Aggregation of analytic hierarchy process models based on similarities in decision makers’ preferences

The analytic hierarchy process is widely used in both individual and group decision making environments. In this paper we investigate its applicability to model a specific class of decentralized decision problems where many decision makers take individual subjective decisions using locally available information. In such subjective decision making environments, it is neither possible nor appropriate to use group preference aggregation techniques to model the problem as a single group decision problem. An approach to identify homogeneous subgroups of decision makers based on similarities in preferences and to aggregate preferences within each subgroup is proposed. This approach is validated using employment preferences of 70 subjects modeled using the analytic hierarchy process.     

Aggregation of utility-based individual preferences for group decision-making

Multi-attribute utility theory (MAUT) elicits an individual decision maker’s preferences for single attributes and develops a utility function by mathematics formulation to add up the preferences of the entire set of attributes when assessing alternatives. A common aggregation method of MAUT for group decisions is the simple additive weighting (SAW) method, which does not consider the different preferential levels and preferential ranks for individual decision makers’ assessments of alternatives in a decision group, and thus seems too intuitive in achieving the consensus and commitment for group decision aggregation. In this paper, the preferential differences denoting the preference degrees among different alternatives and preferential priorities denoting the favorite ranking of the alternatives for each decision maker are both considered and aggregated to construct the utility discriminative values for assessing alternatives in a decision group. A comparative analysis is performed to compare the proposed approach to the SAW model, and a satisfaction index is used to investigate the satisfaction levels of the final two resulting group decisions. In addition, a feedback interview is conducted to understand the subjective perceptions of decision makers while examining the results obtained from these two approaches for the second practical case. Both investigation results show that the proposed approach is able to achieve a more satisfying and agreeable group decision than that of the SAW method.     

Geometry of decision making and the vector space formulation of the analytic hierarchy process

We extend the conventional Analytic Hierarchy Process (AHP) to an Euclidean vector space and develop formulations for aggregation of the alternative preferences with the criteria preferences. Relative priorities obtained from such a formulation are almost identical with the ones obtained using conventional AHP. Each decision is represented by a preference vector indicating the orientation of the decision maker's mind in the decision space spanned by the decision alternatives. This adds a geometric meaning to the decision making processes. We utilise the measure of similarity between any two decision makers and apply it for analysing decisions in a homogeneous group. We propose an aggregation scheme for calculating the group preference from individual preferences using a simple vector addition procedure that satisfies Pareto optimality condition. The results agree very well with the ones of conventional AHP.     

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.     

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.     

Combining expert judgment: On the performance of trimmed mean vote aggregation procedures in the presence of strategic voting

Analytic group decision techniques for selecting a subset of alternatives range between multicriteria decision analysis techniques such as multiattribute utility theory and the analytic hierarchy process to voting techniques where each member of the decision group submits a ranking of the alternatives, and these individual rankings are then aggregated into an overall ranking. The obvious advantage of voting is that it bypasses the rather intensive data generation requirements of multicriteria techniques. In this paper we compare the performance of trimmed mean rank-order aggregation procedures in the case where a subset of the individuals in the group charged with the decision vote strategically. We employ a Monte Carlo simulation experiment on a specific decision instance and find that trimmed mean aggregation compares favorably with other procedures.     

A fuzzy ranking method with range reduction techniques

For ranking alternatives based on pairwise comparisons, current analytic hierarchy process (AHP) methods are difficult to use to generate useful information to assist decision makers in specifying their preferences. This study proposes a novel method incorporating fuzzy preferences and range reduction techniques. Modified from the concept of data envelopment analysis (DEA), the proposed approach is not only capable of treating incomplete preference matrices but also provides reasonable ranges to help decision makers to rank decision alternatives confidently.     

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.     

Decision-Making with Sugeno Integrals

This paper clarifies the connection between multiple criteria decision-making and decision under uncertainty in a qualitative setting relying on a finite value scale. While their mathematical formulations are very similar, the underlying assumptions differ and the latter problem turns out to be a special case of the former. Sugeno integrals are very general aggregation operations that can represent preference relations between uncertain acts or between multifactorial alternatives where attributes share the same totally ordered domain. This paper proposes a generalized form of the Sugeno integral that can cope with attributes having distinct domains via the use of qualitative utility functions. It is shown that in the case of decision under uncertainty, this model corresponds to state-dependent preferences on consequences of acts. Axiomatizations of the corresponding preference functionals are proposed in the cases where uncertainty is represented by possibility measures, by necessity measures, and by general order-preserving set-functions, respectively. This is achieved by weakening previously proposed axiom systems for Sugeno integrals.     

Linguistic group decision-making: opinion aggregation and measures of consensus

Group decision-making is a crucial activity, necessary in many aspects of our civilization. In many cases, due to inherent complexity, experts cannot express their opinion or preferences using exact numbers, thus resorting to a qualitative preference such as linguistic labels. Another complicating factor is the fact that very seldom all individuals in a group share the same opinion about the alternatives. This creates the need to aggregate all the differing individual opinions into a group opinion. Moreover, it is desirable to be able to assess the level of agreement among the experts; termed consensus. This paper presents a methodology for aggregating experts’ judgements, presented as linguistic labels, into a group opinion with a measure of the group consensus. The aggregation model allows weighted experts to express a degree of optimism or upward bias in their opinions. Then the paper presents two models of calculating the consensus based on the individual expert opinions and the group aggregated opinion.     

社会网络环境下基于“信任—知识”模型的风险性大群体应急决策方法

针对大群体应急决策专家之间信任关系及其传递引发的决策风险,以及由于大群体中个体偏好差异较大导致生成独立聚集等问题。首先,提出一个“信任—知识模型”对决策专家之间的信任关系进行集成和传递,并根据决策专家的信任风险偏好得出决策专家之间的信任知识度网络;其次,利用Louvain算法对信任知识度网络进行聚类,高效快速的获得若干个聚集,并用社会网络分析技术确定每个决策者和聚集的权重;然后对每个聚集中的决策者偏好进行集结,并综合决策者给出的信息对备选决策方案进行排序。最后,通过案例分析和对比验证了所提方法的合理性与有效性。     

PROMETHEE-MD-2T method for project selection

Project selection is a real problem of multicriteria group decision making (MCGDM) where each decision maker expresses his/her preferences depending on the nature of the alternatives and on his/her own knowledge over them. Thus, information, as much quantitative as qualitative, coexists. The traditional methods of MCGDM developed for project selection usually discriminates in favour of quantitative information at the expense of qualitative information, and this is due to the capability to integrate this first type of information inside their procedure. In this article, two new multicriteria 2-tuple group decision methods called “Preference Ranking Organisation Method for Enrichment Evaluation Multi Decision maker 2-Tuple-I and II” (PROMETHEE-MD-2T-I and II) are presented. They are able to integrate inside their procedure both quantitative and qualitative information in an uncertain context. This has been performed by integrating a 2-tuple linguistic representation model dealing with non-homogeneous and imprecise information data made up by valued intervals, numerical and linguistic values into the aggregation operators of Promethee methods. Although they have been developed for project selection problems, these proposed methods can be applied to all kinds of decision-making problems with heterogeneous and multigranular information.     

An extension of TOPSIS for group decision making

An extension of TOPSIS (technique for order performance by similarity to ideal solution), a multi-attribute decision making (MADM) technique, to a group decision environment is investigated. TOPSIS is a practical and useful technique for ranking and selection of a number of externally determined alternatives through distance measures. To get a broad view of the techniques used, we provide a few options for the operations, such as normalization, distance measures and mean operators, at each of the corresponding steps of TOPSIS. In addition, the preferences of more than one decision maker are internally aggregated into the TOPSIS procedure. Unlike in previous developments, our group preferences are aggregated within the procedure. The proposed model is indeed a unified process and it will be readily applicable to many real-world decision making situations without increasing the computational burden. In the final part, the effects of external aggregation and internal aggregation of group preferences for TOPSIS with different computational combinations are compared using examples. The results have demonstrated our model to be both robust and efficient.     

基于群体冲突的模糊偏好关系大群体决策方法

在大群体决策中,针对每一个决策者都有一个关于决策方案的模糊偏好关系的决策问题,提出了一种基于冲突的模糊偏好关系大群体决策方法。该方法首先考虑了复杂大群体的偏好差异,对决策者偏好进行聚类分析,形成不同的聚集,然后通过熵权法确定聚集的权重,集结成大群体模糊偏好关系,再对聚集内及聚集间进行冲突分析,通过一个迭代算子进行冲突消解,以达到一定冲突范围内的群体模糊偏好关系。最后通过一个算例说明了方法的有效性。     

Qualitative operators for reasoning maps: Evaluating multi-criteria options with networks of reasons

Cognitive/causal maps have been widely used as a powerful way of capturing decision-makers’ views about a problem, representing it as a cause–effect discourse. Several ways of making causal inferences from this type of model have been proposed in the Operational Research and Artificial Intelligence literatures, but none, as far as we are aware, has attempted to use a causal map structure to perform a multi-criteria evaluation of decision alternatives. Recently, we have proposed a new multi-criteria method, denominated as a Reasoning Map, which permits the use of decision-makers’ reasoning, structured as a network of means-and-ends (a particular type of causal map) to perform such an evaluation. In this manner, the model resembles the way that people talk and think about decisions in practice. The method also pays explicit attention to the cognitive limitations of decision-makers in providing preference information. Thus it employs qualitative assessment of preferences, utilises aggregation operators for qualitative data and provides also qualitative outputs. In this paper we discuss and evaluate possible ways of aggregating qualitative performance information in Reasoning Maps.     

NP-hardness results for the aggregation of linear orders into median orders

Given a collection Π of individual preferences defined on a same finite set of candidates, we consider the problem of aggregating them into a collective preference minimizing the number of disagreements with respect to Π and verifying some structural properties. We study the complexity of this problem when the individual preferences belong to any set containing linear orders and when the collective preference must verify different properties, for instance transitivity. We show that the considered aggregation problems are NP-hard for different types of collective preferences (including linear orders, acyclic relations, complete preorders, interval orders, semiorders, quasi-orders or weak orders), if the number of individual preferences is sufficiently large.     

DIS-CARD: a new method of multiple criteria sorting to classes with desired cardinality

In this paper, we present a new preference disaggregation method for multiple criteria sorting problems, called DIS-CARD. Real-life experience indicates the need of considering decision making situations in which a decision maker (DM) specifies a desired number of alternatives to be assigned to single classes or to unions of some classes. These situations require special methods for multiple criteria sorting subject to desired cardinalities of classes. DIS-CARD deals with such a problem, using the ordinal regression approach to construct a model of DM’s preferences from preference information provided in terms of exemplary assignments of some reference alternatives, together with the above desired cardinalities. We develop a mathematical model for incorporating such preference information via mixed integer linear programming (MILP). Then, we adapt the MILP model to two types of preference models: an additive value function and an outranking relation. Illustrative example is solved to illustrate the methodology.     

Group decision making based on intuitionistic multiplicative aggregation operators

Preference relations are the most common techniques to express decision maker’s preference information over alternatives or criteria. To consistent with the law of diminishing marginal utility, we use the asymmetrical scale instead of the symmetrical one to express the information in intuitionistic fuzzy preference relations, and introduce a new kind of preference relation called the intuitionistic multiplicative preference relation, which contains two parts of information describing the intensity degrees that an alternative is or not priority to another. Some basic operations are introduced, based on which, an aggregation principle is proposed to aggregate the intuitionistic multiplicative preference information, the desirable properties and special cases are further discussed. Choquet Integral and power average are also applied to the aggregation principle to produce the aggregation operators to reflect the correlations of the intuitionistic multiplicative preference information. Finally, a method is given to deal with the group decision making based on intuitionistic multiplicative preference relations.     

Interactive group decision-making procedure using weak strength of preference

In the paper, we present an interactive decision procedure for aggregating group members’ preferences which are specified in incomplete ways. A group consensus under incomplete information is not usually reached at a single step since less-specific preferences on attribute weights and performance scores make a clear selection of best alternative more difficult. To circumvent these difficulties, a measure, indicating the strength of preference between alternatives, is derived to help each of decision makers change his/her preference structure. To make preference changes based on the measure effective, we present a solution method to convert an intractable nonlinear programme into a linear one.     

Robust multi-criteria ranking with additive value models and holistic pair-wise preference statements

We consider a problem of ranking alternatives based on their deterministic performance evaluations on multiple criteria. We apply additive value theory and assume the Decision Maker’s (DM) preferences to be representable with general additive monotone value functions. The DM provides indirect preference information in form of pair-wise comparisons of reference alternatives, and we use this to derive the set of compatible value functions. Then, this set is analyzed to describe (1) the possible and necessary preference relations, (2) probabilities of the possible relations, (3) ranges of ranks the alternatives may obtain, and (4) the distributions of these ranks. Our work combines previous results from Robust Ordinal Regression, Extreme Ranking Analysis and Stochastic Multicriteria Acceptability Analysis under a unified decision support framework. We show how the four different results complement each other, discuss extensions of the main proposal, and demonstrate practical use of the approach by considering a problem of ranking 20 European countries in terms of 4 criteria reflecting the quality of their universities.