Dealing with the problem of null weights and scores in Fuzzy Analytic Hierarchy Process

  • Francisco Rodrigues Lima-JuniorEmail author
  • Luiz Cesar Ribeiro Carpinetti
Methodologies and Application


Fuzzy Analytic Hierarchy Process (Fuzzy AHP) has been widely adopted to support decision making problems. The Fuzzy AHP approach based on the synthetic extent analysis is the most applied approach to calculate the values of the criteria weights from fuzzy comparative matrices. The min operator is used to calculate the weights based on values of degree of possibility. If any of the degrees of possibilities is zero, the output of this operator will also be zero. Thus, the criterion weight or alternative score will be set to zero. If not prevented, this problem may lead to a distorted rank. Despite the fact that there are other propositions based on synthetic extent analysis method, none of the studies found in the literature investigate how the problem of null weights and scores can be avoided. This paper investigates different approaches of the Fuzzy AHP method to evaluate whether they can avoid the problem of null weights and scores without affecting the consistency of the results. Five different approaches based on synthetic extent analysis method were implemented and evaluated. Tests were performed considering 12 decision problems. The results indicated that the Fuzzy AHP approach proposed by Ahmed and Kilic is the most appropriate to overcome the problem of null weight of criteria and scores of alternatives without affecting the consistency of the results. Other benefits of using this approach are the simplicity of the computational implementation and better ability to differentiate the importance of the criteria when the weight values are very close.


Fuzzy sets Fuzzy AHP Synthetic extent analysis Null weights Multicriteria decision making 



This work was supported by CAPES (Agency for supporting human resource development in high education institutions).


This study was funded by CAPES—Agency for supporting human resource development in high education institutions (Phd scholarship).

Compliance with ethical standards

Conflict of interest

The authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Management and EconomicsFederal Technological University of ParanáCuritibaBrazil
  2. 2.Production Engineering DepartmentSão Carlos School of Engineering– University of São PauloSão CarlosBrazil

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