An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems
The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/Ma...
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| Format: | Article |
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MDPI AG
2023-08-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/15/3383 |
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| author | Salma Iqbal Naveed Yaqoob Muhammad Gulistan |
| author_facet | Salma Iqbal Naveed Yaqoob Muhammad Gulistan |
| author_sort | Salma Iqbal |
| collection | DOAJ |
| description | The linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, we can solve DM from the set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi></mrow></semantics></math></inline-formula>-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-cut analysis bestowing the degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example. |
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| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T00:22:56Z |
| publishDate | 2023-08-01 |
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| series | Mathematics |
| spelling | doaj.art-ec11395e8ea14b1595fdb87a63d491182023-11-18T23:15:55ZengMDPI AGMathematics2227-73902023-08-011115338310.3390/math11153383An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming ProblemsSalma Iqbal0Naveed Yaqoob1Muhammad Gulistan2Department of Mathematics and Statistics, Riphah International University, Sector I-14, Islamabad 44000, PakistanDepartment of Mathematics and Statistics, Riphah International University, Sector I-14, Islamabad 44000, PakistanDepartment of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2R3, CanadaThe linear Diophantine fuzzy set notion is the main foundation of the interactive method of tackling nonlinear fractional programming problems that is presented in this research. When the decision maker (DM) defines the degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> level sets, the max-min problem is solved in this interactive technique using Zimmermann’s min operator method. By using the updating technique of degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>, we can solve DM from the set of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>α</mi></mrow></semantics></math></inline-formula>-cut optimal solutions based on the membership function and non-membership function. Fuzzy numbers based on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula>-cut analysis bestowing the degree <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula> given by DM can first be used to classify fuzzy Diophantine inside the coefficients. After this, a crisp multi-objective non-linear fractional programming problem (MONLFPP) is created from a Diophantine fuzzy nonlinear programming problem (DFNLFPP). Additionally, the MONLFPP can be reduced to a single-objective nonlinear programming problem (NLPP) using the idea of fuzzy mathematical programming, which can then be solved using any suitable NLPP algorithm. The suggested approach is demonstrated using a numerical example.https://www.mdpi.com/2227-7390/11/15/3383nonlinear programming problemsfuzzy setslinear Diophantine fuzzy setsLDF-nonlinear programming problems |
| spellingShingle | Salma Iqbal Naveed Yaqoob Muhammad Gulistan An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems Mathematics nonlinear programming problems fuzzy sets linear Diophantine fuzzy sets LDF-nonlinear programming problems |
| title | An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems |
| title_full | An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems |
| title_fullStr | An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems |
| title_full_unstemmed | An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems |
| title_short | An Investigation of Linear Diophantine Fuzzy Nonlinear Fractional Programming Problems |
| title_sort | investigation of linear diophantine fuzzy nonlinear fractional programming problems |
| topic | nonlinear programming problems fuzzy sets linear Diophantine fuzzy sets LDF-nonlinear programming problems |
| url | https://www.mdpi.com/2227-7390/11/15/3383 |
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