Consistency Of Pairwise Comparison Matrices Research Articles

Generating a Set of Consistent Pairwise Comparison Test Matrices in AHP using Particle Swarm Optimization

Pairwise Comparison Matrix (PCM) play a crucial role in multi-criteria decision-making, especially within the framework of the Analytic Hierarchy Process (AHP). It is essential for an expert to systematically assess and compare alternatives, considering various criteria as part of the decision-making process. The AHP relies on consistency to ensure the accuracy of pairwise comparisons made by decision makers, increasing the overall integrity of the decision-making process. The aim of this paper is to present an approach based on Particle Swarm Optimization (PSO) as a metaheuristic method to generate a set of consistent pairwise comparison matrices. Numerical results are presented with different sizes of matrices showing the effectiveness of our algorithm in producing acceptable matrices for the benefit of expets.

A metaheuristic approach to improve consistency of the pairwise matrix in AHP

In this paper, we are interested in modifying inconsistent pairwise comparison matrix which is a critical step in the AHP methodology, where decision makers have to improve the consistency by revising the process. To this end, we propose an improved genetic algorithm (GA) to allow decision makers to find an appropriate matrix and adjust the consistency of their judgment without loss of original comparison matrix. Numerical results with different dimensions of matrices taken randomly show the effectiveness of these strategy to improve and identify the consistency of pairwise matrix which mean that GAs are a very good tool to generate the consistent pairwise comparison matrices with different number of criteria.

Applying statistical approach to check the consistency of pairwise comparison matrices during software requirements prioritization process

Applying statistical approach to check the consistency of pairwise comparison matrices during software requirements prioritization process

A decision making model based on the leading principal submatrices of a reciprocal preference relation

A completed comparison matrix is the basic tool in the analytic hierarchy process (AHP). In practice, the process of forming a comparison matrix is complex and it is worth being investigated carefully. In this study, by decomposing pairwise comparison process of alternatives, the leading principal submatrices (LPSMs) of a completed comparison matrix are used as the basis for decision analysis. Based on the particle swarm optimization (PSO), a novel method for improving consistency of inconsistent comparison matrices is proposed. The fitness function is constructed by considering the acceptable consistency of pairwise comparison matrices and the similarity degree between the initial and the adjusted decision-making information. A new algorithm is elaborated on for solving a decision making problem with multiplicative reciprocal matrices. Numerical results are reported to show the advantages of the proposed model by comparing with the other methods. The observations reveal that the proposed method and algorithm are effective when dealing with inconsistent comparison matrices and the corresponding decision making problems.

A system approach to decision support on basis of hierarchical and network models

A methodology and a system approach to decision support based on hierarchical and network models are proposed, which include methods for estimating and improving the consistency of pairwise comparison matrices (PCM), calculating fuzzy local weights for model elements based on fuzzy PCM, calculating confidence intervals for local weights, finding aggregated weights and aggregated belief functions for model elements, estimating the sensitivity of the results, tools for analyzing the effectiveness of the methods and an expert evaluation modeling system. A decision support system is built on basis of the proposed methodology. The proposed methodology makes it possible to increase the reliability of solutions of practical weakly structured decision-making problems by using more efficient methods at each of the stages of solving the problem. Reliability of results of application of the methodology while solving a practical decision-making problem can be assessed by means of sensitivity analysis using the stability indices of the computed local and global rankings of the decision alternatives.

A hybrid approach for supplier selection based on revised data envelopment analytic hierarchy process

For most managers, purchasing decisions are key decisions in the achievement of business organisation's strategic objectives. Appropriate supplier selection is necessary as it fulfils these important objectives of the organisation by maximising overall value to the purchaser, reducing uncertainty in supply, and by forming a long term, reliable relationship between buyers and suppliers. Supplier selection being affected by several conflicting factors requires an appropriate multi-criteria decision making approach. One of the recent approaches used for supplier selection is a hybrid data envelopment analytic hierarchy process (DEAHP) approach. However, this approach suffers from serious drawbacks as it uses counter intuitive priority vectors for inconsistent pair wise comparison matrices and also at times, it generate illogical weights for consistent pair wise comparison matrices as well. The objective of this paper is to present a revised DEAHP (RDEAHP) approach to overcome these drawbacks of DEAHP. The uniqueness of the research lies in the application of RDEAHP approach for supplier selection problem.

Strong reciprocity and strong consistency in pairwise comparison matrix with fuzzy elements

The decision making problem considered in this paper is to rank n alternatives from the best to the worst, using the information given by the decision maker in the form of an $$n\times n$$ pairwise comparison matrix. Here, we deal with pairwise comparison matrices with fuzzy elements. Fuzzy elements of the pairwise comparison matrix are applied whenever the decision maker is not sure about the value of his/her evaluation of the relative importance of elements in question. We investigate pairwise comparison matrices with elements from abelian linearly ordered group (alo-group) over a real interval. The concept of reciprocity and consistency of pairwise comparison matrices with fuzzy elements have been already studied in the literature. Here, we define stronger concepts, namely the strong reciprocity and strong consistency of pairwise comparison matrices with fuzzy intervals as the matrix elements (PCF matrices), derive the necessary and sufficient conditions for strong reciprocity and strong consistency and investigate their properties as well as some consequences to the problem of ranking the alternatives.

Towards a Quantitative Model of Cloud Computing Risks and Benefits

Migrating to the cloud is the main direction of enterprise IT optimization today. Many research papers confirm that cloud computing provides economic benefits, because it enhances flexibility and reduces costs. In other studies, cloud-specific risks are identified and their impact on the customer business is evaluated. However, most often, benefits and risks are considered separately. Model that allows simultaneously evaluate these factors is proposed here. Key factors of tangible and intangible benefits and risks are identified that allows to estimate joint impact of costs and risks on cloud adoption. Simple rules that help to quantify these factors and compute consistent pairwise comparison matrices are also proposed. Usage of proposed method is demonstrated with simple example.

AHP and weak consistency in the evaluation of works of art - a case study of a large problem

The paper describes an evaluation methodology based on Saaty's AHP, that relaxes the classical Saaty's consistency condition and works with the concept of weak consistency. Weak consistency is seen as a minimum requirement on the consistency of experts' preferences. The relationship of weak consistency and the linguistic level of consistency description using the Saaty's scale is discussed in the paper. The benefits of using weak consistency with large problems are presented - for data input, for the flexibility of the mathematical model and to facilitate further adjustments of the evaluation methodology. In the case of pairwise comparison matrices with ordered categories, the fulfilment of weak consistency can be checked during the data input phase. This way the weak consistency of pairwise comparison matrices can be achieved even for large numbers of categories - unlike the full consistency in Saaty's sense. The paper also provides a case study of a practical application of the proposed evaluation methodology - the mathematical model for the evaluation of creative work outcomes of Czech Art Colleges. The case presented here combines evaluation based on objective criteria with peer review and suggests a possible solution to the problem of arts evaluation for funding purposes.

Linking validation: A search for coherency within the Supermatrix

The consistency check within each pairwise comparison matrix is an important step in an Analytic Network Process (ANP) decision. In an ANP network there is both the ability and the need to test for additional levels of consistency or coherency among the priority vectors. Examples are used to highlight cases where a Supermatrix with priority vectors that were obtained from either perfect or nearly perfect consistent pairwise comparison matrices generates suboptimal decisions. Simulations are used to further demonstrate the frequency of these occurrences in general ANP networks. A form of cross validation within the Supermatrix called linking validation is developed and demonstrated. The linking validation method allows decision makers to use the priority vectors within the Supermatrix to validate other priority vectors within the Supermatrix. The linking validation method involves generating linking estimates. The linking estimates are compared against each other to identify the most incoherent priority vector by calculating the Linking Coherency Index (LCI) scores. The decision maker can then update the specified priority vector and repeat this process until the LCI-score for every linking estimate is below the given threshold. The use of linking validation to test for coherency further improves the validity of ANP models.

Development of a model and modification of the hierarchy analysis method for energy efficiency level estimation

The components of the information technology for energy efficiency level estimation of municipal buildings were investigated. As a result of the research, the following was carried out: 1. Decomposition of the solution of the factors ranking problem in the hierarchy was performed. 2. A multi-layer model for energy efficiency estimation was developed and the factors ranking process was described. 3. The hierarchy analysis method was modified through: • forming the consistent pairwise comparison matrices of indicators in the individual factors A F =(a F ij ); • developing a consistent comparison matrix of factors A F =(a F ij ); • determining a global criterion based on factor-indicator estimates. Implementation of the proposed method that allows, despite a fairly large dimension of the array of indicators within a given factor, to conduct correct pairwise comparisons with achieving a specified consistency level (1Y≤10 %) and determine the relative values of indicators {ω i 1 , ω i 2 , ........., ω i k } that can be trusted is of particular interest and at the same time has a positive impact. Using the proposed method allows to organize, algorithmize and adjust the procedure of expert assessment of unlike factors and improve the quality of the results obtained to form the decision-making process.

Ensuring reliability of the weighting vector: Weak consistent pairwise comparison matrices

In the context of Pairwise Comparison Matrices (PCMs) defined over abelian linearly ordered group, ⊙-consistency and ⊙-transitivity represent a full coherence of the Decision Maker (DM) and the minimal logical requirement that DM's preferences should satisfy, respectively. Moreover, the ⊙-mean vector wm⊙ is proposed as weighting vector for the decision elements related to the PCM. In this paper, we investigate the effects of ⊙-inconsistency of a ⊙-transitive PCM on wm⊙ and, in order to ensure its reliability as weighting vector, we provide the notion of weak ⊙-consistency; it is weaker than ⊙-consistency and stronger than ⊙-transitivity, and ensures that vectors associated with a PCM, by means of a strictly increasing synthesis functional, are reliable for assigning a preference order on the set of related decision elements. The ⊙-mean vector wm⊙ is associated with a PCM by means of one of these functionals. Finally, we introduce an order relation on the rows of the PCM, that is a simple order if and only if the condition of weak ⊙-consistency is satisfied; the simple order allows us to easily determine the actual ranking on the set of related decision elements.

Consistency issues of interval pairwise comparison matrices

In this paper, we investigate the consistency issues of interval pairwise comparison matrices in detail. Using logarithmic Manhattan distance to define the deviation degree of a pairwise comparison matrix to consistent pairwise comparison matrices, we propose a new consistency index of pairwise comparison matrices. Based on this consistency index of pairwise comparison matrices, we develop a consistency index of interval pairwise comparison matrices. Several desired properties of the proposed consistency indexes are presented. Furthermore, linear programming (LP) models are developed to compute the consistency indexes. Then, we propose a LP-based consistency improving model, which optimally preserves original pairwise comparison information in improving consistency. Meanwhile, considering the uncertainty plays an important role in the consistency index of interval pairwise comparison matrices, this consistency improving model is extended to simultaneously manage the uncertain degree in interval pairwise comparison matrices. Finally, we discuss the consistency-based prioritization method, and propose the strong consistency index of interval pairwise comparison matrices.

Deriving consistent pairwise comparison matrices in decision making methodologies based on linear programming method

Pairwise comparison matrix (PCM) with crisp or fuzzy elements should satisfy consistency requirements when it is used in analytic hierarchy process (AHP) or in fuzzy AHP methodologies. An algorithm...

Deriving consistent pairwise comparison matrices in decision making methodologies based on linear programming method

Pairwise comparison matrix (PCM) with crisp or fuzzy elements should satisfy consistency requirements when it is used in analytic hierarchy process (AHP) or in fuzzy AHP methodologies. An algorithm has been presented to obtain a new modified consistent PCM for the corresponding inconsistent original one. The algorithm sets a linear programming problem based on all of the constraints. To obtain the optimum eigenvector of the middle value of the new PCM, segment tree is used to gradually approach the greatest lower bound of distance with the original PCM. As to obtain the lower value and upper value of the new PCM, a theory is proposed to reduce adding uncertainty factors and could maximum maintain the similarity with original PCM. The experiments for crisp elements show that the proposed approach can preserve more the original information than references. The experiments for fuzzy elements show that our method can effectively reduce inconsistency and obtain suitable modified fuzzy PCMs.

Cross-Weight Evaluation for Pairwise Comparison Matrices

Data envelopment analysis (DEA) methodology has recently been applied to weight derivation in the analytic hierarchy process (AHP). This paper proposes a cross-weight evaluation technique, which is similar to the cross-efficiency evaluation in DEA, for priority determination in the AHP. The cross-weight evaluation produces true weights for perfectly consistent pairwise comparison matrices and logical weights for inconsistent pairwise comparison matrices. Numerical examples are examined to illustrate the advantages and potential applications of the cross-weight evaluation in priority determination in the AHP.

A heuristic method to rectify intransitive judgments in pairwise comparison matrices

This paper investigates the effects of intransitive judgments on the consistency of pairwise comparison matrices. Statistical evidence regarding the occurrence of intransitive judgements in pairwise matrices of acceptable consistency is gathered by using a Monte–Carlo simulation, which confirms that relatively high percentage of comparison matrices, satisfying Saaty’s CR criterion are ordinally inconsistent. It is also shown that ordinal inconsistency does not necessarily decrease in the group aggregation process, in contrast with cardinal inconsistency. A heuristic algorithm is proposed to improve ordinal consistency by identifying and eliminating intransitivities in pairwise comparison matrices. The proposed algorithm generates near-optimal solutions and outperforms other tested approaches with respect to computation time.

A linear programming approximation to the eigenvector method in the analytic hierarchy process

Eigenvector method (EM) is a well-known approach to deriving priorities from pairwise comparison matrices in the analytic hierarchy process (AHP), which requires the solution of a set of nonlinear eigenvalue equations. This paper proposes an approximate solution approach to the EM to facilitate its computation. We refer to the approach as a linear programming approximation to the EM, or LPAEM for short. As the name implies, the LPAEM simplifies the nonlinear eigenvalue equations as a linear programming for solution. It produces true weights for perfectly consistent pairwise comparison matrices. Numerical examples are examined to show the validity and effectiveness of the proposed LPAEM and its significant advantages over a recently developed linear programming method entitled LP-GW-AHP in rank preservation.

Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups

Characterizations of consistent pairwise comparison matrices over abelian linearly ordered groups

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

Abstract 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.