are robust implications of social effects. (JEL: C40, D01, O33)     

Approximate weighting method for multiattribute decision problems with imprecise parameters

A multiattribute decision problem with imprecise parameters refers to one in which at least one of the parameters such as attribute weights and value scores is not represented by precise numerical values. Some well-known types of incomplete attribute weights are chosen and analyzed to find their extreme points. In doing so, we show that their coefficients matrix, by itself or by the change of variables, belongs to a class of M-matrix which enables us to find its extreme points readily due to the inverse-positive property.The knowledge of extreme points not only helps us to prioritize alternatives but also supports iterative exploration of decision-maker’s preference by investigating modified extreme points caused by additional preference information. A wide range of eligible attribute weights, however, often fail to result in the best alternative or a complete ranking of alternatives. To address this situation, we consider an approximate weighting method, so called the minimizing squared deviations from extreme points (MSD) which locates the attribute weights at the barycenter of a weight set. Accordingly, the MSD approach extends the rank order centroid (ROC) weighting method which is known to outperform other approximate weighting methods in case of ranked attribute weights. The evidence of the MSD’s superiority over a linear program-based weighting method is verified via simulation analysis under different forms of incomplete attribute weights.     

基于三维密度算子的群体DEMATEL指标权重确定方法

针对现有DEMATEL指标权重确定方法大多基于个体决策,且未考虑群体决策评价标度不一致的情况,提出一种新的基于三维密度算子的群体DEMATEL指标权重确定方法。首先,定义了不同评价标度的转换函数,以此将群体DEMATEL矩阵进行一致化处理;其次,给出一种群体DEMATEL矩阵的聚类方法,在此基础上利用三维密度算子对其进行集结;最后,依据DEMATEL方法识别出指标的中心度和原因度,并计算各指标的权重。文末通过一个应用实例验证了所提方法的可行性与有效性。实例结果表明,该方法由于能够较好解决群体决策评价标度不一致的问题,还能够充分考虑群体决策的共识度,因此可使指标权重结果更为客观合理且更为可靠。     

A qualitative perspective to deriving weights from pairwise comparison matrices

Deriving weights from pairwise comparison matrices (PCM) is a highly researched topic. The analytic hierarchy process (AHP) traditionally uses the eigenvector method for the purpose. Numerous other methods have also been suggested. A distinctive feature of all these methods is that they associate a quantitative meaning to the judgemental information given by the decision-maker. In contrast, the verbal scale used in AHP to capture judgements does not associate such a quantitative meaning. Though this issue of treating judgements qualitatively is recognized in the extant literature on multi-criteria decision making, unfortunately, there is no research effort so far in the AHP literature. Deriving motivation from the application of data envelopment analysis (DEA) for deriving weights, it is proposed in this paper that DEA models developed to deal with a mix of qualitative and quantitative factors can be used to derive weights from PCMs by treating judgements as qualitative factors. The qualitative DEA model is discussed and illustrated in this paper.     

Least Squares Model Averaging

This paper considers the problem of selection of weights for averaging across least squares estimates obtained from a set of models. Existing model average methods are based on exponential Akaike information criterion (AIC) and Bayesian information criterion (BIC) weights. In distinction, this paper proposes selecting the weights by minimizing a Mallows criterion, the latter an estimate of the average squared error from the model average fit. We show that our new Mallows model average (MMA) estimator is asymptotically optimal in the sense of achieving the lowest possible squared error in a class of discrete model average estimators. In a simulation experiment we show that the MMA estimator compares favorably with those based on AIC and BIC weights. The proof of the main result is an application of the work of Li (1987).     

A three-stage hybrid approach for weight assignment in MADM

How to determine weights for attributes is one of the key issues in multiple attribute decision making (MADM). This paper aims to investigate a new approach for determining attribute weights based on a data envelopment analysis (DEA) model without explicit inputs (DEA-WEI) and minimax reference point optimisation. This new approach first considers a set of preliminary weights and the most favourite set of weights for each alternative or decision making unit (DMU) and then aggregates these weight sets to find the best compromise weights for attributes with the interests of all DMUs taken into account fairly and simultaneously. This approach is intended to support the solution of such MADM problems as performance assessment and policy analysis where (a) the preferences of decision makers (DMs) are either unclear and partial or difficult to acquire and (b) there is a need to consider the best "will" of each DMU. Two case studies are conducted to show the property of this new proposed approach and how to use it to determine weights for attributes in practice. The first case is about the assessment of research strengths of EU-28 member countries under certain measures and the second is for analysing the performances of Chinese Project 985 universities, where the weights of the attributes need to be assigned in a fair and unbiased manner.     

我国研究型大学核心竞争力的评价指标体系研究

本文借用企业核心竞争力的理论,结合研究型大学的特点,设计出了一个用以评价、分析我国研究型大学核心竞争力的指标体系,利用AHP法与专家评判相结合的方法对指标体系进行了评价,并通过实例分析表明该指标体系和评价模型的通用性和科学性。     

授权决策的期权特征与模型解析

本文利用实物期权方法对授权决策的核心问题——授权时机和授权对象的选择——进行了研究。首先分析了授权决策的期权特征,对授权决策中存在的实物期权进行了总结;然后构建了授权决策的实物期权模型,分别计算出了授权决策的期权价值和授权时员工对公司的人力资本价值,由此得出了一个可比性的授权决策标准,即临界业绩水平,从而简化了决策指标,提高了信息的利用率,使企业可以对特质各异的候选人进行准确比较,并分三种授权决策情形分别给出了决策方案,最后基于数值算例作出了进一步的解析。     

Dynamic pricing and revenue management process in Internet retailing under uncertainty: An integrated real options approach

Competition and demand volatility often cause modern enterprises to be confronted by uncertain environments. When a firm manages revenue in such competitive and risky environments, the optimization of pricing and capacity allocation, subject to a fixed time and capacity, becomes a complicated problem. Many previous papers concerning revenue management (RM) and pricing require that the firm possesses the ability to know the demand curve (or demand distribution) and set prices on it to maximize profits. However, this assumption may not be the case in some industries. Therefore, this paper focuses on the dynamic lead indicators rather than assumptive lag indicators to establish a concise and flexible decision model for practical use. This paper provides an integrated real options (IRO) approach with analytic hierarchy process (AHP) for the auction RM problem under competitive/dynamic pricing and revenue uncertainty in Internet retailing. A numerical example is also presented to illustrate that the IRO approach can generate better decisions than the naı¨ve (or risk unawareness) approach in revenue quality of safety and profitability. The new perspective and approach proposed by this paper can be extended to other RM fields whenever both profitability and risk are critical to decision making.     

THE RELATIONSHIP BETWEEN THE ANALYTIC HIERARCHY PROCESS AND THE ADDITIVE VALUE FUNCTION*

The relationship between the analytical hierarchy process (AHP), a multiple-criteria decision method based on the properties of the eigenvalues of positive reciprocal matrices, and the additive value function is studied. The AHP is compared with the standard method of building an additive value function. The advantages and disadvantages of each method are discussed and a new method that uses elements of both is proposed. The proposed method relies on the AHP for building the unidimensional value functions and incorporates the systematic use of pairwise comparisons into the standard method of estimating the weights of the additive value function. Subject Area: Decision Processes.     

多层网络视角下金融机构关联性的演化特征研究

基于股票收益率间Pearson相关性、Kendall秩相关性以及Tail相关性，构建了金融机构多层网络模型，其中三种相关性分别对应多层网络中Pearson层、Kendall层和Tail层。根据2010年10月至2018年3月期间我国上市金融机构的数据，实证分析了金融机构多层网络结构演化特征。实证分析结果表明：Kendall层和Tail层中的平均权重均高于Pearson层的平均权重，Pearson层和Kendall层中的平均权重变化趋势比较相似，但前者波动幅度明显大于后者；Pearson层与Kendall层的边独特性变化趋势非常相似，而Tail层的与它们有较大的差别；任意两层的度相关性均为正相关，但随时间波动均较为剧烈；Pearson层与Kendall层的相似性整体上较高，Tail层与Pearson层和Kendall层的相似性整体上均较低；多层网络中平均权重、边独特性、度相关性和相似性等结构指标变化趋势与股票市场行情相关。     

Best-worst multi-criteria decision-making method

In this paper, a new method, called best-worst method (BWM) is proposed to solve multi-criteria decision-making (MCDM) problems. In an MCDM problem, a number of alternatives are evaluated with respect to a number of criteria in order to select the best alternative(s). According to BWM, the best (e.g. most desirable, most important) and the worst (e.g. least desirable, least important) criteria are identified first by the decision-maker. Pairwise comparisons are then conducted between each of these two criteria (best and worst) and the other criteria. A maximin problem is then formulated and solved to determine the weights of different criteria. The weights of the alternatives with respect to different criteria are obtained using the same process. The final scores of the alternatives are derived by aggregating the weights from different sets of criteria and alternatives, based on which the best alternative is selected. A consistency ratio is proposed for the BWM to check the reliability of the comparisons. To illustrate the proposed method and evaluate its performance, we used some numerical examples and a real-word decision-making problem (mobile phone selection). For the purpose of comparison, we chose AHP (analytic hierarchy process), which is also a pairwise comparison-based method. Statistical results show that BWM performs significantly better than AHP with respect to the consistency ratio, and the other evaluation criteria: minimum violation, total deviation, and conformity. The salient features of the proposed method, compared to the existing MCDM methods, are: (1) it requires less comparison data; (2) it leads to more consistent comparisons, which means that it produces more reliable results.