经济优化分析方法的研究及扩展
|
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。
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 paper
1. 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.
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。
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 paper
1. 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.
引文
[1]张守一,《数量经济学概论》[M].辽宁人民出版社,1985.
[2]威韦尔,《政治经济学某些学说的数学解释》[M].人民出版社,1983,29.
[3]飞利浦.汉斯.佛朗西斯著,彭立志译,《计量经济学入门》[M].上海财经大学出版社,2005.
[4]程祖瑞,《经济学数学化导论》[M].中国社会科学出版社,2003年。
[5]佛雷德里克.希利尔,马克.S.希利尔著,任建标译,《数据、模型、与决策》[M].中国财政经济出版社,2004年。
[6]焦宝聪,陈兰平,《运筹学的思想方法及应用》[M].北京大学出版社,2008年。
[7]徐玖平,胡知能,《运筹学——数据.模型.决策》[M].科学出版社,2006年。
[8]李荣均,《模糊多准则决策理论与应用》[M].科学出版社,2002年。
[9]方述诚、汪定伟,《模糊数学与模糊优化》[M].科学出版社,1997年。
[10]刘宝碇,彭锦,《不确定理论及其公理体系》[J].第一届不确定系统年会论文集,大庆,2003年8月,1-11.
[11]汪培庄,《模糊关系方程极小解的个数》[J].模糊数学,1984,4：67-73.
[12]罗承忠,《模糊集引论(上)》[M].北京,北京：师范大学出版社,1989年。
[13]夏少刚《运筹学-经济优化方法与模型》[M].北京：清华大学出版社,2005年9月。
[14]夏少刚,《线性规划联合算法的理论和应用》[J].运筹与管理,2004年,13(1)：11-16.
[15]江道琪、何建坤、陈松华,《实用线性规划方法及其支持系统》[M].北京：清华大学出版社,2006年2月,58-64.
[16]范贻昌,《实用管理运筹学》[M],天津：天津大学出版社,1995.
[17]《运筹学》教材编写组,《运筹学》(第三版)[M].北京：清华大学出版社,2005年。
[18]胡运权,《运筹学教程》,[M].北京：清华大学出版社,1998年。
[19]摩特J J,爱尔玛拉巴S E.《运筹学手册》(基础和基本原理)[M].上海： 上海科学技术出版社,1987年。
[20]杨德权等,《线性经济系统管理创新的数量方法研究》[J].预测,2002年,21(6)：32-35.
[21]夏少刚,《有界变量线性规划的一种简易解法》[J].运筹与管理,2005年,14(6)： 12-18.
[22]俞玉森,《数学规划的原理和方法》[M].武汉：华中工学院出版社,1985年。
[23]宋学锋,《运筹学》[M].南京：东南大学出版社,2003年。
[24]龙巧云,《线性规划问题局部灵敏度分析的一个新方法》[J].湘潭师范学院学报,湖南：2008年,30(1)：11-13.
[25]佛雷德里克.S.、希利尔杰拉尔德.J.利伯曼,胡运权等译《运筹学导论》(第8版)[M].北京：清华大学出版社,2007年8月,96-245.
[26]刁在筠,《运筹学》[M].北京：高等教育出版社,1996年。
[27]晏晓焰、李秾,《线性规划问题初始基可行解的一种求法》[J].数值计算与计算机应用,1984年,(4)：276—280.
[28]夏少刚、李敏,《求基可行解一种概率意义下的多项式算法》[J].运筹学学报,1998年2月,(4)：39—47.
[29]孙可钦,《线性规划两阶段法的改进算法》[J].运筹与管理,2000年,9(1)：91-93.
[30]夏少刚,《对求线性规划可行基的一种方法的修正》[J].运筹与管理,2000年,9(2)：91-93.
[31]夏少刚、丛春霞,《直接求线性规划可行解的一种方法》[J].运筹与管理,2005年,14(5)：1-6.
[32]夏少刚、刘心,《线性规划增减约束条件的灵敏度分析》[J].运筹与管理,2007年,16(2)：1-5.
[33]夏少刚、刘心,《线性规划求基可行解的一种方法》,[J].运筹与管理,2008年,4：9-12.
[34]夏少刚、纪凤兰,《二次规划的一种简易算法》[J].运筹与管理,2006年,15(2)： 13-17.
[35]魏权龄等,《数学规划与优化设计》[M].北京：国防工业出版社,1984年,318-340.
[36]Wayne L. Winston,扬振凯等译,《运筹学应用范例与解法》(第4版)[M].北京：清华大学出版社,2006年8月,296-321.
[37]曹炳元,《应用模糊数学与系统》[M].北京,科学出版社,2005年10月。
[38]陈水利、李敬功、王向公,《模糊集理论及其应用》[M].北京,科学出版社,2004年9月。
[39]许仁忠,《模糊数学及其在经济管理中的应用》[M].西南财经大学出版社,1987年7月。
[40]杨纶标、高英仪,《模糊数学原理及其应用》[M].广州,华南理工大学出版社,2004年2月。
[41]闫立梅、韩海山,《具有模糊系数的可能性线性规划》[J].内蒙古民族大学学报(自然科学版),vol.20.No.2 All r.2005年,125-127.
[42]张成,《模糊凸分析及其在模糊规划中的应用》[D].博士学位论文,大连,大连理工大学应用数学系,2004年5月。
[43]汪培庄、韩立岩,《应用模糊数学》[M].北京,北京经济学院出版社,1989年。
[44]乔光、王光远,《模糊随机规划理论》[M].北京,科学出版社,1996年。
[45]陈守煜、刘金禄,《多目标模糊优选权敏感分析及应用》[J].大连理工大学学报,2000年,42(4)：477-479.
[46]唐焕文,秦学志实用最大化方法[M].大连理工大学出版社,2000.
[47]索尔.加斯著,王建华等译,《线性规划方法与应用》[M].北京,高等教育出版社,1990年。
[48]张洪涛、董晓敏、任咏红,《具有关系不等式约束规划问题》[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
[49]李安贵、张志宏等,《模糊数学及其应用》[M].北京,冶金工业出版社,1994年6月。
[50]郭方芳,《关于模糊优化与模糊空间的某些研究结果》[D].大连,(博士论文),大连理工大学,2004年。
[51]哈维.M.瓦格纳著,邓三瑞、王元超、秋同、俞鸿森等译,《运筹学原理与应用》[M].北京,国防工业出版社,2001年,66-115.
[52]黄政书,《具有模糊系数的模糊线性规划问题》[J].应用数学,1995年,8：96-101.
[53]张文修,《模糊数学基础》[M].西安,西安交通大学出版社,1984年。
[54]汪培庄、李洪兴,《模糊系统与模糊计算机》[M].科学出版社,1996年。
[55]刘晋寅、吴孟达,《模糊理论及其应用》[M].国防科技大学出版社,1998年。
[56]刘宝碇、赵瑞清,《不确定规划及进一步的研究问题》[J].指挥技术学院学报,1998年10月,Vol.10,No.6.
[57]宋业新、胡伟文、张建军,《具有模糊系数约束的线性规划》[J].海军工程大学学报,2004年2月,16(1)：40-44.
[58]王立新著,王迎军译,《模糊系统与模糊控制》[M].北京,清华大学出版社,2003年6月,305-313.
[59]程理民、吴江、张玉林,《运筹学模型与方法教程》[M].北京,清华大学出版社,2004,6-36.
[60]马仲蕃等,《数学规划讲义》[M].北京,人民大学出版社,1984.
[61]中山大学等,《运筹学》[M].成都,四川大学出版社,1989.
[62]卢向华,《运筹学教程》[M].北京,高等教育出版社,1992.
[63]盛昭瀚等,《最优化方法基础教程》[M].南京,东南大学出版社,1992.
[64]李颖、黄晓霖、王书宁,《运用改进线性规划算法求解分片线性方程组》[J].北京,清华大学学报(自然科学版),2009年8月(网络预览)。
[65]黄政书,《模糊线性规划的灵敏度分析》[J].福建师范大学学报(自然科学版),1986年02期。
[66]李小军,《线性规划的灵敏度分析》[J].广州师范学院学报(自然科学版),2000年03期。
[67]解心江,《减少约束条件的灵敏度分析》[J].农业系统科学与综合研究,2002年03期。
[68]李刚、黄玉林,《模糊线性规划的对偶算法》[J].山东轻工业学院学报,2005年01期。
[69]陈宝林,《最优化理论与算法》[M].北京,清华大学出版社,2005年10月,122-157.
[70]黄红选、韩继业,《数学规划》[M].北京,清华大学出版社,2006年3月,79-108.
[71]Lai, Y.J. and C.L. Hwang, Fuzzy Mathematical Programming Methods andApplications, Springer-Verlag, Berlin Heidelberg,1992.
[72]R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Science 1970,17B:141-164.
[73]H. Zimmarmann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems,1978,1:46-51.
[74]Rommelfanger, H. and R. Slowinski. "Fuzzy Linear programming With Single or Multiple Objective Functions". In R. Slowiski(ed.), Fuzzy Sets in Decision Analysis, Operations Research and statistics. The handbooks of fuzzy sets series. Boston-Dordrecht-London:Kluwer Academic publ.,1998,179-213.
[75]Campos L, Verdegay J L. Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems,1989.32:1-11.
[76]Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,197630:38-48.
[77]Sanchez E. Inverse of fuzzy relations application to possibility distribution and medical diagnosis. Fuzzy Sets and Systems.1979,2:75-86
[78]Czoala E.,Predryca W. Fuzzy relation equations on a finite set. Fuzzy Sets and Systems,1982,7:89-101.
[79]Higashi M. Klim G J. Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems,1984,13:65-82.
[80]Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl.1991,159:72-87.
[81]Wang H F,Ho C W, Hsieh M J. Diagnosis on gastric cancer with fuzzy pattern recognition. J systems Eng.1992,2:151-163
[82]Lu J, Fang S C. Solving nonlinear optimization problems with relation equation constraints. Fuzzy Sets and Systems.2001,119:1-20.
[83]Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
[84]JAROSLAV RAMIK. Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making. 2005,4:25-39.
[85]HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling. 2004,40.:329-336.
[86]HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems. 2003,139:363-377.
[87]HSIEN-CHUNG WU. Duality Theory in Fuzzy Optimization and Decision Making.2004,3:345-365
[88]C.R. Bactor. S. Chandra. On duality in linear programming under fuzzy environment. Fuzzy sets and systems.2002,125:317-325. (2002).
[89]Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems. 2003,135:151-177.
[90]Levy A B. Solution sensitivity for general principles. [J] SIAMJ Control OPTM 2001,40(1):1-38.
[91]Izmalilov A F. Solldlv M V. A note on solution sensitivity for Karush-Kuhn-Tucker systems. [J] Mathematical Methods or Operations Research.2005,61:347-363.
[92]Castillo E, Conejo A J, Castillo C. Ortigosa, Perturbation approach to sensitivity analysis in mathematical programming. [J]. Journal or Optimization Theory and Application.2006,128(1):49-74.
[93]Izmalilov A F. Sensitivity of solutions to systems of optimality conditions under violation of constraint qualifications. [J]. Computational Mathematics and Mathematical Physics.2007,47(4):533-554.
[94]Dantzig G. Linear Programming and Extensions [M].Princeton, New Jersey: Princeton University Press,1963.
[95]Lin Y, Cryer C W. An alternative direction implicit algorithm for the solution of linear complementary problems arising form boundary problems. [J] Appl. Math. Optim.1985,13:1-17
[96]Visweswaran V, Floudas C A. A global optimization algorithm (GOP) for certain classes of nonconvex NLP's:Ⅱ [J]. Application of theory and test computers and chemical engineering,1990,14:1417-1434.
[97]Lemke C E. A method of solution for quadratic problems [J]. Management Sci. 1972,8:442-455
[98]Fuju T, Kojima M. Semi-definite programming relaxation for non-convex quadratic programs [J]. Journal of Global Optimization,1997,10:367-380.
[99]Lin Y Y, Pang J S. Iterative methods for large convex quadratic programs. A survey [J]. SIAM, J. Control Optimization,1987,25:383-411.
[100]Ulrichraber. A simplicial Branch-and-bounded method for solving non-convex All-quadratic programs [J]. Journal of Global Optimization,1998,13:417-432
[101]Nickel R H, Tolle J W. A sequential programming algorithm for solving large sparse nonlinear programs [J]. Journal of Optimization Theory and Its Application,1989.60:453-473.
[102]Cao Bing-Yuan. New model with T-fuzzy variations in linear programming. Fuzzy Sets and Systems.1996,78:289-292.
[103]R.E Bellman and L.A. zadeh, Decision-making in a fuzzy environment, Management Sciences Ser. B 1970,17:141-164
[104]JJ Buckley possibility linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems 1988,26:135-138.
[105]Hsien-chung Wu. Duality theory in Fuzzy linear programming problems with Fuzzy coefficient. Fuzzy Optimization and Decision Making.2003,2:61-73.
[106]WAYNY L. WINSTON, Operations Research-Mathematical Programming THIRD EDITION,北京，清华大学出版社，2004.1
[107]Walk,M., Theory of Duality in Mathematical Programming, Springer-Verlag, Wien-New York.1989.
[108]J. L. Verdegar, A dual approach to solve the fuzzy linear programming problem, Fuzzy Set and Systems,1984,14:31-141.
[109]JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making. 2005,4:25-39.
[110]HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling. 2004,40:329-336.
[111]HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems. 2003,139:363-377.
[112]C.R. Bactor. S. Chandra. On duality in linear programming under fuzzy environment. Fuzzy sets and systems.2002,125.:317-325.
[113]Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems. 2003,135.:151-177.
[114]HEINECHJ. ROMMEL FANGER. The Advantage of Fuzzy Optimization Models in Practical Use. Fuzzy Optimization and Decision Making. 2004,3:295-309.
[115]HSIEN-CHUNG WU. The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. European Journal of Operation Research.2005,1-14.
[116]Yihua Zhong, Yong Shi. Duality in fuzzy multi-criteria and multi-constraint level linear programming:a parametric approach, Fuzzy Sets and Systems, 2002,132:335-346.
[117]Hichihashi H, Tanaka.H. Formulation and analysis of fuzzy linear programming problem with interval coefficients, [J]. Japan Indus. Management Assoc. 1988,40:320-329.
[118]Ramik J, Rommelfanger H. Fuzzy mathematical programming based on new inequality relation, [J]. Fuzzy Set and Systems.1996,81:77-87.
[119]Liou T S, Wang M J. Ranking fuzzy numbers with integral value, [J].Fuzzy Sets and Systems.1992,50:247-255.
[120]Liu X W. Measuring the Satisfaction of constraints in fuzzy linear programming, [J].Fuzzy Sets and Systems.2001,122:263-275.
[121]Erceg. MA. Metric in fuzzy set theory, [J].Math Anal. Appl.1979,69:205-230.
[122]Sanchez E. Resolution of composite fuzzy relation equations, [J]. Inform and Control,1976,30:38-48.
[123]Loetamonphong J, Fang S C, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems,2001,118:509-517.
[124]Cozogala E, Predrycz W, Fuzzy relation equations on a finite set, Fuzzy and Systems,1982,7:89-101.
[125]Bryson, A. E., Y. C. Ho, Applied Optimal Control, Hemisphere, NY.1975.
[126]Isidori, A. Nonlinear Control Systems, Springer-Verlag, Berlin.1989.
[127]Luo C Z. Research solution set of a fuzzy relation equation, [J]. Math. Anal. Appl.1984,103:524-532.
[128]Ching-Lai Hawang, Kwangsun Yoon. Multiple Attribute Decision Making Methods and Applications. Berlin:Springer-Verlag,1981.
①张守一,数量经济学概论,辽宁人民出版社,1985,14.
①威韦尔,政治经济学某些学说的数学解释,人民出版社,1983,29.
②张守一,数量经济学概论,辽宁人民出版社,1985,9-10.
③张守一,数量经济学概论,辽宁人民出版社,1985,9-10.
④飞利浦.汉斯.佛朗西斯著彭立志译,计量经济学入门,上海财经大学出版社,2005,19-25.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,45-46.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,70-72.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,76-77.
①弗雷德里克.S.希利尔,马克.S.希利尔著,任建标译,数据、模型、与决策,中国财政经济出版社,2004,148.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,129-130.
①李荣均模糊多准则决策理论与应用,科学出版社,2002.
② Lai, Y.J. and C.L. Hwang, Fuzzy Mathematical Programming Methods and Applications, Springer-Verlag, Berlin Heidelberg,1992.
③ R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Science 1970,17B:141-164.
①H. Zimmarmann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems,1978,1:46-51.
②方述诚,汪定伟,模糊数学与模糊优化,科学出版社,1997.
③ Rommelfanger, H. and R. Slowinski. "Fuzzy Linear programming With Single or Multiple Objective Functions". In R. Slowiski(ed.), Fuzzy Sets in Decision Analysis, Operations Research and statistics. The handbooks of fuzzy sets series. Boston-Dordrecht-London:Kluwer Academic publ.,1998,179-213.
④刘宝碇,彭锦,不确定理论及其公理体系,第一届不确定系统年会论文集,大庆,2003.8.1-11.
①黄政书,《模糊线性规划的灵敏度分析》,[J].福建师范大学学报(自然科学版),1986年02期。
②李小军,《线性规划的灵敏度分析》,[J].广州师范学院学报(自然科学版),2000年03期。
③解心江,《减少约束条件的灵敏度分析》,[J].农业系统科学与综合研究,2002年03期。
④龙巧云,《线性规划问题局部灵敏度分析的一个新方法》,[J].湘潭师范学院学报,湖南：2008年,30(1)：11-13.
⑤夏少刚 刘心,《线性规划增减约束条件的灵敏度分析》,[J].运筹与管理,2007年,16(2)：1-5.
⑥夏少刚 刘心,线性规划求基可行解的一种方法,《运筹与管理》,2008年,4：9-12.
⑦ Campos L, Verdegay J L. Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems,1989.32:1-11.
① Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,1976 30:38-48.
② Sanchez E. Inverse of fuzzy relations application to possibility distribution and medical diagnosis. Fuzzy Sets and Systems.1979,2:75-86
③ Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,1976 30:38-48.
④ Czoala E.,Predryca W. Fuzzy relation equations on a finite set. Fuzzy Sets and Systems, 1982,7:89-101.
⑤ Higashi M. Klim G J. Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems, 1984,13:65-82.
⑥汪培庄,模糊关系方程极小解的个数。模糊数学。1984,4：67-73.
⑦罗承忠,模糊集引论(上),北京,北京师范大学出版社,1989.
① Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
② Wang H F,Ho C W, Hsieh M J. Diagnosis on gastric cancer with fuzzy pattern recognition. J systems Eng.1992,2:151-163.
③ Lu J, Fang S C. Solving nonlinear optimization problems with relation equation constraints. Fuzzy Sets and Systems.2001,119:1-20.
④ Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
⑤ Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
⑥ Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
⑦张洪涛、董晓敏、任咏红,具有关系不等式约束规划问题,[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
① Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
②张洪涛、董晓敏、任咏红,具有关系不等式约束规划问题,[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
③ JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
④李刚、黄玉林,《模糊线性规划的对偶算法》,[J].山东轻工业学院学报,2005年01期。
⑤李颖、黄晓霖、王书宁,《运用改进线性规划算法求解分片线性方程组》[J].北京,清华大学学报(自然科学版),2009年8月(网络预览)。
①《运筹学》教材编写组,《运筹学》(第三版)[M].北京：清华大学出版社,2005.
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
①夏少刚线性规划联合算法的理论和应用[J].运筹与管理,2004,13(1)：11-16.
①夏少刚有界变量线性规划的一种简易解法[J].运筹与管理,2005,14(6)；12-18..
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
①范贻昌实用管理运筹学[M].天津：天津大学出版社,1995.
②夏少刚运筹学-经济优化方法与模型[M].北京：清华大学出版社,2005,9.
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
① Dantzig G. Linear Programming and Extensions [M].Princeton, New Jersey:Princeton University Press,1963.
②运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
③刁在筠,运筹学[M].北京：高等教育出版社,1996.
④晏晓焰,李秾.线性规划问题初始基可行解的一种求法[J].数值计算与计算机应用,1984(4)：276—280.
⑤夏少刚,丛春霞.直接求线性规划可行解的一种方法[J].运筹与管理,2005,14(5)：1—6.
⑥夏少刚,刘心.线性规划增减约束条件的灵敏度分析[J].运筹与管理,2007,16(2)：1—5.
⑦夏少刚,纪凤兰.二次规划的一种简易算法[J].运筹与管理,2006,15(2)：13—17.
①运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
①夏少刚,纪凤兰.二次规划的一种简易算法[J].运筹与管理,2006,15(2)：13—17.
② Lin Y, Cryer C W. An alternative direction implicit algorithm for the solution of linear complementary problems arising form boundary problems.[J] Appl. Math. Optim.1985,13:1-17
③ Visweswaran V, Floudas C A. A global optimization algorithm (GOP) for certain classes of nonconvex NLP's:[J]. Application of theory and test computers and chemical engineering, 1990,14:1417-1434.
④ Lemke C E. A method of solution for quadratic problems [J]. Management Sci.1972,8:442-455
⑤ Ulrichraber. A simplicial Branch-and-bounded method for solving non-convex All-quadratic programs [J]. Journal of Global Optimization,1998,13:417-432
⑥ Nickel R H, Tolle J W. A sequential programming algorithm for solving large sparse nonlinear programs [J]. Journal of Optimization Theory and Its Application,1989.60:453-473.
①魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
②运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
③魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
④魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
⑤魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
①魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
②魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
①曹炳元应用模糊数学与系统,北京,科学出版社,2005,10.
②罗承忠,模糊集引论(上),北京,北京师范大学出版社,1989.
① HSIEN-CHUNG WU. Duality Theory in Fuzzy Optimization and Decision Making.2004, 3.345-365.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40.329-336.
① Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems.2003,135.151-177.
② JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4.25-39.
①陈水利、李敬功、王向公模糊集理论及其应用,北京,科学出版社,2004.9.
①Cao Bing-Yuan. New model with T-fuzzy variations in linear programming. Fuzzy Sets and Systems.1996,78:289-292.
①许仁忠模糊数学及其在经济管理中的应用,西南财经大学出版社,1987.7.
① R.E Bellman and L.A. zadeh, Decision-making in a fuzzy environment, Management Sciences Ser. B 1970,17:141-164
①陈水利、李敬功、王向公模糊集理论及其应用,北京,科学出版社,2004.9.
① JJ Buckley possibility linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems 1988,26:135-138.
①杨纶标、高英仪模糊数学原理及其应用, 广州,华南理工大学出版社,2004,2.
①杨纶标、高英仪,模糊数学原理及其应用, 广州,华南理工大学出版社,2004.2.
①杨纶标、高英仪,《模糊数学原理及其应用》, 广州,华南理工大学出版社,2004.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
②同上
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40:329-336.
② HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems.2003,139:363-377.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems.2003,139:363-377.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40:329-336.
[2]威韦尔,《政治经济学某些学说的数学解释》[M].人民出版社,1983,29.
[3]飞利浦.汉斯.佛朗西斯著,彭立志译,《计量经济学入门》[M].上海财经大学出版社,2005.
[4]程祖瑞,《经济学数学化导论》[M].中国社会科学出版社,2003年。
[5]佛雷德里克.希利尔,马克.S.希利尔著,任建标译,《数据、模型、与决策》[M].中国财政经济出版社,2004年。
[6]焦宝聪,陈兰平,《运筹学的思想方法及应用》[M].北京大学出版社,2008年。
[7]徐玖平,胡知能,《运筹学——数据.模型.决策》[M].科学出版社,2006年。
[8]李荣均,《模糊多准则决策理论与应用》[M].科学出版社,2002年。
[9]方述诚、汪定伟,《模糊数学与模糊优化》[M].科学出版社,1997年。
[10]刘宝碇,彭锦,《不确定理论及其公理体系》[J].第一届不确定系统年会论文集,大庆,2003年8月,1-11.
[11]汪培庄,《模糊关系方程极小解的个数》[J].模糊数学,1984,4：67-73.
[12]罗承忠,《模糊集引论(上)》[M].北京,北京：师范大学出版社,1989年。
[13]夏少刚《运筹学-经济优化方法与模型》[M].北京：清华大学出版社,2005年9月。
[14]夏少刚,《线性规划联合算法的理论和应用》[J].运筹与管理,2004年,13(1)：11-16.
[15]江道琪、何建坤、陈松华,《实用线性规划方法及其支持系统》[M].北京：清华大学出版社,2006年2月,58-64.
[16]范贻昌,《实用管理运筹学》[M],天津：天津大学出版社,1995.
[17]《运筹学》教材编写组,《运筹学》(第三版)[M].北京：清华大学出版社,2005年。
[18]胡运权,《运筹学教程》,[M].北京：清华大学出版社,1998年。
[19]摩特J J,爱尔玛拉巴S E.《运筹学手册》(基础和基本原理)[M].上海： 上海科学技术出版社,1987年。
[20]杨德权等,《线性经济系统管理创新的数量方法研究》[J].预测,2002年,21(6)：32-35.
[21]夏少刚,《有界变量线性规划的一种简易解法》[J].运筹与管理,2005年,14(6)： 12-18.
[22]俞玉森,《数学规划的原理和方法》[M].武汉：华中工学院出版社,1985年。
[23]宋学锋,《运筹学》[M].南京：东南大学出版社,2003年。
[24]龙巧云,《线性规划问题局部灵敏度分析的一个新方法》[J].湘潭师范学院学报,湖南：2008年,30(1)：11-13.
[25]佛雷德里克.S.、希利尔杰拉尔德.J.利伯曼,胡运权等译《运筹学导论》(第8版)[M].北京：清华大学出版社,2007年8月,96-245.
[26]刁在筠,《运筹学》[M].北京：高等教育出版社,1996年。
[27]晏晓焰、李秾,《线性规划问题初始基可行解的一种求法》[J].数值计算与计算机应用,1984年,(4)：276—280.
[28]夏少刚、李敏,《求基可行解一种概率意义下的多项式算法》[J].运筹学学报,1998年2月,(4)：39—47.
[29]孙可钦,《线性规划两阶段法的改进算法》[J].运筹与管理,2000年,9(1)：91-93.
[30]夏少刚,《对求线性规划可行基的一种方法的修正》[J].运筹与管理,2000年,9(2)：91-93.
[31]夏少刚、丛春霞,《直接求线性规划可行解的一种方法》[J].运筹与管理,2005年,14(5)：1-6.
[32]夏少刚、刘心,《线性规划增减约束条件的灵敏度分析》[J].运筹与管理,2007年,16(2)：1-5.
[33]夏少刚、刘心,《线性规划求基可行解的一种方法》,[J].运筹与管理,2008年,4：9-12.
[34]夏少刚、纪凤兰,《二次规划的一种简易算法》[J].运筹与管理,2006年,15(2)： 13-17.
[35]魏权龄等,《数学规划与优化设计》[M].北京：国防工业出版社,1984年,318-340.
[36]Wayne L. Winston,扬振凯等译,《运筹学应用范例与解法》(第4版)[M].北京：清华大学出版社,2006年8月,296-321.
[37]曹炳元,《应用模糊数学与系统》[M].北京,科学出版社,2005年10月。
[38]陈水利、李敬功、王向公,《模糊集理论及其应用》[M].北京,科学出版社,2004年9月。
[39]许仁忠,《模糊数学及其在经济管理中的应用》[M].西南财经大学出版社,1987年7月。
[40]杨纶标、高英仪,《模糊数学原理及其应用》[M].广州,华南理工大学出版社,2004年2月。
[41]闫立梅、韩海山,《具有模糊系数的可能性线性规划》[J].内蒙古民族大学学报(自然科学版),vol.20.No.2 All r.2005年,125-127.
[42]张成,《模糊凸分析及其在模糊规划中的应用》[D].博士学位论文,大连,大连理工大学应用数学系,2004年5月。
[43]汪培庄、韩立岩,《应用模糊数学》[M].北京,北京经济学院出版社,1989年。
[44]乔光、王光远,《模糊随机规划理论》[M].北京,科学出版社,1996年。
[45]陈守煜、刘金禄,《多目标模糊优选权敏感分析及应用》[J].大连理工大学学报,2000年,42(4)：477-479.
[46]唐焕文,秦学志实用最大化方法[M].大连理工大学出版社,2000.
[47]索尔.加斯著,王建华等译,《线性规划方法与应用》[M].北京,高等教育出版社,1990年。
[48]张洪涛、董晓敏、任咏红,《具有关系不等式约束规划问题》[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
[49]李安贵、张志宏等,《模糊数学及其应用》[M].北京,冶金工业出版社,1994年6月。
[50]郭方芳,《关于模糊优化与模糊空间的某些研究结果》[D].大连,(博士论文),大连理工大学,2004年。
[51]哈维.M.瓦格纳著,邓三瑞、王元超、秋同、俞鸿森等译,《运筹学原理与应用》[M].北京,国防工业出版社,2001年,66-115.
[52]黄政书,《具有模糊系数的模糊线性规划问题》[J].应用数学,1995年,8：96-101.
[53]张文修,《模糊数学基础》[M].西安,西安交通大学出版社,1984年。
[54]汪培庄、李洪兴,《模糊系统与模糊计算机》[M].科学出版社,1996年。
[55]刘晋寅、吴孟达,《模糊理论及其应用》[M].国防科技大学出版社,1998年。
[56]刘宝碇、赵瑞清,《不确定规划及进一步的研究问题》[J].指挥技术学院学报,1998年10月,Vol.10,No.6.
[57]宋业新、胡伟文、张建军,《具有模糊系数约束的线性规划》[J].海军工程大学学报,2004年2月,16(1)：40-44.
[58]王立新著,王迎军译,《模糊系统与模糊控制》[M].北京,清华大学出版社,2003年6月,305-313.
[59]程理民、吴江、张玉林,《运筹学模型与方法教程》[M].北京,清华大学出版社,2004,6-36.
[60]马仲蕃等,《数学规划讲义》[M].北京,人民大学出版社,1984.
[61]中山大学等,《运筹学》[M].成都,四川大学出版社,1989.
[62]卢向华,《运筹学教程》[M].北京,高等教育出版社,1992.
[63]盛昭瀚等,《最优化方法基础教程》[M].南京,东南大学出版社,1992.
[64]李颖、黄晓霖、王书宁,《运用改进线性规划算法求解分片线性方程组》[J].北京,清华大学学报(自然科学版),2009年8月(网络预览)。
[65]黄政书,《模糊线性规划的灵敏度分析》[J].福建师范大学学报(自然科学版),1986年02期。
[66]李小军,《线性规划的灵敏度分析》[J].广州师范学院学报(自然科学版),2000年03期。
[67]解心江,《减少约束条件的灵敏度分析》[J].农业系统科学与综合研究,2002年03期。
[68]李刚、黄玉林,《模糊线性规划的对偶算法》[J].山东轻工业学院学报,2005年01期。
[69]陈宝林,《最优化理论与算法》[M].北京,清华大学出版社,2005年10月,122-157.
[70]黄红选、韩继业,《数学规划》[M].北京,清华大学出版社,2006年3月,79-108.
[71]Lai, Y.J. and C.L. Hwang, Fuzzy Mathematical Programming Methods andApplications, Springer-Verlag, Berlin Heidelberg,1992.
[72]R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Science 1970,17B:141-164.
[73]H. Zimmarmann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems,1978,1:46-51.
[74]Rommelfanger, H. and R. Slowinski. "Fuzzy Linear programming With Single or Multiple Objective Functions". In R. Slowiski(ed.), Fuzzy Sets in Decision Analysis, Operations Research and statistics. The handbooks of fuzzy sets series. Boston-Dordrecht-London:Kluwer Academic publ.,1998,179-213.
[75]Campos L, Verdegay J L. Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems,1989.32:1-11.
[76]Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,197630:38-48.
[77]Sanchez E. Inverse of fuzzy relations application to possibility distribution and medical diagnosis. Fuzzy Sets and Systems.1979,2:75-86
[78]Czoala E.,Predryca W. Fuzzy relation equations on a finite set. Fuzzy Sets and Systems,1982,7:89-101.
[79]Higashi M. Klim G J. Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems,1984,13:65-82.
[80]Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl.1991,159:72-87.
[81]Wang H F,Ho C W, Hsieh M J. Diagnosis on gastric cancer with fuzzy pattern recognition. J systems Eng.1992,2:151-163
[82]Lu J, Fang S C. Solving nonlinear optimization problems with relation equation constraints. Fuzzy Sets and Systems.2001,119:1-20.
[83]Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
[84]JAROSLAV RAMIK. Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making. 2005,4:25-39.
[85]HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling. 2004,40.:329-336.
[86]HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems. 2003,139:363-377.
[87]HSIEN-CHUNG WU. Duality Theory in Fuzzy Optimization and Decision Making.2004,3:345-365
[88]C.R. Bactor. S. Chandra. On duality in linear programming under fuzzy environment. Fuzzy sets and systems.2002,125:317-325. (2002).
[89]Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems. 2003,135:151-177.
[90]Levy A B. Solution sensitivity for general principles. [J] SIAMJ Control OPTM 2001,40(1):1-38.
[91]Izmalilov A F. Solldlv M V. A note on solution sensitivity for Karush-Kuhn-Tucker systems. [J] Mathematical Methods or Operations Research.2005,61:347-363.
[92]Castillo E, Conejo A J, Castillo C. Ortigosa, Perturbation approach to sensitivity analysis in mathematical programming. [J]. Journal or Optimization Theory and Application.2006,128(1):49-74.
[93]Izmalilov A F. Sensitivity of solutions to systems of optimality conditions under violation of constraint qualifications. [J]. Computational Mathematics and Mathematical Physics.2007,47(4):533-554.
[94]Dantzig G. Linear Programming and Extensions [M].Princeton, New Jersey: Princeton University Press,1963.
[95]Lin Y, Cryer C W. An alternative direction implicit algorithm for the solution of linear complementary problems arising form boundary problems. [J] Appl. Math. Optim.1985,13:1-17
[96]Visweswaran V, Floudas C A. A global optimization algorithm (GOP) for certain classes of nonconvex NLP's:Ⅱ [J]. Application of theory and test computers and chemical engineering,1990,14:1417-1434.
[97]Lemke C E. A method of solution for quadratic problems [J]. Management Sci. 1972,8:442-455
[98]Fuju T, Kojima M. Semi-definite programming relaxation for non-convex quadratic programs [J]. Journal of Global Optimization,1997,10:367-380.
[99]Lin Y Y, Pang J S. Iterative methods for large convex quadratic programs. A survey [J]. SIAM, J. Control Optimization,1987,25:383-411.
[100]Ulrichraber. A simplicial Branch-and-bounded method for solving non-convex All-quadratic programs [J]. Journal of Global Optimization,1998,13:417-432
[101]Nickel R H, Tolle J W. A sequential programming algorithm for solving large sparse nonlinear programs [J]. Journal of Optimization Theory and Its Application,1989.60:453-473.
[102]Cao Bing-Yuan. New model with T-fuzzy variations in linear programming. Fuzzy Sets and Systems.1996,78:289-292.
[103]R.E Bellman and L.A. zadeh, Decision-making in a fuzzy environment, Management Sciences Ser. B 1970,17:141-164
[104]JJ Buckley possibility linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems 1988,26:135-138.
[105]Hsien-chung Wu. Duality theory in Fuzzy linear programming problems with Fuzzy coefficient. Fuzzy Optimization and Decision Making.2003,2:61-73.
[106]WAYNY L. WINSTON, Operations Research-Mathematical Programming THIRD EDITION,北京，清华大学出版社，2004.1
[107]Walk,M., Theory of Duality in Mathematical Programming, Springer-Verlag, Wien-New York.1989.
[108]J. L. Verdegar, A dual approach to solve the fuzzy linear programming problem, Fuzzy Set and Systems,1984,14:31-141.
[109]JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making. 2005,4:25-39.
[110]HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling. 2004,40:329-336.
[111]HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems. 2003,139:363-377.
[112]C.R. Bactor. S. Chandra. On duality in linear programming under fuzzy environment. Fuzzy sets and systems.2002,125.:317-325.
[113]Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems. 2003,135.:151-177.
[114]HEINECHJ. ROMMEL FANGER. The Advantage of Fuzzy Optimization Models in Practical Use. Fuzzy Optimization and Decision Making. 2004,3:295-309.
[115]HSIEN-CHUNG WU. The Karush-Kuhn-Tucker optimality conditions in an optimization problem with interval-valued objective function. European Journal of Operation Research.2005,1-14.
[116]Yihua Zhong, Yong Shi. Duality in fuzzy multi-criteria and multi-constraint level linear programming:a parametric approach, Fuzzy Sets and Systems, 2002,132:335-346.
[117]Hichihashi H, Tanaka.H. Formulation and analysis of fuzzy linear programming problem with interval coefficients, [J]. Japan Indus. Management Assoc. 1988,40:320-329.
[118]Ramik J, Rommelfanger H. Fuzzy mathematical programming based on new inequality relation, [J]. Fuzzy Set and Systems.1996,81:77-87.
[119]Liou T S, Wang M J. Ranking fuzzy numbers with integral value, [J].Fuzzy Sets and Systems.1992,50:247-255.
[120]Liu X W. Measuring the Satisfaction of constraints in fuzzy linear programming, [J].Fuzzy Sets and Systems.2001,122:263-275.
[121]Erceg. MA. Metric in fuzzy set theory, [J].Math Anal. Appl.1979,69:205-230.
[122]Sanchez E. Resolution of composite fuzzy relation equations, [J]. Inform and Control,1976,30:38-48.
[123]Loetamonphong J, Fang S C, Optimization of fuzzy relation equations with max-product composition, Fuzzy Sets and Systems,2001,118:509-517.
[124]Cozogala E, Predrycz W, Fuzzy relation equations on a finite set, Fuzzy and Systems,1982,7:89-101.
[125]Bryson, A. E., Y. C. Ho, Applied Optimal Control, Hemisphere, NY.1975.
[126]Isidori, A. Nonlinear Control Systems, Springer-Verlag, Berlin.1989.
[127]Luo C Z. Research solution set of a fuzzy relation equation, [J]. Math. Anal. Appl.1984,103:524-532.
[128]Ching-Lai Hawang, Kwangsun Yoon. Multiple Attribute Decision Making Methods and Applications. Berlin:Springer-Verlag,1981.
①张守一,数量经济学概论,辽宁人民出版社,1985,14.
①威韦尔,政治经济学某些学说的数学解释,人民出版社,1983,29.
②张守一,数量经济学概论,辽宁人民出版社,1985,9-10.
③张守一,数量经济学概论,辽宁人民出版社,1985,9-10.
④飞利浦.汉斯.佛朗西斯著彭立志译,计量经济学入门,上海财经大学出版社,2005,19-25.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,45-46.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,70-72.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,76-77.
①弗雷德里克.S.希利尔,马克.S.希利尔著,任建标译,数据、模型、与决策,中国财政经济出版社,2004,148.
①程祖瑞,经济学数学化导论,中国社会科学出版社,2003,129-130.
①李荣均模糊多准则决策理论与应用,科学出版社,2002.
② Lai, Y.J. and C.L. Hwang, Fuzzy Mathematical Programming Methods and Applications, Springer-Verlag, Berlin Heidelberg,1992.
③ R.E. Bellman and L.A. Zadeh, Decision making in a fuzzy environment, Management Science 1970,17B:141-164.
①H. Zimmarmann, Fuzzy programming and linear programming with several objective functions, Fuzzy Sets and Systems,1978,1:46-51.
②方述诚,汪定伟,模糊数学与模糊优化,科学出版社,1997.
③ Rommelfanger, H. and R. Slowinski. "Fuzzy Linear programming With Single or Multiple Objective Functions". In R. Slowiski(ed.), Fuzzy Sets in Decision Analysis, Operations Research and statistics. The handbooks of fuzzy sets series. Boston-Dordrecht-London:Kluwer Academic publ.,1998,179-213.
④刘宝碇,彭锦,不确定理论及其公理体系,第一届不确定系统年会论文集,大庆,2003.8.1-11.
①黄政书,《模糊线性规划的灵敏度分析》,[J].福建师范大学学报(自然科学版),1986年02期。
②李小军,《线性规划的灵敏度分析》,[J].广州师范学院学报(自然科学版),2000年03期。
③解心江,《减少约束条件的灵敏度分析》,[J].农业系统科学与综合研究,2002年03期。
④龙巧云,《线性规划问题局部灵敏度分析的一个新方法》,[J].湘潭师范学院学报,湖南：2008年,30(1)：11-13.
⑤夏少刚 刘心,《线性规划增减约束条件的灵敏度分析》,[J].运筹与管理,2007年,16(2)：1-5.
⑥夏少刚 刘心,线性规划求基可行解的一种方法,《运筹与管理》,2008年,4：9-12.
⑦ Campos L, Verdegay J L. Linear programming problems and ranking of fuzzy numbers. Fuzzy Sets and Systems,1989.32:1-11.
① Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,1976 30:38-48.
② Sanchez E. Inverse of fuzzy relations application to possibility distribution and medical diagnosis. Fuzzy Sets and Systems.1979,2:75-86
③ Sanchez E. Resolution of Composite fuzzy relation equations. Inform and Control,1976 30:38-48.
④ Czoala E.,Predryca W. Fuzzy relation equations on a finite set. Fuzzy Sets and Systems, 1982,7:89-101.
⑤ Higashi M. Klim G J. Resolution of finite fuzzy relation equations. Fuzzy Sets and Systems, 1984,13:65-82.
⑥汪培庄,模糊关系方程极小解的个数。模糊数学。1984,4：67-73.
⑦罗承忠,模糊集引论(上),北京,北京师范大学出版社,1989.
① Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
② Wang H F,Ho C W, Hsieh M J. Diagnosis on gastric cancer with fuzzy pattern recognition. J systems Eng.1992,2:151-163.
③ Lu J, Fang S C. Solving nonlinear optimization problems with relation equation constraints. Fuzzy Sets and Systems.2001,119:1-20.
④ Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
⑤ Fang S C, Li G. Solving fuzzy relation equations with a linear objective function. Fuzzy Sets and Systems.1999,103:107-113.
⑥ Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
⑦张洪涛、董晓敏、任咏红,具有关系不等式约束规划问题,[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
① Wang P Z. Latticed linear programming and fuzzy relation inequalities. J math Anal Appl. 1991,159:72-87.
②张洪涛、董晓敏、任咏红,具有关系不等式约束规划问题,[J].大连,辽宁师范大学学报(自然科学版),2003年,26(3)：231-233.
③ JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
④李刚、黄玉林,《模糊线性规划的对偶算法》,[J].山东轻工业学院学报,2005年01期。
⑤李颖、黄晓霖、王书宁,《运用改进线性规划算法求解分片线性方程组》[J].北京,清华大学学报(自然科学版),2009年8月(网络预览)。
①《运筹学》教材编写组,《运筹学》(第三版)[M].北京：清华大学出版社,2005.
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
①夏少刚线性规划联合算法的理论和应用[J].运筹与管理,2004,13(1)：11-16.
①夏少刚有界变量线性规划的一种简易解法[J].运筹与管理,2005,14(6)；12-18..
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
①范贻昌实用管理运筹学[M].天津：天津大学出版社,1995.
②夏少刚运筹学-经济优化方法与模型[M].北京：清华大学出版社,2005,9.
①江道琪,何建坤,陈松华实用线性规划方法及其支持系统[M],北京：清华大学出版社,2006.2,58-64.
① Dantzig G. Linear Programming and Extensions [M].Princeton, New Jersey:Princeton University Press,1963.
②运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
③刁在筠,运筹学[M].北京：高等教育出版社,1996.
④晏晓焰,李秾.线性规划问题初始基可行解的一种求法[J].数值计算与计算机应用,1984(4)：276—280.
⑤夏少刚,丛春霞.直接求线性规划可行解的一种方法[J].运筹与管理,2005,14(5)：1—6.
⑥夏少刚,刘心.线性规划增减约束条件的灵敏度分析[J].运筹与管理,2007,16(2)：1—5.
⑦夏少刚,纪凤兰.二次规划的一种简易算法[J].运筹与管理,2006,15(2)：13—17.
①运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
①夏少刚,纪凤兰.二次规划的一种简易算法[J].运筹与管理,2006,15(2)：13—17.
② Lin Y, Cryer C W. An alternative direction implicit algorithm for the solution of linear complementary problems arising form boundary problems.[J] Appl. Math. Optim.1985,13:1-17
③ Visweswaran V, Floudas C A. A global optimization algorithm (GOP) for certain classes of nonconvex NLP's:[J]. Application of theory and test computers and chemical engineering, 1990,14:1417-1434.
④ Lemke C E. A method of solution for quadratic problems [J]. Management Sci.1972,8:442-455
⑤ Ulrichraber. A simplicial Branch-and-bounded method for solving non-convex All-quadratic programs [J]. Journal of Global Optimization,1998,13:417-432
⑥ Nickel R H, Tolle J W. A sequential programming algorithm for solving large sparse nonlinear programs [J]. Journal of Optimization Theory and Its Application,1989.60:453-473.
①魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
②运筹学教材编写组.运筹学(第三版)[M].北京：清华大学出版社,2005.
③魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
④魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
⑤魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
①魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
②魏权龄等,数学规划与优化设计[M].北京：国防工业出版社,1984.318-340
①曹炳元应用模糊数学与系统,北京,科学出版社,2005,10.
②罗承忠,模糊集引论(上),北京,北京师范大学出版社,1989.
① HSIEN-CHUNG WU. Duality Theory in Fuzzy Optimization and Decision Making.2004, 3.345-365.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40.329-336.
① Masahiro Inuiguchi, Jaroslar Ramik, Tetsuzo Tanino, ctal. Satisfying solutions and duality in interval and fuzzy linear programming. Fuzzy sets and systems.2003,135.151-177.
② JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4.25-39.
①陈水利、李敬功、王向公模糊集理论及其应用,北京,科学出版社,2004.9.
①Cao Bing-Yuan. New model with T-fuzzy variations in linear programming. Fuzzy Sets and Systems.1996,78:289-292.
①许仁忠模糊数学及其在经济管理中的应用,西南财经大学出版社,1987.7.
① R.E Bellman and L.A. zadeh, Decision-making in a fuzzy environment, Management Sciences Ser. B 1970,17:141-164
①陈水利、李敬功、王向公模糊集理论及其应用,北京,科学出版社,2004.9.
① JJ Buckley possibility linear programming with triangular fuzzy numbers. Fuzzy Sets and Systems 1988,26:135-138.
①杨纶标、高英仪模糊数学原理及其应用, 广州,华南理工大学出版社,2004,2.
①杨纶标、高英仪,模糊数学原理及其应用, 广州,华南理工大学出版社,2004.2.
①杨纶标、高英仪,《模糊数学原理及其应用》, 广州,华南理工大学出版社,2004.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
②同上
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40:329-336.
② HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems.2003,139:363-377.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① JAROSLAV RAMIK Duality in Fuzzy Linear Programming:Some New Concepts and Results. Fuzzy Optimization and Decision-Making.2005,4:25-39.
① HSIEN-CHUNG WU. Duality theorems in Fuzzy mathematical programming problems based on the concept of necessity. Fuzzy Sets and Systems.2003,139:363-377.
① HSIEN-CHUNG WU. Fuzzy Optimization Problem Based on the Embedding Theorem and Possibility Measures. Mathematical and Computer Modeling.2004,40:329-336.