Generalized multigranulation rough sets and optimal granularity selection

Transcription

1 Granu. Comput. DOI /s ORIGINAL PAPER Generaized mutigranuation rough sets and optima granuarity seection Weihua Xu 1 Wentao Li 2 Xiantao Zhang 1 Received: 27 September 2016 / Accepted: 3 March 2017 Springer Internationa Pubishing Switzerand 2017 Abstract Mutigranuation rough set theory is a desirabe direction in the fied of rough set, in which upper and ower approximations are approximated by mutipe granuar structures. However, cassic mutigranuation rough set is studied from two kinds of quaitative combination rues which were generated by pessimistic and optimistic viewpoints, respectivey. The two combination rues seem to ack of practicabiity since one is too restrictive and the other too reaxed. To overcome this disadvantage, we propose a generaized mutigranuation rough set mode in this paper. First, we discuss upper and ower approximation sets of a generaized mutigranuation rough set by introducing a support characteristic function and an information eve. Then, as one of the most important probems in granuar computing, we carefuy study how to seect optima granuarity in generaized mutigranuation rough sets. Furthermore, agorithms of optima granuarity seection are constructed, by which we can provide an efficient approach to compute the optima granuarity based on generaized mutigranuation rough sets. Finay, an iustrative exampe is given to show the effectiveness of the proposed approach. The main contribution of this paper is to construct the mode of the optima partice size seection on account of the generaized muti granuarity, and overcome the imitation of the cassica muti granuarity. Keywords Mutigranuation rough set Optima granuarity seection Quaity function Support characteristic function * Weihua Xu chxuwh@gmai.com 1 2 Schoo of Sciences, Chongqing University of Technoogy, Chongqing , Peope s Repubic of China Department of Mathematics, Harbin Institute of Technoogy, Harbin , Peope s Repubic of China 1 Introduction Rough set theory proposed by Pawak (1982), is an extension of the cassica set theory. This theory coud be regarded as a mathematica and soft computing too to hande vagueness, imprecision and uncertainty in data anaysis. The methodoogy has received great attention in recent years, and it has been successfuy appied in many science and engineering fieds, such as pattern recognition, data mining, image processing, medica diagnosis and so on. Rough set theory was deveoped by practica needs to characterize, interpret, present, process indiscernibiity of individuas (objects). The discernibiity is typicay characterized by an equivaence reation. A rough set is the resut of two approximating crisp sets, which are ower and upper approximation sets, using equivaence casses. The key idea of rough set theory is the use of some known knowedge to approximate the inaccurate and uncertain knowedge in information systems. However, partition or indiscernibiity reation in Pawak s origina rough set theory, is sti restrictive for many appications. To overcome such unreasonabeness, the dominance-based rough set approach has been proposed by Greco et a. (1999). On the other hand, the generaization of rough sets is an interesting topic not ony in mathematica point of view but aso in practica point of view. Aong this direction, rough sets have been generaized under simiarity reations (Inuiguchi and Tanino 2001), covers (Bonikowski et a. 1998) and genera reations (Inuiguchi and Tanino 2002; Shen and Jensen 2007; Yao 1996, 1998; Yao and Lin 1996). Those resuts demonstrate a diversity of generaizations. Moreover, the introduction of fuzziness into rough set approaches (Lu et a. 2016) has attracted researchers to obtain more reaistic and usefu toos. Vo.:( )

2 Granu. Comput. In 1985, Zadeh first expored the concept of granuar computing (Zadeh 1997) between 1996 and He thought that information granues refer to pieces, casses, and groups into which compex information are divided in accordance with the characteristics and processes of the understanding and decision-making. Currenty, granuar computing has been viewed as an emerging computing paradigm of information processing. It concerns the processing of compex information entities caed information granues (Li et a. 2015; Pedrycz 2013; Pedrycz and Bargiea 2002, 2012; Xu and Li 2014). Information granues, as encountered in natura anguage, are impicit in their nature. To make them fuy operationa so that they become effectivey used in the anaysis and design of inteigent systems (Pedrycz 2013), we need to make information granues expicit. This is possibe through a prudent formaization avaiabe within the ream of granuar computing. In a genera sense, by information granue, one regards a coection of eements drawn together by their coseness (resembance, proximity, functionaity, etc.) articuated in terms of some usefu spatia, tempora or functiona reationships. Subsequenty, Granuar Computing is about representing, constructing and processing information granues. Resuts of computing competed in the setting of Granuar Computing come in the form of information granues. Information granues are buiding bocks refective of domain knowedge about a probem. From the perspective of granuar computing (Liu et a. 2016), an equivaence reation on the universe can be regarded as a granuarity, and the corresponding partition can be regarded as a granuar structure (Qian et a. 2009). Hence, the cassic rough set theory is based on a singe granuarity (ony one equivaence reation). However, the rough set may be associated with mutipe granuar structures (Apooni et a. 2016), which can be divided into two cases as foows: Case 1 If there exists at east one granuar structure such that eements surey beongs to a given concept, then we say that an eement surey beong to the concept. Case 2 If there exists at east one granuar structure such that eements possiby beongs to a given concept, then we say that an eement possiby beong to the concept. Currenty, Yao and Deng (2014) proposes a framework of quantitative rough sets based on subsethood measures. A specific quantitative rough set mode is defined by a particuar cass of subsethood measures satisfying a set of axioms. Consequenty, the framework enabes us to cassify and unify existing generaized rough set modes [e.g., decision-theoretic rough sets (Xu and Wang 2016), probabiistic rough sets, and variabe precision rough sets], to investigate imitations of existing modes, and to deveop new modes. Actuay, an attribute subset (Guo and Zheng 2014) induces an equivaence reation, and a partition formed by the equivaence reation can be regard as a granuarity. Using a finer granuar structure formed through combining two known granuarities induced from two attribute subsets to describe a target concept, this combination destroys the origina granuar structure. Qian and Liang extended Pawak s singe granuation rough set mode to a mutipe granuation rough set mode (Qian and Liang 2006; Qian et a. 2010), where the set approximations were defined using muti equivaence reations on the universe. Moreover, many researchers have extended the mutigranuation rough sets (Dou et a. 2012). Xu et a. (2014) deveoped a mutigranuation fuzzy rough set mode, mutigranuation rough sets based on toerance reations (Xu et a. 2013), a mutigranuation rough set mode in ordered information systems (Xu et a. 2012) and a mutigranuation fuzzy rough set in a fuzzy toerance approximation space (Xu et a. 2011). Yang et a. proposed the hierarchica structure properties of the mutigranuation rough sets (Yang et a. 2012), mutigranuation rough set in incompete information system (Yang et a. 2012a, b), and a test cost sensitive mutigranuation rough set mode (Yang and Qi 2013). Lin et a. (2012) presented a neighborhood-based mutigranuation rough set. She et a. (2012) expored the topoogica structures and the properties of mutigranuation rough sets. Qian et a. (2014) introduced three kinds of mutigranuation decision-theoretic rough set modes. Li et a. (2014) extended mutigranuation decision-theoretic rough sets by considering dominance reations in ordered information system (Li and Xu 2015), and investigated reationships between mutigranuation and cassica T-fuzzy rough sets. Yao et a. proposed a unified framework to cassify and compare existing studies. And an underying principe is to expain rough sets in a mutigranuation space through rough sets derived using individua equivaence reations (Yao and She 2016). Feng and Mi (2015) studied variabe precision mutigranuation fuzzy decision-theoretic rough sets in an information system. A nove membership degree based on singe granuation rough sets and two operators based on this membership degree were investigated in their study. Zhang et a. (2015) estabished four kinds of constructive methods of rough approximation operators from existing rough sets and studied the non-dua mutigranuation rough sets and hybrid mutigranuation rough sets. Tan et a. (2016) empoyed the beief and pausibiity functions from evidence theory to characterize the set approximations and attribute reductions in mutigranuation rough set theory in incompete information systems, and an attribute reduction agorithm for mutigranuation rough sets was proposed based on evidence theory. Lin et a. (2015) proposed a twograde fusion approach invoved in the evidence theory and mutigranuation rough set theory based on a we-defined distance function among granuation structures, and presented three types of covering based mutigranuation

3 Granu. Comput. rough sets whose set approximations were defined by different covering approximation operators (Lin et a. 2013). Li et a. investigated the reationship between mutigranuation rough sets and concept attices via rue acquisition (Li et a. 2016; Yang et a. 2009). Kumar and Inbarani appied rough set based data mining techniques for medica data to discover ocay frequent diseases (Senthi Kumar and Hannah Inbarani 2015). Huang et a. (2014) deveoped a new mutigranuation rough set mode that was caed intuitionistic fuzzy mutigranuation rough set (IFMGRS) and three types of IFMGRSs that are generaizations of three existing intuitionistic fuzzy rough set modes buit. Liu et a. proposed four types of muti-granuation covering rough set (MGCRS) modes under covering approximation space (Wang et a. 2017a), where a target concept was approximated by empoying the maxima or minima descriptors of objects in a given universe of discourse (Liu et a. 2014). Lin et a. (2014) presented a new feature seection method that seects distinguishing features by fusing neighborhood muti-granuation, and first used neighborhood rough sets as an effective granuar computing too. Yang et a. (2014) first expored the updating of the mutigranuation rough approximations. Qian et a. (2014b) deveop a new mutigranuation rough set mode based on Seeking common ground whie eiminating differences (SCED) strategy, caed pessimistic mutigranuation rough sets based decision. Liang et a. (2012) proposed an efficient rough feature seection agorithm for arge-scae data sets, which was stimuated from muti-granuation rough sets. In mutigranuation rough set theory, optimistic mutigranuation (Wang et a. 2017b) and pessimistic mutigranuation are two basic ways of research. For the ower approximation of a mutigranuation rough set, the view of optimistic mutigranuation refects that there exists at east one granuar structure such that eements surey beong to a given concept, and the view of pessimistic mutigranuation shows that eements surey beong to a given concept in each granuar structure. It is easy to notice that both optimistic and pessimistic conditions are too strict to a widerange of conditions. So, we wi introduce a parameter, namey an information eve to propose a generaized mutigranuation rough set mode. The ower approximation of a concept is the set that a of the eements support the concept based on the information eve is not ess than the given parameter in the mutigranuation perspective. The motivation of this paper is as foows: three aspects. (1) How to generaize the cassica mutigranuarity rough sets to the generaized mutigranuarity rough sets. (2) how to seect the proper granuarity is an important issue. It offers a systematic and theoretic framework for feature seection. (3) How to discover knowedge in hierarchicay organized information tabes is of particuar importance in rea ife data mining. In this paper, to describe a nove granuation perspective, we wi estabish a specia mutigranuation rough set mode, and discuss the methods of optima granuarity seection in this generaized mutigranuation rough set mode. This paper is organized as foows. In Sect. 2, some preiminary concepts of optimistic and pessimistic mutigranuation rough set theories are briefy reviewed. In Sect. 3, we introduce a support characteristic function and propose the generaized mutigranuation rough set mode. In Sect. 4, Moreover, measures and properties of the generaized mutigranuation rough set are carefuy investigated. How to seect the optima granuarity is discussed in generaized mutigranuation rough set. In Sect. 5, we consider agorithms of the optima granuarity seection in the new rough set mode. In Sect. 6, an iustrative exampe is given to show the effectiveness of the proposed approach. Finay, we concude our contribution with a summary and an outook for the further research. 2 Cassic mutigranuation rough sets In an information system, the equivaence cass of an object with respect to an attribute subset of A is a granuarity from the viewpoint of granuar computing. A partition of the universe is a granuar structure. Rough set proposed by Pawak is a singe granuation rough set mode, and the granuar structure in this mode is induced by the indiscernibiity reation of the attribute set. In genera, the above cases cannot aways be satisfied or required in practica probems. In the three cases referred in reference (Qian et a. 2010), there are imitations in singe granuation rough set for addressing practica probems with mutipe partitions, and mutigranuation rough set can now be used to effectivey sove these probems. Under those circumstances, we must describe a target concept through mutipe binary reations on the universe according to a user s requirements or targets of probem soving. In the iterature (Qian et a. 2009, 2010; Dang and Qian 2009; Yao 2000), to appy rough set theory to practica probems widey, mutigranuation rough set mode has been studied based on mutipe equivaence reations. Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A. Then P i or U P i is referred to as a granuation. The equivaence cass of an object x with respect to P i is defined as [x] Pi ={y U f (x, a) =f (y, a)} (a P i ). The ower and upper approximation sets of X with respect to singe P i are defined as foows: P i (X) ={x U [x] Pi X}, P i (X) ={x U [x] Pi X }.

4 Granu. Comput. Considering further studies on mutigranuation rough set, we now review the two basic forms of mutigranuation rough set mode. Definition 2.1 (Qian et a. 2010) Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). The optimistic mutigranuation ower and upper approximation sets of X with respect to singe P are defined as foows: P(X) OM ={x U ([x] Pi X), i }; P(X) OM ={x U ([x] Pi X ), i }, where means the ogica operator or, which represents that the aternative conditions are satisfied, and means the ogica operator and, which represents that a of the conditions are satisfied. The set X is definabe if and ony if P(X) OM = P(X) OM. Otherwise, X is rough. P(X) OM and P(X) OM are referred to as optimistic ower and upper approximation sets, respectivey. From the above definition, the operators and can be exchanged between the optimistic ower approximation set and the optimistic upper approximation set. Corresponding to optimistic mutigranuation rough set, pessimistic mutigranuation rough set mode can be defined in the foowing. Definition 2.2 (Qian et a. 2010) Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). The pessimistic mutigranuation ower and upper approximation sets of X with respect to singe P are defined as foows: P(X) PM ={x U ([x] Pi X), i }, P(X) PM ={x U ([x] Pi X ), i }. The set X is definabe if and ony if P(X) PM = P(X) PM. Otherwise, X is rough. P(X) PM and P(X) PM are referred to as pessimistic ower and upper approximation sets, respectivey. The uncertainty of a concept in a mutigranuation rough set mode is aso due to the existence of a boundary region. The greater the boundary of a concept is, the ower its accuracy is, and the coarser the concept is. Simiar to the measures in the Pawak rough set mode, the accuracy and roughness measures in optimistic mutigranuation rough set and pessimistic mutigranuation rough set were defined in the same way (Qian et a. 2010). As generaizations of the Pawak rough set mode, we ony show the reations among optimistic mutigranuation rough set, pessimistic mutigranuation rough set and singe granuation rough set in the foowing. Proposition 2.1 (Qian et a. 2010) Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). The foowing properties hod: (1) P(X) OM = P i (X); (2) P(X) OM = P i (X); (3) P(X) PM = P i (X); (4) P(X) PM = P i (X); (5) P(X) PM P(X) OM ; (6) P(X) OM P(X) PM. In addition, there are many reated properties as we as proof pease refer to (Qian and Liang 2006; Qian et a. 2010). 3 Generaized mutigranuation rough sets To present and iustrate the generaized mutigranuation rough set mode, we first define the support characteristic function. Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). A characteristic function S P i(x) describes the incusion reation X between the cass [x] Pi, which is defined in the foowing: { S P i 1, = [x]pi X; (i ). 0, ese. We then ca S P i(x) the support characteristic function of x. X It shows whether x supports the concept X or not precisey with respect to P i. From the support characteristic function for any P i A (i = 1, 2,, ) and X U, the number of equivaence casses [x] Pi that satisfies [x] Pi X can be computed by S P i.

5 Granu. Comput. At the same time, for any x U, the number of equivaence casses [x] Pi that satisfies [x] Pi X can be represented by (1 S P i ). By the support characteristic function, the ower and upper approximation sets in optimistic mutigranuation rough set and pessimistic mutigranuation rough set can be represented, respectivey, with the foowing formuas: { P(X) OM = x U } > 0 ; { P(X) OM = x U (1 ). } = 1. { P(X) PM = x U } = 1 ; { P(X) PM = x U (1 ). } > 0. From the view of granuar computing, one can note that cassica mutigranuation rough set might not aways be effective in practice. Optimistic mutigranuation rough set might be so oose that the approximation sets cannot describe the concepts as precisey as possibe. Additionay, pessimistic mutigranuation rough set might be too strict to depict concepts on the universe. In optimistic mutigranuation rough set, we consider the case that an object x supports the concept X precisey if there exists at east one P i P such that [x] Pi X. This mode coud bring in a arge amount of useess information for the concept described. The descriptions and information in optimistic mutigranuation rough set coud be redundant and cannot show the nature of the concept. Furthermore, some usefu information wi be ost because this mode demands that any object x can possiby describe a concept X in terms of mutigranuation satisfying [x] Pi X for a P i. In practica appications, the object can possiby describe a concept by most of the granuations. Conversey, an object x supporting the concept X precisey means that [x] Pi X must hod for a P i with respect to mutigranuation in pessimistic mutigranuation rough set. This approach aso causes disadvantages in practice. If x supports the concept, then a granuations must be considered. This approach is so strict that some information and descriptions which are not very effective can be ignored. Thus, we can introduce the parameter β, i.e., the information eve, to reaize that the objects support a concept with respect to the majority granuations. The higher the information eve β is, the stricter our requirements are. Our requirements can be empoyed to depict the concept better. Next, we wi propose a nove mutigranuation rough set mode with a parameter β (0.5, 1]. Definition 3.1 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P, } P i A(i = 1, 2,, ). A characteristic function S P i describes the incusion reation between the cass [x] Pi. For any β (0.5, 1], generaized ower and upper approximation sets of X with respect to P are defined as foows: { P(X) β = x U } β ; { P(X) β = x U (1 ). } > 1 β. The set X is definabe if and ony if P(X) β = P(X) β. Otherwise, X is rough. We ca β the information eve with respect to P. For cassic rough set mode, the roughness or uncertainty in an information system is aso due to the existence of a boundary region of concepts in generaized mutigranuation rough set. The boundary region of a concept with respect to P in generaized mutigranuation rough set is defined by Bn(X) GM = P(X) β P(X) β. Objects in approximation sets and boundary regions are changed corresponding to the information eve β. Additionay, we can have the foowing interpretations to approximation sets and boundary regions in generaized mutigranuation rough set. The ower approximation set of a concept X is the set of a of the eements that can surey support the concept X on the basis of an information eve not ess than β in terms of the mutigranuation. The upper approximation set of a concept X is the set of a of the eements that can possiby support the concept X on the basis of an information eve not ess than 1 β in terms of the mutigranuation. The boundary region of a concept X with respect to P is the set of a of the eements that cannot surey support either X or X on the basis of an information eve β.

6 Granu. Comput. That is, Bn(X) GM is the set of a of the eements that cause the uncertainty of X in an information system with respect to P on the basis of the information eve β. From the above, we can easiy compare the approximation sets in optimistic mutigranuation rough set, pessimistic mutigranuation rough set and generaized mutigranuation rough set. Next, we wi study the reations between generaized mutigranuation rough set, optimistic mutigranuation rough set and pessimistic mutigranuation rough set, and discuss some important properties of the approximation operators in generaized mutigranuation rough set. Figure 1 shows the reationships of the ower and upper approximations in generaized mutigranuation rough set, optimistic mutigranuation rough set and pessimistic mutigranuation rough set, respectivey. Proposition 3.1 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). For any β (0.5, 1], the ower and upper approximation operators in generaized mutigranuation rough set have the foowing reations with those in optimistic mutigranuation rough set and pessimistic mutigranuation rough set (Fig. 1): (1) P(X) PM P(X) β P(X) OM ; (2) P(X) OM P(X) β P(X) PM. Proof (1) For any x and β, one can prove P(X) PM P(X) β through x P(X) PM 1. As β (0.5, 1], so 1 β, that is to say x P(X) β. Simiary, for any β (0.5, 1], x P(X) β β>0 x P(X) OM. Then P(X) β P(X) OM. This item is proved. (2) This item can be obtained simiary. Lemma 3.1 For any a 1, a 2, b 1, b 2 [0, 1], the foowing inequaities hod: (1) a 1 b 1 + a 2 b 2 (a 1 + a 2 ) (b 1 + b 2 ); (2) a 1 b 1 + a 2 b 2 (a 1 + a 2 ) (b 1 + b 2 ), where and represent the operators minimum and maximum, respectivey. Proof (1) Because a 1 b 1 a 1, a 1 b 1 b 1 and a 2 b 2 a 2, a 2 b 2 b 2, we have that a 1 b 1 + a 2 b 2 (a 1 + a 2 ). a 1 b 1 + a 2 b 2 b 1 + b 2.Then we have a 1 b 1 + a 2 b 2 (a 1 + a 2 ) (b 1 + b 2 ). (2) This item can be obtained simiary. Proposition 3.2 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). For any β (0.5, 1]. The foowing properties are true. (1a) P( X) β = P(X) β ; (1b) P( X) β = P(X) β. (2a) P(X) β X; (2b) X P(X) β. (3a) P( ) β = P( ) β = ; (3b) P(U) β = P(U) β = U. (4a) P(X Y) β P(X) β P(Y) β ; (4b) P(X Y) β P(X) β P(Y) β. (5a) P(X Y) β P(X) β P(Y) β ; (5b) P(X Y) β P(X) β P(Y) β. (6a) X Y P(X) β P(Y) β ; (6b) X Y P(X) β P(Y) β. (7a) X Y = P(X) β P(Y) β = ; (7b) X Y = P(X) β P(Y) β =. Proof (1a) For any x U, Fig. 1 Reationships of upper and ower approximations x P(X) β (1 ). > 1 β,

7 Granu. Comput. we have that x P(X) β (1 ). 1 β. β P( X) β. Then this item is proved. Item (1b) can be proved simiary. (2a) For any x P(X) β, we have β>0. There exists i such that [x] Pi X. Thus, we can get x X. (2b) By the duaity and item (2a), we have P(X) β = P( X) β X. Thus, X P(X) β. (3a) (3b) From item (2) in Proposition 3.1, S P i(x) =0 and S P i(x) =1 ( X U), then we can have U { P( ) β = x U (x) = 0 } = 0 β =, { P(U) β = x U (x) U = 1 } = 1 β = U. From the duaity, we can easiy have P( ) β = P( U) β = P(U) β = U =. P(U) β = P( ) β = P( ) β = =U. (4a) For any x P(X Y) β, we can get X Y (x) = Y (x) By Lemma 3.1, we have β. Y (x) Y (x). Then, we have the foowing Y (x) β ( ) ( β Y (x) ) β x P(X) β x P(Y) β. Therefore x P(X) β P(Y) β. (4b) From the duaity, this item can be proved by item (4a) in this proposition. (5a) From Proposition 3.1, for any x U, x P(X) β P(Y) β, it means that x P(X) β P(Y) β, ( By Lemma 3.1, one can have Then, β ) ( Y (x) β Y (x) β. Y (x) Y (x). X Y (x) Y (x) It means that x P(X Y) β. (5b) From duaity, we can get it easiy. (6a) For any x P(X) β, we have β. As X Y, we have S P i (x). Then we can get Y Y (x) Thus, x P(Y) β is obtained. Then this item is proved. (6b) Simiary, x P(X) β, we have (1 ) > 1 β. As X Y, we have S P i (x). Then Y (1 Y (x)) β. β. Additionay, x P(Y) β is obtained. (7a) From X Y =, we can directy obtain that X Y. We then have P(X) β P( Y) β. Moreover, from the duaity and items (2a) (2b) in this proposition, we have P( Y) β = P(Y) β P(Y) β. Thus, we can deveop that P(X) β P(Y) β. That is to say P(X) β P(Y) β =. This item is proved. (7b) It is easy to get this item. Remark 1 The properties P(P(X) β ) β = P(X) β = P(P(X) β ) β and P(P(X) β ) β = P(X) β = P(P(X) β ) β do not hod in generaized mutigranuation rough set. ) (1 ) > 1 β.

8 Granu. Comput. For different information eves α and β, the foowing properties can be obtained. Proposition 3.3 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). For any α β and α, β (0.5, 1], the foowing properties are true. (1) P(X) β P(X) α ; (2) P(X) α P(X) β. Proof (1) From Definition 3.1, on can have that for any x P(X) β β α x P(X) α. (2) From the duaity in Proposition 3.2 and item (1), this item can be proved easiy. Proposition 3.4 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). Additionay, Q ={Q i Q i P i, i = 1, 2,, } is a set for which some attributes have been removed from the corresponding granuations. For any β (0.5, 1], the foowing properties hod: (1) Q(X) β P(X) β ; (2) P(X) β Q(X) β. Proof As Q ={Q i Q i P i, i = 1, 2,, }, so Q i may be the empty set. If Q i =, then we denote [x] = U. This finding is ogica and reasonabe because a of the objects are indistinguishabe with no attributes being considered. (1) From the above assumptions, one can obtain that [x] Pi [x] Qi hods for any x U and i is obvious. Furthermore, S P i SQ i. Thus, S P i =1 SQ i(x) =1. Moreover, for any x U, we X can obtain that x Q(X) β SQ i β SQ i β x P(X) β. (2) This item can be proved using the duaity. For any β (0.5, 1], P(X) β = P(X) β Q(X) β = Q(X) β. Next, we wi discuss some measures in generaized mutigranuation rough sets to further study the new mode. Definition 3.2 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). The accuracy and roughness of X are defined as α(x) P β = P(X) β P(X) β, ρ(x) P β = 1 P(X) β P(X) β. The reationships of accuracies and roughness among generaized, optimistic and pessimistic mutigranuation rough sets are described as foows: (1) α(x) PM α(x) P β α(x) OM; (2) ρ(x) PM ρ(x) P β ρ(x) OM. Where α(x) OM = P(X) OM, α(x) P(X) PM = P(X) PM, ρ(x) OM P(X) OM = PM 1 P(X) OM, ρ(x) P(X) PM = 1 P(X) PM. OM P(X) PM Proposition 3.5 Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). For any X, Y U and β (0.5, 1], the foowing properties hod: (1) X is more accurate than Y with respect to P under β, if and ony if α(x) P β α(y)p β ; (2) X is more rough than Y with respect to P under β, if and ony if ρ(x) P β ρ(y)p β. For different information eves α, β (0.5, 1]. If α β, α(x) P α α(x)p β and ρ(x)p α ρ(x)p β hod. In Pawak rough sets mode, a parameter caed the dependent degree is used to iustrate the importance of a condition attribute subset with respect to the decision attributes in a target information system. This parameter can be defined simiary in generaized mutigranuation rough set, as foows: Let I =(U, A, V, f ) be an information system, X U and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ), U d ={D 1, D 2,, D r }. The dependent degree of P with respect to d under the information eve β is defined by r γ(p, d) β = k=1 P(D k ) β = U r k=1 P(D k ) β. U

9 Granu. Comput. From Definition 3.3, the dependent degrees in pessimistic mutigranuation rough set, generaized mutigranuation rough set and optimistic mutigranuation rough set have the foowing reations. γ(p, d) PM γ(p, d) β γ(p, d) OM, where r k=1 γ(p, d) PM = P(D k ) PM r k=1, γ(p, d) U OM = P(D k ) OM. U 4 Optima granuation seection in generaized mutigranuation rough sets It is important to seect the optima granuation in detai corresponding a suitabe information system. In this section, we investigate optima granuation seection with different requirements for the generaized mutigranuation rough sets. Let I =(U, A {d}, V, f ) be a target information system, P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). We say that I is granuar consistent if for any P i P, [x] Pi [x] d hods. Otherwise, I is a granuar inconsistent information system. For a granuar consistent information system I =(U, A {d}, V, f ) and Q P, if the dependent degree of Q with respect to d under the eve β: γ(q, d) β = γ(p, d) β, we ca Q is a granuarity seection. R Q, if the dependent degree of R with respect to d under the eve β: γ(q {R}, d) β γ(p, d) β, then Q is an optima granuation seection of I. Let I =(U, A {d}, V, f ) be a granuar inconsistent information system, and P ={P 1, P 2,, P }, P i A (i = 1, 2,, ). The ower and upper approximation granuar distribution functions of I are denoted as foows: f (P) β =(P(D 1 ) β, P(D 2 ) β,, P(D r ) β ), f (P) β =(P(D 1 ) β, P(D 2 ) β,, P(D r ) β ). The ower approximation granuar distribution function figures a of the certain knowedge representations, and the upper approximation granuar distribution function shows a of the possibe knowedge representations in the sense of mutigranuation. Moreover, optima granuarity seection can be acquired by considering these representations in terms of mutigranuation. Definition 4.1 Let I =(U, A {d}, V, f ) be a granuar inconsistent information system, and P ={P 1, P 2,, P }. P i A(i = 1, 2,, ), Q ={P i i }, and R ={P i i < }. (1) If f (P) β = f (Q) β, we say that Q is a ower distribution granuation seection of I. Moreover, if Q is a ower distribution granuation seection and no proper R of Q is a ower distribution granuation seection, then Q is caed a ower distribution optima granuation seection of I. (2) If f (P) β = f (Q) β, we say that Q is a upper distribution granuation seection of I. Moreover, if Q is a upper distribution granuation seection and no proper R of Q is a upper distribution granuation seection, then Q is caed a upper distribution optima granuation seection of I. Let I =(U, A {d}, V, f ) be a granuar inconsistent information system, and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). The ower and upper approximation granuar quaity functions of I are denoted as foows: r σ P β = k=1 P(D k ) β, U r λ P β = k=1 P(D k ) β. U The ower approximation granuar quaity function ays out the number of objects in a of the certain knowedge representations, and the upper approximation granuar quaity function deivers the number of objects in a of the possibe knowedge representations in the sense of mutigranuation. Definition 4.2 Let I =(U, A {d}, V, f ) be a granuar inconsistent information system, and P ={P 1, P 2,, P }. P i A(i = 1, 2,, ), Q ={P i i }, and R ={P i i < }. (1) If σ P β = σq, then we say that Q is a ower quaity granuation seection of I. Moreover, if Q is a ower quaity β granuar and no proper R of Q is a ower quaity granuation seection, then Q is referred to as a ower quaity optima granuation seection of I. (2) If λ P β = λq, we say that Q is an upper quaity granuation seection of I. Moreover, if Q is an upper quaity β granuation seection and no proper R of Q is an upper quaity granuation seection, then Q is referred to as an upper quaity optima granuation seection of I. The optima granuation seections defined in Definitions 4.1 and 4.2 are entirey the same when the considered target information system is granuar consistent. Additionay, the method of computing the significance of every granuation and every condition attribute in significant granuations is the same as considering σ P β = σq β. Thus, this method can aso be used in granuar

10 Granu. Comput. inconsistent information system, and the resuts obtained are ower approximation quaity granuation seection. Proposition 4.1 LetI =(U, A {d}, V, f ) be a granuar inconsistent information system, and P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). (1) The ower approximation distribution optima granuation seection is equivaent to the ower quaity optima granuation seection in generaized mutigranuation rough set. (2) The upper approximation distribution optima granuation seection is equivaent to the upper quaity optima granuation seection in generaized mutigranuation rough set. Proof Assume that Q P. (1) f (Q) β = f (P) β σ Q β = σp is obvious. β For any D i U d, we can have that Q(D i ) β P(D i ) β. Then we can get σ Q β σp β. If σq β = σp β, for any D i U d, we can have Q(D i ) β = P(D i ) β. Otherwise, if there exists D i0 U d such that Q(D i0 ) β P(D i0 ) β, then σ Q β <σp β. Therefore, σ Q β = σp β f (Q) β = f (P) β. (2) f (Q) β = f (P) β λ Q β = λp is obvious. β For any D i U d, we can have that P(D i ) β Q(D i ) β. Then we can get λ Q β λp β. If there exists D i 0 U d such that P(D i0 ) β P(D i0 ) β, λ P β <λq β. Therefore, λ Q β = λp β f (Q) β = f (P) β. When β = 1, generaized mutigranuation rough set can be degenerated into pessimistic mutigranuation rough set. That is to say, pessimistic mutigranuation rough set is a specia case of generaized mutigranuation rough set whie β = 1. Thus, the above proposition hods for pessimistic mutigranuation rough set. From the proof of the proposition, we can easiy obtain that this proposition aso hods for optimistic mutigranuation rough set. However, it woud not hod for other rough set modes such as variabe precision being considered in the sense of mutigranuation. The ower and upper quaity consistent reductions provide an easy and quick way to check reductions in computing by programs on computers. From the above, we can know that the higher the information eve β is, the stricter our requirements are. The reason is that when β = 1, generaized mutigranuation rough set degenerated into pessimistic mutigranuation rough set. Optimistic mutigranuation rough set needs ony at east one granue that supports the concept, whie pessimistic mutigranuation rough set needs a granue to support the concept. Information eve β must be subordinated to the majority principe, so β need to satisfy β [0.5, 1]. The bigger the β the finer the information granuar, the resuts wi be more accurate. The smaer the β the thicker the information granuar, the resuts wi be more macroscopic. Therefore, β can be artificiay adjusted according to our needs. For a granuar consistent information system, from the perspective of dependence, optima granuarity seection eve: to find the smaest subset Q of P and Q meets γ(q, d) β = γ(p, d) β. Thus, Q is the optima granuarity of the information system. The dependent degree of P with regard to d can be observed in an overa and systematic way. Furthermore, it aso improves the computationa efficiency. For a granuar inconsistent information system, from the perspective of the ower distribution functions, the criterion for optima granuarity seection is to find the smaest subset Q of P and make Q satisfy f (P) β = f (Q) β. So Q becomes the optima granuarity of the information system. Each decision cass ower approximate distribution can be observed under the attribute set P and information eve β. The upper approximation distribution is simiary anayzed. From the perspective of ower approximation granuar quaity functions, the eve of optima granuarity seection is to find the smaest subset Q of P and such that σ P β = σq β. Subsequenty, Q is the optima granuarity of the information system. The proportion of ower approximation of decision casses in the tota objects can be considered from the macroscopic point of view. This approach can effectivey promote the performance of computing works. For the upper approximation granuar quaity function, the situation is anaogous to the ower approximation. 5 Agorithm According to the above theory, one can seect the optima granuation in generaized mutigranuation rough sets. In this section, we present the agorithm of optima granuation seection. And a rea-ife case study is given to show effectiveness of the proposed approach. We outine the optima granuation seection process of the ower and upper distribution in Agorithm 1. And the optima granuation seection process of ower and upper quaity are shown in Agorithm 2.

11 Granu. Comput.

12 Granu. Comput. In Agorithm 1, it shoud be noted that Q 1 and Q 2 take a of the subsets of P. Time compexity anaysis of Agorithm 1 Let I =(U, A {d}, V, f )) be a target information system. The number of objects and attributes are denoted by N and A. The number of objects and attributes in the i-th granue P i are denoted by N i and K i (i {1, 2,, }, ΣN i = N and ΣK i = A ), respectivey. U d ={D 1, D 2,, D r } is the decision casses. We take a variabe t i to stand for the time compexity in an impementation. In the next, we can anayze the time compexity of Agorithm 1 step by step. The time compexity to do initiaized setting and input the information tabe, the information eve, and the granue set is 0, then the anaysis to do the initia settings is finished. The time compexity to cacuate U/d is denoted by t 1 =(N 1)+(N 2)+ + 1 = N (N 1) 2, the time compexity to cacuate U P i (i = 1, 2,, ) is t 2 = N A 2. So the time is t 3 = t 1 + t 2 = N (N 1) 2 + N A 2. The next to judge whether the information system is granuar consistent is t 4 = N. The time to obtain ower and upper approximations in generaized mutigranuation rough sets is t 5 = 2 N r + 2 N. The first four steps cacuate the ower and upper approximation granuar distribution functions of I, ower and upper distribution optima granuation seection. The time compexity of methods are denoted by t 6 = 2 1 (N 2 + N A N r + 2 N) 2 = 2 1 (N A + 4 N r + 5N 2). From the above anaysis, we can know that the maximum time compexity of the main part in the Agorithm 1 is t 1 main = t 3 + t 4 + t 5 + t 6 = N 2 2 +( ) N r +( ) N +( ) N A + N As r (1 r N) is the number of decision casses, and (1 A ) is the number of granuar structure, the maximum compexity of the main agorithm is approximatey O((2 A ) N 2 A ). The expanation of the Fig. 2: In Agorithm 1, the ower distribution optima granuarity seection and upper distribution optima granuarity seection are accordance with this fow graph. Input a target information system (an information tabe I =(U, A {d}, V, f )), an information eve β, and the granue set P ={P 1, P 2,, P }, P i A(i = 1, 2,, ). We first cacuate U / d and U P i to make the judgement of whether this information system is granuar consistent or not. If it is granuar inconsistent, we cannot do the ower and upper distribution optima Fig. 2 The program fow graph of ower and upper distribution optima granuation seection granuarity seection. If it is granuar inconsistent, we then compute the ower and upper approximations of each decision cass and its compement in generaized mutigranuation rough set. After that, we get the ower and upper approximation granuar distribution functions of I which are f (P) β and f (P) β, respectivey. According to the optima granuarity seection of ower distribution and upper distribution in Definition 4.1, we can obtain the ower and upper distribution granuarity seections, respectivey. Among the obtained ower and upper distribution granuarity seections, we can further get the ower distribution optima granuarity seection and upper distribution optima granuarity seection. In Agorithm 2, it shoud be noted that Q 1 and Q 2 take a of the subsets of P. Time compexity anaysis of Agorithm 2 Let I =(U, A {d}, V, f )) be a target information system. The number of objects and attributes are denoted by N and A. The number of objects and attributes in the i-th granue P i are denoted by N i and K i (i {1, 2,, }, ΣN i = N and ΣK i = A ), respectivey. U d ={D 1, D 2,, D r } is the decision casses. We take a variabe t i to stand for the time compexity in an impementation. Next, we can anayze the time compexity of Agorithm 2. The time compexity to do initiaized setting and input the information tabe, the information eve, and the granue set is 0, then the anaysis to do initia settings is finished. The time compexity to cacuate U / d is denoted by t 1 =(N 1)+(N 2)+ + 1 = N (N 1) 2, the time compexity to cacuate U P i (i = 1, 2,, ) is t 2 = N A 2. So the time to finish them is

13 Granu. Comput. t 3 = t 1 + t 2 = N (N 1) 2 + N A 2. The time to obtain ower and upper approximations in generaized mutigranuation rough sets is t 4 = 2 N r + 2 N. The first three steps cacuate the ower and upper approximation granuar distribution functions of I, ower and upper distribution optima granuarity seection. To judge whether the information system is granuar consistent is t 5 = N. The time compexity of methods are denoted by t 6 = 2 1 (N A N r + 5N 2) 3 2. From the above anaysis, we can know that the maximum time compexity of the main part in Agorithm 2 is t 2 main = t 3 + t 4 + t 5 + t 6 =( ) N r +( ) N +( ) N A N (N 1)+N The maximum compexity of the main agorithm is approximatey O((3 2 A 1 + 2) N 2 A ). The expanation of Fig. 3: In Agorithm 2, the ower quaity optima granuarity seection and upper quaity optima granuarity seection are in accordance with this fow graph. Unike Agorithm 1, we can get the optima granuarity seection when it is granuar consistent, so is it inconsistent, we can get the ower quaity optima granuarity seection. After cacuating U / d and U P i (i = 1, 2,, ), we obtain the characteristic functions of D k and D k. Then we further get the ower and upper approximation granuar quaity functions, respectivey. After getting the ower approximation granuar quaity function, we need to make the judgement of whether this information system is granuar consistent. If it is granuar consistent, we can do the optima granuarity seection. If it is granuar inconsistent, we then compute the ower and upper approximations of each decision cass and its compement in generaized mutigranuation rough set. After that, we get the ower and upper approximation granuar distribution functions of I which are σ P β = r k=1 P(D k) β and U λ P β = r k=1 P(D k) β, respectivey. According to Definition 4.2, U we can obtain the ower and upper quaity granuarity seections, respectivey. Among the obtained ower and upper quaity granuarity seections, we can further get the ower quaity optima granuarity seection and upper quaity optima granuarity seection. Mutigranuation rough set can be very usefu in many case, especiay in handing probem in information system. 6 Case study Suppose that Tabe 1 is an information system I =(U, C {d}, V, f ) which concerns the achievements of some students and U ={x 1, x 2,, x 20 } is a universe incuding twenty students in a schoo, a 1 (Chinese), a 2 (Mathematics), a 3 (Engish), a 4 (History), a 5 (Geography), a 6 (Poitics), a 7 (Physics), a 8 (Chemistry), a 9 (Bioogy) are the conditiona attributes of the system, and d(decision) is the decision attribute given by the experts according to the achievements of these students. We use 1 to express that the student is exceent and 0 to express that the student is not exceent. However, in the coege entrance examination, junior coeges ony seect exceent students from the view of three major subjects (Chinese, Mathematics, Engish). Universities of undergraduate eve seect exceent students from the view of arts (History, Geography, Poitics) and from the view of science (Physics, Chemistry, Bioogy). From the point of view of arts, seecting the attribute set {a 4, a 5, a 6 } is better than the attribute set {a 7, a 8, a 9 }. And from the point of view of science, seecting the attribute set {a 7, a 8, a 9 } is better than the attribute set {a 4, a 5, a 6 }. So we can get the foowing three granuations. P 1 ={a 1, a 2, a 3 }, P 2 ={a 4, a 5, a 6 }, P 3 ={a 7, a 8, a 9 }. Fig. 3 The program fow graph of ower and upper quaity optima granuarity seection If we consider ony one of these conditions, we can obtain that

14 Granu. Comput. Tabe 1 A target information system U a 1 a 2 a 3 a 4 a 5 a 6 a 7 a 8 a 9 d x x x x x x x x x x x x x x x x x x x x U P 1 = {{x 1, x 6, x 15, x 20 }, {x 2, x 12 }, {x 3, x 5 }, {x 8, x 9, x 14, x 19 }, {x 4, x 18 }, {x 7, x 10, x 16 }, {x 11, x 13, x 17 }}; U P 2 = {{x 1, x 8, x 16, x 19 }, {x 2, x 7, x 11 }, {x 3, x 4 }, {x 5, x 14 }, {x 6, x 10 }, {x 12, x 13, x 17, x 18 }, {x 9, x 15, x 20 }}; U P 3 = {{x 1 }, {x 3, x 5, x 6 }, {x 10, x 14 }, {x 2, x 12, x 20 }, {x 4, x 9, x 15, x 18 }, {x 7, x 8, x 19 }, {x 11, x 13 }, {x 16, x 17 }}; U d ={D 1, D 2 } = {{x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }}. First, according to existence [x] P [x] d,it is easy to check that Tabe 1 is a granuar inconsistent information system. Next, on the basis of generaized muti granuarity, how to choose the best partice size of the two agorithms is expained. The nonempty subsets of P are denoted by P ={P 1, P 2, P 3 }, Q 1 ={P 1, P 2 }, Q 2 ={P 1, P 3 }, Q 3 ={P 2, P 3 }, Q 4 ={P 1 }, Q 5 ={P 2 }, Q 6 ={P 3 }. We set the information eve β = 2 3. The cacuation steps of the agorithm 1 are as foows: Step 1. From Definition 3.1, we can cacuate the generaized ower and upper approximations of the decision cass D 1 as foows: P(D 1 ) 2 3 ={x 1, x 8, x 14, x 19 }; P(D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. The generaized ower and upper approximations of the decision cass D 2 are P(D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; P(D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 9, x 10, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 20 }. Step 2. According to the definition of distribution function, we can obtain the ower and upper approximation granuar distribution functions as foows. f (P ) 2 3 =(P (D 1 ) 2 3, P (D 2 ) 2 3 ) = ({x 1, x 8, x 14, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f ( P) 2 3 =( P(D 1 ) 2 3, P(D 2 ) 2 3 ) = ({x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 9, x 10, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 20 }). Step 3. For Q 1, Q 2, Q 3, Q 4, Q 5, Q 6, we can get the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 :

15 Granu. Comput. For Q 1 ={P 1, P 2 }, we can get the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 1 (D 1 ) 2 3 ={x 8, x 19 }; Q 1 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 1 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; Q 1 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 10, x 11, x 12, x 13, x 15, x 17, x 18, x 20 }. For Q 2 ={P 1, P 3 }, we can cacuate the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 2 (D 1 ) 2 3 ={x 14 }; Q 2 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 2 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; Q 2 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 20 }. For Q 3 ={P 2, P 3 }, we can compute the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 3 (D 1 ) 2 3 ={x 1 }; Q 3 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 3 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; Q 3 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 9, x 11, x 12, x 13, x 15, x 17, x 18, x 20 }. For Q 4 ={P 1 }, we can get the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 4 (D 1 ) 2 3 ={x 8, x 9, x 14, x 19 }; Q 4 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 4 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; Q 4 (D 2 ) 2 3 ={x 1, x 2, x 3, x 4, x 5, x 6, x 7, x 10, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 20 }. For Q 5 ={P 2 }, we can receive the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 5 (D 1 ) 2 3 ={x 1, x 8, x 16, x 19 }; Q 5 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 5 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, For Q 6 ={P 3 }, we can obtain the generaized mutigranuation ower and upper approximations of the decision casses D 1 and D 2 : Q 6 (D 1 ) 2 3 ={x 1, x 10, x 14 }; Q 6 (D 1 ) 2 3 ={x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }. Q 6 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }; Q 6 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 8, x 9, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 19, x 20 }. Step 4. For Q 1, Q 2, Q 3, Q 4, Q 5, Q 6, the ower approximation granuar distribution functions are computed as foows: f (Q 1 ) 2 3 =(Q 1 (D 1 ) 2 3, Q 1 (D 2 ) 2 3 ) = ({x 8, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f (Q 2 ) 2 3 =(Q 2 (D 1 ) 2 3, Q 2 (D 2 ) 2 3 ) = ({x 14 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f (Q 3 ) 2 3 =(Q 3 (D 1 ) 2 3, Q 3 (D 2 ) 2 3 ) = ({x 1 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f (Q 4 ) 2 3 =(Q 4 (D 1 ) 2 3, Q 4 (D 2 ) 2 3 ) = ({x 8, x 9, x 14, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f (Q 5 ) 2 3 =(Q 5 (D 1 ) 2 3, Q 5 (D 2 ) 2 3 ) = ({x 1, x 8, x 16, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). f (Q 6 ) 2 3 =(Q 6 (D 1 ) 2 3, Q 6 (D 2 ) 2 3 ) = ({x 1, x 10, x 14 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 17, x 18, x 20 }). Step 5. For Q 1, Q 2, Q 3, Q 4, Q 5, Q 6, the upper approximation granuar distribution functions are computed as foows: f (Q 1 ) 2 3 =(Q 1 (D 1 ) 2 3, Q 1 (D 2 ) 2 3 ) = ({x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 10, x 11, x 12, x 13, x 15, x 17, x 18, x 20 }). f (Q 2 ) 2 3 =(Q 2 (D 1 ) 2 3, Q 2 (D 2 ) 2 3 ) = ({x 1, x 8, x 9, x 10, x 14, x 15, x 16, x 19 }, {x 2, x 3, x 4, x 5, x 6, x 7, x 11, x 12, x 13, x 15, x 16, x 17, x 18, x 20 }). x 18, x 20 }; Q 5 (D 2 ) 2 3 ={x 2, x 3, x 4, x 5, x 6, x 7, x 9, x 10, x 11, x 12, x 13, x 14, x 15, x 17, x 18, x 20 }.