Soft Set Theory And Its Application In Decision Making

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.