Picture Fuzzy Logic and Its Applications in Decision Making Problems provides methodological frameworks and the latest empirical research findings in the field of picture fuzzy operators, and their applications in scientific research and real-world engineering problems. Although fuzzy logic can be applied in a number of different areas, many researchers and developers are not yet familiar with how picture fuzzy operators can be applied to a variety of advanced decision-making problems. Picture fuzzy set is a more powerful tool than fuzzy set or intuitionistic fuzzy set to tackle uncertainty in a variety real-world modeling applications. Picture fuzzy set is actually the generalization of intuitionistic fuzzy set, and intuitionistic fuzzy set is the generalization of fuzzy set. In this book, the picture fuzzy sets are investigated, and different types of operators are defined to solve a number of important decision making and optimization problems. The hybrid operator on picture fuzzy set based on the combination of picture fuzzy weighted averaging operators and picture fuzzy weighted geometric operators is developed and named Hybrid Picture Fuzzy Weighted Averaging Geometric (H-PFWAG) operator. Another operator is developed for interval-valued picture fuzzy environment, which is named Hybrid Interval-Valued Picture Fuzzy Weighted Averaging Geometric (H-IVPFWAG) operator. These two operators are then demonstrated as solutions to Multiple-Attribute Decision-Making (MADM) problems. The picture fuzzy soft weighted aggregation operators (averaging and geometric) are defined, and these are applied to develop a multi-criteria group decision making system. The Dombi operator in the picture fuzzy environment is then defined and applied to solve MADM problems. Based on the Dombi operator, several other operators are defined. These are the picture fuzzy Dombi aggregation operators, including picture fuzzy Dombi weighted averaging operator, picture fuzzy Dombi order weighted averaging operator, picture fuzzy Dombi hybrid averaging operator, picture fuzzy Dombi weighted geometric operator, picture fuzzy Dombi order weighted geometric operator, and picture fuzzy Dombi hybrid geometric operator. Each of these operators are used to solve MADM problems. An extension picture fuzzy set known as m-polar picture fuzzy set is proposed and investigated along with many properties of m-polar picture fuzzy Dombi weighted averaging and geometric operators; each of these operators are applied to MADM problems. Another extension of the picture fuzzy set is the interval-valued picture fuzzy uncertain linguistic environment. In this set, interval-valued picture fuzzy uncertain linguistic weighted averaging and geometric operators are developed, and interval-valued picture fuzzy uncertain linguistic Dombi weighted aggregation operators are utilized in the MADM process. In the complex picture fuzzy environment, the authors demonstrate some complex picture fuzzy weighted aggregation operators to be used in solving MADM problems. Another approach called MABAC with picture fuzzy numbers is studied and developed as a multi-attribute group decision making model. Furthermore, the picture fuzzy linear programming problem (PFLPP) is initiated, in which the parameters are picture fuzzy numbers (PFNs). The picture fuzzy optimization method is applied for solving the PFLPP. This concept is used to solve the picture fuzzy multi-objective programming problem (PFMOLPP) under the picture fuzzy environment.
1. 1 Introduction to picture fuzzy sets and operators - p. 1 2. 1.1: Introduction - p. 1 3. 1.2: Preliminaries - p. 3 4. 1.3: Relation on picture fuzzy set - p. 13 5. 1.4: Picture fuzzy graph - p. 15 6. 1.5: Arithmetics on picture fuzzy set - p. 17 7. 1.6: Ordering of PFN - p. 19 8. 1.7: Similarity measures between picture fuzzy sets - p. 20 9. 1.8: Convex combination of picture fuzzy sets - p. 23 10. 1.9: Picture fuzzy averaging operators - p. 25 11. 1.10: Implication operator on picture fuzzy set - p. 27 12. 1.11: Topological operators on picture fuzzy set - p. 30 13. 1.12: Dombi operations on PFNs - p. 32 14. 1.13: Picture fuzzy Dombi aggregation operator - p. 35 15. 1.14: Conclusion - p. 42 16. References - p. 42 17. 2 Picture fuzzy hybrid weighted operators and their application in the decision-making process - p. 47 18. 2.1: Introduction - p. 47 19. 2.2: Preliminaries - p. 48 20. 2.3: Aggregation operators with PFN information - p. 49 21. 2.4: Interval-valued picture fuzzy approach - p. 57 22. 2.5: MCDM based on the proposed operators - p. 59 23. 2.6: Sensitivity analysis for the parameter γ - p. 65 24. 2.7: Conclusion - p. 67 25. References - p. 68 26. 3 Multicriteria group decision-making process based on a picture fuzzy soft parameterized environment - p. 71 27. 3.1: Introduction - p. 71 28. 3.2: Basic concept of PFSS and PFSN - p. 72 29. 3.3: Picture fuzzy soft weighted average operators - p. 74 30. 3.4: Model for MCGDM method using picture fuzzy soft information - p. 85 31. 3.5: Case study - p. 86 32. 3.6: Comparative studies - p. 88 33. 3.7: Advantages of the approach - p. 90 34. 3.8: Conclusions - p. 90 35. References - p. 91 36. 4 Picture fuzzy Dombi operators and their applications in multiattribute decision-making processes - p. 95 37. 4.1: Introduction - p. 95 38. 4.2: Preliminaries - p. 96 39. 4.3: Picture fuzzy Dombi weighted average operators - p. 97 40. 4.4: Picture fuzzy Dombi weighted geometric operators - p. 105 41. 4.5: The MADM model based on PFN - p. 108 42. 4.6: Numerical results - p. 109 43. 4.7: Analysis on the effect of parameter ϱ on decision making results - p. 111 44. 4.8: Comparative analysis - p. 111 45. 4.9: Conclusions - p. 111 46. References - p. 112 47. 5 Picture fuzzy Dombi prioritized operators and their application in decision-making processes - p. 115 48. 5.1: Introduction - p. 115 49. 5.2: Preliminaries - p. 116 50. 5.3: Picture fuzzy Dombi prioritized weighted arithmetic aggregation operators - p. 117 51. 5.4: Picture fuzzy Dombi prioritized geometric aggregation operators - p. 123 52. 5.5: Model for MADM using picture fuzzy Dombi operator - p. 129 53. 5.6: Numerical example and comparative analysis - p. 130 54. 5.7: Analysis on the effect of parameter ϱ on decision making results - p. 132 55. 5.8: Comparative analysis - p. 133 56. 5.9: Conclusions - p. 135 57. References - p. 135 58. 6 Picture fuzzy power Dombi operators and their utilization in decision-making problems - p. 137 59. 6.1: Introduction - p. 137 60. 6.2: Basic definitions and terminologies - p. 138 61. 6.3: Picture fuzzy power Dombi averaging operators - p. 139 62. 6.4: Dombi power geometric AOs with PFNs - p. 146 63. 6.5: MADM approach for PFNs - p. 151 64. 6.6: Case study and comparative analysis - p. 153 65. 6.7: Conclusion - p. 156 66. References - p. 157 67. 7 m-Polar picture fuzzy Dombi operators and their applications in multicriteria decision-making processes - p. 159 68. 7.1: Introduction - p. 159 69. 7.2: Preliminaries - p. 160 70. 7.3: Dombi operations on mPFNs - p. 162 71. 7.4: mPoPFN Dombi arithmetic operators - p. 164 72. 7.5: mPoPFN Dombi geometric operators - p. 170 73. 7.6: Model for MADM using mPoPF data - p. 177 74. 7.7: Numerical example - p. 178 75. 7.8: Conclusion - p. 181 76. References - p. 182 77. 8 Picture fuzzy MABAC approach and its application in multi-attribute group decision-making processes - p. 185 78. 9 Linear programming problem in a picture fuzzy environment - p. 197 79. 9.1: Introduction - p. 197 80. 9.2: Preliminaries - p. 198 81. 9.3: Some results - p. 204 82. 9.4: Methodology for solving FPFLPP with LR flat PFNs - p. 210 83. 9.5: Numerical example of FPFLPP - p. 213 84. 9.6: Conclusions - p. 219 85. References - p. 219 86. 10 Multiobjective linear programming problem in a picture fuzzy environment - p. 221 87. 11 Picture fuzzy goal programming problem - p. 241 88. 12 Picture fuzzy linear assignment problem and its application in multicriteria group decision-making problems - p. 261 89. Index - p. 277
Picture Fuzzy Logic and Its Applications in Decision Making Problems provides methodological frameworks and the latest empirical research findings in the field of picture fuzzy operators, and their applications in scientific research and real-world engineering problems. Although fuzzy logic can be applied in a number of different areas, many researchers and developers are not yet familiar with how picture fuzzy operators can be applied to a variety of advanced decision-making problems. Picture fuzzy set is a more powerful tool than fuzzy set or intuitionistic fuzzy set to tackle uncertainty in a variety real-world modeling applications. Picture fuzzy set is actually the generalization of intuitionistic fuzzy set, and intuitionistic fuzzy set is the generalization of fuzzy set. In this book, the picture fuzzy sets are investigated, and different types of operators are defined to solve a number of important decision making and optimization problems. The hybrid operator on picture fuzzy set based on the combination of picture fuzzy weighted averaging operators and picture fuzzy weighted geometric operators is developed and named Hybrid Picture Fuzzy Weighted Averaging Geometric (H-PFWAG) operator. Another operator is developed for interval-valued picture fuzzy environment, which is named Hybrid Interval-Valued Picture Fuzzy Weighted Averaging Geometric (H-IVPFWAG) operator. These two operators are then demonstrated as solutions to Multiple-Attribute Decision-Making (MADM) problems. The picture fuzzy soft weighted aggregation operators (averaging and geometric) are defined, and these are applied to develop a multi-criteria group decision making system. The Dombi operator in the picture fuzzy environment is then defined and applied to solve MADM problems. Based on the Dombi operator, several other operators are defined. These are the picture fuzzy Dombi aggregation operators, including picture fuzzy Dombi weighted averaging operator, picture fuzzy Dombi order weighted averaging operator, picture fuzzy Dombi hybrid averaging operator, picture fuzzy Dombi weighted geometric operator, picture fuzzy Dombi order weighted geometric operator, and picture fuzzy Dombi hybrid geometric operator. Each of these operators are used to solve MADM problems. An extension picture fuzzy set known as m-polar picture fuzzy set is proposed and investigated along with many properties of m-polar picture fuzzy Dombi weighted averaging and geometric operators; each of these operators are applied to MADM problems. Another extension of the picture fuzzy set is the interval-valued picture fuzzy uncertain linguistic environment. In this set, interval-valued picture fuzzy uncertain linguistic weighted averaging and geometric operators are developed, and interval-valued picture fuzzy uncertain linguistic Dombi weighted aggregation operators are utilized in the MADM process. In the complex picture fuzzy environment, the authors demonstrate some complex picture fuzzy weighted aggregation operators to be used in solving MADM problems. Another approach called MABAC with picture fuzzy numbers is studied and developed as a multi-attribute group decision making model. Furthermore, the picture fuzzy linear programming problem (PFLPP) is initiated, in which the parameters are picture fuzzy numbers (PFNs). The picture fuzzy optimization method is applied for solving the PFLPP. This concept is used to solve the picture fuzzy multi-objective programming problem (PFMOLPP) under the picture fuzzy environment.
1. 1 Introduction to picture fuzzy sets and operators - p. 1 2. 1.1: Introduction - p. 1 3. 1.2: Preliminaries - p. 3 4. 1.3: Relation on picture fuzzy set - p. 13 5. 1.4: Picture fuzzy graph - p. 15 6. 1.5: Arithmetics on picture fuzzy set - p. 17 7. 1.6: Ordering of PFN - p. 19 8. 1.7: Similarity measures between picture fuzzy sets - p. 20 9. 1.8: Convex combination of picture fuzzy sets - p. 23 10. 1.9: Picture fuzzy averaging operators - p. 25 11. 1.10: Implication operator on picture fuzzy set - p. 27 12. 1.11: Topological operators on picture fuzzy set - p. 30 13. 1.12: Dombi operations on PFNs - p. 32 14. 1.13: Picture fuzzy Dombi aggregation operator - p. 35 15. 1.14: Conclusion - p. 42 16. References - p. 42 17. 2 Picture fuzzy hybrid weighted operators and their application in the decision-making process - p. 47 18. 2.1: Introduction - p. 47 19. 2.2: Preliminaries - p. 48 20. 2.3: Aggregation operators with PFN information - p. 49 21. 2.4: Interval-valued picture fuzzy approach - p. 57 22. 2.5: MCDM based on the proposed operators - p. 59 23. 2.6: Sensitivity analysis for the parameter γ - p. 65 24. 2.7: Conclusion - p. 67 25. References - p. 68 26. 3 Multicriteria group decision-making process based on a picture fuzzy soft parameterized environment - p. 71 27. 3.1: Introduction - p. 71 28. 3.2: Basic concept of PFSS and PFSN - p. 72 29. 3.3: Picture fuzzy soft weighted average operators - p. 74 30. 3.4: Model for MCGDM method using picture fuzzy soft information - p. 85 31. 3.5: Case study - p. 86 32. 3.6: Comparative studies - p. 88 33. 3.7: Advantages of the approach - p. 90 34. 3.8: Conclusions - p. 90 35. References - p. 91 36. 4 Picture fuzzy Dombi operators and their applications in multiattribute decision-making processes - p. 95 37. 4.1: Introduction - p. 95 38. 4.2: Preliminaries - p. 96 39. 4.3: Picture fuzzy Dombi weighted average operators - p. 97 40. 4.4: Picture fuzzy Dombi weighted geometric operators - p. 105 41. 4.5: The MADM model based on PFN - p. 108 42. 4.6: Numerical results - p. 109 43. 4.7: Analysis on the effect of parameter ϱ on decision making results - p. 111 44. 4.8: Comparative analysis - p. 111 45. 4.9: Conclusions - p. 111 46. References - p. 112 47. 5 Picture fuzzy Dombi prioritized operators and their application in decision-making processes - p. 115 48. 5.1: Introduction - p. 115 49. 5.2: Preliminaries - p. 116 50. 5.3: Picture fuzzy Dombi prioritized weighted arithmetic aggregation operators - p. 117 51. 5.4: Picture fuzzy Dombi prioritized geometric aggregation operators - p. 123 52. 5.5: Model for MADM using picture fuzzy Dombi operator - p. 129 53. 5.6: Numerical example and comparative analysis - p. 130 54. 5.7: Analysis on the effect of parameter ϱ on decision making results - p. 132 55. 5.8: Comparative analysis - p. 133 56. 5.9: Conclusions - p. 135 57. References - p. 135 58. 6 Picture fuzzy power Dombi operators and their utilization in decision-making problems - p. 137 59. 6.1: Introduction - p. 137 60. 6.2: Basic definitions and terminologies - p. 138 61. 6.3: Picture fuzzy power Dombi averaging operators - p. 139 62. 6.4: Dombi power geometric AOs with PFNs - p. 146 63. 6.5: MADM approach for PFNs - p. 151 64. 6.6: Case study and comparative analysis - p. 153 65. 6.7: Conclusion - p. 156 66. References - p. 157 67. 7 m-Polar picture fuzzy Dombi operators and their applications in multicriteria decision-making processes - p. 159 68. 7.1: Introduction - p. 159 69. 7.2: Preliminaries - p. 160 70. 7.3: Dombi operations on mPFNs - p. 162 71. 7.4: mPoPFN Dombi arithmetic operators - p. 164 72. 7.5: mPoPFN Dombi geometric operators - p. 170 73. 7.6: Model for MADM using mPoPF data - p. 177 74. 7.7: Numerical example - p. 178 75. 7.8: Conclusion - p. 181 76. References - p. 182 77. 8 Picture fuzzy MABAC approach and its application in multi-attribute group decision-making processes - p. 185 78. 9 Linear programming problem in a picture fuzzy environment - p. 197 79. 9.1: Introduction - p. 197 80. 9.2: Preliminaries - p. 198 81. 9.3: Some results - p. 204 82. 9.4: Methodology for solving FPFLPP with LR flat PFNs - p. 210 83. 9.5: Numerical example of FPFLPP - p. 213 84. 9.6: Conclusions - p. 219 85. References - p. 219 86. 10 Multiobjective linear programming problem in a picture fuzzy environment - p. 221 87. 11 Picture fuzzy goal programming problem - p. 241 88. 12 Picture fuzzy linear assignment problem and its application in multicriteria group decision-making problems - p. 261 89. Index - p. 277
