New algorithms for solving fuzzy linear fractional programming problems by ranking functions

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.

<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>

In fuzzy sets, the elements of a universal set may belong to that set completely, may not belong to that set, or may belong partially. Therefore, fuzzy sets are called generalizations of crisp sets. Due to the diverse nature of problems in real-world situations, fuzzy set theory is found to be unable to address some of the other features of ambiguity. Therefore, researchers extended fuzzy sets to picture fuzzy sets, orthopair, intuitionistic, Pythagorean, Fermatean, neutrosophic, spherical fuzzy sets, etc. Real-world problems can be solved using the method of fractional programming. Decision-makers are trying to get good results using generalized fuzzy numbers in fractional programming problems; hence, this review paper provides a specific framework for generalized fuzzy sets and generalized fuzzy fractional programming problems. Furthermore, it is a platform for researchers who want to work on generalized fuzzy parameter-based fractional programming problems. This research paper summarizes all the basic concepts and prior research on fractional programming and generalized fuzzy linear fractional programming problems.

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

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 difficulty in solving either multi-objective linear programming (MOLP) problems or fuzzy multi-objective linear programming (FMOLP) problems is the trade-off among objectives. To deal with this difficulty, we proposed a new algorithm for solving FMOLP problems by using zero-sum game. First, FMOLP problem given is converted to a crisp MOLP problem by using ranking function, and then a payoff matrix is constructed to find the weights of each objective function of the MOLP problem. After that, each weight is multiplied with the corresponding objective function, a single-objective LP problem is obtained and thus, FMOLP problem is solved. The proposed algorithm is illustrated by numerical examples for the FMLOP problems having fuzzy constraints or not.

We present an approach for ordering fuzzy numbers, then we utilize it to suggest two models for solving the fuzzy linear programming (FLP) problem. In this research we present a new approach for ordering two fuzzy numbers and show some practical advantages of it over other methods. We also utilize the suggested approach to introduce two techniques to solve the FLP problem. In the first technique, we convert the fuzzy linear program into a crisp bicriteria linear program. In this new model the fuzzy objective function is replaced by two crisp objectives to be optimized, and each fuzzy constraint is replaced by two or three crisp constraints depending on the level of certainty /spl alpha/-whether /spl alpha/ has or has not been assigned. In the second technique we find the possibility distribution of the objective function over all possible feasible decisions directly.

In this paper, shortcomings and limitations of the existing methods for solving fuzzy linear programming (FLP) problems are pointed out. To overcome the limitations as well as to resolve the shortcomings, a new method is proposed for solving FLP problems. To show the advantage of the proposed method over existing methods, a FLP problem is solved by the existing methods as well as the proposed method, and the obtained results are compared.

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 this paper, we propose a new model for linear fractional programming problem with absolute value functions. The major contribution of this paper is that transformation of linear fractional programming problem into separate linear programming problems with some theorems and then solution of these problems by popular algorithm. This work also used to simplex type algorithms to arrive at an optimal solution for a linear programming problem with absolute value. Moreover, we compare this method with an existing method. Numerical experiments are also given to illustrate the assertions.

This research is about the development of a dynamic programming model for solving fuzzy linear programming problems. Initially, fuzzy dynamic linear programming model FDLP is developed. This research revises the established dynamic programming model for solving linear programming problems in a crisp environment. The mentioned approach is upgraded to address the problem in an uncertain environment. Dynamic programming model can either be passing forward or backward. In the proposed approach backward dynamic programming approach is adopted to address the problem. It is then followed by implementing the proposed method on the education system of Pakistan. The education system of Pakistan comprises of the Primary, Middle, Secondary, and Tertiary education stages. The problem is to maximize the efficiency of the education system while achieving the targets with minimum usage of the constrained resources. Likewise the model tries to maximize the enrollment in the Primary, Middle, Secondary and Tertiary educational categories, subject to the total available resources in a fuzzy uncertain environment. The solution proposes that the enrollment can be increased by an amount 9997130, by increasing the enrollment in the Middle and Tertiary educational categories. Thus the proposed method contributes to increase the objective function value by 30%. Moreover, the proposed solutions violate none of the constraints. In other words, the problem of resources allocation in education system is efficiently managed to increase efficiency while remaining in the available constrained resources. The motivation behind using the dynamic programming methodology is that it always possesses a numerical solution, unlike the other approaches having no solution at certain times. The proposed fuzzy model takes into account uncertainty in the linear programming modeling process and is more robust, flexible and practicable.

In this paper, we discusses the problem that involving the conflict fuzzy multi-objective in transportation planning which is one of special case in multi-objective linear programming. To find the simultaneously optimal solution of the problem, we use the Interactive Fuzzy Multi-Objective Linear Programming (IFMOLP) method. This method can reduce the fuzzy multi-objective linear programming problem into deterministic single objective linear programming which can be solved using simplex method. Beside that , with IFMOLP method, decision maker (DM) can establish interactively the goal from the objective function to produce the pareto optimal solution so that.

<p>There exist several methods for solving fuzzy linear or nonlinear programming problems under positivity fuzzy variables and restricted fuzzy coefficients. Due to the limitation of these methods, they can’t be applied for solving fully fuzzy linear or non- linear programming problems with unrestricted fuzzy coefficients and fuzzy variables. In this paper an efficient method to find the fuzzy optimal solution for fully fuzzy quadratic programming (FFQP) problem with unrestricted variables and parameters has been proposed. All the coefficients and decision variables of both objective functions and the constraints are triangular fuzzy numbers. The proposed method is based on converted FFQP problem into crisp quadratic programming (CQP) problem. Finally an illustrative numerical example has been given to clarify the proposed solution method.</p>