{"title":"带有双极最大-T 模糊关系式约束的可分离程序设计问题","authors":"Ali Abbasi Molai","doi":"10.1016/j.fss.2024.108944","DOIUrl":null,"url":null,"abstract":"<div><p>One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.</p></div>","PeriodicalId":55130,"journal":{"name":"Fuzzy Sets and Systems","volume":null,"pages":null},"PeriodicalIF":3.2000,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Separable programming problem with bipolar max-T fuzzy relation equation constraints\",\"authors\":\"Ali Abbasi Molai\",\"doi\":\"10.1016/j.fss.2024.108944\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><p>One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.</p></div>\",\"PeriodicalId\":55130,\"journal\":{\"name\":\"Fuzzy Sets and Systems\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":3.2000,\"publicationDate\":\"2024-03-18\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Fuzzy Sets and Systems\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0165011424000903\",\"RegionNum\":1,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"COMPUTER SCIENCE, THEORY & METHODS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Fuzzy Sets and Systems","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0165011424000903","RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
摘要
可分离程序设计问题是非线性程序设计问题中最重要的一类,因为它在理论和实践中都有广泛的应用。本文使用 max-T 组成算子(其中 T 是连续的阿基米德 t 准则)研究了这一类受双极性模糊关系方程系统约束的问题。其可行解集结构由两个向量决定,这两个向量分别称为可行域的下限和上限向量。研究表明,在最大连续和阿基米德 t-norm 条件下,存在一个最优解,其分量就是下限或上限向量的相应分量。根据这一有趣的性质,提出了一些充分条件来检测其某些最优成分或最优解之一。当确定了问题的某些最优成分后,这些条件可以减少原始问题的维数。缩小后的问题的目标函数等同于以一种特定的形式重写。这种形式指导我们根据特征矩阵和构成目标函数的单变量函数的升序或降序设计一个值矩阵。在矩阵上扩展一种方法,以找到(简化)问题的最优解。值得注意的是,一些问题完全可以利用充分条件求解。
Separable programming problem with bipolar max-T fuzzy relation equation constraints
One of the most important classes of nonlinear programming problems is separable programming problem due to its applications in theory and practice. This paper studies this class of the problems subject to a system of bipolar fuzzy relation equations using the max-T composition operator, where T is a continuous and Archimedean t-norm. Its feasible solution set structure is determined by two vectors called lower and upper bound vector of its feasible domain. It is shown that there exists an optimal solution for the problem with max-continuous and Archimedean t-norm such that its components are the corresponding components of either the lower or upper bound vector. Based on the interesting property, some sufficient conditions are proposed to detect some of its optimal components or one of its optimal solutions. These conditions can reduce the dimensions of the original problem when some its optimal components are determined. The objective function of the reduced problem is equivalently rewritten in a specific form. This form guides us to design a value matrix based on the characteristic matrix and ascending or descending of univariate functions constituting the objective function. An approach is extended on the matrix to find the optimal solution of the (reduced) problem. It is important to note that a number of problems can completely be solved using the sufficient conditions.
期刊介绍:
Since its launching in 1978, the journal Fuzzy Sets and Systems has been devoted to the international advancement of the theory and application of fuzzy sets and systems. The theory of fuzzy sets now encompasses a well organized corpus of basic notions including (and not restricted to) aggregation operations, a generalized theory of relations, specific measures of information content, a calculus of fuzzy numbers. Fuzzy sets are also the cornerstone of a non-additive uncertainty theory, namely possibility theory, and of a versatile tool for both linguistic and numerical modeling: fuzzy rule-based systems. Numerous works now combine fuzzy concepts with other scientific disciplines as well as modern technologies.
In mathematics fuzzy sets have triggered new research topics in connection with category theory, topology, algebra, analysis. Fuzzy sets are also part of a recent trend in the study of generalized measures and integrals, and are combined with statistical methods. Furthermore, fuzzy sets have strong logical underpinnings in the tradition of many-valued logics.