Soft Computing

, Volume 23, Issue 1, pp 19–26 | Cite as

The complexity analysis of solving the max-product fuzzy relation equation with LU decomposition

  • Dechao Li
  • Jiaheng Shi


\(L\circ U\)-factorization was recently used to solve the max-product fuzzy relation equation by Molai (Inf Sci 234:86–96, 2013). Considering the forward and backward substitutions play an important role in this method, this paper firstly amend the forward and backward substitutions for solving max-product fuzzy relation equation with \(L\circ U\)-factorization. And then, the computational complexities of improved forward and backward substitutions are analyzed in detail. Finally, we find that the \(L\circ U\)-factorization acts as splitting an irredundant covering of max-product fuzzy relation equation into two parts. It therefore cannot change the fact that finding the solutions of max-product fuzzy relation equation with \(L\circ U\)-factorization is an NP-hard problem. On the contrary, the computational expense will linearly increase with the number of minimal solutions of \(L\circ \mathbf {y}=\mathbf {b}\).


Fuzzy relation equation Max-product composition \(L\circ U\)-factorization Solutions set Computational complexity 



The authors are very appreciative to the anonymous referees and the Editor-in-Chief for their valuable comments that helped us to improve our paper quality. This work was supported by the National Natural Science Foundation of China (Grant No. 61673352).

Compliance with ethical standards

Conflict of interest

Author declares that he has no conflict of interest.

Human and animal rights

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


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.School of Mathematics, Physics and Information ScienceZhejiang Ocean UniversityZhoushanChina
  2. 2.Key Laboratory of Oceanographic Big Data Mining and Application of Zhejiang ProvinceZhoushanChina

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