TOPSIS Method Based on Correlation Coefficient and Entropy Measure for Linguistic Pythagorean Fuzzy Sets and Its Application to Multiple Attribute Decision Making

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Introduction
e multiple attribute decision making (MADM) refers to the process in which some alternatives with respect to multiple attributes are rated and ranked [1][2][3][4][5].e multiple attribute decision-making processes happen in various elds [6][7][8][9].In the initial stage of decision-making evolution, crisp numbers are usually used to rate the alternatives [10].With the growing complexity of decision-making problems, human beings do not have the capability to express the vague and ambiguous information [11][12][13].To model this kind of information, fuzzy set (FS) was initially put forward in [14].
e FS only expresses the membership degree (MD), but it does not model the dissenting opinions, namely, nonmembership degree (NMD).To overcome this defect, the concept of intuitionistic fuzzy sets (IFSs) [15] was proposed, which is composed of MD and NMD. e sum of MD and NMD in an IFS should satisfy that MD + NMD ≤ 1.To improve the modeling capability, Pythagorean fuzzy set (PFS) [16,17] was put forward by Yager et al., in which the MD and the NMD must satisfy that MD 2 + NMD 2 ≤ 1.Because of its superiority, the concept of PFS has attracted much attention from researchers [18][19][20][21][22].It has been widely applied to various elds, such as pattern recognition [23,24], medical diagnosis [25,26], risk attitudinal assignment [27], and fault diagnosis [28].
In some situations, qualitative terms are preferred to quantify the evaluation information of the objects [29][30][31][32][33].A fuzzy linguistic approach was proposed by Zadeh [34].It can model the evaluation information of decision makers (DMs) as the qualitative form [35,36].To extend the modeling capability of IFSs into the qualitative setting, Zhang [37] proposed the linguistic intuitionistic fuzzy set (LIFS), which is produced by combining IFS with linguistic term set (LTS) [38][39][40].In the LIFSs, each linguistic intuitionistic fuzzy number (LIFN) consists of MD and NMD whose values are expressed using the linguistic terms in the LTS.e MD and NMD should satisfy that I(MD) + I(NMD) ≤ τ, where I(MD) and I(NMD) denote the subscripts of MD and NMD and τ is the cardinality of LTS.Inspired by the LIFS, Garg put forward the linguistic Pythagorean fuzzy sets (LPFSs) [41], in which each linguistic Pythagorean fuzzy number (LPFN) also uses two linguistic terms to express its MD and NMD. e main difference between LPFS and LIFS is that the square sum of the subscripts of MD and NMD in the LPFN is less than or equal to the square of the cardinality of the LTS.As shown in Figure 1, the distribution area of the LPFN is larger than that of the LIFN.e LIFN may only fall into the blue area, while the LPFN may fall into the blue area or yellow area.us, an LIFN is also an LPFN, while an LPFN may not be an LIFN.It indicates that the LPFS can provide DMs with more freedom to express their information.It has drawn the attention of researchers.For example, Lin et al. [42] put forward some novel interactional operational laws for computing LPFSs, based on which, the partitioned Bonferroni mean was extended for fusing LPFSs.
As for the MADM methods for LPFSs, there exist some deficiencies that are analyzed as follows: (1) ere exist few MADM methods that have been proposed to handle LPFSs.Until now, only Garg [41] and Lin et al. [42] proposed some aggregation operators for fusing LPFSs so as to rank the alternatives.e aggregation operators are noncompensatory MADM methods, which simply aggregate all the attribute values into a single value and cannot achieve trade-offs among conflicting attributes.
(2) e TOPSIS method [43][44][45] is a classical tool used to solve MADM problems.It belongs to the compensatory MADM methods.It has been extended to solve IFSs [46], interval-valued IFSs [47,48], and PFSs [49][50][51].However, the data structure contained in LPFSs is different from that of the previous fuzzy information.erefore, the previous TOPSIS methods cannot be directly used for LPFSs.Moreover, to the best of our knowledge, there are no research results that report the use of the TOPSIS method in the LPFSs.
(3) e classical TOPSIS method uses the distance measure to get ideal solutions [52].However, the TOPSIS method based on distance measure cannot distinguish ideal solutions in some special cases [53].
It is necessary to improve the TOPSIS method using other measure, such as correlation coefficient.
Moreover, the attribute weight information in the MADM problems is usually unknown.us, how to determine the attribute weight information is also a big challenge.
To overcome the above deficiencies, we propose a novel TOPSIS method for LPFSs, which combines the correlation coefficient and entropy measure.Our contributions can be summarized as follows: (1) We propose the equations of correlation coefficient and weighted correlation coefficient for LPFSs based on MD, NMD, and hesitance degree, and then their properties are discussed.(2) Two entropy measures are developed to measure the uncertainty degree of LPFSs considering MD and NMD, and then their properties are discussed.ese entropy measures are used to derive the attribute weight information for the MADM problems under the LPFS environment.(3) e proposed correlation coefficient and entropy measure are used to modify the classical TOPSIS method for proposing a novel TOPSIS method for LPFSs, which is called LPF-TOPSIS.e LPF-TOPSIS method is used to solve the MADM problems with unknown attribute weight information.(4) A case concerning the selection of firewall productions is studied to demonstrate the application of the proposed LPF-TOPSIS method.en, a case concerning the security evaluation of computer systems is provided to compare the proposed LPF-TOPSIS method with the existing decision-making methods based on aggregation operators [41,42].
Our study is different from the existing studies [41,42] in the following aspects: (1) e attribute weight information in the existing studies is given by DMs in advance, while in our study, the attribute weight information is determined using entropy measures of LPFSs.(2) e correlation coefficient and entropy measure have not been developed for LPFSs in the existing studies.Both of them are achieved in our study, and their properties are discussed.erefore, our study can help to enrich the theoretical basis of LPFSs and promote their wide applications in various fields.
(3) e aggregator operators in the existing studies are noncompensatory MADM methods, which are inefficient to solve MADM problems with conflicting attributes.e proposed LPF-TOPSIS method in our study is a compensatory MADM method.It can efficiently achieve trade-offs among conflicting attributes.
e remainder of this paper is organized as follows: Section 2 provides some basic knowledge about LPFSs and classical TOPSIS method.
e correlation coefficient and two entropy measures are devised for LPFSs in Section 3. A novel LPF-TOPSIS method based on correlation coefficient and entropy measure is developed in Section 4. A case concerning the selection of firewall productions is provided and comparative analyses are performed in Section 5. Finally, some valuable conclusions are drawn in Section 6.

Preliminaries
In this section, we briefly review some basic knowledge about LPFSs and classical TOPSIS method.

Linguistic Pythagorean Fuzzy Sets.
e LPFS was proposed by Garg [41] by combining PFS with LTS.Its definition is given as follows.
Definition 1 (see [41]).Given a reference set with a positive integer τ, then the mathematical form of an LPFS on X is defined as where s p (x) and s q (x) are two functions, which return the MD and NMD of the element x belonging to the LPFS A.
Each pair of MD and NMD (s p (x), s q (x)) is simplified as α � (s p , s q ), which is named an linguistic Pythagorean fuzzy number (LPFN).In the LPFNs, the MD s p and NMD s q should satisfy that 0 ≤ p ≤ τ, 0 ≤ q ≤ τ, and , then s h is named as the hesitance degree (HD) of this LPFN.
Garg [41] also put forward a comparison method consisting of score function and accuracy function to compare LPFNs.
Definition 2 (see [41]).Given an LPFN A � (s p , s q ) with s p and s q being the elements of a continuous LTS , then the score function is defined as and its accuracy function is defined as

√
. Suppose that there are two LPFNs A and B, if S(A) > S(B), then it can be considered as A > B. If S(A) � S(B), then their accuracy function values should be further compared.If To aggregate LPFNs A i � (s p i , s q i ), i � 1, 2, . . ., n, Garg [41] devised the linguistic Pythagorean fuzzy weighted averaging (LPFWA) and linguistic Pythagorean fuzzy weighted geometric (LPFWG) aggregation operators as follows.

Classical TOPSIS Method.
e TOPSIS is a classical MADM method based on ideal solutions [54].Its designing idea is that the optimal alternative should be closest to the positive ideal solution (PIS) and furthest away from the negative ideal solution (NIS) [55].ere is an MADM problem composed of m alternatives c i , i � 1, 2, . . ., m and n attributes u j , j � 1, 2, . . ., n.All the evaluation information of the alternatives concerning their attributes compose a decision matrix X � (x ij ) m×n .
e weight vector of attributes is ω � (ω 1 , ω 2 , . . ., ω n ) T .It satisfies 0 ≤ ω j ≤ 1 and  n j�1 ω j � 1. e decision-making process of the classical TOPSIS method can be summarized as follows: Step 1: normalize the decision matrix X � (x ij ) m×n to be R � (r ij ) m×n using Step 2: determine the PIS and NIS as Complexity where Ω b is a set containing benefit-type attributes and Ω c is a set containing cost-type attributes.
Step 3: calculate the distance between each alterative and PIS as and the distance between each alternative and NIS as Step 4: calculate the relative closeness of each alternative as Step 5: rank all the alternatives according to their values of relative closeness and select the one with the highest relative closeness as the optimal one.

Correlation Coefficient and Entropy Measure
In this section, the correlation coefficient is developed for LPFSs to measure the correlation degree and two entropy measures are proposed for LPFSs to measure the uncertainty degree.e distance measure cannot effectively determine the proximity of each alternative to ideal solutions in some special cases.In our study, we plan to replace the distance measure in the TOPSIS method with the correlation coefficient.e correlation coefficient is applied to measure the correlation degree between each alternative and ideal solution.
e attribute weight information in the MADM problems is usually unknown.e entropy measures can reflect the fluctuation of LPFSs in terms of the uncertainty degree.
erefore, the entropy measures can be used to derive the attribute weight information for the MADM problems under the LPFS environment.

Correlation Coefficients and eir
Properties.Suppose that there are a reference set then its information energy is defined as where s h A (x i ) denotes the hesitance degree of the ith LPFN.e correlation of between two LPFSs is defined as e correlation of LPFSs has the following properties: (

1) C(A, A) � T(A) (2) C(A, B) � C(B, A)
Based on the above information energy and correlation, the correlation coefficient of LPFSs is defined as follows.
LPFSs on X, then the correlation coefficient between them is defined as , then the correlation coefficient between them satisfies the following conditions: (2) According to Definition 5, it can be seen that 0 ≤ K 1 (A, B). en, we need to prove that K 1 (A, B) ≤ 1: According to Cauchy-Schwarz inequality
In the real applications, the attributes usually have different weights.us, the weight of each element x i in the reference set X should be considered when the correlation coefficient is computed.Suppose that ω � (ω 1 , ω 2 , . . ., ω n ) is the weight vector of the elements in X, which satisfies that ω i ≥ 0 and  n i�1 ω i � 1, then the weighted correlation coefficient between A and B is defined as is the weight vector of the elements in X, then the weighted correlation coefficient between them satisfies the following conditions:
(2) It can be noted that 0 ≤ K 2 (A, B).We need to prove that K 2 (A, B) ≤ 1.According to (4), we have According to Cauchy-Schwarz inequality, we have en, we have (3) It is straightforward.

Entropy Measures and eir Properties.
For the MADM problems, the weights of attributes are usually unknown due to the time pressure, the DMs' limited capability, and the 6 Complexity complexity of the considered problems.e entropy measure is an efficient tool to derive the weights of attributes [56].Ye [57] devised two entropy measures to determine the weights of attributes in the MADM problems with intervalvalued IFSs.Xia and Xu [58] put forward the entropy and cross-entropy measures for IFSs, based on which, the weights of attributes are computed.Xu and Xia [59] put forward the entropy and cross entropy for hesitant fuzzy sets and used the entropy measure to determine the weight vector of attributes.Nevertheless, the entropy measure of LPFSs has not been proposed.To measure the uncertainty degree of LPFSs, we propose two entropy measures for LPFSs and discuss their properties.
, and an LPFS A � 〈x i ,  s p A (x i ), s q A (x i )〉 | x i ∈ X}, two entropy measures for the LPFS are defined as where I(s p A (x i )) and I(s q A (x i )) denote the subscripts of the MD and NMD of the ith element x i .

Theorem 3. Given two LPFSs
, the entropy measures of them satisfy the following properties:

Proof
(1) To make the proof process be easy, let and y � I(s q A (x i )), then equations ( 19) and ( 20) can be transformed as us, to prove that equations ( 17) and ( 18) satisfy the property (11), we need to prove that the functions F 1 (x, y) and F 2 (x, y) are increasing with respect to its parameter x and decreasing for the parameter y.Taking the partial derivative of the functions F 1 (x, y) and F 2 (x, y) with respect to x and y yields To obtain the critical point of the functions F 1 (x, y) and F 2 (x, y), let zF 1 (x, y)/zx � 0, zF 1 (x, y)/zy � 0, zF 2 (x, y)/zx � 0, and zF 2 (x, y)/zy � 0. By solving these four equations, the critical point can be obtained as From equations ( 22), (24), and ( 26), we have zF 1 (x, y) zx ≥ 0 for x ≤ y and zF 1 (x, y) zx ≤ 0 for x ≥ y, zF 2 (x, y) zx ≥ 0 for x ≤ y and zF 2 (x, y) zx ≤ 0 for x ≥ y. ( us, it can be noted that the functions F 1 (x, y) and F 2 (x, y) are increasing with respect to x for x ≤ y and decreasing for x ≥ y.Similarly, it can be derived that zF 1 (x, y) zy ≤ 0 for x ≤ y and zF 1 (x, y) zy ≥ 0 for x ≥ y, zF 2 (x, y) zy ≤ 0 for x ≤ y and zF 2 (x, y) zy ≥ 0 for x ≥ y. ( en, it can be seen that the functions F 1 (x, y) and F 2 (x, y) are decreasing with respect to y for x ≤ y and increasing for x ≥ y.Let us now consider equations (19) and (20) with A ≤ B. Assume that the reference set X is divided into two parts X 1 and X 2 satisfying according to the monotonicity of the functions F 1 (x, y) and F 2 (x, y).(19) and (20).Hence,

Novel TOPSIS Method Based on Correlation Coefficient and Entropy Measure
In this section, a novel TOPSIS method based on correlation coefficient and entropy measure is given to handle the MADM problems under the LPFS environment.

Problem Description.
Consider an MADM problem that is composed of m alternatives denoted as C � c 1 , c 2 , . . ., c m   and n attributes denoted as U � u 1 , u 2 , . . ., u n  . e weights of attributes are expressed as ω � (ω 1 , ω 2 , . . ., ω n ), which satisfy ω j ≥ 0 and  n j�1 ω j � 1.Each attribute of each alternative is evaluated using an LPFN: 8 Complexity where s p ij and s q ij are the elements in a continuous LTS All the attribute values of the alternatives can form a linguistic Pythagorean fuzzy decision matrix (LPFDM) R � (r ij ) m×n � ((S p ij , S q ij )) m×n where i � 1, 2, . . ., m, j � 1, 2, . . ., n.
Because of the time pressure, limited professional knowledge of DMs, and complexity of problems, it is difficult to provide the weight information of attributes in advance.To derive the weights of attributes, we use the proposed entropy measures to compute the weight of each attribute as where ω j ∈ [0, 1],  n j�1 ω j � 1, and H j is calculated as and H j ∈ [0, 1] for j � 1, 2, . . ., n.

e Decision-Making
Procedure.According to the above analysis, the decision-making procedure of our proposed LPF-TOPSIS based on correlation coefficient and entropy measure is summarized as follows: Step 1: in the MADM problems, there are benefit-type and cost-type attributes.erefore, the LPFDM R � (r ij ) m×n should be normalized as R � (r ij ) m×n , where Step 2: use equation ( 30) to determine the weight vector of attributes.
Step 3: according to the LPFDM R � (r ij ) m×n , the PIS and NIS among the alternatives are determined as where r + j � (s p max j , s q min j ) � (max 1≤i≤m s p ij , min 1≤i≤m s q ij ) and r − j � (s p min j , s q max j ) � (min 1≤i≤m s p ij , max 1≤i≤m s q ij ).
Step 4: according to equation ( 14), the weighted correlation coefficient value between each alternative and linguistic Pythagorean fuzzy PIS is computed as where I(s h max min j ) is the HD of the LPFN r + j and I(s h min max j ) is the HD of the LPFN r − j .
Step 5: based on the weighted correlation coefficients, the relative closeness of each alternative can be calculated as Step 6: all the alternatives are ranked according to their values of relative closeness and the alternative with the highest value of relative closeness is selected as the optimal one.
e decision-making procedure of the proposed LPF-TOPSIS method based on correlation coefficient and entropy measure is depicted as shown in Figure 2. It can be noted that the proposed LPF-TOPSIS method can solve the MADM problems with unknown attribute weight information using entropy measures.It uses the weighted correlation coefficient to replace the distance measure so as to improve the decision results of the TOPSIS method.erefore, our study can not only enrich the theoretical basis related to the LPFSs but also promote the wide applications of LPFSs in various fields.

Case Study and Comparative Analyses
In this section, a practical problem concerning the selection of firewall productions is provided to show the implementation process of the proposed LPF-TOPSIS method and the security evaluation of computer systems is given to compare the proposed LPF-TOPSIS method with the previous decision-making methods.

Case Study: e Selection of Firewall Productions.
e firewall refers to a security gateway that protects the computer networks from malicious intrusions [60][61][62].ere are many kinds of firewall productions in the market, which have different advantages and disadvantages.Hence, how to select an appropriate firewall production according to their factors is a typical MADM problem.

Example 1.
e intranet of an enterprise is usually attacked by malicious intrusions.To enhance the security of the intranet, this enterprise plans to purchase the firewall production and deploys it between the intranet and extranet for blocking illegal accesses.Currently, there are five kinds of firewall productions available to be chosen, which are denoted as C � c 1 , c 2 , c 3 , c 4 , c 5  .When selecting the firewall production, the enterprise pays attention to the factors, which are the promotion (u 1 ), configuration simplicity (u 2 ), security level (u 3 ), and maintenance service level (u 4 ).e weights of these attributes are ω � (ω 1 , ω 2 , ω 3 , ω 4 ), which satisfy that ω j ≥ 0 and  4 j�1 ω j � 1.To evaluate the firewall productions with respect to their factors, the enterprise chooses a pair of linguistic terms from the following LTS: S � s 0 � extremely poor, s 1 � very poor, s 2 � poor, s 3 � slightly poor, s 4 � fair, s 5 � slightly good, s 6 � good, s 7 � very good, s 8 � extremely good, (37) to express their evaluation information as LPFNs.All the LPFNs form an LPFDM R � (r ij ) 5×4 as shown in Table 1.
People expect to select then productions having low price but high quality.ere is no exception to the firewall productions.For all the attributes, the higher the attribute values are, the better the alternatives are.Hence, all the attributes are benefit-type.
Step 1: since all the attributes are benefit-type, it is unnecessary to normalize the LPFDM R � (r ij ) 5×4 .
Step 2: using equation ( 30), the weights of the attributes can be computed as ω � (0.2190, 0.3229, 0.2636, 0.1945). ( Step 3: according to the LPFDM, the linguistic Pythagorean fuzzy PIS and NIS can be obtained as Step 4: using equation ( 34), the weighted correlation coefficient between each alternative and the linguistic Pythagorean fuzzy PIS is computed as 10 Complexity Using equation (35), the weighted correlation coefficient between each alternative and the linguistic Pythagorean fuzzy NIS is computed as Step 5: using equation ( 36), the relative closeness of each alternative is calculated as Step 6: according to the values of relative closeness, all the alternatives can be ranked as en, the optimal one is c 5 .

Comparative Analysis.
In this subsection, an example concerning the security evaluation of computer systems is provided to compare the proposed LPF-TOPSIS method with some decision-making methods, which are the proposed TOPSIS method based on distance measure and the existing decision-making methods based on aggregation operators [41,42].

Example 2.
e Information Technology Security Evaluation Center is an authoritative organization, which provides security evaluation services for companies.
ey are authorized to evaluate four computer systems C � c 1 , c 2 , c 3 , c 4   according to their factors, which are the hardware security (u 1 ), system software security (u 2 ), application software security (u 3 ), and data security (u 4 ).For these four factors, the higher the values are, the better the alternatives are.Hence, all of these attributes are benefittype.Each alternative concerning each attribute is evaluated using the LPFNs, whose MD and NMD are from the following LTS: S � s 0 � extremely poor, s 1 � very poor, s 2 � poor, s 3 � slightly poor, s 4 � fair, s 5 � slightly good, s 6 � good, s 7 � very good, s 8 � extremely good.

(44)
All the LPFNs can form an LPFDM R � (r ij ) 4×4 as shown in Table 2. e Proposed LPF-TOPSIS Method.e proposed LPF-TOPSIS method is used to handle Example 2, which involves the following steps: Step 1: since all the attributes are benefit-type, it is unnecessary to normalize the LPFDM R � (r ij ) 4×4 .
Step 2: using equation ( 30), the weights of the attributes can be computed as ω � (0.2681, 0.3068, 0.2088, 0.2163). (45) Step 3: according to the LPFDM, the linguistic Pythagorean fuzzy PIS and NIS can be obtained as Step 4: using equation ( 34), the weighted correlation coefficient between each alternative and the linguistic Pythagorean fuzzy PIS is computed as Using equation (35), the weighted correlation coefficient between each alternative and the linguistic Pythagorean fuzzy NIS is computed as Step 5: using equation ( 36), the relative closeness of each alternative is calculated as Step 6: According to the values of relative closeness, all the alternatives can be ranked as Step 7: end.

e TOPSIS Method Based on Distance
Measure. e TOPSIS method based on distance measure computes the relative closeness values of alternatives using the distance measure between each alternative and ideal solution.According to the literature [63], we first propose the distance measure for LPFSs.
Based on the distance measure, the relative closeness of each alternative can be calculated as where d + i (r ij , R + ) and d − i (r ij , R − ) are the distance measure between the alternative and PIS and distance measure between the alternative and NIS, respectively.e decision-making process of the TOPSIS method based on distance measure is given below: Steps 1-3: see Steps 1-3 of the proposed LPF-TOPSIS method.
Step 4: using equation ( 51), the distance measure between each alternative and the linguistic Pythagorean fuzzy PIS is computed as and the distance measure between each alternative and the linguistic Pythagorean fuzzy NIS is computed as Step 5: using equation ( 52), the relative closeness of each alternative is computed as Step 6: according to the values of relative closeness, all the alternatives can be ranked as Step 7: end.[41].

e Existing Decision-Making Methods Based on LPFWA and LPFWG Operators
e existing decisionmaking methods based on LPFWA and LPFWG operators [41] are used to solve Example 2: Steps 1-2: see Steps 1-2 of the proposed LPF-TOPSIS method.
Step 3: use the LPFWA defined in equation ( 2 Step 5: according to the score function values, all the alternatives can be ranked as when the LPFWA operator is used.
when the LPFWG operator is used.

e Existing Decision-Making Methods Based on LPFWIPBM and LPFWIPGBM Operators
Steps 1-2: see Steps 1-2 of the proposed LPF-TOPSIS method.
when the LPFWIPGBM operator is used.
e ranking results of the proposed LPF-TOPSIS method and existing decision-making methods [41,42] are summarized as shown in Table 3.
From Table 3, it can be noted that all the decision-making methods obtain the same best solution, and the proposed LPF-TOPSIS method and the existing decision-making methods based on LPFWA and LPFWIPBM operators derive the same ranking result.It indicates that the proposed LPF-TOPSIS method is practicable.However, the proposed LPF-TOPSIS method shows different ranking results from the TOPSIS method based on distance measure and existing decisionmaking methods based on LPFWG and LPFWIPGBM operators.e reasons can be analyzed as follows: (1) e proposed LPF-TOPSIS method can efficiently achieve trade-offs among conflicting attributes, which usually exist in the MADM problems (2) In Example 2, the correlation coefficient performs better than the distance measure when they are used to distinguish ideal solutions

Conclusions
In this paper, a novel LPF-TOPSIS method is proposed to solve the MCDM problems under the LPFS environment.e correlation coefficient measure is used to replace the distance measure so as to improve the decision results of the LPF-TOPSIS method.Two entropy measures are also put forward to measure the uncertainty degree of LPFSs, based on which, the attribute weight information can be derived.A case study concerning the selection of firewall productions is provided to demonstrate the application of the proposed LPF-TOPSIS method.e comparative analyses can test the effectiveness and advantages of the proposed LPF-TOPSIS method.e advantages of our study are summarized as follows: (1) e existing decision-making methods based on aggregation operators [41,42] simply aggregate a series of LPFSs into a single one.ey do not consider the relationships among attributes.By contrast, our proposed LPF-TOPSIS method can efficiently achieve trade-offs among conflicting attributes.
(2) In this study, two novel entropy measures are proposed to determine the unknown attribute weight information.By contrast, the attribute weight information in the studies [41,42] is provided by DMs in advance.(3) Compared with the TOPSIS method based on distance measure, the proposed LPF-TOPSIS method uses the correlation coefficient to measure the proximity between each alternative and ideal solution.It is an innovative way to make the decisions under the LPFS environment.
However, our study still has some limitations as follows: (1) It is insufficient to only consider the evaluation information from individual DMs when making the decision results (2) In our proposed LPF-TOPSIS method, the attribute weight information determining method is an objective method, which neglects the preference information of DMs In the future, we will apply the concept of LPFSs into the group MADM problems.Moreover, various decisionmaking methods will be extended to handle LPFSs.
Figure 2: e decision-making procedure of the LPF-TOPSIS method.
two LPFSs on X, then the distance measure between them is defined as

Table 1 :
e LPFDM for the selection of firewall productions.

Table 3 :
Ranking results for various decision-making methods.