Research On Decision Making Models And Uncertainty Based On Double-quantitative Rough Approximations

As a decision making model considering double quantification,double-quantitative rough approximation has better fault tolerance than probabilistic rough set model considering only relative quantitative information and graded rough set model considering only absolute quantitative information.Based on various requirements,many generalized rough set models have been developed to alleviate the limitations of generic Pawlak rough set theory and tackle different categories of information systems.One of the limitations is that rough set models based on equivalence relation are only applicable to discrete data information systems,and not suitable for dealing with real-valued continuous data without any prior processing.Another limitation is that “classical” rough sets do not consider the quantitative information about the degree of overlap between equivalence classes and the basic set,so they cannot cope well with the quantification problems.In order to overcome the two limitations of the classical rough set model,it is necessary to construct different double-quantitative rough set models based on non-equivalence relations to meet requirements of quantitative information in practical relations and to deal with different types information systems.The main objective of this paper is to present new decision making models based on double-quantitative rough approximations,and discuss uncertainty measures of double-quantitative decision-theoretic rough set(Dq-DTRS)model,which including the following four aspects:Firstly,two kinds of consistency levels are introduced from the perspective of double quantification in ordered information system,namely relative quantitative consistency level and absolute quantitative consistency level.The single-quantitative consistency rough set models based on these two kinds of quantitative consistency levels and their basic properties with the relevant three-way decision rules are discussed respectively in an ordered information system.Moreover,two kinds of double-quantitative consistency rough set models and their basic properties with the relevant decision rules based on these two kinds of quantitative consistency level are introduced.A consistency analysis of decision making in a practical case study is used to illustrate and interpret the double-quantitative consistency rough set models and the related decision rules in the ordered information system.The obvious shortcomings of dominance-based rough set approach(DRSA)without quantitative information are compared to illustrate the advantages of the two quantitative consistency levels in practical application.Secondly,a distance-based fuzzy similarity relation is constructed by using the distance formula between any two objects in the information system,then the framework of double-quantitative rough fuzzy set models with logic operations are proposed,and the influence of parameters on decision rules is studied.It should be noted that the models(Db-Sq-RFS and Db-Dq-RFS)proposed in this study are quite different from the typical fuzzy rough set model and its generalizations.The main differences are shown as follows.(1)The method of how to obtain the fuzzy similarity relation is not discussed in classical fuzzy rough set model.However,this study provides a visual and systematic formation process for obtaining the distance-based fuzzy similarity relation.(2)The set type of the obtained approximations in this paper is completely different from the set type of the approximations in classical fuzzy rough set model.In classical fuzzy rough set model,the membership functions for upper and lower approximations are obtained firstly,which means that the upper and lower approximations are fuzzy sets.However,the upper and lower approximations defined in this study are classical sets,and it means that we use two classical sets(upper and lower approximations)to approximate a given fuzzy set.(3)The quantitative information is not reflected on classical fuzzy rough set model.While the Db-Sq-RFS models can reflect one kind of quantitative information,and the Db-DqRFS models can reflect two kinds of quantitative information in their upper and lower approximations.Thirdly,the uncertainty measure methods of relative and absolute quantitative information in the Dq-DTRS model are studied from the perspective of information theory.Since the Dq-DTRS model was proposed,there have been few research on the uncertainty analysis in the model.Unlike other types of uncertainty measures in the rough set model,the upper and lower approximations of all subsets of the universe of discourse are taken into consideration in this paper.The two kinds of quantitative information are granulated in Dq-DTRS model firstly,the information representation characteristics and basic properties of granulated quantitative information are further investigated,then the methods of measuring the uncertainty of the quantitative information in the model,such as granularity of quantitative information,discernibility,rough entropy and information entropy,are proposed.The relationship between these measures is discussed and their respective important properties are proved,it provides a theoretical basis for uncertainty measures of quanttiative information.Finally,the uncertainty measure of the four disjoint regions in Dq-DTRS models is investigated by introducing a fuzziness formula for rough set,that is,fuzziness,and then the changing regularities of fuzziness of disjoint regions are described in DqI-DTRS model and DqII-DTRS model along with the variation of two parameters and the grade,respectively.In addition,three kinds of incremental information for Dq-DTRS model,namely useful incremental information,useless incremental information and error-correction incremental information are presented being formed with regard to the changes of boundary regions,and also the related assessment methods for these special types of incremental information are discussed in the form of several important theorems.