Transportation Problem (TP) is a very important problem which has been vastly studied in Operations Research domain. There are some classical methods to find the initial basic feasible solution (IBFS) which minimize the total shipping cost of (TP) such as north-west corner method (NWCM), minimum cost method (MCM) and Vogel’s approximation method (VAM) which the best one of them. In this paper, we suggest a new amendment to (VAM) to find (IBFS) of (TP), which is an iterative method and the results will be near the optimal solution and in some cases equal to the optimal solution. In the numerical experiences we compare the results of the new approach with other classical methods to verify the efficiency of the new method. The proposed method is very effective and well-suited for use in solving these problems of various sizes.

Transportation Problem (TP) is singular of the paradigms in the Linear Programming Problems (LPP). The TP in Operations Research represent vastly applied optimization. (TP) has some goals, like reducing transportation costs or reducing transportation time, etc. Whereas meeting both supply level and request level requirements. Transportation problem plays a major role in industry, trade, logistics, etc. To get the most possible profit, organizations are always looking for better ways to reduce cost and improve revenue. To solve the transportation problems, it is always required to find an initial basic feasible solution (IBFS) for get the optimal solution. The Vogel’s Approximation Method (VAM) is the important known traditional methods for obtaining an IBFS of TP. In this work, we introduce a new modification to the VAM for finding an IBFS for the transportation problems almost nearer to the optimal solve. Proposed modification is illustrated with solved numerical examples. A comparison study was also conducted with the results of classic methods. This modified approach most of times give better solution and very nearer to the optimal solve, furthermore, occasionally gives the optimal solve. This method is clear, easy to comprehend.

The trust region method (TRM) is a very important technique to solve both of linear and nonlinear systems of equations. In this work, a new modified algorithm of a TRM with adaptive radius is introduced in purpose of solving systems of nonlinear equations. At each iteration, the new algorithm changes the trust region radius (TRR) automatically to reduce the subproblems resolving number when the current radius is rejected. The global convergence results of the new procedure under some appropriate conditions is established. The numerical effects indicate that the suggested algorithm is interesting and robustness.

The projection technique is one of the famous method and highly useful to solve the optimization problems and nonlinear systems of equations. In this work, a new projection approach for solving systems of nonlinear monotone equation is proposed combining with the conjugate gradient direction because of their low storage. The new algorithm can be used to solve the large-scale nonlinear systems of equations and satisfy the sufficient descent condition. The new algorithm generates appropriate direction then employs a good line search along this direction to reach a new point. If this point solves the problem then the algorithm stops, otherwise, it constructs a suitable hyperplane that strictly separate the current point from the solution set. The next iteration is obtained by projection the new point onto the separating hyperplane. We proved that the line search of the new projection algorithm is well defined. Furthermore, we established the global convergence under some mild conditions. The numerical experiment indicates that the new method is effective and very well.

The assignment problems (AP) are an important part of linear programming problems (LPP) that deal with the allocation of different resources for different activities based on one to one. The assignment problem is established in a variety positions when decision makers need to determine the optimal allocation and this means assigning only one task to one person to achieve maximum profits or imports or achieve less time or less cost based on the type of problem. In this work, a new technique has been provided to find an optimal solution for the assignment problems of maximization objective function. Comparing the proposed technique results with the Hungarian method indicates that the new technique has easier and less steps to find the optimal solution and thus the time is reduced and the effort is largely reduced.