A New Idea to Evaluate Networking Problem and MCGDM Problem in Parametric Interval Valued Pythagorean Arena

Chakraborty, Avishek; Mondal, Sankar Prasad; Alam, Shariful; Pamucar, Dragan; Marinkovic, Dragan

doi:https://doi.org/10.1155/2022/7369045

Discrete Dynamics in Nature and Society

On this page

Abstract Introduction Conclusion Data Availability Conflicts of Interest References Copyright Related Articles

Special Issue

Uncertain Optimization Problems in Sustainable Engineering

View this Special Issue

Research Article | Open Access

Volume 2022 | Article ID 7369045 | https://doi.org/10.1155/2022/7369045

A New Idea to Evaluate Networking Problem and MCGDM Problem in Parametric Interval Valued Pythagorean Arena

Avishek Chakraborty,¹Sankar Prasad Mondal,²Shariful Alam,³Dragan Pamucar,⁴and Dragan Marinkovic⁵

Academic Editor: Ewa Pawluszewicz

Received23 Sept 2021

Revised14 Nov 2021

Accepted15 Mar 2022

Published27 Apr 2022

Abstract

In this article, the concept of parametric interval valued Pythagorean number (PIVPN) has been introduced, which is an extended version of Pythagorean number (PN). Here, a new score and accuracy function have been innovated in the PIVPN environment along with the De-Pythagorean value concept. The new tool and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision making problem (MCGDM) problem. In order to solve the MCGDM problem, we have prepared Parametric Interval valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO) and Parametric Interval valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO) operator in PIVPN environment. Finally, sensitivity analysis and industrious comprehensive numerical simulations have been performed to identify the reliability, efficiency, and usefulness of this novel work. In this article, we have shown that PIVPNs are a more well-organized representation to grip a real-life problem, and they can handle inconsistent conditions in a better compatible way in comparison to the other existing methods.

1. Introduction

Fuzzy set theory [1] plays a quintessential role in modern science, medical diagnoses, engineering, and technical problem, but there is a basic note of interrogation as to how we can relate or use the impreciseness concept in our computational mathematical modeling. Myriad researchers from the distinct field have defined many approaches to define it and have given numerous recommendations like triangular [2], trapezoidal [3], pentagonal [4] fuzzy number, and their perspectives to use the theory of uncertainty. Further, researchers developed the concept of Intuitionistic fuzzy number (IFN) [5], introducing the perception of both belongingness and nonbelongingness membership functions. After that, triangular [6] and trapezoidal [7] IFN’s are formulated and applied in the distinct domain of the mathematical field. Further, the idea of interval-valued IFN [8] is manifested, which is the logical extension of IFN. After that, the concept of neutrosophic set (NS) [9] is established in the research domain which is the extension of IFN. In the case of NS theory, it actually deals with three components, namely, truth, false, and hesitation, whereas IFN contains two components, namely membership and nonmembership functions. Apart from this literature survey, researchers observed one important point. For instance, someone considered their liking towards any item is 0.7 whereas malaise is 0.4, and then the definition of intuitionistic fuzzy number has been violated. Thus, researchers developed the idea of Pythagorean number [10], which is the modified version of the intuitionistic fuzzy number considering the nonlinearity portions of the membership functions. Further, aggregation operators [11], generalized averaging aggregation operators [12], and confidence level-based aggregation operators [13] are developed by researchers day by day. Also, strategic decision-making approach [14], interval-valued Pythagorean number [15], some new results on Pythagorean set [16], decision-making approach on Pythagorean set [17], new novel score and accuracy function development [18], and linear programming method based on score value [19] have been developed by several researchers from different aspects. Apart from it, several researchers are worked in this field till now, and they proposed lots of concepts [20–25] related to multicriteria decision-making, score value, correlation coefficient, accuracy function of Pythagorean or intervalued Pythagorean number. There is plentiful literary evidence to classify some basic uncertain parameters. It should be adumbrated that there is no such unique representation of this uncertainty parameter. It can be optimized according to the choice of decision-makers for solving problems & can be varied as well as presented as different applications. Here, we present some information about the parameter of uncertainty showing how they vary from other concepts maneuvering the idea of uncertainty by utilizing some definitions, flowcharts, and diagrams. In this paper, we recommend that the researchers take the uncertainty parameter as PIVPN.

In this current era, the MCGDM problem is a predominant topic in the advance science and engineering field. The information of several parameters is precisely unspecified to create a model in real-life circumstances. Involving such cases, there is an occurrence of the notion of impreciseness. Uncertainty too forms the chief theme of the MCGDM problem, but the complication of such a method is how we can measure the uncertainty. Buyukozkan and Guleryuz [26] have adopted MCGDM approach to select better smart phones; Wang et al. [27] manifested intuitionistic linguistic operators on MCGDM problems; You et. al. [28] focused on an entropy based MCGDM method in interval-valued IFN; interval valued intuitionistic work in the MCGDM zone has been surveyed by Wang et al. [29]; Chakraborty A. [30–33] introduced MCGDM based problems in distinct domain, Chaio [34] derived MCGDM methodology based on type 2 fuzzy linguistic judgments; Wibowo [35] introduced MCGDM model in human resources management information projects; Chuu [36] established MCGDM model in manufacturing selection problem; Liu and Liu [37] proposed MCGDM model using Power Bonferroni Mean operator in IFN domain; Liu and Jiang [38] incorporated MCDM problem in interval valued intuitionistic set; Kumar and Garg [39] introduced TOPSIS in interval valued domain; Garg and Kumar [40] proposed advanced TOPSIS method in using SPA theory in IFS domain; Du et al. [41] focused on MADM problem in IVPN environment; Li et al. [42] described MADM model in IVPN environment. Apart from this literature survey, the concept of the Pythagorean set has also been fruitfully applied in several domains like operation research, graph theory, matrix evolution, etc. Luqman et al. [43] incorporated the Pythagorean concept in risk management problems using Digraph and matrix concept; Akram et al. [44] proposed LP model in the Pythagorean arena with equality constraints; Akram et al. [44, 45] manifested L-R type Pythagorean LP approaches with mixed and equality constraints; Enayattabar et al. [46] introduced the concept of shortest path problem in IVPN environment; Di Caprio et al. [47] focused on ant colony algorithm based on Pythagorean arena; Ebrahimnejad et al. [48] developed advanced bee colony algorithm in IVPN domain; Akram et al. [49] incorporated the concept of a planar graph in the Pythagorean environment which plays an essential role in graph theory problem.

From the available literature, it is observed that all the works which had been done earlier are mainly based on a general Pythagorean set or in an IVPN environment. Researchers have not explored the parametric representation of IVPN and its utility and usefulness in the domain of decision-making. In this article, we focused on the parametric representation of IVPN and constructed a few operators, de-Pythagorean skills in this environment that have been successfully and effectively applied to some real-life problems in the field of decision making. In this research article, we have developed two different operators, namely PIVPWAMO and PIVPWGMO, in an uncertain environment and introduced a new de-Pythagorean technique along with a new score and accuracy function for the crispification of PIVPN. The newly developed tools and techniques have been fruitfully applied to two realistic problems, namely the networking critical path model (CPM) problem and the multicriteria group decision-making (MCGDM) problem. Finally, sensitivity analysis and industrious comprehensive numerical simulations has been performed to identify the reliability, efficiency, and usefulness of this novel work.

1.1. Motivation

Although a good number of research works have already been toiled in the area of Pythagorean number that has been effectively and successfully applied to enrich MCDM/MADM/MCGDM techniques but only a few work is available in the area of IVPN. It is fascinating to note that many researchers have written Pythagorean numbers where they deploy the extension concept of FN and IFN. But why can we not structure the concept of the Pythagorean number using a simple parametric interval valued concept? Also, what will be the mathematical structure of PIVPN, and how can we relate it to real-life problems? Moreover, can we tackle the networking problem in PIVPN environment? If all the entities of decision matrices are considered as PIVPN, then how can we find out the best alternative! How can we perform a rigorous sensitivity and numerical analysis of an MCGDM model in a PIVPN environment? In this research article, an attempt is made to address all these issues and we proposed an efficient layout of MCGDM and networking when the decision-maker scored the feasible alternatives in the form of PIVPN. Different classifications of uncertain parameters and the conceptual workflow have been presents in Figure 1.

1.2. Novelties

The novelties of the work are described as follows:(i)Formulation of PIVPN based on the concept of IVPN(ii)Establishment of De-Pythagorean value using a noble technique to crispify a PIVPN into a real number.(iii)Realistic application of PIVPN into MCGDM and Networking model.(iv)Apply PIVPWAMO and PIVPWGMO in decision matrices to evaluate ranking.(v)Well planned sensitivity analysis and comprehensive numerical simulations have been performed in order to understand the effect of weights on the PIVPN in the MCGDM problem.

2. Mathematical Preliminaries

Definition 1. Interval Number: An interval number X is denoted by and defined as , where R real number set and and generally denoted the left and right range of the interval, respectively.

Lemma 1. The interval can also be represented as for

Definition 2. Fuzzy Set: A set , generally defined as , denoted by the pair , where and , then set is called a fuzzy set.

Definition 3. Intuitionistic Fuzzy Set (IFS): [2] in the universal discourse which is denoted generically by is said to be a IFS if where is called the truth membership function, is called the indeterminacy membership function. exhibits the following relation:The degree of hesitation is between membership function and nonmembership function is defined as

Definition 4. Pythagorean Fuzzy Set (PFS): [4] in the universal discourse which is denoted generically by is said to be a Pythagorean fuzzy set if where is called the truth membership function which represents the degree of confidence, is called the indeterminacy membership function which represents the degree of falsity. Geometrical representation of PFN is shown in Figure 2. Here, exhibits the following relation:The degree of hesitation is between the membership function and the nonmembership function is defined as . Zhang and Xu [6] defined the score function as where for two Pythagorean fuzzy numbers and if(i), then (ii), then (iii), then and accuracy function is defined as where

Definition 5. Interval Valued Pythagorean fuzzy Set (IVPFS): A set “A” in X is defined as follows:where and
Also,
For each element its hesitation interval relative to A is defined as follows:

Definition 6. Triangular Interval Valued Pythagorean fuzzy Set (TIVPFS): A TIVPFS is denoted as where , and also and . Here are the maximum degree of the membership function of the triangular fuzzy number and , respectively, and are the minimum degree of the nonmembership function of the triangular fuzzy number and , respectively.
The upper and lower membership function is defined by the following:and non membership functions are defined as follows:

3. Parametric Interval Valued Pythagorean Number and Its Properties

Definition 7. Parametric Interval Valued Pythagorean Number: PIVPN is defined as follows:where
Also,

3.1. De-Pythagorean of Parametric Interval Valued Pythagorean Number

A PIVPN can be converted into a crisp number C as follows:where is an index parameter of the decision-maker and . That is, the crisp value of an IVPN fully depends on the parameter

Example 1. There is an IVPN written in the parametric form as Then the value of the crisp number when index is 1.48.
Let be two IVPNs whose De-Pythagorean values are ; then the relation between them is defined as follows:We have computed some ranking results using the De-Pythagorean value in Table 1 for different IVPNs.

3.2. Properties and Operators in Parametric Interval Valued Pythagorean Number

Let us consider two IVPNs where and 0 then, we have the following:

3.2.1. Addition

3.2.2. Subtraction

3.2.3. Multiplication

3.2.4. Multiplication by a Constant

3.2.5. Division

3.2.6. Inverse

3.2.7. Example

If

Find

Solution:

3.3. Operators

We consider some operators used for interval valued parametric Pythagorean fuzzy numbers.

3.3.1. Parametric Interval Valued Pythagorean Weighted Arithmetic Mean Operator (PIVPWAMO)

Assuming where is family of PIVPN, λ is a set of PIVPN. A mapping PIVPWAMO such that , then PIVPWAMO where is the weight vector assigned with the PIVPN and , also each .

3.3.2. Parametric Interval Valued Pythagorean Weighted Geometric Mean Operator (PIVPWGMO)

Assuming where is family of PIVPN and λ is a set of PIVPN. A mapping PIVPWGMO such that , then

PIVPWGMO where is the weight vector assigned with the PIVPN and , also each .

4. Score and Accuracy Function IVPFN

A score function of IVPFS is mainly depends on hesitation indicator and score indicator. Let be an IVPFN, and a new score function depends on the following:(i) are called hesitation indicators.(ii) , are called benefit degrees.(iii) , are called nonbeneficiary degrees.

Sois the score function where

If then and if

The new accuracy function is defined as follows:where .

Let are two IVPFN then If(1)(2)(3)(i)(ii)(iii)

5. Critical Path Problem in Pythagorean Fuzzy Environment

The full form of CPM method is the critical path method which is a project management technique for progress planning that defines critical and noncritical tasks with the goal of preventing time frame problems. The critical path is defined as the (time) longest possible path from the start point to endpoint of the chart. Each project has at least one critical path. Each critical path consists of a list of activities on which the project manager should focus the most if he wants to ensure timely completion of the project. The date of the completion of the last task on the critical path is also the date of the completion of the project. For critical tasks to be applied, the total time reserve and thus free time reserve is equal to zero, i.e., that the start delay of this task or to extend its duration will affect the final date of the project. The critical path is reflected in the schedule and project management in all phases of the project lifecycle. Activity: It is any portion of a project that has a definite beginning and ending and may use some resources such as time, labor, material, equipment, etc. Event or Node: Beginning and ending points of activities denoted by circles are called nodes or events. Critical Path: The sequence of activities in a network is called the critical path. It is the longest path in the network, from the starting event to the ending event, and defines the minimum time required to complete the project. Forward pass method: Based on fixed occurrence time of the initial network event, the forward pass computation yields the earliest start and earliest finish times for each activity and, indirectly, the earliest expected occurrence time for each event.(1)The computations begin from the start node and move to the end node. To accomplish this, the forward pass computations start with an assumed earliest occurrence time of zero for the initial project event.(2)For any activity , let denote the earliest time of event , then .(3)If more than one activity enters an event, the earliest start time for that event is computed as for all activities emanating from node i entering into j. Backward pass method: The latest occurrence event time specifies the time by which all activities entering into that event must be completed without delaying the total project. These are computed by reversing the method of calculation used for the earliest event times.(1)To compute backward pass, starting from the final node, the computation proceeds from right to left, up to the initial event.(2)For any activity , let denote the latest finished time of event i, then .(3)If more than one activity enters an event the latest finish time for that event is computed as for all activities emanating from node j entering into i.

In our developed technique, we consider all the activities are IVPN to calculate the and ; we utilize the concept of a new score function which can compare the finite number of distinct IVPFN to estimate the minimum and maximum value. If the score values are equal, then estimate the IVPFN using the new accuracy function defined as

5.1. Flowchart

Flowchart of critical path problem in Pythagorean fuzzy environment is presented in Figure 3.

5.2. Numerical Problem

Table 2 and Figure 4 present a numerical problem implemented in this study.(1)Sketch the project network(2)Evaluate the and (3)Compute the critical path and total project duration. Step-1 Step-2 After computation and we have a network as follows (Figure 5):

In step-1, we draw the network diagram of the given problem in Figure 3, and further after computation of forward pass and backward pass, we finally get Figure 4, which is the final diagram with node allocation. Hence, the expected project duration −0.879 and Critical Path- 1 3 7 11

5.3. Comparative Analysis

In Table 3 we compare our work with other previous score function and accuracy function described by the authors [15, 18, 19] for finding the alternatives.

5.3.1. Discussion

We have compared the outcome of our proposed method with the outcome of other similar research work that has been presented in Table 2. Here, we observed that some score and accuracy functions fail to calculate the critical path as negativity arises in calculating and which is impossible for a networking problem. It is to be further noted that our proposed approach can find out a critical path that gives us a minimum project time duration in comparison to the approach of Garg [19]. Hence, we can conclude that it is a better and modified approach which can solve critical path problem in networking domain.

6. Proposed MCGDM Problem in Pythagorean Environment

In this MCGDM arena, we shall always focus on finding out the best feasible alternatives by disjunctive kind of factors, which is basically called the criteria. Though, it is not a very easy job to find out the attribute value in the format of crisp numbers due to the presence of impreciseness. All the information of the decision matrix is of the interval valued Pythagorean parametric form in nature. In this article, we actually consider an MCGDM, where there will be finite number of distinctive decision-makers available and according to their viewpoint we need to find out the best alternative. To do so, we construct an algorithm based on IVPWAMO and the normalization approach using the conception of IVPWGMO such that we can tackle the impreciseness problem very easily and using the conception of De-Pythagorean value, we shall choose the best alternative among all of them.

6.1. Illustration of the MCGDM Problem

We consider the problem as follows:

Let be “m” number of distinctive alternatives and be “n” number of distinctive attribute values, respectively. Let be the weight set associated with the attributes where each 0 and also satisfies the relation We also consider the set of decision-maker associated with alternatives whose weight vector is defined as where each 0 and also satisfies the relation . This weight vector will be selected according to the decision-makers quality of judgment, knowledge, thinking power, etc. The flowchart of the proposed methodology is presented in Figure 6. Step-1 Creation of Decision Matrices First, we formulate the decision matrices for each decision maker’s choice related to alternatives versus attribute functions. We consider the elements of the matrices in IVPN in parametric form, so all are members of the Pythagorean fuzzy number set. The related matrix is described as follows: Step-2 Creation of Single decision matrix To create the single group decision matrix, we utilize the operator PIVPWAMO for each decision matrix . Thus, we get the modified matrix as follows: Step-3 Creation of Final matrix according to weight Priority To obtain a single Column decision matrix, we all use the operator PIVPWGMO for each Column, and hence we get the decision matrix as follows: Step-4 Ranking Now, we consider the De-Pythagorean value and convert the matrix (3) into a crisp one such that we can evaluate the best alternative corresponding to the best attributes.

6.2. Illustrative Example

Let us consider a problem associated with four distinctive products and their three different attributes. Generally, lots of commodities are available in the market, having different types of components with different quality and facilities. Now in this problem, there are four different kinds of decision-makers available. We choose the problem as follows:

Comodity 1, Comodity 2, Comodity 3, Comodity 4 are the different kinds of alternatives. Quality, longibity, Price, Service are the four different attributes. There are three distinct types of decision-makers classified according to their ages Young aged people Middle-aged people Old aged people having corresponding weight set and we also consider the weight function related to the attribute function We consider a verbal matrix for the designer to assist the decision-maker in the creation of his decision matrix. The verbal phrases are listed in Table 4 and the verbal matrix of this MCGDM problem is shown in Table 5. Step-1 All the members of the matrices are PIVPN in nature. So, the decision matrices are as follows: For decision maker For decision maker For decision maker Step-2Now, we utilized our parametric interval valued Pythagorean weighted arithmetic mean operator (PIVPWAMO)to calculate all individual decision matrices and we get the final weighted normalized matrix as follows: Step-3 Now, we utilized our parametric interval valued Pythagorean weighted geometric mean operator (PIVPWGMO) considering the weight factors of attributes to calculate the matrix M in step 2, and we get the following: Step-4 Now, we apply our developed De-Pythagorean value to convert the fuzzy number into a crisp one considering , and then we have the final decision matrix M as follows:

From the final decision matrix we can say that the ranking will be .

6.3. Comparison of Work

We compare our work with other previously established work described by the authors 31,8,32, for finding the best alternatives whenever . The comparison result has been presented in Table 6.

From Table 5, we notice that has been declared the best alternative by our model, which matches with the outcomes of other researchers, namely Ye [7], Garg [18], Li et al. [42], although we have taken the parametric interval valued form of Pythagorean number which is more generalized from in compare to the established forms. Thus, we may conclude that PIVPN is a more robust structure that is reliable in the decision-making process.

6.4. Sensitivity Analysis

The main target of performing the sensitivity analysis is to check the efficiency, effectiveness, and stability of the proposed work. Here, we performed a well-organized sensitivity analysis by exchanging the weights of the attributes and values of the parameter keeping values of other terms unchanged. For different values of the change of ranking have been shown in Table 7.

6.4.1. Graphical Chart

Here, we computed the graphical chart of comparison of work in Figures 7–9 which shows the ranking of the alternatives for different values of and .Here, Series-1, Series-2, Series-3, and Series-4 denote the alternatives Commodity 1, Commodity 2, Commodity 3, and Commodity 4, respectively.

7. Numerical Analysis

In this section, we consider our sensitivity results and we did a numerical analysis whenever the value of Row ( actually varies from 0.05 to 0.95. For different three sets of values of , we got the following figures numerically (Figures 10–12),

7.1. Discussion

After the rigorous sensitivity analysis, numerical simulations, and graphical analysis, we have observed that in Figure 8, alternatives 3, 1 change their position in a range of (0.55, 0.6) and become the best one. Now, if we observe Figure 9, then we can conclude that alternatives 3, 1 change their position in a range of (0.6, 0.62) and become the best one. Lastly, if we observe Figure 10, then we can conclude that the alternatives 3, 1 change their position in a range of (0.6, 0.68) and becomes the best one. Finally, from the sensitivity analysis, we can conclude that the ranking of the alternatives depends on the range of the parameter, which plays an essential role in this proposed method.

7.2. Advantages and Limitations of the Proposed Method

(1)After this scientific discussion mentioned above, we have noticed that the existing uncertain parameters such as FS, IFS, PFS can be easily obtained from PIVPNs, and hence PIVPNs are a more well-organized representation to grip a problem more effectively where the existing theories fail. Moreover, PIVPNs can grab the inconsistent condition in a more compatible way rather than other existing methods. Henceforth, the complete studies under the PIVPN environment are effective in solving MCGDM problems.(2)Through sensitivity analysis, numerical simulation, and graphical representation of the result, we have noticed that if we vary the weights of the attributes within a certain range, then the best alternatives may change their positions within the given range. That is, it is basically case dependent which shows the limitation of this proposed technique.

8. Conclusion

The concept of PIVPN is logical and pragmatic and it has practical usefulness in the current research arena. In this research article, we have been extended the conception of the Pythagorean fuzzy set into PIVPN and developed some logical operators related to it, namely PIVPWAMO and PIVPWGMO. We also developed a score and accuracy function in the PIVPN environment and solved a critical path problem and performed a comparison with the other existing method to observe the advantage of this proposed model. Also, marketing based real life MCGDM problem is described in this article to find out the best alternatives whenever all the decision matrices members are in the PIVPN form. Applying the concept of developed operators, we solved this problem and we utilized the concept of De-Pythagorean value to short out the ranking of the alternatives. Finally, we performed a rigorous sensitivity analysis, numerical simulation, and comparison of work to check the utility, reliability, usefulness, and advantages of this proposed novel work.

In a future study, we can extend some theoretical ideas of PIVPN and can form the structure of exponential operational law, and logarithmic operational law in the PIVPN environment. Further, we can apply it to different fields like cloud computing, image segmentation, pattern recognition, medical diagnosis, banking policy making problems, economic and social science based several real-life problems, etc.

Data Availability

The data used to support the findings of this study are included in this article. However, the reader may contact the corresponding author for more details on the data.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

L. A. Zadeh, “Fuzzy sets,” Information and Control, vol. 8, no. 5, pp. 338–353, 1965.
View at: Publisher Site | Google Scholar
O. F. Gorcun, S. Senthil, S. Senthil, and H. Küçükönder, “Evaluation of tanker vehicle selection using a novel hybrid fuzzy MCDM technique,” Decision Making: Applications in Management and Engineering, vol. 4, no. 2, pp. 140–162, 2021.
View at: Publisher Site | Google Scholar
M. Bakır, Ö. Atalık, and Ö. Atalık, “Application of fuzzy AHP and fuzzy MARCOS approach for the evaluation of E-service quality in the airline industry,” Decision Making: Applications in Management and Engineering, vol. 4, no. 1, pp. 127–152, 2021.
View at: Publisher Site | Google Scholar
D. Pamucar and F. Ecer, “Prioritizing the weights of the evaluation criteria under fuzziness: the fuzzy full consistency method - fucom-f,” Facta Universitatis – Series: Mechanical Engineering, vol. 18, no. 3, pp. 419–437, 2020.
View at: Publisher Site | Google Scholar
K. T. Atanassov, “Intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 20, no. 1, pp. 87–96, 1986.
View at: Publisher Site | Google Scholar
T. Milosevic, D. Pamucar, and P. Chatterjee, “Model for selecting a route for the transport of hazardous materials using a fuzzy logic system,” Military Technical Courier, vol. 69, no. 2, pp. 355–390, 2021.
View at: Google Scholar
J. Ye, “Prioritized aggregation operators of trapezoidal intuitionistic fuzzy sets and their application to multicriteria decision-making,” Neural Computing & Applications, vol. 25, no. 6, pp. 1447–1454, 2014.
View at: Publisher Site | Google Scholar
K. Atanassov and G. Gargov, “Interval valued intuitionistic fuzzy sets,” Fuzzy Sets and Systems, vol. 31, no. 3, pp. 343–349, 1989.
View at: Publisher Site | Google Scholar
F. Smarandache, A Unifying Field in Logics Neutrosophy: Neutrosophic Probability, Set and Logic, American Research Press, New Mexico , USA, 1998.
R. R. Yager and A. M. Abbasov, “Pythagorean membership grades, complex numbers, and decision making,” International Journal of Intelligent Systems, vol. 28, no. 5, pp. 436–452, 2013.
View at: Publisher Site | Google Scholar
R. R. Yager, “On ordered weighted averaging aggregation operators in multi-criteriadecision making,” IEEE Transactions on Systems, Man and Cybernetics, vol. 18, no. 1, pp. 183–190, 1988.
View at: Google Scholar
D. Pamucar, “Normalized weighted Geometric Dombi Bonferoni Mean Operator with interval grey numbers: application in multicriteria decision making,” Reports in Mechanical Engineering, vol. 1, no. 1, pp. 44–52, 2020.
View at: Publisher Site | Google Scholar
H. Garg, “Confidence levels based Pythagorean fuzzy aggregation operators and its application to decision-making process,” Computational & Mathematical Organization Theory, vol. 23, no. 4, pp. 546–571, 2017.
View at: Publisher Site | Google Scholar
H. Garg, “Some methods for strategic decision-making problems with immediate probabilities in pythagorean fuzzy environment,” International Journal of Intelligent Systems, vol. 33, no. 4, pp. 687–712, 2018.
View at: Publisher Site | Google Scholar
X. Zhang, “Multicriteria Pythagorean fuzzy decision analysis: a hierarchical QUALIFLEX approach with the closeness index-based ranking methods,” Information Sciences, vol. 330, pp. 104–124, 2016.
View at: Publisher Site | Google Scholar
X. Peng and Y. Yang, “Some results for pythagorean fuzzy sets,” International Journal of Intelligent Systems, vol. 30, no. 11, pp. 1133–1160, 2015.
View at: Publisher Site | Google Scholar
N. Sumera, A. Samina, and A. Muhammad, “A novel approach to decision-making with pythagorean fuzzy information,” Mathematics, vol. 6, no. 6, p. 95, 2018.
View at: Google Scholar
H. Garg, “A novel accuracy function under interval-valued Pythagorean fuzzy environment for solving multicriteria decision making problem,” Journal of Intelligent and Fuzzy Systems, vol. 31, no. 1, pp. 529–540, 2016.
View at: Publisher Site | Google Scholar
H. Garg, “A linear programming method based on an improved score function for interval-valued pythagorean fuzzy numbers and its application to decision-making,” International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, vol. 26, no. 1, pp. 67–80, 2018.
View at: Publisher Site | Google Scholar
X. Peng and H. Yuan, “Fundamental properties of pythagorean fuzzy aggregation operators,” Fundamenta Informaticae, vol. 147, no. 4, pp. 415–446, 2016.
View at: Publisher Site | Google Scholar
P. Ren, Z. Xu, and X. Gou, “Pythagorean fuzzy TODIM approach to multicriteria decision making,” Applied Soft Computing, vol. 42, pp. 246–259, 2016.
View at: Publisher Site | Google Scholar
W. Lang, X. Zhang, and M. Liu, “The maximizing deviation method based on interval-valued Pythagorean fuzzy weighted aggregating operator for multiple criteria group decision analysis,” Discrete Dynamics in Nature and Society, vol. 2015, Article ID 746572, 15 pages, 2015.
View at: Publisher Site | Google Scholar
Ç. Karamaşa, D. Karabasevic, D. Stanujkic, A. R. Kookhdan, A. R. Mishra, and M. Ertürk, “An extended single-valued neutrosophic AHP and MULTIMOORA method to evaluate the optimal training aircraft for flight training organizations,” Facta Universitatis – Series: Mechanical Engineering, vol. 19, no. 3, pp. 555–578, 2021.
View at: Publisher Site | Google Scholar
H. Garg, “A novel improved accuracy function for interval valued pythagorean fuzzy sets and its applications in the decision-making process,” International Journal of Intelligent Systems, vol. 32, no. 12, pp. 1247–1260, 2017.
View at: Publisher Site | Google Scholar
Z. Ali, T. Mahmood, T. Mahmood, K. Ullah, and Q. Khan, “Einstein geometric aggregation operators using a novel complex interval-valued pythagorean fuzzy setting with application in green supplier chain management,” Reports in Mechanical Engineering, vol. 2, no. 1, pp. 105–134, 2021.
View at: Publisher Site | Google Scholar
G. Buyukozkan and S. Guleryuz, “Multi criteria group decision making approach for smart phone selection using intuitionistic fuzzy TOPSIS,” International Journal of Computational Intelligence Systems, vol. 9, no. 4, pp. 709–725, 2016.
View at: Publisher Site | Google Scholar
X.-F. Wang, J.-Q. Wang, and W.-E. Yang, “Multi-criteria group decision making method based on intuitionistic linguistic aggregation operators,” Journal of Intelligent and Fuzzy Systems, vol. 26, no. 1, pp. 115–125, 2014.
View at: Publisher Site | Google Scholar
X. You, T. Chen, and Q. Yang, “Approach to multicriteria group decision-making problems based on the best-worst-method and ELECTRE method,” Symmetry, vol. 8, no. 95, pp. 1–16, 2016.
View at: Publisher Site | Google Scholar
J.-q. Wang, Z.-q. Han, and H.-y. Zhang, “Multi-criteria group decision-making method based on intuitionistic interval fuzzy information,” Group Decision and Negotiation, vol. 23, no. 4, pp. 715–733, 2014.
View at: Publisher Site | Google Scholar
A. Chakraborty, S. P. Mondal, S. Alam et al., “Disjunctive representation of triangular bipolar neutrosophic numbers, de-bipolarization technique and application in multicriteria decision-making problems,” Symmetry, vol. 11, no. 7, p. 932, 2019.
View at: Publisher Site | Google Scholar
A. Chakraborty, B. Banik, S. P. Mondal, and S. Alam, “Arithmetic and geometric operators of pentagonal neutrosophic number and its application in mobile communication service based MCGDM problem,” Neutrosophic Sets and Systems, vol. 32, pp. 61–79, 2020.
View at: Google Scholar
A. Chakraborty, S. P. Mondal, S. Alam, and A. Dey, “Classification of trapezoidal bipolar neutrosophic number, de-bipolarization technique and its execution in cloud service-based MCGDM problem,” Complex & Intelligent Systems, vol. 7, no. 1, pp. 145–162, 2021.
View at: Publisher Site | Google Scholar
S. Maity, A. Chakraborty, S. Dey, S. Mondal, and S. Alam, “A comprehensive study of a backlogging EOQ model with nonlinear heptagonal densefuzzyenvironment, Rairo Operations Research,” EDP Science, vol. 54, Article ID 2018114, 2018.
View at: Publisher Site | Google Scholar
K. P. Chaio, “The multicriteria group decision making methodology using type 2 fuzzy linguistic judgments,” Applied Soft Computing, vol. 47, pp. 189–211, 2016.
View at: Google Scholar
S. Wibowo, S. Grandhi, and H. Deng, “Multicriteria group decision making for selecting human resources management information systems projects,” in Proceedings of the 2016 IEEE 11th Conference on Industrial Electronics and Applications (ICIEA), IEEE Xplore, Hefei, China, June 2016.
View at: Publisher Site | Google Scholar
S. J. Chuu, “Selecting the advanced manufacturing technology using fuzzy multiple attributes group decision making with multiple fuzzy information,” Computers & Industrial Engineering, vol. 57, no. 3, pp. 1033–1042, 2007.
View at: Google Scholar
P. Liu and X. Liu, “Multiattribute group decision making methods based on linguistic intuitionistic fuzzy power Bonferroni mean operators,” Complexity, vol. 2017, Article ID 3571459, pp. 1–15, 2017.
View at: Publisher Site | Google Scholar
Y. Liu and W. Jiang, “A new distance measure of interval-valued intuitionistic fuzzy sets and its application in decision making,” Soft Computing, vol. 24, no. 9, pp. 6987–7003, 2020.
View at: Publisher Site | Google Scholar
K. Kumar and H. Garg, “TOPSIS method based on the connection number of set pair analysis under interval-valued intuitionistic fuzzy set environment,” Computational and Applied Mathematics, vol. 37, no. 2, pp. 1319–1329, 2018.
View at: Publisher Site | Google Scholar
H. Garg and K. Kumar, “A novel exponential distance and its based TOPSIS method for interval-valued intuitionistic fuzzy sets using connection number of SPA theory,” Artificial Intelligence Review, vol. 53, no. 1, pp. 595–624, 2020.
View at: Publisher Site | Google Scholar
Y. Du, F. Hou, W. Zafar, Q. Yu, and Y. Zhai, “A novel method for multiattribute decision making with interval-valued pythagorean fuzzy linguistic information,” International Journal of Intelligent Systems, vol. 32, no. 10, pp. 1085–1112, 2017.
View at: Publisher Site | Google Scholar
Z. Li, G. Wei, and H. Gao, “Methods for multiple attribute decision making with interval-valued pythagorean fuzzy information,” Mathematics, vol. 6, no. 11, p. 228, 2018.
View at: Publisher Site | Google Scholar
A. Luqman, M. Akram, and J. C. R. Alcantud, “Digraph and matrix approach for risk evaluations under Pythagorean fuzzy information,” Expert Systems with Applications, vol. 170, p. 114518, 2021.
View at: Publisher Site | Google Scholar
M. Akram, I. Ullah, M. G. Alharbi, and S. A. Edalatpanah, “Methods for solving L R -type pythagorean fuzzy linear programming problems with mixed constraints,” Mathematical Problems in Engineering, vol. 2021, no. 4, pp. 1–29, 2021.
View at: Publisher Site | Google Scholar
M. Akram, I. Ullah, T. Allahviranloo, and S. A. Edalatpanah, “LR-type fully Pythagorean fuzzy linear programming problems with equality constraints,” Journal of Intelligent and Fuzzy Systems, vol. 41, no. 1, pp. 1975–1992, 2021.
View at: Publisher Site | Google Scholar
M. Enayattabar, A. Ebrahimnejad, and H. Motameni, “Dijkstra algorithm for shortest path problem under interval-valued Pythagorean fuzzy environment,” Complex & Intelligent Systems, vol. 5, no. 2, pp. 93–100, 2019.
View at: Publisher Site | Google Scholar
D. Di Caprio, A. Ebrahimnejad, H. Alrezaamiri, and F. J. Santos-Arteaga, “A novel ant colony algorithm for solving shortest path problems with fuzzy arc weights,” Alexandria Engineering Journal, vol. 61, no. 5, pp. 3403–3415, 2022.
View at: Publisher Site | Google Scholar
A. Ebrahimnejad, M. Enayattabr, H. Motameni, and H. Garg, “Modified artificial bee colony algorithm for solving mixed interval-valued fuzzy shortest path problem,” Complex & Intelligent Systems, vol. 7, no. 3, pp. 1527–1545, 2021.
View at: Publisher Site | Google Scholar
M. Akram, J. Dar, and A. Farooq, “Planar graphs under pythagorean fuzzy environment,” Mathematics, vol. 6, no. 12, p. 278, 2018.
View at: Publisher Site | Google Scholar

Copyright

Copyright © 2022 Avishek Chakraborty et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

PDF Download Citation

Download other formats

Order printed copies

Views

299

Downloads

309

Citations