Abstract
In this paper, fuzzy relational inequalities with addition-Łukasiewicz-product compositions are first introduced for describing the urban sewage treatment systems. It’s well known that a classical fuzzy relational system usually consists of two composed operations. The most commonly used composed operations consists of max–min and max-product. But in our introduced fuzzy relational system, there exist three composed operations, i.e., addition-Łukasiewicz-product. Due to the specific composed operations, the construction for the corresponding solution set is much different. For a consistent fuzzy relation inequality system, composed by addition-Łukasiewicz-product, there exist some maximal solutions and a unique minimum solution. Especially, the number of the maximal solutions might be infinite. In fact, the solution set can be constructed by means of these maximal solutions and the sole minimum solution. Properties of the solutions to an addition-Łukasiewicz-product system is investigated. Following the presented properties, we further study the corresponding single-level and bilevel optimization problems. Two effective resolution approaches are developed and demonstrated by some numerical experiments.
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