软集理论及其在决策中的应用研究
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摘要
在现实世界中,事物以及事物之间的关系是极其复杂的,由于客观存在的随机性、模糊性以及某些事物或现象体现的不充分性,导致人们对事物的认识往往是不精确、不完全的,具有一定程度的不确定性。不确定性普遍存在于经济、工程、环境、社会科学和商业管理等领域的许多重要问题中。概率论、模糊集理论和区间数学是常用的处理不确定性问题的数学工具,其中概率论和模糊集理论是分别处理随机性和模糊性的有力工具。软集理论是Molodtsov于1999年从参数化角度提出的一种新的处理不确定性问题的数学工具,模糊集合可以看作是一种特殊的软集合。目前,软集理论已经成功应用到许多领域如运筹学、测度论、博弈论、企业竞争力的综合评价、文本分类、数据挖掘、农村土地使用权的估价、信用资料的处理、外贸进出口量的预测、医疗诊断、洪水预测、决策等等。同时,软集合理论方面的研究也在不断完善,大体可分为三个方向:一是继续探讨软集合本身所具有的性质,如定义新的运算、提出软集合的子类等;二是在软集合上讨论各种代数结构,如软群、软环、软模、软BCK/BCI代数、软BCH代数等;三是将软集合与已有的各种处理不确定性的方法相结合,如模糊软集合、直觉模糊软集合、区间值模糊软集合等。
本文继续探讨软集合的理论和应用,提出了软P-超群和软超模,结合软集合和2型模糊集合定义了2型模糊软集合,并研究其在决策中的应用。同时阐明了决策、推理、知识编译之间的关系,并指出基于扩展规则的知识编译方法的不足,提出了两种有效的启发式策略分别用于指导待扩展子句和变量的选择,降低了知识编译后目标子句集的规模,进而提高了在线推理的效率。本文的创新成果具体如下:
首先,我们讨论了软集合的代数超结构,代数超结构是代数结构的扩展。我们定义了软P-超群、正规软P-超群、软子P-超群、正规软子P-超群,并且在它们的基础上讨论了软集合的各种运算,得到了一些相关性质。同时,我们研究了软P-超群的同态和同构,证明了软P-超群的三个同构定理。超模是另一种代数超结构,我们继续讨论了软集合的代数超结构,给出软超模、软子超模的定义并得到一些基本性质。相应地,利用软超模的同态与同构我们证明了软超模的三个同构定理。进一步,利用正规模糊子超模的概念,我们证明了软超模的三个模糊同构定理。
其次,我们结合软集合与2型模糊集合,提出了2型模糊软集合,作为模糊软集合的扩展。2型模糊集合是在模糊集合的基础上提出的,它比模糊集合具有更强的表达能力。因此,2型模糊软集合比模糊软集合具有更强的处理不确定性问题的能力。我们定义了2型模糊软集合的交、并、补等基本运算,并且证明一些基本定律的成立。同时,利用2型模糊软集合的水平软集合的概念,提出了一种基于2型模糊软集合的灵活决策方法。其优点主要体现在如下两个方面:一是不用直接处理2型模糊软集合,降低了计算的复杂性;二是该算法具有灵活性,可以根据决策者的偏好来得到不同的决策结果。
最后,我们提出了基于启发式策略的扩展规则知识编译方法。知识编译将推理过程分为两个阶段:离线编译阶段和在线推理阶段。离线编译阶段得到的知识库规模对之后的在线推理的效率起着至关重要的作用。我们对基于扩展规则的知识编译方法进行深入研究后发现,该方法对待扩展子句的选择没有考虑子句之间的内在关系,而且在选择变量进行扩展时,没有使用任何启发式策略,只是采用顺序扩展的方式。为了减小编译后的子句集规模,我们提出了两种启发式策略MCN策略和MO策略分别用于指导扩展过程中子句与变量的选择。实验结果表明,MCN策略和MO策略都可以大幅度减小编译后的子句集规模,两种启发式策略同时使用,效果更为明显。
The objects and the relation among them are very complicated in real world. Because of the randomness and fuzziness existing objectively, and insufficiency revealed themselves by objects or phenomena, it leads the knowledge for world to be often imprecise and incomplete, and manifest uncertainty to some extent. Uncertainties are pervasive in many complicated problems in engineering, economics, environment, medical science and social science. Theory of probability, theory of fuzzy sets and the interval mathematics can be considered as mathematical tools for dealing with uncertainties, where theory of probability and theory of fuzzy sets are the most appropriate theories for dealing with the randomness and fuzziness, respectively. Soft set theory, introduced by Molodtsov from parametrization perspective in 1999, has been considered as an effective mathematical tool for modeling uncertainties. Fuzzy sets can be considered as a special case of soft sets. Recently, soft set theory has been applied to many different fields, such as operational research, game theory, measurement theory, business competitive capacity evaluation, classification of the natural textures, rural land usage right evaluation, personal credit evaluation, forecasting the export and import volume, medical diagnosis, flood alarm model, decision making, and so on. At the same time, researches on theoretical aspect of soft sets are progressing rapidly. This work can be classified into three classes. The first class is to continue to study the properties of soft sets. The second class is to discuss the algebraic structures of soft sets, such as soft groups, soft rings, soft BCK/BCI-algebras, BCH-algebras, and so on. The third class is to combine soft sets with other theories for dealing with uncertainties, for example, fuzzy soft sets, interval-valued fuzzy soft sets and intuitionistic fuzzy soft sets.
The thesis is to continue to study the theory and applications of soft sets. We present the soft polygroups and soft hypermodules, define the type-2 fuzzy soft sets by the combination of soft sets and type-2 fuzzy sets, and consider the application of type-2 fuzzy soft sets in decision making. Also, we discuss the relation among decision making, automated reasoning and knowledge compilation, and point out the disadvantages existing in the algorithm of knowledge compilation using extension rule. Furthermore, we propose two heuristic strategies, MCN and MO, to lead the chooses of relevant clause and variable, respectively, in order to reduce the times of using extension rule, and further decrease the size of the compiled knowledge base. The contributions of this thesis are concretely as follows:
Firstly, we consider the algebraic hyperstructures of soft sets. Algebraic hyperstructures are the generalization of algebraic structures. Polygroup is a kind of algebraic hyperstructures. We introduce soft polygroups, normal soft polygroups, soft subpolygroups, normal soft subpolygroups, consider several operations on them, and obtain some related results. Also, we define the homomorphism and isomorphism of soft polygroups, and establish three isomorphism theorems of soft polygroups. Hypermodule is also a kind of algebraic hyperstructures. We study the soft hypermodules and soft subhypermodules, and investigate some basic properties. Accordingly, we derive three isomorphism theorems of soft hypermodules by using the homomorphism and isomorphism of soft hypermodules. By using normal fuzzy subhypermodules, three fuzzy isomorphism theorems of soft hypermodules are established.
Secondly, we present type-2 fuzzy soft sets which are based on the combination of type-2 fuzzy sets and soft sets, as a generalization of fuzzy soft sets. The concept of a type-2 fuzzy set was introduced by Zadeh as an extension of the concept of an ordinary fuzzy set. Type-2 fuzzy sets have more powerful expressiveness than ordinary fuzzy sets. Therefore, type-2 fuzzy soft sets have more power for handling uncertainty than fuzzy soft sets. We define some operations and prove some basic laws on type-2 fuzzy soft sets. Moreover, we propose a flexible approach to decision making based on type-2 fuzzy soft sets by using level soft sets of type-2 fuzzy soft sets. The advantages of the approach are mainly twofold. First, it is simpler and easier for application in practical problems, because we do not need to treat type-2 fuzzy soft sets directly in decision making but only deal with the related the crisp level soft sets after choosing certain threshold pairs. Second, it can be seen as an adjustable approach to type-2 fuzzy soft sets based decision making because the final optimal decision is in relation to the decision criteria used by decision makers.
Finally, we propose knowledge compilation using extension rule based on heuristic strategies. Knowledge compilation approach splits the reasoning process into two phases: an off-line compilation phase and an on-line query-answering phase. The size of the compiled knowledge base is crucial to the effectiveness in on-line reasoning phase. After a deep research on the approach of knowledge compilation using extension rule, we found that the approach does not consider the relation among clauses, when it chooses a clause to extend. Also, when it chooses a variable to extend, the approach does not consider any heuristic strategy and extends variable sequentially. In order to decrease the size of the compiled knowledge base, we propose two heuristic strategies, MCN and MO, to lead the chooses of relevant clause and variable, respectively. Experimental results indicate that the MCN and MO play a great role in minimizing the size of the compiled knowledge base. When MCN and MO are used together, the efficiency is better.
本文继续探讨软集合的理论和应用,提出了软P-超群和软超模,结合软集合和2型模糊集合定义了2型模糊软集合,并研究其在决策中的应用。同时阐明了决策、推理、知识编译之间的关系,并指出基于扩展规则的知识编译方法的不足,提出了两种有效的启发式策略分别用于指导待扩展子句和变量的选择,降低了知识编译后目标子句集的规模,进而提高了在线推理的效率。本文的创新成果具体如下:
首先,我们讨论了软集合的代数超结构,代数超结构是代数结构的扩展。我们定义了软P-超群、正规软P-超群、软子P-超群、正规软子P-超群,并且在它们的基础上讨论了软集合的各种运算,得到了一些相关性质。同时,我们研究了软P-超群的同态和同构,证明了软P-超群的三个同构定理。超模是另一种代数超结构,我们继续讨论了软集合的代数超结构,给出软超模、软子超模的定义并得到一些基本性质。相应地,利用软超模的同态与同构我们证明了软超模的三个同构定理。进一步,利用正规模糊子超模的概念,我们证明了软超模的三个模糊同构定理。
其次,我们结合软集合与2型模糊集合,提出了2型模糊软集合,作为模糊软集合的扩展。2型模糊集合是在模糊集合的基础上提出的,它比模糊集合具有更强的表达能力。因此,2型模糊软集合比模糊软集合具有更强的处理不确定性问题的能力。我们定义了2型模糊软集合的交、并、补等基本运算,并且证明一些基本定律的成立。同时,利用2型模糊软集合的水平软集合的概念,提出了一种基于2型模糊软集合的灵活决策方法。其优点主要体现在如下两个方面:一是不用直接处理2型模糊软集合,降低了计算的复杂性;二是该算法具有灵活性,可以根据决策者的偏好来得到不同的决策结果。
最后,我们提出了基于启发式策略的扩展规则知识编译方法。知识编译将推理过程分为两个阶段:离线编译阶段和在线推理阶段。离线编译阶段得到的知识库规模对之后的在线推理的效率起着至关重要的作用。我们对基于扩展规则的知识编译方法进行深入研究后发现,该方法对待扩展子句的选择没有考虑子句之间的内在关系,而且在选择变量进行扩展时,没有使用任何启发式策略,只是采用顺序扩展的方式。为了减小编译后的子句集规模,我们提出了两种启发式策略MCN策略和MO策略分别用于指导扩展过程中子句与变量的选择。实验结果表明,MCN策略和MO策略都可以大幅度减小编译后的子句集规模,两种启发式策略同时使用,效果更为明显。
The objects and the relation among them are very complicated in real world. Because of the randomness and fuzziness existing objectively, and insufficiency revealed themselves by objects or phenomena, it leads the knowledge for world to be often imprecise and incomplete, and manifest uncertainty to some extent. Uncertainties are pervasive in many complicated problems in engineering, economics, environment, medical science and social science. Theory of probability, theory of fuzzy sets and the interval mathematics can be considered as mathematical tools for dealing with uncertainties, where theory of probability and theory of fuzzy sets are the most appropriate theories for dealing with the randomness and fuzziness, respectively. Soft set theory, introduced by Molodtsov from parametrization perspective in 1999, has been considered as an effective mathematical tool for modeling uncertainties. Fuzzy sets can be considered as a special case of soft sets. Recently, soft set theory has been applied to many different fields, such as operational research, game theory, measurement theory, business competitive capacity evaluation, classification of the natural textures, rural land usage right evaluation, personal credit evaluation, forecasting the export and import volume, medical diagnosis, flood alarm model, decision making, and so on. At the same time, researches on theoretical aspect of soft sets are progressing rapidly. This work can be classified into three classes. The first class is to continue to study the properties of soft sets. The second class is to discuss the algebraic structures of soft sets, such as soft groups, soft rings, soft BCK/BCI-algebras, BCH-algebras, and so on. The third class is to combine soft sets with other theories for dealing with uncertainties, for example, fuzzy soft sets, interval-valued fuzzy soft sets and intuitionistic fuzzy soft sets.
The thesis is to continue to study the theory and applications of soft sets. We present the soft polygroups and soft hypermodules, define the type-2 fuzzy soft sets by the combination of soft sets and type-2 fuzzy sets, and consider the application of type-2 fuzzy soft sets in decision making. Also, we discuss the relation among decision making, automated reasoning and knowledge compilation, and point out the disadvantages existing in the algorithm of knowledge compilation using extension rule. Furthermore, we propose two heuristic strategies, MCN and MO, to lead the chooses of relevant clause and variable, respectively, in order to reduce the times of using extension rule, and further decrease the size of the compiled knowledge base. The contributions of this thesis are concretely as follows:
Firstly, we consider the algebraic hyperstructures of soft sets. Algebraic hyperstructures are the generalization of algebraic structures. Polygroup is a kind of algebraic hyperstructures. We introduce soft polygroups, normal soft polygroups, soft subpolygroups, normal soft subpolygroups, consider several operations on them, and obtain some related results. Also, we define the homomorphism and isomorphism of soft polygroups, and establish three isomorphism theorems of soft polygroups. Hypermodule is also a kind of algebraic hyperstructures. We study the soft hypermodules and soft subhypermodules, and investigate some basic properties. Accordingly, we derive three isomorphism theorems of soft hypermodules by using the homomorphism and isomorphism of soft hypermodules. By using normal fuzzy subhypermodules, three fuzzy isomorphism theorems of soft hypermodules are established.
Secondly, we present type-2 fuzzy soft sets which are based on the combination of type-2 fuzzy sets and soft sets, as a generalization of fuzzy soft sets. The concept of a type-2 fuzzy set was introduced by Zadeh as an extension of the concept of an ordinary fuzzy set. Type-2 fuzzy sets have more powerful expressiveness than ordinary fuzzy sets. Therefore, type-2 fuzzy soft sets have more power for handling uncertainty than fuzzy soft sets. We define some operations and prove some basic laws on type-2 fuzzy soft sets. Moreover, we propose a flexible approach to decision making based on type-2 fuzzy soft sets by using level soft sets of type-2 fuzzy soft sets. The advantages of the approach are mainly twofold. First, it is simpler and easier for application in practical problems, because we do not need to treat type-2 fuzzy soft sets directly in decision making but only deal with the related the crisp level soft sets after choosing certain threshold pairs. Second, it can be seen as an adjustable approach to type-2 fuzzy soft sets based decision making because the final optimal decision is in relation to the decision criteria used by decision makers.
Finally, we propose knowledge compilation using extension rule based on heuristic strategies. Knowledge compilation approach splits the reasoning process into two phases: an off-line compilation phase and an on-line query-answering phase. The size of the compiled knowledge base is crucial to the effectiveness in on-line reasoning phase. After a deep research on the approach of knowledge compilation using extension rule, we found that the approach does not consider the relation among clauses, when it chooses a clause to extend. Also, when it chooses a variable to extend, the approach does not consider any heuristic strategy and extends variable sequentially. In order to decrease the size of the compiled knowledge base, we propose two heuristic strategies, MCN and MO, to lead the chooses of relevant clause and variable, respectively. Experimental results indicate that the MCN and MO play a great role in minimizing the size of the compiled knowledge base. When MCN and MO are used together, the efficiency is better.
引文
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