Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: An application to biomedical problems

Information Sciences 517 (2020) 315–339

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Information Sciences journal homepage: www.elsevier.com/locate/ins

Intuitionistic fuzzy TOPSIS method based on CVPIFRS models: An application to biomedical problems Li Zhang a, Jianming Zhan a,∗, Yiyu Yao b a b

Department of Mathematics, Hubei Minzu University, Enshi, 445000, China Department of Computer Science, University of Regina, Regina, Saskatchewan S4S 0A2, Canada

a r t i c l e

i n f o

Article history: Received 21 January 2019 Revised 5 November 2019 Accepted 3 January 2020 Available online 7 January 2020 Keywords: Multi-attribute decision-making TOPSIS method CVPIFRS model Variable precision IF logical operator

a b s t r a c t In order to obtain the weights of a set of criteria by means of real-world data, an effective method based on the covering-based variable precision intuitionistic fuzzy rough set (CVPIFRS) models is presented. By combining the CVPIFRS models with the idea of TOPSIS, we propose a decision-making method to effectively settle the complex and changeable bone transplant selections, which is one of typical multi-attribute decision-making (MADM) problems. The sensitivity analysis of the proposed method shows that the approach is highly flexible and can be applied to a wide range of environments by adjusting the values of the intuitionistic fuzzy (IF) variable precision, together with the choice of different IF logical operators. Through a comparison of the proposed method and some existing MADM methods, it is shown that our method is more effective in dealing with these complex and changeable bone transplant selections issues. © 2020 Elsevier Inc. All rights reserved.

1. Introduction With continuous development of computer and information technology, industrial enterprises can easily obtain a large amount of data and information [25]. Data-driven methods and techniques are significant tools and have been widely used in decision-making [20]. They play a key role in addressing uncertainty and incompleteness. Multiple attributes problems (MAPs) are a typical class of problems of data mining and knowledge management. A number of multi-attribute decisionmaking (MADM) related methods have been proposed [3,31]. Health care is an area where the importance of MAPs cannot be underestimated. For example, bone transplant materia selections are a difficult problem in medical and health care. When obtained information is vague and inaccurate, selecting works are somewhat difficult to be implemented. In this paper, by taking the advantages of rough set theory (RST) [29] to handle fuzzy and inaccurate data, we propose an IF TOPSIS decision-making method based on covering-based variable precision intuitionistic fuzzy rough sets (CVPIFRS) for solving the complicated bone transplant selecting problems.

∗

Corresponding author. E-mail addresses: [email protected] (L. Zhang), [email protected] (J. Zhan), [email protected] (Y. Yao).

https://doi.org/10.1016/j.ins.2020.01.003 0020-0255/© 2020 Elsevier Inc. All rights reserved.

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Fig. 1. A brief overview of the development of RS models.

1.1. A brief introduction to rough set models Research in RST has made great progress in both theory and applications. The advantages of RST-based data processing have been received extensive attentions. Many researchers have extended the theory in diverse ways and applied the extended models to deal with plenty of complicated MADM problems. Yao [36,37] introduced extended models of rough sets by using neighborhoods of objects defined by binary relations. When a binary relation is not an equivalence relation, the family of neighborhoods of objects is not a partition but a covering of a universal set of objects. Zhu [48], Zhu and Wang [49] and Wang et al. [33] studied several types of rough set approximations based on coverings, as well as attribute reductions in coverings based rough set (CRS) models. By combining RST with the fuzzy set theory (FST), Dubois and Prade [10] introduced fuzzy rough set (FRS) and rough fuzzy set (RFS) models. Based on fuzzy coverings, i.e., a natural extension of the covering, many fusion models between the CRS models and the FRS models have been investigated [7,35,39,43]. Zhang et al. [42] proposed a flexible and efficient IF TOPSIS method based on coverings to aid decision-makers to resolve MAPs. Since many existing rough set models are sensitive to noisy data, variable precision rough set (VPRS) models have been proposed and studied [26,50], which are suitable for classification problems involving vague or uncertain information. Many researchers generalized this model and made many fruitful results. Based on the study of FRS, Fernández Salido and Murakami [12] developed related methods by using FRSs techniques to deal with uncertain data due to measurement errors. Covering-based variable precision fuzzy rough set (CVPFRS) models have been introduced [40]. Hu et al. [18] investigated the relationship between the VPRS model and the robust model. The notions of IF rough sets and VPRSs are highly useful in the research of intelligent systems. Gong and Zhang [15] presented an extension of the RST through the VPRS theory with the intuitionistic fuzzy rough set (IFRS) theory. Zhang [44] introduced the concept of an IF covering as a natural extension of the fuzzy covering and investigated the properties of covering-based IF rough sets (CIFRSs). In this paper, based on the work of Zhang et al. [42], we generalize the RST by combining IF theory, VPRS theory and IF covering theory. The main contribution is to propose a flexible, efficient, and widely applicable MADM method based on improved RST models. Fig. 1 provides a brief overview of the development of rough set (RS) models. Fusion models of rough sets and other classic MADM methods are a novel attempt. 1.2. A brief introduction to MADM methods At present, there are many methods for determining attribute weighting. According to different sources of raw data when calculating weighting, these methods can be divided into two categories: subjective weighting method and objective weighting method. Among them, the subjective weighting method adopts a qualitative method to obtain weighting values by experts based on subjective judgments. Commonly used subjective weighting methods include analytic hierarchy process, comprehensive scoring method, fuzzy evaluation method, exponential weighting method and power coefficient method. Subjective weighting method is an early and mature method of research. The advantage of the subjective weighting method is that the expert can reasonably determine the ordering of each attribute according to the actual decision-making problem and the expert’s own knowledge and experience, so as not to appear that the attribute weighting is contrary to the actual importance of the attribute. However, the decision-making or evaluation results have strong subjective randomness, poor objectivity, and increase the burden on decision-makers. The application has great limitations. In view of the various deficiencies of the subjective weighting method, people have proposed the objective weighting method. The raw data of the objective weighting method is formed by the actual data of each attribute in the decisionmaking scheme. The original information of the objective weighting method comes directly from the objective environment. The process of the objective weighting method to process information is to deeply explore the interrelationship and influence of each attribute, and then determine the attribute weighting according to the degree of contact of each attribute or the amount of information provided by each attribute. Principal component analysis, entropy method, dispersion and mean

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317

square error method and multi-objective programming method are commonly used objective weighting methods. The objective weighting method mainly determines the weighting based on the relationship between the original data. Therefore, the objectivity of the weighting is strong, and the burden of the decision maker is not increased. The method has a strong mathematical theoretical basis. However, this method does not consider the subjective intentions of the decision makers, so the determined weighting may be inconsistent with people’s subjective wishes and cause confusion. In this method, decision makers are less likely to participate and do not consider the subjective intentions of decision makers. he characteristics of the two kinds of weighting methods are different. The subjective weighting evaluation method measures the relative importance of each index according to the experience of experts. It has certain subjective randomness and is greatly interfered by human factors. It is difficult to obtain accurate evaluation when there are many evaluation indicators. The objective weighting method comprehensively considers the mutual relationship between the indicators, and determines the weighting coefficient according to the initial information amount provided by each index, which can achieve the accuracy of the evaluation result, but when the index is large, the calculation amount is very large. With the advancement of science and technology, MADM has drawn attentions from many researchers. A variety of MADM [41] methods have emerged. Three families of tools have been introduced, i.e., the outranking methods, the interactive approaches, and the utility-based methods. With respect to the utility-based methods, decision-makers rank alternatives according to the total value which is composited by the evaluated values of every alternative under diverse attributes by means of the information fusion techniques. The AHP methods, the TOPSIS methods [16,17,23,30], and the weighted sum methods based on aggregation operators [21,34,45] are three typical and popular tools in the utility based approaches. Among the three classical methods, the TOPSIS method is an effective and beneficial approach to MAPs. Many researchers studied this useful tool [8,27] and extended it in several ways. By means of the TOPSIS method, a fuzzy MADM issue was reformed into a truthful one and the unblurred problem [32]. With the fuzzy integral, fuzzy MADM issues were changed into the non-fuzzy MADM issues [2]. However, when using the TOPSIS tool, a compromise solution cannot reach the goal that the solution should be the remotest one from the negative ideal solution. For this reason, Hadi-Vencheh and Mirjaberi [16] introduced a fuzzy inferior ratio method that considers the relative significance of the distances from the positive and negative ideal points, which has some advantages over TOPSIS. Under IF environment, a generalized TOPSIS method has achieved many highly effective results [30]. Zhang et al. [42] introduced a promoted IF TOPSIS tool by taking advantages of rough sets and TOPSIS method, which is effective to solve many complex MAPs and can be applied in many environments. In this paper, we aim to present an IF TOPSIS method based on CVPIFS models in which a objective weighting method is proposed. 1.3. The motivation and contribution of our research There are two basic motivations for the present study. The first motivation is to take advantages of RST for dealing with the uncertain information. However, classical RST is limited in applications due to its strict conditions. In order to increase the application range of RST, it is valuable to constantly improve and develop it. Through our research, we find that the introduction of IF theory makes many problems involving hesitation degree. Based on the four IF neighborhoods proposed by Zhang et al. [42], we enrich the concept of RST by combining IF theory with VPRS theory through different logical operators. The second motivation is an application to MADM. After investigating many existing MADM methods, we find that some of them have limitations in applications [21,34,45]. For example, aggregation operators [34], the generalized OWA operators [21], and other generalized aggregation operators for IF sets [45] may not valid in some complicated IF environments. We analyze these methods from theories and numerical examples to explain the reasons for their limitations. The weighting of attributes of many MADM methods are given by experts. The values of the weighting with respect to the attributes have impact on the final results. If some experts lack rational judgment or lack of experience, the weighting values of attributes will not truly reflect the importance of the attributes. In this situation, we cannot find an alternative that meets the actual needs. Aiming at searching for an objective and effective way to obtain the weighting of all attributes, we introduce a MADM method that takes advantages of the approximate accuracy of the proposed models. This method not only provides us with a tool for calculating weighting with respect to attributes, but also enables us to effectively solve many difficult problems. The contributions of this paper are summarized as follows. RST is an important tool for dealing with uncertain information. Due to its requirement of equivalence relations, crisp sets, and so on, it is greatly limited in some applications. Due to the large amount of real-valued data in real life, the classical RST cannot solve these problems. IF set theory is an effective tool for solving these problems. Therefore, this paper generalizes the equivalence relation of rough sets to IF neighborhood relations in an IF system. Although RST is an effective tool to solve problems under uncertainty, it is sensitive to noisy data. In order to solve these problems, this paper combines the idea of variable precision with the CIFRS models and proposes CVPIFRS models. Our proposed CVPIFRS models combine the advantages of IF theory with VPRS theory. The CVPIFRS models not only enrich the concept of RST, but also increase the scope of applications of RST. Based on the proposed CVPIFRS models, an objective method of calculating weights is proposed. The obtained weight is more objective. The final result is more suitable for the actual needs. By combining the CVPIFRS models with the idea of TOPSIS method, an MADM method is suggested in this paper. Experimental analysis shows that our approach is highly

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flexible and can be applied to a wide range of environments by means of different logical operators. In addition, through adjusting the value of ϑ, the proposed method can effectively solve many complicated problems that the existing methods [21,34,45] cannot deal with. 1.4. The structure of the present paper Section 2 introduces basic theories and related concepts. In Section 3, we analyze the inadequacies of some MADM methods from theories and numerical examples. We also construct four types of CVPIFSs by means of four types of IF neighborhoods. Some significant properties of our proposed models and the relationships among our proposed models and other types of RST models are investigated in Section 3. In Section 4, an effective method based on CVPIFSs is given. Through comparative analysis and sensitive analysis, the flexibility and effectiveness of our approach are demonstrated by a practical problem in selecting materials for bone transplantation selections. Finally, several conclusions are given in Section 5. 2. Basic terminologies This section recalls some important and basic theories about IF sets, IF logical operators and IF neighborhoods. It is worth noting that E is a universe, IF (E ) denotes the family of all IF sets. And we have (∼N H )(e ) = N (H (e )), where H ∈ IF (E ), e ∈ E. In this paper,  is a non-empty indicator set and L = {(β1 , β2 ) ∈ [0, 1] × [0, 1]|0 ≤ β1 + β2 ≤ 1} [5]. For each e ∈ E, H (e ) = (μH (e ) , νH (e ) ), where μH(e) and ν H(e) are the degrees of membership and non-membership of H(e), H ∈ IF (E ) (μH(e) , ν H(e) ∈ [0, 1]). Definition 2.1 [9]. A mapping N : L → L , which is decreasing and satisfying the condition that N (0L ) = 1L and N (1L ) = 0L , is named an IF negator. If Ns ((β1 , β2 )) = (β2 , β1 ), then Ns is a standard IF negator. N is an involutive IF negator when N (N (β )) = β where β ∈ L✠ . Definition 2.2 [9]. An IF triangular norm (IF t-norm) is a mapping T: L✠ × L✠ → L✠ which satisfies the following conditions: (1) T (1L , ε ) = ε , for each ε ∈ L✠ ; (2) T (ε , η ) = T (η, ε ), for each ε , η ∈ L✠ ; (3) If α1 ≤L α2 , β1 ≤L β2 , then T (α1 , β1 ) ≤L T (α2 , β2 ), for each α 1 , α 2 , β 1 , β 2 ∈ L✠ ; (4) T (α , T (β , γ )) = T (β , T (α , γ )), for all α , β , γ ∈ L✠ . Definition 2.3 [9]. An IF triangular conorm (IF t-conorm) which is a mapping ⊥: L✠ × L✠ → L✠ , satisfies the listed conditions below: (1) ⊥ (ε , 0L ) = ε , for each ε ∈ L✠ ; (2) ⊥ (ε , η ) =⊥ (η, ε ), for each ε , η ∈ L✠ ; (3) If α1 ≤L α2 , β1 ≤L β2 , then ⊥ (α1 , β1 ) ≤L ⊥ (α2 , β2 ), for each α 1 , α 2 , β 1 , β 2 ∈ L✠ ; (4) ⊥ (α , ⊥ (β , γ )) =⊥ (β , ⊥ (α , γ )), for all α , β , γ ∈ L✠ . Definition 2.4 [4]. If a mapping I : L × L → L meets the following requirements: (1) I (0L , 0L ) = 1L , I (0L , 1L ) = 1L , I (1L , 1L ) = 1L , I (1L , 0L ) = 0L ; (2) If α1 ≤L α2 , then I (α1 , β ) ≥L I (α2 , β ), for each α 1 , α 2 , β ∈ L✠ ; (3) If α1 ≤L α2 , then I (β , α1 ) ≤L I (β , α2 ), for each α 1 , α 2 , β ∈ L✠ ; then I is called an IF implicator. Based on an IF t-norm T, if an IF implicator IT satisfies IT (α , β ) = sup{η|η ∈ L , T (α , η ) ≤L β}, then IT is called an IF R-implicator. Definition 2.5 [9]. An IF coimplicator σ is a mapping: L✠ × L✠ → L✠ satisfying the listed following conditions: (1) σ (0L , 0L ) = 0L , σ (1L , 0L ) = 0L , σ (1L , 1L ) = 0L , σ (0L , 1L ) = 1L ; (2) If α1 ≤L α2 , then σ (α1 , β ) ≥L σ (α2 , β ), for each α 1 , α 2 , β ∈ L✠ ; (3) If α1 ≤L α2 , then σ (β , α1 ) ≤L σ (β , α2 ), for each α 1 , α 2 , β ∈ L✠ . Based on results from our previous studies [42], we summarize properties of IF neighborhoods. Definition 2.6 [42]. Assume that (E, C ) is a finite intuitionistic fuzzy covering approximation space (IFCAS). Let I1 and T1 be an IF R-implicator and an IF t-norm, respectively. For each e, f ∈ E, four types of IF neighborhoods NγC (γ = 1, 2, 3, 4 ) of e are defined as follows:

N1C (e )( f ) =



I1 (K (e ), K ( f ));

K∈C

N2C (e )( f ) =



T1 (K (e ), K ( f ));

K∈md (C,e )

N3C (e )( f ) =



I1 (K (e ), K ( f ));

K∈MD(C,e )

N4C (e )( f ) =



K∈C

T1 (K (e ), K ( f )),

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

319

Table 1 The description of (E, C ). E/C

C1

C2

C3

C4

C5

e1 e2 e3 e4 e5 e6

(0.1060,0.8040) (0.4060,0.5797) (1,0) (1,0) (0.2551,0.6693) (0.4984,0.4314)

(0.2543,0.2399) (0.4733,0.4909) (0.1386,0.3510) 0.3500,0.4173) (0.2575,0.4018) (0.8407,0.0760)

(1,0) (1,0) (0.0759,0.1320) (0.4308,0.5561) (0.9072,0.1012) (1,0)

(0.0782,0.4359) (1,0) (0.4694,0.3212) (0.0119,0.0154) (1,0) (0.1622,0.1690)

(0.4505,0.1835) (0.2630,0.2963) (0.4749,0.3176) (0.7133,0.2802) (0.1524,0.0811) (0.5258,0.4294)

where C is an IF covering, md (C, e ) and MD(C, e ) are the IF minimal and maximal descriptions of e, respectively. Three important IF relations were reviewed in [42]. The IF relation R is called symmetric when R(e, f ) = R( f, e ), for each e, f ∈ E. If R(e, e ) = 1L , the R is named reflexive. The IF relation R is called T-transitive, if for each e, f, g ∈ E, T (R(e, f ), R( f, g)) ≤L R(e, g). Definition 2.7 [21]. A generalized OWA operator with IF sets is listed below:

 g( b1 , b2 , · · · , bn ) =

n 

⎛

 1k ω j bkj

=⎝ 1−

j=1

where

n j=1

n



1 − μkb j

ω j

 1k  ,

j=1

1−

n

1 − ( 1 − νb j )

k ωj

 1k ⎞ ⎠,

j=1

ω j = 1, k ∈ (0, +∞ ) and the b j = (μb j , νb j ) is the jth largest one of all IF sets cj ( j = 1, 2, · · · , m ).

Definition 2.8 [34]. Three IF aggregation operators, which are introduced by Xu, are listed as follows:   (1) IF WAω (b1 , b2 , · · · , bn ) = (1 − nj=1 (1 − μb j )ω j , nj=1 (νb j )ω j ); m  ωj (2) IF OWAω (b1 , b2 , · · · , bn ) = (1 − j=1 (1 − μb(δ ) )ω j , m j=1 (νb(δ ) ) ); j j  n (3) IF HAω,ω (b1 , b2 , · · · , bn ) = (1 − j=1 (1 − μb˙ )ω j , mj=1 (νb˙ )ω j ). (δ j )

(δ j )

Definition 2.9 [45]. Three generalized aggregation operators for IF sets are shown as follows: 1 1   (1) GIF WAω (b1 , b2 , · · · , bn ) = ((1 − nj=1 (1 − μkb )ω j ) k , (1 − nj=1 (1 − (1 − (1 − νb j )k )ω j ) k ); j 1 1   (2) GIF OWAω (b1 , b2 , · · · , bn ) = ((1 − nj=1 (1 − μkb )ω j ) k , (1 − nj=1 (1 − (1 − νb(δ ) )k )ω j ) k ); (3) GIF HAω,ω (b1 , b2 , · · · , bn ) = (1 −

n

(δ j )

k j=1 ((1 − μ ˙

b (δ ) j

)ω j ) k , ( 1 − 1

m

j

k j=1 (1 − (1 − ν ˙

b (δ ) j

))ω j ) k ). 1

3. CVPIFRS models We demonstrate some inadequacy in existing studies [21,34,45] for solving many MADM problems. This enables us to introduce an MADM method based on CVPIFRS models. Remark 3.1. From the analysis of the methods [21,34,45], we find that these methods are invalid for solving the problems in which for each alternative ei (i = 1, 2, · · · , n ) there exists at least one attribute Cj ( j = 1, 2, · · · , m ) such that the value of C j (ei ) = 1L . In this situation, using these methods, the results of alternatives ei is IF WAω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = IF OWAω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = IF HAω,ω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = g(C1 (ei ), C2 (ei ), · · · , Cm (ei )) = GIF WAω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = GIF OWAω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = GIF HAω,ω (C1 (ei ), C2 (ei ), · · · , Cm (ei )) = 1L . It is noted that whatever the values of weights of criteria are given, the ranking results remain the same, that is, e1 = e2 = e3 = · · · = en . Under this situation, the decision-makers can not choose the optimal alternative. In the following, an example is given to illustrate it. Example 3.2. In a hospital, a patient wants to choose a kind of nanomaterials for surgery. In this hospital, 6 kinds of nanomaterials are provided. The doctors need to select a suitable nanomaterial to make the operation from five characteristics to combine 5 characteristics of 6 kinds of nanomaterials. Let E = {e1 , e2 , · · · , e6 } be a set of 6 nanomaterials, C = {C1 , C2 , · · · , C5 } be the set of five characteristics for the nanomaterials. Here, C1 , , C5 denote chain flexibility, thermal stability, light stability, oxidation resistance and biocompatibility, respectively. In order to make the operation successfully, the hospital demands that for each kind of nanomaterial ei there exists at least one characteristic Cj such that the value of C j (ei ) = 1L . It is clear that C is an IF covering. The information of (E, C ) is listed as Table 1: Let ω = (ω1 , ω2 , ω3 , ω4 , ω5 ) = (0.1, 0.15, 0.2, 0.15, 0.15, 0.25 ) be a weight vector of characteristic Cj ( j = 1, 2, · · · , 5 ). By means of these methods in [21,34,45], the ranking orders of 6 kinds of nanomaterials are all e1 = e2 = e3 = e4 = e5 = e6 . In this way, the doctors can not help the patient to choose the optimal kind of nanomaterials to make an operation. At the same time, the acquisition of the weight value is a question worth pondering, as we can see that the weights of attributes

320

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 2 The description of (E, C ). E/C

C1

C2

C3

C4

C5

e1 e2 e3 e4 e5

(0.1060,0.8040) (0.6797,0.0844) (1,0) (1,0) (0.1190,0.8001)

(0.4593,0.3530) (0.5472,0.2221) (0.1386,0.3510) 0.1493,0.5132) (0.2575,0.4018)

(1,0) (1,0) (0.2853,0.7001) (0.5497,0.3692) (0.9072,0.1012)

(0.1299,0.2348) (1,0) (0.4694,0.3212) (0.0119,0.0154) (1,0)

(0.4505,0.1835) (0.0838,0.3685) (0.2290,0.6256) (0.7133,0.2802) (0.1524,0.0811)

of many MADM methods are given by experts. It is noted that the values of the weights with respect to the attributes have impact on our results. In particular, if these experts lack rational judgment or lack of experience, the weight value of the attribute will be greatly deviated from the value of the original importance of the attribute. In order to solve the complicated MADM problems, we need to search an effective too! Based on our analysis, we have raised two key questions as follows: (1) How to construct a formula to calculate the values of the weights of criteria according to the characteristics of the information system? (2) How to put forward a method to solve the MADM problems which can not be solved by these methods [21,34,45] effectively? Aiming at the above two problems, the corresponding four kinds of CVPIFRS models are proposed based on the four types of IF neighborhood operators. Secondly, by means of the concept of approximation precision, we construct an objective method for calculating attribute weights. Finally, based on our proposed CVPIFRS models, a scientific and effective IF TOPSIS method is proposed. In the following, four types of CVPIFRS models are given and the related properties of these models are investigated. Definition 3.3. Suppose that (E, C ) is a finite IFCAS. Let I2 , σ , T2 , ⊥ and N be an IF implicator, IF coimplicator, IF t-norm, IF t-conorm and IF negator, respectively. For each e ∈ E, H ∈ IF (E ), two pairs of approximations of the γ th (γ = 1, 2, 3, 4 ) type of CVPIFRS of H with ϑ ∈ L✠ (ϑ = 1L ) are defined as follows:

aprϑNC ,I (H )(e ) = γ

2

ϑ

aprNγC ,T2 (H )(e ) = aprϑNC ,⊥ (H )(e ) = γ

ϑ

aprNγC ,σ (H )(e ) =





I2 NγC (e )( f ), ϑ ∨ H ( f ) ,

(3-1)

f ∈E







T2 NγC (e )( f ), N (ϑ ) ∧ H ( f ) ,

(3-2)

f ∈E



⊥ N (NγC (e )( f )), ϑ ∨ H ( f ) ,

(3-3)



σ N (NγC (e )( f )), N (ϑ ) ∧ H ( f ) .

(3-4)

f ∈E

 f ∈E

The approximations (aprϑC

Nγ ,I2

(H ), aprϑNC ,T (H )) and (aprϑNC ,⊥ (H ), aprϑNC ,σ (H )) are called the first pair and the second pair γ

γ

γ

2

of approximations of the γ th type of CVPIFRS (γ -CVPIFRS) of H, respectively. The symbols apr ϑC

Nγ ,I2

ϑ

, aprϑ , aprϑC N C ,T γ

2

Nγ ,⊥

and

aprNC ,σ are called the I2 -lower, T2 -upper, ⊥-lower and σ -upper approximation operators of CVPIFRS models on U, respecγ

tively. Example 3.4. In a finite IFCAS (E, C ), let E = {e1 , e2 , · · · , e5 }, C = {C1 , C2 , · · · , C5 }, γ = 1, ϑ = (0.5, 0.4 ), I1 = IT ,1 I2 = ITw ,2 T2 = Tω ,3 ⊥ = ⊥w 4 , σ = σ⊥w 5 and N = Ns .6 The information of (E, C ) is given as Table 2. By means of the formulas (3-1) and (3-2), the results of N1C (ei ) (i = 1, 2, · · · , 5 ) are listed as Table 3. Suppose that H = (0.1060e,0.8040 ) + (0.5472e,0.2221 ) + (0.2853e,0.7001 ) + (0.0119e,0.0154 ) + (0.1524e,0.0811 ) . By the formulas (3-1) and 1 2 3 4 5 (3-2) in Definition 3.6, the results are listed as

aprN(0C.,5I,0.4) (H ) = 1

1 2 3 4 5 6

2

( 0.5, 0.4 ) e1

+

(0.5472, 0.2221 ) e2

+

( 0.5, 0.4 ) e3

+

(0.5, 0.0154 ) e4

+

(0.5, 0.0811 ) e5

;

IT (β1 , β2 ) = (min(1, 1 + μβ2 − μβ1 , 1 + νβ1 − νβ2 ), max(0, νβ2 − νβ1 )), where β 1 , β 2 ∈ L✠ and β1 = (μβ1 , νβ1 ), β2 = (μβ2 , νβ2 ); ITw (β1 , β2 ) = (min(1, 1 + μβ2 − μβ1 , 1 + νβ1 − νβ2 ), max(0, νβ2 + μβ1 − 1 )); Tω (β1 , β2 ) = (max(0, μβ1 + μβ2 − 1 ), min(1, νβ1 − μβ2 , 1 + νβ2 − μβ1 )); ⊥w (β1 , β!2 ) = (min(1, 1 − νβ1 + μβ2 , 1 − νβ2 + μβ1 ), 1 − min(1, 2 − νβ1 − νβ2 )); σ⊥w (β1 , β2 ) = (max(0, μβ2 + νβ1 − 1 ), min(1, νβ2 + 1 − νβ1 , μβ1 + 1 − νβ2 )); Ns (β1 ) = (Ns (1 − νβ1 ), 1 − Ns (μβ1 )) = (νβ1 , μβ1 ), where Ns (b) = 1 − b, b ∈ [0, 1].

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

321

Table 3 The description of N1C (ei ). N1C (ei )/E N1C N1C N1C N1C N1C

( 0. 5,0. 4 )

aprNC ,T 1

2

( e1 ) ( e2 ) ( e3 ) ( e4 ) ( e5 )

1

e3

e4

e5

(1,0) (0.1299,0.7196) (0.1060,0.8040) (0.1060,0.8040) (0.1299,0.2348)

(0.6333,0.1850) (1,0) (0.6797,0.0844) 0.3705,0.0883) (0.7126,0.2874)

(0.2853,0.7001) (0.2853,0.7001) (1,0) (0.5157,0.3454) (0.3781,0.5989)

(0.5497,0.3692) (0.0119,0.3692) (0.5425,0.1622) (1,0) (0.0119,0.2680)

(0.7019,0.1012) (0.2843,0.7157) (0.1190,0.8001) (0.1190,0.8001) (1,0)

e1

( 0.5, 0.4 ) e1

1

( 0. 5,0. 4 )

e2

(0.1060, 0.8040 )

(H ) =

aprN(0C.,5⊥,0.4) (H ) = aprNC ,σ

e1

+

e1

( 0.4, 0.5 ) e2

(0.5472, 0.2221 )

(0.106, 0.1960 )

(H ) =

+

e2 +

( 0.4, 0.5 ) e2

+ +

+

(0.2853, 0.7001 ) e3

( 0.5, 0.4 ) e3

+

(0.0119, 0.5 ) e4

(0.5, 0.0154 )

(0.2853, 0.2999 ) e3

+

e4 +

+

(0.1524, 0.5 ) e5

(0.5, 0.0811 )

(0.0119, 0.5 ) e4

+

e5 +

;

;

(0.1524, 0.5 ) e5

.

Through our research of these results, we observe that H is 1-CVPIFRS model. In a similar manner, the related results of H when γ = 2, 3, 4 will be obtained, here we omit it. Next, we study the models related to γ -CVPIFRS (γ = 1, 2, 3, 4 ) models. Remark 3.5. (1) If for each f ∈ E, H ( f ) ≥L ϑ and IF neighborhood NγC (γ = 1, 2, 3, 4 ) is substituted with an IF relation R, then the formula (3-1) turns into the form defined in [47]; If N (ϑ ) ≤L H ( f ), then the formula (3-2) will boil down to the model defined in [47]. (2) If ϑ = 0L , the models defined in Definition 3.3 become the following forms:

aprϑNC ,I (H )(e ) = γ

2

ϑ

aprNγC ,T2 (H )(e ) = aprϑNC ,⊥ (H )(e ) = γ

ϑ

aprNγC ,σ (H )(e ) =







I2 NγC (e )( f ), H ( f ) ,

(3-5)

f ∈E



T2 NγC (e )( f ), H ( f ) ,

(3-6)

f ∈E





⊥ N (NγC (e )( f )), H ( f ) ,

(3-7)



σ N (NγC (e )( f )), H ( f ) ,

(3-8)

f ∈E

 f ∈E

If the IF neighborhood NγC is substituted with an IF relation R, i.e., NγC (e )( f ) = R(e, f ), then the formulas (3-5) to (38) change into the following models:

aprϑR,I (H )(e ) = 2

ϑ

aprR,T2 (H )(e ) =



I2 (R(e, f ), H ( f ) ),

(3-9)

T2 (R(e, f ), H ( f ) ),

(3-10)

⊥(N (R(e, f )), H ( f ) ),

(3-11)

σ (N (R(e, f )), H ( f ) ).

(3-12)

f ∈E

 f ∈E

aprϑR,⊥ (H )(e ) = ϑ

aprR,σ (H )(e ) =

 f ∈E

 f ∈E

It is noted that the models (3-9) and (3-10) are the models raised by Zhou et al. in [47]. (3) Under the condition of (2), the formulas (3-5) and (3-6) will degenerate into the presented models [6] in the fuzzy environment. (4) When σ (ei , e j ) = N (I (N (ei )), N (e j ))), I (ei , e j ) = I⊥,N (ei , e j ) =⊥ (N (ei ), e j ), the formula (3-8) changes into the  form: aprϑ (H )(e ) = f ∈U T2 (R(e, f ), H ( f )) which was raised in [47]. N C ,σ γ

322

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

(5) If H is a crisp set i.e., for each v ∈ E, H (v ) = 0L or H (v ) = 1L , then

aprϑNC ,I (H )(e ) = γ

2



H ( f )=0L



ϑ

aprNγC ,T2 (H )(e ) =

H ( f )=1L



aprϑNC ,⊥ (H )(e ) = γ

H ( f )=0L



ϑ

aprNγC ,σ (H )(e ) =

H ( f )=1L

I2 (NγC (e )( f ), ϑ ),

(3-13)

T2 (NγC (e )( f ), N (ϑ )),

(3-14)

⊥(N (NγC (e )( f )), ϑ ),

(3-15)

σ (N (NγC (e )( f )), N (ϑ )).

(3-16)

It is noted that, in the fuzzy environment, the formulas (3-13) to (3-16) change into the following models:

aprϑNC ,I (H )(e ) = γ

2

ϑ

aprNγC ,T2 (H )(e ) = aprϑNC ,⊥ (H )(e ) = γ

ϑ

aprNγC ,σ (H )(e ) =



I2 (NγC (e )( f ), ϑ ),

(3-17)

T2 (NγC (e )( f ), N (ϑ )),

(3-18)

⊥(N (NγC (e )( f )), ϑ ),

(3-19)

σ (N (NγC (e )( f )), N (ϑ )).

(3-20)

H ( f )=0

 H ( f )=1

 H ( f )=0

 H ( f )=1

(6) In the fuzzy environment, if an IF implicator I2 , an IF coimplicator σ , an IF t-norm T2 , an IF t-conorm ⊥ and an IF





negator N are replaced by a fuzzy implicator I2 , a fuzzy coimplicator σ , a fuzzy t-norm T2 , a fuzzy t-conorm ⊥ and

a fuzzy negator N , respectively. When the IF neighborhood NγC is replaced by a fuzzy relation R, the formulas (3-1) to (3-4) change into the following forms:

aprϑR,I (H )(e ) = 2

ϑ

aprR,T (H )(e ) = 2

aprϑR,⊥ (H )(e ) = ϑ

aprR,σ (H )(e ) =





I2 (R(e, f ), H ( f )),

(3-21)

f ∈E





T2 (R(e, f ), H ( f )),

(3-22)

f ∈E





⊥ (N (R(e, f )), H ( f )),

(3-23)

f ∈E





σ (N (R(e, f )), H ( f )),

(3-24)

f ∈E

which were proposed in [46]. When ϑ = 0, the models defined in Definition 3.3 degenerate into the forms in [38]. In this situation, the I2 -lower and T2 -upper approximation operators proposed in Definition 3.3 turn into the models in [6]. Next, we mainly study the properties and relationships of the models raised in Definition 3.3 . In this section, since the proofs of the following propositions are simple and cumbersome, then we omit the details. Proposition 3.6. For each H1 , H2 ∈ IF (E ) and γ = 1, 2, 3, 4, the following statements hold: (1) aprϑC (E ) = E, aprϑC (E ) = E, aprϑ (E ) = N (γ ), aprϑNC ,σ (E ) = ∅; N C ,T Nγ ,I2

Nγ ,⊥

γ

γ

2

7 , aprϑ (∅ ) = ϑ , aprϑC (∅ ) = N (γ ), aprϑC (∅ ) = ∅; (∅ ) = ϑ Nγ ,T2 Nγ ,σ Nγ ,I2 NγC ,⊥     (3) aprϑC ( i∈ Hi ) = i∈ aprϑC (Hi ), aprϑC ( i∈ Hi ) = i∈ aprϑC (Hi ); Nγ ,I2 Nγ ,I2 Nγ ,I2 Nγ ,I2     ϑ ϑ ϑ ϑ (4) aprNC ,T ( i∈ Hi ) = i∈ aprNC ,T (Hi ), aprNC ,T ( i∈ Hi ) = i∈ aprNC ,T (Hi ); (2) aprϑC

γ

7

2

γ

2

γ

(e ) = ϑ . Here, ϑ  is called a constant IF set. For each ei ∈ U, ϑ i

2

γ

2

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

(5) aprϑC

Nγ ,⊥

(

ϑ

(6) aprNC ,σ (



i∈ Hi )



γ

i∈ Hi )

= =





(7) If H1 ⊆H2 , then aprϑC

Nγ

γ ,I2

i∈ Hi )

=

γ

2



ϑ

i∈ apr N C ,⊥ (Hi ); γ



ϑ

i∈ apr NγC ,σ (Hi ); γ

2

γ

γ

ϑ1

2

γ

γ

2

γ

2

2

(H ) ⊆ aprϑN2C ,⊥ (H ), aprϑN2C ,σ (H ) ⊆ aprϑN1C ,σ (H ); N C ,⊥

 ∪ H ) = aprϑ (ϑ

Nγ ,I2

=

(H ) ⊆ aprϑN2C ,I (H ), aprϑN2C ,T (H ) ⊆ aprϑN1C ,T (H );

NC

(12) aprϑC

Nγ



γ

ϑ1

 ∪ H ) = aprϑ (ϑ

aprϑC

ϑ

γ

(11) aprϑC

Nγ ,⊥

ϑ

i∈ Hi )

(H1 ) ⊆ aprϑNC ,I (H2 ), aprϑNC ,T (H1 ) ⊆ aprϑNC ,T (H2 );

(10) If ϑ1 ≤L ϑ2 , then apr

(13) aprϑC



(H1 ) ⊆ aprϑNC ,⊥ (H2 ), aprϑNC ,σ (H1 ) ⊆ aprϑNC ,σ (H2 ); ,⊥

(9) If ϑ1 ≤L ϑ2 , then apr

Nγ ,I2

ϑ

i∈ apr NγC ,σ (Hi ), apr NγC ,σ (

Nγ ,I2

(8) If H1 ⊆H2 , then aprϑC

ϑ

i∈ apr N C ,⊥ (Hi ), apr N C ,⊥ ( γ γ

323

γ

NγC ,I2

ϑ

γ

(H ), aprNC ,T γ

2

γ

γ

 (N (ϑ ) ∩ H ) = aprϑNC ,T (H ); γ

2

 (H ), aprϑNC ,σ (N (ϑ ) ∩ H ) = aprϑNC ,σ (H ); ,⊥

NC

γ

γ

γ

) = aprϑ ) = (ϑ =β , then aprϑ (β ) =  (β (β ∨ β ) when I2 is an IF R-implicator; especially, if ϑ N C ,I N C ,⊥ ,⊥ γ

) = ϑ ; (ϑ

γ

2

) = (N (   (β ϑ ) ∧ β ); If β = N (ϑ ), then aprϑNC ,T (N (ϑ ) ) = N (ϑ ); (14) aprϑ N C ,T γ

2

(15) aprϑC

Nγ ,I2

(16) aprϑC

Nγ

Nγ ,I2

2

γ

γ

γ

2

(H ) =∼N (aprϑNC ,T (∼N H ))), aprϑNC ,T (H ) =∼N (aprϑNC ,⊥ (∼N H ))); ,⊥ γ

γ

2

2

is an IF R-implicator, then aprϑC

(17) If I2 0L , aprϑC

γ

(H ) =∼N (aprϑNC ,σ (∼N H )), aprϑNC ,σ (H ) =∼N (aprϑNC ,I (∼N H ));

Nγ ,I2

(aprϑNC ,I (H )) = aprϑNC ,I (H ); γ

γ

2

γ

; especially, if ϑ = (H ) ⊆ aprϑNC ,I (aprϑNC ,I (H )) ⊆ aprϑNC ,I (H ) ∪ ϑ γ

γ

2

γ

2

2

2

 (aprϑNC ,T (H )) = N (ϑ ) ∩ aprϑNC ,T (H ); especially, if ϑ = 0L , aprϑNC ,T (aprϑNC ,T (H )) = aprϑNC ,T (H ); (18) aprϑ N C ,T γ

(19) aprϑC

Nγ

γ

2

γ

2

γ

2

2

γ

γ

2

(aprϑNC ,⊥ (H )) = ϑ˜ ∪ aprϑNC ,⊥ (H ); especially, if ϑ = 0L , aprϑNC ,⊥ (aprϑNC ,⊥ (H )) = aprϑNC ,⊥ (H ). ,⊥ γ

γ

γ

γ

2

γ

Since the properties of the four types of IF neighborhoods are not always the same, then in the following, we will discuss the properties of four types of CVPIFRS models respectively. Proposition 3.7. For γ = 1, 2, 3, the following statements hold:  ∪ H, if I is an IF R-implicator; (1) aprϑC (H ) ⊆ ϑ 2 Nγ ,I2

(2) aprϑC

Nγ ,⊥

(H ) ⊆ ϑ˜ ∪ H;

 (ϑ ) ∩ H ⊆ aprϑNC ,T (H ); (3) N γ

(4) aprϑC

Nγ ,I2

2

 ∪ aprϑ (aprϑNC ,I (H )) = ϑ (H ) with I2 is an IF R-implicator and γ = 1. N C ,I γ

γ

2

2

Proposition 3.8. For γ = 4, the following statements hold:  ∪ H, if I is an IF R-implicator; (1) aprϑC (H ) ⊆ ϑ 2 Nγ ,I2

(2) aprϑC

Nγ ,⊥

(H ) ⊆ ϑ˜ ∪ H;

 (ϑ ) ∩ H ⊆ aprϑNC ,T (H ); (3) N γ

2

(4) aprϑC

(1E \d )(e ) = aprϑNC ,I (1E \e )(d );

(5) aprϑC

(1d −→I2 ϑ˜ )(e ) = aprϑNC ,I (1e −→I2 ϑ˜ )(d );

Nγ ,I2 Nγ ,I2

γ

2

γ

2

(6) aprϑ (1d )(e ) = aprϑNC ,T (1e )(d ); N C ,T γ

2

γ

2

γ

2

(7) aprϑ (1d ∩T2 ϑ˜ )(e ) = aprϑNC ,T (1e ∩T2 ϑ˜ )(d ). N C ,T γ

2

In the following, the relationships among four types of CVPIFRS models are researched. Proposition 3.9. The relationships of I2 -lower, T2 -upper approximation operators of four types of CVPIFRS models are listed as follows: (1) aprϑC (H ) ⊆ aprϑC (H ) ⊆ aprϑC (H ); N4 ,I2

(2) aprϑC

N4 ,I2

N3 ,I2

N1 ,I2

(H ) ⊆ aprϑNC ,I (H ) ⊆ aprϑNC ,I (H ); 2

2

1

2

(H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ); (3) aprϑ N C ,T 1

2

3

2

4

2

1

2

2

2

4

2

(3) aprϑ (H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ). N C ,T

324

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 4 Several types of IF neighborhood operators based on IF coverings. Groups

Operators

Groups

Operators

a1 a2 b c d e f1 f2

N1C , N1C1 , N1C3 , N1C∩ N1C3 N3C1 N3C3 N4C3 N2C , N2C1 N3C , N3C2 , N3C∩ N1C2

g h1 h2 i j k l m

N1C4 N4C , N4C2 , N4C∩ N2C2 N2C4 N3C4 N4C4 N4C1 N2C∩

Table 5 The classifications of the first pair of approximation operators of several CVPIFRS models. Number

IF neighborhoods

the first pair of approximation operators

1

a1

(aprϑNC ,I , aprϑN1C ,T2 ), (aprϑC1 1

(aprϑ

2

C

N13 ,I2

2 3 4 5

ϑ

2

, aprNC3 ,T )

b

(aprϑC1

, aprNC1 ,T )

c

(aprϑ

, aprNC3 ,T )

N3 ,I2 C

N33 ,I2

(aprϑC3

N4 ,I2

e

7

1

(aprϑC3

d

6

, aprNC3 ,T

), (aprϑ

a2

N2 ,I2

f1

2

ϑ

3

ϑ

3

ϑ

2

, aprNC1 ,T )

(aprϑNC ,I , aprN3C ,T2 ), (aprϑC2

, aprNC2 ,T )

2

2

3

2

N2 ,I2

ϑ

2

3

ϑ

N3 ,I2

g

(aprϑC4

10

h1

(aprϑNC ,I , aprϑN4C ,T2 ), (aprϑC2

N1 ,I2 N1 ,I2 4

(aprϑ

C N4∩

,I2

1

2

, aprNC∩ ,T ) ϑ

4

(aprϑC4

, aprNC4 ,T )

(aprϑC4

, aprNC4 ,T )

(aprϑC4

, aprNC4 ,T )

(aprϑC1

, aprNC1 ,T )

N3 ,I2 N4 ,I2 N4 ,I2

2

2

N4 ,I2

ϑ

, aprNC2 ,T ) 4

2

2

i

N2 ,I2

3

2

(aprϑC2

l

16

1

ϑ

, aprNC4 ,T ) ϑ

N2 ,I2

2

ϑ

, aprNC2 ,T )

h2

k

15

2

ϑ

3

9

14

2

(aprϑNC ,I , aprϑN2C ,T2 ), (aprϑC1

(aprϑC2

j

2

1

2

f2

13

ϑ

, aprNC∩ ,T )

2

, aprNC3 ,T ) 4

1

N1∩ ,I2

8

12

C

, aprNC1 ,T )

2

(aprϑNC∩ ,I , aprϑNC∩ ,T2 )

11

ϑ

N1 ,I2

ϑ

, aprNC2 ,T ) 2

ϑ

2

ϑ

3

ϑ

4

ϑ

4

2

2

2

2

2

(aprϑNC∩ ,I , aprϑNC∩ ,T2 )

m

2

2

2

Proposition 3.10. The relationships of ⊥-lower, σ -upper approximation operators of four types of CVPIFRS models are listed as follows: (1) aprϑC (H ) ⊆ aprϑC (H ) ⊆ aprϑC (H ); N4 ,⊥

(2) aprϑC N4

N3 ,⊥

N1 ,⊥

(H ) ⊆ aprϑNC ,⊥ (H ) ⊆ aprϑNC ,⊥ (H ); ,⊥ 2

1

1

3

4

1

2

4

(H ) ⊆ aprϑNC ,σ (H ) ⊆ aprϑNC ,σ (H ); (3) aprϑ N C ,σ

(4) aprϑ (H ) ⊆ aprϑNC ,σ (H ) ⊆ aprϑNC ,σ (H ). N C ,σ



In addition, the IF extensions of the derived coverings of C are shown below:

C1 = ∪{md (C, e )|e ∈ E }, C2 = ∪{MD(C, e )|e ∈ E }, C3 = {N1C (e )|e ∈ E },C4 = {N4C (e )|e ∈ E }, C∩ = C \ {A ∈ C |(∃C ⊆ C \{A} )(A =



∩C )}, C∪ = C \ {A ∈ C |(∃C ⊆ C \{A} )(A = ∪C )}, where md (C, e ), MD(C, e ) are the IF minimal and maximal descriptions of e defined in [42]. Based on the six IF coverings, several types of IF neighborhoods are listed as Table 4. It is easy to check that the 16 groups of IF neighborhoods are the natural extensions of the 16 groups of fuzzy neighborhoods in [6]. Then the comparison of 16 groups of IF neighborhoods is similar to the comparison of the fuzzy neighborhoods in [6]. The relationships of several types of IF neighborhood operators based on IF coverings are listed as Fig. 2. Based on the 16 groups of IF neighborhoods, several types of approximation operators of CVPIFRS models are constructed. By the relationship among 16 groups of IF neighborhoods, the relationships of 24 types of approximation operators of CVPIFRS models are given as Figs. 3. It is noticed that the relationships among the 16 groups of the ⊥-lower and σ -upper approximation operators are the same as the relationships of the 16 groups of I2 -lower and T2 -upper approximation operators of our pro-

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

325

Table 6 The classifications of the second pair of approximation operators of several CVPIFRS models. Number

IF neighborhoods

the second pair of approximation operators

1´

a1

(aprϑNC ,⊥ , aprϑN1C ,σ ), (aprϑC1 1

(aprϑ 2´

a2

3´

b

4´

c

5´

d

6´

e

7´

C

N13 ,⊥

f1

, aprNC3 ,σ 1

), (aprϑ

ϑ

(aprϑC3 , aprNC3 ,σ ) N2 ,⊥

2

N3 ,⊥

3

N3 ,⊥

3

N4 ,⊥

4

f2 g

´ 10

h1

i

´ 13

j

´ 14

k

´ 15

l

´ 16

m

1

2

N2 ,⊥

2

3

N3 ,⊥

3

(aprϑNC ,⊥ , aprϑN3C ,σ ), (aprϑC2 , aprϑNC2 ,σ ) 3

(aprϑC2 , aprϑNC2 ,σ ) N1 ,⊥

1

N1 ,⊥

1

(aprϑC4 , aprϑNC4 ,σ )

(aprϑNC ,⊥ , aprϑN4C ,σ ), (aprϑC2 , aprϑNC2 ,σ ) 4

´ 12

ϑ

(aprϑNC ,⊥ , aprϑN2C ,σ ), (aprϑC1 , aprϑNC1 ,σ )

(aprϑNC∩ ,⊥ , aprϑNC∩ ,σ ) h2

1

, aprNC∩ ,σ )

(aprϑC3 , aprϑNC3 ,σ )

4

´ 11

N1∩ ,⊥

(aprϑC3 , aprϑNC3 ,σ )

3

9´

C

, aprNC1 ,σ )

(aprϑC1 , aprϑNC1 ,σ )

(aprϑNC∩ ,⊥ , aprϑNC∩ ,σ ) 8´

ϑ

N1 ,I2

ϑ

N4 ,⊥

4

4

(aprϑC2 , aprϑNC2 ,σ ) N2 ,⊥

2

N2 ,⊥

2

N3 ,⊥

3

N4 ,⊥

4

N4 ,⊥

4

(aprϑC4 , aprϑNC4 ,σ ) (aprϑC4 , aprϑNC4 ,σ )

(aprϑC4 , aprϑNC4 ,σ ) (aprϑC1 , aprϑNC1 ,σ )

(aprϑNC∩ ,⊥ , aprϑNC∩ ,σ ) 2

2

Fig. 2. The relationships among 16 groups of IF neighborhoods.

posed models, respectively. In Figs. 2 and 3, K → T represents the K!⊆T. It is noted that K → T denotes that the I2 -lower approximation operator K is included by the I2 -lower approximation operator T. The red arrow K → T denotes the T2 -upper approximation operator K is included by the T2 -upper approximation operator T. From Fig. 3, the first pair of approximation operators of 1-CVPIFRS are more accurate than the other types of approximation operators presented in this paper. Then, in the next section, the first pair of approximation operators of 1-CVPIFRS models will be used to make a decision. Next, we study the relationship between our proposed model and other existing covering-based generalized IF rough set (CIFRS) models.

326

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

Fig. 3. The relationships among 16 groups of the approximation operators of our proposed models.

Definition 3.11. [42] Suppose that I2 and T1 are an IF R-implicator and an IF t-norm, respectively. For each e, f1 , f2 ∈ E, γ th type of CIFRS approximation operators are defined below:

C γ (H )(e ) =



T1 (NγC ( f1 )(e ),

f 1 ∈E

C γ (H )(e ) =







C γ (H )(e ) =



I2 (NγC ( f1 )( f2 ), H ( f2 )));

f 2 ∈E

I2 (NγC ( f1 )(e ),

f 1 ∈E

C γ (H )(e ) =





T1 (NγC ( f1 )( f2 ), H ( f2 )));

f 2 ∈E

I2 (NγC ( f1 )(e ),



f 1 ∈E

f 2 ∈E





T1 (NγC ( f1 )(e ),

f 1 ∈E

I2 (NγC ( f1 )( f2 ), H ( f2 ))); I2 (NγC ( f1 )( f2 ), H ( f2 ))).

f 2 ∈E

∪H = N  Proposition 3.12. If I2 is an IF R-implicator, ϑ (ϑ ) ∩ H and NγC (γ = 1, 2, 3, 4 ) is symmetric, the following statements are established:



∪H = N  C γ (H ) ⊆ aprϑNC ,I (H ) ⊆ ϑ (ϑ ) ∩ H ⊆ aprϑNγC ,T2 (H ) ⊆ C γ (H ). γ

2





Proposition 3.13. If ϑ = 0L , for each H ∈ IF (E ), C γ (H ) ⊆ aprϑC

Nγ ,I2

(H ) ⊆ H ⊆ aprϑNC ,T (H ) ⊆ C γ (H ) where γ = 1, 2, 3, 4. γ

2

∪H = N  Proposition 3.14. If ϑ (ϑ ) ∩ H, the following statements hold:

ϑ ∪H = N  (1) C 4 (H ) ⊆ apr C (H ) ⊆ aprϑC (H ) ⊆ aprϑC (H ) ⊆ ϑ (ϑ ) ∩ H ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ N4 ,I2



C 4 ( H );



C 4 (H ) ⊆ aprϑC

(2)

N4 ,I2



N3 ,I2

N1 ,I2

1

2

3

2

4

∪H = N  (H ) ⊆ aprϑNC ,I (H ) ⊆ aprϑNC ,I (H ) ⊆ ϑ (ϑ ) ∩ H ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ 2

2

1

2

1

2

2

2

4

C 4 ( H ). Proposition 3.15. If ϑ = 0L , then the following statements are established:

(1) C 4 (H ) ⊆ aprϑC

N4 ,I2

(2) C 4 (H ) ⊆ aprϑC

N4 ,I2

2



(H ) ⊆ aprϑNC ,I (H ) ⊆ aprϑNC ,I (H ) ⊆ H ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ C 4 (H ); 3

2

2

2

1

2

1

2

1

2

3

2

4

2



(H ) ⊆ aprϑNC ,I (H ) ⊆ aprϑNC ,I (H ) ⊆ H ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ aprϑNC ,T (H ) ⊆ C 4 (H ). 1

2

2

2

Since the N4C is symmetric, we can obtain the following conclusions. Proposition 3.16. Let γ = 4, for each H ∈ IF (E ), then the following statements are true:

4

2

2

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

327

Table 7 The description of (E, C ).

(1) apr (2) apr

0L

N4C ,I2 0L

N4C ,T2

0

E/C

C1

C2



Cm−1

Cm

e1 e2 e3  en−1 en

κ 11 κ 21 κ 31

κ 12 κ 22 κ 32

     

κ1(m−1) κ2(m−1) κ3(m−1)

κ 1m κ 2m κ 3m





κ(n−1)1 κ n1

κ(n−1)2 κ n2



κ(n−1)(m−1) κn(m−1)



κ(n−1)m κ nm



0

(H ) = aprNLC,I (aprNLC,I (H )) = C 4 (H ); 4

2

4

0

2

0



(H ) = aprNLC,T (aprNLC,T (H )) = C 4 (H ). 4

2

4

2

4. An approach to MADM with evaluation of IF information based on CVPIFRS models 4.1. Decision-making background As we all know, once some human organs are damaged, they cannot be replaced in the era of poor medical backwardness. In the past, if a person was amputated, he or she could never stand. However, with the development of medical care, nanotechnology will provide individuals with a pair of legs to help them walk like a normal person. When nanomaterials were born, they attracted widespread attention in the material world with their unusual characteristics. This is because nanomaterials own some features that are significantly different from traditional materials. Nano-topological scaffolds formed by composite materials provide excellent compatibility conditions for seed cell adhesion and proliferation due to their good biocompatibility, ductility and thermal stability, and are widely used in the construction of engineered tissues. The survival and proliferation of chondrocytes on the surface of composite nanoscaffolds compared with conventional scaffolds stronger is expected to be the ideal stent for nasal septal cartilage. For example, the diffusion rate of gas through the nanomaterial is several thousand times faster than the diffusion rate through the general material. In the daily life, bone diseases such as bone infections, bone tumors and bone injuries are frequent, while the source of autologous bone tissue is limited, resulting in a large lack of bone sources. Compared with traditional materials, the biomimetic properties, excellent physical and chemical properties of nanomaterials are more conducive to cell adhesion, stimulate new bone growth and play an important role in remodeling bone tissue. With the advancement of medical technology, the classes and functions of nano-synthetic materials are becoming more and more abundant. Faced with variety of nanomaterials, it is necessary to choose the optimal bio-synthesis nanomaterial to treat the patient. Obviously, this is a typical type of MADM problems. In order to deal with those MADM problems, the TOPSIS method was put forward. However, in some references, the weights of criteria are given by some experts. Sometimes the experts’ assessments are subjective and irrational. Since then, based on our proposed models, a method for computing the weight of each criterion is given by a formula. In this way, the weight of criteria is objective. Then, we extend the IF TOPSIS method based on our proposed models to solve many MADM problems. Skeletal transplantation is a typical type of MADM problems. In a bone transplant hospital, a patient needs a bone transplant. How to choose a nanomaterial is the key to surgery. Firstly, many experts will judge the nanomaterials according to their experience. After many experts’ diagnoses, a patient will be given an operation. It is obvious that these problems are MADM problems in the medical field. As we all know, most of data obtained are vague or uncertain. Hence, using many traditional methods, we can not efficiently make a decision for the patient. However, our proposed method can help the patient to do this work. Let E = {ei : i = 1, 2, · · · , n} be n alternatives, C = {C j : j = 1, 2, · · · , m} be m criteria and ϑ ∈ L✠ be the variable precision. Here, Cj (ei ) represents the value of the alternative ei for the criterion Cj with a lots of experts’ diagnose. Let C j (ei ) = (μC j (ei ) , νC j (ei ) ) where Cj (ei ) ∈ L✠ and μC j (ei ) , νC j (ei ) are the degrees of membership and non-membership of Cj (ei ), respectively. Suppose that for each alternative ei , there at least exists one criterion Cj such that the value of alternative ei for the criterion Cj is equal to 1L . Obviously, C is an IF covering of U. In the following, using our proposed method, the weight of each criterion is obtained. Then, by means of the weight of every criterion, we apply the IF TOPSIS method proposed in this paper to solve the complex MADM problems. 4.2. Decision-making methodology Firstly, according to many authoritative experts, an MADM matrix is given as Table 7. Here, Ci (e j ) = κi j ∈ L . Let γ = 1, by Definition 2.6, the values of N1C (ei ) (i = 1, 2, 3, · · · , n ) are obtained. Furthermore, the value of ϑ is given. Then by the formulas (3-1) and (3-2) the approximation operators of Cj ( j = 1, 2, 3, · · · , m) are calculated and shown as Tables 8 and 9. Here, aprϑC

N1 ,I2

(C j )(ei ) = λi j and aprϑNC ,T (C j )(ei ) = ξi j where λij , ξ ij ∈ L✠ . 1

2

328

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 8 The description of aprϑNC ,I (C j ). 2

1

E/aprϑNC ,I (C j )

aprϑNC ,I (C1 )

aprϑNC ,I (C2 )



aprϑNC ,I (Cm−1 )

aprϑNC ,I (Cm )

e1 e2 e3  en−1 en

λ11 λ21 λ31

λ12 λ22 λ32

     

λ1(m−1) λ2(m−1) λ3(m−1)

λ1 m λ2 m λ3 m

2

1

2

1



1

2



λ(n−1)1 λn 1

λ(n−1)2 λn 2

1

2

1



2



λ(n−1)(m−1) λn(m−1)

λ(n−1)m λnm

Table 9 ϑ The description of aprNC ,T2 (C j ). 1

ϑ

ϑ

ϑ

ϑ

ϑ

E/aprNC ,T2 (C j )

aprNC ,T2 (C1 )

aprNC ,T2 (C2 )



aprNC ,T2 (Cm−1 )

aprNC ,T2 (Cm )

e1 e2 e3  en−1 en

ξ 11 ξ 21 ξ 31

ξ 12 ξ 22 ξ 32

     

ξ1(m−1) ξ2(m−1) ξ3(m−1)

ξ 1m ξ 2m ξ 3m

1

1



1



ξ(n−1)1 ξ n1

ξ(n−1)2 ξ n2

1

1





ξ(n−1)(m−1) ξn(m−1)

ξ(n−1)m ξ nm

Table 10 The description of weighting matrix. E/C

C1

C2



Cm−1

Cm

e1 e2 e3  en−1 en

r11 r21 r31  r(n−1)1 rn1

r12 r22 r32  r(n−1)2 rn2

     

r1(m−1) r2(m−1) r3(m−1)  r(n−1)(m−1) rn(m−1)

r1m r2m

Then, using the formula in [19], the distance between aprϑC

N1 ,I2

as:





1  d aprϑNC ,I (C j ), aprNC ,I2 (C j ) = 1 |E | 1 2 ϑ

 |μ

aprϑC

N ,I 2 1

(C j )(ei )

κ 3m

 r(n−1)m rnm

(C j ) and aprϑNC ,T (C j ) for each criterion Cj is calculated 1

− μaprϑ

N C ,T 2 1

(C j )(ei )

| + |νaprϑC

N ,I 2 1

(C j )(ei )

− νaprϑ

N C ,T 2 1

(C j )(ei )

|

4

ei ∈E

+

2

 max |μaprϑ C

N ,I 2 1

(C j )(ei ) − μaprϑ

N C ,T 2 1

|, |νaprϑC (C j )(ei )

N ,I 2 1

2

(C j )(ei ) − νaprϑ

N C ,T 2 1

| (C j )(ei )

⎤ ⎥ ⎥. (4-2-1) ⎦

The precision degree of Cj is computed according to the following formula:



ϑ



b j = 1 − d aprϑNC ,I (C j ), aprNC ,T2 (C j ) . 1

(4-2-2)

1

2

Then, the weight of criterion Cj is obtained by the following formula:

b

ω j = m j

k=1

bk

.

(4-2-3)

Furthermore, by the formula

wα = (1 − (1 − μα )w , (να )w ),

(4-2-4)

α ∈ L✠ ,

where w ∈ [0, 1] and the weighting matrix is calculated as Table 10. Here, ri j = ω j × C j (ei ), where ωj ∈ [0, 1] is the weight of criterion Cj and C j (ei ) = κ ji ∈ L . The positive and negative ideal solutions are obtained according to the following formulas: + + B+ = {b+ 1 , b2 , · · · , bm } =

− − B− = {b− 1 , b2 , · · · , bm } =





max r ji |C j ∈ B , min r ji |C j ∈ C 1≤i≤n

1≤i≤n





min r ji |C j ∈ B , max r ji |C j ∈ C

1≤i≤n

1≤i≤n



,

(4-2-5)



,

(4-2-6)

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

329

Table 11 The description of (E, C ). E/C

C1

C2

C3

C4

C5

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(0.1060,0.8040) (0.6797,0.0844) (1,0) (1,0) (0.1190,0.8001) (0.4984,0.4314) (0.9597,0.0106) (1,0) (0.5853,0.2638) (1,0) (0.7513,0.1361) (0.2551,0.6693) (0.4060,0.5797) (1,0) (0.8509,0.1450)

(0.4593,0.3530) (0.5472,0.2221) (0.1386,0.3510) (0.1493,0.5132) (0.2575,0.4018) (0.8407,0.0760) (0.2543,0.2399) (0.8143,0.1233) (0.2435,0.1839) (0.7293,0.24) (0.3500,0.4173) (0.1966,0.0497) (0.2511,0.7027) (0.3160,0.6448) (0.4733,0.4909)

(1,0) (1,0) (0.2853,0.7001) (0.5497,0.3692) (0.9072,0.1012) (1,0) (0.7572,0.1897) (1,0) (0.3804,0.4039) (0.5678,0.0965) (0.0759,0.1320) (1,0) (0.4308,0.5561) (0.7792,0.1752) (1,0)

(0.1299,0.2348) (1,0) (0.4694,0.3212) (0.0119,0.0154) (1,0) (0.1622,0.1690) (1,0) (0.3112,0.6317) (1,0) (0.1656,0.4509) (1,0) (0.2630,0.2963) (1,0) (0.6892,0.1890) (0.7482,0.1868)

(0.4505,0.1835) (0.0838,0.3685) (0.2290,0.6256) (0.7133,0.2802) (0.1524,0.0811) (0.5258,0.4294) (0.5383,0.3757) (0.9961,0) (0.0782,0.4359) (0.4427,0.4468) (0.1067,0.3063) (0.6619,0.3085) (0.0046,0.5108) (0.4749,0.3176) (0.8173,0.0948)

where B and C represent the set of benefit and cost criteria, respectively. It is noted that j = 1, 2, · · · , m. According to the formula in [28], the difference between the alternative ei and the positive ideal solution B+ is measured as follows:

D ∗ ( ei ) = d1 ( ei , B+ ) =

 m

1 |μb+j − μri j | + |νb+j − νri j | + |πb+j − πri j | , 2

(4-2-7)

j=1

here, πi j = 1 − μri j − νri j and πb+ = 1 − μb+ − νb+ . j

j

j

In a similar way, the difference between the alternative ei and the negative ideal solution B− is measured as follows:

D∗ ( ei ) = d1 ( ei , B− ) =

 m

1 |μb−j − μri j | + |νb−j − νri j | + |πb−j − πri j | , 2

(4-2-8)

j=1

where πb− = 1 − μb− − νb− . j

j

j

Finally, according to the idea of [16], the intimate function value of δ ∗ (ei ) for alternative ei is obtained as follows:

δ ∗ ( ei ) =

D ∗ ( ei ) D∗ ( ei ) − , D∗ D∗

(4-2-9)

where, D∗ = max1≤i≤n D∗ (ei ) and D ∗ = min1≤i≤n D ∗ (ei ). According the value of δ ∗ (ei ), a ranking order for all alternatives is given. Then, the optimal alternative is selected according to the ranking order. 4.3. Procedure for decision-making method In this section, an algorithm of our proposed method is given as follows: Input An MADM matrix (E, C ) and the value of ϑ. Output A ranking order of all alternatives. Step 1: Calculate the upper and lower approximation according to the formulas (3-1) and (3-2). Step 2: According to the formulas (4-2-1) to (4-2-3), the weight of every criterion is computed. Step 3: According to the formulas (4-2-4) to (4-2-6), the positive and negative ideal solutions are obtained, respectively. Step 4: According to the formulas (4-2-7) to (4-2-9), the intimate function value of δ ∗ (ei ) for alternative ei is obtained. According to the value of δ ∗ (ei ), the ranking order of all alternatives is obtained. 4.4. An example In a bone transplant hospital, a patient with amputated legs needs to choose a nanomaterial for surgery. Before surgery, experts should combine 5 characteristics of 15 kinds of nanomaterials to select a suitable nanomaterial to make the operation. Let E = {e1 , e2 , · · · , e15 } be a set of 15 nanomaterials which is provided for a hospital, C = {C1 , C2 , · · · , C5 } be the set of five characteristics for the nanomaterials. Here, C1 , , C5 denote chain flexibility, thermal stability, light stability, oxidation resistance and biocompatibility, respectively. Here, C j ∈ IF (E ) and for each ei ∈ E, Cj (ei ) ∈ L✠ represents the efficiency value of nanomaterial ei w.r.t. characteristic Cj where j = 1, 2, · · · , 5 and i = 1, 2, · · · , 15. Suppose that for each nanomaterial ei , there at least exists one characteristic Cj such that C j (ei ) = 1L . Then, the matrix of (E, C ) is listed as Table 11. The hospital requirements the variable precision ϑ = (0.3, 0.4 ).

330

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 12a The description of N1C (ei ). N1C (ei )/E N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C N1C

( e1 ) ( e2 ) ( e3 ) ( e4 ) ( e5 ) ( e6 ) ( e7 ) ( e8 ) ( e9 ) (e10 ) (e11 ) (e12 ) (e13 ) (e14 ) (e15 )

e1

e2

e3

e4

e5

(1,0) (0.1299,0.7196) (0.106,0.804) (0.106,0.804) (0.1299,0.2348) (0.6067,0.3726) (0.1299,0.7934) (0.106,0.804) (0.1299,0.5402) (0.106,0.804) (0.1299,0.6679) (0.6967,0.3033) (0.1299,0.2348) (0.106,0.804) (0.2551,0.659)

(0.6333,0.185) (1,0) (0.6797,0.0844) (0.3705,0.0883) (0.7126,0.2874) (0.558,0.1461) (0.5455,0.0738) (0.0877,0.3685) (0.9618,0.0.382) (0.6411,0.0844) (0.9284,0.0622) (0.4219,0.1724) (1,0) (0.6089,0.0844) (0.2665,0.2737)

(0.2853,0.7001) (0.2853,0.7001) (1,0) (0.5157,0.3454) (0.3781,0.5989) (0.2853,0.7001) (0.4964,0.5104) (0.2329,0.7001) (0.4694,0.3212) (0.3964,0.6036) (0.4319,0.5681) (0.2853,0.7001) (0.4694,0.3212) (0.4751,0.5249) (0.2853,0.7001)

(0.5497,0.3692) (0.0119,0.3692) (0.5425,0.1622) (1,0) (0.0119,0.268) (0.3086,0.4372) (0.0119,0.2733) (0.335,0.3899) (0.0119,0.3293) (0.42,0.2732) (0.0119,0.2732) (0.5365,0.4635) (0.0119,0.0154) (0.3227,0.194) (0.2637,0.3692)

(0.7019,0.1012) (0.2843,0.7157) (0.119,0.8001) (0.119,0.8001) (1,0) (0.4168,0.3687) (0.1593,0.7895) (0.119,0.8001) (0.4637,0.5363) (0.119,0.8001) (0.336,0.664) (0.4905,0.3521) (0.713,0.2204) (0.119,0.8001) (0.2681,0.6551)

Table 12b The description of N1C (ei ). N1C (ei )/E

e6

e7

e8

e9

e10

N1C (e1 ) N1C (e2 ) N1C (e3 ) N1C (e4 ) N1C (e5 ) N1C (e6 ) N1C (e7 ) N1C (e8 ) N1C (e9 ) N1C (e10 ) N1C (e11 ) N1C (e12 ) N1C (e13 ) N1C (e14 ) N1C (e15 )

(0.7541,0.2459) (0.1622,0.347) (0.4984,0.4314) (0.4984,0.4314) (0.1622,0.3483) (1,0) (0.1622,0.4208) (0.4984,0.4314) (0.1622,0.169) (0.4984,0.4314) (0.1622,0.2953) (0.8639,0.1209) (0.1622,0.169) (0.473,0.4314) (0.414,0.3346)

(0.7572,0.1922) (0.7071,0.1897) (0.9597,0.0106) (0.825,0.0955) (0.7054,0.2946) (0.4136,0.1897) (1,0) (0.44,0.3757) (0.944,0.0.056) (0.525,0.0932) (0.9043,0.0694) (0.7572,0.1902) (1,0) (0.9383,0.0581) (0.7191,0.2809)

(0.6031,0.3969) (0.3112,0.6371) (0.6895,0.3105) (0.3837,0.6063) (0.3112,0.6317) (0.5373,0.4927) (0.3112,0.6317) (1,0) (0.3112,0.6317) (0.8192,0.1808) (0.3112,0.6317) (0.6646,0.3354) (0.3112,0.6317) (0.5573,0.4427) (0.5551,0.4449)

(0.3804,0.4039) (0.3804,0.4039) (0.5853,0.2638) (0.3649,0.2638) (0.4732,0.3548) (0.3804,0.4039) (0.5399,0.2532) (0.0821,0.4359) (1,0) (0.5142,0.3074) (0.7281,0.2719) (0.3804,0.4039) (0.9496,0) (0.5853,0.2638) (0.2609,0.4039)

(0.5678,0.2633) (0.1656,0.4509) (0.6962,0.1297) (0.5645,0.4355) (1656,0.4509) (0.5678,0.2819) (0.1656,0.4509) (0.4466,0.4468) (0.1656,0.4509) (1,0) (0.1656,0.4509) (0.5678,0.1903) (0.1656,0.4509) (0.4764,0.2619) (0.4174,0.352)

Table 12c The description of N1C (ei ). N1C (ei )/E

e11

e12

e13

e14

e15

N1C (e1 ) N1C (e2 ) N1C (e3 ) N1C (e4 ) N1C (e5 ) N1C (e6 ) N1C (e7 ) N1C (e8 ) N1C (e9 ) N1C (e10 ) N1C (e11 ) N1C (e12 ) N1C (e13 ) N1C (e14 ) N1C (e15 )

(0.0759,0.132) (0.0759,0.1952) (0.7513,0.1361) (0.3934,0.1361) (0.1687,0.2252) (0.0759,0.3413) (0.3187,0.1774) (0.0759,0.3036) (0.6955,0.2334) (0.5081,0.1773) (1,0) (0.0759,0.3676) (0.6451,0) (0.2967,0.1361) (0.0759,0.2115)

(0.7373,0.125) (0.263,0.5849) (0.2551,0.6693) (0.2551,0.6693) (0.263,0.2963) (0.3559,0.2379) (0.263,0.6587) (0.2551,0.6693) (0.263,0.4055) (0.2551,0.6693) (0.263,0.5332) (1,0) (0.263,0.2963) (0.2551,0.6693) (0.4042,0.5243)

(0.4308,0.5561) (0.4308,0.5561) (0.406,0.5797) (0.2913,0.5797) (0.5236,0.4549) (0.3733,0.6267) (0.7572,0.5691) (0.4309,0.5797) (0.0085,0.5188) (0.4812,0.5797) (0.406,0.4436) (0.5564,0.653) (0.3427,0) (1,0.5797) (0.1873,0.5561)

(0.7082,0.2918) (0.5773,0.4227) (0.7062,0.2938) (0.7616,0.1736) (0.6892,0.243) (0.4312,0.5688) (0.5951,0.4049) (0.4785,0.5215) (0.5391,0.4609) (0.5867,0.4048) (0.6892,0.2275) (0.4049,0.5951) (0.6892,0.189) (1,0) (0.6576,0.2228)

(0.8621,0.1379) (0.7312,0.2688) (0.8509,0.145) (0.8286,0.1714) (0.7482,0.1868) (0.5851,0.4149) (0.7482,0.251) (0.6324,0.3676) (0.693,0.307) (0.744,0.2509) (0.7482,0.1868) (0.5588,0.4412) (0.7482,0.1868) (0.8509,0.145) (1,0)

In the following, two numerical examples based on different logical operators are given to show that our proposed method is feasible and effective in the medical field. Case 1 : Let I1 = I2 = IT , T2 = T 8 and N = Ns , by means of the formulas (3-1-1) and (3-1-2) with γ = 1, the results are listed as Tables 12a, 12b, 12c, 13, 14. Using the formulas (4-2-1) to (4-2-3), the weight of each criterion is obtained as Table 15.

8

T (β1 , β2 ) = (max(0, μβ1 + μβ2 − 1 ), min(1, νβ1 + νβ2 )).

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

331

Table 13 The description of (E, aprN(0C.,3I,0.4) (Ck )). 1

T

E/aprN(0C.,3I,0.4) (Ck )

aprN(0C.,3I,0.4) (C1 )

aprN(0C.,3I,0.4) (C2 )

aprN(0C.,3I,0.4) (C3 )

aprN(0C.,3I,0.4) (C4 )

aprN(0C.,3I,0.4) (C5 )

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(0.3,0.4) (0.6797,0.0844) (1,0) (1,0) (0.3,0.4) (0.4984,0.4) (0.9597,0.0106) (1,0) (0.5853,0.2638) (1,0) (0.7513,0.1361) (0.3,0.4) (0.4060,0.4) (1,0) (0.8509,0.1450)

(0.4593,0.3530) (0.5472,0.2221) (0.3,0.3510) (0.3,0.4) (0.3,0.4) (0.8407,0.0760) (0.3,0.2399) (0.8143,0.1233) (0.3,0.1839) (0.7293,0.24) (0.3500,0.4) (0.3,0.0497) (0.3,0.4) (0.3160,0.4) (0.4733,0.4)

(1,0) (1,0) (0.3,0.4) (0.5497,0.3692) (0.8988,0.1012) (1,0) (0.7572,0.1897) (1,0) (0.3804,0.4) (0.5678,0.0965) (0.3,0.1320) (1,0) (0.4308,0.4) (0.7792,0.1752) (1,0)

(0.3,0.2348) (1,0) (0.4694,0.3212) (0.3,0.0154) (1,0) (0.3,0.1690) (1,0) (0.3112,0.4) (1,0) (0.3,0.4) (1,0) (0.3,0.2963) (1,0) (0.6892,0.1890) (0.7482,0.1868)

(0.4505,0.1835) (0.3,0.3685) (0.3,0.4) (0.7133,0.2802) (0.3,0.0811) (0.5258,0.4) (0.5383,0.3757) (0.9961,0) (0.3,0.4) (0.4427,0.4) (0.3,0.3063) (0.6619,0.3085) (0.3,0.4) (0.4749,0.3176) (0.8173,0.0948)

1

T

1

T

1

T

1

T

T

1

1

T

Table 14 ( 0.3,0.4 ) (Ck )). The description of (E, aprNC ,T 1

( 0.3,0.4 )

E/aprNC ,T 1

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(Ck )

( 0.3,0.4 )

aprNC ,T 1

(C1 )

(0.7169,0.2028) (0.6797,0.0844) (1,0) (1,0) (0.6829,0.243) (0.5678,0.2003) (0.9597,0.0106) (1,0) (0.9037,0.0666) (1,0) (0.864,0.08) (0.7169,0.1903) (0.9597,0.0106) (1,0) (0.8509,0.1450)

( 0.3,0.4 )

aprNC ,T 1

(C2 )

(0.5948,0.1747) (0.5472,0.2221) (0.5038,0.2505) (0.3391,0.3104) (0.3,0.346) (0.8407,0.0760) (0.3,0.2399) (0.8143,0.1233) (0.509,0.1839) (0.7293,0.24) (0.4756,0.2843) (0.7046,0.0497) (0.5472,0.1839) (0.3716,0.2980) (0.4733,0.4)

( 0.3,0.4 )

aprNC ,T 1

( 0.3,0.4 )

(C3 )

aprNC ,T 1

(1,0) (1,0) (0.8509,0.0844) (0.8286,0.0883) (0.9072,0.1012) (1,0) (0.7572,0.0738) (1,0) (0.9618,0.0382) (0.8192,0.0844) (0.9284,0.0622) (1,0) (1,0) (0.8509,0.0844) (1,0)

(C4 )

(0.7572,0.1012) (1,0) (0.9597,0.0106) (0.825,0.0154) (1,0) (0.558,0.1461) (1,0) (0.44,0.3036) (1,0) (0.6411,0.0844) (1,0) (0.7572,0.1724) (1,0) (0.9383,0.0581) (0.7482,0.1868)

( 0.3,0.4 )

aprNC ,T 1

(C5 )

(0.6794,0.1823) (0.5485,0.3636) (0.6856,0.2398) (0.7133,0.2662) (0.5655,0.0811) (0.5334,0.4) (0.5655,0.3458) (0.9961,0) (0.5103,0.4) (0.4427,0.1808) (0.5655,0.2816) (0.6619,0.3085) (0.5655,0.2816) (0.6682,0.2389) (0.8173,0.0948)

Table 15 The description of ωi . Weight

ω1

ω2

ω3

ω4

ω5

The value of ω i

0.1975

0.2067

0.1924

0.1982

0.2052

Through the weight of each criterion obtained, by means of the formula (4-2-4), the weighting matrix is listed as Table 16. Since the five characteristics are benefit attributes, using the formulas (4-2-5) and (4-2-6), the value of positive and negative ideal solutions are listed below: ,0 ) B+ = (1C,0 ) + (0.3160C,0.5376 ) + (1C,0 ) + (1C,0 ) + (0.6795 ; C 1

2

3

4

5

B− = (0.0219C,0.9578 ) + (0.0304C,0.9297 ) + (0.0151C,0.9337 ) + (0.0024C,0.9130 ) + (0.0009C,0.9083 ) . 1 2 3 4 5 The distance between each alternative and the positive ideal solution is calculated as Table 17 by means of the formula (4-2-7). According to the formula (4-2-8), the distance between each alternative and the negative ideal solution is obtained as Table 18. Finally, using the formula (4-2-9), the intimate function value of δ ∗ (ei ) for alternative ei is obtained as Table 19. According to the value of δ ∗ (ei ), we rank the 15 nanomaterials. The ranking of them is listed as: e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 . Thus, the nanomaterial e8 is selected. Case 2 : Let I1 = IT , I2 = ITw , T2 = Tω and N = Ns , by means of formulas (3-1-1) and (3-1-2) with γ = 1, the IF neighborhoods of N1C (ei ) are the same as Tables 12a–12c. The approximation operators of each criterion are obtained as Tables 20 and 21. Using the formulas (4-2-1) to (4-2-3), the weight of each criterion is obtained as Table 22.

332

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 16 The description of weighting matrix. E/C

C1

C2

C3

C4

C5

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(0.0219,0.9578) (0.2014,0.6137) (1,0) (1,0) (0.0247,0.9569) (0.1274,0.8470) (0.4697,0.4074) (1,0) (0.1596,0.7686) (1,0) (0.2403,0.6744) (0.0565,0.9238) (0.0980,0.8979) (1,0) (0.3133,0.6829)

(0.1194,0.8063) (0.1511,0.7327) (0.0304,0.8054) (0.0329,0.8712) (0.0597,0.8282) (0.316,0.587) (0.0589,0.7444) (0.294,0.6487) (0.0561,0.7046) (0.2367,0.7445) (0.0852,0.8347) (0.0442,0.5376) (0.0580,0.9297) (0.0755,0.9133) (0.1241,0.8632)

(1,0) (1,0) (0.0626,0.9337) (0.1423,0.8256) (0.3670,0.6436) (1,0) (0.2384,0.7263) (1,0) (0.0880,0.8400) (0.1490,0.6377) (0.0151,0.6773) (1,0) (0.1027,0.8932) (0.2522,0.7153) (1,0)

(0.0272,0.7503) (1,0) (0.118,0.7984) (0.0024,0.4373 (1,0) (0.0345,0.703) (1,0) (0.0712,0.913) (1,0) (0.0352,0.8540) (1,0) (0.0587,0.7858) (1,0) (0.2068,0.7188) (0.2392,0.7171)

(0.1156,0.7062) (0.0178,0.8148) (0.052,0.9083) (0.2261,0.7703) (0.0334,0.5973) (0.1419,0.8408) (0.1466,0.8108) (0.6795,0) (0.0166,0.8434) (0.113,0.8477) (0.0229,0.7845) (0.1995,0.7856) (0.0009,0.8713) (0.1238,0.7903) (0.2944,0.6167)

Table 17 The description of D ∗ (ei ). D ∗ ( ei )\E

e1

e2

e3

e4

e5

e6

e7

e8

0.5864 e9 0.5682

0.3652 e10 0.5716

0.6005 e11 0.5619

0.5897 e12 0.5903

0.5091 e13 0.6106

0.5464 e14 0.5409

0.4722 e15 0.4803

0.2071

D∗ (μi )\E

e1

e2

e3

e4

e5

e6

e7

e8

The value of D ∗ (ei ) D∗ ( ei )\E The value of D ∗ (ei )

0.2887 e9 0.3130

0.5151 e10 0.3275

0.2614 e11 0.3481

0.3702 e12 0.3433

0.3508 e13 0.2429

0.3528 e14 0.3138

0.4176 e15 0.3736

0.6407

The value of D (ei ) D ∗ (ei )\U The value of D ∗ (ei ) ∗

Table 18 The description of D∗ (ei ).

Table 19 The description of δ ∗ (ei ).

δ ∗ ( e i )\E

e1

e2

e3

e4

e5

e6

e7

e8

The value of δ ∗ (ei ) δ ∗ ( e i )\E The value of δ ∗ (ei )

−2.3815 e9 −2.2555

−0.9598 e10 −2.2494

−2.4922 e11 −2.1703

−2.2702 e12 −2.3148

−1.9110 e13 −2.5697

−2.0882 e14 −2.1224

−1.6289 e15 −1.7363

0

Through the weight of each criterion obtained, by means of the formula (4-2-4), the weighting matrix is listed as Table 23. Since the five characteristics are benefit attributes, using the formulas (4-2-5) and (4-2-6), the value of positive and negative ideal solutions are listed as: ,0 ) B+ = (1C,0 ) + (0.3157C,0.5379 ) + (1C,0 ) + (1C,0 ) + (0.6741 ; C 1

2

3

4

5

B− = (0.0219C,0.9577 ) + (0.030C,0.9297 ) + (0.0151C,0.9337 ) + (0.0024C,0.9118 ) + (0.0009C,0.9085 ) . 1 2 3 4 5 The distance between each alternative and the positive ideal solution is calculated as Table 24 by means of the formula (4-2-7). According to the formula (4-2-8), the distance between each alternative and the negative ideal solution is obtained as Table 25. Finally, using the formula (4-2-9), the intimate function value of δ ∗ (ei ) for alternative ei is obtained as Table 26. The ranking of 15 nanomaterials is listed as: e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 . Thus, the nanomaterial e8 is selected.

Remark 4.1. Through our research, we make a conclusion as follows: (1) The ranking order of the nanomaterials is different using different logical operators. But, the optimal nanomaterial is the same that is e8 . This shows that our approach is feasible and flexible. The decision-makers can according to their actual situation to choose the logical operators when dealing with the complex problems using our proposed method. (2) It is easy to verify that the algorithmic complexity of our proposed method is o(mn2 ).

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339

333

Table 20 The description of (E, aprN(0C.,3I,0.4) (Ck )). 1

Tw

E/aprN(0C.,3I,0.4) (Ck )

aprN(0C.,3I,0.4) (C1 )

aprN(0C.,3I,0.4) (C2 )

aprN(0C.,3I,0.4) (C3 )

aprN(0C.,3I,0.4) (C4 )

aprN(0C.,3I,0.4) (C5 )

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(0.3,0.4) (0.6797,0.0844) (1,0) (1,0) (0.3,0.4) (0.4984,0.4) (0.9597,0.0106) (1,0) (0.5853,0.2638) (1,0) (0.7513,0.1361) (0.3,0.4) (0.4060,0.4) (1,0) (0.8509,0.1450)

(0.4593,0.3530) (0.5472,0.2221) (0.3,0.3510) (0.3,0.4) (0.3,0.4) (0.8407,0.0760) (0.3,0.2399) (0.8143,0.1233) (0.3,0.1839) (0.7293,0.24) (0.3500,0.4) (0.3,0.0497) (0.3,0.4) (0.3160,0.4) (0.4733,0.4)

(1,0) (1,0) (0.3,0.4) (0.5497,0.3692) (0.8988,0.1012) (1,0) (0.7572,0.1897) (1,0) (0.3804,0.4) (0.5678,0.0965) (0.3,0.1320) (1,0) (0.4308,0.4) (0.7792,0.1752) (1,0)

(0.3,0.2348) (1,0) (0.4694,0.3212) (0.3,0.0154) (1,0) (0.3,0.1690) (1,0) (0.3112,0.4) (1,0) (0.3,0.4) (1,0) (0.3,0.2963) (1,0) (0.6892,0.1890) (0.7482,0.1868)

(0.4505,0.1835) (0.3,0.3685) (0.3,0.4) (0.7133,0.2802) (0.3,0.0811) (0.5258,0.4) (0.5383,0.3757) (0.9961,0) (0.3,0.4) (0.4427,0.4) (0.3,0.3063) (0.6619,0.3085) (0.3,0.4) (0.4749,0.3176) (0.8173,0.0948)

1

Tw

1

Tw

1

Tw

1

Tw

1

Tw

1

Tw

Table 21 ( 0.3,0.4 ) The description of (E, aprNC ,Tw (Ck )). 1

( 0.3,0.4 )

E/aprNC ,Tw 1

(Ck )

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

( 0.3,0.4 )

aprNC ,Tw 1

(C1 )

(0.7169,0.2534) (0.6797,0.0844) (1,0) (1,0) (0.6892,0.3052) (0.5678,0.4) (0.9597,0.0106) (1,0) (0.9037,0.0666) (1,0) (0.864,0.1063) (0.7169,0.2534) (0.9597,0.0106) (1,0) (0.8509,0.1450)

( 0.3,0.4 )

aprNC ,Tw 1

(C2 )

(0.5948,0.3124) (0.5472,0.2221) (0.5038,0.2820) (0.3391,0.3104) (0.3,0.4) (0.8407,0.0760) (0.3,0.2399) (0.8143,0.1233) (0.509,0.1839) (0.7293,0.24) (0.4756,0.2937) (0.7046,0.0497) (0.5472,0.2221) (0.3716,0.3016) (0.4733,0.4)

( 0.3,0.4 )

aprNC ,Tw 1

( 0.3,0.4 )

(C3 )

aprNC ,Tw 1

(1,0) (1,0) (0.8509,0.1497) (0.8286,0.1714) (0.9072,0.1012) (1,0) (0.7572,0.1012) (1,0) (0.9618,0.0382) (0.8192,0.0965) (0.9284,0.0716) (1,0) (1,0) (0.8509,0.1491) (1,0)

(C4 )

(0.7572,0.2348) (1,0) (0.9597,0.0403) (0.825,0.0154) (1,0) (0.558,0.1690) (1,0) (0.44,0.4) (1,0) (0.6411,0.3589) (1,0) (0.7572,0.2428) (1,0) (0.9383,0.0617) (0.7482,0.1868)

( 0.3,0.4 )

aprNC ,Tw 1

(0.6794,0.1835) (0.5485,0.3636) (0.6856,0.2439) (0.7133,0.262) (0.5655,0.0811) (0.5334,0.4) (0.5383,0.3466) (0.9961,0) (0.5103,0.1808) (0.8153,0.1808) (0.5655,0.3036) (0.6619,0.3085) (0.5655,0.3466) (0.6682,0.2439) (0.8173,0.0948)

Table 22 The description of ωi . Weight

ω1

ω2

ω3

ω4

ω5

The value of ω i

0.1979

0.2065

0.1923

0.2011

0.2021

(C5 )

Table 23 The description of weighting matrix. E/C

C1

C2

C3

C4

C5

e1 e2 e3 e4 e5 e6 e7 e8 e9 e10 e11 e12 e13 e14 e15

(0.0219,0.9577) (0.2017,0.6131) (1,0) (1,0) (0.0248,0.9568) (0.1276,0.8467) (0.4704,0.4066) (1,0) (0.1599,0.7682) (1,0) (0.2407,0.6739) (0.0566,0.9236) (0.0980,0.8977) (1,0) (0.3138,0.6824)

(0.1193,0.8065) (0.1510,0.7329) (0.0303,0.8055) (0.0328,0.8713) (0.0596,0.8283) (0.3157,0.5873) (0.0588,0.7446) (0.2937,0.6490) (0.0560,0.7049) (0.2365,0.7447) (0.0851,0.8348) (0.0442,0.5379) (0.0580,0.9297) (0.0754,0.9134) (0.1240,0.8633)

(1,0) (1,0) (0.0626,0.9337) (0.1423,0.8256) (0.3670,0.6437) (1,0) (0.2383,0.7264) (1,0) (0.0880,0.8400) (0.1490,0.6378) (0.0151,0.6774) (1,0) (0.1027,0.8933) (0.2521,0.7153) (1,0)

(0.0276,0.7472) (1,0) (0.1196,0.7958) (0.0024,0.4321) (1,0) (0.0350,0.6994) (1,0) (0.0722,0.9118) (1,0) (0.0357,0.8520) (1,0) (0.0595,0.7830) (1,0) (0.2094,0.7153) (0.2422,0.7137)

(0.1140,0.7098) (0.0175,0.8173) (0.0512,0.9095) (0.2232,0.7732) (0.0329,0.6018) (0.1400,0.8429) (0.1446,0.8205) (0.6741,0) (0.0163,0.8455) (0.1115,0.8497) (0.0225,0.7873) (0.1968,0.7884) (0.0009,0.8730) (0.1221,0.7931) (0.2908,0.6211)

334

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 24 The description of D ∗ (ei ). D ∗ ( ei )\E

e1

e2

e3

e4

e5

e6

e7

e8

0.5880 e9 0.5662

0.3634 e10 0.5720

0.6001 e11 0.5602

0.5907 e12 0.5913

0.5064 e13 0.6185

0.5473 e14 0.5407

0.4702 e15 0.4809

0.2095

D∗ ( ei )\E

e1

e2

e3

e4

e5

e6

e7

e8

The value of D ∗ (ei ) D∗ ( ei )\E The value of D ∗ (ei )

0.2883 e9 0.3155

0.5176 e10 0.3274

0.2621 e11 0.3504

0.3715 e12 0.3427

0.3520 e13 0.2456

0.3529 e14 0.3146

0.4199 e15 0.3734

0.6389

The value of D (ei ) D ∗ ( ei )\E The value of D ∗ (ei ) ∗

Table 25 The description of D∗ (ei ).

Table 26 The description of δ ∗ (ei ).

δ ∗ ( e i )\E

e1

e2

e3

e4

e5

e6

e7

e8

The value of δ ∗ (ei ) δ ∗ ( e i )\E The value of δ ∗ (ei )

−2.3557 e9 −2.2090

−0.9248 e10 −2.2181

−2.4544 e11 −2.1255

−2.2383 e12 −2.2860

−1.8668 e13 −2.5202

−2.0599 e14 −2.0884

−1.5874 e15 −1.7113

0

Table 27 The description of the ranking order of 15 nanomaterials for the case 1. The values of ϑ

The ranking order of 15 nanomaterials

(0,1) (0.1,0.9) (0.2,0.8) (0.3,0.7) (0.4,0.6) (0.5,0.5) (0.6,0.4) (0.7,0.3) (0.8,0.2) (0.9,0.1) (0.9,0)

e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e12 e4 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e12 e4 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e12 e4 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13

Table 28 The description of the ranking order of 15 nanomaterials for the case 2. The values of ϑ

The ranking order of 15 nanomaterials

(0,1) (0.1,0.9) (0.2,0.8) (0.3,0.7) (0.4,0.6) (0.5,0.5) (0.6,0.4) (0.7,0.3) (0.8,0.2) (0.9,0.1) (0.9,0)

e8 e2 e7 e15 e5 e6 e11 e14 e9 e10 e12 e4 e1 e3 e13 e8 e2 e7 e15 e5 e6 e11 e14 e9 e10 e12 e4 e1 e3 e13 e8 e2 e7 e15 e5 e6 e11 e14 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13

4.5. Sensitivity analysis According to known research, we know that the existence of variable precision value ϑ can effectively deal with various types of noise data of intuitionistic fuzzy rough sets. So, what role does it play in our approach? Whether the variable precision values will affect our decision results. To solve this problem, in this section, we select 10 different values of ϑ to study whether the change of the value of ϑ will affect the ranking results of two cases on the basis of different logical operators. The results of two cases with different values of ϑ are given as Tables 27 and 28. It is noted that v1 = {(0, 1 ), (0.1, 0.9 ), (0.2, 0.8 ), (0.3, 0.7 ), (0.4, 0.6 ), (0.5, 0.5 ), (0.6, 0.4 )}, v2 = {(0.7, 0.3 ), (0.8, 0.2 ) , (0.9, 0.1 ), (0.9, 0 )} are two families of IF sets. When ϑ ∈ v1 , the ranking results are the same. Also, when ϑ ∈ v2 , the ranking results are also the same.

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Fig. 4. The comparison among the results of our proposed method in the case 1 with different values of ϑ.

Fig. 5. The comparison among the outcomes of our prosed method in the case 2 with different values of ϑ.











It is noted that v1 = {(0, 1 ), (0.1, 0.9 )}, v2 = {(0.2, 0.8 )}, v3 = {(0.3, 0.7 ), (0.4, 0.6 ), (0.5, 0.5 ), (0.6, 0.4 )} and v4 =

{(0.7, 0.3 ), (0.8, 0.2 ), (0.9, 0.1 ), (0.9, 0 )} are four families of IF sets. When ϑ ∈







v 1 , v2 , v3



v4 ,

and the ranking orders are shown as Fig. 5. From Tables 27 and 28 and Fig. 4 and 5, we find that although the result of case 1 (or case 2) is changing with the changes of the value of ϑ, the optimal nanomaterial is the same that is nanomaterial e8 . For the case 1, it is easy to find that the sorting results of most nanomaterials remain the same and only the ranking results of e10 , e9 and e4 , e12 are different. With the value of ϑ increases, the ranking order of all nanomaterials tends to be stable. When the value of ϑ is not less than (0.7,0.3), the ranking order is remain the same, that is, e8 e2 e7 e15 e5 e6 e14 e10 e9 e4 e12 e1 e3 e13 . For the case 2, also the ranking orders e8 e2 e7 e15 e5 e6 and e1 e3 e13 are remain the same, but the ranking of nanomaterials e14 e9 e10 e12 e4 is changing. But when the value of ϑ gradually increases, the sorting result tends to be stable. From Table 28 and Fig. 5, we obtain that when the value of ϑ is not less than (0.7,0.3), the ranking result of all nanomaterials remain the same that is e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 . Above all, we make a conclusion that the difference in the ranking result is due to the change of the weight value caused by the selection of different variable precision of ϑ. It is easy to check that the value of weight of each criterion will change with the value of ϑ changes. Although only the change of ϑ will affect the ranking order of all nanomaterials, when the value of ϑ is not less than (0.7,0.3), the ranking order of all nanomaterials remains unchanged. That is to say, the existence of variable precision values will have a certain degree of influence on the sorting result, but the impact on the selection of the optimal alternative is negligible. 4.6. Comparative analysis For the MADM problems, many researchers proposed a lot of methods such as the IF OWA operators in [21], the IFHA operator method in [34], the GIFWA operator method, the GIFOWA operator method, the GIFHA operator method in [45], and so on. After the research of these methods, we find that some of them have some defects in solving these complex MADM problems. In order to make a comparison among our proposed method and the other methods in [14,21,30,34,42,45], under the situation in two cases, the results of these methods are listed as Table 29 and Figs. 6, 7, 8. It is noted that the IF TOPSIS method (1) in [1] denotes that the results are calculated using the weights given by the case 1 through the IF

336

L. Zhang, J. Zhan and Y. Yao / Information Sciences 517 (2020) 315–339 Table 29 The ranking results of different methods. Methods

The ranking results of 15 nanomaterials

The The The The The The The The The The The The Our Our

e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 e8 e2 e7 e15 e5 e6 e4 e11 e12 e14 e10 e9 e1 e3 e13 e8 e2 e7 e15 e5 e6 e4 e11 e14 e12 e10 e9 e1 e3 e13 e8 e15 e7 e2 e14 e4 e10 e6 e11 e9 e5 e1 e12 e3 e13 e15 e14 e8 e3 e7 e1 e4 e12 e10 e13 e11 e5 e2 e6 e9 e8 e2 e7 e15 e5 e6 e14 e11 e10 e9 e4 e12 e1 e3 e13 e8 e2 e7 e15 e5 e6 e14 e11 e9 e10 e4 e12 e1 e3 e13

IF OWA operators in [21] IFHA operators in [34] GIFWA operators in [45] GIFOWA operators in [45] GIFHA operators in [45] IFEIWA operators in [14] IFEIOWA operators in [14] IFEIHA operators in [14] IF TOPSIS method (1) in [1] IF TOPSIS method (2) in [1] extended IF TOPSIS method in [30] (I, T )-CIFRS method in [42] proposed method in case 1 proposed method in case 2

= e14 = e14 = e14 = e14 = e14 = e14 = e14 = e14

= e15 = e15 = e15 = e15 = e15 = e15 = e15 = e15

20 Our proposed method (case 1) Our proposed method (case 2)

Ranking

15

The IF TOPSIS method (1)

10 5 0

0

5

10

15

10

15

Alternatives 20 Our proposed method (case 1) Our proposed method (case 2)

Ranking

15

The IF TOPSIS method (2)

10 5 0

0

5 Alternatives

Fig. 6. The comparison among the outcomes of our proposed method in two cases with the IF TOPSIS method in [1].

20

Our proposed method (case 1) Our proposed method (case 2)

Ranking

15

The extended IF TOPSIS method

10 5 0

0

5

10

15

10

15

Alternatives 25 Our proposed method (case 1) Our proposed method (case 2)

20 Ranking

The (I,T)−CIFRS method

15 10 5 0

0

5 Alternatives

Fig. 7. The comparison among the outcomes of our proposed method in two cases with the extended IF TOPSIS method in [30] and the (I, T )-CIFRS method in [42].

TOPSIS method in [1]. Similarly, The IF TOPSIS method (2) in [1] denotes that the results are computed using the weights given by the case 2 through the IF TOPSIS method in [1]. Through Table 29 and Figs. 6, 7, 8, we make a conclusion as follows: (1) Under the situation as Example 4.1, the ranking orders of the methods in [14,21,34,45] are the same, that is, e1 = e2 = e3 = e4 = · · · = e15 . Using these methods, we can not choose the optimal nanomaterial. In other words, using these methods is invalid for solving the complicated MADM issues.

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337

Fig. 8. The comparison among the outcomes of our proposed method in two cases with the other methods in [14,21,34,45].

Table 30 The Spearman rank correlation coefficient value rs among our method and existing methods. rs The The The The Our Our

method (1) in [1] method (2) in [1] method in [30] method in [42] proposed method in case 1 proposed method in case 2

The method (1) in [1]

The method (2) in [1]

The method in [30]

The method in [42]

Case 1

Case 2

− 0.9964 0.8 −0.1286 0.925 0.9214

− − 0.8286 −0.125 0.9429 0.9393

− − − −0.0607 0.85 0.8393

− − − − −0.2179 −0.175

− − − − − 0.9964

− − − − − −

(2) The ranking results of our two cases are different. The difference in sorting results is due to the difference in logical operators. However, the optimal nanomaterial is the same, that is, e8 . This shows that our method can be applied to solve many complex MADM problems. It is noted that our approach is very flexible in which decision-makers can choose logical operators according the actual situation. (3) Although the ranking orders of our proposed method (two cases) and the IF TOPSIS method in [1] are different, the optimal nanomaterial is the same e8 . Special emphasis is that our weights of criteria is derived from a formula. The ranking order of all nanomaterials obtained by our method are more objective than the ranking order of all nanomaterials of the method in [1]. The values of the weights of criteria in [1] are given according to the experts’ suggestion. Sometimes, due to lack of experience or lack of rationality, it leads to misjudgment of information and make the results non-objective. In comparison, the optimal results obtained by our method are more objective and fair than the results of IF TOPSIS method in [1]. (4) The optimal nanomaterials of extended IF TOPSIS method in [30] and our proposed method are the same, i.e., e8 . However, the optimal nanomaterial of the (I, T )-CIFRS method in [42] is e15 . Based on the ranking results in Table 29, we further use the Spearman rank correlation analysis to show the degree of consistency among several existing decision methods [1,30,42] and our method. In general, the Spearman rank correlation coefficient between the two methods is greater than 0.8, which is said to be positively correlated or highly consistent. Based on Table 30, we find that the correlation coefficient between our method (Cases 1 and 2) and methods in [1,30] is higher than 0.8. This shows that our method and methods in [1,30] are highly similar. In addition, we see that the correlation coefficients among method in [42] and other methods are all negatively correlated. This shows that method in [42] has certain problems. In particular, we observe that the correlation coefficient among our method (Cases 1 and 2) and methods in [1] are higher than the correlation coefficient among method in [30] and methods in [1], and the correlation coefficient among our method (Cases 1 and 2) and method in [42] is lower than the correlation coefficient between method in [30] and method in [42]. This shows that our method is relatively better than method in [30]. In the following, we will give a discussion with respect to these methods in [14,21,30,34,42,45] and our proposed method. Remark 4.2. (1) Although these methods in [14,21,34,45] are valuable for solving some MADM problems, they have some shortcomings. Through our study, we find that these methods in [14,21,34,45] are invalid for solving some problems. For example, for each alternative ei (i = 1, 2, · · · , n ) there at least exists one attribute Cj ( j = 1, 2, · · · , m ) such that the value of C j (ei ) = 1L . Using these methods in [14,21,34,45], the ranking order of all nanomaterials is e1 = e2 = e3 = e4 = e5 = e6 = e7 = e8 = e9 = e10 = e11 = e12 = e13 = e14 = e15 . That is, using these methods, the decision makers can not choose the best nanomaterial to make operation. (2) For the IF TOPSIS method in [1], the weights of the criteria are given by experts or other tools for subjective weight calculation.

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(3) The optimal nanomaterials of the extended IF TOPSIS method in [30] and our raised method are the same, i.e., e8 . It is pointed that they are using different methods to obtain the weights of criteria. For the extended IF TOPSIS method in [30], they are using the maximizing methods to compute the optimal weight. While our proposed method takes advantage of the concept of approximation precision for CVPIFRS models to obtain the weights of criteria. (4) The best optimal nanomaterials of (I, T )-CIFRS method in [42] and our proposed method have some differences. The reason is that for (I, T )-CIFRS method in [42], they just learn from some idea of TOPSIS method not completely following the principle of the classical TOPSIS method. For example, for TOPSIS method, we need to calculate the distance between each alternative and the negative ideal solution (positive ideal solution) but the (I, T )-CIFRS method in [42] not to compute distance between each alternative and the negative ideal solution (positive ideal solution). It is noted that our proposed method not only computes distance between each alternative and the negative ideal solution (positive ideal solution) but also takes advantage of the rough sets to compute the weights of the criteria. Remark 4.3. The advantages of our approach can be stated in the following aspects: (1) The results obtained by our method are objective, accurate and close to actual needs. The reason is that the weight given by means of our proposed method is based on the characteristics of data and computed by formulas. It is noted that the weight of each criterion is not given by human; (2) Our approach is highly flexible and can be applied to a wide range of environments by means of different logical operators and through adjusting the value of ϑ; (3) Our proposed method can effectively solve many complicated problems that the methods in [21,34,45] can not deal with. In other words, the method put forward in this paper can be widely applied in MADM. 5. Conclusion Based on the works of Zhang et al. [42] and Zhao et al. [46], we construct four types of CVPIFSs as extensions of the RST models. The CVPIFRS models can be used to deal with many uncertain and vague data in an IF environment. These models generalize the equivalence relation to a more general IF neighborhood relationship, generalize the equivalence class to the IF covering and combine the idea of VPRS models to generalize the rough set models. Since these generalized models weaken many of the restrictive conditions of rough sets, they are more applicable. In Section 4, the approximation operators of CVPIFRS models are more accurate than the approximation operators of the models proposed in [42] under some special conditions. Due to the symmetry of N4C , apr

0L

N4C ,I2



= C 4 , apr

0L

N4C ,T2



= C 4 . Based on different logical operators, the proposed

models not only retain some important properties of the rough sets, but also have their own unique properties, such as duality. By means of the CVPIFRS models, many vague and inaccurate information in an IF environment can be processed. We take the advantages of the CVPIFRS models to propose an effective MADM method to select the optimal solution for the bone transplantation problem. Our proposed method introduces a good tool to obtain the weighs of attributes by combining approximate accuracy of CVPIFRS models. This method can avoid the problems caused by experts misjudgment. This approach allows decision-makers to make decisions based on obtained data without the support of experts. Different from some existing TOPSIS methods, IF TOPSIS based on CVPIFRS models that consider the relative significance of the distances from the positive and negative ideal points in an IF environment. The effectiveness, flexibility, and broad applicability of our approach are demonstrated by a comparative analysis and a sensitivity analysis. The decision-maker can choose different IF logical operators and IF variable precision values according to his or her actual situations and personal preferences to choose the optimal alternative. This phenomenon shows that our methodology is more flexible and adaptable than those methods in [21,34,45]. In the future work, we can investigate the CVPIFRS models from the aspects of residual lattice and topology based on the work in [22]. We can extend these models to the Pythagorean fuzzy (PF) environment [13,23] and generalize these IF neighborhoods to a general IF relation. By following the research about the decision-making method [13,24] in a PF environment, based on the idea of our given approach, we can introduce a decision-making tool for solving many complex problems in a PF environment. By drawing on the ideas of DUrso and Massari [11], we can take advantages of the extensions of RST to linear fit and regression analysis of many uncertain data. By combining our proposed CVPIFRS models with data driven technology [25], we can develop an intelligent software based on the characteristics of the data. Through these improved rough set models and MADM methods, more difficult problems can be effectively solved in the future. Declaration of Competing Interest The authors declared that they have no conflicts of interest to this work. Acknowledgments The authors are extremely grateful to the editor and five reviewers for their valuable comments and helpful suggestions which helped to improve the presentation of this paper. The research was partially supported by the NNSFC (grant numbers 6186601 11961025 11561023 11461025) and a Discovery Grant from NSERC, Canada.

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