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Title: Optimal Solutions for Linear Programming Problems using Linear Programming Relaxation

Summary: This thesis presents a new approach to solving linear programming problems by relaxing them into smaller sub-problems that can be solved more efficiently using standard linear programming techniques. The proposed method is tested on a set of benchmark problems and provides better solutions compared to traditional LP relaxation methods in terms of solution quality and computational efficiency.

Introduction: Linear programming (LP) is a powerful optimization tool, but solving LP problems can be computationally expensive for large-scale problems. To address this issue, many researchers have proposed relaxation techniques. This thesis proposes a new approach to solving LP problems using linear programming relaxation.

Literature Review: The literature on LP relaxation is extensive and covers various topics. Early works include the simplex method, which remains widely used today. However, the simplex method has limitations, such as its inability to handle non-convex constraints. To address these limitations, researchers have proposed other relaxation techniques, including the revised simplex method, modified simplex method, and graphical method. These methods have been shown to provide better solutions compared to the simplex method in some cases.

Methodology: The proposed method relaxes the original LP problem into smaller sub-problems that can be solved using standard LP techniques. A combination of linear programming relaxation techniques is used to solve each sub-problem, and the solution to each sub-problem is combined to obtain an approximate solution to the original problem.

Results: The proposed method is tested on a set of benchmark problems varying in size. Results show that the proposed method provides better solutions compared to traditional LP relaxation methods in terms of solution quality and computational efficiency. Additionally, the method is compared with other state-of-the-art LP relaxation methods, and it outperforms them in many cases.

Conclusion: In this thesis, a new approach to solving linear programming problems using linear programming relaxation is proposed. The proposed method demonstrates effectiveness in providing better solutions compared to traditional LP relaxation methods in terms of solution quality and computational efficiency. The results have practical implications for solving large-scale LP problems efficiently.