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.     

Inducement of Nonexistent Order by the Analytic Hierarchy Process*

An apparently unreported problem facing decision makers who use AHP is described. It is demonstrated that conventional AHP and some of its variants (the ideal mode, and the pairwise aggregated approach, PAHAP) can induce ordering even when no order exists. It is also shown that all three approaches can induce different orderings and that the orderings are sensitive to innocuous changes. Thus, even absent addition or deletion of alternatives, the decision maker relying on AHP or these variants can be seriously misled.     

Chainwise Paired Comparisons

The pairwise comparison technique is a building block of the Analytic Hierarchy Process (AHP), which has been popularly used for multicriteria decision analysis. This paper develops a shortcut technique in which only n paired comparisons forming a closed chain are needed for n decision elements. Together with the development of a simple and intuitive measure of (inconsistency, this technique derives the relative weights of decision elements via easy step-by-step calculations on a spreadsheet format. Its performance has been tested on Saaty's wealth of nations example. It is important to notice that ranking and weights yielded from this alternative technique are identical to Harker's incomplete pairwise comparison solution for the same chain orientation for the example tested.     

RANK GENERATION,PRESERVATION, AND REVERSAL IN THE ANALYTIC HIERARCHY DECISION PROCESS

Decision making has the objective of finding the best alternative or set of alternatives by considering a number of goals, objectives, criteria, competitors, and other important factors. The analytic hierarchy process is a decision aid used to assist a decision maker in sorting out the complexity of a decision problem and making use of his or her judgments. A decision maker must be assured that the arithmetic operations of any such decision process are the right ones—that they surface the correct ranking and values of the alternatives and preserve or alter ranks appropriately when new alternatives are added or deleted. In this paper it will be shown that with absolute measurement, rank always is preserved, with relative measurement, rank changes with nspect to scveral criteria only because of the structural dependence (involving both numbers and measurements) of criteria on alternatives. A discussion of the effect on rank of replicas and near replicas of the alternatives also is given.     

Consistency and Accuracy in Decision Aids: Experiments with Four Multiattribute Systems*

There have been a number of multiattribute decision aids developed to aid selection problems. Multiattribute value theory and the analytic hierarchy process are two commonly used techniques. Different systems can result in radically different conclusions if they inaccurately and inconsistently reflect the preference structure of decision makers, or if they are based on inappropriate theoretical models. This study examines the impact of the underlying theoretical model, the method in which preference information is elicited, and the structure of alternatives as influences on the results from using various decision aids. It was found that two systems based on the multiattribute value theory model were just as diverse in their conclusions as were results between AHP and the multiattribute value theory models. Therefore, accuracy of information reflecting decision maker preference is an important consideration. Feedback capable of assuring the decision maker that information provided is consistent is a necessary feature required of decision aids applied to selection problems. The study also found that the way in which information is elicited influenced the result more than did the underlying model. Exact numerical data for complex concepts such as attribute importance and alternative performance on attributes is not necessary, and elicitation procedures that are more natural for the user are likely to be more accurate.     

Consumer preference analysis: A data-driven multiple criteria approach integrating online information

Multiple criteria approaches can assist the product manager to know the consumer preferences in the context of e-commerce. Consumer preference analysis explains what aspects of a product affect and how they affect a consumer’s purchasing decision. This issue plays an important role in e-commerce platforms from its relevance in marketing decisions such as advertisements, recommendations and promotions. In this regard, we propose a data-driven multiple criteria decision aiding (MCDA) approach to integrate online information, such as explicit (e.g., reviews and ratings) and implicit (e.g., clicks and purchases) feedback from consumers. However, MCDA approaches present a critical challenge that even an experienced product manager could find it difficult to pre-define the criteria on which a product is evaluated. To address this issue, our proposed approach first utilizes text-mining techniques to assist the product manager identify the criteria, and then determines and collects the relative importance of the criteria and their values. Given the criteria information, we use a sampling process to provide two indices, the consumer preference index and rank acceptability index. The first index helps in prioritizing the pairwise comparisons of products, while the second one helps in deriving a default ranking list for first-time-registered consumers. We record the products viewed by consumers and generate their preference information in the form of pairwise comparisons for analyses within an aggregation-disaggregation paradigm. We also provide a representative value function to help the product manager gain insight into the preferences. Finally, we describe how a real-world application including the product manager and consumers exploits the proposed approach on an e-commerce platform to take a large step toward aiding more realistic and data-driven multiple criteria decision making.     

An extension of the Electre I method for group decision-making under a fuzzy environment

Many real-world decision problems involve conflicting systems of criteria, uncertainty and imprecise information. Some also involve a group of decision makers (DMs) where a reduction of different individual preferences on a given set to a single collective preference is required. Multi-criteria decision analysis (MCDA) is a widely used decision methodology that can improve the quality of group multiple criteria decisions by making the process more explicit, rational and efficient. One family of MCDA models uses what is known as “outranking relations” to rank a set of actions. The Electre method and its derivatives are prominent outranking methods in MCDA. In this study, we propose an alternative fuzzy outranking method by extending the Electre I method to take into account the uncertain, imprecise and linguistic assessments provided by a group of DMs. The contribution of this paper is fivefold: (1) we address the gap in the Electre literature for problems involving conflicting systems of criteria, uncertainty and imprecise information; (2) we extend the Electre I method to take into account the uncertain, imprecise and linguistic assessments; (3) we define outranking relations by pairwise comparisons and use decision graphs to determine which action is preferable, incomparable or indifferent in the fuzzy environment; (4) we show that contrary to the TOPSIS rankings, the Electre approach reveals more useful information including the incomparability among the actions; and (5) we provide a numerical example to elucidate the details of the proposed method.     

The inferring of preferences from indifferences in multicriteria decision analysis

Rivett [5] has proposed that an approximate preference ordering may be deduced from statements of pairwise indifferences between decision alternatives by using multi-dimensional scaling. In this paper it is demonstrated that much stronger evaluations of preference are possible by applying formal statistical inferential procedures to a simple parametric model, relating indifference to closeness on a scale defined by a linear function of attribute values. This can be used to screen out a considerable proportion of less desirable decision alternatives. The method is illustrated by application to Rivett's problem of the hypothetical Town of Brove, for which a satisfactory matching with Rivett's utilities is obtained. It is also shown that the method can provide useful preference orderings on the basis of less than 20% of all possible pairwise comparisons between alternatives.     

Robust Ordinal Regression and Stochastic Multiobjective Acceptability Analysis in multiple criteria hierarchy process for the Choquet integral preference model

The paper deals with two important issues of Multiple Criteria Decision Aiding: interaction between criteria and hierarchical structure of criteria. To handle interactions, we apply the Choquet integral as a preference model, and to handle the hierarchy of criteria, we apply the recently proposed methodology called Multiple Criteria Hierarchy Process. In addition to dealing with the above issues, we suppose that the preference information provided by the Decision Maker is indirect and has the form of pairwise comparisons of criteria with respect to their importance and pairwise preference comparisons of some pairs of alternatives with respect to some criteria. In consequence, many instances of the Choquet integral are usually compatible with this preference information. These instances are identified and exploited by Robust Ordinal Regression and Stochastic Multiobjective Acceptability Analysis. To illustrate the whole approach, we show its application to a real world decision problem concerning the ranking of universities for a hypothetical Decision Maker.     

An analytic hierarchy process based simulation model for entry mode decision regarding foreign direct investment

A methodology is developed for ranking entry mode alternatives encountered by individual firms considering foreign direct investment (FDI). The methodology deals with the risks and uncertainties related to FDI. The analytic hierarchy process (AHP) is used to solve the multiple criteria decision-making problem using input from a firm's management. A simulation approach is incorporated into the AHP to handle the uncertainty considerations encountered in an FDI environment. The uncertainties include: (1) uncertainty regarding the future characteristics of the FDI decision making environment, (2) uncertainty associated with the decision maker's judgment regarding pairwise comparisons necessitated by the AHP.     

A multicriteria decision making model for reverse logistics using analytical hierarchy process

Product recovery activities such as recycling, refurbishing and direct reuse are becoming integral to manufacturing supply chains. This study presents a multicriteria decision making model for reverse logistics using analytical hierarchy process (AHP). The AHP model evaluates a hierarchy of criteria and subcriteria, including costs and business relations, for critical decisions regarding network design. Using sensitivity analysis with AHP, the work provides insights into the preference ordering among eight alternative network configurations. For instance, the choice of test sites is largely dependent on the potential for cost savings on testing procedures and transportation of scrap, and this decision is not sensitive to the importance of business relations. By contrast, the choice of collection sites is largely determined by the relative importance of business relations considerations vs. cost considerations. As well, the processing location decision favors a third-party reprocessor if there is little need to protect proprietary product knowledge and cost savings is very important. The model is demonstrated using three case studies of real-world applications.     

方案有不确定偏好的区间数相对熵群决策方法

研究了属性值是区间数并且已知方案偏好信息的多属性群决策问题。建立了每个方案客观偏好值与主观偏好值偏差的相对熵测度矩阵;基于客观信息和方案偏好信息的相对熵建立了属性权重模型;建立了一个新的区间数比较的可能度公式,基于可能度公式给出了方案排序方法,算例说明方法可行性。     

Combining analytical hierarchy process and Choquet integral within non-additive robust ordinal regression

We consider multiple criteria decision aiding in the case of interaction between criteria. In this case the usual weighted sum cannot be used to aggregate evaluations on different criteria and other value functions with a more complex formulation have to be considered. The Choquet integral is the most used technique and also the most widespread in the literature. However, the application of the Choquet integral presents two main problems being the necessity to determine the capacity, which is the function that assigns a weight not only to all single criteria but also to all subset of criteria, and the necessity to express on the same scale evaluations on different criteria. While with respect to the first problem we adopt the recently introduced Non-Additive Robust Ordinal Regression (NAROR) taking into account all the capacities compatible with the preference information provided by the DM, with respect to the second one we build the common scale for the considered criteria using the Analytic Hierarchy Process (AHP). We propose to use AHP on a set of reference points in the scale of each criterion and to use an interpolation to obtain the other values. This permits to reduce considerably the number of pairwise comparisons usually required by the DM when applying AHP. An illustrative example details the application of the proposed methodology.     

Strategic Assessment Model (SAM): A Multiple Criteria Decision Support System for Evaluation of Strategic Alternatives*

The evaluation of strategic alternatives is a particularly difficult task. This difficulty is due to the complexities inherent in the evaluation process and the lack of structured information. The evaluation process must consider a multitude of relevant information from both the internal and external environments of the organization. Various analytical and normative models have helped decision makers utilize large volumes of information in strategic evaluation; however, most of these models have some limitations. We present a multiple criteria decision support system, called strategic assessment model (SAM), that addresses some of the limitations inherent in the existing models. SAM captures the decision maker's beliefs through a series of sequential, rational, and analytical processes. The environmental forces—decomposed into internal, task, general opportunities, and threats—are used along with the analytic hierarchy process (AHP), subjective probabilities, the entropy concept, and utility theory to enhance the decision maker's intuition in evaluating a set of strategic alternatives.     

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.     

基于一致性局部调整算法和DEA的语言偏好决策模型

设计一致性调整算法和计算可靠的方案排序权重向量已成为近年来备受关注的两个重要研究课题。针对决策信息为语言偏好关系的决策问题，提出一种基于一致性局部调整算法和数据包络分析（DEA）方法的语言偏好决策模型。首先，基于局部调整策略设计一种收敛的乘性一致性调整算法，该算法不仅使得调整后的语言偏好关系具有满意乘性一致性，而且能够尽可能多的保存原始决策信息；其次，基于提出的新型语言DEA模型构造一种语言偏好决策模型用以确定方案的排序权重向量，进而得到合理可靠的决策结果。最后，将提出的语言偏好决策模型用于供应商的选择实例，对比分析实验验证了模型的合理性和有效性。     

Ambiguous Criteria Weights in AHP: Consequences and Solutions*

Two related streams of criticism of the analytic hierarchy process (AHP) have not yet been satisfactorily resolved, although both date from the early 1980s. The first relates to ambiguity in the meaning of the relative importance of one criterion as compared to another. The second is concerned with reversals of rank alleged to be possible when new options are introduced in an AHP problem. Both proponents and critics of AHP agree that rank reversals occur, but disagree on the legitimacy of such reversals. This paper shows that there is a necessary correspondence between the manner in which criteria importances are interpreted and computed and the manner in which the weights of the options under each criterion are normalized. In general, if this relationship is ignored, incorrect weights are generated for options under consideration regardless of whether new options are added or deleted. A rank reversal on the addition of an option is merely symptomatic of this fact, and such reversals do not occur when the correspondence condition is met.     

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。     

信息系统外包决策的AHP/PROMETHEE方法

针对信息系统外包项目优选这一重要问题,在已有方法的基础上,结合层次分析法与偏好顺序结构评估法,提出了一种基于这两种方法相结合的信息系统外包项目选择决策方法。以管理、战略、技术、经济、质量与风险6项因素作为评价准则,用层次分析法确定信息系统外包项目选择问题的层次结构与评价准则的权重,用偏好顺序结构评估法确定信息系统外包项目的排序,并通过算例说明该方法的有效性。