Abstract
\(L\circ U\)-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 \(L\circ \mathbf {y}=\mathbf {b}\).
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Acknowledgements
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).
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Li, D., Shi, J. The complexity analysis of solving the max-product fuzzy relation equation with LU decomposition. Soft Comput 23, 19–26 (2019). https://doi.org/10.1007/s00500-018-3325-4
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DOI: https://doi.org/10.1007/s00500-018-3325-4