Estimation of missing judgments in AHP pairwise matrices using a neural network-based model

Selecting relevant features to make a decision and expressing the relationships between these features is not a simple task. The decision maker must precisely define the alternatives and criteria which are more important for the decision making process. The Analytic Hierarchy Process (AHP) uses hierarchical structures to facilitate this process. The comparison is realized using pairwise matrices, which are filled in according to the decision maker judgments. Subsequently, matrix consistency is tested and priorities are obtained by calculating the matrix principal eigenvector. Given an incomplete pairwise matrix, two procedures must be performed: first, it must be completed with suitable values for the missing entries and, second, the matrix must be improved until a satisfactory level of consistency is reached. Several methods are used to fill in missing entries for incomplete pairwise matrices with correct comparison values. Additionally, once pairwise matrices are complete and if comparison judgments between pairs are not consistent, some methods must be used to improve the matrix consistency and, therefore, to obtain coherent results. In this paper a model based on the Multi-Layer Perceptron (MLP) neural network is presented. Given an AHP pairwise matrix, this model is capable of completing missing values and improving the matrix consistency at the same time.     

A hybrid inexact-stochastic water management model

An inexact-stochastic water management (ISWM) model is proposed and applied to a case study of water quality management within an agricultural system. The model is based on an inexact chance-constrained programming (ICCP) method, which improves upon the existing inexact and stochastic programming approaches by allowing both distribution information in B and uncertainties in A and C to be effectively incorporated within its optimization process. In its solution process, the ICCP model (under a given pi level) is first transformed into two deterministic submodels, which correspond to the upper and lower bounds for the desired objective function value. This transformation process is based on an interactive algorithm, which is different from normal interval analysis or best/worst case analysis. Interval solutions, which are feasible and stable in the given decision space, can then be obtained by solving the two submodels sequentially. Thus, decision alternatives can be generated by adjusting decision variable values within their solution intervals. The obtained ICCP solutions are also useful for decision makers to obtain insight regarding tradeoffs between environmental and economic objectives and between increased certainties and decreased safeties (or increased risks). Results of the case study indicate that useful solutions for the planning of agricultural activities in the water quality management system have been obtained. A number of decision alternatives have been generated and analyzed based on projected applicable conditions. Generally, some alternatives can be considered when water quality objective is given priority, while the others may provide compromises between environmental and economic considerations. The above alternatives represent various options between environmental and economic tradeoffs. Willingness to accept low agricultural income will guarantee meeting the water quality objectives. A strong desire to acquire high agricultural income will run into the risk of violating water quality constraints.     

偏好信息为模糊互反判断矩阵的模糊多属性决策法

研究只有部分权重信息且决策者对方案的偏好信息以模糊互反判断矩阵形式给出的模糊多属性决策问题。提出了一种基于目标规划模型的模糊多属性决策方法。该法首先基于模糊互反判断矩阵，利用转换函数将决策信息一致化，建立了一个目标规划模型．通过求解该模型确定属性的权重，然后运用加性加权法求出各方案的模糊综合属性值，并利用已有的三角模糊数排序公式求得决策方案的排序。文章最后把该法应用于解决风险投资领域中的项目评估问题。     

Combining compound linguistic ordinal scale and cognitive pairwise comparison in the rectified fuzzy TOPSIS method for group decision making

Group decision making is the process to explore the best choice among the screened alternatives under predefined criteria with corresponding weights from assessment of a group of decision makers. The Fuzzy TOPSIS taking an evaluated fuzzy decision matrix as input is a popular tool to analyze the ideal alternative. This research, however, finds that the classical fuzzy TOPSIS produces a misleading result due to some inappropriate definitions, and proposes the rectified fuzzy TOPSIS addressing two technical problems. As the decision accuracy also depends on the evaluation quality of the fuzzy decision matrix comprising rating scores and weights, this research applies compound linguistic ordinal scale as the fuzzy rating scale for expert judgments, and cognitive pairwise comparison for determining the fuzzy weights. The numerical case of a robot selection problem demonstrates the hybrid approach leading to the much reliable result for decision making, comparing with the conventional fuzzy Analytic Hierarchy Process and TOPSIS.     

Multi-criteria decision making with fuzzy linguistic preference relations

Although the analytic hierarchy process (AHP) and the extent analysis method (EAM) of fuzzy AHP are extensively adopted in diverse fields, inconsistency increases as hierarchies of criteria or alternatives increase because AHP and EAM require rather complicated pairwise comparisons amongst elements (attributes or alternatives). Additionally, decision makers normally find that assigning linguistic variables to judgments is simpler and more intuitive than to fixed value judgments. Hence, Wang and Chen proposed fuzzy linguistic preference relations (Fuzzy LinPreRa) to address the above problem. This study adopts Fuzzy LinPreRa to re-examine three numerical examples. The re-examination is intended to compare our results with those obtained in earlier works and to demonstrate the advantages of Fuzzy LinPreRa. This study demonstrates that, in addition to reducing the number of pairwise comparisons, Fuzzy LinPreRa also increases decision making efficiency and accuracy.     

An improved TOPSIS with weighted hesitant vague information

The selection of the best alternatives in project management has attracted increasing attention due to the uncertain environment. Vague TOPSIS is one of the powerful methods to solve this problem. In this work, firstly, a method of measuring the similarities of vague set which can take the uncertainty preference into account is raised and comparison between methods has been made to verify its effectiveness. Then, a vague set based TOPSIS in group decision is proposed to aid the decision making in project management. In this method, the weights of the experts for different criteria in the group decision are completely unknown and are calculated with the similarities of the judgments by them in the project. Finally, a computation example is shown to illustrate the method.     

Multicriteria Group Decision Making: Optimal Priority Synthesis from Pairwise Comparisons

We propose an optimization based hybrid approach for multicriteria group decision making that takes advantage of aggregating individual judgments and aggregating individual priority methods. Different error structures and criteria with particular emphasis on weighted least square measure and weighted least logarithmic square measure for additive and multiplicative error structures are studied. We evaluate also the statistical properties of the derived group preference and new measures to assess consistency. The proposed methods provide an iterative and flexible framework for guiding group decision making.The authors thank Professor P.I. Yu and two anonymous reviewers for insightful comments and helpful suggestions.     

Supporting complex group decisions: A probabilistic multi-dimensional scaling approach

Groups often face complex decisions; decisions in which the decision alternatives are not clearly defined and the criteria for choosing an alternative are subject to dispute within the group. We present a Group Decision Support System that will use judgments from the group to visualize the decision problem in a probabilistic geometric space. In this geometric representation, actual decision alternatives and an ideal alternative—an artificial alternative that identifies the ideal solution to the group's decision dilemma—are portrayed as distributions in a multi-dimensional space. Dispersions of the distributions measure the uncertainties of the decision process. The psychometric theory used to develop the probabilistic geometric representation is described. Preliminary research is presented which demonstrates that geometric representations of this type help groups both to understand better the decision they face and to find better solutions.     

The Analytic Hierarchy Process in an uncertain environment: A simulation approach

Traditionally, decision makers were forced to converge ambiguous judgments to a single point estimate in order to describe a pairwise relationship between alternatives relative to some criterion for use in the Analytic Hierarchy Process (AHP). Since many circumstances exist which make such a convergence difficult, confidence in the outcome of an ensuing AHP synthesis may be reduced. Likewise, when a group of decision makers cannot arrive at a consensus regarding a judgment, some members of the group may simply lose confidence in the overall synthesis if they lack faith in some of the judgments. The AHP utilizes point estimates in order to derive the relative weights of criteria, sub-criteria, and alternatives which govern a decision problem. However, when point estimates are difficult to determine, distributions describing feasible judgments may be more appropriate. Using simulation, we will demonstrate that levels of confidence can be developed, expected weights can be calculated and expected ranks can be determined. It will also be shown that the simulation approach is far more revealing than traditional sensitivity analysis.     

考虑决策者心理行为的直觉语言多属性决策方法

针对应用直觉语言集来表达决策信息的语言多属性决策问题,在考虑决策者有限理性的心理行为基础上,提出一种决策方法。该方法通过比较每个属性下方案之间的得分函数和精确函数, 构建方案的收益-损失分析矩阵。在考虑决策者参照依赖和损失规避心理行为基础上,计算每个方案相对于其它方案在每个属性下的收益-损失值优先度;在此基础上,计算备选方案的综合优先度, 并根据其大小对方案进行排序择优。最后通过一个算例验证所提出方法的有效性和合理性。     

Global Priority Estimation in Multiperson Decision Making

The analytic hierarchy process generalized to synthesizing local preferences across a complex hierarchy structure into global priorities in multiperson decision making is considered in various approaches. They include classic eigenvector synthesis and multiplicative method, and several other suggested techniques, such as three-dimensional eigenvectors, and simultaneous linear and nonlinear estimations by whole hierarchy structure in synthetic optimizing procedures. A numerical example from marketing research with many criteria, subcriteria, alternatives, and respondents is presented. The quality of approximation compared by different approaches shows that the best results are produced by the nonlinear synthetic priority techniques. The techniques described have been successfully used in dozens of real-world projects and are very helpful for practical managerial decision making in complex hierarchies. Dr. Stan Lipovetsky is Senior Research Director, GfK Custom Research North America, Marketing Science, Research Center for Excellence. The author thanks Professor T. Rapcsak and two reviewers for insightful suggestions that improved the paper.     

Classifying efficient alternatives in SMAA using cross confidence factors

Stochastic multicriteria acceptability analysis (SMAA) is a family of methods for aiding multicriteria group decision making. These methods are based on exploring the weight space in order to describe the preferences that make each alternative the most preferred one. The main results of the analysis are rank acceptability indices, central weight vectors and confidence factors for different alternatives. The rank acceptability indices describe the variety of different preferences resulting in a certain rank for an alternative; the central weight vectors represent the typical preferences favouring each alternative; and the confidence factors measure whether the criteria data are sufficiently accurate for making an informed decision.In some cases, when the problem involves a large number of efficient alternatives, the analysis may fail to discriminate between them. This situation is revealed by low confidence factors. In this paper we develop cross confidence factors, which are based on computing confidence factors for alternatives using each other’s central weight vectors. The cross confidence factors can be used for classifying efficient alternatives into sets of similar and competing alternatives. These sets are related to the concept of reference sets in Data Envelopment Analysis (DEA), but generalized for stochastic models. Forming these sets is useful when trying to identify one or more most preferred alternatives, or suitable compromise alternatives. The reference sets can also be used for evaluating whether criteria need to be measured more accurately, and at which alternatives the measurements should be focused. This may cause considerable savings in measurement costs. We demonstrate the use of the cross confidence factors and reference sets using a real-life example.     

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 analysis of distributions of priority values from alternative comparison scales within AHP

The analytic hierarchy process (AHP) introduced by T.L. Saaty is a well known and popular method of multi-criteria decision making. Central to this method are the pairwise comparisons between criteria (and decision alternatives) made using a 9-unit scale. The appropriateness of Saaty's original one-to-nine (1–9) scale has been the subject of much debate and cause for concern. This paper contrasts the appropriateness of the 1–9 scale with other alternative 9-unit scales also used in AHP, by looking at the probability distributions of the associated priority values. For large problems, estimated probability distributions are found for the priority values through using the method of Parzen Windows.     

权重信息未知对方案有偏好的vague集多指标决策

就指标权重未知,且对方案有偏好的vague集多指标决策问题,提出了通过使决策者的主观偏好值与属性值的相离度最小来建立最优化模型,从而获得指标的权重.通过将vague值转化为模糊值来建立模糊值矩阵,由模糊值矩阵按各指标对应值的大小对方案进行排序,形成多个线性序,进而由线性序来构造模糊优先矩阵,对其进行截割,得到最优方案.最后通过一个实例说明此方法的具体决策过程.     

Prioritization of e-learning forms: a multicriteria methodology

Strategic planning of e-learning implementation includes decision making about the most suitable form of implementing e-learning on different levels in an institution. Decision making about e-learning implementation has been covered as consisting of four phases: (1) intelligence, (2) design, (3) choice and (4) implementation. During the Intelligence phase we have precisely identified our central decision problem and have conducted situation analysis. In the Design phase we have developed alternatives and established criteria and subcriteria. The questionnaire about the importance of the advantages and goals of e-learning implementation and about criteria and subcriteria significant for decision making was created. Essential for the survey was use of Croatian e-learning experts that are familiar with higher education (HE) environment. Further, we connected these findings with the results of the factor analysis which was performed on the survey. The results of the factor analysis have served as input in the multicriteria decision model (AHP) that we have developed in the Choice phase. In the implementation phase we have solved the problem of prioritisation of e-learning options with the help of multi-criteria modelling in the process of group decision making. In this article, firstly we will present and analyze the results of the survey. Secondly, the outputs of factor analysis will be stated and compared with the model used in the questionnaire. Finally, the structure of AHP model will be given and the results of the quantitative evaluation of the model will be presented.     

Effects of proactive decision making on life satisfaction

Proactive decision making, a concept recently introduced to behavioral operational research and decision analysis, addresses effective decision making during its phase of generating alternatives. It is measured on a scale comprising six dimensions grouped into two categories: proactive personality traits and proactive cognitive skills. Personality traits are grounded on theoretical constructs such as proactive attitude and proactive behavior; cognitive skills reflect value-focused thinking and decision quality. These traits and skills have been used to explain decision satisfaction, although their antecedents and other consequences have not yet been the subject of rigorous hypotheses and testing.This paper embeds proactive decision making within a model of three possible consequences. We consider—and empirically test—decision satisfaction, general self-efficacy, and life satisfaction by conducting three studies with 1300 participants. We then apply structural equation modeling to show that proactive decision making helps to account for life satisfaction, an explanation mediated by general self-efficacy and decision satisfaction. Thus proactive decision making fosters greater belief in one's abilities and increases satisfaction with one's decisions and with life more generally. These results imply that it is worthwhile to help individuals enhance their decision-making proactivity.Demonstrating the positive effects of proactive decision making at the individual level underscores how important the phase of generating alternatives is, and it also highlights the merit of employing “decision quality” principles and being proactive during that phase. Hence the findings presented here confirm the relevance of OR, and of decision-analytic principles, to the lives of ordinary people.     

Performance evaluation: An integrated method using data envelopment analysis and fuzzy preference relations

In a multi-attribute decision-making (MADM) context, the decision maker needs to provide his preferences over a set of decision alternatives and constructs a preference relation and then use the derived priority vector of the preference to rank various alternatives. This paper proposes an integrated approach to rate decision alternatives using data envelopment analysis and preference relations. This proposed approach includes three stages. First, pairwise efficiency scores are computed using two DEA models: the CCR model and the proposed cross-evaluation DEA model. Second, the pairwise efficiency scores are then utilized to construct the fuzzy preference relation and the consistent fuzzy preference relation. Third, by use of the row wise summation technique, we yield a priority vector, which is used for ranking decision-making units (DMUs). For the case of a single output and a single input, the preference relation can be directly obtained from the original sample data. The proposed approach is validated by two numerical examples.     

基于相对熵的区间Pythagorean模糊多属性AQM决策方法及其应用

针对决策信息为区间Pythagorean模糊数,属性权重不完全确定的多属性决策问题,提出了一种基于相对熵的AQM决策方法。首先,提出区间Pythagorean模糊数的相对熵,计算了各方案与区间Pythagorean模糊正理想方案和负理想方案间的相对熵,据此构建了基于方案相对满意度最大的非线性规划属性权重确定模型;其次,针对每个属性,利用新的区间Pythagorean模糊数得分函数计算方案的0-1优先关系矩阵,依据AQM方法对所有0-1优先关系矩阵进行融合得到合成0-1优先关系矩阵,并确定了方案的综合度,由此获得方案的排序。最后,以软件开发项目的选取为实例说明了该方法的可行性和有效性。