(483) basic principles in linear programming المبادئ الأساسية في البرمجة الخطية - amer, roshdi (INP, 2018)
This study examines the basic principles of linear programming as one of the most significant mathematical tools used in operations research, quantitative analysis, and managerial and economic decision-making. The study is based on the assumption that many economic and administrative problems involve limited resources combined with competing objectives and constraints, making it necessary to develop analytical methods capable of determining optimal resource allocation. The study begins by presenting the theoretical foundations of linear programming and explains that it is a mathematical technique designed to maximize or minimize an objective function subject to a set of linear constraints associated with available resources. The fundamental components of a linear programming model include decision variables, an objective function, constraints, and non-negativity conditions requiring variables to assume positive or zero values. The study further discusses the mathematical and geometrical representation of linear programming models and explains the concept of feasible solutions as combinations of values satisfying all imposed restrictions. Particular attention is given to the feasible region, which consists of all points meeting the model requirements. The study emphasizes that the geometrical characteristics of linear programming are closely associated with mathematical concepts such as convex sets and feasible regions. It also highlights that optimal solutions generally occur at extreme points or corner points of the feasible region. Moreover, the study addresses practical applications of linear programming in areas such as production planning, transportation systems, distribution management, resource allocation, industrial scheduling, and economic planning. It explains that the effectiveness of linear programming depends largely on the proper formulation of problems and the validity of the assumptions underlying the model. The study concludes that linear programming constitutes an effective scientific tool for improving resource utilization and supporting planning and decision-making processes through quantitative approaches that enhance efficiency and promote optimal allocation of available resources.