Solvability and unique solvability of max–min fuzzy equations

Gavalec, Martin

doi:10.1016/s0165-0114(01)00108-7

Cited by 33 publications

(26 citation statements)

References 15 publications

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“…This result clarifies some complexity issues discussed in Chen and Wang (2007). It also provides an alternative interpretation of the method proposed by Gavalec (2001).…”

Section: Discussionsupporting

confidence: 76%

“…In this paper, following the idea of Li and Fang (2008), a necessary and sufficient condition will be presented in a concise manner for a system of max-min equations to have a unique minimal solution. Although this condition coincides with that presented by Gavalec (2001) in its very essence, it does not rely on the idempotency property of the minimum operator, i.e., min(x, x) = x, ∀ x ∈ [0, 1], and hence can be directly extended to max-T equations with T being a continuous t-norm. Besides, an analogous necessary and sufficient condition is presented for a system of max-min equations to have a unique solution.…”

Section: Introductionmentioning

confidence: 56%

“…See Liguori (1984, 1985), Di , Guo et al (1988), Li (1991), Cechlárová (1995), Gavalec (2001), and Chen and Wang (2007). However, this problem has not been fully understood for max-min equations while little is known for general max-T equations.…”

Section: Introductionmentioning

confidence: 99%

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On the unique solvability of fuzzy relational equations

Fang

2011

Fuzzy Optim Decis Making

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The solution set of a consistent system of fuzzy relational equations with max-min composition can be characterized by one maximum solution and a finite number of minimal solutions. A polynomial-time method of O(mn) complexity is proposed to determine whether such a system has a unique minimal solution and/or a unique solution, where m, n are the dimensions of the input data. The proposed method can be extended to examining a system of fuzzy relational equations with max-T composition where T is a continuous triangular norm.

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“…This result clarifies some complexity issues discussed in Chen and Wang (2007). It also provides an alternative interpretation of the method proposed by Gavalec (2001).…”

Section: Discussionsupporting

confidence: 76%

Section: Introductionmentioning

confidence: 56%

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On the unique solvability of fuzzy relational equations

Fang

2011

Fuzzy Optim Decis Making

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“…Furthermore, the condition that all the variables are super-essential can only guarantee that the uniqueness of the minimal solution. Some discussion on the unique solvability of sup-T M equations can be found in Sessa (1984), Liguori (1984, 1985), Di , Cechlárová (1990Cechlárová ( , 1995, Li (1990) and Gavalec (2001).…”

Section: Non-archimedean Property and Constrained Irredundant Coveringsmentioning

confidence: 99%

“…The unique solvability of a system of sup-T equations, as well as the existence of the minimum solution, was investigated by Di Sessa (1983, 1988), Sessa (1984), Liguori (1984, 1985), Cechlárová (1990Cechlárová ( , 1995, Li (1990), Gavalec (2001) and Gavalec and Plávka (2003). Various estimates of the number of the minimal solutions can be found in Czogała et al (1982), Wang et al (1984), Shi (1987), Peeva (1992Peeva ( , 2006 and Kyosev (2004, 2007).…”

Section: Introductionmentioning

confidence: 99%

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

Fang

2008

Fuzzy Optim Decis Making

121

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This paper provides a thorough investigation on the resolution of a finite system of fuzzy relational equations with sup-T composition, where T is a continuous triangular norm. When such a system is consistent, although we know that the solution set can be characterized by a maximum solution and finitely many minimal solutions, it is still a challenging task to find all minimal solutions in an efficient manner. Using the representation theorem of continuous triangular norms, we show that the systems of sup-T equations can be divided into two categories depending on the involved triangular norm. When the triangular norm is Archimedean, the minimal solutions correspond one-to-one to the irredundant coverings of a set covering problem. When it is non-Archimedean, they only correspond to a subset of constrained irredundant coverings of a set covering problem. We then show that the problem of minimizing a linear objective function subject to a system of sup-T equations can be reduced into a 0-1 integer programming problem in polynomial time. This work generalizes most, if not all, known results and provides a unified framework to deal with the problem of resolution and optimization of a system of sup-T equations. Further generalizations and related issues are also included for discussion.

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2016

Insight Into Fuzzy Modeling

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Solvability and unique solvability of max–min fuzzy equations

Cited by 33 publications

References 15 publications

On the unique solvability of fuzzy relational equations

On the unique solvability of fuzzy relational equations

On the resolution and optimization of a system of fuzzy relational equations with sup-T composition

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