Methods based on pairwise comparison matrices (PCMs) form a significant part of multicriteria decision making (MCDM) methods. These methods are based on structuring pairwise comparisons (PCs) of objects from a finite set of objects into a PCM and deriving priorities of objects that represent the relative importance of each object with respect to all other objects in the set. However, the crisp PCMs are not able to capture uncertainty stemming from subjectivity of human thinking and from incompleteness of information about the problem that are often closely related to MCDM problems. That is why the fuzzy extension of methods based on PCMs has been of great interest. In order to derive fuzzy priorities of objects from a fuzzy PCM (FPCM), standard fuzzy arithmetic is usually applied to the fuzzy extension of the methods originally developed for crisp PCMs. %Fuzzy extension of the methods based on PCMs usually consists in simply replacing the crisp PCs in the given model by fuzzy PCs and applying standard fuzzy arithmetic to obtain the desired fuzzy priorities. However, such approach fails in properly handling uncertainty of preference information contained in the FPCM. Namely, reciprocity of the related PCs of objects in a FPCM and invariance of the given method under permutation of objects are violated when standard fuzzy arithmetic is applied to the fuzzy extension. This leads to distortion of the preference information contained in the FPCM and consequently to false results. Thus, the first research question of the thesis is: ``Based on a FPCM of objects, how should fuzzy priorities of these objects be determined so that they reflect properly all preference information available in the FPCM?'' This research question is answered by introducing an appropriate fuzzy extension of methods originally developed for crisp PCMs. That is, such fuzzy extension that does not violate reciprocity of the related PCs and invariance under permutation of objects, and that does not lead to a redundant increase of uncertainty of the resulting fuzzy priorities of objects. Fuzzy extension of three different types of PCMs is examined in this thesis  multiplicative PCMs, additive PCMs with additive representation, and additive PCMs with multiplicative representation. In particular, construction of PCMs, verifying consistency, and deriving priorities of objects from PCMs are studied in detail for each type of these PCMs. First, wellknown and in practice most often applied methods based on crisp PCMs are reviewed. Afterwards, fuzzy extensions of these methods proposed in the literature are reviewed in detail and their drawbacks regarding the violation of reciprocity of the related PCs and of invariance under permutation of objects are pointed out. It is shown that these drawbacks can be overcome by properly applying constrained fuzzy arithmetic instead of standard fuzzy arithmetic to the computations. In particular, we always have to look at a FPCM as a set of PCMs with different degrees of membership to the FPCM, i.e. we always have to consider only PCs that are mutually reciprocal. Constrained fuzzy arithmetic allows us to impose the reciprocity of the related PCs as a constraint on arithmetic operations with fuzzy numbers, and its appropriate application also guarantees invariance of the methods under permutation of objects. Finally, new fuzzy extensions of the methods are proposed based on constrained fuzzy arithmetic and it is proved that these methods do not violate the reciprocity of the related PCs and are invariant under permutation of objects. Because of these desirable properties, fuzzy priorities of objects obtained by the methods proposed in this thesis reflect the preference information contained in fuzzy PCMs better in comparison to the fuzzy priorities obtained by the methods based on standard fuzzy arithmetic. Beside the inability to capture uncertainty, methods based on PCMs are also not able to cope with situations where it is not possible or reasonable to obtain complete preference information from DMs. This problem occurs especially in the situations involving largedimensional PCMs. When dealing with incomplete largedimensional PCMs, compromise between reducing the number of PCs required from the DM and obtaining reasonable priorities of objects is of paramount importance. This leads to the second research question: ``How can the amount of preference information required from the DM in a largedimensional PCM be reduced while still obtaining comparable priorities of objects?'' This research question is answered by introducing an efficient twophase method. Specifically, in the first phase, an interactive algorithm based on weakconsistency condition is introduced for partially filling an incomplete PCM. This algorithm is designed in such a way that minimizes the number of PCs required from the DM and provides sufficient amount of preference information at the same time. The weakconsistency condition allows for providing ranges of possible intensities of preference for every missing PC in the incomplete PCM. Thus, at the end of the first phase, a PCM containing intervals for all PCs that were not provided by the DM is obtained. Afterward, in the second phase, the methods for obtaining fuzzy priorities of objects from fuzzy PCMs proposed in this thesis within the answer to the first research question are applied to derive interval priorities of objects from this incomplete PCM. The obtained interval priorities cover all weakly consistent completions of the incomplete PCM and are very narrow. The performance of the method is illustrated by a reallife case study and by simulations that demonstrate the ability of the algorithm to reduce the number of PCs required from the DM in PCMs of dimension 15 and greater by more than 60\% on average while obtaining interval priorities comparable with the priorities obtainable from the hypothetical complete PCMs.
MCDM methods based on pairwise comparison matrices and their fuzzy extension / Krejčí, Jana.  (2017), pp. 1233.
MCDM methods based on pairwise comparison matrices and their fuzzy extension
Krejčí, Jana
20170101
Abstract
Methods based on pairwise comparison matrices (PCMs) form a significant part of multicriteria decision making (MCDM) methods. These methods are based on structuring pairwise comparisons (PCs) of objects from a finite set of objects into a PCM and deriving priorities of objects that represent the relative importance of each object with respect to all other objects in the set. However, the crisp PCMs are not able to capture uncertainty stemming from subjectivity of human thinking and from incompleteness of information about the problem that are often closely related to MCDM problems. That is why the fuzzy extension of methods based on PCMs has been of great interest. In order to derive fuzzy priorities of objects from a fuzzy PCM (FPCM), standard fuzzy arithmetic is usually applied to the fuzzy extension of the methods originally developed for crisp PCMs. %Fuzzy extension of the methods based on PCMs usually consists in simply replacing the crisp PCs in the given model by fuzzy PCs and applying standard fuzzy arithmetic to obtain the desired fuzzy priorities. However, such approach fails in properly handling uncertainty of preference information contained in the FPCM. Namely, reciprocity of the related PCs of objects in a FPCM and invariance of the given method under permutation of objects are violated when standard fuzzy arithmetic is applied to the fuzzy extension. This leads to distortion of the preference information contained in the FPCM and consequently to false results. Thus, the first research question of the thesis is: ``Based on a FPCM of objects, how should fuzzy priorities of these objects be determined so that they reflect properly all preference information available in the FPCM?'' This research question is answered by introducing an appropriate fuzzy extension of methods originally developed for crisp PCMs. That is, such fuzzy extension that does not violate reciprocity of the related PCs and invariance under permutation of objects, and that does not lead to a redundant increase of uncertainty of the resulting fuzzy priorities of objects. Fuzzy extension of three different types of PCMs is examined in this thesis  multiplicative PCMs, additive PCMs with additive representation, and additive PCMs with multiplicative representation. In particular, construction of PCMs, verifying consistency, and deriving priorities of objects from PCMs are studied in detail for each type of these PCMs. First, wellknown and in practice most often applied methods based on crisp PCMs are reviewed. Afterwards, fuzzy extensions of these methods proposed in the literature are reviewed in detail and their drawbacks regarding the violation of reciprocity of the related PCs and of invariance under permutation of objects are pointed out. It is shown that these drawbacks can be overcome by properly applying constrained fuzzy arithmetic instead of standard fuzzy arithmetic to the computations. In particular, we always have to look at a FPCM as a set of PCMs with different degrees of membership to the FPCM, i.e. we always have to consider only PCs that are mutually reciprocal. Constrained fuzzy arithmetic allows us to impose the reciprocity of the related PCs as a constraint on arithmetic operations with fuzzy numbers, and its appropriate application also guarantees invariance of the methods under permutation of objects. Finally, new fuzzy extensions of the methods are proposed based on constrained fuzzy arithmetic and it is proved that these methods do not violate the reciprocity of the related PCs and are invariant under permutation of objects. Because of these desirable properties, fuzzy priorities of objects obtained by the methods proposed in this thesis reflect the preference information contained in fuzzy PCMs better in comparison to the fuzzy priorities obtained by the methods based on standard fuzzy arithmetic. Beside the inability to capture uncertainty, methods based on PCMs are also not able to cope with situations where it is not possible or reasonable to obtain complete preference information from DMs. This problem occurs especially in the situations involving largedimensional PCMs. When dealing with incomplete largedimensional PCMs, compromise between reducing the number of PCs required from the DM and obtaining reasonable priorities of objects is of paramount importance. This leads to the second research question: ``How can the amount of preference information required from the DM in a largedimensional PCM be reduced while still obtaining comparable priorities of objects?'' This research question is answered by introducing an efficient twophase method. Specifically, in the first phase, an interactive algorithm based on weakconsistency condition is introduced for partially filling an incomplete PCM. This algorithm is designed in such a way that minimizes the number of PCs required from the DM and provides sufficient amount of preference information at the same time. The weakconsistency condition allows for providing ranges of possible intensities of preference for every missing PC in the incomplete PCM. Thus, at the end of the first phase, a PCM containing intervals for all PCs that were not provided by the DM is obtained. Afterward, in the second phase, the methods for obtaining fuzzy priorities of objects from fuzzy PCMs proposed in this thesis within the answer to the first research question are applied to derive interval priorities of objects from this incomplete PCM. The obtained interval priorities cover all weakly consistent completions of the incomplete PCM and are very narrow. The performance of the method is illustrated by a reallife case study and by simulations that demonstrate the ability of the algorithm to reduce the number of PCs required from the DM in PCMs of dimension 15 and greater by more than 60\% on average while obtaining interval priorities comparable with the priorities obtainable from the hypothetical complete PCMs.File  Dimensione  Formato  

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