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<p>In real-world decision-making (DM) environments, transportation problems often involve multiple conflicting objectives and inherent uncertainty. This paper tackles a Multi-Objective Transportation Problem (MOTP) under uncertain conditions by employing a novel method based on Fermatean Cubic Fuzzy Numbers (FCFNs). FCFNs offer a richer and more flexible outline for representing imprecise, vague and hesitant information compared to traditional fuzzy sets (FS). To effectively rank and compare fuzzy alternatives, a modified score function is developed, which enhances discrimination capability within the Fermian cubic fuzzy environment (FCFE). The propounded fuzzy mathematical model is formulated using a Fuzzy Programming technique, allowing the integration of multiple goals and uncertain parameters into a unified framework. For this programming method, we use two software i.e LINGO 21.0 and MATLAB. These choices are motivated by LINGO’s optimization efficiency for large real life transportation problems and MATLAB’s advanced capability for handling fuzzy computations and custom score functions. By using this software,The solution methodology is implemented using LINGO 21.0 software, which efficiently handles the multi-objective nature and computational difficulty of the problem. A numerical example is presented to illustrate the effectiveness and practicality of the proposed model. The minimum transportation time is 0.0002197979, the transportation cost is 0.0004549125, and the carbon emission cost is 0.0002257991, resulting in a total minimum objective value of Z = 0.0009005095. When the Fuzzy Goal Programming Approach (FGPA) is applied under the FCFE scenario, the optimal transportation time, transportation cost, and carbon emission cost are obtained as 0.0001571088, 0.0002497209, and 0.0002423366, respectively. Consequently, the total minimum objective value is Z = 0.0006491663. The results show that the approach effectively captures the uncertainty and supports better DM in complex transportation systems.</p>

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Fermatean Cubic Fuzzy Score Function in Multi-objective Transportation Problems

  • 期刊论文

  • 发表日期：

    2026年1月4日

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  • Ritu Bhadana,
  • Tarun Kumar,
  • M.K. Sharma

摘要

In real-world decision-making (DM) environments, transportation problems often involve multiple conflicting objectives and inherent uncertainty. This paper tackles a Multi-Objective Transportation Problem (MOTP) under uncertain conditions by employing a novel method based on Fermatean Cubic Fuzzy Numbers (FCFNs). FCFNs offer a richer and more flexible outline for representing imprecise, vague and hesitant information compared to traditional fuzzy sets (FS). To effectively rank and compare fuzzy alternatives, a modified score function is developed, which enhances discrimination capability within the Fermian cubic fuzzy environment (FCFE). The propounded fuzzy mathematical model is formulated using a Fuzzy Programming technique, allowing the integration of multiple goals and uncertain parameters into a unified framework. For this programming method, we use two software i.e LINGO 21.0 and MATLAB. These choices are motivated by LINGO’s optimization efficiency for large real life transportation problems and MATLAB’s advanced capability for handling fuzzy computations and custom score functions. By using this software,The solution methodology is implemented using LINGO 21.0 software, which efficiently handles the multi-objective nature and computational difficulty of the problem. A numerical example is presented to illustrate the effectiveness and practicality of the proposed model. The minimum transportation time is 0.0002197979, the transportation cost is 0.0004549125, and the carbon emission cost is 0.0002257991, resulting in a total minimum objective value of Z = 0.0009005095. When the Fuzzy Goal Programming Approach (FGPA) is applied under the FCFE scenario, the optimal transportation time, transportation cost, and carbon emission cost are obtained as 0.0001571088, 0.0002497209, and 0.0002423366, respectively. Consequently, the total minimum objective value is Z = 0.0006491663. The results show that the approach effectively captures the uncertainty and supports better DM in complex transportation systems.