Shahzaib Ashraf received MS degree in mathematics from International Islamic University, Islamabad, Pakistan. Currently he is a Ph.D. scholar at Department of Mathematics, Abdul Wali Khan University, Mardan, Pakistan. Also served the Abdul Wali Khan University as visiting Lecturer.Research Interests Applications of fuzzy systems and related topics, logical algebras
PurposeThis is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (... more PurposeThis is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (IHFN) is relaxed by the membership functions of Pythagorean probabilistic hesitant fuzzy number (PyPHFN), so the range of domain value of PyPHFN is greatly expanded. The paper aims to develop a novel decision-making technique based on aggregation operators under PyPHFNs. For this, the authors propose Algebraic operational laws using algebraic norm for PyPHFNs. Furthermore, a list of aggregation operators, namely Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy weighted geometric (PyPHFWG) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted average (PyPHFOWA) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted geometric (PyPHFOWG) operator, Pythagorean probabilistic hesitant fuzzy hybrid weighted average (PyPHFHWA) operator and Pythagorean probabilistic hesitant fuzzy hybrid weighted geom...
Industrial control system (ICS) attacks are usually targeted attacks that use the ICS entry appro... more 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 presente...
Wind energy is one of the most significant renewable energy sources due to its widespread availab... more Wind energy is one of the most significant renewable energy sources due to its widespread availability, low environmental impact, and great cost-effectiveness. The effective design of ideal wind energy extraction areas to generate electricity is one of the most critical issues in the exploitation of wind energy. The appropriate site selection for wind power plants is based on the concepts and criteria of sustainable environmental advancement, resulting in a low-cost and renewable energy source, as well as cost-effectiveness and job creation. The aim of this article is to introduce the idea of q-rung orthopair hesitant fuzzy rough set (q-ROHFRS) as a robust fusion of q-rung orthopair fuzzy set, hesitant fuzzy set, and rough set. A q-ROHFRS is a new approach towards modeling uncertainties in the multi-criteria decision making (MCDM). Various key properties of q-ROHFRS and some elementary operations on q-ROHFRSs are established. A list of novel q-rung orthopair hesitant fuzzy rough wei...
The paper aim is to resolve the issue of ranking to the fuzzy numbers in decision analysis, artif... more The paper aim is to resolve the issue of ranking to the fuzzy numbers in decision analysis, artificial intelligence and optimization. In the literature lot of ideologies have been established for ranking to the fuzzy numbers, that ideologies have some restrictions and limitations. In this paper, we proposed a method based on cubic picture fuzzy information's, for ranking to defeat the existing restrictions. Further introduced some cubic picture fuzzy algebraic and cubic picture fuzzy algebraic* aggregated operators for aggregated the information. Finally, a multi-attribute decision making problem is assumed as a practical application to establish the appropriateness and suitability of the proposed ranking approach.
In this article we used picture linguistic fuzzy Choquet integral weighted averaging (PLFCIWA) op... more In this article we used picture linguistic fuzzy Choquet integral weighted averaging (PLFCIWA) operator for multi-attribute group decision making problem, where the attribute weight information are completely unknown. Firstly, we utilized the PLFCIWA operator to aggregate the total preference value of each alternative by decision makers. An optimization model are used to obtain the weighting vector of the criteria, using the basic ideal of traditional grey relational analysis (GRA) method. The degree of grey relation between every alternative and positive-ideal solution and negative-ideal solution is computed. Then, a relative relational degree is defined to find the ranking order of all alternatives by calculating the degree of grey relation to both the positive-ideal solution (PIS) and negative ideal solution (NIS) simultaneously. Finally, a descriptive example is solved to satisfy effectiveness and practicality of the proposed techniques.
As a generalization of Pythagorean fuzzy sets and picture fuzzy sets, spherical fuzzy sets provid... more As a generalization of Pythagorean fuzzy sets and picture fuzzy sets, spherical fuzzy sets provide decision makers more flexible space in expressing their opinions. Preference relations have received widespread acceptance as an efficient tool in representing decision makers’ preference over alternatives in the decision-making process. In this paper, some new preference relations are investigated based on the spherical fuzzy sets. Firstly, the deficiency of the existing operating laws is elaborated in detail and three cases are described to identify the accuracy of the proposed operating laws in the context of t-spherical fuzzy environment. Also, a novel score function is proposed to obtain the consistent value in ranking of the alternatives. The backbone of this research, t-spherical fuzzy preference relation, consistent t-spherical fuzzy preference relations, incomplete t-spherical fuzzy preference relations, consistent incomplete t-spherical fuzzy preference relations, and accepta...
Having encouraged by the linguistic term in decision models, it is proposed a method of multi att... more Having encouraged by the linguistic term in decision models, it is proposed a method of multi attribute group decision making. This amalgamates the idea of picture fuzzy sets and linguistic term sets to discourse the situations where the real-life problems fail to express in numerical form. Firstly, it is introduced the concept of picture fuzzy linguistic number and comparison rules for ranking the alternatives are discussed. Further the aggregation operators based on picture fuzzy linguistic information are introduced. Finally, it is introduced a technique to obtain satisfactory results about real-life complex problems, and it is given a descriptive example to discuss the reliability and effectiveness of the suggested technique by using group decision criteria. Subject classification: 03E72
As a new extension of a cubic set, the notion of a cubic picture fuzzy set is introduced. The pro... more As a new extension of a cubic set, the notion of a cubic picture fuzzy set is introduced. The propose work is separated into two portions. Firstly, establish the concept of cubic picture fuzzy set and explore associated properties. Secondly, establish internal (external) cubic picture fuzzy sets and define P-order and R-order union and intersection. Deliver some examples to support of established P-order and R-order union and intersection of internal (external) cubic picture fuzzy sets.
Spherical fuzzy set is the generalized structure over existing structures of fuzzy sets to deals... more Spherical fuzzy set is the generalized structure over existing structures of fuzzy sets to deals with uncertainty and imprecise information in decision-making problems. Viewing the effectiveness of the spherical fuzzy set, we developed a decision-making algorithm to deal with multi-criteria decision-making problems. In this paper, we extend operational laws to propose spherical fuzzy Choquet integral weighted averaging (SFCIWA) operator based on spherical fuzzy numbers. Further, the proposed SFCIWA operator is applied to multi-attribute group decision-making problems. Also, we propose the GRA method to aggregate the spherical fuzzy infor mation. To implement the proposed models, we provide some numerical applications of group decision-making problems. Also compared with the previous model, we conclude that the proposed technique is more effective and reliable.
The problem of energy crisis and environmental pollution has been mitigated by the generation and... more The problem of energy crisis and environmental pollution has been mitigated by the generation and use of wind power; however, the choice of locations for wind power plants is a difficult task because the decision-making process includes political, socioeconomic, and environmental aspects. Thus, several adverse consequences have been created by the choice of suboptimal locations. The objective of this paper is to address the integrated qualitative and quantitative multicriteria decision-making framework for the selection of wind power plant locations. Spherical fuzzy sets are the latest extension of the ordinary fuzzy sets. The main characteristic of the spherical fuzzy sets is satisfying the condition that the squared sum of the positive, neutral, and negative grades must be at least zero and at most one. In this research, we establish novel operational laws based on the Yager t-norm and t-conorm under spherical fuzzy environments (SFE). Furthermore, based on these Yager operational...
A single valued neutrosophic set (SVNS) is a useful tool to portray uncertainty in multi attribut... more A single valued neutrosophic set (SVNS) is a useful tool to portray uncertainty in multi attribute decisionmaking. In this article, we develop hybrid averaging and hybrid geometric aggregation operator using sine trigonometric function to handle uncertainty in single valued Neutrosophic information, which are, sine trigonometricsingle valued neutrosophic hybrid weighted averaging (ST-SVNHWA) operator and , sine trigonometric-single valued neutrosophic hybrid weighted geometric (ST-SVNHWG) operator. We investigate properties, namely, idempotancy, monotonicity and boundedness for the proposed operators. Moreover, we give an algorithm to solve multi criteria decision-making issues which involve SVN information with ST-SVNHWA and STSVNHWG operators. Finally, an illustrative example of agricultural land selection is provided to verify the effectiveness. Sensitivity and comparative analyses are also implemented to assess the stability and validity of our method.
The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easil... more The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easily handle imprecise and uncertain information. The method of aggregation plays a great role in decision-making problems, particularly when there are more conflicting criteria. The purpose of this article is to present novel operational laws based on the Yager t-norm and t-conorm under spherical hesitant fuzzy information. Furthermore, based on the Yager operational laws, we develop the list of Yager weighted averaging and Yager weighted geometric aggregation operators. The basic fundamental properties of the proposed operators are given in detail. We design an algorithm to address the uncertainty and ambiguity information in multi-criteria group decision making (MCGDM) problems. Finally, a numerical example related to Parkinson disease is presented for the proposed model. To show the supremacy of the proposed algorithms, a comparative analysis of the proposed techniques with some existing...
The emergency response to the health care management in the hospital do not have enough systems f... more The emergency response to the health care management in the hospital do not have enough systems for providing medical service to the COVID19 patients (e.g., scheduled or nonemergency). Therefore, in this paper, we developed an emergency decision support model for consideration of patients care and admission scheduling (PCAS). The complex decision support model assigns a set of patients into a number of restricted resources like rooms, time slots, and beds depending on satisfying a number of predefined constraints such as disease severity, waiting time, and disease types. This is a crucial issue with multi‐criteria decision making (MCDM). In this paper, we first begin an assessment into the admission and care to tackle this issue and collect four factors effecting the admission and care of COVID‐19 patients that form a system of criteria. While there is a lot of vague and uncertain data that can be effectively depicted for these indicators by the spherical hesitant fuzzy set, then, we implement a strong MCDM method based on list of aggregation operators to address the patients' hospital admission and care. Last of all, a numerical real‐life application about PCAS is provided to demonstrate the validity of the proposed approaches along with relevant discussions, the merits of proposed approaches are also analyzed by validity test. The proposed methodology has been shown to help hospitals manage the admissions and care of COVID‐19 patients in a flexible manner.
Spherical fuzzy sets have recently become more popular in various fields. It was proposed as a ge... more Spherical fuzzy sets have recently become more popular in various fields. It was proposed as a generalization of picture fuzzy sets and Pythagorean fuzzy sets in order to deal with uncertainty and fuzziness information. This paper presents a multi-attribute group decision making method based on novel sine aggregation operators to help decision makers choose the optimal alternative. Moreover, the well-known sine trigonometry function preserves the periodic and symmetric nature about the origin, and hence, it satisfies the decision makers preferences over the multi-time phase parameters. Keeping these features and the importance of the spherical fuzzy (SF) sets, the objective of this paper is to present some robust sine trigonometric (ST) operation laws for SF sets. Associated with these laws, we define some series of new aggregation operators (AOs) named as ST-weighted averaging and geometric operators to aggregate the spherical fuzzy information. Afterward, we present group decision making techniques to solve the multi-attribute group decision making problems based on proposed AOs and illustrate with a numerical example of an internet finance soft power evaluation problem to validate it. Also, we conduct some comparison analysis to study the reasonability and practicality of the proposed method.
Handbook of Research on Emerging Applications of Fuzzy Algebraic Structures
The main objective of the chapter is to introduce a series of picture fuzzy weighted averaging an... more The main objective of the chapter is to introduce a series of picture fuzzy weighted averaging and geometric aggregation operators by using t-norm and t-conorm. In this chapter, they discussed generalized form of weighted averaging and geometric aggregation operator for picture fuzzy information. Further, the proposed aggregation operators of picture fuzzy number are applied to multi-attribute group decision making problems. To implement the proposed models, they provide some numerical applications of group decision making problems. Also compared with previous model, they conclude that the proposed technique is more effective and reliable.
PurposeThis is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (... more PurposeThis is mainly because the restrictive condition of intuitionistic hesitant fuzzy number (IHFN) is relaxed by the membership functions of Pythagorean probabilistic hesitant fuzzy number (PyPHFN), so the range of domain value of PyPHFN is greatly expanded. The paper aims to develop a novel decision-making technique based on aggregation operators under PyPHFNs. For this, the authors propose Algebraic operational laws using algebraic norm for PyPHFNs. Furthermore, a list of aggregation operators, namely Pythagorean probabilistic hesitant fuzzy weighted average (PyPHFWA) operator, Pythagorean probabilistic hesitant fuzzy weighted geometric (PyPHFWG) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted average (PyPHFOWA) operator, Pythagorean probabilistic hesitant fuzzy ordered weighted geometric (PyPHFOWG) operator, Pythagorean probabilistic hesitant fuzzy hybrid weighted average (PyPHFHWA) operator and Pythagorean probabilistic hesitant fuzzy hybrid weighted geom...
Industrial control system (ICS) attacks are usually targeted attacks that use the ICS entry appro... more 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 presente...
Wind energy is one of the most significant renewable energy sources due to its widespread availab... more Wind energy is one of the most significant renewable energy sources due to its widespread availability, low environmental impact, and great cost-effectiveness. The effective design of ideal wind energy extraction areas to generate electricity is one of the most critical issues in the exploitation of wind energy. The appropriate site selection for wind power plants is based on the concepts and criteria of sustainable environmental advancement, resulting in a low-cost and renewable energy source, as well as cost-effectiveness and job creation. The aim of this article is to introduce the idea of q-rung orthopair hesitant fuzzy rough set (q-ROHFRS) as a robust fusion of q-rung orthopair fuzzy set, hesitant fuzzy set, and rough set. A q-ROHFRS is a new approach towards modeling uncertainties in the multi-criteria decision making (MCDM). Various key properties of q-ROHFRS and some elementary operations on q-ROHFRSs are established. A list of novel q-rung orthopair hesitant fuzzy rough wei...
The paper aim is to resolve the issue of ranking to the fuzzy numbers in decision analysis, artif... more The paper aim is to resolve the issue of ranking to the fuzzy numbers in decision analysis, artificial intelligence and optimization. In the literature lot of ideologies have been established for ranking to the fuzzy numbers, that ideologies have some restrictions and limitations. In this paper, we proposed a method based on cubic picture fuzzy information's, for ranking to defeat the existing restrictions. Further introduced some cubic picture fuzzy algebraic and cubic picture fuzzy algebraic* aggregated operators for aggregated the information. Finally, a multi-attribute decision making problem is assumed as a practical application to establish the appropriateness and suitability of the proposed ranking approach.
In this article we used picture linguistic fuzzy Choquet integral weighted averaging (PLFCIWA) op... more In this article we used picture linguistic fuzzy Choquet integral weighted averaging (PLFCIWA) operator for multi-attribute group decision making problem, where the attribute weight information are completely unknown. Firstly, we utilized the PLFCIWA operator to aggregate the total preference value of each alternative by decision makers. An optimization model are used to obtain the weighting vector of the criteria, using the basic ideal of traditional grey relational analysis (GRA) method. The degree of grey relation between every alternative and positive-ideal solution and negative-ideal solution is computed. Then, a relative relational degree is defined to find the ranking order of all alternatives by calculating the degree of grey relation to both the positive-ideal solution (PIS) and negative ideal solution (NIS) simultaneously. Finally, a descriptive example is solved to satisfy effectiveness and practicality of the proposed techniques.
As a generalization of Pythagorean fuzzy sets and picture fuzzy sets, spherical fuzzy sets provid... more As a generalization of Pythagorean fuzzy sets and picture fuzzy sets, spherical fuzzy sets provide decision makers more flexible space in expressing their opinions. Preference relations have received widespread acceptance as an efficient tool in representing decision makers’ preference over alternatives in the decision-making process. In this paper, some new preference relations are investigated based on the spherical fuzzy sets. Firstly, the deficiency of the existing operating laws is elaborated in detail and three cases are described to identify the accuracy of the proposed operating laws in the context of t-spherical fuzzy environment. Also, a novel score function is proposed to obtain the consistent value in ranking of the alternatives. The backbone of this research, t-spherical fuzzy preference relation, consistent t-spherical fuzzy preference relations, incomplete t-spherical fuzzy preference relations, consistent incomplete t-spherical fuzzy preference relations, and accepta...
Having encouraged by the linguistic term in decision models, it is proposed a method of multi att... more Having encouraged by the linguistic term in decision models, it is proposed a method of multi attribute group decision making. This amalgamates the idea of picture fuzzy sets and linguistic term sets to discourse the situations where the real-life problems fail to express in numerical form. Firstly, it is introduced the concept of picture fuzzy linguistic number and comparison rules for ranking the alternatives are discussed. Further the aggregation operators based on picture fuzzy linguistic information are introduced. Finally, it is introduced a technique to obtain satisfactory results about real-life complex problems, and it is given a descriptive example to discuss the reliability and effectiveness of the suggested technique by using group decision criteria. Subject classification: 03E72
As a new extension of a cubic set, the notion of a cubic picture fuzzy set is introduced. The pro... more As a new extension of a cubic set, the notion of a cubic picture fuzzy set is introduced. The propose work is separated into two portions. Firstly, establish the concept of cubic picture fuzzy set and explore associated properties. Secondly, establish internal (external) cubic picture fuzzy sets and define P-order and R-order union and intersection. Deliver some examples to support of established P-order and R-order union and intersection of internal (external) cubic picture fuzzy sets.
Spherical fuzzy set is the generalized structure over existing structures of fuzzy sets to deals... more Spherical fuzzy set is the generalized structure over existing structures of fuzzy sets to deals with uncertainty and imprecise information in decision-making problems. Viewing the effectiveness of the spherical fuzzy set, we developed a decision-making algorithm to deal with multi-criteria decision-making problems. In this paper, we extend operational laws to propose spherical fuzzy Choquet integral weighted averaging (SFCIWA) operator based on spherical fuzzy numbers. Further, the proposed SFCIWA operator is applied to multi-attribute group decision-making problems. Also, we propose the GRA method to aggregate the spherical fuzzy infor mation. To implement the proposed models, we provide some numerical applications of group decision-making problems. Also compared with the previous model, we conclude that the proposed technique is more effective and reliable.
The problem of energy crisis and environmental pollution has been mitigated by the generation and... more The problem of energy crisis and environmental pollution has been mitigated by the generation and use of wind power; however, the choice of locations for wind power plants is a difficult task because the decision-making process includes political, socioeconomic, and environmental aspects. Thus, several adverse consequences have been created by the choice of suboptimal locations. The objective of this paper is to address the integrated qualitative and quantitative multicriteria decision-making framework for the selection of wind power plant locations. Spherical fuzzy sets are the latest extension of the ordinary fuzzy sets. The main characteristic of the spherical fuzzy sets is satisfying the condition that the squared sum of the positive, neutral, and negative grades must be at least zero and at most one. In this research, we establish novel operational laws based on the Yager t-norm and t-conorm under spherical fuzzy environments (SFE). Furthermore, based on these Yager operational...
A single valued neutrosophic set (SVNS) is a useful tool to portray uncertainty in multi attribut... more A single valued neutrosophic set (SVNS) is a useful tool to portray uncertainty in multi attribute decisionmaking. In this article, we develop hybrid averaging and hybrid geometric aggregation operator using sine trigonometric function to handle uncertainty in single valued Neutrosophic information, which are, sine trigonometricsingle valued neutrosophic hybrid weighted averaging (ST-SVNHWA) operator and , sine trigonometric-single valued neutrosophic hybrid weighted geometric (ST-SVNHWG) operator. We investigate properties, namely, idempotancy, monotonicity and boundedness for the proposed operators. Moreover, we give an algorithm to solve multi criteria decision-making issues which involve SVN information with ST-SVNHWA and STSVNHWG operators. Finally, an illustrative example of agricultural land selection is provided to verify the effectiveness. Sensitivity and comparative analyses are also implemented to assess the stability and validity of our method.
The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easil... more The concept of spherical hesitant fuzzy set is a mathematical tool that have the ability to easily handle imprecise and uncertain information. The method of aggregation plays a great role in decision-making problems, particularly when there are more conflicting criteria. The purpose of this article is to present novel operational laws based on the Yager t-norm and t-conorm under spherical hesitant fuzzy information. Furthermore, based on the Yager operational laws, we develop the list of Yager weighted averaging and Yager weighted geometric aggregation operators. The basic fundamental properties of the proposed operators are given in detail. We design an algorithm to address the uncertainty and ambiguity information in multi-criteria group decision making (MCGDM) problems. Finally, a numerical example related to Parkinson disease is presented for the proposed model. To show the supremacy of the proposed algorithms, a comparative analysis of the proposed techniques with some existing...
The emergency response to the health care management in the hospital do not have enough systems f... more The emergency response to the health care management in the hospital do not have enough systems for providing medical service to the COVID19 patients (e.g., scheduled or nonemergency). Therefore, in this paper, we developed an emergency decision support model for consideration of patients care and admission scheduling (PCAS). The complex decision support model assigns a set of patients into a number of restricted resources like rooms, time slots, and beds depending on satisfying a number of predefined constraints such as disease severity, waiting time, and disease types. This is a crucial issue with multi‐criteria decision making (MCDM). In this paper, we first begin an assessment into the admission and care to tackle this issue and collect four factors effecting the admission and care of COVID‐19 patients that form a system of criteria. While there is a lot of vague and uncertain data that can be effectively depicted for these indicators by the spherical hesitant fuzzy set, then, we implement a strong MCDM method based on list of aggregation operators to address the patients' hospital admission and care. Last of all, a numerical real‐life application about PCAS is provided to demonstrate the validity of the proposed approaches along with relevant discussions, the merits of proposed approaches are also analyzed by validity test. The proposed methodology has been shown to help hospitals manage the admissions and care of COVID‐19 patients in a flexible manner.
Spherical fuzzy sets have recently become more popular in various fields. It was proposed as a ge... more Spherical fuzzy sets have recently become more popular in various fields. It was proposed as a generalization of picture fuzzy sets and Pythagorean fuzzy sets in order to deal with uncertainty and fuzziness information. This paper presents a multi-attribute group decision making method based on novel sine aggregation operators to help decision makers choose the optimal alternative. Moreover, the well-known sine trigonometry function preserves the periodic and symmetric nature about the origin, and hence, it satisfies the decision makers preferences over the multi-time phase parameters. Keeping these features and the importance of the spherical fuzzy (SF) sets, the objective of this paper is to present some robust sine trigonometric (ST) operation laws for SF sets. Associated with these laws, we define some series of new aggregation operators (AOs) named as ST-weighted averaging and geometric operators to aggregate the spherical fuzzy information. Afterward, we present group decision making techniques to solve the multi-attribute group decision making problems based on proposed AOs and illustrate with a numerical example of an internet finance soft power evaluation problem to validate it. Also, we conduct some comparison analysis to study the reasonability and practicality of the proposed method.
Handbook of Research on Emerging Applications of Fuzzy Algebraic Structures
The main objective of the chapter is to introduce a series of picture fuzzy weighted averaging an... more The main objective of the chapter is to introduce a series of picture fuzzy weighted averaging and geometric aggregation operators by using t-norm and t-conorm. In this chapter, they discussed generalized form of weighted averaging and geometric aggregation operator for picture fuzzy information. Further, the proposed aggregation operators of picture fuzzy number are applied to multi-attribute group decision making problems. To implement the proposed models, they provide some numerical applications of group decision making problems. Also compared with previous model, they conclude that the proposed technique is more effective and reliable.
Uploads
Papers by Shahzaib Ashraf