Fuzzy Relation Posynomial Geometric Programming

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.