<p>The structure of the Linear plus Linear Fractional Programming Problem (LLFPP) is inherently complex and NP-complete, as it may possess multiple local optimal solutions. Consequently, achieving the global optimal solution is generally challenging. In fields, such as management science, game theory, and industry, there are numerous problems whose mathematical models are represented as LLFPPs. Thus, research on these types of problems is highly valuable. In this study, we propose an approach to address the fully fuzzy LLFPP, i.e., an LLFPP with fuzzy coefficients and fuzzy decision variables. The approach involves transforming the fuzzy problem into a Bi-Objective Linear plus Linear Fractional Programming Problem (BOLLFPP) using the concept of <InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40815_2025_2070_Article_IEq1.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="14" /> </InlineMediaObject> <EquationSource Format="TEX">$\alpha $</EquationSource> <EquationSource Format="MATHML"><math> <mi>α</mi> </math></EquationSource> </InlineEquation>-cut. This bi-objective problem can be solved using various existing methods available in the literature. However, most of these methods rely on the first-order Taylor approximation to handle non-linearity, which often compromises accuracy and can be considered a significant limitation. In contrast to existing methods, this article employs a weighted sum approach to convert the bi-objective problem into a single-objective programming problem. In this single-objective formulation, the objective function is the sum of a linear function and two fractional functions. Ultimately, the solution requires the simultaneous consideration of seven linear programming problems. It is demonstrated that the obtained solution is at least an <InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="40815_2025_2070_Article_IEq2.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="10" /> </InlineMediaObject> <EquationSource Format="TEX">$\epsilon $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ϵ</mi> </math></EquationSource> </InlineEquation>-efficient solution for the bi-objective problem. To illustrate the effectiveness of the proposed method, two examples are solved, and comparisons with a genetic algorithm are provided to highlight the accuracy of the approach.</p>

A Linearization to the Fully Fuzzy Linear plus Linear Fractional Program

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发表日期：

2025年9月3日

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Mojtaba Borza,
Azmin Sham Rambely

摘要

The structure of the Linear plus Linear Fractional Programming Problem (LLFPP) is inherently complex and NP-complete, as it may possess multiple local optimal solutions. Consequently, achieving the global optimal solution is generally challenging. In fields, such as management science, game theory, and industry, there are numerous problems whose mathematical models are represented as LLFPPs. Thus, research on these types of problems is highly valuable. In this study, we propose an approach to address the fully fuzzy LLFPP, i.e., an LLFPP with fuzzy coefficients and fuzzy decision variables. The approach involves transforming the fuzzy problem into a Bi-Objective Linear plus Linear Fractional Programming Problem (BOLLFPP) using the concept of $\alpha $ $α$ -cut. This bi-objective problem can be solved using various existing methods available in the literature. However, most of these methods rely on the first-order Taylor approximation to handle non-linearity, which often compromises accuracy and can be considered a significant limitation. In contrast to existing methods, this article employs a weighted sum approach to convert the bi-objective problem into a single-objective programming problem. In this single-objective formulation, the objective function is the sum of a linear function and two fractional functions. Ultimately, the solution requires the simultaneous consideration of seven linear programming problems. It is demonstrated that the obtained solution is at least an $\epsilon $ $ϵ$ -efficient solution for the bi-objective problem. To illustrate the effectiveness of the proposed method, two examples are solved, and comparisons with a genetic algorithm are provided to highlight the accuracy of the approach.