【作者】 杨吉会；

【导师】 曹炳元；

【作者基本信息】 汕头大学 ， 基础数学， 2007， 博士

【摘要】 1965年，美国著名控制论专家扎德教授提出了模糊集的概念，标志了模糊数学这一学科的诞生。经过各国学者几十年的努力，该学科已经形成了模糊集合、模糊系统、模糊拓扑、模糊微积分、模糊代数、模糊控制、模糊图论、模糊逻辑、模糊规划、模糊决策等诸多分支。几何规划诞生于1961年，是运筹学中优化理论的一个研究方向。四十多年来，几何规划不仅在理论上得到了长足的发展，在应用上也取得了众多辉煌成就。本论文结合模糊关系方程与几何规划理论，研究了模糊关系正项几何规划中的若干问题。模糊关系几何规划是作者和曹炳元教授提出的，该问题属于模糊优化中的一个新的研究方向。模糊优化是运筹学和模糊数学这两个学科互相融合的产物，研究的范围非常广泛，包括有模糊线性规划、模糊二次规划、模糊目标规划、模糊几何规划、模糊整数规划、模糊动态规划、模糊鞍点规划、模糊关系规划等，并且研究的范围一直在不断扩大。1987年，曹炳元在IFSA(国际Fuzzy系统协会)第二次大会上提出了“模糊几何规划”，此后又用该理论成功解决了供电系统中变电站的供电半径的优选和废水处理方案的优选问题，从而丰富了模糊优化理论的内容。目前已有中国、印度、加拿大、英国、古巴、比利时和台湾等多个国家和地区的10余名学者加入了“Fuzzy几何规划”研究行列，本文对这些工作作了简单介绍。在曹老师的工作基础上，2005年，作者和曹老师提出了模糊关系几何规划，这是对模糊几何规划理论的扩展，并且系统研究了模糊关系正项几何规划中的若干问题，它以模糊关系方程和模糊几何规划、目标模糊型模糊关系线性规划为基础，包括有：单项模糊关系正项几何规划；目标函数为逻辑式的模糊关系正项几何规划；目标模糊型模糊关系正项几何规划等；给出了这些规划的算法，并且将软计算技术引入到了求解模糊关系规划问题中。本文共由五章组成，在第一章首先介绍了模糊集合的基本概念和理论。第二章介绍了模糊关系方程理论，包括模糊关系方程的几种求解方法，以及应用软计算技术求解模糊关系方程的方法，这就为研究模糊关系正项几何规划提供了理论基础和计算方法。第三章概括介绍了几何规划和模糊几何规划目前的研究进展和应用情况，第四章介绍了模糊关系规划的最简单情形：几种类型的模糊关系线性规划和它们的求解方法，引入了求解该类规划的软计算方法。第五章系统介绍了几种类型的模糊关系正项几何规划及其算法。作者的工作主要分布在第二章的2．2．2节，第三章的3．1，3．4，3．5节，第四章和第五章。重点是第五章的工作：该章提出了Fuzzy关系正项几何规划的框架，正待在实践中接受检验。更多还原

【Abstract】 In 1965, Zadeh, a well-known expert from the United States in cybernetics, produced the concept of fuzzy sets. It symbolized the birth of the fuzzy math-ematics academics. Through decades of efforts by the researchers all over theworld, the subject have been including fuzzy sets, fuzzy systems, fuzzy topol-ogy, fuzzy calculus, fuzzy algebra, fuzzy control, fuzzy graph theory, fuzzy logic, fuzzy optimization, fuzzy decision making and many other branches. Amongthem geometric programming was founded in 1961, it is a research direction ofoptimization theory in operations research. During the past 40 years, it was notonly developed by leaps and bounds in theory, but also many brilliant achieve-ments were made in practice. Combining the two theories of fuzzy relationequations and geometric programming, the author researches some problemsof the fuzzy relation posynomial geometric programming. He presented fuzzyrelation geometric programming together with Prof. Bingyuan Cao, which is anew research direction in fuzzy optimization. Fuzzy optimization is a mutualproduct of operations research and fuzzy mathematics, its scope of researchwidespread extremely, including fuzzy linear programming, fuzzy quadratic pro-gramming, fuzzy multi-objective programming, fuzzy geometric programming, fuzzy integer programming, fuzzy dynamic programming, fuzzy saddle pointprogramming, fuzzy relation programming, etc. The scope is continuously ex-panding. In 1987, Prof. Cao proposed " fuzzy geometric programming" in the2nd IFSA (International Fuzzy Systems Association) conference, subsequently, the reasonable siting issues with the transformer power supply system and theoptimum scheme for waste water disposal in power plants have been solvedsuccessfully through his theory. His work plays a role in enrichment of fuzzyoptimization. Currently, more than 10 scholars, who come from China, India, Canada, England, Cuba, Belgium, Taiwan of China and other countries and re-gions, have joined the research team of "fuzzy geometric programming". Whichis introduced briefly in this dissertation. Based on the theory of Prof. Cao, in 2005, the author and Prof. Caoproposed fuzzy relation geometric programming, expanding the theory of fuzzygeometric programming. This dissertation has systemically studied some prob-lems of the fuzzy relation posynomial geometric programming. On the basisof theory in fuzzy relation equation, fuzzy geometric programming and fuzzyrelation linear programming, including the nomomial fuzzy relation posynomialgeometric programming, the fuzzy relation posynomial geometric programmingwith logic formula objective function, the fuzzy relation posynomial geomet-ric programming with fuzzy objective function, the author gives algorithms tothose programming. Moreover, he introduces soft computing into the solutionto the fuzzy relation programming.This dissertation consists of five chapters. In chapter 1, some basic con-cepts and theories of fuzzy sets have been demonstrated firstly. In chapter 2, the theory of fuzzy relation equations have been presented, including meth-ods to fuzzy relation equations, and several applied examples of fuzzy relationequations. Moreover, the soft computing have also been introduced into solvingthe fuzzy relation equations, which lays theory basis and computing method onresearching fuzzy relation posynomial geometric programming. In chapter 3, the current progress and applied circumstance of geometric programming andfuzzy geometric programming have been introduced. In chapter 4, several typesof fuzzy relation linear programming have been shown, as a special situationof fuzzy relation programming, and the algorithms have also been given to theprogramming. In this chapter, the soft computing method is included to fuzzyrelation linear programming. Chapter 5 several types and algorithms of fuzzyrelation posynomial geometric programming have been proposed.Author’s work shows in 2.2.2 section of Chapter 3.1, 3.4, 3.5 section ofChapter 3, Chapter 4 and Chapter 5. Chapter 5 is a focus in this dissertation, and in this chapter the frame has been produced in fuzzy relation posynomialgeometric programming. The theory has been treated in practice.This work is supported by the National Natural Science Foundation of China (NSFC)(No. 70271047 and No. 79670012) and "211" Project Founda-tion of Shantou University and Li Jiacheng Science Development Foundation ofShantou University.更多还原