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1
Content available REDUCE – A Python Module for Reducing Inconsistency in Pairwise Comparison Matrices
Kuraś Paweł, Strzałka Dominik, Kowal Bartosz, Mazurek Jiří
Advances in Science and Technology. Research Journal
|
2023
|
Vol. 17, no 4
227--234
EN
This paper introduces REDUCE, a Python module designed to minimize inconsistency in multiplicative pairwise comparisons (PC), a fundamental technique in Multi-Criteria Decision Making (MCDM). Pairwise comparisons are extensively used in various fields, including engineering science and numerical simulation methods, to compare different options based on a set of criteria. However, human errors in perception and judgment often lead to inconsistencies in pairwise comparison matrices (PCM). REDUCE addresses this issue by implementing several algorithms that identify and correct inaccurate data in PCMs, thereby reducing the inconsistency ratio. These algorithms do not require expert intervention, making REDUCE a valuable tool for both scientific research and small to medium enterprises that may not have access to costly commercial software or dedicated decision-making experts. The main functionality of the module is incorporating iterative algorithms for inconsistency reduction. The REDUCE library, written in Python and utilizing auxiliary libraries such as NumPy, SciPy, and SymPy, offers 21 functions categorized into data input helpers, consistency ratio (CR) reduction algorithms, PCM indexes, and support functions. Performance testing indicates that the library can efficiently handle matrices of varying sizes, particularly those ranging from 3x3 to 10x10, and its use significantly accelerates the process compared to spreadsheets, especially when dealing with large quantities of matrices. The library has already been used in several research papers and application tools, and its availability as a free resource opens up opportunities for small and medium-sized enterprises to leverage multi-criteria decision-making methods. Currently, there are no publicly available libraries for this solution. The authors believe that the proposed module may contribute to do better decision-making process in pairwise comparisons, not only for the circle of scientists but also for small and medium enterprises that usually cannot afford expensive commercial software and do not employ full-time experts in decision-making as they rely on the experience of their employees and free online resources. It should also contributes to the transition to Industry 4.0 and advances research in fields such as fuzzy logic, preference programming, and constructive consistent approximations.
2
Content available On estimation of priority vectors derived from inconsistent pairwise comparison matrices
Kazibudzki Paweł Tadeusz
Journal of Applied Mathematics and Computational Mechanics
|
2022
|
Vol. 21, nr 4
52--59
EN
The most critical and purely heuristic assumption about priority vector estimation on the basis of pairwise comparisons is that which states a positive relationship between the consistency of decision makers’ judgments and the quality of estimates of their priorities. As this issue constitutes the area of interest of the Multi-Criteria Decision Making theory in relation to AHP, it’s examined in this paper via Monte Carlo simulations from the perspective of a new measure of PCM consistency i.e. Index of Square Logarithm Deviations. It needs to be emphasized that such problems of applied mathematics have been already studied via computer simulations as the only way of this phenomenon examination.
3
Content available Satisfaction of the condition of order preservation : a simulation study
Mazurek Jiří, Kułakowski Konrad
Operations Research and Decisions
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2020
|
Vol. 30, No. 2
77--89
EN
We examine the satisfaction of the condition of order preservation (COP) concerning different levels of inconsistency for randomly generated multiplicative pairwise comparison matrices (MPCMs) of the order from 3 to 9, where a priority vector is derived both by the eigenvalue (eigenvector) method (EV) and the geometric mean (GM) method. Our results suggest that the GM method and the EV method preserve the COP almost identically, both for the less inconsistent matrices (with Saaty’s consistency index below 0.10), and the more inconsistent matrices (Saaty’s consistency index equal to or greater than 0.10). Further, we find that the frequency of the COP violations grows (almost linearly) with the increasing inconsistency of MPCMs measured by Koczkodaj’s inconsistency index and Saaty’s consistency index, respectively, and we provide graphs to illustrate these relationships.
4
Content available remote On Orthogonal Projections on the Space of Consistent Pairwise Comparisons Matrices
Koczkodaj Waldemar W., Smarzewski Ryszard, Szybowski Jacek
Fundamenta Informaticae
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2020
|
Vol. 172, nr 4
379--397
EN
In this study, the orthogonalization process for different inner products is applied to pairwise comparisons. Properties of consistent approximations of a given inconsistent pairwise comparisons matrix are examined. A method of a derivation of a priority vector induced by a pairwise comparison matrix for a given inner product has been introduced. The mathematical elegance of orthogonalization and its universal use in most applied sciences has been the motivating factor for this study. However, the finding of this study that approximations depend on the inner product assumed, is of considerable importance.
5
Content available Some notes on the properties of inconsistency indices in pairwise comparisons
Mazurek J.
Operations Research and Decisions
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2018
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Vol. 28, No. 1
27--42
EN
Pairwise comparisons are an important tool of modern (multiple criteria) decision making. Since human judgments are often inconsistent, many studies have focused on the means of expressing and measuring this inconsistency, and several inconsistency indices have been proposed as an alternative to Saaty’s inconsistency index, CI, and consistency ratio, CR, for reciprocal pairwise comparison matrices. The aims of this paper are threefold: firstly, a row inconsistency index (RIC) is proposed and the properties of this index are examined. Secondly, a comparison of selected inconsistency indices for a corner pairwise comparison matrix is provided. Last, but not least, another axiom about the upper bound on the value of an inconsistency index is postulated, and a set of selected inconsistency indices is examined with respect to this axiom. Numerical examples complete the paper.
6
Content available Determining an estimate of an equivalence relation for moderate and large sized sets
Klukowski L.
Operations Research and Decisions
|
2017
|
Vol. 27, No. 2
45--58
EN
This paper presents two approaches to determining estimates of an equivalence relation on the basis of pairwise comparisons with random errors. Obtaining such an estimate requires the solution of a discrete programming problem which minimizes the sum of the differences between the form of the relation and the comparisons. The problem is NP hard and can be solved with the use of exact algorithms for sets of moderate size, i.e. about 50 elements. In the case of larger sets, i.e. at least 200 comparisons for each element, it is necessary to apply heuristic algorithms. The paper presents results (a statistical preprocessing), which enable us to determine the optimal or a near-optimal solution with acceptable computational cost. They include: the development of a statistical procedure producing comparisons with low probabilities of errors and a heuristic algorithm based on such comparisons. The proposed approach guarantees the applicability of such estimators for any size of set.
7
Content available Redefinition of triad’s inconsistency and its impact on the consistency measurement of pairwise comparison matrix
Kazibudzki P.
Journal of Applied Mathematics and Computational Mechanics
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2016
|
Vol. 15, nr 1
71--78
EN
There is a theory which meets a prescription of the efficient and effective multicriteria decision making support system called the Analytic Hierarchy Process (AHP). It seems to be the most widely used approach in the world today, as well as the most validated methodology for decision making. The consistency measurement of human judgments appears to be the crucial problem in this concept. This research paper redefines the idea of the triad’s consistency within the pairwise comparison matrix (PCM) and proposes a few seminal indices for PCM consistency measurement. The quality of new propositions is then studied with application of computer simulations coded and run in Wolfram Mathematica 9.0.
8
Content available remote Recent Advances on Inconsistency Indices for Pairwise Comparisons : A Commentary
Brunelli M.
Fundamenta Informaticae
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2016
|
Vol. 144, nr 3/4
321--332
EN
This paper recalls the definition of consistency for pairwise comparison matrices and briefly presents the concept of inconsistency index in connection to other aspects of the theory of pairwise comparisons. By commenting on a recent contribution by Koczkodaj and Szwarc, it will be shown that the discussion on inconsistency indices is far from being over, and the ground is still fertile for debates.
9
Content available remote Important Facts and Observations about Pairwise Comparisons (the special issue edition)
Koczkodaj W. W., Redlarski G., Szybowski J., Wajch E., Mikhailov L., Soltys M., Tamazian G., Yuen K. K. F.
Fundamenta Informaticae
|
2016
|
Vol. 144, nr 3/4
291--307
EN
This study has been inspired by numerous requests for clarification from researchers who often confuse Saaty’s Analytic Hierarchy Process (AHP) with the pairwise comparisons (PC) method, taking AHP as the only representation of PC. This study should be regarded as an interpretation and clarification of past investigations of PC. In addition, this article is a reflection on general PC research at a higher level of abstraction: the philosophy of science. It delves into the foundations and implications of pairwise comparisons. Some results of this study are based on a recently published work by Koczkodaj and Szwarc. Finally, open problems have also been reported for future research.
10
Content available remote Continuous Pairwise Comparisons
Saaty T. L.
Fundamenta Informaticae
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2016
|
Vol. 144, nr 3/4
213--221
EN
One often assumes that comparisons are discrete and carried out in a matrix of numbers. However, our eyes and other senses perform comparisons in a continuous way by making many simultaneously. Here the mathematics of pairwise comparisons is generalized to the continuous case. It is more likely that in evolution, little creatures had and now still have a "feeling" about the environment in which they find themselves by sensing many things at once. Through evolution, sensing has been transformed to thinking and to discrete comparisons. This paper presents some material from other works written about continuous comparisons.
11
Ocena porównawcza systemów logistycznych z wykorzystaniem metody AHP
Jankowski J., Ziemba P.
Logistyka
|
2015
|
nr 4
3785--3794, CD 2
PL
W związku z ewoluowaniem sposobu funkcjonowania przedsiębiorstw w kierunku organizacji informacyjnych oraz z rosnącą wagą logistyki, szerokie zastosowanie w przedsiębiorstwach znajdują systemy informatyczne zarządzania, ułatwiające w znacznym stopniu działania logistyczne. Wśród nich najważniejszą klasę stanowią systemy ERP. W artykule podjęto temat oceny systemów informatycznych zarządzania pod względem wspierania przez nie działań dystrybucyjnych i zarządzania odbiorcami oraz zarządzania potrzebami materiałowymi. W analizie skoncentrowano się na systemach wspierających funkcjonowanie logistyki w małych i średnich przedsiębiorstwach. Ocenie poddano systemy wytwarzane w Polsce, a jako punkt odniesienia ujęto system SAP ERP. Do oceny porównawczej zastosowano hierarchiczny proces decyzyjny AHP, umożliwiający wielokryterialne porównania parami poszczególnych systemów. W wyniku przeprowadzonej oceny porównawczej stwierdzono, że spośród rozpatrywanych systemów polskich producentów najlepszy pod względem rozpatrywanych kryteriów jest system Comarch Optima, który pod pewnymi względami dorównuje światowemu potentatowi, czyli systemowi SAP ERP.
EN
In connection with the evolution of the way of enterprises functioning in the direction of the information organization and the increasing importance of logistics, widely used in enterprises are management information systems, which largely facilitate the logistical activities. Among them the most important class is the ERP systems. The article deals with the issue of evaluation of management information systems in terms of supporting activities of distribution management, customer relationship management and the materials management. The analysis focused on supporting the functioning of logistics systems in small and medium-sized enterprises. We considered information systems constructed in Polish, additionally as a point of reference was included the SAP ERP system. The Analytic Hierarchy Process was applied to comparative assessment. It enables multicriteria pairwise comparison of individual systems. As a result of the comparative assessment we found that among Polish information systems, the best in terms of criteria considered is Comarch Optima, which certain evaluations in terms of criteria are equal to the SAP ERP system.
12
Content available remote Fast Convergence of Distance-based Inconsistency in Pairwise Comparisons
Koczkodaj W. W., Kosiek M., Szybowski J., Xu D.
Fundamenta Informaticae
|
2015
|
Vol. 137, nr 3
355--367
EN
This study presents theoretical proof and empirical evidence of the reduction algorithm convergence for the distance-based inconsistency in pairwise comparisons. Our empirical research shows that the convergence very quick. It usually takes less than 10 reductions to bring the inconsistency of the pairwise comparisons matrix below the assumed threshold of 1/3 (sufficient for most applications). We believe that this is the first Monte Carlo study demonstrating such results for the convergence speed of inconsistency reduction in pairwise comparisons.
13
Content available remote On Axiomatization of Inconsistency Indicators for Pairwise Comparisons
Koczkodaj W. W., Szwarc R.
Fundamenta Informaticae
|
2014
|
Vol. 132, nr 4
485--500
EN
This study examines the notion of inconsistency in pairwise comparisons for providing an axiomatization for it. It also proposes two inconsistency indicators for pairwise comparisons. The primary motivation for the inconsistency reduction is expressed by a computer industry concept “garbage in, garbage out”. The quality of the output depends on the quality of the input.
14
Content available Determining preference orders based on a majority margin matrix
Bury H., Wagner D.
Operations Research and Decisions
|
2013
|
Vol. 23, No. 2
25--41
EN
When determining a group ranking, the information about experts’ opinions may be sometimes incomplete. In such cases, usually only the outranking matrix L or the majority margin matrix ΔL is available. Debord [2] presented a technique for constructing the set of experts’ opinions based on the majority margin matrix. The method proposed – simple and efficient – provides a set of experts’ opinions consistent with the given majority margin matrix.
15
Content available remote Using the analytic hierarchy process in evaluating decision alternatives
Cabała P.
Operations Research and Decisions
|
2010
|
Vol. 20, No. 1
5-23
EN
In this paper the method of the Analytic Hierarchy Process (AHP) is described. At the beginning the general assumptions of the method are characterized and discussed. These are related to assumptions held within General Systems Theory. Then the problems of pairwise comparisons of elements, with its use of a specific scale, as well as the resulting reciprocal matrix are presented. There are many ways of estimating the eigenvectors of this matrix. These eigenvectors reflect weights of preferences. Despite the fact that we are able to evaluate the consistency of judgements the problem of acceptable weights still remains. Therefore, by way of an illustration, the method for the sensitivity analysis of preferences is also discussed in the paper.
16
Content available remote Medical Knowledge Mining from Image Data : Synthesis of Medical Image Assessments for Early Stroke Detection
Koczkodaj W. W., Przelaskowski A., Szopinski K. T.
Machine Graphics and Vision
|
2010
|
Vol. 19, No. 3
283-298
EN
The key issue of this study is synthesis of medical images and expert knowledge for early detection of a medical condition, such as stroke or cancer. Such synthesis is a missing link for making decisions during the diagnostic process. Knowledge mining in image databases can be enhanced by computing the relative importance of image features using pairwise comparisons. Computed weights can be systematically used for synthesis of various image features present in the same or different images.
17
Content available remote About Business Decision Making by A Consistency-Driven Pairwise Comparisons Method
Koczkodaj W. W., LeBrasseur R., Wassilew A., Tadeusziewicz R.
Journal of Applied Computer Science
|
2009
|
Vol. 17, nr 1
55-70
EN
Writing this paper has been inspired by the most recent economic crises in the world. Better decision making methods are more needed now than ever before. This study presents an innovate approach to the assessment of management capability in businesses. It is based on the consistency-driven pairwise comparisons method. A proposed conceptual model of performance is flexible and adaptable to different requirements and preconditions (e.g., grant or loan applications). Considering the complexity, a hierarchical structure is used and an inconsistency analysis is performed for all the levels of the structure. The pairwise comparisons method synthesizes together performance assessments assessed at two levels (in our case; there may be more levels in general). The method of consistency-driven pairwise comparisons can be combined with other quantitative and qualitative assessment methods (including brainstorming and Delphi method). Non measurable criteria which often bypassed in other approaches, can be included in the presented model. The consistency-driven pairwise comparisons method contributes to the reliability of assessment through the consistency analysis and solid statistical studies show the accuracy improvement. The management capability merit index (MC-merit) and a procedure for computing it are introduced.
18
Content available remote Completion and clarification to the paper "Tests for relation type - equivalence or tolerance - in finite set of elements"
Klukowski L.
Control and Cybernetics
|
2007
|
Vol. 36, no 2
467-468
19
Content available remote Tests for relation type - equivalence or tolerance - in a finite set of elements
Klukowski L.
Control and Cybernetics
|
2006
|
Vol. 35, no 2
369-384
EN
The statistical procedure for determination of the type of relation - equivalence or tolerance - in a finite set of elements, estimated on the basis of pairwise comparisons with random errors, is presented. The procedure consists of two tests based on Chebyshev's inequality for variance of a random variable; the test statistic is a mixture of some random variables. An example of application of the procedure - determination of relation type in the set of functions expressing profitability of treasury securities sold at auctions in Poland - is presented, too.
20
Content available remote Methods of determining group opinion using pairwise comparisons. Analysis of properties and application aspects
Bury H., Petriczek G., Wagner D.
Bulletin of the Polish Academy of Sciences. Technical Sciences
|
2000
|
Vol. 48, nr 2
145-165
EN
The problem considered in the paper is formulated as follows. Given a finite set of elements called alternatives, set of properties or attributes and a set of experts. Their task is to estimate the importance of alternatives under consideration wit h respect to the chosen criterion, i.e. to evaluate estimates of alternative weights. Experts can be asked to give their judgements on all the alternatives analyzed or to perform pairwise comparisons only. In the paper the latter case is considered only. Moreover, it is assumed that transitivity conditions do not hold. Methods to solve the problem are analysed. Numerical examples illustrating the application of methods investigated are given.
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