Research on Evaluating Algorithms for the Service Quality of Wireless Sensor Networks Based on Interval-Valued Intuitionistic Fuzzy EDAS and CRITIC Methods

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Introduction
In order to improve the accuracy of real-life decisionmaking, Zadeh [1] initially designed the fuzzy sets (FSs).Atanassov [2] designed the intuitionistic fuzzy sets (IFSs), which could be a generalization of FSs.In IFSs, there are three mathematical functions expressing the degrees of membership, nonmembership, and hesitancy.And they must satisfy the only condition that their sum of three degrees cannot exceed one.Gou et al. [3] pointed out a novel exponential operational law about IFNs and offered a method which was utilized to aggregate intuitionistic fuzzy information.He et al. [4] integrated the power averaging operators with IFSs and defined several intuitionistic fuzzy power interaction aggregation operators.Zhang and He [5] defined the extensions of intuitionistic fuzzy geometric interaction operators by using the t-norm and the corresponding t-conorm means.Li and Wu [6] presented the intuitionistic fuzzy cross-entropy distance and the GRA.Liang et al. [7] extended the MABAC method to IFSs by utilizing the novel distance measures.Khan and Lohani [8] put forward a novel similarity measure about IFNs depending on the distance measure of the double sequence of bounded variation.Chen et al. [9] developed the novel MCDM method based on the TOPSIS method and similarity measures in the context of IFSs.Li et al. [10] developed a grey target decision-making method in the form of IFNs on the basis of grey relational analysis [11].Garg [12] developed some intuitionistic fuzzy averaging operators by taking the degrees of hesitation between the membership mathematical functions into consideration.Gupta et al. [13] extended the fuzzy entropy [14] to IFSs with axiomatic justification and proposed the importance of parameter alpha.Bao et al. [15] put forward the prospect theory and the evidential reasoning method under IFSs.Gan and Luo [16] employed a hybrid method on the basis of DEMATEL and IFSs.Gupta et al. [17] modified the superiority and inferiority ranking (SIR) method and combined it under IFSs.Krishankumar et al. [18] developed IFSP (intuitionistic fuzzy set-based PROMETHEE) which was a novel ranking method.Luo and Wang [19] combined IFSs with the VIKOR method relying on a novel distance measure by taking the IFSs into consideration.Hao et al. [20] presented the novel intuitionistic fuzzy MADM method depending on the decision theory.Zhang et al. [21] defined the intuitionistic fuzzy TOPSIS method based on CVPIFRS models with an application to biomedical problems.Garg [22] developed the generalized intuitionistic fuzzy entropy-based approach for solving multiattribute decision-making problems with unknown attribute weights.Liu et al. [23] presented some novel intuitionistic fuzzy operators by extending the BM operator on the basis of the Dombi operations [24] and designed some MAGDM methods.Jin et al. [25] developed two group decision-making (GDM) methods which could obtain the normalized intuitionistic fuzzy priority weights from the designed IFPRs on the basis of the order consistency and the multiplicative consistency.Wu et al. [26] gave VIKOR algorithms for assessing the financing risk about rural tourism projects under IVIFSs.Wu et al. [27] designed the algorithms for evaluating the competitiveness of tourist destination with some IVIF Hamy mean operators.Wu et al. [28] proposed some IVIF Dombi Heronian mean operators for evaluating the ecological tourism value.Chen and Kuo [29] presented the novel MADM method using the nonlinear programming (NLP) model with hyperbolic tangent function and IVIFSs.Lu and Wei [30] proposed the TODIM method for social-integrationbased rural performance appraisal under IVIFSs and integrated the ELECTRE method with IFSs to tackle some MCDM issues.Garg and Kumar [31] defined the group decision-making approach based on possibility degree measures and the linguistic intuitionistic fuzzy aggregation operators using Einstein norm operations.Garg and Arora [32] proposed the generalized intuitionistic fuzzy soft power aggregation operator based on the t-norm and its application in multicriteria decision-making.
Keshavarz Ghorabaee et al. [33] designed the evaluation based on distance from average solution (EDAS) to solve multicriteria inventory classification (MCIC) issues.In recent years, this method was enriched by the related extensions.For example, Ghorabaee et al. [34] modified such an EDAS method to tackle supplier selection issues.Keshavarz Ghorabaee et al. [35] presented the EDAS method with normal distribution to tackle stochastic issues.Peng and Liu [36] designed the neutrosophic soft MADM algorithms on the basis of EDAS and defined the similarity measure.Kahraman et al. [37] integrated the EDAS method with IFSs to select the solid waste disposal site.He et al. [38] designed the EDAS model for MAGDM with PULTSs.Keshavarz Ghorabaee et al. [39] made some comparative analyses about the phenomenon of order reversal depending on EDAS and TOPSIS.Wang et al. [40] proposed the EDAS model for MAGDM under the 2tuple linguistic neutrosophic environment.Li et al. [41] defined the EDAS for MAGDM issues under the q-rung orthopair fuzzy environment.Feng et al. [42] integrated the EDAS with the extended hesitant fuzzy linguistic environment.Karasan and Kahraman [43] designed the interval-valued neutrosophic EDAS to decision-making issues.
Unfortunately, we failed to find the work of the EDAS method based on the CRITIC method with IVIFSs in the existing literature.So, investigating the EDAS method with IVIFSs is essential.e fundamental objective of our research is to develop an original method which can be more effectively to address some MAGDM issues in the context of the EDAS method and IVIFSs.Hence, the highlights of this work are illustrated subsequently.Above all, the EDAS method is extended to the IVIFSs.In addition, because the DMs are restrained by their knowledge, it is tricky to assign the criteria weights directly.Hence, the CRITIC method is utilized to decide each attribute's weight.Last but not the least, an empirical application is offered to demonstrate this novel approach, and several comparative analyses are offered to demonstrate some merits of the novel approach.
However, there are no studies on the EDAS method for MAGDM under IVIFSs in the existing literature.erefore, it is necessary to pay attention to this issue.e innovativeness of the paper can be summarized as follows: (1) the EDAS method is modified by IVIFSs; (2) the interval-valued intuitionistic fuzzy EDAS (IVIF-EDAS) method is designed to solve the MAGDM issues with IVIFSs; (3) a case study for evaluating the service quality of wireless sensor networks is designed to prove the developed method; and (4) some comparative studies are given to verify the rationality of the IVIF-EDAS method.
e reminder of our essay proceeds as follows.Some fundamental knowledge of IVIFSs is concisely reviewed in Section 2. e extended EDAS method is integrated with IVIFSs, and the calculating procedures are simply depicted in Section 3.An empirical application for assessing the service quality of wireless sensor networks is given to show the superiority of this approach, and some comparative analyses are offered to prove some merits of such a method in Section 4. At last, we make an overall conclusion of such a work in Section 5.

Preliminaries
Definition 1 (see [2]). e interval-valued IFSs (IVIFSs) on X are the object of the form where  μ I (x) ⊂ [0, 1] is the "membership degree of I" and  ] I (x) ⊂ [0, 1] is named the "nonmembership degree of I," and  μ I (x) and  ] I (x) meet the mathematical condition: Definition 2 (see [44]).Let ) be two IVIFNs; the operation formula of them can be defined as follows: 2 Mathematical Problems in Engineering ( Derived from Definition 2, the following properties of the operation laws can be obtained: (1) Definition 3 (see [45]).Let ) be IVIFNs; the score and accuracy values of I 1 and I 2 can be defined as follows: For two IFNs I 1 and I 2 , regarding Definition 3, (1) If s(I 1 ) < s(I 2 ), then Under the context of the IVIFNs, some aggregation operators will be introduced in this chapter, including the interval-valued intuitionistic fuzzy WA (IVIFWA) operator and the interval-valued intuitionistic fuzzy WG (IVIFWG) operator.

The EDAS Method with IVIFNs
Integrating the EDAS method with IVIFSs, we build the IVIF-EDAS method in which the assessment values are given by IVIFNs.e calculating procedures of the developed method can be described subsequently.
Let Z � Z 1 , Z 2 , . . ., Z n   be the attribute set and z � z 1 ,  z 2 , . . ., z n } be the attribute weight Z j , where be a discrete collection of alternatives.And Q � (q ij ) m×n is the overall IVIFN decision matrix; q ij means the value of alternative Y i regarding attribute Z j .Subsequently, the specific calculating procedures will be depicted.
Step 1: set up each decision maker's IVIFN decision matrix Q (k) � (q k ij ) m×n , and calculate the overall IVIFN decision matrix Q � (q ij ) m×n : where q k ij is the assessment value of alternative Y i (i � 1, 2, . . ., m) on the basis of the attribute Z j (j � 1, 2, . . ., n) and the decision maker D k (k � 1, 2, . . ., l).
Step 2: normalize the overall IVIFN decision matrix Step 3: utilize the CRITIC method to determine the weighting matrix of attributes.
CRiteria Importance through Intercriteria Correlation (CRITIC) method will be designed in this part which is utilized to decide attributes' weights. is method was initially put forward by Diakoulaki et al. [46] which took the correlations between attributes into consideration.Subsequently, the calculating procedures of this method will be presented: (1) Depending on the normalized overall IVIFN decision matrix Q N � (q N ij ) m×n , the correlation coefficient between attributes can be calculated: where (2) Calculate attributes' standard deviation: where q N j � 1/m  m i�1 S(q N ij ).
Step 4: calculate the average solution (AV) regarding all designed attributes: Step 5: depending on the AV's results, the positive distance from average (PDA) and negative distance from average (NDA) can be defined: Step 6: calculate SP i and SN i which express the weighted sum of PDA and NDA: Step 7: depending on the above calculated results, SP i and SN i can be normalized as Step 8: calculate the appraisal score AS i regarding every alternative's NSP i and NSN i : Step 9: according to AS i , all the alternatives can be ranked.e higher the value of AS i is, the optimal alternative will be selected.(3) Since the traffic distributes are nonuniform in space and time, a method for the efficient transmission in a dynamic traffic is required.(4) ere exists nonuniform replacement of the nodes as well as dynamic and changeful topology in many applications of the WSN.It is necessary to provide an efficient deployment method to satisfy the requirement of effectively covering.In this chapter, an empirical example for evaluating the service quality of wireless sensor networks which considered the complex MAGDM issues [47][48][49][50][51][52][53][54] will be provided by making use of the IVIF-EDAS method.us, in such a section, we present a numerical example to assess computer network systems with IVIFNs in order to show the designed method.ere are five wireless sensor networks A i (i � 1, 2, 3, 4, 5) to select.e expert group selects four attributes to evaluate these five wireless sensor networks: ① G 1 is the product quality factor; ② G 2 is the technology factor; ③ G 3 is the delivery factor; and ④ G 4 is the price factor.Taking its own business development into consideration, a company wants to choose a wireless sensor network.ere are five potential wireless sensor networks Y i (i � 1, 2, 3, 4, 5).In order to select the optimal wireless sensor network, the expert group invites five experts D � D 1 , D 2 , D 3 , D 4 , D 5   (expert's weight d � (1/5, 1/5, 1/5, 1/5, 1/5)) to assess these wireless sensor networks.All experts give their assessment information depending on the four subsequently attributes: ① Z 1 is the traffic convenience; ② Z 2 is the product price; ③ Z 3 is the green environmental protection ability; and ④ Z 4 is the service quality.Evidently, Z 2 is the g cost attribute, while Z 1 , Z 3 , and Z 4 are the benefit attributes.To obtain the optimal wireless sensor network, the calculating procedures are involved:

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Step 3: decide the attribute weights z j (j � 1, 2, . . ., n) by making use of the CRITIC method as recorded in Table 8.
Step 4: depending on the calculated results of Table 8, the value of average solution (AV) can be obtained on the basis of all proposed attributes by equations ( 19) and ( 20) (see Table 9).
Step 5: relying on the results of AV, the PDA and NDA can be calculated by utilizing equations ( 21) and (22) (see Tables 10 and 11).
Step 6: on the basis of equation (23)   Mathematical Problems in Engineering Step 9: according to the AS, all the alternatives can be ranked; the higher the value of AS is, the optimal alternative will be selected.Evidently, the rank of these five alternatives is Furthermore, our presented method is compared with the modified VIKOR method with IVIFSs [55] In the end, our presented method is compared with GRA-based IVIFSs [56].en, we can obtain the calculation result.
e grey relational grades of each alternative are calculated as c 1 � 0.8398, c 2 � 1.0000, c 3 � 0.8307, c 4 � 0.8821, and c 5 � 0.8672.erefore, the ranking order of alternatives is Eventually, the results of dissimilar methods are recorded in Table 12.
Derived from Table 12, it is evident that the optimal wireless sensor network is Y 2 in the mentioned methods, while the worst choice is Y 1 in most situations.In other words, these methods' ranking results are slightly different.Different methods could effectively tackle MAGDM issues from different kinds of angles.IVIFWA and IVIFWG operators emphasize to aggregate evaluation information.
e modified VIKOR method with IVIFSs emphasizes the closest to the ideal solution and the farthest to the worst solution.e GRA-based IVIFSs emphasize the degree of similarity or difference between two sequences on the basis of the relation.However, our developed method emphasizes to calculate the expected function from the average solution.Compared with the aforementioned methods, it is more practical and effective since the procedures of calculation are simpler, and it is more convenient to apply to the practical situations.

Conclusion
In this paper, IVIF-EDAS method is developed to tackle the MAGDM issues based on the description of the EDAS method and some fundamental notions of IVIFSs.To begin with, the fundamental information of IVIFSs is simply introduced.After that, the IVIFWA and IVIFWG operators are utilized to integrate the IVIFNs.Subsequently, relying on the CRITIC method, the attributes' weights are decided.In addition, applying the EDAS method to the IVIFSs, a new method is designed, and the calculating procedures are listed in detail.Finally, an application for assessing the service quality of wireless sensor networks has been given to show the superiority of this novel method, and comparative analysis between the IVIF-EDAS method and some other methods could also be made to further verify some merits of such a method.In our future works, the EDAS method and the CRITIC method will be extensively applied in different uncertain and ambiguous environments [57][58][59][60][61][62][63][64][65][66][67].
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Table 8 :
e attribute weights z j .

Table 9 :
e value of average solution.

Table 10 :
e results of PDA ij .

Table 11 :
e results of NDA ij .

Table 12 :
Evaluation results of dissimilar methods.