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1

On estimation of priority vectors derived from inconsistent pairwise comparison matrices

Kazibudzki Paweł Tadeusz

Journal of Applied Mathematics and Computational Mechanics

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2022

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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.

2

Remarks about geometric scale in the analytic hierarchy process

Starczewski T.

Journal of Applied Mathematics and Computational Mechanics

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2018

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Vol. 17, nr 3

71--82

EN

The Analytic Hierarchy Process (AHP) is perhaps the most popular approach to decision-making problems of prioritization. The basis of the AHP is pairwise comparison, which is used to compare alternatives. This comparisons are provided by decision makers usually as linguistic expressions which are next converted to numbers from a fixed set called a scale. The influence of the scale on the quality of prioritization was investigated in a number of papers. One of the most important types of judgment scale is the Geometric Scale. Its elements depend on specific parameters. In this paper, the impact of the choice of this scale’s parameters on errors in priority vectors and on values of the inconsistency indices is studied via Monte Carlo simulations.

3

Remarks on the impact of the adopted scale on the priority estimation quality

Starczewski T.

Journal of Applied Mathematics and Computational Mechanics

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2017

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Vol. 16, nr 3

105--116

EN

The Analytic Hierarchy Process (AHP) is the method that supports people’s decisions in the multi-criteria decision making problems. In this method the decision process is based on pairwise comparing of every two possible alternatives. The decision maker (DM) compares alternatives by choosing an appropriate “linguistic phrase” or a number from a proper set. This set of “linguistic phrases” and/or the numbers connected with them are referred to as the priority scale. There are several different scales that are described in literature and used in AHP practice. In dependence of the scale chosen by the DM, the final decisions might differ. In the AHP it is assumed that DMs make mistakes over comparing pairs of alternatives, but it was also observed that the assumed scale increases these errors as well. In our paper, we investigate the impact of the adopted scale to the number and magnitude of errors in the final decision. Our results show that the choice of the scale has a big impact on the final decision, so it is crucial part of AHP. It turns out that scales with bigger resource of options result in better evaluations of priority vectors.

4

Usprawnianie podejmowania decyzji w budownictwie dzięki zróżnicowaniu sposobów wykorzystywania porównywania parami

Dytczak M., Ginda G., Jastrząbek B.

Materiały Budowlane

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2016

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nr 6

94--96

PL

Złożoność uwarunkowań otoczenia powoduje, że rozwiązywane zagadnienia decyzyjne w budownictwie wymagają uwzględniania informacji o niedoskonałym charakterze. W tym celu wykorzystywane są różne narzędzia. Jednym z nich są porównania parami (PP), które stały się podstawą wielu metod wspomagania decyzji. Są one również używane wspólnie z innymi narzędziami oraz ułatwiają ich implementację. Takie sposoby stosowania wzbogacają inne narzędzia, a także poszerzają potencjał aplikacyjny porównań parami (PP). W artykule omówiono zaawansowane sposoby stosowania PP, które okazują się użyteczne w przypadku podejmowania decyzji w budownictwie i dziedzinach pokrewnych.

EN

The complexity of surrounding environment results in a need for utilizing imperfect information while solving decision making problems in civil engineering. Diverse pair-wise comparison-based tools are often applied to make use of imperfect information. They are applied both in a standalone mode as well as together with other tools. Their common application with other tools provides necessary means for mutual enhancement in application potential of them and the other tools they are used together with. The utilization modes which prove to be useful in solving decision making problemin civil engineering and related fields are therefore presented in the paper.

5

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

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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.

6

The limit of inconsistency reduction in pairwise comparisons

Koczkodaj W. W., Szybowski J.

International Journal of Applied Mathematics and Computer Science

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2016

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Vol. 26, no. 3

721--729

EN

This study provides a proof that the limit of a distance-based inconsistency reduction process is a matrix induced by the vector of geometric means of rows when a distance-based inconsistent pairwise comparisons matrix is transformed into a consistent PC matrix by stepwise inconsistency reduction in triads. The distance-based inconsistency indicator was defined by Koczkodaj (1993) for pairwise comparisons. Its convergence was analyzed in 1996 (regretfully, with an incomplete proof) and finally completed in 2010. However, there was no interpretation provided for the limit of convergence despite its considerable importance. This study also demonstrates that the vector of geometric means and the right principal eigenvector are linearly independent for the pairwise comparisons matrix size greater than three, although both vectors are identical (when normalized) for a consistent PC matrix of any size.

7

Wielokryterialne wspomaganie decyzji. Przegląd metod i zastosowań

Trzaskalik T.

Zeszyty Naukowe. Organizacja i Zarządzanie / Politechnika Śląska

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2014

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z. 74

239--263

EN

The universality of the decision situations and the importance of many of them contributed to the development of multicriteria decision support methods. These are mainly the two-stage methods – the analytical and synthesis stage are distinguished. Among the presented methods there are AHP, ANP, ELECTRE, PROMETHEE, verbal methods, TOPSIS, BIPOLAR, and interactive methods in terms of risk. The aim of the paper is to present the most frequently used multicriteria discrete methods and the examples of their applications.

PL

Powszechność sytuacji, w których podejmujemy decyzje, oraz wielka waga wielu z nich przyczyniły się do rozwoju wielokryterialnych metod wspomagania decyzji. Są to głównie metody dwuetapowe – wyróżniamy etap analityczny i etap syntezy. Wśród przedstawionych metod znajdują się m.in. metody AHP, ANP, ELECTRE, PROMETHEE, metody werbalne, TOPSIS, BIPOLAR oraz metody interaktywne w warunkach ryzyka. Celem artykułu jest prezentacja najczęściej wykorzystywanych wielokryterialnych metod dyskretnych oraz możliwości ich zastosowań.

8

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

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2009

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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.