Efficient Solution Strategy for Transportation Linear Fractional Programming under Fuzzy Conditions

Sultan S. Alodhaibi; Moodi Abdulrahman Abdullah Al- Rajeh; Florentin Smarandache; Hamiden Khalifa

Authors

Sultan S. Alodhaibi Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia Author https://orcid.org/0000-0003-4439-1878
Moodi Abdulrahman Abdullah Al- Rajeh Department of Mathematics, College of Science, Qassim University, Buraydah 51452, Saudi Arabia Author https://orcid.org/0009-0002-1325-0060
Florentin Smarandache Mathematics, Physics and Natural Science Division, The University of New Mexico, Gallup, MN, USA Author https://orcid.org/0000-0002-5560-5926
Hamiden Khalifa Department of Operations and Management Research, Faculty of Graduate Studies and Statistical Research, Cairo University, Giza 12613, Egypt Author https://orcid.org/0000-0002-8269-8822

Keywords:

Piecewise quadratic fuzzy numbers, Interval-valued modeling, Linear fractional transportation model, Transformation techniques, Multi-objective optimization, LINGO 20.0, Efficient solutions, Optimal compromise strategy

Abstract

Linear fractional programming (LFP) serves as an effective optimization framework in which the objective function is formulated as the ratio of two linear expressions. In numerous daily-life situations, the parameters involved in the objective function may not be defined precisely, leading to the incorporation of fuzzy representations. Transportation problems with fractional objectives are commonly described through system representations involving a limited number of lumps interconnected by curves. This study investigates a transportation model with a linear fractional objective where the parameters are characterized using fully piecewise quadratic fuzzy (PQF) numbers (PQFNs). To address the inherent uncertainty, a closeness-based interval approximation of these fuzzy numbers is introduced, allowing the original problem to be converted into an interval-valued LFP model. By applying the Charnes–Cooper transformation, this formulation is further reduced to an interval-valued linear transportation problem. This is then converted into a multi-objective optimization problem. Order relations on interval-valued outcomes are used to represent preferences of decision makers, considering lower and upper bounds, midpoints, and interval widths. An appropriate weighting scheme is then used to arrive at an ideal cooperation outcome. An example of the applicability and computational efficiency of the proposed approach is illustrated mathematically. Concluding the paper, the main findings and possible areas for further research are given.

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