| Summary: | The notions of fuzzy set (FS) and intuitionistic fuzzy set (IFS) make a major contribution to dealing with practical situations in an indeterminate and imprecise framework, but there are some limitations. Pythagorean fuzzy set (PFS) is an extended form of the IFS, in which degree of truthness and degree of falsity meet the condition <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msup><mover accent="true"><mi>Θ</mi><mo>˘</mo></mover><mn>2</mn></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mi mathvariant="fraktur">K</mi><mn>2</mn></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn></mrow></semantics></math></inline-formula>. Another extension of PFS is a <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-rung orthopair fuzzy set (<inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROFS), in which truthness degree and falsity degree meet the condition <inline-formula><math display="inline"><semantics><mrow><mn>0</mn><mo>≤</mo><msup><mover accent="true"><mi>Θ</mi><mo>˘</mo></mover><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>+</mo><msup><mi mathvariant="fraktur">K</mi><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></msup><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>≤</mo><mn>1</mn><mo>,</mo><mrow><mo>(</mo><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover><mo>≥</mo><mn>1</mn><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, so they can characterize the scope of imprecise information in more comprehensive way. <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROFS theory is superior to FS, IFS, and PFS theory with distinguished characteristics. This study develops a few aggregation operators (AOs) for the fusion of <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROF information and introduces a new approach to decision-making based on the proposed operators. In the framework of this investigation, the idea of a generalized parameter is integrated into the <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROFS theory and different generalized <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROF geometric aggregation operators are presented. Subsequently, the AOs are extended to a “group-based generalized parameter”, with the perception of different specialists/decision makers. We developed <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROF geometric aggregation operator under generalized parameter and <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROF geometric aggregation operator under group-based generalized parameter. Increased water requirements, in parallel with water scarcity, force water utilities in developing countries to follow complex operating techniques for the distribution of the available amounts of water. Reducing water losses from water supply systems can help to bridge the gap between supply and demand. Finally, a decision-making approach based on the proposed operator is being built to solve the problems under the <inline-formula><math display="inline"><semantics><mover accent="true"><mi mathvariant="fraktur">q</mi><mo>´</mo></mover></semantics></math></inline-formula>-ROF environment. An illustrative example related to water loss management has been given to show the validity of the developed method. Comparison analysis between the proposed and the existing operators have been performed in term of counter-intuitive cases for showing the liability and dominance of proposed techniques to the existing one is also considered.
|
|---|
