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The most convenient models of Solid Transportation (ST) problems have been justly considered a kind of uncertainty in their parameters such as fuzzy, grey, stochastic, etc. and usually, they suggest solving the main problems by solving some crisp equivalent model/models based on their proposed approach such as using ranking functions, embedding problems etc. Furthermore, there exist some shortcomings in formulating the main model for the realistic situations, since it omitted the flexibility conditions in their studies. Hence, to overcome these shortages, we formulate these conditions for the mentioned these problems with fuzzy flexible constraints, where there are no exact predictions for the values of the resources. In particluar, numerical investigation shows that each increasing for the values of the supply and demand is not effective for improving the objective function.  The value of the objective function is sensitive when supply and demand change, so we conduct a new study to diversify the value of the objective function, due to changes in resource and demand levels simultaneously.

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In this paper, we deal with a fully fuzzy linear programming (FFLP) when the constraints are described as equality and inequality. With respect to Hadi method which is a new and a comfortable ranking method for ordering the trapezoidal fuzzy numbers, we introduce a new ranking function. We show that this function has some smooth properties when we use it for new classes of the trapezoidal fuzzy numbers which we called them k-scale trapezoidal fuzzy numbers. The k- scale trapezoidal fuzzy numbers are in fact a generalization of symmetric trapezoidal fuzzy numbers. Based on this ranking function, a new method is proposed to find the fuzzy solution for solving k-scale FFLP. Numerical examples are providing to illustrate the method.

 

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The conjugate gradient method (CGM) stands out as one of the most rapidly growing and effective approaches for addressing unconstrained optimization problems. In recent years, significant efforts have been dedicated to adapting the CGM for tackling nonlinear optimization challenges. This research article introduces a new modification of the Fletcher–Reeves (FR) conjugate gradient projection method. The proposed method is characterized by its sufficient descent property, and its global convergence has been established under specific assumptions. Numerical experiments conducted on a range of functions from the CUTEr collection demonstrate the potential and effectiveness of the proposed methods.