Judgment Matrices Research Articles

Assigning Method for Decision Power Based on Linguisitc 2-tuple Judgment Matrices

Decision power is very important in group decision making, which effects the final decision making result. When there exists uncertainty in group decision making, it is easy for an expert to express his/her preferences using fuzzy linguistic term such as ‘good’, ‘very good’. The linguistic 2-tuple representation model was selected to represent fuzzy linguistic term for its accuracy in representing fuzzy terms and less loss of information in processing. To obtain the objective decision power in group decision making where the experts express their opinions using linguistic 2-tuple judgment matrices, the paper put forward a new approach to assign decision power based on element consistency level. And the method for calculating the contribution degree of each expert was also provided. The proposed method thought of calculating the different weight to aggregate different element according to the element consistent level. An illustrated example was used to demonstrate the proposed method. And a GDSS based on the method was also developed to simplify the application in VB.NET. The GDSS was applied to building non-financial performance system for listed company in small and medium enterprise board in China, which showed its reasonability.

A Fuzzy-AHP Method to Enterprise’s Partner Selection under the Green Operation Strategy

To solve partner selection problems under enterprise’s green operation strategy nowadays, a new fuzzy-AHP method based on the AHP and basic theory of the triangular fuzzy number was given. Moreover, the evaluation hierarchy structure and corresponding evaluation process were described. Compared with traditional AHP approach, the proposed model combines subjective analysis with quantitative analysis more reasonably, synthesizes group opinions more adequately. The method can be practiced easily and widely, mainly because of the model’s simple, normative arithmetic formulas and the consistency of triangular-fuzzy-number-based judgment matrices. The results of a case study on selecting fabric suppliers in a middle-scale clothing enterprise indicate that, by applying the method-based software system, fair and reasonable conclusions under group decision-making environment can be acquired clearly.

Application of Synthetic Relative Measuring Method in On-Line Monitoring for Capacitive Equipment

On-line measurement for dielectric loss angle can effectively monitor the insulation condition of capacitive equipment in power system. Synthetic relative measuring method not only markedly overcomes the shortcomings of traditional absolute measuring method but also greatly improves the accuracy of dielectric loss angle measurement. However, the papers nowadays only focus on the application of synthetic relative measuring method based on two or three pieces of capacitive equipment, which does not have the characteristic of generality. In the paper, the principle of synthetic relative measuring method is given. The example of application for synthetic relative method based on three and four pieces of capacitive equipment run in the same phase is taken to present the failure judgment matrices for N pieces of equipment. According to these matrices, the fault condition of N pieces of capacitive equipment can be watched, which is more general. Finally some problems needing attention and two diagnostic methods used in diagnostic system are put forward in the paper to guide the application of synthetic relative measuring method on local.

Risk Evaluation for Maxi Horizontal Directional Drilling Crossing Projects

Horizontal directional drilling (HDD) has become increasingly used for the construction of large diameter (more than 600 mm or 24 in.) pipeline projects throughout China. Due to the large investment of capital and resources, it is important to conduct a risk evaluation prior to actual construction. However, there are relatively few appropriate analyses or investigations available for such applications. This paper addresses the risk assessment for Maxi HDD projects. The combination of the fuzzy comprehensive evaluation method and analytical hierarchy process was adopted as the basic model. The corresponding risk indices were constructed considering the factors affecting the projects and based on discussions with experts who have conducted HDD crossing projects. Structured judgment matrices and membership matrices were then directly developed from the index systems, and the weight vectors, subordinated vectors, and risk values determined by the use of Matlab software. Finally, the risk level of the projects was obtained by implementing the maximum membership degree law. The synthesis of the fuzzy comprehensive evaluation method and analytical hierarchy process has been demonstrated to provide a theoretical basis for the risk evaluation and establishment of risk control measures for Maxi HDD projects.

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 Wei's 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.

On qualitatively consistent, transitive and contradictory judgment matrices emerging from multiattribute decision procedures

Several known and newly introduced classes of positive reciprocal matrices emerging from pairwise comparisons in multiattribute decision problems are studied in the paper. Mainly qualitative features in connection with consistency and inconsistency are considered in order to extend the range of the available analytical methods regarding pairwise comparisons. By using graph representation of positive reciprocal matrices, graph theoretic approach is applied for the argumentation. The applied notions and the theorems developed in the paper can be useful for eliminating the illogical data that may occur during pairwise comparisons.

Multicriteria optimization in a fuzzy environment: The fuzzy analytic hierarchy process

In the paper the fuzzy extension of the Analytic Hierarchy Process (AHP) based on fuzzy numbers, and its application in solving a practical problem, are considered. The paper advocates the use of contradictory test to check the fuzzy user preferences during fuzzy AHP decision-making process. We also propose consistency check and deriving priorities from inconsistent fuzzy judgment matrices to be included in the process, in order to check if the fuzzy approach can be applied in the AHP for the problem considered. An aggregation of local priorities obtained at different levels into composite global priorities for the alternatives based on weighted-sum method is also discussed. The contradictory fuzzy judgment matrix is analyzed. Our theoretical consideration has been verified by an application of commercially available Super Decisions program (developed for solving multi-criteria optimization problems using AHP approach) on the problem previously treated in the literature. The obtained results are compared with those from the literature. The conclusions are given and the possibilities for further work in the field are pointed out.

Further analysis on matrix operation of AHP

This paper is major on discussing the paper of Barzilai, Cook and Golany published in Operations Research Letters to discuss consistent weights for judgment matrices of the relative importance of alternatives. We point out that they misuse the reciprocal operation to imply meaningless alternatives such that their challenge to conventional AHP is not well founded. Our findings should result in the development of both the theoretical and practical applications.

On consistency of the weighted arithmetical mean complex judgement matrix

The weighted arithmetical mean complex judgement matrix (WAMCJM) is the most common method for aggregating group opinions, but it has a shortcoming, namely the WAMCJM of the perfectly consistent judgement matrices given by experts canot guarantee its perfect consistency. An upper bound of the WAMCJM's consistency is presented. Simultaneously, a compatibility index of judging the aggregating extent of group opinions is also introduced. The WAMCJM is of acceptable consistency and is proved provided the compatibilities of all judgement matrices given by experts are smaller than the threshold value of acceptable consistency. These conclusions are important to group decision making.

Approach to Group Decision Making Based on Interval-Valued Intuitionistic Judgment Matrices

Abstract In this article, the ordered weighted aggregation operator and hybrid aggregation operator are developed for aggregating interval-valued intuitionistic preference information. Interval-valued intuitionistic judgment matrix and its score matrix and accuracy matrix are defined. Some of their desirable properties are investigated in detail. The relationships among interval-valued intuitionistic judgment matrix, intuitionistic judgment matrix, and complement judgment matrix, are discussed. On the basis of the arithmetic aggregation operator and hybrid aggregation operator, an approach to group decision making with interval-valued intuitionistic judgment matrices is given. Finally, a practical example is provided to verify the effectiveness of the developed approach.

FARSJUM, A FUZZY SYSTEM FOR RANKING SPARSE JUDGMENT MATRICES: A CASE STUDY IN SOCCER TOURNAMENTS

The establishment of the priorities from pairwise comparison matrices is the major constituent of the Analytic Hierarchy Process (AHP). However, the number of pairwise comparisons necessary in real problems often becomes overwhelming. In such cases, generally the experts are not able to answer all questions and consequently sparse judgment matrix is generated which caused "equal ranks" and "rank reversal" based on AHP method. In this paper, a new ranking system (FARSJUM) is developed for such sparse judgment matrix. The system is constructed on fuzzy rules and fuzzy reasoning methods. The numerical example of world cup soccer tournament is brought to clarify the performance of the developed system comparing with AHP method in ranking the sparse judgment matrices.

Fuzzy analytic hierarchy process based group decision support system to select and evaluate new manufacturing technologies

Manufacturing organizations often make complex decisions in regards to investment in new manufacturing technologies (NMT). These NMT are assessed based on attributes such as flexibility, quality etc., which are hard to quantify. This fact calls upon the need for a structured decision support systems that can adequately represent qualitative and subjective assessments. The gist of this paper is to propose an integrated fuzzy analytic hierarchy process (AHP) based approach to facilitate the selection and evaluation of NMT in the presence of intangible attributes and uncertainty. Three important issues that are pertinent and critical to fuzzy AHP, namely contradiction in user preferences, deriving priorities from inconsistent fuzzy judgment matrices, and group decision-making are addressed in detail. Also, an attempt is made to compare fuzzy preference programming (FPP) and two-stage logarithmic goal programming (TLGP) based fuzzy prioritization methods. Based on the comparative study, it is concluded that FPP is preferred over TLGP to select and evaluate NMT. Numerical examples are provided to illustrate the aforementioned issues.

Data envelopment analysis for weight derivation and aggregation in the analytic hierarchy process

Data envelopment analysis (DEA) is proposed in this paper to generate local weights of alternatives from pair-wise comparison judgment matrices used in the analytic hierarchy process (AHP). The underlying assumption behind the approach is explained, and some salient features are explored. It is proved that DEA correctly estimates the true weights when applied to a consistent matrix formed using a known set of weights. DEA is further proposed to aggregate the local weights of alternatives in terms of different criteria to compute final weights. It is proved further that the proposed approach, called DEAHP in this paper, does not suffer from rank reversal when an irrelevant alternative(s) is added or removed.

A common framework for deriving preference values from pairwise comparison matrices

Pairwise comparison is commonly used to estimate preference values of finite alternatives with respect to a given criterion. We discuss 18 estimating methods for deriving preference values from pairwise judgment matrices under a common framework of effectiveness: distance minimization and correctness in error free cases. We point out the importance of commensurate scales when aggregating all the columns of a judgment matrix and the desirability of weighting the columns according to the preference values. The common framework is useful in differentiating the strength and weakness of the estimated methods. Some comparison results of these 18 methods on two sets of judgment matrices with small and large errors are presented. We also give insight regarding the underlying mathematical structure of some of the methods. Scope and purpose Pairwise comparison is commonly used to estimate preference values of finite alternatives with respect to a given criterion. This is part of the model structure of the analytical hierarchy process, a widely used multicriteria decision-making methodology. The main difficulty is to reconcile the inevitable inconsistency of the pairwise comparison matrix elicited from the decision makers in real-world applications. We discuss 18 estimating methods for deriving preference values from pairwise judgment matrices under a common framework of effectiveness: the common concepts of minimizing aggregated deviation and correctness in error free cases. The common framework is useful in differentiating the strength and weakness of these methods. For each of these methods, we point out their individual strength in decisional effectiveness. Some comparison results of these 18 methods on two sets of judgment matrices with small and large errors are presented. We also give insight regarding the underlying mathematical structure of some of the methods. We recommend the simple geometric mean method with the stronger feature of distance minimization and the simple normalized column sum method that is based on the simple ideas of commensurate unit and column sum. These two methods have closed-form formulas for easy calculation and good performance on both sets of judgment matrices with small and large errors.

Inconsistent and contradictory judgements in pairwise comparison method in the AHP

The aim of this paper is to show the relationship between inconsistent and contradictory matrices of data obtained as a result of the pairwise comparison of factors in the sense of the analytic hierarchy process. The consistency check is performed to ensure that judgements are neither random nor illogical. This paper shows that even if a matrix will pass a consistency test successfully, it can be contradictory. Moreover an algorithm of checking contradictions is proposed. Scope and purpose A consistency of judgements in the pairwise comparison method in the analytic hierarchy process can be measured using the consistency index of judgement matrices. The consistency index is based on a judgement matrix rank. Depending on its value judgement matrices can be classified into two sets, namely a set of consistent and a set of inconsistent matrices. The paper introduces a new class of judgement matrices which is defined as contradictory and shows that a judgement matrix can be very closed to fully consistent in sense of the assumed consistency index but still to be contradictory.

Two approaches to improving the consistency of complementary judgement matrix

By the transformation relations between complementary judgement matrix and reciprocal judgement matrix, this paper proposes two methods for improving the consistency of complementary judgement matrix and gives two simple practical iterative algorithms. These two algorithms are easy to implement on computer, and the modified complementary judgement matrices remain most information that original matrix contains. Thus the methods supplement and develop the theory and methodology for improving consistency of complementary judgement matrix.

On consistency of the weighted geometric mean complex judgement matrix in AHP

The weighted geometric mean method (WGMM) is the most common group preference aggregation method in the Analytic Hierarchy Process. This paper reports on research concerning the consistency of WGMM and proves that the weighted geometric mean complex judgement matrix (WGMCJM) is of acceptable consistency. In Saaty's (T.L. Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980) opinion, a consistency ratio (CR) of 0.1 or less is acceptable under the condition that all judgement matrices given by experts for the same problem of decision-making are of acceptable consistency. Accordingly, a theoretic basis has been developed for the application of the WGMM in group decision making.

On fuzzy solutions to the pairwise comparison

The paper concerns a fuzzy extension of the pairwise comparison method introduced by Saaty in order to rank a finite number of factors, attributes or criteria in multicriteria group decision making. The approach can be used when an expert can provide his preference in a fuzzy form rather than in a crisp one, however the method applies in case of mixed fuzzy crisp opinions. It can be used in cases with lacking judgements as well. Two main approaches to solve so-called normal equations with fuzzy right-hand sides are considered and compared, namely ones based on implicit and explicit solutions to the problem. The pairwise comparison method with crisp judgement matrices is recalled. To solve the pairwise comparison problem a pseudoinverse approach is used in each considered case. Numerical results are included. Finally paper concludes with further possible extensions, remarks and comments, and indicates an area of application.

Extracting Consistent Weight Ratio Matrices from Inconsistent Judgment Matrices

Extracting Consistent Weight Ratio Matrices from Inconsistent Judgment Matrices

A statistical approach to the analytic hierarchy process with interval judgements. (I). Distributions on feasible regions

Abstract This paper addresses the problem of extracting preferences for alternatives from interval judgement matrices in the Analytic Hierarchy Process (AHP). The method of Arbel for extracting such preferences, which is based on the assumption that the interval judgements specify a feasible region in the weight space of the alternatives, is critically appraised from a statistical perspective and some new ideas emanating from this approach are developed and discussed. In particular it is proposed that a distribution for the weights on the feasible region, which is both tractable and meaningful, be adopted. The mean of the distribution can then be used as an assessment of the overall ranking of the alternatives and quantities of interest, such as probabilities and marginal distributions, can immediately be quantified. Two specific distributions on the feasible region, the uniform distribution and the distribution of random convex combinations with coefficients which are uniform spacings, are examined in some detail and the ideas which emerge are illustrated by means of selected examples.