| Summary: | In this study, by generalizing the notion of fuzzy bi-ideals of ordered semirings, the notion of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-ideals is established. We prove that <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-ideals are fuzzy bi-ideals but that the converse is not true, and an example is provided to support this proof. A condition is given under which fuzzy bi-ideals of ordered semirings coincide with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-ideals. An equivalent condition and certain correspondences between bi-ideals and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-ideals are presented. Moreover, the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><mi>κ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-lower part of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-ideals is described and depicted in terms of several classes of ordered semirings. Furthermore, it is shown that the ordered semiring is bi-simple if and only if it is <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mo>∈</mo><mo>,</mo><mo>∈</mo><mo>∨</mo><mrow><mo>(</mo><msup><mi>κ</mi><mo>*</mo></msup><mo>,</mo><msub><mi>q</mi><mi>κ</mi></msub><mo>)</mo></mrow><mo>)</mo></mrow></semantics></math></inline-formula>-fuzzy bi-simple.
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