On the maximum number of feasible ranking sequences in multi-criteria decision making problems

In many decision problems the focus is on ranking a set of m alternatives in terms of a number, say n, of decision criteria. Given are the performance values of the alternatives for each one of the criteria and the weights of importance of the criteria. This paper demonstrates that if one assumes that the criteria weights are changeable, then the number of all possible rankings may be significantly less than the upper limit of m!. As a matter of fact, this paper demonstrates that the number of possible rankings is a function of the number of alternatives and the number of criteria. These findings are important from a sensitivity analysis point of view or when a group decision making environment is considered.     

The group consensus based evidential reasoning approach for multiple attributive group decision analysis

Many multiple attribute decision analysis problems include both quantitative and qualitative attributes with various kinds of uncertainties such as ignorance, fuzziness, interval data, and interval belief degrees. An evidential reasoning (ER) approach developed in the 1990s and in recent years can be used to model these problems. In this paper, the ER approach is extended to group consensus (GC) situations for multiple attributive group decision analysis problems. In order to construct and check the GC, a compatibility measure between two belief structures is developed first. Considering two experts’ utilities, the compatibility between their assessments is naturally constructed using the compatibility measure. Based on the compatibility between two experts’ assessments, the GC at a specific level that may be the attribute level, the alternative level, or the global level, can be constructed and reached after the group analysis and discussion within specified times. Under the condition of GC, we conduct a study on the forming of group assessments for alternatives, the achievement of the aggregated utilities of assessment grades, and the properties and procedure of the extended ER approach. An engineering project management software selection problem is solved by the extended ER approach to demonstrate its detailed implementation process, and its validity and applicability.     

A new data envelopment analysis method for priority determination and group decision making in the analytic hierarchy process

The DEAHP method for weight deviation and aggregation in the analytic hierarchy process (AHP) has been found flawed and sometimes produces counterintuitive priority vectors for inconsistent pairwise comparison matrices, which makes its application very restrictive. This paper proposes a new data envelopment analysis (DEA) method for priority determination in the AHP and extends it to the group AHP situation. In this new DEA methodology, two specially constructed DEA models that differ from the DEAHP model are used to derive the best local priorities from a pairwise comparison matrix or a group of pairwise comparison matrices no matter whether they are perfectly consistent or inconsistent. The new DEA method produces true weights for perfectly consistent pairwise comparison matrices and the best local priorities that are logical and consistent with decision makers (DMs)’ subjective judgments for inconsistent pairwise comparison matrices. In hierarchical structures, the new DEA method utilizes the simple additive weighting (SAW) method for aggregation of the best local priorities without the need of normalization. Numerical examples are examined throughout the paper to show the advantages of the new DEA methodology and its potential applications in both the AHP and group decision making.     

An attribute weight based feedback model for multiple attributive group decision analysis problems with group consensus requirements in evidential reasoning context

In an evidential reasoning context, a group consensus (GC) based approach can model multiple attributive group decision analysis problems with GC requirements. The predefined GC is reached through several rounds of group analysis and discussion (GAD) in the approach. However, the GAD with no guidance may not be the most appropriate way to reach the predefined GC because several rounds of GAD will spend a lot of time of all experts and yet cannot help them to effectively emphasize on the assessments which primarily damage the GC. In this paper, an attribute weight based feedback model is constructed to effectively identify the assessments primarily damaging the GC and accelerate the GC convergence. Considering important attributes with the weights more than or at least equal to the mean of the weights of all attributes, the feedback model constructs identification rules to identify the assessments damaging the GC for the experts to renew. In addition, a suggestion rule is introduced to generate appropriate recommendations for the experts to renew their identified assessments. The identification rules are constructed at three levels including the attribute, alternative and global levels. The feedback model is used to solve an engineering project management software selection problem to demonstrate its detailed implementation process, its validity and applicability, and its advantages compared with the GC based approach.     

Mining categorical sequences from data using a hybrid clustering method

The identification of different dynamics in sequential data has become an every day need in scientific fields such as marketing, bioinformatics, finance, or social sciences. Contrary to cross-sectional or static data, this type of observations (also known as stream data, temporal data, longitudinal data or repeated measures) are more challenging as one has to incorporate data dependency in the clustering process. In this research we focus on clustering categorical sequences. The method proposed here combines model-based and heuristic clustering. In the first step, the categorical sequences are transformed by an extension of the hidden Markov model into a probabilistic space, where a symmetric Kullback–Leibler distance can operate. Then, in the second step, using hierarchical clustering on the matrix of distances, the sequences can be clustered. This paper illustrates the enormous potential of this type of hybrid approach using a synthetic data set as well as the well-known Microsoft dataset with website users search patterns and a survey on job career dynamics.     

A novel group ranking model for revealing sequence and quantity knowledge

The aggregation of individuals’ preferences into a consensus ranking is a group ranking problem which has been widely utilized in various applications, such as decision support systems, recommendation systems, and voting systems. Gathering the comparison of preferences and aggregate them to gain consensuses is a conventional issue. For example, b > c ? d ? a indicates that b is favorable to c, and c (d) is somewhat favorable but not fully favorable to d (a), where > and ? are comparators, and a, b, c, and d are items. Recently, a new type of ranking model was proposed to provide temporal orders of items. The order, b&c → a, means that b and c can occur simultaneously and are also before a. Although this model can derive the order ranking of items, the knowledge about quantity-related items is also of importance to approach more real-life circumstances. For example, when enterprises or individuals handle their portfolios in financial management, two considerations, the sequences and the amount of money for investment objects, should be raised initially. In this study, we propose a model for discovering consensus sequential patterns with quantitative linguistic terms. Experiments using synthetic and real datasets showed the model’s computational efficiency, scalability, and effectiveness.     

A new method for ranking fuzzy numbers and its application to group decision making

In this paper, a new method for comparing fuzzy numbers based on a fuzzy probabilistic preference relation is introduced. The ranking order of fuzzy numbers with the weighted confidence level is derived from the pairwise comparison matrix based on 0.5-transitivity of the fuzzy probabilistic preference relation. The main difference between the proposed method and existing ones is that the comparison result between two fuzzy numbers is expressed as a fuzzy set instead of a crisp one. As such, the ranking order of n fuzzy numbers provides more information on the uncertainty level of the comparison. Illustrated by comparative examples, the proposed method overcomes certain unreasonable (due to the violation of the inequality properties) and indiscriminative problems exhibited by some existing methods. More importantly, the proposed method is able to provide decision makers with the probability of making errors when a crisp ranking order is obtained. The proposed method is also able to provide a probability-based explanation for conflicts among the comparison results provided by some existing methods using a proper ranking order, which ensures that ties of alternatives can be broken.     

An evidential reasoning based consensus model for multiple attribute group decision analysis problems with interval-valued group consensus requirements

With the aim of modeling multiple attribute group decision analysis problems with group consensus (GC) requirements, a GC based evidential reasoning approach and further an attribute weight based feedback model are sequentially developed based on an evidential reasoning (ER) approach. In real situations, however, giving precise (crisp) assessments for alternatives is often too restrictive and difficult for experts, due to incompleteness or lack of information. Experts may also find it difficult to give appropriate assessments on specific attributes, due to limitation or lack of knowledge, experience and provided data about the problem domain. In this paper, an ER based consensus model (ERCM) is proposed to deal with these situations, in which experts’ assessments are interval-valued rather than precise. Correspondingly, predefined interval-valued GC (IGC) requirements need to be reached after group analysis and discussion within specified times. Also, the process of reaching IGC is accelerated by a feedback mechanism including identification rules at three levels, consisting of the attribute, alternative and global levels, and a suggestion rule. Particularly, recommendations on assessments in the suggestion rule are constructed based on recommendations on their lower and upper bounds detected by the identification rule at a specific level. A preferentially developed industry selection problem is solved by the ERCM to demonstrate its detailed implementation process, validity, and applicability.     

Fuzzy preference orderings in group decision making

In this paper, some use of fuzzy preference orderings in group decision making is discussed. First, fuzzy preference orderings are defined as fuzzy binary relations satisfying reciprocity and max-min transitivity. Then, particularly in the case where individual preferences are represented by utility functions (utility values), group fuzzy preference orderings of which fuzziness is caused by differences or diversity of individual opinions are defined. Those orderings might be useful for proceeding the group decision making process smoothly, in the same manner as the extended contributive rule method.     

Avoiding non-optimal management decisions by applying a three-person inspection game

An inspection game models a conflict situation between an inspector and an inspectee. The mathematical analysis aims to generate optimal behavior of the inspectee under the assumption that an undesirable action of the inspectee could otherwise be carried out strategically. In this paper the controller’s (inspector’s) particular job is to audit a manager’s (inspectee’s) decision and to submit a report to the company’s top managers for examination. Thus, a conflict as regards the choice of behavioral actions of the manager, the controller and the top management impends. Based on Fandel and Trockel (2011a) this modified inspection game is discussed here for the first time as a three-person game in the context of a manager’s faulty decision that will unnecessarily add to the company’s costs and that the top management understandably wishes to minimize. We will first examine the conditions under which a Nash equilibrium occurs in this three-person game in which poor management, poor monitoring and poor revision coincide. We will then examine the effects that the penalties and bonuses exert on the Nash equilibrium solution. We will find that penalties and bonuses can neutralize each other in their effects on the improved decision making by the manager and the controller.     

Self-consistency of decision rules for group decision making

The author treats, in this paper, a group of decision makers, where each of them already has preference on a given set of alternatives but the group as a whole does not have a decision rule to make their group decision, yet. Then, the author examines which decision rules are appropriate. As a criterion of “appropriateness” the author proposes the concepts of self-consistency and universal self-consistency of decision rules. Examining the existence of universally self-consistent decision rules in two cases: (1) decision situations with three decision makers and two alternatives, and (2) those with three decision makers and three alternatives, the author has found that all decision rules are universally self-consistent in the case (1), whereas all universally self-consistent decision rules have one and just one vetoer in the essential cases in (2). The result in the case (2) implies incompatibility of universal self-consistency with symmetry. An example of applications of the concept of self-consistency to a bankruptcy problem is also provided in this paper, where compatibility of self-consistency with symmetry in a particular decision situation is shown.     

Generalized weighted exponential proportional aggregation operators and their applications to group decision making

We present a new aggregation operator called the generalized ordered weighted exponential proportional averaging (GOWEPA) operator, which is based on an optimal model. We study some properties and different families of the GOWEPA operator. We also generalize the GOWEPA operator. The key advantage of the GOWEPA operator is that it is an aggregation operator with theoretic basis on aggregation. Moreover, we propose an orness measure of the GOWEPA operator and indicate some properties of this orness measure. Furthermore, we introduce the least exponential squares method (LESM) to determine the GOWEPA operator weights based on its orness measure. In the end, we develop an application of the new approach in a case of group decision making in investment selection.     

The uncertain probabilistic OWA distance operator and its application in group decision making

In this paper, we present the uncertain probabilistic ordered weighted averaging distance (UPOWAD) operator. Its main advantage is that it uses distance measures in a unified framework between the probability and the OWA operator that considers the degree of importance of each concept in the aggregation. Moreover, it is able to deal with uncertain environments represented in the form of interval numbers. We study some of its main properties and particular cases such as the uncertain probabilistic distance (UPD) and the uncertain OWA distance (UOWAD) operator. We end the paper by presenting an application to a group decision making problem regarding the selection of robots.     

Aggregating crisp values into intuitionistic fuzzy number for group decision making

This paper presents a multiple attribute group decision making model based on aggregating crisp values into intuitionistic fuzzy numbers. First, each alternative is evaluated with respect to their attributes, whose values are provided by decision maker as crisp numbers. Second, to make a reasonable normalization of attribute values in the group decision making environment, a maximum grade and a minimum grade are added to the attribute values. These normalized attribute values are then aggregated (per attribute) into an induced intuitionistic fuzzy number. Each alternative is then evaluated according to the induced intuitionistic fuzzy number. To show the major technical advances in this paper, comparisons with other methods are also made. Finally, an experimental analysis for supplier selection is given to illustrate the reasonableness and efficiency of the introduced method.     

一种基于LPA集结算子的群体一致性程度的计算方法

针对基于语言评价信息的多指标群决策分析中群体一致性程度计算问题,依据近年来最新发展的二元语义及其算子和PA算子,给出了一种用于集结语言评价信息的L PA算子,并基于L PA算子,进一步给出了关于群体一致性程度的计算步骤.最后给出了一个算例.     

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.     

A least deviation method for priority derivation in group decision making with incomplete reciprocal preference relations

In this paper, based on the transfer relationship between reciprocal preference relation and multiplicative preference relation, we proposed a least deviation method (LDM) to obtain a priority vector for group decision making (GDM) problems where decision-makers' (DMs') assessments on alternatives are furnished as incomplete reciprocal preference relations with missing values. Relevant theorems are investigated and a convergent iterative algorithm about LDM is developed. Using three numerical examples, the LDM is compared with the other prioritization methods based on two performance evaluation criteria: maximum deviation and maximum absolute deviation. Statistical comparative study, complexity of computation of different algorithms, and comparative analyses are provided to show its advantages over existing approaches.     

A method based on PSO and granular computing of linguistic information to solve group decision making problems defined in heterogeneous contexts

Group decision making is a type of decision problem in which multiple experts acting collectively, analyze problems, evaluate alternatives, and select a solution from a collection of alternatives. As the natural language is the standard representation of those concepts that humans use for communication, it seems natural that they use words (linguistic terms) instead of numerical values to provide their opinions. However, while linguistic information is readily available, it is not operational and thus it has to be made usable though expressing it in terms of information granules. To do so, Granular Computing, which has emerged as a unified and coherent framework of designing, processing, and interpretation of information granules, can be used. The aim of this paper is to present an information granulation of the linguistic information used in group decision making problems defined in heterogeneous contexts, i.e., where the experts have associated importance degrees reflecting their ability to handle the problem. The granulation of the linguistic terms is formulated as an optimization problem, solved by using the particle swarm optimization, in which a performance index is maximized by a suitable mapping of the linguistic terms on information granules formalized as sets. This performance index is expressed as a weighted aggregation of the individual consistency achieved by each expert.     

Logarithmic least squares method to priority for group decision making with incomplete fuzzy preference relations

The aim of this paper is to present a logarithmic least squares method (LLSM) to priority for group decision making with incomplete fuzzy preference relations. We give a reasonable definition of multiplicative consistent for incomplete fuzzy preference relation. We develop the acceptable fuzzy consistency ratio (FCR for short), which is simple and similar to Saaty’s consistency ratio CR for multiplicative fuzzy preference relations. We also extend the LLSM method to the case of individual preference relation with complete information. Finally, some examples are illustrated to show that our method is simple, efficient, and can be performed on computer easily.