Yinfeng Xu

Consensus models for AHP group decision making under row geometric mean prioritization method

The consistency measure is a vital basis for consensus models of group decision making using preference relations, and includes two subproblems: individual consistency measure and consensus measure. In the analytic hierarchy process (AHP), the decision makers express their preferences using judgement matrices (i.e., multiplicative preference relations). Also, the geometric consistency index is suggested to measure the individual consistency of judgement matrices, when using row geometric mean prioritization method (RGMM), one of the most extended AHP prioritization procedures. This paper further defines the consensus indexes to measure consensus degree among judgement matrices (or decision makers) for the AHP group decision making using RGMM. By using Chiclana et al.s consensus framework, and by extending Xu and Weis individual consistency improving method, we present two AHP consensus models under RGMM. Simulation experiments show that the proposed two consensus models can improve the consensus indexes of judgement matrices to help AHP decision makers reach consensus. Moreover, our proposal has two desired features: (1) in reaching consensus, the adjusted judgement matrix has a better individual consistency index (i.e., geometric consistency index) than the corresponding original judgement matrix; (2) this proposal satisfies the Pareto principle of social choice theory.

The OWA-based consensus operator under linguistic representation models using position indexes

When using linguistic approaches to solve decision problems, we need linguistic representation models. The symbolic model, the 2-tuple fuzzy linguistic representation model and the continuous linguistic model are three existing linguistic representation models based on position indexes. Together with these three linguistic models, the corresponding ordered weighted averaging operators, such as the linguistic ordered weighted averaging operator, the 2-tuple ordered weighted averaging operator and the extended ordered weighted averaging operator, have been developed, respectively. In this paper, we analyze the internal relationship among these operators, and propose a consensus operator under the continuous linguistic model (or the 2-tuple fuzzy linguistic representation model). The proposed consensus operator is based on the use of the ordered weighted averaging operator and the deviation measures. Some desired properties of the consensus operator are also presented. In particular, the consensus operator provides an alternative consensus model for group decision making. This consensus model preserves the original preference information given by the decision makers as much as possible, and supports consensus process automatically, without moderator.

Combinatorial Optimization and Applications

The Seventh Annual International Conference on Combinatorial Optimization and Applications, abbreviated as COCOA 2013, was held during December 12–14, 2013 in Chengdu, China. Some of the best papers for COCOA’2013 were invited to be published in this special issue of Theoretical Computer Science. The nine selected papers are across computational geometry, computational biology, on-line algorithms, graph theory, parameterized complexity and social networks. The first paper is “An Inductive Construction of Minimally Rigid Body–Hinge Simple Graphs”, by Yuki Kobayashi et al. The authors showed that a minimally rigid body–hinge simple graph can be constructed with five elementary operations in polynomial time. The second paper “Mining Hidden Links in Social Networks to Achieve Equilibrium”, by Huan Ma et al., focuses on mining interesting links in a complex social network. While the problem is NP-hard, interesting empirical results are obtained. In the third paper “A Loopless Algorithm for Generating Multiple Binary Tree Sequences Simultaneously”, Ro-Yu Wu et al. studied the problem of generating binary trees using two extra LCand RC-sequences. A loopless algorithm is presented. In the fourth paper “Following a Curve with the Discrete Frechet Distance”, Tim Wylie and Binhai Zhu studied the problem of fitting a polygonal curve with a set of given points under the discrete Frechet distance. Several different versions of the problems are studied, some are in P and some are NP-complete. In the fifth paper “Touring a Sequence of Disjoint Polygons: Complexity and Extension”, Arash Ahadi, Amirhossein Mozafari and Alireza Zarei proved that the problem of touring a set of disjoint polygons with a shortest path between a given source and a given sink is NP-hard, solving a long-standing open problem. The previous NP-hardness result only holds when the polygons could be overlapping. In the sixth paper “Circular Convex Bipartite Graphs: Feedback Vertex Sets”, Tian Liu et al. showed that the famous NP-complete feedback vertex set problem is polynomially solvable on a circular convex bipartite graph. The solution is obtained by using Turing reductions. In the seventh paper “Approximating the Maximum Multiple RNA Interaction Problem”, Weitian Tong et al. presented new approximation algorithms for two variations of the maximum multiple RNA interaction problem. In the eighth paper “Online Bin Covering: Expectations vs. Guarantees”, Marie Christ, Lene Favrholdt and Kim Larsen analyzed the performance of two classic online algorithms HARMONIC and NEXT-FIT for the bin covering problem. Several different performance measures are used. In the ninth paper “Parameterized and Approximation Algorithms for Finding two Disjoint Matchings”, Zhi-Zhong Chen, Ying Fan and Lusheng Wang studied the problem of computing two disjoint matching in weighted and unweighted graphs. They gave an FPT algorithm for the problem on unweighted graph and an approximation algorithm for the problem on a weighted graph. We thank Sichuan University for hosting the conference, all the authors for submitting their contributions and the timely work of all the reviewers. Finally, we thank Giorgio Ausiello for his assistance and the program committee of COCOA 2013 (who helped select and review the papers).

On consistency measures of linguistic preference relations

Inspired by the concept of deviation measure between two linguistic preference relations, this paper further defines the deviation measure of a linguistic preference relation to the set of consistent linguistic preference relations. Based on this, we present a consistency index of linguistic preference relations and develop a consistency measure method for linguistic preference relations. This method is performed to ensure that the decision maker is being neither random nor illogical in his or her pairwise comparisons using the linguistic label set. Using this consistency measure, we discuss how to deal with inconsistency in linguistic preference relations, and also investigate the consistency properties of collective linguistic preference relations. These results are of vital importance for group decision making with linguistic preference relations.

A comparative study of the numerical scales and the prioritization methods in AHP

Abstract In the analytic hierarchy process (AHP), a decision maker first gives linguistic pairwise comparisons, then obtains numerical pairwise comparisons by selecting certain numerical scale to quantify them, and finally derives a priority vector from the numerical pairwise comparisons. In particular, the validity of this decision-making tool relies on the choice of numerical scale and the design of prioritization method. By introducing a set of concepts regarding the linguistic variables and linguistic pairwise comparison matrices (LPCMs), and by defining the deviation measures of LPCMs, we present two performance measure algorithms to evaluate the numerical scales and the prioritization methods. Using these performance measure algorithms, we compare the most common numerical scales (the Saaty scale, the geometrical scale, the Ma–Zheng scale and the Salo–Hamalainen scale) and the prioritization methods (the eigenvalue method and the logarithmic least squares method). In addition, we also discuss the parameter of the geometrical scale, develop a new prioritization method, and construct an optimization model to select the appropriate numerical scales for the AHP decision makers. The findings in this paper can help the AHP decision makers select suitable numerical scales and prioritization methods.

Computing the Numerical Scale of the Linguistic Term Set for the 2-Tuple Fuzzy Linguistic Representation Model

When using linguistic approaches to solve decision problems, we need the techniques for computing with words (CW). Together with the 2-tuple fuzzy linguistic representation models (i.e., the Herrera and Martinez model and the Wang and Hao model), some computational techniques for CW are also developed. In this paper, we define the concept of numerical scale and extend the 2-tuple fuzzy linguistic representation models under the numerical scale. We find that the key of computational techniques based on linguistic 2-tuples is to set suitable numerical scale with the purpose of making transformations between linguistic 2-tuples and numerical values. By defining the concept of the transitive calibration matrix and its consistent index, this paper develops an optimization model to compute the numerical scale of the linguistic term set. The desired properties of the optimization model are also presented. Furthermore, we discuss how to construct the transitive calibration matrix for decision problems using linguistic preference relations and analyze the linkage between the consistent index of the transitive calibration matrix and one of the linguistic preference relations. The results in this paper are pretty helpful to complete the fuzzy 2-tuple representation models for CW.

Consistency and consensus measures for linguistic preference relations based on distribution assessments

In this paper, we propose the concept of distribution assessments in a linguistic term set, and study the operational laws of linguistic distribution assessments. The weighted averaging operator and the ordered weighted averaging operator for linguistic distribution assessments are presented. We also develop the concept of distribution linguistic preference relations, whose elements are linguistic distribution assessments. Further, we study the consistency and consensus measures for group decision making based on distribution linguistic preference relations. Two desirable properties of the proposed measures are shown. A consensus model also has been developed to help decision makers improve the consensus level among distribution linguistic preference relations. Finally, illustrative numerical examples are given. The results in this paper provide a theoretic basis for the application of linguistic distribution assessments in group decision making.

Linguistic multiperson decision making based on the use of multiple preference relations

In multiperson decision making situations, it is quite natural that different decision makers who may have different background and knowledge will provide their preferences by different kinds of preference relations. This paper proposes a linguistic multiperson decision making model (LMDMM) based on linguistic preference relations, integrating fuzzy preference relations, different types of multiplicative preference relations and multigranular linguistic preference relations. In the LMDMM, several transformation functions are first obtained to relate fuzzy preference relations and different types of multiplicative preference relations with multigranular linguistic preference relations. Then, we design the selection process of the LMDMM, based on the use of the fuzzy majority and the extended ordered weighted averaging operator (or the 2-tuple ordered weighted averaging operator), and discuss conditions under which the proposed selection process satisfies the social choice axioms. At last, we analyze the internal consistency of the proposed transformation functions. The results in this paper are helpful to complete Chiclana et al.s fuzzy decision model [Integrating three representation models in fuzzy multipurpose decision making based on fuzzy preference relations, Fuzzy Sets and Systems 97 (1998) 33-48].

Selecting the Individual Numerical Scale and Prioritization Method in the Analytic Hierarchy Process: A 2-Tuple Fuzzy Linguistic Approach

The validity of the priority vector used in the analytic hierarchy process (AHP) relies on two factors: the selection of a numerical scale and the selection of a prioritization method. The traditional AHP selects only one numerical scale (e.g., the Saaty scale) and one prioritization method (e.g., the eigenvector method) for each particular problem. For this traditional selection approach, there is disagreement on which numerical scale and prioritization method is better in deriving a priority vector. In fact, the best numerical scale and the best prioritization method both rely on the content of the pairwise comparison data provided by the AHP decision makers. By defining a set of concepts regarding the scale function and the linguistic pairwise comparison matrices (LPCMs) of the priority vector and by using LPCMs to unify the format of the input and output of AHP, this paper extends the AHP prioritization process under the 2-tuple fuzzy linguistic model. Based on the extended AHP prioritization process, we present two performance measure criteria to evaluate the effect of the numerical scales and prioritization methods. We also use the performance measure criteria to develop a 2-tuple fuzzy linguistic multicriteria approach to select the best numerical scales and the best prioritization methods for different LPCMs. In this paper, we call this type of selection the individual selection of the numerical scale and prioritization method. We also compare this individual selection with traditional selection by using both random and real data and show better results with individual selection.

Minimum-Cost Consensus Models Under Aggregation Operators

In group decision making, consensus models are decision aid tools and help experts modify their individual opinions to reach a closer agreement. Based on the concept of minimum-cost consensus, this paper proposes a novel framework to achieve minimum-cost consensus under aggregation operators. Analytical results indicate that the proposed framework reduces to the consensus model of Ben-Arieh when the selected aggregation operator is the ordered weighted averaging (OWA) operator with weight vector (1/2, ..., 0, ..., 1/2)T. Furthermore, this paper closely examines the minimum-cost consensus models with a linear cost function under the common aggregation operators (e.g., the weighted averaging operator and the OWA operator). Linear-programming-based approaches are also developed to solve these models. The results of this paper significantly contribute to efforts to develop the consensus model of Ben-Arieh et al.