Incomplete Pairwise Comparison Research Articles

How to choose a completion method for pairwise comparison matrices with missing entries: An axiomatic result

Since there exist several completion methods to estimate the missing entries of pairwise comparison matrices, practitioners face a difficult task in choosing the best technique. Our paper contributes to this issue: we consider a special set of incomplete pairwise comparison matrices that can be represented by a weakly connected directed acyclic graph, and study whether the derived weights are consistent with the partial order implied by the underlying graph. According to previous results from the literature, two popular procedures, the incomplete eigenvector and the incomplete logarithmic least squares methods fail to satisfy the required property. Here, the recently introduced lexicographically optimal completion combined with any of these weighting methods is shown to avoid ordinal violation in the above setting. Our finding provides a powerful argument for using the lexicographically optimal completion to determine the missing elements in an incomplete pairwise comparison matrix.

A new SMAA-based methodology for incomplete pairwise comparison matrices: evaluating production errors in the automotive sector

Analysing and mitigating errors in production processes is a primary objective of companies in the automotive sector. Unfortunately, due to inaccurate or partially missing information, comparing errors is often very difficult, resulting from the experts’ provision of incomplete pairwise comparison matrices. In the literature, several techniques have been developed to complete such matrices. These techniques merely estimate what the decision makers or experts would have entered according to known entries. In this article, we propose a new methodology based on the stochastic multi-objective acceptability analysis; we apply it to vary the missing entries of the pairwise comparison matrix, thus providing the probability that an alternative/criterion will attain a given rank. This approach gives a complete view of the possible outcomes because it represents all possible decision maker mindsets. We present a case study carried out in a multinational automotive industry where we apply our methodology for evaluating errors in the production process.

On the coincidence of optimal completions for small pairwise comparison matrices with missing entries

AbstractIncomplete pairwise comparison matrices contain some missing judgements. A natural approach to estimate these values is provided by minimising a reasonable measure of inconsistency after unknown entries are replaced by variables. Two widely used inconsistency indices for this purpose are Saaty’s inconsistency index and the geometric inconsistency index, which are closely related to the eigenvector and the logarithmic least squares priority deriving methods, respectively. The two measures are proven to imply the same optimal filling for incomplete pairwise comparison matrices up to order four but not necessarily for order at least five.

PyMissingAHP: An Evolutionary Algorithm for Filling Missing Values in Incomplete Pairwise Comparisons Matrices with Real or Fuzzy Numbers via Mono and Multiobjective Approaches

PyMissingAHP: An Evolutionary Algorithm for Filling Missing Values in Incomplete Pairwise Comparisons Matrices with Real or Fuzzy Numbers via Mono and Multiobjective Approaches

Some methods to derive the priority weights from the best–worst method matrix and weight efficiency test in view of incomplete pairwise comparison matrix

Some methods to derive the priority weights from the best–worst method matrix and weight efficiency test in view of incomplete pairwise comparison matrix

Enhancing Consistency of Pairwise Comparisons on the Base of Linear Algebraic Equations

A problem of improving consistency of pairwise comparisons matrices in application to ranking given alternatives is considered in the paper. But it can be shown that consistency is not the only issue as to the quality of pairwise comparisons. Given an arbitrary positive square matrix, we can obtain an ideally consistent pairwise comparison matrix with the same Perronian vector. Therefore, the quality of experts’ judgements is an issue of great importance as well.Technically, an approach to improving consistency of pairwise comparisons on the basis of solving a linear algebraic equations system is suggested. The system contains two groups of equations. One of them represents experts’ judgments, and the other is related to demands of cardinal consistency. Such a system can be over- or maybe underdetermined, and it typically can be inconsistent. Then a pseudo-solution can be obtained by means of pseudo-inverse Moore-Penrose matrix.For improving the quality of pairwise comparisons, it appears urgent to take into account reliabilities of certain judgements by giving them appropriate weight coefficients.Some numerical examples are provided in the paper. The first is a simple basic example without any serious inconsistencies. The second illustrates as to treat incomplete pairwise comparison matrices. And the latest illustrates possible expert’s manipulation, when an expert wants to secure the winning of a certain alternative whereas they don’t want to postulate the advantage of this alternative implicitly, and this results in the order violation. It is illustrated how introducing weight coefficients of equations can help counteract such manipulations.

Incomplete pairwise comparison matrices: Ranking top women tennis players

The method of pairwise comparisons is frequently applied for ranking purposes. This article aims to rank top women tennis players based on their win/lose ratios. Incomplete pairwise comparison matrices (PCMs) were constructed from data obtained from the WTA (Women’s Tennis Association) homepage. The database contains head-to-head results from the period between 1973 and 2022 for 28 players who had the position No. 1 in the official rankings of WTA. The weight vector was calculated from the incomplete PCM with the logarithmic least squares method and the eigenvector method. The results are not surprising: Serena Williams, Steffi Graf, and Martina Navratilova stand in the first three positions, and Martina Hingis, Kim Clijsters, and Justine Henin follow them. We also tested the frequently used probability-based Bradley-Terry method and found high rank-correlation values. Using graph representations, the results gave us a deeper insight into the properties of incomplete PCMs. Special attention was given to the nontransitive triads. A data modification was necessary to remove ties in order to apply the commonly used tests. The results indicate that ordinally nontransitive triads are not significant in the data we analysed.

Worker knowledge sharing in logistics services: case of a Brazilian aircraft manufacturer

Purpose - This paper assesses factors that enable tacit knowledge sharing among logistic workers at Brazilian aircraft manufacturer.Design/methodology/approach - The proposal methodology is supported by a mixed methods approach to examine how organizations can develop favorable contexts to explore knowledge sharing relations among workers in the Logistics area of a Brazilian aircraft manufacturer. The proposal has a qualitative phase based on local interviews, informal conversations and on-site observations. The quantitative phase entails a ranking of the different factors that support knowledge sharing in terms of their relative importance to each other, using the Analytic Hierarchy Process with Incomplete Pairwise Comparison (AHP-IPC).Findings - Our overall results indicated that Understanding of Work Instruction and Registration in Work Instruction were perceived as the most important processes among the logistic workers, followed by Conversation among Workers. Based on the qualitative phase analysis it was possible to observe that the factors Incentive, Work Instruction, Zero Defect and Training were considered relevant.Research limitations/implications - This study was limited to the logistics department of a Brazilian aircraft manufacturer and its logistic workers. Practical implications - The findings contribute to support managers to deal knowledge sharing through efficient communication, collaboration and thrust among workers in the company.Originality/value - This is the first study in a Logistics department of a Brazilian aircraft manufacturer applying the proposal methodology based on questionnaires in order to capture the workers perception of the resilience and knowledge sharing in the shop floor and it is very relevant due the complexity of the operation in a logistic area of an aircraft manufacturer. Keywords: Knowledge Sharing, Logistic, Aircraft Manufacturer, Brazil.

On the derivation of weights from incomplete pairwise comparisons matrices via spanning trees with crisp and fuzzy confidence levels

In this paper, we propose a new method for the derivation of a priority vector from an incomplete pairwise comparisons (PC) matrix. We assume that each entry of a PC matrix provided by an expert is also evaluated in terms of the expert's confidence in a particular judgment. Then, from corresponding graph representations of a given PC matrix, all spanning trees are found. For each spanning tree, a unique priority vector is obtained with the weight corresponding to the confidence levels of entries that constitute this tree. At the end, the final priority vector is obtained through an aggregation of priority vectors achieved from all spanning trees. Confidence levels are modeled by real (crisp) numbers and triangular fuzzy numbers. Numerical examples and comparisons with other methods are also provided. Last, but not least, we introduce a new formula for an upper bound of the number of spanning trees, so that a decision maker gains knowledge (in advance) on how computationally demanding the proposed method is for a given PC matrix.

Incomplete pairwise comparison matrices based on graphs with average degree approximately 3

A crucial, both from theoretical and practical points of view, problem in preference modelling is the number of questions to ask from the decision maker. We focus on incomplete pairwise comparison matrices based on graphs whose average degree is approximately 3 (or a bit more), i.e., each item is compared to three others in average. In the range of matrix sizes we considered, n=5,6,7,8,9,10, this requires from 1.4n to 1.8n edges, resulting in completion ratios between 33% (n=10) and 80% (n=5). We analyze several types of union of two spanning trees (three of them building on additional ordinal information on the ranking), 2-edge-connected random graphs and 3-(quasi-)regular graphs with minimal diameter (the length of the maximal shortest path between any two vertices). The weight vectors are calculated from the natural extensions, to the incomplete case, of the two most popular weighting methods, the eigenvector method and the logarithmic least squares. These weight vectors are compared to the ones calculated from the complete matrix, and their distances (Euclidean, Chebyshev and Manhattan), rank correlations (Kendall and Spearman) and similarity (Garuti, cosine and dice indices) are computed in order to have cardinal, ordinal and proximity views during the comparisons. Surprisingly enough, only the union of two star graphs centered at the best and the second best items perform well among the graphs using additional ordinal information on the ranking. The union of two edge-disjoint spanning trees is almost always the best among the analyzed graphs.

Introducing a Novel Method for Smart Expansive Systems’ Operation Risk Synthesis

In different areas of human activity, the need to choose optimal (rational) options for actions from the proposed alternatives inevitably arises. In the case of retrospective statistical data, risk analysis is a convenient tool for solving the problem of choice. However, when planning the growth and development of complex systems, a new approach to decision-making is needed. This article discusses the concept of risk synthesis when comparing alternative options for the development of a special class of complex systems, called smart expansive systems, by the authors. “Smart” in this case implies a system capable of ensuring a balance between its growth and development, considering possible external and internal risks and limitations. Smart expansive systems are considered in a quasi-linear approximation and in stationary conditions of problem-solving. In general, when the alternative to comparison is not the object itself, but some scalar way of determining risks, the task of selecting the objects most at risk is reduced to assessing the weights of factors affecting the integral risk. As a result, there is a complex task of analyzing the risks of objects, solved through the amount by which the integral risk can be minimized. Risks are considered as anti-potentials of the system development, being retarders of the reproduction rate of the system. The authors give a brief description of a smart expansive system and propose approaches to modeling the type of functional dependence of the integral risk of functioning of such a system on many risks, measured, as a rule, in synthetic scales of pairwise comparisons. The solution to the problem of reducing the dimension of influencing factors (private risks) using the vector compression method (in group and inter-scale formulations) is described. This article presents an original method for processing matrices of incomplete pairwise comparisons with indistinctly specified information, based on the idea of constructing reference-consistent solutions. Examples are provided of how the vector compression method can be applied to solve practical problems.

A decision-making model with sequential incomplete additive pairwise comparisons

A sequential modelling of pairwise comparisons exhibits the capability to effectively cope with some complexity in a decision making process. When the complexity of decision making problems leads to incomplete pairwise comparisons, a feasible approach to optimal solutions is worth investigations. In this paper, we propose the idea that a missing value is mainly dependent on the neighbour ones in a sequential of comparisons; and develop a decision making model. First, additive pairwise comparisons with missing values are captured by proposing a sequential model and some properties are studied. Second, an optimization model for completing an incomplete additive reciprocal matrix (ARM) is recalled. Based on the sequential model, the missing values of incomplete ARMs are estimated using particle swarm optimization (PSO). Different with the previous studies, the permutations of alternatives are considered to give various estimations of a missing value. Third, a granularity-based method is proposed to improve additive consistency of a completed ARM; and the best granularity level is chosen. Finally, a decision-making algorithm is established, where the opinions of decision makers (DMs) are expressed as incomplete ARMs. Some cases are studied to illustrate the finding that multi-optimal solutions could exist in a decision making problem. The novel phenomenon is in agreement with the case that different orders of comparing alternatives may yield different results due to the inconsistency of human-originated information.

Inconsistency thresholds for incomplete pairwise comparison matrices

Pairwise comparison matrices are increasingly used in settings where some pairs are missing. However, there exist few inconsistency indices for similar incomplete data sets and no reasonable measure has an associated threshold. This paper generalises the famous rule of thumb for the acceptable level of inconsistency, proposed by Saaty, to incomplete pairwise comparison matrices. The extension is based on choosing the missing elements such that the maximal eigenvalue of the incomplete matrix is minimised. Consequently, the well-established values of the random index cannot be adopted: the inconsistency of random matrices is found to be the function of matrix size and the number of missing elements, with a nearly linear dependence in the case of the latter variable. Our results can be directly built into decision-making software and used by practitioners as a statistical criterion for accepting or rejecting an incomplete pairwise comparison matrix.

Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter

Pairwise comparisons have become popular in the theory and practice of preference modelling and quantification. In case of incomplete data, the arrangements of known comparisons are crucial for the quality of results. We focus on decision problems where the set of pairwise comparisons can be chosen and it is designed completely before the decision making process, without any further prior information. The objective of this paper is to provide recommendations for filling patterns of incomplete pairwise comparison matrices based on their graph representation. The proposed graphs are regular and quasi-regular ones with minimal diameter (longest shortest path). Regularity means that each item is compared to others for the same number of times, resulting in a kind of symmetry. A graph on an odd number of vertices is called quasi-regular, if the degree of every vertex is the same odd number, except for one vertex whose degree is larger by one. We draw attention to the diameter, which is missing from the relevant literature, in order to remain the closest to direct comparisons. If the diameter of the graph of comparisons is as low as possible (among the graphs of the same number of edges), we can decrease the cumulated errors that are caused by the intermediate comparisons of a long path between two items. Contributions of this paper include a list containing (quasi-)regular graphs with diameter 2 and 3 up until 24 vertices. Extensive numerical tests show that the recommended graphs indeed lead to better weight vectors compared to various other graphs with the same number of edges. It is also revealed by examples that neither regularity nor small diameter is sufficient on its own, both properties are needed. Both theorists and practitioners can utilize the results, given in several formats in the appendix: plotted graph, adjacency matrix, list of edges, ‘Graph6’ code.

Constrained Eigenvalue Minimization of Incomplete Pairwise Comparison Matrices by Nelder-Mead Algorithm

Pairwise comparison matrices play a prominent role in multiple-criteria decision-making, particularly in the analytic hierarchy process (AHP). Another form of preference modeling, called an incomplete pairwise comparison matrix, is considered when one or more elements are missing. In this paper, an algorithm is proposed for the optimal completion of an incomplete matrix. Our intention is to numerically minimize a maximum eigenvalue function, which is difficult to write explicitly in terms of variables, subject to interval constraints. Numerical simulations are carried out in order to examine the performance of the algorithm. The results of our simulations show that the proposed algorithm has the ability to solve the minimization of the constrained eigenvalue problem. We provided illustrative examples to show the simplex procedures obtained by the proposed algorithm, and how well it fills in the given incomplete matrices.

Incomplete analytic hierarchy process with minimum weighted ordinal violations

ABSTRACT Incomplete pairwise comparison matrices offer a natural way of expressing preferences in decision-making processes. Although ordinal information is crucial, there is a bias in the literature: cardinal models dominate. Ordinal models usually yield nonunique solutions; therefore, an approach blending ordinal and cardinal information is needed. In this work, we consider two cascading problems: first, we compute ordinal preferences, maximizing an index that combines ordinal and cardinal information; then, we obtain a cardinal ranking by enforcing ordinal constraints. Notably, we provide a sufficient condition (that is likely to be satisfied in practical cases) for the first problem to admit a unique solution and we develop a provably polynomial-time algorithm to compute it. The effectiveness of the proposed method is analyzed and compared with respect to other approaches and criteria at the state of the art.

New inconsistency indicators for incomplete pairwise comparisons matrices

We introduce two new inconsistency measures for the incomplete pairwise comparisons matrices and show several examples of their calculation. We also carry out a comparative analysis of the new inconsistency indices with the existing ones based on the Monte Carlo simulation.

Base-criterion on multi-criteria decision-making method and its applications

ABSTRACTIn this paper, Base-criterion method (BCM) is proposed to solve multi-criteria decision-making problems. According to BCM, instead of executing a pairwise comparisons between all criteria or executing a pairwise comparisons between each of the best and worst criteria to other criteria, one of the criteria is chosen by the decision-maker as a base-criterion (e.g. preferential, selective) and then pairwise comparisons between base-criterion and other criteria are obtained. Then, a max-min problem is formulated and solved to determine the weight of the criteria. In this way, the pairwise comparisons to obtain weights of the criteria are fully consistent. In other ability of BCM, we find the lost pairwise comparisons by using the BCM framework so that the pairwise comparisons matrix remains consistent. To illustrate the reliability of the proposed method and evaluate its performance to determine the weights, we have used two numerical examples. The outcomes of numerical examples indicate that the BCM has high accuracy and better consistency ratio than the other multi-criteria decision-making(MCDM) methods. The cause of high consistency ratio in the BCM is the dependency of pairwise comparisons between criteria. Moreover, we have used two other numerical examples to illustrate the function of the BCM in order to find out the missing comparisons in incomplete pairwise comparison matrix. We have shown the BCM framework able to find lost comparisons in the worst terms of the incomplete pairwise comparison matrix. The outcomes of the proposed article show that BCM performance is significantly better than AHP and BWM methods with respect to the consistency ratio, and it requires fewer comparison data and has the ability to calculate missing pairwise comparisons.

EXTENDED CONSISTENCY ANALYSIS FOR PAIRWISE COMPARISON METHOD

Pairwise comparison (PC) is a widely used scientific technique to compare criteria or alternatives in pairs in order to express the decision maker’s judgments without the need for a unique common measurement unit between criteria. The method constructs a PC matrix by requesting the assessments of the decision maker(s) in the judgment acquisition phase and calculates an inconsistency measure to determine whether the judgments are adequately consistent with each other before subsequent phases. Although the method requires the decision maker to make all judgments in a PC matrix, it does not force him/her to make a judgment for each element of the matrix. If any judgment in a PC matrix is absent, for this reason, the judgment acquisition phase yields an incomplete PC matrix rather than a complete one. Missing judgments are calculated by multiplication of the judgments made by the decision maker. If the judgements of the decision maker are transitive and well-proportioned, missing judgments will not disturb the consistency of the resulting PC matrix. In other words, consistency of a PC matrix relies on the judgments made by the decision maker. Since the current consistency analysis procedure is designed for complete PC matrices, the suitability for evaluating the inconsistency of incomplete PC matrices is questionable. Probability density functions of random PC matrices with altering numbers of missing judgments show distinct features, indicating an incomplete PC matrix and a complete PC matrix do not come from the same probability function, and their mean consistency index (RI) is different. Consequently, we propose an extended consistency analysis procedure to evaluate the consistency of incomplete PC matrices.

A DEMATEL-based completion method for incomplete pairwise comparison matrix in AHP

Pairwise comparison matrix as a crucial component of AHP, presents the prefer- ence relations among alternatives. However, in many cases, the pairwise comparison matrix is difficult to complete, which obstructs the subsequent operations of the clas- sical AHP. In this paper, based on DEMATEL which has ability to derive the total relation matrix from direct relation matrix, a new completion method for incomplete pairwise comparison matrix is proposed. The proposed method provides a new per- spective to estimate the missing values with explicit physical meaning. Besides, the proposed method has low computational cost. This promising method has a wide application in multi-criteria decision-making.