Optimization of fuzzy relational equations with max-av composition

Wu, Yan-Kuen

doi:10.1016/j.ins.2007.02.037

Cited by 49 publications

(29 citation statements)

References 30 publications

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“…See Fodor (1991), De Cooman andKerre (1994), Han and Li (2005), Wang and Xiong (2005) and Han et al (2006) for details. More general composite operators were also investigated in CuninghameGreen (1979CuninghameGreen ( , 1995, Kawaguchi and Miyakoshi (1998), Cheng and Peng (1988), Noskoá (2005), and Wu (2007).…”

Section: Discussionmentioning

confidence: 99%

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

Fang

2008

Fuzzy Optim Decis Making

122

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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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Section: Discussionmentioning

confidence: 99%

“…and Wu (2007) considered this problem under the max-average composition. However, it was pointed out by Zimmermann (2007) that one of the algorithms proposed by may not lead to the optimal solution in some cases.…”

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

122

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“…Although the authors of this paper update the upper bound of the branch and bound method -when a better bound is generated and causes the number of the nodes in this method to be reduced -it still requires a relatively time-consuming process to solve the branch and bound problem. Wu [14] considered the fuzzy relation equation with max-average composition. He proposed six rules that are very effective in reducing the problem size .…”

Section: Minimizementioning

confidence: 99%

A New Algorithm for Optimization of the Fuzzy Relation Equation With Maxalgebraic Sum Composition

Zharfi¹,

Mirzazadeh²

2011

The South African Journal of Industrial Engineering

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This paper considers an optimization problem with a linear objective function under the constraints expressed by a system of fuzzy relation equations using max-as (Algebraic Sum) composition. First, some properties of minimal solutions of the system with fuzzy relation equations and max-as composition are shown. Then, a new algorithm for solving the optimization problem is derived. The numerical examples have been provided to illustrate the theoretical results. OPSOMMINGHierdie artikel bestudeer 'n optimiseringsprobleem met 'n lineêre doelwitfunksie en wasige randvoorwaardes met 'n algebraïese somsamestelling. Aanvanklik word sommige eienskappe van die minimale oplossings van die wasige vergelykings en die algebraïese samestelling getoon. Daarna word 'n nuwe algoritme vir die oplossing van die optimiseringsprobleem afgelei. Numeriese voorbeelde word voorsien om die teoretiese resultate te ondersteun.

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“…Following the idea of Fang and Li [14] , the linear optimization problem subject to a system of sup-T P equations was discussed by Loetamonphong and Fang [17] , Guu and Wu [18] , and Ghodousian and Khorram [19] . Furthermore, this problem was investigated under various composite operations by Wu and Guu [20] , Khorram and Ghodousian [21] , Wu [22] , and Abbasi Molai and Khorram [23] . Some other generalizations on this issue can be found in [24][25].…”

Section: Introductionmentioning

confidence: 99%

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

Fang

2009

J Syst Sci Complex

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This paper shows that the problem of minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm T can be reduced to a 0-1 linear fractional optimization problem in polynomial time. Consequently, parametrization techniques, e.g., Dinkelbach's algorithm, can be applied by solving a classical set covering problem in each iteration. Similar reduction can also be performed on the sup-T equation constrained optimization problems with an objective function being monotone in each variable separately. This method could be extended as well to the case in which the triangular norm is non-Archimedean.

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Optimization of fuzzy relational equations with max-av composition

Cited by 49 publications

References 30 publications

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

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

A New Algorithm for Optimization of the Fuzzy Relation Equation With Maxalgebraic Sum Composition

Minimizing a linear fractional function subject to a system of sup-T equations with a continuous Archimedean triangular norm

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