Double ‐ Quantitative Generalized Multi ‐ Granulation Set ‐ Pair Dominance Rough Sets in Incomplete Ordered Information System

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Introduction
The acceleration of the information era makes it possible to acquire and process diversified feature data. However, in reality, due to the limitations of data acquisition technology and errors in data measurement, the acquired information contains incomplete and inaccurate data in most cases, thus increasing the requirements for data analysis tools. As we know, the uncertain knowledge hidden in these systems is of great significance in decision-making. In reality, there are many problems that require the consideration both relative and absolute information. For example, in the comprehensive evaluation of graduate students, a school assesses students' scientific research ability according to the quality and quantity of articles published. Inspired by the above, we consider decision-making in IOIS with double-quantitative RS theory.
Rough sets (RS) theory is an effective mathematical tool proposed by Pawlak in 1982 to address uncertainties and imprecisions [1]. It has been successfully applied in feature selection [2], safety monitoring data classification [3], decision making [4][5][6], information fusion [7], uncertainty analysis [8,9], medical diagnosis [10], and other fields [11][12][13][14]. Classical RS require strict inclusion relations between their equivalent classes and sets, and because there is no fault-tolerant mechanism, they are limited. Accordingly, scholars have proposed a series of extended RS models . Yao et al. [15] presented a decision theory rough sets (DTRS) model by introducing Bayes risk decision theory into RS. In 1993, Ziarko [16] introduced parameter β into the RS model and proposed a variable precision rough sets (VPRS) model that reflects the relative quantitative information of approximation space and has a certain fault-tolerant ability. Considering the importance of overlapping information of equivalence classes and sets, Yao et al. [17] put forward graded rough sets (GRS) in 1996. GRS can allow certain errors and can reflect the absolute quantization information of approximation space. From the perspective of information quantization, it is of great significance to combine absolute quantization information with relative quantization information [18][19][20][21][22][23][24][25][26][27][28][29]. Zhang et al. [18] performed a comparative study of VPRS and GRS and put forward two double-quantitative RS models, which enriched rough sets theory. Li et al. [19] studied two double-quantitative DTRS models and verified the validities of the models through a medical diagnosis case. Moreover, Fan et al. [20] introduced fuzzy sets into the double-quantitative problem, constructed upper and lower approximation sets based on logical operators, which can solve the pattern recognition problem in big data. Since RS model based on equivalence relation is suitable only for discrete data, discrete preprocessing of data will cause some information to be lost and reduce classification accuracy. To address the abovementioned problem, literature [21] explored a distance-based double-quantitative rough fuzzy sets model that can settle the problem of information loss. Guo et al. [22,23] studied double-quantitative RS theory under fuzzy relation and presented a local logic disjunctive double-quantitative RS model based on local RS, which provided an effective approach for decision-making. In addition, dominance relation often plays a considerable role in practical applications, and it was necessary to establish the ordered information system (OIS) [24][25][26] through the dominance relation. Some scholars have investigated various double-quantitative RS models based on logical combinations of VPRS and GRS in OIS [27][28][29]. The above studies rarely deal with decision-making problem in multi-granulation environment. However, in many cases, multi-granulation is needed to describe the concept of the target.
From the granular computing perspective, Qian et al. [30,31] first proposed the multi-granulation rough sets model (MRS) and extended the optimistic multi-granulation rough sets and pessimistic multi-granulation rough sets, which attracted extensive attention from scholars [32,33]. Subsequently, Xu et al. [34][35][36] explored a generalized multi-granulation rough sets (GMRS) model. Then, they introduced generalized multi-granulation into the DTRS and GRS, provided two RS models, and compared them with GMRS before demonstrating the advantages of the model through examples. Literature [37] established three double-quantitative DTRS in multi-granulation approximate space, but there are still some problems, such as double-quantitative multi-granulation DTRS under dynamic granulation and the practical applications.
The abovementioned RS models are all under the complete information system (IS), but in reality, the IS encountered are often incomplete, such as data integration [38], data mining [39], fault diagnosis [40], uncertainty measurement [41], and many others [42][43][44]. Therefore, a large number of studies on Incomplete Information System (IIS) have emerged [45][46][47][48][49][50][51][52][53][54][55][56][57][58][59][60][61]. Kryszkiewicz [45] described an RS model by introducing the tolerance relation in IIS. Furthermore, Stefanowski et al. [48] defined the similarity relation in IIS. In the light of granular computing, Zhai et al. [49,50] introduced the tolerance relation into VPRS, combined with the multi-granulation, and gave the optimistic and pessimistic variable precision MRS model under the IIS. Then, they expanded the single granulation RS model based on the extended dominance relation and the limited dominance relation. Yao et al. made a series of studies [51][52][53] about MRS in IIS. Based on maximal consistent block, a new optimistic MRS and a new pessimistic MRS in IIS were established. Then, they investigated the granular space reductions in the variable precision MRS model. Lin and Xu [54] constructed optimistic MRS and pessimistic MRS in interval-valued information system. As the ranking of values plays a significant role in many practical applications, Scholars have extended the method to various types of IOIS [55][56][57][58][59]. Huang [58] introduced the idea of set-pair analysis and proposed a set-pair dominance degree RS model, which can overcome the shortcomings of the dominance relation in existing IOIS. Literature [60] mainly investigated attribute reduction methods in IOIS. However, the above researches have not applied the rough sets theory based on the double quantification approach to the decision-making in IOIS. To settle the problem, we extend the double quantification rough sets model. The paper studies decision-making in IIS, and provide better decision results for decision makers without preprocessing the data. The works of this paper provide a method for decision-making research of IIS and expand its application scope.
The main contributions of this article are shown below.
(1) To better extract effective information in IOIS, we combine the set-pair dominance relation and generalized multi-granulation. In consideration that VPRS can reflect the relative quantization information of approximation space, we propose generalized multi-granulation set-pair dominance variable precision rough sets (GM-SPD-VPRS), and the related properties are discussed.
(2) Based on the proposed GM-SPD-VPRS, considering the GRS reflect absolute quantization information of approximation space, we defined generalized multi-granulation set-pair dominance graded rough sets (GM-SPD-GRS). In addition, we give the corresponding properties.
(3) To better reflect the relative and absolute quantization information in IOIS, the lower and upper approximation sets of GM-SPD-VPRS and GM-SPD-GRS are fused and combined with logical operators, and five double-quantitative generalized multi-granulation set-pair dominance relation rough sets (DQGM-SPD-RS I to DQGM-SPD-RS V ) are obtained.
(4) Taking DQGM-SPD-RS I model as an example, the Algorithm 1 is given. In addition, the validity of the novel models is proved by a medical diagnosis case. It can also guide people to select a suitable model.
The rest of this paper is organized below. In Section 2, some basic concepts related to RS, IOIS, and set-pair dominance RS are recalled. In Section 3, GM-SPD-VPRS and GM-SPD-GRS are proposed in IOIS, and the related properties are given and proven. In Section 4, DQGM-SPD-RS I , DQGM-SPD-RS II , DQGM-SPD-RS III , DQGM-SPD-RS IV , and DQGM-SPD-RS V are proposed to reflect the relative and absolute quantization information better in IOIS. Moreover, taking into account extreme circumstances, DQGM-SPD-RS I and DQGM-SPD-RS II are discussed with optimistic and pessimistic extremes, respectively. Subsequently, an algorithm to calculate rough regions of DQGM-SPD-RS I is constructed. In Section 5, the validity of the novel models is illustrated through a medical example. Finally, this paper concludes a brief summary and looks forward to the future research in Section 6.

Preliminaries
Definition 1 (VPRS). [ K K R X R X is called the lower and upper approximation sets of GRS, where ( ) K R X is a set of equivalent classes with at most K elements that do not belong to X, and ( ) K R X is a set with more than K equivalent classes belonging to X, which can reflect the absolute quantitative information in approximate space.
In particular, when 0 K  , the GRS model degenerates into the RS model, that is, Moreover, it should be noted that Similarly, the positive region, the upper boundary region, the lower boundary region, and the negative region of X with respect to graded K can be defined: Approximation accuracy describes the ratio of correct decisions in possible decisions when using knowledge R to classify the objects.

Definition 4 (OIS). [24,25] Let
 IS ( , , , ) U AT V f be a complete information system, AT is non-empty finite attribute set, and V is a set of all attribute values, where a V is the value range of attribute a and has a partial order relation and : the set-pair dominance degree of x and y is denoted as: M x y n is the strong dominance degree of x and y,  2 | ( , )|/ S N x y n is the weak dominance degree of x and y, and  3 | ( , )|/ S K x y n is the disadvantage degree of x and y. In addition: . In order to calculate the possible advantage of the objects in the weak dominance, the concept of the joint dominance rate is proposed.
Additionally, it can be seen that the joint dominance ratio of the attribute values of objects is compared by means of the average value rather than unknown value  . Therefore, the subjectivity is not too strong.
Definition 7 (Set-pair dominance relation). [58] Assume that  IOIS ( , , , ) U AT V f ; the set-pair dominance relation can be defined as: Proof. This can be proved easily from Definition 8. □ otherwise, X is rough if It should be pointed out that with the increase of  , the lower approximation set becomes larger, and the upper approximation becomes smaller. In addition, when  =0, GM-SPD-VPRS model degenerates into GM-SPD-RS. Namely, the GM-SPD-VPRS model is an extension of GMRS model, set-pair dominance relation RS model and VPRS model, and it correspondingly meets some basic properties. (11) The proofs are similar to (8) and (9). □ Based on Definition 9, the rough regions of X with regard to at information level  are as follows: It should be pointed out that the K in the GM-SPD-GRS model is different from the k in the classical GRS. k in the classical GRS is used to measure the relation between the equivalence classes and the target set, while K in the GM-SPD-GRS model is used to measure the relationship between the dominance class and the target set.
According to the Definition 10, we can find that with the increase of k, the lower approximation set becomes larger and the upper approximation becomes smaller. In addition, when k = 0, GM-SPD-GRS model degenerates into GM-SPD-RS. Namely, the GM-SPD-GRS is an extension of GMRS model, set-pair dominance relation RS model and GRS model, and it correspondingly meets some basic properties. The details are expressed as follows. (1) Proof.

(5) For any
It is similar to (5).
(7) According to supporting characteristic function x . Thus, for any

DQGM-SPD-RS Models
The GM-SPD-VPRS and GM-SPD-GRS models provide two new methods for decision-making under IOIS. In addition, these two models expand the RS model in the light of relative quantization and absolute quantization, respectively, which have their own quantization advantages. Aiming at the decision-making in IOIS, we introduce the double-quantitative idea into GM-SPD-VPRS model and GM-SPD-GRS model. In [23], Li et al. fused absolute and relative quantitative approximation operators, and two novel scenarios were generated. On the other hand, [27,[33][34][35][36] introduced logical operators into different RS models, enriching RS theory. Therefore, we can get six DQGM-SPD-RS models by fusing and combining them with logical disjunction and logical conjunction operators: (1) Lower approximation sets quantify relative quantitative information, and upper approximation sets quantify absolute quantitative information.
(2) Lower approximation sets quantify absolute quantitative information, and upper approximation sets quantify relative quantitative information.
(3) Lower and upper approximation sets quantify information by employing a logical disjunction operator.
(4) Lower approximation sets and upper approximation sets quantify information by employing a logical conjunction operator.
(5) Lower approximation sets quantify information by employing a logical disjunction operator, and upper approximation sets quantify information by employing a logical conjunction operator.
(6) Lower approximation sets quantify information by employing a logical conjunction operator, and upper approximation sets quantify information by employing a logical disjunction operator.
As observed above, the RS from the sixth model make the lower approximation smaller and the upper approximation larger. From the perspective of view of RS, knowledge becomes rougher. Therefore, this paper discusses only the first five models. (

AT VT f be an IOIS; considering the extremes of optimism and pessimism, double-quantitative optimistic multi-granulation set-pair dominance relation rough sets (DQOM-SPD-RS I ) and double-quantitative pessimistic multi-granulation set-pair dominance rough sets (DQPM-SPD-RS
Proof. This can be demonstrated according to Definition 11. □ According to the DQGM-SPD-RS I , DQOM-SPD-RS I , and DQPM-SPD-RS I obtained above, the following properties hold. (2) Proof. This can be demonstrated based on Definition 11 and Proposition 2. □ According to Definition 11, the rough regions can be obtained: (1 ( )) Proof. This can be demonstrated directly according to Definition 12. □ According to the DQGM-SPD-RS II , DQOM-SPD-RS II , and DQPM-SPD-RS II obtained above, it has the following properties.
Proof. This can be demonstrated based on Definition 12 and Proposition 3. □ According to Definition 12, the rough regions can be achieved below: ( ) ( ( ) ( )); Based on Definition 13, Proposition 4 can be obtained. Proposition 4. The DQGM-SPD-RS III obtained by the logical disjunction operator can also be expressed below: According to Definition 13, the rough regions can be obtained: (1 ( ))) (1 ) Negative region decision rule (N III ): Upper boundary region decision rule (UB III ): Lower boundary region decision rule (LB III ): , ) U AT VT f , the approximation sets of the DQGM-SPD-RS IV can be defined as follows: (1 ( )) (1 ( )) Based on Definition 14, Proposition 5 can be obtained. Proposition 5. The DQGM-SPD-RS IV obtained by the logical conjunction operator can also be expressed below: }; Proof. It's easy to prove. Based on Definition 14, the rough regions can be obtained: (1 ( ))) (1 ) Negative region decision rule (N IV ): Based on Definition 15, Proposition 6 can be obtained.

Proposition 6.
The DQGM-SPD-RS V achieved by the logical disjunction operator and logical conjunction operator can also be expressed below:

Rough Regions under the DQGM-SPD-RS I Model
According to the decision rules given in the five models above, Algorithm 1 is given, taking the DQGM-SPD-RS I model as an example in this section. Through Algorithm 1, the rough regions under the DQGM-SPD-RSI model can be calculated.  It can be seen that the difference between the above algorithm and the other four models' algorithms is mainly in steps 14 to 17, which is the essence of each model. Therefore, the algorithm steps of DQGM-SPD-RS I are given in this paper, and the algorithms of the other four models are similar.

Example Analysis
In IOIS, five DQGM-SPD-RS models have been defined in the previous section, and which provide a way for decision-making in IOIS. Inspired by the literature [37], we think that medical example is close to real life. It is interesting to demonstrate our models with medical example. It should be noted that the data in this paper are derived from the literature [37], we select the first 20 objects and randomly set up an IIS.  Table 1 gives the specific conditions of 20 patients in detail.   D stand for no cold and cold, respectively. We denote 2 D as the object set, namely,  2 X D . In addition, we assume that   1 .
According to Algorithm 1, Tables 2-4 show the classification and statistical results of dominance classes under different granular sets, respectively.