#### Abstract

Industrial control system (ICS) attacks are usually targeted attacks that use the ICS entry approach to get a foothold within a system and move laterally throughout the organization. In recent decades, powerful attacks such as Stuxnet, Duqu, Flame, and Havex have served as wake-up calls for industrial units. All organizations are faced with the rise of security challenges in technological innovations. This paper aims to develop aggregation operators that can be used to address the decision-making problems based on a spherical fuzzy rough environment. Meanwhile, some interesting properties of idempotence, boundedness, and monotonicity for the proposed operators are analyzed. Moreover, we use this newly constructed framework to select ICS security suppliers and validate its acceptability. Furthermore, a different test has been performed based on a new operator to strengthen the suggested approach. Additionally, comparative analysis based on the novel extended TOPSIS method is presented to demonstrate the superiority of the proposed technique. The results show that the conventional approach has a larger area for information representation, better adaptability to the evaluation environment, and higher reliability of the evaluation results.

#### 1. Introduction

Many governments are launching initiatives to encourage the implementation of electronic and manufacturing innovations, including Germany's industry 4.0 systems, the United States' reindustrialization, and China's “Made in China 2025” strategy to advance next-generation information technology. All of these have has motivated the continuous development of industrial control systems (ICS). All of these have prompted the development of industrial control systems (ICSs) to proceed. Information technology, while adding new development strength to ICS, it also introduces new security flaws. ICS plays a crucial role in the national economy and people's livelihood of crucial national infrastructure. Furthermore, in recent years, a number of notable ICS attacks, such as Stuxnet, Duqu, Flame, and Havex, have raised the concern for authorities and industry sectors. As a result, many manufacturers are working to keep an increasingly open ICS safe. And evaluating ICS security suppliers and selecting the best one are a vital part of management decisions. A typical multi attribute decision-making (MADM) problem is selecting an ICS security supplier. MAGDM theories and approaches have stimulated the interest of many researchers in the field of operational research and decision sciences, and significant achievements have been highlighted [1–8]. Scientists have developed several techniques to address the MADM problem, such as the TODIM (an acronym in Portuguese for Interative Multi-criteria Decision Making) method [9], MABAC (multi-attributive border approximation area comparison) method [10], EDAS (evaluation based on distance from average solution) method [11], VIKOR (VIekriterijumsko KOmpromisno Rangiranje) method [12], GRA (gray relative analysis) method [13], CODAS (combinative distance-based assessment) method [14], GLDS method [15], QUALIFLEX (qualitative flexible multiple criteria) method [16], and so on. However, as technology advances, experts have observed that the decision-making environment has become more complicated, increasing ambiguity and uncertainty. Trying to solve MADM problems solely with standard tools would fall short of the practical needs [17]. Therefore, several researchers address MADM challenges by combining fuzzy numbers with classical MADM approaches [18]. In this study by Zadeh [19] the fuzzy set syntax was introduced. The membership degree explains “obviously yes” and “certainly no” in its feature. Afterward, the concept of FS has been continually explored and expanded. Atanassov [20] introduced intuitionistic fuzzy sets (IFS) and a novel extension of IFSs known as Pythagorean fuzzy sets (PFSs) [21]. This is well known that the limitation of IFSs is the sum of membership and non-membership that must be less than or equal to one, but PFSs fulfill the criteria that the square sum of membership and non-membership must not be greater than one. PFS provides a broader range of information than IFSs, and PFS-based MAGDM approaches have become a novel and active study field [22–25]. Yager [26] introduced the concept q-rung orthopair fuzzy (q-ROF) set as an extension to the conventional IFS set. The limitation of the q-ROF set is that the sum of the qth-power of membership and non-membership is less than or equal to 1. Cong and Kreinovich [27] established picture FSs and described operations and relations on them. PFSs deal with three functions: membership, neutral, and non-membership. Manemaran et al. [28] developed a temporal picture fuzzy soft set (PFSS) and addressed its related concepts. Khan et al. [29] presented generalized PFSSs and their application in DM systems. Further, Ashraf et al. [30] introduced the idea of cubic PFSs. Abdullah et al. [31] implemented the cubic PFS approach to address the MADM problem and show it with a numerical example of a petroleum circulation center evaluation problem to demonstrate the usage and application of the proposed ranking technique. It should observe that both IFSs and PFS still have some limitations, although they have been efficient when dealing with complex fuzzy information in some practical applications. The novel idea of spherical fuzzy sets (SFSs) was introduced by Ashraf et al. [8], an advanced tool of FSs, IFSs, and PFSs. They investigated the fundamental characteristics of SFSs and compared them with those of PFSs. Graphically, the spaces of spherical and picture membership grades are investigated. They implemented the aforementioned concept as a practical application to demonstrate the problem of evaluating a food circulation center. Kutlu [32] extended the TOPSIS (Technique for Order of Preference by Similarity to Ideal Solution) method to spherical fuzzy TOPSIS and presented an illustrative example in the MADM problem. Kahraman [33] described q-spherical fuzzy sets and discussed their application in the MADM problem. Zeng [34] discussed T-spherical fuzzy Einstein interactive aggregation operators and their application to selecting photovoltaic cells. Emergency decision making based on SFSs to address the uncertainty in COVID-19 situation is discussed in [6, 7].

The rough set (RS) theory, proposed by Professor Pawlak in 1982, is an essential mathematical tool for dealing with ambiguous, inconsistent, and incomplete data and information [35]. The fuzzy rough set (FRS) can be combined with RSs to manage information with continuous attributes and investigate data inconsistencies. The FRS model has shown to be highly effective in many application areas because it is a powerful tool for analyzing inconsistent and ambiguous information. FRS set theory is a rough set theory extension that handles continuous numerical attributes [36]. The significance of the FRS theory may be observed in a variety of applications. Pan et al. [37] established the additive consistent fuzzy preference relation to improve the rough set model of the fuzzy preference relation. Li et al. [38] suggested a practical FRS approach for robust feature selection. Feng et al. [39] reduced multi granulation using uncertainty measures based on FRSs, eliminating the negative and positive regions. Sun et al. [40] used a constructive technique to provide three multi granulation FRSs over two universes. In this study by Liu [41], in the framework of interval-valued fuzzy and fuzzy probabilistic approximation space models, a decision-theoretic RS model was investigated. Zhang used axiomatic and constructive techniques to integrate rough set theory and interval-valued fuzzy set theory and [42] presented a novel paradigm based on FRSs with extended interval values. Zhang et al. [43] offered FR based feature selection based on information entropy to minimize heterogeneous data. By combining granular variable precision FRSs and general fuzzy relations, Wang and Hu [44] provided a random set of fuzzy relationships. Vluymans et al. [45] introduced a new type of classifier for unbalanced multi-instance data based on FRS theory. Shaheen et al. [46] described the application of generalized hesitant fuzzy rough sets (GHFRS) in risk analysis. Khan et al. [47] addressed the use of a probabilistic hesitant FRS in a decision support system. Tang et al. [48] proposed the decision-theoretic rough set model with q-rung orthopair fuzzy information, as well as its application in evaluating stock investments. Liang et al. [49] suggested q-Rung orthopair fuzzy sets on decision-theoretic rough sets for three-way decisions under group DM. Zhang et al. [50] proposed group DM using incomplete q-rung orthopair fuzzy preference relations. For MADM, Hussain et al. [51] presented a covering-based q-rung orthopair fuzzy rough set model hybrid with the TOPSIS approach. In practice, these extensions of the q-rung orthopair fuzzy rough set successfully handle DMs' evaluation values in MAGDM problems. Some decision-making techniques are discussed in [53–58].

In this research, motivated by the above discussion, we plan to introduce a new concept of spherical fuzzy rough set (SFRSs). Also, the hybrid technique of aggregation operators and expanded TOPSIS procedure under spherical fuzzy rough setting is presented, in order to benefit of the advantages of the TOPSIS method and SFSs. As a consequence, the SFS is the generalized structure of fuzzy set like IF set and PyF set. Thus SFRS is capable of handling more uncertainty than FS, IF set, PyF set, and rough set. Therefore, in this paper, a novel improved TOPSIS-based method and novel algebraic norm-based aggregation operators are established to address with such circumstances of unknown weight information of both DMs and criteria weights and to solve the MAGDM problem after computing all the weights. In order to solve the decision-making problems (DMPs), choosing the ideal opinion, which is better connected to each DMs matrix, is quite essential. In the presented procedure, ideal opinion is nominated under the SFR average method. Generalized distance measure is established to find the differences between two SFRSs. In the presented SFR TOPSIS and SFR aggregation operators for solving MAGDM problems, generalized distance measures-based entropy measure is introduced to find out the criteria weights under SFR information used in this paper. The challenges in the industrial control systems security and sustainability are addressed by using proposed advance methodology.

In summary, the main contributions of the present study are follows:(1)Novel idea of spherical fuzzy rough aggregation operators namely spherical fuzzy rough weighted averaging, spherical fuzzy rough ordered weighted averaging and spherical fuzzy rough hybrid weighted averaging operators is introduced and their basic operational laws are investigated.(2)A case study of industrial control security service provider selection is also presented to demonstrate the applicability of the established operators.(3)To validate the findings, different test on aggregation operators are implemented.(4)Finally, comparisons with the spherical fuzzy rough TOPSIS method are made to interpret the outcomes. The ranking of the obtained results is presented graphically.

#### 2. Preliminaries

In this section, we will put forward some basic literature concerning IFS, PFS, SFS, relation and rough sets, which will be helpful for subsequent sections.

*Definition 1. *(see [20]). Consider a universal set . An IFS on a set is defined aswhere and denotes the MemD and NMemD of an object to the set such that For an alternative , is known as the degree of hesitancy.

*Definition 2. *(see [27]). Assume a universal set . A PFS on a set is of the form as follows:where and denotes the positive membership (PM), neutral membership (NeuM) and negative membership (NM) of an object to the set such that For an alternative , is known as the degree of hesitancy.

*Definition 3. *(see [8]). Assume a universal set . A SFS on a set is of the form as follows:where , and denote the PM, NeuM and NM of an object to the set such that For an alternative , the degree of hesitancy is given as .

Consider denotes known as a SP values (SPV) if there is no confusion. The family of subsets of SFS is represented by .

*Definition 4. *(see [8]). Let and be any two SFVs. Then the basic operations are given as, for .(i).(ii),(iii) if for all and ,(iv),(v),(vi),(vii).

*Definition 5. *Suppose a universal set and let be any binary relation. Then(i) is reflexive if , ;(ii) is symmetric if , then ;(iii) is transitive if , and , then .

*Definition 6. *Let be a universal set and be an arbitrary binary relation on set . Then denotes the set value mapping given as , whereThe set is said to be a successor neighborhood of an element with respect to . The pair represents the approximation space with respect to Consider an approximation space and be any subset of , then lower and upper approximation of is represented and defined as follows:Therefore, the pair is called rough set. Then the mapping denotes the approximation operators.

#### 3. Spherical Fuzzy Rough Sets

SFS is the most significant generalization of IFS, PyFS and PFS which provide more space for experts for assigning values to PM, NeuM and NM. Here we will present the hybrid notion of SFS and rough sets to get the novel concept of SF rough set. The new score and accuracy mapping are defined for the developed model and studied their desirable properties of the presented model with detail.

*Definition 7. *Suppose a universal set and let be SF relation. Then(i) is reflexive if , , and , ;(ii) is symmetric if , , , and ;(iii) is transitive if , , , and .Alternatively, the relation , is transitive if it holds the following:

For all and (a), and ,(b), and ,(c), and .

*Definition 8. *Suppose a universal set and let be any SF relation. The pair denotes a SF approximation space. Let be any subset of i.e. . Then based on SF approximation space , the upper and lower approximations of are denoted by and and are given as follows:wheresuch that and. As and are SFSs, so are upper and lower approximation operators. Therefore, the pair is called SF rough set. For simplicity is denoted as known as SF rough number (SFRN).

*Example 1. *Consider a fixed set . Let be an SF approximation space and let be a SF relation which is given in Table 1,

Consider the professional experts presented the optimum decision normal object in the form of SFS, that is,Now to calculate and , we haveSo, .

Likewise, we can calculate the others values of upper and lower approximation:

*Definition 9. *Assume two SFRNs and . Then some basic operations on them are given as follows:(i);(ii);(iii);(iv);(v);(vi)for ;(vii) for;(viii)where denotes the complement of SF rough approximation operators , i.e. and (ix)iff.

*Definition 10. *Let the SFRN . Then the score function defined as:The accuracy of the SFRN for is defined as:Consider two SFRNs and , then(i)If , then ,(ii)If , then ,(iii)If , then(a)If , then ,(b)If , then ,(c)If , then .

Proposition 1. *Assume for any two SFRNs and with respect to SF approximation space . Then the following properties are hold for SFRNs.*(i)*, where is the complement of ;*(ii)*(iii)**;*(iv)*;*(v)*If , then ;*(vi)*;*(vii)*.*

#### 4. Spherical Fuzzy Rough Averaging Aggregation Operators

Aggregation operators have the ability to aggregative the collective information of several professional experts into a single value. Here we will put forward the hybrid study of SFRS and averaging aggregation operators to get SFR averaging aggregation operators and presented their desirable properties.

##### 4.1. Spherical Fuzzy Rough Weighted Averaging Aggregation Operators

This subsection is devoted for the study of SPRWA aggregation operators and presented the fundamental properties of SFRWA operators.

*Definition 11. *Let be the collection of SFRNs. Assume be the weight vectors with and . Then the SFRWA aggregation operators are given as follows:The aggregation result for the above definition is described in Theorem 1.

Theorem 1. *Consider be the collections of SFRNs. Consider the weight vectors with and . Then the aggregation result for SFRWA operator is given as follows:*

*Proof. *To get the required proof, we will use mathematical induction.

By using the operation law, we haveSuppose, thenThus, the result holds for.

Now suppose that result is true for Let us show that the result holds for , thus we haveThis show that the result is true for . Therefore, the result is hold for all

Based on Theorem 1, and are SFRNs. Therefore, and are also SFRNs. Thus, it is clear that S is also a SFRN based on SF approximation space.

Some fundamental and desirable properties of SFRWA operator are presented in Theorem 2.

Theorem 2. *Let be the collections of SFRNs with with and . Then SFRWA operator satisfies the following the properties.*(i)*Idempotency: if where, then *(ii)*Monotonicity: consider that be another family of SFRVs with and , then *(iii)*Boundedness: let and , then *

*Proof. *(i)Idempotency: as , where For all . Therefore, Hence,(ii)Monotonicity: since and and and , Further, Next, Similarly, we can show that Thus, from the above calculation, it is cleared that Therefore,(iii)Boundedness: proof is easy and follow from and .

##### 4.2. Spherical Fuzzy Rough Ordered Weighted Averaging Aggregation Operators

This subsection is devoted for the study of SPROWA aggregation operators, which weigh the ordered position of the argument. Then we have presented the fundamental properties of SFROWA operators.

*Definition 12. *Let be the collection of SFRNs. Assume be the weight vectors with and . Then the SFROWA aggregation operators are given as follows:The aggregation result for the above definition is described in Theorem 3.

Theorem 3. *Consider be the collections of SFRNs. Consider the weight vectors with and . Then the aggregation result for SFROWA operator given is as follows:where denotes the largest value of permutation from the collection .*

*Proof. *Proof is easy and followed from Theorem 1.

Some fundamental and desirable properties of SFROWA operator are presented in Theorem 4.

Theorem 4. *Let be the collections of SFRNs with with and . Then SFROWA operator satisfies the following the properties.*(i)*Idempotency: if where. Then *(ii)*Monotonicity: consider that be another family of SFRVs with and , then*(iii)*Boundedness: let and, then *

*Proof. *Proofs directly followed from Theorem 2.

##### 4.3. Spherical Fuzzy Rough Hybrid Averaging Aggregation Operators

SFRHA operators are the significant generalization of SFRWA and SFROWA aggregation operators because it has the ability to weight both the ordered position and the argument itself. In this subsection, we will investigate the study of SPRHA aggregation operators and presented the fundamental properties of SFRHA aggregation operators.

*Definition 13. *Let be the collection of SFRNs with weight vector such that and .Consider be the associated weight vectors with and . Then the SFRHA aggregation operators are given as follows:The aggregation result for above definition is described in Theorem 3.

Theorem 5. *Consider be the collections of SFRNs with weight vector such that and . Consider *