The paper proposed a methodology for obtaining a set of efficient solutions for a model which is multi-level multiobjective quadratic with fractional objectives and constraints having trapezoidal fuzzy number (MLMOQFP-TrFN) as coefficients. The model consists of r-levels with several objectives involved to be solved under a set of quadratic constraints. The proposed approach starts with the solution process of the top level and other levels are solved in succession but depending on the solution of the previous levels. The solution process of each level comprises mainly three stages. In the beginning, the Rouben Ranking Function is used to convert the rth-level of fuzzy model into a deterministic or crisp one. After that, the crisp form is reconstructed to get a non-fractional model with the help of an iterative parametric approach. Further, in the last, non-fractional model which is still having multiple objectivesis reconstructed to form a model having only one objective with ɛ -constraint method and is lastly solved by following the solution of (r-1)th- level to get a desired set of efficient solution. Such programming models are very useful in day to day life such as in economic planning, industrial activities, waste management, neural networking, unmanned aerial and underwater vehicle management, agricultural yield improvement, transportation problems with maximizing profits and minimizing wastage of material and cost and so on. An algorithm depicting all the steps of solution approach is also presented to reflect a clear idea for the approach. In addition, a numerical regarding the presentation of complete approach that is studied is given in the end.

中文翻译：

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一种基于Rouben排序函数的梯形模糊数多级多目标二次分数规划的求解方法

本文提出了一种方法，用于为具有分数目标和以梯形模糊数（MLMOQFP-TrFN）作为系数的约束的多级多目标二次模型获得一组有效解。该模型由 r 级组成，其中涉及在一组二次约束下求解的多个目标。所提出的方法从顶层的求解过程开始，其他层次依次求解，但取决于先前层次的求解。每个层次的求解过程主要包括三个阶段。一开始，使用 Rouben Ranking Function 将第 r 级模糊模型转换为确定性或清晰模型。之后，在迭代参数方法的帮助下，重建清晰的形式以获得非分数模型。更远，最后，将仍然有多个目标的非分数模型用ɛ-约束方法重构为只有一个目标的模型，最后通过第（r-1）级的解得到所需的集合的有效解决方案。这样的编程模型在日常生活中非常有用，例如经济规划、工业活动、废物管理、神经网络、无人机和水下航行器管理、农业产量提高、利润最大化和材料和成本浪费最小化的运输问题以及很快。还提出了一种描述解决方法的所有步骤的算法，以反映该方法的清晰思想。此外，最后给出了一个关于所研究的完整方法的呈现的数值。对仍具有多个目标的非分数模型采用 ɛ 约束方法进行重构，形成只有一个目标的模型，最后按照第 (r-1) 级的解进行求解，得到所需的一组有效解。这样的编程模型在日常生活中非常有用，例如经济规划、工业活动、废物管理、神经网络、无人机和水下航行器管理、农业产量提高、利润最大化和材料和成本浪费最小化的运输问题以及很快。还提出了一种描述解决方法的所有步骤的算法，以反映该方法的清晰思想。此外，最后给出了一个关于所研究的完整方法的呈现的数值。对仍具有多个目标的非分数模型采用 ɛ 约束方法进行重构，形成只有一个目标的模型，最后按照第 (r-1) 级的解进行求解，得到所需的一组有效解。这样的编程模型在日常生活中非常有用，例如经济规划、工业活动、废物管理、神经网络、无人机和水下航行器管理、农业产量提高、利润最大化和材料和成本浪费最小化的运输问题以及很快。还提出了一种描述解决方法的所有步骤的算法，以反映该方法的清晰思想。此外，最后给出了一个关于所研究的完整方法的呈现的数值。