Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms
Nonlinear optimization problem subjected to fuzzy relational equations defined by Dubois-Prade family of t-norms
复制标题
DOI：
10.1016/j.cie.2018.03.038
复制
发表时间：
2018-05
期刊：
影响因子：
--
通讯作者：
A. Ghodousian;Marjan Naeeimi;A. Babalhavaeji
中科院分区：
文献类型：
--
作者：
A. Ghodousian;Marjan Naeeimi;A. Babalhavaeji
在模糊集理论中，三角模（简称t-模）和三角余模（简称t-余模）为模糊集的交、并运算提供了通用的模型，起着关键的作用。许多作者提出了各种连续和不连续的t-模。尽管t-范数有很多不同的形式，但大多数著名的连续t-范数都是阿基米德的（例如Frank、Yager、Hamacher、Sugeno-Weber和Schweizer-Sklar家族）。一个有趣的非阿基米德连续t-模族是由Dubois和Prade引入的。本文研究了一类约束条件由特殊的模糊关系方程组构成的非线性优化问题。在这种类型的约束下，FRE被定义为max-Dubois-Prade复合。首先，我们研究可行解集的分解。然后给出了判定解集可行或不可行的充分必要条件。同时，还介绍了简化问题的一些步骤.由于FREs的可行解集是非凸的，传统的非线性规划方法可能无法直接用于解决问题。因此，为了克服这一困难，遗传算法（GA）的一些理论性质的基础上设计的问题。结果表明，该算法保留了新生成的解决方案的可行性。此外，提出了一种方法来产生可行的max-Dubois-Prade FREs作为测试问题。这些测试问题被用来评估我们的算法的性能。最后，将该算法与相关文献进行了比较.所得到的结果证实了所提出的算法在解决此类非线性问题的高性能。
In fuzzy set theory, triangular norms (t-norm for short) and triangular co-norms (t-conorm for short) play a key role by providing generic models for intersection and union operations on fuzzy sets. Various continuous and discontinuous t-norms have been proposed by many authors. Despite variation in the t-norms, most of the well-known continuous t-norms are Archimedean (for example, Frank, Yager, Hamacher, Sugeno-Weber and Schweizer-Sklar family). An interesting family of non-Archimedean continuous t-norms was introduced by Dubois and Prade. This paper is an attemp to study a nonlinear optimization problem whose constraints are formed as a special system of fuzzy relational equations (FRE). In this type of constraint, FREs are defined with max-Dubois-Prade composition. Firstly, we investigate the resolution of the feasible solutions set. Then, some necessary and sufficient conditions are presented to determine the feasibility or infeasibility of the solutions set. Also, some procedures are introduced for simplifying the problem. Since the feasible solutions sets of FREs are non-convex, conventional nonlinear programming methods may not be directly employed to solve the problem. Therefore, in order to overcome this difficulty, a genetic algorithm (GA) is designed based on some theoretical properties of the problem. It is shown that the proposed algorithm preserves the feasibility of new generated solutions. Moreover, a method is presented to generate feasible max-Dubois-Prade FREs as test problems. These test problems are used to evaluate the performance of our algorithm. Finally, the algorithm are compared with some related works. The obtained results confirm the high performance of the proposed algorithm in solving such nonlinear problems.