序方法与多属性决策研究
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摘要
序关系是决策分析的基础内容之一,在决策分析中,把对象作比较,排次序是经常遇到的,从关系的角度来看,这就是一种序关系。由全序关系推广而成的各种形式的序关系和公理化的序理论在解决一些实际问题中发挥着越来越重要的作用。随着管理科学、生物学、经济学、系统工程学、数学等领域中的各种各样的非线性问题的出现,非线性分析成为现代科学中最重要的研究方向之一,而半序方法作为研究非线性问题的基本方法之一,对研究非线性问题起着非常重要的作用。
本文对序方法和多属性决策问题进行研究,得到了一些重要的新结果。主要内容如下：
第一章介绍非线性分析中的半序方法与多属性决策的历史背景、研究现状以及本文所需要的预备知识；介绍本文研究的主要内容及研究意义。
第二章介绍了数学及经济管理科学中主要的序关系及其性质。介绍了锥与半序、楔与拟序之间的关系。
第三章用序方法研究了一些非线性算子问题。首先,用半序方法研究了非线性算子不动点的存在性及唯一性,得到了相应的不动点定理；然后,提出了n次合理扩张算子的新概念,并对其不动点的存在性进行了讨论,得到了一些新的不动点定理,同时推广了若干重要的结论；最后,用半序方法及单调迭代技巧讨论了一类非线性算子方程组解的存在性并研究了它们的最小最大解和最大最小解。
第四章讨论楔与拟偏好的关系,并在拟偏好意义下研究了一类算子方程解的存在性及有关集值增算子问题。
第五章定义了风险型有序加权平均(R-OWA)算子和风险型有序加权几何平均(R-OWGA)算子,并对其性质进行了讨论。同时,基于有序信息集成算子,研究了风险型多属性决策的新方法。
第六章在偏好关系之间距离的基础上,给出了偏好结构中的偏好关系占优关联度和被占优关联度的新概念,利用这些新概念研究了序数信息条件下的多属性决策方法。
第七章研究了多属性决策中的等级偏好优序法。这章首先对优序法进行了介绍；其次给出了等级偏好占优关联系数和等级偏好被占优关联系数的新概念,并利用等级偏好关联系数代替优序法中的优序数,从而提出了等级偏好优序法,并在计算机上进行了实现,于是使得该方法在解决实际决策问题中发挥了很大的作用;最后,探讨了等级偏好优序法的拓展情况,并对属性权重不完全已知的多属性决策问题进行了研究。
第八章利用等级偏好优序法建立了深基坑支护结构选型模型。利用等级偏好优序法建立了源范例检索模型,在该源范例检索模型中,通过海明距离比较范例库中的源范例与目标范例的相似程度的大小来找出与目标范例最相似的源范例,该范例的基坑支护结构类型即为目标范例的可优先考虑的支护结构类型。通过工程实例证明了本章提出的深基坑支护结构选型模型是可靠适用的。
第九章对本论文的主要内容和主要结论进行归纳总结,提出今后需要进一步完善和深入研究的方向。
Order relation is one of the basis for decision making analysis, in the decision making analysis, an object for comparison and ordering are frequently encountered, which is an order relation from the viewpoint of the relationship. The various order relations and axiomatized order theories promoted by linear order play an important role in some practical problems. With the emergence of a wide variety of nonlinear problems in management science, biology, economics, engineering, mathematics and so on, nonlinear analysis has become one of the most important researches in modern science, while the partial order method, one of the basic method to research nonlinear problems, plays a very important role.
In this dissertation, there are some important new results by studying on the order methods and multi-attribute decision making. The main contents are as follows:
The first chapter introduces the historical background, current situation and the required prior knowledge in this dissertation about the partial order in nonlinear analysis and multi-attribute decision making; describes the main issues of this study and research significance of this dissertation.
The second chapter describes the main order relations in mathematics and economic management science and the nature of these order relations; the relationship between the cone and partial order, wedge and quasi-order are described in this chapter.
Chapter three studies some nonlinear operator problems with the order methods. First of all, there comes to the corresponding fixed point theorems through studying the existence of fixed points for nonlinear operator with partial order method. Then, it proposes the new concept of n times reasonable expansive operator and discusses its existence of fixed points to get some new fixed point theorems, promoting some important conclusions. At last, it conducts a discussion on the existence of solutions of a class of nonlinear equations with partial order method and monotone iterative technique and studys their Min-max solution and Max-min solution.
The fourth chapter addresses the relationship between the wedge and quasi-preferences, and studies the existence of solutions to a class of operator equations and problems about set-valued increasing operator with the sense of quasi-preferences.
Chapter five defines the Risk Ordered Weighted Averaging (R-OWA) operator and the Risk Ordered Weighted Geometric Averaging (R-OWGA) operator and their properties are discussed. At the same time, it studies a new method of the risk multi-attribute decision making based on ordered information integration operator.
Chapter six gives the new concepts of the preference dominating relational index and the preference be dominated relational index in preference structure based on the distance between the preference relations and on this basis this chapter studies multi-attribute decision making method with the only ordinal information.
Chapter seven studies the rank preference optimal ordering method in the multi-attribute decision making. This chapter first describes the optimal ordering method and puts forward the concept of rank preferences correlation coefficient, then proposes the rank preference optimal ordering method by using rank preferences dominating correlation coefficient instead of the precedence order number in optimal ordering method. The rank preference optimal ordering method was achieved on the computer, therefore makes the method to play a significant role in solving practical decision problems. Finally, the studies on the expansion of the rank preference optimal ordering method are carried out.
Chapter eight sets up a lectotype decision model of retaining & protecting structures for deep foundation pit by use of the rank preference optimal ordering method. Using the rank preference optimal ordering method, the base case indexing model is proposed. In the model, the degree of similarity between the base case and the target case of deep foundation pit is obtained by the Hamming distance in order to find the deep foundation pit base case which is the most similar to the target case in the base cases of deep foundation pit. The retaining & protecting structure of the deep foundation pit base case which is the most similar to the target case in the base cases of deep foundation pit shall be the priority structure that can be considered for the terget case. The engineering example presented in this chapter proves that the lectotype decision model of retaining & protecting structures is reliable and applicable.
Chapter nine sums up the main content in this dissertation and the main conclusions then puts forward the need for further refinement and in-depth research directions in the future.
本文对序方法和多属性决策问题进行研究,得到了一些重要的新结果。主要内容如下：
第一章介绍非线性分析中的半序方法与多属性决策的历史背景、研究现状以及本文所需要的预备知识；介绍本文研究的主要内容及研究意义。
第二章介绍了数学及经济管理科学中主要的序关系及其性质。介绍了锥与半序、楔与拟序之间的关系。
第三章用序方法研究了一些非线性算子问题。首先,用半序方法研究了非线性算子不动点的存在性及唯一性,得到了相应的不动点定理；然后,提出了n次合理扩张算子的新概念,并对其不动点的存在性进行了讨论,得到了一些新的不动点定理,同时推广了若干重要的结论；最后,用半序方法及单调迭代技巧讨论了一类非线性算子方程组解的存在性并研究了它们的最小最大解和最大最小解。
第四章讨论楔与拟偏好的关系,并在拟偏好意义下研究了一类算子方程解的存在性及有关集值增算子问题。
第五章定义了风险型有序加权平均(R-OWA)算子和风险型有序加权几何平均(R-OWGA)算子,并对其性质进行了讨论。同时,基于有序信息集成算子,研究了风险型多属性决策的新方法。
第六章在偏好关系之间距离的基础上,给出了偏好结构中的偏好关系占优关联度和被占优关联度的新概念,利用这些新概念研究了序数信息条件下的多属性决策方法。
第七章研究了多属性决策中的等级偏好优序法。这章首先对优序法进行了介绍；其次给出了等级偏好占优关联系数和等级偏好被占优关联系数的新概念,并利用等级偏好关联系数代替优序法中的优序数,从而提出了等级偏好优序法,并在计算机上进行了实现,于是使得该方法在解决实际决策问题中发挥了很大的作用;最后,探讨了等级偏好优序法的拓展情况,并对属性权重不完全已知的多属性决策问题进行了研究。
第八章利用等级偏好优序法建立了深基坑支护结构选型模型。利用等级偏好优序法建立了源范例检索模型,在该源范例检索模型中,通过海明距离比较范例库中的源范例与目标范例的相似程度的大小来找出与目标范例最相似的源范例,该范例的基坑支护结构类型即为目标范例的可优先考虑的支护结构类型。通过工程实例证明了本章提出的深基坑支护结构选型模型是可靠适用的。
第九章对本论文的主要内容和主要结论进行归纳总结,提出今后需要进一步完善和深入研究的方向。
Order relation is one of the basis for decision making analysis, in the decision making analysis, an object for comparison and ordering are frequently encountered, which is an order relation from the viewpoint of the relationship. The various order relations and axiomatized order theories promoted by linear order play an important role in some practical problems. With the emergence of a wide variety of nonlinear problems in management science, biology, economics, engineering, mathematics and so on, nonlinear analysis has become one of the most important researches in modern science, while the partial order method, one of the basic method to research nonlinear problems, plays a very important role.
In this dissertation, there are some important new results by studying on the order methods and multi-attribute decision making. The main contents are as follows:
The first chapter introduces the historical background, current situation and the required prior knowledge in this dissertation about the partial order in nonlinear analysis and multi-attribute decision making; describes the main issues of this study and research significance of this dissertation.
The second chapter describes the main order relations in mathematics and economic management science and the nature of these order relations; the relationship between the cone and partial order, wedge and quasi-order are described in this chapter.
Chapter three studies some nonlinear operator problems with the order methods. First of all, there comes to the corresponding fixed point theorems through studying the existence of fixed points for nonlinear operator with partial order method. Then, it proposes the new concept of n times reasonable expansive operator and discusses its existence of fixed points to get some new fixed point theorems, promoting some important conclusions. At last, it conducts a discussion on the existence of solutions of a class of nonlinear equations with partial order method and monotone iterative technique and studys their Min-max solution and Max-min solution.
The fourth chapter addresses the relationship between the wedge and quasi-preferences, and studies the existence of solutions to a class of operator equations and problems about set-valued increasing operator with the sense of quasi-preferences.
Chapter five defines the Risk Ordered Weighted Averaging (R-OWA) operator and the Risk Ordered Weighted Geometric Averaging (R-OWGA) operator and their properties are discussed. At the same time, it studies a new method of the risk multi-attribute decision making based on ordered information integration operator.
Chapter six gives the new concepts of the preference dominating relational index and the preference be dominated relational index in preference structure based on the distance between the preference relations and on this basis this chapter studies multi-attribute decision making method with the only ordinal information.
Chapter seven studies the rank preference optimal ordering method in the multi-attribute decision making. This chapter first describes the optimal ordering method and puts forward the concept of rank preferences correlation coefficient, then proposes the rank preference optimal ordering method by using rank preferences dominating correlation coefficient instead of the precedence order number in optimal ordering method. The rank preference optimal ordering method was achieved on the computer, therefore makes the method to play a significant role in solving practical decision problems. Finally, the studies on the expansion of the rank preference optimal ordering method are carried out.
Chapter eight sets up a lectotype decision model of retaining & protecting structures for deep foundation pit by use of the rank preference optimal ordering method. Using the rank preference optimal ordering method, the base case indexing model is proposed. In the model, the degree of similarity between the base case and the target case of deep foundation pit is obtained by the Hamming distance in order to find the deep foundation pit base case which is the most similar to the target case in the base cases of deep foundation pit. The retaining & protecting structure of the deep foundation pit base case which is the most similar to the target case in the base cases of deep foundation pit shall be the priority structure that can be considered for the terget case. The engineering example presented in this chapter proves that the lectotype decision model of retaining & protecting structures is reliable and applicable.
Chapter nine sums up the main content in this dissertation and the main conclusions then puts forward the need for further refinement and in-depth research directions in the future.
引文
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