【作者】 刘心；

【导师】 夏少刚；

【作者基本信息】 东北财经大学 ， 数量经济， 2009， 博士

【摘要】 经济优化方法与模型是数量经济学的基本内容之一。经济优化方法作为研究经济问题的重要方法,在数量经济学中从不同的侧面丰富和发展了经济问题的计算方法和实践。自从二十世纪四十年代以来,经济优化方法在理论和算法上已逐渐趋向成熟,在实践上有着广泛的应用。其中,我们发现对于有些优化后分析的方法还有待进一步研究,以满足实际需要及减少其人工或计算机的计算量。虽然关于优化后分析(亦称灵敏度分析)方法,国内外有许多学者进行研究,但研究的方法和理论出发点各不相同。另一方面,由于优化方法的理论基础已日臻成熟和计算机各种应用软件的使用,所以,有关这方面的理论和算法的研究反而日见减少。但理论与算法是应用的基础,因此,此领域的研究仍有广阔的空间。同时人们在处理实际问题时还经常会遇到大量的不确定性,像模糊性、随机性等。而这些不确定性因素所带来的问题,用传统的数学规划方法一般难以得到很好的解决。清华大学刘宝碇教授曾指出：“从不确定理论内容延伸来讲,需要更深入的数学理论分析；从不确定规划模型的扩充来讲,需要进一步研究不确定环境下的动态规划和多层规划。从另外一个侧面来看,寻求不确定规划的最优性条件或建立对偶理论以及如何进行灵敏度分析,都是具有挑战性的课题；从不确定规划的计算效率来讲,需要设计更有效的基于启发式算法的求解方法；从应用角度来看,可以进一步考虑在模式识别、排队系统、环境保护、质量控制、风险分析等领域的应用。”从而可见,模糊规划理论是不确定规划理论研究的一个重要方面,对模糊规划的深入研究将进一步丰富不确定规划的理论。但是,目前对于模糊规划的对偶理论、KKT条件等的研究尚不多见,甚至还没有什么进展。这样关于模糊线性规划的对偶理论的研究就成了重要的研究课题。本文从研究及扩展某些经济优化方法和理论入手,主要做了两部分工作：第一部分,从线性规划问题的优化后分析的方法入手,对线性规划增减约束条件的灵敏度分析,求初始基可行解的方法进行了深入的研究。对于灵敏度分析,给出了目前少有研究的减少约束条件的灵敏度分析方法及其理论依据,并将此方法应用于求解带有上界约束的线性规划问题；对于初始基可行解,给出了通过增加一个特殊约束,然后再去掉该约束,结果却可得到一个基可行解的方法,然后,将这种增减约束条件的思想方法应用于求解二次规划问题,使用该方法,可以使二次规划的单纯形算法,从算法到收敛条件均加以改进,得到更简易的程序和收敛准则。第二部分,对线性规划问题优化后分析的理论进行扩展,将线性规划的对偶问题模糊化。首先通过介绍模糊线性规划问题的基本概念及其与经典线性规划问题之间的关系,对多种模糊线性规划模型进行了概括和梳理,总结得出各种有关模糊线性规划模型；其次,对模糊不等式型的线性规划问题的对偶理论进行了研究。给出了模糊不等式型对偶规划的模型,总结出了构成模糊对偶规划一般规则,证明了模糊不等式型的对称性对偶定理；最后,把经典LP问题中的重要结果在模糊系数型的FLP问题中进行了推广,得到并推导证明了基于模糊系数型的模糊线性规划对偶问题的对称定理和互补松弛定理。全文共分六章：第1章引论。论述了有关优化后分析的国内外研究现状及选题背景和意义；并具体说明了本论文的研究思路和结构安排及论文主要创新点和需要进一步研究的问题。第2章线性规划增减约束条件的灵敏度分析。本章在一般灵敏度分析的基础上,给出了线性规划增加约束条件的灵敏度分析的方法及减少约束条件的灵敏度分析的方法,特别是对于减少约束条件的线性规划问题,分别讨论了含有辅助变量与不含辅助变量时减少约束条件时的求得最优解的方法。最后以增加、减少约束条件作为手段,利用它求解变量带有上界约束的线性规划问题。第3章线性规划求基可行解的一种方法。本章首先介绍了求初始基可行解的一般方法：“大M法”、“二阶段法”及求初始基可行解的简化方法。然后提出了求初始基可行解的一种新的方法。该方法通过增加一个特殊约束,贯彻对偶单纯形法检验数全非正的思想,迭代求优；然后再去掉该约束,结果却可得到一个基可行解。上述过程经简化处理后,增减约束可以不必出现,它仅使单纯形表矩阵增加几次初等变换而已,足见其方法之简捷及有效性。最后,将此思想方法应用于求解二次规划问题。使用该方法,可以使二次规划的单纯形算法,从算法到收敛条件均加以改进,得到更简易的程序和收敛准则。第4章模糊线性规划及模型。本章首先介绍了模糊线性规划问题的基本概念及其与经典线性规划问题之间的关系。其次,对多种模糊线性规划模型进行了概括和梳理,总结得到以下模型：1、FLP(Ⅰ-a)：≤模糊型(模糊≤型)。2、FLP(Ⅰ-b)：模糊目标与模糊≤型。3、FLP(Ⅱ-a)：右端系数模糊型,即b型。4、FLP(Ⅱ-b)：目标函数模糊型,即C型。5、FLP(Ⅱ-c)：约束系数模糊型,即A,b型。6、FLP(Ⅱ-d)：全模糊系数型,即A,b,C型。最后,重点介绍了模糊线性规划的常用算法：1、WERNER的对称模型算法。2、Zimmermann的对称模型算法。3、模型FLP(Ⅱ-a)的算法。4、可能性线性规划问题FLP(Ⅱ-d)型算法。第5章模糊不等式型的FLP对偶理论。本章对模糊不等式型的线性规划问题的对偶理论进行了研究。给出了对称模糊对偶规划与非对称对模糊对偶规划之间的关系模型；提出了由对称型模糊对偶规划推出非对称型模糊对偶规划情形及由非对称形推出对称形模糊对偶规划情形的方法；总结出了构成模糊对偶规划一般规则；证明了模糊不等式型的对称性对偶定理。第6章模糊系数型的FLP对偶理论。本章首先介绍了模糊系数型的FLP问题的最优解定义与性质以及模糊系数型的对偶模糊线性规划问题的最优解概念、性质及其强弱对偶定理,主要研究了基于模糊关系的模糊系数型的线性规划对偶理论,对经典LP问题中的重要结果进行了推广,得到并推导证明了模糊线性规划对偶问题的对称定理和互补松弛定理。本文的创新之处：1.在第二章中,对于增加和减少约束条件的灵敏度分析问题,给出了目前少有研究的减少约束条件的灵敏度分析方法及原理。分别给出了含有辅助变量与不含辅助变量时减少约束条件时的求得最优解的方法,并举例说明此方法简单实用,且具有实际应用价值。2.在第三章中,给出了求初始基可行解的一种全新的方法。该方法通过增加一个特殊约束,贯彻对偶单纯形法检验数全非正的思想,迭代求优；然后再去掉该约束,结果却可得到一个基可行解。该方法的运用使得增减约束已不限于灵敏度分析的范畴,而是大大的扩展了,成为处理某些问题的有效手段。3.在第五章中,提出了模糊不等式型的线性规划问题的对偶理论的对偶规划模型,给出了对称模糊对偶规划与非对称模糊对偶规划之间的关系模型,总结得出构成模糊对偶规划一般规则,证明了模糊不等式型的对称性对偶定理。4.在第六章中,研究了基于模糊关系的模糊系数型的线性规划对偶理论,对经典LP问题中的重要结果进行了推广,得到并推导证明了模糊线性规划对偶问题的对称定理和互补松弛定理。更多还原

【Abstract】 Economic optimized methods and models are parts of the basic contents of Quantitative Economics. As the important methods of studying economic problems in economic, optimized analysis methods enrich and develop the measurement and practice of economic relation from deferent aspects in quantity economics. Since 1940s, economic optimized analysis methods are gradually mature in theory and algorithm and widely applied in practice. But, some optimized analysis methods need further improving to meet practical needs and decrease its artificial and computer measurement.Although many internal and foreign scholars are studying optimized analysis methods (also called sensitivity analysis),the methods of study and starting points of theory are different. On the other hand, theory basis of optimized analysis methods are becoming mature and the various applying software of computer are used, so the study of this theory and measurement is decreasing. The theory and measurement are the basis of application. Therefore there is wide space of the study in this field.Meanwhile, people often meet amount of the undefined such as confusion and randomness in dealing with practical problems. And the problems which these undefined factors bring about can’t be well solved by the traditional mathematical methods. As professor Liu Baoding from Qing Hua University pointed out:From the extension of undefined theory, it needs deeper mathematical theory analysis; from the expansion of undefined program model, it needs further study of dynamic program and poly-storied program under undefined circumstance. From the other aspect, it is a challenging question of study to search for optimized conditions or set up duality theory and how to analyzing sensitivity; from the calculation efficiency of undefined program, it needs designing more efficient resolution method based on elicitation method algorithm; from the aspect of application, it is further considered to be applied in model identification, lining system, environment protection,quality control, risk analysis.’It is obvious that fuzzy program theory is an important part of undefined program theory, the deep study of fuzzy program will further enrich the undefined program theory. But, at present, it is rare to study duality theory and KKT condition of fuzzy program, even no progress. Therefore, the study of duality theory of fuzzy linear programming become an important question for study.This paper does two main parts of the work from improving some economic optimization methods.The first part researches the method that finding an initial basic feasible solution by the sensitivity analysis of increasing or decreasing restrains of LP (linear programming) from the optimized analysis method of the linear programming problem. It supplies the sensitivity analysis method by decreasing restrains which is scarcely studied recently, and applies the method to solve the linear programming problem with the upper bounds constraint for the sensitivity analysis. It supplies the method by increasing a special restrain and then removing the restrain to solve the initial basic feasible solution, and applies the thought and the way to simplex method of quadratic programming, improve the algorithm and convergent conditions to obtain the more simple procedure and convergence criterions by this way.The second part researches the duality of LP with fuzzy mathematics. First, it introduces the basic concepts of FLP (fuzzy linear programming) and the relationship between classical L P, generalizes various models of FLP, and summarizes various fuzzy programming models. Secondly, it studies duality theorem of linear programming with fuzzy≤. It gives the dual programming model with fuzzy≤, summarizes the general rule constituting fuzzy dual programming, and proves symmetric fuzzy duality with fuzzy≤.Finally, the important results of classic LP are popularized in FLP with fuzzy coefficient type.This paper is constituted of six chapters.Chapter I Introduction. It discusses domestic and foreign status quo about the optimized analysis and the background and significance of the topics, it introduces the related knowledge about LP and FLP, specifies the ideas, structural arrangements, the major innovation points and the need for the problems deserving further study.Chapter II The sensitivity analysis of increasing or decreasing restrain conditions for LP. The chapter supplies the sensitivity analysis methods of increasing restrain conditions and decreasing restrain conditions based on general sensitivity analysis, especially discusses the optimal solutions method with and without auxiliary variables for LP with decreasing restrain conditions. Finally it solves LP with upper limited variables by the way of increasing or decreasing restrain conditions. ChapterⅢA method of solving basic feasible solution for LP. The chapter first introduces the general method of solving initial basic feasible solution M-Method, Two-Phase Method and the simplified method seeking initial basic feasible solutions. And then it proposes a new method of solving initial basic feasible solutions. The method solves the optimal solutions by iteration by increasing a special restrain, carries out the thought that simplex method for dual all non-positive tests numbers, then get rid of the restrain, as a result, a basic feasible solution can be obtained. Increasing or decreasing restrain can be omitted after the process is simplified. It only makes simplex table for matrix increase several elementary transformation. It proves this method is easy simple and efficient. Finally, it applies the thought way to simplex method of quadratic programming, improving the algorithm and convergent conditions and obtaining the more simple procedure and convergence criterions by this way.Chapter IV FLP and models. The chapter first introduces the basic concepts about FLP and the relationship with classic LP. Secondly, a variety of FLP models are generalized, getting the following models-FLP(Ⅰ-a):≤fuzzy type, FLP(Ⅰ-b):fuzzy object and≤fuzzy type, FLP(Ⅱ-a):fuzzy coefficients of the right side b type, FLP(Ⅱ-b):fuzzy object (?)type, FLP(Ⅱ-c):fuzzy restraint coefficient A,b type and FLP(Ⅱ-d):the whole fuzzy coefficient (?),(?),(?)type. Finally, it introduces the common algorithm for FLP-WERNER symmetric model, Zimmermann symmetric model, the algorithm for FLP (Ⅱ-a) and the possibility of FLP (Ⅱ-d)-type algorithm.Chapter V A study of duality theorem of FLP for fuzzy≤fuzzy type. The chapter studies duality theorem of FLP for fuzzy≤fuzzy type, supplies the relationship model between symmetric and non-symmetric fuzzy duality, and puts forward the method that deriving non-symmetric fuzzy duality by symmetric fuzzy duality and symmetric fuzzy duality by non-symmetric fuzzy duality, summarizes the general rule constituting FLP, and proves symmetric duality theorem for≤fuzzy type.Chapter VI FLP duality theorem for fuzzy coefficient type. The chapter first introduces the definition and the nature of the optimal solution, then introduces the definition and the nature of the optimal solution of DFLP (dual fuzzy linear programming) with fuzzy coefficient type, strong and weak duality theorem. It mainly studies duality theorem for fuzzy coefficient type based on fuzzy relationship, extends classical LP results, and derives and proves FLP duality theorem and the complementary slackness theorem.Innovations of this paper1. Supplies the sensitivity analysis method of decreasing restrains which is few currently for the sensitivity analysis of increasing or decreasing restrains of LP, discuss the optimal solutions method with and without auxiliary variables for LP with decreasing restrain conditions, and illustrate that the method is simple and practical by giving examples.2. Propose a new method of solving initial basic feasible solution. The method solves the optimal solutions by iteration by increasing a special restrain, carries out the thought that simplex method for dual all non-positive tests numbers, then get rid of the restrain, as a result, a basic feasible solution can be obtained. The usage of method doesn’t makes increasing or decreasing restraint not limited to the scope of sensitivity analysis, but greatly expanded and become an effective means of dealing with certain issues.3. Propose duality model for duality theorem of FLP with≤type, supplies the general rule of constituting DFLP, and prove symmetric duality theorem for≤type.4. Research duality theorem for FLP with fuzzy coefficient type based on fuzzy relation, extend the important results of classic LP, and derive and prove the symmetric theorem and the complementary slackness theorem of FLP.更多还原