A Survey on Generalized Fuzzy Parameter-Based Fractional Programming Problem

In this paper, we propose a method of solving the fully fuzzy linear fractional programming problems, Express all the parameters and variables are triangular fuzzy numbers. Convert all the triangular fuzzy numbers in their parametric form, we convert the fractional programming problem in to a single objective linear programming problem in parametric form. We put new fuzzy arithmetic and fuzzy ranking, we obtain the optimal solution the given fully fuzzy linear fractional programming problem without converting to its equivalent crisp linear programming problem. A numerical example is provided to illustrate the efficiency of the proposed method.

There are very few methods in literature to deal with fully fuzzy linear fractional programming (FFLFP) problems. In this paper, it is pointed out that in the existing methods (Journal of Applied Mathematics and Computing, 27 (2008), 227-242; Yugoslav Journal of Operations Research, 22 (2012), 41-50) for solving FFLFP problems, an inappropriate ranking property is used which may lead to the erroneous results. To resolve the flaw of the existing methods, a new method is proposed for solving FFLFP problems, and is illustrated with the help of a numerical problem.

In this paper, an efficient approach is presented to solve fully fuzzy linear fractional programming problems. In this approach, first by Alpha-cuts on the objective function and the constraints, the problem is transforming into a bi-objective linear programming problem, and then the resulting problem is converted into two crisp linear programming problems. A numerical example is presented to illustrate the proposed approach. The obtained results demonstrate the acceptable performance of the proposed solution approach.

The problem of fully fuzzy programming was one of the problems that many treatments have appeared for due to ambiguity and inaccuracy of its ambiguous coefficients. In this study, finding the optimal numerical solution for fully fuzzy linear fractional programming problems (FFLFPP) using the ranking functions was obtained. The proposed pentagonal ranking function (PPRF) multiplied by γ5 has been derived, as well as the proposed heptagonal ranking function (PHRF) multiplied by γ5 has been derived too. Two laws were obtained the first one was the novel ranking membership pentagonal and the second one was the heptagonal function then utilized to solve FFLFPP after converting the problems into FFLPP by utilizing the complementary method, then converting them to crisp linear programming problems and finding the optimal numerical solution for problems by using the proposed ranking functions. The paper included numerical example to illustrate the applied technique and the accuracy of the solution, and a comparison with available techniques were add the novelty of this study.

<abstract><p>The primary goal of optimization theory is to formulate solutions for real-life challenges that play a fundamental role in our daily lives. One of the most significant issues within this framework is the Linear Fractional Programming Problem (LFrPP). In practical situations, such as production planning and financial decision-making, it is often feasible to express objectives as a ratio of two distinct objectives. To enhance the efficacy of these problems in representing real-world scenarios, it is reasonable to utilize fuzzy sets for expressing variables and parameters. In this research, we have worked on the Fully Fuzzy Linear Fractional Linear Programming Problem (FFLFrLPP). In our approach to problem-solving, we simplified the intricate structure of the FFLFrLPP into a crisp Linear Programming Problem (LPP) while accommodating the inherent fuzziness. Notably, unlike literature, our proposed technique avoided variable transformation, which is highly competitive in addressing fuzzy-based problems. Our methodology also distinguishes itself from the literature in preserving fuzziness throughout the process, from problem formulation to solution. In this study, we conducted a rigorous evaluation of our proposed methodology by applying it to a selection of numerical examples and production problems sourced from the existing literature. Our findings revealed significant improvements in performance when compared to established solution approaches. Additionally, we presented comprehensive statistical analyses showcasing the robustness and effectiveness of our algorithms when addressing large-scale problem instances. This research underscores the innovative contributions of our methods to the field, further advancing the state-of-the-art in problem-solving techniques.</p></abstract>

This chapter has highlighted the study of the fuzzy linear fractional programming problem that considered coefficients of objective function and constraints as trapezoidal fuzzy number. To obtain an optimal solution, utilization of ranking functions has been employed to trapezoidal fuzzy number to convert the fuzzy FLPP to crisp FLPP, and for further computation, the crisp fractional LPP is solved with the Charan Cooper method to transform fractional programming into LPP and the same crisp FLPP using complementary LPP. An illustration from real life has been used to demonstrate the effectiveness of our suggested strategy.

The fuzzy set theory is being applied massively in many fields these days. One of these is linear programming problems. Kheirfam and Hasani (Sensitivity analysis for fuzzy linear programming problems with fuzzy variables, Advanced Modeling and Optimization, 12 (2010) 257-272), proposed a method for the sensitivity analysis for fuzzy linear programming (FLP) problems with fuzzy variables. There are some important cases that are not considered by Kheirfam and Hasani. In this paper, those cases are considered and numerical examples are solved.

In addition to the well-known fuzzy sets, a novel type of fuzzy sets called as spherical fuzzy set (SFS) is recently introduced in the literature. SFS is the generalized structure over existing structures of fuzzy sets (intuitionistic fuzzy sets, Pythagorean fuzzy sets, and neutrosophic sets) based on three dimensions (truth, falsehood, and indeterminacy) to provide a wider choice for decision makers (DMs). Although the SFS is introduced recently, the topic attracts the attention of academicians at a remarkable rate. Hence, a comprehensive literature review of recent and state-of-the-art papers is vital to draw a framework of the past, and to shed light on future directions. To do so, a systematic review methodology is followed and 43 academic papers include SFS are selected and reviewed. The papers are then analyzed and categorized based on the titles and abstracts to construct a useful foundation of past research. Finally, trends and gaps in the literature are identified to clarify and to suggest future research opportunities in fuzzy logic area.

An optimization problem where the goal function is a proportion between two functions is known as fractional programming, and the goal is to maximize or minimize the ratio. This paper discusses a methodology to resolve Fuzzy Fractional Programming Problem as a Fuzzy Linear Complementarity Problem. The study seeks to emphasize the key characteristics, make some new observation and motivate further application in linear complementarity problem. The constraints of the Fractional Programming problem is taken as a Fuzzy Linear Programming problem and further it transformed in to Fuzzy Linear Complementarity Problem by taking various – cut levels. Utilizing Interval numbers operations the given problem is resolved. A numerical instance is provided to show how the suggested strategy works.

As the extension of fuzzy set, intuitionistic fuzzy set, Pythagorean fuzzy set, and picture fuzzy set, the spherical fuzzy set is characterized by three functions expressing the positive-membership degree, the neutral-membership degree and the negative-membership degree which the sum squares of them is equal or less than 1. In this work, we shall present some novel Dice similarity measures of spherical fuzzy sets and the generalized Dice similarity measures of spherical fuzzy sets and indicates that the Dice similarity measures and asymmetric measures (projection measures) are the special cases of the generalized Dice similarity measures in some parameter values. Then, we propose the generalized Dice similarity measures-based multiple attribute group decision making models with spherical fuzzy information. Then, we apply the generalized Dice similarity measures between spherical fuzzy sets to multiple attribute group decision making. Finally, an illustrative example is given to demonstrate the efficiency of the similarity measures for selecting the desirable ERP system.

On the polynomial mixed 0–1 fractional programming problems

Using different dominance criteria in stochastic fuzzy linear multiobjective programming: A case of fuzzy weighted objective function

The extensions of ordinary fuzzy sets such as intuitionistic fuzzy sets (IFS), Pythagorean fuzzy sets (PFS), and neutrosophic sets (NS), whose membership functions are based on three dimensions, aim to describe expert’s judgments more informatively and explicitly. Introduction of generalized three dimensional spherical fuzzy sets (SFS) including some essential differences from the other fuzzy sets is presented in the literature with their arithmetic, aggregation, and defuzzfication operations [1]. This study summarizes the previously introduced spherical fuzzy sets and as an application spherical fuzzy CODAS method will be applied to the warehouse location selection problem.

Design for six sigma (DFSS) is a structured approach for designing new products, processes and/or services or redesigning the existing ones. It builds quality into the product during the development and ensures the customer expectations are understood, met, or exceed. In DFSS, variation is aimed to be minimized by applying the techniques of the six sigma approach. Some of these techniques are process control and process capability analysis. Vagueness and impreciseness in these analyses such as the statistical distribution of the considered process can be handled by the fuzzy set theory. Recently, ordinary fuzzy sets have been extended to several new types of fuzzy sets such as Pythagorean fuzzy sets, spherical fuzzy sets, picture fuzzy sets, and intuitionistic fuzzy sets. In this paper, we propose a new extension of intuitionistic fuzzy sets, which is called Penthagorean fuzzy sets and employ it in the process capability analysis of a production process. Penthagorean fuzzy process capability indices are developed and applied to an Autonomous Underwater Vehicle (AUV) design example.

Cosine similarity and information measures of [formula omitted] spherical fuzzy sets: Application in selecting migration destination